0.25$ for regular BNS and BHNS mass configurations, which perfectly describes the second channel of the kilonova emission sources from the postmerger torus producing bright and blue transients in a few hours after merger.","The results for LBH cases are different regarding the composition which is discussed in Sec.","REF and REF .","Focusing on more details in our results show that the remnant BH spin plays an important role in outflows features.","The models labeled as M1.0-0.14-a0.98, M1.0-0.14-a0.9, M1.0-0.14-a0.6 and M1.0-0.14-a0.2 all have identical initial parameters except for BH spin.","As shown in Table REF the mass of ejecta is hugely affected by the BH spin.","The mass increases monotonically by the BH spin and the difference may reach to more than three orders of magnitude between the low-spin case M1.0-0.14-a0.2 and extremely high-spin case M1.0-0.14-a0.98.","This is consistent with previous GRMHD numerical studies observing that high spin BH cases with robust magnetically collimated jets produce more massive outflows ([36]).","Similar observation about massive ejecta has been reported by [24] for viscous accretion disks with highly spinning black holes.","The significant mass-loss is due to energy release by accretion happening deeper in the gravitational potential and closer to the BH singularity.","This conclusion may let us rule out the merger scenarios with zero or low-spin remnants from the parameter estimation for bright kilonova observations.","Moreover, the 2D profiles of the electron fraction given in Fig.","REF and the average $Y_e$ in Table.","REF , as well as the histograms in Fig.","REF derived from SkyNet code for r-process nucleosynthesis for models with LBH ($M_{BH} = 1.0 M_{\\odot }$ ) and different spins show a dramatic change in the outflow compositions.","Generally, the outflows are more neutron-rich for models with higher BH spin (with average $Y_e < 0.25$ ), which leads to more opaque lanthanide-rich material.","This observation is in contrast with [40] and [24], which both found that the electron fraction will increase as the BH spin increases for stellar mass BH.","The argument for their observation is, when the accretion happens in a deeper gravitational potential, it releases more energy and heat the inner part of the disk, therefore the weak interactions' equilibrium condition change in a way to release more neutrinos for more effective neutrino cooling.","This causes producing more proton-rich plasma.","However, in our equation of state, the partial trapping of neutrinos is taken into account (see the details in [40] for the calculation of trapped neutrino pressure), so the weak interaction rates are affected by neutrino trapping.","In fact, a closer look into some quantities such as neutrino luminosity and neutrino emissivity show that the highest spin case, M1.0-0.14-a0.98, goes through a less effective neutrino cooling during the evolution compared with M1.0-0.14-a0.9 case.","However, more cases with LBH and stellar mass BH configurations with different spins are needed in future studies to investigate this effect more closely.","Similarly for velocity, we observe significant changes imposed by BH spin.","Fig.","REF shows the high spin cases such as M1.0-0.14-a0.98 and M1.0-0.14-a0.9 generate outflows with higher and broader range of velocities.","However, the average velocity does not increase with BH spin monotonically as our highest spin case M1.0-0.14-a0.98 has the average velocity slightly lower than the M1.0-0.14-a0.9 case.","From the geometrical point of view, as shown in Fig.","REF all cases with different spins have very broad range of ejected angles.","Almost all the cases have a distinguishable peak around the equator $\\theta \\sim 80^{\\circ }-100^{\\circ }$ , however it is interesting to mention that the symmetry is somehow broken for some cases, i.e.","significantly more wind ejected from the northern hemisphere than the southern hemisphere and vise versa.","This geometrical pattern is almost visible for all the cases regardless of their BH spins."],["Prograde versus retrograde","For a complete comparison, we perform similar analysis between one prograde case with moderate spin $a=0.6$ labeled as M6.0-0.14-a0.6 and one retrograde case with spin $a=-0.6$ labeled as M6.0-0.14-aR0.6.","Both cases have BH mass equals to $6M_{\\odot }$ and disk mass equals to $0.14M_{\\odot }$ , however the initial disk parameters such the inner radius and the constant specific angular momentum for each case is different for each model based on the nature of the Fishbone-Mocrief solution (see [50] for possible retrograde disk solutions).","So, the disks are not identical in term of size and compactness.","The comparison between these two models show that the retrograde case generates much lower mass (by one order of magnitude), and faster ejecta.","However the higher velocity can be explained by that fact this disk is initially larger than the prograde case (larger initial specific FM angular momentum) and therefore less bounded to the BH, and produce higher speed ejecta.","On the other hand, as shown in Figs.","REF -REF the retrograde case has a narrow range of velocities and more symmetric geometry with more outflows leaving the grid around the equator compared with the prograde case.","Overall, one may conclude that the negative spin can provide distinguishable changes in the amount of the outflows, its velocity and geometry.","Figure: Electron fraction profiles for models with different BH spins at final time snapshot (t∼0.247t \\sim 0.247s).Figure: Density profiles for models with different BH mass and disk mass at final time snapshot.Figure: The mass distribution versus polar angle, comparison for cases with spin and mass configurations as Fig.","."],["Effects of BH and Disk Masses","We found out the postmerger remnant disk's mass has a significant impact on the outflow properties as well.","However the changes are less dramatic compared to the spin effects.","The outflow becomes more massive by more than one order of magnitude as move from M2.0-0.05-0.9 case with $M_{disk} = 0.05 M_{\\odot }$ to the highest disk mass case M5.0-0.3-0.9 with $M_{disk} = 0.3 M_{\\odot }$ .","This obviously suggests that mergers with massive remnants is more likely to have observable EM counterpart.","Such scenarios are possible from BNS mergers and BHNS mergers with low mass ratio, softer neutron star's equation of state and high spin BH.","The histogram plots in Figs.","REF -REF show the massive disk M5.0-0.3-0.9 produce massive ejecta over a broad range of velocities $0.01-0.3$ c, but also with higher mass distribution over the lower velocities which makes the larger part of ejecta from this case slower than the others.","The broken symmetry in the outflows geometry is still visible in Fig.","REF , and overall we observe the outflows are ejected over a broad range of angles except the poles.","This feature is not affected by the disk mass and/or BH mass.","Moreover, we observe the composition of the winds is affected by the mass of the BH quite significantly.","As mentioned in Sec.","REF , for the LBH cases, the ejecta contains more neutron-rich material with lower average electron fraction $Y_e < 0.25$ , while for the systems with stellar mass central BH, the outflow is dominated by less neutron-rich material ($Y_e > 0.25$ ) and therefore more lanthanide-free composition.","As a result, one might expect to observe only red and IR transients from accretion disk systems with central LBH.","This feature can be considered as an important signature to estimate the mass of the central postmerger remnant, and therefore an evidence for the existence of the primordial BH progenitors for such systems ([12]).","At this point, one might wonder how realistic these masses and geometries are compared with the remnant disks from merger simulations.","Our selected models can present different BNS and BHNS with different parameters.","[68] and [13] derived fitting formula calculating the masses of ejecta and remnant disk for BNS systems based on numerical simulations results.","Later, [51] suggested a simpler version of this formula based on the same data base.","Similar analytic fitting formulae were proposed by [29], [46], [30] for BHNS systems.","Using this data base and reversing this analysis to estimate the merger parameters from the black hole and accretion disk's masses, we can make the following statement about our models.","For instance, the M5.0-0.3-a0.9 presents the remnants from a BHNS merger with mass ratio $Q \\sim 3.3$ , with $M_{BH}=4M_\\odot $ , $M_{NS}=1.2M_\\odot $ , and $R_{NS} = 13.2$ km.","Such NS can be explained by a stiff equation of state for a non-rotating spherical star such as DD2 or BHB.","Our FM model gives a disk with mass of $M_{disk}=0.3M_{\\odot }$ and the radius of the maximum density at $r\\sim 100$ km.","Model M2.65-0.1-a0.9, resembles the remnants from a BNS merger with identical mass $M_{NS}=1.2 M_{\\odot }$ and $R_{NS} = 11.7$ km.","Such NS can be explained by a soft equation of state for a non-rotating spherical star such as SRO and APR.","The FM solution creates a disk with mass of $M_{disk}=0.1 M_{\\odot }$ and the radius of the maximum density at $r \\sim 44$ km.","Comparing against literature, for compact binary mergers, the remnant disk's outer radius is usually around 100-200km with maximum density around 50-60km depending on the initial parameters of the merger ([17], [23]).","Since in the FM solution, the size of the disk is scaled by the mass of the BH, this solution may create disks with larger radius.","Therefore, our disk models resembling BHNS mergers are larger and less compact compared with final disks from merger simulations.","This difference can cause deviations in the mass, velocity and composition of the outflows launching from the disks with different geometries, and non-axisymmetric configurations expected for disks from mergers.","Regardless of BH spin and mass configurations, we see similar pattern in the ejecta's mass evolved during time.","Fig.","REF shows the evolution of the cumulative ejecta's mass launched from selected models.","As illustrated in this figure, the ejecta is being developed shortly after the simulations start, increases exponentially during the first half of the evolution, and then continues increasing with a slower pace for the rest of the simulation.","This exponential growth in outflow mass at early time is consistent with the GRMHD model reported by [26], and it is a characteristic of the magnetized disk models compared with the $\\alpha $ -disk models.","The detailed discussion over outflow measurements and how it is affected by the computational grid resolution and tracer setups is given in the appendix .","Figure: The mass distribution versus polar angle, comparing cases with different disk and black hole mass configurations.Figure: Evolution of the cumulative outflow mass launched from the disk measured by tracers.","Time is given in the code unit in this figure."],["Kilonovae peak properties","In order to investigate some important kilonovae properties based on measured outflows, we use the approximations given by [34] and [20].","The time $t_{peak}$ at which the peak occurs, the luminosity $L_{peak}$ at this time, and the corresponding temperature $T_{peak}$ are estimated as: $t_{peak} = 4.9~d \\times \\left( \\frac{M_{ej}}{10^{-2}M_\\odot } \\right) ^{\\frac{1}{2}} \\left( \\frac{\\kappa }{10~cm^2g^{-1}} \\right) ^{\\frac{1}{2}} \\left( \\frac{v_{ej}}{0.1} \\right)^{-\\frac{1}{2}},$ $\\begin{aligned}L_{peak} = 2.5 \\times 10^{40}~\\text{erg/s}~\\times \\left( \\frac{M_{ej}}{10^{-2}M_\\odot } \\right) ^{1-\\frac{\\alpha }{2}} \\\\\\left( \\frac{\\kappa }{10~cm^2g^{-1}} \\right) ^{-\\frac{\\alpha }{2}} \\left( \\frac{v_{ej}}{0.1} \\right)^{-\\frac{\\alpha }{2}},\\\\\\end{aligned}$ $\\begin{aligned}T_{peak} = 2200~K \\times \\left(\\frac{M_{ej}}{10^{-2}M_\\odot } \\right) ^{-\\frac{\\alpha }{8}} \\\\\\left( \\frac{\\kappa }{10~cm^2g^{-1}} \\right) ^{-\\frac{\\alpha +2}{8}} \\left( \\frac{v_{ej}}{0.1} \\right)^{-\\frac{\\alpha -2}{8}},\\end{aligned}$ where $\\kappa = 1 cm^2g^{-1}$ is the average opacity suggested by [34] for less opaque material produced by weak r-process, and $\\alpha = 1.3$ .","The results of these peak values are given in table REF .","Based on these analytic fitting formulae, the majority of our models power EM transients with their peaks in a few hours after merger and the luminosity around $\\sim 10^{40}-10^{41} erg/s$ .","Table: Properties of the outflows and kilonovae peaks."],["Lightcurves and r-process nucleosynthesis","In order to estimate the possible lightcurves powered by the ejecta from different models, we consider the same approach as [46].","As it is shown by [20] this approximation agrees well with the results given by the radiative transfer simulations for the dynamical ejecta.","However, this analytic fitting overestimates the luminosity at the peak time (see Fig.8 from [20] for comparisons).","Therefore, we use the [34] fitting formula to estimate the peak values for a better approximation.","The luminosity curve is given by $L_{bol}(t) = (1+\\theta _{ej}) \\epsilon _{th} \\dot{ \\epsilon }_{0} M_{ej} {\\left\\lbrace \\begin{array}{ll} \\frac{t}{t_c} \\left( \\frac{t}{1~d} \\right)^{-\\alpha }, & t \\leqslant t_c \\\\\\left( \\frac{t}{1~d} \\right)^{-\\alpha }, & t > t_c\\end{array}\\right.","}$ where $\\dot{\\epsilon }=1.58 \\times 10^{10}~erg~g^{-1}~s^{-1}$ is the specific heating rate, $\\epsilon _{th}$ is the efficiency of thermalization introduced by [59], $0.5 < \\epsilon _{th} < 1$ and $t_c$ is derived by $t_c = \\sqrt{\\frac{\\theta _{ej} \\kappa M_{ej}}{2\\phi _{ej}(v_{max}-v_{min})}},$ with $v_{max}$ and $v_{min}$ are the maximum and minimum speed of the ejecta.","$\\theta _{ej}$ and $\\phi _{ej}$ are geometrical parameters of the outflows.","It was shown by [20] that if we assume the ejecta as homogeneously distributed material moving with velocity $v_{ej}$ in a $\\rho -z$ plane, the polar and azimuthal opening angles $\\theta _{ej}$ and $\\phi _{ej}$ are related to the velocity components with: $\\theta _{ej} \\approx \\frac{2^{4/3} v^2_{\\rho } - 2^{2/3} \\left(v^2_{\\rho } \\left(3v_z + \\sqrt{9v^2_z+4v^2_{\\rho }}\\right) \\right)^{2/3}}{\\left(v^5_{\\rho } \\left(3v_z+\\sqrt{9v^2_z+4v^2_{\\rho }}\\right)\\right)^{1/3}},$ $\\phi _{ej} = 4 \\theta _{ej} + \\frac{\\pi }{2}.$ Fig.","REF shows the predicted lighcurves from this method.","As one expects, M1.0-0.14-a0.98 and M5.0-0.3-a0.9 cases with the highest mass outflows produce brighter transients.","There are two pairs of models launching outflows with similar features and therefore generates lightcurves following each other closely, M2.65-0.1-a0.9 and M1.0-0.14-a0.9, and the other pair is M2.0-0.05-a0.9 and M1.0-0.14-a0.6.","Our models suggest that although bright blue transients is possible to be observed from light BH-accretion disk systems, they may not be distinguishable from BNS mergers.","Overall, the possible degeneracy make the mergers parameter estimation very challenging from kilonova observations.","Figure: Luminosity versus time estimated by Kawaguchi's method for different cases.As the next post-process analysis, we have applied the r-process nucleosynthesis on the simulations outputs using the open source SkyNet code ([55]) to measure the nuclear abundances.","The results of these simulation are illustrated in Fig.","REF by taking the average over all the tracers and compared with the Solar system abundances.","The results show the 2nd and 3rd r-process abundance peaks are resolved well in our models.","We would like to emphasise again that the composition analysis discussed in sections.","REF and REF for Figs.","REF and REF are from these skynet simulations, and these figures showing the composition of the ejecta at the onset of the r-process.","Figure: Nuclear abundances as a function of mass number A foreach simulation, based on average of tracers sampling the outflows."],["Comparison with previous studies","There are two major mechanisms driving postmerger outflows, one is neutrino heating and the other is magnetically driven turbulence.","The analytical estimation by [64] indicated that in a post-merger accretion disk neutrino heating timescale is short enough to drive a wind within the lifetime of the disk.","On the other hand, in the case of a weakly magnetized accretion disk, MRI can be triggered in the disk's dynamical timescale, resulting in magnetic field amplification and turbulence.","The plasma is being accreted to the BH and wind is originated from the disk, while the angular momentum is transported by magnetic turbulence effects ([4]).","[74] has shown that the expected outflow mass from a BNS postmerger remnant disk has to be around $\\sim 0.03 M_\\odot $ to power emissions observed in AT2017gfo (kilonova emission observed right after GW170817) as predicted by the r-process nucleosynthesis.","They also showed that both optical and near-infrared emissions are simultaneously reproduced by the ejecta with a medium $Y_e$ of 0.25.","The recent Bayesian analysis by [70] constrained a mass ratio of $M_w/M_d = 2.81$ (the ratio of wind mass to dynamical ejecta mass) to reproduce the observed AT2017gfo kilonova lightcurves while it is also consistent with the observed r-process elements abundance measured in the Solar system.","While the most perfect way to study kilonovae in a numerical simulation is evolving BNS and/or BHNS systems for late inspiral, merger and postmerger phases, measuring dynamical and postmeger ejecta, most recent studies focused only on one side of the scenario.","In the case of disk winds, ideally we want to evolve the magnetized remnant torus in a high-resolution 3D grid for a long period ($\\sim 10$ seconds), however due to the limit of computational resources, long-term 2D simulations with the $\\alpha -$ viscosity prescription as the main mechanism to produce the outflows were initially preferred [24].","Nonetheless, there are a few studies in the literature, which evolved magnetized remnants with neutrino cooling in fully GRMHD simulations within a multi-second timescales [26], [37].","Generally, in the outflow studies, some groups considered analytical disk as the initial data configuration and then perturbed them by weak magnetic field (similar to our study in this paper) or $\\alpha $ viscosity, while others deal with remnant disk from a BHNS or BNS simulations, applying Cowling approximation (frozen space-time) at $t \\sim 10-15$ ms after merger.","At this point, it would be interesting to compare our results with these studies, as well as the predicted values from the kilonova observations.","For obvious reasons, the outflow properties are highly case dependent.","The wind mass reported by different groups typically varied from $10^{-5}$ to $10^{-2} M_{\\odot }$ .","We found most of our cases produce $\\sim 10^{-4}-10^{-3} M_{\\odot }$ , which are lower by one to two orders of magnitude compared with the predicted values for AT2017gfo observation.","The outflow mass might be underestimated as a results of various reasons.","The full discussion on our tracer's setup and alternative measurements is given in Appendix .","Comparing our models with similar cases in the literature, our results confirm the conclusion made by [44] for $\\alpha $ -viscosity disk models.","As they mentioned in the case of the most mass massive disk $a0.8_{M0.3}$ (disk mass$\\sim 0.3 M_{\\odot }$ and BH spin $a=0.8$ ), the outflow mass increases by almost factor of 4 compared to their $\\sim 0.1 M_{\\odot }$ case, but the average velocity becomes slightly smaller.","For similar cases, we observed the massive disk, M5.0-0.3-a0.9, provides the highest outflow mass compared to the intermediate mass cases such as M2.65-0.1-0.9, but in our models the difference is quite dramatic (about one order of magnitude), while the velocity is slightly bigger.","However, the outflow of their $a0.8_{M0.3}$ case is still much higher than our M5.0-0.3-a0.9 by factor of 2.","Our M2.65-0.1-a0.9 case can be compared with HS-Therm and HS-Magn models from [40].","This case is initially moderately magnetized and the comparison of this case with Table(2) from the same reference indicates that the outflows properties are scaled by the initial magnetic field strength, i. e. the outflow mass and velocity increase as the initial magnetic field increases.","Also, [40] found that the average electron fraction in the outflows depended on the disk magnetisation, especially at the earlier time.","Though, the average electron fraction we calculated for M2.65-0.1-a0.9 case is closer to the highly magnetized case 'HS-Magn' from this reference.","Here, we measured the averaged electron fraction around $0.1<Y_e<0.3$ , which is consistent with the GRMHD model in [26], and it is relatively lower than the averaged value reported by [35] ($0.3 < Y_e <0.39$ ) for viscous disks.","Our results also show that outflow mass increases sharply at early times, matching well with GRMHD model, while the viscous disks have this large growth at much later times in [26].","Though, the outflow masses are still lower in our models, which can be caused by the time of evolution and differences in disk parameters and equation of state (more analysis is given in the Appendix REF ).","The outflow velocity measurements agree well with the previous studies and the theoretical models' prediction for the disk winds.","The velocity range is around $0.1-0.2$ c and is generally higher for systems with higher spins.","Regarding the geometry, the outflows launch over a wide range of angle which agrees with results reported in [26] and [22].","However, our results show less equatorial symmetry compared with these studies."],["Parameter estimation, binary population synthesis and spin effects","Based on population synthesis studies, the GW from BNS mergers are more likely to be observed with EM counterparts ([57]).","As already discussed in the literature there are some features in the kilonova emissions, which help to distinguish the BHNS from BNS mergers and make some parameter estimations.","For instance, [44] claimed that there is a rapid decline in the lightcurves from a BHNS postmerger ejecta (or equivalently, a prompt collapse for a BNS merger).","Unfortunately, our current models do not allow us to investigate different possible scenarios for BNS mergers, such as creation of a magnetized and diferentially rotating HMNS with different lifetimes, which can affect the amount of ejected matter significantly ( [16]).","However, even the results from [44] show that the light curves of models with high spin BH and HMNS with 30ms lifetime are quite similar, and this degeneracy introduces a challenge in using kilonova observations to estimate the lifetime of a HMNS remnant.","For the BHNS merger cases, two signals GW200105 and GW200115 have been detected by the LIGO-Virgo collaborations so far.","The masses of GW200115 and GW200105 are 8.9M and 1.9M, and 5.7M and 1.5M, respectively.","The spin of the black hole in GW200115 is not tightly constrained but it is estimated to be $\\sim -0.5-0.04$ and most likely misaligned.","The dimensionless spin magnitude of the black hole in GW200105 is estimated to be $< 0.2$ and its direction is unconstrained ([1]).","The spins of the remnant central BHs are not constrained up to certain points, but they are likely around $\\sim 0.38$ and $0.43$ for GW200105 and GW200115 respectively ([9]) However, no EM counterpart have been observed from these mergers.","The discussion given by [72] indicated that only BHNS with highly spinning black holes can generate massive dynamical ejecta (the difference may exceed three orders of magnitude from low spin ($\\sim 0.2$ ) to high spin $\\sim 0.9$ ).","This high spin can be obtained by two mechanisms during a BHNS binary formation: i) inheritance due to weak core-envelope coupling of the stellar progenitors, and ii) accretion from the companion star during stable mass transfer.","However, population synthesis studies of merger rates of BHNSs suggest that the majority of binaries will not result in observable EM counterparts.","Studies by [21] found that only a fraction ($\\sim 20\\%$ ) of BHNS binaries gain a high dimensionless BH spin from their stellar progenitors and produce massive ejecta during merger.","For postmerger outflows, we obtained similar trends in our results, i.e.","the disk with lower spin BH generates lower mass ejecta, up to two orders of magnitude compared to high spin cases.","We also concluded that the ejecta mass may become lower for the retrograde accretion.","However the mass of the postmerger remnant disk can be another important factor for launching the ejecta, which is highly dependent on the mass ratio, NS equation of state and BH spin ([32], [56]).","Simulations done by [56] show that the massive remnant disk is likely to be originated from BHNS mergers with high spin BH.","Overall, our models confirm that almost not much ejecta and no EM counterparts are expected to be observed from these BHNS systems, unless for BHs with high spins and lower mass ratio cases providing massive postmerger disks."],["Outflow measurements and other important physical components","In this section we present a discussion about important numerical and physical elements which are ignored in our simulations and can affect our outflow measurements significantly.","First, it is worth mentioning that the accuracy of our outflow studies are affected by the current version of neutrino and equation of state treatments.","Neutrino emissions can affect the disk's thermal and composition evolution significantly at the early time of evolution after merger ([39], [26]).","Therefore, a more advanced neutrino treatment such as leakage or neutrino radiation transport schemes along with composition evolution are needed to provide a better accuracy.","Our equation of state does not include nuclei heavier than Helium, however [35] and [22] showed the inclusion of these nuclei in the equation of state may cause significant changes in the outflow's mass and velocity.","In addition, the numerical scheme and grid resolution typically have impacts on the modelling of the kilonova emission.","In the study by [60] the authors compared the dynamical ejecta properties measured from 4th-order and regular 2nd-order finite difference schemes for magnetized BNS simulations.","They found the second-order scheme overestimates the amount of proton-rich shock-heated ejecta.","Also, [11] performed convergence studies on the outflows from a BNS merger simulation, and they found out the highest resolution resolve the shock-heated plasma more accurately.","The postmerger ejecta can be affected in a similar way.","Fully-resolved MRI effects requires high accuracy in numerical scheme and high resolution grid, and both can impact the magnetically-driven viscous heating effects which leads to changes in the composition and other features of the disk and the outflows (see Appendix REF about the resolution test.)","For the sake of electromagnetic lightcurve estimations, we use here a simple fitting formula, which can be correct only up to the order of magnitude.","In our simple approach we assumed a constant value for the average opcacity of the ejected matter, which in reality varies a lot depending on the geometry and composition of Lanthanide/Actinide-rich matter.","In order to predict the lightcurves and spectrum of the emissions accurately, one needs to apply a radioactive transfer code on the ejecta.","Such studies have been done recently e.g.","by [78].","One crucial thermal component in studying outflows is the inclusion of the r-process heating and neutrino cooling terms in the long-term hydro simulations (see the discussion and eqs.","(20-21) from [31]).","Recent study by [35] for long-term 2D viscous disk simulations indicated that adding these terms to hydro equations after one second evolution increases the outflow mass by 30%.","Another important physics to be considered for kilonova studies is neutrino's flavour oscillations.","A few groups recently studied the effects of fast flavour instability (FFI) on the disk outflows in their simulations ([54], [25]).","Neutrino flavor oscillation is reported to have moderate (or large, in some cases), impacts on the mass ejection, average velocity, and average electron fraction.","Even with very accurate numerical evolution and measurements, there are possible ways to make our results different from the observations.","First, the measured abundances can be largely affected by the nuclear uncertainties in the r-process models, especially at late-time emissions, $t>100$ days ([52]).","Second, in a realistic merger and post-merger scenario, the emissions from disk winds can be affected by the dynamical ejecta.","According to the discussion by [44], dynamical ejecta is generally faster and more neutron-rich and therefore more opaque, which acts like a 'Lanthanide curtain’, masking the emissions originating from the wind ejecta.","They observed that for an edge-on orientation, the dynamical ejecta partially blocks the wind's optical flux by an order of magnitude."],["Conclusions and Summary","We have carried out two-dimensional simulations of several black hole-accretion disk models with different initial setups using the GRMHD HARM-COOL code.","These simulations include both magnetic field evolution and neutrino emission effects, as well as realistic nuclear equation of state.","The initial magnetic field has been seeded in poloidal loops configuration confined within the plasma.","We evolved these models about $t \\sim 200$ ms, and used particle tracers to measure the outflows properties.","We observed there is strong correlation between BH spin and the outflows properties.","Generally, disks with higher mass and BH spin generate faster and more massive outflows.","We observed our models generate winds with moderate velocity (v/c $\\sim 0.1-0.24$ ) and a broad range of electron fraction ($Y_e \\sim 0.01-0.5$ ), which are consistent with previous studies on GRMHD postmerger accretion disk simulations.","We have included a few cases with lower mass black holes representing BH formed from primordial black hole and neutron star mergers, and we found out such cases generate more neutron-rich ejecta in comparison with regular BHNS and BNS mergers.","Generally, the outflow masses measured by tracers are lower compared with long viscous disk and GRMHD simulations.","We have investigated the accuracy of the outflow measurements with tracer method by altering the tracer's density threshold and extraction radius, and also grid resolution (explained in the Appendix ).","We found that our measurements for ejecta's mass can be affected up to $50\\%$ by this alterations.","However, we consider this method to be more accurate than the unbound matter estimation from the geodesic and Bernoulli criteria.","The general properties of postmerger ejecta derived from our simulations give us the opportunity to estimate the luminosity lightcurves of possible radioactively powered transients using analytic fitting formula.","We found the luminosity peaks within the range of $\\sim 10^{40}-10^{42}$ erg/s, brighter peaks for cases with higher ejecta mass, which agrees with previous studies for neutrino-driven disk wind models.","Applying the r-process nucleosynthesis code on our results and comparing against the Solar-system abundances showed that the 2nd and the 3rd r-process abundance peaks are resolved well in our models.","In the end, we should point out that for future studies we need to improve our numerical method by including a more sophisticated neutrino treatment such as leakage or transport schemes, as well as composition evolution to capture all the neutrino absorption/emission features.","Moreover, including heavy nuclei in the equation of state are required to achieve a more realistic scenario with better accuracy.","The authors thank Oleg Korobkin for helpful discussion and advice over the course of this project.","This work was supported by grant No.","2019/35/B/ST9/04000 from the Polish National Science Center, Poland.","We also acknowledge the support from the PL-Grid infrastructure under computational grant plgmicrophysics."],["Outflow measurements: methods and accuracy","In this appendix first, we investigate the alternative methods to measure the outflow mass and compare them with the results from the tracer method.","In the second part, we perform several test simulations for a single case to study the effects of the resolution and tracer's setup parameters on the outflow measurements.","At the end, we leave a comment regarding the evolution's time and its effect on our measurements."],["Unbound matter estimation: Geodesic and Bernoulli criteria versus tracer methods","As mentioned, in this study we use the tracer particle technique to measure the outflows.","In this method, all the particle trajectories leaving the outer boundary (or a large radius close to the outer boundary) of the computational grid during the evolution are marked as outflow winds.","For the results discussed in Sec.","we set this radius at $r=800~r_g$ .","The details of the tracer method's implementation in HARM-COOL are given in [40].","The tracer particles provide the information on how different quantities such as density, temperature and electron fraction vary while tracking the particles.","This information is used for the postprocess calculations of the element abundances from the r-process nucleosynthesis presented in Sec.","REF .","In addition, a detailed picture of wind's geometry can be obtained from the trajectories of the outflowing particles.","Despite the fact that tracers are useful tools for outflow studies, the amount of measured outflows can be possibly underestimated if the system is not evolved long enough to provide enough time for the unbound mass to leave the grid.","In fact, $t \\approx 200$ ms we have for these simulations can not be considered as a very long evolution to study the disk winds.","On the other hand, the evolution of the disk ejecta can be affected by several other factors including the artificial atmosphere controls over the low density regions and the tracers' initial setup parameters.","To test the accuracy of our tracers measurements, we apply the other alternative methods from the literature to identify the unbound matter.","Referring to the discussion given by [31], the most accurate way to study outflows is a long-term (multi-second) 3D simulation where all the r-process heating and neutrino cooling terms are included in the source terms of the hydrodynamic evolution equations (see Eqs.","(20-21) from the same reference for more details).","However it is still possible to obtain a reasonable estimation for the outflow mass based on the energy criteria from a short-term simulation.","We try three different criteria introduced in this paper and compare them with the results from tracers for our M2.65-0.1-a0.9 case: 1- The geodesic criterion $u_t<-1$ , which is not suitable for hot disk outflows, because it ignores the thermal energy of the fluid and underestimates the unbound mass significantly.","2- The Bernoulli crtieria $hu_t < -h_{\\infty }$ , which includes the thermal energy but ignores the cooling effects caused by neutrino losses, and therefore overestimates the unbound mass.","3- The improved version of the Bernoulli criterion, which includes the r-process heating and neutrino losses by a simple approximation given by $hu_t/h_{\\infty }(0.9968+0.0085Y_e) < -1.$ Here $h$ is the enthalpy given by $h=1 + \\epsilon + P/\\rho $ with $\\epsilon $ represent the specific internal energy, $h_{\\infty }$ is the asymptotic enthalpy for $\\rho \\rightarrow 0$ and $\\epsilon \\rightarrow 0$ (for our case $h_{\\infty }=1$ ) and $Y_e$ is the electron fraction.","Generally, the improved version of the Bernoulli's criterion can be considered as a more realistic estimation to flag the unbound mass for different merger and post-merger scenarios.","The comparison for ejected mass computed from tracer method and estimation from geodesic, Bernoulli and improved Bernoulli criteria calculated at a single time snapshot in the middle of the simulation at $t \\sim 100$ ms for $M2.65-0.1-a0.9$ case is given in Table REF .","These results show that the outflow mass computed from tracers is lower by about factor of four compared with the improved version of Bernoulli criterion.","However, we should remind ourselves that this criterion is still considered as an approximation, and can not be taken as an accurate measurement of the unbound mass.","Table: Ejected mass: tracers versus geodesic, Bernoulli and improved Bernoulli criteria (at t∼100t \\sim 100ms snapshot).The volume integral is taken from r=400r g r=400~r_g (∼1500\\sim 1500km) to the outer boundary."],["Tracer's Setup: Grid Resolution, density threshold and extraction radius effects","Investigating the effects of the computational gird resolution and tracers setup parameters, we perform several tests for our M2.65-0.1-a0.9 model, and compare their results with the standard M2.65-0.1-a0.9-Std case discussed in Sec. .","This is our closest case for a BNS postmerger scenario.","The list of tests and the measured ejecta mass and velocity are given in Table REF .","For Trac-Low-RhoMin test we alter the threshold density identifying the active tracers during the evolution and reduce this value by two orders of magnitude.","The original threshold for the standard case was about $\\sim 10^{-4}$ of the maximum density.","Trac-1500km is designed to study the effect of the radius where we extract the information about the outflows.","This radius has been set to $r=800~r_g$ , which is almost 3000km for the standard case.","Finally High-Res test which has the identical setup as M2.65-0.1-a0.9-Std, but with higher grid resolution 480*426 along the radial and angular directions respectively.","Figs.","REF and REF show the cumulative outflow mass versus time and the mass distribution versus velocity respectively for these tests.","The simulation with lower density threshold measures slightly larger mass outflows, with larger mass distribution over the high velocity ranges.","This makes the average velocity for this test significantly higher than the other cases.","Extraction radius is another important parameter to be explored.","Our results show that more massive and slower ejecta are measured by the smaller extraction radius.","However, this mass is still lower by more than factor of two compared to the unbound mass measured by the improved Bernoulli criterion reported in Table REF .","Specifying the extraction radius can be a challenging job, and it can affect some quantities' measurements such as velocity.","In the literature, a smaller extraction radius $\\sim 200-500$ km has been used to measure the dynamical ejecta or the ejecta from magnetized postmerger HMNS with tracers (see for example [11] and [16]), and each tracer has to satisfy the geodesic or Bernoulli condition to identify the unbound mass.","However the disk wind is a different scenario; the recent studies by [35] showed that if the r-process heating source terms are included in the evolution of the postmerger ejecta during a multi-second simulation, the ejecta's velocity need to be measured at a much larger distance.","This study claimed that the ejecta is being continuously accelerated as a result of the r-process heating and it reaches to its asymptotic value around $r \\sim 40,000$ km.","The High-Res test shows that our measured mass can be affected by the grid resolution up to some extend.","For this particular case we have the ejacta mass increased by about 12%.","Fig.","REF shows that the outflows with higher velocities are overestimated in the lower resolution.","The standard resolution of our simulations are high enough to resolve the fastest growing MRI mode almost everywhere in the torus (see Fig.","(2) from [40] for a similar case).","However, fully resolved MRI modes and capturing all the transport effects of turbulence requires a very high grid resolution.","Even high resolution GRMHD simulations reported only a qualitative convergence in capturing all the turbulence's heating effects caused by the magnetorotational instability ([48]).","Obviously, higher resolution simulations provide higher accuracy for resolving the heating effects of the turbulent plasma and produce more massive outflows.","In conclusion, the results of these tests explain that our outflows measurements are affected by the resolution and the tracers setup parameters up to $50\\%$ , but still unable to explain the entire huge gap between the ejected mass from our model and the value predicted from GW170917 kilonova observation.","Table: Different resolution and tracers setups for outflow measurements for M2.65-0.1-0.9 case.Figure: The outflows mass distribution versus velocity for M2.65-0.1-a0.9 case with different resolution and tracer setups."],["Other important factors for outflow measurements","One might argue that the numerical treatments can impose quantitative and qualitative effects on the outflows.","Though, our results are less likely to be suffered from the numerical treatment over the atmosphere.","In HARM we apply the velocity control only over low-density atmosphere with very high velocity.","The disk outflow is dominated by subrelativistic plasma, and therefore, is not influenced significantly by the artificial atmosphere adjustments.","The evolution time is another key factor we should take into account.","We evolved our models for about $\\sim 200$ ms, while the viscous timescale for a thick postmerger accretion disk with $H/R \\sim 0.3$ is around $0.3$ s. The evolution of about a few viscous timescales is required to capture all the magnetically and thermally driven effects for launching the outflows.","On the other hand, the evolution time is perhaps not long enough, so not all the unbound matter have enough time to leave the grid, though the longer evolution is somehow pointless for our simulation as it is impossible to maintain the magnetic field in a long two-dimensional simulation due to the anti-dynamo theorem.","Even from thermal evolution point of view, the timescale of the evolution can be very effective based on studies by [26].","They showed that a 3D GRMHD model ejects mass in two ways: one is the MHD-driven outflow at the earlier time of evolution when the torus is NDAF (neutrino-dominated accretion flow), and the second way is late-time, thermally driven wind, which occurs when the disk becomes advection-domianted (ADAF).","They showed that the total amount of unbound mass ejected can reach to $0.013 M_{\\odot }$ , which is around 40% of the initial disk.","Half of this mass lost over the first second of the evolution and the rest has launched thereafter.","[40] investigated these types of outflows in two separate models: thermally-driven winds being ejected from an initially weakly magnetized disk and magnetically-driven winds launching from strongly magnetized disk.","Computing the time averaged mass loss rate over the outer boundary estimated the final mass loss to be in the range of $2-16\\%$ of the initial disk.","In comparison, our models are initially moderately magnetized (with $\\beta = 50$ ), so they are neither purely thermally-driven nor purely MHD-driven outflows from the very beginning.","However, the cumulative outflow mass in Fig.","REF shows that the outflow mass increases in a constant and slower pace at the later time of the evolution.","Assuming this mass loss rate for later times, by extrapolation, we estimate to obtain about $\\sim 0.0035 M_{\\odot }$ and $\\sim 0.02 M_{\\odot }$ mass loss after about 1 second and 9 seconds of evolution respectively; The latter would be around $20\\%$ of the initial disk's mass for M2.65-0.1-a0.9 case."]],"string":"[\n [\n \"Studying Postmerger Outflows from Magnetized Neutrino-cooled Accretion\\n Disks\"\n ],\n [\n \"Abstract Neutrino-cooled accretion flow around a spinning black hole, produced by a compact binary merger is a promising scenario for jet formation and launching magnetically-driven outflows.\",\n \"Based on GW170817 gravitational wave detection by LIGO and Virgo observatories followed by electromagnetic counterparts, this model can explain the central engine of the short duration gamma ray bursts (GRB) and kilonova radiations.\",\n \"Using the open-source GRMHD HARM-COOL code, we evolved several 2D magnetized accretion disk-black hole models with realistic equation of state in the fixed curved space-time background.\",\n \"We applied particle tracer technique to measure the properties of the outflows.\",\n \"The disk and black hole's initial parameters are chosen in a way to represent different possible post-merger scenarios of the merging compact objects.\",\n \"Our simulations show a strong correlation between black hole's spin and ejected mass.\",\n \"Generally, mergers producing massive disks and rapidly spinning black holes launch stronger outflows.\",\n \"We observed our models generate winds with moderate velocity ($v/c \\\\sim 0.1-0.2$), and broad range of electron fraction.\",\n \"We use these results to estimate the luminosity and light curves of possible radioactively powered transients emitted by such systems.\",\n \"We found the luminosity peaks within the range of $10^{40}-10^{42}$ erg/s which agrees with previous studies for disk wind outflows.\"\n ],\n [\n \"Introduction\",\n \"Observation of gravitational waves accompanied by electromagnetic emissions from black hole-neutron star (BHNS) and binary neutron star (BNS) mergers provide treasured information about the physics of the dense matter ([28]), compact binary formation ([8]) and the origin of heavy elements ([45], [65], [77]).\",\n \"In a post-merger black hole-accretion disk remnants, energy can be channeled into ultra-relativistic outflow needed to explain GRB properties( [61], [6]) by hot, dense accretion flow.\",\n \"In such systems, the plasma cools by neutrino emission continuously ([66], [19], [42]).\",\n \"On the other hand, magnetic fields can also extract energy from the disk and black hole spin, by generating the magnetically-driven winds and Poynting flux-dominated jets ([7]).\",\n \"Moreover, it can support the heating process of the plasma through the viscous effects of the MRI ([3]).\",\n \"The neutron-rich dynamical ejecta outflows during merger, and post-merger winds are considered as possible sources explaining the observation of kilonovae, afterglows and other electromagnetic (EM) counterparts following the gravitational wave detections, ([59], [58], [20], [80]).\",\n \"Kilonovae are transient emissions in the optical or near-infrared band, powered by the radioactive decay of the elements produced through r-process nucleosynthesis ([75]).\",\n \"Two distinguishable sources are suggested to explain kilonova emissions: 1- High-speed, neutron-rich dynamical ejecta expelled from the tidal tail during merger; 2- Less neutron-rich material with moderate velocity ejected from post-merger accretion disk due to magnetic viscous heating and/or neutrino absorption heating effects.\",\n \"The emissions from former source are dimmer and long-lasting near infrared ([53], [58], [34], [73]), while the latter produces optically brighter and bluer transients ([34], [64], [44]).\",\n \"However, depending on the merger's scenario, it is possible to have multiple channels for outflow formation; in more recent studies by [11] it is shown that the decay of the fast free neutrons in the outermost layers of dynamical ejecta can power a UV/optical kilonova precursor on $\\\\approx 1$ hour timescale.\",\n \"In the case of a BNS merger, the long-lived postmerger hypermassive neutron star (HMNS) is considered as another channel of neutrino and magnetically-driven outflow formations ( [15], [16]).\",\n \"The observation of the electromagnetic counterpart of the gravitational wave signal from the BNS merger GW170817 gives us a unique opportunity to investigate the r-process nucleosynthesis directly.\",\n \"The identification of strontium in these emissions ([77]) is considered as the first evidence of rapid nuclear process and heavy elements production in the BNS merger scenario.\",\n \"The observed kilonova light-curves and spectra suggest that models with two components consistent with lanthanide-poor and lanthanide-rich ejecta provide a good fit to the data.\",\n \"These models indicate the ejecta mass of $M_{ej}/M_{\\\\odot } = 0.03-0.06$ , with postmerger disk ejeta as a dominant component ([14], [74], [47]).\",\n \"The theoretical models suggest that the magnitude, color, and duration of a kilonova are significantly affected by the mass, velocity, and composition of the ejected matter, and the properties of these outflows can themselves be related to the properties of the merging compact objects, such as mass ratio, compactness and equation of state ([5], [34]).\",\n \"In a post-merger scenario, the jet formation, disk evolution and ejected matter can be impacted by the magnetic field strength and geometry.\",\n \"[76] studied the long evolution of a BHNS merger where the neutron star is initially magnetized by asymmetric magnetic field dipole configurations.\",\n \"[10] evolved a long-term ($\\\\sim 4$ s) post-merger black hole-accretion disk system with different magnetic field configurations, concluding the ejected matter is slightly more massive for poloidal field configuration compared with purely toroidal field.\",\n \"[40] also reported that disk configurations with higher initial magnetic field strength and higher spin BH generate massive, lanthanide-poor outflows.\",\n \"Moreover, the total mass lost through wind outflows depends on the mass of the remnant torus as well.\",\n \"Larger and more massive disks generates stronger matter outflows ([45]).\",\n \"However, only a long-term 3D GRMHD simulations of a posmerger system can capture all the disk's and outflows features accurately, including the geometry and thermal evolution.\",\n \"[26] and [37] have evolved multi-second GRMHD simulations of black hole-disk remnants for BNS and BHNS mergers respectively.\",\n \"Both studies concluded that the neutrino cooling is only effective at the earlier time after merger, and the disk eventually becomes advective, hence the mass loss rate and ejecta's composition are expected to be affected by long evolution of the disk.\",\n \"Predicting the properties of matter outflows in BNS and BHNS mergers is crucial to carry out detailed analysis of post-merger electromagnetic emissions.\",\n \"[53] and [34] obtained the analytical fits for the mass, kinetic energy, and the velocities of the dynamical ejected material to approximate the main properties of kilonovae according to BNS numerical simulations.\",\n \"[20] suggested fitting formulae to derive the dynamical ejecta's characteristics based on the results from over 170 BNS merger numerical simulations.\",\n \"Most recently, [38] introduced a novel method on estimating the individual masses and equation of state of BNS systems from r-process abundance signatures and numerical simulations using Bayesian analysis.\",\n \"[67] is another recent paper in the Bayesian framework, which improves the estimation of the gravitational wave source parameters using the kilonova's lightcurves in the case of BHNS systems.\",\n \"In order to investigate the properties of outflows, it is important to have a reliable method to determine the unbound matter and its evolution.\",\n \"The accurate measurement of the outflow's quantities and geometry can be captured only by 3D hydrodynamics simulations including the r-process heating and neutrino cooling terms in evolution equations ([49], [31]).\",\n \"However, [18] and [31] presented improved versions of Bernoulli's criteria to include the effects of these heating and cooling process approximately, to have a realistic estimation of the unbound matter.\",\n \"In this paper, we utilize the numerical simulations to investigate the evolution and different properties of the disk and the outflows in the presence of magnetic fields and neutrino cooling.\",\n \"We mainly focus on measuring the outflows’ properties such as composition, velocity and mass to estimate the kilonova’s luminosity and lightcurve.\",\n \"We also investigate how the initial analytic disk parameters can affect the features of the post-merger outflows by altering disk's mass and black hole's spin and mass.\",\n \"The range of parameters are chosen to mimic real post-merger remnant disks.\",\n \"This paper is organized as follows, in Sec.\",\n \"we briefly explain the numerical framework and initial setup of our simulations.\",\n \"In Sec.\",\n \"we present the general properties of the outflows and predict the possible kilonova lightcurves powered by each model.\",\n \"We give a discussion over comparison of our results with the previous studies in the literature and also physical and numerical limits of our study in Sec. .\",\n \"The summary and conclusion is given in Sec.\",\n \", and finally, in the appendix we present the results from several tests examining the accuracy of our tracer method for outflow measurements.\",\n \"We use a developed version of the general relativistic magnetohydrodynamic (GRMHD) code HARM ([33], [62]) explained in [71].\",\n \"This version of HARM has been developed to include realistic equation of state as well as neutrino treatment.\",\n \"HARM is a finite-volume code with HLL shock capturing scheme, which solves the partial differential GRMHD equations in the standard Valencia conservation formalism [63], for continuity equation, energy and momentum conservation, $\\\\begin{aligned}\\\\left( \\\\rho u_{\\\\mu } \\\\right)_{;\\\\mu } =0 \\\\\\\\T^{\\\\mu }_{\\\\nu ; \\\\mu } = 0,\\\\end{aligned}$ and induction equation, $\\\\begin{aligned}\\\\partial _t \\\\left( \\\\sqrt{-g} B^i \\\\right) = -\\\\partial _j \\\\left(\\\\sqrt{-g} \\\\left(b^j u^i - b^i u^j\\\\right)\\\\right).\\\\end{aligned}$ Here $T$ is the energy-momentum tensor with both matter and electromagnetic contributions $T^{\\\\mu \\\\nu } = T^{\\\\mu \\\\nu }_{gas} + T^{\\\\mu \\\\nu }_{EM}$ , $u_\\\\mu $ is the four-velocity, $B$ is the magnetic field and $b^i \\\\equiv (B^i + B^i u^{\\\\mu } g_{i \\\\mu }u^i)/u^t$ .\",\n \"The metric is frozen and fixed to the Kerr metric.\",\n \"The hydro equations are evolved in the modified spherical Kerr-Schild coordinates, where the following radial and angular maps are applied to decrease the grid spacing close to the black hole and the equatorial plane, respectively to improve the accuracy.\",\n \"$\\\\begin{aligned}& r_{KS} = R_0 + e^{r_{MKS}} \\\\\\\\& \\\\theta = \\\\pi x^{[2]} + \\\\frac{(1-h)}{2}sin(2\\\\pi x^{[2]})\\\\end{aligned}$ The coordinate parameter $h$ is set to $0.3$ for all models in this paper.\"\n ],\n [\n \"Initial Setup\",\n \"This work can be considered as a follow-up studies of [40] including a wider range of parameters for disk setup.\",\n \"We choose the initial disk mass in the range of [0.05,0.3]$M_{\\\\odot }$ to match with a realist post-merger remnant disks from a black hole-neutron star (BHNS) or neutron star binary system (BNS) [51].\",\n \"In order to investigate the spin effects on the disk ejecta, we alter the spin of the black hole (BH) for a few cases of prograde and one retrograde case.\",\n \"The mass of the BH has been set to $\\\\sim 2.65M_{\\\\odot }$ for BNS case, and $~5-6M_{\\\\odot }$ for the BHNS cases.\",\n \"In addition, we have a few cases of light black holes (LBH) $1-2M_{\\\\odot }$ , which are predicted to be formed through the collisions of low mass primordial black holes with galactic neutron stars in galactic dark matter halos [2].\",\n \"The details of our different models are given in table REF .\",\n \"The initial state of the torus is derived from the analytic solution of the Fisbone-Mocrief (FM) [27] disk model around a Kerr black hole.\",\n \"The FM disk is defined by $r_{in}$ the inner radius of the disk, $r_{max}$ the radius of maximum pressure, and $a$ dimensionless spin of the black hole.\",\n \"The FM disk free parameters in geometrical units, and the equivalent constant specific angular momentum $l_{FM}$ for each model are given in table REF .\",\n \"The initial magnetic field is confined within the torus with pure poloidal configuration for all the cases defined by vector potential: $A_{\\\\phi } = \\\\frac{\\\\overline{\\\\rho }}{\\\\rho _{max}} - 0.2,$ where $\\\\overline{\\\\rho }$ is the density averaged over density at grid point and the neighboring cells, and $\\\\rho _{max}$ is the maximum density.\",\n \"The strength of the magnetic field is set to $\\\\beta =50$ parameter, where $\\\\beta $ is defined as the ratio of the gas pressure to the magnetic pressure $\\\\beta \\\\equiv P_{g}/P_{B}$ .\",\n \"The resolution of the grid is 384x300 for radial and polar angular direction, respectively for all the models.\"\n ],\n [\n \"Equation of State and Neutrino Treatment\",\n \"For the nucleus equation of state we follow the same approach as [40] to have the plasma composed of free protons, neutrons, electron–positron pairs, and helium nuclei.\",\n \"The fraction of each species is determined by the equilibrium condition assumed between the reactions of electron–positron capture on nucleons, and neutron decays based on prescription given by [69].\",\n \"The gas is in equilibrium, so that the ratio of protons to neutrons satisfies the balance between forward and backward of the following weak nuclear reactions: $p + e^- \\\\rightarrow n + \\\\nu _e$ , $p + \\\\overline{\\\\nu }_e \\\\rightarrow n + e^+$ , $p + e^- + \\\\overline{\\\\nu }_e \\\\rightarrow n$ , $n + e^+ \\\\rightarrow p + \\\\overline{\\\\nu }_e$ , $n + \\\\nu _e \\\\rightarrow p + e^-$ , and $n \\\\rightarrow p + e^- + \\\\overline{\\\\nu }_e$ .\",\n \"We can define the proton-to-baryon number density ratio $Y_p$ (equivalently the electron fraction, for charge neutrality condition) as: $Y_e = (n_{e^-} - n_{e^+})/n_b$ , where $n_b$ is the baryon number density, $n_{e^-}$ and $n_{e^+}$ are electron and positron number densities respectively.\",\n \"In our simulations $Y_e$ is not evolved by time but it determines after each time step by the equilibrium conditions.\",\n \"This quantity is used to study the composition of the outflow winds.\",\n \"Based on the discussion in [79] and [44] the r-process nucleosynthesis is only effective when $Y_e < 0.4$ , and the lanthanide elements are more likely to be produced when $Y_e < 0.25$ in the ejected matter.\",\n \"Therefore the accuracy of $Y_e < 0.25$ measurement is crucial to predict the kilonova's features.\",\n \"For the neutrino treatment, we applied the same approach as [40] where the absorption optical depth for different neutrino species is given by an approximation ([19]), $\\\\tau _{a,\\\\nu _i} = \\\\frac{H}{4 \\\\frac{7}{8} \\\\sigma T^4} q_{a,\\\\nu _i},$ where $q_{a,\\\\nu _i}$ is the absorption rate derived by summation over the absorption rates from the weak interactions mentioned above, $T$ is temperature, $\\\\sigma $ is the Stefan-Bolzmann constant, and $H$ is disk's height.\",\n \"The lepton flavour considered here for neutrino species are electron, $\\\\mu $ and $\\\\tau $ .\",\n \"The scattering optical depth is estimated as: $\\\\begin{aligned}& \\\\tau _{s} = \\\\tau _{s,p} + \\\\tau _{s,n} \\\\\\\\& = 24.28 \\\\times 10^{-5} \\\\left[ \\\\left(\\\\frac{kT}{m_e c^2})\\\\right)^2 H \\\\left( C_{s,p}n_p + C_{s,n}n_n \\\\right)\\\\right],\\\\end{aligned}$ where $C_{s,p}=[4(C_V-1)^2+5\\\\alpha ^2]/24, C_{s,n}=(1+5 \\\\alpha ^2)/24, C_V=1/2+2~sin^2 \\\\theta _C$ , with $\\\\alpha =1.25$ and $sin^2 \\\\theta _C = 0.23$ The neutrino cooling rate is computed as: $Q_{\\\\nu } = \\\\frac{(7/8) \\\\sigma T^4}{(3/4)} \\\\sum _{i=e,\\\\mu ,\\\\tau }\\\\frac{1}{0.5(\\\\tau _{a,i}+\\\\tau _{s}) + 1/\\\\sqrt{3} +1/(3\\\\tau _{\\\\tau _{a,i}})}$ The detailed discussions over nuclear equation of state and neutrino treatment implemented in HARM-COOL code are given in [43], [41], [40].\",\n \"Table: Different disk setup for numerical simulations.\"\n ],\n [\n \"Outflows: Mass, Composition, Velocity and Geometry\",\n \"Measuring the general properties of the outflows such as mass, velocity and composition, with high accuracy is a crucial task to predict the observable kilonovae lightcurves.\",\n \"Using the particle tracer technique would allow us to learn about the geometry of the outflows as well.\",\n \"In this section, we present our quantitative measurements for outflow properties using the tracer method, focusing on the BH spin effects and also BH and disk masses' effects on these properties.\",\n \"At the end of this section, we use these results to estimate the time, luminosity and temperature peaks, and lightcurves of the kilonova emitting from each model.\",\n \"The total outflow mass, average velocity and electron fraction, measured by tracers at $r=800 r_g$ , are given in Table REF .\",\n \"Figs REF -REF show mass distribution histograms versus velocity, electron fraction and polar angle $\\\\theta $ for models with different spin and mass configurations.\",\n \"A quick observation from these quantities show that all the magnetized-neutrino cooled disk models generate ejecta with moderate velocity $v \\\\sim 0.1-0.2$ c, and high electron fraction $Y_e > 0.25$ for regular BNS and BHNS mass configurations, which perfectly describes the second channel of the kilonova emission sources from the postmerger torus producing bright and blue transients in a few hours after merger.\",\n \"The results for LBH cases are different regarding the composition which is discussed in Sec.\",\n \"REF and REF .\",\n \"Focusing on more details in our results show that the remnant BH spin plays an important role in outflows features.\",\n \"The models labeled as M1.0-0.14-a0.98, M1.0-0.14-a0.9, M1.0-0.14-a0.6 and M1.0-0.14-a0.2 all have identical initial parameters except for BH spin.\",\n \"As shown in Table REF the mass of ejecta is hugely affected by the BH spin.\",\n \"The mass increases monotonically by the BH spin and the difference may reach to more than three orders of magnitude between the low-spin case M1.0-0.14-a0.2 and extremely high-spin case M1.0-0.14-a0.98.\",\n \"This is consistent with previous GRMHD numerical studies observing that high spin BH cases with robust magnetically collimated jets produce more massive outflows ([36]).\",\n \"Similar observation about massive ejecta has been reported by [24] for viscous accretion disks with highly spinning black holes.\",\n \"The significant mass-loss is due to energy release by accretion happening deeper in the gravitational potential and closer to the BH singularity.\",\n \"This conclusion may let us rule out the merger scenarios with zero or low-spin remnants from the parameter estimation for bright kilonova observations.\",\n \"Moreover, the 2D profiles of the electron fraction given in Fig.\",\n \"REF and the average $Y_e$ in Table.\",\n \"REF , as well as the histograms in Fig.\",\n \"REF derived from SkyNet code for r-process nucleosynthesis for models with LBH ($M_{BH} = 1.0 M_{\\\\odot }$ ) and different spins show a dramatic change in the outflow compositions.\",\n \"Generally, the outflows are more neutron-rich for models with higher BH spin (with average $Y_e < 0.25$ ), which leads to more opaque lanthanide-rich material.\",\n \"This observation is in contrast with [40] and [24], which both found that the electron fraction will increase as the BH spin increases for stellar mass BH.\",\n \"The argument for their observation is, when the accretion happens in a deeper gravitational potential, it releases more energy and heat the inner part of the disk, therefore the weak interactions' equilibrium condition change in a way to release more neutrinos for more effective neutrino cooling.\",\n \"This causes producing more proton-rich plasma.\",\n \"However, in our equation of state, the partial trapping of neutrinos is taken into account (see the details in [40] for the calculation of trapped neutrino pressure), so the weak interaction rates are affected by neutrino trapping.\",\n \"In fact, a closer look into some quantities such as neutrino luminosity and neutrino emissivity show that the highest spin case, M1.0-0.14-a0.98, goes through a less effective neutrino cooling during the evolution compared with M1.0-0.14-a0.9 case.\",\n \"However, more cases with LBH and stellar mass BH configurations with different spins are needed in future studies to investigate this effect more closely.\",\n \"Similarly for velocity, we observe significant changes imposed by BH spin.\",\n \"Fig.\",\n \"REF shows the high spin cases such as M1.0-0.14-a0.98 and M1.0-0.14-a0.9 generate outflows with higher and broader range of velocities.\",\n \"However, the average velocity does not increase with BH spin monotonically as our highest spin case M1.0-0.14-a0.98 has the average velocity slightly lower than the M1.0-0.14-a0.9 case.\",\n \"From the geometrical point of view, as shown in Fig.\",\n \"REF all cases with different spins have very broad range of ejected angles.\",\n \"Almost all the cases have a distinguishable peak around the equator $\\\\theta \\\\sim 80^{\\\\circ }-100^{\\\\circ }$ , however it is interesting to mention that the symmetry is somehow broken for some cases, i.e.\",\n \"significantly more wind ejected from the northern hemisphere than the southern hemisphere and vise versa.\",\n \"This geometrical pattern is almost visible for all the cases regardless of their BH spins.\"\n ],\n [\n \"Prograde versus retrograde\",\n \"For a complete comparison, we perform similar analysis between one prograde case with moderate spin $a=0.6$ labeled as M6.0-0.14-a0.6 and one retrograde case with spin $a=-0.6$ labeled as M6.0-0.14-aR0.6.\",\n \"Both cases have BH mass equals to $6M_{\\\\odot }$ and disk mass equals to $0.14M_{\\\\odot }$ , however the initial disk parameters such the inner radius and the constant specific angular momentum for each case is different for each model based on the nature of the Fishbone-Mocrief solution (see [50] for possible retrograde disk solutions).\",\n \"So, the disks are not identical in term of size and compactness.\",\n \"The comparison between these two models show that the retrograde case generates much lower mass (by one order of magnitude), and faster ejecta.\",\n \"However the higher velocity can be explained by that fact this disk is initially larger than the prograde case (larger initial specific FM angular momentum) and therefore less bounded to the BH, and produce higher speed ejecta.\",\n \"On the other hand, as shown in Figs.\",\n \"REF -REF the retrograde case has a narrow range of velocities and more symmetric geometry with more outflows leaving the grid around the equator compared with the prograde case.\",\n \"Overall, one may conclude that the negative spin can provide distinguishable changes in the amount of the outflows, its velocity and geometry.\",\n \"Figure: Electron fraction profiles for models with different BH spins at final time snapshot (t∼0.247t \\\\sim 0.247s).Figure: Density profiles for models with different BH mass and disk mass at final time snapshot.Figure: The mass distribution versus polar angle, comparison for cases with spin and mass configurations as Fig.\",\n \".\"\n ],\n [\n \"Effects of BH and Disk Masses\",\n \"We found out the postmerger remnant disk's mass has a significant impact on the outflow properties as well.\",\n \"However the changes are less dramatic compared to the spin effects.\",\n \"The outflow becomes more massive by more than one order of magnitude as move from M2.0-0.05-0.9 case with $M_{disk} = 0.05 M_{\\\\odot }$ to the highest disk mass case M5.0-0.3-0.9 with $M_{disk} = 0.3 M_{\\\\odot }$ .\",\n \"This obviously suggests that mergers with massive remnants is more likely to have observable EM counterpart.\",\n \"Such scenarios are possible from BNS mergers and BHNS mergers with low mass ratio, softer neutron star's equation of state and high spin BH.\",\n \"The histogram plots in Figs.\",\n \"REF -REF show the massive disk M5.0-0.3-0.9 produce massive ejecta over a broad range of velocities $0.01-0.3$ c, but also with higher mass distribution over the lower velocities which makes the larger part of ejecta from this case slower than the others.\",\n \"The broken symmetry in the outflows geometry is still visible in Fig.\",\n \"REF , and overall we observe the outflows are ejected over a broad range of angles except the poles.\",\n \"This feature is not affected by the disk mass and/or BH mass.\",\n \"Moreover, we observe the composition of the winds is affected by the mass of the BH quite significantly.\",\n \"As mentioned in Sec.\",\n \"REF , for the LBH cases, the ejecta contains more neutron-rich material with lower average electron fraction $Y_e < 0.25$ , while for the systems with stellar mass central BH, the outflow is dominated by less neutron-rich material ($Y_e > 0.25$ ) and therefore more lanthanide-free composition.\",\n \"As a result, one might expect to observe only red and IR transients from accretion disk systems with central LBH.\",\n \"This feature can be considered as an important signature to estimate the mass of the central postmerger remnant, and therefore an evidence for the existence of the primordial BH progenitors for such systems ([12]).\",\n \"At this point, one might wonder how realistic these masses and geometries are compared with the remnant disks from merger simulations.\",\n \"Our selected models can present different BNS and BHNS with different parameters.\",\n \"[68] and [13] derived fitting formula calculating the masses of ejecta and remnant disk for BNS systems based on numerical simulations results.\",\n \"Later, [51] suggested a simpler version of this formula based on the same data base.\",\n \"Similar analytic fitting formulae were proposed by [29], [46], [30] for BHNS systems.\",\n \"Using this data base and reversing this analysis to estimate the merger parameters from the black hole and accretion disk's masses, we can make the following statement about our models.\",\n \"For instance, the M5.0-0.3-a0.9 presents the remnants from a BHNS merger with mass ratio $Q \\\\sim 3.3$ , with $M_{BH}=4M_\\\\odot $ , $M_{NS}=1.2M_\\\\odot $ , and $R_{NS} = 13.2$ km.\",\n \"Such NS can be explained by a stiff equation of state for a non-rotating spherical star such as DD2 or BHB.\",\n \"Our FM model gives a disk with mass of $M_{disk}=0.3M_{\\\\odot }$ and the radius of the maximum density at $r\\\\sim 100$ km.\",\n \"Model M2.65-0.1-a0.9, resembles the remnants from a BNS merger with identical mass $M_{NS}=1.2 M_{\\\\odot }$ and $R_{NS} = 11.7$ km.\",\n \"Such NS can be explained by a soft equation of state for a non-rotating spherical star such as SRO and APR.\",\n \"The FM solution creates a disk with mass of $M_{disk}=0.1 M_{\\\\odot }$ and the radius of the maximum density at $r \\\\sim 44$ km.\",\n \"Comparing against literature, for compact binary mergers, the remnant disk's outer radius is usually around 100-200km with maximum density around 50-60km depending on the initial parameters of the merger ([17], [23]).\",\n \"Since in the FM solution, the size of the disk is scaled by the mass of the BH, this solution may create disks with larger radius.\",\n \"Therefore, our disk models resembling BHNS mergers are larger and less compact compared with final disks from merger simulations.\",\n \"This difference can cause deviations in the mass, velocity and composition of the outflows launching from the disks with different geometries, and non-axisymmetric configurations expected for disks from mergers.\",\n \"Regardless of BH spin and mass configurations, we see similar pattern in the ejecta's mass evolved during time.\",\n \"Fig.\",\n \"REF shows the evolution of the cumulative ejecta's mass launched from selected models.\",\n \"As illustrated in this figure, the ejecta is being developed shortly after the simulations start, increases exponentially during the first half of the evolution, and then continues increasing with a slower pace for the rest of the simulation.\",\n \"This exponential growth in outflow mass at early time is consistent with the GRMHD model reported by [26], and it is a characteristic of the magnetized disk models compared with the $\\\\alpha $ -disk models.\",\n \"The detailed discussion over outflow measurements and how it is affected by the computational grid resolution and tracer setups is given in the appendix .\",\n \"Figure: The mass distribution versus polar angle, comparing cases with different disk and black hole mass configurations.Figure: Evolution of the cumulative outflow mass launched from the disk measured by tracers.\",\n \"Time is given in the code unit in this figure.\"\n ],\n [\n \"Kilonovae peak properties\",\n \"In order to investigate some important kilonovae properties based on measured outflows, we use the approximations given by [34] and [20].\",\n \"The time $t_{peak}$ at which the peak occurs, the luminosity $L_{peak}$ at this time, and the corresponding temperature $T_{peak}$ are estimated as: $t_{peak} = 4.9~d \\\\times \\\\left( \\\\frac{M_{ej}}{10^{-2}M_\\\\odot } \\\\right) ^{\\\\frac{1}{2}} \\\\left( \\\\frac{\\\\kappa }{10~cm^2g^{-1}} \\\\right) ^{\\\\frac{1}{2}} \\\\left( \\\\frac{v_{ej}}{0.1} \\\\right)^{-\\\\frac{1}{2}},$ $\\\\begin{aligned}L_{peak} = 2.5 \\\\times 10^{40}~\\\\text{erg/s}~\\\\times \\\\left( \\\\frac{M_{ej}}{10^{-2}M_\\\\odot } \\\\right) ^{1-\\\\frac{\\\\alpha }{2}} \\\\\\\\\\\\left( \\\\frac{\\\\kappa }{10~cm^2g^{-1}} \\\\right) ^{-\\\\frac{\\\\alpha }{2}} \\\\left( \\\\frac{v_{ej}}{0.1} \\\\right)^{-\\\\frac{\\\\alpha }{2}},\\\\\\\\\\\\end{aligned}$ $\\\\begin{aligned}T_{peak} = 2200~K \\\\times \\\\left(\\\\frac{M_{ej}}{10^{-2}M_\\\\odot } \\\\right) ^{-\\\\frac{\\\\alpha }{8}} \\\\\\\\\\\\left( \\\\frac{\\\\kappa }{10~cm^2g^{-1}} \\\\right) ^{-\\\\frac{\\\\alpha +2}{8}} \\\\left( \\\\frac{v_{ej}}{0.1} \\\\right)^{-\\\\frac{\\\\alpha -2}{8}},\\\\end{aligned}$ where $\\\\kappa = 1 cm^2g^{-1}$ is the average opacity suggested by [34] for less opaque material produced by weak r-process, and $\\\\alpha = 1.3$ .\",\n \"The results of these peak values are given in table REF .\",\n \"Based on these analytic fitting formulae, the majority of our models power EM transients with their peaks in a few hours after merger and the luminosity around $\\\\sim 10^{40}-10^{41} erg/s$ .\",\n \"Table: Properties of the outflows and kilonovae peaks.\"\n ],\n [\n \"Lightcurves and r-process nucleosynthesis\",\n \"In order to estimate the possible lightcurves powered by the ejecta from different models, we consider the same approach as [46].\",\n \"As it is shown by [20] this approximation agrees well with the results given by the radiative transfer simulations for the dynamical ejecta.\",\n \"However, this analytic fitting overestimates the luminosity at the peak time (see Fig.8 from [20] for comparisons).\",\n \"Therefore, we use the [34] fitting formula to estimate the peak values for a better approximation.\",\n \"The luminosity curve is given by $L_{bol}(t) = (1+\\\\theta _{ej}) \\\\epsilon _{th} \\\\dot{ \\\\epsilon }_{0} M_{ej} {\\\\left\\\\lbrace \\\\begin{array}{ll} \\\\frac{t}{t_c} \\\\left( \\\\frac{t}{1~d} \\\\right)^{-\\\\alpha }, & t \\\\leqslant t_c \\\\\\\\\\\\left( \\\\frac{t}{1~d} \\\\right)^{-\\\\alpha }, & t > t_c\\\\end{array}\\\\right.\",\n \"}$ where $\\\\dot{\\\\epsilon }=1.58 \\\\times 10^{10}~erg~g^{-1}~s^{-1}$ is the specific heating rate, $\\\\epsilon _{th}$ is the efficiency of thermalization introduced by [59], $0.5 < \\\\epsilon _{th} < 1$ and $t_c$ is derived by $t_c = \\\\sqrt{\\\\frac{\\\\theta _{ej} \\\\kappa M_{ej}}{2\\\\phi _{ej}(v_{max}-v_{min})}},$ with $v_{max}$ and $v_{min}$ are the maximum and minimum speed of the ejecta.\",\n \"$\\\\theta _{ej}$ and $\\\\phi _{ej}$ are geometrical parameters of the outflows.\",\n \"It was shown by [20] that if we assume the ejecta as homogeneously distributed material moving with velocity $v_{ej}$ in a $\\\\rho -z$ plane, the polar and azimuthal opening angles $\\\\theta _{ej}$ and $\\\\phi _{ej}$ are related to the velocity components with: $\\\\theta _{ej} \\\\approx \\\\frac{2^{4/3} v^2_{\\\\rho } - 2^{2/3} \\\\left(v^2_{\\\\rho } \\\\left(3v_z + \\\\sqrt{9v^2_z+4v^2_{\\\\rho }}\\\\right) \\\\right)^{2/3}}{\\\\left(v^5_{\\\\rho } \\\\left(3v_z+\\\\sqrt{9v^2_z+4v^2_{\\\\rho }}\\\\right)\\\\right)^{1/3}},$ $\\\\phi _{ej} = 4 \\\\theta _{ej} + \\\\frac{\\\\pi }{2}.$ Fig.\",\n \"REF shows the predicted lighcurves from this method.\",\n \"As one expects, M1.0-0.14-a0.98 and M5.0-0.3-a0.9 cases with the highest mass outflows produce brighter transients.\",\n \"There are two pairs of models launching outflows with similar features and therefore generates lightcurves following each other closely, M2.65-0.1-a0.9 and M1.0-0.14-a0.9, and the other pair is M2.0-0.05-a0.9 and M1.0-0.14-a0.6.\",\n \"Our models suggest that although bright blue transients is possible to be observed from light BH-accretion disk systems, they may not be distinguishable from BNS mergers.\",\n \"Overall, the possible degeneracy make the mergers parameter estimation very challenging from kilonova observations.\",\n \"Figure: Luminosity versus time estimated by Kawaguchi's method for different cases.As the next post-process analysis, we have applied the r-process nucleosynthesis on the simulations outputs using the open source SkyNet code ([55]) to measure the nuclear abundances.\",\n \"The results of these simulation are illustrated in Fig.\",\n \"REF by taking the average over all the tracers and compared with the Solar system abundances.\",\n \"The results show the 2nd and 3rd r-process abundance peaks are resolved well in our models.\",\n \"We would like to emphasise again that the composition analysis discussed in sections.\",\n \"REF and REF for Figs.\",\n \"REF and REF are from these skynet simulations, and these figures showing the composition of the ejecta at the onset of the r-process.\",\n \"Figure: Nuclear abundances as a function of mass number A foreach simulation, based on average of tracers sampling the outflows.\"\n ],\n [\n \"Comparison with previous studies\",\n \"There are two major mechanisms driving postmerger outflows, one is neutrino heating and the other is magnetically driven turbulence.\",\n \"The analytical estimation by [64] indicated that in a post-merger accretion disk neutrino heating timescale is short enough to drive a wind within the lifetime of the disk.\",\n \"On the other hand, in the case of a weakly magnetized accretion disk, MRI can be triggered in the disk's dynamical timescale, resulting in magnetic field amplification and turbulence.\",\n \"The plasma is being accreted to the BH and wind is originated from the disk, while the angular momentum is transported by magnetic turbulence effects ([4]).\",\n \"[74] has shown that the expected outflow mass from a BNS postmerger remnant disk has to be around $\\\\sim 0.03 M_\\\\odot $ to power emissions observed in AT2017gfo (kilonova emission observed right after GW170817) as predicted by the r-process nucleosynthesis.\",\n \"They also showed that both optical and near-infrared emissions are simultaneously reproduced by the ejecta with a medium $Y_e$ of 0.25.\",\n \"The recent Bayesian analysis by [70] constrained a mass ratio of $M_w/M_d = 2.81$ (the ratio of wind mass to dynamical ejecta mass) to reproduce the observed AT2017gfo kilonova lightcurves while it is also consistent with the observed r-process elements abundance measured in the Solar system.\",\n \"While the most perfect way to study kilonovae in a numerical simulation is evolving BNS and/or BHNS systems for late inspiral, merger and postmerger phases, measuring dynamical and postmeger ejecta, most recent studies focused only on one side of the scenario.\",\n \"In the case of disk winds, ideally we want to evolve the magnetized remnant torus in a high-resolution 3D grid for a long period ($\\\\sim 10$ seconds), however due to the limit of computational resources, long-term 2D simulations with the $\\\\alpha -$ viscosity prescription as the main mechanism to produce the outflows were initially preferred [24].\",\n \"Nonetheless, there are a few studies in the literature, which evolved magnetized remnants with neutrino cooling in fully GRMHD simulations within a multi-second timescales [26], [37].\",\n \"Generally, in the outflow studies, some groups considered analytical disk as the initial data configuration and then perturbed them by weak magnetic field (similar to our study in this paper) or $\\\\alpha $ viscosity, while others deal with remnant disk from a BHNS or BNS simulations, applying Cowling approximation (frozen space-time) at $t \\\\sim 10-15$ ms after merger.\",\n \"At this point, it would be interesting to compare our results with these studies, as well as the predicted values from the kilonova observations.\",\n \"For obvious reasons, the outflow properties are highly case dependent.\",\n \"The wind mass reported by different groups typically varied from $10^{-5}$ to $10^{-2} M_{\\\\odot }$ .\",\n \"We found most of our cases produce $\\\\sim 10^{-4}-10^{-3} M_{\\\\odot }$ , which are lower by one to two orders of magnitude compared with the predicted values for AT2017gfo observation.\",\n \"The outflow mass might be underestimated as a results of various reasons.\",\n \"The full discussion on our tracer's setup and alternative measurements is given in Appendix .\",\n \"Comparing our models with similar cases in the literature, our results confirm the conclusion made by [44] for $\\\\alpha $ -viscosity disk models.\",\n \"As they mentioned in the case of the most mass massive disk $a0.8_{M0.3}$ (disk mass$\\\\sim 0.3 M_{\\\\odot }$ and BH spin $a=0.8$ ), the outflow mass increases by almost factor of 4 compared to their $\\\\sim 0.1 M_{\\\\odot }$ case, but the average velocity becomes slightly smaller.\",\n \"For similar cases, we observed the massive disk, M5.0-0.3-a0.9, provides the highest outflow mass compared to the intermediate mass cases such as M2.65-0.1-0.9, but in our models the difference is quite dramatic (about one order of magnitude), while the velocity is slightly bigger.\",\n \"However, the outflow of their $a0.8_{M0.3}$ case is still much higher than our M5.0-0.3-a0.9 by factor of 2.\",\n \"Our M2.65-0.1-a0.9 case can be compared with HS-Therm and HS-Magn models from [40].\",\n \"This case is initially moderately magnetized and the comparison of this case with Table(2) from the same reference indicates that the outflows properties are scaled by the initial magnetic field strength, i. e. the outflow mass and velocity increase as the initial magnetic field increases.\",\n \"Also, [40] found that the average electron fraction in the outflows depended on the disk magnetisation, especially at the earlier time.\",\n \"Though, the average electron fraction we calculated for M2.65-0.1-a0.9 case is closer to the highly magnetized case 'HS-Magn' from this reference.\",\n \"Here, we measured the averaged electron fraction around $0.1<Y_e<0.3$ , which is consistent with the GRMHD model in [26], and it is relatively lower than the averaged value reported by [35] ($0.3 < Y_e <0.39$ ) for viscous disks.\",\n \"Our results also show that outflow mass increases sharply at early times, matching well with GRMHD model, while the viscous disks have this large growth at much later times in [26].\",\n \"Though, the outflow masses are still lower in our models, which can be caused by the time of evolution and differences in disk parameters and equation of state (more analysis is given in the Appendix REF ).\",\n \"The outflow velocity measurements agree well with the previous studies and the theoretical models' prediction for the disk winds.\",\n \"The velocity range is around $0.1-0.2$ c and is generally higher for systems with higher spins.\",\n \"Regarding the geometry, the outflows launch over a wide range of angle which agrees with results reported in [26] and [22].\",\n \"However, our results show less equatorial symmetry compared with these studies.\"\n ],\n [\n \"Parameter estimation, binary population synthesis and spin effects\",\n \"Based on population synthesis studies, the GW from BNS mergers are more likely to be observed with EM counterparts ([57]).\",\n \"As already discussed in the literature there are some features in the kilonova emissions, which help to distinguish the BHNS from BNS mergers and make some parameter estimations.\",\n \"For instance, [44] claimed that there is a rapid decline in the lightcurves from a BHNS postmerger ejecta (or equivalently, a prompt collapse for a BNS merger).\",\n \"Unfortunately, our current models do not allow us to investigate different possible scenarios for BNS mergers, such as creation of a magnetized and diferentially rotating HMNS with different lifetimes, which can affect the amount of ejected matter significantly ( [16]).\",\n \"However, even the results from [44] show that the light curves of models with high spin BH and HMNS with 30ms lifetime are quite similar, and this degeneracy introduces a challenge in using kilonova observations to estimate the lifetime of a HMNS remnant.\",\n \"For the BHNS merger cases, two signals GW200105 and GW200115 have been detected by the LIGO-Virgo collaborations so far.\",\n \"The masses of GW200115 and GW200105 are 8.9M and 1.9M, and 5.7M and 1.5M, respectively.\",\n \"The spin of the black hole in GW200115 is not tightly constrained but it is estimated to be $\\\\sim -0.5-0.04$ and most likely misaligned.\",\n \"The dimensionless spin magnitude of the black hole in GW200105 is estimated to be $< 0.2$ and its direction is unconstrained ([1]).\",\n \"The spins of the remnant central BHs are not constrained up to certain points, but they are likely around $\\\\sim 0.38$ and $0.43$ for GW200105 and GW200115 respectively ([9]) However, no EM counterpart have been observed from these mergers.\",\n \"The discussion given by [72] indicated that only BHNS with highly spinning black holes can generate massive dynamical ejecta (the difference may exceed three orders of magnitude from low spin ($\\\\sim 0.2$ ) to high spin $\\\\sim 0.9$ ).\",\n \"This high spin can be obtained by two mechanisms during a BHNS binary formation: i) inheritance due to weak core-envelope coupling of the stellar progenitors, and ii) accretion from the companion star during stable mass transfer.\",\n \"However, population synthesis studies of merger rates of BHNSs suggest that the majority of binaries will not result in observable EM counterparts.\",\n \"Studies by [21] found that only a fraction ($\\\\sim 20\\\\%$ ) of BHNS binaries gain a high dimensionless BH spin from their stellar progenitors and produce massive ejecta during merger.\",\n \"For postmerger outflows, we obtained similar trends in our results, i.e.\",\n \"the disk with lower spin BH generates lower mass ejecta, up to two orders of magnitude compared to high spin cases.\",\n \"We also concluded that the ejecta mass may become lower for the retrograde accretion.\",\n \"However the mass of the postmerger remnant disk can be another important factor for launching the ejecta, which is highly dependent on the mass ratio, NS equation of state and BH spin ([32], [56]).\",\n \"Simulations done by [56] show that the massive remnant disk is likely to be originated from BHNS mergers with high spin BH.\",\n \"Overall, our models confirm that almost not much ejecta and no EM counterparts are expected to be observed from these BHNS systems, unless for BHs with high spins and lower mass ratio cases providing massive postmerger disks.\"\n ],\n [\n \"Outflow measurements and other important physical components\",\n \"In this section we present a discussion about important numerical and physical elements which are ignored in our simulations and can affect our outflow measurements significantly.\",\n \"First, it is worth mentioning that the accuracy of our outflow studies are affected by the current version of neutrino and equation of state treatments.\",\n \"Neutrino emissions can affect the disk's thermal and composition evolution significantly at the early time of evolution after merger ([39], [26]).\",\n \"Therefore, a more advanced neutrino treatment such as leakage or neutrino radiation transport schemes along with composition evolution are needed to provide a better accuracy.\",\n \"Our equation of state does not include nuclei heavier than Helium, however [35] and [22] showed the inclusion of these nuclei in the equation of state may cause significant changes in the outflow's mass and velocity.\",\n \"In addition, the numerical scheme and grid resolution typically have impacts on the modelling of the kilonova emission.\",\n \"In the study by [60] the authors compared the dynamical ejecta properties measured from 4th-order and regular 2nd-order finite difference schemes for magnetized BNS simulations.\",\n \"They found the second-order scheme overestimates the amount of proton-rich shock-heated ejecta.\",\n \"Also, [11] performed convergence studies on the outflows from a BNS merger simulation, and they found out the highest resolution resolve the shock-heated plasma more accurately.\",\n \"The postmerger ejecta can be affected in a similar way.\",\n \"Fully-resolved MRI effects requires high accuracy in numerical scheme and high resolution grid, and both can impact the magnetically-driven viscous heating effects which leads to changes in the composition and other features of the disk and the outflows (see Appendix REF about the resolution test.)\",\n \"For the sake of electromagnetic lightcurve estimations, we use here a simple fitting formula, which can be correct only up to the order of magnitude.\",\n \"In our simple approach we assumed a constant value for the average opcacity of the ejected matter, which in reality varies a lot depending on the geometry and composition of Lanthanide/Actinide-rich matter.\",\n \"In order to predict the lightcurves and spectrum of the emissions accurately, one needs to apply a radioactive transfer code on the ejecta.\",\n \"Such studies have been done recently e.g.\",\n \"by [78].\",\n \"One crucial thermal component in studying outflows is the inclusion of the r-process heating and neutrino cooling terms in the long-term hydro simulations (see the discussion and eqs.\",\n \"(20-21) from [31]).\",\n \"Recent study by [35] for long-term 2D viscous disk simulations indicated that adding these terms to hydro equations after one second evolution increases the outflow mass by 30%.\",\n \"Another important physics to be considered for kilonova studies is neutrino's flavour oscillations.\",\n \"A few groups recently studied the effects of fast flavour instability (FFI) on the disk outflows in their simulations ([54], [25]).\",\n \"Neutrino flavor oscillation is reported to have moderate (or large, in some cases), impacts on the mass ejection, average velocity, and average electron fraction.\",\n \"Even with very accurate numerical evolution and measurements, there are possible ways to make our results different from the observations.\",\n \"First, the measured abundances can be largely affected by the nuclear uncertainties in the r-process models, especially at late-time emissions, $t>100$ days ([52]).\",\n \"Second, in a realistic merger and post-merger scenario, the emissions from disk winds can be affected by the dynamical ejecta.\",\n \"According to the discussion by [44], dynamical ejecta is generally faster and more neutron-rich and therefore more opaque, which acts like a 'Lanthanide curtain’, masking the emissions originating from the wind ejecta.\",\n \"They observed that for an edge-on orientation, the dynamical ejecta partially blocks the wind's optical flux by an order of magnitude.\"\n ],\n [\n \"Conclusions and Summary\",\n \"We have carried out two-dimensional simulations of several black hole-accretion disk models with different initial setups using the GRMHD HARM-COOL code.\",\n \"These simulations include both magnetic field evolution and neutrino emission effects, as well as realistic nuclear equation of state.\",\n \"The initial magnetic field has been seeded in poloidal loops configuration confined within the plasma.\",\n \"We evolved these models about $t \\\\sim 200$ ms, and used particle tracers to measure the outflows properties.\",\n \"We observed there is strong correlation between BH spin and the outflows properties.\",\n \"Generally, disks with higher mass and BH spin generate faster and more massive outflows.\",\n \"We observed our models generate winds with moderate velocity (v/c $\\\\sim 0.1-0.24$ ) and a broad range of electron fraction ($Y_e \\\\sim 0.01-0.5$ ), which are consistent with previous studies on GRMHD postmerger accretion disk simulations.\",\n \"We have included a few cases with lower mass black holes representing BH formed from primordial black hole and neutron star mergers, and we found out such cases generate more neutron-rich ejecta in comparison with regular BHNS and BNS mergers.\",\n \"Generally, the outflow masses measured by tracers are lower compared with long viscous disk and GRMHD simulations.\",\n \"We have investigated the accuracy of the outflow measurements with tracer method by altering the tracer's density threshold and extraction radius, and also grid resolution (explained in the Appendix ).\",\n \"We found that our measurements for ejecta's mass can be affected up to $50\\\\%$ by this alterations.\",\n \"However, we consider this method to be more accurate than the unbound matter estimation from the geodesic and Bernoulli criteria.\",\n \"The general properties of postmerger ejecta derived from our simulations give us the opportunity to estimate the luminosity lightcurves of possible radioactively powered transients using analytic fitting formula.\",\n \"We found the luminosity peaks within the range of $\\\\sim 10^{40}-10^{42}$ erg/s, brighter peaks for cases with higher ejecta mass, which agrees with previous studies for neutrino-driven disk wind models.\",\n \"Applying the r-process nucleosynthesis code on our results and comparing against the Solar-system abundances showed that the 2nd and the 3rd r-process abundance peaks are resolved well in our models.\",\n \"In the end, we should point out that for future studies we need to improve our numerical method by including a more sophisticated neutrino treatment such as leakage or transport schemes, as well as composition evolution to capture all the neutrino absorption/emission features.\",\n \"Moreover, including heavy nuclei in the equation of state are required to achieve a more realistic scenario with better accuracy.\",\n \"The authors thank Oleg Korobkin for helpful discussion and advice over the course of this project.\",\n \"This work was supported by grant No.\",\n \"2019/35/B/ST9/04000 from the Polish National Science Center, Poland.\",\n \"We also acknowledge the support from the PL-Grid infrastructure under computational grant plgmicrophysics.\"\n ],\n [\n \"Outflow measurements: methods and accuracy\",\n \"In this appendix first, we investigate the alternative methods to measure the outflow mass and compare them with the results from the tracer method.\",\n \"In the second part, we perform several test simulations for a single case to study the effects of the resolution and tracer's setup parameters on the outflow measurements.\",\n \"At the end, we leave a comment regarding the evolution's time and its effect on our measurements.\"\n ],\n [\n \"Unbound matter estimation: Geodesic and Bernoulli criteria versus tracer methods\",\n \"As mentioned, in this study we use the tracer particle technique to measure the outflows.\",\n \"In this method, all the particle trajectories leaving the outer boundary (or a large radius close to the outer boundary) of the computational grid during the evolution are marked as outflow winds.\",\n \"For the results discussed in Sec.\",\n \"we set this radius at $r=800~r_g$ .\",\n \"The details of the tracer method's implementation in HARM-COOL are given in [40].\",\n \"The tracer particles provide the information on how different quantities such as density, temperature and electron fraction vary while tracking the particles.\",\n \"This information is used for the postprocess calculations of the element abundances from the r-process nucleosynthesis presented in Sec.\",\n \"REF .\",\n \"In addition, a detailed picture of wind's geometry can be obtained from the trajectories of the outflowing particles.\",\n \"Despite the fact that tracers are useful tools for outflow studies, the amount of measured outflows can be possibly underestimated if the system is not evolved long enough to provide enough time for the unbound mass to leave the grid.\",\n \"In fact, $t \\\\approx 200$ ms we have for these simulations can not be considered as a very long evolution to study the disk winds.\",\n \"On the other hand, the evolution of the disk ejecta can be affected by several other factors including the artificial atmosphere controls over the low density regions and the tracers' initial setup parameters.\",\n \"To test the accuracy of our tracers measurements, we apply the other alternative methods from the literature to identify the unbound matter.\",\n \"Referring to the discussion given by [31], the most accurate way to study outflows is a long-term (multi-second) 3D simulation where all the r-process heating and neutrino cooling terms are included in the source terms of the hydrodynamic evolution equations (see Eqs.\",\n \"(20-21) from the same reference for more details).\",\n \"However it is still possible to obtain a reasonable estimation for the outflow mass based on the energy criteria from a short-term simulation.\",\n \"We try three different criteria introduced in this paper and compare them with the results from tracers for our M2.65-0.1-a0.9 case: 1- The geodesic criterion $u_t<-1$ , which is not suitable for hot disk outflows, because it ignores the thermal energy of the fluid and underestimates the unbound mass significantly.\",\n \"2- The Bernoulli crtieria $hu_t < -h_{\\\\infty }$ , which includes the thermal energy but ignores the cooling effects caused by neutrino losses, and therefore overestimates the unbound mass.\",\n \"3- The improved version of the Bernoulli criterion, which includes the r-process heating and neutrino losses by a simple approximation given by $hu_t/h_{\\\\infty }(0.9968+0.0085Y_e) < -1.$ Here $h$ is the enthalpy given by $h=1 + \\\\epsilon + P/\\\\rho $ with $\\\\epsilon $ represent the specific internal energy, $h_{\\\\infty }$ is the asymptotic enthalpy for $\\\\rho \\\\rightarrow 0$ and $\\\\epsilon \\\\rightarrow 0$ (for our case $h_{\\\\infty }=1$ ) and $Y_e$ is the electron fraction.\",\n \"Generally, the improved version of the Bernoulli's criterion can be considered as a more realistic estimation to flag the unbound mass for different merger and post-merger scenarios.\",\n \"The comparison for ejected mass computed from tracer method and estimation from geodesic, Bernoulli and improved Bernoulli criteria calculated at a single time snapshot in the middle of the simulation at $t \\\\sim 100$ ms for $M2.65-0.1-a0.9$ case is given in Table REF .\",\n \"These results show that the outflow mass computed from tracers is lower by about factor of four compared with the improved version of Bernoulli criterion.\",\n \"However, we should remind ourselves that this criterion is still considered as an approximation, and can not be taken as an accurate measurement of the unbound mass.\",\n \"Table: Ejected mass: tracers versus geodesic, Bernoulli and improved Bernoulli criteria (at t∼100t \\\\sim 100ms snapshot).The volume integral is taken from r=400r g r=400~r_g (∼1500\\\\sim 1500km) to the outer boundary.\"\n ],\n [\n \"Tracer's Setup: Grid Resolution, density threshold and extraction radius effects\",\n \"Investigating the effects of the computational gird resolution and tracers setup parameters, we perform several tests for our M2.65-0.1-a0.9 model, and compare their results with the standard M2.65-0.1-a0.9-Std case discussed in Sec. .\",\n \"This is our closest case for a BNS postmerger scenario.\",\n \"The list of tests and the measured ejecta mass and velocity are given in Table REF .\",\n \"For Trac-Low-RhoMin test we alter the threshold density identifying the active tracers during the evolution and reduce this value by two orders of magnitude.\",\n \"The original threshold for the standard case was about $\\\\sim 10^{-4}$ of the maximum density.\",\n \"Trac-1500km is designed to study the effect of the radius where we extract the information about the outflows.\",\n \"This radius has been set to $r=800~r_g$ , which is almost 3000km for the standard case.\",\n \"Finally High-Res test which has the identical setup as M2.65-0.1-a0.9-Std, but with higher grid resolution 480*426 along the radial and angular directions respectively.\",\n \"Figs.\",\n \"REF and REF show the cumulative outflow mass versus time and the mass distribution versus velocity respectively for these tests.\",\n \"The simulation with lower density threshold measures slightly larger mass outflows, with larger mass distribution over the high velocity ranges.\",\n \"This makes the average velocity for this test significantly higher than the other cases.\",\n \"Extraction radius is another important parameter to be explored.\",\n \"Our results show that more massive and slower ejecta are measured by the smaller extraction radius.\",\n \"However, this mass is still lower by more than factor of two compared to the unbound mass measured by the improved Bernoulli criterion reported in Table REF .\",\n \"Specifying the extraction radius can be a challenging job, and it can affect some quantities' measurements such as velocity.\",\n \"In the literature, a smaller extraction radius $\\\\sim 200-500$ km has been used to measure the dynamical ejecta or the ejecta from magnetized postmerger HMNS with tracers (see for example [11] and [16]), and each tracer has to satisfy the geodesic or Bernoulli condition to identify the unbound mass.\",\n \"However the disk wind is a different scenario; the recent studies by [35] showed that if the r-process heating source terms are included in the evolution of the postmerger ejecta during a multi-second simulation, the ejecta's velocity need to be measured at a much larger distance.\",\n \"This study claimed that the ejecta is being continuously accelerated as a result of the r-process heating and it reaches to its asymptotic value around $r \\\\sim 40,000$ km.\",\n \"The High-Res test shows that our measured mass can be affected by the grid resolution up to some extend.\",\n \"For this particular case we have the ejacta mass increased by about 12%.\",\n \"Fig.\",\n \"REF shows that the outflows with higher velocities are overestimated in the lower resolution.\",\n \"The standard resolution of our simulations are high enough to resolve the fastest growing MRI mode almost everywhere in the torus (see Fig.\",\n \"(2) from [40] for a similar case).\",\n \"However, fully resolved MRI modes and capturing all the transport effects of turbulence requires a very high grid resolution.\",\n \"Even high resolution GRMHD simulations reported only a qualitative convergence in capturing all the turbulence's heating effects caused by the magnetorotational instability ([48]).\",\n \"Obviously, higher resolution simulations provide higher accuracy for resolving the heating effects of the turbulent plasma and produce more massive outflows.\",\n \"In conclusion, the results of these tests explain that our outflows measurements are affected by the resolution and the tracers setup parameters up to $50\\\\%$ , but still unable to explain the entire huge gap between the ejected mass from our model and the value predicted from GW170917 kilonova observation.\",\n \"Table: Different resolution and tracers setups for outflow measurements for M2.65-0.1-0.9 case.Figure: The outflows mass distribution versus velocity for M2.65-0.1-a0.9 case with different resolution and tracer setups.\"\n ],\n [\n \"Other important factors for outflow measurements\",\n \"One might argue that the numerical treatments can impose quantitative and qualitative effects on the outflows.\",\n \"Though, our results are less likely to be suffered from the numerical treatment over the atmosphere.\",\n \"In HARM we apply the velocity control only over low-density atmosphere with very high velocity.\",\n \"The disk outflow is dominated by subrelativistic plasma, and therefore, is not influenced significantly by the artificial atmosphere adjustments.\",\n \"The evolution time is another key factor we should take into account.\",\n \"We evolved our models for about $\\\\sim 200$ ms, while the viscous timescale for a thick postmerger accretion disk with $H/R \\\\sim 0.3$ is around $0.3$ s. The evolution of about a few viscous timescales is required to capture all the magnetically and thermally driven effects for launching the outflows.\",\n \"On the other hand, the evolution time is perhaps not long enough, so not all the unbound matter have enough time to leave the grid, though the longer evolution is somehow pointless for our simulation as it is impossible to maintain the magnetic field in a long two-dimensional simulation due to the anti-dynamo theorem.\",\n \"Even from thermal evolution point of view, the timescale of the evolution can be very effective based on studies by [26].\",\n \"They showed that a 3D GRMHD model ejects mass in two ways: one is the MHD-driven outflow at the earlier time of evolution when the torus is NDAF (neutrino-dominated accretion flow), and the second way is late-time, thermally driven wind, which occurs when the disk becomes advection-domianted (ADAF).\",\n \"They showed that the total amount of unbound mass ejected can reach to $0.013 M_{\\\\odot }$ , which is around 40% of the initial disk.\",\n \"Half of this mass lost over the first second of the evolution and the rest has launched thereafter.\",\n \"[40] investigated these types of outflows in two separate models: thermally-driven winds being ejected from an initially weakly magnetized disk and magnetically-driven winds launching from strongly magnetized disk.\",\n \"Computing the time averaged mass loss rate over the outer boundary estimated the final mass loss to be in the range of $2-16\\\\%$ of the initial disk.\",\n \"In comparison, our models are initially moderately magnetized (with $\\\\beta = 50$ ), so they are neither purely thermally-driven nor purely MHD-driven outflows from the very beginning.\",\n \"However, the cumulative outflow mass in Fig.\",\n \"REF shows that the outflow mass increases in a constant and slower pace at the later time of the evolution.\",\n \"Assuming this mass loss rate for later times, by extrapolation, we estimate to obtain about $\\\\sim 0.0035 M_{\\\\odot }$ and $\\\\sim 0.02 M_{\\\\odot }$ mass loss after about 1 second and 9 seconds of evolution respectively; The latter would be around $20\\\\%$ of the initial disk's mass for M2.65-0.1-a0.9 case.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05628"}}},{"rowIdx":81,"cells":{"text":{"kind":"list like","value":[["FactorJoin: A New Cardinality Estimation Framework for Join Queries"],["Abstract Cardinality estimation is one of the most fundamental and challenging problems in query optimization.","Neither classical nor learning-based methods yield satisfactory performance when estimating the cardinality of the join queries.","They either rely on simplified assumptions leading to ineffective cardinality estimates or build large models to understand the data distributions, leading to long planning times and a lack of generalizability across queries.","In this paper, we propose a new framework FactorJoin for estimating join queries.","FactorJoin combines the idea behind the classical join-histogram method to efficiently handle joins with the learning-based methods to accurately capture attribute correlation.","Specifically, FactorJoin scans every table in a DB and builds single-table conditional distributions during an offline preparation phase.","When a join query comes, FactorJoin translates it into a factor graph model over the learned distributions to effectively and efficiently estimate its cardinality.","Unlike existing learning-based methods, FactorJoin does not need to de-normalize joins upfront or require executed query workloads to train the model.","Since it only relies on single-table statistics, FactorJoin has small space overhead and is extremely easy to train and maintain.","In our evaluation, FactorJoin can produce more effective estimates than the previous state-of-the-art learning-based methods, with 40x less estimation latency, 100x smaller model size, and 100x faster training speed at comparable or better accuracy.","In addition, FactorJoin can estimate 10,000 sub-plan queries within one second to optimize the query plan, which is very close to the traditional cardinality estimators in commercial DBMS."],["Introduction","Cardinality estimation (CardEst) is a critical component of modern database query optimizers.","The goal of CardEst is to estimate the result size of each query operator (i.e., filters and joins), allowing the optimizer to select the most efficient join ordering and physical operator implementations.","blackAn ideal CardEst method satisfies several properties: it is effective at generating high-quality query plans, efficient so that it minimizes estimation latency, and easy to deploy in that it has a small model size, fast training times, and the ability to scale with the number of tables and generalize to new queries.","Unfortunately, existing CardEst techniques do not satisfy at least one of these properties.","The reason behind these failures can largely be reduced to the handling of two related problems: (1) how to characterize attribute correlations and (2) how to model the distribution of join-keys, which directly determine the size of joins.","Background: Many deployed CardEst approaches are still based on the Selinger model [60] and assume attribute independence, where the model ignores correlations between attributes, and join-key uniformity, blackwhere the model further assumes that join keys have uniformly distributed values.","These assumptions provide a simple way of estimating the cardinality of join queries using only single-table statistics (e.g., histograms) over single attributes, which are easy to create and maintain.","blackAs a result, the Selinger-based CardEst techniques are extremely efficient and easy to deploy but at the cost of effectiveness because most real-world datasets contain complex attribute correlations and skewed join-key distributions.","To overcome the limitations of these assumptions, many commercial database systems try to blackrelax them to increase their effectiveness.","blackFor example, past works [26], [29], [7] propose and some systems (e.g.","Oracle [8]) employ join-histograms that relax the join-key uniformity assumption.","Specifically, these methods build frequency histograms on the join keys and then construct the join distribution by “multiplying” the histograms to estimate the join size (see Figure REF (b)).","In this way, these methods can more precisely capture the join-key distributions to generate more effective estimates of the join cardinality.","However, they still assume that the join keys are uniformly distributed within each bin of the histogram.","Figure: Existing CardEst approaches for handling join queries.","(a) The Selinger model writes the cardinality of a two-table join query as the product of filter selectivity P A ,P B P_A, P_B on both tables and the estimated join size |A⋈B||A \\bowtie B|.","Using join-key uniformity, this is estimated as the product of table sizes |A|*|B||A| * |B| divided by the maximal number of unique values of the join keys.","(b) The join-histogram bin the domain of join keys Id 1 ,Id 2 Id_1, Id_2 as histograms, assumes join-key uniformity within each bin.","(c) The learned data-driven methods de-normalize J=A⋈BJ = A \\bowtie B and learn the distribution P J P_J to estimate the cardinality of QQ.","(d) Our framework captures the correlation between join keys and filter predicates to accurately estimate |Q||Q| without de-normalization.Similarly, there have been works on relaxing the attribute independence assumption.","For example, multi-dimensional histograms [9], [14], [15], [52], [66], [43], self-tuning histograms [4], [61], [31], [12], and singular value decomposition [58], have been proposed to capture the correlation between attributes.","However, their effectiveness and/or efficiency are still not satisfactory [74], [18].","More recently, learning-based methods which learn the underlying data distributions [33], [23], [77], [74] have been proposed to more accurately and compactly represent the attribute correlations within a single table, but these still have efficiency concerns that we detail below.","Specifically, the learned data-driven methods [13], [64], [73], [70], [77], [25] analyze data and build distributions for all join patterns in a database.","They need to de-normalize the joined tables and add a potentially exponential number of extra columns.","Then, they build distributions over the de-normalized tables to characterize all attribute correlations and handle joins (see also Figure REF (c).","blackThis allows the learned data-driven methods to be highly accurate for join estimates at the cost of slow training time and large model size (i.e., worse deployability).","Alternatively, learned query-driven methods [33], [63], [42] circumvent the join-key uniformity assumption by building supervised models to map the join queries to their cardinalities.","However, they require an impractical number of executed queries to train their models, which is unavailable to new DB instances and do not generalize well to unseen queries.","Our Approach: Surprisingly, we are not aware of any work which tries to combine the advances of learning-based methods for accurately characterizing correlations with the idea of join-histograms.","In this work, we develop a novel CardEst framework called FactorJoin that combines these two approaches, i.e., using join-histograms to efficiently handle joins coupled with learning-based methods to accurately capture attribute correlation.","blackThis is not a straightforward extension.","blackAs we will show, adapting the idea of join-histograms while preserving the benefits of using learning-based methods for attribute correlations is non-trivial.","FactorJoin only builds sophisticated models to capture the correlations within a single table.","The choice of the model to capture correlation is orthogonal to the techniques of FactorJoin, however, we do require the model to be able to provide conditional distributions of one or more keys.","In our current implementation of FactorJoin, we use blacksampling [41] and Bayesian Networks [34] as they are fast to train and execute.","These single-table conditional distributions are then combined using a factor graph model [44].","Specifically, FactorJoin translates a join query $Q$ into a factor graph over single-table data distributions.","This allows us to formulate the problem of estimating the cardinality of $Q$ as a well-studied inference problem [34], [44], [37] on this factor graph.","Conceptually, there exist some similarities to the join-histogram technique.","However, our formulation allows FactorJoin to generalize to cyclic and self-joins.","It also improves the computation efficiency by relying on existing inference algorithms [46], [35], and facilitates scaling to arbitrary join sizes.","Finally, in contrast to the idea of join-histogram and many other techniques in the space, we propose to actually not estimate the expected join cardinality, but rather use a probabilistic upper bound.","As shown by others [5], [3], [1], [22], under-estimation is often worse than over-estimation, and tailoring the estimate to a probabilistic upper bound can significantly improve the end-to-end query performance.","As a result, FactorJoin is able to estimate the cardinality of join queries using only single table statistics with similar or better estimation effectiveness than the state-of-the-art (SOTA) learning-based techniques.","When compared with the learned data-driven methods, FactorJoin has a much smaller model size, faster training/updating speed, and easier for system deployment because we do not need to denormalize the join tables or add exponentially many extra columns.","In addition, FactorJoin supports all forms of equi-joins, including cyclic joins and self joins, as well as complex base table filter predicates, including disjunctive filter clauses and string pattern matching predicates blackbecause it can flexibly plug-in various types of base-table estimators to support them.","These queries are not supported by the existing learned data-driven methods [73], [25], [77], which require the join template to be a tree.","Unlikely the learned query-driven methods [33], [63], [42], FactorJoin does not depend on the executed query workload, so it is robust against data updates and workload shifts, can be quickly adapted to a new DB instance, and is generalizable to new queries.","However, in presence of query workload, FactorJoin can incorporate this information to further optimize the model construction process.","We integrate FactorJoin into Postgres' query optimizer and evaluate its end-to-end query performance on two well-established and challenging real-world CardEst benchmarks: STATS-CEB [18] and IMDB-JOB [38].","On the STATS-CEB, FactorJoin achieves near-optimal performance in end-to-end query time.","It has comparable performance to the previous SOTA learned data-driven model (FLAT [77]) in terms of estimation effectiveness but with 40x lower estimation latency, 100x smaller model sizes, and 100x faster training times.","On the IMDB-JOB, our framework achieves the SOTA performance in end-to-end query time (query execution plus planning time).","Specifically, the pure execution time of our framework is comparable to the previous SOTA method (pessimistic estimator [5]), but our estimation latency is 100x lower, making our overall end-to-end query time significantly faster.","Specifically, our framework can estimate 10,000 sub-plan queries in one second to optimize the query plan, which is close to the planning time of the traditional CardEst method used in Postgres.","We also carry out a series of controlled ablation studies to demonstrate the robustness and advantages of different technical novelties of FactorJoin.","In summary, our main contributions are: $\\bullet $ We formulate the problem of estimating the cardinality of join queries in its general form as a factor graph inference problem involving only single-table data distributions (Section ).","$\\bullet $ We propose a new binning and upper bound-based algorithm to approximate the factor graph inference (Section ).","$\\bullet $ We design blackattribute causal relation exploration and progressive estimation of sub-plan queries techniques to improve the efficiency of our framework (Section ).","$\\bullet $ We conduct extensive experiments to show the advantages of our framework (Section ).","Before describing the details of FactorJoin, we begin with a detailed description of existing CardEst approaches."],["Background and analysis","In this section, we first define the CardEst problem and then analyze existing CardEst methods and how our approach differs from them."],["Single table CardEst: Let $A$ be a table with $n$ attributes $A_1, \\ldots , A_n$ .","A single table selection query $Q(A)$ on $A$ can be viewed a conjunction or disjunction of filter predicates over each attribute, e.g.","$Q(A) = (A_1 \\in [0, 5]) \\ \\vee $ $(A_2$ LIKE `%An%'$) \\wedge (A_5 \\le -10)$ .","Let $|Q(A)|$ denote the cardinality of $Q(A)$ , i.e.","the number of tuples in $A$ satisfying $Q(A)$ .","Let $P_A(Q(A))$ denotes the selectivity (probability) of tuples satisfying $Q(A)$ .","Then, we have $|Q(A)| = P_A(Q(A)) * |A|$ , where $|A|$ denotes total number of tuples in $A$ .","CardEst for multi-table join queries: blackConsider a database instance $\\mathcal {D}$ with m tables $A, B, \\ldots , M$ , a query $Q$ consists of a join graph representing the join conditions among the selected tables and a set of base-table filter predicates.","blackFor simplicity, we abuse the notation and use $Q(I)$ to denote the filter predicates of $Q$ on table $I$ .","Let $\\Omega $ denote the denormalized table resulting from the join conditions of $Q$ .","Then, the cardinality $|Q| = P_{\\Omega }(Q(A), Q(B), \\ldots , Q(M)) * |\\Omega |$ , where $Q(I)$ can be empty set if table $I$ is not touched by $Q$ .","Estimating $Q$ is particularly challenging because there can exponential number of join patterns in the database, each associated with a unique $\\Omega $ and distinct probability distribution $P_{\\Omega }$ .","Therefore, to accurately estimate different join queries, the CardEst methods need to capture exponential many complicated data distributions."],["Analysis of existing ","Since the focus of this paper is multi-table join queries, we defer the discussion of single-table estimators to Section and focus on analyzing different approaches for handling join queries.","We can categorize these approaches into four classes: traditional, learned data-driven, learned query-driven, and bound-based.","We summarized their characteristics in Table REF and provide their details as follows.","Traditional methods: The sampling-based methods [41], [39], [76], [40] construct a small sample on each table and join these samples to estimate the cardinality of join queries.","The traditional histogram-based methods generally adopt the attribute independence and join uniformity assumptions to decompose the join queries as a combination of single table estimates.","Taking a query $Q$ joining two tables $A$ and $B$ as an example, these assumptions will simplify the cardinality $|Q| = P_{\\Omega }(Q(A), Q(B)) * |\\Omega |$ as $P_A(Q(A)) * P_B(Q(B)) * \\overline{|\\Omega |}$ , where $P_A(Q(A))$ is the estimated single table selectivity.","There exist two widely-used approaches to estimate the size for the denormalized join table $\\overline{|\\Omega |}$ .","First, the Selinger models [60], [10], [59] assume that join keys have uniformly distributed values, collect the number of distinct values (NDV) on the join keys from both tables, and estimate $\\overline{|\\Omega |}$ as $|A| * |B| / max\\lbrace NDV(A), NDV(B)\\rbrace $ .","Second, the join-histogram methods [26], [29], [7] first bin the domain of the join keys as histograms and assume join uniformity assumption for the values in each histogram bin.","Then they apply the distinct values methods for each bin and sum over all bins.","Our work follows the convention of the join-histogram methods but avoids the simplifying assumptions, which can not be trivially achieved.","Naive approaches such as building the full multi-dimensional histograms can resolve the attribute independence assumption but would introduce unaffordable storage overhead.","Many follow-up works [27], [30], [28] explore different histogram strategies to reduce the error caused by the join uniformity assumption.","They either take an impractical amount of time to construct the histograms or cannot provide high-quality estimates.","In short, the traditional methods combine the estimates from single tables to estimate the join queries.","This approach is very efficient, easy to train, update and maintain, thus perfect for system deployment.","However, their simplifying assumptions can generate erroneous estimates and poor query plans.","Learned data-driven methods: These methods circumvent the simplifying assumptions used in traditional methods by understanding the data distributions for the exponential number of join templates in a DB instance.","Specifically, some methods [13], [64], [65] denormalize the join of all tables in a database and use Bayesian networks to model the distribution.","The current SOTA approaches for handling join queries try to model the data distributions of the denormalized join tables by using fanout-based techniques [23], [77], [70], [73], [67].","These methods denormalize some tables and add a possibly exponential number of fanout columns, which are used to formulate the distributions for each join template.","Using this method, the learned data-driven methods can produce effective estimations for join queries and high-quality query plans [18].","However, these models generally have a long training time, large model size, slow estimation speed, and unscalable performance with the number of joins in the query.","Furthermore, the learned data-driven approaches currently can not handle self-joins or cyclic-joins and are very ineffective in processing complicated filter predicates such as string pattern matching queries.","Learned query-driven methods: These methods analyze the executed query workload, map the featurized join query directly to its cardinality using supervised models such as neural networks [33], [54], xgboost [11], tree-LSTM [63], and deep ensembles [42].","The query-driven methods generally have low estimation latency but the estimation effectiveness is highly dependent on the training workloadblack; we put a semi-check mark ([scale=0.3,fill=black](0,.3) – (.2,0) – (1,.7) – (.25,.15) – cycle (0.75,0.2) – (0.77,0.2) – (0.6,0.7) – cycle;) for these methods in Table REF .","To achieve reasonable estimation quality, they generally need an excessive amount of executed training queries [18], which are unavailable for new DB instances.","Furthermore, the query-driven models need to retrain in the case of data update or query workload shift.","An impractical amount of new executed queries are again needed for this re-training process.","Bound-based methods: Traditional CardEst methods in modern DBMS tend to severely under-estimate the cardinality, thus sometimes choosing significantly more expensive query plans [55], [3], [5].","To avoid under-estimation, the bound-based methods [5], [3], [1], [22] use information theory to provide an upper bound on the cardinality.","These bounds are very effective to help generate high-quality query plans as they can avoid expensive join orders and physical operators [55], [18] However, these methods do not build single-table estimators to understand the data distributions.","Instead, in presence of base-table filters, they need to materialize the filtered tables and populate the bound during query run-time, which can produce very high latency and overhead.","Therefore, despite the bound-based methods providing meaningful insights into understanding joins, they can not be practically deployed in a DBMS.","Summary: None of the existing approaches can simultaneously satisfy the desired properties of CardEst, namely effective, efficient and appropriate for system deployment.","However, each of them contains some advantageous techniques, inspiring us to build FactorJoin that can bridge the gaps amongst all these CardEst categories and unify their merits."],["New framework for join queries","Given this background on previous CardEst approaches, we now introduce the high-level operation of FactorJoin.","In Section REF , we elaborate on FactorJoin's core technique: accurately estimating join queries with only single-table statistics using factor graph inference techniques.","Then, in Section REF , we analyze the complexity of this inference procedure and motivate several optimizations in FactorJoin.","Finally, we provide a workflow overview in Section REF ."],["Problem formulation","We first illustrate the main idea with a simple example of a two-table join query.","Then, we formalize the join query estimation problem as a PGM inference problem over single table distributions.","Two-table join query example: Figure REF illustrates a query $Q$ joining tables $A$ and $B$ on the inner joinIn this paper, we only discuss inner and equality joins.","FactorJoin naturally supports left, right, and outer joins.","We leave support for blacknon-equal joins as future work.","condition $A.id = B.Aid$ with base table filter predicates $Q(A)$ and $Q(B)$ .","Table $A$ first goes through the filter $Q(A)$ , resulting in an intermediate table $A|Q(A)$ (records in $A$ that satisfy the filter $Q(A)$ ).","The same procedure is applied to table $B$ .","Then, the query $Q$ will match the value of join keys $A.id$ and $B.Aid$ from these two intermediate tables.","Specifically, the value $a$ appears 8 times in table $A|Q(A)$ and 6 times in table $B|Q(B)$ , resulting in value $a$ appearing 48 times in the join result.","Therefore, we can calculate the cardinality of this query $Q$ as $8\\times 6 + 4\\times 5 + 3 \\times 5 = 83$ .","We can formulate the above procedure for calculating $Q$ as a statistical equation in Equation REF , where $D(A.id)$ denotes the domain of all unique values of $A.id$ .","We observe that only single-table distributions $P_A$ and $P_B$ are required to accurately compute the cardinality of this join query.","Also in this equation, $P_{A}(A.id \\!","= \\!","v | Q(A)) * |Q(A)|$ equals to $P_{A}(A.id \\!","= \\!","v \\wedge Q(A)) * |A|$ , which is exactly what single-table CardEst methods estimate.","Thus, we can accurately calculate the cardinalities of two-table join queries using single-table estimators.","Note that the summation over the domain of join key $\\mathcal {D}(A.id)$ has the same complexity as computing the join.","Therefore, we need to approximate this calculation for FactorJoin to be practical.","The details of our approximation schemes are given in Section REF and Section .","$|Q| = \\sum _{v \\in \\mathcal {D}(A.id)} & P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)|$ Figure: Formulating a join query CardEst into a factor graph inference problem.","Figure (i) shows a SQL query Q 2 Q_2.","Figure (ii) visualizes the join template of Q 2 Q_2.","Figure (iii) presents the factor graph model to compute Q 2 Q_2's cardinality.Formal problem formulation: A general join query can involve a combination of different forms of joins (e.g., chain, star, self, or cyclic joins) so its counterpart equation (as Equation REF ) can be very difficult to derive and compute; blackwe have put these equations/derivations in the supplementary material.","We provide a generalizable formulation that automatically decomposes join queries into single-table estimations using the factor graph model.","Figure REF -(i) shows a SQL query $Q_2$ joining four tables $A, B, C, D$ .","We visualize its join template as a graph in Figure REF -(ii), where each dashed rectangle represents a table, each ellipse (node) represents a join-key in $Q_2$ , and each solid line (edge) represents an equi-join relation between two join keys connected by it.","Note that both sides of an equi-join relation represent the semantically equivalent join keys.","We call them equivalent key group variables.","In Figure REF -(ii), there are three connected components and thus three equivalent key group variables $V_1, \\ldots , V_3$ .","$V_1$ represents $A.id$ and $B.Aid$ in this group since $Q_2$ contains the join condition $A.id = B.Aid$ .","The exact computation required to compute the cardinality of the SQL query in Figure REF extends Equation REF to sum over the domain $\\mathcal {D}$ of each join key.","black $|Q_2| = & \\sum _{a1 \\in \\mathcal {D}(A.id)} \\ \\ \\ \\ \\ \\sum _{a2 \\in \\mathcal {D}(A.id2)} \\ \\ \\ \\ \\ \\sum _{c \\in \\mathcal {D}(C.id)} \\nonumber \\\\& P_{A}(A.id = a1, \\ A.id2 = a2 \\ | \\ Q_2(A)) \\ast |Q_2(A)| \\nonumber \\\\& \\ast P_{B}(B.Aid = a1, \\ B.Cid = c \\ | \\ Q_2(B)) \\ast |Q_2(B)| \\nonumber \\\\& \\ast P_{C}(C.Aid2 = a2, \\ C.id = c \\ | \\ Q_2(C)) \\ast |Q_2(C)| \\nonumber \\\\& \\ast P_{D}(D.Cid = c \\ | \\ Q_2(D)) \\ast |Q_2(D)|$ This computation is extremely expensive — assuming the domains of each id being $n$ , a naive implementation would have the complexity $O(n^3)$ .","We can improve this complexity to $O(n^2)$ by representing this computation as an equivalent factor graph model [44], which is a particular class of probabilistic graphical models (PGMs) [34].","We have 2 in the exponent because the maximum number of join keys in $Q_2$ is two in the table $A$ .","This is still very expensive, but in Section REF , we describe how we can use principled approximations made possible by the factor graph formulation to make the inference complexity practical.","A factor graph is a bipartite graph with two types of nodes: variable nodes, and factor nodes representing an unnormalized probability distribution w.r.t.","the variables connected to it.","Figure REF -(iii) shows the constructed factor graph $\\mathcal {F}$ for computing the cardinality of $Q_2$ .","Specifically, $\\mathcal {F}$ contains a variable node for each equivalent key group variable $V_i$ , and a factor node for each table touched by $Q_2$ .","A factor node is connected to a variable node if the variable represents a key in the table.","In this case, the factor node representing table $A$ is connected to $V_1$ (equivalent to $A.id$ ) and $V_2$ (equivalent to $A.id2$ ).","Each factor node maintains an unnormalized probability distribution for the variable nodes connected to it, for e.g., node $A$ maintains $P_A(V_1, V_2|(Q_2(A)))*|Q_2(A)|$ , which is the same as the distribution $P_A(A.id, A.id2|(Q_2(A)))*|Q_2(A)|$ used in Equation REF .","The factor graph model utilizes the graph structure to compute the sum using the well-studied inference algorithms.","We can more formally state this relationship between computing the join cardinalities and factor graphs as the following lemma, whose proof is provided in the supplementary material.","Lemma 1 Given a join graph $\\mathcal {G}$ representing a query $Q$ , there exists a factor graph $\\mathcal {F}$ such that the variable nodes in $\\mathcal {F}$ are the equivalent key group variables of $\\mathcal {G}$ and each factor node represents a table $I$ touched by $Q$ .","A factor node is connected to a variable node if and only if this variable represents a join-key in table $I$ .","The potential function of a factor node is defined as table $I$ 's probability distribution of the connected variables (join keys) conditioned on the filter predicates $Q(I)$ .","Then, calculating the cardinality of $Q$ is equivalent to computing the partition function of $\\mathcal {F}$ ."],["Inference complexity analysis","Inference on factor graphs is a well-studied problem in the domain of PGMs [46], [34], [44].","Popular approaches to solving this problem are variable elimination (VE) [34] and belief propagation algorithms [35].","In FactorJoin's implementation, we use the VE, which first determines an optimal order of variables and then sums over the distributions defined by the factor graph model in this order.","The complexity of VE is $O(N*|D|^{max(|JK|)})$ , where $N$ is the number of equivalent key groups (3 in Figure REF ), $|D|$ is the largest domain size of all join keys, and $max(|JK|)$ is the maximum number of join keys in a single table (2 in Figure REF ).","Intuitively, this complexity is because the factor nodes need to understand the joint distribution of all join keys in one table.","Thus, the largest factor node in the factor graph model maintains a probability distribution of size $|D|^{max(|JK|)}$ .","This complexity is not practical for real-world queries, as $|D|$ can be millions, and $max(|JK|)$ can be larger than 4 in real-world DB instances, such as IMDB [38].","Therefore, instead of calculating the exact cardinality, we propose a new PGM inference algorithm for FactorJoin that can estimate an upper bound on cardinality.","Past works [5], [3], [1] have shown that cardinality upper bounds can help avoid very expensive query plans since under-estimating a large result can sometimes be catastrophic.","For instance, an optimizer might choose to do a nested loop join if a CardEst method under-estimates that the result would easily fit in memory.","This would lead to a disastrous plan if the actual result size is much larger than estimated.","blackOur approximate inference algorithm performs two approximations: 1) reducing the domain size $|D|$ using binning and 2) decreasing the exponent, $max(|JK|)$ , by approximating the distribution of attributes in a single table.","blackSpecifically, for binning, we first partition the domain of all join keys into $k$ bins.","Instead of summing over the entire domain, we only need to sum over $k$ summarized values (probabilistic bounds) of each bin.","The details are provided in Section .","blackSecond, we approximate the causal relation among all attributes within a table as a tree structure.","This allows us to factorize the $max(|JK|)$ -dimensional joint distribution of all join keys in a table as a product of two-dimensional conditional distributions.","This procedure is known as structure learning in Bayesian networks in the PGM domain [6].","Past work on single table cardinality estimation has shown that such tree approximations do not decrease the modeling accuracy significantly [70].","Further details are provided in Section REF .","Thereafter, we can directly run the VE inference algorithm on the binned domains and factorized two-dimensional distributions to estimate the cardinality upper bound of $Q$ .","With these two approximate inference techniques, the complexity of estimating the cardinality bound is reduced to $O(N*k^2)$ , which is very efficient as both $N$ and $k$ are very small in practice."],["Workflow overview","The workflow of FactorJoin contains two phases: offline training and online inference (Figure REF ).","During the offline phase, our framework analyzes the DB instance, bins the domain of join keys, explores the causal relation, and builds CardEst models for every single table.","When a join query $Q$ comes in during the online phase, FactorJoin formulates the estimation of this query as a factor graph involving only single-table distributions, estimates them using (pre-trained) single-table estimators, runs PGM inference, and generates a probabilistic bound for $Q$ .","We provide the details as follows.","Offline training phase: Given a new DB instance, FactorJoin first analyzes its DB schema and data tables to get all possible join relations between different join-keys.","We consider two join-keys to be semantically equivalent if there exists a join relation between them.","After identifying all groups of equivalent join-keys, we bin the domains of all join-keys in each group.","FactorJoin can also optionally use query workload information to optimize this binning procedure (details provided in Section REF ).","Based on the data tables and the binned domains, we explore the blackcausal relation of these attributes and model them as a tree structure (details are provided in Section REF ).","Then, we build the CardEst models to understand the data distribution of every single table.","In principle, any single-table CardEst method that is able to provide conditional distributions can be adapted into FactorJoin.","In practice, we implement two methods: traditional random sampling and the learned data-driven BayesCard method [70].","The sampling method is extremely flexible to use and can easily support any complex filter predicates with disjunctions, string pattern matching, or any user-defined functions.","Alternatively, similar to most data-driven methods, BayesCard can only work with queries filtering numeric and categorical attributes, but BayesCard has been shown to consistently produce accurate, fast, and robust estimations for various data distributions [70].","Users of FactorJoin can switch between different single-table estimators based on their objectives and knowledge about the DB instances.","Figure: Workflow of FactorJoin framework.Online inference phase: When a join query $Q$ comes in during run-time, FactorJoin will first parse its join graph and construct the corresponding factor graph $\\mathcal {F}$ (as in Figure REF ).","Then, it uses the trained single-table estimators to estimate the join-key distributions conditioned on the filter predicates of $Q$ ; FactorJoin will use these distributions in the corresponding factor nodes of $\\mathcal {F}$ .","Finally, we run the approximate version of VE inference algorithms on $\\mathcal {F}$ to derive the estimated/probabilistic cardinality bound.","We note that the CardEst methods working inside a query optimizer will estimate all sub-plan queries of a target query to decide the best query plan.","Since these sub-plan queries contain a large amount of overlapping information, we provide a progressive estimation algorithm to avoid computation redundancy in Section REF ."],["Probabilistic bound algorithm","In this section, we design a new probabilistic bound algorithm based on binning to calculate the factor graph inference problem.","Specifically, we first explain the main ideas with a two-table join example and formulate the algorithm details within FactorJoin in Section REF .","Then, we discuss how to optimize the bin selection using data and query information in Section REF ."],["Algorithm details","As described in Section , the main objective of our probabilistic bound-based algorithm is to reduce the domain size of the join keys, i.e., reducing the complexity of summation in Equation REF .","Thus, instead of running the inference algorithm over the entire domain, FactorJoin only needs to sum over the binned domain.","Two-table join query example: We use the previous example query $Q$ from Section REF to illustrate the core idea of our probabilistic upper bound algorithm based on binning.","The objective is to estimate an upper bound on $Q$ , whose exact computation requires summing over the entire domain of $A.id$ according to Equation REF .","Assume that we have a set of $k$ bins $\\lbrace bin_1, \\ldots , bin_k \\rbrace $ partitioning the domain of $A.id$ and we apply this same set of bins to $B.Aid$ .","blackWe need to make sure that a given value in the domain of $A.id$ and $B.Aid$ will always belong to the bin with the same indexes.","Then, Equation REF can be equivalently written as Equation 3.","We can then replace the sum over all values in a bin $bin_i$ with a probabilistic bound on $bin_i$ to estimate an upper bound for $Q$ as Equation REF .","$|Q| &= \\sum _{i = 1}^{k} \\sum _{v \\in bin_i} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\\\&\\lesssim \\sum _{i = 1}^{k} && Probabilistic\\_bound(A, B, bin_i)$ Motivated by a bound based on the most frequent value (MFV) [22], we use a simple probabilistic bound method to derive $Probabilistic\\_bound(A, B, bin_i)$ for a particular bin.","Specifically, assuming that value $\\lbrace a, b, c , e, f\\rbrace $ of $A.id$ and $B.Aid$ are binned into $bin_1$ as shown in Figure REF .","We know the summation of all values in $bin_1$ equals to black$8\\times 6 + 4 \\times 5 + 3 \\times 5 = 83$ .","This summation has a dominating term of $8 \\times 6$ , because the count of MFV of $bin_1$ is 8 for $A.id$ (denoted as $V^{*}_1(A.id)$ ) and 6 for $B.Aid$ (denoted as $V^{*}_1(B.Aid)$ ) so each value can appear at most $8 \\times 6$ times in the denormalized table after the join.","Since we know the total count of values in $bin_1$ for $A.id$ is 16, there can be at most $16/8 = 2$ MFVs.","Similarly, there can be at most black4 MFVs in $bin_1$ for $B.Aid$ .","Therefore, we have the summation of all values in $bin_1$ is upper bounded by black$min(2, 4) \\times 8 \\times 6 = 96$ .","We formally represent the aforementioned procedure in Equation REF , where $V^{*}_i(A.id)$ and $P_A(A.id \\in bin_i|Q(A)) * |Q(A)|$ are the MFV count and estimated total count of $bin_i$ for $A.id$ .","Our bound is probabilistic because $P_A(A.id \\in bin_i|Q(A)) * |Q(A)|$ is estimated with a single table CardEst method, which may have some errors.","$|Q| & \\lesssim \\sum _{i = 1}^{k} min(\\frac{P_A(A.id \\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\nonumber \\\\& \\frac{P_B(B.Aid \\in bin_i|Q(B)) * |Q(B)|}{V^{*}_i(B.Aid)}) * V^{*}_i(A.id) * V^{*}_i(B.Aid)$ Figure: The bound-based algorithm; each bin is summarized by the most frequent value count and total count.Working with PGM inference: Next, we will explain how to generalize this probabilistic bound to make it work inside the factor graph VE inference algorithm mentioned in Section REF .","During the offline training, FactorJoin computes the MFV counts $V^*$ for all bins of each join-key.","During the online phase, FactorJoin estimates the probability distribution for the binned domain of all join-keys in each table $I$ .","Then, FactorJoin puts $I$ 's distribution $P_{I}$ and MFV counts $V^*(I)$ into the corresponding factor node.","Recall that the VE algorithm generates an optimal elimination order of all variables and sums over the domain of these variables sequentially in this order.","Since at each step, the variable elimination algorithm sums out only one variable, its probabilistic bound can be derived using Equation REF over multiple equivalent join keys.","Thus, this algorithm does not need to sum over the entire value domain of a variable but can only sum over the probabilistic bounds in the binned domain instead, which greatly reduces the complexity.","Analysis: This probabilistic bound algorithm significantly reduces the domain size of join keys and enables practical approximate PGM inference.","Further, this algorithm upper-bounds the cardinality most of the time and the bound is very tight with even a small number of bins, such as $k=100$ .","Therefore, as we empirically verify in our evaluation, the estimated cardinality bound is very effective."],["Bin selection optimizations","In our probabilistic bound algorithm, different bins can result in drastically different bound tightness.","blackThere are two decisions: how many bins to use for each join key group, and what values to put in the same bin.","Deciding $k$ based on query workloads: blackThe number of bins, $k$ , has a significant effect on the performance of FactorJoin: fewer bins aggregate more distinct values in the join key domain to each bin and are thus less accurate but more efficient.","An approach to tune $k$ is to set different values of $k_i$ for different equivalent key groups $Gr_i$ .","Suppose that users provide a budget $K$ with $K = \\sum k_i$ .","If FactorJoin has access to the query workload, it can analyze the join patterns and count the number of times $n_i$ each $Gr_i$ appears in the workload.","Our framework can adaptively set a larger $k_i$ for the frequently visited $Gr_i$ and vice versa.","In our implementation, we use a simple heuristic — set $k_i = K * n_i/\\sum _j n_j$ .","In this way, we can optimize the modeling capacity of FactorJoin to make sure that it is spent on the important joins.","Binning strategy: For the probabilistic bound algorithm described in Section , we observe that the upper bound on a particular bin $bin_i$ can be very loose if the MFV count $V^*_i$ is a large outlier in $bin_i$ .","Taking the two table join query as an example, if $bin_i$ contains only one value that appears 100 times in $A.id$ but $10,000$ values that only appear once in $B.Aid$ , then the bound could be 100 times larger than the actual cardinality.","blackCommon binning strategies used in DBMS histograms such as equal-depth or equal-width bins can be catastrophic in such cases.","Instead, we want the variance of the join key counts in each bin to be low.","This strategy has also been proposed in different contexts [28], [30], but the key challenge is that the same bins will be built for all join keys within an equivalent key group, i.e.","join keys that share the same semantics.","For example, for a primary key (e.g.","“title.id” in IMDB JOB [38]), any binning strategy will have zero variance because of uniqueness.","However, the same bins will be applied for its equivalent foreign keys (e.g.","“movie_id”) on other tables, which may lead to a large variance depending on how often each value repeats in the other tables.","blackWe design a greedy binning selection algorithm (GBSA) to optimize for bins with low variance value counts across all tables, as illustrated in Algorithm REF .","Jointly minimizing the variance of one bin for all join keys has exponential complexity.","Therefore, GBSA uses a greedy algorithm to iteratively minimize the bin variance for all join keys.","blackAt a high level, GBSA first optimizes the minimal variance bins with half the binning budget of $k/2$ on the domain of one join key (lines 2-4).","blackThen, it recursively updates these bins by minimizing the variance of other join keys using the other half of the budget (lines 5-13).","blackWe put the details of GBSA in the supplementary material.","blackDiscussion: blackAs shown in Section REF , GBSA has significantly better performance than naive binning strategies.","In the extreme case, if the value counts have zero variance for all equivalent join keys, then our bound can output the exact cardinality.","Admittedly, this binning strategy does have the drawback that after applying the filter predicates, the join key frequencies will change and their variance may become higher within a bin.","However, we empirically evaluate that for real-world datasets, this phenomenon does not have a severe impact on FactorJoin.","In future work, we will explore an enhanced binning strategy that can efficiently address this issue.","[t] blackGreedy Bin Selection Algorithm (GBSA) Input: Equivalent key groups $Gr_1, \\ldots , Gr_m$ , where $Gr_i = \\lbrace Id_i^1, \\ldots , Id_i^{|Gr_i|} \\rbrace $ ; Column data $\\mathcal {D}(Id_i^j)$ of all join keys in the DB instance $\\mathcal {D}$ ; Number of bins $k_i$ for each group $Gr_i$ .","[1] $Gr_i \\in \\lbrace Gr_1, \\ldots , Gr_m\\rbrace $ $Bin(Gr_i) \\leftarrow $ [] $Gr_i^{\\prime } \\leftarrow sort\\_key\\_based\\_on\\_domain\\_size(\\mathcal {D}, Gr_i)$ $Bin(Gr_i) \\leftarrow get\\_min\\_variance\\_bins(\\mathcal {D}(Gr_i^{\\prime }[1]), k_i/2)$ remain_bins $\\leftarrow k_i/2$ $j \\in \\lbrace 2, \\ldots , |Gr_i^{\\prime }| \\rbrace $ binned_data $\\leftarrow apply\\_bin\\_to\\_data(\\mathcal {D}(Gr_i^{\\prime }[j]), Bin(Gr_i))$ bin_variance $\\leftarrow calculate\\_variance$ (binned_data) arg_sort_idx $\\leftarrow arg\\_sort\\_decreasing(bin\\_variance)$ $p \\in arg\\_sort\\_idx[1:remain\\_bins/2]$ $Bin(Gr_i) \\leftarrow min\\_variance\\_dichotomy(Bin(Gr_i)[p]$ , binned_data[p]) remain_bins $\\leftarrow $ remain_bins/2 return $\\lbrace Bin(Gr_i) | i, \\ldots , m\\rbrace $ black"],["Incremental model updates","black FactorJoin is friendly for DBs with frequent data changes because we only need to incrementally update the single table statistics.","In the following, we will discuss the algorithm details in the scenario of data insertion and data deletion can be handled similarly.","Consider we have trained a stale FactorJoin model $\\mathcal {F}$ on data $\\mathcal {D}$ and would like to incrementally update $\\mathcal {F}$ with the inserted data $\\mathcal {D}^{\\prime }$ .","FactorJoin needs to identify the inserted tuple values from $\\mathcal {D}^{\\prime }$ for every join key and put them into the original bins optimized on $\\mathcal {D}$ .","Then, it will update the total and most-frequent value count in each bin.","At last, FactorJoin can incrementally update the base table CardEst models with off-the-shelf tools (e.g.","BayesCard provides an efficient and effective approach to incrementally update the single-table models using $\\mathcal {D}^{\\prime }$ [70]; materializing a new sample on $\\mathcal {D} \\bigcup \\mathcal {D}^{\\prime }$ incrementally updates the sampling-based estimator).","Therefore, the incremental update of FactorJoin does not need to normalize the joined tables nor need to use executed queries.","It is also empirically verified to be very efficient and effective.","However, during incremental updates, the FactorJoin models keep the same set of bins, which is optimized on the previous data and thus might not be optimal after data insertion.","This might cause slight degrade in performance and the user can choose to retrain the model in case of massive data updates."],["Improving ","We describe two techniques to improve the modeling and estimation efficiency of FactorJoin.","First, we model the joint distribution of attributes on a single table as a tree structure and use it to simplify the PGM inference calculations.","Next, we define a systematic way of reusing estimates of intermediate sub-plan queries to estimate a larger query.","This is only possible because we decompose cardinality estimates into single table sub-components."],["blackExploring attributes causal relation","Recall from blackSection REF , the factor graph computation to calculate the cardinality bound is $O(N*k^{max(|JK|)})$ , where $N$ is the number of tables, $k$ is the number of bins, and $max(|JK|)$ is the maximum number of join keys in a table.","The exponent term, $max(|JK|)$ , represents the factor nodes requiring the joint distribution between all join keys on a single table.","As $max(|JK|)$ can sometimes be large (e.g., in IMDB it can be up to 4), we need to reduce the dimensionality of the distributions defined in factor nodes to ensure an efficient CardEst procedure.","Fortunately, real-world data has a lot of correlations that can be used to simplify this joint distribution.","blackConsider a table $A$ with 6 attributes, $\\lbrace id_1, id_2, id_3, id_4 attr_1, attr_2\\rbrace $ , four of which are join keys.","The factor graph computation blackmay need to estimate quantities such as: $P_A(id_1, id_2, id_3, id_4 | Q(A))$ blackThis requires storing a 4 dimensional probability distribution, and even with binning the domains of all ids into $k$ bins, it still requires $k^4$ space and inference time complexity.","This joint distribution can be represented as a graph where each attribute is a node, and there are edges between every pair of attributes.","We will use the dependencies and relationships between the attributes and model them as a Bayesian network (BN) [34].","Specifically, we will assign each edge in the joint distribution graph a weight proportional to the mutual information between the two attributes connected by the edge.","Then, we will remove edges with the least mutual information until only a tree structure graph remains to represent the joint probability over these attributes.","blackThis process is known as the Chow-Liu algorithm for decomposing joint distributions [6].","The factorized tree representation of the joint distribution can let us approximate Equation REF with an equation of the form: $P_A(id_1 | Q(A)) * P_A(id_2 | Q(A)) * P_A(id_3 | id_1) * P_A(id_4 | id_3)$ , which is much more efficient to compute because it only involves a product over at most a two-dimensional distribution.","This dimension reduction can cause some errors.","But it has been shown empirically that the factorized distributions are accurate approximations of the original distribution for most real-world datasets [70].","Using such an optimization, instead of storing the $max(|JK|)$ -dimensional distributions in the factor nodes, the factor graph only needs to store a series of one or two-dimensional data distributions without a significant decrease in estimation accuracy.","Therefore, the overall complexity for running approximate inference on this factor graph becomes $O(N*k^2)$ in both storage and inference speed.","Since $k$ is relatively small (around one hundred), estimating one query using FactorJoin is very efficient.","blackIn principle, FactorJoin can also use other methods for the single table distribution inference.","This may involve traditional methods, such as sampling, or more powerful graph-structured factorizations of the distributions — which can increase modeling accuracy at the cost of slower estimation speed.","Thus, users of FactorJoin can make trade-offs between estimation accuracy and efficiency.","We leave exploring these trade-offs as future work."],["Progressive estimates of sub-plan queries","In order to optimize the plan for one query, the CardEst method needs to estimate the cardinalities of hundreds or thousands of sub-plan queries.","Estimating all these queries independently is very inefficient because they contain a lot of redundant computation.","Similar to the traditional CardEst methods deployed in real DBMSes, FactorJoin supports estimating all sub-plan queries of one query progressively, reusing the computation as much as possible.","At a high level, the progressive estimation algorithm will estimate all sub-plan queries in a bottom-up fashion.","FactorJoin first estimates all two-table join sub-plan queries, then it estimates the three-table join sub-plan queries, which will contain the two-table joins as their sub-structures, and so on.","Joining Factor Graphs: The key insight is that when we estimate the cardinality of a two-table join, we can also combine their factor graphs into a new factor graph for the joined table.","Specifically, as described in Section , each base table is represented as a factor node in the factor graph.","For instance, the factor node representing $A$ will store the unnormalized probability distribution and the MFV counts for the binned domain of join keys in $A$ .","When estimating the cardinality of joining $A$ and $C$ , our approximate inference algorithm in Equation REF computes the bound over all bins and sums them.","These bounds essentially define an unnormalized probability distribution over all join keys on the denormalized table $A \\bowtie C$ .","We can derive a bound on the new MFV counts for a join-key $A \\bowtie C$ as the multiplication of the corresponding MFV counts in $A$ and $C$ .","Therefore, we can cache this new probability distribution and MFV counts in a new factor node representing $A \\bowtie C$ .","Thus, when estimating the cardinality of $A \\bowtie B \\bowtie C$ we can consider it as a join of two single tables $(A \\bowtie C$ ) and $B$ .","Since all sub-plans can essentially be considered a join of two sub-plans that we have already estimated, our inference algorithm only needs to compute the join of blacktwo-factor nodes for each sub-plan query.","Therefore, the progressive estimation algorithm has no redundant computation and maximizes efficiency.","In experiments, with this algorithm, FactorJoin can estimate the cardinality of $10,000$ sub-plan queries in one second, which is more than ten times faster than estimating all these queries independently.","Note that the progressive estimation algorithm is only possible to implement for FactorJoin and the traditional methods because it requires decomposing join queries into single-table estimates.","None of the existing learned methods can apply this method since their models are built to model the distributions of specific join patterns."],["Experimental Evaluation","In this section, we empirically demonstrate the advantages of our framework over existing CardEst methods.","We first introduce the datasets, baselines, and experimental environment in Section REF .","Then, we explore the following questions: $\\bullet $ Overall performance (Section REF ): how much improvement can FactorJoin achieve in terms of end-to-end query time?","What are the model size and training time compared to SOTA methods?","$\\bullet $ Detailed analysis (Section REF ): Why does FactorJoin attain good overall performance?","blackWhen does it perform better or worse than existing baselines?","How will it perform in terms of data updates?","$\\bullet $ Ablation study (Section REF ): How much does each optimization technique of FactorJoin contribute to the overall performance?"],["Experimental Setup","Datasets: We evaluate the performance of our CardEst framework on two well-established benchmarks: STATS-CEB [18] and IMDB-JOB [38], whose statistics are summarized in Table REF .","The STATS-CEB benchmark uses the real-world dataset STATS, which is an anonymized dump of user-contributed content on Stack Exchange (Stack Overflow and related websites).","This dataset consists of 8 tables with 34 columns and more than one million rows.","It contains a large number of attribute correlations and skewed data distributions.","The STATS-CEB query workload contains 146 queries involving 70 different join templates with real-world semantics.","The true cardinalities of these queries range from 200 to $2 \\cdot 10^{10}$ , with some queries taking more than one hour to execute on Postgres.","The STATS-CEB benchmark only has star or chain join templates and numerical or categorical filtered attributes — therefore it is possible to evaluate all existing CardEst baselines on it.","The IMDB-JOB benchmark uses the real-world Internet movie database (IMDB).","This dataset contains 21 tables with more than fifty million rows.","The IMDB-JOB query workload contains 113 queries with 33 different join templates.","Some queries have more than $10,000$ sub-plan queries, posing a significant challenge to the efficiency of CardEst methods.","Because IMDB-JOB contains cyclic joins and string pattern matching filters, the JoinHist and existing learned data-driven methods do not support this benchmark.","The queries in these two benchmarks cover a wide range of joins, filter predicates, cardinalities, and runtimes.","To the best of our knowledge, they represent the most comprehensive and challenging real-world benchmarks for CardEst evaluation.","Baselines: We compare our framework with the most competitive baselines in each CardEst method category.","1) PostgreSQL [10] refers to the histogram-based CardEst method used in PostgreSQL.","2) JoinHist [7] is the classical join-histogram method whose details are explained in Section REF .","3) WJSample [40] uses a random walk based-method called wander join to generate samples from the join of multiple tables.","Its superior performance over traditional sampling-based methods has been demonstrated in recent studies [40], [76], [56].","blackWe limit the amount of time spent for each random walk of WJSample so the overall time spent for estimating cardinalities is comparable to other methods.","If we allow WJSample longer inference time, it can generate more effective query plans — but it is impractical since its total estimation time will exceed the query execution time.","4) MSCN [33] is a learned query-driven method based on a multi-set convolutional network model.","On both datasets, we train MSCN on roughly $100K$ sub-plan training queries that have similar distributions to the testing query workloads.","5) BayesCard [70], 6) DeepDB [25], and 7) FLAT [77] are the learned data-driven CardEst methods.","They use fanout methods to understand the data distributions of all join templates.","Then, they apply different distribution models (Bayesian networks, sum-product networks [57], factorized sum-product networks [72], respectively) to capture the data distribution on single tables or denormalized join tables.","We test these methods on STATS-CEB using the optimally-tuned model parameters, as used in [18].","8) PessEst [5] is the SOTA bound-based method, which leverages randomized hashing and data sketching to tighten the bound for join queries.","Its estimation has been verified to be very effective in real-world DBMSes.","We use the best-tuned hyperparameters to evaluate PessEst on both benchmarks.","9) U-Block [22] uses top-k statistics to estimate cardinality bounds.","The original paper designed a new plan enumeration strategy to work with this bound and showed promising results on IMDB-JOB.","However, for a fair comparison, we only evaluate the standalone U-Block CardEst method.","10) TrueCard outputs true cardinality for given queries with no estimation latency.","This baseline does not refer to any specific method but represents the optimal CardEst performance.","We omit comparisons with many other CardEst methods [4], [61], [31], [12], [62], [68], [21], [32], [39], [20], [73], [69], [11] blackbecause the baselines above have demonstrated clear advantages over these methods in various aspects [18].","In addition, some newly proposed methods [63], [67], [42] do not have open-source implementations yet.","We implemented FactorJoin in Python mainly using the NumPy package.","Regarding the hyperparameters of FactorJoin, we assume no knowledge of the query workload and set the bin size $k$ to 100 for both datasets.","We use the BN-based CardEst method [70] as the single-table estimators for STATS-CEB benchmark because this method has been shown to be very accurate, fast, and robust for single-table estimations.","For IMDB-JOB, we use random sampling with $1\\%$ sampling rate as the single-table estimator because this benchmark contains very complicated filter predicates such as disjunctions and string pattern matching, which are not supported by the learned data-driven methods.","We discuss and compare other hyperparameters of our framework in Section REF .","Table: Summary of STATS-CEB and IMDB-JOB benchmark.Figure: Overall performance on STATS-CEB and IMDB-JOB.Environment: blackWe use the open-source implementation of all the baselines and tune their parameters to achieve the best possible performance.","All experiments are performed on a Ubuntu server with Intel Core Processor CPU with 20 cores, 57GB DDR4 main memory, and 300GB SSD.","For end-to-end evaluation, we integrate our framework along with all the baselines into the query optimizer of PostgreSQL 13.1 following the procedure described in the recent work [18], [17].","We build the primary key and foreign key indices and disable parallel execution in PostgreSQL.","Specifically, we inject into PostgreSQL all sub-plan query cardinalities estimated by each method, so the PostgreSQL optimizer uses the injected cardinalities to optimize the query plan, and record the end-to-end query runtime (planning plus execution time).","To eliminate the effects of cache, we first execute each workload multiple times.","Then, we execute the queries three times for each method and report the average blackand the standard errors."],["Overall performance","The most straightforward criterion to evaluate the performance of a CardEst method is to test how much each method improves the end-to-end query performance inside the query optimizer.","Therefore, we compare the end-to-end performance along with the model size and training time of existing baselines on STATS-CEB and IMDB-JOB.","We summarize the overall performance of all baselines in Figure REF .","Recall that JoinHist, BayesCard, DeepDB, and FLAT can not support the IMDB-JOB benchmark.","In addition, we do not provide the model size and training time for Postgres, TrueCard, JoinHist, WJSample, and U-Block as they are negligible.","Table: End-to-end performance on STATS-CEB.Performance on STATS-CEB: In addition to Figure REF , we provide a detailed comparison of query execution and planning time (sum over all queries in the STATS-CEB) in Table REF , with the relative end-to-end improvement over Postgres shown in the last column, i.e.","(Postgres time - method time) / Postgres time.","FactorJoin achieves the best end-to-end query runtime ($19,116s$ ), which is close to optimal performance ($18,456s$ for TrueCard).","When compared with the traditional methods (Postgres, JoinHist, and WJSample), FactorJoin has significantly better query execution time, indicating that our estimates are much more effective at generating high-quality query plans.","Meanwhile, the planning time of our framework is close to Postgres.","Judging from the pure query execution time in Table REF , FactorJoin has comparable estimation effectiveness as the previous SOTA method FLAT on this benchmark.","This learned data-driven method uses the fanout method to understand the distribution of all join patterns, which is accurate but very inefficient.","Specifically, FactorJoin can achieve better end-to-end performance than FLAT, with $10 \\times $ smaller planning time, $160 \\times $ smaller model size, and $240 \\times $ faster training time.","When compared to the other learned methods: BayesCard, DeepDB, and MSCN, FactorJoin can simultaneously achieve better query execution and planning time, smaller model size, and faster training time.","Therefore, we empirically verify that instead of modeling the distributions of all join patterns, accurately decomposing the join query into single-table estimates can generate equally effective but more efficient estimation with much smaller model size and training time.","FactorJoin and the SOTA bound-based method PessEst can both generate very effective estimates.","This observation verifies that a bound-based method can produce effective estimates because it can help avoid very expensive query plans [5], [3], [1], [18].","However, the planning time of PessEst (1135s) is undesirable because it needs to materialize tables after applying the filters and populate the upper bound during run-time.","FactorJoin has $35 \\times $ less planning latency since instead of using exact bounds, we use probabilistic bounds derived from single-table estimates, which is much more efficient.","Table: End-to-end performance on IMDB-JOB.Performance on IMDB-JOB: Table REF shows a detailed end-to-end time comparison for the IMDB-JOB workload.","FactorJoin achieves the best end-to-end query runtime ($1324s$ ) amongst all baselines.","Similar observations hold as in the STATS-CEB workload when compared to the traditional methods.","Like STATS-CEB, PessEst achieves the best performance in terms of pure execution time but has a much larger overhead and planning latency.","FactorJoin has comparable performance as PessEst in terms of pure execution time.","However, our planning time is $50 \\times $ shorter than PessEst, making our framework much faster in overall end-to-end query time.","Specifically, our framework can estimate $10,000$ sub-plan queries within one second, which is close to Postgres.","However, unlike in STATS-CEB where we have near-optimal performance, there still exists a large gap between FactorJoin (1324s) and the optimal TrueCard (375s).","There are two possible explanations.","First, our probabilistic bound error can accumulate with the number of joins.","Compared to STATS-CEB, the IMDB-JOB contains more than twice the number of joins in the queries.","Therefore, the relative performance of our framework drops severely when shifting from STATS-CEB to IMDB-JOB.","Second, the base-table filters in IMDB-JOB contain a large amount of highly-selective predicates, making our sampling-based single-table CardEst less accurate.","These observations suggest room for improving FactorJoin, which we discuss further in the future work section.","Summary: FactorJoin achieves the best performance among all the baselines on both benchmarks.","Specifically, FactorJoin is as effective as the previous SOTA methods, and simultaneously as efficient and practical as the traditional CardEst methods.","Figure: Relative estimation errors on STATS-CEB."],["Detailed Analysis","In this section, we first study why FactorJoin is able to achieve SOTA performance by analyzing the tightness of our probabilistic bound.","Then, we exhaustively compare the CardEst method's end-to-end performance on queries with different runtimes to investigate when can FactorJoin perform better or worse than the competitive baselines.","For compactness, here, we only report the results of the most representative methods (Postgres, FLAT, PessEst) on STATS-CEB; the remaining results are in the appendix.","Bound tightness: We show the relative errors between the baselines' estimates and the true cardinalities (estimate / true) for all sub-plan queries on STATS-CEB in Figure REF .","Overall, all three SOTA methods significantly outperform Postgres in terms of estimation accuracy.","PessEst generates exact upper bounds and never underestimates.","FLAT uses a much larger model to understand the distributions of join patterns and produces the most accurate estimates.","FactorJoin can output an upper bound on cardinality for more than $90\\%$ of the sub-plan queries.","Most of the marginal underestimates are very close to the true cardinality, which can still generate relatively effective plans.","Our generated bounds are slightly tighter than the bounds from PessEst, which explains why our query plans achieve slightly better results.","Detailed comparision: We sort the 146 queries of STATS-CEB based on their Postgres end-to-end runtime and cluster them into 6 different runtime intervals.","Figure REF reports relative improvements of the most competitive baselines over Postgres for each query on the left and each cluster of queries on the right.","For the very short-running queries (which represent an OLTP-like workload), Postgres is the best amongst all baselines.","These baselines perform worse because the estimation latency plays a significant role in these queries.","We observe that improving the estimation accuracy of these queries has a very limited effect on the query plan quality.","This also explains why the optimal TrueCard only marginally outperforms Postgres on queries with less than $1s$ of runtime.","However, these queries only contribute a negligible proportion of total runtime in STATS-CEB.","The query optimizer should fall back to default traditional CardEst methods for OLTP-like workload.","FLAT and PessEst are significantly worse than FactorJoin because their planning latency surpasses the execution time, which overshadows the minor improvement in query plans.","For the extremely long-running queries, the advantages of the SOTA methods over Postgres gradually appear.","The reason is that estimation latency becomes increasingly insignificant for queries with a long execution time.","Thus, FLAT and PessEst which spend a long time planning can generate significantly better query plans.","FactorJoin has comparable performance on these queries.","Figure: Per query performance of STATS-CEB.blackIncremental updates: Following the same evaluation setup as previous work [18], we train a stale FactorJoin model on STATS data created before 2014 (roughly $50\\%$ ) and insert the rest of the data to incrementally update this model.","We provide the end-to-end query time of the updated FactorJoin and the update time on STATS-CEB queries in Table REF .","Since FactorJoin only needs to update single-table statistics, it is extremely efficient, i.e.","taking only $2.5s$ to update millions of tuples.","This update speed up to $168 \\times $ faster than the learned data-driven methods with better end-to-end query performance after model update.","We cannot fairly compare with the learned query-driven methods because there does not exist a training query workload after the data insertion.","It worth noticing that the update times reported for other methods in Table REF include the time to re-compute the denormalized join tables for the inserted data, so numbers are larger than the ones reported in the original paper [18].","However, we do see a slight drop in the end-to-end improvement when compared to the FactorJoin trained on the entire dataset ($43.4\\%$ in Table REF versus $45.9\\%$ in Table REF ).","This small performance difference is due to the fact that the bins are decided on the data before 2014 and remain fixed during incremental updates.","Therefore, after inserting the new data, the min-variance property of the bins could be violated, resulting in less accurate predictions.","Table: blackIncremental update performance on STATS-CEB."],["Ablation study","We conduct a series of ablation studies to analyze different optimization techniques inside FactorJoin.","Specifically, we first investigate how the number of bins (k) affects the overall performance.","Next, we analyze the effectiveness of our new bin selection algorithm (GBSA in Section REF ) over the traditional equal-width and equal-depth algorithms.","Then, we study the performance of plugging-in different single-table CardEst methods into FactorJoin.","Due to space limitations, we only provide the results on STATS-CEB.","At last, we investigate how removing each of the simplifying assumptions improves the performance of the original joining histogram.","Number of bins (k): We train and evaluate FactorJoin with five different number of bins (1, 10, 50, 100, 200).","Figure REF shows the end-to-end query time (A), bound tightness (B), estimation latency per query (C), training time (D), and model size (E) of FactorJoin.","For bound tightness, we report the $50\\%, 95\\%, 99\\%$ percentiles of all queries' relative error: estimated/true, as the one used in Figure REF .","Figure REF -(A-B) shows that larger numbers of bins ($k$ ) will generate tighter cardinality bounds and thus more effective query plans.","Figure REF -(C-E) shows that more bins also increase estimation latency, training time, and model size, as expected.","There are several interesting observations from these results.","$\\bullet $ According to Figure REF -(B), with $k=1$ (i.e.","bin the entire domain of each join key into one bin) FactorJoin does not generate tight bounds but it still outperforms Postgres baseline by $24.9\\%$ .","This highlights the importance of upper bounds over underestimation and verifies the effectiveness of bound-based algorithms.","$\\bullet $ Increasing $k$ can substantially increase the bound tightness.","However, at some point (from $k=100$ to $k=200$ ) the framework stops achieving better end-to-end performance.","One explanation is that this fine-grained improvement in estimation accuracy can generate a slightly better query plan but the slower estimation latency cancels out the improvements in query plans.","$\\bullet $ The model size increases quadratically with $k$ because FactorJoin has $O(N*k^2)$ complexity in both storage and inference.","However, estimation latency increases linearly with $k$ because the overall inference time is dominated by the single-table CardEst methods' inference time, whose complexity is linear with $k$ .","Figure: Performance for different number of bins.Table: Performance of different binning techniques.Different bin selection algorithms: Recall that in Section REF , we design a new algorithm (GBSA) to optimize the binning process for FactorJoin.","Here, we compare it with the traditional equal-width and equal-depth binning algorithms.","We set $k = 100$ for all three algorithms.","They have approximately the same estimation latency, training time, and model size.","Thus, we only report the end-to-end performance and bound tightness in Figure REF .","We observe that GBSA generates much tighter upper bounds, leading to significantly better end-to-end performance.","This demonstrates the effectiveness of our GBSA algorithm.","Varying single-table CardEst methods: To compare the importance of different single-table CardEst methods in FactorJoin, we tried three different estimators: 1) Bayesian network (blackBayesCard) for single-table, which is the SOTA learned single-table method [70], 2) Sampling, which draws a random sample (5%) from single tables on-the-fly and estimates the cardinalities of filter predicates, and 3) TrueScan which scans and filters the tables during query time and calculate the true cardinalities.","The end-to-end performance of different single table estimators with bin size $k=100$ is shown in Table REF .","The blackBayesCard method performs significantly better than Sampling because of more accurate estimates and slightly faster estimation speeds.","blackTrueScan produces an exact upper bound on the cardinalities and thus generates more effective query plans.","However, the estimation latency is too high, making its overall end-to-end performance worse than blackBayesCard.","Table: FactorJoin, varying single-table CardEst methods.Improvement over joining histograms: Since FactorJoin follows the convention of JoinHist, we investigate how much each component of FactorJoin help improve this method.","Specifically, apart from the original JoinHist method, we evaluate and compare the following variants.","We incorporate our new probabilistic bound into JoinHist to remove its join uniformity assumption (denote as with Bound).","We incorporate into JoinHist the histograms learned from BayesNet that represent conditional distributions of join keys given the filter predicates (denote as with Conditional).","This variant avoids the attribute independent assumption.","FactorJoin reduces to the JoinHist with both bound and conditional techniques on non-cyclic join templates.","The results in Table REF demonstrate that we can achieve significant improvement over the JoinHist method by removing either or all of its simplifying assumptions.","Summary: With a reasonable bin size $k$ , FactorJoin generates tight bounds and effective query plans.","The GBSA bin selection algorithm is very effective at generating high-quality plans.","Single-table estimators do have an impact on FactorJoin's overall performance and we should choose one that can generate effective and efficient estimates.","Overall, all different settings/hyperparameters of FactorJoin can significantly outperform Postgres as well as all existing SOTA methods, demonstrating the robustness and stability of FactorJoin.","Table: Performance comparison with JoinHist."],["Related work","We briefly review literatures on single-table CardEst methods and the learned query optimizers.","Single-table CardEst: Machine learning approaches have been use to solve single-table CardEst problem, which can achieve accurate estimation with very low latency and overhead [74], [70], [77].","Specifically, the data-driven learned methods build statistical models to understand data distributions, such as deep autoregressive models [74], [19], Bayesian networks [70], [16], [13], sum-product-networks [25], factorized sum-product-networks [77], and normalizing flow models [67].","As for the query-driven methods, the first approach using neural networks for CardEst has been proposed for UDF predicates [36].","Later on, a semi-automatic alternative [47] and a regression-based model [2] were used for general predicates.","Recently, more sophisticated supervised models such as multi-set convolutional networks [33], XG-boost [11], tree-LSTM [63], and deep ensembles [42], are used to provide accurate estimation.","These works are orthogonal to our framework and in principle, we can support plugging in any one of them for single table CardEst.","Learned query optimizer: Apart from CardEst, there exist a large number of works that use ML to solve other tasks of query optimizer, such as cost estimation and join order selection.","Learned cost estimation methods use tree convolution [51] and tree-LSTM [63] to encode a query plan as a tree and map the encoding to its estimated costs.","Active learning [45] and zero-shot learning [24] are proposed to tackle this problem from a new perspective.","Some deep reinforcement learning approaches [75], [50] have been proposed to determine the optimal join order.","Recently, many methods [49], [48], [53], [71] propose to learn the entire query optimizer end-to-end without a clear separation of these components."],["Conclusions","In this paper, we propose FactorJoin, a framework for cardinality estimation of join queries.","It combines classical join-histogram methods with learned single table cardinality estimates into a factor graph model.","This framework converts CardEst problem into an inference problem over the factor graph involving only single-table distributions.","We further propose several optimizations to make this problem tractable for large join graphs.","Our experiments show that FactorJoin generates effective and efficient estimates and is suitable for system deployment on large real-world join benchmarks.","Specifically, FactorJoin produces more effective estimates than the previous SOTA learned methods, with 40x less estimation latency, 100x smaller model size, 100x faster updating speed, and 100x faster training speed, while matching or exceeding them in terms of query execution time.","We believe this work points out a new direction for estimating join queries, which would enable truly practical learned CardEst methods as a system component.","Equations and proofs Equations for decomposing joins into single-tables In this section, we provide the case-by-case equations on how to accurately formulate join estimation problem into single-table CardEst problem without any simplified assumptions.","Two-table join case: Assume that we have two tables $A$ and $B$ with join keys $A.id$ and $B.Aid$ and a query $Q = Q(A) \\mathbin {\\bowtie }\\rule [0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\hspace{-3.05542pt}\\bowtie \\hspace{-3.05542pt}$$$Q(B)$ where $ Q(A)$ is the filter on table $ A$ and same for $ Q(B)$.","The join condition is $ A.id = B.Aid$.","Then, the join cardinality of $ Q$ can be expressed as follows where $ D(A.id)$ represents the domain of $ A$.","{\\begin{@align}{1}{-1}|Q| = \\sum _{v \\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)|\\end{@align}}\\smallskip \\underline{\\textbf {Chain join cases:}}\\emph {\\textbf {Case1 A.id = B.Aid = C.Aid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, and $ C.Aid$.","There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.Aid = C.Aid$.Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| &= \\sum _{v \\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)|\\end{@align}}\\emph {\\textbf {Case2 A.id = B.Aid, B.id = C.Bid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, $ B.id$, and $ C.Bid$.","There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.id = C.Bid$.Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| = &\\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\nonumber \\\\&P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)|\\end{@align}}\\smallskip \\underline{\\textbf {Self join case:}}Assume we have one table $ A$ with join keys $ A.id, A.id2$.","There exists a query $ Q = Q(A) Q(A')$ on join condition $ A.id = A.id2$.","Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| = \\sum _{v \\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& P_{A}(A.id_2 = v | Q(A^{\\prime })) * |Q(A^{\\prime })|\\end{@align}}\\smallskip \\underline{\\textbf {Cyclic join case:}}Assume we have two tables $ A$ and $ B$ with join keys $ A.id, A.id2$ and $ B.Aid, B.Aid2$.","There exists a query $ Q = Q(A) Q(B)$ on join condition $ A.id = B.Aid AND A.id2 = B.Aid2$.","Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| = & \\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(A.id_2)} P_{A}(A.id = v_1| Q(A)) * |Q(A)| * \\nonumber \\\\&P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& P_{A}(A.id_2 = v_2| Q(A^{\\prime })) * |Q(A^{\\prime })| \\nonumber \\\\= & \\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(A.id_2)} P_{B}(B.Aid = v_1| Q(B)) * |Q(B)| \\nonumber \\\\&P_{A}(A.id = v_1, A.id_2 = v_2| Q(A)) * |Q(A)| * \\nonumber \\\\&P_{B}(B.Aid_2 = v_2 | Q(B^{\\prime })) * |Q(B^{\\prime })|\\end{@align}}\\subsection {Proof of lamma 1}\\begin{lemma}Given a join graph \\mathcal {G} representing a query Q, there exists a factor graph \\mathcal {F} such that the variable nodes in \\mathcal {F} are the equivalent key group variables of \\mathcal {G} and each factor node represents a table I touched by Q.","A factor node is connected to a variable node if and only if this variable represents a join key in table I.","The potential function of a factor node is defined as table I^{\\prime }s probability distribution of the connected variables (join keys) conditioned on the filter predicates Q(I).","Then, calculating the cardinality of Q is equivalent to computing the partition function of \\mathcal {F}.\\end{lemma}\\smallskip \\underline{\\textbf {Proof:}}Assume that we have a query $ Q$ joining $ m$ tables $ A, B, ..., M$.We represent $ Q$ as a join graph $ G$, where each node represents a join key and each edge represents a join relation between two keys connected by it.","We further define hyper-nodes in $ G$ as a set of nodes whose corresponding join key lies in the same table.","Thus, each hyper-nodes naturally represents a table in $ Q$.","A visualization of such $ G$ can be found in Figure~\\ref {fig: formulation}.Each connected components of $ G$ connects a group of join keys with equal join relation, suggesting that they share the same semantics.","Therefore, we consider all join keys in a connected component as a equivalent key group variable, denoted as $ Vi$.Therefore, $ G$ defines $ n$ equivalent variables $ V1, ..., Vn$.Then, assume that we have a set of unnormalized single table distributions $ PA(VA|Q(A)) * |Q(A)|, ..., PM(VM|Q(M)) * |Q(M)|$ and $ PA(VA'|Q(A')) * |Q(A')|, ..., PM(VM'|Q(M')) * |Q(M')|$.Each $ VI$ presents a set of equivalent variables that represent a join key in table $ I$.","The $ I'$ is the duplicated table introduced by query $ Q$ only if there exists a cyclic join situation.Therefore, the cardinality of $ Q$ can be writen as:{\\begin{@align}{1}{-1}|Q| = & \\sum _{v_1 \\in D(V_1)} \\sum _{v_2 \\in D(V_2)} \\cdots \\sum _{v_n \\in D(V_n)} (P_{A}(V_A| Q(A)) * |Q(A)|) * \\nonumber \\\\&(P_{B}(V_B | Q(B)) * |Q(B)|) * \\cdots (P_{M}(V_M | Q(M)) * |Q(M)|) * \\nonumber \\\\& (P_{A}(V_{A^{\\prime }}| Q(A^{\\prime })) * |Q(A^{\\prime })|) * \\cdots (P_{M^{\\prime }}(V_{M^{\\prime }} | Q(M^{\\prime })) * |Q(M^{\\prime })|)\\end{@align}}Next, let^{\\prime }s construct a factor graph $ F$, such that the variable nodes in $ F$ are the equivalent key group variables $ Vi$ of $ G$ and the factor nodes in $ F$ represent the tables $ A, ..., M$ touched by $ Q$ and the duplicated tables if exist $ A', ..., M'$.","A factor node representing table $ I$ is connected to a variable node if and only if this variable represents a join key in table $ I$.The potential function of a factor node is defined as the unnormalized probability distribution $ PI(VI|Q(I)) * |Q(I)|$.The partition function of $ F$ is defined exactly the same as Equation~\\ref {equ: lemma1}.Therefore, instead of summing a nested loop of $ n$ variables $ Vi$ by brute-force, the partition function (cardinality) can be computed using the well-established variable elimination and belief propogation techniques in PGM domain.\\subsection {Probabilistic bound}\\emph {FactorJoin} approximate the exact cardinality computation with a probabilistic bound based inference algorithm.","An example on two table join queries can found below.","In this section we discuss how to derive an upper bound ($ Probabilistic_bound(A, B, bini)$ in this equation) for different case of join.","{\\begin{@align}{3}{2}|Q| &= \\sum _{i = 1}^{k} \\sum _{v \\in bin_i} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\nonumber \\\\&\\lesssim \\sum _{i = 1}^{k} && Probabilistic\\_bound(A, B, bin_i)\\end{@align}}\\smallskip \\underline{\\textbf {Case1: Joining two tablesA.id = B.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$.","We have the following bound:{\\begin{@align}{3}{2}|Q| & = \\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min( &&\\frac{P_A(A.id \\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\nonumber \\\\& && \\frac{P_B(B.Aid \\in bin_i|Q(B)) * |Q(B)|}{V^{*}_i(B.Aid)}) * \\nonumber \\\\& && V^{*}_i(A.id) * V^{*}_i(B.Aid)\\end{@align}}\\smallskip \\underline{\\textbf {Case2: Joining three tables A.id = B.Aid = C.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(C.Aid)$.","We have the following bound:{\\begin{@align}{1}{-1}|Q| & = \\sum _{v \\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& \\hspace{20.0pt} P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min\\lbrace \\frac{P_{A}(A.id \\in bin_i | Q(A)) * |Q(A)|}{V_i^{*}(A.id)}, \\nonumber \\\\& \\hspace{50.0pt} \\frac{P_{B}(B.Aid \\in bin_i | Q(B)) * |Q(B)|}{V_i^{*}(B.Aid)} \\nonumber \\\\& \\hspace{50.0pt} \\frac{P_{C}(C.Aid \\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Aid) }\\rbrace * \\nonumber \\\\& \\hspace{50.0pt} {V_i^{*}(A.id)} * {V_i^{*}(B.Aid) } * {V_i^{*}(C.Aid)}\\end{@align}}\\smallskip \\underline{\\textbf {Case3: Joining three tables A.id = B.Aid, B.id = C.Bid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(B.id)$, and $ Vi*(C.Bid)$.We have the following bound, where $ Upper(A B)$ is derived from Equation~\\ref {equ: bound2join}.","{\\begin{@align}{1}{-1}|Q| &= \\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\nonumber \\\\&\\hspace{60.0pt} P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& \\hspace{60.0pt} P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)| \\nonumber \\\\&\\le \\sum _{i = 1}^{k} min\\lbrace \\frac{Upper(Q(A) \\mathbin {\\bowtie }\\rule [0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\hspace{-3.05542pt}\\bowtie \\hspace{-3.05542pt}\\bowtie }{\\rule }[0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\end{@align}Q(B))}{V_i^{*}(A.id) * V_i^{*}(B.id)}, \\nonumber \\\\& \\hspace{50.0pt} \\frac{P_{C}(C.Aid \\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Bid)}\\rbrace * \\nonumber \\\\& \\hspace{50.0pt} * V_i^{*}(A.id) * V_i^{*}(B.id) * {V_i^{*}(C.Bid)}$ Case4: Self join of one table $A.id = A.id2$ : Let $V_i^{*}(A.id) = MAX_{v \\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(A.id_2)$ .","We have the following bound: $|Q| & = \\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& && P_{A}(A.id_2 = v | Q(A^{\\prime })) * |Q(A^{\\prime })| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min( &&\\frac{P_A(A.id \\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\nonumber \\\\& && \\frac{P_A(A.id_2 \\in bin_i|Q(A^{\\prime })) * |Q(A^{\\prime })|}{V^{*}_i(A.id_2)}) * \\nonumber \\\\& && V^{*}_i(A.id) * V^{*}_i(A.id_2)$ Case5: Cyclic join of two tables $A.id = B.Aid$ , $A.id_2 = B.Aid_2$ : Let $V_i^{*}(A.id) = MAX_{v \\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(B.Aid)$ , $V_i^{*}(A.id_2)$ , and $V_i^{*}(B.Aid_2)$ .","We have the following bound: $|Q| &= \\sum _{ v_1 \\in D(A.id)} \\sum _{v_2 \\in D(A.id_2)} P_{A}(A.id = v_1, A.id_2 = v_2 | Q(A)) * \\nonumber \\\\& \\hspace{30.0pt} |Q(A)| * P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& \\hspace{30.0pt} P_{A}(A.id_2 = v_2| Q(A^{\\prime })) * |Q(A^{\\prime })| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min\\lbrace \\frac{Upper(Q(A) \\mathbin {\\bowtie }\\rule [0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\hspace{-3.05542pt}\\bowtie \\hspace{-3.05542pt}\\bowtie }{\\rule }[0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}$ Q(B)Vi*(A.id2) * Vi*(B.Aid) , Upper(Q(A') Q(B))Vi*(A.id) * Vi*(B.Aid2)} * Vi*(A.id2) * Vi*(B.Aid) * Vi*(A.id) * Vi*(B.Aid2) Greedy bin selection Algorithm details We observe that the bound on a particular bin $bin_i$ can be very loose if the MFV count $V^*_i$ is a large outlier in $bin_i$ .","Taking the two table join query as an example, if $bin_i$ contains only one value that appears 100 times in $A.id$ but $10,000$ values that only appear once in $B.Aid$ , then the bound could be 100 times larger than the actual cardinality.","The existing equal-width or equal-depth binning strategy can generate very large estimation errors, so we design a new binning strategy called the greedy bin selection algorithm (GBSA) to optimize our bound tightness.","The objective of GBSA is to minimize the variance of the value counts within $bin_i$ .","In the extreme case, if the value counts have zero variance for all join keys in one equivalent key group, then our bound can output the exact cardinality (if with perfect single table CardEst models).","However, as the same bin partition will be applied for all equivalent join keys, minimizing the value counts variance of $bin_i$ on the domain of one key may result in a bad bin for other keys.","Jointly minimizing the variance of one bin for all join keys has exponential complexity.","There, GBSA uses a greedy algorithm to iteratively minimize the bin variance for all join keys.","At a high level, GBSA first optimizes the minimal variance bins with half number of binning budget $k/2$ on the domain of one join key.","Then, it recursively updates these bins by minimizing the variance of other join keys using the rest half of the budget.","The details of GBSA are provided in Algorithm REF .","[t] Greedy Bin Selection Algorithm (GBSA) Input: Equivalent key groups $Gr_1, \\ldots , Gr_m$ , where $Gr_i = \\lbrace Id_i^1, \\ldots , Id_i^{|Gr_i|} \\rbrace $ ; Column data $\\mathcal {D}(Id_i^j)$ of all join keys in the DB instance $\\mathcal {D}$ ; Number of bins $k_i$ for each group $Gr_i$ .","[1] $Gr_i \\in \\lbrace Gr_1, \\ldots , Gr_m\\rbrace $ $Bin(Gr_i) \\leftarrow $ [] $Gr_i^{\\prime } \\leftarrow sort\\_key\\_based\\_on\\_domain\\_size(\\mathcal {D}, Gr_i)$ $Bin(Gr_i) \\leftarrow get\\_min\\_variance\\_bins(\\mathcal {D}(Gr_i^{\\prime }[1]), k_i/2)$ remain_bins $\\leftarrow k_i/2$ $j \\in \\lbrace 2, \\ldots , |Gr_i^{\\prime }| \\rbrace $ binned_data $\\leftarrow apply\\_bin\\_to\\_data(\\mathcal {D}(Gr_i^{\\prime }[j]), Bin(Gr_i))$ bin_variance $\\leftarrow calculate\\_variance$ (binned_data) arg_sort_idx $\\leftarrow arg\\_sort\\_decreasing(bin\\_variance)$ $p \\in arg\\_sort\\_idx[1:remain\\_bins/2]$ $Bin(Gr_i) \\leftarrow min\\_variance\\_dichotomy(Bin(Gr_i)[p]$ , binned_data[p]) remain_bins $\\leftarrow $ remain_bins/2 return $\\lbrace Bin(Gr_i) | i, \\ldots , m\\rbrace $ FactorJoin analyzes the schema of DB instance $\\mathcal {D}$ to derive $m$ equivalent key groups $Gr_1, \\ldots , Gr_m$ , each of which contains $|Gr_i|$ join keys $Id_i^1, \\ldots , Id_i^{|Gr_i|}$ .","Let $\\mathcal {D}(Id_i^j)$ represents the column data (domain) of join key $Id_i^j$ .","GBSA will derive the sub-optimal binning with $k_i$ bins $Bin(Gr_i) = \\lbrace bin_1, \\ldots , bin_{k_i} \\rbrace $ for each the equivalent key group $Gr_i$ (line 1).","We explain the procedure of this algorithm for binning one group $Gr_i$ (line 2-14).","First, GBSA sorts the join keys $\\lbrace d_i^1, \\ldots , Id_i^{|Gr_i|}\\rbrace $ in decreasing order based on their domain size (line 3) to get $Gr_i^{\\prime }=\\lbrace Id_i^{1^{\\prime }}, \\ldots , Id_i^{|Gr_i|^{\\prime }}\\rbrace $ .","We apply half of the binning budget to generate $k_i/2$ minimal variance bins on the domain of $d_i^{1^{\\prime }}$ (line 4).","Note that minimal variance bins on a single attribute can be easily obtained by sorting the value counts of $\\mathcal {D}(d_i^{1^{\\prime }})$ and applying equal-depth binning over the sorted values.","The remaining number of bins available $remain_bins$ is $k_i/2$ (line 5).","Then for each rest of the join key $d_i^j$ , GBSA applies the current bins $Bin(Gr_i)$ to its data column $\\mathcal {D}(d_i^j)$ , calculates the variance for each bin, and sorts these bins in decreasing order (line 6-9).","For the top $remain_bins/2$ bins with the largest variance, we dichotomize each bin into two bins to minimize the variance on join key $d_i^j$ (lines 10 -12).","We iterative the above procedure until all join keys are optimized.","According to our evaluation results, the GBSA has a significant impact on improving our probabilistic bound tightness and estimation effectiveness.","Figure: Per query performance of STATS-CEB.Figure: Per query performance of STATS-CEB.","Additional experiments and details In this section, we provide the detailed comparision of all baselines on STATS-CEB and IMDB-JOB benchmarks.","Performance on STATS-CEB: We sort the 146 queries of STATS-CEB based on their Postgres end-to-end runtime and cluster them into 6 different runtime intervals.","Figure REF reports relative improvements of all competitive baselines over Postgres for each query and the overall performance comparision on each cluster of queries on the last figure.","For the very short-running queries (which represent an OLTP-like workload), Postgres is the best among all baselines.","These baselines perform worse because the estimation latency plays a significant role in these queries.","We observe that improving estimation accuracy has a very limited effect on the query plan quality of short-running queries.","This also explains why the optimal TrueCard only marginally outperforms Postgres on queries with less than $2s$ of runtime.","Overall, FactorJoin has the best performance among all baselines except for Postgres.","For the extremely long-running queries, the advantage of the learning-based methods over Postgres gradually appear.","The reason is that estimation latency becomes increasingly insignificant for queries with a long execution time.","FactorJoin has comparable performance as the SOTA learning-based methods on these queries.","Performance on IMDB-JOB: Since a large proportion of the queries in IMDB-JOB workload are short-running queries, Postgres has a better performance than all baselines for more than half of the queries.","Similar to STATS-CEB queries, we do not observe any performance gain for using the learned CardEst method over Postgres on queries with less on $1s$ runtime.","In fact, the optimal TrueCard also barely improves the Postgres.","This suggests that the learned methods shoulf fall back to Postgres for OLTP workloads.","Similar to STATS-CEB, FactorJoin also has a significantly better performance over all other baselines except Postgres."],["Equations for decomposing joins into single-tables","In this section, we provide the case-by-case equations on how to accurately formulate join estimation problem into single-table CardEst problem without any simplified assumptions.","Two-table join case: Assume that we have two tables $A$ and $B$ with join keys $A.id$ and $B.Aid$ and a query $Q = Q(A) \\mathbin {\\bowtie }\\rule [0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\hspace{-3.05542pt}\\bowtie \\hspace{-3.05542pt}$$$Q(B)$ where $ Q(A)$ is the filter on table $ A$ and same for $ Q(B)$.","The join condition is $ A.id = B.Aid$.","Then, the join cardinality of $ Q$ can be expressed as follows where $ D(A.id)$ represents the domain of $ A$.","{\\begin{@align}{1}{-1}|Q| = \\sum _{v \\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)|\\end{@align}}\\smallskip \\underline{\\textbf {Chain join cases:}}\\emph {\\textbf {Case1 A.id = B.Aid = C.Aid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, and $ C.Aid$.","There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.Aid = C.Aid$.Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| &= \\sum _{v \\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)|\\end{@align}}\\emph {\\textbf {Case2 A.id = B.Aid, B.id = C.Bid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, $ B.id$, and $ C.Bid$.","There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.id = C.Bid$.Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| = &\\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\nonumber \\\\&P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)|\\end{@align}}\\smallskip \\underline{\\textbf {Self join case:}}Assume we have one table $ A$ with join keys $ A.id, A.id2$.","There exists a query $ Q = Q(A) Q(A')$ on join condition $ A.id = A.id2$.","Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| = \\sum _{v \\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& P_{A}(A.id_2 = v | Q(A^{\\prime })) * |Q(A^{\\prime })|\\end{@align}}\\smallskip \\underline{\\textbf {Cyclic join case:}}Assume we have two tables $ A$ and $ B$ with join keys $ A.id, A.id2$ and $ B.Aid, B.Aid2$.","There exists a query $ Q = Q(A) Q(B)$ on join condition $ A.id = B.Aid AND A.id2 = B.Aid2$.","Then, the join cardinality can be expressed as follows:{\\begin{@align}{1}{-1}|Q| = & \\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(A.id_2)} P_{A}(A.id = v_1| Q(A)) * |Q(A)| * \\nonumber \\\\&P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& P_{A}(A.id_2 = v_2| Q(A^{\\prime })) * |Q(A^{\\prime })| \\nonumber \\\\= & \\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(A.id_2)} P_{B}(B.Aid = v_1| Q(B)) * |Q(B)| \\nonumber \\\\&P_{A}(A.id = v_1, A.id_2 = v_2| Q(A)) * |Q(A)| * \\nonumber \\\\&P_{B}(B.Aid_2 = v_2 | Q(B^{\\prime })) * |Q(B^{\\prime })|\\end{@align}}\\subsection {Proof of lamma 1}\\begin{lemma}Given a join graph \\mathcal {G} representing a query Q, there exists a factor graph \\mathcal {F} such that the variable nodes in \\mathcal {F} are the equivalent key group variables of \\mathcal {G} and each factor node represents a table I touched by Q.","A factor node is connected to a variable node if and only if this variable represents a join key in table I.","The potential function of a factor node is defined as table I^{\\prime }s probability distribution of the connected variables (join keys) conditioned on the filter predicates Q(I).","Then, calculating the cardinality of Q is equivalent to computing the partition function of \\mathcal {F}.\\end{lemma}\\smallskip \\underline{\\textbf {Proof:}}Assume that we have a query $ Q$ joining $ m$ tables $ A, B, ..., M$.We represent $ Q$ as a join graph $ G$, where each node represents a join key and each edge represents a join relation between two keys connected by it.","We further define hyper-nodes in $ G$ as a set of nodes whose corresponding join key lies in the same table.","Thus, each hyper-nodes naturally represents a table in $ Q$.","A visualization of such $ G$ can be found in Figure~\\ref {fig: formulation}.Each connected components of $ G$ connects a group of join keys with equal join relation, suggesting that they share the same semantics.","Therefore, we consider all join keys in a connected component as a equivalent key group variable, denoted as $ Vi$.Therefore, $ G$ defines $ n$ equivalent variables $ V1, ..., Vn$.Then, assume that we have a set of unnormalized single table distributions $ PA(VA|Q(A)) * |Q(A)|, ..., PM(VM|Q(M)) * |Q(M)|$ and $ PA(VA'|Q(A')) * |Q(A')|, ..., PM(VM'|Q(M')) * |Q(M')|$.Each $ VI$ presents a set of equivalent variables that represent a join key in table $ I$.","The $ I'$ is the duplicated table introduced by query $ Q$ only if there exists a cyclic join situation.Therefore, the cardinality of $ Q$ can be writen as:{\\begin{@align}{1}{-1}|Q| = & \\sum _{v_1 \\in D(V_1)} \\sum _{v_2 \\in D(V_2)} \\cdots \\sum _{v_n \\in D(V_n)} (P_{A}(V_A| Q(A)) * |Q(A)|) * \\nonumber \\\\&(P_{B}(V_B | Q(B)) * |Q(B)|) * \\cdots (P_{M}(V_M | Q(M)) * |Q(M)|) * \\nonumber \\\\& (P_{A}(V_{A^{\\prime }}| Q(A^{\\prime })) * |Q(A^{\\prime })|) * \\cdots (P_{M^{\\prime }}(V_{M^{\\prime }} | Q(M^{\\prime })) * |Q(M^{\\prime })|)\\end{@align}}Next, let^{\\prime }s construct a factor graph $ F$, such that the variable nodes in $ F$ are the equivalent key group variables $ Vi$ of $ G$ and the factor nodes in $ F$ represent the tables $ A, ..., M$ touched by $ Q$ and the duplicated tables if exist $ A', ..., M'$.","A factor node representing table $ I$ is connected to a variable node if and only if this variable represents a join key in table $ I$.The potential function of a factor node is defined as the unnormalized probability distribution $ PI(VI|Q(I)) * |Q(I)|$.The partition function of $ F$ is defined exactly the same as Equation~\\ref {equ: lemma1}.Therefore, instead of summing a nested loop of $ n$ variables $ Vi$ by brute-force, the partition function (cardinality) can be computed using the well-established variable elimination and belief propogation techniques in PGM domain.\\subsection {Probabilistic bound}\\emph {FactorJoin} approximate the exact cardinality computation with a probabilistic bound based inference algorithm.","An example on two table join queries can found below.","In this section we discuss how to derive an upper bound ($ Probabilistic_bound(A, B, bini)$ in this equation) for different case of join.","{\\begin{@align}{3}{2}|Q| &= \\sum _{i = 1}^{k} \\sum _{v \\in bin_i} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\nonumber \\\\&\\lesssim \\sum _{i = 1}^{k} && Probabilistic\\_bound(A, B, bin_i)\\end{@align}}\\smallskip \\underline{\\textbf {Case1: Joining two tablesA.id = B.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$.","We have the following bound:{\\begin{@align}{3}{2}|Q| & = \\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min( &&\\frac{P_A(A.id \\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\nonumber \\\\& && \\frac{P_B(B.Aid \\in bin_i|Q(B)) * |Q(B)|}{V^{*}_i(B.Aid)}) * \\nonumber \\\\& && V^{*}_i(A.id) * V^{*}_i(B.Aid)\\end{@align}}\\smallskip \\underline{\\textbf {Case2: Joining three tables A.id = B.Aid = C.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(C.Aid)$.","We have the following bound:{\\begin{@align}{1}{-1}|Q| & = \\sum _{v \\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& \\hspace{20.0pt} P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min\\lbrace \\frac{P_{A}(A.id \\in bin_i | Q(A)) * |Q(A)|}{V_i^{*}(A.id)}, \\nonumber \\\\& \\hspace{50.0pt} \\frac{P_{B}(B.Aid \\in bin_i | Q(B)) * |Q(B)|}{V_i^{*}(B.Aid)} \\nonumber \\\\& \\hspace{50.0pt} \\frac{P_{C}(C.Aid \\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Aid) }\\rbrace * \\nonumber \\\\& \\hspace{50.0pt} {V_i^{*}(A.id)} * {V_i^{*}(B.Aid) } * {V_i^{*}(C.Aid)}\\end{@align}}\\smallskip \\underline{\\textbf {Case3: Joining three tables A.id = B.Aid, B.id = C.Bid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(B.id)$, and $ Vi*(C.Bid)$.We have the following bound, where $ Upper(A B)$ is derived from Equation~\\ref {equ: bound2join}.","{\\begin{@align}{1}{-1}|Q| &= \\sum _{v_1 \\in D(A.id)} \\sum _{v_2 \\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\nonumber \\\\&\\hspace{60.0pt} P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& \\hspace{60.0pt} P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)| \\nonumber \\\\&\\le \\sum _{i = 1}^{k} min\\lbrace \\frac{Upper(Q(A) \\mathbin {\\bowtie }\\rule [0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\hspace{-3.05542pt}\\bowtie \\hspace{-3.05542pt}\\bowtie }{\\rule }[0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\end{@align}Q(B))}{V_i^{*}(A.id) * V_i^{*}(B.id)}, \\nonumber \\\\& \\hspace{50.0pt} \\frac{P_{C}(C.Aid \\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Bid)}\\rbrace * \\nonumber \\\\& \\hspace{50.0pt} * V_i^{*}(A.id) * V_i^{*}(B.id) * {V_i^{*}(C.Bid)}$ Case4: Self join of one table $A.id = A.id2$ : Let $V_i^{*}(A.id) = MAX_{v \\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(A.id_2)$ .","We have the following bound: $|Q| & = \\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\nonumber \\\\& && P_{A}(A.id_2 = v | Q(A^{\\prime })) * |Q(A^{\\prime })| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min( &&\\frac{P_A(A.id \\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\nonumber \\\\& && \\frac{P_A(A.id_2 \\in bin_i|Q(A^{\\prime })) * |Q(A^{\\prime })|}{V^{*}_i(A.id_2)}) * \\nonumber \\\\& && V^{*}_i(A.id) * V^{*}_i(A.id_2)$ Case5: Cyclic join of two tables $A.id = B.Aid$ , $A.id_2 = B.Aid_2$ : Let $V_i^{*}(A.id) = MAX_{v \\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(B.Aid)$ , $V_i^{*}(A.id_2)$ , and $V_i^{*}(B.Aid_2)$ .","We have the following bound: $|Q| &= \\sum _{ v_1 \\in D(A.id)} \\sum _{v_2 \\in D(A.id_2)} P_{A}(A.id = v_1, A.id_2 = v_2 | Q(A)) * \\nonumber \\\\& \\hspace{30.0pt} |Q(A)| * P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\nonumber \\\\& \\hspace{30.0pt} P_{A}(A.id_2 = v_2| Q(A^{\\prime })) * |Q(A^{\\prime })| \\nonumber \\\\& \\le \\sum _{i = 1}^{k} min\\lbrace \\frac{Upper(Q(A) \\mathbin {\\bowtie }\\rule [0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}\\hspace{-3.05542pt}\\bowtie \\hspace{-3.05542pt}\\bowtie }{\\rule }[0.15ex]{.22em}{.6pt}\\unknown.","{\\rule [0.9ex]{.22em}{.6pt}}$ Q(B)Vi*(A.id2) * Vi*(B.Aid) , Upper(Q(A') Q(B))Vi*(A.id) * Vi*(B.Aid2)} * Vi*(A.id2) * Vi*(B.Aid) * Vi*(A.id) * Vi*(B.Aid2)"],["Greedy bin selection Algorithm details","We observe that the bound on a particular bin $bin_i$ can be very loose if the MFV count $V^*_i$ is a large outlier in $bin_i$ .","Taking the two table join query as an example, if $bin_i$ contains only one value that appears 100 times in $A.id$ but $10,000$ values that only appear once in $B.Aid$ , then the bound could be 100 times larger than the actual cardinality.","The existing equal-width or equal-depth binning strategy can generate very large estimation errors, so we design a new binning strategy called the greedy bin selection algorithm (GBSA) to optimize our bound tightness.","The objective of GBSA is to minimize the variance of the value counts within $bin_i$ .","In the extreme case, if the value counts have zero variance for all join keys in one equivalent key group, then our bound can output the exact cardinality (if with perfect single table CardEst models).","However, as the same bin partition will be applied for all equivalent join keys, minimizing the value counts variance of $bin_i$ on the domain of one key may result in a bad bin for other keys.","Jointly minimizing the variance of one bin for all join keys has exponential complexity.","There, GBSA uses a greedy algorithm to iteratively minimize the bin variance for all join keys.","At a high level, GBSA first optimizes the minimal variance bins with half number of binning budget $k/2$ on the domain of one join key.","Then, it recursively updates these bins by minimizing the variance of other join keys using the rest half of the budget.","The details of GBSA are provided in Algorithm REF .","[t] Greedy Bin Selection Algorithm (GBSA) Input: Equivalent key groups $Gr_1, \\ldots , Gr_m$ , where $Gr_i = \\lbrace Id_i^1, \\ldots , Id_i^{|Gr_i|} \\rbrace $ ; Column data $\\mathcal {D}(Id_i^j)$ of all join keys in the DB instance $\\mathcal {D}$ ; Number of bins $k_i$ for each group $Gr_i$ .","[1] $Gr_i \\in \\lbrace Gr_1, \\ldots , Gr_m\\rbrace $ $Bin(Gr_i) \\leftarrow $ [] $Gr_i^{\\prime } \\leftarrow sort\\_key\\_based\\_on\\_domain\\_size(\\mathcal {D}, Gr_i)$ $Bin(Gr_i) \\leftarrow get\\_min\\_variance\\_bins(\\mathcal {D}(Gr_i^{\\prime }[1]), k_i/2)$ remain_bins $\\leftarrow k_i/2$ $j \\in \\lbrace 2, \\ldots , |Gr_i^{\\prime }| \\rbrace $ binned_data $\\leftarrow apply\\_bin\\_to\\_data(\\mathcal {D}(Gr_i^{\\prime }[j]), Bin(Gr_i))$ bin_variance $\\leftarrow calculate\\_variance$ (binned_data) arg_sort_idx $\\leftarrow arg\\_sort\\_decreasing(bin\\_variance)$ $p \\in arg\\_sort\\_idx[1:remain\\_bins/2]$ $Bin(Gr_i) \\leftarrow min\\_variance\\_dichotomy(Bin(Gr_i)[p]$ , binned_data[p]) remain_bins $\\leftarrow $ remain_bins/2 return $\\lbrace Bin(Gr_i) | i, \\ldots , m\\rbrace $ FactorJoin analyzes the schema of DB instance $\\mathcal {D}$ to derive $m$ equivalent key groups $Gr_1, \\ldots , Gr_m$ , each of which contains $|Gr_i|$ join keys $Id_i^1, \\ldots , Id_i^{|Gr_i|}$ .","Let $\\mathcal {D}(Id_i^j)$ represents the column data (domain) of join key $Id_i^j$ .","GBSA will derive the sub-optimal binning with $k_i$ bins $Bin(Gr_i) = \\lbrace bin_1, \\ldots , bin_{k_i} \\rbrace $ for each the equivalent key group $Gr_i$ (line 1).","We explain the procedure of this algorithm for binning one group $Gr_i$ (line 2-14).","First, GBSA sorts the join keys $\\lbrace d_i^1, \\ldots , Id_i^{|Gr_i|}\\rbrace $ in decreasing order based on their domain size (line 3) to get $Gr_i^{\\prime }=\\lbrace Id_i^{1^{\\prime }}, \\ldots , Id_i^{|Gr_i|^{\\prime }}\\rbrace $ .","We apply half of the binning budget to generate $k_i/2$ minimal variance bins on the domain of $d_i^{1^{\\prime }}$ (line 4).","Note that minimal variance bins on a single attribute can be easily obtained by sorting the value counts of $\\mathcal {D}(d_i^{1^{\\prime }})$ and applying equal-depth binning over the sorted values.","The remaining number of bins available $remain_bins$ is $k_i/2$ (line 5).","Then for each rest of the join key $d_i^j$ , GBSA applies the current bins $Bin(Gr_i)$ to its data column $\\mathcal {D}(d_i^j)$ , calculates the variance for each bin, and sorts these bins in decreasing order (line 6-9).","For the top $remain_bins/2$ bins with the largest variance, we dichotomize each bin into two bins to minimize the variance on join key $d_i^j$ (lines 10 -12).","We iterative the above procedure until all join keys are optimized.","According to our evaluation results, the GBSA has a significant impact on improving our probabilistic bound tightness and estimation effectiveness.","Figure: Per query performance of STATS-CEB.Figure: Per query performance of STATS-CEB."],["Additional experiments and details","In this section, we provide the detailed comparision of all baselines on STATS-CEB and IMDB-JOB benchmarks.","Performance on STATS-CEB: We sort the 146 queries of STATS-CEB based on their Postgres end-to-end runtime and cluster them into 6 different runtime intervals.","Figure REF reports relative improvements of all competitive baselines over Postgres for each query and the overall performance comparision on each cluster of queries on the last figure.","For the very short-running queries (which represent an OLTP-like workload), Postgres is the best among all baselines.","These baselines perform worse because the estimation latency plays a significant role in these queries.","We observe that improving estimation accuracy has a very limited effect on the query plan quality of short-running queries.","This also explains why the optimal TrueCard only marginally outperforms Postgres on queries with less than $2s$ of runtime.","Overall, FactorJoin has the best performance among all baselines except for Postgres.","For the extremely long-running queries, the advantage of the learning-based methods over Postgres gradually appear.","The reason is that estimation latency becomes increasingly insignificant for queries with a long execution time.","FactorJoin has comparable performance as the SOTA learning-based methods on these queries.","Performance on IMDB-JOB: Since a large proportion of the queries in IMDB-JOB workload are short-running queries, Postgres has a better performance than all baselines for more than half of the queries.","Similar to STATS-CEB queries, we do not observe any performance gain for using the learned CardEst method over Postgres on queries with less on $1s$ runtime.","In fact, the optimal TrueCard also barely improves the Postgres.","This suggests that the learned methods shoulf fall back to Postgres for OLTP workloads.","Similar to STATS-CEB, FactorJoin also has a significantly better performance over all other baselines except Postgres."]],"string":"[\n [\n \"FactorJoin: A New Cardinality Estimation Framework for Join Queries\"\n ],\n [\n \"Abstract Cardinality estimation is one of the most fundamental and challenging problems in query optimization.\",\n \"Neither classical nor learning-based methods yield satisfactory performance when estimating the cardinality of the join queries.\",\n \"They either rely on simplified assumptions leading to ineffective cardinality estimates or build large models to understand the data distributions, leading to long planning times and a lack of generalizability across queries.\",\n \"In this paper, we propose a new framework FactorJoin for estimating join queries.\",\n \"FactorJoin combines the idea behind the classical join-histogram method to efficiently handle joins with the learning-based methods to accurately capture attribute correlation.\",\n \"Specifically, FactorJoin scans every table in a DB and builds single-table conditional distributions during an offline preparation phase.\",\n \"When a join query comes, FactorJoin translates it into a factor graph model over the learned distributions to effectively and efficiently estimate its cardinality.\",\n \"Unlike existing learning-based methods, FactorJoin does not need to de-normalize joins upfront or require executed query workloads to train the model.\",\n \"Since it only relies on single-table statistics, FactorJoin has small space overhead and is extremely easy to train and maintain.\",\n \"In our evaluation, FactorJoin can produce more effective estimates than the previous state-of-the-art learning-based methods, with 40x less estimation latency, 100x smaller model size, and 100x faster training speed at comparable or better accuracy.\",\n \"In addition, FactorJoin can estimate 10,000 sub-plan queries within one second to optimize the query plan, which is very close to the traditional cardinality estimators in commercial DBMS.\"\n ],\n [\n \"Introduction\",\n \"Cardinality estimation (CardEst) is a critical component of modern database query optimizers.\",\n \"The goal of CardEst is to estimate the result size of each query operator (i.e., filters and joins), allowing the optimizer to select the most efficient join ordering and physical operator implementations.\",\n \"blackAn ideal CardEst method satisfies several properties: it is effective at generating high-quality query plans, efficient so that it minimizes estimation latency, and easy to deploy in that it has a small model size, fast training times, and the ability to scale with the number of tables and generalize to new queries.\",\n \"Unfortunately, existing CardEst techniques do not satisfy at least one of these properties.\",\n \"The reason behind these failures can largely be reduced to the handling of two related problems: (1) how to characterize attribute correlations and (2) how to model the distribution of join-keys, which directly determine the size of joins.\",\n \"Background: Many deployed CardEst approaches are still based on the Selinger model [60] and assume attribute independence, where the model ignores correlations between attributes, and join-key uniformity, blackwhere the model further assumes that join keys have uniformly distributed values.\",\n \"These assumptions provide a simple way of estimating the cardinality of join queries using only single-table statistics (e.g., histograms) over single attributes, which are easy to create and maintain.\",\n \"blackAs a result, the Selinger-based CardEst techniques are extremely efficient and easy to deploy but at the cost of effectiveness because most real-world datasets contain complex attribute correlations and skewed join-key distributions.\",\n \"To overcome the limitations of these assumptions, many commercial database systems try to blackrelax them to increase their effectiveness.\",\n \"blackFor example, past works [26], [29], [7] propose and some systems (e.g.\",\n \"Oracle [8]) employ join-histograms that relax the join-key uniformity assumption.\",\n \"Specifically, these methods build frequency histograms on the join keys and then construct the join distribution by “multiplying” the histograms to estimate the join size (see Figure REF (b)).\",\n \"In this way, these methods can more precisely capture the join-key distributions to generate more effective estimates of the join cardinality.\",\n \"However, they still assume that the join keys are uniformly distributed within each bin of the histogram.\",\n \"Figure: Existing CardEst approaches for handling join queries.\",\n \"(a) The Selinger model writes the cardinality of a two-table join query as the product of filter selectivity P A ,P B P_A, P_B on both tables and the estimated join size |A⋈B||A \\\\bowtie B|.\",\n \"Using join-key uniformity, this is estimated as the product of table sizes |A|*|B||A| * |B| divided by the maximal number of unique values of the join keys.\",\n \"(b) The join-histogram bin the domain of join keys Id 1 ,Id 2 Id_1, Id_2 as histograms, assumes join-key uniformity within each bin.\",\n \"(c) The learned data-driven methods de-normalize J=A⋈BJ = A \\\\bowtie B and learn the distribution P J P_J to estimate the cardinality of QQ.\",\n \"(d) Our framework captures the correlation between join keys and filter predicates to accurately estimate |Q||Q| without de-normalization.Similarly, there have been works on relaxing the attribute independence assumption.\",\n \"For example, multi-dimensional histograms [9], [14], [15], [52], [66], [43], self-tuning histograms [4], [61], [31], [12], and singular value decomposition [58], have been proposed to capture the correlation between attributes.\",\n \"However, their effectiveness and/or efficiency are still not satisfactory [74], [18].\",\n \"More recently, learning-based methods which learn the underlying data distributions [33], [23], [77], [74] have been proposed to more accurately and compactly represent the attribute correlations within a single table, but these still have efficiency concerns that we detail below.\",\n \"Specifically, the learned data-driven methods [13], [64], [73], [70], [77], [25] analyze data and build distributions for all join patterns in a database.\",\n \"They need to de-normalize the joined tables and add a potentially exponential number of extra columns.\",\n \"Then, they build distributions over the de-normalized tables to characterize all attribute correlations and handle joins (see also Figure REF (c).\",\n \"blackThis allows the learned data-driven methods to be highly accurate for join estimates at the cost of slow training time and large model size (i.e., worse deployability).\",\n \"Alternatively, learned query-driven methods [33], [63], [42] circumvent the join-key uniformity assumption by building supervised models to map the join queries to their cardinalities.\",\n \"However, they require an impractical number of executed queries to train their models, which is unavailable to new DB instances and do not generalize well to unseen queries.\",\n \"Our Approach: Surprisingly, we are not aware of any work which tries to combine the advances of learning-based methods for accurately characterizing correlations with the idea of join-histograms.\",\n \"In this work, we develop a novel CardEst framework called FactorJoin that combines these two approaches, i.e., using join-histograms to efficiently handle joins coupled with learning-based methods to accurately capture attribute correlation.\",\n \"blackThis is not a straightforward extension.\",\n \"blackAs we will show, adapting the idea of join-histograms while preserving the benefits of using learning-based methods for attribute correlations is non-trivial.\",\n \"FactorJoin only builds sophisticated models to capture the correlations within a single table.\",\n \"The choice of the model to capture correlation is orthogonal to the techniques of FactorJoin, however, we do require the model to be able to provide conditional distributions of one or more keys.\",\n \"In our current implementation of FactorJoin, we use blacksampling [41] and Bayesian Networks [34] as they are fast to train and execute.\",\n \"These single-table conditional distributions are then combined using a factor graph model [44].\",\n \"Specifically, FactorJoin translates a join query $Q$ into a factor graph over single-table data distributions.\",\n \"This allows us to formulate the problem of estimating the cardinality of $Q$ as a well-studied inference problem [34], [44], [37] on this factor graph.\",\n \"Conceptually, there exist some similarities to the join-histogram technique.\",\n \"However, our formulation allows FactorJoin to generalize to cyclic and self-joins.\",\n \"It also improves the computation efficiency by relying on existing inference algorithms [46], [35], and facilitates scaling to arbitrary join sizes.\",\n \"Finally, in contrast to the idea of join-histogram and many other techniques in the space, we propose to actually not estimate the expected join cardinality, but rather use a probabilistic upper bound.\",\n \"As shown by others [5], [3], [1], [22], under-estimation is often worse than over-estimation, and tailoring the estimate to a probabilistic upper bound can significantly improve the end-to-end query performance.\",\n \"As a result, FactorJoin is able to estimate the cardinality of join queries using only single table statistics with similar or better estimation effectiveness than the state-of-the-art (SOTA) learning-based techniques.\",\n \"When compared with the learned data-driven methods, FactorJoin has a much smaller model size, faster training/updating speed, and easier for system deployment because we do not need to denormalize the join tables or add exponentially many extra columns.\",\n \"In addition, FactorJoin supports all forms of equi-joins, including cyclic joins and self joins, as well as complex base table filter predicates, including disjunctive filter clauses and string pattern matching predicates blackbecause it can flexibly plug-in various types of base-table estimators to support them.\",\n \"These queries are not supported by the existing learned data-driven methods [73], [25], [77], which require the join template to be a tree.\",\n \"Unlikely the learned query-driven methods [33], [63], [42], FactorJoin does not depend on the executed query workload, so it is robust against data updates and workload shifts, can be quickly adapted to a new DB instance, and is generalizable to new queries.\",\n \"However, in presence of query workload, FactorJoin can incorporate this information to further optimize the model construction process.\",\n \"We integrate FactorJoin into Postgres' query optimizer and evaluate its end-to-end query performance on two well-established and challenging real-world CardEst benchmarks: STATS-CEB [18] and IMDB-JOB [38].\",\n \"On the STATS-CEB, FactorJoin achieves near-optimal performance in end-to-end query time.\",\n \"It has comparable performance to the previous SOTA learned data-driven model (FLAT [77]) in terms of estimation effectiveness but with 40x lower estimation latency, 100x smaller model sizes, and 100x faster training times.\",\n \"On the IMDB-JOB, our framework achieves the SOTA performance in end-to-end query time (query execution plus planning time).\",\n \"Specifically, the pure execution time of our framework is comparable to the previous SOTA method (pessimistic estimator [5]), but our estimation latency is 100x lower, making our overall end-to-end query time significantly faster.\",\n \"Specifically, our framework can estimate 10,000 sub-plan queries in one second to optimize the query plan, which is close to the planning time of the traditional CardEst method used in Postgres.\",\n \"We also carry out a series of controlled ablation studies to demonstrate the robustness and advantages of different technical novelties of FactorJoin.\",\n \"In summary, our main contributions are: $\\\\bullet $ We formulate the problem of estimating the cardinality of join queries in its general form as a factor graph inference problem involving only single-table data distributions (Section ).\",\n \"$\\\\bullet $ We propose a new binning and upper bound-based algorithm to approximate the factor graph inference (Section ).\",\n \"$\\\\bullet $ We design blackattribute causal relation exploration and progressive estimation of sub-plan queries techniques to improve the efficiency of our framework (Section ).\",\n \"$\\\\bullet $ We conduct extensive experiments to show the advantages of our framework (Section ).\",\n \"Before describing the details of FactorJoin, we begin with a detailed description of existing CardEst approaches.\"\n ],\n [\n \"Background and analysis\",\n \"In this section, we first define the CardEst problem and then analyze existing CardEst methods and how our approach differs from them.\"\n ],\n [\n \"Single table CardEst: Let $A$ be a table with $n$ attributes $A_1, \\\\ldots , A_n$ .\",\n \"A single table selection query $Q(A)$ on $A$ can be viewed a conjunction or disjunction of filter predicates over each attribute, e.g.\",\n \"$Q(A) = (A_1 \\\\in [0, 5]) \\\\ \\\\vee $ $(A_2$ LIKE `%An%'$) \\\\wedge (A_5 \\\\le -10)$ .\",\n \"Let $|Q(A)|$ denote the cardinality of $Q(A)$ , i.e.\",\n \"the number of tuples in $A$ satisfying $Q(A)$ .\",\n \"Let $P_A(Q(A))$ denotes the selectivity (probability) of tuples satisfying $Q(A)$ .\",\n \"Then, we have $|Q(A)| = P_A(Q(A)) * |A|$ , where $|A|$ denotes total number of tuples in $A$ .\",\n \"CardEst for multi-table join queries: blackConsider a database instance $\\\\mathcal {D}$ with m tables $A, B, \\\\ldots , M$ , a query $Q$ consists of a join graph representing the join conditions among the selected tables and a set of base-table filter predicates.\",\n \"blackFor simplicity, we abuse the notation and use $Q(I)$ to denote the filter predicates of $Q$ on table $I$ .\",\n \"Let $\\\\Omega $ denote the denormalized table resulting from the join conditions of $Q$ .\",\n \"Then, the cardinality $|Q| = P_{\\\\Omega }(Q(A), Q(B), \\\\ldots , Q(M)) * |\\\\Omega |$ , where $Q(I)$ can be empty set if table $I$ is not touched by $Q$ .\",\n \"Estimating $Q$ is particularly challenging because there can exponential number of join patterns in the database, each associated with a unique $\\\\Omega $ and distinct probability distribution $P_{\\\\Omega }$ .\",\n \"Therefore, to accurately estimate different join queries, the CardEst methods need to capture exponential many complicated data distributions.\"\n ],\n [\n \"Analysis of existing \",\n \"Since the focus of this paper is multi-table join queries, we defer the discussion of single-table estimators to Section and focus on analyzing different approaches for handling join queries.\",\n \"We can categorize these approaches into four classes: traditional, learned data-driven, learned query-driven, and bound-based.\",\n \"We summarized their characteristics in Table REF and provide their details as follows.\",\n \"Traditional methods: The sampling-based methods [41], [39], [76], [40] construct a small sample on each table and join these samples to estimate the cardinality of join queries.\",\n \"The traditional histogram-based methods generally adopt the attribute independence and join uniformity assumptions to decompose the join queries as a combination of single table estimates.\",\n \"Taking a query $Q$ joining two tables $A$ and $B$ as an example, these assumptions will simplify the cardinality $|Q| = P_{\\\\Omega }(Q(A), Q(B)) * |\\\\Omega |$ as $P_A(Q(A)) * P_B(Q(B)) * \\\\overline{|\\\\Omega |}$ , where $P_A(Q(A))$ is the estimated single table selectivity.\",\n \"There exist two widely-used approaches to estimate the size for the denormalized join table $\\\\overline{|\\\\Omega |}$ .\",\n \"First, the Selinger models [60], [10], [59] assume that join keys have uniformly distributed values, collect the number of distinct values (NDV) on the join keys from both tables, and estimate $\\\\overline{|\\\\Omega |}$ as $|A| * |B| / max\\\\lbrace NDV(A), NDV(B)\\\\rbrace $ .\",\n \"Second, the join-histogram methods [26], [29], [7] first bin the domain of the join keys as histograms and assume join uniformity assumption for the values in each histogram bin.\",\n \"Then they apply the distinct values methods for each bin and sum over all bins.\",\n \"Our work follows the convention of the join-histogram methods but avoids the simplifying assumptions, which can not be trivially achieved.\",\n \"Naive approaches such as building the full multi-dimensional histograms can resolve the attribute independence assumption but would introduce unaffordable storage overhead.\",\n \"Many follow-up works [27], [30], [28] explore different histogram strategies to reduce the error caused by the join uniformity assumption.\",\n \"They either take an impractical amount of time to construct the histograms or cannot provide high-quality estimates.\",\n \"In short, the traditional methods combine the estimates from single tables to estimate the join queries.\",\n \"This approach is very efficient, easy to train, update and maintain, thus perfect for system deployment.\",\n \"However, their simplifying assumptions can generate erroneous estimates and poor query plans.\",\n \"Learned data-driven methods: These methods circumvent the simplifying assumptions used in traditional methods by understanding the data distributions for the exponential number of join templates in a DB instance.\",\n \"Specifically, some methods [13], [64], [65] denormalize the join of all tables in a database and use Bayesian networks to model the distribution.\",\n \"The current SOTA approaches for handling join queries try to model the data distributions of the denormalized join tables by using fanout-based techniques [23], [77], [70], [73], [67].\",\n \"These methods denormalize some tables and add a possibly exponential number of fanout columns, which are used to formulate the distributions for each join template.\",\n \"Using this method, the learned data-driven methods can produce effective estimations for join queries and high-quality query plans [18].\",\n \"However, these models generally have a long training time, large model size, slow estimation speed, and unscalable performance with the number of joins in the query.\",\n \"Furthermore, the learned data-driven approaches currently can not handle self-joins or cyclic-joins and are very ineffective in processing complicated filter predicates such as string pattern matching queries.\",\n \"Learned query-driven methods: These methods analyze the executed query workload, map the featurized join query directly to its cardinality using supervised models such as neural networks [33], [54], xgboost [11], tree-LSTM [63], and deep ensembles [42].\",\n \"The query-driven methods generally have low estimation latency but the estimation effectiveness is highly dependent on the training workloadblack; we put a semi-check mark ([scale=0.3,fill=black](0,.3) – (.2,0) – (1,.7) – (.25,.15) – cycle (0.75,0.2) – (0.77,0.2) – (0.6,0.7) – cycle;) for these methods in Table REF .\",\n \"To achieve reasonable estimation quality, they generally need an excessive amount of executed training queries [18], which are unavailable for new DB instances.\",\n \"Furthermore, the query-driven models need to retrain in the case of data update or query workload shift.\",\n \"An impractical amount of new executed queries are again needed for this re-training process.\",\n \"Bound-based methods: Traditional CardEst methods in modern DBMS tend to severely under-estimate the cardinality, thus sometimes choosing significantly more expensive query plans [55], [3], [5].\",\n \"To avoid under-estimation, the bound-based methods [5], [3], [1], [22] use information theory to provide an upper bound on the cardinality.\",\n \"These bounds are very effective to help generate high-quality query plans as they can avoid expensive join orders and physical operators [55], [18] However, these methods do not build single-table estimators to understand the data distributions.\",\n \"Instead, in presence of base-table filters, they need to materialize the filtered tables and populate the bound during query run-time, which can produce very high latency and overhead.\",\n \"Therefore, despite the bound-based methods providing meaningful insights into understanding joins, they can not be practically deployed in a DBMS.\",\n \"Summary: None of the existing approaches can simultaneously satisfy the desired properties of CardEst, namely effective, efficient and appropriate for system deployment.\",\n \"However, each of them contains some advantageous techniques, inspiring us to build FactorJoin that can bridge the gaps amongst all these CardEst categories and unify their merits.\"\n ],\n [\n \"New framework for join queries\",\n \"Given this background on previous CardEst approaches, we now introduce the high-level operation of FactorJoin.\",\n \"In Section REF , we elaborate on FactorJoin's core technique: accurately estimating join queries with only single-table statistics using factor graph inference techniques.\",\n \"Then, in Section REF , we analyze the complexity of this inference procedure and motivate several optimizations in FactorJoin.\",\n \"Finally, we provide a workflow overview in Section REF .\"\n ],\n [\n \"Problem formulation\",\n \"We first illustrate the main idea with a simple example of a two-table join query.\",\n \"Then, we formalize the join query estimation problem as a PGM inference problem over single table distributions.\",\n \"Two-table join query example: Figure REF illustrates a query $Q$ joining tables $A$ and $B$ on the inner joinIn this paper, we only discuss inner and equality joins.\",\n \"FactorJoin naturally supports left, right, and outer joins.\",\n \"We leave support for blacknon-equal joins as future work.\",\n \"condition $A.id = B.Aid$ with base table filter predicates $Q(A)$ and $Q(B)$ .\",\n \"Table $A$ first goes through the filter $Q(A)$ , resulting in an intermediate table $A|Q(A)$ (records in $A$ that satisfy the filter $Q(A)$ ).\",\n \"The same procedure is applied to table $B$ .\",\n \"Then, the query $Q$ will match the value of join keys $A.id$ and $B.Aid$ from these two intermediate tables.\",\n \"Specifically, the value $a$ appears 8 times in table $A|Q(A)$ and 6 times in table $B|Q(B)$ , resulting in value $a$ appearing 48 times in the join result.\",\n \"Therefore, we can calculate the cardinality of this query $Q$ as $8\\\\times 6 + 4\\\\times 5 + 3 \\\\times 5 = 83$ .\",\n \"We can formulate the above procedure for calculating $Q$ as a statistical equation in Equation REF , where $D(A.id)$ denotes the domain of all unique values of $A.id$ .\",\n \"We observe that only single-table distributions $P_A$ and $P_B$ are required to accurately compute the cardinality of this join query.\",\n \"Also in this equation, $P_{A}(A.id \\\\!\",\n \"= \\\\!\",\n \"v | Q(A)) * |Q(A)|$ equals to $P_{A}(A.id \\\\!\",\n \"= \\\\!\",\n \"v \\\\wedge Q(A)) * |A|$ , which is exactly what single-table CardEst methods estimate.\",\n \"Thus, we can accurately calculate the cardinalities of two-table join queries using single-table estimators.\",\n \"Note that the summation over the domain of join key $\\\\mathcal {D}(A.id)$ has the same complexity as computing the join.\",\n \"Therefore, we need to approximate this calculation for FactorJoin to be practical.\",\n \"The details of our approximation schemes are given in Section REF and Section .\",\n \"$|Q| = \\\\sum _{v \\\\in \\\\mathcal {D}(A.id)} & P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)|$ Figure: Formulating a join query CardEst into a factor graph inference problem.\",\n \"Figure (i) shows a SQL query Q 2 Q_2.\",\n \"Figure (ii) visualizes the join template of Q 2 Q_2.\",\n \"Figure (iii) presents the factor graph model to compute Q 2 Q_2's cardinality.Formal problem formulation: A general join query can involve a combination of different forms of joins (e.g., chain, star, self, or cyclic joins) so its counterpart equation (as Equation REF ) can be very difficult to derive and compute; blackwe have put these equations/derivations in the supplementary material.\",\n \"We provide a generalizable formulation that automatically decomposes join queries into single-table estimations using the factor graph model.\",\n \"Figure REF -(i) shows a SQL query $Q_2$ joining four tables $A, B, C, D$ .\",\n \"We visualize its join template as a graph in Figure REF -(ii), where each dashed rectangle represents a table, each ellipse (node) represents a join-key in $Q_2$ , and each solid line (edge) represents an equi-join relation between two join keys connected by it.\",\n \"Note that both sides of an equi-join relation represent the semantically equivalent join keys.\",\n \"We call them equivalent key group variables.\",\n \"In Figure REF -(ii), there are three connected components and thus three equivalent key group variables $V_1, \\\\ldots , V_3$ .\",\n \"$V_1$ represents $A.id$ and $B.Aid$ in this group since $Q_2$ contains the join condition $A.id = B.Aid$ .\",\n \"The exact computation required to compute the cardinality of the SQL query in Figure REF extends Equation REF to sum over the domain $\\\\mathcal {D}$ of each join key.\",\n \"black $|Q_2| = & \\\\sum _{a1 \\\\in \\\\mathcal {D}(A.id)} \\\\ \\\\ \\\\ \\\\ \\\\ \\\\sum _{a2 \\\\in \\\\mathcal {D}(A.id2)} \\\\ \\\\ \\\\ \\\\ \\\\ \\\\sum _{c \\\\in \\\\mathcal {D}(C.id)} \\\\nonumber \\\\\\\\& P_{A}(A.id = a1, \\\\ A.id2 = a2 \\\\ | \\\\ Q_2(A)) \\\\ast |Q_2(A)| \\\\nonumber \\\\\\\\& \\\\ast P_{B}(B.Aid = a1, \\\\ B.Cid = c \\\\ | \\\\ Q_2(B)) \\\\ast |Q_2(B)| \\\\nonumber \\\\\\\\& \\\\ast P_{C}(C.Aid2 = a2, \\\\ C.id = c \\\\ | \\\\ Q_2(C)) \\\\ast |Q_2(C)| \\\\nonumber \\\\\\\\& \\\\ast P_{D}(D.Cid = c \\\\ | \\\\ Q_2(D)) \\\\ast |Q_2(D)|$ This computation is extremely expensive — assuming the domains of each id being $n$ , a naive implementation would have the complexity $O(n^3)$ .\",\n \"We can improve this complexity to $O(n^2)$ by representing this computation as an equivalent factor graph model [44], which is a particular class of probabilistic graphical models (PGMs) [34].\",\n \"We have 2 in the exponent because the maximum number of join keys in $Q_2$ is two in the table $A$ .\",\n \"This is still very expensive, but in Section REF , we describe how we can use principled approximations made possible by the factor graph formulation to make the inference complexity practical.\",\n \"A factor graph is a bipartite graph with two types of nodes: variable nodes, and factor nodes representing an unnormalized probability distribution w.r.t.\",\n \"the variables connected to it.\",\n \"Figure REF -(iii) shows the constructed factor graph $\\\\mathcal {F}$ for computing the cardinality of $Q_2$ .\",\n \"Specifically, $\\\\mathcal {F}$ contains a variable node for each equivalent key group variable $V_i$ , and a factor node for each table touched by $Q_2$ .\",\n \"A factor node is connected to a variable node if the variable represents a key in the table.\",\n \"In this case, the factor node representing table $A$ is connected to $V_1$ (equivalent to $A.id$ ) and $V_2$ (equivalent to $A.id2$ ).\",\n \"Each factor node maintains an unnormalized probability distribution for the variable nodes connected to it, for e.g., node $A$ maintains $P_A(V_1, V_2|(Q_2(A)))*|Q_2(A)|$ , which is the same as the distribution $P_A(A.id, A.id2|(Q_2(A)))*|Q_2(A)|$ used in Equation REF .\",\n \"The factor graph model utilizes the graph structure to compute the sum using the well-studied inference algorithms.\",\n \"We can more formally state this relationship between computing the join cardinalities and factor graphs as the following lemma, whose proof is provided in the supplementary material.\",\n \"Lemma 1 Given a join graph $\\\\mathcal {G}$ representing a query $Q$ , there exists a factor graph $\\\\mathcal {F}$ such that the variable nodes in $\\\\mathcal {F}$ are the equivalent key group variables of $\\\\mathcal {G}$ and each factor node represents a table $I$ touched by $Q$ .\",\n \"A factor node is connected to a variable node if and only if this variable represents a join-key in table $I$ .\",\n \"The potential function of a factor node is defined as table $I$ 's probability distribution of the connected variables (join keys) conditioned on the filter predicates $Q(I)$ .\",\n \"Then, calculating the cardinality of $Q$ is equivalent to computing the partition function of $\\\\mathcal {F}$ .\"\n ],\n [\n \"Inference complexity analysis\",\n \"Inference on factor graphs is a well-studied problem in the domain of PGMs [46], [34], [44].\",\n \"Popular approaches to solving this problem are variable elimination (VE) [34] and belief propagation algorithms [35].\",\n \"In FactorJoin's implementation, we use the VE, which first determines an optimal order of variables and then sums over the distributions defined by the factor graph model in this order.\",\n \"The complexity of VE is $O(N*|D|^{max(|JK|)})$ , where $N$ is the number of equivalent key groups (3 in Figure REF ), $|D|$ is the largest domain size of all join keys, and $max(|JK|)$ is the maximum number of join keys in a single table (2 in Figure REF ).\",\n \"Intuitively, this complexity is because the factor nodes need to understand the joint distribution of all join keys in one table.\",\n \"Thus, the largest factor node in the factor graph model maintains a probability distribution of size $|D|^{max(|JK|)}$ .\",\n \"This complexity is not practical for real-world queries, as $|D|$ can be millions, and $max(|JK|)$ can be larger than 4 in real-world DB instances, such as IMDB [38].\",\n \"Therefore, instead of calculating the exact cardinality, we propose a new PGM inference algorithm for FactorJoin that can estimate an upper bound on cardinality.\",\n \"Past works [5], [3], [1] have shown that cardinality upper bounds can help avoid very expensive query plans since under-estimating a large result can sometimes be catastrophic.\",\n \"For instance, an optimizer might choose to do a nested loop join if a CardEst method under-estimates that the result would easily fit in memory.\",\n \"This would lead to a disastrous plan if the actual result size is much larger than estimated.\",\n \"blackOur approximate inference algorithm performs two approximations: 1) reducing the domain size $|D|$ using binning and 2) decreasing the exponent, $max(|JK|)$ , by approximating the distribution of attributes in a single table.\",\n \"blackSpecifically, for binning, we first partition the domain of all join keys into $k$ bins.\",\n \"Instead of summing over the entire domain, we only need to sum over $k$ summarized values (probabilistic bounds) of each bin.\",\n \"The details are provided in Section .\",\n \"blackSecond, we approximate the causal relation among all attributes within a table as a tree structure.\",\n \"This allows us to factorize the $max(|JK|)$ -dimensional joint distribution of all join keys in a table as a product of two-dimensional conditional distributions.\",\n \"This procedure is known as structure learning in Bayesian networks in the PGM domain [6].\",\n \"Past work on single table cardinality estimation has shown that such tree approximations do not decrease the modeling accuracy significantly [70].\",\n \"Further details are provided in Section REF .\",\n \"Thereafter, we can directly run the VE inference algorithm on the binned domains and factorized two-dimensional distributions to estimate the cardinality upper bound of $Q$ .\",\n \"With these two approximate inference techniques, the complexity of estimating the cardinality bound is reduced to $O(N*k^2)$ , which is very efficient as both $N$ and $k$ are very small in practice.\"\n ],\n [\n \"Workflow overview\",\n \"The workflow of FactorJoin contains two phases: offline training and online inference (Figure REF ).\",\n \"During the offline phase, our framework analyzes the DB instance, bins the domain of join keys, explores the causal relation, and builds CardEst models for every single table.\",\n \"When a join query $Q$ comes in during the online phase, FactorJoin formulates the estimation of this query as a factor graph involving only single-table distributions, estimates them using (pre-trained) single-table estimators, runs PGM inference, and generates a probabilistic bound for $Q$ .\",\n \"We provide the details as follows.\",\n \"Offline training phase: Given a new DB instance, FactorJoin first analyzes its DB schema and data tables to get all possible join relations between different join-keys.\",\n \"We consider two join-keys to be semantically equivalent if there exists a join relation between them.\",\n \"After identifying all groups of equivalent join-keys, we bin the domains of all join-keys in each group.\",\n \"FactorJoin can also optionally use query workload information to optimize this binning procedure (details provided in Section REF ).\",\n \"Based on the data tables and the binned domains, we explore the blackcausal relation of these attributes and model them as a tree structure (details are provided in Section REF ).\",\n \"Then, we build the CardEst models to understand the data distribution of every single table.\",\n \"In principle, any single-table CardEst method that is able to provide conditional distributions can be adapted into FactorJoin.\",\n \"In practice, we implement two methods: traditional random sampling and the learned data-driven BayesCard method [70].\",\n \"The sampling method is extremely flexible to use and can easily support any complex filter predicates with disjunctions, string pattern matching, or any user-defined functions.\",\n \"Alternatively, similar to most data-driven methods, BayesCard can only work with queries filtering numeric and categorical attributes, but BayesCard has been shown to consistently produce accurate, fast, and robust estimations for various data distributions [70].\",\n \"Users of FactorJoin can switch between different single-table estimators based on their objectives and knowledge about the DB instances.\",\n \"Figure: Workflow of FactorJoin framework.Online inference phase: When a join query $Q$ comes in during run-time, FactorJoin will first parse its join graph and construct the corresponding factor graph $\\\\mathcal {F}$ (as in Figure REF ).\",\n \"Then, it uses the trained single-table estimators to estimate the join-key distributions conditioned on the filter predicates of $Q$ ; FactorJoin will use these distributions in the corresponding factor nodes of $\\\\mathcal {F}$ .\",\n \"Finally, we run the approximate version of VE inference algorithms on $\\\\mathcal {F}$ to derive the estimated/probabilistic cardinality bound.\",\n \"We note that the CardEst methods working inside a query optimizer will estimate all sub-plan queries of a target query to decide the best query plan.\",\n \"Since these sub-plan queries contain a large amount of overlapping information, we provide a progressive estimation algorithm to avoid computation redundancy in Section REF .\"\n ],\n [\n \"Probabilistic bound algorithm\",\n \"In this section, we design a new probabilistic bound algorithm based on binning to calculate the factor graph inference problem.\",\n \"Specifically, we first explain the main ideas with a two-table join example and formulate the algorithm details within FactorJoin in Section REF .\",\n \"Then, we discuss how to optimize the bin selection using data and query information in Section REF .\"\n ],\n [\n \"Algorithm details\",\n \"As described in Section , the main objective of our probabilistic bound-based algorithm is to reduce the domain size of the join keys, i.e., reducing the complexity of summation in Equation REF .\",\n \"Thus, instead of running the inference algorithm over the entire domain, FactorJoin only needs to sum over the binned domain.\",\n \"Two-table join query example: We use the previous example query $Q$ from Section REF to illustrate the core idea of our probabilistic upper bound algorithm based on binning.\",\n \"The objective is to estimate an upper bound on $Q$ , whose exact computation requires summing over the entire domain of $A.id$ according to Equation REF .\",\n \"Assume that we have a set of $k$ bins $\\\\lbrace bin_1, \\\\ldots , bin_k \\\\rbrace $ partitioning the domain of $A.id$ and we apply this same set of bins to $B.Aid$ .\",\n \"blackWe need to make sure that a given value in the domain of $A.id$ and $B.Aid$ will always belong to the bin with the same indexes.\",\n \"Then, Equation REF can be equivalently written as Equation 3.\",\n \"We can then replace the sum over all values in a bin $bin_i$ with a probabilistic bound on $bin_i$ to estimate an upper bound for $Q$ as Equation REF .\",\n \"$|Q| &= \\\\sum _{i = 1}^{k} \\\\sum _{v \\\\in bin_i} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\\\\\\\&\\\\lesssim \\\\sum _{i = 1}^{k} && Probabilistic\\\\_bound(A, B, bin_i)$ Motivated by a bound based on the most frequent value (MFV) [22], we use a simple probabilistic bound method to derive $Probabilistic\\\\_bound(A, B, bin_i)$ for a particular bin.\",\n \"Specifically, assuming that value $\\\\lbrace a, b, c , e, f\\\\rbrace $ of $A.id$ and $B.Aid$ are binned into $bin_1$ as shown in Figure REF .\",\n \"We know the summation of all values in $bin_1$ equals to black$8\\\\times 6 + 4 \\\\times 5 + 3 \\\\times 5 = 83$ .\",\n \"This summation has a dominating term of $8 \\\\times 6$ , because the count of MFV of $bin_1$ is 8 for $A.id$ (denoted as $V^{*}_1(A.id)$ ) and 6 for $B.Aid$ (denoted as $V^{*}_1(B.Aid)$ ) so each value can appear at most $8 \\\\times 6$ times in the denormalized table after the join.\",\n \"Since we know the total count of values in $bin_1$ for $A.id$ is 16, there can be at most $16/8 = 2$ MFVs.\",\n \"Similarly, there can be at most black4 MFVs in $bin_1$ for $B.Aid$ .\",\n \"Therefore, we have the summation of all values in $bin_1$ is upper bounded by black$min(2, 4) \\\\times 8 \\\\times 6 = 96$ .\",\n \"We formally represent the aforementioned procedure in Equation REF , where $V^{*}_i(A.id)$ and $P_A(A.id \\\\in bin_i|Q(A)) * |Q(A)|$ are the MFV count and estimated total count of $bin_i$ for $A.id$ .\",\n \"Our bound is probabilistic because $P_A(A.id \\\\in bin_i|Q(A)) * |Q(A)|$ is estimated with a single table CardEst method, which may have some errors.\",\n \"$|Q| & \\\\lesssim \\\\sum _{i = 1}^{k} min(\\\\frac{P_A(A.id \\\\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\\\nonumber \\\\\\\\& \\\\frac{P_B(B.Aid \\\\in bin_i|Q(B)) * |Q(B)|}{V^{*}_i(B.Aid)}) * V^{*}_i(A.id) * V^{*}_i(B.Aid)$ Figure: The bound-based algorithm; each bin is summarized by the most frequent value count and total count.Working with PGM inference: Next, we will explain how to generalize this probabilistic bound to make it work inside the factor graph VE inference algorithm mentioned in Section REF .\",\n \"During the offline training, FactorJoin computes the MFV counts $V^*$ for all bins of each join-key.\",\n \"During the online phase, FactorJoin estimates the probability distribution for the binned domain of all join-keys in each table $I$ .\",\n \"Then, FactorJoin puts $I$ 's distribution $P_{I}$ and MFV counts $V^*(I)$ into the corresponding factor node.\",\n \"Recall that the VE algorithm generates an optimal elimination order of all variables and sums over the domain of these variables sequentially in this order.\",\n \"Since at each step, the variable elimination algorithm sums out only one variable, its probabilistic bound can be derived using Equation REF over multiple equivalent join keys.\",\n \"Thus, this algorithm does not need to sum over the entire value domain of a variable but can only sum over the probabilistic bounds in the binned domain instead, which greatly reduces the complexity.\",\n \"Analysis: This probabilistic bound algorithm significantly reduces the domain size of join keys and enables practical approximate PGM inference.\",\n \"Further, this algorithm upper-bounds the cardinality most of the time and the bound is very tight with even a small number of bins, such as $k=100$ .\",\n \"Therefore, as we empirically verify in our evaluation, the estimated cardinality bound is very effective.\"\n ],\n [\n \"Bin selection optimizations\",\n \"In our probabilistic bound algorithm, different bins can result in drastically different bound tightness.\",\n \"blackThere are two decisions: how many bins to use for each join key group, and what values to put in the same bin.\",\n \"Deciding $k$ based on query workloads: blackThe number of bins, $k$ , has a significant effect on the performance of FactorJoin: fewer bins aggregate more distinct values in the join key domain to each bin and are thus less accurate but more efficient.\",\n \"An approach to tune $k$ is to set different values of $k_i$ for different equivalent key groups $Gr_i$ .\",\n \"Suppose that users provide a budget $K$ with $K = \\\\sum k_i$ .\",\n \"If FactorJoin has access to the query workload, it can analyze the join patterns and count the number of times $n_i$ each $Gr_i$ appears in the workload.\",\n \"Our framework can adaptively set a larger $k_i$ for the frequently visited $Gr_i$ and vice versa.\",\n \"In our implementation, we use a simple heuristic — set $k_i = K * n_i/\\\\sum _j n_j$ .\",\n \"In this way, we can optimize the modeling capacity of FactorJoin to make sure that it is spent on the important joins.\",\n \"Binning strategy: For the probabilistic bound algorithm described in Section , we observe that the upper bound on a particular bin $bin_i$ can be very loose if the MFV count $V^*_i$ is a large outlier in $bin_i$ .\",\n \"Taking the two table join query as an example, if $bin_i$ contains only one value that appears 100 times in $A.id$ but $10,000$ values that only appear once in $B.Aid$ , then the bound could be 100 times larger than the actual cardinality.\",\n \"blackCommon binning strategies used in DBMS histograms such as equal-depth or equal-width bins can be catastrophic in such cases.\",\n \"Instead, we want the variance of the join key counts in each bin to be low.\",\n \"This strategy has also been proposed in different contexts [28], [30], but the key challenge is that the same bins will be built for all join keys within an equivalent key group, i.e.\",\n \"join keys that share the same semantics.\",\n \"For example, for a primary key (e.g.\",\n \"“title.id” in IMDB JOB [38]), any binning strategy will have zero variance because of uniqueness.\",\n \"However, the same bins will be applied for its equivalent foreign keys (e.g.\",\n \"“movie_id”) on other tables, which may lead to a large variance depending on how often each value repeats in the other tables.\",\n \"blackWe design a greedy binning selection algorithm (GBSA) to optimize for bins with low variance value counts across all tables, as illustrated in Algorithm REF .\",\n \"Jointly minimizing the variance of one bin for all join keys has exponential complexity.\",\n \"Therefore, GBSA uses a greedy algorithm to iteratively minimize the bin variance for all join keys.\",\n \"blackAt a high level, GBSA first optimizes the minimal variance bins with half the binning budget of $k/2$ on the domain of one join key (lines 2-4).\",\n \"blackThen, it recursively updates these bins by minimizing the variance of other join keys using the other half of the budget (lines 5-13).\",\n \"blackWe put the details of GBSA in the supplementary material.\",\n \"blackDiscussion: blackAs shown in Section REF , GBSA has significantly better performance than naive binning strategies.\",\n \"In the extreme case, if the value counts have zero variance for all equivalent join keys, then our bound can output the exact cardinality.\",\n \"Admittedly, this binning strategy does have the drawback that after applying the filter predicates, the join key frequencies will change and their variance may become higher within a bin.\",\n \"However, we empirically evaluate that for real-world datasets, this phenomenon does not have a severe impact on FactorJoin.\",\n \"In future work, we will explore an enhanced binning strategy that can efficiently address this issue.\",\n \"[t] blackGreedy Bin Selection Algorithm (GBSA) Input: Equivalent key groups $Gr_1, \\\\ldots , Gr_m$ , where $Gr_i = \\\\lbrace Id_i^1, \\\\ldots , Id_i^{|Gr_i|} \\\\rbrace $ ; Column data $\\\\mathcal {D}(Id_i^j)$ of all join keys in the DB instance $\\\\mathcal {D}$ ; Number of bins $k_i$ for each group $Gr_i$ .\",\n \"[1] $Gr_i \\\\in \\\\lbrace Gr_1, \\\\ldots , Gr_m\\\\rbrace $ $Bin(Gr_i) \\\\leftarrow $ [] $Gr_i^{\\\\prime } \\\\leftarrow sort\\\\_key\\\\_based\\\\_on\\\\_domain\\\\_size(\\\\mathcal {D}, Gr_i)$ $Bin(Gr_i) \\\\leftarrow get\\\\_min\\\\_variance\\\\_bins(\\\\mathcal {D}(Gr_i^{\\\\prime }[1]), k_i/2)$ remain_bins $\\\\leftarrow k_i/2$ $j \\\\in \\\\lbrace 2, \\\\ldots , |Gr_i^{\\\\prime }| \\\\rbrace $ binned_data $\\\\leftarrow apply\\\\_bin\\\\_to\\\\_data(\\\\mathcal {D}(Gr_i^{\\\\prime }[j]), Bin(Gr_i))$ bin_variance $\\\\leftarrow calculate\\\\_variance$ (binned_data) arg_sort_idx $\\\\leftarrow arg\\\\_sort\\\\_decreasing(bin\\\\_variance)$ $p \\\\in arg\\\\_sort\\\\_idx[1:remain\\\\_bins/2]$ $Bin(Gr_i) \\\\leftarrow min\\\\_variance\\\\_dichotomy(Bin(Gr_i)[p]$ , binned_data[p]) remain_bins $\\\\leftarrow $ remain_bins/2 return $\\\\lbrace Bin(Gr_i) | i, \\\\ldots , m\\\\rbrace $ black\"\n ],\n [\n \"Incremental model updates\",\n \"black FactorJoin is friendly for DBs with frequent data changes because we only need to incrementally update the single table statistics.\",\n \"In the following, we will discuss the algorithm details in the scenario of data insertion and data deletion can be handled similarly.\",\n \"Consider we have trained a stale FactorJoin model $\\\\mathcal {F}$ on data $\\\\mathcal {D}$ and would like to incrementally update $\\\\mathcal {F}$ with the inserted data $\\\\mathcal {D}^{\\\\prime }$ .\",\n \"FactorJoin needs to identify the inserted tuple values from $\\\\mathcal {D}^{\\\\prime }$ for every join key and put them into the original bins optimized on $\\\\mathcal {D}$ .\",\n \"Then, it will update the total and most-frequent value count in each bin.\",\n \"At last, FactorJoin can incrementally update the base table CardEst models with off-the-shelf tools (e.g.\",\n \"BayesCard provides an efficient and effective approach to incrementally update the single-table models using $\\\\mathcal {D}^{\\\\prime }$ [70]; materializing a new sample on $\\\\mathcal {D} \\\\bigcup \\\\mathcal {D}^{\\\\prime }$ incrementally updates the sampling-based estimator).\",\n \"Therefore, the incremental update of FactorJoin does not need to normalize the joined tables nor need to use executed queries.\",\n \"It is also empirically verified to be very efficient and effective.\",\n \"However, during incremental updates, the FactorJoin models keep the same set of bins, which is optimized on the previous data and thus might not be optimal after data insertion.\",\n \"This might cause slight degrade in performance and the user can choose to retrain the model in case of massive data updates.\"\n ],\n [\n \"Improving \",\n \"We describe two techniques to improve the modeling and estimation efficiency of FactorJoin.\",\n \"First, we model the joint distribution of attributes on a single table as a tree structure and use it to simplify the PGM inference calculations.\",\n \"Next, we define a systematic way of reusing estimates of intermediate sub-plan queries to estimate a larger query.\",\n \"This is only possible because we decompose cardinality estimates into single table sub-components.\"\n ],\n [\n \"blackExploring attributes causal relation\",\n \"Recall from blackSection REF , the factor graph computation to calculate the cardinality bound is $O(N*k^{max(|JK|)})$ , where $N$ is the number of tables, $k$ is the number of bins, and $max(|JK|)$ is the maximum number of join keys in a table.\",\n \"The exponent term, $max(|JK|)$ , represents the factor nodes requiring the joint distribution between all join keys on a single table.\",\n \"As $max(|JK|)$ can sometimes be large (e.g., in IMDB it can be up to 4), we need to reduce the dimensionality of the distributions defined in factor nodes to ensure an efficient CardEst procedure.\",\n \"Fortunately, real-world data has a lot of correlations that can be used to simplify this joint distribution.\",\n \"blackConsider a table $A$ with 6 attributes, $\\\\lbrace id_1, id_2, id_3, id_4 attr_1, attr_2\\\\rbrace $ , four of which are join keys.\",\n \"The factor graph computation blackmay need to estimate quantities such as: $P_A(id_1, id_2, id_3, id_4 | Q(A))$ blackThis requires storing a 4 dimensional probability distribution, and even with binning the domains of all ids into $k$ bins, it still requires $k^4$ space and inference time complexity.\",\n \"This joint distribution can be represented as a graph where each attribute is a node, and there are edges between every pair of attributes.\",\n \"We will use the dependencies and relationships between the attributes and model them as a Bayesian network (BN) [34].\",\n \"Specifically, we will assign each edge in the joint distribution graph a weight proportional to the mutual information between the two attributes connected by the edge.\",\n \"Then, we will remove edges with the least mutual information until only a tree structure graph remains to represent the joint probability over these attributes.\",\n \"blackThis process is known as the Chow-Liu algorithm for decomposing joint distributions [6].\",\n \"The factorized tree representation of the joint distribution can let us approximate Equation REF with an equation of the form: $P_A(id_1 | Q(A)) * P_A(id_2 | Q(A)) * P_A(id_3 | id_1) * P_A(id_4 | id_3)$ , which is much more efficient to compute because it only involves a product over at most a two-dimensional distribution.\",\n \"This dimension reduction can cause some errors.\",\n \"But it has been shown empirically that the factorized distributions are accurate approximations of the original distribution for most real-world datasets [70].\",\n \"Using such an optimization, instead of storing the $max(|JK|)$ -dimensional distributions in the factor nodes, the factor graph only needs to store a series of one or two-dimensional data distributions without a significant decrease in estimation accuracy.\",\n \"Therefore, the overall complexity for running approximate inference on this factor graph becomes $O(N*k^2)$ in both storage and inference speed.\",\n \"Since $k$ is relatively small (around one hundred), estimating one query using FactorJoin is very efficient.\",\n \"blackIn principle, FactorJoin can also use other methods for the single table distribution inference.\",\n \"This may involve traditional methods, such as sampling, or more powerful graph-structured factorizations of the distributions — which can increase modeling accuracy at the cost of slower estimation speed.\",\n \"Thus, users of FactorJoin can make trade-offs between estimation accuracy and efficiency.\",\n \"We leave exploring these trade-offs as future work.\"\n ],\n [\n \"Progressive estimates of sub-plan queries\",\n \"In order to optimize the plan for one query, the CardEst method needs to estimate the cardinalities of hundreds or thousands of sub-plan queries.\",\n \"Estimating all these queries independently is very inefficient because they contain a lot of redundant computation.\",\n \"Similar to the traditional CardEst methods deployed in real DBMSes, FactorJoin supports estimating all sub-plan queries of one query progressively, reusing the computation as much as possible.\",\n \"At a high level, the progressive estimation algorithm will estimate all sub-plan queries in a bottom-up fashion.\",\n \"FactorJoin first estimates all two-table join sub-plan queries, then it estimates the three-table join sub-plan queries, which will contain the two-table joins as their sub-structures, and so on.\",\n \"Joining Factor Graphs: The key insight is that when we estimate the cardinality of a two-table join, we can also combine their factor graphs into a new factor graph for the joined table.\",\n \"Specifically, as described in Section , each base table is represented as a factor node in the factor graph.\",\n \"For instance, the factor node representing $A$ will store the unnormalized probability distribution and the MFV counts for the binned domain of join keys in $A$ .\",\n \"When estimating the cardinality of joining $A$ and $C$ , our approximate inference algorithm in Equation REF computes the bound over all bins and sums them.\",\n \"These bounds essentially define an unnormalized probability distribution over all join keys on the denormalized table $A \\\\bowtie C$ .\",\n \"We can derive a bound on the new MFV counts for a join-key $A \\\\bowtie C$ as the multiplication of the corresponding MFV counts in $A$ and $C$ .\",\n \"Therefore, we can cache this new probability distribution and MFV counts in a new factor node representing $A \\\\bowtie C$ .\",\n \"Thus, when estimating the cardinality of $A \\\\bowtie B \\\\bowtie C$ we can consider it as a join of two single tables $(A \\\\bowtie C$ ) and $B$ .\",\n \"Since all sub-plans can essentially be considered a join of two sub-plans that we have already estimated, our inference algorithm only needs to compute the join of blacktwo-factor nodes for each sub-plan query.\",\n \"Therefore, the progressive estimation algorithm has no redundant computation and maximizes efficiency.\",\n \"In experiments, with this algorithm, FactorJoin can estimate the cardinality of $10,000$ sub-plan queries in one second, which is more than ten times faster than estimating all these queries independently.\",\n \"Note that the progressive estimation algorithm is only possible to implement for FactorJoin and the traditional methods because it requires decomposing join queries into single-table estimates.\",\n \"None of the existing learned methods can apply this method since their models are built to model the distributions of specific join patterns.\"\n ],\n [\n \"Experimental Evaluation\",\n \"In this section, we empirically demonstrate the advantages of our framework over existing CardEst methods.\",\n \"We first introduce the datasets, baselines, and experimental environment in Section REF .\",\n \"Then, we explore the following questions: $\\\\bullet $ Overall performance (Section REF ): how much improvement can FactorJoin achieve in terms of end-to-end query time?\",\n \"What are the model size and training time compared to SOTA methods?\",\n \"$\\\\bullet $ Detailed analysis (Section REF ): Why does FactorJoin attain good overall performance?\",\n \"blackWhen does it perform better or worse than existing baselines?\",\n \"How will it perform in terms of data updates?\",\n \"$\\\\bullet $ Ablation study (Section REF ): How much does each optimization technique of FactorJoin contribute to the overall performance?\"\n ],\n [\n \"Experimental Setup\",\n \"Datasets: We evaluate the performance of our CardEst framework on two well-established benchmarks: STATS-CEB [18] and IMDB-JOB [38], whose statistics are summarized in Table REF .\",\n \"The STATS-CEB benchmark uses the real-world dataset STATS, which is an anonymized dump of user-contributed content on Stack Exchange (Stack Overflow and related websites).\",\n \"This dataset consists of 8 tables with 34 columns and more than one million rows.\",\n \"It contains a large number of attribute correlations and skewed data distributions.\",\n \"The STATS-CEB query workload contains 146 queries involving 70 different join templates with real-world semantics.\",\n \"The true cardinalities of these queries range from 200 to $2 \\\\cdot 10^{10}$ , with some queries taking more than one hour to execute on Postgres.\",\n \"The STATS-CEB benchmark only has star or chain join templates and numerical or categorical filtered attributes — therefore it is possible to evaluate all existing CardEst baselines on it.\",\n \"The IMDB-JOB benchmark uses the real-world Internet movie database (IMDB).\",\n \"This dataset contains 21 tables with more than fifty million rows.\",\n \"The IMDB-JOB query workload contains 113 queries with 33 different join templates.\",\n \"Some queries have more than $10,000$ sub-plan queries, posing a significant challenge to the efficiency of CardEst methods.\",\n \"Because IMDB-JOB contains cyclic joins and string pattern matching filters, the JoinHist and existing learned data-driven methods do not support this benchmark.\",\n \"The queries in these two benchmarks cover a wide range of joins, filter predicates, cardinalities, and runtimes.\",\n \"To the best of our knowledge, they represent the most comprehensive and challenging real-world benchmarks for CardEst evaluation.\",\n \"Baselines: We compare our framework with the most competitive baselines in each CardEst method category.\",\n \"1) PostgreSQL [10] refers to the histogram-based CardEst method used in PostgreSQL.\",\n \"2) JoinHist [7] is the classical join-histogram method whose details are explained in Section REF .\",\n \"3) WJSample [40] uses a random walk based-method called wander join to generate samples from the join of multiple tables.\",\n \"Its superior performance over traditional sampling-based methods has been demonstrated in recent studies [40], [76], [56].\",\n \"blackWe limit the amount of time spent for each random walk of WJSample so the overall time spent for estimating cardinalities is comparable to other methods.\",\n \"If we allow WJSample longer inference time, it can generate more effective query plans — but it is impractical since its total estimation time will exceed the query execution time.\",\n \"4) MSCN [33] is a learned query-driven method based on a multi-set convolutional network model.\",\n \"On both datasets, we train MSCN on roughly $100K$ sub-plan training queries that have similar distributions to the testing query workloads.\",\n \"5) BayesCard [70], 6) DeepDB [25], and 7) FLAT [77] are the learned data-driven CardEst methods.\",\n \"They use fanout methods to understand the data distributions of all join templates.\",\n \"Then, they apply different distribution models (Bayesian networks, sum-product networks [57], factorized sum-product networks [72], respectively) to capture the data distribution on single tables or denormalized join tables.\",\n \"We test these methods on STATS-CEB using the optimally-tuned model parameters, as used in [18].\",\n \"8) PessEst [5] is the SOTA bound-based method, which leverages randomized hashing and data sketching to tighten the bound for join queries.\",\n \"Its estimation has been verified to be very effective in real-world DBMSes.\",\n \"We use the best-tuned hyperparameters to evaluate PessEst on both benchmarks.\",\n \"9) U-Block [22] uses top-k statistics to estimate cardinality bounds.\",\n \"The original paper designed a new plan enumeration strategy to work with this bound and showed promising results on IMDB-JOB.\",\n \"However, for a fair comparison, we only evaluate the standalone U-Block CardEst method.\",\n \"10) TrueCard outputs true cardinality for given queries with no estimation latency.\",\n \"This baseline does not refer to any specific method but represents the optimal CardEst performance.\",\n \"We omit comparisons with many other CardEst methods [4], [61], [31], [12], [62], [68], [21], [32], [39], [20], [73], [69], [11] blackbecause the baselines above have demonstrated clear advantages over these methods in various aspects [18].\",\n \"In addition, some newly proposed methods [63], [67], [42] do not have open-source implementations yet.\",\n \"We implemented FactorJoin in Python mainly using the NumPy package.\",\n \"Regarding the hyperparameters of FactorJoin, we assume no knowledge of the query workload and set the bin size $k$ to 100 for both datasets.\",\n \"We use the BN-based CardEst method [70] as the single-table estimators for STATS-CEB benchmark because this method has been shown to be very accurate, fast, and robust for single-table estimations.\",\n \"For IMDB-JOB, we use random sampling with $1\\\\%$ sampling rate as the single-table estimator because this benchmark contains very complicated filter predicates such as disjunctions and string pattern matching, which are not supported by the learned data-driven methods.\",\n \"We discuss and compare other hyperparameters of our framework in Section REF .\",\n \"Table: Summary of STATS-CEB and IMDB-JOB benchmark.Figure: Overall performance on STATS-CEB and IMDB-JOB.Environment: blackWe use the open-source implementation of all the baselines and tune their parameters to achieve the best possible performance.\",\n \"All experiments are performed on a Ubuntu server with Intel Core Processor CPU with 20 cores, 57GB DDR4 main memory, and 300GB SSD.\",\n \"For end-to-end evaluation, we integrate our framework along with all the baselines into the query optimizer of PostgreSQL 13.1 following the procedure described in the recent work [18], [17].\",\n \"We build the primary key and foreign key indices and disable parallel execution in PostgreSQL.\",\n \"Specifically, we inject into PostgreSQL all sub-plan query cardinalities estimated by each method, so the PostgreSQL optimizer uses the injected cardinalities to optimize the query plan, and record the end-to-end query runtime (planning plus execution time).\",\n \"To eliminate the effects of cache, we first execute each workload multiple times.\",\n \"Then, we execute the queries three times for each method and report the average blackand the standard errors.\"\n ],\n [\n \"Overall performance\",\n \"The most straightforward criterion to evaluate the performance of a CardEst method is to test how much each method improves the end-to-end query performance inside the query optimizer.\",\n \"Therefore, we compare the end-to-end performance along with the model size and training time of existing baselines on STATS-CEB and IMDB-JOB.\",\n \"We summarize the overall performance of all baselines in Figure REF .\",\n \"Recall that JoinHist, BayesCard, DeepDB, and FLAT can not support the IMDB-JOB benchmark.\",\n \"In addition, we do not provide the model size and training time for Postgres, TrueCard, JoinHist, WJSample, and U-Block as they are negligible.\",\n \"Table: End-to-end performance on STATS-CEB.Performance on STATS-CEB: In addition to Figure REF , we provide a detailed comparison of query execution and planning time (sum over all queries in the STATS-CEB) in Table REF , with the relative end-to-end improvement over Postgres shown in the last column, i.e.\",\n \"(Postgres time - method time) / Postgres time.\",\n \"FactorJoin achieves the best end-to-end query runtime ($19,116s$ ), which is close to optimal performance ($18,456s$ for TrueCard).\",\n \"When compared with the traditional methods (Postgres, JoinHist, and WJSample), FactorJoin has significantly better query execution time, indicating that our estimates are much more effective at generating high-quality query plans.\",\n \"Meanwhile, the planning time of our framework is close to Postgres.\",\n \"Judging from the pure query execution time in Table REF , FactorJoin has comparable estimation effectiveness as the previous SOTA method FLAT on this benchmark.\",\n \"This learned data-driven method uses the fanout method to understand the distribution of all join patterns, which is accurate but very inefficient.\",\n \"Specifically, FactorJoin can achieve better end-to-end performance than FLAT, with $10 \\\\times $ smaller planning time, $160 \\\\times $ smaller model size, and $240 \\\\times $ faster training time.\",\n \"When compared to the other learned methods: BayesCard, DeepDB, and MSCN, FactorJoin can simultaneously achieve better query execution and planning time, smaller model size, and faster training time.\",\n \"Therefore, we empirically verify that instead of modeling the distributions of all join patterns, accurately decomposing the join query into single-table estimates can generate equally effective but more efficient estimation with much smaller model size and training time.\",\n \"FactorJoin and the SOTA bound-based method PessEst can both generate very effective estimates.\",\n \"This observation verifies that a bound-based method can produce effective estimates because it can help avoid very expensive query plans [5], [3], [1], [18].\",\n \"However, the planning time of PessEst (1135s) is undesirable because it needs to materialize tables after applying the filters and populate the upper bound during run-time.\",\n \"FactorJoin has $35 \\\\times $ less planning latency since instead of using exact bounds, we use probabilistic bounds derived from single-table estimates, which is much more efficient.\",\n \"Table: End-to-end performance on IMDB-JOB.Performance on IMDB-JOB: Table REF shows a detailed end-to-end time comparison for the IMDB-JOB workload.\",\n \"FactorJoin achieves the best end-to-end query runtime ($1324s$ ) amongst all baselines.\",\n \"Similar observations hold as in the STATS-CEB workload when compared to the traditional methods.\",\n \"Like STATS-CEB, PessEst achieves the best performance in terms of pure execution time but has a much larger overhead and planning latency.\",\n \"FactorJoin has comparable performance as PessEst in terms of pure execution time.\",\n \"However, our planning time is $50 \\\\times $ shorter than PessEst, making our framework much faster in overall end-to-end query time.\",\n \"Specifically, our framework can estimate $10,000$ sub-plan queries within one second, which is close to Postgres.\",\n \"However, unlike in STATS-CEB where we have near-optimal performance, there still exists a large gap between FactorJoin (1324s) and the optimal TrueCard (375s).\",\n \"There are two possible explanations.\",\n \"First, our probabilistic bound error can accumulate with the number of joins.\",\n \"Compared to STATS-CEB, the IMDB-JOB contains more than twice the number of joins in the queries.\",\n \"Therefore, the relative performance of our framework drops severely when shifting from STATS-CEB to IMDB-JOB.\",\n \"Second, the base-table filters in IMDB-JOB contain a large amount of highly-selective predicates, making our sampling-based single-table CardEst less accurate.\",\n \"These observations suggest room for improving FactorJoin, which we discuss further in the future work section.\",\n \"Summary: FactorJoin achieves the best performance among all the baselines on both benchmarks.\",\n \"Specifically, FactorJoin is as effective as the previous SOTA methods, and simultaneously as efficient and practical as the traditional CardEst methods.\",\n \"Figure: Relative estimation errors on STATS-CEB.\"\n ],\n [\n \"Detailed Analysis\",\n \"In this section, we first study why FactorJoin is able to achieve SOTA performance by analyzing the tightness of our probabilistic bound.\",\n \"Then, we exhaustively compare the CardEst method's end-to-end performance on queries with different runtimes to investigate when can FactorJoin perform better or worse than the competitive baselines.\",\n \"For compactness, here, we only report the results of the most representative methods (Postgres, FLAT, PessEst) on STATS-CEB; the remaining results are in the appendix.\",\n \"Bound tightness: We show the relative errors between the baselines' estimates and the true cardinalities (estimate / true) for all sub-plan queries on STATS-CEB in Figure REF .\",\n \"Overall, all three SOTA methods significantly outperform Postgres in terms of estimation accuracy.\",\n \"PessEst generates exact upper bounds and never underestimates.\",\n \"FLAT uses a much larger model to understand the distributions of join patterns and produces the most accurate estimates.\",\n \"FactorJoin can output an upper bound on cardinality for more than $90\\\\%$ of the sub-plan queries.\",\n \"Most of the marginal underestimates are very close to the true cardinality, which can still generate relatively effective plans.\",\n \"Our generated bounds are slightly tighter than the bounds from PessEst, which explains why our query plans achieve slightly better results.\",\n \"Detailed comparision: We sort the 146 queries of STATS-CEB based on their Postgres end-to-end runtime and cluster them into 6 different runtime intervals.\",\n \"Figure REF reports relative improvements of the most competitive baselines over Postgres for each query on the left and each cluster of queries on the right.\",\n \"For the very short-running queries (which represent an OLTP-like workload), Postgres is the best amongst all baselines.\",\n \"These baselines perform worse because the estimation latency plays a significant role in these queries.\",\n \"We observe that improving the estimation accuracy of these queries has a very limited effect on the query plan quality.\",\n \"This also explains why the optimal TrueCard only marginally outperforms Postgres on queries with less than $1s$ of runtime.\",\n \"However, these queries only contribute a negligible proportion of total runtime in STATS-CEB.\",\n \"The query optimizer should fall back to default traditional CardEst methods for OLTP-like workload.\",\n \"FLAT and PessEst are significantly worse than FactorJoin because their planning latency surpasses the execution time, which overshadows the minor improvement in query plans.\",\n \"For the extremely long-running queries, the advantages of the SOTA methods over Postgres gradually appear.\",\n \"The reason is that estimation latency becomes increasingly insignificant for queries with a long execution time.\",\n \"Thus, FLAT and PessEst which spend a long time planning can generate significantly better query plans.\",\n \"FactorJoin has comparable performance on these queries.\",\n \"Figure: Per query performance of STATS-CEB.blackIncremental updates: Following the same evaluation setup as previous work [18], we train a stale FactorJoin model on STATS data created before 2014 (roughly $50\\\\%$ ) and insert the rest of the data to incrementally update this model.\",\n \"We provide the end-to-end query time of the updated FactorJoin and the update time on STATS-CEB queries in Table REF .\",\n \"Since FactorJoin only needs to update single-table statistics, it is extremely efficient, i.e.\",\n \"taking only $2.5s$ to update millions of tuples.\",\n \"This update speed up to $168 \\\\times $ faster than the learned data-driven methods with better end-to-end query performance after model update.\",\n \"We cannot fairly compare with the learned query-driven methods because there does not exist a training query workload after the data insertion.\",\n \"It worth noticing that the update times reported for other methods in Table REF include the time to re-compute the denormalized join tables for the inserted data, so numbers are larger than the ones reported in the original paper [18].\",\n \"However, we do see a slight drop in the end-to-end improvement when compared to the FactorJoin trained on the entire dataset ($43.4\\\\%$ in Table REF versus $45.9\\\\%$ in Table REF ).\",\n \"This small performance difference is due to the fact that the bins are decided on the data before 2014 and remain fixed during incremental updates.\",\n \"Therefore, after inserting the new data, the min-variance property of the bins could be violated, resulting in less accurate predictions.\",\n \"Table: blackIncremental update performance on STATS-CEB.\"\n ],\n [\n \"Ablation study\",\n \"We conduct a series of ablation studies to analyze different optimization techniques inside FactorJoin.\",\n \"Specifically, we first investigate how the number of bins (k) affects the overall performance.\",\n \"Next, we analyze the effectiveness of our new bin selection algorithm (GBSA in Section REF ) over the traditional equal-width and equal-depth algorithms.\",\n \"Then, we study the performance of plugging-in different single-table CardEst methods into FactorJoin.\",\n \"Due to space limitations, we only provide the results on STATS-CEB.\",\n \"At last, we investigate how removing each of the simplifying assumptions improves the performance of the original joining histogram.\",\n \"Number of bins (k): We train and evaluate FactorJoin with five different number of bins (1, 10, 50, 100, 200).\",\n \"Figure REF shows the end-to-end query time (A), bound tightness (B), estimation latency per query (C), training time (D), and model size (E) of FactorJoin.\",\n \"For bound tightness, we report the $50\\\\%, 95\\\\%, 99\\\\%$ percentiles of all queries' relative error: estimated/true, as the one used in Figure REF .\",\n \"Figure REF -(A-B) shows that larger numbers of bins ($k$ ) will generate tighter cardinality bounds and thus more effective query plans.\",\n \"Figure REF -(C-E) shows that more bins also increase estimation latency, training time, and model size, as expected.\",\n \"There are several interesting observations from these results.\",\n \"$\\\\bullet $ According to Figure REF -(B), with $k=1$ (i.e.\",\n \"bin the entire domain of each join key into one bin) FactorJoin does not generate tight bounds but it still outperforms Postgres baseline by $24.9\\\\%$ .\",\n \"This highlights the importance of upper bounds over underestimation and verifies the effectiveness of bound-based algorithms.\",\n \"$\\\\bullet $ Increasing $k$ can substantially increase the bound tightness.\",\n \"However, at some point (from $k=100$ to $k=200$ ) the framework stops achieving better end-to-end performance.\",\n \"One explanation is that this fine-grained improvement in estimation accuracy can generate a slightly better query plan but the slower estimation latency cancels out the improvements in query plans.\",\n \"$\\\\bullet $ The model size increases quadratically with $k$ because FactorJoin has $O(N*k^2)$ complexity in both storage and inference.\",\n \"However, estimation latency increases linearly with $k$ because the overall inference time is dominated by the single-table CardEst methods' inference time, whose complexity is linear with $k$ .\",\n \"Figure: Performance for different number of bins.Table: Performance of different binning techniques.Different bin selection algorithms: Recall that in Section REF , we design a new algorithm (GBSA) to optimize the binning process for FactorJoin.\",\n \"Here, we compare it with the traditional equal-width and equal-depth binning algorithms.\",\n \"We set $k = 100$ for all three algorithms.\",\n \"They have approximately the same estimation latency, training time, and model size.\",\n \"Thus, we only report the end-to-end performance and bound tightness in Figure REF .\",\n \"We observe that GBSA generates much tighter upper bounds, leading to significantly better end-to-end performance.\",\n \"This demonstrates the effectiveness of our GBSA algorithm.\",\n \"Varying single-table CardEst methods: To compare the importance of different single-table CardEst methods in FactorJoin, we tried three different estimators: 1) Bayesian network (blackBayesCard) for single-table, which is the SOTA learned single-table method [70], 2) Sampling, which draws a random sample (5%) from single tables on-the-fly and estimates the cardinalities of filter predicates, and 3) TrueScan which scans and filters the tables during query time and calculate the true cardinalities.\",\n \"The end-to-end performance of different single table estimators with bin size $k=100$ is shown in Table REF .\",\n \"The blackBayesCard method performs significantly better than Sampling because of more accurate estimates and slightly faster estimation speeds.\",\n \"blackTrueScan produces an exact upper bound on the cardinalities and thus generates more effective query plans.\",\n \"However, the estimation latency is too high, making its overall end-to-end performance worse than blackBayesCard.\",\n \"Table: FactorJoin, varying single-table CardEst methods.Improvement over joining histograms: Since FactorJoin follows the convention of JoinHist, we investigate how much each component of FactorJoin help improve this method.\",\n \"Specifically, apart from the original JoinHist method, we evaluate and compare the following variants.\",\n \"We incorporate our new probabilistic bound into JoinHist to remove its join uniformity assumption (denote as with Bound).\",\n \"We incorporate into JoinHist the histograms learned from BayesNet that represent conditional distributions of join keys given the filter predicates (denote as with Conditional).\",\n \"This variant avoids the attribute independent assumption.\",\n \"FactorJoin reduces to the JoinHist with both bound and conditional techniques on non-cyclic join templates.\",\n \"The results in Table REF demonstrate that we can achieve significant improvement over the JoinHist method by removing either or all of its simplifying assumptions.\",\n \"Summary: With a reasonable bin size $k$ , FactorJoin generates tight bounds and effective query plans.\",\n \"The GBSA bin selection algorithm is very effective at generating high-quality plans.\",\n \"Single-table estimators do have an impact on FactorJoin's overall performance and we should choose one that can generate effective and efficient estimates.\",\n \"Overall, all different settings/hyperparameters of FactorJoin can significantly outperform Postgres as well as all existing SOTA methods, demonstrating the robustness and stability of FactorJoin.\",\n \"Table: Performance comparison with JoinHist.\"\n ],\n [\n \"Related work\",\n \"We briefly review literatures on single-table CardEst methods and the learned query optimizers.\",\n \"Single-table CardEst: Machine learning approaches have been use to solve single-table CardEst problem, which can achieve accurate estimation with very low latency and overhead [74], [70], [77].\",\n \"Specifically, the data-driven learned methods build statistical models to understand data distributions, such as deep autoregressive models [74], [19], Bayesian networks [70], [16], [13], sum-product-networks [25], factorized sum-product-networks [77], and normalizing flow models [67].\",\n \"As for the query-driven methods, the first approach using neural networks for CardEst has been proposed for UDF predicates [36].\",\n \"Later on, a semi-automatic alternative [47] and a regression-based model [2] were used for general predicates.\",\n \"Recently, more sophisticated supervised models such as multi-set convolutional networks [33], XG-boost [11], tree-LSTM [63], and deep ensembles [42], are used to provide accurate estimation.\",\n \"These works are orthogonal to our framework and in principle, we can support plugging in any one of them for single table CardEst.\",\n \"Learned query optimizer: Apart from CardEst, there exist a large number of works that use ML to solve other tasks of query optimizer, such as cost estimation and join order selection.\",\n \"Learned cost estimation methods use tree convolution [51] and tree-LSTM [63] to encode a query plan as a tree and map the encoding to its estimated costs.\",\n \"Active learning [45] and zero-shot learning [24] are proposed to tackle this problem from a new perspective.\",\n \"Some deep reinforcement learning approaches [75], [50] have been proposed to determine the optimal join order.\",\n \"Recently, many methods [49], [48], [53], [71] propose to learn the entire query optimizer end-to-end without a clear separation of these components.\"\n ],\n [\n \"Conclusions\",\n \"In this paper, we propose FactorJoin, a framework for cardinality estimation of join queries.\",\n \"It combines classical join-histogram methods with learned single table cardinality estimates into a factor graph model.\",\n \"This framework converts CardEst problem into an inference problem over the factor graph involving only single-table distributions.\",\n \"We further propose several optimizations to make this problem tractable for large join graphs.\",\n \"Our experiments show that FactorJoin generates effective and efficient estimates and is suitable for system deployment on large real-world join benchmarks.\",\n \"Specifically, FactorJoin produces more effective estimates than the previous SOTA learned methods, with 40x less estimation latency, 100x smaller model size, 100x faster updating speed, and 100x faster training speed, while matching or exceeding them in terms of query execution time.\",\n \"We believe this work points out a new direction for estimating join queries, which would enable truly practical learned CardEst methods as a system component.\",\n \"Equations and proofs Equations for decomposing joins into single-tables In this section, we provide the case-by-case equations on how to accurately formulate join estimation problem into single-table CardEst problem without any simplified assumptions.\",\n \"Two-table join case: Assume that we have two tables $A$ and $B$ with join keys $A.id$ and $B.Aid$ and a query $Q = Q(A) \\\\mathbin {\\\\bowtie }\\\\rule [0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\hspace{-3.05542pt}\\\\bowtie \\\\hspace{-3.05542pt}$$$Q(B)$ where $ Q(A)$ is the filter on table $ A$ and same for $ Q(B)$.\",\n \"The join condition is $ A.id = B.Aid$.\",\n \"Then, the join cardinality of $ Q$ can be expressed as follows where $ D(A.id)$ represents the domain of $ A$.\",\n \"{\\\\begin{@align}{1}{-1}|Q| = \\\\sum _{v \\\\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)|\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Chain join cases:}}\\\\emph {\\\\textbf {Case1 A.id = B.Aid = C.Aid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, and $ C.Aid$.\",\n \"There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.Aid = C.Aid$.Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| &= \\\\sum _{v \\\\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)|\\\\end{@align}}\\\\emph {\\\\textbf {Case2 A.id = B.Aid, B.id = C.Bid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, $ B.id$, and $ C.Bid$.\",\n \"There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.id = C.Bid$.Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| = &\\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)|\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Self join case:}}Assume we have one table $ A$ with join keys $ A.id, A.id2$.\",\n \"There exists a query $ Q = Q(A) Q(A')$ on join condition $ A.id = A.id2$.\",\n \"Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| = \\\\sum _{v \\\\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& P_{A}(A.id_2 = v | Q(A^{\\\\prime })) * |Q(A^{\\\\prime })|\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Cyclic join case:}}Assume we have two tables $ A$ and $ B$ with join keys $ A.id, A.id2$ and $ B.Aid, B.Aid2$.\",\n \"There exists a query $ Q = Q(A) Q(B)$ on join condition $ A.id = B.Aid AND A.id2 = B.Aid2$.\",\n \"Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| = & \\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(A.id_2)} P_{A}(A.id = v_1| Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& P_{A}(A.id_2 = v_2| Q(A^{\\\\prime })) * |Q(A^{\\\\prime })| \\\\nonumber \\\\\\\\= & \\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(A.id_2)} P_{B}(B.Aid = v_1| Q(B)) * |Q(B)| \\\\nonumber \\\\\\\\&P_{A}(A.id = v_1, A.id_2 = v_2| Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&P_{B}(B.Aid_2 = v_2 | Q(B^{\\\\prime })) * |Q(B^{\\\\prime })|\\\\end{@align}}\\\\subsection {Proof of lamma 1}\\\\begin{lemma}Given a join graph \\\\mathcal {G} representing a query Q, there exists a factor graph \\\\mathcal {F} such that the variable nodes in \\\\mathcal {F} are the equivalent key group variables of \\\\mathcal {G} and each factor node represents a table I touched by Q.\",\n \"A factor node is connected to a variable node if and only if this variable represents a join key in table I.\",\n \"The potential function of a factor node is defined as table I^{\\\\prime }s probability distribution of the connected variables (join keys) conditioned on the filter predicates Q(I).\",\n \"Then, calculating the cardinality of Q is equivalent to computing the partition function of \\\\mathcal {F}.\\\\end{lemma}\\\\smallskip \\\\underline{\\\\textbf {Proof:}}Assume that we have a query $ Q$ joining $ m$ tables $ A, B, ..., M$.We represent $ Q$ as a join graph $ G$, where each node represents a join key and each edge represents a join relation between two keys connected by it.\",\n \"We further define hyper-nodes in $ G$ as a set of nodes whose corresponding join key lies in the same table.\",\n \"Thus, each hyper-nodes naturally represents a table in $ Q$.\",\n \"A visualization of such $ G$ can be found in Figure~\\\\ref {fig: formulation}.Each connected components of $ G$ connects a group of join keys with equal join relation, suggesting that they share the same semantics.\",\n \"Therefore, we consider all join keys in a connected component as a equivalent key group variable, denoted as $ Vi$.Therefore, $ G$ defines $ n$ equivalent variables $ V1, ..., Vn$.Then, assume that we have a set of unnormalized single table distributions $ PA(VA|Q(A)) * |Q(A)|, ..., PM(VM|Q(M)) * |Q(M)|$ and $ PA(VA'|Q(A')) * |Q(A')|, ..., PM(VM'|Q(M')) * |Q(M')|$.Each $ VI$ presents a set of equivalent variables that represent a join key in table $ I$.\",\n \"The $ I'$ is the duplicated table introduced by query $ Q$ only if there exists a cyclic join situation.Therefore, the cardinality of $ Q$ can be writen as:{\\\\begin{@align}{1}{-1}|Q| = & \\\\sum _{v_1 \\\\in D(V_1)} \\\\sum _{v_2 \\\\in D(V_2)} \\\\cdots \\\\sum _{v_n \\\\in D(V_n)} (P_{A}(V_A| Q(A)) * |Q(A)|) * \\\\nonumber \\\\\\\\&(P_{B}(V_B | Q(B)) * |Q(B)|) * \\\\cdots (P_{M}(V_M | Q(M)) * |Q(M)|) * \\\\nonumber \\\\\\\\& (P_{A}(V_{A^{\\\\prime }}| Q(A^{\\\\prime })) * |Q(A^{\\\\prime })|) * \\\\cdots (P_{M^{\\\\prime }}(V_{M^{\\\\prime }} | Q(M^{\\\\prime })) * |Q(M^{\\\\prime })|)\\\\end{@align}}Next, let^{\\\\prime }s construct a factor graph $ F$, such that the variable nodes in $ F$ are the equivalent key group variables $ Vi$ of $ G$ and the factor nodes in $ F$ represent the tables $ A, ..., M$ touched by $ Q$ and the duplicated tables if exist $ A', ..., M'$.\",\n \"A factor node representing table $ I$ is connected to a variable node if and only if this variable represents a join key in table $ I$.The potential function of a factor node is defined as the unnormalized probability distribution $ PI(VI|Q(I)) * |Q(I)|$.The partition function of $ F$ is defined exactly the same as Equation~\\\\ref {equ: lemma1}.Therefore, instead of summing a nested loop of $ n$ variables $ Vi$ by brute-force, the partition function (cardinality) can be computed using the well-established variable elimination and belief propogation techniques in PGM domain.\\\\subsection {Probabilistic bound}\\\\emph {FactorJoin} approximate the exact cardinality computation with a probabilistic bound based inference algorithm.\",\n \"An example on two table join queries can found below.\",\n \"In this section we discuss how to derive an upper bound ($ Probabilistic_bound(A, B, bini)$ in this equation) for different case of join.\",\n \"{\\\\begin{@align}{3}{2}|Q| &= \\\\sum _{i = 1}^{k} \\\\sum _{v \\\\in bin_i} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\\\nonumber \\\\\\\\&\\\\lesssim \\\\sum _{i = 1}^{k} && Probabilistic\\\\_bound(A, B, bin_i)\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Case1: Joining two tablesA.id = B.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$.\",\n \"We have the following bound:{\\\\begin{@align}{3}{2}|Q| & = \\\\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min( &&\\\\frac{P_A(A.id \\\\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\\\nonumber \\\\\\\\& && \\\\frac{P_B(B.Aid \\\\in bin_i|Q(B)) * |Q(B)|}{V^{*}_i(B.Aid)}) * \\\\nonumber \\\\\\\\& && V^{*}_i(A.id) * V^{*}_i(B.Aid)\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Case2: Joining three tables A.id = B.Aid = C.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(C.Aid)$.\",\n \"We have the following bound:{\\\\begin{@align}{1}{-1}|Q| & = \\\\sum _{v \\\\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& \\\\hspace{20.0pt} P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min\\\\lbrace \\\\frac{P_{A}(A.id \\\\in bin_i | Q(A)) * |Q(A)|}{V_i^{*}(A.id)}, \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} \\\\frac{P_{B}(B.Aid \\\\in bin_i | Q(B)) * |Q(B)|}{V_i^{*}(B.Aid)} \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} \\\\frac{P_{C}(C.Aid \\\\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Aid) }\\\\rbrace * \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} {V_i^{*}(A.id)} * {V_i^{*}(B.Aid) } * {V_i^{*}(C.Aid)}\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Case3: Joining three tables A.id = B.Aid, B.id = C.Bid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(B.id)$, and $ Vi*(C.Bid)$.We have the following bound, where $ Upper(A B)$ is derived from Equation~\\\\ref {equ: bound2join}.\",\n \"{\\\\begin{@align}{1}{-1}|Q| &= \\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&\\\\hspace{60.0pt} P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& \\\\hspace{60.0pt} P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)| \\\\nonumber \\\\\\\\&\\\\le \\\\sum _{i = 1}^{k} min\\\\lbrace \\\\frac{Upper(Q(A) \\\\mathbin {\\\\bowtie }\\\\rule [0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\hspace{-3.05542pt}\\\\bowtie \\\\hspace{-3.05542pt}\\\\bowtie }{\\\\rule }[0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\end{@align}Q(B))}{V_i^{*}(A.id) * V_i^{*}(B.id)}, \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} \\\\frac{P_{C}(C.Aid \\\\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Bid)}\\\\rbrace * \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} * V_i^{*}(A.id) * V_i^{*}(B.id) * {V_i^{*}(C.Bid)}$ Case4: Self join of one table $A.id = A.id2$ : Let $V_i^{*}(A.id) = MAX_{v \\\\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(A.id_2)$ .\",\n \"We have the following bound: $|Q| & = \\\\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& && P_{A}(A.id_2 = v | Q(A^{\\\\prime })) * |Q(A^{\\\\prime })| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min( &&\\\\frac{P_A(A.id \\\\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\\\nonumber \\\\\\\\& && \\\\frac{P_A(A.id_2 \\\\in bin_i|Q(A^{\\\\prime })) * |Q(A^{\\\\prime })|}{V^{*}_i(A.id_2)}) * \\\\nonumber \\\\\\\\& && V^{*}_i(A.id) * V^{*}_i(A.id_2)$ Case5: Cyclic join of two tables $A.id = B.Aid$ , $A.id_2 = B.Aid_2$ : Let $V_i^{*}(A.id) = MAX_{v \\\\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(B.Aid)$ , $V_i^{*}(A.id_2)$ , and $V_i^{*}(B.Aid_2)$ .\",\n \"We have the following bound: $|Q| &= \\\\sum _{ v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(A.id_2)} P_{A}(A.id = v_1, A.id_2 = v_2 | Q(A)) * \\\\nonumber \\\\\\\\& \\\\hspace{30.0pt} |Q(A)| * P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& \\\\hspace{30.0pt} P_{A}(A.id_2 = v_2| Q(A^{\\\\prime })) * |Q(A^{\\\\prime })| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min\\\\lbrace \\\\frac{Upper(Q(A) \\\\mathbin {\\\\bowtie }\\\\rule [0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\hspace{-3.05542pt}\\\\bowtie \\\\hspace{-3.05542pt}\\\\bowtie }{\\\\rule }[0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}$ Q(B)Vi*(A.id2) * Vi*(B.Aid) , Upper(Q(A') Q(B))Vi*(A.id) * Vi*(B.Aid2)} * Vi*(A.id2) * Vi*(B.Aid) * Vi*(A.id) * Vi*(B.Aid2) Greedy bin selection Algorithm details We observe that the bound on a particular bin $bin_i$ can be very loose if the MFV count $V^*_i$ is a large outlier in $bin_i$ .\",\n \"Taking the two table join query as an example, if $bin_i$ contains only one value that appears 100 times in $A.id$ but $10,000$ values that only appear once in $B.Aid$ , then the bound could be 100 times larger than the actual cardinality.\",\n \"The existing equal-width or equal-depth binning strategy can generate very large estimation errors, so we design a new binning strategy called the greedy bin selection algorithm (GBSA) to optimize our bound tightness.\",\n \"The objective of GBSA is to minimize the variance of the value counts within $bin_i$ .\",\n \"In the extreme case, if the value counts have zero variance for all join keys in one equivalent key group, then our bound can output the exact cardinality (if with perfect single table CardEst models).\",\n \"However, as the same bin partition will be applied for all equivalent join keys, minimizing the value counts variance of $bin_i$ on the domain of one key may result in a bad bin for other keys.\",\n \"Jointly minimizing the variance of one bin for all join keys has exponential complexity.\",\n \"There, GBSA uses a greedy algorithm to iteratively minimize the bin variance for all join keys.\",\n \"At a high level, GBSA first optimizes the minimal variance bins with half number of binning budget $k/2$ on the domain of one join key.\",\n \"Then, it recursively updates these bins by minimizing the variance of other join keys using the rest half of the budget.\",\n \"The details of GBSA are provided in Algorithm REF .\",\n \"[t] Greedy Bin Selection Algorithm (GBSA) Input: Equivalent key groups $Gr_1, \\\\ldots , Gr_m$ , where $Gr_i = \\\\lbrace Id_i^1, \\\\ldots , Id_i^{|Gr_i|} \\\\rbrace $ ; Column data $\\\\mathcal {D}(Id_i^j)$ of all join keys in the DB instance $\\\\mathcal {D}$ ; Number of bins $k_i$ for each group $Gr_i$ .\",\n \"[1] $Gr_i \\\\in \\\\lbrace Gr_1, \\\\ldots , Gr_m\\\\rbrace $ $Bin(Gr_i) \\\\leftarrow $ [] $Gr_i^{\\\\prime } \\\\leftarrow sort\\\\_key\\\\_based\\\\_on\\\\_domain\\\\_size(\\\\mathcal {D}, Gr_i)$ $Bin(Gr_i) \\\\leftarrow get\\\\_min\\\\_variance\\\\_bins(\\\\mathcal {D}(Gr_i^{\\\\prime }[1]), k_i/2)$ remain_bins $\\\\leftarrow k_i/2$ $j \\\\in \\\\lbrace 2, \\\\ldots , |Gr_i^{\\\\prime }| \\\\rbrace $ binned_data $\\\\leftarrow apply\\\\_bin\\\\_to\\\\_data(\\\\mathcal {D}(Gr_i^{\\\\prime }[j]), Bin(Gr_i))$ bin_variance $\\\\leftarrow calculate\\\\_variance$ (binned_data) arg_sort_idx $\\\\leftarrow arg\\\\_sort\\\\_decreasing(bin\\\\_variance)$ $p \\\\in arg\\\\_sort\\\\_idx[1:remain\\\\_bins/2]$ $Bin(Gr_i) \\\\leftarrow min\\\\_variance\\\\_dichotomy(Bin(Gr_i)[p]$ , binned_data[p]) remain_bins $\\\\leftarrow $ remain_bins/2 return $\\\\lbrace Bin(Gr_i) | i, \\\\ldots , m\\\\rbrace $ FactorJoin analyzes the schema of DB instance $\\\\mathcal {D}$ to derive $m$ equivalent key groups $Gr_1, \\\\ldots , Gr_m$ , each of which contains $|Gr_i|$ join keys $Id_i^1, \\\\ldots , Id_i^{|Gr_i|}$ .\",\n \"Let $\\\\mathcal {D}(Id_i^j)$ represents the column data (domain) of join key $Id_i^j$ .\",\n \"GBSA will derive the sub-optimal binning with $k_i$ bins $Bin(Gr_i) = \\\\lbrace bin_1, \\\\ldots , bin_{k_i} \\\\rbrace $ for each the equivalent key group $Gr_i$ (line 1).\",\n \"We explain the procedure of this algorithm for binning one group $Gr_i$ (line 2-14).\",\n \"First, GBSA sorts the join keys $\\\\lbrace d_i^1, \\\\ldots , Id_i^{|Gr_i|}\\\\rbrace $ in decreasing order based on their domain size (line 3) to get $Gr_i^{\\\\prime }=\\\\lbrace Id_i^{1^{\\\\prime }}, \\\\ldots , Id_i^{|Gr_i|^{\\\\prime }}\\\\rbrace $ .\",\n \"We apply half of the binning budget to generate $k_i/2$ minimal variance bins on the domain of $d_i^{1^{\\\\prime }}$ (line 4).\",\n \"Note that minimal variance bins on a single attribute can be easily obtained by sorting the value counts of $\\\\mathcal {D}(d_i^{1^{\\\\prime }})$ and applying equal-depth binning over the sorted values.\",\n \"The remaining number of bins available $remain_bins$ is $k_i/2$ (line 5).\",\n \"Then for each rest of the join key $d_i^j$ , GBSA applies the current bins $Bin(Gr_i)$ to its data column $\\\\mathcal {D}(d_i^j)$ , calculates the variance for each bin, and sorts these bins in decreasing order (line 6-9).\",\n \"For the top $remain_bins/2$ bins with the largest variance, we dichotomize each bin into two bins to minimize the variance on join key $d_i^j$ (lines 10 -12).\",\n \"We iterative the above procedure until all join keys are optimized.\",\n \"According to our evaluation results, the GBSA has a significant impact on improving our probabilistic bound tightness and estimation effectiveness.\",\n \"Figure: Per query performance of STATS-CEB.Figure: Per query performance of STATS-CEB.\",\n \"Additional experiments and details In this section, we provide the detailed comparision of all baselines on STATS-CEB and IMDB-JOB benchmarks.\",\n \"Performance on STATS-CEB: We sort the 146 queries of STATS-CEB based on their Postgres end-to-end runtime and cluster them into 6 different runtime intervals.\",\n \"Figure REF reports relative improvements of all competitive baselines over Postgres for each query and the overall performance comparision on each cluster of queries on the last figure.\",\n \"For the very short-running queries (which represent an OLTP-like workload), Postgres is the best among all baselines.\",\n \"These baselines perform worse because the estimation latency plays a significant role in these queries.\",\n \"We observe that improving estimation accuracy has a very limited effect on the query plan quality of short-running queries.\",\n \"This also explains why the optimal TrueCard only marginally outperforms Postgres on queries with less than $2s$ of runtime.\",\n \"Overall, FactorJoin has the best performance among all baselines except for Postgres.\",\n \"For the extremely long-running queries, the advantage of the learning-based methods over Postgres gradually appear.\",\n \"The reason is that estimation latency becomes increasingly insignificant for queries with a long execution time.\",\n \"FactorJoin has comparable performance as the SOTA learning-based methods on these queries.\",\n \"Performance on IMDB-JOB: Since a large proportion of the queries in IMDB-JOB workload are short-running queries, Postgres has a better performance than all baselines for more than half of the queries.\",\n \"Similar to STATS-CEB queries, we do not observe any performance gain for using the learned CardEst method over Postgres on queries with less on $1s$ runtime.\",\n \"In fact, the optimal TrueCard also barely improves the Postgres.\",\n \"This suggests that the learned methods shoulf fall back to Postgres for OLTP workloads.\",\n \"Similar to STATS-CEB, FactorJoin also has a significantly better performance over all other baselines except Postgres.\"\n ],\n [\n \"Equations for decomposing joins into single-tables\",\n \"In this section, we provide the case-by-case equations on how to accurately formulate join estimation problem into single-table CardEst problem without any simplified assumptions.\",\n \"Two-table join case: Assume that we have two tables $A$ and $B$ with join keys $A.id$ and $B.Aid$ and a query $Q = Q(A) \\\\mathbin {\\\\bowtie }\\\\rule [0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\hspace{-3.05542pt}\\\\bowtie \\\\hspace{-3.05542pt}$$$Q(B)$ where $ Q(A)$ is the filter on table $ A$ and same for $ Q(B)$.\",\n \"The join condition is $ A.id = B.Aid$.\",\n \"Then, the join cardinality of $ Q$ can be expressed as follows where $ D(A.id)$ represents the domain of $ A$.\",\n \"{\\\\begin{@align}{1}{-1}|Q| = \\\\sum _{v \\\\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)|\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Chain join cases:}}\\\\emph {\\\\textbf {Case1 A.id = B.Aid = C.Aid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, and $ C.Aid$.\",\n \"There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.Aid = C.Aid$.Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| &= \\\\sum _{v \\\\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)|\\\\end{@align}}\\\\emph {\\\\textbf {Case2 A.id = B.Aid, B.id = C.Bid:}}Assume we have three tables $ A$, $ B$ and $ C$ with join keys $ A.id$, $ B.Aid$, $ B.id$, and $ C.Bid$.\",\n \"There exists query $ Q = Q(A) Q(B) Q(C)$ on join condition $ A.id = B.Aid AND B.id = C.Bid$.Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| = &\\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)|\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Self join case:}}Assume we have one table $ A$ with join keys $ A.id, A.id2$.\",\n \"There exists a query $ Q = Q(A) Q(A')$ on join condition $ A.id = A.id2$.\",\n \"Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| = \\\\sum _{v \\\\in D(A.id)} &P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& P_{A}(A.id_2 = v | Q(A^{\\\\prime })) * |Q(A^{\\\\prime })|\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Cyclic join case:}}Assume we have two tables $ A$ and $ B$ with join keys $ A.id, A.id2$ and $ B.Aid, B.Aid2$.\",\n \"There exists a query $ Q = Q(A) Q(B)$ on join condition $ A.id = B.Aid AND A.id2 = B.Aid2$.\",\n \"Then, the join cardinality can be expressed as follows:{\\\\begin{@align}{1}{-1}|Q| = & \\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(A.id_2)} P_{A}(A.id = v_1| Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& P_{A}(A.id_2 = v_2| Q(A^{\\\\prime })) * |Q(A^{\\\\prime })| \\\\nonumber \\\\\\\\= & \\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(A.id_2)} P_{B}(B.Aid = v_1| Q(B)) * |Q(B)| \\\\nonumber \\\\\\\\&P_{A}(A.id = v_1, A.id_2 = v_2| Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&P_{B}(B.Aid_2 = v_2 | Q(B^{\\\\prime })) * |Q(B^{\\\\prime })|\\\\end{@align}}\\\\subsection {Proof of lamma 1}\\\\begin{lemma}Given a join graph \\\\mathcal {G} representing a query Q, there exists a factor graph \\\\mathcal {F} such that the variable nodes in \\\\mathcal {F} are the equivalent key group variables of \\\\mathcal {G} and each factor node represents a table I touched by Q.\",\n \"A factor node is connected to a variable node if and only if this variable represents a join key in table I.\",\n \"The potential function of a factor node is defined as table I^{\\\\prime }s probability distribution of the connected variables (join keys) conditioned on the filter predicates Q(I).\",\n \"Then, calculating the cardinality of Q is equivalent to computing the partition function of \\\\mathcal {F}.\\\\end{lemma}\\\\smallskip \\\\underline{\\\\textbf {Proof:}}Assume that we have a query $ Q$ joining $ m$ tables $ A, B, ..., M$.We represent $ Q$ as a join graph $ G$, where each node represents a join key and each edge represents a join relation between two keys connected by it.\",\n \"We further define hyper-nodes in $ G$ as a set of nodes whose corresponding join key lies in the same table.\",\n \"Thus, each hyper-nodes naturally represents a table in $ Q$.\",\n \"A visualization of such $ G$ can be found in Figure~\\\\ref {fig: formulation}.Each connected components of $ G$ connects a group of join keys with equal join relation, suggesting that they share the same semantics.\",\n \"Therefore, we consider all join keys in a connected component as a equivalent key group variable, denoted as $ Vi$.Therefore, $ G$ defines $ n$ equivalent variables $ V1, ..., Vn$.Then, assume that we have a set of unnormalized single table distributions $ PA(VA|Q(A)) * |Q(A)|, ..., PM(VM|Q(M)) * |Q(M)|$ and $ PA(VA'|Q(A')) * |Q(A')|, ..., PM(VM'|Q(M')) * |Q(M')|$.Each $ VI$ presents a set of equivalent variables that represent a join key in table $ I$.\",\n \"The $ I'$ is the duplicated table introduced by query $ Q$ only if there exists a cyclic join situation.Therefore, the cardinality of $ Q$ can be writen as:{\\\\begin{@align}{1}{-1}|Q| = & \\\\sum _{v_1 \\\\in D(V_1)} \\\\sum _{v_2 \\\\in D(V_2)} \\\\cdots \\\\sum _{v_n \\\\in D(V_n)} (P_{A}(V_A| Q(A)) * |Q(A)|) * \\\\nonumber \\\\\\\\&(P_{B}(V_B | Q(B)) * |Q(B)|) * \\\\cdots (P_{M}(V_M | Q(M)) * |Q(M)|) * \\\\nonumber \\\\\\\\& (P_{A}(V_{A^{\\\\prime }}| Q(A^{\\\\prime })) * |Q(A^{\\\\prime })|) * \\\\cdots (P_{M^{\\\\prime }}(V_{M^{\\\\prime }} | Q(M^{\\\\prime })) * |Q(M^{\\\\prime })|)\\\\end{@align}}Next, let^{\\\\prime }s construct a factor graph $ F$, such that the variable nodes in $ F$ are the equivalent key group variables $ Vi$ of $ G$ and the factor nodes in $ F$ represent the tables $ A, ..., M$ touched by $ Q$ and the duplicated tables if exist $ A', ..., M'$.\",\n \"A factor node representing table $ I$ is connected to a variable node if and only if this variable represents a join key in table $ I$.The potential function of a factor node is defined as the unnormalized probability distribution $ PI(VI|Q(I)) * |Q(I)|$.The partition function of $ F$ is defined exactly the same as Equation~\\\\ref {equ: lemma1}.Therefore, instead of summing a nested loop of $ n$ variables $ Vi$ by brute-force, the partition function (cardinality) can be computed using the well-established variable elimination and belief propogation techniques in PGM domain.\\\\subsection {Probabilistic bound}\\\\emph {FactorJoin} approximate the exact cardinality computation with a probabilistic bound based inference algorithm.\",\n \"An example on two table join queries can found below.\",\n \"In this section we discuss how to derive an upper bound ($ Probabilistic_bound(A, B, bini)$ in this equation) for different case of join.\",\n \"{\\\\begin{@align}{3}{2}|Q| &= \\\\sum _{i = 1}^{k} \\\\sum _{v \\\\in bin_i} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\\\nonumber \\\\\\\\&\\\\lesssim \\\\sum _{i = 1}^{k} && Probabilistic\\\\_bound(A, B, bin_i)\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Case1: Joining two tablesA.id = B.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$.\",\n \"We have the following bound:{\\\\begin{@align}{3}{2}|Q| & = \\\\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& && P_{B}(B.Aid = v | Q(B)) * |Q(B)| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min( &&\\\\frac{P_A(A.id \\\\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\\\nonumber \\\\\\\\& && \\\\frac{P_B(B.Aid \\\\in bin_i|Q(B)) * |Q(B)|}{V^{*}_i(B.Aid)}) * \\\\nonumber \\\\\\\\& && V^{*}_i(A.id) * V^{*}_i(B.Aid)\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Case2: Joining three tables A.id = B.Aid = C.Aid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(C.Aid)$.\",\n \"We have the following bound:{\\\\begin{@align}{1}{-1}|Q| & = \\\\sum _{v \\\\in D(A.id)} P_{A}(A.id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& \\\\hspace{20.0pt} P_{B}(B.Aid = v | Q(B)) * |Q(B)| * P_{C}(C.Aid = v | Q(C)) * |Q(C)| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min\\\\lbrace \\\\frac{P_{A}(A.id \\\\in bin_i | Q(A)) * |Q(A)|}{V_i^{*}(A.id)}, \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} \\\\frac{P_{B}(B.Aid \\\\in bin_i | Q(B)) * |Q(B)|}{V_i^{*}(B.Aid)} \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} \\\\frac{P_{C}(C.Aid \\\\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Aid) }\\\\rbrace * \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} {V_i^{*}(A.id)} * {V_i^{*}(B.Aid) } * {V_i^{*}(C.Aid)}\\\\end{@align}}\\\\smallskip \\\\underline{\\\\textbf {Case3: Joining three tables A.id = B.Aid, B.id = C.Bid:}}Let $ Vi*(A.id) = MAXv bini |A.id = v|$ be the most frequent value (MFV) count of $ A.id$ in a bin $ bini$, and same for $ Vi*(B.Aid)$, $ Vi*(B.id)$, and $ Vi*(C.Bid)$.We have the following bound, where $ Upper(A B)$ is derived from Equation~\\\\ref {equ: bound2join}.\",\n \"{\\\\begin{@align}{1}{-1}|Q| &= \\\\sum _{v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(B.id)} P_{A}(A.id = v_1 | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\&\\\\hspace{60.0pt} P_{B}(B.Aid = v_1, B.id = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& \\\\hspace{60.0pt} P_{C}(C.Bid = v_2 | Q(C)) * |Q(C)| \\\\nonumber \\\\\\\\&\\\\le \\\\sum _{i = 1}^{k} min\\\\lbrace \\\\frac{Upper(Q(A) \\\\mathbin {\\\\bowtie }\\\\rule [0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\hspace{-3.05542pt}\\\\bowtie \\\\hspace{-3.05542pt}\\\\bowtie }{\\\\rule }[0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\end{@align}Q(B))}{V_i^{*}(A.id) * V_i^{*}(B.id)}, \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} \\\\frac{P_{C}(C.Aid \\\\in bin_i | Q(C)) * |Q(C)|}{V_i^{*}(C.Bid)}\\\\rbrace * \\\\nonumber \\\\\\\\& \\\\hspace{50.0pt} * V_i^{*}(A.id) * V_i^{*}(B.id) * {V_i^{*}(C.Bid)}$ Case4: Self join of one table $A.id = A.id2$ : Let $V_i^{*}(A.id) = MAX_{v \\\\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(A.id_2)$ .\",\n \"We have the following bound: $|Q| & = \\\\sum _{v = D(A.id)} && P_{A}(A.Id = v | Q(A)) * |Q(A)| * \\\\nonumber \\\\\\\\& && P_{A}(A.id_2 = v | Q(A^{\\\\prime })) * |Q(A^{\\\\prime })| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min( &&\\\\frac{P_A(A.id \\\\in bin_i|Q(A)) * |Q(A)|}{V^{*}_i(A.id)}, \\\\nonumber \\\\\\\\& && \\\\frac{P_A(A.id_2 \\\\in bin_i|Q(A^{\\\\prime })) * |Q(A^{\\\\prime })|}{V^{*}_i(A.id_2)}) * \\\\nonumber \\\\\\\\& && V^{*}_i(A.id) * V^{*}_i(A.id_2)$ Case5: Cyclic join of two tables $A.id = B.Aid$ , $A.id_2 = B.Aid_2$ : Let $V_i^{*}(A.id) = MAX_{v \\\\in bin_i} |A.id = v|$ be the most frequent value (MFV) count of $A.id$ in a bin $bin_i$ , and same for $V_i^{*}(B.Aid)$ , $V_i^{*}(A.id_2)$ , and $V_i^{*}(B.Aid_2)$ .\",\n \"We have the following bound: $|Q| &= \\\\sum _{ v_1 \\\\in D(A.id)} \\\\sum _{v_2 \\\\in D(A.id_2)} P_{A}(A.id = v_1, A.id_2 = v_2 | Q(A)) * \\\\nonumber \\\\\\\\& \\\\hspace{30.0pt} |Q(A)| * P_{B}(B.Aid = v_1, B.Aid_2 = v_2 | Q(B)) * |Q(B)| * \\\\nonumber \\\\\\\\& \\\\hspace{30.0pt} P_{A}(A.id_2 = v_2| Q(A^{\\\\prime })) * |Q(A^{\\\\prime })| \\\\nonumber \\\\\\\\& \\\\le \\\\sum _{i = 1}^{k} min\\\\lbrace \\\\frac{Upper(Q(A) \\\\mathbin {\\\\bowtie }\\\\rule [0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}\\\\hspace{-3.05542pt}\\\\bowtie \\\\hspace{-3.05542pt}\\\\bowtie }{\\\\rule }[0.15ex]{.22em}{.6pt}\\\\unknown.\",\n \"{\\\\rule [0.9ex]{.22em}{.6pt}}$ Q(B)Vi*(A.id2) * Vi*(B.Aid) , Upper(Q(A') Q(B))Vi*(A.id) * Vi*(B.Aid2)} * Vi*(A.id2) * Vi*(B.Aid) * Vi*(A.id) * Vi*(B.Aid2)\"\n ],\n [\n \"Greedy bin selection Algorithm details\",\n \"We observe that the bound on a particular bin $bin_i$ can be very loose if the MFV count $V^*_i$ is a large outlier in $bin_i$ .\",\n \"Taking the two table join query as an example, if $bin_i$ contains only one value that appears 100 times in $A.id$ but $10,000$ values that only appear once in $B.Aid$ , then the bound could be 100 times larger than the actual cardinality.\",\n \"The existing equal-width or equal-depth binning strategy can generate very large estimation errors, so we design a new binning strategy called the greedy bin selection algorithm (GBSA) to optimize our bound tightness.\",\n \"The objective of GBSA is to minimize the variance of the value counts within $bin_i$ .\",\n \"In the extreme case, if the value counts have zero variance for all join keys in one equivalent key group, then our bound can output the exact cardinality (if with perfect single table CardEst models).\",\n \"However, as the same bin partition will be applied for all equivalent join keys, minimizing the value counts variance of $bin_i$ on the domain of one key may result in a bad bin for other keys.\",\n \"Jointly minimizing the variance of one bin for all join keys has exponential complexity.\",\n \"There, GBSA uses a greedy algorithm to iteratively minimize the bin variance for all join keys.\",\n \"At a high level, GBSA first optimizes the minimal variance bins with half number of binning budget $k/2$ on the domain of one join key.\",\n \"Then, it recursively updates these bins by minimizing the variance of other join keys using the rest half of the budget.\",\n \"The details of GBSA are provided in Algorithm REF .\",\n \"[t] Greedy Bin Selection Algorithm (GBSA) Input: Equivalent key groups $Gr_1, \\\\ldots , Gr_m$ , where $Gr_i = \\\\lbrace Id_i^1, \\\\ldots , Id_i^{|Gr_i|} \\\\rbrace $ ; Column data $\\\\mathcal {D}(Id_i^j)$ of all join keys in the DB instance $\\\\mathcal {D}$ ; Number of bins $k_i$ for each group $Gr_i$ .\",\n \"[1] $Gr_i \\\\in \\\\lbrace Gr_1, \\\\ldots , Gr_m\\\\rbrace $ $Bin(Gr_i) \\\\leftarrow $ [] $Gr_i^{\\\\prime } \\\\leftarrow sort\\\\_key\\\\_based\\\\_on\\\\_domain\\\\_size(\\\\mathcal {D}, Gr_i)$ $Bin(Gr_i) \\\\leftarrow get\\\\_min\\\\_variance\\\\_bins(\\\\mathcal {D}(Gr_i^{\\\\prime }[1]), k_i/2)$ remain_bins $\\\\leftarrow k_i/2$ $j \\\\in \\\\lbrace 2, \\\\ldots , |Gr_i^{\\\\prime }| \\\\rbrace $ binned_data $\\\\leftarrow apply\\\\_bin\\\\_to\\\\_data(\\\\mathcal {D}(Gr_i^{\\\\prime }[j]), Bin(Gr_i))$ bin_variance $\\\\leftarrow calculate\\\\_variance$ (binned_data) arg_sort_idx $\\\\leftarrow arg\\\\_sort\\\\_decreasing(bin\\\\_variance)$ $p \\\\in arg\\\\_sort\\\\_idx[1:remain\\\\_bins/2]$ $Bin(Gr_i) \\\\leftarrow min\\\\_variance\\\\_dichotomy(Bin(Gr_i)[p]$ , binned_data[p]) remain_bins $\\\\leftarrow $ remain_bins/2 return $\\\\lbrace Bin(Gr_i) | i, \\\\ldots , m\\\\rbrace $ FactorJoin analyzes the schema of DB instance $\\\\mathcal {D}$ to derive $m$ equivalent key groups $Gr_1, \\\\ldots , Gr_m$ , each of which contains $|Gr_i|$ join keys $Id_i^1, \\\\ldots , Id_i^{|Gr_i|}$ .\",\n \"Let $\\\\mathcal {D}(Id_i^j)$ represents the column data (domain) of join key $Id_i^j$ .\",\n \"GBSA will derive the sub-optimal binning with $k_i$ bins $Bin(Gr_i) = \\\\lbrace bin_1, \\\\ldots , bin_{k_i} \\\\rbrace $ for each the equivalent key group $Gr_i$ (line 1).\",\n \"We explain the procedure of this algorithm for binning one group $Gr_i$ (line 2-14).\",\n \"First, GBSA sorts the join keys $\\\\lbrace d_i^1, \\\\ldots , Id_i^{|Gr_i|}\\\\rbrace $ in decreasing order based on their domain size (line 3) to get $Gr_i^{\\\\prime }=\\\\lbrace Id_i^{1^{\\\\prime }}, \\\\ldots , Id_i^{|Gr_i|^{\\\\prime }}\\\\rbrace $ .\",\n \"We apply half of the binning budget to generate $k_i/2$ minimal variance bins on the domain of $d_i^{1^{\\\\prime }}$ (line 4).\",\n \"Note that minimal variance bins on a single attribute can be easily obtained by sorting the value counts of $\\\\mathcal {D}(d_i^{1^{\\\\prime }})$ and applying equal-depth binning over the sorted values.\",\n \"The remaining number of bins available $remain_bins$ is $k_i/2$ (line 5).\",\n \"Then for each rest of the join key $d_i^j$ , GBSA applies the current bins $Bin(Gr_i)$ to its data column $\\\\mathcal {D}(d_i^j)$ , calculates the variance for each bin, and sorts these bins in decreasing order (line 6-9).\",\n \"For the top $remain_bins/2$ bins with the largest variance, we dichotomize each bin into two bins to minimize the variance on join key $d_i^j$ (lines 10 -12).\",\n \"We iterative the above procedure until all join keys are optimized.\",\n \"According to our evaluation results, the GBSA has a significant impact on improving our probabilistic bound tightness and estimation effectiveness.\",\n \"Figure: Per query performance of STATS-CEB.Figure: Per query performance of STATS-CEB.\"\n ],\n [\n \"Additional experiments and details\",\n \"In this section, we provide the detailed comparision of all baselines on STATS-CEB and IMDB-JOB benchmarks.\",\n \"Performance on STATS-CEB: We sort the 146 queries of STATS-CEB based on their Postgres end-to-end runtime and cluster them into 6 different runtime intervals.\",\n \"Figure REF reports relative improvements of all competitive baselines over Postgres for each query and the overall performance comparision on each cluster of queries on the last figure.\",\n \"For the very short-running queries (which represent an OLTP-like workload), Postgres is the best among all baselines.\",\n \"These baselines perform worse because the estimation latency plays a significant role in these queries.\",\n \"We observe that improving estimation accuracy has a very limited effect on the query plan quality of short-running queries.\",\n \"This also explains why the optimal TrueCard only marginally outperforms Postgres on queries with less than $2s$ of runtime.\",\n \"Overall, FactorJoin has the best performance among all baselines except for Postgres.\",\n \"For the extremely long-running queries, the advantage of the learning-based methods over Postgres gradually appear.\",\n \"The reason is that estimation latency becomes increasingly insignificant for queries with a long execution time.\",\n \"FactorJoin has comparable performance as the SOTA learning-based methods on these queries.\",\n \"Performance on IMDB-JOB: Since a large proportion of the queries in IMDB-JOB workload are short-running queries, Postgres has a better performance than all baselines for more than half of the queries.\",\n \"Similar to STATS-CEB queries, we do not observe any performance gain for using the learned CardEst method over Postgres on queries with less on $1s$ runtime.\",\n \"In fact, the optimal TrueCard also barely improves the Postgres.\",\n \"This suggests that the learned methods shoulf fall back to Postgres for OLTP workloads.\",\n \"Similar to STATS-CEB, FactorJoin also has a significantly better performance over all other baselines except Postgres.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05526"}}},{"rowIdx":82,"cells":{"text":{"kind":"list like","value":[["Deep Inelastic Scattering with Application to Nuclear Targets: Lectures\n at the 1985 Los Alamos School on Relativistic Dynamics and Quark Nuclear\n Physics"],["Abstract This paper is essentially a verbatim reconstruction of lectures that I gave at the Los Alamos School on Relativistic Dynamics and Quark Nuclear Physics in 1985.","They were published in the school proceedings, but the book is not widely available.","The Los Alamos School took place at the height of the first wave of interest in the quark substructure of nuclei, stimulated by the 1983 discovery of the EMC Effect.","Interest in this subject has been increasing for years and the prospect of a dedicated Electron Ion Collider within the decade guarantees even greater attention to quarks and gluons in nuclei among both theorists and experimentalists.","Recently, to my surprise, I learned that copies of my old lectures have been circulating and been found useful by the relatively few people who know about them.","The are, of course, dated: experiments have far outstripped what was available 37 years ago and theory has progressed too.","However, the rest frame derivation of the parton model, the derivation and discussion of the convolution formalism for nucleons, nucleon correlations, and other, virtual, constituents of nuclei, and sections on scaling violation and the operator product expansion have aged pretty well and seem to still be useful.","With the help and encouragement of Richard Milner, I have recreated the LaTeX files necessary to post the 1985 Lectures on the arXiv, making them available to the nuclear and particle physics community.","Apart from correcting some typographical errors, I have made no attempt to edit, improve, or update these lectures.","I hope readers will nevertheless find them useful."],["MOTTO","“Looking for the quarks in the nucleus is like looking for the Mafia in Sicily: Everyone knows they're there, but its hard to find the evidence.” (Anonymous) This paper is essentially a verbatim reconstruction of lectures that I gave at the Los Alamos School on Relativistic Dynamics and Quark Nuclear Physics in 1985.","They were published in the school proceedings[1], but the book is not widely available.","The Los Alamos School took place at the height of the first wave of interest in the quark substructure of nuclei, stimulated by the 1983 discovery of the EMC Effect[2].","Interest in this subject has been increasing for years and the prospect of a dedicated Electron Ion Collider within the decade guarantees even greater attention to quarks and gluons in nuclei among both theorists and experimentalists.","Recently, to my surprise, I learned that copies of my old lectures have been circulating and been found useful by the relatively few people who know about them.","The are, of course, dated: experiments have far outstripped what was available 37 years ago and theory has progressed too.","However, the rest frame derivation of the parton model, the derivation and discussion of the convolution formalism for nucleons, nucleon correlations, and other, virtual, constituents of nuclei, and sections on scaling violation and the operator product expansion have aged pretty well and seem to still be useful.","With the help and encouragement of Richard Milner, I have recreated the LaTeX files necessary to post the 1985 Lectures on the arXiv, making them available to the nuclear and particle physics community.","Apart from correcting some typographical errors, I have made no attempt to edit, improve, or update these lectures.","I hope readers will nevertheless find them useful.","These lectures are addressed to a very specific audience: graduate students and young researchers in theoretical nuclear physics.","They contain very little that cannot be found elsewhere in the vast literature on the subject, but they summarize a particular viewpoint which is both straightforward and fairly correct.","My intention is to provide a reasonably thorough introduction to inclusive, inelastic electron scattering and enough advanced material to excite the reader to go on by him/herself.","The reader is assumed to have some familiarity with relativistic quantum mechanics and with the more elementary side of quantum field theory.","Some previous exposure to QCD $-$ quarks, color, gluons, SU(3) and the like $-$ will help.","But; previous exposure to renormalization theory, the renormalization group and the operator product expansion is not assumed.","I have tried to avoid the temptation to oversimplify $-$ the subject has its subtleties and it is not possible to do good work in the field without understanding them.","More simplified treatments can be found in books by Feynman (Photon-Hadron Interactions, (W. A. Benjamin, New York (1972)) and by Close (An Introduction to Quarks and Hadrons (Academic Press, New York, 1979)).","Equally, I have tried to avoid too much formalism.","I hope the power of the more advanced methods developed in §5 and applied in §6 will encourage the reader to pursue the subject more formally in the future.","More advanced treatments can be found in tho lecture notes of D. J.","Gross (in Methods in Field Theory, Proc.","of 1975 Les Houches Summer School, (ed., R. Balian and J. Zinn-Justin, North-llolland, Amsterdam) and C.H.","Llewellyn Smith (Topics in Quantum Chronomdynamics, in Quantum Flavordynamics, Quantum Chromodynamics and Unified Theories (ed., K. T. Mahanthappa and J. Randa, Plenum Press, New York, 1980) in the recent books by C. Itzykson and J. Zuber (Introduction to Quantum Field Theory, McGraw-Hill, New York, 1980) and T. P. Cheng and L.F. Li Gauge Theories of Elementary Particle Physics, (Clarendon Press, Oxford, 1984).","In the interest of time, I have had to eliminate all mention of inclusive, inelastic scattering of neutrinos and other related processes such as electron-positron annihilatiqn and Drell-Yan production of lepton pairs.","These too can be found treated in detail elsewhere in the literature.","Finally I would like to thank Frank Close, Chris Llewellyn-Smith, Dick Roberts and especially Graham Ross for collaboration on many of the ideas presented here.","I would also like to thank Gerry Garvey and Mikkel Johnson for making it possible for me to attend the school, and Roger Gilson for preparing the manuscript.","The quark description of hadrons is now universally accepted, but its implications for nuclear physics are anything but clear.","Nuclear structure and low energy nuclear reactions are well-described by a variety of semi-phenomenological theories which leave little room for insight from thinking about quark and gluon dynamics.","The reason is obvious: the energies involved in most nuclear phenomena are so low compared to the natural scale of quark dynamics that quark and gluon degrees-of freedom are effectively frozen into nucleon and meson “quasiparticles\".","To support this, consider first that the string tension in QCD is $\\sim $ 1 GeV /fm, so it costs $\\sim $ 1 GeV to separate a quark from two others (or from an antiquark) by an additional fermi beyond their equilibrium separation; and second, that the N – $\\Delta $ mass difference is 300 MeV, so it costs $\\sim $ 300 MeV to flip a quark's spin relative to the two others which together with it form a nucleon.","Both energies are much larger than typical nuclear energy scales.","Thus, $\\Delta $ s are relegated to a relatively minor role in the description of nuclei, and quarks (as distinct from nucleons and mesons) are even less important.","It is not even clear, as a matter of principle, how to establish the necessity of a fundamentally “quarkic\" description of nuclei.","Because color is confined, it is almost always possible to find an entirely equivalent hadronic description of some quark-dynamical process.","For example, there has been much interest in isolating so-called “hidden color\" components in the deuteron wave function.","These are of the form [(q$^3$ )$^8$ –(q$^3$ )$^8$ ]$^1$ : although the entire six quark system is a color singlet, each group of three quarks is coupled to a color octet.","Such admixtures have been invoked, for example, to resolve discrepancies in models of deuteron photodisintegration.","It is trivial to show, however, that any such configuration can be rewritten, after a change of coupling transformation, as a sum of configurations in which each set of three quarks is coupled to a color singlet, i.e., an ordinary baryon.","[The proof is merely to note that $\\epsilon _{abc}$ is the only covariant, invariant tensor in SU(3).]","So any hidden color state can be equally well represented as a sum over conventional baryon-baryon configurations, and any phenomenon ascribed to hidden color could be explained equivalently by a sufficiently clever theorist who had never heard of quarks and color.","The quark theorist might triumph in the end if their description were simpler and more economical: one “hidden color\" state might do as well as some complex superposition of baryon resonances.","But this is not the “smoking gun\" enthusiasts are seeking.","Similar observations could have been made about low energy hadron physics twenty years ago, and one therefore wonders how the quark description of hadrons was established in the first place, which brings me to the subject of these lectures.","Of course, quarks were recognized as a convenient bookkeeping device as early as 1964, but the need for a quark dynamics for hadrons was not compelling until a set of experiments was performed at SLAC in the late 1960s.","High energy electrons were scattered inelastically from nucleons in close analogy to the $\\alpha $ -particle scattering experiments performed by Rutherford in the early 1900s.","In both cases, the experimenters were surprised to find that the scattering cross section remained large even at high momentum transfer indicating the presence of point-like scattering centers within the target.","Further studies with electrons and then with neutrinos established the spin, charge and baryon number of the point-like objects within hadrons and showed they were quarks.","It was also recognized that these experiments not only detect quarks but directly measure their distribution within the target baryon.","During the early 1970s, it became clear that only a non-Abelian gauge field theory, quantum chromodynamics (QCD), could explain the qualitative features of quark dynamics.","One of its triumphs was the quantitative prediction of the details of inelastic electron scattering in the very high momentum transfer, or deep inelastic limit.","Despite the complexity of QCD, the simple Rutherford picture remains valid: the scattering electron measures the quark distribution in the target.","These lectures explore the use of deep inelastic electron scattering as a probe of the quark distribution in nuclei.","For more than a decade it apparently was thought that the quark distribution in a nucleus was simply given by the quark distribution in so many neutrons and protons, corrected for the fact that they are in motion within the nucleus (Fermi motion).","Little attention was paid to quark distributions in nuclei.","It came as a surprise to particle and nuclear theorists alike when the first careful comparison of a nuclear target (iron) with a nucleon (actually a deuteron) presented at the 1982 Paris Conference by the European Muon Collaboration (EMC), showed a $\\sim $ 15% difference in a region where Fermi motion effects are thought to be negligible.","The “EMC effect\" as it is known, was immediately confirmed by experiments at SLAC and elsewhere.","Theorists quickly presented a variety of explanations of the EMC effect.","Now, three years later, there are several well-developed schools of thought, much controversy, and only a little agreement among partisans about the origin of the effect.","Most explanations are based on a “convolution model\" formalism in which one supposes the nucleus contains, in addition to nucleons, some small admixture of more exotic constituents (pions, multiquark bags, $\\Delta $ s, $\\alpha $ clusters ... ), and then adds up their quark distributions.","Unfortunately, this approach has a hitch: the assumption that quark distributions in constituents add incoherently is not justified and in many cases probably wrong.","Another approach, known as “rescaling\" suggested by the scale transformation properties of QCD, avoids the questionable assumption of convolution models by dealing directly and solely with the quark distribution of the nucleus.","The result is simple and striking: the EMC effect shows that the typical length scale associated with quark propagation in the nuclear ground state is longer than the corresponding length scale in the nucleon.","For a particle physicist, one of the most interesting features of this approach is its implications for QCD at finite density.","Nuclei provide samples of quark/nuclear matter at a variety of mean densities (as $A$ increases, the surface to volume ratio goes to zero so the mean density grows).The $A$ dependence of the EMC effect closely reflects the variations in mean nuclear density, indicating that the quark length scale in quark/nuclear matter grows with density.","In these lectures, I have tried to avoid detailed analysis of models of the EMC effect, although critics of the “rescaling\" approach, to which I am devoted, will point out that I present it in considerable detail.","In fact, much of the rescaling analysis is model independent and essential for a modern education in the subject.","In §1, I introduce the kinematic variables in coordinate and momentum space.","The whole discussion is set in the laboratory or target rest frame.","Certain general tools like dispersion relations are reviewed and summarized there.","In §2, I present the parton model from a somewhat unfamiliar point-of-view, one which experts will recognize as imitating the more formal operator product expansion analysis used in QCD.","The reason for this approach is to keep as close to coordinate space as possible since considerable insight into the parton model and the EMC effect comes from an easy fluency between coordinate and momentum space.","In §3, I summarize the data, as far as I know it, and use the parton model to interpret it.","§4, is dedicated to convolution models.","Since so much work has been based on these models, I have tried to present them in detail, illustrated by the case in which the constituents are the nucleons themselves (Fermi motion), and let the readers judge their utility for themselves.","In §5, I return to fundamentals.","QCD changed our understanding of inelastic scattering and corrected and extended the parton model.","I have tried to introduce the QCD analysis with as little excess formalism as possible, though for a real working knowledge the reader will have to learn more about gauge field theory and the renormalization group elsewhere.","Finally, in §6, I use the powerful methods developed in §5 to help give a new way of looking at the EMC effect, and draw some surprisingly simple conclusions.","We are interested in the process $eA \\rightarrow e^\\prime X$ where $A$ is a nucleus, the proton and neutron being important special cases, and $X$ is an unobserved hadronic final state.","The electron and nucleus are assumed unpolarized, although polarization dependent effects can be handled in the same fashion.","The process is known as inelastic electron scattering or inclusive electroproduction.","To lowest order in $\\alpha $ , the process is described [3] by one photon exchange (Fig.","REF ): $A \\propto \\bar{u}(k^\\prime )\\gamma ^\\mu u(k)\\frac{1}{q^2}\\mathinner {\\langle {X|J_\\mu (0)|p}\\rangle } \\ , \\qquad \\mathrm {(1.1)}$ where $J_\\mu (0)$ is the hadronic electromagnetic current operator.","The differential cross-section for scattering in which $X$ is not observed is proportional to $\\Sigma _X |A|^2 (2 \\pi )^4 \\delta ^4(p+q-p_X)$ , or $d \\sigma \\propto l^{\\mu \\nu } W_{\\mu \\nu } \\ , \\qquad \\mathrm {(1.2)}$ where $l^{\\mu \\nu } = \\frac{1}{2} Tr {k}^\\prime \\gamma ^\\mu {k} \\gamma ^\\nu = 2({k^\\prime }^\\mu k^\\nu + {k^\\prime }^\\nu k^\\mu - g^{\\mu \\nu } k \\cdot k^\\prime ) \\qquad \\mathrm {(1.3)}$ (we ignore the electron mass), and $W_{\\mu \\nu } \\equiv \\frac{1}{4 \\pi } \\Sigma _X (2 \\pi )^4 \\delta ^4(p+q-p_X) \\mathinner {\\langle {p|J_\\mu (0)|X}\\rangle } \\mathinner {\\langle {X|J_\\nu (0)|p}\\rangle } \\ .","\\qquad \\mathrm {(1.4)}$ Figure: Inclusive inelastic electron scattering via one photon exchange.","EE, E ' E^\\prime and θ\\theta are defined in the target rest frame.$W_{\\mu \\nu }$ contains all reference to hadronic states.","It is represented graphically in Fig.","REF .","Figure: W μν W_{\\mu \\nu } which determines the cross section for electron scattering and also, the imaginary part of forward, virtual Compton scattering.Eq.","(1.4) may be simplified by replacing $(2 \\pi )^4 \\delta ^4(p+q-p_X) \\equiv \\int d^4 \\xi \\ e^{i(p+q-p_X) \\cdot \\xi } \\ , \\qquad \\mathrm {(1.5)}$ translating $J_\\mu (0)$ to the space-time point $\\xi $ , and using completeness ($\\Sigma _X|X><X| = 1$ ): $W_{\\mu \\nu } = \\frac{1}{4 \\pi } \\int d^4 \\xi \\ e^{iq \\cdot \\xi } \\mathinner {\\langle {p|J_\\mu (\\xi )J_\\nu (0)|p}\\rangle }_c \\ .","\\qquad \\mathrm {(1.6)}$ The subscript $c$ on the matrix element denotes “connected\" and ensures that vacuum-to-vacuum transitions of the form $\\mathinner {\\langle {0|J_\\mu (\\xi ) J_\\nu (0)|0}\\rangle }\\mathinner {\\langle {p|p}\\rangle }$ are excluded.","The current product in Eq.","(1.6) can be replaced by a commutator $W_{\\mu \\nu } =\\frac{1}{4 \\pi } \\int d^4 \\xi \\ e^{iq \\cdot \\xi } \\mathinner {\\langle {p|[J_\\mu (\\xi ),J_\\nu (0)]|p}\\rangle }_c\\qquad \\mathrm {(1.7)}$ because the term we have subtracted vanishes for stable targets if $q^0 > 0$ .","In quantum field theory, it is most convenient to deal with time-ordered products of operators.","Thus, $T_{\\mu \\nu } =i \\int d^4 \\xi \\ e^{iq \\cdot \\xi } \\mathinner {\\langle {p|T(J_\\mu (\\xi )J_\\nu (0))|p}\\rangle }_c\\qquad \\mathrm {(1.8)}$ is the amplitude [4] for forward scattering of a virtual photon of momentum $q$ from a hadronic target of momentum $p$ (Fig.","REF ).","Figure: Forward virtual Compton scattering.It is easily seen that $W_{\\mu \\nu }$ is the imaginary part of $T_{\\mu \\nu }$ (when $q^0$ is taken to have a small positive imaginary part) $W_{\\mu \\nu } = \\frac{1}{2 \\pi } IT_{\\mu \\nu } \\ .","\\qquad \\mathrm {(1.9)}$ So, inclusive electroproduction is intimately related to virtual Compton scattering.","$W_{\\mu \\nu }$ and $T_{\\mu \\nu }$ can be decomposed in terms of a pair of Lorentz invariant “structure functions\" $W_{1,2}$ ($T_{1,2}$ ): $W_{\\mu \\nu } &= - \\left(g_{\\mu \\nu } - \\frac{q_\\mu q_\\nu }{q^2} \\right) W_1(q^2,\\nu ) \\nonumber \\\\&+ \\frac{1}{M^2_T} \\left( p_\\mu - \\frac{M_T \\nu }{q^2} q_\\mu \\right)\\left( p_\\nu - \\frac{M_T \\nu }{q^2} q_\\nu \\right) W_2(q^2,\\nu ) $ and likewise for $T_{\\mu \\nu }$ .","$W_{1,2}$ are functions of the of the Lorentz invariants $q^2$ and $p \\cdot q = M_T \\nu $ ($q^2 = -4EE^\\prime \\sin ^2 \\frac{\\theta }{2}$ and $\\nu = E-E^\\prime $ in the target rest frame) where $M_T$ is the target mass.","No other terms are allowed by Lorentz invariance, current conservation ($q^\\mu W_{\\mu \\nu } = W_{\\mu \\nu } q^\\nu = 0$ ) and parity.","The structure functions depend on the squared four momentum transfer $q^2 \\equiv - Q^2 = \\nu ^2 - {\\cal {Q}}^2$ which is spacelike (so $Q^2 > 0$ ) and the energy transfer in the laboratory, $\\nu = q^0$ , which is positive.","The squared mass of the final hadronic state, $X$ , is $(p + q)^2 = M_T^2 + 2M_T \\nu - Q^2 \\equiv W^2$ and is greater than $M_T^2$ .","Thus, the “scaling\" variable, $x_T \\equiv Q^2 /2M_T \\nu $ must be between 0 and 1.","Often experimentalists and theorists prefer to use a uniform scaling variable $x = x_N = Q^2 /2M\\nu $ ($M \\equiv M_N$ is the nucleon mass) for all targets.","The reason for this is that the structure functions of different nuclei look to first order like the sum of structure functions for $A$ independent nucleons.","They are therefore very small for $x > 1$ regardless of $A$ .","It is easy to get confused between the scaling variable intrinsic to a target of mass $M_T$ , $x_T \\equiv Q^2 /2M_T \\nu $ , which is bounded between 0 and 1, and the uniform scaling variable $x =x_N = Q^2/2M$ , which ranges from zero to $M_T/M$ .","Note that $M_T/M \\unknown.","A$ for a nucleus of mass number $A$ , so the distinction between the two variables is quite significant for nuclear targets.","Bjorken suggested [5] that in the limit of large $Q^2$ at fixed $x_T$ (now known as the Bjorken or “deep\" inelastic limit) $W_1$ and $\\frac{\\nu }{M_T} W_2$ should become functions of $x_T$ alone: $\\lim _{Bj} W_1 (q^2,\\nu ) &= F_1(x_T)\\nonumber \\\\\\lim _{Bj} \\frac{\\nu }{M_T} W_2(q^2,\\nu ) &= F_2(x_T) \\ ,$ which is approximately verified by experiment.","This phenomenon is known as “Bjorken scaling\" or just “scaling\" for short.","The kinematic range of inclusive scattering is shown in Fig.","REF .","Note that the $Q^2 \\rightarrow 0$ limit gives photoproduction.","Figure: Kinematic variables for inelastic electron scattering.The virtual forward Compton amplitude, $T_{\\mu \\nu }$ , possesses simple properties when regarded as an analytic function of $\\nu $ at fixed $q^2$ .","These are special cases of the general results of dispersion theory, most of which are forgotten [6].","Since I will need these properties in the subsequent chapters, I will review them here.","For pedagogical simplicity I will ignore spin and analyze a hypothetical virtual “Compton\" amplitude $T( q^2 , \\nu )$ for a scalar “photon\" scattering from a proton or neutron defined by $T(q^2,\\nu ) \\equiv i \\int d^4 \\xi \\ e^{i q \\cdot \\xi } \\mathinner {\\langle {p|T(J(\\xi )J(0))|p}\\rangle } \\ .","\\qquad \\mathrm {(1.12)}$ (The generalization to other targets is straightforward.)","The generalization to the physically interesting case of $T_{\\mu \\nu }$ will be quoted at the end.","$T(q^2 , \\nu )$ is a real analytic function of $\\nu $ at fixed $q^2$ , $T(q^2,\\nu ^\\star ) = T^\\star (q^2,\\nu )\\nonumber \\ ,$ and is crossing symmetric $T(q^2,\\nu ) = T(q^2,-\\nu ) \\nonumber \\ .$ The fundamental assumption of dispersion theory is that scattering amplitudes are analytic except at values of the kinematic variables which allow intermediate states to be physical (i.e., on shell).","This can be proven to all orders of perturbation theory [7], but must be regarded as an assumption in QCD where the quanta of the perturbation theory (quarks and gluons) are not physical states.","When $\\nu \\ge - q^2/2M$ , the virtual photon-target system can form a physical hadronic intermediate state so $T(q^2,\\nu )$ has a cut along the positive real-$\\nu $ axis.","More precisely, $T(q^2,\\nu )$ has a pole at $\\nu = -q^2/2M$ corresponding to elastic scattering ($ep \\rightarrow e^\\prime p^\\prime $ ) and a cut beginning at pion production threshold.","$T(q^2,\\nu )$ also has a cut on the negative real axis corresponding to the “crossed\" process $p \\rightarrow \\gamma + x$ which is physically allowed when $\\nu \\le -|q|^2/2M$ .","The discontinuity across the right hand cut is ${\\rm disc}\\ T(q^2,\\nu ) = 2i \\,IT(q^2,\\nu ) = 4\\pi i W(q^2,\\nu ) \\ .","\\qquad \\mathrm {(1.15)}$ These analytic properties are summarized in Fig.","REF .","Figure: The complex ν\\nu -plane.","The contour shown figures in the derivation of dispersion relations.Using Cauchy's theorem on the contour shown in Fig.","REF , it is possible to derive a “dispersion relation\" for $T(q^2,\\nu )$ , $T(q^2,\\nu ) = 4 \\int _{-q^2/2M}^{\\infty } \\frac{d \\nu ^\\prime \\nu ^\\prime }{{\\nu ^\\prime }^2 - \\nu ^2} W(q^2,\\nu ^\\prime ) \\ .","\\qquad \\mathrm {(1.16)}$ Here, $\\nu $ is complex ($\\nu ^\\prime $ is real) and the physical Compton amplitude is obtained by letting it approach the positive real axis from above $\\nu \\rightarrow \\nu _R + i \\epsilon $ .","If $W(q^2,\\nu ^\\prime )$ falls too slowly as $ \\nu ^\\prime \\rightarrow \\infty $ then the integral may not converge.","In that case one can derive a weaker, “subtracted\" dispersion relation.","Formally, take Eq.","(1.16) for two different choices of $\\nu $ and subtract, e.g.","$T(q^2,\\nu ) = T(q^2,0) +4 \\nu \\int _{-q^2/2M}^{\\infty } \\frac{d \\nu ^\\prime }{\\nu ^\\prime ({\\nu ^\\prime }^2 - \\nu ^2)} W(q^2,\\nu ^\\prime ) \\ .","\\qquad \\mathrm {(1.17)}$ The integral is better behaved at large $\\nu ^\\prime $ .","This procedure can be continued as far as necessary and works as long as $W(q^2,\\nu )$ is polynomial bounded as $\\nu \\rightarrow \\infty $ , which we will assume (see below).","Modulo subtractions, the real part of $T$ can be calculated if $W$ is known, thus dispersion relations can be (and have been) verified experimentally.","It is convenient to change variables to $\\omega \\equiv -2M \\nu /q^2$ in Eq.","(1.16), $T(q^2,\\omega ) = 4 \\int _1^\\infty \\frac{\\omega ^\\prime d \\omega ^\\prime }{{\\omega ^\\prime }^2 - \\omega ^2} W(q^2,\\omega ^\\prime ) \\ .","\\qquad \\mathrm {(1.18)}$ Note that $T(q^2,\\omega )$ is analytic in the circle of radius 1 about $\\omega = 0$ and may therefore be expanded in a Taylor series for $|\\omega | < 1$ : $T(q^2,\\omega ) = 4 \\sum _{n\\,\\rm even} M^n(q^2) \\omega ^n \\ .","\\qquad \\mathrm {(1.19)}$ with $M^n(q^2) &= \\int _1^\\infty d \\omega ^\\prime {\\omega ^\\prime }^{-n-1} W(q^2,\\omega ^\\prime ) \\nonumber \\\\&= \\int _0^1 dx\\ x^{n-1} W(q^2,x) \\ .$ The physically interesting case of $T_{\\mu \\nu }$ is summarized by dispersion relations for the two invariant amplitudes $T_{1,2}(q^2,\\omega )$ .","$T_1(q^2,\\omega )$ requires subtraction $T_1(q^2,\\omega ) &= T_1(q^2,0) + 4 \\omega ^2 \\int d \\omega ^\\prime \\frac{W_1(q^2,\\omega ^\\prime )}{\\omega ^\\prime ({\\omega ^\\prime }^2-\\omega ^2)} \\nonumber \\\\\\frac{\\nu T_2}{M_T}(q^2,\\omega )&= 4 \\omega \\int _1^\\infty d \\omega ^\\prime \\frac{\\frac{\\nu W_2}{M}(q^2,\\omega ^\\prime )}{{\\omega ^\\prime }^2 - \\omega ^2} \\ .$ Most introductory treatments of deep inelastic lepton scattering are formulated in the “infinite momentum frame\" where the target is boosted to some arbitrarily large momentum $P_\\infty $ .","I find this approach both unnecessary and misleading and prefer instead to work in the target rest frame where I suppose nuclear physicists also feel at home.","Of course, the physics is frame independent, at least if one is careful enough [8].","In the rest frame of a target with mass $M_T$ , the Bjorken limit takes on a particularly simple form.","We choose the negative $z$ -axis to lie along the virtual photon direction: $q = (\\nu ,0,0,-\\sqrt{\\nu ^2+Q^2}) \\ .","\\qquad \\mathrm {(1.22)}$ As $Q^2 \\rightarrow \\infty $ with $x$ fixed, $Q^2/\\nu ^2 \\rightarrow 0$ so $q \\rightarrow (\\nu ,0,0,-\\nu -M_Tx_T) \\ .","\\qquad \\mathrm {(1.23)}$ The significance of this form is most transparent if we introduce “light-cone\" coordinates, $q^\\pm = \\frac{1}{\\sqrt{2}}(q^0 \\pm q^3) \\ .","\\qquad \\mathrm {(1.24)}$ [Note: $a \\cdot b = a^+b^- + a^-b^+ - {\\bf a}_\\perp \\cdot {\\bf b}_\\perp $ , $a^2 = 2a^+a^- - {\\bf a}^2_\\perp $ , so $g^{+-} = g^{-+} = 1$ and $a_\\mp =a^\\pm $ .","To avoid confusion, I will stick to contravariant indices.]","In the Bjorken limit $q^- \\rightarrow \\infty $ but $q^+ \\rightarrow -M_Tx_T/\\sqrt{2}$ , i.e., $q^+$ remains finite.","Note $M_Tx_T = Mx = Q^2/2\\nu $ , so the limiting value of $q^+$ is independent of $M_T$ .","We will frequently discuss the important distant scales which contribute to electroproduction.","The distance referred to is the space-time separation, $\\xi _\\lambda $ , between the points at which the currents $J_\\nu $ and $J_\\nu $ act.","Note that $\\xi _\\lambda $ is only defined in the Compton amplitude, $T_{\\mu \\nu }$ , or its imaginary part $W_{\\mu \\nu }$ , not in the electroproduction amplitude $A$ itself (Eq.","1.1).","Since $\\xi $ and $q$ appear as conjugate variables in the definition of $W_{\\mu \\nu }$ and $q \\cdot \\xi \\equiv q^+\\xi ^- + q^- \\xi ^+$ , $q^- \\rightarrow \\infty $ forces $\\xi ^+ \\rightarrow 0$ , but $q^+=-Mx/\\sqrt{2}$ requires only $|\\xi ^-|\\lesssim \\sqrt{2}/Mx$ .","The first of these relations, $\\xi ^+ \\rightarrow 0$ , follows from general theorems on Fourier transforms [9].","Let $\\tilde{f}(q^-) = \\int d \\xi ^+ e^{i q^- \\cdot \\xi ^+} f(\\xi ^+)\\ .","\\qquad \\mathrm {(1.25)}$ If ${f}(\\xi ^+)$ is smooth (infinitely differentiable everywhere) and well-behaved as $|\\xi ^+| \\rightarrow \\infty $ , then $\\tilde{f}(q^-)$ vanishes faster than any power of $q^-$ as $q^- \\rightarrow \\infty $ .","But, if $f(\\xi ^+)$ has singularities, then the $q^- \\rightarrow \\infty $ limit is dominated by the behavior of $f(\\xi ^+)$ near the singularities.","Suppose, for example, $f(\\xi ^+) =0$ for $\\xi ^+ \\le 0$ , then as $q^- \\rightarrow \\infty $ $\\tilde{f}(q^-) \\sim \\frac{i}{q^-} f(0) - (\\frac{i}{q^-})^2 f^\\prime (0) + ...\\ , \\qquad \\mathrm {(1.26)}$ as easily obtained from integration by parts.","The integrand in Eq.","(1.7) defining $W_{\\mu \\nu }$ is singular at $\\xi ^+ =0$ since it vanishes when $\\xi ^2 <0$ , so the $q^- \\rightarrow \\infty $ limit is dominated by $\\xi ^+ \\approx 0$ .","The second relation, $|\\xi ^-|\\lesssim \\sqrt{2}/Mx$ , is more subtle.","After all the dust settles (in $§$ 2) the structure functions will be given by Fourier transforms in $q^+=-Mx/\\sqrt{2}$ of a smooth function of $\\xi ^-$ .","The Fourier transform, however, is only conditionally convergent at large $\\xi ^-$ .","It diverges like $1/\\sqrt{x}$ or $1/x$ (for “valence\" or “ocean\" quarks, see $§2$ as $x \\rightarrow 0$ , so the oscillation in the exponential acts as a large $\\xi ^-$ cutoff.","The larger $\\sqrt{2}/Mx$ , the larger range of $\\xi ^-$ contributes to the structure function.","For a more complete discussion with examples see the Heidelberg talk by Llewellyn Smith in [10].","Since the commutator in Eq.","(1.7) is causal, $\\xi ^2 = 2\\xi ^+\\xi ^- - {\\xi }^2_\\perp $ is positive.","Hence, $\\xi ^+ \\rightarrow 0$ with $\\xi ^-$ finite requires $\\xi _\\perp \\rightarrow 0$ .","All components of $\\xi ^\\lambda $ except $\\xi ^-$ vanish in the Bjorken limit.","Thus, deep inelastic scattering is not, as is often incorrectly remarked, a short distance ($\\xi _\\lambda \\rightarrow 0$ ) phenomenon; it is instead a light-cone ($\\xi ^2 \\rightarrow 0$ ) dominated process.","This distinction is crucial to understanding the dynamics and yields considerable insight into nuclear effects in inclusive electron scattering.","Together $\\xi ^+ \\rightarrow 0$ and $|\\xi ^-| \\lesssim \\sqrt{2}/Mx$ imply $|\\xi ^0|\\lesssim 1/Mx$ and $|\\xi ^3| \\lesssim 1/Mx$ .","$W_{\\mu \\nu }$ measures a current-current correlation function in the target ground state (c.f.","Eq.","(1.7)).","In the Bjorken limit, the correlation probed becomes light-like, but may extend to very large spatial distances (and times) in the small $x$ limit.","Note that the ranges of $\\xi ^0$ and $\\xi ^3$ probed by the correlated pair of currents in the target rest frame are independent of the target mass because $1/Mx=2\\nu /Q^2$ .","The relation between $x$ and $|\\xi ^3|$ will be crucial to our analysis of nuclear effects in leptoproduction.","It is often convenient to use different structure functions to describe $W_{\\mu \\nu }$ .","A particularly important pair are $W_L$ and $W_T$ defined by $W_T &= W_1 \\nonumber \\\\W_L&= \\left( 1 + \\frac{\\nu ^2}{Q^2} \\right) W_2 - W_1 \\ .$ $W_T$ and $W_L$ arise when one considers the polarization of the virtual photon, $\\epsilon _\\mu $ .","The leptonic current produces a flux of virtual photons which may be transverse, $\\epsilon ^\\mu _{T_1} &= (0, 0, 1, 0) \\nonumber \\\\\\epsilon ^\\mu _{T_2}&= (0, 1, 0, 0) \\ ,$ or longitudinal $\\epsilon ^\\mu _L = \\frac{1}{\\sqrt{Q^2}}(\\sqrt{\\nu ^2+Q^2}, 0, 0, -\\nu )\\ , \\qquad \\mathrm {(1.29)}$ in the target rest frame in which $q^\\mu $ is given by Eq.","(1.21).","Note $\\epsilon \\cdot q =0$ and $\\epsilon ^2_{T_j} = - \\epsilon ^2_L = -1$ .","$W_T$ and $W_L$ are the components of $W_{\\mu \\nu }$ which couple to $\\epsilon _T$ and $\\epsilon _L$ , respectively: $W_T &\\equiv \\epsilon ^\\mu _{T_j} W_{\\mu \\nu } \\epsilon ^\\nu _{T_j} \\nonumber \\\\W_L&\\equiv \\epsilon ^\\mu _{L} W_{\\mu \\nu } \\epsilon ^\\nu _{L} \\ .$ They are proportional to the total cross sections for absorption of a transverse or longitudinal polarized virtual photon, respectively, $\\sigma _{T,L}(q^2,\\nu ) = \\frac{4 \\pi ^2 \\alpha }{k} W_{T,L}(q^2,\\nu ) \\qquad \\mathrm {(1.31)}$ where $k= (W^2-M^2_T)/2M_T$ .","$\\sigma _L$ and $\\sigma _T$ are positive, so $W_1 \\ge 0$ and $(1+\\nu ^2/Q^2)W_2 \\ge W_1$ .","As $q^2 \\rightarrow 0$ , $\\sigma _L$ vanishes and $\\sigma _T$ approaches the photoproduction cross section for real photons.","Experimentalists measure $W_1$ and $W_2$ by comparing cross sections at fixed $q^2$ and $\\nu $ , but different values of $E$ , $E^\\prime $ and $\\theta $ : $\\frac{d^2 \\sigma }{dE^\\prime d \\Omega } = \\frac{4 \\alpha ^2{E^\\prime }^2 \\cos ^2 \\theta /2}{q^4} \\left[W_2(q^2,\\nu )+2W_1(q^2,\\nu ) \\tan ^2 \\theta /2 \\right]\\ .","\\qquad \\mathrm {(1.32)}$ In practice, it is convenient to write the differential cross section in terms of $\\sigma _L$ , $\\sigma _T$ and a parameter $\\epsilon $ : $\\epsilon ^{-1} \\equiv 1+2 \\left( 1+ \\frac{\\nu ^2}{Q^2} \\right) \\tan ^2 \\theta /2\\ , \\qquad \\mathrm {(1.33)}$ $\\frac{d^2 \\sigma }{dE^\\prime d \\Omega } = \\frac{\\alpha }{4 \\pi ^2 Q^2} \\frac{kE^\\prime }{E} \\left( \\frac{2}{1- \\epsilon } \\right) [\\sigma _T(q^2,\\nu ) + \\epsilon \\sigma _L(q^2,\\nu )]\\ .","\\qquad \\mathrm {(1.34)}$ Apart from kinematic factors, $d^2\\sigma $ is a linear function of $\\epsilon $ .","A linear fit to the data yields $\\sigma _L + \\sigma _T \\propto \\nu W_2$ as the intercept at $\\epsilon = 1$ , and $R \\equiv \\sigma _L/\\sigma _T$ as the slope.","$R$ is difficult to measure.","Data sets from different spectrometer settings and different beam energies must be combined, which introduces systematic uncertainties.","At high beam energies $\\epsilon \\approx 1$ so the experiments are not sensitive to $R$ .","Anyone who studies inelastic electron scattering should begin with the parton model of Bjorken and Feynman.","There are many fine sources from which to learn it [8]; one or more should be studied in conjunction with these lectures.","I, too, will describe the parton model, but quickly and maintaining as much contact with coordinate space as possible.","If you have never seen the parton model before, the derivation presented here will appear difficult and rather formal.","The parton model is often misused.","To learn how to not misuse it one must approach the model who more formally, e.g, via the operator product expansion (OPE).","Anyone who intends to do research in this field is strongly advised to study the OPE and the renormalization group in QCD beforehand [11], but I will only touch briefly on them.","It is conventional to motivate the parton model by arguing that in some sense interactions can be ignored near the light cone.","There is no realistic theory in which this is true, though in QCD it is approximately true.","In $§5$ , we will develop a more precise language for such matters.","Here, I will only state the assumptions which lead to the model.","The first assumption of the parton model is that the current $J_\\mu $ couples to quarks (as opposed to fundamental scalars, etc.).","Then the contributions to the forward virtual Compton amplitude can be classified by the flow and interactions of quark lines.","The second assumption is that at large values of $Q^2$ the currents, but not the states, may be treated as in free field theory.","Thus, final state interactions (Fig.","REF (b)) and vertex corrections (Fig.","REF (c)) are ignored.","Figure: Some contributions to forward virtual Compton scattering in QCD: (a) The parton model diagram; (b) final state interactions; (c) vertex corrections; (d) interference.This leaves the one and two particle contributions shown in Figs.","REF (a) and (d), respectively.","Methods similar to those we shall apply to Fig.","REF (a) show that the contribution of Fig.","REF (d) vanish faster by a power of $Q^2$ , so I will ignore them henceforth.","This leaves only Fig.","REF (a): the elastic and incoherent scattering of each quark in the target, i.e., “quasielastic\" scattering.","It is important to keep in mind the place of the parton model in QCD.","It is valid modulo logarithms: quantities which scale — that is, they become functions of $x$ -alone in the parton model — will be modulated by powers of $\\ln Q^2$ when QCD interactions are included.","Quantities which vanish like a power of $Q^2$ in the parton model may vanish only like a power of $\\ln Q^2$ in QCD.","Figure: Parton model for W μν W_{\\mu \\nu }.The structure function, $W_{\\mu \\nu }$ , in the parton model is obtained by placing the intermediate state in Fig.","REF (a) on-shell.","The struck quark and the remnants of the target appear separately as physical intermediate states as shown in Fig.","REF .","This, of course, is wrong $-$ quarks are confined by non-perturbative effects in QCD.","It is assumed that the processes which neutralize quark quantum numbers do not affect the dominant terms at large $Q^2$ .","The justification for this is that those non-perturbative, confining effects in QCD which have been studied, while strong, appear to vanish rapidly with $Q^2$ .","Until the non-perturbative aspects of QCD are better understood this will remain a major, though reasonable, assumption.","The current $J_\\mu (\\xi )$ reduces to $\\overline{\\psi }(\\xi ) {\\cal {Q}} \\gamma _\\mu \\psi (\\xi )$ in a free quark model, where $\\cal {Q}$ is the quark charge matrix: ${\\cal {Q}} =$ diag$(2/3,-1/3,-1/3,...)$ for $u, d, s,...$ The current commutator in $W_{\\mu \\nu }$ reduces to [12] $[J_\\mu (\\xi ),J_\\nu (0)] = \\overline{\\psi }(\\xi ) \\gamma _\\mu S(\\xi ) \\gamma _\\nu {\\cal {Q}}^2 \\psi (0) - \\overline{\\psi }(0) \\gamma _\\nu S(-\\xi ) \\gamma _\\mu {\\cal {Q}}^2 \\psi (\\xi ) \\qquad \\mathrm {(2.1)}$ where $S(\\xi )$ is the anticommutator function [3] $S(\\xi ) = \\lbrace \\psi (\\xi ),\\overline{\\psi }(0) \\rbrace = -\\gamma ^\\rho \\partial /\\partial \\xi ^\\rho \\Delta (\\xi ) = S(-\\xi ) \\qquad \\mathrm {(2.2)}$ and $\\Delta (\\xi ) = \\frac{1}{2 \\pi } \\delta (\\xi ^2) \\epsilon (\\xi ^0) + .... \\qquad \\mathrm {(2.3)}$ The additional terms in $\\Delta (\\xi )$ vanish for zero quark mass.","Since quark masses generate only ${\\cal O}(m^2/Q^2$ ) corrections in the Bjorken limit, we ignore them from now on.","The product of operators in Eq.","(2.1) is singular when $\\xi _\\lambda \\rightarrow 0$ , but the singularity is a $C$ -number $-$ namely the term removed by normal ordering $-$ so it doesn't contribute to a connected matrix element.","Thus, $\\mathinner {\\langle {p|\\overline{\\psi }(\\xi )\\psi (0)|p}\\rangle }_c &= \\mathinner {\\langle {p|:\\overline{\\psi }(\\xi )\\psi (0):|p}\\rangle } \\nonumber \\\\&= - \\mathinner {\\langle {p|\\psi (0)\\overline{\\psi }(\\xi )|p}\\rangle } $ where the second equality is a consequence of normal ordering.","The Lorentz structure of Eq.","(2.1) is simplified by the identity $\\gamma _\\mu \\gamma _\\rho \\gamma _\\nu &\\equiv S_{\\mu \\nu \\rho \\sigma } \\gamma ^\\sigma - \\epsilon _{\\mu \\nu \\rho \\sigma } \\gamma ^\\sigma \\gamma ^5 \\nonumber \\\\S_{\\mu \\nu \\rho \\sigma } &\\equiv g_{\\mu \\rho } g_{\\nu \\sigma } + g_{\\mu \\sigma } g_{\\nu \\rho } - g_{\\mu \\nu } g_{\\rho \\sigma } \\ .","$ The $\\epsilon $ -term does not contribute because it is impossible to construct a pseudovector from $\\xi _\\lambda $ and $p_\\lambda $ .","Putting together the pieces, we find $\\lim _{Bj} W_{\\mu \\nu } &= - \\frac{S_{\\mu \\nu \\rho \\sigma }}{8 \\pi ^2} \\int d^4 \\xi \\ e^{i q \\cdot \\xi } \\left[\\frac{\\partial }{\\partial \\xi _\\rho } \\delta (\\xi ^2) \\epsilon (\\xi ^0) \\right] \\nonumber \\\\&\\times \\mathinner {\\langle {p|\\overline{\\psi }(\\xi ) \\gamma ^\\sigma {\\cal {Q}}^2\\psi (0) - \\overline{\\psi }(0) \\gamma ^\\sigma {\\cal {Q}}^2 \\psi (\\xi )|p}\\rangle }_c \\ .$ The $\\lim _{Bj}$ reminds us that the parton model assumptions which went into Eq.","(2.6 ) are only (approximately) valid as $Q^2 \\rightarrow \\infty $ at fixed $x$ .","Integrating by parts and introducing light-cone coordinates in the target rest frame $\\lim _{Bj} W_{\\mu \\nu } &= \\lim _{q^- \\rightarrow \\infty } \\frac{S_{\\mu \\nu \\rho \\sigma } i q^\\rho }{8 \\pi ^2} \\int d \\xi ^+ d \\xi ^- d^2 \\xi _\\perp e^{i q^+ \\xi ^- + i q^- \\xi ^+} \\nonumber \\\\&\\times \\delta (2\\xi ^+\\xi ^- - {\\xi }^2_\\perp ) \\epsilon (\\xi ^+ + \\xi ^-) \\nonumber \\\\&\\times \\mathinner {\\langle {p|\\overline{\\psi }(\\xi ) \\gamma ^\\sigma {\\cal {Q}}^2\\psi (0) - \\overline{\\psi }(0) \\gamma ^\\sigma {\\cal {Q}}^2 \\psi (\\xi )|p}\\rangle }_c \\ .$ We have dropped the term in which $\\partial /\\partial \\xi _\\rho $ acts on the matrix element since it generates at most a factor $p^\\rho $ or $\\xi ^\\rho \\mu ^2$ ($\\mu ^2$ is some mass characteristic of the target) both of which are negligible with respect to $q^\\rho $ in the Bjorken limit.","[Note $\\xi ^\\rho $ Fourier transforms into $q^\\rho /q^2$ .]","The form of $S_{\\mu \\nu \\rho \\sigma }$ implies that the coefficient of $-g_{\\mu \\nu }$ in $W_{\\mu \\nu }$ equals half the trace of $W_{\\mu \\nu }$ , or referring to Eqs.","(1.10) and (1.11), $F_1 = \\frac{1}{2} \\left( 3F_1 - \\frac{1}{2 x_T} F_2 \\right) \\qquad \\mathrm {(2.8)}$ which requires $F_1 = \\frac{1}{2 x_T} F_2 \\qquad \\mathrm {(2.9)}$ or $\\lim _{Bj} \\sigma _L/\\sigma _T = 0 \\qquad \\mathrm {(2.10)}$ which is the famous Callan-Gross relation and follows from the quark spin being $1/2$ .","It is well verified experimentally.","Even though $R$ is difficult to measure and still a subject of debate, all experiments agree that for $Q^2 >1$ GeV$^2$ , $R < 0.2$ .","Eq.","(2.7) can be reduced to a one-dimensional integral.","First, we use the $\\delta -$ function to perform the $\\xi ^2_\\perp $ integral leaving $F_2 &= 2 x_T \\lim _{q^- \\rightarrow \\infty } \\frac{iq^-}{8 \\pi } \\int d \\xi ^+ d \\xi ^- e^{i(q^+\\xi ^- + q^-\\xi ^+)} \\nonumber \\\\&\\times [\\theta (\\xi ^+)\\theta (\\xi ^-) - \\theta (-\\xi ^+)\\theta (-\\xi ^-)] \\nonumber \\\\&\\times \\mathinner {\\langle {p|\\overline{\\psi }(\\xi ) \\gamma ^+ {\\cal {Q}}^2 \\psi (0) - \\overline{\\psi }(0) \\gamma ^+ {\\cal {Q}}^2 \\psi (\\xi )|p}\\rangle }_c\\bigg \\vert _{{\\xi }^2_\\perp = 2 \\xi ^+ \\xi ^-} \\ ,$ then integrate by parts on $\\xi ^+$ keeping only the leading term at large $q^-$ $F_2 = \\frac{x_T}{4 \\pi } \\int d \\xi ^- e^{i q^+ \\xi ^-} \\mathinner {\\langle {p|\\overline{\\psi }(\\xi ^-) \\gamma ^+ {\\cal {Q}}^2 \\psi (0) - \\overline{\\psi }(0) \\gamma ^+ {\\cal {Q}}^2 \\psi (\\xi ^-)|p}\\rangle }_c \\bigg \\vert _{\\xi ^+ = {\\bf \\xi }_\\perp =0} \\ .\\qquad \\mathrm {(2.12)}$ So, $F_2$ is a dimensionless function of $q^+=-M_Tx_T/\\sqrt{2}$ , i.e.","$F_2 \\rightarrow F_2(x_T)$ , which is Bjorken scaling.","Before converting Eq.","(2.12) into the most familiar parton model form, we should note that $F_2(x_T)$ measures a particular quark correlation function in the target ground state.","The first term in Eq.","(2.12), for example, measures the amplitude to remove a quark from the target at some point, $\\xi _1^\\mu $ , and replace it at $\\xi _2^\\mu $ with $\\xi _2^\\mu - \\xi _1^\\mu \\equiv \\xi ^\\mu $ , $\\xi ^+ = {\\bf \\xi }_\\perp =0$ and $|\\xi ^-|\\lesssim 1/q^+$ , leaving the target in the ground state.","The shape of the structure function teaches us about a correlation function in the target ground state.","One must be careful, however, not to use one's intuition from non-relativistic quantum mechanics: this is not an equal time correlation function, but instead a light-cone correlation function.","More about these later.","To further simplify (2.12) we must study light-cone $\\gamma -$ matrices.","The matrices $P^\\pm = \\frac{1}{2} \\gamma ^\\mp \\gamma ^\\pm = \\frac{1}{2} (1 \\pm \\alpha ^3) \\qquad \\mathrm {(2.13)}$ are projection matrices: $P^+ = P^- =1$ , ${P^\\pm }^2 = P^\\pm $ , $P^\\pm P^\\mp =0$ , and if we define $\\psi _\\pm = P^\\pm \\psi \\ , \\qquad \\mathrm {(2.14)}$ then (2.12) may be written $F_2 = \\frac{x_T}{2 \\sqrt{2} \\pi } \\int d \\xi ^- e^{i q^+ \\xi ^-} \\mathinner {\\langle {p|\\psi ^\\dagger _+(\\xi ^-) {\\cal {Q}}^2 \\psi _+(0) + \\psi _+(\\xi ^-) {\\cal {Q}}^2 \\psi ^\\dagger _+(0)|p}\\rangle }_c \\bigg \\vert _{\\xi ^+ = {\\bf \\xi }_\\perp =0} \\ .\\qquad \\mathrm {(2.15)}$ where we used Eq.","(2.4) to interchange the quark fields in the second term.","If we now insert a complete set of states between quark fields, translate the $\\xi ^-$ dependence out of $\\psi _+$ or $\\psi ^\\dagger _+$ , integrate over $\\xi ^-$ and sum explicitly over quark flavors ($a = u,d,s,....$ ), we get $F^T_2(x_T) = x_T \\sum _a {\\cal {Q}}^2_a \\sum _n \\frac{1}{\\sqrt{2}} \\delta (p^++q^+-p^+_n)\\lbrace |\\mathinner {\\langle {n|\\psi _{a+}|p}\\rangle }|^2 + |\\mathinner {\\langle {n|\\psi ^\\dagger _{a+}|p}\\rangle }|^2 \\rbrace \\ .","\\qquad \\mathrm {(2.16)}$ I have added a superscript $T$ to $F_2$ to remind us that $F_2$ depends on the target.","For a target of mass $M_T$ , $q^+ = -x_TM_T/\\sqrt{2}$ where $x_T = Q^2/2M_T \\nu $ , and $p^+ = M_T/\\sqrt{2}$ (we are working in the target rest frame), so $F^T_2(x_T) = x_T \\sum _a {\\cal {Q}}^2_a (f_{a/T}(x_T) + f_{\\bar{a}/T}(x_T)) \\ , \\qquad \\mathrm {(2.17)}$ where $f_{a/T}(x_T) &= \\frac{1}{\\sqrt{2}} \\sum _n \\delta (p^+-x_Tp^+-p_n^+) |\\mathinner {\\langle {n|\\psi _{a+}|p}\\rangle }|^2 \\nonumber \\\\f_{\\bar{a}/T}(x_T) &= \\frac{1}{\\sqrt{2}} \\sum _n \\delta (p^+-x_Tp^+-p_n^+) |\\mathinner {\\langle {n|\\psi ^\\dagger _{a+}|p}\\rangle }|^2 \\ .$ This is the familiar parton model, except it is written in the target rest frame rather than the “infinite momentum frame\".","$f_{a/T}(x_T)$ is the probability (per unit $x_T$ ) to remove from the target a quark of flavor $a$ with “momentum\" (i.e.","$p^+$ ) fraction $x_T$ , leaving behind a physical state ($\\mathinner {|{n}\\rangle }$ ) with $p^+_n = (1-x_T)p^+$ .","Similarly, $f_{\\bar{a}/T}(x_T)$ is the probability (per unit $x_T$ ) to remove an antiquark with $p^+$ -fraction $x_T$ leaving behind a physical state with $p^+_n = (1-x_T)p^+$ .","$f_{a/T}(x_T)$ $f_{\\bar{a}/T}(x_T)$ are shown graphically in Figs.","REF (b) and (c).","Figure: (a) The virtual-quark hadron scattering amplitude; (b) The quark distribution function f a/T (x T )f_{a/T}(x_T) .","Note that k + k^+ is fixed but k - k^- and 𝐤 ⊥ {\\bf k}_\\perp [Notice that $p^+$ appears in the target rest frame formulation where the “infinite momentum\" $P_\\infty $ appears in the more familiar formulation.]","$f_{a/T}(x_T)$ and $f_{\\bar{a}/T}(x_T)$ obey important positivity, spectral and normalization constraints.","The state $n$ in Eq.","(2.18) is physical and must have $p^+_n>0$ , i.e., $E_n > |\\bf p_n|$ , thus $f_{a/T}(x_T) = f_{\\bar{a}/T}(x_T) =0$ , for $x_T \\ge 1$ or $x \\ge M_T/M$ ($\\approx A$ for a nuclear target).","For $0<x_T<1$ , $f_{a/T}(x_T)$ defined by Eq.","(2.18) is manifestly positive.","Next, let us consider $f_{a/T}(x_T)$ for $x_T<0$ .","Returning to Eq.","(2.15) $f_{a/T}(x_T) &= \\frac{1}{2 \\sqrt{2} \\pi } \\int d \\xi ^- e^{i q^+ \\xi ^-} \\mathinner {\\langle {p|\\psi ^\\dagger _{a+}(\\xi ^-) \\psi _{a+}(0)|p}\\rangle }_c \\bigg \\vert _{\\xi ^+ = {\\bf \\xi }_\\perp =0} \\nonumber \\\\&= - \\frac{1}{2 \\sqrt{2} \\pi } \\int d \\xi ^- e^{i q^+ \\xi ^-} \\mathinner {\\langle {p|\\psi ^\\dagger _{a+}(0) \\psi _{a+}(\\xi ^-)|p}\\rangle }_c \\bigg \\vert _{\\xi ^+ = {\\bf \\xi }_\\perp =0} $ where we've used Eq.","(2.4).","Replacing $\\xi ^-$ by $-\\xi ^-$ and translating the matrix element we find $f_{a/T}(-x_T) = - f_{\\bar{a}/T}(x_T) \\ .","\\qquad \\mathrm {(2.20)}$ Although $f_{a/T}(x_T)$ does not vanish for $x<0$ (it does vanish for $x_T<-1$ ) it is determined by $f_{\\bar{a}/T}(x_T)$ for $x>0$ .","The literature is very confused on this point: there is a lot of talk about the need to “prove\" that “no partons can go backwards in the infinite momentum frame\", i.e., $f_{a/T}(x_T) \\equiv 0$ for $x_T <0$ .","We see here that $f_{a/T}(x_T)$ is defined in such a way that measurements in the physical region $0<x_T<1$ determine it everywhere.","For a more extensive discussion of this issue, see Ref [13].","Now let us integrate $f_{a/T}(x_T)$ over all $x_T$ .","Using $x_T = -\\sqrt{2}q^+/M_T$ and Eq.","(2.19) and Eq.","(2.20) we find $\\int ^\\infty _{-\\infty } dx_T f_{a/T}(x_T) &= \\int _0^1 dx_T(f_{a/T}(x_T) - f_{{\\bar{a}}/T}(x_T)) \\nonumber \\\\&= \\frac{1}{M_T} \\mathinner {\\langle {p|\\psi ^\\dagger _{a+}(0)\\psi _{a+}(0)|p}\\rangle }_c \\nonumber \\\\&= N_{a/T} - N_{{\\bar{a}}/T} $ where the last step follows from the fact that $\\psi ^\\dagger _{a+} \\psi _{a+} = \\frac{1}{\\sqrt{2}}j^+_a$ and $j_a^\\mu $ is a conserved current whose expectation value measures the number of quarks (minus the number of antiquarks) of flavor $a$ : $\\mathinner {\\langle {p|j^\\mu _a|p}\\rangle } = 2 p^\\mu (N_{a/T} - N_{{\\bar{a}}/T}$ ).","Clearly, we may interpret $f_{a/T}(x_T)$ ($f_{\\bar{a}/T}$ ) as a probability per unit $x_T$ to find a quark (antiquark) of flavor $a$ with $k^+ = x_T p^+$ in the target T $\\frac{d P_{a/T}}{d x_T} = f_{a/T}(x_T) \\ .","\\qquad \\mathrm {(2.22)}$ From this interpretation follows a host of parton model sum rules for the structure functions.","The most important for our purpose is the “momentum sum rule\", $\\int _0^1 dx_T\\,\\,x_T (f_{a/T}(x_T) + f_{\\bar{a}/T}(x_T)) = \\epsilon _{a/T} + \\epsilon _{\\bar{a}/T} \\qquad \\mathrm {(2.23)}$ where $\\epsilon _{a/T}$ ($\\epsilon _{\\bar{a}/T}$ ) is the fraction of the target's $p^+$ carried by quarks (antiquarks) of flavor $a$ .","The derivation mimics the derivation of Eq.","(2.21) except the quark stress tensor $i \\bar{\\psi }_a \\gamma ^\\mu \\partial ^\\nu \\psi _a$ appears instead of $j^\\mu _a$ .","If hadrons contained only quarks $\\sum _a \\epsilon _{a/T}$ would be 1.","Instead, it is typically $\\sim \\frac{1}{2}$ (at large $Q^2$ ) indicating that substantial momentum and energy are carried by other, neutral quanta, namely gluons.","If two targets are related by a symmetry, their quark distributions are similarly related.","Isospin relates the neutron and proton and gives $f_{u/n}(x) &= f_{d/p}(x) \\nonumber \\\\f_{d/n}(x) &= f_{u/p}(x) \\nonumber \\\\f_{s/n}(x) &= f_{s/p}(x),\\ {\\rm etc.}","$ For completeness, I record the structure functions for electron and (charged current) neutrino scattering from nucleon targets assuming isospin symmetry: $F^{ep}_2(x) &= x \\left[ \\frac{4}{9}(f_{u/p}(x) + f_{\\bar{u}/p}(x)) + \\frac{1}{9}(f_{d/p}(x) + f_{\\bar{d}/p}(x)) + f_{s/p}(x) + f_{\\bar{s}/p}(x)) \\right] \\nonumber \\\\F^{en}_2(x) &= x \\left[ \\frac{4}{9}(f_{d/p}(x) + f_{\\bar{d}/p}(x)) + \\frac{1}{9}(f_{u/p}(x) + f_{\\bar{u}/p}(x)) + f_{s/p}(x) + f_{\\bar{s}/p}(x)) \\right] \\nonumber \\\\F^{\\nu p}_2 (x) &= F^{\\bar{\\nu }n}_2(x) = 2x [f_{d/p}(x) + f_{\\bar{u}/p}(x)] \\nonumber \\\\F^{\\nu n}_2 (x) &= F^{\\bar{\\nu }p}_2(x) = 2x [f_{u/p}(x) + f_{\\bar{d}/p}(x)] \\ .","$ I have ignored heavy quarks ($c, b, t$ ) and the Cabibbo angle ($\\sin ^2 \\theta _c \\cong 0.05$ ) and used the shorthand notation $\\nu p$ ($\\bar{\\nu }p$ ) for $ \\nu p \\rightarrow \\mu ^- X$ ($\\bar{\\nu }p \\rightarrow \\mu ^+ X$ ), etc.","I have also left out the (important) parity violating structure function, $F_3$ , which arises in neutrino scattering.","One important implication of Eqs.","(2.25) is that in the absence of any nuclear effect in deuterium one would expect $F^{ed}_2(x) &= x \\frac{5}{9} \\left[ f_{u/p}(x) + f_{\\bar{u}/p}(x) + f_{d/p}(x) + f_{\\bar{d}/p}(x) \\right]+ \\frac{2}{9}\\left[ (f_{s/p}(x) + f_{\\bar{s}/p}(x)) \\right] \\nonumber \\\\&\\cong \\frac{5}{18} \\left[ F^{\\nu p}_2(x) + F^{\\bar{\\nu }p}_2(x) \\right] \\ , $ because strange quarks are suppressed in the nucleon and weighted by $2/5$ relative to non-strange quarks.","This relation allows one to look for a nuclear effect in the deuteron in a model independent manner (see.","Ref. [15]).","To further simplify the structure functions of nucleons, it is customary to distinguish between quarks which must be present to account for the target's quantum numbers ($u$ and $d$ quarks for nucleons) known as “valence\" quarks and those which may be present in pairs due to relativistic effects, known as “ocean quarks\".","Thus, for an isospin averaged nucleon $f_{u/N}(x) &= f_{d/N}(x) = f_V(x) + f_O(x) \\nonumber \\\\f_{\\bar{u}/N}(x) &= f_{\\bar{d}/N}(x) = f_O(x) \\ .","$ Strange and heavier quarks may also be present.","Typically, one assumes $f_{s/N}(x) &= f_{\\bar{s}/N}(x) \\le f_O(x) \\nonumber \\\\f_{c/N}(x) &= f_{\\bar{c}/N}(x) \\approx 0 \\ ,\\ {\\rm etc.}","\\nonumber $ Before leaving this general discussion of the parton model, it is useful to relate the quark distribution function to the amplitude for quark-target scattering.","We define the (connected) virtual quark-target forward scattering amplitude by $\\chi _{a/T} \\equiv \\int d^4 \\xi e^{-i k \\cdot \\xi } \\mathinner {\\langle {p|T(\\overline{\\psi }_a(\\xi )\\psi _a(0))|p}\\rangle }_c \\qquad \\mathrm {(2.28)}$ as illustrated in Fig.","REF (a).","$\\chi $ is a matrix in color and Dirac spaces, but we have suppressed those indices.","The distribution function $f_{a/T}(x_T)$ can be projected out of $\\chi _{a/T}$ by integrating over all components of $k$ except $k^+$ which is held fixed, $k^+=x_Tp^+$ , and tracing the Dirac indices with $\\gamma ^+$ : $f_{a/T}(x_T) = \\int \\frac{d^4 k}{(2 \\pi )^4} \\delta \\left(\\frac{k^+}{p^+}-x_T \\right) {\\rm Tr} [\\gamma ^+\\chi _{a/T}(k,p)] \\ .","\\qquad \\mathrm {(2.29)}$ It is easy to verify that Eqs.","(2.28)$-$ (2.29) lead to Eq.","(2.18) provided one used Eq.","(2.4) to relate the $T$ -product to the ordinary product.","It is often convenient, when studying electroproduction from nuclei, to define quark distribution functions depending on a universal variable $x = Q^2/2M_N \\nu $ .","To preserve their probabilistic interpretation, it is necessary to rescale them: $F_{a/T}(x) \\equiv \\frac{d P_{a/T}}{dx} = \\frac{d P_{a/T}}{d x_T} \\frac{dx_T}{dx}=\\frac{M}{M_T} f_{a/T}(x_T) \\ .","\\qquad \\mathrm {(2.30)}$ (See Eq. 2.22.)","Then, $\\int _0^{M_T/M} dx (F_{a/T}(x) - F_{\\bar{a}/T}(x)) = N_{a/T} - N_{\\bar{a}/T} \\qquad \\mathrm {(2.31)}$ and $\\int _0^{M_T/M} x dx (F_{a/T}(x) + F_{\\bar{a}/T}(x)) = \\frac{M_T}{M} (\\epsilon _{a/T} + \\epsilon _{\\bar{a}/T}) \\ .","\\qquad \\mathrm {(2.32)}$ At the same time, it is convenient to introduce a sructure function per nucleon, $\\overline{F}^T_2(x)$ $\\overline{F}^T_2 (x) \\equiv x \\sum _a {\\cal {Q}}_a^2 (\\overline{F}_{a/T}(x) + \\overline{F}_{\\bar{a}/T}(x)) \\qquad \\mathrm {(2.33)}$ where $\\overline{F}_{a/T}(x) = F_{a/T}(x)/A$ .","In the analysis of nuclear targets, I will try to preserve this notation: lower case $(f_{a/T}(x_T))$ for intrinsically defined distribution functions as functions of $x_T$ , upper case for functions of $x$ and barred upper case for functions of $x$ “per nucleon\".","Also, the label “A\" will denote a nucleus, “T\" a generic target and “a\" a quark of flavor $a$ .","Note that $\\overline{F}^A_2(x,q^2)$ is defined so that it would reduce to the (isospin weighted) nucleon structure function $\\frac{Z}{A}F^P_2(x,q^2) + \\frac{N}{A}F^N_2(x,q^2)$ if the nucleons in the nucleus were non-interacting.","At this point, it would be appropriate to discuss the phenomenology of the neutron and proton structure functions; however, no time would be left for nuclear targets.","So the reader will have to consult the references [14] for more information about nucleons.","Here, I will mention only a few properties of importance for future work.","Near $x = 0$ both $F_{O/N}(x)$ and $F_{V/N}(x)$ are expected to diverge: $F_{O/N} \\sim 1/x$ and $F_{V/N}(x) \\sim 1/\\sqrt{x}$ .","As $x \\rightarrow 1$ , $F_{O/N}/F_{V/N} \\rightarrow 0$ .","The neutron and proton structure functions differ significantly at large $x$ leading to the observation that $f_{d/p}/f_{u/p} \\rightarrow 0$ as $x \\rightarrow 1$ .","Some quark distributions extracted from electron and neutrino scattering experiments are shown in Fig.","REF .","Figure: Examples of quark, antiquark and gluon distributions in the nucleon, from F. Dydak .Before 1982, little attention had been paid to deep inelastic lcpton scattering from nuclei.","Interest in the subject was awakened by experimental results from CERN.","In order to put the rest of these lectures in proper context it is necessary to present those results, discuss the (dis)agreemcnt among experiments and present the most rudimentary parton model analysis of the data, which already has important implications for the quark substructure of nuclei.","The basic parton model of Bjorken and Feynman and the material of §2 are the only prerequisites for this analysis.","It will be impossible, however, to avoid some reference to QCD, asymptotic freedom and other issues which will be introduced later in these lectures.","Figure: EMC data on F ¯ 2 Fe /F 2 D \\overline{F}^{Fe}_2/F^D_2 .The first precise comparison of deep inelastic scattering from a nuclear target with scattering from a nucleon was made by the EMC Collaboration [2].","They compared iron and deuterium targets.","They assumed that the weakly bound deuteron approximates the isospin averaged nucleon.","Subsequent analysis by Bodek and Simon [15] has confirmed this assumption.","In the absence of any nuclear effect at large $Q^2$ one expects $F^A_2(x,Q^2) = AF^D_2(x,Q^2)/2$ up to a small correction for the neutron excess in the nucleus.","To display deviations from this naive expectation and to minimize systematic errors, it is conventional to plot $S^A (x,Q^2) \\equiv 2F^A_2(x,Q^2)/AF^D_2(x,Q^2)$ .","The deviation of $S^{\\rm Fe}(x,Q^2)$ from unity shown in Fig.","REF caught nearly everyone by surprise [16] The effect at large $x$ was particularly surprising because $\\overline{F}^A_2(x, Q^2)/ F^N_2(x, Q^2)$ must go to infinity as $x \\rightarrow 1$ .","[The denominator vanishes for $x > 1$ , the numerator vanishes only for $x > A$ .]","The “EMC effect\", as the deviation of $S^{\\rm Fe}$ from unity came to be known, was quickly confirmed by a reanalysis of old SLAC data with iron, aluminum and deuterium targets [17].","Last year a dedicated SLAC experiment (E-139) measured the EMC effect on a sequence of nuclear targets [18].","Their data are shown in Fig.","REF .","Figure: SLAC data on σ A /σ D \\sigma ^A/\\sigma ^D .","The fits are from .The SLAC and EMC data agree reasonably well for $x > 0.3$ but the enhancement seen at low-$x$ by EMC was not observed at SLAC.","Recently, there has been input from other groups.","The BCDMS collaboration at CERN [19], has measured deep inelastic muon scattering from iron, nitrogen and deuterium targets.","Their iron data are restricted by detector geometry to $x \\ge 0.2$ where they agree with both SLAC and EMC.","All available data on iron are shown in Fig.","REF .","The nitrogen data include a point at $x = 0.1$ which is significantly above 1 but below the trend of the EMC iron data.","[See Fig.","REF ] Figure: Compilation of data on F ¯ 2 Fe /F ¯ 2 D \\overline{F}^{\\rm Fe}_2/\\overline{F}^D_2 .Figure: Comparison of BCDMS nitrogen data with SLAC carbon data .Several neutrino detector collaborations have measured ratios of cross sections from nuclear targets [20].","In general, their results are subject to larger statistical and systematic uncertainties than the electron and muon data.","It is fair to say, however, that at low-$x$ , where their statistics are best, the neutrino experiments fail to confirm the enhancement seen by the EMC group.","For example, the CDHS group presented some data on $S^{\\rm Fe}$ at this year's Moriond meeting [21].","Their data are also shown in Fig.","REF .","Figure: CDHS (neutrino scattering) measurement of F ¯ 2 Fe /F 2 N \\overline{F}^{Fe}_2/F^N_2 .It is of great importance to sort out the apparent disagreement among experiments at low-$x$ .","This can only be done for certain by future experiments.","However, we can catalog some of the possibilities while we wait.","To help, some of the salient features of the experiments are summarized in Table I.","Among the possibilities are: 1.","Systematic errors: The EMC collaboration quote a considerable systematic error in the slope of the straight line fit to their data.","They also quote a 7% systematic uncertainty in the normalization [2].","Rotating their data to smaller slope and lowering it by 7% improves its agreement with SLAC and BCDMS considerably.","Merlo quoted a systematic error on the $x = 0.03$ CDHS point equal to the statistical error.","Moving the CDHS point up by that amount largely removes the discrepancy.","2.","An $A$ dependence of $R (\\sigma _L/\\sigma _T$ ): The SLAC E-139 experiment measured $R$ on several nuclear targets and found it large ($R \\cong 0.2$ ) and possibly $A$ dependent.","If they extract $S^A$ from their data with an $A$ dependent $R$ it agrees better with the EMC measurement.","This reconciliation is only superficial, however.","It generates a much greater disagreement in the measurement of $\\overline{F}^{\\rm Fe}_1/F^D_1$ .","[EMC do not, in fact, measure $F_1$ because the beam energy is high ($\\epsilon \\approx 1$ , see Eq.","(1.34)) but at EMC energies $Q^2$ is quite large ($Q^2 \\ge 9$ GeV$^2$ for all $x-$ bins) and $R$ is expected to be very small at large $Q^2$ (even in QCD where the Callan-Gross relation is not exact) so $\\overline{F}^A_1/F^N_1 \\cong \\overline{F}^A_2/F^N_2$ .]","3.","Strong $Q^2$ , $x$ and $A$ dependence at low-$x$ : Perhaps the differences among experiments are due to the fact that their bins average in different ways over a rapidly varying function.","The likely source of this variation is “shadowing\" which is expected to be important at low$-x$ and will be discussed further in $§6.4$ .","Table: Summary of experiments.For the remainder of these lectures, I will assume that the EMC data for $x > 0.3$ are correct, but for $x < 0.3$ , I will assume the truth lies somewhere between the EMC and SLAC results.","The parton model ideas developed in the previous section can be applied directly to the nuclear structure function [22].","In the parton model and the Bjorken limit, $\\overline{F}^A_2(x)$ is independent of $Q^2$ .","In QCD, the notion of parton distribution functions and other aspects of the parton model are preserved, but $\\overline{F}^A_2(x)$ and the parton distribution functions develop a weak but important $Q^2$ -dependence.","To allow for this, we occasionally keep the $Q^2$ -label explicit $\\overline{F}^A_2(x,Q^2) = x \\sum _a {\\cal {Q}}^2_a (\\overline{F}_{a/A}(x,Q^2) + \\overline{F}_{\\bar{a}/A}(x,Q^2)) \\qquad \\mathrm {(3.1)}$ and the difference between a nucleus and deuterium is defined by $\\overline{\\Delta }_A(x,Q^2) \\equiv \\overline{F}^A_2(x,Q^2) - \\overline{F}^D_2(x,Q^2) \\ .\\qquad \\mathrm {(3.2)}$ $\\overline{\\Delta }$ depends only on the difference of quark distributions in nucleus $A$ and deuterium.","In the valence parton model described in $§2$ , $\\overline{\\Delta }_A(x,Q^2) = x \\left( \\frac{5}{9}\\delta \\overline{F}_{V/A}(x,Q^2) + \\frac{4}{3} \\delta \\overline{F}_{O/A}(x,Q^2) \\right) \\qquad \\mathrm {(3.3)}$ and $\\delta \\overline{F}_{V/A} \\equiv \\overline{F}_{V/A} - \\overline{F}_{V/D}$ , etc.","Figure: The difference of the structure function per nucleon of iron an deuterium .The EMC data for $\\overline{\\Delta }_{\\rm Fe}$ are shown in Fig.","REF , which was constructed from the published EMC iron data and the data on $S^{\\rm Fe}$ .","Experimentalists caution that systematic errors are more treacherous in the difference $\\overline{\\Delta }_{\\rm Fe}$ than in the ratio $S^{\\rm Fe}$ .","Nevertheless, the qualitative features of $\\overline{\\Delta }_{\\rm Fe}$ are probably reliable (modulo our caveats about $x < 0.3$ ) and they tell us what has happened to the quark distributions in passing from deuterium to iron.","The EMC effect — at least as seen in the EMC data — has several different aspects which can be extracted from a parton model analysis of the data.","The same analysis applied to the SLAC data leads to somewhat different conclusions which I will mention along the way: 1.","The valence quarks in iron are “degraded\" — shifted to lower $p^+$ — relative to those in deuterium.","[All experiments agree on this.]","2.","There is an increase in the number of ocean quark pairs in iron compared to deuterium.","[The increase is large if we believe the EMC data, smaller or even absent if one accepts the SLAC data at low-$x$ .]","3.","The fraction of momentum ($p^+$ ) per nucleon on quarks and antiquarks in iron relative to deuterium can be extracted.","[It increases slightly if one believes the EMC data but the SLAC data are ambiguous.]","These results hold for other nuclei as well in proportion to the size of the measured “EMC effect\".","The extraction makes use of parton model sum rules and positivity constraints.","For $x > 0.35$ , $\\overline{F}_{O/N}(x)$ (the ocean quark distribution in the isolated nucleon) is known to be negligible, thus $\\delta \\overline{F}_{O/N}(x) \\ge 0$ for $x > 0.35$ .","Since $\\overline{\\Delta }_A(x) < 0$ for $x > 0.35$ , we see from Eq.","(3.3) that $\\delta \\overline{F}_{V/A}(x)$ must be negative for $x > 0.35$ .","The valence quark distribution is conserved, i.e., $\\int _0^A dx~\\delta \\overline{F}_{V/A}(x) = 0 \\qquad \\mathrm {(3.4)}$ because the number of valence quarks (per nucleon) in all nuclei is three.","$\\delta \\overline{F}_{V/A}(x)$ must therefore be positive for some $x$ values where it has not been observed.","The SLAC data show $\\delta \\overline{F}_{V/A}(x)$ turning positive for $x>0.8$ by which point $F_{V/A}(x)$ is very small.","I assume that $\\int _{0.8}^A dx~\\delta F_{V/A}(x)$ is negligible.","This leaves $x< 0.35$ as the region in which $\\delta \\overline{F}_{V/A}(x)$ is positive.","In $§1$ we saw that $x$ is conjugate to $1/M \\xi ^3$ in the laboratory, where $\\xi ^3$ is the spatial separation of the currents in the correlation function defining $F_{a/A}(x)$ .","Thus, the shift to lower $x$ is indicative of a shift, to longer range quark correlations in the target ground state.","This result, is independent of whatever “microscopic\" explanation of the EMC effect might eventually be forthcoming.","Whether it is attributed to “dynamical rescaling\", $N$ -quark bags, quark percolation, or more prosaic sources like pion admixtures in the nuclear wavefunction or other binding effects, the EMC effect directly measures an increased quark light-cone correlation length in nuclei.","For more discussion of the space-time interpretation of the EMC effect, see [10].","In retrospect, it is not surprising that measures of the quark correlation length increase in nuclei [23].","It is believed that quark/nuclear matter, regarded as a function of density at zero temperature, undergoes a deconfining phase transition at some $\\rho _{\\rm critical}$ .","For densities below $\\rho _{\\rm critical}$ , quarks are confined in nucleons but for densities above $\\rho _{\\rm critical}$ , they move about more or less freely in a degenerate quark gas.","One support for this is that QCD is known to become asymptotically free at large chemical potential (equivalent to high density), so at high enough density a quark gas will become free.","We identify the nucleon as the zero density limit of quark matter.","As $A$ increases, the mean density of the nucleus increases (as the surface to volume ratio goes to zero), so we may regard the increased quark correlation length in iron as a consequence of its increased mean density and as a precursor of a deconfining phase transition where the correlation length would become very large.","This view of the EMC effect is supported by the $A$ dependence observed at SLAC which correlates very closely with nuclear densities.","It is discussed at length in $§6.3$ .","Point 2, the measurement of ocean quark pairs, is obtained by examining $\\int dx \\delta \\overline{F}_{O/A}(x)$ over the range of the measured data ($x_{\\rm min}<x<x_{\\rm max}$ ): $\\int _{x_{\\rm min}}^{x_{\\rm max}} dx~\\delta \\overline{F}_{O/A}(x) = \\frac{3}{4} \\int _{x_{\\rm min}}^{x_{\\rm max}} \\frac{dx}{x} \\overline{\\Delta }_A(x) + \\frac{5}{12} \\int _0^{x_{\\rm min}} dx~\\delta \\overline{F}_{V/A}(x) \\qquad \\mathrm {(3.5)}$ where we have used Eq.","(3.4) and assumed $\\delta \\overline{F}_{V/A}(x)$ is negligible for $x_{\\rm max}<x<A$ .","If $\\delta \\overline{F}_{V/A}(x)$ does not change sign twice, that is, if it remains positive for $x<x_{\\rm min}$ , then $\\int _{x_{\\rm min}}^{x_{\\rm max}} dx~\\delta \\overline{F}_{O/A}(x) > \\frac{3}{4} \\int _{x_{\\rm min}}^{x_{\\rm max}} \\frac{dx}{x} \\overline{\\Delta }_A(x) \\ .\\qquad \\mathrm {(3.6)}$ Eq.","(3.6) could fail only if $\\overline{F}_{V/A}(x)$ behaves as shown in Fig.","REF [24].","Figure: Behavior of F ¯ 2 Fe /F ¯ 2 D \\overline{F}^{\\rm Fe}_2/\\overline{F}^D_2 required by the EMC data at low-x-x if only valence quarks are involved in the effect.Shadowing is expected to produce a depletion in $\\overline{F}^A_2(x,Q^2)$ at low-$x$ and low $Q^2$ , but it should affect primarily the ocean quark distribution and a shadowing of the valence quarks sufficient to invalidate Eq.","(3.6) at large $Q^2$ would be surprising.","According to Eq.","(3.6), the ocean quarks are enhanced to the extent that the $x^{-1}$ weighted integral of $\\overline{\\Delta }_A$ is $>0$ .","If one accepts the EMC data, the effect is quite large: $\\int _{x_{\\rm min}}^{x_{\\rm max}} dx~\\delta \\overline{F}_{O/{\\rm Fe}}(x) > 1.5 \\int _{x_{\\rm min}}^{x_{\\rm max}}dx\\, F_{O/N}(x) \\ .","\\qquad \\mathrm {(3.7)}$ If we ignore the small contributions from $x > x_{\\rm max}$ , and $x < x_{\\rm min}$ , the EMC data give $5.0 \\pm 1.5 \\times 10^{-2}$ for the first term on the right.","The second term is small and positive (see Eq.","(3.4) and subsequent discussion) so we obtain a bound $\\delta \\epsilon _{\\rm Fe} > 5.0 \\pm 1.5 \\times 10^{-2}$ .","The SLAC data, on the other hand, do not give a positive $\\overline{\\Delta }_A(x)$ for $x < 0.3$ so the sign of $\\delta \\epsilon _{\\rm Fe}$ cannot be determined although its magnitude is certainly small [25] At the level of the parton model, these features of the data appear logically independent.","Some models, notably the rescaling model, are able to correlate a modest increase in the number of ocean quarks with the degradation (i.e., shift to lower $x$ ) of the valence quarks.","Other models are able to account for only one feature of the data or invoke several effects in concert to account for the different aspects of the EMC effect.","It is intuitively appealing to regard inclusive electroproduction from nuclei as a two step process.","First, the nuclear wave function is decomposed into some basis of constituents, nucleons in the first instance, nucleons and pions in more elaborate schemes, and later perhaps including more exotic objects like $\\Delta $ s, multiquark configurations, and so on.","Then, the structure functions of the constituents are added incoherently to give the structure function of the whole nucleus.","This is the “convolution\" model\" [26].","The simplest version, which includes only nucleons, gives what are known as “Fermi motion\" corrections to the free nucleon structure function [27].","These were calculated long before the present excitement about electroproduction from nuclei.","More recently, the model has been extended to more exotic constituents [28], [29], [30] in an attempt to “explain\" the EMC effect.","There is no adequate derivation of the convolution model.","The parton analyses of the $§2$ will provide a framework in which the assumptions which lead to the convolution model may be analyzed and criticized.","The two steps of the convolution model are summarized diagrammatically in Fig.","REF .","The nucleus, with baryon number $A$ and momentum $P$ , contains a constituent, label $T$ and momentum $p$ , which in turn contains a quark, flavor $a$ and momentum $k$ .","The quark absorbs the virtual photon while the fragments of the nucleus and the constituent propagate into the final state without interaction or interference.","Other diagrams in which fragments of the nucleus or the constituent $T$ interact or interfere are ignored.","Some are shown in Fig.","REF .","Figure: Contributions which are ignored in convolution models.Superficially, they resemble the diagrams of Fig.","6 (b)–(d) which were ignored in the parton model, but there is an important difference: Fig.","6(b)–(d) can be dropped because $Q^2 \\rightarrow \\infty $ , so $\\xi ^2 \\rightarrow 0$ and the operator product expansion in QCD can be used to prove that they are O($1/Q^2$ ) compared to Fig.","6(a).","There is no analogous large mass scale characterizing the process enclosed in the dashed line in Fig.","REF , so there is no a priori justification for ignoring the additional processes of Fig.","REF .","Furthermore, the fragments of the nucleus and the constituent have a long time, $\\xi ^0 \\lesssim 1/M x$ , to interact while awaiting the return of the active quark.","Nevertheless, for certain constituents under favorable kinematic conditions (e.g., for nucleons in the weak binding limit, or for pions near $x = 0$ ) [10] ignoring final state interactions and interference may be justifiable.","For the moment, we will simply ignore the problem and proceed.","The quark distribution for nucleus $A$ with four-momentum $P$ is defined in analogy to Eq.","(2.29): $f_{a/A}(x_A) = \\int \\frac{d^4k}{(2 \\pi )^4} \\delta \\left( \\frac{k^+}{P^+} - x_A \\right) {\\rm Tr}[\\gamma ^+ \\chi _{a/A}(k,P)] \\qquad \\mathrm {(4.1)}$ where $x_A = Q^2/2M_A \\nu $ .","The content of Fig.","REF is a convolution form for $\\chi _{a/A}(k,P)$ : $\\chi _{a/A}(k,P) = \\sum _T \\int \\frac{d^4p}{(2 \\pi )^4} \\chi _{a/T}(k,p) \\chi _{T/A}(p,P) \\qquad \\mathrm {(4.2)}$ where the sum covers all constituents of the nucleus.","We have treated the constituent $T$ as a scalar (omitting a sum over the spins of $T$ ).","The generalization to spin $1/2$ is straightforward.","Note that the quark and/or constituent legs in Fig.","REF have been properly included in Eq.","(4.2): From Eq.","(2.28) it is apparent that $\\chi _{a/T}(\\chi _{T/A)})$ is untruncated in the quark (constituent) legs but truncated in the constituent (nucleus) legs.","Only the $+$ components of momentum really matter in Eq.","(4.1)-(4.2).","To make this manifest, we substitute the identity: $1 = \\int dy\\ \\delta (y_A-p^+/P^+) \\int dz_T\\ \\delta (z_T-k^+/p^+) \\qquad \\mathrm {(4.3)}$ where the ranges of $y_A$ and $z_T$ integrals are yet to be determined.","After some algebra we find $f_{a/A}(x_A) = &\\sum _T \\int dy_Adz_T~\\delta (y_Az_T-x_A)\\int \\frac{d^4p}{(2 \\pi )^4} \\delta (y_A-p^+/P^+) \\chi _{T/A}(p,P)\\times \\\\\\times &\\int d^4k~\\delta (z_T-k^+/p^+) {\\rm Tr}[\\gamma ^+ \\chi _{a/T}(k,p)]\\ .","\\nonumber $ The $k-$ integration defines a Lorentz invariant function of $z$ and $p^2$ which is the off-shell generalization of $f_{a/T}(x_T)$ defined in Eq.","(2.29): $f_{a/T}(z_T,p^2)\\equiv \\int \\frac{d^4k}{(2 \\pi )^4} \\delta (z_T-k^+/p^+) Tr[\\gamma ^+ \\chi _{a/T}(k,p)]\\ .","\\qquad \\mathrm {(4.5)}$ Let us save the $p^2$ integral for last by inserting $\\int dp^2_0~\\delta (p^2-p^2_0) = 1$ .","Then the $d^4p$ integration gives $f_{a/A}(x_A) = \\sum _T \\int dy_A dz_T \\delta (y_Az_T-x_A) \\int d p^2_0f_{a/T}(z_T,p^2_0)f_{T/A}(p^2_0,y_A) \\qquad \\mathrm {(4.6)}$ where $f_{T/A}(p^2_0,y_A) = \\int \\frac{d^4p}{(2 \\pi )^4}~\\delta (p^2-p^2_0)~\\delta (y_A-p^+/P^+) \\chi _{T/A}(p,P) \\ .","\\qquad \\mathrm {(4.7)}$ Note that $f_{T/A}(p^2_0,y_A)$ is the probability to find a constituent $T$ in nucleus $A$ with momentum fraction $y_A$ of the nucleus's $P=M_A/\\sqrt{2}$ and invariant mass $p^2_0$ , whereas $f_{a/T}(z_T,p^2_0)$ is the probability to find a quark of flavor $a$ with momentum fraction $z_T$ in an off shell target with invariant mass $p^2_0$ .","So $f_{a/T}(z_T)$ , which we have defined in Eq.","2.18, would correspond to $f_{a/T}(z_T,M^2_T)$ , but $f_{T/A}(y_A) = \\int dp^2_0 f_{T/A}(p^2_0,y_A)$ .","In practice, no one uses a $p^2_0$ dependent quark distribution $f_{a/T}(z_T,p^2_0)$ in convolution model calculations.","The rationale for this must be that $f_{T/A}(p^2_0,y_A)$ peaks very strongly at some $\\overline{p}^2$ so $f_{a/T}(z,p^2_0)$ can be replaced by $f_{a/T}(z,\\overline{p}^2)$ .","Then the $p^2_0$ integration in Eq.","(4.7) can be performed, but it is an additional assumption that $f_{a/T}(z_T,\\overline{p}^2) = f_{a/T}(z_T,M^2_T)$ unless of course $\\overline{p}^2 \\approx M^2_T$ .","The result $f_{a/A}(x_A) = \\sum _T \\int dy_A dz_T \\delta (y_Az_T-x_A) f_{a/T}(z_T)f_{T/A}(y_A) \\qquad \\mathrm {(4.8)}$ together with the equations which define $f_{a/T}(z_T)$ and $f_{T/A}(y_A)$ comprise the convolution model.","The range of the $y_A$ and $z_T$ integrations can be determined by arguments identical to those used in $§2.2$ to show $0<x_A<1$ .","The result is $0<y_A$ , $z_T<1$ .","The parton distribution functions are often written as functions of Bjorken's variable $x= M_A x_A/M$ .","As given in Eq.","2.30, $F_{a/A}(x) =(M/M_A)f_{a/A}(Mx/M_A)$ .","If the constituents in question are nucleons and the binding is not too strong, then $\\chi _{N/A}(p,P)$ does peak strongly and near mass shell (compared to the scale for variation in $f_{a/T}(z_T,p^2)$ which is $\\Lambda \\sim 200-300$ MeV).","However, the model is applied to other constituents which are far off shell.","Pion contributions are believed to be largest for $p^0 \\approx 0$ and $|\\bf p| \\approx $ 300-400 MeV [31] giving $\\overline{p}^2 \\approx -(0.1-0.2)$ GeV compared to $m^2_{\\pi } \\cong 0.02$ GeV.","Whether the distribution of quarks in a pion so far off shell is the same as the distribution on shell is anybody's guess.","In any case, advocates of pion and other convolution based models ignore any $p^2$ dependence of the constituents' quark distributions.","The assumptions leading to a convolution model are arguable at best and probably can only be supported on a case by case basis.","Despite this, I believe they have substantial value as qualitative guides to nuclear effects and especially as a formalism for treating Fermi motion and other “trivial\" sources of nuclear modifications of structure functions.","Here is a convenient summary of the convolution model formulas: $f_{a/A}(x_A) &= \\sum _T f_{a/T/A}(x_A) \\nonumber \\\\&= \\sum _T \\int dy_A dz_T \\delta (y_Az_T-x_A) f_{a/T}(z_T)f_{T/A}(y_A) $ $f_{a/T}(z_T) &= \\frac{d P_{a/T}}{d z_T} \\, 0<z_T<1, \\ z_T=k^+/p^+ \\ , \\nonumber \\\\f_{T/A}(y_A) &= \\frac{d P_{T/A}}{d y_A} \\, 0<y_A<1, \\ y_A=p^+/P^+ \\ , $ $\\int _0^1 dz_Tf_{a/T}(z_T) = N_{a/T} \\ \\ \\ &\\ \\ \\ \\int _0^1 dy_Af_{T/A}(y_A) = N_{T/A} \\nonumber \\\\\\int _0^1 dz_T z_T f_{a/T}(z_T) = \\epsilon _{a/T} \\ \\ \\ &\\ \\ \\ \\int _0^1 dy_A y_A f_{T/A}(y_A) = \\epsilon _{T/A} $ $\\int _0^1 dx_A f_{a/A}(x_A) &= \\sum _T N_{a/T} N_{T/A} \\nonumber \\\\\\int _0^1 dx_A x_A f_{a/A}(x_A) &= \\sum _T \\epsilon _{a/T} \\epsilon _{T/A} \\nonumber $ where $N_{T/A}$ is the number of constituents of type $T$ in nucleus $A$ and $\\epsilon _{T/A}$ is the fraction of the nucleus's $P^+$ carried by constituents $T$ .","It is useful to introduce quark and constituent distributions depending on Bjorken's variable $x = M_Ax_A/M$ and $y=M_Ay_A/M$ .","We leave $z_T$ as is since the constituent $T$ is in general not at rest and $M_T$ plays no special role.","Thus, $x=\\sqrt{2}k^+/M$ and $F_{T/A}(y)=dP_{T/A}/dy=\\frac{M}{M_A}f_{T/A}(y_A)$ .","Then, Eq.","(4.8) becomes $F_{a/A}(x) &= \\sum _T \\int _0^{M_A/M}dy\\int _0^1 dz_T \\delta (x-yz_T)f_{a/T}(z_T)F_{T/A}(y)\\\\&=\\sum _T\\int _x^{M_A/M}\\frac{dy}{y}f_{a/T}(x/y)F_{T/A}(y) $ and the sum rules analogous to Eqs.","(4.11) are $\\int _0^{M_A/M} dx \\overline{F}_{a/A}(x) &= \\sum _T N_{a/T}N_{T/A}/A \\\\\\int _0^{M_A/M} dx \\overline{F}_{a/A}(x) &= \\frac{M_A}{MA} \\sum _T \\epsilon _{a/T} \\epsilon _{T/A} \\\\$ where $\\overline{F}_{a/A}=F_{a/A}/A$ is the quark distribution per nucleon.","Throughout this discussion, I have been careful not to make the approximation $M_A \\cong MA$ although this is a reasonable approximation in most cases.","Convolution has some unexpected and important kinematic effects on the quark distribution of some constituents.","Consider, for example, the contribution of pions in a model like Erikson and Thomas [31]: $p^0 \\approx 0$ , but $|{\\bf p}| \\approx 300 - 400$ MeV, so $y_{\\pi } \\sim |{\\bf p}| \\cos \\theta /M$ which is typically less than $\\sim 1/3$ .","The valence quark distribution in the pion is expected on theoretical grounds and found experimentally to be quite “hard\", $f_{V/\\pi }(z_\\pi ) \\sim (1-z_\\pi )$ .","Convolution converts this into $F_{V/\\pi /A}(x) = \\int _x^{M_A/M} \\frac{dy}{y} F_{\\pi /A}(y) f_{V/\\pi }(x/y)$ which is negligible for $x$ much larger than $1/3$ .","So the hard pion distribution has been mapped to small $x$ .","No such fate befalls the nucleon: $p^0 \\approx M$ , so $y_N \\approx 1 + p \\cos \\theta /M$ which peaks near $y_N =1$ .","Nevertheless, the motion of nucleons does affect the quark distribution in the nucleus even if it is composed of quarks alone.","The most obvious source of a nuclear effect in deep inelastic scattering comes from the fact that the nucleons in the nucleus are in motion.","As a “baseline\" model for nuclear targets we assume the nucleus consists exclusively of nucleons and that the quark distribution in the nucleons are the same as in isolation.","One might naively think if their kinetic energies are small with respect to $\\nu $ , the motion could be neglected as $\\nu \\rightarrow \\infty $ .","This is not correct, as we shall see.","We begin with $f_{N/A}(y_A) = \\int \\frac{d^4p}{(2 \\pi )^4} \\delta (y_A-p^+/P^+) \\chi _{N/A}(p,P) \\qquad \\mathrm {(4.15)}$ where $\\chi _{N/A}(p,P)$ is the forward, possibly virtual, nucleon nucleus scattering amplitude defined in analogy to Eq.","(2.28).","$\\chi _{N/A}(p,P) = \\int d^4 \\zeta e^{-ip \\cdot \\zeta } \\mathinner {\\langle {P|T(\\Phi ^+(\\zeta )\\Phi (0))|P}\\rangle }_c\\qquad \\mathrm {(4.16)}$ where $\\Phi $ is a nucleon interpolating field.","$\\Phi $ is not uniquely defined, and when $p^2 \\ne M^2_N$ different choices yield different results.","This reflects an inherent uncertainty when one attempts to use field-theoretic methods to manipulate composite objects.","Spin has been suppressed in Eq.","(4.16).","If we substitute Eq.","(4.16) into Eq.","(4.15) and perform as many $\\zeta $ and $p$ integrations as possible we obtain a form analogous to Eq.","(2.18): $f_{N/A}(y_A) = \\sum _n \\delta (y_A-1+P^+_n/P^+)|\\mathinner {\\langle {n|\\Phi |P}\\rangle }|^2 \\qquad \\mathrm {(4.17)}$ where the sum is on all states which can be obtained by removing a single nucleon from the nucleus leaving behind a state with $P^+_n = (1-y_A)P^+$ .","The problem is how to calculate $|\\mathinner {\\langle {n|\\Phi |P}\\rangle }|^2$ .","This must be done inclusively, i.e., all possible states {$\\mathinner {|{n}\\rangle }$ } must be included and they must be physical states.","A variety of approximations can be made but great care must be taken to ensure that the number and $p^+$ sum rules (Eqs.","(4.11)) remain valid.","The most naive approach, to replace the nucleus by an independent particle model in which nucleons occupy energy eigenstates in some potential, obeys neither sum rule and must be altered in some ad hoc way before it can be used in this context.","When this form for $f_{N/A}(y_A)$ is inserted into Eq.","(4.12) to obtain $F_{a/N/A}(x)$ we obtain an independent nucleon model for $F_{a/A}(x)$ which includes what are generally known as “Fermi motion corrections\": $\\overline{F}_{a/N/A}(x) = \\int _x^{M_A/M} \\frac{dy}{y} f_{a/N}(x/y) \\overline{F}_{N/A}(y) \\qquad \\mathrm {(4.18)}$ (Reminder: $F_{N/A}(y)=dP_{T/A}/dy=\\frac{M}{M_A}f_{N/A}(y_A)$ , $\\overline{F}_{a/N/A}=F_{a/N/A}/A$ and $\\overline{F}_{N/A}=F_{N/A}/A.$ ) $F_{a/N/A}$ has several important features independent of the explicit form of $\\overline{F}_{N/A}$ .","First, as already noted $\\overline{F}_{a/N/A}(x)/F_{a/N}(x)$ diverges as $x \\rightarrow 1$ , so Fermi motion corrections to the ratio are large and positive near $x \\sim 1$ .","The divergence of the ratio is deceptive because both $\\overline{F}_{a/N/A}(x)$ and $F_{a/N}(x)$ are very small for $x \\sim 1$ .","Second, Fermi motion corrections cannot change the number of quarks of any flavor: $\\int _0^{M_A/M}(\\overline{F}_{a/N/A}(x) - F_{a/N}(x)) = 0$ , which can be obtained from Eq.","(4.13) with $T = N$ and $N_{T/A} = A$ .","Third, the effect of nuclear binding appears to be to decrease slightly the $p^+$ carried by the quarks (and antiquarks) even though the quark distribution in the nucleon is not altered.","This can he seen from Eq.","(4.14): $M_A/MA = 1 -\\delta $ where $\\delta $ is the binding energy per nucleon in units of the nucleon mass, and $\\epsilon _{N/A} = 1$ (the nucleons carry all the nucleus $P^+$ in this simple model), so $\\int _0^{M_A/M} dx x[\\overline{F}_{a/N/A}(x)-F_{a/N}(x)] = -\\delta \\epsilon _{a/N}$ .","This goes in the right direction toward explaining the EMC effect but explicit calculations with “realistic\" nuclear wave functions fail to get a large enough shift in the valence quark distribution and get the wrong shape (i.e.","$x-$ dependence) of the effect.","Also, the model cannot produce an increase in the ocean quark pairs.","Recently, it has been claimed that a model of the form we have been discussing can account for the valence quark part of the EMC effect [32].","In that model, $\\epsilon _{N/A} < 1$ so $P^+$ has been lost, presumably to the constituents responsible for nuclear binding, and the quark content of those constituents has not been included in the calculation.","It is interesting to explore the effect of binding in a simple model.","Let us assume the nucleons form a relativistic, degenerate free Fermi gas (FG) with Fermi momentum $k_F$ .","Then $|\\mathinner {\\langle {n|\\Phi |P}\\rangle }|^2 \\equiv \\frac{dN}{d^3p} = \\frac{3A}{4 \\pi k^3_F} \\theta (k_F-|{\\bf p}|) \\ .","\\qquad \\mathrm {(4.19)}$ The constant $3A/4\\pi k^3_F$ is chosen so $\\int ^{k_F} dN = A$ .","The bound nucleons must have an effective mass $M^\\star < M_N$ in this model otherwise the sum over the energies of the nucleons would exceed $M_A$ .","Substituting into Eq.","(4.18) and evaluating the $d^3p$ integral, $f^{FG}_{N/A}(y_A) = \\frac{3}{4} \\frac{AM_A}{k_F} \\left( 1-\\frac{M^2_A}{k^2_F} \\left(\\frac{y^2_A-M^{\\star 2}/M^2_A}{2 y_A} \\right)^2 \\right) \\ , \\ y_-<y_A<y_+ \\qquad \\mathrm {(4.20)}$ where $y_\\pm $ are the values for which $f^{FG}_{N/A}(y_\\pm )=0$ .","One can check that $f^{FG}_{N/A}(y_A)$ satisfies both $\\int _0^1 dy_A f^{FG}_{N/A}(y_A)=A$ and $\\int _0^1 dy_A y_A f^{FG}_{N/A}(y_A)=1$ provided $M^\\star $ is chosen so the energy of the Fermi gas is $M_A$ .","For non-relativistic nucleons ($M^\\star = M_A/A + O(k^3_F/M)$ ) a quadratic approximation suffices $\\overline{F}^{FG}_{N/A}(y) = \\frac{M}{AM_A} f^{FG}_{N/A}(y_A) \\approx \\frac{3}{4 \\lambda } \\left( 1-\\frac{(y-\\eta )^2}{\\lambda ^2} \\right) \\ , \\ \\eta -\\lambda <y<\\eta + \\lambda \\ , \\qquad \\mathrm {(4.21)}$ where $\\lambda =k_F/M$ and $\\eta =M_A/MA\\ (\\le 1)$ .","Since $\\overline{F}^{FG}_{N/A}(y)$ has maximum height $\\sim 1/\\lambda $ and width $\\sim \\lambda $ and since it is convoluted with a smooth function $f_{a/N}(x/y)$ in Eq.","(4.18), it is convenient to approximate it as a generalized function $F^{FG}_{N/A}(y) \\cong \\delta (y-\\eta ) + \\frac{\\lambda ^2}{10} \\delta ^{^{\\prime \\prime }}(y-\\eta ) \\qquad \\mathrm {(4.22)}$ for $\\lambda <<1$ .","Note that $F^{FG}_{N/A}(y)$ in this form satisfies the required sum rules trivially.","In particular, $\\int _0^{M_A/M} dy y F^{FG}_{N/A}(y)= \\eta = M_A/MA$ which leads to the decrease in the quark's momentum noted above.","Substituting into Eq.","(4.18), we obtain $\\overline{F}^{FG}_{a/N/A}(x) = \\frac{1}{\\eta } f_{a/N}(x/\\eta ) + \\frac{\\lambda ^2}{10} d^2 /dy^2 \\frac{1}{y} f_{a/n}(x/y) |_{y=\\eta } \\ .","\\qquad \\mathrm {(4.23)}$ In this model the effects of Fermi smearing are small (for $k_F = 300$ MeV, $\\lambda ^2/10 \\approx 0.01$ , for typical nuclei $1-\\eta \\le 0.01$ ).","A sample calculation of $\\overline{F}_{a/N/A}(x)/F_{a/N}(x)$ is shown in Fig.","REF .","It clearly cannot account for the EMC effect.","Figure: Estimates of Fermi motion effects in a Fermi gas model for both valence and ocean quarks.Within the context of convolution models, there are only two alternatives: either other constituents must be present (e.g., pions, $\\Delta $ 's, 6 quark bags, etc.)","[28], [29] or the structure function of the nucleons must be modified by the nuclear medium [30] I will not discuss either alternative in these lectures.","The interested reader should consult the references for some work in these directions.","Instead, I will describe a framework for analyzing the EMC effect which emerges from the scaling properties of QCD.","In interacting field theories the structure functions $F_1$ and $F_2$ depend on both $x$ and $Q^2$ even at very large $Q^2$ .","The $Q^2-$ dependence will give us a new handle on the distance scales characterizing the target.","To understand the $Q^2-$ dependence we must take another excursion into formalism.","At present, the $x-$ dependence of $F_2(x,Q^2 )$ at fixed $Q^2$ cannot be predicted for any hadronic target in a rigorous fashion.","It depends on the non-perturbative dynamics which confines quarks.","However, given $F_2(x,Q^2)$ at some large $Q^2 = Q^2_0$ , QCD perturbation theory enables one to predict $F_2(x,Q^2)$ at all $Q^2 > Q^2_0$ and at $Q^2 < Q^2_0$ down to some minimum which appears to be of order 1 GeV$^2$ .","In QCD, $F_2(x,Q^2)$ depends logarithmically on $Q^2$ at large $Q^2$ .","The logarithmic $Q^2-$ dependence has been verified experimentally and constitutes one of the major quantitative tests of the theory.","In addition to $\\ln Q^2$ corrections there are expected to be $O(1/Q^2)$ and higher order corrections which become important at low-$Q^2$ .","In this chapter, I first give a heuristic “derivation\" of the logarithmic $Q^2-$ dependence of quark distributions.","Next, I will describe some of the formalism behind the $Q^2-$ dependence.","Then, I catalogue and discuss $O(1/Q^2)$ corrections.","Finally, I compare the quark description of hadrons in QCD with more naive quark models.","This analysis applies equally well to any target, so the target subscripts $T$ and $A$ will generally be suppressed .","Until further notice the scaling variable is $x_T$ with $0 < x_T < 1$ .","In QCD, quarks are coupled to gluons in much the same fashion that electrons are coupled to photons in QED.","The fundamental vertex is shown in Fig.","REF (a).","Instead of the electric charge $e$ , one has $g\\lambda ^a_{ij}/2$ , where $g$ is the QCD coupling ($\\alpha _c \\equiv g^2 /4 \\pi $ ).","$\\lambda ^a_{ij}$ are Gell-Mann's matrices (normalized so ${\\rm tr}\\,\\lambda ^2 = 2$ ) describing the coupling of quark with color $i$ to quark with color $j$ ($i,j = 1,2,3$ ) by emitting a gluon with color $a$ ($a = 1, ... 8$ ).","In QED it is well-known that the electric and magnetic fields of a relativistic electron are predominantly transverse and look like the fields of a photon [33].","This is the basis of the Weizsäcker-Williams or “equivalent photon\" approximation in QED.","A careful study of the analogous process in QCD will lead to the logarithmic scaling violations we seek [34].","The equivalent photon approximation is usually formulated in a frame in which the electron is extremely relativistic (moving for definiteness, in the $x$ -direction) $-$ the “infinite momentum frame\".","The number of photons associated with the electron depends on the energy of the photon ($E_\\gamma = x E_e$ ) and the impact parameter at which one probes the electron's electromagnetic field: $\\frac{d N_\\gamma }{dx dA} \\sim \\frac{\\alpha }{\\pi ^2 b^2 x} + {\\rm terms\\ less\\ singular\\ in\\ b\\ and\\ x} \\qquad \\mathrm {(5.1)}$ where $dA=2\\pi b db$ and the variables are defined in Fig.","REF .","Figure: Weizsäcker-Williams kinematics.The proper interpretation of Eq.","(5.1) is that a measurement sensitive to the intensity of the electron's electromagnetic field performed at impact parameter $b$ and absorbing energy $E_\\gamma =x E_e$ could not differentiate between a passing electron and the equivalent number of photons.","Eq.","(5.1) is derived, for example, by Jackson.","The derivation neglects the recoil of the electron, which must lose energy when a photon with $E_\\gamma /E_e =x$ is absorbed.","This is reasonable for $x \\approx 0$ .","For larger $x$ , one must include recoil and electron spin effects, which requires calculation of Feynman graphs.","The lowest order graph corresponding to the measurements I've described is that of Fig.","REF (a).","The calculations can be found in Ref. [35].","The result is $\\frac{1}{x} \\rightarrow \\frac{1}{2x} (1+(1-x)^2)$ .","QED and QCD are identical to this order except $e \\rightarrow g \\lambda ^a_{ij}$ .","So the number of gluons (summed over all colors) with momentum fraction $x$ at impact parameter $b$ in a quark (averaged over colors) is $\\frac{dN_g}{dxdb} \\sim \\frac{2 \\alpha _c}{\\pi b} \\frac{1}{3} {\\rm Tr} \\sum _a \\left( \\frac{\\lambda ^a}{2} \\right)^2 \\left[\\frac{1+(1-x)^2}{2x} \\right] \\qquad \\mathrm {(5.2)}$ to leading order in $b$ and $\\alpha _c$ .","Since $\\frac{1}{3} {\\rm Tr} \\sum _a \\left( \\frac{\\lambda ^a}{2} \\right)^2 = \\frac{4}{3} \\qquad \\mathrm {(5.3)}$ we have $\\frac{dN_g}{dxdb} = \\frac{8 \\alpha _c}{3 \\pi b} \\left[\\frac{1+(1-x)^2}{2x} \\right] \\ .","\\qquad \\mathrm {(5.4)}$ Figure: Feynman diagrams corresponding to Weizsäcker-Williams processes: (a) An (electron) quark radiating a photon (gluon); (b) An electron (quark) “radiating\" an electron (quuark); Vertex correction required at x=1x=1.If one probes a quark (or electron) by delivering a momentum transfer $Q_\\perp = \\sqrt{Q^2}$ , then $b_{\\rm min}=1/\\sqrt{Q^2}$ plays the role of a resolution: all gluons (or photons in QED) with $b \\ge b_{\\rm min}$ will appear distinct from the quark (electron), but the rest ($b < b_{\\rm min}$ ) will be lumped into what one calls the quark (electron) when probed with that resolution.","This is evident, for example, in the Weizsäcker-Williams calculation of bremsstrahlung (per unit $x$ ) is $\\approx x E_e \\int _{b_{\\rm min}}^\\infty db \\frac{dN_\\gamma }{dx db}$ with $b_{\\rm min} = 1/\\sqrt{Q^2_\\perp }$ .","Thus, $\\frac{d N_g}{dx d \\ln Q^2/Q^2_0} = \\frac{2 \\alpha _c}{3 \\pi } \\frac{1+(1-x)^2}{x} \\qquad \\mathrm {(5.5)}$ is the rate of change with $t \\equiv \\ln Q^2/Q^2_0$ of the probability to find a gluon of momentum fraction $x$ in a quark.","(Note that the logarithm of $Q^2$ arose when $b\\propto 1/\\sqrt{Q^2}$ was substituted for $db/b$ in Eq.","5.4 and that $Q^2_0$ is an arbitrary scale for the logarithm.)","In the following, I will often use the variable $t=\\ln (Q^2/Q_0^2)$ .","So we define $P_{g/q}(x,t)=dN_g/dxdt$ .","($P_{g/q}$ appears to be independent of $t$ , but in QCD $\\alpha _c$ depends logarithmically on $t$ .)","When a quark radiates a gluon it leaves behind a quark of lower momentum.","So we can determine the rate of change with $t$ of the probability that a measurement on a quark with energy $E_q$ will detect a quark with $E^\\prime _q = xE_q$ : $P_{q/q}(x,t) = \\frac{dN_g(1-x)}{dxdt} = \\frac{2 \\alpha _c}{3 \\pi } \\frac{1+x^2}{1-x}\\qquad \\mathrm {(5.6)}$ corresponding to the Feynman graphs of Fig.","REF (b).","The three gluon coupling of QCD shown in Fig.","REF (b) leads to a probability for a gluon to contain gluons of lower momentum, and the $q\\bar{q}q$ vertex of Fig.","REF (a) leads to a probability for a gluon to contain quark-antiquark pairs.","If we confine our attention to valence quarks we need not study these other processes since they produce only ocean $q \\bar{q}$ pairs.","Eqs.","(5.5) and (5.6) lead to the notion of a quark distribution “evolving\" with increased $t \\sim \\ln Q^2$ .","As $\\ln Q^2$ grows, valence quarks emit gluons, gluons in turn emit quark-antiquark pairs and more gluons.","This “evolutionary\" view of the $Q^2$ dependence of quark (and gluon) distributions was first suggested by Kogut and Susskind [36] and was developed for QCD by Altarelli and Parisi [34].","The idea that the definition of a quark or gluon depends upon the mass or distance scale at which one probes it is a familiar one in quantum field theory.","It is related to renormalization.","A quark (or electron) propagating in isolation couples to quantum fluctuations in the vacuum.","When we perform some measurement on the quark we lump some of the fluctuations (and all of the very short distance ones which lead to divergences) into the definition of the quark and treat the remainder as radiative corrections.","To do this we must introduce a mass (or length) scale ($\\mu $ ) known as the “renormalization point\" into the theory, which distinguishes, roughly speaking, those fluctuations we treat explicitly ( $\\Delta x > \\mu ^{-1}$ ) from those we incorporate into the definition of the particles ($\\Delta x< \\mu ^{-1}$ ).","It is well-known that $\\mu $ is arbitrary - nothing physical depends on it - but the scale invariance of QCD allows us to trade $\\mu ^2$ -dependence for $Q^2$ -dependence in a fashion which will be outlined briefly below ($§$ 5.2).","Up to now, we've worked in the infinite momentum frame.","$x$ measures the $z$ -component of a daughter particle's momentum relative to its parent: $p^3_\\infty = xP^3_\\infty $ .","The subscript \"$\\infty $ \" denotes quantities measured in the infinite momentum frame.","The particles are not far from mass shell and have limited transverse momenta (we can choose $P^3_\\infty > > \\sqrt{Q^2}$ ) so $p^0_\\infty \\cong x P^3_\\infty $ ) and $p^+_\\infty = x P^+_\\infty \\ .","\\qquad \\mathrm {(5.7)}$ The ratio $p^+/P^+$ is invariant under Lorentz transformation in the along the $z$ -axis.","This follows from the Lorentz transformation in the form $p^{\\pm \\,\\prime }= e^{\\pm y}p^\\pm $ , ${\\mathbf {p}}^\\prime _\\perp = {\\bf p}_\\perp $ , where $\\beta = \\tanh y$ is the relative velocity of theframes along the $z$ -axis.","Thus, Eq.","(5.7) implies $p^+ = x P^+ \\qquad \\mathrm {(5.8)}$ in the laboratory, and we can identify $x$ as the variable we've been using all along.","$P_{q/q}(x,t)$ appears to be singular at $x = 1$ .","This presents a problem because we would like the number of valence quarks in a target to be independent of $Q^2$ , $\\frac{d}{dt} \\int _0^1 dx \\frac{dN_q}{dx} = \\int _0^1 dx P_{q/q}(x,t) = 0 \\,.\\qquad \\mathrm {(5.9)}$ With Eq.","(5.6) as it stands, the integral diverges.","The resolution of this problem requires methods outside the scope of these lectures [38].","The proper prescription is to interpret $1/(1-x)$ as a distribution: $1/(1-x) \\rightarrow 1/(1-x)_+$ , defined by $\\int _0^1dx \\frac{f(x)}{(1-x)_+}\\equiv \\int _0^1 dx(f(x)-f(1))/(1-x) \\qquad \\mathrm {(5.10)}$ and to add a $\\delta $ -function to $P_{q/q}(x,t)$ , so that Eq.","(5.9) is satisfied.","Since $\\int _0^1 dx (1+x^2)/(1-x)_+ = -\\frac{3}{2}$ we find $P_{q/q}(x,t) = \\frac{2 \\alpha _c}{3 \\pi } \\left( \\frac{1+x^2}{(1-x)_+} + \\frac{3}{2} \\delta (x-1) \\right) \\ .","\\qquad \\mathrm {(5.11)}$ The rate of change of the quark distributions in a target will be determined by the functions $P_{q/q}(x, t)$ and $P_{q/g}(x,t)$ and by the quark and gluon distributions at $t$ : A quark or gluon with momentum fraction $y$ will be observed to consist of quarks with momentum fraction $zy$ according to $P_{q/q}(x, t)$ and $P_{q/g}(x,t)$ respectively.","Clearly, evolution is described mathematically by convolution in precise analogy to $§4.2$ : $\\frac{df_{\\rm N. S.}}{dt} = \\frac{\\alpha _c}{2 \\pi } \\int _x^1 \\frac{dy}{y} \\overline{P}_{q/q} (x/y)f_{\\rm N. S.}(y,t) \\qquad \\mathrm {(5.12)}$ where $P_{q/q} \\equiv \\frac{\\alpha _c}{2 \\pi } \\overline{P}_{q/q}$ The subscript $\\rm N. S.$ denotes non-singlet and indicates a quark distribution with non-trivial flavor quantum numbers such as the valence quark distribution of $§2$ .","A non-singlet distribution is one which cannot be populated by the quark-antiquark pairs evolved from gluons.","Singlet distributions, on the other hand, satisfy coupled integro-differential equations in which the gluon distribution appears as well.","To describe the evolution of $F_2(x,Q^2)$ it is necessary to solve these differential equations involving valence and ocean quark distributions and a gluon distribution $f_g(x,t)$ .","Starting values of all distributions at $t=$ 0, i.e., $Q^2 = Q^2_0$ , are required as input.","The general features of the $t-$ dependence are clear from the Weizäcker-Williams approximation.","As $t$ increases, valence quarks lose momentum by emitting gluons — i.e., probed with higher resolution more of a quark momentum appears to be carried by its gluon field.","Gluons, in turn, lose momentum to quark-antiquark pairs.","The net result, with increasing $t$ , is a transfer of the valence quark momenta to newly created pairs.","The gluons are caught in the middle - valence quarks emit gluons but the gluons turn into $q\\bar{q}$ pairs.","Not surprisingly, the process saturates: at very large $\\ln Q^2$ the fraction of the momentum of any target carried by gluons saturates at $\\approx $ 0.47 [38].","The $Q^2$ -dependence of $F_2(x,Q^2)$ is illustrated schematically in Fig.","REF .","Close, Roberts and Ross were struck by the similarity of Fig.","REF to the shape of the EMC effect at fixed $Q^2$ , and were led to the “rescaling\" analysis which is the subject of the next chapter.","Figure: Q 2 Q^2-dependence of the structure function (dF 2 (x,Q 2 )/dlnQ 2 )(dF_2(x,Q^2)/d \\ln Q^2) at Q 2 Q^2 = 3.2 GeV 2 ^2 using F 2 (x,Q 2 )F_2(x,Q^2) from SLAC data .Integrodifferential equations like Eq.","(5.12) are not easily solved directly.","We can gain considerable insight, though, by taking moments in $x$ on both sides and using the special properties of convolutions.","Let $M^n_{\\rm N. S.} = \\int _0^1 dx x^{n-1} f_{\\rm N. S.}(x,t) \\ .","\\qquad \\mathrm {(5.13)}$ Then, it is easy to show from Eq.","(5.12), that $\\frac{dM^n_{\\rm N. S.}}{dt} = \\frac{\\alpha _c}{2 \\pi }B_nM^n_{\\rm N. S.}(t) \\qquad \\mathrm {(5.14)}$ where $B_n = \\int _0^1 dz z^{n-1} \\overline{P}_{q/q} (z) \\ .","\\qquad \\mathrm {(5.15)}$ Eq.","(5.14) is easier to solve.","The character of the solution depends on the $t$ -dependence of $\\alpha _c$ , which I have not yet specified.","In quantum field theories, the effective coupling has a definite and calculable $t$ -dependence [39].","In free field theory $\\alpha _c = 0$ and $M^n_{\\rm N. S.}$ is constant - corresponding to exact Bjorken scaling.","In some model field theories the effective coupling becomes constant at large $t$ , $\\lim _{t \\rightarrow \\infty } \\alpha _c(t) \\rightarrow \\alpha _0$ [40].","This behavior is known as “ultraviolet fixed point\".","In such a theory, the solution of Eq.","(5.14) at large $Q^2$ is $M^n_{\\rm N. S.}(Q^2) \\sim M^n_{\\rm N. S.} (Q^2_0) \\left(\\frac{Q^2}{Q^2_0}\\right)^{\\frac{\\alpha _0 B_n}{2 \\pi }} \\qquad \\mathrm {(5.16)}$ i.e.","scaling is violated by powers of $Q^2$ .","The coefficient $B_n$ is known as the “anomolous dimension\".","In QCD, the effective coupling vanishes as $t \\rightarrow \\infty $ butit vanishes too slowly to use the approximation $\\alpha _c=0$ at large $t$ : $\\alpha _c(t)$ can be shown to have an expansion of the form: $\\frac{\\alpha (0)}{\\alpha _c(t)} = 1 + b \\alpha _c(0)t + O(\\alpha _c(0)^2) \\ , \\qquad \\mathrm {(5.17)}$ or to leading order, $\\alpha _c(t) \\sim \\frac{\\alpha _c(0)}{1+bt\\alpha _c(0)} \\ .","\\qquad \\mathrm {(5.18)}$ The coefficient $b$ is $b = \\frac{1}{4 \\pi } \\left( 11-\\frac{2}{3}N_f \\right) \\ , \\qquad \\mathrm {(5.19)}$ where $N_f$ is the number of quark flavors with masses small compared with $\\sqrt{Q^2}$ , in practice $N_f \\sim 3-4$ .","Eq.","(5.18) can be rewritten $\\alpha _c(Q^2) \\sim \\frac{1}{b \\ln Q^2/\\Lambda ^2} \\qquad \\mathrm {(5.20)}$ where $\\Lambda ^2 = Q^2_0 \\exp (-1/b \\alpha _c(0))$ .","This well-known behavior of $\\alpha _c(Q^2)$ is known as asymptotic freedom.","It is very special to non-Abelian gauge field theories and it makes it possible to calculate with QCD at large $Q^2$ [41].","If $\\alpha _c(t)$ behaves as in Eq.","(5.18), the moments evolve logarithmically: $M^n_{\\rm N. S.}(t) = M^n_{\\rm N. S.}(0)(\\alpha _c(0)/\\alpha _c(t))^{B_n/2 \\pi b} \\ .","\\qquad \\mathrm {(5.21)}$ A more familiar form of Eq.","(5.21) is obtained by changing variables to $Q^2$ and taking the logarithm $\\ln M^n_{\\rm N. S.}(Q^2) \\sim \\ln M^n_{\\rm N. S.}(Q^2_0) + \\frac{B_n\\alpha _c(Q^2_0)}{2 \\pi b} \\ln Q^2/Q^2_0 \\ , \\qquad \\mathrm {(5.22)}$ where we've replaced $\\ln (1+bt\\alpha _c(0)) \\sim bt\\alpha _c(0)$ since we are working to lowest order in $\\alpha _c$ .","The coefficients {$B_n$ }, still known as (non-singlet) anomolous dimensions, can be computed from Eqs.","(5.11) and (5.15) $B_n = \\frac{4}{3} \\left\\lbrace \\frac{1}{n(n+1)} - 2 \\sum _{j=2}^n \\frac{1}{j} -\\frac{1}{2} \\right\\rbrace \\ .","\\qquad \\mathrm {(5.23)}$ $B_1$ is zero, which corresponds to the fact that the number of valence quarks does not change with $Q^2$ .","All the others are negative, so all other moments decrease monotonically with $Q^2$ at large $Q^2$ .","Figure: Testing QCD in inelastic lepton scattering.","At large Q 2 Q^2, [M n ] 2πb/B n [M^n]^{2 \\pi b/B_n} lies on a straight line with a predicted slope when plotted versus lnQ 2 \\ln Q^2.In principle, a particularly simple way to test QCD is to plot the moments of the valence quark distribution versus $\\ln Q^2$ : the slope is predicted by QCD up to a single parameter, $\\alpha (Q^2_0)$ or $\\Lambda $ .","In practice, the situation is quite complicated: $F_2(x,Q^2)$ is not well-measured at all $x$ , so moments are hard to compute; non-singlet quark distributions must be extracted from $F_2(x,Q^2)$ in order to apply the simple analysis I've described; $\\alpha _c(t)$ is not so small where there is good data (remember $d \\sigma /dE^\\prime d\\Omega $ falls like $1/Q^4$ ) so higher corrections to Eq.","(5.17) must be included at least at the low-$t$ end.","Nevertheless, the moments have been extracted and compared with theory and the agreement is excellent as shown in Fig.","REF .","For our purposes, it is important to remember that the logarithms of the moments depend linearly on $\\ln Q^2$ and that the slope is negative and independent of the target.","To get more insight into the $Q^2$ -dependence of deep inelastic structure functions it is necessary to return to coordinate space.","In $§2$ we derived the parton model by studying the spacetime dependence of the product of currents.","This gave us considerable insight into the relation between the $x$ -dependence of $F_2(x)$ and the quark correlation function in the target ground state.","Now I want to look for similar insight into the $Q^2-$ dependence we've found in QCD and to make sure our earlier insight is not spoiled by the interactions we've added.","(It isn't.)","The starting point is Wilson's operator product expansion [42] (OPE).","The idea is that the product of operators simplifies in the limit that their arguments coincide: $\\lim _{\\xi ^\\mu \\rightarrow 0} A(\\xi )B(0) \\sim \\sum _{\\lbrace \\beta \\rbrace } C_{\\lbrace \\beta \\rbrace }(\\xi )O_{\\lbrace \\beta \\rbrace } (0) \\qquad \\mathrm {(5.24)}$ where $C_{\\lbrace \\beta \\rbrace }$ are $C$ -number functions and $O_{\\lbrace \\beta \\rbrace }$ are finite local operators.","$C_{\\lbrace \\beta \\rbrace }$ may depend on any other parameters in the theory such as masses and coupling constants, in addition to $\\xi $ ; $\\lbrace \\beta \\rbrace $ are all labels which might occur in such an expansion (Lorentz and internal symmetry indices, etc.).","The content of Eq.","(5.24) is that the singularities are factored out from the operators and that the terms in the expansion can be organized in decreasing order of singularity as $\\xi _\\mu \\rightarrow 0$ .","Eq.","(5.24) is useful only for $\\xi ^\\mu \\approx 0$ , the short distance limit.","This will translate into information about the Compton amplitude in the limit $Q^2 \\rightarrow \\infty $ and $\\omega \\rightarrow 0$ ($\\omega = 1/x$ ).","Deep inelastic scattering probes the light-cone and requires $\\omega \\ge 1$ , but the dispersion relations discussed in $§1.2$ will provide the necessary connection between the two regimes.","It is easy to check that Eq.","(5.24) works in free field theories.","For example, 1.","In a free, massless, scalar field $\\phi (\\xi )\\phi (0) &= \\mathinner {\\langle {0|\\phi (\\xi )\\phi (0)|0}\\rangle }+ : \\phi (\\xi )\\phi (0): \\nonumber \\\\&= \\frac{i}{4 \\pi ^2(-\\xi ^2+i \\epsilon \\xi ^0)} I \\nonumber \\\\&+ \\sum _{n=0}^{\\infty } \\frac{\\xi ^{\\mu _1}....\\xi ^{\\mu _n}}{n!","}: \\phi (0) \\overleftarrow{\\partial }_{\\mu _1},...\\overleftarrow{\\partial }_{\\mu _n} \\phi (0): $ $I$ is the identity, and $:\\phi (0)...\\phi (0):$ are the regular operators.","Their “coefficient functions\" are $C$ -numbers.","Note that the first term in Eq.","(5.25) diverges as $\\xi _\\mu \\rightarrow 0$ but the rest vanish with successfully higher powers of $\\xi _\\mu $ .","2.","In a free massless Dirac theory let $J(\\xi ) = : \\overline{\\psi }(\\xi )\\psi (\\xi ):$ then in analogy to Eq.","(2.1) et seq.","$[J(\\xi ),J(0)] &= \\frac{1}{2 \\pi }[\\partial ^\\rho \\delta (\\xi ^2)\\epsilon (\\xi ^0)] \\sum _{n=0}^{\\infty } \\frac{\\xi ^{\\mu _1}...\\xi ^{\\mu _n}}{n!}","\\nonumber \\\\&: \\overline{\\psi }(0)\\overleftarrow{\\partial }_{\\mu _1}...\\overleftarrow{\\partial }_{\\mu _n} \\gamma _\\rho \\psi (0) \\nonumber \\\\&- \\overline{\\psi }(0) \\overrightarrow{\\partial }_{\\mu _1}...\\overrightarrow{\\partial }_{\\mu _n} \\gamma _\\rho \\psi (0): \\ .","$ In a free field theory, the Taylor expansions in Eq's.","(5.25) and (5.26) can of course be summed to give a “bilocal operator\" like :$\\overline{\\psi }(\\xi )\\gamma _\\rho \\psi (0)$ : but in interacting theories the coefficient function will differ slightly for each term in the Taylor expansion preventing resummation.","This, it will turn out, makes the difference between exact Bjorken scaling as in free field theory, and logarithmic scaling violation as in QCD.","It is clear from these examples that the form the operator product expansion takes in a particular field theory depends upon the procedure necessary to remove the singularity which arises when one tries to bring operators to the same space time point.","In free field theories the singular piece is a $C-$ number and can be isolated by normal ordering the operator product.","In interacting field theories, the divergences are worse and normal ordering does not suffice.","Instead, it is necessary to renormalize the operators.","There is a certain arbitrariness in renormalization: to define finite operators it is necessary to introduce a mass-scale $\\mu ^2$ — loosely speaking, it is the scale at which the operator's matrix elements have the values they would have in free field theory — but $\\mu ^2$ is arbitrary, nothing physical can depend on it.","Nevertheless, both the operators $O_{\\lbrace \\beta \\rbrace }$ and the coefficient functions $C_{\\lbrace \\beta \\rbrace }$ will separately depend on the mass scale introduced by the necessity of renormalization.","To allow for this we replace $C_{\\lbrace \\beta \\rbrace }(\\xi ) $ by $C_{\\lbrace \\beta \\rbrace }(\\xi ,\\mu ^2)$ and $O_{\\lbrace \\beta \\rbrace }(0)$ by $O^{(\\mu ^2)}_{\\lbrace \\beta \\rbrace }(0)$ .","In anything physically measurable, the $\\mu ^2$ dependence of $C_{\\lbrace \\beta \\rbrace }(\\xi ,\\mu ^2)$ will cancel that of $O_{\\lbrace \\beta \\rbrace }^{(\\mu ^2)}(0)$ .","$\\mu ^2$ is known as the “renormalization point\".","To make use of the OPE, it is convenient to make explicit the factors of $\\xi _\\mu $ which accompany an operator carrying Lorentz indices.","To this end we rewrite Eq.","(5.24) as $A(\\xi )B(0) \\sim \\sum _{\\lbrace \\beta \\rbrace } C^\\prime _{\\lbrace \\beta \\rbrace }(\\xi ^2,\\mu ^2)\\xi _{\\mu _1}.....\\xi _{\\mu _{n_{\\beta }}} O^{\\lbrace \\mu ^2\\rbrace \\mu _1...{\\mu _{n_{\\beta }}}}_{\\lbrace \\beta \\rbrace } \\ .","\\qquad \\mathrm {(5.27)}$ The operators $O_{\\lbrace \\beta \\rbrace }$ can always be defined so they are symmetric traceless in all Lorentz indices: $O^{\\mu _1..\\mu _i..\\mu _j..{\\mu _{n_{\\beta }}}} = O^{\\mu _1..\\mu _j..\\mu _i..{\\mu _{n_{\\beta }}}}$ and $g_{\\mu _i \\mu _j}O^{\\mu _1..\\mu _i..\\mu _j..{\\mu _{n_{\\beta }}}} =0$ [43].","Then, $n_\\beta $ is called the “spin\" of the operators [44].","If $A$ and $B$ carry Lorentz indices then the form of Eq.","(5.27) becomes more complicated.","To avoid writing complicated equations, I will suppress the indices on the currents $J_\\mu $ and $J_\\nu $ or equivalently study deep inelastic scattering by a particle coupled to a hypothetical scalar current $J(\\xi )=\\overline{\\psi }(\\xi )\\psi (\\xi )$ .","Let us expand the product $T(J(\\xi )J(0))$ in the fashion of Eq.","(5.27) and calculate the contributions to $T(q^2,\\omega )$ (see Eq.","(1.12)).","We must carry out the Fourier transform $\\tilde{C}_{\\mu _1...{\\mu _{n_{\\beta }}}}(q,\\mu ^2) \\equiv \\int d^4 \\xi e^{i q \\cdot \\xi } C^\\prime _{\\lbrace \\beta \\rbrace }(\\xi ^2,\\mu ^2)\\xi _{\\mu _1}...\\xi _{\\mu _{n_{\\beta }}} \\ .","\\qquad \\mathrm {(5.28)}$ The Lorentz indices on $\\tilde{C}$ can only be in the form of $q_{\\mu _k}$ or $g_{\\mu _i \\mu _j}$ .","The latter vanish when contracted with the traceless operator $O_{\\lbrace \\beta \\rbrace }$ .","In effect, then $\\tilde{C}_{\\mu _1...{\\mu _{n_{\\beta }}}}(q,\\mu ^2) = \\frac{q_{\\mu _1}...q_{{\\mu _{n_{\\beta }}}}}{(q^2)^{n_\\beta }} (-1)^{n_\\beta } \\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2) \\ .","\\qquad \\mathrm {(5.29)}$ The phase and the factors of $(q^2)^{n_\\beta }$ have been introduced for later convenience.","Note that $\\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2)$ has the same dimension as $\\int d^4 \\xi e^{i q \\cdot \\xi } C^\\prime _{\\lbrace \\beta \\rbrace }(\\xi ^2,\\mu ^2)$ .","Now, $T(q^2,\\omega )$ can be written as $T(q^2,\\omega ) \\sim \\sum _{\\lbrace \\beta \\rbrace }\\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2) (-1)^{n_\\beta } \\frac{q_{\\mu _1}...q_{{\\mu _{n_{\\beta }}}}}{(q^2)^{n_\\beta }} \\mathinner {\\langle {p|O_{\\lbrace \\beta \\rbrace }^{(\\mu ^2)\\mu _1..{\\mu _{n_{\\beta }}}}|p}\\rangle }_c \\ .","\\qquad \\mathrm {(5.30)}$ The matrix element in Eq.","(5.30) must carry Lorentz indices $\\mu _1...{\\mu _{n_{\\beta }}}$ .","We can write $\\mathinner {\\langle {p|O_{\\lbrace \\beta \\rbrace }^{(\\mu ^2)\\mu _1..{\\mu _{n_{\\beta }}}}|p}\\rangle } = \\Theta _{\\lbrace \\beta \\rbrace }(\\mu ^2)(p^{\\mu _1}...p^{{\\mu _{n_{\\beta }}}}+....) \\ , \\qquad \\mathrm {(5.31)}$ where the other terms are determined by the fact that $O_{\\lbrace \\beta \\rbrace }$ is symmteric and traceless.","The terms required to remove traces all contain at least one factor of $g^{\\mu _i\\mu _j}$ .","Thus, for example, if $n_\\beta =2$ , one has $p^{\\mu _1}p^{\\mu _2}-\\frac{M^2_p}{4}g^{\\mu _1\\mu _2}$ .","Combining Eqs.","(5.30) and (5.31) we find $T(q^2,\\omega ) \\sim \\sum _\\beta \\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2) \\Theta _{\\lbrace \\beta \\rbrace }(\\mu ^2) \\left[ (\\frac{\\omega }{2})^{n_\\beta } + O(\\frac{1}{q^2}) \\right] \\ , \\qquad \\mathrm {(5.32)}$ ($\\omega \\equiv -2p \\cdot q/q^2$ ) because $p^\\alpha p^\\beta q_\\alpha q_\\beta /(q^2)^2 = \\omega ^2/4$ but $g^{\\alpha \\beta }q_\\alpha q_\\beta /(q^2)^2 = 1/q^2$ .","In fact, only even powers of $\\omega $ occur in Eq.","(5.32) because $T(q^2,\\omega )$ is crossing symmetric (see $§1.2$ ).","Since $\\omega $ ($\\cong 1/x$ ) scales in the Bjorken limit, the importance of any particular operator $O_{\\lbrace \\beta \\rbrace }$ is determined by the large-$q^2$ behavior of $\\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2)$ .","This can be calculated to all orders in perturbation theory using the methods of the renormalization group.","The first step is to determine the dimension of $\\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2)$ .","We define the “naive\" or “canonical\" dimension (or, for short, simply dimension) of an operator to be the units in which it is measured as a power of mass (with $\\hbar = c= 1$ ).","We use the notation $[O_{\\lbrace \\beta \\rbrace }] = d_{\\lbrace \\beta \\rbrace }$ .","Here are some examples: $[J_\\mu ]=3$ because $\\int d^3x J_0 =$ charge which is dimensionless; $[\\phi ]=1$ because the action, $S=\\int d^4 x \\frac{1}{2}(\\partial \\phi )^2$ , is dimensionless; likewise $[\\psi ] = \\frac{3}{2}$ .","Using these dimensions and remembering $\\mathinner {|{p}\\rangle }$ is covariantly normed, $[\\mathinner {|{p}\\rangle }]=-1$ , we see that $T(q^2,\\omega )$ is dimensionless.","Let us suppose the operator $O_{\\lbrace \\beta \\rbrace }$ has dimension $d_\\beta $ , then from Eq.","(5.27), $[C^\\prime _{\\lbrace \\beta \\rbrace }]=6-d_\\beta -n_\\beta \\qquad \\mathrm {(5.33)}$ and $[\\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2)] = 2-d_\\beta +n_\\beta \\ .","\\qquad \\mathrm {(5.34)}$ T he dimensions of $\\tilde{C}_{\\lbrace \\beta \\rbrace }$ could, in principle, be provided by $q^2$ , $\\mu ^2$ or quark masses.","Weinberg's theorem [45] can be used to show that at large $-q^2$ to each order in perturbation theory quark masses can be ignored provided $\\mu ^2$ is fixed and not zero.","Also, in each order of perturbation theory the $\\mu ^2$ dependence of a renormalizable field theory is at most logarithmic, reflecting at worst logarithmic divergences in such theories.","So we can write $\\lim _{q^2 \\rightarrow -\\infty }\\tilde{C}_{\\lbrace \\beta \\rbrace }(q^2,\\mu ^2) = \\left( \\frac{1}{q^2} \\right)^{d_\\beta -n_\\beta -2} c_{\\lbrace \\beta \\rbrace }(\\ln q^2/\\mu ^2) \\ .","\\qquad \\mathrm {(5.35)}$ The next step is to calculate the leading large $-q^2$ dependence of $c_{\\lbrace \\beta \\rbrace }(\\ln q^2/\\mu ^2)$ to all orders in perturbation theory.","In each order, $c_{\\lbrace \\beta \\rbrace }$ grows like a power of $\\ln q^2/\\mu ^2$ .","When summed to all orders the logarithms may yield a power, i.e., $c_{\\lbrace \\beta \\rbrace }$ may depend exponentially on its argument.","The actual calculation of the asymptotic behavior of $c_{\\lbrace \\beta \\rbrace }$ requires renormalization group methods beyond the scope of these lectures.","But as will be seen below, we have performed an equivalent calculation using Weizsäcker-Williams methods in $§5.1$ .","If $c_{\\lbrace \\beta \\rbrace }$ does not go like an exponential of its argument, then operators with $d_\\beta -n_\\beta =2$ give rise to Bjorken scaling modulo powers of $\\ln q^2$ .","$d_\\beta -n_\\beta $ plays such a central role in the analysis of large $q^2$ effects that it is given a name, “twist\", $t_\\beta =d_\\beta -n_\\beta \\ .","\\qquad \\mathrm {(5.36)}$ Operators with $t_\\beta < 2$ would give contributions which diverge in the Bjorken limit, but the only operator with $t_\\beta <2$ which can couple to the product of the two currents is the identity, $t_I=0$ , and the identity has no connected matrix elements.","Operators with $t_\\beta >2$ give contributions to $T(q^2,\\omega )$ which vanish in the Bjorken limit (provided $c_{\\lbrace \\beta \\rbrace }$ is not exponential in its argument).","Only even twists occur in the expansion of two currents in the limit of zero quark mass [46] so the next important case is $t_\\beta = 4$ .","Twist-4 or $O(1/q^2)$ corrections to scaling are a rich and fascinating, if technically complicated subject, in themselves [47].","The twist-two operators in QCD come in two classes: quark operators $O_{n,a}^{(\\mu ^2)\\mu _1..\\mu _n} = S \\overline{\\psi }_a \\gamma ^{\\mu _1} \\partial ^{\\mu _1}D^{\\mu _2}...D^{\\mu _n} \\psi _a^{(\\mu ^2)} \\ , \\qquad \\mathrm {(5.37)}$ where $D_\\mu $ is the (color) gauge covariant derivative [48], “a\" labels flavor and $S$ symbolizes the operation of making $O_{n,a}$ traceless and symmetric; and gluon operators which we needn't write out, since they contribute only to singlet distributions.","It is easy to check that $O_{n,a}$ indeed has $t=2$ for all $n$ .","It should be emphasized that it requires an infinite tower of operators of increasing spin to describe $T(q^2,\\omega )$ .","This is not at all surprising.","First of all, if the sum in Eq.","(5.30) stopped at some $n_{\\rm max}$ , $T(q^2,\\omega )$ would be a polynomial in $\\omega $ with no cut on the real axis for $|\\omega |>1$ .","Second, we know that deep inelastic electroproduction probes the light cone, not just short distances.","The OPE is a short distance expansion ($\\xi _\\mu \\rightarrow 0$ ) and no finite number of terms in the short distance expansion gives information about light-like separation.","To summarize: the OPE approach leads to a simultaneous expansion of $T(q^2,\\omega )$ in $q^2$ and $\\omega $ .","The expansion in $q^2$ is an asymptotic expansion and is ordered by the twist quantum number of the operators, the expansion in $\\omega $ is a Taylor expansion (which converges for $|\\omega |<1$ , see $§1$ ) and is ordered by the spin of the operators.","Ignoring gluon operators $-$ which is adequate if we are interested in non-singlet quark distributions alone $-$ we have $T_{\\rm N. S.}(q^2,\\omega ) = \\sum _{\\rm n\\ even} c^n_{\\rm N. S.}(\\ln q^2/\\mu ^2) \\Theta ^n_{\\rm N. S.} (\\mu ^2) \\omega ^n + {\\rm higher\\ twist} \\ , \\qquad \\mathrm {(5.38)}$ where we have replaced the generic label $\\beta $ by the spin ($n$ ) of the 2 twist-two, non-singlet quark operator.","$c^n_{\\rm N. S.}(\\ln q^2/\\mu ^2)$ depends only on $\\ln q^2/\\mu ^2$ and explicit calculation shows that it equals 4 when $q^2 = \\mu ^2$ .","Comparing Eq.","(5.38) with the Taylor expansion of $T(q^2,\\omega )$ developed in $§1$ , (Eq.","1.18), we identify $M^n_{\\rm N. S.}(q^2) = \\frac{1}{4} c^n_{\\rm N. S.}(\\ln q^2/\\mu ^2) \\Theta ^n_{\\rm N. S.}(\\mu ^2) \\ {\\rm for\\ n\\ even} \\ .","\\qquad \\mathrm {(5.39)}$ At this point, we can make a connection with the “evolutionary\" approach of $§5.1$ and comment on the derivation of the parton model in QCD.","The ordering of contributions at large $Q^2$ by twist has given a result, Eq.","(5.38), which looks like scaling modulo logarithms.","This is deceptive because in general $c^n_{\\rm N. S.}$ depends exponentially on its argument giving rise to power law violations of scaling.","Only in asymptotically free theories like QCD does $c^n_{\\rm N. S.}$ go like a power of its argument, giving scaling up to logarithms.","The derivation of the dependence of $c^n_{\\rm N. S.}$ on $\\ln q^2/\\mu ^2$ is outside the scope of these lectures, but the result $c^n_{\\rm N. S.}(\\ln Q^2/Q^2_0) = (\\alpha _c(0)/\\alpha _c(t))^{B_n/2 \\pi b} \\qquad \\mathrm {(5.40)}$ (where $\\mu ^2 \\equiv Q^2_0$ ) should not be surprising since it converts Eq.","(5.39) into Eq.","(5.21) which we already derived using the more heuristic, Weizäcker-Williams approach.","Comparing Eqs.","(5.39), (5.40) and (5.21) we see that the moments of the structure functions are directly related to the matrix elements of specific local operators, $M^n_{\\rm N. S.}(q^2)=\\Theta ^n_{\\rm N. S.}(q^2)$ where $q^2$ is the mass-scale at which the operator is $O^{\\mu _1...\\mu _n}_{n,\\rm N. S.}$ is renormalized.","The quark operators $O_{n,a}$ which determine the moments are the (gauge invariant) terms in the Taylor expansion of the operator product $\\overline{\\psi }(\\xi ) \\gamma _\\rho \\psi (0)$ which determined the quark distribution function in the parton model.","So we see that the modification of the parton model required by the interactions in QCD is that each moment of the quark distribution (i.e., each term in the Taylor expansion) scales modulo a slightly different power of $\\ln q^2$ .","Earlier, I remarked that the normalization point, $\\mu ^2$ , was arbitrary, that nothing physical could depend on it.","There is no contradiction here.","Eq.","(5.30) is independent of $\\mu ^2$ , specifically $&\\Theta ^n_{\\rm N. S.}(\\mu ^2)d/d\\mu ^2 c^n_{\\rm N. S.}(\\ln q^2/\\mu ^2) \\nonumber \\\\&= -c^n_{\\rm N. S.}(\\ln q^2/\\mu ^2) d/d\\mu ^2 \\Theta ^n_{\\rm N. S.} (\\mu ^2) \\ .","$ But, $c^n_{\\rm N. S.}$ depends only on $\\ln q^2/\\mu ^2$ , so $d/d\\mu ^2 c^n_{\\rm N. S.}(\\ln q^2/\\mu ^2) = -d/dq^2 c^n_{\\rm N. S.}(\\ln q^2/\\mu ^2) \\qquad \\mathrm {(5.42)}$ and therefore $d/dq^2 M^n_{\\rm N. S.}(q^2) = d/d \\mu ^2 \\Theta ^n_{M.S.","}(\\mu ^2) |_{\\mu ^2=q^2} \\ , \\qquad \\mathrm {(5.43)}$ so the renormalization point dependence of the operator matrix elements determines the $q^2-$ dependence of the moment of the structure function.","Combining the OPE analysis with the evolutionary picture of the previous section we recognize that the renormalization point introduced in the OPE analysis is the same as the the transverse resolution in the Weizäcker-Williams method.","In both cases, it is necessary to define how much of the gluon field is to be lumped into the definition of a quark.","Whatever way you look at it, the fact that this definition changes with $Q^2$ gives rise to the logarithmic scaling violation of QCD.","In addition to the $\\ln Q^2$ corrections we have uncovered, there are expected to be $O(1/Q^2)$ and higher order $O(1/Q^{2n})$ corrections which become important at small $Q^2.$ The $O(1/Q^2)$ corrections take several forms.","They are easy to distinguish using the language of OPE.","First are “target mass\" corrections.","As will become clear, this is an unfortunate name.","These are apparent $O(M^2_T/Q^2)$ terms which arise from the $g^{\\mu _i \\mu _j}$ factors in the matrix elements of traceless operators.","It has been shown [49] that these corrections may be completely absorbed by replacing the variable $x_T$ by the variable $\\xi = -q^+/P^+$ everywhere in the definition of $f_{a/T}(x_T)$ .","This isn't surprising since $q^+$ is the variable which emerged automatically in the derivation (see, e.g., Eq.","(2.18)).","$\\xi $ is written in many forms: $\\xi &= (\\sqrt{\\nu ^2 + Q^2}-\\nu )/M_T \\ , \\\\\\xi &= 2 x_T/(1+ \\sqrt{1+Q^2/\\nu ^2}) \\ , \\\\\\xi &= 2x_T/(1+\\sqrt{1+4 M^2_Tx^2_T/Q^2}) \\ .","$ It is clear from Eq.","(5.44b) that the large mass corrections do not, in fact, depend on the target mass [50]!","They are kinematic corrections (which do not grow like $A^2$ for nuclear targets), and are well-understood.","Second are “quark mass\" corrections.","These are important for heavy quarks ($c,b,t$ ) but negligible for up and down quarks whose masses, $m_{u,d} < 20$ MeV, are tiny.","Finally are the dynamical $O(1/Q^2)$ corrections associated with operators of twist-4.","Although they are complicated, twist-4 corrections to inelastic electron (and neutrino) scattering have been completely analyzed in QCD [47].","Typically, they are small because the natural mass scale associated with a target is one upon its radius (once “target mass\" corrections have been incorporated via $\\xi -$ scaling), $1/R \\sim 1$ fm$^{-1} \\sim $ 200 MeV.","It is therefore not surprising that scaling, modulo logarithms and using the $\\xi $ variable, sets in at a very low value of $Q^2$ ($<$ 1 GeV$^2$ ).","For precisely this reason higher twist contributions to $F_2(x,Q^2)$ , which measure matrix elements of interesting local operators, are hard to extract from available experimental data.","We have seen that the quark, antiquark and gluon content of a hadron changes with the scale at which it is probed.","In more naive quark models the nucleon, for example, is treated as (approximately) three quarks in some confining “bag\" with no reference to the scale at which this description might hold.","Certainly, if the nucleon were three quarks at some scale $\\mu ^2_0$ ($\\mu ^2_0 \\sim 1$ GeV$^2$ ) then it would become more complicated, containing antiquarks and glue at larger scales $Q^2 > \\mu ^2_0$ by virtue of QCD radiation.","In the early days of QCD it was recognized that if the nucleon's quark, antiquark and gluon distributions measured at large $Q^2$ were evolved back to lower $Q^2$ , then quark-antiquark pairs and gluons are reabsorbed into the valence quarks, so that at some $\\mu ^2_0 \\sim $ 1 GeV$^2$ all of the $q \\bar{q}$ pairs and most of the glue would be gone leaving a nucleon made of three quarks alone.","G. Ross and I checked this quantitatively in the M.I.T.","version of the bag model [52].","Using QCD evolution to second order, which is necessary because $\\alpha (\\mu ^2_0)$ is not small, we found that measured non-singlet nucleon structure functions evolved backwards to a $\\mu ^2_0$ of order 1 GeV$^2$ indeed gave valence quark $x-$ distributions in agreement with earlier bag calculations [53].","$\\mu ^2_0$ is then interpreted as a parameter of the quark model: It is the mass scale (or resolution) at which quark fields should be defined in order that the nucleon should be made of three quarks.","A recent reevaluation of this program with modern values for structure functions and the QCD $\\Lambda $ parameter (c.f.","Eq.","(5.20)) gave $\\mu ^2_0 \\cong 0.75$ GeV$^2$ [58].","Notice that the structure function predicted by simple quark models cannot be compared directly with experimental measurements of $F_2$ at $Q^2=\\mu ^2_0$ because at such a low $Q^2$ higher twist effects are large but have not been included in the quark model calculations.","It seems best to regard quark models as models for the twist-two matrix elements at a renormalization point $\\mu ^2_0$ , which must then be evolved to $Q^2 >> \\mu ^2_0$ in order to be compared with experiment.","Close, Roberts and Ross [55] realized that the scale ($Q^2$ ) dependence of quark distribution functions in QCD could be used to parametrize and, to some extent, explain the $A$ dependence of structure functions [56].","In $§2$ , we learned that the shift in the valence quarks observed in nuclei could be understood as an increase in the quark correlation length in the nuclear ground state.","The increase in ocean quark pairs appeared to be an independent phenomenon.","In the QCD inspired analysis, I shall describe both aspects of the EMC effect have a single origin: a dynamical change in scale of the twist-2 matrix elements in nuclei.","In the last chapter, we saw that QCD evolution reduced the momentum on valence quarks and increased the number of pairs.","Suitably adapted, evolution can explain the EMC effect.","This method of analysis has come to be known as “dynamical rescaling\" or simply “rescaling\".","Its virtues are first, it gives a unified description of all aspects of the EMC effect; second, it avoids the dubious assumptions of the constituent convolution models of $§4$ ; and third, it gives us insight into the reason other superficially quite different “explanations\" of the EMC effect work.","Its drawback is that it does not provide a microscopic enough explanation to satisfy most of us: it is not clear exactly what the quarks and gluons are doing differently in a nucleus which gives rise to the effect.","In this chapter, I will work from the general toward the specific.","First, I will merely use QCD as an aid to present the data in a new way.","This presentation will lead to a surprising conclusion and suggest rescaling as a mechanism behind the EMC effect.","Then, I will analyze rescaling in some detail.","Next, I will describe a calculation of the $A$ dependence motivated by, but perhaps more general than rescaling [10].","Finally, I will close with some remarks about shadowing and future experiments.","These have little to do with QCD and less to do with rescaling, but they follow naturally upon the discussion of $A$ dependence.","In $§2$ , we compared the structure functions of different nuclei at fixed $Q^2$ , as functions of $x$ .","This is the way the data come from the experimenters.","QCD provides an alternative.","Consider the moments: $M^n_A(Q^2) \\equiv \\int _0^A dx x^{n-2} \\overline{F}_2^A(x,Q^2) \\qquad \\mathrm {(6.1)}$ ($\\overline{F}^A_2=\\frac{1}{A}F^A_2$ ).","According to Eq.","(5.21), the moments are monotonically falling functions of $Q^2$ .","(This analysis like that of $§5$ is restricted to non-singlet structure functions but a similar conclusion applies as well to singlets.","): $M^n_A(Q^2) = \\left( \\frac{\\alpha _c(Q^2)}{\\alpha _c(Q^2_o)} \\right)^{d_n} M^n_A(Q^2_0) \\ , \\qquad \\mathrm {(6.2)}$ where $d_n \\equiv -B_n/2 \\pi b>0$ .","If QCD is correct, and if $Q^2$ is large enough so leading order in perturbation theory suffices and $O(1/Q^2)$ corrections are negligible, then $\\ln M^n_A(Q^2)$ must lie on a straight line when plotted versus $\\ln [\\alpha _c(Q^2_0)/\\alpha _c(Q^2)]$ and he slope must be $-d_n$ .","Such a plot is shown schematically in Fig.","REF for two different targets with baryon numbers $A$ and $A^\\prime $ .","Figure: Rescaling a single moment.At fixed $Q^2$ , the EMC effect appears as the observation that $M^n_{A^\\prime }(Q^2)<M^n_A(Q^2)$ for $A^\\prime > A$ .","On the other hand, it is clear that there is a value of $Q^2$ , call it $Q^{\\prime \\,2}$ , such that $M^n_{A^\\prime }(Q^{\\prime \\,2})=M^n_A(Q^2)$ .","As illustrated in Fig.","REF , $Q^{\\prime \\,2} < Q^2$ .","The value of $Q^{\\prime \\,2}$ , in principle, depends on $A, A^\\prime $ and $Q^2$ and on $n$ .","Let us define $\\xi ^n_{AA^\\prime }(Q^2)$ (not to be confused with the coordinate $\\xi _\\lambda $ or the modified scaling variable $\\xi $ of $§5$ ) so $M^n_{A^\\prime }(Q^2) = M^n_A(\\xi ^n_{AA^\\prime }(Q^2)Q^2) \\ .","\\qquad \\mathrm {(6.3)}$ From its definition $\\xi ^n_{AA^\\prime }(Q^2)>1 \\ \\ {\\rm for\\ } A^\\prime > A \\qquad \\mathrm {(6.4)}$ and $\\xi ^n_{AA^{\\prime \\prime }}(Q^2) &= \\xi ^n_{AA^\\prime }(Q^2) \\xi ^n_{A^\\prime A^{\\prime \\prime }}(Q^2) \\nonumber \\\\\\xi ^n_{AA^\\prime }(Q^2) &= 1/\\xi ^n_{A^\\prime A}(Q^2) \\ .","$ The $Q^2-$ dependence of $\\xi ^n_{AA^\\prime }(Q^2)$ is determined entirely by QCD, and is independent of $A$ and $A^\\prime $ .","Consider Eq.","(6.2) first for $A^\\prime $ at $Q^2_0$ and $Q^2$ $M^n_{A^\\prime }(Q^2) = \\left( \\frac{\\alpha _c(Q^2)}{\\alpha _c(Q^2_o)} \\right)^{d_n} M^n_{A^\\prime }(Q^2_0) \\qquad \\mathrm {(6.6)}$ and then for $A$ at $\\xi ^n_{AA^\\prime }(Q^2_0)Q^2_0$ and $\\xi ^n_{AA^\\prime }(Q^2)Q^2$ : $M^n_{A}(\\xi ^n_{AA^\\prime }(Q^2)Q^2) = \\left( \\frac{\\alpha _c(\\xi ^n_{AA^\\prime }(Q^2)Q^2)}{\\alpha _c(\\xi ^n_{AA^\\prime }(Q^2_0)Q^2_0)} \\right)^{d_n} M^n_{A}(\\xi ^n_{AA^\\prime }(Q^2_0)Q^2_0) \\ .","\\qquad \\mathrm {(6.7)}$ Now use Eq.","(6.3) to eliminate all reference to moments $\\frac{\\alpha _c(\\xi ^n_{AA^\\prime }(Q^2)Q^2)}{\\alpha _c(\\xi ^n_{AA^\\prime }(Q^2_0)Q^2_0)} = \\frac{\\alpha _c(Q^2)}{\\alpha _c(Q^2_0)} \\ .","\\qquad \\mathrm {(6.8)}$ To lowest order in $\\alpha _c$ , $\\alpha _c(\\xi Q^2) = \\alpha _c(Q^2)/(1+ \\alpha _c(Q^2)b \\ln \\xi ) \\ , \\qquad \\mathrm {(6.9)}$ which, together with Eq.","(6.8), implies $\\xi ^n_{AA^\\prime }(Q^2) = [\\xi ^n_{AA^\\prime }(Q^2_0)]^{\\alpha _c(Q^2_0)/\\alpha _c(Q^2)} \\ .","\\qquad \\mathrm {(6.10)}$ The extension to next order in $\\alpha _c$ is quoted in [58].","Suppose $Q^2_0 < Q^2$ , then $\\alpha _c(Q^2_0)/\\alpha _c(Q^2)>1$ and $1< \\xi ^n_{AA^\\prime }(Q^2_0) < \\xi ^n_{AA^\\prime }(Q^2) \\ \\ {\\rm for \\ \\ } Q^2_0<Q^2 \\ .","\\qquad \\mathrm {(6.11)}$ The implication of this result is that a small change of $Q^2$ -scale at low $Q^2$ gets magnified into a large change when observed at large $Q^2$ .","Eqs.","(6.4), (6.5) and (6.10) summarize the properties of $\\xi ^n_{AA^\\prime }(Q^2)$ which can be determined from general considerations alone.","The surprise came when Close, Roberts and Ross used this method to analyze the EMC data and found that to a good approximation $\\xi ^n_{AA^\\prime }(Q^2)$ appears to be independent of $n$: $\\xi ^n_{AA^\\prime }(Q^2) \\Rightarrow \\xi _{AA^\\prime }(Q^2)$ , at least for the values of $n$ sensitive to the $x$ -range of the EMC data.","Actually CRR did not construct moments but made an equivalent observation about the structure function itself.","Namely, if $\\xi ^n_{AA^\\prime }(Q^2)$ is independent of $n$ then the structure functions themselves as functions of $x$ are related by a universal scale change in $Q^2$ : $\\overline{F}^{A^\\prime }_2(x,Q^2) = \\overline{F}^A_2(x,\\xi _{AA^\\prime }(Q^2) Q^2) \\ .","\\qquad \\mathrm {(6.12)}$ Eq.","(6.12) has become known as “rescaling\".","CRR were led to it by the observation we referred to in $§5$ , that the EMC effect resembles QCD evolution.","In fact, the EMC data are not in complete agreement with Eq.","(6.12).","The excess at low $x$ is more than can be produced by the amount of evolution that is required to fit the depletion at large $x$ .","Their analysis, with $\\xi _{DFe} \\cong 2$ at $Q^2 \\approx $ 20 GeV$^2$ is shown in Fig.","REF .","Figure: Rescaling the EMC iron data .The newer SLAC data on iron and deuterium data have a smaller enhancement at low-$x$ and a lower cross over point (where $\\overline{F}^{\\rm Fe}_2/\\overline{F}^{D}_2 =1$ ), both of which improve the agreement with the rescaling analysis [57], [58].","The reader might wish to look back to Fig.","REF to see the present state of the rescaling fits.","Several comments and caveats are in order: 1.","QCD is subtle: Changing the scale creates quark-antiquark pairs.","If, after all the discussion of $§5$ , this still seems unreasonable, perhaps it would help to remember that a similar thing happens in a Bogoliubov transformation.","By redefining the vacuum, annihilation and creation operators get mixed up with one another and a state which originally contained only particles, appears after the transformation, to contain both particles and antiparticles.","2.","Rescaling predicts that the EMC effect should vanish at $x \\approx 0.2$ where QCD evolution vanishes (see Fig.","REF ).","The EMC data cross unity at $x \\approx 0.35$ in disagreement with this prediction.","Once again, however, the SLAC data look better: $\\overline{F}^{A}_2/\\overline{F}^{D}_2$ is definitely below unity for $x>0.3$ .","A careful test must await better data at small-$x$ .","3.","Rescaling cannot work for $x \\rightarrow 1$ , or equivalently for $n \\rightarrow \\infty $ .","At $x \\rightarrow 1$ , the structure function of the nucleon vanishes but that of a nucleus does not.","Eq.","(6.12) fails to reproduce this behavior.","The reason for this failure will become clear soon.","4.","Rescaling does not mean that an iron nucleus can be “mapped\" to a nucleon by a universal change of scale.","The complete description of a nucleus requires matrix elements of all twists.","The gross structure of the EMC effect only involves a few operators of twist-two.","In $§5.4$ we learned to associate a mass scale $\\mu ^2_0$ with the quark model description of a hadron.","It is the scale at which, approximately, the hadron consists of valence quarks alone.","It is an intrinsic characteristic of each hadron.","The measured structure function of the nucleon is consistant with this notion with $\\mu ^2_0 \\cong 0.75$ GeV$^2$ .","Clearly, any nucleus related to the nucleon by rescaling, as in Eq.","(6.12), also admits a valence quark description, but at a shifted mass scale $\\mu _A$ : $\\mu ^2_A = \\frac{\\mu ^2_0}{\\xi _{NA}(\\mu ^2_0)} \\ .","\\qquad \\mathrm {(6.13)}$ Since $\\xi _{NA}>1$ , $\\mu ^2_A$ is less than $\\mu ^2_0$ .","This can be checked by the following argument: At any fixed $Q^2$ , the nucleus appears more highly evolved than the nucleon (more pairs, softer valence quarks).","So to reabsorb all pairs it is necessary to devolve the nucleus further than the nucleon.","It has become necessary to express Eq.","(6.13) as a ratio of length scales.","Let us define $\\lambda _A = c/\\mu _A$ , $\\lambda _N = c/\\mu _0$ , then $\\lambda _A/\\lambda _N = \\mu _0/\\mu _A = \\sqrt{\\xi _{NA}(\\mu ^2_0)} \\ \\qquad \\mathrm {(6.14)}$ $\\xi (\\mu ^2_0)$ is typically much smaller than $\\xi (Q^2)$ , because of Eq.","(6.10).","For example, $\\xi _{\\rm Fe\\,N}(20\\ {\\rm GeV^2}) \\cong 2.0$ , but $\\xi _{\\rm Fe\\,N}(\\mu ^2_0) \\cong 1.33$ making $\\lambda _{\\rm Fe}/\\lambda _N \\cong 1.15$ .","Thus, an increase of only $\\sim $ 15% in the intrinsic length scale of the twist-two matrix elements for iron compared to the nucleon can give rise to the EMC effect.","If there is anything fundamental about rescaling, rather than being merely an accident, the basic relation must be defined at the intrinsic to the target, $\\mu ^2_0$ , not at some arbitrary $Q^2$ [55].","At $\\mu ^2_0$ , however, the structure function contains large contributions from higher twists, so an equation like Eq.","(6.12) with $Q^2$ replaced by $\\mu ^2_0$ cannot be written down.","Instead, we must write relations between matrix elements of twist-two operators using the formalism of $§5.2$ .","We define $\\mathinner {\\langle {P|O_{n,a}^{(\\mu ^2)\\mu _1..\\mu _n}|P}\\rangle } = P^{\\mu _1}...P^{\\mu _n}\\Theta ^A_{n,a}(\\mu ^2) \\qquad \\mathrm {(6.15)}$ in analogy to Eq.","(5.31).","Then the operator equivalent of Eq.","(6.12) is $A^{n-2} \\Theta ^A_{n,a}(\\mu ^2_A) = A^{\\prime \\,{n-2}} \\Theta ^{A^\\prime }_{n,a} (\\mu ^2_{A^\\prime }) \\ .","\\qquad \\mathrm {(6.16)}$ This is the basic rescaling between targets.","The powers of $A$ and $A^\\prime $ are only kinematic.","It leads to Eq.","(6.12) at large $Q^2$ provided $\\xi _{AA^\\prime }(Q^2)$ is obtained from $\\mu _A/\\mu _A^\\prime $ via QCD evolution (Eq.","(6.10) plus higher order improvements) and provided the moments can be reliably evolved from $\\mu _A^2$ to $Q^2$ using perturbative QCD.","This turns out to be an important proviso.","In [52], we found that only the moments with $n<8$ could be reliably devolved from large $Q^2$ to a $\\mu ^2_0$ as small as 1 GeV$^2$ .","The problem is that higher order (in $\\alpha _c$ ) corrections typically contain $\\ln n$ factors making perturbation theory worse for large $n$ .","Suppose, then, that Eq.","(6.16) were exact for all $n$ .","Nevertheless, at large $Q^2$ only the moments with $n<8$ would be likely to show uniform scaling.","This explains why rescaling fails near $x=1$ (as noted earlier) since large $n$ moments probe exclusively large $x$ .","In fact, one can estimate the $x$ values for which rescaling should be reliable when observed at large $Q^2$ by using the Mellin transform relation [58].","$n \\sim \\ln \\alpha _s/ \\ln x \\qquad \\mathrm {(6.17)}$ which for $n=8$ and $\\alpha _s =0.2$ gives $x \\sim 0.8$ as the upper limit of reliability.","Turning this argument around one can see that exact uniform rescaling for all $x$ at large $Q^2$ would be very hard to understand, since it would imply a complicated and non-uniform relation among twist-two matrix elements with $n>8$ at $\\mu ^2_0$ .","The task for someone trying to understand the EMC effect from the point of rescaling, then, is two-fold.","First, one must explain why rescaling should be uniform at the intrinsic scale, i.e.","why should $\\lambda _A/\\lambda _{A^\\prime }$ be independent of $n$ ?","Second, one must predict the $A$ dependence of the rescaling parameter, i.e.","of $\\lambda _A/\\lambda _N$ .","Close, Roberts, Ross and I have argued that rescaling is a rather natural prediction of quark models with only one length scale.","An example is the MIT bag model with only $u$ and $d$ quarks, which may be taken to be massless, so the only dimensionful parameter is the bag constant $B$ .","Such models are very close in spirit to QCD itself in which the only dimensionful parameter is $\\Lambda $ .","In a model like this, quarks carrying momentum $p$ confined within a radius $\\lambda $ transform into quarks carrying momentum $p^\\prime =(\\lambda /\\lambda ^\\prime )p$ when the confinement scale is changed to $\\lambda ^\\prime $ - the dimensionless quantity $p \\lambda $ is constant.","The intrinsic scale $\\mu ^2$ is then proportional to $p^2$ , there being no other scale in the problem.","It is hard to turn this heuristic argument into a proof of uniform rescaling.","That would probably require a consistent formulation of perturbative QCD (including renormalization) in a bag model, something which exists only in fragments [62].","On the other hand, as Llewellyn Smith has noted, it seems clear that other models such as the non-relativistic quark model have no hope of giving rescaling unless the quark masses are assumed (rather unnaturally) to scale with the inverse confinement radius [10].","A satisfactory theoretical understanding of rescaling will have to await a more powerful QCD-based theory of confinement.","The second task $-$ determining the $A$ dependence of $\\lambda _A/\\lambda _N$ $-$ is more straightforward.","It is the subject of the next Section.","It hardly needs saying that the EMC effect derives from the proximity of nucleons within the nucleus, and that it would vanish if one could arrange that that the nucleus was very dilute.","Fortunately, nature has given us one very dilute nucleus $-$ the deuteron $-$ and Bodek and Simon [15] have shown that $\\overline{F}^D_2(x,Q^2)/F^N_2(x,Q^2)$ is very close to unity.","It seems reasonable to assume, therefore, that the magnitude of the EMC effect should be proportional to the probability that nucleons approach each other or overlap within the nucleus.","Close, Roberts, Ross and I [57], [58] defined the simplest measure of this effect we could imagine: we defined an “overlapping\" volume for a nucleus which is the integral over the nucleus of the two body density $\\rho _A({\\bf r_1},{\\bf r_2})$ multiplied by an overlap factor $V_0(|{\\bf r_1-r_2}|)$ which we took to be the overlapping volume of two spheres of radius $a$ : $V_0(d) &= 1-\\frac{3}{4} \\left(\\frac{d}{a} \\right) + \\frac{1}{16}\\left(\\frac{d}{a} \\right)^3 {\\ } &d \\le 2a \\nonumber \\\\&= 0 {\\ \\ \\ \\ \\ \\ } &d>2a \\ .","$ Thus, the overlapping volume per nucleon is $V_A =(A-1) \\int d^3{\\bf r_1} d^3 {\\bf r_2} \\rho _A({\\bf r_1},{\\bf r_2}) V_0(|{\\bf r_1-r_2}|) \\ .","\\qquad \\mathrm {(6.19)}$ $\\rho _A({\\bf r_1},{\\bf r_2})$ is normalized to $\\int d^3{\\bf r_1} d^3 {\\bf r_2} \\rho _A({\\bf r_1},{\\bf r_2})=1$ .","Saturation of the nuclear density at large $A$ implies $\\rho _A \\sim 1/A^2$ .","With this behavior of $\\rho _A$ and the finite integral of $V_0$ it is easy to see that $V_A$ saturates at large $A$ , i.e.","$\\lim _{A \\rightarrow \\infty } V_A =$ constant.","The choice of a geometrical form for $V_0(|{\\bf r_1-r_2}|)$ was in fact quite arbitrary.","Any function which goes to unity as ${\\bf r_1-r_2} \\rightarrow 0$ and to zero when $|{\\bf r_1-r_2}|>2R_{\\rm nucleon}$ and which respects the three dimensional geometry of the problem would do.","We calculated the overlapping volume for nucleons with $a$ chosen so $a_{\\rm RMS} = 0.9$ fm ($a_{\\rm RMS}=\\sqrt{\\frac{3}{5}}a$ , so $a=1.16$ fm).","$\\rho _A({\\bf r_1},{\\bf r_2})$ was written in terms of the single particle density $\\rho _A({\\bf r})$ and a correlation function $f({\\bf r_1-r_2})$ : $\\rho _A({\\bf r_1-r_2}) = \\rho _A({\\bf r_1}) \\rho _A({\\bf r_2}) f({\\bf r_1-r_2}) \\ .","\\qquad \\mathrm {(6.20)}$ $\\rho _A({\\bf r})$ was taken from experimental measurements of nuclear charge densities.","$f({\\bf r_1-r_2})$ should, in principle, depend on $A$ but there is little or no direct information on it from experiment.","We took $f({\\bf r_1-r_2})$ from models of nuclear matter, the most realistic probably being one based on a Reid soft core potential [60] shown in Fig.","REF .","Figure: Choice of the correlation function f(r)f(r): (a) no correlation; (b) Fermi gas correlation; (c) Reid soft core correlation.We then assumed that the effective confinement size in a nucleus $A$ will be intermediate between that for an isolated nucleon and some limiting value $\\lambda _{\\rm tot}$ associated with two totally overlapping nucleons.","We assumed a linear interpolation in $V_A$ leading to $\\frac{\\lambda _A}{\\lambda _N} = 1 + V_A \\left( \\frac{\\lambda _{\\rm tot}}{\\lambda _N}-1 \\right) \\ .","\\qquad \\mathrm {(6.21)}$ $\\lambda _{\\rm tot}$ might be viewed as a parameter.","Instead, we estimated its value from the bag model, where a spherical baryon number two system must have volume at least twice as large as a single nucleon (otherwise it would be stable against decay into two nucleons), so $R_6 \\ge 2^{1/3}R_3$ .","This led us to take $\\lambda _{\\rm tot}/\\lambda _N = 2^{1/3}$ , and to the values of $\\lambda _A/\\lambda _N$ given in Table REF .","Table: Values of the confinement size relative to that for the free nucleon for a range of nuclei.","The three values (a), (b) and (c) correspond to three choices of the correlation function f(r)f(r) shown in Fig. .","ξ NA \\xi _{NA} (20 GeV 2 ^2) is shown for the Reid correlation function.","The others are similar.These translate into values of $\\xi _{NA}$ (20 GeV$^2$ ) which are much larger, and these in turn give predictions for the EMC effect in a variety of nuclei.","The predictions of the rescaling model are compared with the SLAC data in Fig.","11.","The agreement is fine for $x < 0.7$ above which Fermi motion becomes important.","One thing which is not obvious from Fig.","11 is that the data correlate well with idiosyncracies in the periodic table.","Fig.","REF shows predictions for several $x$ values and $Q^2$ = 4.98 GeV$^2$ compared with SLAC data.","Figure: AA dependence of the EMC effect at fixed xx.","The data are squared with error bars.","The predictions of rescaling are solid dots .The fluctuations in the rescaling predictions reflect variations in nuclear densities, e.g.","$^4$ He is more tightly bound than $^3$ He, and generally follows the data.","Some predictions for the future are shown in Fig.","REF .","Figure: AA dependence at x=0.6x=0.6, Q 2 ≈10Q^2 \\approx 10 GeV 2 ^2 predicted by rescaling.Several comments are appropriate before leaving the discussion of $A$ dependence: 1.","Llewellyn Smith has argued that the calculation of the $A$ dependence we have given is much more general than the rescaling model.","Any scheme which fits the EMC effect in iron and in which the effect is linear in $V_A$ will agree as well with the SLAC data.","This is qualitatively true but the assumption of linearity is non-trivial.","We argued that $\\lambda _A/\\lambda _N$ should be linear in the overlapping volume.","Had we instead (erroneously) assumed $\\xi _{NA}(Q^2=20\\ GeV^2)$ to be linear in $V_A$ the $A$ dependence would have come out wrong.","2.","The rescaling model is not related to convolution models nor does it ascribe the EMC effect to any particular exotic component in the nuclear wavefunction (e.g., six quark bags).","The scale change might originate from a change in the size of individual nucleons, from quark percolation between nucleons, or from multiquark or meson admixtures in the nuclear wavefunction.","Personally, I suspect that to the extent that these notions can be well-defined they will turn out to correspond to the same underlying physics — partial deconfinement.","3.","The rescaling fit to the $A$ -dependence depends on several parameters: the nucleon radius $a$ , the formula chosen to relate $V_A$ to $\\lambda _A/\\lambda _N$ , the scale $\\mu _0$ and the choice of two nucleon correlation function.","In fact, none of these were treated as free parameters in [58].","Instead, they were fixed at “reasonable\" values from other aspects of hadron dynamics.","Of course, the precise choice of parameters is not particularly central to the explanation of the EMC effect.","4.","The value of $V_A$ is very large for large $A$ .","It is $\\approx $ 0.72 for $^{208}$ Pb with the Reid correlation function.","It has been remarked that such a large value of $V_A$ is “unreasonable\" and in some sense contradicted by the many successes of conventional nuclear physics.","The overlapping volume of nucleons with $a_{\\rm rms}$ = 0.9 fm is very large.","That is a fact, not a shortcoming of the rescaling model.","As for the idea that large $V_A$ is inconsistent with nuclear physics in general - I believe it is an unwarranted concern: It costs $\\approx $ 1 GeV (the string tension) to separate colored sources by l fm.","It even costs 300 MeV to flip a quark spin.","These energetic considerations, not naive classical, geometrical considerations determine the quantum mechanics of nuclei.","It has long been expected that at very low values of $x$ deep inelastic electron scattering from nuclei would behave like hadron scattering from nuclei and exhibit “shadowing\" [61].","At high energies hadron nucleus total cross sections grow like $A^{2/3}$ .","The simple explanation of this effect is that the incoming hadron doesn't “see\" the nucleons at the back of the nucleus.","They are in the shadow of those in front.","The cross section then grows like $\\pi R^2 \\sim A^{2/3}$ .","Real photon-nucleus interactions are shadowed [62].","This finds a simple explanation in the framework of vector meson dominance: the hadronic interactions of a real photon are well-approximated by supposing the photon converts with probability $\\sim \\!\\!\\alpha $ into a vector meson ($\\rho $ , $\\omega $ or $\\phi $ ) which then interacts hadronically and experiences shadowing.","Vector meson dominance fails to explain inelastic lepton scattering at large $Q^2$ , or at least an infinite tower of vector states with precisely tuned couplings is required, but the prejudice that shadowing occurs there too is strong.","Figure: Cartoons representing the ξ 3 \\xi ^3 values probed by deep inelastic lepton scattering at low-xx from an iron target.My own thoughts on shadowing are still in flux.","Since much of the future experimental work in this field will be carried out in the kinematic regime where shadowing might be important, I would like to outline the problem here.","Perhaps some reader will solve it!","The kinematic relation $\\xi ^3<1/Mx \\qquad \\mathrm {(6.22)}$ tells us that for very small values of $x$ the struck quark may propagate over very large distances in the target.","For $ x \\approx 0.5$ , $\\xi ^3 \\le 0.4$ fm, but for $x \\approx 0.05$ , $\\xi ^3 \\le 4$ fm and for $x \\approx 0.005$ , $\\xi ^3 \\le 40$ fm!","The parton model diagram dominant at large $Q^2$ is shown relative to an iron nucleus in Fig.","REF where the struck quark is associated with a single nucleon.","Other possibilities represent EMC-like corrections.","The question is whether a quark propagating over such distances is or is not strongly absorbed in the nuclear medium.","Hadron interactions at very high energies are predominantly absorptive, but the state which is propagating in Fig.","REF is not an ordinary hadron.","In particular, the propagating quark, which is a color triplet is always close to an antiquark [63] which is a color antitriplet that may neutralize some or most of its strong interactions.","Furthermore, the invariant mass of the $q \\bar{q}$ pair is $-Q^2$ .","Suppose for the sake of discussion, we characterize quark propagation in the nuclear medium (accompanied by an antiquark as in Fig.","REF ) by an absorption length $\\lambda $ which may depend on $Q^2$ .","For a given nucleus when $1/Mx< \\lambda (Q^2)$ there is little shadowing.","Shadowing sets in when $1/Mx \\sim \\lambda (Q^2)$ , but when $1/Mx$ exceeds twice the radius of the nucleus ($R_A$ ) shadowing saturates: decreasing $x$ further does not put more matter along the propagating quark's path.","So there are two $x$ -values characterizing shadowing, $x_0(Q^2) = 1/M \\lambda (Q^2)$ marking its onset and $x(A) = 1/2MR_A$ marking its saturation.","For $x < x(A)$ , only the front skin of the nucleus down to a depth $\\lambda (Q^2)$ participates in the scattering, so shadowing increases with $A$ but goes away as $\\lambda (Q^2) \\rightarrow \\infty $ .","All of this can be summarized by the purely phenomenological formula $\\overline{F}^A_2(x,Q^2) = \\frac{F^N_2(x,Q^2)}{\\left[ 1 + \\frac{x_0(Q^2)}{x+x(A)} \\right] } \\qquad \\mathrm {(6.23)}$ in which for simplicity I have ignored all other nuclear effects such as rescaling and Fermi motion.","Eq.","(6.23) does not satisfy the quark number sum rule.","It is not intended to be valid at all $x$ , only for $x \\sim 0$ where shadowing may be important.","According to Eq.","(6.23) there is an asymptotic shadowing curve for nuclear matter ($A \\rightarrow \\infty $ , $x(A) \\rightarrow 0$ ), shown for example in Fig.","REF .","Finite nuclei track along $\\overline{F}^\\infty _2(x,Q^2)$ until $x \\sim z(A)$ below which they depart above $\\overline{F}^\\infty _2$ .","A determination of $\\lambda (Q^2)$ is crucial [64].","Unfortunately, it is largely unknown.","Naively, one might expect $\\lambda (Q^2)$ to be a typical meson mean free path, corresponding to a cross section of order 30 mb.","This is too naive.","The propagating $q \\bar{q}$ pair in Fig.","REF must have invariant mass $-Q^2$ .","This can be generated if both quark and antiquark have transverse momentum of order $\\frac{1}{2}\\sqrt{Q^2}$ or by a large mismatch in their longitudinal momentum.","In the former case, the transverse dimension of the $q \\bar{q}$ system is very small, $\\sim 1/\\sqrt{Q^2}$ .","The system looks like a small color dipole and has a small absorption cross section.","In the latter case, the transverse dimensions are not small, color screening is not so effective and the absorption cross section may be large.","The importance of shadowing depends on which region of phase space dominates.","This is not yet known.","But at least some of the shadowing seen at $Q^2 =0$ should disappear at large $Q^2$ .","The behavior of $\\overline{F}^A_2(x,Q^2)$ at small $x$ expected if shadowing indeed disappears at large $Q^2(x(Q^2) \\rightarrow 0$ ) as shown in Fig.","REF .","Figure: Scenarios for shadowing.","(a) and (b) compared to two different values of λ(Q 2 )\\lambda (Q^2) with λ a >λ b \\lambda _a > \\lambda _b.The discussion in this section has been very qualitative and phenomenological.","Eq.","(6.23) should be taken with a grain of salt.","There is much to do on the subject of shadowing and little of substance to report here.","Nevertheless, I thought it might be appropriate to end by whetting the reader's appetite for the next round of experiments."]],"string":"[\n [\n \"Deep Inelastic Scattering with Application to Nuclear Targets: Lectures\\n at the 1985 Los Alamos School on Relativistic Dynamics and Quark Nuclear\\n Physics\"\n ],\n [\n \"Abstract This paper is essentially a verbatim reconstruction of lectures that I gave at the Los Alamos School on Relativistic Dynamics and Quark Nuclear Physics in 1985.\",\n \"They were published in the school proceedings, but the book is not widely available.\",\n \"The Los Alamos School took place at the height of the first wave of interest in the quark substructure of nuclei, stimulated by the 1983 discovery of the EMC Effect.\",\n \"Interest in this subject has been increasing for years and the prospect of a dedicated Electron Ion Collider within the decade guarantees even greater attention to quarks and gluons in nuclei among both theorists and experimentalists.\",\n \"Recently, to my surprise, I learned that copies of my old lectures have been circulating and been found useful by the relatively few people who know about them.\",\n \"The are, of course, dated: experiments have far outstripped what was available 37 years ago and theory has progressed too.\",\n \"However, the rest frame derivation of the parton model, the derivation and discussion of the convolution formalism for nucleons, nucleon correlations, and other, virtual, constituents of nuclei, and sections on scaling violation and the operator product expansion have aged pretty well and seem to still be useful.\",\n \"With the help and encouragement of Richard Milner, I have recreated the LaTeX files necessary to post the 1985 Lectures on the arXiv, making them available to the nuclear and particle physics community.\",\n \"Apart from correcting some typographical errors, I have made no attempt to edit, improve, or update these lectures.\",\n \"I hope readers will nevertheless find them useful.\"\n ],\n [\n \"MOTTO\",\n \"“Looking for the quarks in the nucleus is like looking for the Mafia in Sicily: Everyone knows they're there, but its hard to find the evidence.” (Anonymous) This paper is essentially a verbatim reconstruction of lectures that I gave at the Los Alamos School on Relativistic Dynamics and Quark Nuclear Physics in 1985.\",\n \"They were published in the school proceedings[1], but the book is not widely available.\",\n \"The Los Alamos School took place at the height of the first wave of interest in the quark substructure of nuclei, stimulated by the 1983 discovery of the EMC Effect[2].\",\n \"Interest in this subject has been increasing for years and the prospect of a dedicated Electron Ion Collider within the decade guarantees even greater attention to quarks and gluons in nuclei among both theorists and experimentalists.\",\n \"Recently, to my surprise, I learned that copies of my old lectures have been circulating and been found useful by the relatively few people who know about them.\",\n \"The are, of course, dated: experiments have far outstripped what was available 37 years ago and theory has progressed too.\",\n \"However, the rest frame derivation of the parton model, the derivation and discussion of the convolution formalism for nucleons, nucleon correlations, and other, virtual, constituents of nuclei, and sections on scaling violation and the operator product expansion have aged pretty well and seem to still be useful.\",\n \"With the help and encouragement of Richard Milner, I have recreated the LaTeX files necessary to post the 1985 Lectures on the arXiv, making them available to the nuclear and particle physics community.\",\n \"Apart from correcting some typographical errors, I have made no attempt to edit, improve, or update these lectures.\",\n \"I hope readers will nevertheless find them useful.\",\n \"These lectures are addressed to a very specific audience: graduate students and young researchers in theoretical nuclear physics.\",\n \"They contain very little that cannot be found elsewhere in the vast literature on the subject, but they summarize a particular viewpoint which is both straightforward and fairly correct.\",\n \"My intention is to provide a reasonably thorough introduction to inclusive, inelastic electron scattering and enough advanced material to excite the reader to go on by him/herself.\",\n \"The reader is assumed to have some familiarity with relativistic quantum mechanics and with the more elementary side of quantum field theory.\",\n \"Some previous exposure to QCD $-$ quarks, color, gluons, SU(3) and the like $-$ will help.\",\n \"But; previous exposure to renormalization theory, the renormalization group and the operator product expansion is not assumed.\",\n \"I have tried to avoid the temptation to oversimplify $-$ the subject has its subtleties and it is not possible to do good work in the field without understanding them.\",\n \"More simplified treatments can be found in books by Feynman (Photon-Hadron Interactions, (W. A. Benjamin, New York (1972)) and by Close (An Introduction to Quarks and Hadrons (Academic Press, New York, 1979)).\",\n \"Equally, I have tried to avoid too much formalism.\",\n \"I hope the power of the more advanced methods developed in §5 and applied in §6 will encourage the reader to pursue the subject more formally in the future.\",\n \"More advanced treatments can be found in tho lecture notes of D. J.\",\n \"Gross (in Methods in Field Theory, Proc.\",\n \"of 1975 Les Houches Summer School, (ed., R. Balian and J. Zinn-Justin, North-llolland, Amsterdam) and C.H.\",\n \"Llewellyn Smith (Topics in Quantum Chronomdynamics, in Quantum Flavordynamics, Quantum Chromodynamics and Unified Theories (ed., K. T. Mahanthappa and J. Randa, Plenum Press, New York, 1980) in the recent books by C. Itzykson and J. Zuber (Introduction to Quantum Field Theory, McGraw-Hill, New York, 1980) and T. P. Cheng and L.F. Li Gauge Theories of Elementary Particle Physics, (Clarendon Press, Oxford, 1984).\",\n \"In the interest of time, I have had to eliminate all mention of inclusive, inelastic scattering of neutrinos and other related processes such as electron-positron annihilatiqn and Drell-Yan production of lepton pairs.\",\n \"These too can be found treated in detail elsewhere in the literature.\",\n \"Finally I would like to thank Frank Close, Chris Llewellyn-Smith, Dick Roberts and especially Graham Ross for collaboration on many of the ideas presented here.\",\n \"I would also like to thank Gerry Garvey and Mikkel Johnson for making it possible for me to attend the school, and Roger Gilson for preparing the manuscript.\",\n \"The quark description of hadrons is now universally accepted, but its implications for nuclear physics are anything but clear.\",\n \"Nuclear structure and low energy nuclear reactions are well-described by a variety of semi-phenomenological theories which leave little room for insight from thinking about quark and gluon dynamics.\",\n \"The reason is obvious: the energies involved in most nuclear phenomena are so low compared to the natural scale of quark dynamics that quark and gluon degrees-of freedom are effectively frozen into nucleon and meson “quasiparticles\\\".\",\n \"To support this, consider first that the string tension in QCD is $\\\\sim $ 1 GeV /fm, so it costs $\\\\sim $ 1 GeV to separate a quark from two others (or from an antiquark) by an additional fermi beyond their equilibrium separation; and second, that the N – $\\\\Delta $ mass difference is 300 MeV, so it costs $\\\\sim $ 300 MeV to flip a quark's spin relative to the two others which together with it form a nucleon.\",\n \"Both energies are much larger than typical nuclear energy scales.\",\n \"Thus, $\\\\Delta $ s are relegated to a relatively minor role in the description of nuclei, and quarks (as distinct from nucleons and mesons) are even less important.\",\n \"It is not even clear, as a matter of principle, how to establish the necessity of a fundamentally “quarkic\\\" description of nuclei.\",\n \"Because color is confined, it is almost always possible to find an entirely equivalent hadronic description of some quark-dynamical process.\",\n \"For example, there has been much interest in isolating so-called “hidden color\\\" components in the deuteron wave function.\",\n \"These are of the form [(q$^3$ )$^8$ –(q$^3$ )$^8$ ]$^1$ : although the entire six quark system is a color singlet, each group of three quarks is coupled to a color octet.\",\n \"Such admixtures have been invoked, for example, to resolve discrepancies in models of deuteron photodisintegration.\",\n \"It is trivial to show, however, that any such configuration can be rewritten, after a change of coupling transformation, as a sum of configurations in which each set of three quarks is coupled to a color singlet, i.e., an ordinary baryon.\",\n \"[The proof is merely to note that $\\\\epsilon _{abc}$ is the only covariant, invariant tensor in SU(3).]\",\n \"So any hidden color state can be equally well represented as a sum over conventional baryon-baryon configurations, and any phenomenon ascribed to hidden color could be explained equivalently by a sufficiently clever theorist who had never heard of quarks and color.\",\n \"The quark theorist might triumph in the end if their description were simpler and more economical: one “hidden color\\\" state might do as well as some complex superposition of baryon resonances.\",\n \"But this is not the “smoking gun\\\" enthusiasts are seeking.\",\n \"Similar observations could have been made about low energy hadron physics twenty years ago, and one therefore wonders how the quark description of hadrons was established in the first place, which brings me to the subject of these lectures.\",\n \"Of course, quarks were recognized as a convenient bookkeeping device as early as 1964, but the need for a quark dynamics for hadrons was not compelling until a set of experiments was performed at SLAC in the late 1960s.\",\n \"High energy electrons were scattered inelastically from nucleons in close analogy to the $\\\\alpha $ -particle scattering experiments performed by Rutherford in the early 1900s.\",\n \"In both cases, the experimenters were surprised to find that the scattering cross section remained large even at high momentum transfer indicating the presence of point-like scattering centers within the target.\",\n \"Further studies with electrons and then with neutrinos established the spin, charge and baryon number of the point-like objects within hadrons and showed they were quarks.\",\n \"It was also recognized that these experiments not only detect quarks but directly measure their distribution within the target baryon.\",\n \"During the early 1970s, it became clear that only a non-Abelian gauge field theory, quantum chromodynamics (QCD), could explain the qualitative features of quark dynamics.\",\n \"One of its triumphs was the quantitative prediction of the details of inelastic electron scattering in the very high momentum transfer, or deep inelastic limit.\",\n \"Despite the complexity of QCD, the simple Rutherford picture remains valid: the scattering electron measures the quark distribution in the target.\",\n \"These lectures explore the use of deep inelastic electron scattering as a probe of the quark distribution in nuclei.\",\n \"For more than a decade it apparently was thought that the quark distribution in a nucleus was simply given by the quark distribution in so many neutrons and protons, corrected for the fact that they are in motion within the nucleus (Fermi motion).\",\n \"Little attention was paid to quark distributions in nuclei.\",\n \"It came as a surprise to particle and nuclear theorists alike when the first careful comparison of a nuclear target (iron) with a nucleon (actually a deuteron) presented at the 1982 Paris Conference by the European Muon Collaboration (EMC), showed a $\\\\sim $ 15% difference in a region where Fermi motion effects are thought to be negligible.\",\n \"The “EMC effect\\\" as it is known, was immediately confirmed by experiments at SLAC and elsewhere.\",\n \"Theorists quickly presented a variety of explanations of the EMC effect.\",\n \"Now, three years later, there are several well-developed schools of thought, much controversy, and only a little agreement among partisans about the origin of the effect.\",\n \"Most explanations are based on a “convolution model\\\" formalism in which one supposes the nucleus contains, in addition to nucleons, some small admixture of more exotic constituents (pions, multiquark bags, $\\\\Delta $ s, $\\\\alpha $ clusters ... ), and then adds up their quark distributions.\",\n \"Unfortunately, this approach has a hitch: the assumption that quark distributions in constituents add incoherently is not justified and in many cases probably wrong.\",\n \"Another approach, known as “rescaling\\\" suggested by the scale transformation properties of QCD, avoids the questionable assumption of convolution models by dealing directly and solely with the quark distribution of the nucleus.\",\n \"The result is simple and striking: the EMC effect shows that the typical length scale associated with quark propagation in the nuclear ground state is longer than the corresponding length scale in the nucleon.\",\n \"For a particle physicist, one of the most interesting features of this approach is its implications for QCD at finite density.\",\n \"Nuclei provide samples of quark/nuclear matter at a variety of mean densities (as $A$ increases, the surface to volume ratio goes to zero so the mean density grows).The $A$ dependence of the EMC effect closely reflects the variations in mean nuclear density, indicating that the quark length scale in quark/nuclear matter grows with density.\",\n \"In these lectures, I have tried to avoid detailed analysis of models of the EMC effect, although critics of the “rescaling\\\" approach, to which I am devoted, will point out that I present it in considerable detail.\",\n \"In fact, much of the rescaling analysis is model independent and essential for a modern education in the subject.\",\n \"In §1, I introduce the kinematic variables in coordinate and momentum space.\",\n \"The whole discussion is set in the laboratory or target rest frame.\",\n \"Certain general tools like dispersion relations are reviewed and summarized there.\",\n \"In §2, I present the parton model from a somewhat unfamiliar point-of-view, one which experts will recognize as imitating the more formal operator product expansion analysis used in QCD.\",\n \"The reason for this approach is to keep as close to coordinate space as possible since considerable insight into the parton model and the EMC effect comes from an easy fluency between coordinate and momentum space.\",\n \"In §3, I summarize the data, as far as I know it, and use the parton model to interpret it.\",\n \"§4, is dedicated to convolution models.\",\n \"Since so much work has been based on these models, I have tried to present them in detail, illustrated by the case in which the constituents are the nucleons themselves (Fermi motion), and let the readers judge their utility for themselves.\",\n \"In §5, I return to fundamentals.\",\n \"QCD changed our understanding of inelastic scattering and corrected and extended the parton model.\",\n \"I have tried to introduce the QCD analysis with as little excess formalism as possible, though for a real working knowledge the reader will have to learn more about gauge field theory and the renormalization group elsewhere.\",\n \"Finally, in §6, I use the powerful methods developed in §5 to help give a new way of looking at the EMC effect, and draw some surprisingly simple conclusions.\",\n \"We are interested in the process $eA \\\\rightarrow e^\\\\prime X$ where $A$ is a nucleus, the proton and neutron being important special cases, and $X$ is an unobserved hadronic final state.\",\n \"The electron and nucleus are assumed unpolarized, although polarization dependent effects can be handled in the same fashion.\",\n \"The process is known as inelastic electron scattering or inclusive electroproduction.\",\n \"To lowest order in $\\\\alpha $ , the process is described [3] by one photon exchange (Fig.\",\n \"REF ): $A \\\\propto \\\\bar{u}(k^\\\\prime )\\\\gamma ^\\\\mu u(k)\\\\frac{1}{q^2}\\\\mathinner {\\\\langle {X|J_\\\\mu (0)|p}\\\\rangle } \\\\ , \\\\qquad \\\\mathrm {(1.1)}$ where $J_\\\\mu (0)$ is the hadronic electromagnetic current operator.\",\n \"The differential cross-section for scattering in which $X$ is not observed is proportional to $\\\\Sigma _X |A|^2 (2 \\\\pi )^4 \\\\delta ^4(p+q-p_X)$ , or $d \\\\sigma \\\\propto l^{\\\\mu \\\\nu } W_{\\\\mu \\\\nu } \\\\ , \\\\qquad \\\\mathrm {(1.2)}$ where $l^{\\\\mu \\\\nu } = \\\\frac{1}{2} Tr {k}^\\\\prime \\\\gamma ^\\\\mu {k} \\\\gamma ^\\\\nu = 2({k^\\\\prime }^\\\\mu k^\\\\nu + {k^\\\\prime }^\\\\nu k^\\\\mu - g^{\\\\mu \\\\nu } k \\\\cdot k^\\\\prime ) \\\\qquad \\\\mathrm {(1.3)}$ (we ignore the electron mass), and $W_{\\\\mu \\\\nu } \\\\equiv \\\\frac{1}{4 \\\\pi } \\\\Sigma _X (2 \\\\pi )^4 \\\\delta ^4(p+q-p_X) \\\\mathinner {\\\\langle {p|J_\\\\mu (0)|X}\\\\rangle } \\\\mathinner {\\\\langle {X|J_\\\\nu (0)|p}\\\\rangle } \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.4)}$ Figure: Inclusive inelastic electron scattering via one photon exchange.\",\n \"EE, E ' E^\\\\prime and θ\\\\theta are defined in the target rest frame.$W_{\\\\mu \\\\nu }$ contains all reference to hadronic states.\",\n \"It is represented graphically in Fig.\",\n \"REF .\",\n \"Figure: W μν W_{\\\\mu \\\\nu } which determines the cross section for electron scattering and also, the imaginary part of forward, virtual Compton scattering.Eq.\",\n \"(1.4) may be simplified by replacing $(2 \\\\pi )^4 \\\\delta ^4(p+q-p_X) \\\\equiv \\\\int d^4 \\\\xi \\\\ e^{i(p+q-p_X) \\\\cdot \\\\xi } \\\\ , \\\\qquad \\\\mathrm {(1.5)}$ translating $J_\\\\mu (0)$ to the space-time point $\\\\xi $ , and using completeness ($\\\\Sigma _X|X><X| = 1$ ): $W_{\\\\mu \\\\nu } = \\\\frac{1}{4 \\\\pi } \\\\int d^4 \\\\xi \\\\ e^{iq \\\\cdot \\\\xi } \\\\mathinner {\\\\langle {p|J_\\\\mu (\\\\xi )J_\\\\nu (0)|p}\\\\rangle }_c \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.6)}$ The subscript $c$ on the matrix element denotes “connected\\\" and ensures that vacuum-to-vacuum transitions of the form $\\\\mathinner {\\\\langle {0|J_\\\\mu (\\\\xi ) J_\\\\nu (0)|0}\\\\rangle }\\\\mathinner {\\\\langle {p|p}\\\\rangle }$ are excluded.\",\n \"The current product in Eq.\",\n \"(1.6) can be replaced by a commutator $W_{\\\\mu \\\\nu } =\\\\frac{1}{4 \\\\pi } \\\\int d^4 \\\\xi \\\\ e^{iq \\\\cdot \\\\xi } \\\\mathinner {\\\\langle {p|[J_\\\\mu (\\\\xi ),J_\\\\nu (0)]|p}\\\\rangle }_c\\\\qquad \\\\mathrm {(1.7)}$ because the term we have subtracted vanishes for stable targets if $q^0 > 0$ .\",\n \"In quantum field theory, it is most convenient to deal with time-ordered products of operators.\",\n \"Thus, $T_{\\\\mu \\\\nu } =i \\\\int d^4 \\\\xi \\\\ e^{iq \\\\cdot \\\\xi } \\\\mathinner {\\\\langle {p|T(J_\\\\mu (\\\\xi )J_\\\\nu (0))|p}\\\\rangle }_c\\\\qquad \\\\mathrm {(1.8)}$ is the amplitude [4] for forward scattering of a virtual photon of momentum $q$ from a hadronic target of momentum $p$ (Fig.\",\n \"REF ).\",\n \"Figure: Forward virtual Compton scattering.It is easily seen that $W_{\\\\mu \\\\nu }$ is the imaginary part of $T_{\\\\mu \\\\nu }$ (when $q^0$ is taken to have a small positive imaginary part) $W_{\\\\mu \\\\nu } = \\\\frac{1}{2 \\\\pi } IT_{\\\\mu \\\\nu } \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.9)}$ So, inclusive electroproduction is intimately related to virtual Compton scattering.\",\n \"$W_{\\\\mu \\\\nu }$ and $T_{\\\\mu \\\\nu }$ can be decomposed in terms of a pair of Lorentz invariant “structure functions\\\" $W_{1,2}$ ($T_{1,2}$ ): $W_{\\\\mu \\\\nu } &= - \\\\left(g_{\\\\mu \\\\nu } - \\\\frac{q_\\\\mu q_\\\\nu }{q^2} \\\\right) W_1(q^2,\\\\nu ) \\\\nonumber \\\\\\\\&+ \\\\frac{1}{M^2_T} \\\\left( p_\\\\mu - \\\\frac{M_T \\\\nu }{q^2} q_\\\\mu \\\\right)\\\\left( p_\\\\nu - \\\\frac{M_T \\\\nu }{q^2} q_\\\\nu \\\\right) W_2(q^2,\\\\nu ) $ and likewise for $T_{\\\\mu \\\\nu }$ .\",\n \"$W_{1,2}$ are functions of the of the Lorentz invariants $q^2$ and $p \\\\cdot q = M_T \\\\nu $ ($q^2 = -4EE^\\\\prime \\\\sin ^2 \\\\frac{\\\\theta }{2}$ and $\\\\nu = E-E^\\\\prime $ in the target rest frame) where $M_T$ is the target mass.\",\n \"No other terms are allowed by Lorentz invariance, current conservation ($q^\\\\mu W_{\\\\mu \\\\nu } = W_{\\\\mu \\\\nu } q^\\\\nu = 0$ ) and parity.\",\n \"The structure functions depend on the squared four momentum transfer $q^2 \\\\equiv - Q^2 = \\\\nu ^2 - {\\\\cal {Q}}^2$ which is spacelike (so $Q^2 > 0$ ) and the energy transfer in the laboratory, $\\\\nu = q^0$ , which is positive.\",\n \"The squared mass of the final hadronic state, $X$ , is $(p + q)^2 = M_T^2 + 2M_T \\\\nu - Q^2 \\\\equiv W^2$ and is greater than $M_T^2$ .\",\n \"Thus, the “scaling\\\" variable, $x_T \\\\equiv Q^2 /2M_T \\\\nu $ must be between 0 and 1.\",\n \"Often experimentalists and theorists prefer to use a uniform scaling variable $x = x_N = Q^2 /2M\\\\nu $ ($M \\\\equiv M_N$ is the nucleon mass) for all targets.\",\n \"The reason for this is that the structure functions of different nuclei look to first order like the sum of structure functions for $A$ independent nucleons.\",\n \"They are therefore very small for $x > 1$ regardless of $A$ .\",\n \"It is easy to get confused between the scaling variable intrinsic to a target of mass $M_T$ , $x_T \\\\equiv Q^2 /2M_T \\\\nu $ , which is bounded between 0 and 1, and the uniform scaling variable $x =x_N = Q^2/2M$ , which ranges from zero to $M_T/M$ .\",\n \"Note that $M_T/M \\\\unknown.\",\n \"A$ for a nucleus of mass number $A$ , so the distinction between the two variables is quite significant for nuclear targets.\",\n \"Bjorken suggested [5] that in the limit of large $Q^2$ at fixed $x_T$ (now known as the Bjorken or “deep\\\" inelastic limit) $W_1$ and $\\\\frac{\\\\nu }{M_T} W_2$ should become functions of $x_T$ alone: $\\\\lim _{Bj} W_1 (q^2,\\\\nu ) &= F_1(x_T)\\\\nonumber \\\\\\\\\\\\lim _{Bj} \\\\frac{\\\\nu }{M_T} W_2(q^2,\\\\nu ) &= F_2(x_T) \\\\ ,$ which is approximately verified by experiment.\",\n \"This phenomenon is known as “Bjorken scaling\\\" or just “scaling\\\" for short.\",\n \"The kinematic range of inclusive scattering is shown in Fig.\",\n \"REF .\",\n \"Note that the $Q^2 \\\\rightarrow 0$ limit gives photoproduction.\",\n \"Figure: Kinematic variables for inelastic electron scattering.The virtual forward Compton amplitude, $T_{\\\\mu \\\\nu }$ , possesses simple properties when regarded as an analytic function of $\\\\nu $ at fixed $q^2$ .\",\n \"These are special cases of the general results of dispersion theory, most of which are forgotten [6].\",\n \"Since I will need these properties in the subsequent chapters, I will review them here.\",\n \"For pedagogical simplicity I will ignore spin and analyze a hypothetical virtual “Compton\\\" amplitude $T( q^2 , \\\\nu )$ for a scalar “photon\\\" scattering from a proton or neutron defined by $T(q^2,\\\\nu ) \\\\equiv i \\\\int d^4 \\\\xi \\\\ e^{i q \\\\cdot \\\\xi } \\\\mathinner {\\\\langle {p|T(J(\\\\xi )J(0))|p}\\\\rangle } \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.12)}$ (The generalization to other targets is straightforward.)\",\n \"The generalization to the physically interesting case of $T_{\\\\mu \\\\nu }$ will be quoted at the end.\",\n \"$T(q^2 , \\\\nu )$ is a real analytic function of $\\\\nu $ at fixed $q^2$ , $T(q^2,\\\\nu ^\\\\star ) = T^\\\\star (q^2,\\\\nu )\\\\nonumber \\\\ ,$ and is crossing symmetric $T(q^2,\\\\nu ) = T(q^2,-\\\\nu ) \\\\nonumber \\\\ .$ The fundamental assumption of dispersion theory is that scattering amplitudes are analytic except at values of the kinematic variables which allow intermediate states to be physical (i.e., on shell).\",\n \"This can be proven to all orders of perturbation theory [7], but must be regarded as an assumption in QCD where the quanta of the perturbation theory (quarks and gluons) are not physical states.\",\n \"When $\\\\nu \\\\ge - q^2/2M$ , the virtual photon-target system can form a physical hadronic intermediate state so $T(q^2,\\\\nu )$ has a cut along the positive real-$\\\\nu $ axis.\",\n \"More precisely, $T(q^2,\\\\nu )$ has a pole at $\\\\nu = -q^2/2M$ corresponding to elastic scattering ($ep \\\\rightarrow e^\\\\prime p^\\\\prime $ ) and a cut beginning at pion production threshold.\",\n \"$T(q^2,\\\\nu )$ also has a cut on the negative real axis corresponding to the “crossed\\\" process $p \\\\rightarrow \\\\gamma + x$ which is physically allowed when $\\\\nu \\\\le -|q|^2/2M$ .\",\n \"The discontinuity across the right hand cut is ${\\\\rm disc}\\\\ T(q^2,\\\\nu ) = 2i \\\\,IT(q^2,\\\\nu ) = 4\\\\pi i W(q^2,\\\\nu ) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.15)}$ These analytic properties are summarized in Fig.\",\n \"REF .\",\n \"Figure: The complex ν\\\\nu -plane.\",\n \"The contour shown figures in the derivation of dispersion relations.Using Cauchy's theorem on the contour shown in Fig.\",\n \"REF , it is possible to derive a “dispersion relation\\\" for $T(q^2,\\\\nu )$ , $T(q^2,\\\\nu ) = 4 \\\\int _{-q^2/2M}^{\\\\infty } \\\\frac{d \\\\nu ^\\\\prime \\\\nu ^\\\\prime }{{\\\\nu ^\\\\prime }^2 - \\\\nu ^2} W(q^2,\\\\nu ^\\\\prime ) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.16)}$ Here, $\\\\nu $ is complex ($\\\\nu ^\\\\prime $ is real) and the physical Compton amplitude is obtained by letting it approach the positive real axis from above $\\\\nu \\\\rightarrow \\\\nu _R + i \\\\epsilon $ .\",\n \"If $W(q^2,\\\\nu ^\\\\prime )$ falls too slowly as $ \\\\nu ^\\\\prime \\\\rightarrow \\\\infty $ then the integral may not converge.\",\n \"In that case one can derive a weaker, “subtracted\\\" dispersion relation.\",\n \"Formally, take Eq.\",\n \"(1.16) for two different choices of $\\\\nu $ and subtract, e.g.\",\n \"$T(q^2,\\\\nu ) = T(q^2,0) +4 \\\\nu \\\\int _{-q^2/2M}^{\\\\infty } \\\\frac{d \\\\nu ^\\\\prime }{\\\\nu ^\\\\prime ({\\\\nu ^\\\\prime }^2 - \\\\nu ^2)} W(q^2,\\\\nu ^\\\\prime ) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.17)}$ The integral is better behaved at large $\\\\nu ^\\\\prime $ .\",\n \"This procedure can be continued as far as necessary and works as long as $W(q^2,\\\\nu )$ is polynomial bounded as $\\\\nu \\\\rightarrow \\\\infty $ , which we will assume (see below).\",\n \"Modulo subtractions, the real part of $T$ can be calculated if $W$ is known, thus dispersion relations can be (and have been) verified experimentally.\",\n \"It is convenient to change variables to $\\\\omega \\\\equiv -2M \\\\nu /q^2$ in Eq.\",\n \"(1.16), $T(q^2,\\\\omega ) = 4 \\\\int _1^\\\\infty \\\\frac{\\\\omega ^\\\\prime d \\\\omega ^\\\\prime }{{\\\\omega ^\\\\prime }^2 - \\\\omega ^2} W(q^2,\\\\omega ^\\\\prime ) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.18)}$ Note that $T(q^2,\\\\omega )$ is analytic in the circle of radius 1 about $\\\\omega = 0$ and may therefore be expanded in a Taylor series for $|\\\\omega | < 1$ : $T(q^2,\\\\omega ) = 4 \\\\sum _{n\\\\,\\\\rm even} M^n(q^2) \\\\omega ^n \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.19)}$ with $M^n(q^2) &= \\\\int _1^\\\\infty d \\\\omega ^\\\\prime {\\\\omega ^\\\\prime }^{-n-1} W(q^2,\\\\omega ^\\\\prime ) \\\\nonumber \\\\\\\\&= \\\\int _0^1 dx\\\\ x^{n-1} W(q^2,x) \\\\ .$ The physically interesting case of $T_{\\\\mu \\\\nu }$ is summarized by dispersion relations for the two invariant amplitudes $T_{1,2}(q^2,\\\\omega )$ .\",\n \"$T_1(q^2,\\\\omega )$ requires subtraction $T_1(q^2,\\\\omega ) &= T_1(q^2,0) + 4 \\\\omega ^2 \\\\int d \\\\omega ^\\\\prime \\\\frac{W_1(q^2,\\\\omega ^\\\\prime )}{\\\\omega ^\\\\prime ({\\\\omega ^\\\\prime }^2-\\\\omega ^2)} \\\\nonumber \\\\\\\\\\\\frac{\\\\nu T_2}{M_T}(q^2,\\\\omega )&= 4 \\\\omega \\\\int _1^\\\\infty d \\\\omega ^\\\\prime \\\\frac{\\\\frac{\\\\nu W_2}{M}(q^2,\\\\omega ^\\\\prime )}{{\\\\omega ^\\\\prime }^2 - \\\\omega ^2} \\\\ .$ Most introductory treatments of deep inelastic lepton scattering are formulated in the “infinite momentum frame\\\" where the target is boosted to some arbitrarily large momentum $P_\\\\infty $ .\",\n \"I find this approach both unnecessary and misleading and prefer instead to work in the target rest frame where I suppose nuclear physicists also feel at home.\",\n \"Of course, the physics is frame independent, at least if one is careful enough [8].\",\n \"In the rest frame of a target with mass $M_T$ , the Bjorken limit takes on a particularly simple form.\",\n \"We choose the negative $z$ -axis to lie along the virtual photon direction: $q = (\\\\nu ,0,0,-\\\\sqrt{\\\\nu ^2+Q^2}) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.22)}$ As $Q^2 \\\\rightarrow \\\\infty $ with $x$ fixed, $Q^2/\\\\nu ^2 \\\\rightarrow 0$ so $q \\\\rightarrow (\\\\nu ,0,0,-\\\\nu -M_Tx_T) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.23)}$ The significance of this form is most transparent if we introduce “light-cone\\\" coordinates, $q^\\\\pm = \\\\frac{1}{\\\\sqrt{2}}(q^0 \\\\pm q^3) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.24)}$ [Note: $a \\\\cdot b = a^+b^- + a^-b^+ - {\\\\bf a}_\\\\perp \\\\cdot {\\\\bf b}_\\\\perp $ , $a^2 = 2a^+a^- - {\\\\bf a}^2_\\\\perp $ , so $g^{+-} = g^{-+} = 1$ and $a_\\\\mp =a^\\\\pm $ .\",\n \"To avoid confusion, I will stick to contravariant indices.]\",\n \"In the Bjorken limit $q^- \\\\rightarrow \\\\infty $ but $q^+ \\\\rightarrow -M_Tx_T/\\\\sqrt{2}$ , i.e., $q^+$ remains finite.\",\n \"Note $M_Tx_T = Mx = Q^2/2\\\\nu $ , so the limiting value of $q^+$ is independent of $M_T$ .\",\n \"We will frequently discuss the important distant scales which contribute to electroproduction.\",\n \"The distance referred to is the space-time separation, $\\\\xi _\\\\lambda $ , between the points at which the currents $J_\\\\nu $ and $J_\\\\nu $ act.\",\n \"Note that $\\\\xi _\\\\lambda $ is only defined in the Compton amplitude, $T_{\\\\mu \\\\nu }$ , or its imaginary part $W_{\\\\mu \\\\nu }$ , not in the electroproduction amplitude $A$ itself (Eq.\",\n \"1.1).\",\n \"Since $\\\\xi $ and $q$ appear as conjugate variables in the definition of $W_{\\\\mu \\\\nu }$ and $q \\\\cdot \\\\xi \\\\equiv q^+\\\\xi ^- + q^- \\\\xi ^+$ , $q^- \\\\rightarrow \\\\infty $ forces $\\\\xi ^+ \\\\rightarrow 0$ , but $q^+=-Mx/\\\\sqrt{2}$ requires only $|\\\\xi ^-|\\\\lesssim \\\\sqrt{2}/Mx$ .\",\n \"The first of these relations, $\\\\xi ^+ \\\\rightarrow 0$ , follows from general theorems on Fourier transforms [9].\",\n \"Let $\\\\tilde{f}(q^-) = \\\\int d \\\\xi ^+ e^{i q^- \\\\cdot \\\\xi ^+} f(\\\\xi ^+)\\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.25)}$ If ${f}(\\\\xi ^+)$ is smooth (infinitely differentiable everywhere) and well-behaved as $|\\\\xi ^+| \\\\rightarrow \\\\infty $ , then $\\\\tilde{f}(q^-)$ vanishes faster than any power of $q^-$ as $q^- \\\\rightarrow \\\\infty $ .\",\n \"But, if $f(\\\\xi ^+)$ has singularities, then the $q^- \\\\rightarrow \\\\infty $ limit is dominated by the behavior of $f(\\\\xi ^+)$ near the singularities.\",\n \"Suppose, for example, $f(\\\\xi ^+) =0$ for $\\\\xi ^+ \\\\le 0$ , then as $q^- \\\\rightarrow \\\\infty $ $\\\\tilde{f}(q^-) \\\\sim \\\\frac{i}{q^-} f(0) - (\\\\frac{i}{q^-})^2 f^\\\\prime (0) + ...\\\\ , \\\\qquad \\\\mathrm {(1.26)}$ as easily obtained from integration by parts.\",\n \"The integrand in Eq.\",\n \"(1.7) defining $W_{\\\\mu \\\\nu }$ is singular at $\\\\xi ^+ =0$ since it vanishes when $\\\\xi ^2 <0$ , so the $q^- \\\\rightarrow \\\\infty $ limit is dominated by $\\\\xi ^+ \\\\approx 0$ .\",\n \"The second relation, $|\\\\xi ^-|\\\\lesssim \\\\sqrt{2}/Mx$ , is more subtle.\",\n \"After all the dust settles (in $§$ 2) the structure functions will be given by Fourier transforms in $q^+=-Mx/\\\\sqrt{2}$ of a smooth function of $\\\\xi ^-$ .\",\n \"The Fourier transform, however, is only conditionally convergent at large $\\\\xi ^-$ .\",\n \"It diverges like $1/\\\\sqrt{x}$ or $1/x$ (for “valence\\\" or “ocean\\\" quarks, see $§2$ as $x \\\\rightarrow 0$ , so the oscillation in the exponential acts as a large $\\\\xi ^-$ cutoff.\",\n \"The larger $\\\\sqrt{2}/Mx$ , the larger range of $\\\\xi ^-$ contributes to the structure function.\",\n \"For a more complete discussion with examples see the Heidelberg talk by Llewellyn Smith in [10].\",\n \"Since the commutator in Eq.\",\n \"(1.7) is causal, $\\\\xi ^2 = 2\\\\xi ^+\\\\xi ^- - {\\\\xi }^2_\\\\perp $ is positive.\",\n \"Hence, $\\\\xi ^+ \\\\rightarrow 0$ with $\\\\xi ^-$ finite requires $\\\\xi _\\\\perp \\\\rightarrow 0$ .\",\n \"All components of $\\\\xi ^\\\\lambda $ except $\\\\xi ^-$ vanish in the Bjorken limit.\",\n \"Thus, deep inelastic scattering is not, as is often incorrectly remarked, a short distance ($\\\\xi _\\\\lambda \\\\rightarrow 0$ ) phenomenon; it is instead a light-cone ($\\\\xi ^2 \\\\rightarrow 0$ ) dominated process.\",\n \"This distinction is crucial to understanding the dynamics and yields considerable insight into nuclear effects in inclusive electron scattering.\",\n \"Together $\\\\xi ^+ \\\\rightarrow 0$ and $|\\\\xi ^-| \\\\lesssim \\\\sqrt{2}/Mx$ imply $|\\\\xi ^0|\\\\lesssim 1/Mx$ and $|\\\\xi ^3| \\\\lesssim 1/Mx$ .\",\n \"$W_{\\\\mu \\\\nu }$ measures a current-current correlation function in the target ground state (c.f.\",\n \"Eq.\",\n \"(1.7)).\",\n \"In the Bjorken limit, the correlation probed becomes light-like, but may extend to very large spatial distances (and times) in the small $x$ limit.\",\n \"Note that the ranges of $\\\\xi ^0$ and $\\\\xi ^3$ probed by the correlated pair of currents in the target rest frame are independent of the target mass because $1/Mx=2\\\\nu /Q^2$ .\",\n \"The relation between $x$ and $|\\\\xi ^3|$ will be crucial to our analysis of nuclear effects in leptoproduction.\",\n \"It is often convenient to use different structure functions to describe $W_{\\\\mu \\\\nu }$ .\",\n \"A particularly important pair are $W_L$ and $W_T$ defined by $W_T &= W_1 \\\\nonumber \\\\\\\\W_L&= \\\\left( 1 + \\\\frac{\\\\nu ^2}{Q^2} \\\\right) W_2 - W_1 \\\\ .$ $W_T$ and $W_L$ arise when one considers the polarization of the virtual photon, $\\\\epsilon _\\\\mu $ .\",\n \"The leptonic current produces a flux of virtual photons which may be transverse, $\\\\epsilon ^\\\\mu _{T_1} &= (0, 0, 1, 0) \\\\nonumber \\\\\\\\\\\\epsilon ^\\\\mu _{T_2}&= (0, 1, 0, 0) \\\\ ,$ or longitudinal $\\\\epsilon ^\\\\mu _L = \\\\frac{1}{\\\\sqrt{Q^2}}(\\\\sqrt{\\\\nu ^2+Q^2}, 0, 0, -\\\\nu )\\\\ , \\\\qquad \\\\mathrm {(1.29)}$ in the target rest frame in which $q^\\\\mu $ is given by Eq.\",\n \"(1.21).\",\n \"Note $\\\\epsilon \\\\cdot q =0$ and $\\\\epsilon ^2_{T_j} = - \\\\epsilon ^2_L = -1$ .\",\n \"$W_T$ and $W_L$ are the components of $W_{\\\\mu \\\\nu }$ which couple to $\\\\epsilon _T$ and $\\\\epsilon _L$ , respectively: $W_T &\\\\equiv \\\\epsilon ^\\\\mu _{T_j} W_{\\\\mu \\\\nu } \\\\epsilon ^\\\\nu _{T_j} \\\\nonumber \\\\\\\\W_L&\\\\equiv \\\\epsilon ^\\\\mu _{L} W_{\\\\mu \\\\nu } \\\\epsilon ^\\\\nu _{L} \\\\ .$ They are proportional to the total cross sections for absorption of a transverse or longitudinal polarized virtual photon, respectively, $\\\\sigma _{T,L}(q^2,\\\\nu ) = \\\\frac{4 \\\\pi ^2 \\\\alpha }{k} W_{T,L}(q^2,\\\\nu ) \\\\qquad \\\\mathrm {(1.31)}$ where $k= (W^2-M^2_T)/2M_T$ .\",\n \"$\\\\sigma _L$ and $\\\\sigma _T$ are positive, so $W_1 \\\\ge 0$ and $(1+\\\\nu ^2/Q^2)W_2 \\\\ge W_1$ .\",\n \"As $q^2 \\\\rightarrow 0$ , $\\\\sigma _L$ vanishes and $\\\\sigma _T$ approaches the photoproduction cross section for real photons.\",\n \"Experimentalists measure $W_1$ and $W_2$ by comparing cross sections at fixed $q^2$ and $\\\\nu $ , but different values of $E$ , $E^\\\\prime $ and $\\\\theta $ : $\\\\frac{d^2 \\\\sigma }{dE^\\\\prime d \\\\Omega } = \\\\frac{4 \\\\alpha ^2{E^\\\\prime }^2 \\\\cos ^2 \\\\theta /2}{q^4} \\\\left[W_2(q^2,\\\\nu )+2W_1(q^2,\\\\nu ) \\\\tan ^2 \\\\theta /2 \\\\right]\\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.32)}$ In practice, it is convenient to write the differential cross section in terms of $\\\\sigma _L$ , $\\\\sigma _T$ and a parameter $\\\\epsilon $ : $\\\\epsilon ^{-1} \\\\equiv 1+2 \\\\left( 1+ \\\\frac{\\\\nu ^2}{Q^2} \\\\right) \\\\tan ^2 \\\\theta /2\\\\ , \\\\qquad \\\\mathrm {(1.33)}$ $\\\\frac{d^2 \\\\sigma }{dE^\\\\prime d \\\\Omega } = \\\\frac{\\\\alpha }{4 \\\\pi ^2 Q^2} \\\\frac{kE^\\\\prime }{E} \\\\left( \\\\frac{2}{1- \\\\epsilon } \\\\right) [\\\\sigma _T(q^2,\\\\nu ) + \\\\epsilon \\\\sigma _L(q^2,\\\\nu )]\\\\ .\",\n \"\\\\qquad \\\\mathrm {(1.34)}$ Apart from kinematic factors, $d^2\\\\sigma $ is a linear function of $\\\\epsilon $ .\",\n \"A linear fit to the data yields $\\\\sigma _L + \\\\sigma _T \\\\propto \\\\nu W_2$ as the intercept at $\\\\epsilon = 1$ , and $R \\\\equiv \\\\sigma _L/\\\\sigma _T$ as the slope.\",\n \"$R$ is difficult to measure.\",\n \"Data sets from different spectrometer settings and different beam energies must be combined, which introduces systematic uncertainties.\",\n \"At high beam energies $\\\\epsilon \\\\approx 1$ so the experiments are not sensitive to $R$ .\",\n \"Anyone who studies inelastic electron scattering should begin with the parton model of Bjorken and Feynman.\",\n \"There are many fine sources from which to learn it [8]; one or more should be studied in conjunction with these lectures.\",\n \"I, too, will describe the parton model, but quickly and maintaining as much contact with coordinate space as possible.\",\n \"If you have never seen the parton model before, the derivation presented here will appear difficult and rather formal.\",\n \"The parton model is often misused.\",\n \"To learn how to not misuse it one must approach the model who more formally, e.g, via the operator product expansion (OPE).\",\n \"Anyone who intends to do research in this field is strongly advised to study the OPE and the renormalization group in QCD beforehand [11], but I will only touch briefly on them.\",\n \"It is conventional to motivate the parton model by arguing that in some sense interactions can be ignored near the light cone.\",\n \"There is no realistic theory in which this is true, though in QCD it is approximately true.\",\n \"In $§5$ , we will develop a more precise language for such matters.\",\n \"Here, I will only state the assumptions which lead to the model.\",\n \"The first assumption of the parton model is that the current $J_\\\\mu $ couples to quarks (as opposed to fundamental scalars, etc.).\",\n \"Then the contributions to the forward virtual Compton amplitude can be classified by the flow and interactions of quark lines.\",\n \"The second assumption is that at large values of $Q^2$ the currents, but not the states, may be treated as in free field theory.\",\n \"Thus, final state interactions (Fig.\",\n \"REF (b)) and vertex corrections (Fig.\",\n \"REF (c)) are ignored.\",\n \"Figure: Some contributions to forward virtual Compton scattering in QCD: (a) The parton model diagram; (b) final state interactions; (c) vertex corrections; (d) interference.This leaves the one and two particle contributions shown in Figs.\",\n \"REF (a) and (d), respectively.\",\n \"Methods similar to those we shall apply to Fig.\",\n \"REF (a) show that the contribution of Fig.\",\n \"REF (d) vanish faster by a power of $Q^2$ , so I will ignore them henceforth.\",\n \"This leaves only Fig.\",\n \"REF (a): the elastic and incoherent scattering of each quark in the target, i.e., “quasielastic\\\" scattering.\",\n \"It is important to keep in mind the place of the parton model in QCD.\",\n \"It is valid modulo logarithms: quantities which scale — that is, they become functions of $x$ -alone in the parton model — will be modulated by powers of $\\\\ln Q^2$ when QCD interactions are included.\",\n \"Quantities which vanish like a power of $Q^2$ in the parton model may vanish only like a power of $\\\\ln Q^2$ in QCD.\",\n \"Figure: Parton model for W μν W_{\\\\mu \\\\nu }.The structure function, $W_{\\\\mu \\\\nu }$ , in the parton model is obtained by placing the intermediate state in Fig.\",\n \"REF (a) on-shell.\",\n \"The struck quark and the remnants of the target appear separately as physical intermediate states as shown in Fig.\",\n \"REF .\",\n \"This, of course, is wrong $-$ quarks are confined by non-perturbative effects in QCD.\",\n \"It is assumed that the processes which neutralize quark quantum numbers do not affect the dominant terms at large $Q^2$ .\",\n \"The justification for this is that those non-perturbative, confining effects in QCD which have been studied, while strong, appear to vanish rapidly with $Q^2$ .\",\n \"Until the non-perturbative aspects of QCD are better understood this will remain a major, though reasonable, assumption.\",\n \"The current $J_\\\\mu (\\\\xi )$ reduces to $\\\\overline{\\\\psi }(\\\\xi ) {\\\\cal {Q}} \\\\gamma _\\\\mu \\\\psi (\\\\xi )$ in a free quark model, where $\\\\cal {Q}$ is the quark charge matrix: ${\\\\cal {Q}} =$ diag$(2/3,-1/3,-1/3,...)$ for $u, d, s,...$ The current commutator in $W_{\\\\mu \\\\nu }$ reduces to [12] $[J_\\\\mu (\\\\xi ),J_\\\\nu (0)] = \\\\overline{\\\\psi }(\\\\xi ) \\\\gamma _\\\\mu S(\\\\xi ) \\\\gamma _\\\\nu {\\\\cal {Q}}^2 \\\\psi (0) - \\\\overline{\\\\psi }(0) \\\\gamma _\\\\nu S(-\\\\xi ) \\\\gamma _\\\\mu {\\\\cal {Q}}^2 \\\\psi (\\\\xi ) \\\\qquad \\\\mathrm {(2.1)}$ where $S(\\\\xi )$ is the anticommutator function [3] $S(\\\\xi ) = \\\\lbrace \\\\psi (\\\\xi ),\\\\overline{\\\\psi }(0) \\\\rbrace = -\\\\gamma ^\\\\rho \\\\partial /\\\\partial \\\\xi ^\\\\rho \\\\Delta (\\\\xi ) = S(-\\\\xi ) \\\\qquad \\\\mathrm {(2.2)}$ and $\\\\Delta (\\\\xi ) = \\\\frac{1}{2 \\\\pi } \\\\delta (\\\\xi ^2) \\\\epsilon (\\\\xi ^0) + .... \\\\qquad \\\\mathrm {(2.3)}$ The additional terms in $\\\\Delta (\\\\xi )$ vanish for zero quark mass.\",\n \"Since quark masses generate only ${\\\\cal O}(m^2/Q^2$ ) corrections in the Bjorken limit, we ignore them from now on.\",\n \"The product of operators in Eq.\",\n \"(2.1) is singular when $\\\\xi _\\\\lambda \\\\rightarrow 0$ , but the singularity is a $C$ -number $-$ namely the term removed by normal ordering $-$ so it doesn't contribute to a connected matrix element.\",\n \"Thus, $\\\\mathinner {\\\\langle {p|\\\\overline{\\\\psi }(\\\\xi )\\\\psi (0)|p}\\\\rangle }_c &= \\\\mathinner {\\\\langle {p|:\\\\overline{\\\\psi }(\\\\xi )\\\\psi (0):|p}\\\\rangle } \\\\nonumber \\\\\\\\&= - \\\\mathinner {\\\\langle {p|\\\\psi (0)\\\\overline{\\\\psi }(\\\\xi )|p}\\\\rangle } $ where the second equality is a consequence of normal ordering.\",\n \"The Lorentz structure of Eq.\",\n \"(2.1) is simplified by the identity $\\\\gamma _\\\\mu \\\\gamma _\\\\rho \\\\gamma _\\\\nu &\\\\equiv S_{\\\\mu \\\\nu \\\\rho \\\\sigma } \\\\gamma ^\\\\sigma - \\\\epsilon _{\\\\mu \\\\nu \\\\rho \\\\sigma } \\\\gamma ^\\\\sigma \\\\gamma ^5 \\\\nonumber \\\\\\\\S_{\\\\mu \\\\nu \\\\rho \\\\sigma } &\\\\equiv g_{\\\\mu \\\\rho } g_{\\\\nu \\\\sigma } + g_{\\\\mu \\\\sigma } g_{\\\\nu \\\\rho } - g_{\\\\mu \\\\nu } g_{\\\\rho \\\\sigma } \\\\ .\",\n \"$ The $\\\\epsilon $ -term does not contribute because it is impossible to construct a pseudovector from $\\\\xi _\\\\lambda $ and $p_\\\\lambda $ .\",\n \"Putting together the pieces, we find $\\\\lim _{Bj} W_{\\\\mu \\\\nu } &= - \\\\frac{S_{\\\\mu \\\\nu \\\\rho \\\\sigma }}{8 \\\\pi ^2} \\\\int d^4 \\\\xi \\\\ e^{i q \\\\cdot \\\\xi } \\\\left[\\\\frac{\\\\partial }{\\\\partial \\\\xi _\\\\rho } \\\\delta (\\\\xi ^2) \\\\epsilon (\\\\xi ^0) \\\\right] \\\\nonumber \\\\\\\\&\\\\times \\\\mathinner {\\\\langle {p|\\\\overline{\\\\psi }(\\\\xi ) \\\\gamma ^\\\\sigma {\\\\cal {Q}}^2\\\\psi (0) - \\\\overline{\\\\psi }(0) \\\\gamma ^\\\\sigma {\\\\cal {Q}}^2 \\\\psi (\\\\xi )|p}\\\\rangle }_c \\\\ .$ The $\\\\lim _{Bj}$ reminds us that the parton model assumptions which went into Eq.\",\n \"(2.6 ) are only (approximately) valid as $Q^2 \\\\rightarrow \\\\infty $ at fixed $x$ .\",\n \"Integrating by parts and introducing light-cone coordinates in the target rest frame $\\\\lim _{Bj} W_{\\\\mu \\\\nu } &= \\\\lim _{q^- \\\\rightarrow \\\\infty } \\\\frac{S_{\\\\mu \\\\nu \\\\rho \\\\sigma } i q^\\\\rho }{8 \\\\pi ^2} \\\\int d \\\\xi ^+ d \\\\xi ^- d^2 \\\\xi _\\\\perp e^{i q^+ \\\\xi ^- + i q^- \\\\xi ^+} \\\\nonumber \\\\\\\\&\\\\times \\\\delta (2\\\\xi ^+\\\\xi ^- - {\\\\xi }^2_\\\\perp ) \\\\epsilon (\\\\xi ^+ + \\\\xi ^-) \\\\nonumber \\\\\\\\&\\\\times \\\\mathinner {\\\\langle {p|\\\\overline{\\\\psi }(\\\\xi ) \\\\gamma ^\\\\sigma {\\\\cal {Q}}^2\\\\psi (0) - \\\\overline{\\\\psi }(0) \\\\gamma ^\\\\sigma {\\\\cal {Q}}^2 \\\\psi (\\\\xi )|p}\\\\rangle }_c \\\\ .$ We have dropped the term in which $\\\\partial /\\\\partial \\\\xi _\\\\rho $ acts on the matrix element since it generates at most a factor $p^\\\\rho $ or $\\\\xi ^\\\\rho \\\\mu ^2$ ($\\\\mu ^2$ is some mass characteristic of the target) both of which are negligible with respect to $q^\\\\rho $ in the Bjorken limit.\",\n \"[Note $\\\\xi ^\\\\rho $ Fourier transforms into $q^\\\\rho /q^2$ .]\",\n \"The form of $S_{\\\\mu \\\\nu \\\\rho \\\\sigma }$ implies that the coefficient of $-g_{\\\\mu \\\\nu }$ in $W_{\\\\mu \\\\nu }$ equals half the trace of $W_{\\\\mu \\\\nu }$ , or referring to Eqs.\",\n \"(1.10) and (1.11), $F_1 = \\\\frac{1}{2} \\\\left( 3F_1 - \\\\frac{1}{2 x_T} F_2 \\\\right) \\\\qquad \\\\mathrm {(2.8)}$ which requires $F_1 = \\\\frac{1}{2 x_T} F_2 \\\\qquad \\\\mathrm {(2.9)}$ or $\\\\lim _{Bj} \\\\sigma _L/\\\\sigma _T = 0 \\\\qquad \\\\mathrm {(2.10)}$ which is the famous Callan-Gross relation and follows from the quark spin being $1/2$ .\",\n \"It is well verified experimentally.\",\n \"Even though $R$ is difficult to measure and still a subject of debate, all experiments agree that for $Q^2 >1$ GeV$^2$ , $R < 0.2$ .\",\n \"Eq.\",\n \"(2.7) can be reduced to a one-dimensional integral.\",\n \"First, we use the $\\\\delta -$ function to perform the $\\\\xi ^2_\\\\perp $ integral leaving $F_2 &= 2 x_T \\\\lim _{q^- \\\\rightarrow \\\\infty } \\\\frac{iq^-}{8 \\\\pi } \\\\int d \\\\xi ^+ d \\\\xi ^- e^{i(q^+\\\\xi ^- + q^-\\\\xi ^+)} \\\\nonumber \\\\\\\\&\\\\times [\\\\theta (\\\\xi ^+)\\\\theta (\\\\xi ^-) - \\\\theta (-\\\\xi ^+)\\\\theta (-\\\\xi ^-)] \\\\nonumber \\\\\\\\&\\\\times \\\\mathinner {\\\\langle {p|\\\\overline{\\\\psi }(\\\\xi ) \\\\gamma ^+ {\\\\cal {Q}}^2 \\\\psi (0) - \\\\overline{\\\\psi }(0) \\\\gamma ^+ {\\\\cal {Q}}^2 \\\\psi (\\\\xi )|p}\\\\rangle }_c\\\\bigg \\\\vert _{{\\\\xi }^2_\\\\perp = 2 \\\\xi ^+ \\\\xi ^-} \\\\ ,$ then integrate by parts on $\\\\xi ^+$ keeping only the leading term at large $q^-$ $F_2 = \\\\frac{x_T}{4 \\\\pi } \\\\int d \\\\xi ^- e^{i q^+ \\\\xi ^-} \\\\mathinner {\\\\langle {p|\\\\overline{\\\\psi }(\\\\xi ^-) \\\\gamma ^+ {\\\\cal {Q}}^2 \\\\psi (0) - \\\\overline{\\\\psi }(0) \\\\gamma ^+ {\\\\cal {Q}}^2 \\\\psi (\\\\xi ^-)|p}\\\\rangle }_c \\\\bigg \\\\vert _{\\\\xi ^+ = {\\\\bf \\\\xi }_\\\\perp =0} \\\\ .\\\\qquad \\\\mathrm {(2.12)}$ So, $F_2$ is a dimensionless function of $q^+=-M_Tx_T/\\\\sqrt{2}$ , i.e.\",\n \"$F_2 \\\\rightarrow F_2(x_T)$ , which is Bjorken scaling.\",\n \"Before converting Eq.\",\n \"(2.12) into the most familiar parton model form, we should note that $F_2(x_T)$ measures a particular quark correlation function in the target ground state.\",\n \"The first term in Eq.\",\n \"(2.12), for example, measures the amplitude to remove a quark from the target at some point, $\\\\xi _1^\\\\mu $ , and replace it at $\\\\xi _2^\\\\mu $ with $\\\\xi _2^\\\\mu - \\\\xi _1^\\\\mu \\\\equiv \\\\xi ^\\\\mu $ , $\\\\xi ^+ = {\\\\bf \\\\xi }_\\\\perp =0$ and $|\\\\xi ^-|\\\\lesssim 1/q^+$ , leaving the target in the ground state.\",\n \"The shape of the structure function teaches us about a correlation function in the target ground state.\",\n \"One must be careful, however, not to use one's intuition from non-relativistic quantum mechanics: this is not an equal time correlation function, but instead a light-cone correlation function.\",\n \"More about these later.\",\n \"To further simplify (2.12) we must study light-cone $\\\\gamma -$ matrices.\",\n \"The matrices $P^\\\\pm = \\\\frac{1}{2} \\\\gamma ^\\\\mp \\\\gamma ^\\\\pm = \\\\frac{1}{2} (1 \\\\pm \\\\alpha ^3) \\\\qquad \\\\mathrm {(2.13)}$ are projection matrices: $P^+ = P^- =1$ , ${P^\\\\pm }^2 = P^\\\\pm $ , $P^\\\\pm P^\\\\mp =0$ , and if we define $\\\\psi _\\\\pm = P^\\\\pm \\\\psi \\\\ , \\\\qquad \\\\mathrm {(2.14)}$ then (2.12) may be written $F_2 = \\\\frac{x_T}{2 \\\\sqrt{2} \\\\pi } \\\\int d \\\\xi ^- e^{i q^+ \\\\xi ^-} \\\\mathinner {\\\\langle {p|\\\\psi ^\\\\dagger _+(\\\\xi ^-) {\\\\cal {Q}}^2 \\\\psi _+(0) + \\\\psi _+(\\\\xi ^-) {\\\\cal {Q}}^2 \\\\psi ^\\\\dagger _+(0)|p}\\\\rangle }_c \\\\bigg \\\\vert _{\\\\xi ^+ = {\\\\bf \\\\xi }_\\\\perp =0} \\\\ .\\\\qquad \\\\mathrm {(2.15)}$ where we used Eq.\",\n \"(2.4) to interchange the quark fields in the second term.\",\n \"If we now insert a complete set of states between quark fields, translate the $\\\\xi ^-$ dependence out of $\\\\psi _+$ or $\\\\psi ^\\\\dagger _+$ , integrate over $\\\\xi ^-$ and sum explicitly over quark flavors ($a = u,d,s,....$ ), we get $F^T_2(x_T) = x_T \\\\sum _a {\\\\cal {Q}}^2_a \\\\sum _n \\\\frac{1}{\\\\sqrt{2}} \\\\delta (p^++q^+-p^+_n)\\\\lbrace |\\\\mathinner {\\\\langle {n|\\\\psi _{a+}|p}\\\\rangle }|^2 + |\\\\mathinner {\\\\langle {n|\\\\psi ^\\\\dagger _{a+}|p}\\\\rangle }|^2 \\\\rbrace \\\\ .\",\n \"\\\\qquad \\\\mathrm {(2.16)}$ I have added a superscript $T$ to $F_2$ to remind us that $F_2$ depends on the target.\",\n \"For a target of mass $M_T$ , $q^+ = -x_TM_T/\\\\sqrt{2}$ where $x_T = Q^2/2M_T \\\\nu $ , and $p^+ = M_T/\\\\sqrt{2}$ (we are working in the target rest frame), so $F^T_2(x_T) = x_T \\\\sum _a {\\\\cal {Q}}^2_a (f_{a/T}(x_T) + f_{\\\\bar{a}/T}(x_T)) \\\\ , \\\\qquad \\\\mathrm {(2.17)}$ where $f_{a/T}(x_T) &= \\\\frac{1}{\\\\sqrt{2}} \\\\sum _n \\\\delta (p^+-x_Tp^+-p_n^+) |\\\\mathinner {\\\\langle {n|\\\\psi _{a+}|p}\\\\rangle }|^2 \\\\nonumber \\\\\\\\f_{\\\\bar{a}/T}(x_T) &= \\\\frac{1}{\\\\sqrt{2}} \\\\sum _n \\\\delta (p^+-x_Tp^+-p_n^+) |\\\\mathinner {\\\\langle {n|\\\\psi ^\\\\dagger _{a+}|p}\\\\rangle }|^2 \\\\ .$ This is the familiar parton model, except it is written in the target rest frame rather than the “infinite momentum frame\\\".\",\n \"$f_{a/T}(x_T)$ is the probability (per unit $x_T$ ) to remove from the target a quark of flavor $a$ with “momentum\\\" (i.e.\",\n \"$p^+$ ) fraction $x_T$ , leaving behind a physical state ($\\\\mathinner {|{n}\\\\rangle }$ ) with $p^+_n = (1-x_T)p^+$ .\",\n \"Similarly, $f_{\\\\bar{a}/T}(x_T)$ is the probability (per unit $x_T$ ) to remove an antiquark with $p^+$ -fraction $x_T$ leaving behind a physical state with $p^+_n = (1-x_T)p^+$ .\",\n \"$f_{a/T}(x_T)$ $f_{\\\\bar{a}/T}(x_T)$ are shown graphically in Figs.\",\n \"REF (b) and (c).\",\n \"Figure: (a) The virtual-quark hadron scattering amplitude; (b) The quark distribution function f a/T (x T )f_{a/T}(x_T) .\",\n \"Note that k + k^+ is fixed but k - k^- and 𝐤 ⊥ {\\\\bf k}_\\\\perp [Notice that $p^+$ appears in the target rest frame formulation where the “infinite momentum\\\" $P_\\\\infty $ appears in the more familiar formulation.]\",\n \"$f_{a/T}(x_T)$ and $f_{\\\\bar{a}/T}(x_T)$ obey important positivity, spectral and normalization constraints.\",\n \"The state $n$ in Eq.\",\n \"(2.18) is physical and must have $p^+_n>0$ , i.e., $E_n > |\\\\bf p_n|$ , thus $f_{a/T}(x_T) = f_{\\\\bar{a}/T}(x_T) =0$ , for $x_T \\\\ge 1$ or $x \\\\ge M_T/M$ ($\\\\approx A$ for a nuclear target).\",\n \"For $0<x_T<1$ , $f_{a/T}(x_T)$ defined by Eq.\",\n \"(2.18) is manifestly positive.\",\n \"Next, let us consider $f_{a/T}(x_T)$ for $x_T<0$ .\",\n \"Returning to Eq.\",\n \"(2.15) $f_{a/T}(x_T) &= \\\\frac{1}{2 \\\\sqrt{2} \\\\pi } \\\\int d \\\\xi ^- e^{i q^+ \\\\xi ^-} \\\\mathinner {\\\\langle {p|\\\\psi ^\\\\dagger _{a+}(\\\\xi ^-) \\\\psi _{a+}(0)|p}\\\\rangle }_c \\\\bigg \\\\vert _{\\\\xi ^+ = {\\\\bf \\\\xi }_\\\\perp =0} \\\\nonumber \\\\\\\\&= - \\\\frac{1}{2 \\\\sqrt{2} \\\\pi } \\\\int d \\\\xi ^- e^{i q^+ \\\\xi ^-} \\\\mathinner {\\\\langle {p|\\\\psi ^\\\\dagger _{a+}(0) \\\\psi _{a+}(\\\\xi ^-)|p}\\\\rangle }_c \\\\bigg \\\\vert _{\\\\xi ^+ = {\\\\bf \\\\xi }_\\\\perp =0} $ where we've used Eq.\",\n \"(2.4).\",\n \"Replacing $\\\\xi ^-$ by $-\\\\xi ^-$ and translating the matrix element we find $f_{a/T}(-x_T) = - f_{\\\\bar{a}/T}(x_T) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(2.20)}$ Although $f_{a/T}(x_T)$ does not vanish for $x<0$ (it does vanish for $x_T<-1$ ) it is determined by $f_{\\\\bar{a}/T}(x_T)$ for $x>0$ .\",\n \"The literature is very confused on this point: there is a lot of talk about the need to “prove\\\" that “no partons can go backwards in the infinite momentum frame\\\", i.e., $f_{a/T}(x_T) \\\\equiv 0$ for $x_T <0$ .\",\n \"We see here that $f_{a/T}(x_T)$ is defined in such a way that measurements in the physical region $0<x_T<1$ determine it everywhere.\",\n \"For a more extensive discussion of this issue, see Ref [13].\",\n \"Now let us integrate $f_{a/T}(x_T)$ over all $x_T$ .\",\n \"Using $x_T = -\\\\sqrt{2}q^+/M_T$ and Eq.\",\n \"(2.19) and Eq.\",\n \"(2.20) we find $\\\\int ^\\\\infty _{-\\\\infty } dx_T f_{a/T}(x_T) &= \\\\int _0^1 dx_T(f_{a/T}(x_T) - f_{{\\\\bar{a}}/T}(x_T)) \\\\nonumber \\\\\\\\&= \\\\frac{1}{M_T} \\\\mathinner {\\\\langle {p|\\\\psi ^\\\\dagger _{a+}(0)\\\\psi _{a+}(0)|p}\\\\rangle }_c \\\\nonumber \\\\\\\\&= N_{a/T} - N_{{\\\\bar{a}}/T} $ where the last step follows from the fact that $\\\\psi ^\\\\dagger _{a+} \\\\psi _{a+} = \\\\frac{1}{\\\\sqrt{2}}j^+_a$ and $j_a^\\\\mu $ is a conserved current whose expectation value measures the number of quarks (minus the number of antiquarks) of flavor $a$ : $\\\\mathinner {\\\\langle {p|j^\\\\mu _a|p}\\\\rangle } = 2 p^\\\\mu (N_{a/T} - N_{{\\\\bar{a}}/T}$ ).\",\n \"Clearly, we may interpret $f_{a/T}(x_T)$ ($f_{\\\\bar{a}/T}$ ) as a probability per unit $x_T$ to find a quark (antiquark) of flavor $a$ with $k^+ = x_T p^+$ in the target T $\\\\frac{d P_{a/T}}{d x_T} = f_{a/T}(x_T) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(2.22)}$ From this interpretation follows a host of parton model sum rules for the structure functions.\",\n \"The most important for our purpose is the “momentum sum rule\\\", $\\\\int _0^1 dx_T\\\\,\\\\,x_T (f_{a/T}(x_T) + f_{\\\\bar{a}/T}(x_T)) = \\\\epsilon _{a/T} + \\\\epsilon _{\\\\bar{a}/T} \\\\qquad \\\\mathrm {(2.23)}$ where $\\\\epsilon _{a/T}$ ($\\\\epsilon _{\\\\bar{a}/T}$ ) is the fraction of the target's $p^+$ carried by quarks (antiquarks) of flavor $a$ .\",\n \"The derivation mimics the derivation of Eq.\",\n \"(2.21) except the quark stress tensor $i \\\\bar{\\\\psi }_a \\\\gamma ^\\\\mu \\\\partial ^\\\\nu \\\\psi _a$ appears instead of $j^\\\\mu _a$ .\",\n \"If hadrons contained only quarks $\\\\sum _a \\\\epsilon _{a/T}$ would be 1.\",\n \"Instead, it is typically $\\\\sim \\\\frac{1}{2}$ (at large $Q^2$ ) indicating that substantial momentum and energy are carried by other, neutral quanta, namely gluons.\",\n \"If two targets are related by a symmetry, their quark distributions are similarly related.\",\n \"Isospin relates the neutron and proton and gives $f_{u/n}(x) &= f_{d/p}(x) \\\\nonumber \\\\\\\\f_{d/n}(x) &= f_{u/p}(x) \\\\nonumber \\\\\\\\f_{s/n}(x) &= f_{s/p}(x),\\\\ {\\\\rm etc.}\",\n \"$ For completeness, I record the structure functions for electron and (charged current) neutrino scattering from nucleon targets assuming isospin symmetry: $F^{ep}_2(x) &= x \\\\left[ \\\\frac{4}{9}(f_{u/p}(x) + f_{\\\\bar{u}/p}(x)) + \\\\frac{1}{9}(f_{d/p}(x) + f_{\\\\bar{d}/p}(x)) + f_{s/p}(x) + f_{\\\\bar{s}/p}(x)) \\\\right] \\\\nonumber \\\\\\\\F^{en}_2(x) &= x \\\\left[ \\\\frac{4}{9}(f_{d/p}(x) + f_{\\\\bar{d}/p}(x)) + \\\\frac{1}{9}(f_{u/p}(x) + f_{\\\\bar{u}/p}(x)) + f_{s/p}(x) + f_{\\\\bar{s}/p}(x)) \\\\right] \\\\nonumber \\\\\\\\F^{\\\\nu p}_2 (x) &= F^{\\\\bar{\\\\nu }n}_2(x) = 2x [f_{d/p}(x) + f_{\\\\bar{u}/p}(x)] \\\\nonumber \\\\\\\\F^{\\\\nu n}_2 (x) &= F^{\\\\bar{\\\\nu }p}_2(x) = 2x [f_{u/p}(x) + f_{\\\\bar{d}/p}(x)] \\\\ .\",\n \"$ I have ignored heavy quarks ($c, b, t$ ) and the Cabibbo angle ($\\\\sin ^2 \\\\theta _c \\\\cong 0.05$ ) and used the shorthand notation $\\\\nu p$ ($\\\\bar{\\\\nu }p$ ) for $ \\\\nu p \\\\rightarrow \\\\mu ^- X$ ($\\\\bar{\\\\nu }p \\\\rightarrow \\\\mu ^+ X$ ), etc.\",\n \"I have also left out the (important) parity violating structure function, $F_3$ , which arises in neutrino scattering.\",\n \"One important implication of Eqs.\",\n \"(2.25) is that in the absence of any nuclear effect in deuterium one would expect $F^{ed}_2(x) &= x \\\\frac{5}{9} \\\\left[ f_{u/p}(x) + f_{\\\\bar{u}/p}(x) + f_{d/p}(x) + f_{\\\\bar{d}/p}(x) \\\\right]+ \\\\frac{2}{9}\\\\left[ (f_{s/p}(x) + f_{\\\\bar{s}/p}(x)) \\\\right] \\\\nonumber \\\\\\\\&\\\\cong \\\\frac{5}{18} \\\\left[ F^{\\\\nu p}_2(x) + F^{\\\\bar{\\\\nu }p}_2(x) \\\\right] \\\\ , $ because strange quarks are suppressed in the nucleon and weighted by $2/5$ relative to non-strange quarks.\",\n \"This relation allows one to look for a nuclear effect in the deuteron in a model independent manner (see.\",\n \"Ref. [15]).\",\n \"To further simplify the structure functions of nucleons, it is customary to distinguish between quarks which must be present to account for the target's quantum numbers ($u$ and $d$ quarks for nucleons) known as “valence\\\" quarks and those which may be present in pairs due to relativistic effects, known as “ocean quarks\\\".\",\n \"Thus, for an isospin averaged nucleon $f_{u/N}(x) &= f_{d/N}(x) = f_V(x) + f_O(x) \\\\nonumber \\\\\\\\f_{\\\\bar{u}/N}(x) &= f_{\\\\bar{d}/N}(x) = f_O(x) \\\\ .\",\n \"$ Strange and heavier quarks may also be present.\",\n \"Typically, one assumes $f_{s/N}(x) &= f_{\\\\bar{s}/N}(x) \\\\le f_O(x) \\\\nonumber \\\\\\\\f_{c/N}(x) &= f_{\\\\bar{c}/N}(x) \\\\approx 0 \\\\ ,\\\\ {\\\\rm etc.}\",\n \"\\\\nonumber $ Before leaving this general discussion of the parton model, it is useful to relate the quark distribution function to the amplitude for quark-target scattering.\",\n \"We define the (connected) virtual quark-target forward scattering amplitude by $\\\\chi _{a/T} \\\\equiv \\\\int d^4 \\\\xi e^{-i k \\\\cdot \\\\xi } \\\\mathinner {\\\\langle {p|T(\\\\overline{\\\\psi }_a(\\\\xi )\\\\psi _a(0))|p}\\\\rangle }_c \\\\qquad \\\\mathrm {(2.28)}$ as illustrated in Fig.\",\n \"REF (a).\",\n \"$\\\\chi $ is a matrix in color and Dirac spaces, but we have suppressed those indices.\",\n \"The distribution function $f_{a/T}(x_T)$ can be projected out of $\\\\chi _{a/T}$ by integrating over all components of $k$ except $k^+$ which is held fixed, $k^+=x_Tp^+$ , and tracing the Dirac indices with $\\\\gamma ^+$ : $f_{a/T}(x_T) = \\\\int \\\\frac{d^4 k}{(2 \\\\pi )^4} \\\\delta \\\\left(\\\\frac{k^+}{p^+}-x_T \\\\right) {\\\\rm Tr} [\\\\gamma ^+\\\\chi _{a/T}(k,p)] \\\\ .\",\n \"\\\\qquad \\\\mathrm {(2.29)}$ It is easy to verify that Eqs.\",\n \"(2.28)$-$ (2.29) lead to Eq.\",\n \"(2.18) provided one used Eq.\",\n \"(2.4) to relate the $T$ -product to the ordinary product.\",\n \"It is often convenient, when studying electroproduction from nuclei, to define quark distribution functions depending on a universal variable $x = Q^2/2M_N \\\\nu $ .\",\n \"To preserve their probabilistic interpretation, it is necessary to rescale them: $F_{a/T}(x) \\\\equiv \\\\frac{d P_{a/T}}{dx} = \\\\frac{d P_{a/T}}{d x_T} \\\\frac{dx_T}{dx}=\\\\frac{M}{M_T} f_{a/T}(x_T) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(2.30)}$ (See Eq. 2.22.)\",\n \"Then, $\\\\int _0^{M_T/M} dx (F_{a/T}(x) - F_{\\\\bar{a}/T}(x)) = N_{a/T} - N_{\\\\bar{a}/T} \\\\qquad \\\\mathrm {(2.31)}$ and $\\\\int _0^{M_T/M} x dx (F_{a/T}(x) + F_{\\\\bar{a}/T}(x)) = \\\\frac{M_T}{M} (\\\\epsilon _{a/T} + \\\\epsilon _{\\\\bar{a}/T}) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(2.32)}$ At the same time, it is convenient to introduce a sructure function per nucleon, $\\\\overline{F}^T_2(x)$ $\\\\overline{F}^T_2 (x) \\\\equiv x \\\\sum _a {\\\\cal {Q}}_a^2 (\\\\overline{F}_{a/T}(x) + \\\\overline{F}_{\\\\bar{a}/T}(x)) \\\\qquad \\\\mathrm {(2.33)}$ where $\\\\overline{F}_{a/T}(x) = F_{a/T}(x)/A$ .\",\n \"In the analysis of nuclear targets, I will try to preserve this notation: lower case $(f_{a/T}(x_T))$ for intrinsically defined distribution functions as functions of $x_T$ , upper case for functions of $x$ and barred upper case for functions of $x$ “per nucleon\\\".\",\n \"Also, the label “A\\\" will denote a nucleus, “T\\\" a generic target and “a\\\" a quark of flavor $a$ .\",\n \"Note that $\\\\overline{F}^A_2(x,q^2)$ is defined so that it would reduce to the (isospin weighted) nucleon structure function $\\\\frac{Z}{A}F^P_2(x,q^2) + \\\\frac{N}{A}F^N_2(x,q^2)$ if the nucleons in the nucleus were non-interacting.\",\n \"At this point, it would be appropriate to discuss the phenomenology of the neutron and proton structure functions; however, no time would be left for nuclear targets.\",\n \"So the reader will have to consult the references [14] for more information about nucleons.\",\n \"Here, I will mention only a few properties of importance for future work.\",\n \"Near $x = 0$ both $F_{O/N}(x)$ and $F_{V/N}(x)$ are expected to diverge: $F_{O/N} \\\\sim 1/x$ and $F_{V/N}(x) \\\\sim 1/\\\\sqrt{x}$ .\",\n \"As $x \\\\rightarrow 1$ , $F_{O/N}/F_{V/N} \\\\rightarrow 0$ .\",\n \"The neutron and proton structure functions differ significantly at large $x$ leading to the observation that $f_{d/p}/f_{u/p} \\\\rightarrow 0$ as $x \\\\rightarrow 1$ .\",\n \"Some quark distributions extracted from electron and neutrino scattering experiments are shown in Fig.\",\n \"REF .\",\n \"Figure: Examples of quark, antiquark and gluon distributions in the nucleon, from F. Dydak .Before 1982, little attention had been paid to deep inelastic lcpton scattering from nuclei.\",\n \"Interest in the subject was awakened by experimental results from CERN.\",\n \"In order to put the rest of these lectures in proper context it is necessary to present those results, discuss the (dis)agreemcnt among experiments and present the most rudimentary parton model analysis of the data, which already has important implications for the quark substructure of nuclei.\",\n \"The basic parton model of Bjorken and Feynman and the material of §2 are the only prerequisites for this analysis.\",\n \"It will be impossible, however, to avoid some reference to QCD, asymptotic freedom and other issues which will be introduced later in these lectures.\",\n \"Figure: EMC data on F ¯ 2 Fe /F 2 D \\\\overline{F}^{Fe}_2/F^D_2 .The first precise comparison of deep inelastic scattering from a nuclear target with scattering from a nucleon was made by the EMC Collaboration [2].\",\n \"They compared iron and deuterium targets.\",\n \"They assumed that the weakly bound deuteron approximates the isospin averaged nucleon.\",\n \"Subsequent analysis by Bodek and Simon [15] has confirmed this assumption.\",\n \"In the absence of any nuclear effect at large $Q^2$ one expects $F^A_2(x,Q^2) = AF^D_2(x,Q^2)/2$ up to a small correction for the neutron excess in the nucleus.\",\n \"To display deviations from this naive expectation and to minimize systematic errors, it is conventional to plot $S^A (x,Q^2) \\\\equiv 2F^A_2(x,Q^2)/AF^D_2(x,Q^2)$ .\",\n \"The deviation of $S^{\\\\rm Fe}(x,Q^2)$ from unity shown in Fig.\",\n \"REF caught nearly everyone by surprise [16] The effect at large $x$ was particularly surprising because $\\\\overline{F}^A_2(x, Q^2)/ F^N_2(x, Q^2)$ must go to infinity as $x \\\\rightarrow 1$ .\",\n \"[The denominator vanishes for $x > 1$ , the numerator vanishes only for $x > A$ .]\",\n \"The “EMC effect\\\", as the deviation of $S^{\\\\rm Fe}$ from unity came to be known, was quickly confirmed by a reanalysis of old SLAC data with iron, aluminum and deuterium targets [17].\",\n \"Last year a dedicated SLAC experiment (E-139) measured the EMC effect on a sequence of nuclear targets [18].\",\n \"Their data are shown in Fig.\",\n \"REF .\",\n \"Figure: SLAC data on σ A /σ D \\\\sigma ^A/\\\\sigma ^D .\",\n \"The fits are from .The SLAC and EMC data agree reasonably well for $x > 0.3$ but the enhancement seen at low-$x$ by EMC was not observed at SLAC.\",\n \"Recently, there has been input from other groups.\",\n \"The BCDMS collaboration at CERN [19], has measured deep inelastic muon scattering from iron, nitrogen and deuterium targets.\",\n \"Their iron data are restricted by detector geometry to $x \\\\ge 0.2$ where they agree with both SLAC and EMC.\",\n \"All available data on iron are shown in Fig.\",\n \"REF .\",\n \"The nitrogen data include a point at $x = 0.1$ which is significantly above 1 but below the trend of the EMC iron data.\",\n \"[See Fig.\",\n \"REF ] Figure: Compilation of data on F ¯ 2 Fe /F ¯ 2 D \\\\overline{F}^{\\\\rm Fe}_2/\\\\overline{F}^D_2 .Figure: Comparison of BCDMS nitrogen data with SLAC carbon data .Several neutrino detector collaborations have measured ratios of cross sections from nuclear targets [20].\",\n \"In general, their results are subject to larger statistical and systematic uncertainties than the electron and muon data.\",\n \"It is fair to say, however, that at low-$x$ , where their statistics are best, the neutrino experiments fail to confirm the enhancement seen by the EMC group.\",\n \"For example, the CDHS group presented some data on $S^{\\\\rm Fe}$ at this year's Moriond meeting [21].\",\n \"Their data are also shown in Fig.\",\n \"REF .\",\n \"Figure: CDHS (neutrino scattering) measurement of F ¯ 2 Fe /F 2 N \\\\overline{F}^{Fe}_2/F^N_2 .It is of great importance to sort out the apparent disagreement among experiments at low-$x$ .\",\n \"This can only be done for certain by future experiments.\",\n \"However, we can catalog some of the possibilities while we wait.\",\n \"To help, some of the salient features of the experiments are summarized in Table I.\",\n \"Among the possibilities are: 1.\",\n \"Systematic errors: The EMC collaboration quote a considerable systematic error in the slope of the straight line fit to their data.\",\n \"They also quote a 7% systematic uncertainty in the normalization [2].\",\n \"Rotating their data to smaller slope and lowering it by 7% improves its agreement with SLAC and BCDMS considerably.\",\n \"Merlo quoted a systematic error on the $x = 0.03$ CDHS point equal to the statistical error.\",\n \"Moving the CDHS point up by that amount largely removes the discrepancy.\",\n \"2.\",\n \"An $A$ dependence of $R (\\\\sigma _L/\\\\sigma _T$ ): The SLAC E-139 experiment measured $R$ on several nuclear targets and found it large ($R \\\\cong 0.2$ ) and possibly $A$ dependent.\",\n \"If they extract $S^A$ from their data with an $A$ dependent $R$ it agrees better with the EMC measurement.\",\n \"This reconciliation is only superficial, however.\",\n \"It generates a much greater disagreement in the measurement of $\\\\overline{F}^{\\\\rm Fe}_1/F^D_1$ .\",\n \"[EMC do not, in fact, measure $F_1$ because the beam energy is high ($\\\\epsilon \\\\approx 1$ , see Eq.\",\n \"(1.34)) but at EMC energies $Q^2$ is quite large ($Q^2 \\\\ge 9$ GeV$^2$ for all $x-$ bins) and $R$ is expected to be very small at large $Q^2$ (even in QCD where the Callan-Gross relation is not exact) so $\\\\overline{F}^A_1/F^N_1 \\\\cong \\\\overline{F}^A_2/F^N_2$ .]\",\n \"3.\",\n \"Strong $Q^2$ , $x$ and $A$ dependence at low-$x$ : Perhaps the differences among experiments are due to the fact that their bins average in different ways over a rapidly varying function.\",\n \"The likely source of this variation is “shadowing\\\" which is expected to be important at low$-x$ and will be discussed further in $§6.4$ .\",\n \"Table: Summary of experiments.For the remainder of these lectures, I will assume that the EMC data for $x > 0.3$ are correct, but for $x < 0.3$ , I will assume the truth lies somewhere between the EMC and SLAC results.\",\n \"The parton model ideas developed in the previous section can be applied directly to the nuclear structure function [22].\",\n \"In the parton model and the Bjorken limit, $\\\\overline{F}^A_2(x)$ is independent of $Q^2$ .\",\n \"In QCD, the notion of parton distribution functions and other aspects of the parton model are preserved, but $\\\\overline{F}^A_2(x)$ and the parton distribution functions develop a weak but important $Q^2$ -dependence.\",\n \"To allow for this, we occasionally keep the $Q^2$ -label explicit $\\\\overline{F}^A_2(x,Q^2) = x \\\\sum _a {\\\\cal {Q}}^2_a (\\\\overline{F}_{a/A}(x,Q^2) + \\\\overline{F}_{\\\\bar{a}/A}(x,Q^2)) \\\\qquad \\\\mathrm {(3.1)}$ and the difference between a nucleus and deuterium is defined by $\\\\overline{\\\\Delta }_A(x,Q^2) \\\\equiv \\\\overline{F}^A_2(x,Q^2) - \\\\overline{F}^D_2(x,Q^2) \\\\ .\\\\qquad \\\\mathrm {(3.2)}$ $\\\\overline{\\\\Delta }$ depends only on the difference of quark distributions in nucleus $A$ and deuterium.\",\n \"In the valence parton model described in $§2$ , $\\\\overline{\\\\Delta }_A(x,Q^2) = x \\\\left( \\\\frac{5}{9}\\\\delta \\\\overline{F}_{V/A}(x,Q^2) + \\\\frac{4}{3} \\\\delta \\\\overline{F}_{O/A}(x,Q^2) \\\\right) \\\\qquad \\\\mathrm {(3.3)}$ and $\\\\delta \\\\overline{F}_{V/A} \\\\equiv \\\\overline{F}_{V/A} - \\\\overline{F}_{V/D}$ , etc.\",\n \"Figure: The difference of the structure function per nucleon of iron an deuterium .The EMC data for $\\\\overline{\\\\Delta }_{\\\\rm Fe}$ are shown in Fig.\",\n \"REF , which was constructed from the published EMC iron data and the data on $S^{\\\\rm Fe}$ .\",\n \"Experimentalists caution that systematic errors are more treacherous in the difference $\\\\overline{\\\\Delta }_{\\\\rm Fe}$ than in the ratio $S^{\\\\rm Fe}$ .\",\n \"Nevertheless, the qualitative features of $\\\\overline{\\\\Delta }_{\\\\rm Fe}$ are probably reliable (modulo our caveats about $x < 0.3$ ) and they tell us what has happened to the quark distributions in passing from deuterium to iron.\",\n \"The EMC effect — at least as seen in the EMC data — has several different aspects which can be extracted from a parton model analysis of the data.\",\n \"The same analysis applied to the SLAC data leads to somewhat different conclusions which I will mention along the way: 1.\",\n \"The valence quarks in iron are “degraded\\\" — shifted to lower $p^+$ — relative to those in deuterium.\",\n \"[All experiments agree on this.]\",\n \"2.\",\n \"There is an increase in the number of ocean quark pairs in iron compared to deuterium.\",\n \"[The increase is large if we believe the EMC data, smaller or even absent if one accepts the SLAC data at low-$x$ .]\",\n \"3.\",\n \"The fraction of momentum ($p^+$ ) per nucleon on quarks and antiquarks in iron relative to deuterium can be extracted.\",\n \"[It increases slightly if one believes the EMC data but the SLAC data are ambiguous.]\",\n \"These results hold for other nuclei as well in proportion to the size of the measured “EMC effect\\\".\",\n \"The extraction makes use of parton model sum rules and positivity constraints.\",\n \"For $x > 0.35$ , $\\\\overline{F}_{O/N}(x)$ (the ocean quark distribution in the isolated nucleon) is known to be negligible, thus $\\\\delta \\\\overline{F}_{O/N}(x) \\\\ge 0$ for $x > 0.35$ .\",\n \"Since $\\\\overline{\\\\Delta }_A(x) < 0$ for $x > 0.35$ , we see from Eq.\",\n \"(3.3) that $\\\\delta \\\\overline{F}_{V/A}(x)$ must be negative for $x > 0.35$ .\",\n \"The valence quark distribution is conserved, i.e., $\\\\int _0^A dx~\\\\delta \\\\overline{F}_{V/A}(x) = 0 \\\\qquad \\\\mathrm {(3.4)}$ because the number of valence quarks (per nucleon) in all nuclei is three.\",\n \"$\\\\delta \\\\overline{F}_{V/A}(x)$ must therefore be positive for some $x$ values where it has not been observed.\",\n \"The SLAC data show $\\\\delta \\\\overline{F}_{V/A}(x)$ turning positive for $x>0.8$ by which point $F_{V/A}(x)$ is very small.\",\n \"I assume that $\\\\int _{0.8}^A dx~\\\\delta F_{V/A}(x)$ is negligible.\",\n \"This leaves $x< 0.35$ as the region in which $\\\\delta \\\\overline{F}_{V/A}(x)$ is positive.\",\n \"In $§1$ we saw that $x$ is conjugate to $1/M \\\\xi ^3$ in the laboratory, where $\\\\xi ^3$ is the spatial separation of the currents in the correlation function defining $F_{a/A}(x)$ .\",\n \"Thus, the shift to lower $x$ is indicative of a shift, to longer range quark correlations in the target ground state.\",\n \"This result, is independent of whatever “microscopic\\\" explanation of the EMC effect might eventually be forthcoming.\",\n \"Whether it is attributed to “dynamical rescaling\\\", $N$ -quark bags, quark percolation, or more prosaic sources like pion admixtures in the nuclear wavefunction or other binding effects, the EMC effect directly measures an increased quark light-cone correlation length in nuclei.\",\n \"For more discussion of the space-time interpretation of the EMC effect, see [10].\",\n \"In retrospect, it is not surprising that measures of the quark correlation length increase in nuclei [23].\",\n \"It is believed that quark/nuclear matter, regarded as a function of density at zero temperature, undergoes a deconfining phase transition at some $\\\\rho _{\\\\rm critical}$ .\",\n \"For densities below $\\\\rho _{\\\\rm critical}$ , quarks are confined in nucleons but for densities above $\\\\rho _{\\\\rm critical}$ , they move about more or less freely in a degenerate quark gas.\",\n \"One support for this is that QCD is known to become asymptotically free at large chemical potential (equivalent to high density), so at high enough density a quark gas will become free.\",\n \"We identify the nucleon as the zero density limit of quark matter.\",\n \"As $A$ increases, the mean density of the nucleus increases (as the surface to volume ratio goes to zero), so we may regard the increased quark correlation length in iron as a consequence of its increased mean density and as a precursor of a deconfining phase transition where the correlation length would become very large.\",\n \"This view of the EMC effect is supported by the $A$ dependence observed at SLAC which correlates very closely with nuclear densities.\",\n \"It is discussed at length in $§6.3$ .\",\n \"Point 2, the measurement of ocean quark pairs, is obtained by examining $\\\\int dx \\\\delta \\\\overline{F}_{O/A}(x)$ over the range of the measured data ($x_{\\\\rm min}<x<x_{\\\\rm max}$ ): $\\\\int _{x_{\\\\rm min}}^{x_{\\\\rm max}} dx~\\\\delta \\\\overline{F}_{O/A}(x) = \\\\frac{3}{4} \\\\int _{x_{\\\\rm min}}^{x_{\\\\rm max}} \\\\frac{dx}{x} \\\\overline{\\\\Delta }_A(x) + \\\\frac{5}{12} \\\\int _0^{x_{\\\\rm min}} dx~\\\\delta \\\\overline{F}_{V/A}(x) \\\\qquad \\\\mathrm {(3.5)}$ where we have used Eq.\",\n \"(3.4) and assumed $\\\\delta \\\\overline{F}_{V/A}(x)$ is negligible for $x_{\\\\rm max}<x<A$ .\",\n \"If $\\\\delta \\\\overline{F}_{V/A}(x)$ does not change sign twice, that is, if it remains positive for $x<x_{\\\\rm min}$ , then $\\\\int _{x_{\\\\rm min}}^{x_{\\\\rm max}} dx~\\\\delta \\\\overline{F}_{O/A}(x) > \\\\frac{3}{4} \\\\int _{x_{\\\\rm min}}^{x_{\\\\rm max}} \\\\frac{dx}{x} \\\\overline{\\\\Delta }_A(x) \\\\ .\\\\qquad \\\\mathrm {(3.6)}$ Eq.\",\n \"(3.6) could fail only if $\\\\overline{F}_{V/A}(x)$ behaves as shown in Fig.\",\n \"REF [24].\",\n \"Figure: Behavior of F ¯ 2 Fe /F ¯ 2 D \\\\overline{F}^{\\\\rm Fe}_2/\\\\overline{F}^D_2 required by the EMC data at low-x-x if only valence quarks are involved in the effect.Shadowing is expected to produce a depletion in $\\\\overline{F}^A_2(x,Q^2)$ at low-$x$ and low $Q^2$ , but it should affect primarily the ocean quark distribution and a shadowing of the valence quarks sufficient to invalidate Eq.\",\n \"(3.6) at large $Q^2$ would be surprising.\",\n \"According to Eq.\",\n \"(3.6), the ocean quarks are enhanced to the extent that the $x^{-1}$ weighted integral of $\\\\overline{\\\\Delta }_A$ is $>0$ .\",\n \"If one accepts the EMC data, the effect is quite large: $\\\\int _{x_{\\\\rm min}}^{x_{\\\\rm max}} dx~\\\\delta \\\\overline{F}_{O/{\\\\rm Fe}}(x) > 1.5 \\\\int _{x_{\\\\rm min}}^{x_{\\\\rm max}}dx\\\\, F_{O/N}(x) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(3.7)}$ If we ignore the small contributions from $x > x_{\\\\rm max}$ , and $x < x_{\\\\rm min}$ , the EMC data give $5.0 \\\\pm 1.5 \\\\times 10^{-2}$ for the first term on the right.\",\n \"The second term is small and positive (see Eq.\",\n \"(3.4) and subsequent discussion) so we obtain a bound $\\\\delta \\\\epsilon _{\\\\rm Fe} > 5.0 \\\\pm 1.5 \\\\times 10^{-2}$ .\",\n \"The SLAC data, on the other hand, do not give a positive $\\\\overline{\\\\Delta }_A(x)$ for $x < 0.3$ so the sign of $\\\\delta \\\\epsilon _{\\\\rm Fe}$ cannot be determined although its magnitude is certainly small [25] At the level of the parton model, these features of the data appear logically independent.\",\n \"Some models, notably the rescaling model, are able to correlate a modest increase in the number of ocean quarks with the degradation (i.e., shift to lower $x$ ) of the valence quarks.\",\n \"Other models are able to account for only one feature of the data or invoke several effects in concert to account for the different aspects of the EMC effect.\",\n \"It is intuitively appealing to regard inclusive electroproduction from nuclei as a two step process.\",\n \"First, the nuclear wave function is decomposed into some basis of constituents, nucleons in the first instance, nucleons and pions in more elaborate schemes, and later perhaps including more exotic objects like $\\\\Delta $ s, multiquark configurations, and so on.\",\n \"Then, the structure functions of the constituents are added incoherently to give the structure function of the whole nucleus.\",\n \"This is the “convolution\\\" model\\\" [26].\",\n \"The simplest version, which includes only nucleons, gives what are known as “Fermi motion\\\" corrections to the free nucleon structure function [27].\",\n \"These were calculated long before the present excitement about electroproduction from nuclei.\",\n \"More recently, the model has been extended to more exotic constituents [28], [29], [30] in an attempt to “explain\\\" the EMC effect.\",\n \"There is no adequate derivation of the convolution model.\",\n \"The parton analyses of the $§2$ will provide a framework in which the assumptions which lead to the convolution model may be analyzed and criticized.\",\n \"The two steps of the convolution model are summarized diagrammatically in Fig.\",\n \"REF .\",\n \"The nucleus, with baryon number $A$ and momentum $P$ , contains a constituent, label $T$ and momentum $p$ , which in turn contains a quark, flavor $a$ and momentum $k$ .\",\n \"The quark absorbs the virtual photon while the fragments of the nucleus and the constituent propagate into the final state without interaction or interference.\",\n \"Other diagrams in which fragments of the nucleus or the constituent $T$ interact or interfere are ignored.\",\n \"Some are shown in Fig.\",\n \"REF .\",\n \"Figure: Contributions which are ignored in convolution models.Superficially, they resemble the diagrams of Fig.\",\n \"6 (b)–(d) which were ignored in the parton model, but there is an important difference: Fig.\",\n \"6(b)–(d) can be dropped because $Q^2 \\\\rightarrow \\\\infty $ , so $\\\\xi ^2 \\\\rightarrow 0$ and the operator product expansion in QCD can be used to prove that they are O($1/Q^2$ ) compared to Fig.\",\n \"6(a).\",\n \"There is no analogous large mass scale characterizing the process enclosed in the dashed line in Fig.\",\n \"REF , so there is no a priori justification for ignoring the additional processes of Fig.\",\n \"REF .\",\n \"Furthermore, the fragments of the nucleus and the constituent have a long time, $\\\\xi ^0 \\\\lesssim 1/M x$ , to interact while awaiting the return of the active quark.\",\n \"Nevertheless, for certain constituents under favorable kinematic conditions (e.g., for nucleons in the weak binding limit, or for pions near $x = 0$ ) [10] ignoring final state interactions and interference may be justifiable.\",\n \"For the moment, we will simply ignore the problem and proceed.\",\n \"The quark distribution for nucleus $A$ with four-momentum $P$ is defined in analogy to Eq.\",\n \"(2.29): $f_{a/A}(x_A) = \\\\int \\\\frac{d^4k}{(2 \\\\pi )^4} \\\\delta \\\\left( \\\\frac{k^+}{P^+} - x_A \\\\right) {\\\\rm Tr}[\\\\gamma ^+ \\\\chi _{a/A}(k,P)] \\\\qquad \\\\mathrm {(4.1)}$ where $x_A = Q^2/2M_A \\\\nu $ .\",\n \"The content of Fig.\",\n \"REF is a convolution form for $\\\\chi _{a/A}(k,P)$ : $\\\\chi _{a/A}(k,P) = \\\\sum _T \\\\int \\\\frac{d^4p}{(2 \\\\pi )^4} \\\\chi _{a/T}(k,p) \\\\chi _{T/A}(p,P) \\\\qquad \\\\mathrm {(4.2)}$ where the sum covers all constituents of the nucleus.\",\n \"We have treated the constituent $T$ as a scalar (omitting a sum over the spins of $T$ ).\",\n \"The generalization to spin $1/2$ is straightforward.\",\n \"Note that the quark and/or constituent legs in Fig.\",\n \"REF have been properly included in Eq.\",\n \"(4.2): From Eq.\",\n \"(2.28) it is apparent that $\\\\chi _{a/T}(\\\\chi _{T/A)})$ is untruncated in the quark (constituent) legs but truncated in the constituent (nucleus) legs.\",\n \"Only the $+$ components of momentum really matter in Eq.\",\n \"(4.1)-(4.2).\",\n \"To make this manifest, we substitute the identity: $1 = \\\\int dy\\\\ \\\\delta (y_A-p^+/P^+) \\\\int dz_T\\\\ \\\\delta (z_T-k^+/p^+) \\\\qquad \\\\mathrm {(4.3)}$ where the ranges of $y_A$ and $z_T$ integrals are yet to be determined.\",\n \"After some algebra we find $f_{a/A}(x_A) = &\\\\sum _T \\\\int dy_Adz_T~\\\\delta (y_Az_T-x_A)\\\\int \\\\frac{d^4p}{(2 \\\\pi )^4} \\\\delta (y_A-p^+/P^+) \\\\chi _{T/A}(p,P)\\\\times \\\\\\\\\\\\times &\\\\int d^4k~\\\\delta (z_T-k^+/p^+) {\\\\rm Tr}[\\\\gamma ^+ \\\\chi _{a/T}(k,p)]\\\\ .\",\n \"\\\\nonumber $ The $k-$ integration defines a Lorentz invariant function of $z$ and $p^2$ which is the off-shell generalization of $f_{a/T}(x_T)$ defined in Eq.\",\n \"(2.29): $f_{a/T}(z_T,p^2)\\\\equiv \\\\int \\\\frac{d^4k}{(2 \\\\pi )^4} \\\\delta (z_T-k^+/p^+) Tr[\\\\gamma ^+ \\\\chi _{a/T}(k,p)]\\\\ .\",\n \"\\\\qquad \\\\mathrm {(4.5)}$ Let us save the $p^2$ integral for last by inserting $\\\\int dp^2_0~\\\\delta (p^2-p^2_0) = 1$ .\",\n \"Then the $d^4p$ integration gives $f_{a/A}(x_A) = \\\\sum _T \\\\int dy_A dz_T \\\\delta (y_Az_T-x_A) \\\\int d p^2_0f_{a/T}(z_T,p^2_0)f_{T/A}(p^2_0,y_A) \\\\qquad \\\\mathrm {(4.6)}$ where $f_{T/A}(p^2_0,y_A) = \\\\int \\\\frac{d^4p}{(2 \\\\pi )^4}~\\\\delta (p^2-p^2_0)~\\\\delta (y_A-p^+/P^+) \\\\chi _{T/A}(p,P) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(4.7)}$ Note that $f_{T/A}(p^2_0,y_A)$ is the probability to find a constituent $T$ in nucleus $A$ with momentum fraction $y_A$ of the nucleus's $P=M_A/\\\\sqrt{2}$ and invariant mass $p^2_0$ , whereas $f_{a/T}(z_T,p^2_0)$ is the probability to find a quark of flavor $a$ with momentum fraction $z_T$ in an off shell target with invariant mass $p^2_0$ .\",\n \"So $f_{a/T}(z_T)$ , which we have defined in Eq.\",\n \"2.18, would correspond to $f_{a/T}(z_T,M^2_T)$ , but $f_{T/A}(y_A) = \\\\int dp^2_0 f_{T/A}(p^2_0,y_A)$ .\",\n \"In practice, no one uses a $p^2_0$ dependent quark distribution $f_{a/T}(z_T,p^2_0)$ in convolution model calculations.\",\n \"The rationale for this must be that $f_{T/A}(p^2_0,y_A)$ peaks very strongly at some $\\\\overline{p}^2$ so $f_{a/T}(z,p^2_0)$ can be replaced by $f_{a/T}(z,\\\\overline{p}^2)$ .\",\n \"Then the $p^2_0$ integration in Eq.\",\n \"(4.7) can be performed, but it is an additional assumption that $f_{a/T}(z_T,\\\\overline{p}^2) = f_{a/T}(z_T,M^2_T)$ unless of course $\\\\overline{p}^2 \\\\approx M^2_T$ .\",\n \"The result $f_{a/A}(x_A) = \\\\sum _T \\\\int dy_A dz_T \\\\delta (y_Az_T-x_A) f_{a/T}(z_T)f_{T/A}(y_A) \\\\qquad \\\\mathrm {(4.8)}$ together with the equations which define $f_{a/T}(z_T)$ and $f_{T/A}(y_A)$ comprise the convolution model.\",\n \"The range of the $y_A$ and $z_T$ integrations can be determined by arguments identical to those used in $§2.2$ to show $0<x_A<1$ .\",\n \"The result is $0<y_A$ , $z_T<1$ .\",\n \"The parton distribution functions are often written as functions of Bjorken's variable $x= M_A x_A/M$ .\",\n \"As given in Eq.\",\n \"2.30, $F_{a/A}(x) =(M/M_A)f_{a/A}(Mx/M_A)$ .\",\n \"If the constituents in question are nucleons and the binding is not too strong, then $\\\\chi _{N/A}(p,P)$ does peak strongly and near mass shell (compared to the scale for variation in $f_{a/T}(z_T,p^2)$ which is $\\\\Lambda \\\\sim 200-300$ MeV).\",\n \"However, the model is applied to other constituents which are far off shell.\",\n \"Pion contributions are believed to be largest for $p^0 \\\\approx 0$ and $|\\\\bf p| \\\\approx $ 300-400 MeV [31] giving $\\\\overline{p}^2 \\\\approx -(0.1-0.2)$ GeV compared to $m^2_{\\\\pi } \\\\cong 0.02$ GeV.\",\n \"Whether the distribution of quarks in a pion so far off shell is the same as the distribution on shell is anybody's guess.\",\n \"In any case, advocates of pion and other convolution based models ignore any $p^2$ dependence of the constituents' quark distributions.\",\n \"The assumptions leading to a convolution model are arguable at best and probably can only be supported on a case by case basis.\",\n \"Despite this, I believe they have substantial value as qualitative guides to nuclear effects and especially as a formalism for treating Fermi motion and other “trivial\\\" sources of nuclear modifications of structure functions.\",\n \"Here is a convenient summary of the convolution model formulas: $f_{a/A}(x_A) &= \\\\sum _T f_{a/T/A}(x_A) \\\\nonumber \\\\\\\\&= \\\\sum _T \\\\int dy_A dz_T \\\\delta (y_Az_T-x_A) f_{a/T}(z_T)f_{T/A}(y_A) $ $f_{a/T}(z_T) &= \\\\frac{d P_{a/T}}{d z_T} \\\\, 0<z_T<1, \\\\ z_T=k^+/p^+ \\\\ , \\\\nonumber \\\\\\\\f_{T/A}(y_A) &= \\\\frac{d P_{T/A}}{d y_A} \\\\, 0<y_A<1, \\\\ y_A=p^+/P^+ \\\\ , $ $\\\\int _0^1 dz_Tf_{a/T}(z_T) = N_{a/T} \\\\ \\\\ \\\\ &\\\\ \\\\ \\\\ \\\\int _0^1 dy_Af_{T/A}(y_A) = N_{T/A} \\\\nonumber \\\\\\\\\\\\int _0^1 dz_T z_T f_{a/T}(z_T) = \\\\epsilon _{a/T} \\\\ \\\\ \\\\ &\\\\ \\\\ \\\\ \\\\int _0^1 dy_A y_A f_{T/A}(y_A) = \\\\epsilon _{T/A} $ $\\\\int _0^1 dx_A f_{a/A}(x_A) &= \\\\sum _T N_{a/T} N_{T/A} \\\\nonumber \\\\\\\\\\\\int _0^1 dx_A x_A f_{a/A}(x_A) &= \\\\sum _T \\\\epsilon _{a/T} \\\\epsilon _{T/A} \\\\nonumber $ where $N_{T/A}$ is the number of constituents of type $T$ in nucleus $A$ and $\\\\epsilon _{T/A}$ is the fraction of the nucleus's $P^+$ carried by constituents $T$ .\",\n \"It is useful to introduce quark and constituent distributions depending on Bjorken's variable $x = M_Ax_A/M$ and $y=M_Ay_A/M$ .\",\n \"We leave $z_T$ as is since the constituent $T$ is in general not at rest and $M_T$ plays no special role.\",\n \"Thus, $x=\\\\sqrt{2}k^+/M$ and $F_{T/A}(y)=dP_{T/A}/dy=\\\\frac{M}{M_A}f_{T/A}(y_A)$ .\",\n \"Then, Eq.\",\n \"(4.8) becomes $F_{a/A}(x) &= \\\\sum _T \\\\int _0^{M_A/M}dy\\\\int _0^1 dz_T \\\\delta (x-yz_T)f_{a/T}(z_T)F_{T/A}(y)\\\\\\\\&=\\\\sum _T\\\\int _x^{M_A/M}\\\\frac{dy}{y}f_{a/T}(x/y)F_{T/A}(y) $ and the sum rules analogous to Eqs.\",\n \"(4.11) are $\\\\int _0^{M_A/M} dx \\\\overline{F}_{a/A}(x) &= \\\\sum _T N_{a/T}N_{T/A}/A \\\\\\\\\\\\int _0^{M_A/M} dx \\\\overline{F}_{a/A}(x) &= \\\\frac{M_A}{MA} \\\\sum _T \\\\epsilon _{a/T} \\\\epsilon _{T/A} \\\\\\\\$ where $\\\\overline{F}_{a/A}=F_{a/A}/A$ is the quark distribution per nucleon.\",\n \"Throughout this discussion, I have been careful not to make the approximation $M_A \\\\cong MA$ although this is a reasonable approximation in most cases.\",\n \"Convolution has some unexpected and important kinematic effects on the quark distribution of some constituents.\",\n \"Consider, for example, the contribution of pions in a model like Erikson and Thomas [31]: $p^0 \\\\approx 0$ , but $|{\\\\bf p}| \\\\approx 300 - 400$ MeV, so $y_{\\\\pi } \\\\sim |{\\\\bf p}| \\\\cos \\\\theta /M$ which is typically less than $\\\\sim 1/3$ .\",\n \"The valence quark distribution in the pion is expected on theoretical grounds and found experimentally to be quite “hard\\\", $f_{V/\\\\pi }(z_\\\\pi ) \\\\sim (1-z_\\\\pi )$ .\",\n \"Convolution converts this into $F_{V/\\\\pi /A}(x) = \\\\int _x^{M_A/M} \\\\frac{dy}{y} F_{\\\\pi /A}(y) f_{V/\\\\pi }(x/y)$ which is negligible for $x$ much larger than $1/3$ .\",\n \"So the hard pion distribution has been mapped to small $x$ .\",\n \"No such fate befalls the nucleon: $p^0 \\\\approx M$ , so $y_N \\\\approx 1 + p \\\\cos \\\\theta /M$ which peaks near $y_N =1$ .\",\n \"Nevertheless, the motion of nucleons does affect the quark distribution in the nucleus even if it is composed of quarks alone.\",\n \"The most obvious source of a nuclear effect in deep inelastic scattering comes from the fact that the nucleons in the nucleus are in motion.\",\n \"As a “baseline\\\" model for nuclear targets we assume the nucleus consists exclusively of nucleons and that the quark distribution in the nucleons are the same as in isolation.\",\n \"One might naively think if their kinetic energies are small with respect to $\\\\nu $ , the motion could be neglected as $\\\\nu \\\\rightarrow \\\\infty $ .\",\n \"This is not correct, as we shall see.\",\n \"We begin with $f_{N/A}(y_A) = \\\\int \\\\frac{d^4p}{(2 \\\\pi )^4} \\\\delta (y_A-p^+/P^+) \\\\chi _{N/A}(p,P) \\\\qquad \\\\mathrm {(4.15)}$ where $\\\\chi _{N/A}(p,P)$ is the forward, possibly virtual, nucleon nucleus scattering amplitude defined in analogy to Eq.\",\n \"(2.28).\",\n \"$\\\\chi _{N/A}(p,P) = \\\\int d^4 \\\\zeta e^{-ip \\\\cdot \\\\zeta } \\\\mathinner {\\\\langle {P|T(\\\\Phi ^+(\\\\zeta )\\\\Phi (0))|P}\\\\rangle }_c\\\\qquad \\\\mathrm {(4.16)}$ where $\\\\Phi $ is a nucleon interpolating field.\",\n \"$\\\\Phi $ is not uniquely defined, and when $p^2 \\\\ne M^2_N$ different choices yield different results.\",\n \"This reflects an inherent uncertainty when one attempts to use field-theoretic methods to manipulate composite objects.\",\n \"Spin has been suppressed in Eq.\",\n \"(4.16).\",\n \"If we substitute Eq.\",\n \"(4.16) into Eq.\",\n \"(4.15) and perform as many $\\\\zeta $ and $p$ integrations as possible we obtain a form analogous to Eq.\",\n \"(2.18): $f_{N/A}(y_A) = \\\\sum _n \\\\delta (y_A-1+P^+_n/P^+)|\\\\mathinner {\\\\langle {n|\\\\Phi |P}\\\\rangle }|^2 \\\\qquad \\\\mathrm {(4.17)}$ where the sum is on all states which can be obtained by removing a single nucleon from the nucleus leaving behind a state with $P^+_n = (1-y_A)P^+$ .\",\n \"The problem is how to calculate $|\\\\mathinner {\\\\langle {n|\\\\Phi |P}\\\\rangle }|^2$ .\",\n \"This must be done inclusively, i.e., all possible states {$\\\\mathinner {|{n}\\\\rangle }$ } must be included and they must be physical states.\",\n \"A variety of approximations can be made but great care must be taken to ensure that the number and $p^+$ sum rules (Eqs.\",\n \"(4.11)) remain valid.\",\n \"The most naive approach, to replace the nucleus by an independent particle model in which nucleons occupy energy eigenstates in some potential, obeys neither sum rule and must be altered in some ad hoc way before it can be used in this context.\",\n \"When this form for $f_{N/A}(y_A)$ is inserted into Eq.\",\n \"(4.12) to obtain $F_{a/N/A}(x)$ we obtain an independent nucleon model for $F_{a/A}(x)$ which includes what are generally known as “Fermi motion corrections\\\": $\\\\overline{F}_{a/N/A}(x) = \\\\int _x^{M_A/M} \\\\frac{dy}{y} f_{a/N}(x/y) \\\\overline{F}_{N/A}(y) \\\\qquad \\\\mathrm {(4.18)}$ (Reminder: $F_{N/A}(y)=dP_{T/A}/dy=\\\\frac{M}{M_A}f_{N/A}(y_A)$ , $\\\\overline{F}_{a/N/A}=F_{a/N/A}/A$ and $\\\\overline{F}_{N/A}=F_{N/A}/A.$ ) $F_{a/N/A}$ has several important features independent of the explicit form of $\\\\overline{F}_{N/A}$ .\",\n \"First, as already noted $\\\\overline{F}_{a/N/A}(x)/F_{a/N}(x)$ diverges as $x \\\\rightarrow 1$ , so Fermi motion corrections to the ratio are large and positive near $x \\\\sim 1$ .\",\n \"The divergence of the ratio is deceptive because both $\\\\overline{F}_{a/N/A}(x)$ and $F_{a/N}(x)$ are very small for $x \\\\sim 1$ .\",\n \"Second, Fermi motion corrections cannot change the number of quarks of any flavor: $\\\\int _0^{M_A/M}(\\\\overline{F}_{a/N/A}(x) - F_{a/N}(x)) = 0$ , which can be obtained from Eq.\",\n \"(4.13) with $T = N$ and $N_{T/A} = A$ .\",\n \"Third, the effect of nuclear binding appears to be to decrease slightly the $p^+$ carried by the quarks (and antiquarks) even though the quark distribution in the nucleon is not altered.\",\n \"This can he seen from Eq.\",\n \"(4.14): $M_A/MA = 1 -\\\\delta $ where $\\\\delta $ is the binding energy per nucleon in units of the nucleon mass, and $\\\\epsilon _{N/A} = 1$ (the nucleons carry all the nucleus $P^+$ in this simple model), so $\\\\int _0^{M_A/M} dx x[\\\\overline{F}_{a/N/A}(x)-F_{a/N}(x)] = -\\\\delta \\\\epsilon _{a/N}$ .\",\n \"This goes in the right direction toward explaining the EMC effect but explicit calculations with “realistic\\\" nuclear wave functions fail to get a large enough shift in the valence quark distribution and get the wrong shape (i.e.\",\n \"$x-$ dependence) of the effect.\",\n \"Also, the model cannot produce an increase in the ocean quark pairs.\",\n \"Recently, it has been claimed that a model of the form we have been discussing can account for the valence quark part of the EMC effect [32].\",\n \"In that model, $\\\\epsilon _{N/A} < 1$ so $P^+$ has been lost, presumably to the constituents responsible for nuclear binding, and the quark content of those constituents has not been included in the calculation.\",\n \"It is interesting to explore the effect of binding in a simple model.\",\n \"Let us assume the nucleons form a relativistic, degenerate free Fermi gas (FG) with Fermi momentum $k_F$ .\",\n \"Then $|\\\\mathinner {\\\\langle {n|\\\\Phi |P}\\\\rangle }|^2 \\\\equiv \\\\frac{dN}{d^3p} = \\\\frac{3A}{4 \\\\pi k^3_F} \\\\theta (k_F-|{\\\\bf p}|) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(4.19)}$ The constant $3A/4\\\\pi k^3_F$ is chosen so $\\\\int ^{k_F} dN = A$ .\",\n \"The bound nucleons must have an effective mass $M^\\\\star < M_N$ in this model otherwise the sum over the energies of the nucleons would exceed $M_A$ .\",\n \"Substituting into Eq.\",\n \"(4.18) and evaluating the $d^3p$ integral, $f^{FG}_{N/A}(y_A) = \\\\frac{3}{4} \\\\frac{AM_A}{k_F} \\\\left( 1-\\\\frac{M^2_A}{k^2_F} \\\\left(\\\\frac{y^2_A-M^{\\\\star 2}/M^2_A}{2 y_A} \\\\right)^2 \\\\right) \\\\ , \\\\ y_-<y_A<y_+ \\\\qquad \\\\mathrm {(4.20)}$ where $y_\\\\pm $ are the values for which $f^{FG}_{N/A}(y_\\\\pm )=0$ .\",\n \"One can check that $f^{FG}_{N/A}(y_A)$ satisfies both $\\\\int _0^1 dy_A f^{FG}_{N/A}(y_A)=A$ and $\\\\int _0^1 dy_A y_A f^{FG}_{N/A}(y_A)=1$ provided $M^\\\\star $ is chosen so the energy of the Fermi gas is $M_A$ .\",\n \"For non-relativistic nucleons ($M^\\\\star = M_A/A + O(k^3_F/M)$ ) a quadratic approximation suffices $\\\\overline{F}^{FG}_{N/A}(y) = \\\\frac{M}{AM_A} f^{FG}_{N/A}(y_A) \\\\approx \\\\frac{3}{4 \\\\lambda } \\\\left( 1-\\\\frac{(y-\\\\eta )^2}{\\\\lambda ^2} \\\\right) \\\\ , \\\\ \\\\eta -\\\\lambda <y<\\\\eta + \\\\lambda \\\\ , \\\\qquad \\\\mathrm {(4.21)}$ where $\\\\lambda =k_F/M$ and $\\\\eta =M_A/MA\\\\ (\\\\le 1)$ .\",\n \"Since $\\\\overline{F}^{FG}_{N/A}(y)$ has maximum height $\\\\sim 1/\\\\lambda $ and width $\\\\sim \\\\lambda $ and since it is convoluted with a smooth function $f_{a/N}(x/y)$ in Eq.\",\n \"(4.18), it is convenient to approximate it as a generalized function $F^{FG}_{N/A}(y) \\\\cong \\\\delta (y-\\\\eta ) + \\\\frac{\\\\lambda ^2}{10} \\\\delta ^{^{\\\\prime \\\\prime }}(y-\\\\eta ) \\\\qquad \\\\mathrm {(4.22)}$ for $\\\\lambda <<1$ .\",\n \"Note that $F^{FG}_{N/A}(y)$ in this form satisfies the required sum rules trivially.\",\n \"In particular, $\\\\int _0^{M_A/M} dy y F^{FG}_{N/A}(y)= \\\\eta = M_A/MA$ which leads to the decrease in the quark's momentum noted above.\",\n \"Substituting into Eq.\",\n \"(4.18), we obtain $\\\\overline{F}^{FG}_{a/N/A}(x) = \\\\frac{1}{\\\\eta } f_{a/N}(x/\\\\eta ) + \\\\frac{\\\\lambda ^2}{10} d^2 /dy^2 \\\\frac{1}{y} f_{a/n}(x/y) |_{y=\\\\eta } \\\\ .\",\n \"\\\\qquad \\\\mathrm {(4.23)}$ In this model the effects of Fermi smearing are small (for $k_F = 300$ MeV, $\\\\lambda ^2/10 \\\\approx 0.01$ , for typical nuclei $1-\\\\eta \\\\le 0.01$ ).\",\n \"A sample calculation of $\\\\overline{F}_{a/N/A}(x)/F_{a/N}(x)$ is shown in Fig.\",\n \"REF .\",\n \"It clearly cannot account for the EMC effect.\",\n \"Figure: Estimates of Fermi motion effects in a Fermi gas model for both valence and ocean quarks.Within the context of convolution models, there are only two alternatives: either other constituents must be present (e.g., pions, $\\\\Delta $ 's, 6 quark bags, etc.)\",\n \"[28], [29] or the structure function of the nucleons must be modified by the nuclear medium [30] I will not discuss either alternative in these lectures.\",\n \"The interested reader should consult the references for some work in these directions.\",\n \"Instead, I will describe a framework for analyzing the EMC effect which emerges from the scaling properties of QCD.\",\n \"In interacting field theories the structure functions $F_1$ and $F_2$ depend on both $x$ and $Q^2$ even at very large $Q^2$ .\",\n \"The $Q^2-$ dependence will give us a new handle on the distance scales characterizing the target.\",\n \"To understand the $Q^2-$ dependence we must take another excursion into formalism.\",\n \"At present, the $x-$ dependence of $F_2(x,Q^2 )$ at fixed $Q^2$ cannot be predicted for any hadronic target in a rigorous fashion.\",\n \"It depends on the non-perturbative dynamics which confines quarks.\",\n \"However, given $F_2(x,Q^2)$ at some large $Q^2 = Q^2_0$ , QCD perturbation theory enables one to predict $F_2(x,Q^2)$ at all $Q^2 > Q^2_0$ and at $Q^2 < Q^2_0$ down to some minimum which appears to be of order 1 GeV$^2$ .\",\n \"In QCD, $F_2(x,Q^2)$ depends logarithmically on $Q^2$ at large $Q^2$ .\",\n \"The logarithmic $Q^2-$ dependence has been verified experimentally and constitutes one of the major quantitative tests of the theory.\",\n \"In addition to $\\\\ln Q^2$ corrections there are expected to be $O(1/Q^2)$ and higher order corrections which become important at low-$Q^2$ .\",\n \"In this chapter, I first give a heuristic “derivation\\\" of the logarithmic $Q^2-$ dependence of quark distributions.\",\n \"Next, I will describe some of the formalism behind the $Q^2-$ dependence.\",\n \"Then, I catalogue and discuss $O(1/Q^2)$ corrections.\",\n \"Finally, I compare the quark description of hadrons in QCD with more naive quark models.\",\n \"This analysis applies equally well to any target, so the target subscripts $T$ and $A$ will generally be suppressed .\",\n \"Until further notice the scaling variable is $x_T$ with $0 < x_T < 1$ .\",\n \"In QCD, quarks are coupled to gluons in much the same fashion that electrons are coupled to photons in QED.\",\n \"The fundamental vertex is shown in Fig.\",\n \"REF (a).\",\n \"Instead of the electric charge $e$ , one has $g\\\\lambda ^a_{ij}/2$ , where $g$ is the QCD coupling ($\\\\alpha _c \\\\equiv g^2 /4 \\\\pi $ ).\",\n \"$\\\\lambda ^a_{ij}$ are Gell-Mann's matrices (normalized so ${\\\\rm tr}\\\\,\\\\lambda ^2 = 2$ ) describing the coupling of quark with color $i$ to quark with color $j$ ($i,j = 1,2,3$ ) by emitting a gluon with color $a$ ($a = 1, ... 8$ ).\",\n \"In QED it is well-known that the electric and magnetic fields of a relativistic electron are predominantly transverse and look like the fields of a photon [33].\",\n \"This is the basis of the Weizsäcker-Williams or “equivalent photon\\\" approximation in QED.\",\n \"A careful study of the analogous process in QCD will lead to the logarithmic scaling violations we seek [34].\",\n \"The equivalent photon approximation is usually formulated in a frame in which the electron is extremely relativistic (moving for definiteness, in the $x$ -direction) $-$ the “infinite momentum frame\\\".\",\n \"The number of photons associated with the electron depends on the energy of the photon ($E_\\\\gamma = x E_e$ ) and the impact parameter at which one probes the electron's electromagnetic field: $\\\\frac{d N_\\\\gamma }{dx dA} \\\\sim \\\\frac{\\\\alpha }{\\\\pi ^2 b^2 x} + {\\\\rm terms\\\\ less\\\\ singular\\\\ in\\\\ b\\\\ and\\\\ x} \\\\qquad \\\\mathrm {(5.1)}$ where $dA=2\\\\pi b db$ and the variables are defined in Fig.\",\n \"REF .\",\n \"Figure: Weizsäcker-Williams kinematics.The proper interpretation of Eq.\",\n \"(5.1) is that a measurement sensitive to the intensity of the electron's electromagnetic field performed at impact parameter $b$ and absorbing energy $E_\\\\gamma =x E_e$ could not differentiate between a passing electron and the equivalent number of photons.\",\n \"Eq.\",\n \"(5.1) is derived, for example, by Jackson.\",\n \"The derivation neglects the recoil of the electron, which must lose energy when a photon with $E_\\\\gamma /E_e =x$ is absorbed.\",\n \"This is reasonable for $x \\\\approx 0$ .\",\n \"For larger $x$ , one must include recoil and electron spin effects, which requires calculation of Feynman graphs.\",\n \"The lowest order graph corresponding to the measurements I've described is that of Fig.\",\n \"REF (a).\",\n \"The calculations can be found in Ref. [35].\",\n \"The result is $\\\\frac{1}{x} \\\\rightarrow \\\\frac{1}{2x} (1+(1-x)^2)$ .\",\n \"QED and QCD are identical to this order except $e \\\\rightarrow g \\\\lambda ^a_{ij}$ .\",\n \"So the number of gluons (summed over all colors) with momentum fraction $x$ at impact parameter $b$ in a quark (averaged over colors) is $\\\\frac{dN_g}{dxdb} \\\\sim \\\\frac{2 \\\\alpha _c}{\\\\pi b} \\\\frac{1}{3} {\\\\rm Tr} \\\\sum _a \\\\left( \\\\frac{\\\\lambda ^a}{2} \\\\right)^2 \\\\left[\\\\frac{1+(1-x)^2}{2x} \\\\right] \\\\qquad \\\\mathrm {(5.2)}$ to leading order in $b$ and $\\\\alpha _c$ .\",\n \"Since $\\\\frac{1}{3} {\\\\rm Tr} \\\\sum _a \\\\left( \\\\frac{\\\\lambda ^a}{2} \\\\right)^2 = \\\\frac{4}{3} \\\\qquad \\\\mathrm {(5.3)}$ we have $\\\\frac{dN_g}{dxdb} = \\\\frac{8 \\\\alpha _c}{3 \\\\pi b} \\\\left[\\\\frac{1+(1-x)^2}{2x} \\\\right] \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.4)}$ Figure: Feynman diagrams corresponding to Weizsäcker-Williams processes: (a) An (electron) quark radiating a photon (gluon); (b) An electron (quark) “radiating\\\" an electron (quuark); Vertex correction required at x=1x=1.If one probes a quark (or electron) by delivering a momentum transfer $Q_\\\\perp = \\\\sqrt{Q^2}$ , then $b_{\\\\rm min}=1/\\\\sqrt{Q^2}$ plays the role of a resolution: all gluons (or photons in QED) with $b \\\\ge b_{\\\\rm min}$ will appear distinct from the quark (electron), but the rest ($b < b_{\\\\rm min}$ ) will be lumped into what one calls the quark (electron) when probed with that resolution.\",\n \"This is evident, for example, in the Weizsäcker-Williams calculation of bremsstrahlung (per unit $x$ ) is $\\\\approx x E_e \\\\int _{b_{\\\\rm min}}^\\\\infty db \\\\frac{dN_\\\\gamma }{dx db}$ with $b_{\\\\rm min} = 1/\\\\sqrt{Q^2_\\\\perp }$ .\",\n \"Thus, $\\\\frac{d N_g}{dx d \\\\ln Q^2/Q^2_0} = \\\\frac{2 \\\\alpha _c}{3 \\\\pi } \\\\frac{1+(1-x)^2}{x} \\\\qquad \\\\mathrm {(5.5)}$ is the rate of change with $t \\\\equiv \\\\ln Q^2/Q^2_0$ of the probability to find a gluon of momentum fraction $x$ in a quark.\",\n \"(Note that the logarithm of $Q^2$ arose when $b\\\\propto 1/\\\\sqrt{Q^2}$ was substituted for $db/b$ in Eq.\",\n \"5.4 and that $Q^2_0$ is an arbitrary scale for the logarithm.)\",\n \"In the following, I will often use the variable $t=\\\\ln (Q^2/Q_0^2)$ .\",\n \"So we define $P_{g/q}(x,t)=dN_g/dxdt$ .\",\n \"($P_{g/q}$ appears to be independent of $t$ , but in QCD $\\\\alpha _c$ depends logarithmically on $t$ .)\",\n \"When a quark radiates a gluon it leaves behind a quark of lower momentum.\",\n \"So we can determine the rate of change with $t$ of the probability that a measurement on a quark with energy $E_q$ will detect a quark with $E^\\\\prime _q = xE_q$ : $P_{q/q}(x,t) = \\\\frac{dN_g(1-x)}{dxdt} = \\\\frac{2 \\\\alpha _c}{3 \\\\pi } \\\\frac{1+x^2}{1-x}\\\\qquad \\\\mathrm {(5.6)}$ corresponding to the Feynman graphs of Fig.\",\n \"REF (b).\",\n \"The three gluon coupling of QCD shown in Fig.\",\n \"REF (b) leads to a probability for a gluon to contain gluons of lower momentum, and the $q\\\\bar{q}q$ vertex of Fig.\",\n \"REF (a) leads to a probability for a gluon to contain quark-antiquark pairs.\",\n \"If we confine our attention to valence quarks we need not study these other processes since they produce only ocean $q \\\\bar{q}$ pairs.\",\n \"Eqs.\",\n \"(5.5) and (5.6) lead to the notion of a quark distribution “evolving\\\" with increased $t \\\\sim \\\\ln Q^2$ .\",\n \"As $\\\\ln Q^2$ grows, valence quarks emit gluons, gluons in turn emit quark-antiquark pairs and more gluons.\",\n \"This “evolutionary\\\" view of the $Q^2$ dependence of quark (and gluon) distributions was first suggested by Kogut and Susskind [36] and was developed for QCD by Altarelli and Parisi [34].\",\n \"The idea that the definition of a quark or gluon depends upon the mass or distance scale at which one probes it is a familiar one in quantum field theory.\",\n \"It is related to renormalization.\",\n \"A quark (or electron) propagating in isolation couples to quantum fluctuations in the vacuum.\",\n \"When we perform some measurement on the quark we lump some of the fluctuations (and all of the very short distance ones which lead to divergences) into the definition of the quark and treat the remainder as radiative corrections.\",\n \"To do this we must introduce a mass (or length) scale ($\\\\mu $ ) known as the “renormalization point\\\" into the theory, which distinguishes, roughly speaking, those fluctuations we treat explicitly ( $\\\\Delta x > \\\\mu ^{-1}$ ) from those we incorporate into the definition of the particles ($\\\\Delta x< \\\\mu ^{-1}$ ).\",\n \"It is well-known that $\\\\mu $ is arbitrary - nothing physical depends on it - but the scale invariance of QCD allows us to trade $\\\\mu ^2$ -dependence for $Q^2$ -dependence in a fashion which will be outlined briefly below ($§$ 5.2).\",\n \"Up to now, we've worked in the infinite momentum frame.\",\n \"$x$ measures the $z$ -component of a daughter particle's momentum relative to its parent: $p^3_\\\\infty = xP^3_\\\\infty $ .\",\n \"The subscript \\\"$\\\\infty $ \\\" denotes quantities measured in the infinite momentum frame.\",\n \"The particles are not far from mass shell and have limited transverse momenta (we can choose $P^3_\\\\infty > > \\\\sqrt{Q^2}$ ) so $p^0_\\\\infty \\\\cong x P^3_\\\\infty $ ) and $p^+_\\\\infty = x P^+_\\\\infty \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.7)}$ The ratio $p^+/P^+$ is invariant under Lorentz transformation in the along the $z$ -axis.\",\n \"This follows from the Lorentz transformation in the form $p^{\\\\pm \\\\,\\\\prime }= e^{\\\\pm y}p^\\\\pm $ , ${\\\\mathbf {p}}^\\\\prime _\\\\perp = {\\\\bf p}_\\\\perp $ , where $\\\\beta = \\\\tanh y$ is the relative velocity of theframes along the $z$ -axis.\",\n \"Thus, Eq.\",\n \"(5.7) implies $p^+ = x P^+ \\\\qquad \\\\mathrm {(5.8)}$ in the laboratory, and we can identify $x$ as the variable we've been using all along.\",\n \"$P_{q/q}(x,t)$ appears to be singular at $x = 1$ .\",\n \"This presents a problem because we would like the number of valence quarks in a target to be independent of $Q^2$ , $\\\\frac{d}{dt} \\\\int _0^1 dx \\\\frac{dN_q}{dx} = \\\\int _0^1 dx P_{q/q}(x,t) = 0 \\\\,.\\\\qquad \\\\mathrm {(5.9)}$ With Eq.\",\n \"(5.6) as it stands, the integral diverges.\",\n \"The resolution of this problem requires methods outside the scope of these lectures [38].\",\n \"The proper prescription is to interpret $1/(1-x)$ as a distribution: $1/(1-x) \\\\rightarrow 1/(1-x)_+$ , defined by $\\\\int _0^1dx \\\\frac{f(x)}{(1-x)_+}\\\\equiv \\\\int _0^1 dx(f(x)-f(1))/(1-x) \\\\qquad \\\\mathrm {(5.10)}$ and to add a $\\\\delta $ -function to $P_{q/q}(x,t)$ , so that Eq.\",\n \"(5.9) is satisfied.\",\n \"Since $\\\\int _0^1 dx (1+x^2)/(1-x)_+ = -\\\\frac{3}{2}$ we find $P_{q/q}(x,t) = \\\\frac{2 \\\\alpha _c}{3 \\\\pi } \\\\left( \\\\frac{1+x^2}{(1-x)_+} + \\\\frac{3}{2} \\\\delta (x-1) \\\\right) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.11)}$ The rate of change of the quark distributions in a target will be determined by the functions $P_{q/q}(x, t)$ and $P_{q/g}(x,t)$ and by the quark and gluon distributions at $t$ : A quark or gluon with momentum fraction $y$ will be observed to consist of quarks with momentum fraction $zy$ according to $P_{q/q}(x, t)$ and $P_{q/g}(x,t)$ respectively.\",\n \"Clearly, evolution is described mathematically by convolution in precise analogy to $§4.2$ : $\\\\frac{df_{\\\\rm N. S.}}{dt} = \\\\frac{\\\\alpha _c}{2 \\\\pi } \\\\int _x^1 \\\\frac{dy}{y} \\\\overline{P}_{q/q} (x/y)f_{\\\\rm N. S.}(y,t) \\\\qquad \\\\mathrm {(5.12)}$ where $P_{q/q} \\\\equiv \\\\frac{\\\\alpha _c}{2 \\\\pi } \\\\overline{P}_{q/q}$ The subscript $\\\\rm N. S.$ denotes non-singlet and indicates a quark distribution with non-trivial flavor quantum numbers such as the valence quark distribution of $§2$ .\",\n \"A non-singlet distribution is one which cannot be populated by the quark-antiquark pairs evolved from gluons.\",\n \"Singlet distributions, on the other hand, satisfy coupled integro-differential equations in which the gluon distribution appears as well.\",\n \"To describe the evolution of $F_2(x,Q^2)$ it is necessary to solve these differential equations involving valence and ocean quark distributions and a gluon distribution $f_g(x,t)$ .\",\n \"Starting values of all distributions at $t=$ 0, i.e., $Q^2 = Q^2_0$ , are required as input.\",\n \"The general features of the $t-$ dependence are clear from the Weizäcker-Williams approximation.\",\n \"As $t$ increases, valence quarks lose momentum by emitting gluons — i.e., probed with higher resolution more of a quark momentum appears to be carried by its gluon field.\",\n \"Gluons, in turn, lose momentum to quark-antiquark pairs.\",\n \"The net result, with increasing $t$ , is a transfer of the valence quark momenta to newly created pairs.\",\n \"The gluons are caught in the middle - valence quarks emit gluons but the gluons turn into $q\\\\bar{q}$ pairs.\",\n \"Not surprisingly, the process saturates: at very large $\\\\ln Q^2$ the fraction of the momentum of any target carried by gluons saturates at $\\\\approx $ 0.47 [38].\",\n \"The $Q^2$ -dependence of $F_2(x,Q^2)$ is illustrated schematically in Fig.\",\n \"REF .\",\n \"Close, Roberts and Ross were struck by the similarity of Fig.\",\n \"REF to the shape of the EMC effect at fixed $Q^2$ , and were led to the “rescaling\\\" analysis which is the subject of the next chapter.\",\n \"Figure: Q 2 Q^2-dependence of the structure function (dF 2 (x,Q 2 )/dlnQ 2 )(dF_2(x,Q^2)/d \\\\ln Q^2) at Q 2 Q^2 = 3.2 GeV 2 ^2 using F 2 (x,Q 2 )F_2(x,Q^2) from SLAC data .Integrodifferential equations like Eq.\",\n \"(5.12) are not easily solved directly.\",\n \"We can gain considerable insight, though, by taking moments in $x$ on both sides and using the special properties of convolutions.\",\n \"Let $M^n_{\\\\rm N. S.} = \\\\int _0^1 dx x^{n-1} f_{\\\\rm N. S.}(x,t) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.13)}$ Then, it is easy to show from Eq.\",\n \"(5.12), that $\\\\frac{dM^n_{\\\\rm N. S.}}{dt} = \\\\frac{\\\\alpha _c}{2 \\\\pi }B_nM^n_{\\\\rm N. S.}(t) \\\\qquad \\\\mathrm {(5.14)}$ where $B_n = \\\\int _0^1 dz z^{n-1} \\\\overline{P}_{q/q} (z) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.15)}$ Eq.\",\n \"(5.14) is easier to solve.\",\n \"The character of the solution depends on the $t$ -dependence of $\\\\alpha _c$ , which I have not yet specified.\",\n \"In quantum field theories, the effective coupling has a definite and calculable $t$ -dependence [39].\",\n \"In free field theory $\\\\alpha _c = 0$ and $M^n_{\\\\rm N. S.}$ is constant - corresponding to exact Bjorken scaling.\",\n \"In some model field theories the effective coupling becomes constant at large $t$ , $\\\\lim _{t \\\\rightarrow \\\\infty } \\\\alpha _c(t) \\\\rightarrow \\\\alpha _0$ [40].\",\n \"This behavior is known as “ultraviolet fixed point\\\".\",\n \"In such a theory, the solution of Eq.\",\n \"(5.14) at large $Q^2$ is $M^n_{\\\\rm N. S.}(Q^2) \\\\sim M^n_{\\\\rm N. S.} (Q^2_0) \\\\left(\\\\frac{Q^2}{Q^2_0}\\\\right)^{\\\\frac{\\\\alpha _0 B_n}{2 \\\\pi }} \\\\qquad \\\\mathrm {(5.16)}$ i.e.\",\n \"scaling is violated by powers of $Q^2$ .\",\n \"The coefficient $B_n$ is known as the “anomolous dimension\\\".\",\n \"In QCD, the effective coupling vanishes as $t \\\\rightarrow \\\\infty $ butit vanishes too slowly to use the approximation $\\\\alpha _c=0$ at large $t$ : $\\\\alpha _c(t)$ can be shown to have an expansion of the form: $\\\\frac{\\\\alpha (0)}{\\\\alpha _c(t)} = 1 + b \\\\alpha _c(0)t + O(\\\\alpha _c(0)^2) \\\\ , \\\\qquad \\\\mathrm {(5.17)}$ or to leading order, $\\\\alpha _c(t) \\\\sim \\\\frac{\\\\alpha _c(0)}{1+bt\\\\alpha _c(0)} \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.18)}$ The coefficient $b$ is $b = \\\\frac{1}{4 \\\\pi } \\\\left( 11-\\\\frac{2}{3}N_f \\\\right) \\\\ , \\\\qquad \\\\mathrm {(5.19)}$ where $N_f$ is the number of quark flavors with masses small compared with $\\\\sqrt{Q^2}$ , in practice $N_f \\\\sim 3-4$ .\",\n \"Eq.\",\n \"(5.18) can be rewritten $\\\\alpha _c(Q^2) \\\\sim \\\\frac{1}{b \\\\ln Q^2/\\\\Lambda ^2} \\\\qquad \\\\mathrm {(5.20)}$ where $\\\\Lambda ^2 = Q^2_0 \\\\exp (-1/b \\\\alpha _c(0))$ .\",\n \"This well-known behavior of $\\\\alpha _c(Q^2)$ is known as asymptotic freedom.\",\n \"It is very special to non-Abelian gauge field theories and it makes it possible to calculate with QCD at large $Q^2$ [41].\",\n \"If $\\\\alpha _c(t)$ behaves as in Eq.\",\n \"(5.18), the moments evolve logarithmically: $M^n_{\\\\rm N. S.}(t) = M^n_{\\\\rm N. S.}(0)(\\\\alpha _c(0)/\\\\alpha _c(t))^{B_n/2 \\\\pi b} \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.21)}$ A more familiar form of Eq.\",\n \"(5.21) is obtained by changing variables to $Q^2$ and taking the logarithm $\\\\ln M^n_{\\\\rm N. S.}(Q^2) \\\\sim \\\\ln M^n_{\\\\rm N. S.}(Q^2_0) + \\\\frac{B_n\\\\alpha _c(Q^2_0)}{2 \\\\pi b} \\\\ln Q^2/Q^2_0 \\\\ , \\\\qquad \\\\mathrm {(5.22)}$ where we've replaced $\\\\ln (1+bt\\\\alpha _c(0)) \\\\sim bt\\\\alpha _c(0)$ since we are working to lowest order in $\\\\alpha _c$ .\",\n \"The coefficients {$B_n$ }, still known as (non-singlet) anomolous dimensions, can be computed from Eqs.\",\n \"(5.11) and (5.15) $B_n = \\\\frac{4}{3} \\\\left\\\\lbrace \\\\frac{1}{n(n+1)} - 2 \\\\sum _{j=2}^n \\\\frac{1}{j} -\\\\frac{1}{2} \\\\right\\\\rbrace \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.23)}$ $B_1$ is zero, which corresponds to the fact that the number of valence quarks does not change with $Q^2$ .\",\n \"All the others are negative, so all other moments decrease monotonically with $Q^2$ at large $Q^2$ .\",\n \"Figure: Testing QCD in inelastic lepton scattering.\",\n \"At large Q 2 Q^2, [M n ] 2πb/B n [M^n]^{2 \\\\pi b/B_n} lies on a straight line with a predicted slope when plotted versus lnQ 2 \\\\ln Q^2.In principle, a particularly simple way to test QCD is to plot the moments of the valence quark distribution versus $\\\\ln Q^2$ : the slope is predicted by QCD up to a single parameter, $\\\\alpha (Q^2_0)$ or $\\\\Lambda $ .\",\n \"In practice, the situation is quite complicated: $F_2(x,Q^2)$ is not well-measured at all $x$ , so moments are hard to compute; non-singlet quark distributions must be extracted from $F_2(x,Q^2)$ in order to apply the simple analysis I've described; $\\\\alpha _c(t)$ is not so small where there is good data (remember $d \\\\sigma /dE^\\\\prime d\\\\Omega $ falls like $1/Q^4$ ) so higher corrections to Eq.\",\n \"(5.17) must be included at least at the low-$t$ end.\",\n \"Nevertheless, the moments have been extracted and compared with theory and the agreement is excellent as shown in Fig.\",\n \"REF .\",\n \"For our purposes, it is important to remember that the logarithms of the moments depend linearly on $\\\\ln Q^2$ and that the slope is negative and independent of the target.\",\n \"To get more insight into the $Q^2$ -dependence of deep inelastic structure functions it is necessary to return to coordinate space.\",\n \"In $§2$ we derived the parton model by studying the spacetime dependence of the product of currents.\",\n \"This gave us considerable insight into the relation between the $x$ -dependence of $F_2(x)$ and the quark correlation function in the target ground state.\",\n \"Now I want to look for similar insight into the $Q^2-$ dependence we've found in QCD and to make sure our earlier insight is not spoiled by the interactions we've added.\",\n \"(It isn't.)\",\n \"The starting point is Wilson's operator product expansion [42] (OPE).\",\n \"The idea is that the product of operators simplifies in the limit that their arguments coincide: $\\\\lim _{\\\\xi ^\\\\mu \\\\rightarrow 0} A(\\\\xi )B(0) \\\\sim \\\\sum _{\\\\lbrace \\\\beta \\\\rbrace } C_{\\\\lbrace \\\\beta \\\\rbrace }(\\\\xi )O_{\\\\lbrace \\\\beta \\\\rbrace } (0) \\\\qquad \\\\mathrm {(5.24)}$ where $C_{\\\\lbrace \\\\beta \\\\rbrace }$ are $C$ -number functions and $O_{\\\\lbrace \\\\beta \\\\rbrace }$ are finite local operators.\",\n \"$C_{\\\\lbrace \\\\beta \\\\rbrace }$ may depend on any other parameters in the theory such as masses and coupling constants, in addition to $\\\\xi $ ; $\\\\lbrace \\\\beta \\\\rbrace $ are all labels which might occur in such an expansion (Lorentz and internal symmetry indices, etc.).\",\n \"The content of Eq.\",\n \"(5.24) is that the singularities are factored out from the operators and that the terms in the expansion can be organized in decreasing order of singularity as $\\\\xi _\\\\mu \\\\rightarrow 0$ .\",\n \"Eq.\",\n \"(5.24) is useful only for $\\\\xi ^\\\\mu \\\\approx 0$ , the short distance limit.\",\n \"This will translate into information about the Compton amplitude in the limit $Q^2 \\\\rightarrow \\\\infty $ and $\\\\omega \\\\rightarrow 0$ ($\\\\omega = 1/x$ ).\",\n \"Deep inelastic scattering probes the light-cone and requires $\\\\omega \\\\ge 1$ , but the dispersion relations discussed in $§1.2$ will provide the necessary connection between the two regimes.\",\n \"It is easy to check that Eq.\",\n \"(5.24) works in free field theories.\",\n \"For example, 1.\",\n \"In a free, massless, scalar field $\\\\phi (\\\\xi )\\\\phi (0) &= \\\\mathinner {\\\\langle {0|\\\\phi (\\\\xi )\\\\phi (0)|0}\\\\rangle }+ : \\\\phi (\\\\xi )\\\\phi (0): \\\\nonumber \\\\\\\\&= \\\\frac{i}{4 \\\\pi ^2(-\\\\xi ^2+i \\\\epsilon \\\\xi ^0)} I \\\\nonumber \\\\\\\\&+ \\\\sum _{n=0}^{\\\\infty } \\\\frac{\\\\xi ^{\\\\mu _1}....\\\\xi ^{\\\\mu _n}}{n!\",\n \"}: \\\\phi (0) \\\\overleftarrow{\\\\partial }_{\\\\mu _1},...\\\\overleftarrow{\\\\partial }_{\\\\mu _n} \\\\phi (0): $ $I$ is the identity, and $:\\\\phi (0)...\\\\phi (0):$ are the regular operators.\",\n \"Their “coefficient functions\\\" are $C$ -numbers.\",\n \"Note that the first term in Eq.\",\n \"(5.25) diverges as $\\\\xi _\\\\mu \\\\rightarrow 0$ but the rest vanish with successfully higher powers of $\\\\xi _\\\\mu $ .\",\n \"2.\",\n \"In a free massless Dirac theory let $J(\\\\xi ) = : \\\\overline{\\\\psi }(\\\\xi )\\\\psi (\\\\xi ):$ then in analogy to Eq.\",\n \"(2.1) et seq.\",\n \"$[J(\\\\xi ),J(0)] &= \\\\frac{1}{2 \\\\pi }[\\\\partial ^\\\\rho \\\\delta (\\\\xi ^2)\\\\epsilon (\\\\xi ^0)] \\\\sum _{n=0}^{\\\\infty } \\\\frac{\\\\xi ^{\\\\mu _1}...\\\\xi ^{\\\\mu _n}}{n!}\",\n \"\\\\nonumber \\\\\\\\&: \\\\overline{\\\\psi }(0)\\\\overleftarrow{\\\\partial }_{\\\\mu _1}...\\\\overleftarrow{\\\\partial }_{\\\\mu _n} \\\\gamma _\\\\rho \\\\psi (0) \\\\nonumber \\\\\\\\&- \\\\overline{\\\\psi }(0) \\\\overrightarrow{\\\\partial }_{\\\\mu _1}...\\\\overrightarrow{\\\\partial }_{\\\\mu _n} \\\\gamma _\\\\rho \\\\psi (0): \\\\ .\",\n \"$ In a free field theory, the Taylor expansions in Eq's.\",\n \"(5.25) and (5.26) can of course be summed to give a “bilocal operator\\\" like :$\\\\overline{\\\\psi }(\\\\xi )\\\\gamma _\\\\rho \\\\psi (0)$ : but in interacting theories the coefficient function will differ slightly for each term in the Taylor expansion preventing resummation.\",\n \"This, it will turn out, makes the difference between exact Bjorken scaling as in free field theory, and logarithmic scaling violation as in QCD.\",\n \"It is clear from these examples that the form the operator product expansion takes in a particular field theory depends upon the procedure necessary to remove the singularity which arises when one tries to bring operators to the same space time point.\",\n \"In free field theories the singular piece is a $C-$ number and can be isolated by normal ordering the operator product.\",\n \"In interacting field theories, the divergences are worse and normal ordering does not suffice.\",\n \"Instead, it is necessary to renormalize the operators.\",\n \"There is a certain arbitrariness in renormalization: to define finite operators it is necessary to introduce a mass-scale $\\\\mu ^2$ — loosely speaking, it is the scale at which the operator's matrix elements have the values they would have in free field theory — but $\\\\mu ^2$ is arbitrary, nothing physical can depend on it.\",\n \"Nevertheless, both the operators $O_{\\\\lbrace \\\\beta \\\\rbrace }$ and the coefficient functions $C_{\\\\lbrace \\\\beta \\\\rbrace }$ will separately depend on the mass scale introduced by the necessity of renormalization.\",\n \"To allow for this we replace $C_{\\\\lbrace \\\\beta \\\\rbrace }(\\\\xi ) $ by $C_{\\\\lbrace \\\\beta \\\\rbrace }(\\\\xi ,\\\\mu ^2)$ and $O_{\\\\lbrace \\\\beta \\\\rbrace }(0)$ by $O^{(\\\\mu ^2)}_{\\\\lbrace \\\\beta \\\\rbrace }(0)$ .\",\n \"In anything physically measurable, the $\\\\mu ^2$ dependence of $C_{\\\\lbrace \\\\beta \\\\rbrace }(\\\\xi ,\\\\mu ^2)$ will cancel that of $O_{\\\\lbrace \\\\beta \\\\rbrace }^{(\\\\mu ^2)}(0)$ .\",\n \"$\\\\mu ^2$ is known as the “renormalization point\\\".\",\n \"To make use of the OPE, it is convenient to make explicit the factors of $\\\\xi _\\\\mu $ which accompany an operator carrying Lorentz indices.\",\n \"To this end we rewrite Eq.\",\n \"(5.24) as $A(\\\\xi )B(0) \\\\sim \\\\sum _{\\\\lbrace \\\\beta \\\\rbrace } C^\\\\prime _{\\\\lbrace \\\\beta \\\\rbrace }(\\\\xi ^2,\\\\mu ^2)\\\\xi _{\\\\mu _1}.....\\\\xi _{\\\\mu _{n_{\\\\beta }}} O^{\\\\lbrace \\\\mu ^2\\\\rbrace \\\\mu _1...{\\\\mu _{n_{\\\\beta }}}}_{\\\\lbrace \\\\beta \\\\rbrace } \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.27)}$ The operators $O_{\\\\lbrace \\\\beta \\\\rbrace }$ can always be defined so they are symmetric traceless in all Lorentz indices: $O^{\\\\mu _1..\\\\mu _i..\\\\mu _j..{\\\\mu _{n_{\\\\beta }}}} = O^{\\\\mu _1..\\\\mu _j..\\\\mu _i..{\\\\mu _{n_{\\\\beta }}}}$ and $g_{\\\\mu _i \\\\mu _j}O^{\\\\mu _1..\\\\mu _i..\\\\mu _j..{\\\\mu _{n_{\\\\beta }}}} =0$ [43].\",\n \"Then, $n_\\\\beta $ is called the “spin\\\" of the operators [44].\",\n \"If $A$ and $B$ carry Lorentz indices then the form of Eq.\",\n \"(5.27) becomes more complicated.\",\n \"To avoid writing complicated equations, I will suppress the indices on the currents $J_\\\\mu $ and $J_\\\\nu $ or equivalently study deep inelastic scattering by a particle coupled to a hypothetical scalar current $J(\\\\xi )=\\\\overline{\\\\psi }(\\\\xi )\\\\psi (\\\\xi )$ .\",\n \"Let us expand the product $T(J(\\\\xi )J(0))$ in the fashion of Eq.\",\n \"(5.27) and calculate the contributions to $T(q^2,\\\\omega )$ (see Eq.\",\n \"(1.12)).\",\n \"We must carry out the Fourier transform $\\\\tilde{C}_{\\\\mu _1...{\\\\mu _{n_{\\\\beta }}}}(q,\\\\mu ^2) \\\\equiv \\\\int d^4 \\\\xi e^{i q \\\\cdot \\\\xi } C^\\\\prime _{\\\\lbrace \\\\beta \\\\rbrace }(\\\\xi ^2,\\\\mu ^2)\\\\xi _{\\\\mu _1}...\\\\xi _{\\\\mu _{n_{\\\\beta }}} \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.28)}$ The Lorentz indices on $\\\\tilde{C}$ can only be in the form of $q_{\\\\mu _k}$ or $g_{\\\\mu _i \\\\mu _j}$ .\",\n \"The latter vanish when contracted with the traceless operator $O_{\\\\lbrace \\\\beta \\\\rbrace }$ .\",\n \"In effect, then $\\\\tilde{C}_{\\\\mu _1...{\\\\mu _{n_{\\\\beta }}}}(q,\\\\mu ^2) = \\\\frac{q_{\\\\mu _1}...q_{{\\\\mu _{n_{\\\\beta }}}}}{(q^2)^{n_\\\\beta }} (-1)^{n_\\\\beta } \\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.29)}$ The phase and the factors of $(q^2)^{n_\\\\beta }$ have been introduced for later convenience.\",\n \"Note that $\\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2)$ has the same dimension as $\\\\int d^4 \\\\xi e^{i q \\\\cdot \\\\xi } C^\\\\prime _{\\\\lbrace \\\\beta \\\\rbrace }(\\\\xi ^2,\\\\mu ^2)$ .\",\n \"Now, $T(q^2,\\\\omega )$ can be written as $T(q^2,\\\\omega ) \\\\sim \\\\sum _{\\\\lbrace \\\\beta \\\\rbrace }\\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2) (-1)^{n_\\\\beta } \\\\frac{q_{\\\\mu _1}...q_{{\\\\mu _{n_{\\\\beta }}}}}{(q^2)^{n_\\\\beta }} \\\\mathinner {\\\\langle {p|O_{\\\\lbrace \\\\beta \\\\rbrace }^{(\\\\mu ^2)\\\\mu _1..{\\\\mu _{n_{\\\\beta }}}}|p}\\\\rangle }_c \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.30)}$ The matrix element in Eq.\",\n \"(5.30) must carry Lorentz indices $\\\\mu _1...{\\\\mu _{n_{\\\\beta }}}$ .\",\n \"We can write $\\\\mathinner {\\\\langle {p|O_{\\\\lbrace \\\\beta \\\\rbrace }^{(\\\\mu ^2)\\\\mu _1..{\\\\mu _{n_{\\\\beta }}}}|p}\\\\rangle } = \\\\Theta _{\\\\lbrace \\\\beta \\\\rbrace }(\\\\mu ^2)(p^{\\\\mu _1}...p^{{\\\\mu _{n_{\\\\beta }}}}+....) \\\\ , \\\\qquad \\\\mathrm {(5.31)}$ where the other terms are determined by the fact that $O_{\\\\lbrace \\\\beta \\\\rbrace }$ is symmteric and traceless.\",\n \"The terms required to remove traces all contain at least one factor of $g^{\\\\mu _i\\\\mu _j}$ .\",\n \"Thus, for example, if $n_\\\\beta =2$ , one has $p^{\\\\mu _1}p^{\\\\mu _2}-\\\\frac{M^2_p}{4}g^{\\\\mu _1\\\\mu _2}$ .\",\n \"Combining Eqs.\",\n \"(5.30) and (5.31) we find $T(q^2,\\\\omega ) \\\\sim \\\\sum _\\\\beta \\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2) \\\\Theta _{\\\\lbrace \\\\beta \\\\rbrace }(\\\\mu ^2) \\\\left[ (\\\\frac{\\\\omega }{2})^{n_\\\\beta } + O(\\\\frac{1}{q^2}) \\\\right] \\\\ , \\\\qquad \\\\mathrm {(5.32)}$ ($\\\\omega \\\\equiv -2p \\\\cdot q/q^2$ ) because $p^\\\\alpha p^\\\\beta q_\\\\alpha q_\\\\beta /(q^2)^2 = \\\\omega ^2/4$ but $g^{\\\\alpha \\\\beta }q_\\\\alpha q_\\\\beta /(q^2)^2 = 1/q^2$ .\",\n \"In fact, only even powers of $\\\\omega $ occur in Eq.\",\n \"(5.32) because $T(q^2,\\\\omega )$ is crossing symmetric (see $§1.2$ ).\",\n \"Since $\\\\omega $ ($\\\\cong 1/x$ ) scales in the Bjorken limit, the importance of any particular operator $O_{\\\\lbrace \\\\beta \\\\rbrace }$ is determined by the large-$q^2$ behavior of $\\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2)$ .\",\n \"This can be calculated to all orders in perturbation theory using the methods of the renormalization group.\",\n \"The first step is to determine the dimension of $\\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2)$ .\",\n \"We define the “naive\\\" or “canonical\\\" dimension (or, for short, simply dimension) of an operator to be the units in which it is measured as a power of mass (with $\\\\hbar = c= 1$ ).\",\n \"We use the notation $[O_{\\\\lbrace \\\\beta \\\\rbrace }] = d_{\\\\lbrace \\\\beta \\\\rbrace }$ .\",\n \"Here are some examples: $[J_\\\\mu ]=3$ because $\\\\int d^3x J_0 =$ charge which is dimensionless; $[\\\\phi ]=1$ because the action, $S=\\\\int d^4 x \\\\frac{1}{2}(\\\\partial \\\\phi )^2$ , is dimensionless; likewise $[\\\\psi ] = \\\\frac{3}{2}$ .\",\n \"Using these dimensions and remembering $\\\\mathinner {|{p}\\\\rangle }$ is covariantly normed, $[\\\\mathinner {|{p}\\\\rangle }]=-1$ , we see that $T(q^2,\\\\omega )$ is dimensionless.\",\n \"Let us suppose the operator $O_{\\\\lbrace \\\\beta \\\\rbrace }$ has dimension $d_\\\\beta $ , then from Eq.\",\n \"(5.27), $[C^\\\\prime _{\\\\lbrace \\\\beta \\\\rbrace }]=6-d_\\\\beta -n_\\\\beta \\\\qquad \\\\mathrm {(5.33)}$ and $[\\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2)] = 2-d_\\\\beta +n_\\\\beta \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.34)}$ T he dimensions of $\\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }$ could, in principle, be provided by $q^2$ , $\\\\mu ^2$ or quark masses.\",\n \"Weinberg's theorem [45] can be used to show that at large $-q^2$ to each order in perturbation theory quark masses can be ignored provided $\\\\mu ^2$ is fixed and not zero.\",\n \"Also, in each order of perturbation theory the $\\\\mu ^2$ dependence of a renormalizable field theory is at most logarithmic, reflecting at worst logarithmic divergences in such theories.\",\n \"So we can write $\\\\lim _{q^2 \\\\rightarrow -\\\\infty }\\\\tilde{C}_{\\\\lbrace \\\\beta \\\\rbrace }(q^2,\\\\mu ^2) = \\\\left( \\\\frac{1}{q^2} \\\\right)^{d_\\\\beta -n_\\\\beta -2} c_{\\\\lbrace \\\\beta \\\\rbrace }(\\\\ln q^2/\\\\mu ^2) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.35)}$ The next step is to calculate the leading large $-q^2$ dependence of $c_{\\\\lbrace \\\\beta \\\\rbrace }(\\\\ln q^2/\\\\mu ^2)$ to all orders in perturbation theory.\",\n \"In each order, $c_{\\\\lbrace \\\\beta \\\\rbrace }$ grows like a power of $\\\\ln q^2/\\\\mu ^2$ .\",\n \"When summed to all orders the logarithms may yield a power, i.e., $c_{\\\\lbrace \\\\beta \\\\rbrace }$ may depend exponentially on its argument.\",\n \"The actual calculation of the asymptotic behavior of $c_{\\\\lbrace \\\\beta \\\\rbrace }$ requires renormalization group methods beyond the scope of these lectures.\",\n \"But as will be seen below, we have performed an equivalent calculation using Weizsäcker-Williams methods in $§5.1$ .\",\n \"If $c_{\\\\lbrace \\\\beta \\\\rbrace }$ does not go like an exponential of its argument, then operators with $d_\\\\beta -n_\\\\beta =2$ give rise to Bjorken scaling modulo powers of $\\\\ln q^2$ .\",\n \"$d_\\\\beta -n_\\\\beta $ plays such a central role in the analysis of large $q^2$ effects that it is given a name, “twist\\\", $t_\\\\beta =d_\\\\beta -n_\\\\beta \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.36)}$ Operators with $t_\\\\beta < 2$ would give contributions which diverge in the Bjorken limit, but the only operator with $t_\\\\beta <2$ which can couple to the product of the two currents is the identity, $t_I=0$ , and the identity has no connected matrix elements.\",\n \"Operators with $t_\\\\beta >2$ give contributions to $T(q^2,\\\\omega )$ which vanish in the Bjorken limit (provided $c_{\\\\lbrace \\\\beta \\\\rbrace }$ is not exponential in its argument).\",\n \"Only even twists occur in the expansion of two currents in the limit of zero quark mass [46] so the next important case is $t_\\\\beta = 4$ .\",\n \"Twist-4 or $O(1/q^2)$ corrections to scaling are a rich and fascinating, if technically complicated subject, in themselves [47].\",\n \"The twist-two operators in QCD come in two classes: quark operators $O_{n,a}^{(\\\\mu ^2)\\\\mu _1..\\\\mu _n} = S \\\\overline{\\\\psi }_a \\\\gamma ^{\\\\mu _1} \\\\partial ^{\\\\mu _1}D^{\\\\mu _2}...D^{\\\\mu _n} \\\\psi _a^{(\\\\mu ^2)} \\\\ , \\\\qquad \\\\mathrm {(5.37)}$ where $D_\\\\mu $ is the (color) gauge covariant derivative [48], “a\\\" labels flavor and $S$ symbolizes the operation of making $O_{n,a}$ traceless and symmetric; and gluon operators which we needn't write out, since they contribute only to singlet distributions.\",\n \"It is easy to check that $O_{n,a}$ indeed has $t=2$ for all $n$ .\",\n \"It should be emphasized that it requires an infinite tower of operators of increasing spin to describe $T(q^2,\\\\omega )$ .\",\n \"This is not at all surprising.\",\n \"First of all, if the sum in Eq.\",\n \"(5.30) stopped at some $n_{\\\\rm max}$ , $T(q^2,\\\\omega )$ would be a polynomial in $\\\\omega $ with no cut on the real axis for $|\\\\omega |>1$ .\",\n \"Second, we know that deep inelastic electroproduction probes the light cone, not just short distances.\",\n \"The OPE is a short distance expansion ($\\\\xi _\\\\mu \\\\rightarrow 0$ ) and no finite number of terms in the short distance expansion gives information about light-like separation.\",\n \"To summarize: the OPE approach leads to a simultaneous expansion of $T(q^2,\\\\omega )$ in $q^2$ and $\\\\omega $ .\",\n \"The expansion in $q^2$ is an asymptotic expansion and is ordered by the twist quantum number of the operators, the expansion in $\\\\omega $ is a Taylor expansion (which converges for $|\\\\omega |<1$ , see $§1$ ) and is ordered by the spin of the operators.\",\n \"Ignoring gluon operators $-$ which is adequate if we are interested in non-singlet quark distributions alone $-$ we have $T_{\\\\rm N. S.}(q^2,\\\\omega ) = \\\\sum _{\\\\rm n\\\\ even} c^n_{\\\\rm N. S.}(\\\\ln q^2/\\\\mu ^2) \\\\Theta ^n_{\\\\rm N. S.} (\\\\mu ^2) \\\\omega ^n + {\\\\rm higher\\\\ twist} \\\\ , \\\\qquad \\\\mathrm {(5.38)}$ where we have replaced the generic label $\\\\beta $ by the spin ($n$ ) of the 2 twist-two, non-singlet quark operator.\",\n \"$c^n_{\\\\rm N. S.}(\\\\ln q^2/\\\\mu ^2)$ depends only on $\\\\ln q^2/\\\\mu ^2$ and explicit calculation shows that it equals 4 when $q^2 = \\\\mu ^2$ .\",\n \"Comparing Eq.\",\n \"(5.38) with the Taylor expansion of $T(q^2,\\\\omega )$ developed in $§1$ , (Eq.\",\n \"1.18), we identify $M^n_{\\\\rm N. S.}(q^2) = \\\\frac{1}{4} c^n_{\\\\rm N. S.}(\\\\ln q^2/\\\\mu ^2) \\\\Theta ^n_{\\\\rm N. S.}(\\\\mu ^2) \\\\ {\\\\rm for\\\\ n\\\\ even} \\\\ .\",\n \"\\\\qquad \\\\mathrm {(5.39)}$ At this point, we can make a connection with the “evolutionary\\\" approach of $§5.1$ and comment on the derivation of the parton model in QCD.\",\n \"The ordering of contributions at large $Q^2$ by twist has given a result, Eq.\",\n \"(5.38), which looks like scaling modulo logarithms.\",\n \"This is deceptive because in general $c^n_{\\\\rm N. S.}$ depends exponentially on its argument giving rise to power law violations of scaling.\",\n \"Only in asymptotically free theories like QCD does $c^n_{\\\\rm N. S.}$ go like a power of its argument, giving scaling up to logarithms.\",\n \"The derivation of the dependence of $c^n_{\\\\rm N. S.}$ on $\\\\ln q^2/\\\\mu ^2$ is outside the scope of these lectures, but the result $c^n_{\\\\rm N. S.}(\\\\ln Q^2/Q^2_0) = (\\\\alpha _c(0)/\\\\alpha _c(t))^{B_n/2 \\\\pi b} \\\\qquad \\\\mathrm {(5.40)}$ (where $\\\\mu ^2 \\\\equiv Q^2_0$ ) should not be surprising since it converts Eq.\",\n \"(5.39) into Eq.\",\n \"(5.21) which we already derived using the more heuristic, Weizäcker-Williams approach.\",\n \"Comparing Eqs.\",\n \"(5.39), (5.40) and (5.21) we see that the moments of the structure functions are directly related to the matrix elements of specific local operators, $M^n_{\\\\rm N. S.}(q^2)=\\\\Theta ^n_{\\\\rm N. S.}(q^2)$ where $q^2$ is the mass-scale at which the operator is $O^{\\\\mu _1...\\\\mu _n}_{n,\\\\rm N. S.}$ is renormalized.\",\n \"The quark operators $O_{n,a}$ which determine the moments are the (gauge invariant) terms in the Taylor expansion of the operator product $\\\\overline{\\\\psi }(\\\\xi ) \\\\gamma _\\\\rho \\\\psi (0)$ which determined the quark distribution function in the parton model.\",\n \"So we see that the modification of the parton model required by the interactions in QCD is that each moment of the quark distribution (i.e., each term in the Taylor expansion) scales modulo a slightly different power of $\\\\ln q^2$ .\",\n \"Earlier, I remarked that the normalization point, $\\\\mu ^2$ , was arbitrary, that nothing physical could depend on it.\",\n \"There is no contradiction here.\",\n \"Eq.\",\n \"(5.30) is independent of $\\\\mu ^2$ , specifically $&\\\\Theta ^n_{\\\\rm N. S.}(\\\\mu ^2)d/d\\\\mu ^2 c^n_{\\\\rm N. S.}(\\\\ln q^2/\\\\mu ^2) \\\\nonumber \\\\\\\\&= -c^n_{\\\\rm N. S.}(\\\\ln q^2/\\\\mu ^2) d/d\\\\mu ^2 \\\\Theta ^n_{\\\\rm N. S.} (\\\\mu ^2) \\\\ .\",\n \"$ But, $c^n_{\\\\rm N. S.}$ depends only on $\\\\ln q^2/\\\\mu ^2$ , so $d/d\\\\mu ^2 c^n_{\\\\rm N. S.}(\\\\ln q^2/\\\\mu ^2) = -d/dq^2 c^n_{\\\\rm N. S.}(\\\\ln q^2/\\\\mu ^2) \\\\qquad \\\\mathrm {(5.42)}$ and therefore $d/dq^2 M^n_{\\\\rm N. S.}(q^2) = d/d \\\\mu ^2 \\\\Theta ^n_{M.S.\",\n \"}(\\\\mu ^2) |_{\\\\mu ^2=q^2} \\\\ , \\\\qquad \\\\mathrm {(5.43)}$ so the renormalization point dependence of the operator matrix elements determines the $q^2-$ dependence of the moment of the structure function.\",\n \"Combining the OPE analysis with the evolutionary picture of the previous section we recognize that the renormalization point introduced in the OPE analysis is the same as the the transverse resolution in the Weizäcker-Williams method.\",\n \"In both cases, it is necessary to define how much of the gluon field is to be lumped into the definition of a quark.\",\n \"Whatever way you look at it, the fact that this definition changes with $Q^2$ gives rise to the logarithmic scaling violation of QCD.\",\n \"In addition to the $\\\\ln Q^2$ corrections we have uncovered, there are expected to be $O(1/Q^2)$ and higher order $O(1/Q^{2n})$ corrections which become important at small $Q^2.$ The $O(1/Q^2)$ corrections take several forms.\",\n \"They are easy to distinguish using the language of OPE.\",\n \"First are “target mass\\\" corrections.\",\n \"As will become clear, this is an unfortunate name.\",\n \"These are apparent $O(M^2_T/Q^2)$ terms which arise from the $g^{\\\\mu _i \\\\mu _j}$ factors in the matrix elements of traceless operators.\",\n \"It has been shown [49] that these corrections may be completely absorbed by replacing the variable $x_T$ by the variable $\\\\xi = -q^+/P^+$ everywhere in the definition of $f_{a/T}(x_T)$ .\",\n \"This isn't surprising since $q^+$ is the variable which emerged automatically in the derivation (see, e.g., Eq.\",\n \"(2.18)).\",\n \"$\\\\xi $ is written in many forms: $\\\\xi &= (\\\\sqrt{\\\\nu ^2 + Q^2}-\\\\nu )/M_T \\\\ , \\\\\\\\\\\\xi &= 2 x_T/(1+ \\\\sqrt{1+Q^2/\\\\nu ^2}) \\\\ , \\\\\\\\\\\\xi &= 2x_T/(1+\\\\sqrt{1+4 M^2_Tx^2_T/Q^2}) \\\\ .\",\n \"$ It is clear from Eq.\",\n \"(5.44b) that the large mass corrections do not, in fact, depend on the target mass [50]!\",\n \"They are kinematic corrections (which do not grow like $A^2$ for nuclear targets), and are well-understood.\",\n \"Second are “quark mass\\\" corrections.\",\n \"These are important for heavy quarks ($c,b,t$ ) but negligible for up and down quarks whose masses, $m_{u,d} < 20$ MeV, are tiny.\",\n \"Finally are the dynamical $O(1/Q^2)$ corrections associated with operators of twist-4.\",\n \"Although they are complicated, twist-4 corrections to inelastic electron (and neutrino) scattering have been completely analyzed in QCD [47].\",\n \"Typically, they are small because the natural mass scale associated with a target is one upon its radius (once “target mass\\\" corrections have been incorporated via $\\\\xi -$ scaling), $1/R \\\\sim 1$ fm$^{-1} \\\\sim $ 200 MeV.\",\n \"It is therefore not surprising that scaling, modulo logarithms and using the $\\\\xi $ variable, sets in at a very low value of $Q^2$ ($<$ 1 GeV$^2$ ).\",\n \"For precisely this reason higher twist contributions to $F_2(x,Q^2)$ , which measure matrix elements of interesting local operators, are hard to extract from available experimental data.\",\n \"We have seen that the quark, antiquark and gluon content of a hadron changes with the scale at which it is probed.\",\n \"In more naive quark models the nucleon, for example, is treated as (approximately) three quarks in some confining “bag\\\" with no reference to the scale at which this description might hold.\",\n \"Certainly, if the nucleon were three quarks at some scale $\\\\mu ^2_0$ ($\\\\mu ^2_0 \\\\sim 1$ GeV$^2$ ) then it would become more complicated, containing antiquarks and glue at larger scales $Q^2 > \\\\mu ^2_0$ by virtue of QCD radiation.\",\n \"In the early days of QCD it was recognized that if the nucleon's quark, antiquark and gluon distributions measured at large $Q^2$ were evolved back to lower $Q^2$ , then quark-antiquark pairs and gluons are reabsorbed into the valence quarks, so that at some $\\\\mu ^2_0 \\\\sim $ 1 GeV$^2$ all of the $q \\\\bar{q}$ pairs and most of the glue would be gone leaving a nucleon made of three quarks alone.\",\n \"G. Ross and I checked this quantitatively in the M.I.T.\",\n \"version of the bag model [52].\",\n \"Using QCD evolution to second order, which is necessary because $\\\\alpha (\\\\mu ^2_0)$ is not small, we found that measured non-singlet nucleon structure functions evolved backwards to a $\\\\mu ^2_0$ of order 1 GeV$^2$ indeed gave valence quark $x-$ distributions in agreement with earlier bag calculations [53].\",\n \"$\\\\mu ^2_0$ is then interpreted as a parameter of the quark model: It is the mass scale (or resolution) at which quark fields should be defined in order that the nucleon should be made of three quarks.\",\n \"A recent reevaluation of this program with modern values for structure functions and the QCD $\\\\Lambda $ parameter (c.f.\",\n \"Eq.\",\n \"(5.20)) gave $\\\\mu ^2_0 \\\\cong 0.75$ GeV$^2$ [58].\",\n \"Notice that the structure function predicted by simple quark models cannot be compared directly with experimental measurements of $F_2$ at $Q^2=\\\\mu ^2_0$ because at such a low $Q^2$ higher twist effects are large but have not been included in the quark model calculations.\",\n \"It seems best to regard quark models as models for the twist-two matrix elements at a renormalization point $\\\\mu ^2_0$ , which must then be evolved to $Q^2 >> \\\\mu ^2_0$ in order to be compared with experiment.\",\n \"Close, Roberts and Ross [55] realized that the scale ($Q^2$ ) dependence of quark distribution functions in QCD could be used to parametrize and, to some extent, explain the $A$ dependence of structure functions [56].\",\n \"In $§2$ , we learned that the shift in the valence quarks observed in nuclei could be understood as an increase in the quark correlation length in the nuclear ground state.\",\n \"The increase in ocean quark pairs appeared to be an independent phenomenon.\",\n \"In the QCD inspired analysis, I shall describe both aspects of the EMC effect have a single origin: a dynamical change in scale of the twist-2 matrix elements in nuclei.\",\n \"In the last chapter, we saw that QCD evolution reduced the momentum on valence quarks and increased the number of pairs.\",\n \"Suitably adapted, evolution can explain the EMC effect.\",\n \"This method of analysis has come to be known as “dynamical rescaling\\\" or simply “rescaling\\\".\",\n \"Its virtues are first, it gives a unified description of all aspects of the EMC effect; second, it avoids the dubious assumptions of the constituent convolution models of $§4$ ; and third, it gives us insight into the reason other superficially quite different “explanations\\\" of the EMC effect work.\",\n \"Its drawback is that it does not provide a microscopic enough explanation to satisfy most of us: it is not clear exactly what the quarks and gluons are doing differently in a nucleus which gives rise to the effect.\",\n \"In this chapter, I will work from the general toward the specific.\",\n \"First, I will merely use QCD as an aid to present the data in a new way.\",\n \"This presentation will lead to a surprising conclusion and suggest rescaling as a mechanism behind the EMC effect.\",\n \"Then, I will analyze rescaling in some detail.\",\n \"Next, I will describe a calculation of the $A$ dependence motivated by, but perhaps more general than rescaling [10].\",\n \"Finally, I will close with some remarks about shadowing and future experiments.\",\n \"These have little to do with QCD and less to do with rescaling, but they follow naturally upon the discussion of $A$ dependence.\",\n \"In $§2$ , we compared the structure functions of different nuclei at fixed $Q^2$ , as functions of $x$ .\",\n \"This is the way the data come from the experimenters.\",\n \"QCD provides an alternative.\",\n \"Consider the moments: $M^n_A(Q^2) \\\\equiv \\\\int _0^A dx x^{n-2} \\\\overline{F}_2^A(x,Q^2) \\\\qquad \\\\mathrm {(6.1)}$ ($\\\\overline{F}^A_2=\\\\frac{1}{A}F^A_2$ ).\",\n \"According to Eq.\",\n \"(5.21), the moments are monotonically falling functions of $Q^2$ .\",\n \"(This analysis like that of $§5$ is restricted to non-singlet structure functions but a similar conclusion applies as well to singlets.\",\n \"): $M^n_A(Q^2) = \\\\left( \\\\frac{\\\\alpha _c(Q^2)}{\\\\alpha _c(Q^2_o)} \\\\right)^{d_n} M^n_A(Q^2_0) \\\\ , \\\\qquad \\\\mathrm {(6.2)}$ where $d_n \\\\equiv -B_n/2 \\\\pi b>0$ .\",\n \"If QCD is correct, and if $Q^2$ is large enough so leading order in perturbation theory suffices and $O(1/Q^2)$ corrections are negligible, then $\\\\ln M^n_A(Q^2)$ must lie on a straight line when plotted versus $\\\\ln [\\\\alpha _c(Q^2_0)/\\\\alpha _c(Q^2)]$ and he slope must be $-d_n$ .\",\n \"Such a plot is shown schematically in Fig.\",\n \"REF for two different targets with baryon numbers $A$ and $A^\\\\prime $ .\",\n \"Figure: Rescaling a single moment.At fixed $Q^2$ , the EMC effect appears as the observation that $M^n_{A^\\\\prime }(Q^2)<M^n_A(Q^2)$ for $A^\\\\prime > A$ .\",\n \"On the other hand, it is clear that there is a value of $Q^2$ , call it $Q^{\\\\prime \\\\,2}$ , such that $M^n_{A^\\\\prime }(Q^{\\\\prime \\\\,2})=M^n_A(Q^2)$ .\",\n \"As illustrated in Fig.\",\n \"REF , $Q^{\\\\prime \\\\,2} < Q^2$ .\",\n \"The value of $Q^{\\\\prime \\\\,2}$ , in principle, depends on $A, A^\\\\prime $ and $Q^2$ and on $n$ .\",\n \"Let us define $\\\\xi ^n_{AA^\\\\prime }(Q^2)$ (not to be confused with the coordinate $\\\\xi _\\\\lambda $ or the modified scaling variable $\\\\xi $ of $§5$ ) so $M^n_{A^\\\\prime }(Q^2) = M^n_A(\\\\xi ^n_{AA^\\\\prime }(Q^2)Q^2) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.3)}$ From its definition $\\\\xi ^n_{AA^\\\\prime }(Q^2)>1 \\\\ \\\\ {\\\\rm for\\\\ } A^\\\\prime > A \\\\qquad \\\\mathrm {(6.4)}$ and $\\\\xi ^n_{AA^{\\\\prime \\\\prime }}(Q^2) &= \\\\xi ^n_{AA^\\\\prime }(Q^2) \\\\xi ^n_{A^\\\\prime A^{\\\\prime \\\\prime }}(Q^2) \\\\nonumber \\\\\\\\\\\\xi ^n_{AA^\\\\prime }(Q^2) &= 1/\\\\xi ^n_{A^\\\\prime A}(Q^2) \\\\ .\",\n \"$ The $Q^2-$ dependence of $\\\\xi ^n_{AA^\\\\prime }(Q^2)$ is determined entirely by QCD, and is independent of $A$ and $A^\\\\prime $ .\",\n \"Consider Eq.\",\n \"(6.2) first for $A^\\\\prime $ at $Q^2_0$ and $Q^2$ $M^n_{A^\\\\prime }(Q^2) = \\\\left( \\\\frac{\\\\alpha _c(Q^2)}{\\\\alpha _c(Q^2_o)} \\\\right)^{d_n} M^n_{A^\\\\prime }(Q^2_0) \\\\qquad \\\\mathrm {(6.6)}$ and then for $A$ at $\\\\xi ^n_{AA^\\\\prime }(Q^2_0)Q^2_0$ and $\\\\xi ^n_{AA^\\\\prime }(Q^2)Q^2$ : $M^n_{A}(\\\\xi ^n_{AA^\\\\prime }(Q^2)Q^2) = \\\\left( \\\\frac{\\\\alpha _c(\\\\xi ^n_{AA^\\\\prime }(Q^2)Q^2)}{\\\\alpha _c(\\\\xi ^n_{AA^\\\\prime }(Q^2_0)Q^2_0)} \\\\right)^{d_n} M^n_{A}(\\\\xi ^n_{AA^\\\\prime }(Q^2_0)Q^2_0) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.7)}$ Now use Eq.\",\n \"(6.3) to eliminate all reference to moments $\\\\frac{\\\\alpha _c(\\\\xi ^n_{AA^\\\\prime }(Q^2)Q^2)}{\\\\alpha _c(\\\\xi ^n_{AA^\\\\prime }(Q^2_0)Q^2_0)} = \\\\frac{\\\\alpha _c(Q^2)}{\\\\alpha _c(Q^2_0)} \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.8)}$ To lowest order in $\\\\alpha _c$ , $\\\\alpha _c(\\\\xi Q^2) = \\\\alpha _c(Q^2)/(1+ \\\\alpha _c(Q^2)b \\\\ln \\\\xi ) \\\\ , \\\\qquad \\\\mathrm {(6.9)}$ which, together with Eq.\",\n \"(6.8), implies $\\\\xi ^n_{AA^\\\\prime }(Q^2) = [\\\\xi ^n_{AA^\\\\prime }(Q^2_0)]^{\\\\alpha _c(Q^2_0)/\\\\alpha _c(Q^2)} \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.10)}$ The extension to next order in $\\\\alpha _c$ is quoted in [58].\",\n \"Suppose $Q^2_0 < Q^2$ , then $\\\\alpha _c(Q^2_0)/\\\\alpha _c(Q^2)>1$ and $1< \\\\xi ^n_{AA^\\\\prime }(Q^2_0) < \\\\xi ^n_{AA^\\\\prime }(Q^2) \\\\ \\\\ {\\\\rm for \\\\ \\\\ } Q^2_0<Q^2 \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.11)}$ The implication of this result is that a small change of $Q^2$ -scale at low $Q^2$ gets magnified into a large change when observed at large $Q^2$ .\",\n \"Eqs.\",\n \"(6.4), (6.5) and (6.10) summarize the properties of $\\\\xi ^n_{AA^\\\\prime }(Q^2)$ which can be determined from general considerations alone.\",\n \"The surprise came when Close, Roberts and Ross used this method to analyze the EMC data and found that to a good approximation $\\\\xi ^n_{AA^\\\\prime }(Q^2)$ appears to be independent of $n$: $\\\\xi ^n_{AA^\\\\prime }(Q^2) \\\\Rightarrow \\\\xi _{AA^\\\\prime }(Q^2)$ , at least for the values of $n$ sensitive to the $x$ -range of the EMC data.\",\n \"Actually CRR did not construct moments but made an equivalent observation about the structure function itself.\",\n \"Namely, if $\\\\xi ^n_{AA^\\\\prime }(Q^2)$ is independent of $n$ then the structure functions themselves as functions of $x$ are related by a universal scale change in $Q^2$ : $\\\\overline{F}^{A^\\\\prime }_2(x,Q^2) = \\\\overline{F}^A_2(x,\\\\xi _{AA^\\\\prime }(Q^2) Q^2) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.12)}$ Eq.\",\n \"(6.12) has become known as “rescaling\\\".\",\n \"CRR were led to it by the observation we referred to in $§5$ , that the EMC effect resembles QCD evolution.\",\n \"In fact, the EMC data are not in complete agreement with Eq.\",\n \"(6.12).\",\n \"The excess at low $x$ is more than can be produced by the amount of evolution that is required to fit the depletion at large $x$ .\",\n \"Their analysis, with $\\\\xi _{DFe} \\\\cong 2$ at $Q^2 \\\\approx $ 20 GeV$^2$ is shown in Fig.\",\n \"REF .\",\n \"Figure: Rescaling the EMC iron data .The newer SLAC data on iron and deuterium data have a smaller enhancement at low-$x$ and a lower cross over point (where $\\\\overline{F}^{\\\\rm Fe}_2/\\\\overline{F}^{D}_2 =1$ ), both of which improve the agreement with the rescaling analysis [57], [58].\",\n \"The reader might wish to look back to Fig.\",\n \"REF to see the present state of the rescaling fits.\",\n \"Several comments and caveats are in order: 1.\",\n \"QCD is subtle: Changing the scale creates quark-antiquark pairs.\",\n \"If, after all the discussion of $§5$ , this still seems unreasonable, perhaps it would help to remember that a similar thing happens in a Bogoliubov transformation.\",\n \"By redefining the vacuum, annihilation and creation operators get mixed up with one another and a state which originally contained only particles, appears after the transformation, to contain both particles and antiparticles.\",\n \"2.\",\n \"Rescaling predicts that the EMC effect should vanish at $x \\\\approx 0.2$ where QCD evolution vanishes (see Fig.\",\n \"REF ).\",\n \"The EMC data cross unity at $x \\\\approx 0.35$ in disagreement with this prediction.\",\n \"Once again, however, the SLAC data look better: $\\\\overline{F}^{A}_2/\\\\overline{F}^{D}_2$ is definitely below unity for $x>0.3$ .\",\n \"A careful test must await better data at small-$x$ .\",\n \"3.\",\n \"Rescaling cannot work for $x \\\\rightarrow 1$ , or equivalently for $n \\\\rightarrow \\\\infty $ .\",\n \"At $x \\\\rightarrow 1$ , the structure function of the nucleon vanishes but that of a nucleus does not.\",\n \"Eq.\",\n \"(6.12) fails to reproduce this behavior.\",\n \"The reason for this failure will become clear soon.\",\n \"4.\",\n \"Rescaling does not mean that an iron nucleus can be “mapped\\\" to a nucleon by a universal change of scale.\",\n \"The complete description of a nucleus requires matrix elements of all twists.\",\n \"The gross structure of the EMC effect only involves a few operators of twist-two.\",\n \"In $§5.4$ we learned to associate a mass scale $\\\\mu ^2_0$ with the quark model description of a hadron.\",\n \"It is the scale at which, approximately, the hadron consists of valence quarks alone.\",\n \"It is an intrinsic characteristic of each hadron.\",\n \"The measured structure function of the nucleon is consistant with this notion with $\\\\mu ^2_0 \\\\cong 0.75$ GeV$^2$ .\",\n \"Clearly, any nucleus related to the nucleon by rescaling, as in Eq.\",\n \"(6.12), also admits a valence quark description, but at a shifted mass scale $\\\\mu _A$ : $\\\\mu ^2_A = \\\\frac{\\\\mu ^2_0}{\\\\xi _{NA}(\\\\mu ^2_0)} \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.13)}$ Since $\\\\xi _{NA}>1$ , $\\\\mu ^2_A$ is less than $\\\\mu ^2_0$ .\",\n \"This can be checked by the following argument: At any fixed $Q^2$ , the nucleus appears more highly evolved than the nucleon (more pairs, softer valence quarks).\",\n \"So to reabsorb all pairs it is necessary to devolve the nucleus further than the nucleon.\",\n \"It has become necessary to express Eq.\",\n \"(6.13) as a ratio of length scales.\",\n \"Let us define $\\\\lambda _A = c/\\\\mu _A$ , $\\\\lambda _N = c/\\\\mu _0$ , then $\\\\lambda _A/\\\\lambda _N = \\\\mu _0/\\\\mu _A = \\\\sqrt{\\\\xi _{NA}(\\\\mu ^2_0)} \\\\ \\\\qquad \\\\mathrm {(6.14)}$ $\\\\xi (\\\\mu ^2_0)$ is typically much smaller than $\\\\xi (Q^2)$ , because of Eq.\",\n \"(6.10).\",\n \"For example, $\\\\xi _{\\\\rm Fe\\\\,N}(20\\\\ {\\\\rm GeV^2}) \\\\cong 2.0$ , but $\\\\xi _{\\\\rm Fe\\\\,N}(\\\\mu ^2_0) \\\\cong 1.33$ making $\\\\lambda _{\\\\rm Fe}/\\\\lambda _N \\\\cong 1.15$ .\",\n \"Thus, an increase of only $\\\\sim $ 15% in the intrinsic length scale of the twist-two matrix elements for iron compared to the nucleon can give rise to the EMC effect.\",\n \"If there is anything fundamental about rescaling, rather than being merely an accident, the basic relation must be defined at the intrinsic to the target, $\\\\mu ^2_0$ , not at some arbitrary $Q^2$ [55].\",\n \"At $\\\\mu ^2_0$ , however, the structure function contains large contributions from higher twists, so an equation like Eq.\",\n \"(6.12) with $Q^2$ replaced by $\\\\mu ^2_0$ cannot be written down.\",\n \"Instead, we must write relations between matrix elements of twist-two operators using the formalism of $§5.2$ .\",\n \"We define $\\\\mathinner {\\\\langle {P|O_{n,a}^{(\\\\mu ^2)\\\\mu _1..\\\\mu _n}|P}\\\\rangle } = P^{\\\\mu _1}...P^{\\\\mu _n}\\\\Theta ^A_{n,a}(\\\\mu ^2) \\\\qquad \\\\mathrm {(6.15)}$ in analogy to Eq.\",\n \"(5.31).\",\n \"Then the operator equivalent of Eq.\",\n \"(6.12) is $A^{n-2} \\\\Theta ^A_{n,a}(\\\\mu ^2_A) = A^{\\\\prime \\\\,{n-2}} \\\\Theta ^{A^\\\\prime }_{n,a} (\\\\mu ^2_{A^\\\\prime }) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.16)}$ This is the basic rescaling between targets.\",\n \"The powers of $A$ and $A^\\\\prime $ are only kinematic.\",\n \"It leads to Eq.\",\n \"(6.12) at large $Q^2$ provided $\\\\xi _{AA^\\\\prime }(Q^2)$ is obtained from $\\\\mu _A/\\\\mu _A^\\\\prime $ via QCD evolution (Eq.\",\n \"(6.10) plus higher order improvements) and provided the moments can be reliably evolved from $\\\\mu _A^2$ to $Q^2$ using perturbative QCD.\",\n \"This turns out to be an important proviso.\",\n \"In [52], we found that only the moments with $n<8$ could be reliably devolved from large $Q^2$ to a $\\\\mu ^2_0$ as small as 1 GeV$^2$ .\",\n \"The problem is that higher order (in $\\\\alpha _c$ ) corrections typically contain $\\\\ln n$ factors making perturbation theory worse for large $n$ .\",\n \"Suppose, then, that Eq.\",\n \"(6.16) were exact for all $n$ .\",\n \"Nevertheless, at large $Q^2$ only the moments with $n<8$ would be likely to show uniform scaling.\",\n \"This explains why rescaling fails near $x=1$ (as noted earlier) since large $n$ moments probe exclusively large $x$ .\",\n \"In fact, one can estimate the $x$ values for which rescaling should be reliable when observed at large $Q^2$ by using the Mellin transform relation [58].\",\n \"$n \\\\sim \\\\ln \\\\alpha _s/ \\\\ln x \\\\qquad \\\\mathrm {(6.17)}$ which for $n=8$ and $\\\\alpha _s =0.2$ gives $x \\\\sim 0.8$ as the upper limit of reliability.\",\n \"Turning this argument around one can see that exact uniform rescaling for all $x$ at large $Q^2$ would be very hard to understand, since it would imply a complicated and non-uniform relation among twist-two matrix elements with $n>8$ at $\\\\mu ^2_0$ .\",\n \"The task for someone trying to understand the EMC effect from the point of rescaling, then, is two-fold.\",\n \"First, one must explain why rescaling should be uniform at the intrinsic scale, i.e.\",\n \"why should $\\\\lambda _A/\\\\lambda _{A^\\\\prime }$ be independent of $n$ ?\",\n \"Second, one must predict the $A$ dependence of the rescaling parameter, i.e.\",\n \"of $\\\\lambda _A/\\\\lambda _N$ .\",\n \"Close, Roberts, Ross and I have argued that rescaling is a rather natural prediction of quark models with only one length scale.\",\n \"An example is the MIT bag model with only $u$ and $d$ quarks, which may be taken to be massless, so the only dimensionful parameter is the bag constant $B$ .\",\n \"Such models are very close in spirit to QCD itself in which the only dimensionful parameter is $\\\\Lambda $ .\",\n \"In a model like this, quarks carrying momentum $p$ confined within a radius $\\\\lambda $ transform into quarks carrying momentum $p^\\\\prime =(\\\\lambda /\\\\lambda ^\\\\prime )p$ when the confinement scale is changed to $\\\\lambda ^\\\\prime $ - the dimensionless quantity $p \\\\lambda $ is constant.\",\n \"The intrinsic scale $\\\\mu ^2$ is then proportional to $p^2$ , there being no other scale in the problem.\",\n \"It is hard to turn this heuristic argument into a proof of uniform rescaling.\",\n \"That would probably require a consistent formulation of perturbative QCD (including renormalization) in a bag model, something which exists only in fragments [62].\",\n \"On the other hand, as Llewellyn Smith has noted, it seems clear that other models such as the non-relativistic quark model have no hope of giving rescaling unless the quark masses are assumed (rather unnaturally) to scale with the inverse confinement radius [10].\",\n \"A satisfactory theoretical understanding of rescaling will have to await a more powerful QCD-based theory of confinement.\",\n \"The second task $-$ determining the $A$ dependence of $\\\\lambda _A/\\\\lambda _N$ $-$ is more straightforward.\",\n \"It is the subject of the next Section.\",\n \"It hardly needs saying that the EMC effect derives from the proximity of nucleons within the nucleus, and that it would vanish if one could arrange that that the nucleus was very dilute.\",\n \"Fortunately, nature has given us one very dilute nucleus $-$ the deuteron $-$ and Bodek and Simon [15] have shown that $\\\\overline{F}^D_2(x,Q^2)/F^N_2(x,Q^2)$ is very close to unity.\",\n \"It seems reasonable to assume, therefore, that the magnitude of the EMC effect should be proportional to the probability that nucleons approach each other or overlap within the nucleus.\",\n \"Close, Roberts, Ross and I [57], [58] defined the simplest measure of this effect we could imagine: we defined an “overlapping\\\" volume for a nucleus which is the integral over the nucleus of the two body density $\\\\rho _A({\\\\bf r_1},{\\\\bf r_2})$ multiplied by an overlap factor $V_0(|{\\\\bf r_1-r_2}|)$ which we took to be the overlapping volume of two spheres of radius $a$ : $V_0(d) &= 1-\\\\frac{3}{4} \\\\left(\\\\frac{d}{a} \\\\right) + \\\\frac{1}{16}\\\\left(\\\\frac{d}{a} \\\\right)^3 {\\\\ } &d \\\\le 2a \\\\nonumber \\\\\\\\&= 0 {\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ } &d>2a \\\\ .\",\n \"$ Thus, the overlapping volume per nucleon is $V_A =(A-1) \\\\int d^3{\\\\bf r_1} d^3 {\\\\bf r_2} \\\\rho _A({\\\\bf r_1},{\\\\bf r_2}) V_0(|{\\\\bf r_1-r_2}|) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.19)}$ $\\\\rho _A({\\\\bf r_1},{\\\\bf r_2})$ is normalized to $\\\\int d^3{\\\\bf r_1} d^3 {\\\\bf r_2} \\\\rho _A({\\\\bf r_1},{\\\\bf r_2})=1$ .\",\n \"Saturation of the nuclear density at large $A$ implies $\\\\rho _A \\\\sim 1/A^2$ .\",\n \"With this behavior of $\\\\rho _A$ and the finite integral of $V_0$ it is easy to see that $V_A$ saturates at large $A$ , i.e.\",\n \"$\\\\lim _{A \\\\rightarrow \\\\infty } V_A =$ constant.\",\n \"The choice of a geometrical form for $V_0(|{\\\\bf r_1-r_2}|)$ was in fact quite arbitrary.\",\n \"Any function which goes to unity as ${\\\\bf r_1-r_2} \\\\rightarrow 0$ and to zero when $|{\\\\bf r_1-r_2}|>2R_{\\\\rm nucleon}$ and which respects the three dimensional geometry of the problem would do.\",\n \"We calculated the overlapping volume for nucleons with $a$ chosen so $a_{\\\\rm RMS} = 0.9$ fm ($a_{\\\\rm RMS}=\\\\sqrt{\\\\frac{3}{5}}a$ , so $a=1.16$ fm).\",\n \"$\\\\rho _A({\\\\bf r_1},{\\\\bf r_2})$ was written in terms of the single particle density $\\\\rho _A({\\\\bf r})$ and a correlation function $f({\\\\bf r_1-r_2})$ : $\\\\rho _A({\\\\bf r_1-r_2}) = \\\\rho _A({\\\\bf r_1}) \\\\rho _A({\\\\bf r_2}) f({\\\\bf r_1-r_2}) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.20)}$ $\\\\rho _A({\\\\bf r})$ was taken from experimental measurements of nuclear charge densities.\",\n \"$f({\\\\bf r_1-r_2})$ should, in principle, depend on $A$ but there is little or no direct information on it from experiment.\",\n \"We took $f({\\\\bf r_1-r_2})$ from models of nuclear matter, the most realistic probably being one based on a Reid soft core potential [60] shown in Fig.\",\n \"REF .\",\n \"Figure: Choice of the correlation function f(r)f(r): (a) no correlation; (b) Fermi gas correlation; (c) Reid soft core correlation.We then assumed that the effective confinement size in a nucleus $A$ will be intermediate between that for an isolated nucleon and some limiting value $\\\\lambda _{\\\\rm tot}$ associated with two totally overlapping nucleons.\",\n \"We assumed a linear interpolation in $V_A$ leading to $\\\\frac{\\\\lambda _A}{\\\\lambda _N} = 1 + V_A \\\\left( \\\\frac{\\\\lambda _{\\\\rm tot}}{\\\\lambda _N}-1 \\\\right) \\\\ .\",\n \"\\\\qquad \\\\mathrm {(6.21)}$ $\\\\lambda _{\\\\rm tot}$ might be viewed as a parameter.\",\n \"Instead, we estimated its value from the bag model, where a spherical baryon number two system must have volume at least twice as large as a single nucleon (otherwise it would be stable against decay into two nucleons), so $R_6 \\\\ge 2^{1/3}R_3$ .\",\n \"This led us to take $\\\\lambda _{\\\\rm tot}/\\\\lambda _N = 2^{1/3}$ , and to the values of $\\\\lambda _A/\\\\lambda _N$ given in Table REF .\",\n \"Table: Values of the confinement size relative to that for the free nucleon for a range of nuclei.\",\n \"The three values (a), (b) and (c) correspond to three choices of the correlation function f(r)f(r) shown in Fig. .\",\n \"ξ NA \\\\xi _{NA} (20 GeV 2 ^2) is shown for the Reid correlation function.\",\n \"The others are similar.These translate into values of $\\\\xi _{NA}$ (20 GeV$^2$ ) which are much larger, and these in turn give predictions for the EMC effect in a variety of nuclei.\",\n \"The predictions of the rescaling model are compared with the SLAC data in Fig.\",\n \"11.\",\n \"The agreement is fine for $x < 0.7$ above which Fermi motion becomes important.\",\n \"One thing which is not obvious from Fig.\",\n \"11 is that the data correlate well with idiosyncracies in the periodic table.\",\n \"Fig.\",\n \"REF shows predictions for several $x$ values and $Q^2$ = 4.98 GeV$^2$ compared with SLAC data.\",\n \"Figure: AA dependence of the EMC effect at fixed xx.\",\n \"The data are squared with error bars.\",\n \"The predictions of rescaling are solid dots .The fluctuations in the rescaling predictions reflect variations in nuclear densities, e.g.\",\n \"$^4$ He is more tightly bound than $^3$ He, and generally follows the data.\",\n \"Some predictions for the future are shown in Fig.\",\n \"REF .\",\n \"Figure: AA dependence at x=0.6x=0.6, Q 2 ≈10Q^2 \\\\approx 10 GeV 2 ^2 predicted by rescaling.Several comments are appropriate before leaving the discussion of $A$ dependence: 1.\",\n \"Llewellyn Smith has argued that the calculation of the $A$ dependence we have given is much more general than the rescaling model.\",\n \"Any scheme which fits the EMC effect in iron and in which the effect is linear in $V_A$ will agree as well with the SLAC data.\",\n \"This is qualitatively true but the assumption of linearity is non-trivial.\",\n \"We argued that $\\\\lambda _A/\\\\lambda _N$ should be linear in the overlapping volume.\",\n \"Had we instead (erroneously) assumed $\\\\xi _{NA}(Q^2=20\\\\ GeV^2)$ to be linear in $V_A$ the $A$ dependence would have come out wrong.\",\n \"2.\",\n \"The rescaling model is not related to convolution models nor does it ascribe the EMC effect to any particular exotic component in the nuclear wavefunction (e.g., six quark bags).\",\n \"The scale change might originate from a change in the size of individual nucleons, from quark percolation between nucleons, or from multiquark or meson admixtures in the nuclear wavefunction.\",\n \"Personally, I suspect that to the extent that these notions can be well-defined they will turn out to correspond to the same underlying physics — partial deconfinement.\",\n \"3.\",\n \"The rescaling fit to the $A$ -dependence depends on several parameters: the nucleon radius $a$ , the formula chosen to relate $V_A$ to $\\\\lambda _A/\\\\lambda _N$ , the scale $\\\\mu _0$ and the choice of two nucleon correlation function.\",\n \"In fact, none of these were treated as free parameters in [58].\",\n \"Instead, they were fixed at “reasonable\\\" values from other aspects of hadron dynamics.\",\n \"Of course, the precise choice of parameters is not particularly central to the explanation of the EMC effect.\",\n \"4.\",\n \"The value of $V_A$ is very large for large $A$ .\",\n \"It is $\\\\approx $ 0.72 for $^{208}$ Pb with the Reid correlation function.\",\n \"It has been remarked that such a large value of $V_A$ is “unreasonable\\\" and in some sense contradicted by the many successes of conventional nuclear physics.\",\n \"The overlapping volume of nucleons with $a_{\\\\rm rms}$ = 0.9 fm is very large.\",\n \"That is a fact, not a shortcoming of the rescaling model.\",\n \"As for the idea that large $V_A$ is inconsistent with nuclear physics in general - I believe it is an unwarranted concern: It costs $\\\\approx $ 1 GeV (the string tension) to separate colored sources by l fm.\",\n \"It even costs 300 MeV to flip a quark spin.\",\n \"These energetic considerations, not naive classical, geometrical considerations determine the quantum mechanics of nuclei.\",\n \"It has long been expected that at very low values of $x$ deep inelastic electron scattering from nuclei would behave like hadron scattering from nuclei and exhibit “shadowing\\\" [61].\",\n \"At high energies hadron nucleus total cross sections grow like $A^{2/3}$ .\",\n \"The simple explanation of this effect is that the incoming hadron doesn't “see\\\" the nucleons at the back of the nucleus.\",\n \"They are in the shadow of those in front.\",\n \"The cross section then grows like $\\\\pi R^2 \\\\sim A^{2/3}$ .\",\n \"Real photon-nucleus interactions are shadowed [62].\",\n \"This finds a simple explanation in the framework of vector meson dominance: the hadronic interactions of a real photon are well-approximated by supposing the photon converts with probability $\\\\sim \\\\!\\\\!\\\\alpha $ into a vector meson ($\\\\rho $ , $\\\\omega $ or $\\\\phi $ ) which then interacts hadronically and experiences shadowing.\",\n \"Vector meson dominance fails to explain inelastic lepton scattering at large $Q^2$ , or at least an infinite tower of vector states with precisely tuned couplings is required, but the prejudice that shadowing occurs there too is strong.\",\n \"Figure: Cartoons representing the ξ 3 \\\\xi ^3 values probed by deep inelastic lepton scattering at low-xx from an iron target.My own thoughts on shadowing are still in flux.\",\n \"Since much of the future experimental work in this field will be carried out in the kinematic regime where shadowing might be important, I would like to outline the problem here.\",\n \"Perhaps some reader will solve it!\",\n \"The kinematic relation $\\\\xi ^3<1/Mx \\\\qquad \\\\mathrm {(6.22)}$ tells us that for very small values of $x$ the struck quark may propagate over very large distances in the target.\",\n \"For $ x \\\\approx 0.5$ , $\\\\xi ^3 \\\\le 0.4$ fm, but for $x \\\\approx 0.05$ , $\\\\xi ^3 \\\\le 4$ fm and for $x \\\\approx 0.005$ , $\\\\xi ^3 \\\\le 40$ fm!\",\n \"The parton model diagram dominant at large $Q^2$ is shown relative to an iron nucleus in Fig.\",\n \"REF where the struck quark is associated with a single nucleon.\",\n \"Other possibilities represent EMC-like corrections.\",\n \"The question is whether a quark propagating over such distances is or is not strongly absorbed in the nuclear medium.\",\n \"Hadron interactions at very high energies are predominantly absorptive, but the state which is propagating in Fig.\",\n \"REF is not an ordinary hadron.\",\n \"In particular, the propagating quark, which is a color triplet is always close to an antiquark [63] which is a color antitriplet that may neutralize some or most of its strong interactions.\",\n \"Furthermore, the invariant mass of the $q \\\\bar{q}$ pair is $-Q^2$ .\",\n \"Suppose for the sake of discussion, we characterize quark propagation in the nuclear medium (accompanied by an antiquark as in Fig.\",\n \"REF ) by an absorption length $\\\\lambda $ which may depend on $Q^2$ .\",\n \"For a given nucleus when $1/Mx< \\\\lambda (Q^2)$ there is little shadowing.\",\n \"Shadowing sets in when $1/Mx \\\\sim \\\\lambda (Q^2)$ , but when $1/Mx$ exceeds twice the radius of the nucleus ($R_A$ ) shadowing saturates: decreasing $x$ further does not put more matter along the propagating quark's path.\",\n \"So there are two $x$ -values characterizing shadowing, $x_0(Q^2) = 1/M \\\\lambda (Q^2)$ marking its onset and $x(A) = 1/2MR_A$ marking its saturation.\",\n \"For $x < x(A)$ , only the front skin of the nucleus down to a depth $\\\\lambda (Q^2)$ participates in the scattering, so shadowing increases with $A$ but goes away as $\\\\lambda (Q^2) \\\\rightarrow \\\\infty $ .\",\n \"All of this can be summarized by the purely phenomenological formula $\\\\overline{F}^A_2(x,Q^2) = \\\\frac{F^N_2(x,Q^2)}{\\\\left[ 1 + \\\\frac{x_0(Q^2)}{x+x(A)} \\\\right] } \\\\qquad \\\\mathrm {(6.23)}$ in which for simplicity I have ignored all other nuclear effects such as rescaling and Fermi motion.\",\n \"Eq.\",\n \"(6.23) does not satisfy the quark number sum rule.\",\n \"It is not intended to be valid at all $x$ , only for $x \\\\sim 0$ where shadowing may be important.\",\n \"According to Eq.\",\n \"(6.23) there is an asymptotic shadowing curve for nuclear matter ($A \\\\rightarrow \\\\infty $ , $x(A) \\\\rightarrow 0$ ), shown for example in Fig.\",\n \"REF .\",\n \"Finite nuclei track along $\\\\overline{F}^\\\\infty _2(x,Q^2)$ until $x \\\\sim z(A)$ below which they depart above $\\\\overline{F}^\\\\infty _2$ .\",\n \"A determination of $\\\\lambda (Q^2)$ is crucial [64].\",\n \"Unfortunately, it is largely unknown.\",\n \"Naively, one might expect $\\\\lambda (Q^2)$ to be a typical meson mean free path, corresponding to a cross section of order 30 mb.\",\n \"This is too naive.\",\n \"The propagating $q \\\\bar{q}$ pair in Fig.\",\n \"REF must have invariant mass $-Q^2$ .\",\n \"This can be generated if both quark and antiquark have transverse momentum of order $\\\\frac{1}{2}\\\\sqrt{Q^2}$ or by a large mismatch in their longitudinal momentum.\",\n \"In the former case, the transverse dimension of the $q \\\\bar{q}$ system is very small, $\\\\sim 1/\\\\sqrt{Q^2}$ .\",\n \"The system looks like a small color dipole and has a small absorption cross section.\",\n \"In the latter case, the transverse dimensions are not small, color screening is not so effective and the absorption cross section may be large.\",\n \"The importance of shadowing depends on which region of phase space dominates.\",\n \"This is not yet known.\",\n \"But at least some of the shadowing seen at $Q^2 =0$ should disappear at large $Q^2$ .\",\n \"The behavior of $\\\\overline{F}^A_2(x,Q^2)$ at small $x$ expected if shadowing indeed disappears at large $Q^2(x(Q^2) \\\\rightarrow 0$ ) as shown in Fig.\",\n \"REF .\",\n \"Figure: Scenarios for shadowing.\",\n \"(a) and (b) compared to two different values of λ(Q 2 )\\\\lambda (Q^2) with λ a >λ b \\\\lambda _a > \\\\lambda _b.The discussion in this section has been very qualitative and phenomenological.\",\n \"Eq.\",\n \"(6.23) should be taken with a grain of salt.\",\n \"There is much to do on the subject of shadowing and little of substance to report here.\",\n \"Nevertheless, I thought it might be appropriate to end by whetting the reader's appetite for the next round of experiments.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05616"}}},{"rowIdx":83,"cells":{"text":{"kind":"list like","value":[["Mediation analysis with the mediator and outcome missing not at random"],["Abstract Mediation analysis is widely used for investigating direct and indirect causal pathways through which an effect arises.","However, many mediation analysis studies are often challenged by missingness in the mediator and outcome.","In general, when the mediator and outcome are missing not at random, the direct and indirect effects are not identifiable without further assumptions.","In this work, we study the identifiability of the direct and indirect effects under some interpretable missing not at random mechanisms.","We evaluate the performance of statistical inference under those assumptions through simulation studies and illustrate the proposed methods via the National Job Corps Study."],["Introduction to mediation analysis and the National Job Corps Study","Mediation analysis is increasingly adopted by researchers in a variety of fields, including epidemiology and social sciences, to test specific theories about the underlying mechanism through which an effect arises.","In a typical mediation analysis, the average treatment effect on an outcome is decomposed into a natural indirect effect (NIE) that operates through the mediator of interest and a natural direct effect (NDE) that operates through other pathways.","The NIE and NDE are identified under the sequential ignorability assumption [14], [15], that is, when there is no unmeasured pretreatment confounding in the treatment-mediator and treatment-outcome relationships and there is no post-treatment confounding or unmeasured pretreatment confounding in the mediator-outcome relationship.","With the identification results, the NIE and NDE can be estimated through various approaches, such as regression [32], weighting [11], [13] and simulation-based strategy [14], [15].","However, missing data are prevalent in empirical research and it is often a concern that the missingness is not at random [20].","Missing not at random occurs if the probability of data being missing depends on unobserved data even conditional on observed data.","The motivation of our study comes from the well-known National Job Corps Study (NJCS).","The NJCS is a randomized nation wide evaluation of Job Corps, which is the largest education and vocational training program administered by the U.S. Department of Labor for 16 to 24 years old youths who are unemployed and disconnected from school.","Past research showed that Job Corps successfully improves those disadvantaged youths' employment and increases their earnings [31], [17], [37].","Given that education and vocational training are the central elements of the program, [27] further proposed to evaluate how much of the Job Corps effect on earnings is mediated by educational and vocational attainment and found a significant positive mediated effect.","However, the analysis is challenged by missingness both in the mediator and the outcome.","In the NJCS study, the assignment to either Job Corps program or the control group was random.","The mediator describes whether or not the subject obtained an education credential or vocational certificate after randomization, and the outcome describes subject's weekly earnings in the fourth year after randomization.","The mediator and outcome information were collected at the 30-months and 48-months follow-up, respectively, through in-person interviews.","And the missing rates of the mediator and the outcome are both higher than $20\\%$ .","Qin et al.","(2019) addressed the missing data problem through non-response weighting, which is a valid approach if the missingness is random given the observed pretreatment covariates.","However, we are concerned that the missingness is likely not at random.","Conceivably, people who failed to obtain an education credential or vocational certificate may be less likely to provide the information compared to people who successfully obtained a credential or certificate.","One could also imagine that people who had no earnings may be less willing to report the amount of earnings in the interview.","As a result, the chance of missingness would depend on the unobserved missing value itself, missing not at random.","In such a scenario, commonly adopted strategies to deal with missing data, such as complete case analysis, multiple imputation or non-response weighting, may fail to provide valid inference.","Missingness not at random presents a challenge for causal inference, because in general, the underlying data distribution can not be identified without further assumptions.","This topic has attracted some attention in the literature.","For example, [7] and [36] studied the identifiability of subgroup treatment effects and average treatment effects, respectively, with covariates missing not at random.","In the context of instrumental variable analysis, methods were developed when the missingness in the covariates [35] or in the outcome [9], [26], [4] is not at random.","Previous research also studied the identifiability of graphical models with a binary outcome missing not at random in longitudinal studies [21], and reviewed missing data research in graphical models [22].","However, limited effort has been made to study the identifiability of causal mediation effects with the mediator and outcome missing not at random.","Considering missingness in the outcome only, [19] utilized an instrumental variable type of covariate to identify the direct and indirect effects when the missingness in the outcome depends on the outcome value itself.","To apply their method, we need a covariate that is associated with the outcome, however, is conditionally independent of the missingness of the outcome.","In many studies, such a covariate may not be available."],["Organization of the paper","In this paper, we study the identification of the causal mediation effects with mediator and outcome missing not at random.","We provide conditions for identification under various interpretable missing not at random assumptions, and also provide examples for some missingness mechanisms where the identification cannot be achieved without further assumptions.","The rest of the paper is organized as follows.","We introduce notation and basic assumptions for causal mediation analysis in section .","In section , we discuss the identification in a simple setup where the missingness exists only in the mediator and depends on the missing mediator value itself.","We then extend the results to the more complicated setup where both the mediator and the outcome have missing data in section .","Expectation-maximization algorithms are used for estimation and in section , extensive simulation studies are conducted to test our theoretical results and to evaluate the performance of the proposed methods.","We apply our method to the NJCS data in section and develop a sensitivity analysis approach to study the robustness of conclusions on the causal mediation effects if the underlying missingness mechanism is beyond the mechanisms that allow identification.","We conclude with a discussion and provide all proofs of the theorems in the Supplementary Material."],["Notation and some basic definitions","Let $A\\perp \\!\\!\\!\\perp B\\mid C$ denote that the random variables $A$ and $B$ are conditionally independent given the random variable $C$ .","Further, the property of completeness [18], [2] will play a key role in our nonparametric identification of the mediation effects.","Define a function $f(A,B)$ to be complete in $B$ if $\\int g(A)f(A,B)d\\nu (A)=0$ implies $g(A)=0$ almost surely for any square-integrable function $g$ .","In the above integral, $\\nu (\\cdot )$ presents a generic measure, which is the Lebesgue measure for a continuous variable and the counting measure for a discrete variable."],["Review of mediation analysis without missing data","Let $T$ denote the binary treatment assignment, with $t=0$ and $t=1$ representing the control condition and the experimental condition, respectively.","Let $ X $ be the vector of measured pre-treatment covariates.","We use $M$ and $Y$ to denote the mediator and the outcome, respectively.","We adopt the potential outcomes framework [23], [29] to define the causal effects of interest and make the stable unit treatment value assumption (SUTVA) that there is no hidden variations of each treatment condition and there is no interference between individuals.","Specifically, we use $M_i(t)$ to denote individual $i$ 's potential mediator value under treatment $t$ for $t=0, 1$ , and use $Y_i(t)$ to denote individual $i$ 's potential outcome value under treatment $t$ for $t=0, 1$ .","$T$ potentially may affect $Y$ through $M$ , hence $Y_i(t)$ can be equivalently written as $Y_i(t,M_i(t))$ .","Given the above notation, the average treatment effect ($\\mathrm {ATE}$ ) on the outcome can be defined as $\\mathbb {E}\\lbrace Y(1)-Y(0)\\rbrace $ where the expectation is taken over the population of interest.","Define the nested potential outcome, $Y_i(1,M_i(0))$ , to describe individual $i$ 's potential outcome under the experimental condition, however, with the mediator counter factually taking its value under the control condition [28].","The $\\mathrm {ATE}$ can be decomposed into [25]: $\\mathrm {ATE} = \\mathrm {NIE}+\\mathrm {NDE},$ where $\\mathrm {NIE} = \\mathbb {E}\\lbrace Y(1,M(1))-Y(1,M(0))\\rbrace $ is the natural indirect effect ($\\mathrm {NIE}$ ) and $\\mathrm {NDE} = \\mathbb {E}\\lbrace Y(1,M(0))-Y(0,M(0))\\rbrace $ is the natural direct effect ($\\mathrm {NDE}$ ).","The $\\mathrm {NIE}$ quantifies the average treatment effect on the outcome transmitted through the treatment induced change in the mediator from $M_i(0)$ to $M_i(1)$ ; and the $\\mathrm {NDE}$ quantifies the direct effect of the treatment on the outcome that does not operate through its impact on the mediator.","We invoke the standard sequential ignorability assumption [14], [15] throughout the paper.","Assumption 1 (Sequential ignorability).","$\\lbrace Y(t^{\\prime }, m), M(t)\\rbrace \\perp \\!\\!\\!\\perp T \\mid X = x $ , and $Y(t^{\\prime },m) \\perp \\!\\!\\!\\perp M(t)\\mid T=t, X = x $ , for $t, t^{\\prime } \\in \\lbrace 0, 1\\rbrace $ and all $ X \\in \\mathcal {X}$ .","In addition, $0 < \\mathbb {P}(T=t\\mid X = x )<1$ and $0 < \\mathbb {P}\\lbrace M(t)=m \\mid T=t, X = x \\rbrace <1$ for $t, t^{\\prime } \\in \\lbrace 0, 1\\rbrace $ , and all $ X \\in \\mathcal {X}$ and $m\\in \\mathcal {M}$ .","The notation $\\mathcal {X}$ and $\\mathcal {M}$ denote the supports of the random variables $X$ and $M$ , respectively.","Assumption REF says that both the treatment assignment and the mediator value under each treatment condition can be viewed as if randomized within levels of the observed pre-treatment covariates.","Under Assumption REF without missing data, we have the following mediation formula [25], [14]: $\\mathbb {E}\\lbrace Y(t, M(t^{\\prime }))\\mid X = x \\rbrace = \\int _{\\mathcal {M}}\\mathbb {E}(Y\\mid M=m, T=t, X = x )\\,d F(m\\mid T=t^{\\prime }, X = x ).$ When there exists missing data, the key for the identification of the $\\mathrm {NIE}$ and $\\mathrm {NDE}$ would be to identify the probabilities $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x )$ and $\\mathbb {P}(M=m\\mid T=t, X = x )$ , or equivalently, the joint probability $\\mathbb {P}(Y=y, M=m\\mid T=t, X = x )$ , from the observable data."],["Missingness only in the mediator","In this section, we consider a simple setup where the mediator has missing values and the outcome is fully observed.","It may happen in studies where the outcome is of primary interest with the mediator being a secondary outcome of interest.","Let $R^M$ be the missingness indicator for $M$ such that $R^M=1$ if $M$ is observed and $R^M=0$ if $M$ is missing.","When $R^M \\perp \\!\\!\\!\\perp (Y,M,T, X )$ , the missingness mechanism is missing completely at random (MCAR), and the complete case analysis is enough to provide consistent estimates of $\\mathbb {P}(Y,M \\mid T, X )$ .","When $R^M \\perp \\!\\!\\!\\perp M \\mid (Y,T, X )$ , the missingness mechanism is missing at random (MAR), and the joint distribution $\\mathbb {P}(Y,M \\mid T, X )$ can be consistently estimated from the observed data.","However, as we explained in section , often we have the concern that the missingness of $M$ may depend on the value of $M$ itself even conditional on other observed data.","In such a case, the missingness mechanism is missing not at random (MNAR).","Since the outcome $Y$ occurs after the mediator $M$ , it is plausible in many studies to assume that the missingness of $M$ is conditionally independent of $Y$ .","Based on the above discussion, we propose the following MNAR assumption when missing only the mediator: Assumption 2 $R^M \\perp \\!\\!\\!\\perp Y\\mid (M,T, X )$ and $Y$ is fully observed.","Assumption REF allows $R^M$ to depend on the value of the mediator $M$ , $T$ and $ X $ .","However, we assume $R^M$ to be conditionally independent of the outcome $Y$ given $M$ , $T$ and $ X $ .","The direct acyclic graphs in Figure REF illustrate the different missingness mechanisms under MCAR, MAR and our proposed MNAR mechanism in Assumption REF , respectively.","Figure: Direct acyclic graphs describing MCAR, MAR and the proposed MNAR missingness mechanisms when the missingness exists only in the mediator.","All graphs condition on X X and allow X X to have arrows to all other variables in the graphs.The following theorem presents the nonparametric identification results under the proposed MNAR assumption: Theorem 1 Under Assumptions REF and REF , if the joint distribution $\\mathbb {P}(Y,M,R^M=1 \\mid T=t, X=x )$ is complete in $Y$ for all $t$ and $x$ , and $\\mathbb {P}(R^M=1\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m, t, x $ , the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ is identifiable, and therefore, the $\\mathrm {NIE}$ and $\\mathrm {NDE}$ are identifiable.","For sufficient conditions on completeness, we refer readers to [8] for a comprehensive discussion.","The assumption of completeness is routinely made in nonparametric identification problems [5], [36], [1].","For discrete $M$ and discrete $Y$ , completeness is equivalent to Rank $(\\Theta _{tx})=J$ , where $\\Theta _{tx}$ is a $J \\times K$ matrix with $\\mathbb {P}_{my1\\mid t, x}$ as the $(m,y)$ th element.","This essentially requires that the number of elements in the support of $Y$ is not smaller than the number of elements in the support of $M$ , and that $M \\lnot \\!\\perp \\!\\!\\!\\perp Y\\mid (T, X , R^M=1)$ or equivalently $M \\lnot \\!\\perp \\!\\!\\!\\perp Y\\mid (T, X )$ .","In fact, under Assumption REF , the identification of $\\mathbb {P}(Y\\mid M, T, X )$ does not rely on the completeness assumption.","The completeness assumption is only needed to identify $\\mathbb {P}(M\\mid T, X )$ .","This implies that in a special case when $M \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ and the completeness assumption is violated, since $\\mathbb {P}(Y\\mid M, T, X ) = \\mathbb {P}(Y\\mid T, X )$ , both the $\\mathrm {NIE}$ and $\\mathrm {NDE}$ can still be identified, and $\\mathrm {NIE} = 0$ and $\\mathrm {NDE} = \\mathrm {ATE} = \\int _{\\mathcal {X}}\\lbrace \\mathbb {E}(Y\\mid T=1, X = x )-\\mathbb {E}(Y\\mid T=0, X = x )\\rbrace \\,d F( x ).$ We also illustrate this point using simulation studies."],["Missing data mechanisms","We now extend the results to the scenario where both the mediator and outcome have missing data.","Further define the notation $R^Y$ to be the missingness indicator for $Y$ such that $R^Y=1$ if $Y$ is observed and $R^Y=0$ otherwise.","When $(R^Y, R^M) \\perp \\!\\!\\!\\perp (Y,M,T, X )$ , the missingness mechanism is MCAR; and when $(R^Y,R^M) \\perp \\!\\!\\!\\perp (Y,M)\\mid (T, X )$ , the missingness mechanism is MAR.","Continuing to allow the missingness of $M$ to depend on the value of $M$ itself and assume the outcomes occurred later (i.e., $Y$ and $R^Y$ ) not to affect variables measured earlier (i.e., $M$ and $R^M$ ), we consider the following MNAR mechanisms in Assumption REF , REF , and REF , respectively.","Assumption 3 $(R^Y, R^M) \\perp \\!\\!\\!\\perp Y\\mid (M,T, X )$ and $R^Y \\perp \\!\\!\\!\\perp M\\mid (R^M,T, X )$ .","Assumption 4 $R^Y \\perp \\!\\!\\!\\perp (M,R^M)\\mid (Y,T, X )$ and $R^M \\perp \\!\\!\\!\\perp (Y,R^Y)\\mid (M,T, X )$ .","Assumption 5 $Y, R^Y$ and $ R^M$ are mutually independent given $(M,T, X )$ .","The direct acyclic graphs in Figure REF illustrate different missingness mechanisms under MCAR, MAR, and MNAR Assumptions REF to REF when the missingness exists both in the mediator and outcome.","The differences among the MNAR mechanisms under Assumption REF to REF are in the missingness mechanisms in $Y$ .","The MNAR mechanism under Assumption REF allows the chance of missingness in $M$ to depend on the value of $M$ itself in addition to the fully observed variables $T$ and $ X $ , and allows the missingness in $Y$ to depend on the missingness in $M$ in addition to $T$ and $ X $ .","It is important to note that the MAR mechanism is a special case of the MNAR mechanism under Assumption REF without allowing $M$ to affect $R^M$ .","In the context of NJCS study, this mechanism says that whether or not people successfully obtained a certificate may have an impact on their willingness to report, and their decision on reporting their certificate status may be associated with their decision to report their earnings or not.","The MNAR mechanism under Assumption REF allows the missingness of $Y$ to depend on the outcome value $Y$ itself instead of $R^M$ as in Assumption REF .","In the NJCS study, it says that the amount of earnings may have an impact on the probability to report earnings, which is also a reasonable concern.","Note that both the missingness in $M$ and the missingness in $Y$ are not at random under Assumption REF .","The MNAR mechanism under Assumption REF allows the missingness in $Y$ to depend on $M$ instead of $R^M$ or $Y$ , another case where both the missingness in $M$ and that in $Y$ are not at random.","In the context of NJCS study, it says that whether or not people successfully obtained a certificate drives the missingness in both $M$ and $Y$ after conditioning on $T$ and $ X $ .","Figure: Direct acyclic graphs describing MCAR, MAR and the proposed MNAR missingness mechanisms when the missingness exists both in the mediator and outcome (all graphs condition on X X allowing X X to have directed arrows to all variables in the graphs)."],["Identification results","We now present the nonparametric identification results under each of the MNAR mechanisms described in Assumptions REF to REF .","Theorem 2 Under Assumptions REF and REF , if the joint distribution $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X =x)$ is complete in $Y$ for all $t$ and $x$ , and $\\mathbb {P}(R^M=1,R^Y=1\\mid Y=y,M=m,T=t, X = x ) > 0$ and $\\mathbb {P}(R^M=0,R^Y=1\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m, t, x $ , the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ is identifiable, and therefore, the NIE and NDE are identifiable.","Under the MNAR mechanism described by Assumption REF , $Y\\perp \\!\\!\\!\\perp (R^M, R^Y)\\mid (M, T, X) $ , and therefore, the identification of $\\mathbb {P}(Y\\mid M, T, X )$ does not rely on the completeness assumption and can be identified by the complete cases, i.e., $\\mathbb {P}(Y\\mid M, T, X ) = \\mathbb {P}(Y\\mid M, T, X , R^Y=1, R^M=1)$ .","The completeness assumption is used only to identify $\\mathbb {P}(M\\mid T, X )$ .","As a result, when $M\\perp \\!\\!\\!\\perp Y\\mid (T, X)$ , the NIE and NDE can still be identified as discussed in Section .","Theorem 3 Under Assumptions REF and REF , if the joint distribution $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X=x )$ is complete in $Y$ for all $t$ and $x$ , and the joint distribution $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X=x )$ is also complete in $M$ for all $t$ and $x$ , and if $\\mathbb {P}(R^M=1,R^Y=1\\mid Y=y,M=m,T=t, X = x ) > 0$ , $\\mathbb {P}(R^M=0,R^Y=1\\mid Y=y,M=m,T=t, X = x ) > 0$ and $\\mathbb {P}(R^M=1,R^Y=0\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m,t, x $ , the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ is identifiable, and therefore, the NIE and NDE are identifiable.","Different from Assumption REF , under the MNAR mechanism described by Assumption REF , $Y$ is not independent of $R^Y$ given $M, T$ and $ X $ , and therefore, the conditional distribution $\\mathbb {P}(Y\\mid M, T, X )$ can no longer be identified by the complete cases.","In fact, the MNAR mechanism under Assumption REF requires completeness in $Y$ and completeness in $M$ to be able to identify both $\\mathbb {P}(Y\\mid M, T, X )$ and $\\mathbb {P}(M\\mid T, X )$ .","As a result, when $M\\perp \\!\\!\\!\\perp Y\\mid (T, X) $ , the NIE and NDE can no longer be identified.","Further, since the completeness is required in $Y$ and required in $M$ , to achieve identification of $\\mathbb {P}(Y,M\\mid T, X )$ under Assumption REF , the number of elements in the support of $Y$ has to be the same as the number of elements in the support of $M$ .","We present an unidentifiable case when $Y$ has more categories than $M$ in the section of the Supplementary Material.","Theorem 4 Under Assumptions REF and REF , if either of the following two conditions holds, the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ is identifiable, and therefore, the NIE and NDE are identifiable: (i) the joint distribution $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X=x )$ is complete in $Y$ for all $t$ and $x$ , and $\\mathbb {P}(R^M=1,R^Y=1\\mid Y=y,M=m,T=t, X = x ) > 0$ , $\\mathbb {P}(R^M=0,R^Y=1\\mid Y=y,M=m,T=t, X = x ) > 0$ and $\\mathbb {P}(R^M=1,R^Y=0\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m,t, x $ ; (ii) the joint distribution $\\mathbb {P}(M,R^M=1,R^Y \\mid T=t, X=x )$ is complete in $R^Y$ for all $t$ and $x$ , and $\\mathbb {P}(R^M=1\\mid R^Y=r^Y, M=m,T=t, X = x )>0$ for all $r^Y, m, t, x $ .","Under the MNAR mechanism described by Assumption REF , $\\mathbb {P}(Y\\mid M, T, X )$ can also be directly identified by the complete cases, i.e., $\\mathbb {P}(Y\\mid M, T, X ) = \\mathbb {P}(Y\\mid M, T, X , R^Y=1, R^M=1)$ .","Different from the identification under MNAR Assumption REF or 4, there are two ways to identify $\\mathbb {P}(M\\mid T, X )$ , one is by utilizing the effect of $M$ on $Y$ if exists, i.e., the condition (i), the other one is by utilizing the effect of $M$ on $R^Y$ if exists, i.e., the condition (ii).","Note that for the second condition (ii) to hold, $M$ needs to be binary given that $R^Y$ is binary.","Assuming $Y \\perp \\!\\!\\!\\perp R^M\\mid (M,T, X )$ , and allowing $M$ to have a impact on $R^M$ , we have shown that the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ can be identified under some completeness assumptions when $R^Y$ only depends on one of $(R^M,Y,M)$ given $T$ and $ X $ .","When $R^Y$ depends on more than one of $(R^M,Y,M)$ given $T$ and $ X $ , the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ cannot be identified without further assumptions.","The discussions and examples for those unidentifiable cases are provided in the section of the Supplemental Material.","So far, we have shown the nonparametric identification results of the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ , NIE and NDE, under various MNAR assumptions.","However, the nonparametric estimation for these quantities may suffer from the curse of dimensionality in practice, especially with a large number of covariates.","Therefore, we adopt a parametric method to obtain likelihood-based inference through the Expectation-Maximization algorithm [6].","The estimation details are provided in the section of the Supplementary Material."],["Simulation","We conducted simulation studies to evaluate the performance of the proposed estimators under each of the MNAR assumptions described in sections and .","In a simple context of a single covariate $X \\sim N(0,1)$ and a randomized $T\\sim \\mathrm {Bernoulli}(0.5)$ , we considered the following four setups representing different relationships in the supports of $M$ and $Y$ : (A) binary $M$ and binary $Y$ , (B) binary $M$ and continuous $Y$ , (C) continuous $M$ and continuous $Y$ , and (D) continuous $M$ and binary $Y$ .","We generated the mediator $M$ as follows: $\\mathrm {logit}~\\mathbb {P}(M=1\\mid T,X)=\\alpha _0+\\alpha _t T+\\alpha _x X$ if binary; and $M \\sim N(\\alpha _0+\\alpha _t T+\\alpha _x X, 0.5^2)$ if continuous.","We then generated the outcome $Y$ according to: $\\mathrm {logit}~\\mathbb {P}(Y=1\\mid M,T,X)=\\beta _0+\\beta _m M+\\beta _t T+\\beta _{mt} M \\cdot T+\\beta _x X$ if binary; and $Y \\sim N(\\beta _0+\\beta _m M+\\beta _t T+\\beta _{mt} M \\cdot T+\\beta _x X, 0.5^2)$ if continuous.","For each of the four setups described above, we considered missingness mechanisms under (I) Assumption REF , (II) Assumption REF , (III) Assumption REF , and (IV) Assumption REF , respectively, that is, sixteen simulation scenarios in total.","Under all MNAR Assumptions REF to REF , $R^M$ is allowed to depends on $M$ , $T$ and $X$ , and therefore, we generated the binary variable $R^M$ as follows: $\\mathrm {logit}~\\mathbb {P}(R^M=1\\mid M,T,X)=\\lambda _0+\\lambda _m M+\\lambda _t T+\\lambda _x X.$ Under (I) Assumption REF , $Y$ is fully observed.","For scenarios (II) to (IV) with missingness in $Y$ , the data generating models for $R^Y$ varied according to different MNAR Assumptions: $\\mathrm {logit}~\\mathbb {P}(R^Y=1\\mid R^M,T,X)=\\gamma _0+\\gamma _{r^M} R^M+\\gamma _t T+\\gamma _x X$ if under (II) Assumption REF ; $\\mathrm {logit}~\\mathbb {P}(R^Y=1\\mid Y,T,X)=\\gamma _0+\\gamma _y Y+\\gamma _t T+\\gamma _x X$ if under (III) Assumption REF ; and $\\mathrm {logit}~\\mathbb {P}(R^Y=1\\mid M,T,X)=\\gamma _0+\\gamma _m M+\\gamma _t T+\\gamma _x X$ if under (IV) Assumption REF .","For each of the sixteen simulation scenarios considered, we tested our theoretical results and evaluated the performance of our method when $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ and when $M \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ .","Table REF presents the specifications of parameter values when $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , and when $M \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , respectively.","We set parameter values in the $R^M$ and $R^Y$ models to generate missing rates in $M$ and $Y$ both to be around $20\\%$ to $25\\%$ , which are similar to the missing rates in our application study.","Table: Specifications of the parameter values.","1, M¬⊥⊥Y∣(T,X)M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X ); 2, M⊥⊥Y∣(T,X)M \\perp \\!\\!\\!\\perp Y\\mid (T, X ); Setting A, Binary MM and Binary YY; Setting B, Binary MM and Continuous YY; Setting C, Continuous MM and Continuous YY; Setting D, Continuous MM and Binary YY.We considered a sample size of 1000, and simulated 500 data sets for each simulation scenario.","We applied the following three methods to compare their results on the estimations of the NIE and NDE: 1) complete case analysis, which provides consistent estimates under MCAR; 2) our proposed methods using the Expectation-Maximization algorithm, which are designed to deal with the MNAR assumptions under concern; and 3) oracle estimators, which are obtained by using the true values of the missing data.","Figure REF presents the boxplots of percentages of biases with respect to the true values for each of the simulation scenarios when $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ across the 500 replications.","And Figure REF presents the boxplots for the corresponding scenarios when $M\\perp \\!\\!\\!\\perp Y\\mid (T, X )$ .","The simulation results are consistent with the results predicted by our theorems.","Under MNAR Assumption REF where the missingness only exists in the mediator, the percentages of biases for both NIE and NDE estimated using our methods are close to zero with slightly larger standard errors than the oracle estimates when $M\\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ and the completeness assumption holds (A.I, B.I, C.I), while both the estimated NIE and NDE from the complete case analysis have substantial biases.","When the completeness assumption is violated even though $M\\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ as in the continuous $M$ and binary $Y$ case (D.I), our method would not recover the underlying truth as expected.","When $M \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , as shown in Figure REF A.I (0) to D.I (0) with $(0)$ indicating that $\\mathrm {NIE}=0$ , NIE and NDE from both our methods and the complete case analysis are consistent.","This is due to the fact that although $\\mathbb {P}(M\\mid T, X )$ cannot be identified, the $\\mathbb {P}(Y\\mid M, T, X )$ can still be identified by the complete cases under Assumption REF .","Under MNAR Assumption REF (A.II to D.II, and A.II(0) to D.II(0)) and MNAR Assumption REF (A.IV to D.IV, and A.IV(0) to D.IV(0)), we reached the same conclusions as those under MNAR Assumption REF .","Under MNAR Assumption REF , when $M\\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , the completeness assumption is violated in both the binary $M$ and continuous $Y$ (B.III) and the continuous $M$ and binary $Y$ (D.III) cases, and our estimators would not recover the underlying truth.","Different from the other MNAR assumptions, when $M \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , both the distributions $\\mathbb {P}(Y\\mid M,T, X )$ and $\\mathbb {P}(M\\mid T, X )$ cannot be identified, we observed biases in estimating the $\\mathrm {NIE}$ and $\\mathrm {NDE}$ using either the complete case analysis or our method.","Figure: Simulation results when M¬⊥⊥Y∣(T,X)M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X ).","A, Binary MM and Binary YY; B, Binary MM and Continuous YY; C, Continuous MM and Continuous YY; D, Continuous MM and Binary YY.","I, Assumption ; II, Assumption ; III, Assumption ; IV, Assumption .","CC, complete case analysis; EM, our proposed Expectation-Maximization algorithm; Oracle, oracle estimators.","Bias (%), {(estimate-truth)/truth}*100.Figure: Simulation results when M⊥⊥Y∣(T,X)M \\perp \\!\\!\\!\\perp Y\\mid (T, X ).","A, Binary MM and Binary YY; B, Binary MM and Continuous YY; C, Continuous MM and Continuous YY; D, Continuous MM and Binary YY.","I, Assumption ; II, Assumption ; III, Assumption ; IV, Assumption .","CC, complete case analysis; EM, our proposed Expectation-Maximization algorithm; Oracle, oracle estimators.","(0), M⊥⊥Y∣(T,X)M \\perp \\!\\!\\!\\perp Y\\mid (T, X ).","Bias (%), {(estimate-truth)/truth}*100; Bias (%) for NIE is defined as (estimate/NDE)*100 because NIE equals 0."],["Data","The data describes 8707 eligible applicants in the mid-1990s who lived in the areas selected for in-person interviews at the baseline.","The subjects were randomized either to the experimental group $(T=1)$ where they could join the Job Corps program soon after randomization, or to the control group $(T=0)$ where they were not provided the Job Corps program for three years [30].","The mediator ($M$ ) was collected at the 30-months follow-up describing subject's educational and vocational attainment, measured by whether or not the subject obtained an education credential or vocational certificate after randomization.","We use $M=1$ to denote that an education credential or vocational certificate was obtained, and $M=0$ otherwise.","The outcome ($Y$ ) was collected at the 48-months follow-up describing the subject's weekly earnings in the fourth year after randomization.","The measured covariates $ X $ include information on gender, age, race, education level, earnings in the year before participating in the study, whether the subject had a child or not, and whether the subject had ever been arrested or not.","There are small portions of the missingness exist in covariates $ X $ , including education level (0.6%), earnings levels in the year before participating in the study (6.4%), and whether the subject had ever been arrested or no (3.5%).","Due to the low missing rates in those categorical covariates, we treat the missingness as another category in the analysis.","The number of subjects with missing information in the mediator or outcome is nontrivial.","The missingness patterns in the mediator and the outcome are described in Table REF for the experimental group and the control group.","We suspect that the missingness may be MNAR in the data.","Besides the potential impacts of $T$ and $ X $ on the missingness, we have the following concerns: (1) conceivably, people who failed to obtain an education credential or vocational certificate ($M=0$ ) may be less likely to report compared to people who successfully obtained an educational credential or vocational certificate ($M=1$ ), that is, $M$ may have a direct effect on $R^M$ ; (2) we are concerned that people who were unwilling to respond to questions at the 30-months follow-up may also be unwilling to respond at the 48-months follow-up, that is, $R^M$ may have a direct impact on $R^Y$ ; (3) people who had no earnings $(Y=0)$ may be less likely to report their earnings compared to people who had earnings $(Y>0)$ , which results in a direct effect of $Y$ on $R^Y$ ; (4) in addition, $M$ occurs before $R^Y$ , and therefore, may potentially have an impact on people's probability of reporting earnings through channels other than $R^M$ and $Y$ .","Concerns $(1)$ and $(2)$ can be addressed by the MNAR mechanism under Assumption REF , concerns $(1)$ and $(3)$ can be addressed by the MNAR mechanism under Assumption REF , and concerns $(1)$ and $(4)$ can be addressed by the MNAR mechanism under Assumption REF .","When incorporating the weekly earnings $Y$ into the prediction of $R^Y$ under MNAR Assumption REF , we assume that it is the binary indicator $\\mathrm {I}(Y>0)$ describing whether $Y$ is positive or not that predicts $R^Y$ .","This is because of the following two considerations: first, there are excessive zero values of the earnings (14.85%) in the data; second, the identification under MNAR Assumption REF requires the form of $Y$ in predicting $R^Y$ to be binary given that our mediator $M$ in this study is binary (see the completeness assumption in Theorem REF ).","Table: Missingness patterns in the mediator and the outcome"],["Models","Our outcome $Y$ , weekly earnings, contains many zero values as well as heavily right skewed positive values.","To address those complications, we adopted two-part models.","Let $Z_i$ be the binary indicator describing whether the earning is greater than 0 or not, i.e., $Z_i=1$ if $Y_i>0$ and $Z_i=0$ if $Y_i=0$ .","We used the following logistic regression to model $Z_{i}$ : $\\mathrm {logit}~\\mathbb {P}(Z_i=1\\mid M_i=m,T_i=t, X _i= x ) = \\delta _0+\\delta _m m+\\delta _t t+\\delta _{mt} m \\cdot t+\\delta _x^ \\textsc {t} x .$ Conditioning on $Z_{i}=1$ , we considered two commonly adopted models, Gamma and log-normal models to fit the positively skewed values of earnings.","As an illustration, we describe the Gamma model here: $Y_i \\mid (Z_i=1, M_i=m, T_i=t, X _i= x ) \\sim \\mathrm {Gamma}\\lbrace \\nu ,\\nu /\\mu _i(m, t, x )\\rbrace $ , where $\\nu $ denotes the shape parameter, $\\nu / \\mu _i(m, t, x )$ is the rate parameter, and the function $\\mu _i(m, t, x )$ is parameterized as $\\mathrm {exp}(\\beta _0+\\beta _m m+\\beta _t t + \\beta _{mt} m\\cdot t+\\beta _x^ \\textsc {t} x ).$ We used a logistic model for $M$ : $\\mathrm {logit}~\\mathbb {P}(M_i=1\\mid T_i=t, X _i= x )=\\alpha _0+\\alpha _{t}t+\\alpha _{x}^ \\textsc {t} x ,$ and a logistic model for $R^M$ allowing an effect from $M$ : $\\mathrm {logit} ~\\mathbb {P}(R^M_i=1\\mid M_i=m, T_i=t, X _i= x ) = \\lambda _0+\\lambda _m m+\\lambda _t t +\\lambda _x^ \\textsc {t} x .$ The model for $R^Y$ varied according to different MNAR assumptions.","Under MNAR Assumption REF , we specified that $\\mathrm {logit} ~\\mathbb {P}(R^Y_i=1\\mid R^M_i=r^M, T_i=t, X _i= x ) = \\gamma _0+\\gamma _{r^M}r^M+\\gamma _t t +\\gamma _x^ \\textsc {t} x ;$ under MNAR Assumption REF , we specified that $\\mathrm {logit} ~\\mathbb {P}(R^Y_i=1\\mid Z_i=z, T_i=t, X _i= x ) = \\gamma _0+\\gamma _{z} z+\\gamma _t t +\\gamma _x^ \\textsc {t} x ;$ and under MNAR Assumption REF , we adopted that $\\mathrm {logit} ~\\mathbb {P}(R^Y_i=1\\mid M_i=m, T_i=t, X _i= x ) = \\gamma _0+\\gamma _{m} m+\\gamma _t t +\\gamma _x^ \\textsc {t} x .$"],["Results","We compared the performance and results using two-part Gamma and two-part log-normal models for the outcome under MNAR Assumptions REF , REF and REF .","Table REF presents the log-likelihoods evaluated at the MLEs for those six models.","Since all six models have the same numbers of parameters, we chose the two-part Gamma model and MNAR mechanism under Assumption REF allowing an impact on $R^M$ from $M$ and an impact on $R^Y$ from $R^M$ , besides the impacts from $T$ and $ X $ .","Below we describe the results based on the two-part Gamma model under MNAR Assumption REF .","Table: Model comparison among models under MNAR Assumptions , and using two-part Gamma and two-part log-normal models for the outcome.","The log-likelihoods are evaluated at the MLEs.The coefficient of $M$ in the $R^M$ model is estimated to be $1.73$ with $95\\%$ CI $(0.34,~3.33)$ - see $\\lambda _m$ in Table REF , which suggests that people who had acquired certificates were more likely to report their certificate status.","The coefficient of $R^M$ in the $R^Y$ model is estimated to be $1.87$ with $95\\%$ CI $(1.76,~2.00)$ - see $\\gamma _{r^M}$ in Table REF , which suggests that people who were willing to report their certificate status were also more likely to report their earnings.","The natural indirect effect is estimated to be $10.94$ with $95\\%$ CI $(7.94,~14.29)$ - see $\\mathrm {NIE}$ in Table REF , which indicates that there was a significant indirect effect of the program assignment on weekly earnings through an educational credential or vocational certificate at the $0.05$ significance level.","The natural direct effect is estimated to be $12.93$ with $95\\%$ CI $(-1.95,~27.64)$ - see NDE in Table REF , which indicates that there was no significant direct effect of the program assignment on weekly earnings at the $0.05$ significance level.","The causal conclusions regarding the indirect effect (NIE) and the direct effect (NDE) are the same between the complete case analysis and our methods, in spite of the significant effect of $M \\rightarrow R^M$ ($\\lambda _{m}$ ) and the significant effect of $R^M \\rightarrow R^Y$ ($\\gamma _{r^M}$ ).","Table: Data analysis results from the Gamma model under MNAR Assumption .","Est, estimate; SE, standard error based on 500 bootstrap samples; CI, confidence interval; λ m \\lambda _m, coefficient of MM in the R M R^M model; γ r M \\gamma _{r^M}, coefficient of R M R^M in the R Y R^Y model."],["Sensitivity analysis","We consider the Gamma model under MNAR Assumption REF from the data analysis as a starting model for building the sensitivity analysis.","It is possible that the missingness of earnings also depends on the value of earnings itself and/or the educational and vocational attainment, in addition to the missingness of the educational and vocational attainment.","The goal is to access the sensitivity of our causal conclusions to the additional impacts on $R^Y$ from $Z$ and/or $M$ .","The revised model for $R^Y$ is as follows: $&&\\mathrm {logit}~\\mathbb {P}(R^Y_i=1\\mid R^M_i=r^M, Z_i=z, M_i=m, T_i=t, X _i= x )\\\\&=&\\gamma _0+\\gamma _{r^M} r^M +\\gamma _z z+\\gamma _m m +\\gamma _t t +\\gamma _x^ \\textsc {t} x,$ where $\\gamma _z$ and $\\gamma _m$ are the sensitivity parameters.","We consider a large effect in the log odds ratio [3] and let both sensitivity parameters to vary among $-2$ , 0 and 2.","When $\\gamma _z=0$ and $\\gamma _m=0$ , it is the same as the MNAR mechanism under Assumption REF that stands out in the data analysis.","Figure: The direct acyclic graph describing the missing data mechanism for the sensitivity analysis.","The graph conditions on X X .The results for the sensitivity analysis are presented in Table REF .","The NIE estimate increases more than 10% in the case where $\\gamma _m=-2$ and $\\gamma _z=2$ , and the case where $\\gamma _m=0$ and $\\gamma _z=2$ .","The NDE estimate decreases more than 10% in the case where $\\gamma _m=0$ and $\\gamma _z=2$ , and increases more than 10% in the case where $\\gamma _m=2$ and $\\gamma _z=2$ .","However, the NIEs are estimated to be positive and significant at the $0.05$ level and the NDEs are estimated to be positive but not significant at the $0.05$ level, for all pairs of values $(\\gamma _m, \\gamma _z)$ considered.","In summary, the causal conclusions on the NIE and NDE are not sensitive to a strong impact of $Z \\rightarrow R^Y$ and/or $M \\rightarrow R^Y$ in addition to the impact of $R^M$ on $R^Y$ .","Table: Sensitivity analysis results from the Gamma model.","Est, estimate; CI, confidence interval based on 500 bootstrap samples; λ m \\lambda _m, coefficient of MM in the R M R^M model; γ r M \\gamma _{r^M}, coefficient of R M R^M in the R Y R^Y model; γ z \\gamma _z (sensitivity parameter), coefficient of ZZ in the R Y R^Y model; γ m \\gamma _m (sensitivity parameter), coefficient of MM in the R Y R^Y model."],["Discussion","Mediation analysis plays an important role in studying the underlying biological or social mechanisms through which an exposure exerts its effect.","When the mediator and outcome are missing not at random, the evaluation of the NIE and NDE can be very challenging.","Motivated by the NJCS, in scenarios where it is reasonable to assume that future variables do not have an impact on variables measured in the past, we explored the identification of the NIE and NDE for some interpretable missing not at random assumptions.","The condition of completeness plays an important role in achieving nonparametric identification.","We proved that the joint distribution $\\mathbb {P}(Y,M\\mid T, X )$ can be identified when the missingness of $M$ may depend on the value of $M$ itself and when $Y$ is either fully observed, or the missingness of $Y$ depends on only one of $(R^M, Y, M)$ in addition to $T$ and $ X $ .","In a special case when $M\\perp \\!\\!\\!\\perp Y\\mid T, X $ and therefore violates the completeness requirement, although $\\mathbb {P}(M\\mid T, X )$ is no longer identifiable, the NIE and NDE can still be identified except when both the missingness of $M$ and the missingness of $Y$ depend on their missing values themselves (i.e., MNAR Assumption REF ).","Based on the comparison of likelihoods, the MNAR mechanism under Assumption REF is more plausible compared to other MNAR assumptions in the NJCS study.","The strong association between $R^M$ and $R^Y$ in the NJCS study might be a result of both being affected by the subjects' unmeasured tendency or willingness to respond to interview questions.","Further, to address the concern that the missingness of $Y$ may also depends on $Y$ itself and $M$ in addition to $R^M$ , we proposed a sensitivity analysis approach and found that our conclusions on the NIE and NDE in the NJCS study are not sensitive to some large impacts on $R^Y$ from $Y$ and $M$ .","Several limitations of the work are to be addressed in future research.","First, in current work, we focused on the identification of the NIE and NDE when the mediator and outcome are MNAR.","In practice, prevalent missingness may exist in both the mediator/outcome and the measured covariates, and the missingness of covariates may also be MNAR.","Identifiability in this more complicated scenario is to be explored.","Second, the NIE and NDE evaluated in the NJCS study should be interpreted as the effects of the program assignment instead of the effects of the program.","According to [31], although control group subjects were barred from participating in the Job Corps program, there were about $30\\%$ of the program group subjects who failed to participate in the program.","One possibility to address this one-sided noncompliance issue is to consider the random program assignment as an instrumental variable and incorporate the instrumental variable into causal mediation analysis [34], [10], [24] for the latent subgroup of compliers, i.e., who would comply with their treatment assignment no matter being assigned to the treatment or the control group.","Third, we considered a setting where there is a single mediator.","In practice, cases with multiple mediators may arise, and the potential missingness mechanisms may vary depending on whether those mediators have a sequential relationship or parallel relationship, and etc.","We leave this topic for future research.","Fourth, the results in this paper invoke the sequential ignorability assumption.","In the NJCS study, although the ignorability of the treatment assignment is guaranteed by the study design, however, the ignorability of the mediator is not guaranteed and could be violated if there exists post-treatment or unmeasured pretreatment confounding.","One approach to partly address this issue is to conduct sensitivity analysis to quantify the potential bias due to the departure from the sequential ignorability assumption [33], [12].","SUPPLEMENTARY MATERIAL Section gives the proofs of the theorems.","Section gives counterexamples for the unidentifiable cases.","Section provides parametric estimation details."],["Proof of Theorem ","The identification of $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x )$ follows from $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x ) = \\mathbb {P}(Y=y\\mid R^M = 1, M=m, T=t, X = x )\\nonumber .$ We will focus on the identification of $\\mathbb {P}(M=m\\mid T=t, X = x )$ .","Define $\\mathbb {P}_{my1|t, x} &=& \\mathbb {P}(M=m, Y=y, R^M=1\\mid T=t, X = x ), \\\\\\mathbb {P}_{+y0\\mid t,x} &=& \\mathbb {P}(Y=y, R^M=0\\mid T=t, X = x ), \\\\\\zeta _{t,x}(m) &=& \\frac{\\mathbb {P}(R^M=0\\mid M=m, T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )}.$ We have $\\mathbb {P}_{my1\\mid t, x} = \\mathbb {P}(M=m, Y=y\\mid T=t, X = x )\\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )$ and $\\mathbb {P}_{+y0\\mid t,x} &= \\int _{m\\in \\mathcal {M}} \\mathbb {P}(M=m, Y=y, R^M=0\\mid T=t, X = x )dm \\nonumber \\\\&=\\int _{m\\in \\mathcal {M}} \\mathbb {P}(M=m, Y=y\\mid T=t, X = x ) \\mathbb {P}(R^M=0\\mid M=m, T=t, X = x )dm \\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my1\\mid t, x} \\frac{\\mathbb {P}(R^M=0\\mid M=m, T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )}dm \\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my1\\mid t, x} \\zeta _{t,x}(m)dm.","\\nonumber $ The existence and uniqueness of solutions $\\zeta _{t,x}(m)$ require that $\\mathbb {P}(Y,M,R^M=1 \\mid T=t, X=x )$ is complete in $Y$ .","For discrete $M$ and discrete $Y$ , the completeness assumption is equivalent to Rank $(\\Theta _{tx})=J$ , where $\\Theta _{tx}$ is a $J \\times K$ matrix with $\\mathbb {P}_{my1\\mid t, x}$ as the $(m,y)$ th element.","For binary $M$ , the rank condition further reduces to $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , which is equivalent to the testable condition $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X ,R^M=1)$ .","For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be no smaller than the dimension of $M$ in general as required by the completeness assumption.","We can subsequently identify $\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )$ once $\\zeta _{t,x}(m)$ is identified.","Then $&&\\mathbb {P}(M=m\\mid T=t, X = x ) \\\\&=&\\frac{\\mathbb {P}(M=m, Y=y\\mid T=t, X = x )}{\\mathbb {P}(Y=y\\mid M=m,T=t, X = x )} \\\\&=& \\frac{\\mathbb {P}_{my1\\mid t, x}}{\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x ) \\mathbb {P}(Y=y\\mid M=m,T=t, X = x )}.$"],["Proof of Theorem ","The identification of $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x )$ follows from $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x ) = \\mathbb {P}(Y=y\\mid R^M=1, R^Y=1, M=m, T=t, X = x ).\\nonumber $ We will focus on the identification of $\\mathbb {P}(M=m\\mid T=t, X = x )$ .","Define $\\mathbb {P}_{my11\\mid t,x} &=& \\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1\\mid T=t, X = x ), \\\\\\mathbb {P}_{+y01\\mid t,x} &=& \\mathbb {P}(Y=y, R^M=0, R^Y=1\\mid T=t, X = x ), \\\\\\zeta _{t,x}(m)&=&\\frac{\\mathbb {P}(R^M=0\\mid M=m,T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )}.$ We have $\\mathbb {P}_{my11\\mid t,x} &=& \\mathbb {P}(M=m, Y=y\\mid T=t, X = x ) \\\\&& \\cdot \\mathbb {P}(R^Y=1\\mid R^M=1,T=t, X = x )\\cdot \\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )$ and $\\mathbb {P}_{+y01\\mid t,x} &= \\int _{m\\in \\mathcal {M}} \\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1\\mid T=t, X = x )dm\\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my11\\mid t,x} \\frac{\\mathbb {P}(R^Y=1\\mid R^M=0,T=t, X = x )\\mathbb {P}(R^M=0\\mid M=m,T=t, X = x )}{\\mathbb {P}(R^Y=1\\mid R^M=1,T=t, X = x )\\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )}dm\\nonumber \\\\&=\\frac{\\mathbb {P}(R^Y=1\\mid R^M=0,T=t, X = x )}{\\mathbb {P}(R^Y=1\\mid R^M=1,T=t, X = x )}\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my11\\mid t,x} \\zeta _{t,x}(m)dm.","\\nonumber $ The existence and uniqueness of solutions $\\zeta _{t,x}(m)$ require that $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X=x )$ is complete in $Y$ .","For discrete $M$ and discrete $Y$ , the completeness assumption is equivalent to Rank $(\\Theta _{tx})=J$ , where $\\Theta _{tx}$ is a $J \\times K$ matrix with $\\mathbb {P}_{my11\\mid t,x}$ as the $(m,y)$ th element.","For binary $M$ , the rank condition further reduces to $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , which is equivalent to the testable condition $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X ,R^M=1,R^Y=1)$ .","For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be no smaller than the dimension of $M$ in general as required by the completeness assumption.","We can subsequently identify $\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )$ once $\\zeta _{t,x}(m)$ is identified.","Then $\\mathbb {P}(M=m\\mid T=t, X = x )&=\\frac{\\mathbb {P}(M=m, Y=y\\mid T=t, X = x )}{\\mathbb {P}(Y=y\\mid M=m,T=t, X = x )}\\nonumber \\\\&= \\frac{\\mathbb {P}_{my11\\mid t,x}}{\\mathbb {P}(Y=y, R^Y=1, R^M=1\\mid M=m,T=t, X = x )}.\\nonumber $"],["Proof of Theorem ","We will discuss the identification of $\\mathbb {P}(M=m,Y=y\\mid T=t, X = x )$ .","Define $\\mathbb {P}_{my11\\mid t,x} &=& \\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1\\mid T=t, X = x ), \\\\\\mathbb {P}_{+y01\\mid t,x} &=& \\mathbb {P}(Y=y, R^M=0, R^Y=1\\mid T=t, X = x ), \\\\\\mathbb {P}_{m+10\\mid t,x} &=& \\mathbb {P}(M=m, R^M=1, R^Y=0\\mid T=t, X = x ), \\\\\\zeta _{t,x}(m)&=&\\frac{\\mathbb {P}(R^M=0\\mid M=m,T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )},\\\\\\eta _{t,x}(y)&=&\\frac{\\mathbb {P}(R^Y=0\\mid Y=y,T=t, X = x )}{\\mathbb {P}(R^Y=1\\mid Y=y,T=t, X = x )}.$ We first have $\\mathbb {P}_{my11|t,x}&=& \\mathbb {P}(M=m, Y=y\\mid T=t, X = x ) \\\\&&\\cdot \\mathbb {P}(R^Y=1\\mid Y=y,T=t, X = x )\\cdot \\mathbb {P}(R^M=1\\mid M=m,T=t, X = x ) .$ We then have $\\mathbb {P}_{+y01\\mid t,x} &= \\int _{m\\in \\mathcal {M}} \\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1\\mid T=t, X = x )dm\\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my11\\mid t,x}\\frac{\\mathbb {P}(R^M=0\\mid M=m,T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )}dm\\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my11\\mid t,x} \\zeta _{t,x}(m)dm \\nonumber $ and $\\mathbb {P}_{m+10\\mid t,x} &= \\int _{y\\in \\mathcal {Y}}\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=0\\mid T=t, X = x )dy\\nonumber \\\\&=\\int _{y\\in \\mathcal {Y}}\\mathbb {P}_{my11\\mid t,x} \\frac{\\mathbb {P}(R^Y=0\\mid Y=y,T=t, X = x )}{\\mathbb {P}(R^Y=1\\mid Y=y,T=t, X = x )}dy\\nonumber \\\\&=\\int _{y\\in \\mathcal {Y}}\\mathbb {P}_{my11\\mid t,x} \\eta _{t,x}(y)dy.","\\nonumber $ The existence and uniqueness of solutions $\\zeta _{t,x}(m)$ require that $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X=x )$ is complete in $Y$ , and the existence and uniqueness of solutions $\\eta _{t,x}(y)$ require that $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X=x )$ is complete in $M$ .","For discrete $M$ and discrete $Y$ , the above completeness assumptions are equivalent to Rank $(\\Theta _{tx})=J$ , where $\\Theta _{tx}$ is a $J \\times J$ matrix with $\\mathbb {P}_{my11\\mid t,x}$ as the $(m,y)$ th element.","For binary $M$ and binary $Y$ , the rank condition reduces to $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ .","For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be the same as the dimension of $M$ in general as required by $\\mathbb {P}(Y,M,R^Y=1,R^M=1\\mid T, X )$ being complete in $M$ and being complete in $Y$ .","We can subsequently identify $\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )$ and $\\mathbb {P}(R^Y=1\\mid Y=y, T=t, X = x )$ once $\\zeta _{t,x}(m)$ and $\\eta _{t,x}(y)$ are identified.","Then $&&\\mathbb {P}(Y=y, M=m\\mid T=t, X = x )\\nonumber \\\\&=&\\frac{\\mathbb {P}_{my11\\mid t,x}}{\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )\\mathbb {P}(R^Y=1\\mid Y=y, T=t, X = x )}.\\nonumber $"],["Under condition $(i)$","The identification of $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x )$ follows from $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x ) = \\mathbb {P}(Y=y\\mid R^M = 1, R^Y=1, M=m, T=t, X = x )\\nonumber .$ We will focus on the identification of $\\mathbb {P}(M=m\\mid T=t, X = x )$ .","Define $\\mathbb {P}_{my11\\mid t,x} &=& \\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1\\mid T=t, X = x ), \\\\\\mathbb {P}_{+y01\\mid t,x} &=& \\mathbb {P}(Y=y, R^M=0, R^Y=1\\mid T=t, X = x ), \\\\\\mathbb {P}_{m+10\\mid t,x} &=& \\mathbb {P}(M=m, R^M=1, R^Y=0\\mid T=t, X = x ), \\\\\\mathbb {P}_{m+11\\mid t,x} &=& \\mathbb {P}(M=m, R^M=1, R^Y=1\\mid T=t, X = x ), \\\\\\zeta _{t,x}(m)&=&\\frac{\\mathbb {P}(R^M=0\\mid M=m,T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )}, \\\\\\eta _{t,x}(m)&=&\\frac{\\mathbb {P}(R^Y=0\\mid M=m,T=t, X = x )}{\\mathbb {P}(R^Y=1\\mid M=m,T=t, X = x )}.$ We first have $\\mathbb {P}_{my11\\mid t,x} &=& \\mathbb {P}(M=m, Y=y \\mid T=t, X = x ) \\\\&&\\cdot \\mathbb {P}(R^Y=1\\mid M=m, T=t, X = x )\\cdot \\mathbb {P}(R^M=1\\mid M=m, T=t, X = x ).$ We then have $\\mathbb {P}_{+y01\\mid t,x} &= \\int _{m\\in \\mathcal {M}} \\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1\\mid T=t, X = x )dm\\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my11\\mid t,x} \\frac{\\mathbb {P}(R^M=0\\mid M=m, T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )}dm\\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{my11\\mid t,x} \\zeta _{t,x}(m)dm \\nonumber $ and $\\mathbb {P}_{m+10\\mid t,x} &= \\int _{y\\in \\mathcal {Y}} \\mathbb {P}(M=m, Y=y, R^M=1, R^Y=0\\mid T=t, X = x )dy\\nonumber \\\\&=\\int _{y\\in \\mathcal {Y}}\\mathbb {P}_{my11\\mid t,x} \\frac{\\mathbb {P}(R^Y=0\\mid M=m, T=t, X = x )}{\\mathbb {P}(R^Y=1\\mid M=m, T=t, X = x )}dy\\nonumber \\\\&=\\eta _{t,x}(m)\\int _{y\\in \\mathcal {Y}}\\mathbb {P}_{my11\\mid t,x}dy \\nonumber \\\\&=\\eta _{t,x}(m)\\mathbb {P}_{m+11\\mid t,x}.","\\nonumber $ Solve $\\eta _{t,x}(m)$ as $\\eta _{t,x}(m) = \\mathbb {P}_{m+10\\mid t,x} / \\mathbb {P}_{m+11\\mid t,x} .$ When the joint distribution of $\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\mid T=t, X=x )$ is complete in $Y$ , the existence and uniqueness of solutions $\\zeta _{t,x}(m)$ can be achieved.","For discrete $M$ and discrete $Y$ , this completeness assumption is equivalent to Rank $(\\Theta _{tx})=J$ , where $\\Theta _{tx}$ is a $J \\times K$ matrix with $\\mathbb {P}_{my11\\mid t,x}$ as the $(m,y)$ th element.","For binary $M$ , the rank condition further reduces to $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X )$ , which is equivalent to the testable condition $M \\lnot \\perp \\!\\!\\!\\perp Y\\mid (T, X ,R^M=1,R^Y=1)$ .","For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be no smaller than the dimension of $M$ in general as required by the completeness assumption.","We can subsequently identify $\\mathbb {P}(R^Y=1\\mid M=m, T=t, X = x )$ and $\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )$ once $\\zeta _{t,x}(m)$ and $\\eta _{t,x}(m)$ are identified.","Then $\\mathbb {P}(M=m\\mid T=t, X = x )& = \\frac{\\mathbb {P}(M=m, Y=y\\mid T=t, X = x )}{\\mathbb {P}(Y=y\\mid M=m,T=t, X = x )} \\nonumber \\\\&= \\frac{\\mathbb {P}_{my11\\mid t,x}}{\\mathbb {P}(Y=y,R^M=1,R^Y=1\\mid M=m,T=t, X = x )}\\nonumber .$ The identification of $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x )$ follows from $\\mathbb {P}(Y=y\\mid M=m, T=t, X = x ) = \\mathbb {P}(Y=y\\mid R^M=1, R^Y=1, M=m, T=t, X = x ),$ and the identification of $\\mathbb {P}(R^Y=r^Y\\mid M=m, T=t, X = x )$ follows from $\\mathbb {P}(R^Y=r^Y\\mid M=m, T=t, X = x ) = \\mathbb {P}(R^Y=r^Y\\mid R^M=1,M=m,T=t, X = x ) .$ We will focus on the identification of $\\mathbb {P}(M=m\\mid T=t, X = x )$ .","Define $\\mathbb {P}_{m1r^Y|t,x} &=& \\mathbb {P}(M=m,R^M=1,R^Y=r^Y\\mid T=t, X = x ), \\\\\\mathbb {P}_{+0r^Y\\mid t,x} &=& \\mathbb {P}(R^M=0,R^Y=r^Y\\mid T=t, X = x ), \\\\\\zeta _{t,x}(m) &=& \\frac{\\mathbb {P}(R^M=0\\mid M=m, T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )}.$ We have $\\mathbb {P}_{m1r^Y\\mid t,x} = \\mathbb {P}(M=m,R^Y=r^Y\\mid T=t, X = x )\\mathbb {P}(R^M=1\\mid M=m,T=t, X = x )$ and $\\mathbb {P}_{+0r^Y\\mid t,x} &= \\int _{m\\in \\mathcal {M}} \\mathbb {P}(M=m, R^M=0, R^Y=r^Y\\mid T=t, X = x )dm \\nonumber \\\\&=\\int _{m\\in \\mathcal {M}} \\mathbb {P}_{m1r^Y|t,x} \\frac{\\mathbb {P}(R^M=0\\mid M=m, T=t, X = x )}{\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )}dm \\nonumber \\\\&=\\int _{m\\in \\mathcal {M}}\\mathbb {P}_{m1r^Y|t,x} \\zeta _{t,x}(m)dm.","\\nonumber $ When the joint distribution of $\\mathbb {P}(M,R^M=1,R^Y \\mid T=t, X=x )$ is complete in $R^Y$ ,the existence and uniqueness of solutions $\\zeta _{t,x}(m)$ can be achieved.","This requires that $M$ is binary since $R^Y$ is binary, and the completeness assumption reduces to $M \\lnot \\perp \\!\\!\\!\\perp R^Y\\mid (T, X )$ , which is equivalent to the testable condition $M \\lnot \\perp \\!\\!\\!\\perp R^Y\\mid (T, X ,R^M=1)$ .","We can subsequently identify $\\mathbb {P}(R^M=1\\mid M=m, T=t, X = x )$ once $\\zeta _{t,x}(m)$ is identified.","Then $\\mathbb {P}(M=m\\mid T=t, X = x )\\nonumber &=\\frac{\\mathbb {P}(M=m, R^Y=r^Y\\mid T=t, X = x )}{\\mathbb {P}(R^Y=r^Y\\mid M=m,T=t, X = x )} \\nonumber \\\\&= \\frac{\\mathbb {P}_{m1r^Y\\mid t,x}}{\\mathbb {P}(R^M=1,R^Y=r^Y\\mid M=m, T=t, X = x )}.\\nonumber $"],["Counterexamples for the Unidentifiable Cases","Consider a binary mediator $M$ and binary outcome $Y$ .","All graphs and probabilities below are conditioning on $T$ and $ X $ .","Define $\\mathbb {P}_{my11} &=& \\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1), \\\\\\mathbb {P}_{+y01} &=& \\mathbb {P}(Y=y, R^M=0, R^Y=1),\\\\\\mathbb {P}_{m+10} &=& \\mathbb {P}(M=m, R^M=1, R^Y=0), \\\\\\mathbb {P}_{++00} &=& \\mathbb {P}(R^M=0, R^Y=0).$ Based on the observable data probabilities, we can directly identify $\\mathbb {P}_{my11}$ , $\\mathbb {P}_{+y01}$ , $\\mathbb {P}_{m+10}$ and $\\mathbb {P}_{++00}$ , and $\\sum ^{1}_{m=0}\\sum ^{1}_{y=0}\\mathbb {P}_{my11}+\\sum _{y=0}^1 \\mathbb {P}_{+y01}+\\sum _{m=0}^1 \\mathbb {P}_{m+10}+ \\mathbb {P}_{++00}=1$ ."],["$(i)$ We present below an unidentified case when {{formula:ca2bf032-22fe-4901-8a91-776f1be6e9e3}} depends on both {{formula:8fa07d9e-4c56-4243-bc2f-9023ccaf307d}} and {{formula:975b0000-5a92-4422-a27b-0023e2a95cb3}} as described by Figure ","Consider the following observable data probabilities: $(\\mathbb {P}_{1111},\\mathbb {P}_{0111},\\mathbb {P}_{1011},\\mathbb {P}_{0011},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{++00})=\\left(\\frac{3}{10},\\frac{1}{10},\\frac{1}{20},\\frac{1}{20},\\frac{1}{10},\\frac{1}{20},\\frac{1}{10},\\frac{1}{20},\\frac{1}{5} \\right)\\nonumber .$ The key to identify $\\mathbb {P}(Y=y, M=m)$ is to identify both $\\mathbb {P}(R^M=1\\mid M=m)$ and $\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)$ in the following formula $\\mathbb {P}(Y=y, M=m)=\\frac{\\mathbb {P}_{my11}}{\\mathbb {P}(R^M=1\\mid M=m)\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)}.\\nonumber $ We now show that the identification of $\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)$ can be achieved.","This is because $\\mathbb {P}_{m+10} &= \\sum _{y\\in \\mathcal {Y}}\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=0)\\nonumber \\\\&=\\sum _{y\\in \\mathcal {Y}}\\mathbb {P}_{my11}\\frac{\\mathbb {P}(R^Y=0\\mid Y=y,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)}.\\nonumber $ By plugging in the two possible values for $m$ and $y$ in the above formula, we have $\\mathbb {P}_{1+10} &= \\mathbb {P}_{1111}\\frac{\\mathbb {P}(R^Y=0\\mid Y=1,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=1,R^M=1)}+\\mathbb {P}_{1011}\\frac{\\mathbb {P}(R^Y=0\\mid Y=0,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=0,R^M=1)}$ and $\\mathbb {P}_{0+10} &= \\mathbb {P}_{0111}\\frac{\\mathbb {P}(R^Y=0\\mid Y=1,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=1,R^M=1)}+\\mathbb {P}_{0011}\\frac{\\mathbb {P}(R^Y=0\\mid Y=0,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=0,R^M=1)}.$ And therefore, $\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)$ can be identified by solving the linear equations (REF ) and (REF ).","Based on the observable data probabilities, $\\mathbb {P}(R^Y=1\\mid Y=1,R^M=1)=\\frac{4}{5}$ and $\\mathbb {P}(R^Y=1\\mid Y=0,R^M=1)=\\frac{2}{3}$ .","We now focus on the identifiability of $\\mathbb {P}(R^M=1\\mid M=m)$ and show that $\\mathbb {P}(R^M=1\\mid M=m)$ cannot be identified without further assumptions.","We have $\\mathbb {P}_{+y01} &= \\sum _{m\\in \\mathcal {M}} \\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1)\\nonumber \\\\&=\\sum _{m\\in \\mathcal {M}}\\mathbb {P}_{my11}\\frac{\\mathbb {P}(R^Y=1\\mid Y=y,R^M=0)\\mathbb {P}(R^M=0\\mid M=m)}{\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)\\mathbb {P}(R^M=1\\mid M=m)}\\nonumber \\\\&= \\frac{\\mathbb {P}(R^Y=1\\mid Y=y,R^M=0)}{\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)} \\sum _{m\\in \\mathcal {M}}\\mathbb {P}_{my11}\\frac{\\mathbb {P}(R^M=0\\mid M=m)}{\\mathbb {P}(R^M=1\\mid M=m)},\\nonumber $ and as a result, $\\mathbb {P}_{+y01}\\frac{\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=y,R^M=0)} &= \\sum _{m\\in \\mathcal {M}}\\mathbb {P}_{my11}\\frac{\\mathbb {P}(R^M=0\\mid M=m)}{\\mathbb {P}(R^M=1\\mid M=m)}.\\nonumber $ By plugging in the two possible values for $m$ and $y$ in the above formula, we have $\\mathbb {P}_{+101}\\frac{\\mathbb {P}(R^Y=1\\mid Y=1,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=1,R^M=0)}&= \\mathbb {P}_{1111}\\frac{\\mathbb {P}(R^M=0\\mid M=1)}{\\mathbb {P}(R^M=1\\mid M=1)}\\nonumber +\\mathbb {P}_{0111}\\frac{\\mathbb {P}(R^M=0\\mid M=0)}{\\mathbb {P}(R^M=1\\mid M=0)}\\nonumber $ and $\\mathbb {P}_{+001}\\frac{\\mathbb {P}(R^Y=1\\mid Y=0,R^M=1)}{\\mathbb {P}(R^Y=1\\mid Y=0,R^M=0)}&= \\mathbb {P}_{1011}\\frac{\\mathbb {P}(R^M=0\\mid M=1)}{\\mathbb {P}(R^M=1\\mid M=1)}\\nonumber +\\mathbb {P}_{0011}\\frac{\\mathbb {P}(R^M=0\\mid M=0)}{\\mathbb {P}(R^M=1\\mid M=0)}.\\nonumber $ Since $\\mathbb {P}(R^Y=1\\mid Y=y,R^M=1)$ are identified from the previous step, the identifiability of $\\mathbb {P}(R^M=1 \\mid M=m)$ depends on the identifiability of $\\mathbb {P}(R^Y=1\\mid Y=y,R^M=0)$ .","We have $&&\\mathbb {P}(R^Y=1\\mid Y=y,R^M=0)\\nonumber \\\\&=& \\frac{\\mathbb {P}(Y=y,R^M=0,R^Y=1)}{\\mathbb {P}(Y=y,R^M=0)}\\nonumber \\\\&=& \\frac{\\mathbb {P}(Y=y,R^M=0,R^Y=1)}{\\mathbb {P}(Y=y,R^M=0,R^Y=0)+\\mathbb {P}(Y=y,R^M=0,R^Y=1)}\\nonumber \\\\&=&\\frac{\\mathbb {P}(Y=y,R^M=0,R^Y=1)}{\\mathbb {P}(Y=y \\mid R^M=0, R^Y=0)\\mathbb {P}(R^M=0,R^Y=0)+\\mathbb {P}(Y=y, R^M=0,R^Y=1)}\\nonumber \\\\&=&\\frac{\\mathbb {P}_{+y01}}{\\mathbb {P}(Y=y \\mid R^M=0, R^Y=0)\\mathbb {P}_{++00}+\\mathbb {P}_{+y01}}.\\nonumber $ In the above expression, $\\mathbb {P}_{+y01}$ and $\\mathbb {P}_{++00}$ are known.","$\\mathbb {P}(Y=y \\mid R^Y=0,R^M=0)$ is not observable or identifiable, and can take any value between 0 and 1.","Different values of $\\mathbb {P}(Y=y \\mid R^Y=0,R^M=0)$ will result in different values of $\\mathbb {P}(R^M=1 \\mid M=m)$ , which in turn will give different values of $\\mathbb {P}(Y=y, M=m)$ .","For example, let $\\mathbb {P}(Y=1 \\mid R^Y=0,R^M=0)=\\frac{5}{6}$ , and the corresponding $\\mathbb {P}(R^Y=1\\mid Y=1,R^M=0)$ and $\\mathbb {P}(R^Y=1\\mid Y=0,R^M=0)$ equal $\\frac{3}{8}$ and $\\frac{3}{5}$ , respectively.","Subsequently, we have $\\mathbb {P}(Y=1,M=1)=\\frac{17}{30}$ , $\\mathbb {P}(Y=0,M=1)=\\frac{17}{150}$ , $\\mathbb {P}(Y=1,M=0)=\\frac{1}{5}$ and $\\mathbb {P}(Y=0,M=0)=\\frac{3}{25}$ .","Alternatively, let $\\mathbb {P}(Y=1 \\mid R^Y=0,R^M=0)=\\frac{7}{8}$ , and the corresponding $\\mathbb {P}(R^Y=1\\mid Y=1,R^M=0)$ and $\\mathbb {P}(R^Y=1\\mid Y=0,R^M=0)$ equal $\\frac{4}{11}$ and $\\frac{2}{3}$ , respectively.","Subsequently, we have $\\mathbb {P}(Y=1,M=1)=\\frac{3}{5}$ , $\\mathbb {P}(Y=0,M=1)=\\frac{3}{25}$ , $\\mathbb {P}(Y=1,M=0)=\\frac{7}{40}$ and $\\mathbb {P}(Y=0,M=0)=\\frac{21}{200}$ .","The two sets of values of $\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\mathbb {P}_{1111},\\mathbb {P}_{0111},\\mathbb {P}_{1011},\\mathbb {P}_{0011},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{++00})$ , and therefore, $\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions.","Consider the following observable data probabilities: $(\\mathbb {P}_{1111},\\mathbb {P}_{0111},\\mathbb {P}_{1011},\\mathbb {P}_{0011},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{++00})=\\left(\\frac{3}{10},\\frac{1}{10},\\frac{1}{20},\\frac{1}{10},\\frac{1}{10},\\frac{1}{40},\\frac{1}{10},\\frac{1}{10},\\frac{5}{40}\\right)\\nonumber .$ Define $&\\mathbb {P}_M=\\mathbb {P}(M=1),\\\\&\\mathbb {P}_{Y1}=\\mathbb {P}(Y=1\\mid M=1),\\\\&\\mathbb {P}_{Y0}=\\mathbb {P}(Y=1\\mid M=0),\\\\&\\mathbb {P}_{R^M1}=\\mathbb {P}(R^M=1\\mid M=1),\\\\&\\mathbb {P}_{R^M0}=\\mathbb {P}(R^M=1\\mid M=0),\\\\&\\mathbb {P}_{R^Y00}=\\mathbb {P}(R^Y=1\\mid M=0,R^M=0),\\\\&\\mathbb {P}_{R^Y01}=\\mathbb {P}(R^Y=1\\mid M=0,R^M=1),\\\\&\\mathbb {P}_{R^Y10}=\\mathbb {P}(R^Y=1\\mid M=1,R^M=0),\\\\&\\mathbb {P}_{R^Y11}=\\mathbb {P}(R^Y=1\\mid M=1,R^M=1).$ Below we study the identifiablity of the above 9 parameters describing the graphical model in Figure REF based on the observable data probabilities.","Note that although there are 9 observable data probabilities, the degree of freedom in the probabilities is only 8 given that they sum up to 1.","The following relationships between the observable data probabilities and the parameters hold, $&\\mathbb {P}_{1111}=\\mathbb {P}_M\\mathbb {P}_{Y1}\\mathbb {P}_{R^M1}\\mathbb {P}_{R^Y11}, \\\\&\\mathbb {P}_{1011}=\\mathbb {P}_M(1-\\mathbb {P}_{Y1})\\mathbb {P}_{R^M1}\\mathbb {P}_{R^Y11},\\\\&\\mathbb {P}_{0111}=(1-\\mathbb {P}_M)\\mathbb {P}_{Y0}\\mathbb {P}_{R^M0}\\mathbb {P}_{R^Y01},\\\\&\\mathbb {P}_{0011}=(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y0})\\mathbb {P}_{R^M0}\\mathbb {P}_{R^Y01},\\\\&\\mathbb {P}_{1+10}=\\mathbb {P}_M\\mathbb {P}_{R^M1}(1-\\mathbb {P}_{R^Y11}),\\\\&\\mathbb {P}_{0+10}=(1-\\mathbb {P}_M)\\mathbb {P}_{R^M0}(1-\\mathbb {P}_{R^Y01}),\\\\&\\mathbb {P}_{+101}=\\mathbb {P}_M\\mathbb {P}_{Y1}(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y10}+(1-\\mathbb {P}_M)\\mathbb {P}_{Y0}(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y00},\\\\&\\mathbb {P}_{+001}=\\mathbb {P}_M(1-\\mathbb {P}_{Y1})(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y10}+(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y0})(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y00}.$ By solving the equations (REF ) to (), we can identify the parameters $\\mathbb {P}_{Y1}$ , $\\mathbb {P}_{Y0}$ , $\\mathbb {P}_{R^Y11}$ , and $\\mathbb {P}_{R^Y01}$ : $&\\mathbb {P}_{Y1}=\\frac{\\mathbb {P}_{1111}}{\\mathbb {P}_{1111}+\\mathbb {P}_{1011}},\\\\&\\mathbb {P}_{Y0}=\\frac{\\mathbb {P}_{0111}}{\\mathbb {P}_{0111}+\\mathbb {P}_{0011}},\\\\&\\mathbb {P}_{R^Y11}=\\frac{\\mathbb {P}_{1111}+\\mathbb {P}_{1011}}{\\mathbb {P}_{1111}+\\mathbb {P}_{1011}+\\mathbb {P}_{1+10}},\\\\&\\mathbb {P}_{R^Y01}=\\frac{\\mathbb {P}_{0111}+\\mathbb {P}_{0011}}{\\mathbb {P}_{0111}+\\mathbb {P}_{0011}+\\mathbb {P}_{0+10}}.$ In addition, we can identify the following products of parameters based on equations () to (): $\\mathbb {P}_M\\mathbb {P}_{R^M1}$ , $(1-\\mathbb {P}_M)\\mathbb {P}_{R^M0}$ , $\\mathbb {P}_M(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y10}$ and $(1-\\mathbb {P}_M)(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y00}$ .","As a result, when $\\mathbb {P}_M$ is known, one can solve for $\\mathbb {P}_{R^M1}$ , $\\mathbb {P}_{R^M0}$ , $\\mathbb {P}_{R^Y10}$ and $\\mathbb {P}_{R^Y00}$ .","Based on the observable data probabilities, $\\mathbb {P}_{Y1}=\\frac{6}{7}$ , $\\mathbb {P}_{Y0}=\\frac{1}{2}$ , $\\mathbb {P}_{R^Y11} = \\frac{7}{9}$ , $\\mathbb {P}_{R^Y01} = \\frac{2}{3}$ and $\\mathbb {P}_M$ is bounded between $\\frac{9}{20}$ and $\\frac{14}{20}$ .","For example, let $\\mathbb {P}_M=\\frac{12}{20}$ , we have $\\mathbb {P}_{R^M1}=\\frac{3}{4}$ , $\\mathbb {P}_{R^M0}=\\frac{3}{4}$ , $\\mathbb {P}_{R^Y10}=\\frac{7}{10}$ , $\\mathbb {P}_{R^Y00}=\\frac{1}{5}$ .","This set of parameter values give us the following joint probabilities of $M$ and $Y$ as $\\mathbb {P}(Y=1,M=1) =\\frac{18}{35} $ , $\\mathbb {P}(Y=0,M=1) = \\frac{3}{35}$ , $\\mathbb {P}(Y=1,M=0) = \\frac{1}{5}$ , and $\\mathbb {P}(Y=0,M=0) = \\frac{1}{5}$ .","Alternatively, let $\\mathbb {P}_M=\\frac{13}{20}$ , we have $\\mathbb {P}_{R^M1}=\\frac{9}{13}$ , $\\mathbb {P}_{R^M0}=\\frac{6}{7}$ , $\\mathbb {P}_{R^Y10}=\\frac{21}{40}$ and $\\mathbb {P}_{R^Y00}=\\frac{2}{5}$ .","This alternative set of parameter values give us the following joint probabilities of $M$ and $Y$ as $\\mathbb {P}(Y=1,M=1) = \\frac{39}{70}$ , $\\mathbb {P}(Y=0,M=1) =\\frac{13}{140} $ , $\\mathbb {P}(Y=1,M=0) = \\frac{7}{40}$ , and $\\mathbb {P}(Y=0,M=0) = \\frac{7}{40}$ .","The two sets of values of $\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\mathbb {P}_{1111},\\mathbb {P}_{0111},\\mathbb {P}_{1011},\\mathbb {P}_{0011},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{++00})$ , and therefore, $\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions.","Consider the following probabilities from the observable data: $(\\mathbb {P}_{1111},\\mathbb {P}_{0111},\\mathbb {P}_{1011},\\mathbb {P}_{0011},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{++00})=\\left(\\frac{18}{96},\\frac{6}{96},\\frac{3}{96},\\frac{2}{96},\\frac{12}{96},\\frac{3}{96},\\frac{15}{96},\\frac{16}{96},\\frac{21}{96}\\right)\\nonumber .$ Define $&\\mathbb {P}_M=\\mathbb {P}(M=1),\\\\&\\mathbb {P}_{Y1}=\\mathbb {P}(Y=1\\mid M=1),\\\\&\\mathbb {P}_{Y0}=\\mathbb {P}(Y=1\\mid M=0),\\\\&\\mathbb {P}_{R^M1}=\\mathbb {P}(R^M=1\\mid M=1),\\\\&\\mathbb {P}_{R^M0}=\\mathbb {P}(R^M=1\\mid M=0),\\\\&\\mathbb {P}_{R^Y00}=\\mathbb {P}(R^Y=1\\mid M=0,Y=0),\\\\&\\mathbb {P}_{R^Y01}=\\mathbb {P}(R^Y=1\\mid M=0,Y=1),\\\\&\\mathbb {P}_{R^Y10}=\\mathbb {P}(R^Y=1\\mid M=1,Y=0),\\\\&\\mathbb {P}_{R^Y11}=\\mathbb {P}(R^Y=1\\mid M=1,Y=1).$ Below we study the identifiablity of the above 9 parameters describing the graphical model in Figure REF based on the observable data probabilities.","Note that although there are 9 observable data probabilities, the degree of freedom in the probabilities is only 8 given that they sum up to 1.","The following relationships between the observable data probabilities and the parameters hold, $&\\mathbb {P}_{1111}=\\mathbb {P}_M\\mathbb {P}_{Y1}\\mathbb {P}_{R^M1}\\mathbb {P}_{R^Y11},\\\\&\\mathbb {P}_{1011}=\\mathbb {P}_M(1-\\mathbb {P}_{Y1})\\mathbb {P}_{R^M1}\\mathbb {P}_{R^Y10},\\\\&\\mathbb {P}_{0111}=(1-\\mathbb {P}_M)\\mathbb {P}_{Y0}\\mathbb {P}_{R^M0}\\mathbb {P}_{R^Y01},\\\\&\\mathbb {P}_{0011}=(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y0})\\mathbb {P}_{R^M0}\\mathbb {P}_{R^Y00},\\\\&\\mathbb {P}_{1+10}=\\mathbb {P}_M\\mathbb {P}_{Y1}\\mathbb {P}_{R^M1}(1-\\mathbb {P}_{R^Y11})+\\mathbb {P}_M(1-\\mathbb {P}_{Y1})\\mathbb {P}_{R^M1}(1-\\mathbb {P}_{R^Y10}),\\\\&\\mathbb {P}_{0+10}=(1-\\mathbb {P}_M)\\mathbb {P}_{Y0}\\mathbb {P}_{R^M0}(1-\\mathbb {P}_{R^Y01})+(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y0})\\mathbb {P}_{R^M0}(1-\\mathbb {P}_{R^Y00}),\\\\&\\mathbb {P}_{+101}=\\mathbb {P}_M\\mathbb {P}_{Y1}(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y11}+(1-\\mathbb {P}_M)\\mathbb {P}_{Y0}(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y01},\\\\&\\mathbb {P}_{+001}=\\mathbb {P}_M(1-\\mathbb {P}_{Y1})(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y10}+(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y0})(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y00}.$ By solving the equations (REF ) to (), we can identify the parameters $\\mathbb {P}_{M}$ , $\\mathbb {P}_{R^M1}$ and $\\mathbb {P}_{R^M0}$ .","Given the observable data probabilities, $\\mathbb {P}_{M} = \\frac{1}{2}$ , $\\mathbb {P}_{R^M1} = \\frac{3}{4}$ and $\\mathbb {P}_{R^M0} =\\frac{1}{2} $ .","However, $\\mathbb {P}_{Y1}$ , $\\mathbb {P}_{Y0}$ , $\\mathbb {P}_{R^Y11}$ , $\\mathbb {P}_{R^Y10}$ , $\\mathbb {P}_{R^Y01}$ and $\\mathbb {P}_{R^Y00}$ are not identifiable.","For example, we can have $\\mathbb {P}_{Y1}=\\frac{3}{4}$ , $\\mathbb {P}_{Y0}=\\frac{1}{2}$ , $\\mathbb {P}_{R^Y11}=\\frac{4}{6}$ , $\\mathbb {P}_{R^Y10}=\\frac{2}{6}$ , $\\mathbb {P}_{R^Y01}=\\frac{3}{6}$ and $\\mathbb {P}_{R^Y00}=\\frac{1}{6}$ , which in turn give us $\\mathbb {P}(Y=1, M=1)=\\frac{3}{8}$ , $\\mathbb {P}(Y=1, M=0)=\\frac{1}{4}$ , $\\mathbb {P}(Y=0, M=1)=\\frac{1}{8}$ and $\\mathbb {P}(Y=0, M=0)=\\frac{1}{4}$ .","Alternatively, we can have $\\mathbb {P}_{Y1}=\\frac{2}{3}$ , $\\mathbb {P}_{Y0}=\\frac{3}{4}$ , $\\mathbb {P}_{R^Y11}=\\frac{3}{4}$ , $\\mathbb {P}_{R^Y10}=\\frac{1}{4}$ , $\\mathbb {P}_{R^Y01}=\\frac{1}{3}$ and $\\mathbb {P}_{R^Y00}=\\frac{1}{3}$ , which in turn give us $\\mathbb {P}(Y=1, M=1)=\\frac{1}{3}$ , $\\mathbb {P}(Y=1, M=0)=\\frac{3}{8}$ , $\\mathbb {P}(Y=0, M=1)=\\frac{1}{6}$ and $\\mathbb {P}(Y=0, M=0)=\\frac{1}{8}$ .","The two sets of values of $\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\mathbb {P}_{1111},\\mathbb {P}_{0111},\\mathbb {P}_{1011},\\mathbb {P}_{0011},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{++00})$ , and therefore, $\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions.","Consider the following probabilities from the observable data: $&&(\\mathbb {P}_{1211},\\mathbb {P}_{1111},\\mathbb {P}_{1011},\\mathbb {P}_{0211},\\mathbb {P}_{0111},\\mathbb {P}_{0011},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{+201},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{++00})\\nonumber \\\\&=&\\left(\\frac{180}{1440},\\frac{60}{1440},\\frac{15}{1440},\\frac{120}{1440},\\frac{30}{1440},\\frac{15}{1440},\\frac{285}{1440},\\frac{195}{1440},\\frac{180}{1440},\\frac{50}{1440},\\frac{20}{1440},\\frac{290}{1440}\\right)\\nonumber .$ Define $&\\mathbb {P}_M=\\mathbb {P}(M=1),\\\\&\\mathbb {P}_{Y^21}=\\mathbb {P}(Y=2\\mid M=1),\\\\&\\mathbb {P}_{Y^11}=\\mathbb {P}(Y=1\\mid M=1),\\\\&\\mathbb {P}_{Y^20}=\\mathbb {P}(Y=2\\mid M=0),\\\\&\\mathbb {P}_{Y^10}=\\mathbb {P}(Y=1\\mid M=0),\\\\&\\mathbb {P}_{R^M1}=\\mathbb {P}(R^M=1\\mid M=1),\\\\&\\mathbb {P}_{R^M0}=\\mathbb {P}(R^M=1\\mid M=0),\\\\&\\mathbb {P}_{R^Y2}=\\mathbb {P}(R^Y=1\\mid Y=2),\\\\&\\mathbb {P}_{R^Y1}=\\mathbb {P}(R^Y=1\\mid Y=1),\\\\&\\mathbb {P}_{R^Y0}=\\mathbb {P}(R^Y=1\\mid Y=0).$ The following relationships between the observable data probabilities and the parameters hold, $&\\mathbb {P}_{1211}=\\mathbb {P}_M\\mathbb {P}_{Y^21}\\mathbb {P}_{R^M1}\\mathbb {P}_{R^Y2},\\\\&\\mathbb {P}_{1111}=\\mathbb {P}_M\\mathbb {P}_{Y^11}\\mathbb {P}_{R^M1}\\mathbb {P}_{R^Y1},\\\\&\\mathbb {P}_{1011}=\\mathbb {P}_M(1-\\mathbb {P}_{Y^21}-\\mathbb {P}_{Y^11})\\mathbb {P}_{R^M1}\\mathbb {P}_{R^Y0},\\\\&\\mathbb {P}_{0211}=(1-\\mathbb {P}_M)\\mathbb {P}_{Y^20}\\mathbb {P}_{R^M0}\\mathbb {P}_{R^Y2},\\\\&\\mathbb {P}_{0111}=(1-\\mathbb {P}_M)\\mathbb {P}_{Y^10}\\mathbb {P}_{R^M0}\\mathbb {P}_{R^Y1},\\\\&\\mathbb {P}_{0011}=(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y^20}-\\mathbb {P}_{Y^10})\\mathbb {P}_{R^M0}\\mathbb {P}_{R^Y0},\\\\&\\mathbb {P}_{+201}=\\mathbb {P}_M\\mathbb {P}_{Y^21}(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y2}+(1-\\mathbb {P}_M)\\mathbb {P}_{Y^20}(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y2},\\\\&\\mathbb {P}_{+101}=\\mathbb {P}_M\\mathbb {P}_{Y^11}(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y1}+(1-\\mathbb {P}_M)\\mathbb {P}_{Y^10}(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y1},\\\\&\\mathbb {P}_{+001}=\\mathbb {P}_M(1-\\mathbb {P}_{Y^21}-\\mathbb {P}_{Y^11})(1-\\mathbb {P}_{R^M1})\\mathbb {P}_{R^Y0}+\\nonumber \\\\&\\hspace{42.67912pt}(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y^20}-\\mathbb {P}_{Y^10})(1-\\mathbb {P}_{R^M0})\\mathbb {P}_{R^Y0},\\\\&\\mathbb {P}_{1+10}=\\mathbb {P}_M\\mathbb {P}_{Y^21}\\mathbb {P}_{R^M1}(1-\\mathbb {P}_{R^Y2})+\\mathbb {P}_M\\mathbb {P}_{Y^11}\\mathbb {P}_{R^M1}(1-\\mathbb {P}_{R^Y1})+\\nonumber \\\\&\\hspace{42.67912pt}\\mathbb {P}_M(1-\\mathbb {P}_{Y^21}-\\mathbb {P}_{Y^11})\\mathbb {P}_{R^M1}(1-\\mathbb {P}_{R^Y0}),\\\\&\\mathbb {P}_{0+10}=(1-\\mathbb {P}_M)\\mathbb {P}_{Y^20}\\mathbb {P}_{R^M0}(1-\\mathbb {P}_{R^Y2})+(1-\\mathbb {P}_M)\\mathbb {P}_{Y^10}\\mathbb {P}_{R^M0}(1-\\mathbb {P}_{R^Y1})+\\nonumber \\\\&\\hspace{42.67912pt}(1-\\mathbb {P}_M)(1-\\mathbb {P}_{Y^20}-\\mathbb {P}_{Y^10})\\mathbb {P}_{R^M0}(1-\\mathbb {P}_{R^Y0}).$ By solving the equations (REF ) to (), we can identify the parameters $\\mathbb {P}_{M}$ , $\\mathbb {P}_{R^M1}$ and $\\mathbb {P}_{R^M0}$ .","Given the observable data probabilities, $\\mathbb {P}_{M}=\\frac{1}{2}$ , $\\mathbb {P}_{R^M1}=\\frac{3}{4}$ and $\\mathbb {P}_{R^M0}=\\frac{1}{2}$ .","However, $\\mathbb {P}_{Y^21}$ , $\\mathbb {P}_{Y^11}$ , $\\mathbb {P}_{Y^20}$ , $\\mathbb {P}_{Y^10}$ , $\\mathbb {P}_{R^Y2}$ , $\\mathbb {P}_{R^Y1}$ and $\\mathbb {P}_{R^Y0}$ are not identifiable.","For example, we can have $\\mathbb {P}_{Y^21}=\\frac{3}{6}$ , $\\mathbb {P}_{Y^11}=\\frac{2}{6}$ , $\\mathbb {P}_{Y^20}=\\frac{2}{4}$ , $\\mathbb {P}_{Y^10}=\\frac{1}{4}$ , $\\mathbb {P}_{R^Y2}=\\frac{2}{3}$ , $\\mathbb {P}_{R^Y1}=\\frac{1}{3}$ and $\\mathbb {P}_{R^Y0}=\\frac{1}{6}$ , which in turn give us $\\mathbb {P}(Y=2, M=1)=\\frac{1}{4}$ , $\\mathbb {P}(Y=1, M=1)=\\frac{1}{6}$ , $\\mathbb {P}(Y=0, M=1)=\\frac{1}{12}$ , $\\mathbb {P}(Y=2, M=0)=\\frac{1}{4}$ , $\\mathbb {P}(Y=1, M=0)=\\frac{1}{8}$ and $\\mathbb {P}(Y=0, M=0)=\\frac{1}{8}$ .","Alternatively, we can have $\\mathbb {P}_{Y^21}=\\frac{5}{8}$ , $\\mathbb {P}_{Y^11}=\\frac{1}{4}$ , $\\mathbb {P}_{Y^20}=\\frac{5}{8}$ , $\\mathbb {P}_{Y^10}=\\frac{3}{16}$ , $\\mathbb {P}_{R^Y2}=\\frac{8}{15}$ , $\\mathbb {P}_{R^Y1}=\\frac{4}{9}$ and $\\mathbb {P}_{R^Y0}=\\frac{2}{9}$ , which in turn give us $\\mathbb {P}(Y=2, M=1)=\\frac{5}{16}$ , $\\mathbb {P}(Y=1, M=1)=\\frac{1}{8}$ , $\\mathbb {P}(Y=0, M=1)=\\frac{1}{16}$ , $\\mathbb {P}(Y=2, M=0)=\\frac{5}{16}$ , $\\mathbb {P}(Y=1, M=0)=\\frac{3}{32}$ and $\\mathbb {P}(Y=0, M=0)=\\frac{3}{32}$ .","The two sets of values of $\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\mathbb {P}_{1211},\\mathbb {P}_{1111},\\mathbb {P}_{1011},\\mathbb {P}_{0211},\\mathbb {P}_{0111},\\mathbb {P}_{0011},\\mathbb {P}_{1+10},\\mathbb {P}_{0+10},\\mathbb {P}_{+201},\\mathbb {P}_{+101},\\mathbb {P}_{+001},\\mathbb {P}_{++00})$ , and therefore, $\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions."],["Parametric Estimation Details","For illustration, we describe the parametric methods in the scenarios considered in Theorem REF when the missingness exists only in the mediator.","Under Assumption REF , the log of the complete-data likelihood is $\\ell _{c}(\\theta ) = \\mathrm {constant}+\\sum ^n_{i=1} &\\log ~\\mathbb {P}(Y_i=y_i\\mid M_i=m_i,T_i=t_i, X_i = x_i )+\\log ~\\mathbb {P}(M_i=m_i \\mid T_i=t_i, X_i = x_i )\\\\& + \\log ~\\mathbb {P}(R^M_i=r^M_i \\mid M_i=m_i,T_i=t_i, X_i = x_i ).$ Under Assumption REF , the observed-data likelihood is $L_{obs}(\\theta )\\propto \\prod _{\\lbrace i:R^M_i=1\\rbrace } &\\mathbb {P}(Y_i=y_i\\mid M_i=m_i,T_i=t_i, X_i = x_i )\\mathbb {P}(M_i=m_i \\mid T_i=t_i, X_i = x_i )\\\\&\\mathbb {P}(R^M_i=1 \\mid M_i=m_i,T_i=t_i, X_i = x_i )\\\\&\\hspace{-39.83368pt}\\prod _{\\lbrace i:R^M_i=0\\rbrace }\\int _{\\mathcal {M}} \\mathbb {P}(Y_i=y_i\\mid M_i=m,T_i=t_i, X_i = x_i )\\mathbb {P}(M_i=m \\mid T_i=t_i, X_i = x_i )\\\\&\\mathbb {P}(R^M_i=0 \\mid M_i=m,T_i=t_i, X_i = x_i )\\ d m.$ When $M$ is categorical, the integral involved in the above expression is reduced to summation.","Since the value of $M$ is missing for some subjects, we implement the Expectation-Maximization algorithm to obtain the Maximum Likelihood Estimates (MLEs) by treating the missing $M$ as a latent variable.","Specifically, in the E-step, we find the conditional expectation of complete-data log likelihood by calculating the conditional expectation of $M_i$ for subjects with missing $M_i$ .","For example, if $M$ is binary, $E\\lbrace I(M_{i}=m)\\mid Y_i,R^M_i=0,T_i, X_i ;\\theta ^{(t)}\\rbrace =\\frac{\\mathbb {P}(Y_i,M_{i}=m,R^M_i=0\\mid T_i, X_i )}{\\sum _{m={0,1}}\\mathbb {P}(Y_i,M_{i}=m,R^M_i=0\\mid T_i, X_i )}.$ When $M$ is continuous, the conditional expectation of complete-data log likelihood may be complicated to calculate.","Therefore, we applied fractional imputation [16] using the idea of importance sampling and weighting method to approximate the conditional expectation.","Specifically, we generate the fractionally imputed data $m^{(1)}_{i}, \\ldots , m^{(S)}_{i}$ from a proposed distribution $h(M_{i}\\mid T_i, X_i )$ for subjects with missing $M_i$ .","Then, we compute the fractional weight for each imputed observation.","The Monte Carlo approximation of the conditional expectation becomes more accurate when $S$ is large: $E\\lbrace \\ell _{ci}(M_{i}=m;\\theta )|Y_i,R^M_i=0,T_i, X_i ;\\theta ^{(t)}\\rbrace \\approx \\frac{1}{S}\\sum \\limits _{j=1}^{S} \\ell _{ci}(M_{i}=m^{(j)}_{i};\\theta ) \\hat{w}(m^{(j)}_{i}),$ where $\\hat{w}(m^{(j)}_{i}) \\propto \\frac{\\mathbb {P}(Y_{i},M_{i}=m^{(j)}_{i},R^M_i=0\\mid T_i, X_i )}{h(M_{i}=m^{(j)}_{i}\\mid T_i, X_i )}$ is the fractional weight for $m^{(j)}_{i}$ that satisfy $\\hat{w}(m^{(j)}_{i}) \\ge 0$ and $\\sum _{j=1}^{S} \\hat{w}(m^{(j)}_{i})=1$ .","We iterate between the E-step and M-step until convergence.","The same estimation methods can be applied to the situation where the missingness exists in both the mediator and outcome.","For subjects with both $M_i$ and $Y_i$ missing, we generate the imputed data sequentially.","For binary $M$ and binary $Y$ , we generate the possible value of ($m_i$ , $y_i$ ).","For binary $M$ and continuous $Y$ , we generate the possible value of $m_i$ and then the fractionally imputed data $y^{(1)}_{i}, \\ldots , y^{(S)}_{i}$ for each possible value of $m_i$ .","For continuous $M$ and continuous $Y$ , we generate the fractionally imputed data $(m^{(1)}_{i},y^{(1)}_{i})$ ,...,$(m^{(S)}_{i},y^{(S)}_{i})$ .","For continuous $M$ and binary $Y$ , we generate the fractionally imputed data $m^{(1)}_{i}, \\ldots , m^{(S)}_{i}$ and then the possible value of $y_i$ for each fractionally imputed $m_i$ .","It is important to note that we can identify the outcome model using the complete cases under MNAR Assumption REF , REF and REF .","Therefore, an alternative approach for those scenarios is to estimate the outcome model first using the complete cases, then estimate the parameters in other models through the Expectation-Maximization algorithm by plugging in the estimated outcome model.","We tried those two slightly different approaches to our simulation settings, both provided consistent results, with the alternative approach enjoying higher computation efficiency as expected.","Note that under MNAR Assumption REF , the alternative approach does not work because the outcome distribution $\\mathbb {P}(Y \\mid M, T, X )$ can not be identified by the complete cases."]],"string":"[\n [\n \"Mediation analysis with the mediator and outcome missing not at random\"\n ],\n [\n \"Abstract Mediation analysis is widely used for investigating direct and indirect causal pathways through which an effect arises.\",\n \"However, many mediation analysis studies are often challenged by missingness in the mediator and outcome.\",\n \"In general, when the mediator and outcome are missing not at random, the direct and indirect effects are not identifiable without further assumptions.\",\n \"In this work, we study the identifiability of the direct and indirect effects under some interpretable missing not at random mechanisms.\",\n \"We evaluate the performance of statistical inference under those assumptions through simulation studies and illustrate the proposed methods via the National Job Corps Study.\"\n ],\n [\n \"Introduction to mediation analysis and the National Job Corps Study\",\n \"Mediation analysis is increasingly adopted by researchers in a variety of fields, including epidemiology and social sciences, to test specific theories about the underlying mechanism through which an effect arises.\",\n \"In a typical mediation analysis, the average treatment effect on an outcome is decomposed into a natural indirect effect (NIE) that operates through the mediator of interest and a natural direct effect (NDE) that operates through other pathways.\",\n \"The NIE and NDE are identified under the sequential ignorability assumption [14], [15], that is, when there is no unmeasured pretreatment confounding in the treatment-mediator and treatment-outcome relationships and there is no post-treatment confounding or unmeasured pretreatment confounding in the mediator-outcome relationship.\",\n \"With the identification results, the NIE and NDE can be estimated through various approaches, such as regression [32], weighting [11], [13] and simulation-based strategy [14], [15].\",\n \"However, missing data are prevalent in empirical research and it is often a concern that the missingness is not at random [20].\",\n \"Missing not at random occurs if the probability of data being missing depends on unobserved data even conditional on observed data.\",\n \"The motivation of our study comes from the well-known National Job Corps Study (NJCS).\",\n \"The NJCS is a randomized nation wide evaluation of Job Corps, which is the largest education and vocational training program administered by the U.S. Department of Labor for 16 to 24 years old youths who are unemployed and disconnected from school.\",\n \"Past research showed that Job Corps successfully improves those disadvantaged youths' employment and increases their earnings [31], [17], [37].\",\n \"Given that education and vocational training are the central elements of the program, [27] further proposed to evaluate how much of the Job Corps effect on earnings is mediated by educational and vocational attainment and found a significant positive mediated effect.\",\n \"However, the analysis is challenged by missingness both in the mediator and the outcome.\",\n \"In the NJCS study, the assignment to either Job Corps program or the control group was random.\",\n \"The mediator describes whether or not the subject obtained an education credential or vocational certificate after randomization, and the outcome describes subject's weekly earnings in the fourth year after randomization.\",\n \"The mediator and outcome information were collected at the 30-months and 48-months follow-up, respectively, through in-person interviews.\",\n \"And the missing rates of the mediator and the outcome are both higher than $20\\\\%$ .\",\n \"Qin et al.\",\n \"(2019) addressed the missing data problem through non-response weighting, which is a valid approach if the missingness is random given the observed pretreatment covariates.\",\n \"However, we are concerned that the missingness is likely not at random.\",\n \"Conceivably, people who failed to obtain an education credential or vocational certificate may be less likely to provide the information compared to people who successfully obtained a credential or certificate.\",\n \"One could also imagine that people who had no earnings may be less willing to report the amount of earnings in the interview.\",\n \"As a result, the chance of missingness would depend on the unobserved missing value itself, missing not at random.\",\n \"In such a scenario, commonly adopted strategies to deal with missing data, such as complete case analysis, multiple imputation or non-response weighting, may fail to provide valid inference.\",\n \"Missingness not at random presents a challenge for causal inference, because in general, the underlying data distribution can not be identified without further assumptions.\",\n \"This topic has attracted some attention in the literature.\",\n \"For example, [7] and [36] studied the identifiability of subgroup treatment effects and average treatment effects, respectively, with covariates missing not at random.\",\n \"In the context of instrumental variable analysis, methods were developed when the missingness in the covariates [35] or in the outcome [9], [26], [4] is not at random.\",\n \"Previous research also studied the identifiability of graphical models with a binary outcome missing not at random in longitudinal studies [21], and reviewed missing data research in graphical models [22].\",\n \"However, limited effort has been made to study the identifiability of causal mediation effects with the mediator and outcome missing not at random.\",\n \"Considering missingness in the outcome only, [19] utilized an instrumental variable type of covariate to identify the direct and indirect effects when the missingness in the outcome depends on the outcome value itself.\",\n \"To apply their method, we need a covariate that is associated with the outcome, however, is conditionally independent of the missingness of the outcome.\",\n \"In many studies, such a covariate may not be available.\"\n ],\n [\n \"Organization of the paper\",\n \"In this paper, we study the identification of the causal mediation effects with mediator and outcome missing not at random.\",\n \"We provide conditions for identification under various interpretable missing not at random assumptions, and also provide examples for some missingness mechanisms where the identification cannot be achieved without further assumptions.\",\n \"The rest of the paper is organized as follows.\",\n \"We introduce notation and basic assumptions for causal mediation analysis in section .\",\n \"In section , we discuss the identification in a simple setup where the missingness exists only in the mediator and depends on the missing mediator value itself.\",\n \"We then extend the results to the more complicated setup where both the mediator and the outcome have missing data in section .\",\n \"Expectation-maximization algorithms are used for estimation and in section , extensive simulation studies are conducted to test our theoretical results and to evaluate the performance of the proposed methods.\",\n \"We apply our method to the NJCS data in section and develop a sensitivity analysis approach to study the robustness of conclusions on the causal mediation effects if the underlying missingness mechanism is beyond the mechanisms that allow identification.\",\n \"We conclude with a discussion and provide all proofs of the theorems in the Supplementary Material.\"\n ],\n [\n \"Notation and some basic definitions\",\n \"Let $A\\\\perp \\\\!\\\\!\\\\!\\\\perp B\\\\mid C$ denote that the random variables $A$ and $B$ are conditionally independent given the random variable $C$ .\",\n \"Further, the property of completeness [18], [2] will play a key role in our nonparametric identification of the mediation effects.\",\n \"Define a function $f(A,B)$ to be complete in $B$ if $\\\\int g(A)f(A,B)d\\\\nu (A)=0$ implies $g(A)=0$ almost surely for any square-integrable function $g$ .\",\n \"In the above integral, $\\\\nu (\\\\cdot )$ presents a generic measure, which is the Lebesgue measure for a continuous variable and the counting measure for a discrete variable.\"\n ],\n [\n \"Review of mediation analysis without missing data\",\n \"Let $T$ denote the binary treatment assignment, with $t=0$ and $t=1$ representing the control condition and the experimental condition, respectively.\",\n \"Let $ X $ be the vector of measured pre-treatment covariates.\",\n \"We use $M$ and $Y$ to denote the mediator and the outcome, respectively.\",\n \"We adopt the potential outcomes framework [23], [29] to define the causal effects of interest and make the stable unit treatment value assumption (SUTVA) that there is no hidden variations of each treatment condition and there is no interference between individuals.\",\n \"Specifically, we use $M_i(t)$ to denote individual $i$ 's potential mediator value under treatment $t$ for $t=0, 1$ , and use $Y_i(t)$ to denote individual $i$ 's potential outcome value under treatment $t$ for $t=0, 1$ .\",\n \"$T$ potentially may affect $Y$ through $M$ , hence $Y_i(t)$ can be equivalently written as $Y_i(t,M_i(t))$ .\",\n \"Given the above notation, the average treatment effect ($\\\\mathrm {ATE}$ ) on the outcome can be defined as $\\\\mathbb {E}\\\\lbrace Y(1)-Y(0)\\\\rbrace $ where the expectation is taken over the population of interest.\",\n \"Define the nested potential outcome, $Y_i(1,M_i(0))$ , to describe individual $i$ 's potential outcome under the experimental condition, however, with the mediator counter factually taking its value under the control condition [28].\",\n \"The $\\\\mathrm {ATE}$ can be decomposed into [25]: $\\\\mathrm {ATE} = \\\\mathrm {NIE}+\\\\mathrm {NDE},$ where $\\\\mathrm {NIE} = \\\\mathbb {E}\\\\lbrace Y(1,M(1))-Y(1,M(0))\\\\rbrace $ is the natural indirect effect ($\\\\mathrm {NIE}$ ) and $\\\\mathrm {NDE} = \\\\mathbb {E}\\\\lbrace Y(1,M(0))-Y(0,M(0))\\\\rbrace $ is the natural direct effect ($\\\\mathrm {NDE}$ ).\",\n \"The $\\\\mathrm {NIE}$ quantifies the average treatment effect on the outcome transmitted through the treatment induced change in the mediator from $M_i(0)$ to $M_i(1)$ ; and the $\\\\mathrm {NDE}$ quantifies the direct effect of the treatment on the outcome that does not operate through its impact on the mediator.\",\n \"We invoke the standard sequential ignorability assumption [14], [15] throughout the paper.\",\n \"Assumption 1 (Sequential ignorability).\",\n \"$\\\\lbrace Y(t^{\\\\prime }, m), M(t)\\\\rbrace \\\\perp \\\\!\\\\!\\\\!\\\\perp T \\\\mid X = x $ , and $Y(t^{\\\\prime },m) \\\\perp \\\\!\\\\!\\\\!\\\\perp M(t)\\\\mid T=t, X = x $ , for $t, t^{\\\\prime } \\\\in \\\\lbrace 0, 1\\\\rbrace $ and all $ X \\\\in \\\\mathcal {X}$ .\",\n \"In addition, $0 < \\\\mathbb {P}(T=t\\\\mid X = x )<1$ and $0 < \\\\mathbb {P}\\\\lbrace M(t)=m \\\\mid T=t, X = x \\\\rbrace <1$ for $t, t^{\\\\prime } \\\\in \\\\lbrace 0, 1\\\\rbrace $ , and all $ X \\\\in \\\\mathcal {X}$ and $m\\\\in \\\\mathcal {M}$ .\",\n \"The notation $\\\\mathcal {X}$ and $\\\\mathcal {M}$ denote the supports of the random variables $X$ and $M$ , respectively.\",\n \"Assumption REF says that both the treatment assignment and the mediator value under each treatment condition can be viewed as if randomized within levels of the observed pre-treatment covariates.\",\n \"Under Assumption REF without missing data, we have the following mediation formula [25], [14]: $\\\\mathbb {E}\\\\lbrace Y(t, M(t^{\\\\prime }))\\\\mid X = x \\\\rbrace = \\\\int _{\\\\mathcal {M}}\\\\mathbb {E}(Y\\\\mid M=m, T=t, X = x )\\\\,d F(m\\\\mid T=t^{\\\\prime }, X = x ).$ When there exists missing data, the key for the identification of the $\\\\mathrm {NIE}$ and $\\\\mathrm {NDE}$ would be to identify the probabilities $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x )$ and $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )$ , or equivalently, the joint probability $\\\\mathbb {P}(Y=y, M=m\\\\mid T=t, X = x )$ , from the observable data.\"\n ],\n [\n \"Missingness only in the mediator\",\n \"In this section, we consider a simple setup where the mediator has missing values and the outcome is fully observed.\",\n \"It may happen in studies where the outcome is of primary interest with the mediator being a secondary outcome of interest.\",\n \"Let $R^M$ be the missingness indicator for $M$ such that $R^M=1$ if $M$ is observed and $R^M=0$ if $M$ is missing.\",\n \"When $R^M \\\\perp \\\\!\\\\!\\\\!\\\\perp (Y,M,T, X )$ , the missingness mechanism is missing completely at random (MCAR), and the complete case analysis is enough to provide consistent estimates of $\\\\mathbb {P}(Y,M \\\\mid T, X )$ .\",\n \"When $R^M \\\\perp \\\\!\\\\!\\\\!\\\\perp M \\\\mid (Y,T, X )$ , the missingness mechanism is missing at random (MAR), and the joint distribution $\\\\mathbb {P}(Y,M \\\\mid T, X )$ can be consistently estimated from the observed data.\",\n \"However, as we explained in section , often we have the concern that the missingness of $M$ may depend on the value of $M$ itself even conditional on other observed data.\",\n \"In such a case, the missingness mechanism is missing not at random (MNAR).\",\n \"Since the outcome $Y$ occurs after the mediator $M$ , it is plausible in many studies to assume that the missingness of $M$ is conditionally independent of $Y$ .\",\n \"Based on the above discussion, we propose the following MNAR assumption when missing only the mediator: Assumption 2 $R^M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (M,T, X )$ and $Y$ is fully observed.\",\n \"Assumption REF allows $R^M$ to depend on the value of the mediator $M$ , $T$ and $ X $ .\",\n \"However, we assume $R^M$ to be conditionally independent of the outcome $Y$ given $M$ , $T$ and $ X $ .\",\n \"The direct acyclic graphs in Figure REF illustrate the different missingness mechanisms under MCAR, MAR and our proposed MNAR mechanism in Assumption REF , respectively.\",\n \"Figure: Direct acyclic graphs describing MCAR, MAR and the proposed MNAR missingness mechanisms when the missingness exists only in the mediator.\",\n \"All graphs condition on X X and allow X X to have arrows to all other variables in the graphs.The following theorem presents the nonparametric identification results under the proposed MNAR assumption: Theorem 1 Under Assumptions REF and REF , if the joint distribution $\\\\mathbb {P}(Y,M,R^M=1 \\\\mid T=t, X=x )$ is complete in $Y$ for all $t$ and $x$ , and $\\\\mathbb {P}(R^M=1\\\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m, t, x $ , the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ is identifiable, and therefore, the $\\\\mathrm {NIE}$ and $\\\\mathrm {NDE}$ are identifiable.\",\n \"For sufficient conditions on completeness, we refer readers to [8] for a comprehensive discussion.\",\n \"The assumption of completeness is routinely made in nonparametric identification problems [5], [36], [1].\",\n \"For discrete $M$ and discrete $Y$ , completeness is equivalent to Rank $(\\\\Theta _{tx})=J$ , where $\\\\Theta _{tx}$ is a $J \\\\times K$ matrix with $\\\\mathbb {P}_{my1\\\\mid t, x}$ as the $(m,y)$ th element.\",\n \"This essentially requires that the number of elements in the support of $Y$ is not smaller than the number of elements in the support of $M$ , and that $M \\\\lnot \\\\!\\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X , R^M=1)$ or equivalently $M \\\\lnot \\\\!\\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ .\",\n \"In fact, under Assumption REF , the identification of $\\\\mathbb {P}(Y\\\\mid M, T, X )$ does not rely on the completeness assumption.\",\n \"The completeness assumption is only needed to identify $\\\\mathbb {P}(M\\\\mid T, X )$ .\",\n \"This implies that in a special case when $M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ and the completeness assumption is violated, since $\\\\mathbb {P}(Y\\\\mid M, T, X ) = \\\\mathbb {P}(Y\\\\mid T, X )$ , both the $\\\\mathrm {NIE}$ and $\\\\mathrm {NDE}$ can still be identified, and $\\\\mathrm {NIE} = 0$ and $\\\\mathrm {NDE} = \\\\mathrm {ATE} = \\\\int _{\\\\mathcal {X}}\\\\lbrace \\\\mathbb {E}(Y\\\\mid T=1, X = x )-\\\\mathbb {E}(Y\\\\mid T=0, X = x )\\\\rbrace \\\\,d F( x ).$ We also illustrate this point using simulation studies.\"\n ],\n [\n \"Missing data mechanisms\",\n \"We now extend the results to the scenario where both the mediator and outcome have missing data.\",\n \"Further define the notation $R^Y$ to be the missingness indicator for $Y$ such that $R^Y=1$ if $Y$ is observed and $R^Y=0$ otherwise.\",\n \"When $(R^Y, R^M) \\\\perp \\\\!\\\\!\\\\!\\\\perp (Y,M,T, X )$ , the missingness mechanism is MCAR; and when $(R^Y,R^M) \\\\perp \\\\!\\\\!\\\\!\\\\perp (Y,M)\\\\mid (T, X )$ , the missingness mechanism is MAR.\",\n \"Continuing to allow the missingness of $M$ to depend on the value of $M$ itself and assume the outcomes occurred later (i.e., $Y$ and $R^Y$ ) not to affect variables measured earlier (i.e., $M$ and $R^M$ ), we consider the following MNAR mechanisms in Assumption REF , REF , and REF , respectively.\",\n \"Assumption 3 $(R^Y, R^M) \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (M,T, X )$ and $R^Y \\\\perp \\\\!\\\\!\\\\!\\\\perp M\\\\mid (R^M,T, X )$ .\",\n \"Assumption 4 $R^Y \\\\perp \\\\!\\\\!\\\\!\\\\perp (M,R^M)\\\\mid (Y,T, X )$ and $R^M \\\\perp \\\\!\\\\!\\\\!\\\\perp (Y,R^Y)\\\\mid (M,T, X )$ .\",\n \"Assumption 5 $Y, R^Y$ and $ R^M$ are mutually independent given $(M,T, X )$ .\",\n \"The direct acyclic graphs in Figure REF illustrate different missingness mechanisms under MCAR, MAR, and MNAR Assumptions REF to REF when the missingness exists both in the mediator and outcome.\",\n \"The differences among the MNAR mechanisms under Assumption REF to REF are in the missingness mechanisms in $Y$ .\",\n \"The MNAR mechanism under Assumption REF allows the chance of missingness in $M$ to depend on the value of $M$ itself in addition to the fully observed variables $T$ and $ X $ , and allows the missingness in $Y$ to depend on the missingness in $M$ in addition to $T$ and $ X $ .\",\n \"It is important to note that the MAR mechanism is a special case of the MNAR mechanism under Assumption REF without allowing $M$ to affect $R^M$ .\",\n \"In the context of NJCS study, this mechanism says that whether or not people successfully obtained a certificate may have an impact on their willingness to report, and their decision on reporting their certificate status may be associated with their decision to report their earnings or not.\",\n \"The MNAR mechanism under Assumption REF allows the missingness of $Y$ to depend on the outcome value $Y$ itself instead of $R^M$ as in Assumption REF .\",\n \"In the NJCS study, it says that the amount of earnings may have an impact on the probability to report earnings, which is also a reasonable concern.\",\n \"Note that both the missingness in $M$ and the missingness in $Y$ are not at random under Assumption REF .\",\n \"The MNAR mechanism under Assumption REF allows the missingness in $Y$ to depend on $M$ instead of $R^M$ or $Y$ , another case where both the missingness in $M$ and that in $Y$ are not at random.\",\n \"In the context of NJCS study, it says that whether or not people successfully obtained a certificate drives the missingness in both $M$ and $Y$ after conditioning on $T$ and $ X $ .\",\n \"Figure: Direct acyclic graphs describing MCAR, MAR and the proposed MNAR missingness mechanisms when the missingness exists both in the mediator and outcome (all graphs condition on X X allowing X X to have directed arrows to all variables in the graphs).\"\n ],\n [\n \"Identification results\",\n \"We now present the nonparametric identification results under each of the MNAR mechanisms described in Assumptions REF to REF .\",\n \"Theorem 2 Under Assumptions REF and REF , if the joint distribution $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X =x)$ is complete in $Y$ for all $t$ and $x$ , and $\\\\mathbb {P}(R^M=1,R^Y=1\\\\mid Y=y,M=m,T=t, X = x ) > 0$ and $\\\\mathbb {P}(R^M=0,R^Y=1\\\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m, t, x $ , the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ is identifiable, and therefore, the NIE and NDE are identifiable.\",\n \"Under the MNAR mechanism described by Assumption REF , $Y\\\\perp \\\\!\\\\!\\\\!\\\\perp (R^M, R^Y)\\\\mid (M, T, X) $ , and therefore, the identification of $\\\\mathbb {P}(Y\\\\mid M, T, X )$ does not rely on the completeness assumption and can be identified by the complete cases, i.e., $\\\\mathbb {P}(Y\\\\mid M, T, X ) = \\\\mathbb {P}(Y\\\\mid M, T, X , R^Y=1, R^M=1)$ .\",\n \"The completeness assumption is used only to identify $\\\\mathbb {P}(M\\\\mid T, X )$ .\",\n \"As a result, when $M\\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X)$ , the NIE and NDE can still be identified as discussed in Section .\",\n \"Theorem 3 Under Assumptions REF and REF , if the joint distribution $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X=x )$ is complete in $Y$ for all $t$ and $x$ , and the joint distribution $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X=x )$ is also complete in $M$ for all $t$ and $x$ , and if $\\\\mathbb {P}(R^M=1,R^Y=1\\\\mid Y=y,M=m,T=t, X = x ) > 0$ , $\\\\mathbb {P}(R^M=0,R^Y=1\\\\mid Y=y,M=m,T=t, X = x ) > 0$ and $\\\\mathbb {P}(R^M=1,R^Y=0\\\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m,t, x $ , the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ is identifiable, and therefore, the NIE and NDE are identifiable.\",\n \"Different from Assumption REF , under the MNAR mechanism described by Assumption REF , $Y$ is not independent of $R^Y$ given $M, T$ and $ X $ , and therefore, the conditional distribution $\\\\mathbb {P}(Y\\\\mid M, T, X )$ can no longer be identified by the complete cases.\",\n \"In fact, the MNAR mechanism under Assumption REF requires completeness in $Y$ and completeness in $M$ to be able to identify both $\\\\mathbb {P}(Y\\\\mid M, T, X )$ and $\\\\mathbb {P}(M\\\\mid T, X )$ .\",\n \"As a result, when $M\\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X) $ , the NIE and NDE can no longer be identified.\",\n \"Further, since the completeness is required in $Y$ and required in $M$ , to achieve identification of $\\\\mathbb {P}(Y,M\\\\mid T, X )$ under Assumption REF , the number of elements in the support of $Y$ has to be the same as the number of elements in the support of $M$ .\",\n \"We present an unidentifiable case when $Y$ has more categories than $M$ in the section of the Supplementary Material.\",\n \"Theorem 4 Under Assumptions REF and REF , if either of the following two conditions holds, the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ is identifiable, and therefore, the NIE and NDE are identifiable: (i) the joint distribution $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X=x )$ is complete in $Y$ for all $t$ and $x$ , and $\\\\mathbb {P}(R^M=1,R^Y=1\\\\mid Y=y,M=m,T=t, X = x ) > 0$ , $\\\\mathbb {P}(R^M=0,R^Y=1\\\\mid Y=y,M=m,T=t, X = x ) > 0$ and $\\\\mathbb {P}(R^M=1,R^Y=0\\\\mid Y=y,M=m,T=t, X = x )>0$ for all $y, m,t, x $ ; (ii) the joint distribution $\\\\mathbb {P}(M,R^M=1,R^Y \\\\mid T=t, X=x )$ is complete in $R^Y$ for all $t$ and $x$ , and $\\\\mathbb {P}(R^M=1\\\\mid R^Y=r^Y, M=m,T=t, X = x )>0$ for all $r^Y, m, t, x $ .\",\n \"Under the MNAR mechanism described by Assumption REF , $\\\\mathbb {P}(Y\\\\mid M, T, X )$ can also be directly identified by the complete cases, i.e., $\\\\mathbb {P}(Y\\\\mid M, T, X ) = \\\\mathbb {P}(Y\\\\mid M, T, X , R^Y=1, R^M=1)$ .\",\n \"Different from the identification under MNAR Assumption REF or 4, there are two ways to identify $\\\\mathbb {P}(M\\\\mid T, X )$ , one is by utilizing the effect of $M$ on $Y$ if exists, i.e., the condition (i), the other one is by utilizing the effect of $M$ on $R^Y$ if exists, i.e., the condition (ii).\",\n \"Note that for the second condition (ii) to hold, $M$ needs to be binary given that $R^Y$ is binary.\",\n \"Assuming $Y \\\\perp \\\\!\\\\!\\\\!\\\\perp R^M\\\\mid (M,T, X )$ , and allowing $M$ to have a impact on $R^M$ , we have shown that the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ can be identified under some completeness assumptions when $R^Y$ only depends on one of $(R^M,Y,M)$ given $T$ and $ X $ .\",\n \"When $R^Y$ depends on more than one of $(R^M,Y,M)$ given $T$ and $ X $ , the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ cannot be identified without further assumptions.\",\n \"The discussions and examples for those unidentifiable cases are provided in the section of the Supplemental Material.\",\n \"So far, we have shown the nonparametric identification results of the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ , NIE and NDE, under various MNAR assumptions.\",\n \"However, the nonparametric estimation for these quantities may suffer from the curse of dimensionality in practice, especially with a large number of covariates.\",\n \"Therefore, we adopt a parametric method to obtain likelihood-based inference through the Expectation-Maximization algorithm [6].\",\n \"The estimation details are provided in the section of the Supplementary Material.\"\n ],\n [\n \"Simulation\",\n \"We conducted simulation studies to evaluate the performance of the proposed estimators under each of the MNAR assumptions described in sections and .\",\n \"In a simple context of a single covariate $X \\\\sim N(0,1)$ and a randomized $T\\\\sim \\\\mathrm {Bernoulli}(0.5)$ , we considered the following four setups representing different relationships in the supports of $M$ and $Y$ : (A) binary $M$ and binary $Y$ , (B) binary $M$ and continuous $Y$ , (C) continuous $M$ and continuous $Y$ , and (D) continuous $M$ and binary $Y$ .\",\n \"We generated the mediator $M$ as follows: $\\\\mathrm {logit}~\\\\mathbb {P}(M=1\\\\mid T,X)=\\\\alpha _0+\\\\alpha _t T+\\\\alpha _x X$ if binary; and $M \\\\sim N(\\\\alpha _0+\\\\alpha _t T+\\\\alpha _x X, 0.5^2)$ if continuous.\",\n \"We then generated the outcome $Y$ according to: $\\\\mathrm {logit}~\\\\mathbb {P}(Y=1\\\\mid M,T,X)=\\\\beta _0+\\\\beta _m M+\\\\beta _t T+\\\\beta _{mt} M \\\\cdot T+\\\\beta _x X$ if binary; and $Y \\\\sim N(\\\\beta _0+\\\\beta _m M+\\\\beta _t T+\\\\beta _{mt} M \\\\cdot T+\\\\beta _x X, 0.5^2)$ if continuous.\",\n \"For each of the four setups described above, we considered missingness mechanisms under (I) Assumption REF , (II) Assumption REF , (III) Assumption REF , and (IV) Assumption REF , respectively, that is, sixteen simulation scenarios in total.\",\n \"Under all MNAR Assumptions REF to REF , $R^M$ is allowed to depends on $M$ , $T$ and $X$ , and therefore, we generated the binary variable $R^M$ as follows: $\\\\mathrm {logit}~\\\\mathbb {P}(R^M=1\\\\mid M,T,X)=\\\\lambda _0+\\\\lambda _m M+\\\\lambda _t T+\\\\lambda _x X.$ Under (I) Assumption REF , $Y$ is fully observed.\",\n \"For scenarios (II) to (IV) with missingness in $Y$ , the data generating models for $R^Y$ varied according to different MNAR Assumptions: $\\\\mathrm {logit}~\\\\mathbb {P}(R^Y=1\\\\mid R^M,T,X)=\\\\gamma _0+\\\\gamma _{r^M} R^M+\\\\gamma _t T+\\\\gamma _x X$ if under (II) Assumption REF ; $\\\\mathrm {logit}~\\\\mathbb {P}(R^Y=1\\\\mid Y,T,X)=\\\\gamma _0+\\\\gamma _y Y+\\\\gamma _t T+\\\\gamma _x X$ if under (III) Assumption REF ; and $\\\\mathrm {logit}~\\\\mathbb {P}(R^Y=1\\\\mid M,T,X)=\\\\gamma _0+\\\\gamma _m M+\\\\gamma _t T+\\\\gamma _x X$ if under (IV) Assumption REF .\",\n \"For each of the sixteen simulation scenarios considered, we tested our theoretical results and evaluated the performance of our method when $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ and when $M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ .\",\n \"Table REF presents the specifications of parameter values when $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , and when $M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , respectively.\",\n \"We set parameter values in the $R^M$ and $R^Y$ models to generate missing rates in $M$ and $Y$ both to be around $20\\\\%$ to $25\\\\%$ , which are similar to the missing rates in our application study.\",\n \"Table: Specifications of the parameter values.\",\n \"1, M¬⊥⊥Y∣(T,X)M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ); 2, M⊥⊥Y∣(T,X)M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ); Setting A, Binary MM and Binary YY; Setting B, Binary MM and Continuous YY; Setting C, Continuous MM and Continuous YY; Setting D, Continuous MM and Binary YY.We considered a sample size of 1000, and simulated 500 data sets for each simulation scenario.\",\n \"We applied the following three methods to compare their results on the estimations of the NIE and NDE: 1) complete case analysis, which provides consistent estimates under MCAR; 2) our proposed methods using the Expectation-Maximization algorithm, which are designed to deal with the MNAR assumptions under concern; and 3) oracle estimators, which are obtained by using the true values of the missing data.\",\n \"Figure REF presents the boxplots of percentages of biases with respect to the true values for each of the simulation scenarios when $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ across the 500 replications.\",\n \"And Figure REF presents the boxplots for the corresponding scenarios when $M\\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ .\",\n \"The simulation results are consistent with the results predicted by our theorems.\",\n \"Under MNAR Assumption REF where the missingness only exists in the mediator, the percentages of biases for both NIE and NDE estimated using our methods are close to zero with slightly larger standard errors than the oracle estimates when $M\\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ and the completeness assumption holds (A.I, B.I, C.I), while both the estimated NIE and NDE from the complete case analysis have substantial biases.\",\n \"When the completeness assumption is violated even though $M\\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ as in the continuous $M$ and binary $Y$ case (D.I), our method would not recover the underlying truth as expected.\",\n \"When $M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , as shown in Figure REF A.I (0) to D.I (0) with $(0)$ indicating that $\\\\mathrm {NIE}=0$ , NIE and NDE from both our methods and the complete case analysis are consistent.\",\n \"This is due to the fact that although $\\\\mathbb {P}(M\\\\mid T, X )$ cannot be identified, the $\\\\mathbb {P}(Y\\\\mid M, T, X )$ can still be identified by the complete cases under Assumption REF .\",\n \"Under MNAR Assumption REF (A.II to D.II, and A.II(0) to D.II(0)) and MNAR Assumption REF (A.IV to D.IV, and A.IV(0) to D.IV(0)), we reached the same conclusions as those under MNAR Assumption REF .\",\n \"Under MNAR Assumption REF , when $M\\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , the completeness assumption is violated in both the binary $M$ and continuous $Y$ (B.III) and the continuous $M$ and binary $Y$ (D.III) cases, and our estimators would not recover the underlying truth.\",\n \"Different from the other MNAR assumptions, when $M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , both the distributions $\\\\mathbb {P}(Y\\\\mid M,T, X )$ and $\\\\mathbb {P}(M\\\\mid T, X )$ cannot be identified, we observed biases in estimating the $\\\\mathrm {NIE}$ and $\\\\mathrm {NDE}$ using either the complete case analysis or our method.\",\n \"Figure: Simulation results when M¬⊥⊥Y∣(T,X)M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ).\",\n \"A, Binary MM and Binary YY; B, Binary MM and Continuous YY; C, Continuous MM and Continuous YY; D, Continuous MM and Binary YY.\",\n \"I, Assumption ; II, Assumption ; III, Assumption ; IV, Assumption .\",\n \"CC, complete case analysis; EM, our proposed Expectation-Maximization algorithm; Oracle, oracle estimators.\",\n \"Bias (%), {(estimate-truth)/truth}*100.Figure: Simulation results when M⊥⊥Y∣(T,X)M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ).\",\n \"A, Binary MM and Binary YY; B, Binary MM and Continuous YY; C, Continuous MM and Continuous YY; D, Continuous MM and Binary YY.\",\n \"I, Assumption ; II, Assumption ; III, Assumption ; IV, Assumption .\",\n \"CC, complete case analysis; EM, our proposed Expectation-Maximization algorithm; Oracle, oracle estimators.\",\n \"(0), M⊥⊥Y∣(T,X)M \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ).\",\n \"Bias (%), {(estimate-truth)/truth}*100; Bias (%) for NIE is defined as (estimate/NDE)*100 because NIE equals 0.\"\n ],\n [\n \"Data\",\n \"The data describes 8707 eligible applicants in the mid-1990s who lived in the areas selected for in-person interviews at the baseline.\",\n \"The subjects were randomized either to the experimental group $(T=1)$ where they could join the Job Corps program soon after randomization, or to the control group $(T=0)$ where they were not provided the Job Corps program for three years [30].\",\n \"The mediator ($M$ ) was collected at the 30-months follow-up describing subject's educational and vocational attainment, measured by whether or not the subject obtained an education credential or vocational certificate after randomization.\",\n \"We use $M=1$ to denote that an education credential or vocational certificate was obtained, and $M=0$ otherwise.\",\n \"The outcome ($Y$ ) was collected at the 48-months follow-up describing the subject's weekly earnings in the fourth year after randomization.\",\n \"The measured covariates $ X $ include information on gender, age, race, education level, earnings in the year before participating in the study, whether the subject had a child or not, and whether the subject had ever been arrested or not.\",\n \"There are small portions of the missingness exist in covariates $ X $ , including education level (0.6%), earnings levels in the year before participating in the study (6.4%), and whether the subject had ever been arrested or no (3.5%).\",\n \"Due to the low missing rates in those categorical covariates, we treat the missingness as another category in the analysis.\",\n \"The number of subjects with missing information in the mediator or outcome is nontrivial.\",\n \"The missingness patterns in the mediator and the outcome are described in Table REF for the experimental group and the control group.\",\n \"We suspect that the missingness may be MNAR in the data.\",\n \"Besides the potential impacts of $T$ and $ X $ on the missingness, we have the following concerns: (1) conceivably, people who failed to obtain an education credential or vocational certificate ($M=0$ ) may be less likely to report compared to people who successfully obtained an educational credential or vocational certificate ($M=1$ ), that is, $M$ may have a direct effect on $R^M$ ; (2) we are concerned that people who were unwilling to respond to questions at the 30-months follow-up may also be unwilling to respond at the 48-months follow-up, that is, $R^M$ may have a direct impact on $R^Y$ ; (3) people who had no earnings $(Y=0)$ may be less likely to report their earnings compared to people who had earnings $(Y>0)$ , which results in a direct effect of $Y$ on $R^Y$ ; (4) in addition, $M$ occurs before $R^Y$ , and therefore, may potentially have an impact on people's probability of reporting earnings through channels other than $R^M$ and $Y$ .\",\n \"Concerns $(1)$ and $(2)$ can be addressed by the MNAR mechanism under Assumption REF , concerns $(1)$ and $(3)$ can be addressed by the MNAR mechanism under Assumption REF , and concerns $(1)$ and $(4)$ can be addressed by the MNAR mechanism under Assumption REF .\",\n \"When incorporating the weekly earnings $Y$ into the prediction of $R^Y$ under MNAR Assumption REF , we assume that it is the binary indicator $\\\\mathrm {I}(Y>0)$ describing whether $Y$ is positive or not that predicts $R^Y$ .\",\n \"This is because of the following two considerations: first, there are excessive zero values of the earnings (14.85%) in the data; second, the identification under MNAR Assumption REF requires the form of $Y$ in predicting $R^Y$ to be binary given that our mediator $M$ in this study is binary (see the completeness assumption in Theorem REF ).\",\n \"Table: Missingness patterns in the mediator and the outcome\"\n ],\n [\n \"Models\",\n \"Our outcome $Y$ , weekly earnings, contains many zero values as well as heavily right skewed positive values.\",\n \"To address those complications, we adopted two-part models.\",\n \"Let $Z_i$ be the binary indicator describing whether the earning is greater than 0 or not, i.e., $Z_i=1$ if $Y_i>0$ and $Z_i=0$ if $Y_i=0$ .\",\n \"We used the following logistic regression to model $Z_{i}$ : $\\\\mathrm {logit}~\\\\mathbb {P}(Z_i=1\\\\mid M_i=m,T_i=t, X _i= x ) = \\\\delta _0+\\\\delta _m m+\\\\delta _t t+\\\\delta _{mt} m \\\\cdot t+\\\\delta _x^ \\\\textsc {t} x .$ Conditioning on $Z_{i}=1$ , we considered two commonly adopted models, Gamma and log-normal models to fit the positively skewed values of earnings.\",\n \"As an illustration, we describe the Gamma model here: $Y_i \\\\mid (Z_i=1, M_i=m, T_i=t, X _i= x ) \\\\sim \\\\mathrm {Gamma}\\\\lbrace \\\\nu ,\\\\nu /\\\\mu _i(m, t, x )\\\\rbrace $ , where $\\\\nu $ denotes the shape parameter, $\\\\nu / \\\\mu _i(m, t, x )$ is the rate parameter, and the function $\\\\mu _i(m, t, x )$ is parameterized as $\\\\mathrm {exp}(\\\\beta _0+\\\\beta _m m+\\\\beta _t t + \\\\beta _{mt} m\\\\cdot t+\\\\beta _x^ \\\\textsc {t} x ).$ We used a logistic model for $M$ : $\\\\mathrm {logit}~\\\\mathbb {P}(M_i=1\\\\mid T_i=t, X _i= x )=\\\\alpha _0+\\\\alpha _{t}t+\\\\alpha _{x}^ \\\\textsc {t} x ,$ and a logistic model for $R^M$ allowing an effect from $M$ : $\\\\mathrm {logit} ~\\\\mathbb {P}(R^M_i=1\\\\mid M_i=m, T_i=t, X _i= x ) = \\\\lambda _0+\\\\lambda _m m+\\\\lambda _t t +\\\\lambda _x^ \\\\textsc {t} x .$ The model for $R^Y$ varied according to different MNAR assumptions.\",\n \"Under MNAR Assumption REF , we specified that $\\\\mathrm {logit} ~\\\\mathbb {P}(R^Y_i=1\\\\mid R^M_i=r^M, T_i=t, X _i= x ) = \\\\gamma _0+\\\\gamma _{r^M}r^M+\\\\gamma _t t +\\\\gamma _x^ \\\\textsc {t} x ;$ under MNAR Assumption REF , we specified that $\\\\mathrm {logit} ~\\\\mathbb {P}(R^Y_i=1\\\\mid Z_i=z, T_i=t, X _i= x ) = \\\\gamma _0+\\\\gamma _{z} z+\\\\gamma _t t +\\\\gamma _x^ \\\\textsc {t} x ;$ and under MNAR Assumption REF , we adopted that $\\\\mathrm {logit} ~\\\\mathbb {P}(R^Y_i=1\\\\mid M_i=m, T_i=t, X _i= x ) = \\\\gamma _0+\\\\gamma _{m} m+\\\\gamma _t t +\\\\gamma _x^ \\\\textsc {t} x .$\"\n ],\n [\n \"Results\",\n \"We compared the performance and results using two-part Gamma and two-part log-normal models for the outcome under MNAR Assumptions REF , REF and REF .\",\n \"Table REF presents the log-likelihoods evaluated at the MLEs for those six models.\",\n \"Since all six models have the same numbers of parameters, we chose the two-part Gamma model and MNAR mechanism under Assumption REF allowing an impact on $R^M$ from $M$ and an impact on $R^Y$ from $R^M$ , besides the impacts from $T$ and $ X $ .\",\n \"Below we describe the results based on the two-part Gamma model under MNAR Assumption REF .\",\n \"Table: Model comparison among models under MNAR Assumptions , and using two-part Gamma and two-part log-normal models for the outcome.\",\n \"The log-likelihoods are evaluated at the MLEs.The coefficient of $M$ in the $R^M$ model is estimated to be $1.73$ with $95\\\\%$ CI $(0.34,~3.33)$ - see $\\\\lambda _m$ in Table REF , which suggests that people who had acquired certificates were more likely to report their certificate status.\",\n \"The coefficient of $R^M$ in the $R^Y$ model is estimated to be $1.87$ with $95\\\\%$ CI $(1.76,~2.00)$ - see $\\\\gamma _{r^M}$ in Table REF , which suggests that people who were willing to report their certificate status were also more likely to report their earnings.\",\n \"The natural indirect effect is estimated to be $10.94$ with $95\\\\%$ CI $(7.94,~14.29)$ - see $\\\\mathrm {NIE}$ in Table REF , which indicates that there was a significant indirect effect of the program assignment on weekly earnings through an educational credential or vocational certificate at the $0.05$ significance level.\",\n \"The natural direct effect is estimated to be $12.93$ with $95\\\\%$ CI $(-1.95,~27.64)$ - see NDE in Table REF , which indicates that there was no significant direct effect of the program assignment on weekly earnings at the $0.05$ significance level.\",\n \"The causal conclusions regarding the indirect effect (NIE) and the direct effect (NDE) are the same between the complete case analysis and our methods, in spite of the significant effect of $M \\\\rightarrow R^M$ ($\\\\lambda _{m}$ ) and the significant effect of $R^M \\\\rightarrow R^Y$ ($\\\\gamma _{r^M}$ ).\",\n \"Table: Data analysis results from the Gamma model under MNAR Assumption .\",\n \"Est, estimate; SE, standard error based on 500 bootstrap samples; CI, confidence interval; λ m \\\\lambda _m, coefficient of MM in the R M R^M model; γ r M \\\\gamma _{r^M}, coefficient of R M R^M in the R Y R^Y model.\"\n ],\n [\n \"Sensitivity analysis\",\n \"We consider the Gamma model under MNAR Assumption REF from the data analysis as a starting model for building the sensitivity analysis.\",\n \"It is possible that the missingness of earnings also depends on the value of earnings itself and/or the educational and vocational attainment, in addition to the missingness of the educational and vocational attainment.\",\n \"The goal is to access the sensitivity of our causal conclusions to the additional impacts on $R^Y$ from $Z$ and/or $M$ .\",\n \"The revised model for $R^Y$ is as follows: $&&\\\\mathrm {logit}~\\\\mathbb {P}(R^Y_i=1\\\\mid R^M_i=r^M, Z_i=z, M_i=m, T_i=t, X _i= x )\\\\\\\\&=&\\\\gamma _0+\\\\gamma _{r^M} r^M +\\\\gamma _z z+\\\\gamma _m m +\\\\gamma _t t +\\\\gamma _x^ \\\\textsc {t} x,$ where $\\\\gamma _z$ and $\\\\gamma _m$ are the sensitivity parameters.\",\n \"We consider a large effect in the log odds ratio [3] and let both sensitivity parameters to vary among $-2$ , 0 and 2.\",\n \"When $\\\\gamma _z=0$ and $\\\\gamma _m=0$ , it is the same as the MNAR mechanism under Assumption REF that stands out in the data analysis.\",\n \"Figure: The direct acyclic graph describing the missing data mechanism for the sensitivity analysis.\",\n \"The graph conditions on X X .The results for the sensitivity analysis are presented in Table REF .\",\n \"The NIE estimate increases more than 10% in the case where $\\\\gamma _m=-2$ and $\\\\gamma _z=2$ , and the case where $\\\\gamma _m=0$ and $\\\\gamma _z=2$ .\",\n \"The NDE estimate decreases more than 10% in the case where $\\\\gamma _m=0$ and $\\\\gamma _z=2$ , and increases more than 10% in the case where $\\\\gamma _m=2$ and $\\\\gamma _z=2$ .\",\n \"However, the NIEs are estimated to be positive and significant at the $0.05$ level and the NDEs are estimated to be positive but not significant at the $0.05$ level, for all pairs of values $(\\\\gamma _m, \\\\gamma _z)$ considered.\",\n \"In summary, the causal conclusions on the NIE and NDE are not sensitive to a strong impact of $Z \\\\rightarrow R^Y$ and/or $M \\\\rightarrow R^Y$ in addition to the impact of $R^M$ on $R^Y$ .\",\n \"Table: Sensitivity analysis results from the Gamma model.\",\n \"Est, estimate; CI, confidence interval based on 500 bootstrap samples; λ m \\\\lambda _m, coefficient of MM in the R M R^M model; γ r M \\\\gamma _{r^M}, coefficient of R M R^M in the R Y R^Y model; γ z \\\\gamma _z (sensitivity parameter), coefficient of ZZ in the R Y R^Y model; γ m \\\\gamma _m (sensitivity parameter), coefficient of MM in the R Y R^Y model.\"\n ],\n [\n \"Discussion\",\n \"Mediation analysis plays an important role in studying the underlying biological or social mechanisms through which an exposure exerts its effect.\",\n \"When the mediator and outcome are missing not at random, the evaluation of the NIE and NDE can be very challenging.\",\n \"Motivated by the NJCS, in scenarios where it is reasonable to assume that future variables do not have an impact on variables measured in the past, we explored the identification of the NIE and NDE for some interpretable missing not at random assumptions.\",\n \"The condition of completeness plays an important role in achieving nonparametric identification.\",\n \"We proved that the joint distribution $\\\\mathbb {P}(Y,M\\\\mid T, X )$ can be identified when the missingness of $M$ may depend on the value of $M$ itself and when $Y$ is either fully observed, or the missingness of $Y$ depends on only one of $(R^M, Y, M)$ in addition to $T$ and $ X $ .\",\n \"In a special case when $M\\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid T, X $ and therefore violates the completeness requirement, although $\\\\mathbb {P}(M\\\\mid T, X )$ is no longer identifiable, the NIE and NDE can still be identified except when both the missingness of $M$ and the missingness of $Y$ depend on their missing values themselves (i.e., MNAR Assumption REF ).\",\n \"Based on the comparison of likelihoods, the MNAR mechanism under Assumption REF is more plausible compared to other MNAR assumptions in the NJCS study.\",\n \"The strong association between $R^M$ and $R^Y$ in the NJCS study might be a result of both being affected by the subjects' unmeasured tendency or willingness to respond to interview questions.\",\n \"Further, to address the concern that the missingness of $Y$ may also depends on $Y$ itself and $M$ in addition to $R^M$ , we proposed a sensitivity analysis approach and found that our conclusions on the NIE and NDE in the NJCS study are not sensitive to some large impacts on $R^Y$ from $Y$ and $M$ .\",\n \"Several limitations of the work are to be addressed in future research.\",\n \"First, in current work, we focused on the identification of the NIE and NDE when the mediator and outcome are MNAR.\",\n \"In practice, prevalent missingness may exist in both the mediator/outcome and the measured covariates, and the missingness of covariates may also be MNAR.\",\n \"Identifiability in this more complicated scenario is to be explored.\",\n \"Second, the NIE and NDE evaluated in the NJCS study should be interpreted as the effects of the program assignment instead of the effects of the program.\",\n \"According to [31], although control group subjects were barred from participating in the Job Corps program, there were about $30\\\\%$ of the program group subjects who failed to participate in the program.\",\n \"One possibility to address this one-sided noncompliance issue is to consider the random program assignment as an instrumental variable and incorporate the instrumental variable into causal mediation analysis [34], [10], [24] for the latent subgroup of compliers, i.e., who would comply with their treatment assignment no matter being assigned to the treatment or the control group.\",\n \"Third, we considered a setting where there is a single mediator.\",\n \"In practice, cases with multiple mediators may arise, and the potential missingness mechanisms may vary depending on whether those mediators have a sequential relationship or parallel relationship, and etc.\",\n \"We leave this topic for future research.\",\n \"Fourth, the results in this paper invoke the sequential ignorability assumption.\",\n \"In the NJCS study, although the ignorability of the treatment assignment is guaranteed by the study design, however, the ignorability of the mediator is not guaranteed and could be violated if there exists post-treatment or unmeasured pretreatment confounding.\",\n \"One approach to partly address this issue is to conduct sensitivity analysis to quantify the potential bias due to the departure from the sequential ignorability assumption [33], [12].\",\n \"SUPPLEMENTARY MATERIAL Section gives the proofs of the theorems.\",\n \"Section gives counterexamples for the unidentifiable cases.\",\n \"Section provides parametric estimation details.\"\n ],\n [\n \"Proof of Theorem \",\n \"The identification of $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x )$ follows from $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x ) = \\\\mathbb {P}(Y=y\\\\mid R^M = 1, M=m, T=t, X = x )\\\\nonumber .$ We will focus on the identification of $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )$ .\",\n \"Define $\\\\mathbb {P}_{my1|t, x} &=& \\\\mathbb {P}(M=m, Y=y, R^M=1\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{+y0\\\\mid t,x} &=& \\\\mathbb {P}(Y=y, R^M=0\\\\mid T=t, X = x ), \\\\\\\\\\\\zeta _{t,x}(m) &=& \\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m, T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )}.$ We have $\\\\mathbb {P}_{my1\\\\mid t, x} = \\\\mathbb {P}(M=m, Y=y\\\\mid T=t, X = x )\\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )$ and $\\\\mathbb {P}_{+y0\\\\mid t,x} &= \\\\int _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}(M=m, Y=y, R^M=0\\\\mid T=t, X = x )dm \\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}(M=m, Y=y\\\\mid T=t, X = x ) \\\\mathbb {P}(R^M=0\\\\mid M=m, T=t, X = x )dm \\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my1\\\\mid t, x} \\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m, T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )}dm \\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my1\\\\mid t, x} \\\\zeta _{t,x}(m)dm.\",\n \"\\\\nonumber $ The existence and uniqueness of solutions $\\\\zeta _{t,x}(m)$ require that $\\\\mathbb {P}(Y,M,R^M=1 \\\\mid T=t, X=x )$ is complete in $Y$ .\",\n \"For discrete $M$ and discrete $Y$ , the completeness assumption is equivalent to Rank $(\\\\Theta _{tx})=J$ , where $\\\\Theta _{tx}$ is a $J \\\\times K$ matrix with $\\\\mathbb {P}_{my1\\\\mid t, x}$ as the $(m,y)$ th element.\",\n \"For binary $M$ , the rank condition further reduces to $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , which is equivalent to the testable condition $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ,R^M=1)$ .\",\n \"For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be no smaller than the dimension of $M$ in general as required by the completeness assumption.\",\n \"We can subsequently identify $\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )$ once $\\\\zeta _{t,x}(m)$ is identified.\",\n \"Then $&&\\\\mathbb {P}(M=m\\\\mid T=t, X = x ) \\\\\\\\&=&\\\\frac{\\\\mathbb {P}(M=m, Y=y\\\\mid T=t, X = x )}{\\\\mathbb {P}(Y=y\\\\mid M=m,T=t, X = x )} \\\\\\\\&=& \\\\frac{\\\\mathbb {P}_{my1\\\\mid t, x}}{\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x ) \\\\mathbb {P}(Y=y\\\\mid M=m,T=t, X = x )}.$\"\n ],\n [\n \"Proof of Theorem \",\n \"The identification of $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x )$ follows from $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x ) = \\\\mathbb {P}(Y=y\\\\mid R^M=1, R^Y=1, M=m, T=t, X = x ).\\\\nonumber $ We will focus on the identification of $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )$ .\",\n \"Define $\\\\mathbb {P}_{my11\\\\mid t,x} &=& \\\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{+y01\\\\mid t,x} &=& \\\\mathbb {P}(Y=y, R^M=0, R^Y=1\\\\mid T=t, X = x ), \\\\\\\\\\\\zeta _{t,x}(m)&=&\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m,T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )}.$ We have $\\\\mathbb {P}_{my11\\\\mid t,x} &=& \\\\mathbb {P}(M=m, Y=y\\\\mid T=t, X = x ) \\\\\\\\&& \\\\cdot \\\\mathbb {P}(R^Y=1\\\\mid R^M=1,T=t, X = x )\\\\cdot \\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )$ and $\\\\mathbb {P}_{+y01\\\\mid t,x} &= \\\\int _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1\\\\mid T=t, X = x )dm\\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\frac{\\\\mathbb {P}(R^Y=1\\\\mid R^M=0,T=t, X = x )\\\\mathbb {P}(R^M=0\\\\mid M=m,T=t, X = x )}{\\\\mathbb {P}(R^Y=1\\\\mid R^M=1,T=t, X = x )\\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )}dm\\\\nonumber \\\\\\\\&=\\\\frac{\\\\mathbb {P}(R^Y=1\\\\mid R^M=0,T=t, X = x )}{\\\\mathbb {P}(R^Y=1\\\\mid R^M=1,T=t, X = x )}\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\zeta _{t,x}(m)dm.\",\n \"\\\\nonumber $ The existence and uniqueness of solutions $\\\\zeta _{t,x}(m)$ require that $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X=x )$ is complete in $Y$ .\",\n \"For discrete $M$ and discrete $Y$ , the completeness assumption is equivalent to Rank $(\\\\Theta _{tx})=J$ , where $\\\\Theta _{tx}$ is a $J \\\\times K$ matrix with $\\\\mathbb {P}_{my11\\\\mid t,x}$ as the $(m,y)$ th element.\",\n \"For binary $M$ , the rank condition further reduces to $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , which is equivalent to the testable condition $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ,R^M=1,R^Y=1)$ .\",\n \"For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be no smaller than the dimension of $M$ in general as required by the completeness assumption.\",\n \"We can subsequently identify $\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )$ once $\\\\zeta _{t,x}(m)$ is identified.\",\n \"Then $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )&=\\\\frac{\\\\mathbb {P}(M=m, Y=y\\\\mid T=t, X = x )}{\\\\mathbb {P}(Y=y\\\\mid M=m,T=t, X = x )}\\\\nonumber \\\\\\\\&= \\\\frac{\\\\mathbb {P}_{my11\\\\mid t,x}}{\\\\mathbb {P}(Y=y, R^Y=1, R^M=1\\\\mid M=m,T=t, X = x )}.\\\\nonumber $\"\n ],\n [\n \"Proof of Theorem \",\n \"We will discuss the identification of $\\\\mathbb {P}(M=m,Y=y\\\\mid T=t, X = x )$ .\",\n \"Define $\\\\mathbb {P}_{my11\\\\mid t,x} &=& \\\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{+y01\\\\mid t,x} &=& \\\\mathbb {P}(Y=y, R^M=0, R^Y=1\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{m+10\\\\mid t,x} &=& \\\\mathbb {P}(M=m, R^M=1, R^Y=0\\\\mid T=t, X = x ), \\\\\\\\\\\\zeta _{t,x}(m)&=&\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m,T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )},\\\\\\\\\\\\eta _{t,x}(y)&=&\\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid Y=y,T=t, X = x )}{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,T=t, X = x )}.$ We first have $\\\\mathbb {P}_{my11|t,x}&=& \\\\mathbb {P}(M=m, Y=y\\\\mid T=t, X = x ) \\\\\\\\&&\\\\cdot \\\\mathbb {P}(R^Y=1\\\\mid Y=y,T=t, X = x )\\\\cdot \\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x ) .$ We then have $\\\\mathbb {P}_{+y01\\\\mid t,x} &= \\\\int _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1\\\\mid T=t, X = x )dm\\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11\\\\mid t,x}\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m,T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )}dm\\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\zeta _{t,x}(m)dm \\\\nonumber $ and $\\\\mathbb {P}_{m+10\\\\mid t,x} &= \\\\int _{y\\\\in \\\\mathcal {Y}}\\\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=0\\\\mid T=t, X = x )dy\\\\nonumber \\\\\\\\&=\\\\int _{y\\\\in \\\\mathcal {Y}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid Y=y,T=t, X = x )}{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,T=t, X = x )}dy\\\\nonumber \\\\\\\\&=\\\\int _{y\\\\in \\\\mathcal {Y}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\eta _{t,x}(y)dy.\",\n \"\\\\nonumber $ The existence and uniqueness of solutions $\\\\zeta _{t,x}(m)$ require that $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X=x )$ is complete in $Y$ , and the existence and uniqueness of solutions $\\\\eta _{t,x}(y)$ require that $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X=x )$ is complete in $M$ .\",\n \"For discrete $M$ and discrete $Y$ , the above completeness assumptions are equivalent to Rank $(\\\\Theta _{tx})=J$ , where $\\\\Theta _{tx}$ is a $J \\\\times J$ matrix with $\\\\mathbb {P}_{my11\\\\mid t,x}$ as the $(m,y)$ th element.\",\n \"For binary $M$ and binary $Y$ , the rank condition reduces to $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ .\",\n \"For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be the same as the dimension of $M$ in general as required by $\\\\mathbb {P}(Y,M,R^Y=1,R^M=1\\\\mid T, X )$ being complete in $M$ and being complete in $Y$ .\",\n \"We can subsequently identify $\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )$ and $\\\\mathbb {P}(R^Y=1\\\\mid Y=y, T=t, X = x )$ once $\\\\zeta _{t,x}(m)$ and $\\\\eta _{t,x}(y)$ are identified.\",\n \"Then $&&\\\\mathbb {P}(Y=y, M=m\\\\mid T=t, X = x )\\\\nonumber \\\\\\\\&=&\\\\frac{\\\\mathbb {P}_{my11\\\\mid t,x}}{\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )\\\\mathbb {P}(R^Y=1\\\\mid Y=y, T=t, X = x )}.\\\\nonumber $\"\n ],\n [\n \"Under condition $(i)$\",\n \"The identification of $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x )$ follows from $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x ) = \\\\mathbb {P}(Y=y\\\\mid R^M = 1, R^Y=1, M=m, T=t, X = x )\\\\nonumber .$ We will focus on the identification of $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )$ .\",\n \"Define $\\\\mathbb {P}_{my11\\\\mid t,x} &=& \\\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{+y01\\\\mid t,x} &=& \\\\mathbb {P}(Y=y, R^M=0, R^Y=1\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{m+10\\\\mid t,x} &=& \\\\mathbb {P}(M=m, R^M=1, R^Y=0\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{m+11\\\\mid t,x} &=& \\\\mathbb {P}(M=m, R^M=1, R^Y=1\\\\mid T=t, X = x ), \\\\\\\\\\\\zeta _{t,x}(m)&=&\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m,T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )}, \\\\\\\\\\\\eta _{t,x}(m)&=&\\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid M=m,T=t, X = x )}{\\\\mathbb {P}(R^Y=1\\\\mid M=m,T=t, X = x )}.$ We first have $\\\\mathbb {P}_{my11\\\\mid t,x} &=& \\\\mathbb {P}(M=m, Y=y \\\\mid T=t, X = x ) \\\\\\\\&&\\\\cdot \\\\mathbb {P}(R^Y=1\\\\mid M=m, T=t, X = x )\\\\cdot \\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x ).$ We then have $\\\\mathbb {P}_{+y01\\\\mid t,x} &= \\\\int _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1\\\\mid T=t, X = x )dm\\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m, T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )}dm\\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\zeta _{t,x}(m)dm \\\\nonumber $ and $\\\\mathbb {P}_{m+10\\\\mid t,x} &= \\\\int _{y\\\\in \\\\mathcal {Y}} \\\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=0\\\\mid T=t, X = x )dy\\\\nonumber \\\\\\\\&=\\\\int _{y\\\\in \\\\mathcal {Y}}\\\\mathbb {P}_{my11\\\\mid t,x} \\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid M=m, T=t, X = x )}{\\\\mathbb {P}(R^Y=1\\\\mid M=m, T=t, X = x )}dy\\\\nonumber \\\\\\\\&=\\\\eta _{t,x}(m)\\\\int _{y\\\\in \\\\mathcal {Y}}\\\\mathbb {P}_{my11\\\\mid t,x}dy \\\\nonumber \\\\\\\\&=\\\\eta _{t,x}(m)\\\\mathbb {P}_{m+11\\\\mid t,x}.\",\n \"\\\\nonumber $ Solve $\\\\eta _{t,x}(m)$ as $\\\\eta _{t,x}(m) = \\\\mathbb {P}_{m+10\\\\mid t,x} / \\\\mathbb {P}_{m+11\\\\mid t,x} .$ When the joint distribution of $\\\\mathbb {P}(Y,M,R^M=1,R^Y=1 \\\\mid T=t, X=x )$ is complete in $Y$ , the existence and uniqueness of solutions $\\\\zeta _{t,x}(m)$ can be achieved.\",\n \"For discrete $M$ and discrete $Y$ , this completeness assumption is equivalent to Rank $(\\\\Theta _{tx})=J$ , where $\\\\Theta _{tx}$ is a $J \\\\times K$ matrix with $\\\\mathbb {P}_{my11\\\\mid t,x}$ as the $(m,y)$ th element.\",\n \"For binary $M$ , the rank condition further reduces to $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X )$ , which is equivalent to the testable condition $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp Y\\\\mid (T, X ,R^M=1,R^Y=1)$ .\",\n \"For continuous $M$ and continuous $Y$ , the dimension of $Y$ needs to be no smaller than the dimension of $M$ in general as required by the completeness assumption.\",\n \"We can subsequently identify $\\\\mathbb {P}(R^Y=1\\\\mid M=m, T=t, X = x )$ and $\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )$ once $\\\\zeta _{t,x}(m)$ and $\\\\eta _{t,x}(m)$ are identified.\",\n \"Then $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )& = \\\\frac{\\\\mathbb {P}(M=m, Y=y\\\\mid T=t, X = x )}{\\\\mathbb {P}(Y=y\\\\mid M=m,T=t, X = x )} \\\\nonumber \\\\\\\\&= \\\\frac{\\\\mathbb {P}_{my11\\\\mid t,x}}{\\\\mathbb {P}(Y=y,R^M=1,R^Y=1\\\\mid M=m,T=t, X = x )}\\\\nonumber .$ The identification of $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x )$ follows from $\\\\mathbb {P}(Y=y\\\\mid M=m, T=t, X = x ) = \\\\mathbb {P}(Y=y\\\\mid R^M=1, R^Y=1, M=m, T=t, X = x ),$ and the identification of $\\\\mathbb {P}(R^Y=r^Y\\\\mid M=m, T=t, X = x )$ follows from $\\\\mathbb {P}(R^Y=r^Y\\\\mid M=m, T=t, X = x ) = \\\\mathbb {P}(R^Y=r^Y\\\\mid R^M=1,M=m,T=t, X = x ) .$ We will focus on the identification of $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )$ .\",\n \"Define $\\\\mathbb {P}_{m1r^Y|t,x} &=& \\\\mathbb {P}(M=m,R^M=1,R^Y=r^Y\\\\mid T=t, X = x ), \\\\\\\\\\\\mathbb {P}_{+0r^Y\\\\mid t,x} &=& \\\\mathbb {P}(R^M=0,R^Y=r^Y\\\\mid T=t, X = x ), \\\\\\\\\\\\zeta _{t,x}(m) &=& \\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m, T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )}.$ We have $\\\\mathbb {P}_{m1r^Y\\\\mid t,x} = \\\\mathbb {P}(M=m,R^Y=r^Y\\\\mid T=t, X = x )\\\\mathbb {P}(R^M=1\\\\mid M=m,T=t, X = x )$ and $\\\\mathbb {P}_{+0r^Y\\\\mid t,x} &= \\\\int _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}(M=m, R^M=0, R^Y=r^Y\\\\mid T=t, X = x )dm \\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}_{m1r^Y|t,x} \\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m, T=t, X = x )}{\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )}dm \\\\nonumber \\\\\\\\&=\\\\int _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{m1r^Y|t,x} \\\\zeta _{t,x}(m)dm.\",\n \"\\\\nonumber $ When the joint distribution of $\\\\mathbb {P}(M,R^M=1,R^Y \\\\mid T=t, X=x )$ is complete in $R^Y$ ,the existence and uniqueness of solutions $\\\\zeta _{t,x}(m)$ can be achieved.\",\n \"This requires that $M$ is binary since $R^Y$ is binary, and the completeness assumption reduces to $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp R^Y\\\\mid (T, X )$ , which is equivalent to the testable condition $M \\\\lnot \\\\perp \\\\!\\\\!\\\\!\\\\perp R^Y\\\\mid (T, X ,R^M=1)$ .\",\n \"We can subsequently identify $\\\\mathbb {P}(R^M=1\\\\mid M=m, T=t, X = x )$ once $\\\\zeta _{t,x}(m)$ is identified.\",\n \"Then $\\\\mathbb {P}(M=m\\\\mid T=t, X = x )\\\\nonumber &=\\\\frac{\\\\mathbb {P}(M=m, R^Y=r^Y\\\\mid T=t, X = x )}{\\\\mathbb {P}(R^Y=r^Y\\\\mid M=m,T=t, X = x )} \\\\nonumber \\\\\\\\&= \\\\frac{\\\\mathbb {P}_{m1r^Y\\\\mid t,x}}{\\\\mathbb {P}(R^M=1,R^Y=r^Y\\\\mid M=m, T=t, X = x )}.\\\\nonumber $\"\n ],\n [\n \"Counterexamples for the Unidentifiable Cases\",\n \"Consider a binary mediator $M$ and binary outcome $Y$ .\",\n \"All graphs and probabilities below are conditioning on $T$ and $ X $ .\",\n \"Define $\\\\mathbb {P}_{my11} &=& \\\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=1), \\\\\\\\\\\\mathbb {P}_{+y01} &=& \\\\mathbb {P}(Y=y, R^M=0, R^Y=1),\\\\\\\\\\\\mathbb {P}_{m+10} &=& \\\\mathbb {P}(M=m, R^M=1, R^Y=0), \\\\\\\\\\\\mathbb {P}_{++00} &=& \\\\mathbb {P}(R^M=0, R^Y=0).$ Based on the observable data probabilities, we can directly identify $\\\\mathbb {P}_{my11}$ , $\\\\mathbb {P}_{+y01}$ , $\\\\mathbb {P}_{m+10}$ and $\\\\mathbb {P}_{++00}$ , and $\\\\sum ^{1}_{m=0}\\\\sum ^{1}_{y=0}\\\\mathbb {P}_{my11}+\\\\sum _{y=0}^1 \\\\mathbb {P}_{+y01}+\\\\sum _{m=0}^1 \\\\mathbb {P}_{m+10}+ \\\\mathbb {P}_{++00}=1$ .\"\n ],\n [\n \"$(i)$ We present below an unidentified case when {{formula:ca2bf032-22fe-4901-8a91-776f1be6e9e3}} depends on both {{formula:8fa07d9e-4c56-4243-bc2f-9023ccaf307d}} and {{formula:975b0000-5a92-4422-a27b-0023e2a95cb3}} as described by Figure \",\n \"Consider the following observable data probabilities: $(\\\\mathbb {P}_{1111},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{++00})=\\\\left(\\\\frac{3}{10},\\\\frac{1}{10},\\\\frac{1}{20},\\\\frac{1}{20},\\\\frac{1}{10},\\\\frac{1}{20},\\\\frac{1}{10},\\\\frac{1}{20},\\\\frac{1}{5} \\\\right)\\\\nonumber .$ The key to identify $\\\\mathbb {P}(Y=y, M=m)$ is to identify both $\\\\mathbb {P}(R^M=1\\\\mid M=m)$ and $\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)$ in the following formula $\\\\mathbb {P}(Y=y, M=m)=\\\\frac{\\\\mathbb {P}_{my11}}{\\\\mathbb {P}(R^M=1\\\\mid M=m)\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)}.\\\\nonumber $ We now show that the identification of $\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)$ can be achieved.\",\n \"This is because $\\\\mathbb {P}_{m+10} &= \\\\sum _{y\\\\in \\\\mathcal {Y}}\\\\mathbb {P}(M=m, Y=y, R^M=1, R^Y=0)\\\\nonumber \\\\\\\\&=\\\\sum _{y\\\\in \\\\mathcal {Y}}\\\\mathbb {P}_{my11}\\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid Y=y,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)}.\\\\nonumber $ By plugging in the two possible values for $m$ and $y$ in the above formula, we have $\\\\mathbb {P}_{1+10} &= \\\\mathbb {P}_{1111}\\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid Y=1,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=1,R^M=1)}+\\\\mathbb {P}_{1011}\\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid Y=0,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=0,R^M=1)}$ and $\\\\mathbb {P}_{0+10} &= \\\\mathbb {P}_{0111}\\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid Y=1,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=1,R^M=1)}+\\\\mathbb {P}_{0011}\\\\frac{\\\\mathbb {P}(R^Y=0\\\\mid Y=0,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=0,R^M=1)}.$ And therefore, $\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)$ can be identified by solving the linear equations (REF ) and (REF ).\",\n \"Based on the observable data probabilities, $\\\\mathbb {P}(R^Y=1\\\\mid Y=1,R^M=1)=\\\\frac{4}{5}$ and $\\\\mathbb {P}(R^Y=1\\\\mid Y=0,R^M=1)=\\\\frac{2}{3}$ .\",\n \"We now focus on the identifiability of $\\\\mathbb {P}(R^M=1\\\\mid M=m)$ and show that $\\\\mathbb {P}(R^M=1\\\\mid M=m)$ cannot be identified without further assumptions.\",\n \"We have $\\\\mathbb {P}_{+y01} &= \\\\sum _{m\\\\in \\\\mathcal {M}} \\\\mathbb {P}(M=m, Y=y, R^M=0, R^Y=1)\\\\nonumber \\\\\\\\&=\\\\sum _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11}\\\\frac{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=0)\\\\mathbb {P}(R^M=0\\\\mid M=m)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)\\\\mathbb {P}(R^M=1\\\\mid M=m)}\\\\nonumber \\\\\\\\&= \\\\frac{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=0)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)} \\\\sum _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11}\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m)}{\\\\mathbb {P}(R^M=1\\\\mid M=m)},\\\\nonumber $ and as a result, $\\\\mathbb {P}_{+y01}\\\\frac{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=0)} &= \\\\sum _{m\\\\in \\\\mathcal {M}}\\\\mathbb {P}_{my11}\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=m)}{\\\\mathbb {P}(R^M=1\\\\mid M=m)}.\\\\nonumber $ By plugging in the two possible values for $m$ and $y$ in the above formula, we have $\\\\mathbb {P}_{+101}\\\\frac{\\\\mathbb {P}(R^Y=1\\\\mid Y=1,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=1,R^M=0)}&= \\\\mathbb {P}_{1111}\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=1)}{\\\\mathbb {P}(R^M=1\\\\mid M=1)}\\\\nonumber +\\\\mathbb {P}_{0111}\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=0)}{\\\\mathbb {P}(R^M=1\\\\mid M=0)}\\\\nonumber $ and $\\\\mathbb {P}_{+001}\\\\frac{\\\\mathbb {P}(R^Y=1\\\\mid Y=0,R^M=1)}{\\\\mathbb {P}(R^Y=1\\\\mid Y=0,R^M=0)}&= \\\\mathbb {P}_{1011}\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=1)}{\\\\mathbb {P}(R^M=1\\\\mid M=1)}\\\\nonumber +\\\\mathbb {P}_{0011}\\\\frac{\\\\mathbb {P}(R^M=0\\\\mid M=0)}{\\\\mathbb {P}(R^M=1\\\\mid M=0)}.\\\\nonumber $ Since $\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=1)$ are identified from the previous step, the identifiability of $\\\\mathbb {P}(R^M=1 \\\\mid M=m)$ depends on the identifiability of $\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=0)$ .\",\n \"We have $&&\\\\mathbb {P}(R^Y=1\\\\mid Y=y,R^M=0)\\\\nonumber \\\\\\\\&=& \\\\frac{\\\\mathbb {P}(Y=y,R^M=0,R^Y=1)}{\\\\mathbb {P}(Y=y,R^M=0)}\\\\nonumber \\\\\\\\&=& \\\\frac{\\\\mathbb {P}(Y=y,R^M=0,R^Y=1)}{\\\\mathbb {P}(Y=y,R^M=0,R^Y=0)+\\\\mathbb {P}(Y=y,R^M=0,R^Y=1)}\\\\nonumber \\\\\\\\&=&\\\\frac{\\\\mathbb {P}(Y=y,R^M=0,R^Y=1)}{\\\\mathbb {P}(Y=y \\\\mid R^M=0, R^Y=0)\\\\mathbb {P}(R^M=0,R^Y=0)+\\\\mathbb {P}(Y=y, R^M=0,R^Y=1)}\\\\nonumber \\\\\\\\&=&\\\\frac{\\\\mathbb {P}_{+y01}}{\\\\mathbb {P}(Y=y \\\\mid R^M=0, R^Y=0)\\\\mathbb {P}_{++00}+\\\\mathbb {P}_{+y01}}.\\\\nonumber $ In the above expression, $\\\\mathbb {P}_{+y01}$ and $\\\\mathbb {P}_{++00}$ are known.\",\n \"$\\\\mathbb {P}(Y=y \\\\mid R^Y=0,R^M=0)$ is not observable or identifiable, and can take any value between 0 and 1.\",\n \"Different values of $\\\\mathbb {P}(Y=y \\\\mid R^Y=0,R^M=0)$ will result in different values of $\\\\mathbb {P}(R^M=1 \\\\mid M=m)$ , which in turn will give different values of $\\\\mathbb {P}(Y=y, M=m)$ .\",\n \"For example, let $\\\\mathbb {P}(Y=1 \\\\mid R^Y=0,R^M=0)=\\\\frac{5}{6}$ , and the corresponding $\\\\mathbb {P}(R^Y=1\\\\mid Y=1,R^M=0)$ and $\\\\mathbb {P}(R^Y=1\\\\mid Y=0,R^M=0)$ equal $\\\\frac{3}{8}$ and $\\\\frac{3}{5}$ , respectively.\",\n \"Subsequently, we have $\\\\mathbb {P}(Y=1,M=1)=\\\\frac{17}{30}$ , $\\\\mathbb {P}(Y=0,M=1)=\\\\frac{17}{150}$ , $\\\\mathbb {P}(Y=1,M=0)=\\\\frac{1}{5}$ and $\\\\mathbb {P}(Y=0,M=0)=\\\\frac{3}{25}$ .\",\n \"Alternatively, let $\\\\mathbb {P}(Y=1 \\\\mid R^Y=0,R^M=0)=\\\\frac{7}{8}$ , and the corresponding $\\\\mathbb {P}(R^Y=1\\\\mid Y=1,R^M=0)$ and $\\\\mathbb {P}(R^Y=1\\\\mid Y=0,R^M=0)$ equal $\\\\frac{4}{11}$ and $\\\\frac{2}{3}$ , respectively.\",\n \"Subsequently, we have $\\\\mathbb {P}(Y=1,M=1)=\\\\frac{3}{5}$ , $\\\\mathbb {P}(Y=0,M=1)=\\\\frac{3}{25}$ , $\\\\mathbb {P}(Y=1,M=0)=\\\\frac{7}{40}$ and $\\\\mathbb {P}(Y=0,M=0)=\\\\frac{21}{200}$ .\",\n \"The two sets of values of $\\\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\\\mathbb {P}_{1111},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{++00})$ , and therefore, $\\\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions.\",\n \"Consider the following observable data probabilities: $(\\\\mathbb {P}_{1111},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{++00})=\\\\left(\\\\frac{3}{10},\\\\frac{1}{10},\\\\frac{1}{20},\\\\frac{1}{10},\\\\frac{1}{10},\\\\frac{1}{40},\\\\frac{1}{10},\\\\frac{1}{10},\\\\frac{5}{40}\\\\right)\\\\nonumber .$ Define $&\\\\mathbb {P}_M=\\\\mathbb {P}(M=1),\\\\\\\\&\\\\mathbb {P}_{Y1}=\\\\mathbb {P}(Y=1\\\\mid M=1),\\\\\\\\&\\\\mathbb {P}_{Y0}=\\\\mathbb {P}(Y=1\\\\mid M=0),\\\\\\\\&\\\\mathbb {P}_{R^M1}=\\\\mathbb {P}(R^M=1\\\\mid M=1),\\\\\\\\&\\\\mathbb {P}_{R^M0}=\\\\mathbb {P}(R^M=1\\\\mid M=0),\\\\\\\\&\\\\mathbb {P}_{R^Y00}=\\\\mathbb {P}(R^Y=1\\\\mid M=0,R^M=0),\\\\\\\\&\\\\mathbb {P}_{R^Y01}=\\\\mathbb {P}(R^Y=1\\\\mid M=0,R^M=1),\\\\\\\\&\\\\mathbb {P}_{R^Y10}=\\\\mathbb {P}(R^Y=1\\\\mid M=1,R^M=0),\\\\\\\\&\\\\mathbb {P}_{R^Y11}=\\\\mathbb {P}(R^Y=1\\\\mid M=1,R^M=1).$ Below we study the identifiablity of the above 9 parameters describing the graphical model in Figure REF based on the observable data probabilities.\",\n \"Note that although there are 9 observable data probabilities, the degree of freedom in the probabilities is only 8 given that they sum up to 1.\",\n \"The following relationships between the observable data probabilities and the parameters hold, $&\\\\mathbb {P}_{1111}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y1}\\\\mathbb {P}_{R^M1}\\\\mathbb {P}_{R^Y11}, \\\\\\\\&\\\\mathbb {P}_{1011}=\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y1})\\\\mathbb {P}_{R^M1}\\\\mathbb {P}_{R^Y11},\\\\\\\\&\\\\mathbb {P}_{0111}=(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y0}\\\\mathbb {P}_{R^M0}\\\\mathbb {P}_{R^Y01},\\\\\\\\&\\\\mathbb {P}_{0011}=(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y0})\\\\mathbb {P}_{R^M0}\\\\mathbb {P}_{R^Y01},\\\\\\\\&\\\\mathbb {P}_{1+10}=\\\\mathbb {P}_M\\\\mathbb {P}_{R^M1}(1-\\\\mathbb {P}_{R^Y11}),\\\\\\\\&\\\\mathbb {P}_{0+10}=(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{R^M0}(1-\\\\mathbb {P}_{R^Y01}),\\\\\\\\&\\\\mathbb {P}_{+101}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y1}(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y10}+(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y0}(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y00},\\\\\\\\&\\\\mathbb {P}_{+001}=\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y1})(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y10}+(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y0})(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y00}.$ By solving the equations (REF ) to (), we can identify the parameters $\\\\mathbb {P}_{Y1}$ , $\\\\mathbb {P}_{Y0}$ , $\\\\mathbb {P}_{R^Y11}$ , and $\\\\mathbb {P}_{R^Y01}$ : $&\\\\mathbb {P}_{Y1}=\\\\frac{\\\\mathbb {P}_{1111}}{\\\\mathbb {P}_{1111}+\\\\mathbb {P}_{1011}},\\\\\\\\&\\\\mathbb {P}_{Y0}=\\\\frac{\\\\mathbb {P}_{0111}}{\\\\mathbb {P}_{0111}+\\\\mathbb {P}_{0011}},\\\\\\\\&\\\\mathbb {P}_{R^Y11}=\\\\frac{\\\\mathbb {P}_{1111}+\\\\mathbb {P}_{1011}}{\\\\mathbb {P}_{1111}+\\\\mathbb {P}_{1011}+\\\\mathbb {P}_{1+10}},\\\\\\\\&\\\\mathbb {P}_{R^Y01}=\\\\frac{\\\\mathbb {P}_{0111}+\\\\mathbb {P}_{0011}}{\\\\mathbb {P}_{0111}+\\\\mathbb {P}_{0011}+\\\\mathbb {P}_{0+10}}.$ In addition, we can identify the following products of parameters based on equations () to (): $\\\\mathbb {P}_M\\\\mathbb {P}_{R^M1}$ , $(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{R^M0}$ , $\\\\mathbb {P}_M(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y10}$ and $(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y00}$ .\",\n \"As a result, when $\\\\mathbb {P}_M$ is known, one can solve for $\\\\mathbb {P}_{R^M1}$ , $\\\\mathbb {P}_{R^M0}$ , $\\\\mathbb {P}_{R^Y10}$ and $\\\\mathbb {P}_{R^Y00}$ .\",\n \"Based on the observable data probabilities, $\\\\mathbb {P}_{Y1}=\\\\frac{6}{7}$ , $\\\\mathbb {P}_{Y0}=\\\\frac{1}{2}$ , $\\\\mathbb {P}_{R^Y11} = \\\\frac{7}{9}$ , $\\\\mathbb {P}_{R^Y01} = \\\\frac{2}{3}$ and $\\\\mathbb {P}_M$ is bounded between $\\\\frac{9}{20}$ and $\\\\frac{14}{20}$ .\",\n \"For example, let $\\\\mathbb {P}_M=\\\\frac{12}{20}$ , we have $\\\\mathbb {P}_{R^M1}=\\\\frac{3}{4}$ , $\\\\mathbb {P}_{R^M0}=\\\\frac{3}{4}$ , $\\\\mathbb {P}_{R^Y10}=\\\\frac{7}{10}$ , $\\\\mathbb {P}_{R^Y00}=\\\\frac{1}{5}$ .\",\n \"This set of parameter values give us the following joint probabilities of $M$ and $Y$ as $\\\\mathbb {P}(Y=1,M=1) =\\\\frac{18}{35} $ , $\\\\mathbb {P}(Y=0,M=1) = \\\\frac{3}{35}$ , $\\\\mathbb {P}(Y=1,M=0) = \\\\frac{1}{5}$ , and $\\\\mathbb {P}(Y=0,M=0) = \\\\frac{1}{5}$ .\",\n \"Alternatively, let $\\\\mathbb {P}_M=\\\\frac{13}{20}$ , we have $\\\\mathbb {P}_{R^M1}=\\\\frac{9}{13}$ , $\\\\mathbb {P}_{R^M0}=\\\\frac{6}{7}$ , $\\\\mathbb {P}_{R^Y10}=\\\\frac{21}{40}$ and $\\\\mathbb {P}_{R^Y00}=\\\\frac{2}{5}$ .\",\n \"This alternative set of parameter values give us the following joint probabilities of $M$ and $Y$ as $\\\\mathbb {P}(Y=1,M=1) = \\\\frac{39}{70}$ , $\\\\mathbb {P}(Y=0,M=1) =\\\\frac{13}{140} $ , $\\\\mathbb {P}(Y=1,M=0) = \\\\frac{7}{40}$ , and $\\\\mathbb {P}(Y=0,M=0) = \\\\frac{7}{40}$ .\",\n \"The two sets of values of $\\\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\\\mathbb {P}_{1111},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{++00})$ , and therefore, $\\\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions.\",\n \"Consider the following probabilities from the observable data: $(\\\\mathbb {P}_{1111},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{++00})=\\\\left(\\\\frac{18}{96},\\\\frac{6}{96},\\\\frac{3}{96},\\\\frac{2}{96},\\\\frac{12}{96},\\\\frac{3}{96},\\\\frac{15}{96},\\\\frac{16}{96},\\\\frac{21}{96}\\\\right)\\\\nonumber .$ Define $&\\\\mathbb {P}_M=\\\\mathbb {P}(M=1),\\\\\\\\&\\\\mathbb {P}_{Y1}=\\\\mathbb {P}(Y=1\\\\mid M=1),\\\\\\\\&\\\\mathbb {P}_{Y0}=\\\\mathbb {P}(Y=1\\\\mid M=0),\\\\\\\\&\\\\mathbb {P}_{R^M1}=\\\\mathbb {P}(R^M=1\\\\mid M=1),\\\\\\\\&\\\\mathbb {P}_{R^M0}=\\\\mathbb {P}(R^M=1\\\\mid M=0),\\\\\\\\&\\\\mathbb {P}_{R^Y00}=\\\\mathbb {P}(R^Y=1\\\\mid M=0,Y=0),\\\\\\\\&\\\\mathbb {P}_{R^Y01}=\\\\mathbb {P}(R^Y=1\\\\mid M=0,Y=1),\\\\\\\\&\\\\mathbb {P}_{R^Y10}=\\\\mathbb {P}(R^Y=1\\\\mid M=1,Y=0),\\\\\\\\&\\\\mathbb {P}_{R^Y11}=\\\\mathbb {P}(R^Y=1\\\\mid M=1,Y=1).$ Below we study the identifiablity of the above 9 parameters describing the graphical model in Figure REF based on the observable data probabilities.\",\n \"Note that although there are 9 observable data probabilities, the degree of freedom in the probabilities is only 8 given that they sum up to 1.\",\n \"The following relationships between the observable data probabilities and the parameters hold, $&\\\\mathbb {P}_{1111}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y1}\\\\mathbb {P}_{R^M1}\\\\mathbb {P}_{R^Y11},\\\\\\\\&\\\\mathbb {P}_{1011}=\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y1})\\\\mathbb {P}_{R^M1}\\\\mathbb {P}_{R^Y10},\\\\\\\\&\\\\mathbb {P}_{0111}=(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y0}\\\\mathbb {P}_{R^M0}\\\\mathbb {P}_{R^Y01},\\\\\\\\&\\\\mathbb {P}_{0011}=(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y0})\\\\mathbb {P}_{R^M0}\\\\mathbb {P}_{R^Y00},\\\\\\\\&\\\\mathbb {P}_{1+10}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y1}\\\\mathbb {P}_{R^M1}(1-\\\\mathbb {P}_{R^Y11})+\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y1})\\\\mathbb {P}_{R^M1}(1-\\\\mathbb {P}_{R^Y10}),\\\\\\\\&\\\\mathbb {P}_{0+10}=(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y0}\\\\mathbb {P}_{R^M0}(1-\\\\mathbb {P}_{R^Y01})+(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y0})\\\\mathbb {P}_{R^M0}(1-\\\\mathbb {P}_{R^Y00}),\\\\\\\\&\\\\mathbb {P}_{+101}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y1}(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y11}+(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y0}(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y01},\\\\\\\\&\\\\mathbb {P}_{+001}=\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y1})(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y10}+(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y0})(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y00}.$ By solving the equations (REF ) to (), we can identify the parameters $\\\\mathbb {P}_{M}$ , $\\\\mathbb {P}_{R^M1}$ and $\\\\mathbb {P}_{R^M0}$ .\",\n \"Given the observable data probabilities, $\\\\mathbb {P}_{M} = \\\\frac{1}{2}$ , $\\\\mathbb {P}_{R^M1} = \\\\frac{3}{4}$ and $\\\\mathbb {P}_{R^M0} =\\\\frac{1}{2} $ .\",\n \"However, $\\\\mathbb {P}_{Y1}$ , $\\\\mathbb {P}_{Y0}$ , $\\\\mathbb {P}_{R^Y11}$ , $\\\\mathbb {P}_{R^Y10}$ , $\\\\mathbb {P}_{R^Y01}$ and $\\\\mathbb {P}_{R^Y00}$ are not identifiable.\",\n \"For example, we can have $\\\\mathbb {P}_{Y1}=\\\\frac{3}{4}$ , $\\\\mathbb {P}_{Y0}=\\\\frac{1}{2}$ , $\\\\mathbb {P}_{R^Y11}=\\\\frac{4}{6}$ , $\\\\mathbb {P}_{R^Y10}=\\\\frac{2}{6}$ , $\\\\mathbb {P}_{R^Y01}=\\\\frac{3}{6}$ and $\\\\mathbb {P}_{R^Y00}=\\\\frac{1}{6}$ , which in turn give us $\\\\mathbb {P}(Y=1, M=1)=\\\\frac{3}{8}$ , $\\\\mathbb {P}(Y=1, M=0)=\\\\frac{1}{4}$ , $\\\\mathbb {P}(Y=0, M=1)=\\\\frac{1}{8}$ and $\\\\mathbb {P}(Y=0, M=0)=\\\\frac{1}{4}$ .\",\n \"Alternatively, we can have $\\\\mathbb {P}_{Y1}=\\\\frac{2}{3}$ , $\\\\mathbb {P}_{Y0}=\\\\frac{3}{4}$ , $\\\\mathbb {P}_{R^Y11}=\\\\frac{3}{4}$ , $\\\\mathbb {P}_{R^Y10}=\\\\frac{1}{4}$ , $\\\\mathbb {P}_{R^Y01}=\\\\frac{1}{3}$ and $\\\\mathbb {P}_{R^Y00}=\\\\frac{1}{3}$ , which in turn give us $\\\\mathbb {P}(Y=1, M=1)=\\\\frac{1}{3}$ , $\\\\mathbb {P}(Y=1, M=0)=\\\\frac{3}{8}$ , $\\\\mathbb {P}(Y=0, M=1)=\\\\frac{1}{6}$ and $\\\\mathbb {P}(Y=0, M=0)=\\\\frac{1}{8}$ .\",\n \"The two sets of values of $\\\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\\\mathbb {P}_{1111},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{++00})$ , and therefore, $\\\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions.\",\n \"Consider the following probabilities from the observable data: $&&(\\\\mathbb {P}_{1211},\\\\mathbb {P}_{1111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0211},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{+201},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{++00})\\\\nonumber \\\\\\\\&=&\\\\left(\\\\frac{180}{1440},\\\\frac{60}{1440},\\\\frac{15}{1440},\\\\frac{120}{1440},\\\\frac{30}{1440},\\\\frac{15}{1440},\\\\frac{285}{1440},\\\\frac{195}{1440},\\\\frac{180}{1440},\\\\frac{50}{1440},\\\\frac{20}{1440},\\\\frac{290}{1440}\\\\right)\\\\nonumber .$ Define $&\\\\mathbb {P}_M=\\\\mathbb {P}(M=1),\\\\\\\\&\\\\mathbb {P}_{Y^21}=\\\\mathbb {P}(Y=2\\\\mid M=1),\\\\\\\\&\\\\mathbb {P}_{Y^11}=\\\\mathbb {P}(Y=1\\\\mid M=1),\\\\\\\\&\\\\mathbb {P}_{Y^20}=\\\\mathbb {P}(Y=2\\\\mid M=0),\\\\\\\\&\\\\mathbb {P}_{Y^10}=\\\\mathbb {P}(Y=1\\\\mid M=0),\\\\\\\\&\\\\mathbb {P}_{R^M1}=\\\\mathbb {P}(R^M=1\\\\mid M=1),\\\\\\\\&\\\\mathbb {P}_{R^M0}=\\\\mathbb {P}(R^M=1\\\\mid M=0),\\\\\\\\&\\\\mathbb {P}_{R^Y2}=\\\\mathbb {P}(R^Y=1\\\\mid Y=2),\\\\\\\\&\\\\mathbb {P}_{R^Y1}=\\\\mathbb {P}(R^Y=1\\\\mid Y=1),\\\\\\\\&\\\\mathbb {P}_{R^Y0}=\\\\mathbb {P}(R^Y=1\\\\mid Y=0).$ The following relationships between the observable data probabilities and the parameters hold, $&\\\\mathbb {P}_{1211}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y^21}\\\\mathbb {P}_{R^M1}\\\\mathbb {P}_{R^Y2},\\\\\\\\&\\\\mathbb {P}_{1111}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y^11}\\\\mathbb {P}_{R^M1}\\\\mathbb {P}_{R^Y1},\\\\\\\\&\\\\mathbb {P}_{1011}=\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y^21}-\\\\mathbb {P}_{Y^11})\\\\mathbb {P}_{R^M1}\\\\mathbb {P}_{R^Y0},\\\\\\\\&\\\\mathbb {P}_{0211}=(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y^20}\\\\mathbb {P}_{R^M0}\\\\mathbb {P}_{R^Y2},\\\\\\\\&\\\\mathbb {P}_{0111}=(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y^10}\\\\mathbb {P}_{R^M0}\\\\mathbb {P}_{R^Y1},\\\\\\\\&\\\\mathbb {P}_{0011}=(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y^20}-\\\\mathbb {P}_{Y^10})\\\\mathbb {P}_{R^M0}\\\\mathbb {P}_{R^Y0},\\\\\\\\&\\\\mathbb {P}_{+201}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y^21}(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y2}+(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y^20}(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y2},\\\\\\\\&\\\\mathbb {P}_{+101}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y^11}(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y1}+(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y^10}(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y1},\\\\\\\\&\\\\mathbb {P}_{+001}=\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y^21}-\\\\mathbb {P}_{Y^11})(1-\\\\mathbb {P}_{R^M1})\\\\mathbb {P}_{R^Y0}+\\\\nonumber \\\\\\\\&\\\\hspace{42.67912pt}(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y^20}-\\\\mathbb {P}_{Y^10})(1-\\\\mathbb {P}_{R^M0})\\\\mathbb {P}_{R^Y0},\\\\\\\\&\\\\mathbb {P}_{1+10}=\\\\mathbb {P}_M\\\\mathbb {P}_{Y^21}\\\\mathbb {P}_{R^M1}(1-\\\\mathbb {P}_{R^Y2})+\\\\mathbb {P}_M\\\\mathbb {P}_{Y^11}\\\\mathbb {P}_{R^M1}(1-\\\\mathbb {P}_{R^Y1})+\\\\nonumber \\\\\\\\&\\\\hspace{42.67912pt}\\\\mathbb {P}_M(1-\\\\mathbb {P}_{Y^21}-\\\\mathbb {P}_{Y^11})\\\\mathbb {P}_{R^M1}(1-\\\\mathbb {P}_{R^Y0}),\\\\\\\\&\\\\mathbb {P}_{0+10}=(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y^20}\\\\mathbb {P}_{R^M0}(1-\\\\mathbb {P}_{R^Y2})+(1-\\\\mathbb {P}_M)\\\\mathbb {P}_{Y^10}\\\\mathbb {P}_{R^M0}(1-\\\\mathbb {P}_{R^Y1})+\\\\nonumber \\\\\\\\&\\\\hspace{42.67912pt}(1-\\\\mathbb {P}_M)(1-\\\\mathbb {P}_{Y^20}-\\\\mathbb {P}_{Y^10})\\\\mathbb {P}_{R^M0}(1-\\\\mathbb {P}_{R^Y0}).$ By solving the equations (REF ) to (), we can identify the parameters $\\\\mathbb {P}_{M}$ , $\\\\mathbb {P}_{R^M1}$ and $\\\\mathbb {P}_{R^M0}$ .\",\n \"Given the observable data probabilities, $\\\\mathbb {P}_{M}=\\\\frac{1}{2}$ , $\\\\mathbb {P}_{R^M1}=\\\\frac{3}{4}$ and $\\\\mathbb {P}_{R^M0}=\\\\frac{1}{2}$ .\",\n \"However, $\\\\mathbb {P}_{Y^21}$ , $\\\\mathbb {P}_{Y^11}$ , $\\\\mathbb {P}_{Y^20}$ , $\\\\mathbb {P}_{Y^10}$ , $\\\\mathbb {P}_{R^Y2}$ , $\\\\mathbb {P}_{R^Y1}$ and $\\\\mathbb {P}_{R^Y0}$ are not identifiable.\",\n \"For example, we can have $\\\\mathbb {P}_{Y^21}=\\\\frac{3}{6}$ , $\\\\mathbb {P}_{Y^11}=\\\\frac{2}{6}$ , $\\\\mathbb {P}_{Y^20}=\\\\frac{2}{4}$ , $\\\\mathbb {P}_{Y^10}=\\\\frac{1}{4}$ , $\\\\mathbb {P}_{R^Y2}=\\\\frac{2}{3}$ , $\\\\mathbb {P}_{R^Y1}=\\\\frac{1}{3}$ and $\\\\mathbb {P}_{R^Y0}=\\\\frac{1}{6}$ , which in turn give us $\\\\mathbb {P}(Y=2, M=1)=\\\\frac{1}{4}$ , $\\\\mathbb {P}(Y=1, M=1)=\\\\frac{1}{6}$ , $\\\\mathbb {P}(Y=0, M=1)=\\\\frac{1}{12}$ , $\\\\mathbb {P}(Y=2, M=0)=\\\\frac{1}{4}$ , $\\\\mathbb {P}(Y=1, M=0)=\\\\frac{1}{8}$ and $\\\\mathbb {P}(Y=0, M=0)=\\\\frac{1}{8}$ .\",\n \"Alternatively, we can have $\\\\mathbb {P}_{Y^21}=\\\\frac{5}{8}$ , $\\\\mathbb {P}_{Y^11}=\\\\frac{1}{4}$ , $\\\\mathbb {P}_{Y^20}=\\\\frac{5}{8}$ , $\\\\mathbb {P}_{Y^10}=\\\\frac{3}{16}$ , $\\\\mathbb {P}_{R^Y2}=\\\\frac{8}{15}$ , $\\\\mathbb {P}_{R^Y1}=\\\\frac{4}{9}$ and $\\\\mathbb {P}_{R^Y0}=\\\\frac{2}{9}$ , which in turn give us $\\\\mathbb {P}(Y=2, M=1)=\\\\frac{5}{16}$ , $\\\\mathbb {P}(Y=1, M=1)=\\\\frac{1}{8}$ , $\\\\mathbb {P}(Y=0, M=1)=\\\\frac{1}{16}$ , $\\\\mathbb {P}(Y=2, M=0)=\\\\frac{5}{16}$ , $\\\\mathbb {P}(Y=1, M=0)=\\\\frac{3}{32}$ and $\\\\mathbb {P}(Y=0, M=0)=\\\\frac{3}{32}$ .\",\n \"The two sets of values of $\\\\mathbb {P}(Y=y, M=m)$ correspond to the same observable data probabilities: $(\\\\mathbb {P}_{1211},\\\\mathbb {P}_{1111},\\\\mathbb {P}_{1011},\\\\mathbb {P}_{0211},\\\\mathbb {P}_{0111},\\\\mathbb {P}_{0011},\\\\mathbb {P}_{1+10},\\\\mathbb {P}_{0+10},\\\\mathbb {P}_{+201},\\\\mathbb {P}_{+101},\\\\mathbb {P}_{+001},\\\\mathbb {P}_{++00})$ , and therefore, $\\\\mathbb {P}(Y=y, M=m)$ can not be uniquely identified without further assumptions.\"\n ],\n [\n \"Parametric Estimation Details\",\n \"For illustration, we describe the parametric methods in the scenarios considered in Theorem REF when the missingness exists only in the mediator.\",\n \"Under Assumption REF , the log of the complete-data likelihood is $\\\\ell _{c}(\\\\theta ) = \\\\mathrm {constant}+\\\\sum ^n_{i=1} &\\\\log ~\\\\mathbb {P}(Y_i=y_i\\\\mid M_i=m_i,T_i=t_i, X_i = x_i )+\\\\log ~\\\\mathbb {P}(M_i=m_i \\\\mid T_i=t_i, X_i = x_i )\\\\\\\\& + \\\\log ~\\\\mathbb {P}(R^M_i=r^M_i \\\\mid M_i=m_i,T_i=t_i, X_i = x_i ).$ Under Assumption REF , the observed-data likelihood is $L_{obs}(\\\\theta )\\\\propto \\\\prod _{\\\\lbrace i:R^M_i=1\\\\rbrace } &\\\\mathbb {P}(Y_i=y_i\\\\mid M_i=m_i,T_i=t_i, X_i = x_i )\\\\mathbb {P}(M_i=m_i \\\\mid T_i=t_i, X_i = x_i )\\\\\\\\&\\\\mathbb {P}(R^M_i=1 \\\\mid M_i=m_i,T_i=t_i, X_i = x_i )\\\\\\\\&\\\\hspace{-39.83368pt}\\\\prod _{\\\\lbrace i:R^M_i=0\\\\rbrace }\\\\int _{\\\\mathcal {M}} \\\\mathbb {P}(Y_i=y_i\\\\mid M_i=m,T_i=t_i, X_i = x_i )\\\\mathbb {P}(M_i=m \\\\mid T_i=t_i, X_i = x_i )\\\\\\\\&\\\\mathbb {P}(R^M_i=0 \\\\mid M_i=m,T_i=t_i, X_i = x_i )\\\\ d m.$ When $M$ is categorical, the integral involved in the above expression is reduced to summation.\",\n \"Since the value of $M$ is missing for some subjects, we implement the Expectation-Maximization algorithm to obtain the Maximum Likelihood Estimates (MLEs) by treating the missing $M$ as a latent variable.\",\n \"Specifically, in the E-step, we find the conditional expectation of complete-data log likelihood by calculating the conditional expectation of $M_i$ for subjects with missing $M_i$ .\",\n \"For example, if $M$ is binary, $E\\\\lbrace I(M_{i}=m)\\\\mid Y_i,R^M_i=0,T_i, X_i ;\\\\theta ^{(t)}\\\\rbrace =\\\\frac{\\\\mathbb {P}(Y_i,M_{i}=m,R^M_i=0\\\\mid T_i, X_i )}{\\\\sum _{m={0,1}}\\\\mathbb {P}(Y_i,M_{i}=m,R^M_i=0\\\\mid T_i, X_i )}.$ When $M$ is continuous, the conditional expectation of complete-data log likelihood may be complicated to calculate.\",\n \"Therefore, we applied fractional imputation [16] using the idea of importance sampling and weighting method to approximate the conditional expectation.\",\n \"Specifically, we generate the fractionally imputed data $m^{(1)}_{i}, \\\\ldots , m^{(S)}_{i}$ from a proposed distribution $h(M_{i}\\\\mid T_i, X_i )$ for subjects with missing $M_i$ .\",\n \"Then, we compute the fractional weight for each imputed observation.\",\n \"The Monte Carlo approximation of the conditional expectation becomes more accurate when $S$ is large: $E\\\\lbrace \\\\ell _{ci}(M_{i}=m;\\\\theta )|Y_i,R^M_i=0,T_i, X_i ;\\\\theta ^{(t)}\\\\rbrace \\\\approx \\\\frac{1}{S}\\\\sum \\\\limits _{j=1}^{S} \\\\ell _{ci}(M_{i}=m^{(j)}_{i};\\\\theta ) \\\\hat{w}(m^{(j)}_{i}),$ where $\\\\hat{w}(m^{(j)}_{i}) \\\\propto \\\\frac{\\\\mathbb {P}(Y_{i},M_{i}=m^{(j)}_{i},R^M_i=0\\\\mid T_i, X_i )}{h(M_{i}=m^{(j)}_{i}\\\\mid T_i, X_i )}$ is the fractional weight for $m^{(j)}_{i}$ that satisfy $\\\\hat{w}(m^{(j)}_{i}) \\\\ge 0$ and $\\\\sum _{j=1}^{S} \\\\hat{w}(m^{(j)}_{i})=1$ .\",\n \"We iterate between the E-step and M-step until convergence.\",\n \"The same estimation methods can be applied to the situation where the missingness exists in both the mediator and outcome.\",\n \"For subjects with both $M_i$ and $Y_i$ missing, we generate the imputed data sequentially.\",\n \"For binary $M$ and binary $Y$ , we generate the possible value of ($m_i$ , $y_i$ ).\",\n \"For binary $M$ and continuous $Y$ , we generate the possible value of $m_i$ and then the fractionally imputed data $y^{(1)}_{i}, \\\\ldots , y^{(S)}_{i}$ for each possible value of $m_i$ .\",\n \"For continuous $M$ and continuous $Y$ , we generate the fractionally imputed data $(m^{(1)}_{i},y^{(1)}_{i})$ ,...,$(m^{(S)}_{i},y^{(S)}_{i})$ .\",\n \"For continuous $M$ and binary $Y$ , we generate the fractionally imputed data $m^{(1)}_{i}, \\\\ldots , m^{(S)}_{i}$ and then the possible value of $y_i$ for each fractionally imputed $m_i$ .\",\n \"It is important to note that we can identify the outcome model using the complete cases under MNAR Assumption REF , REF and REF .\",\n \"Therefore, an alternative approach for those scenarios is to estimate the outcome model first using the complete cases, then estimate the parameters in other models through the Expectation-Maximization algorithm by plugging in the estimated outcome model.\",\n \"We tried those two slightly different approaches to our simulation settings, both provided consistent results, with the alternative approach enjoying higher computation efficiency as expected.\",\n \"Note that under MNAR Assumption REF , the alternative approach does not work because the outcome distribution $\\\\mathbb {P}(Y \\\\mid M, T, X )$ can not be identified by the complete cases.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05577"}}},{"rowIdx":84,"cells":{"text":{"kind":"list like","value":[["A Formalization of Doob's Martingale Convergence Theorems in mathlib"],["Abstract We present the formalization of Doob's martingale convergence theorems in the mathlib library for the Lean theorem prover.","These theorems give conditions under which (sub)martingales converge, almost everywhere or in $L^1$.","In order to formalize those results, we build a definition of the conditional expectation in Banach spaces and develop the theory of stochastic processes, stopping times and martingales.","As an application of the convergence theorems, we also present the formalization of L\\'evy's generalized Borel-Cantelli lemma.","This work on martingale theory is one of the first developments of probability theory in mathlib, and it builds upon diverse parts of that library such as topology, analysis and most importantly measure theory."],["=1 calc decorations.pathmorphing curve/.style=settings=1,to path=() .. controls ($()!{pos}!()!","{height}!270:()$ ) and ($()!1-{pos}!()!","{height}!270:()$ ) .. (), settings/.code=quiver/.cd,1 , quiver/.cd,pos/.initial=0.35,height/.initial=0 tail reversed/.code=tikzcd to 2tail/.code=Implies[reversed] 2tail reversed/.code=Implies no body/.style=/tikz/dash pattern=on 0 off 1mm"]],"string":"[\n [\n \"A Formalization of Doob's Martingale Convergence Theorems in mathlib\"\n ],\n [\n \"Abstract We present the formalization of Doob's martingale convergence theorems in the mathlib library for the Lean theorem prover.\",\n \"These theorems give conditions under which (sub)martingales converge, almost everywhere or in $L^1$.\",\n \"In order to formalize those results, we build a definition of the conditional expectation in Banach spaces and develop the theory of stochastic processes, stopping times and martingales.\",\n \"As an application of the convergence theorems, we also present the formalization of L\\\\'evy's generalized Borel-Cantelli lemma.\",\n \"This work on martingale theory is one of the first developments of probability theory in mathlib, and it builds upon diverse parts of that library such as topology, analysis and most importantly measure theory.\"\n ],\n [\n \"=1 calc decorations.pathmorphing curve/.style=settings=1,to path=() .. controls ($()!{pos}!()!\",\n \"{height}!270:()$ ) and ($()!1-{pos}!()!\",\n \"{height}!270:()$ ) .. (), settings/.code=quiver/.cd,1 , quiver/.cd,pos/.initial=0.35,height/.initial=0 tail reversed/.code=tikzcd to 2tail/.code=Implies[reversed] 2tail reversed/.code=Implies no body/.style=/tikz/dash pattern=on 0 off 1mm\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05578"}}},{"rowIdx":85,"cells":{"text":{"kind":"list like","value":[["Bayesian inference for partial orders from random linear extensions:\n power relations from 12th Century Royal Acta"],["Abstract We give a new class of models for time series data in which actors are listed in order of precedence.","We model the lists as a realisation of a queue in which queue-position is constrained by an underlying social hierarchy.","We model the hierarchy as a partial order so that the lists are random linear extensions.","We account for noise via a random queue-jumping process.","We give a marginally consistent prior for the stochastic process of partial orders based on a latent variable representation for the partial order.","This allows us to introduce a parameter controlling partial order depth and incorporate actor-covariates informing the position of actors in the hierarchy.","We fit the model to witness lists from Royal Acta from England, Wales and Normandy in the eleventh and twelfth centuries.","Witnesses are listed in order of social rank, with any bishops present listed as a group.","Do changes in the order in which the bishops appear reflect changes in their personal authority?","The underlying social order which constrains the positions of bishops within lists need not be a complete order and so we model the evolving social order as an evolving partial order.","The status of an Anglo-Norman bishop was at the time partly determined by the length of time they had been in office.","This enters our model as a time-dependent covariate.","We fit the model, estimate partial orders and find evidence for changes in status over time.","We interpret our results in terms of court politics.","Simpler models, based on bucket orders and vertex-series-parallel orders, are rejected.","We compare our results with a stochastic process extension of the Plackett-Luce model."],["Observation model (likelihood)","Let $y = (o_i)_{i=1}^N$ be our data, which is a list of witness lists.","Under a Plackett-Luce model, the likelihood is $L(y | \\lambda , \\beta ) & = \\prod _{i=1}^N \\prod _{j=1}^{n_i} \\frac{\\exp (\\lambda _{o_{ij}, t_i} + \\beta _{r_{o_{ij}}})}{\\sum _{k=j}^{n_i} \\exp (\\lambda _{o_{ik}, t_i} + \\beta _{r_{o_{ik}}})} = \\prod _{i=1}^N \\prod _{j=1}^{n_i} \\frac{\\exp (f_{ij})}{\\sum _{k=j}^{n_i} \\exp (f_{ik})}\\\\l(y | \\lambda , \\beta ) = \\log L(y | \\lambda , \\beta ) & = \\sum _{i=1}^N \\sum _{j=1}^{n_i} \\left( f_{ij} - \\log \\left( \\sum _{k=j}^{n_i} \\exp (f_{ik})\\right)\\right)$ where $f_{ij} = \\lambda _{o_{ij}, t_i} + \\beta _{r_{o_{ij}}}$ , $o_{ij}$ is the j-th bishop of the i-th list, $t_i$ is the time of the i-th list, $r_{bishop}$ is the seniority rank of the bishop.","Here $\\lambda $ is an $n \\times T$ matrix, where $n$ is the total number of bishops (59 here), $T$ is the number of years we are considering (1155 - 1180 = 76 here).","$\\beta $ is a vector of length $n_d$ , which is the number of dioceses (16 here)."],["Priors","The prior on all the parameters is $p(\\lambda , \\beta , \\sigma , \\theta ) = p(\\lambda | \\sigma , \\theta ) p(\\beta ) p(\\sigma ) p(\\theta )$ The prior we put on $\\lambda $ is an AR(1) process as follows.","For all $i, t \\ge 1$ ,we have $\\lambda _{i, t} = \\theta \\lambda _{i, t-1} + \\epsilon _{i, t}$ where $\\epsilon _{i, t} \\sim \\mathcal {N}(0, \\sigma ^2)$ i.i.d., with $\\theta \\in (0, 1)$ and $\\sigma > 0$ .","We let the initial states of the process to have the stationary distribution of the process, i.e.","$\\lambda _{i,0} \\sim \\mathcal {N} \\left( 0, \\frac{\\sigma ^2}{1 - \\theta ^2} \\right) \\text{i.i.d.}","\\quad \\forall i \\in \\lbrace 1, \\dots , n\\rbrace $ However there is a non-identifiability issue with this setup.","Hence we constrain the $\\lambda $ so that $\\sum _{i=1}^n \\lambda _{i,t} = 0 \\forall t$ by considering the following process instead.","$\\lambda _{,t} = \\theta \\lambda _{,t-1} + M \\epsilon _{,t}$ where $M = I_n - \\frac{1}{n} \\mathbf {1} \\mathbf {1}$ with $\\mathbf {1}$ being a column of ones.","Notice the matrix $M$ projects the $\\epsilon $ to a subspace where they are centred.","Hence observe that if $\\lambda _{,t-1}$ is centred, then so is $\\lambda _{,t}$ .","Since the degree of freedom now for each time $t$ is $n-1$ , we only need to consider the prior on $n-1$ $\\lambda $ 's for each time.","Without loss of generality, we consider the prior on the first $n-1$ rows of $\\lambda $ and the prior becomes $p(\\lambda | \\theta , \\sigma ) & = p(\\lambda _{,0} | \\theta , \\sigma )\\prod _{t=1}^T p(\\lambda _{,t} | \\lambda _{, t-1}, \\theta , \\sigma ) \\\\p(\\lambda _{,t} | \\lambda _{,t-1}, \\theta , \\sigma ) & = \\mathcal {N}\\left( \\lambda _{1:(n-1), t} | \\theta \\lambda _{1:(n-1), t-1}, M_{n-1,} M_{n-1,}^T \\sigma ^2 \\right) \\\\p(\\lambda _{,0}| \\theta , \\sigma ) & = \\mathcal {N} \\left( \\lambda _{1:(n-1), 0} | 0, M_{n-1,} M_{n-1,}^T \\frac{\\sigma ^2}{1 - \\theta ^2} \\right)$ where $M_{n-1,}$ is the first $n-1$ rows of $M$ .","The prior on $\\beta $ is a standard multivariate normal, namely $p(\\beta ) = \\mathcal {N}(0, I_{n_d})$ However, there is the same identifiability issue and we also constrain them to be centred.","Hence the resulting prior becomes $p(\\beta ) = \\mathcal {N}(\\beta _{1:(n_d - 1)} | 0, N_{n_d - 1, } N_{n_d - 1, }^T)$ where $N_{n_d - 1, }$ is the first $n_d - 1$ rows of the matrix $N = I_{n_d} - \\frac{1}{n_d} \\mathbf {1}\\mathbf {1}^T$ .","Finally, we specify the prior on $\\theta $ and $\\sigma $ .","We chose a $U(0, 1)$ for $\\theta $ and a $\\mathcal {G}(2, 2)$ for $\\sigma $ ."],["The MCMC algorithm","Here the Metropolis-Hastings algorithm is used.","At each MCMC step, we propose a move for $\\theta $ , $\\sigma $ , each of the entries of the $\\beta $ vector except the last element, and the $\\lambda $ matrix except the last row.","More formally, for $k \\ge 1$ , we proposal distributions are $\\theta ^{(k)} & \\sim U(\\theta ^{(k-1)} - w_{\\theta }, \\theta ^{(k-1)} + w_{\\theta }) \\\\\\sigma ^{(k)} & \\sim U(\\sigma ^{(k-1)} - w_{\\sigma }, \\sigma ^{(k-1)} + w_{\\sigma }) \\\\\\beta _j^{(k)} & \\sim \\mathcal {N}(\\beta _j^{(k-1)}, \\sigma _{\\beta }^2) \\quad \\forall j \\in \\lbrace 1, \\dots , n_d-1\\rbrace , \\quad \\beta _{n_d}^{(k)} = - \\sum _{j=1}^{n_d - 1} \\beta _j^{(k)} \\\\\\lambda _{i,t}^{(k)} & \\sim \\mathcal {N}(\\lambda _{i,t}^{(k-1)}, \\sigma _{\\lambda }^2) \\quad \\forall i \\in \\lbrace 1, \\dots , n-1\\rbrace , \\quad \\lambda _{n,t}^{(k)} = - \\sum _{i=1}^{n - 1} \\lambda _{i,t}^{(k)} \\quad \\forall t \\in \\lbrace 1, \\dots , T\\rbrace $ where in our case, the hyperparameters are chosen as $w_{\\theta } = 0.05, w_{\\sigma } = 0.05, \\sigma _{\\theta } = 0.5, \\sigma _{\\lambda } = 0.5$ .","The initial states are $\\theta ^{(0)} = 0.95, \\sigma ^{(0)} = 0.5, \\beta = \\mathbf {0}, \\lambda =\\mathbf {0}$ .","In total, there are 77500 MCMC steps and we sub-sample every 10 steps, which results in 7750 samples.","When doing the analysis, the first 2000 samples are removed as burn-in, which leaves 5750 samples.","In my MCMC run, i separated the whole run into 10 mini-runs.","The acceptance probabilities for the parameters for the mini-runs has means ($\\pm $ standard deviations) $\\theta : 0.164 (\\pm 0.014), \\sigma : 0.212 (\\pm 0.035), \\beta : 0.506 (\\pm 0.006), \\lambda : 0.577 (\\pm 0.056)$ .","The ESS for the parameters are ($\\theta $ : 26, $\\sigma $ : 18.7, $\\beta $ : 176 - 5532, $\\lambda $ : 41 - 1062)"]],"string":"[\n [\n \"Bayesian inference for partial orders from random linear extensions:\\n power relations from 12th Century Royal Acta\"\n ],\n [\n \"Abstract We give a new class of models for time series data in which actors are listed in order of precedence.\",\n \"We model the lists as a realisation of a queue in which queue-position is constrained by an underlying social hierarchy.\",\n \"We model the hierarchy as a partial order so that the lists are random linear extensions.\",\n \"We account for noise via a random queue-jumping process.\",\n \"We give a marginally consistent prior for the stochastic process of partial orders based on a latent variable representation for the partial order.\",\n \"This allows us to introduce a parameter controlling partial order depth and incorporate actor-covariates informing the position of actors in the hierarchy.\",\n \"We fit the model to witness lists from Royal Acta from England, Wales and Normandy in the eleventh and twelfth centuries.\",\n \"Witnesses are listed in order of social rank, with any bishops present listed as a group.\",\n \"Do changes in the order in which the bishops appear reflect changes in their personal authority?\",\n \"The underlying social order which constrains the positions of bishops within lists need not be a complete order and so we model the evolving social order as an evolving partial order.\",\n \"The status of an Anglo-Norman bishop was at the time partly determined by the length of time they had been in office.\",\n \"This enters our model as a time-dependent covariate.\",\n \"We fit the model, estimate partial orders and find evidence for changes in status over time.\",\n \"We interpret our results in terms of court politics.\",\n \"Simpler models, based on bucket orders and vertex-series-parallel orders, are rejected.\",\n \"We compare our results with a stochastic process extension of the Plackett-Luce model.\"\n ],\n [\n \"Observation model (likelihood)\",\n \"Let $y = (o_i)_{i=1}^N$ be our data, which is a list of witness lists.\",\n \"Under a Plackett-Luce model, the likelihood is $L(y | \\\\lambda , \\\\beta ) & = \\\\prod _{i=1}^N \\\\prod _{j=1}^{n_i} \\\\frac{\\\\exp (\\\\lambda _{o_{ij}, t_i} + \\\\beta _{r_{o_{ij}}})}{\\\\sum _{k=j}^{n_i} \\\\exp (\\\\lambda _{o_{ik}, t_i} + \\\\beta _{r_{o_{ik}}})} = \\\\prod _{i=1}^N \\\\prod _{j=1}^{n_i} \\\\frac{\\\\exp (f_{ij})}{\\\\sum _{k=j}^{n_i} \\\\exp (f_{ik})}\\\\\\\\l(y | \\\\lambda , \\\\beta ) = \\\\log L(y | \\\\lambda , \\\\beta ) & = \\\\sum _{i=1}^N \\\\sum _{j=1}^{n_i} \\\\left( f_{ij} - \\\\log \\\\left( \\\\sum _{k=j}^{n_i} \\\\exp (f_{ik})\\\\right)\\\\right)$ where $f_{ij} = \\\\lambda _{o_{ij}, t_i} + \\\\beta _{r_{o_{ij}}}$ , $o_{ij}$ is the j-th bishop of the i-th list, $t_i$ is the time of the i-th list, $r_{bishop}$ is the seniority rank of the bishop.\",\n \"Here $\\\\lambda $ is an $n \\\\times T$ matrix, where $n$ is the total number of bishops (59 here), $T$ is the number of years we are considering (1155 - 1180 = 76 here).\",\n \"$\\\\beta $ is a vector of length $n_d$ , which is the number of dioceses (16 here).\"\n ],\n [\n \"Priors\",\n \"The prior on all the parameters is $p(\\\\lambda , \\\\beta , \\\\sigma , \\\\theta ) = p(\\\\lambda | \\\\sigma , \\\\theta ) p(\\\\beta ) p(\\\\sigma ) p(\\\\theta )$ The prior we put on $\\\\lambda $ is an AR(1) process as follows.\",\n \"For all $i, t \\\\ge 1$ ,we have $\\\\lambda _{i, t} = \\\\theta \\\\lambda _{i, t-1} + \\\\epsilon _{i, t}$ where $\\\\epsilon _{i, t} \\\\sim \\\\mathcal {N}(0, \\\\sigma ^2)$ i.i.d., with $\\\\theta \\\\in (0, 1)$ and $\\\\sigma > 0$ .\",\n \"We let the initial states of the process to have the stationary distribution of the process, i.e.\",\n \"$\\\\lambda _{i,0} \\\\sim \\\\mathcal {N} \\\\left( 0, \\\\frac{\\\\sigma ^2}{1 - \\\\theta ^2} \\\\right) \\\\text{i.i.d.}\",\n \"\\\\quad \\\\forall i \\\\in \\\\lbrace 1, \\\\dots , n\\\\rbrace $ However there is a non-identifiability issue with this setup.\",\n \"Hence we constrain the $\\\\lambda $ so that $\\\\sum _{i=1}^n \\\\lambda _{i,t} = 0 \\\\forall t$ by considering the following process instead.\",\n \"$\\\\lambda _{,t} = \\\\theta \\\\lambda _{,t-1} + M \\\\epsilon _{,t}$ where $M = I_n - \\\\frac{1}{n} \\\\mathbf {1} \\\\mathbf {1}$ with $\\\\mathbf {1}$ being a column of ones.\",\n \"Notice the matrix $M$ projects the $\\\\epsilon $ to a subspace where they are centred.\",\n \"Hence observe that if $\\\\lambda _{,t-1}$ is centred, then so is $\\\\lambda _{,t}$ .\",\n \"Since the degree of freedom now for each time $t$ is $n-1$ , we only need to consider the prior on $n-1$ $\\\\lambda $ 's for each time.\",\n \"Without loss of generality, we consider the prior on the first $n-1$ rows of $\\\\lambda $ and the prior becomes $p(\\\\lambda | \\\\theta , \\\\sigma ) & = p(\\\\lambda _{,0} | \\\\theta , \\\\sigma )\\\\prod _{t=1}^T p(\\\\lambda _{,t} | \\\\lambda _{, t-1}, \\\\theta , \\\\sigma ) \\\\\\\\p(\\\\lambda _{,t} | \\\\lambda _{,t-1}, \\\\theta , \\\\sigma ) & = \\\\mathcal {N}\\\\left( \\\\lambda _{1:(n-1), t} | \\\\theta \\\\lambda _{1:(n-1), t-1}, M_{n-1,} M_{n-1,}^T \\\\sigma ^2 \\\\right) \\\\\\\\p(\\\\lambda _{,0}| \\\\theta , \\\\sigma ) & = \\\\mathcal {N} \\\\left( \\\\lambda _{1:(n-1), 0} | 0, M_{n-1,} M_{n-1,}^T \\\\frac{\\\\sigma ^2}{1 - \\\\theta ^2} \\\\right)$ where $M_{n-1,}$ is the first $n-1$ rows of $M$ .\",\n \"The prior on $\\\\beta $ is a standard multivariate normal, namely $p(\\\\beta ) = \\\\mathcal {N}(0, I_{n_d})$ However, there is the same identifiability issue and we also constrain them to be centred.\",\n \"Hence the resulting prior becomes $p(\\\\beta ) = \\\\mathcal {N}(\\\\beta _{1:(n_d - 1)} | 0, N_{n_d - 1, } N_{n_d - 1, }^T)$ where $N_{n_d - 1, }$ is the first $n_d - 1$ rows of the matrix $N = I_{n_d} - \\\\frac{1}{n_d} \\\\mathbf {1}\\\\mathbf {1}^T$ .\",\n \"Finally, we specify the prior on $\\\\theta $ and $\\\\sigma $ .\",\n \"We chose a $U(0, 1)$ for $\\\\theta $ and a $\\\\mathcal {G}(2, 2)$ for $\\\\sigma $ .\"\n ],\n [\n \"The MCMC algorithm\",\n \"Here the Metropolis-Hastings algorithm is used.\",\n \"At each MCMC step, we propose a move for $\\\\theta $ , $\\\\sigma $ , each of the entries of the $\\\\beta $ vector except the last element, and the $\\\\lambda $ matrix except the last row.\",\n \"More formally, for $k \\\\ge 1$ , we proposal distributions are $\\\\theta ^{(k)} & \\\\sim U(\\\\theta ^{(k-1)} - w_{\\\\theta }, \\\\theta ^{(k-1)} + w_{\\\\theta }) \\\\\\\\\\\\sigma ^{(k)} & \\\\sim U(\\\\sigma ^{(k-1)} - w_{\\\\sigma }, \\\\sigma ^{(k-1)} + w_{\\\\sigma }) \\\\\\\\\\\\beta _j^{(k)} & \\\\sim \\\\mathcal {N}(\\\\beta _j^{(k-1)}, \\\\sigma _{\\\\beta }^2) \\\\quad \\\\forall j \\\\in \\\\lbrace 1, \\\\dots , n_d-1\\\\rbrace , \\\\quad \\\\beta _{n_d}^{(k)} = - \\\\sum _{j=1}^{n_d - 1} \\\\beta _j^{(k)} \\\\\\\\\\\\lambda _{i,t}^{(k)} & \\\\sim \\\\mathcal {N}(\\\\lambda _{i,t}^{(k-1)}, \\\\sigma _{\\\\lambda }^2) \\\\quad \\\\forall i \\\\in \\\\lbrace 1, \\\\dots , n-1\\\\rbrace , \\\\quad \\\\lambda _{n,t}^{(k)} = - \\\\sum _{i=1}^{n - 1} \\\\lambda _{i,t}^{(k)} \\\\quad \\\\forall t \\\\in \\\\lbrace 1, \\\\dots , T\\\\rbrace $ where in our case, the hyperparameters are chosen as $w_{\\\\theta } = 0.05, w_{\\\\sigma } = 0.05, \\\\sigma _{\\\\theta } = 0.5, \\\\sigma _{\\\\lambda } = 0.5$ .\",\n \"The initial states are $\\\\theta ^{(0)} = 0.95, \\\\sigma ^{(0)} = 0.5, \\\\beta = \\\\mathbf {0}, \\\\lambda =\\\\mathbf {0}$ .\",\n \"In total, there are 77500 MCMC steps and we sub-sample every 10 steps, which results in 7750 samples.\",\n \"When doing the analysis, the first 2000 samples are removed as burn-in, which leaves 5750 samples.\",\n \"In my MCMC run, i separated the whole run into 10 mini-runs.\",\n \"The acceptance probabilities for the parameters for the mini-runs has means ($\\\\pm $ standard deviations) $\\\\theta : 0.164 (\\\\pm 0.014), \\\\sigma : 0.212 (\\\\pm 0.035), \\\\beta : 0.506 (\\\\pm 0.006), \\\\lambda : 0.577 (\\\\pm 0.056)$ .\",\n \"The ESS for the parameters are ($\\\\theta $ : 26, $\\\\sigma $ : 18.7, $\\\\beta $ : 176 - 5532, $\\\\lambda $ : 41 - 1062)\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05524"}}},{"rowIdx":86,"cells":{"text":{"kind":"list like","value":[["Recurrent Vision Transformers for Object Detection with Event Cameras"],["Abstract We present Recurrent Vision Transformers (RVTs), a novel backbone for object detection with event cameras.","Event cameras provide visual information with sub-millisecond latency at a high-dynamic range and with strong robustness against motion blur.","These unique properties offer great potential for low-latency object detection and tracking in time-critical scenarios.","Prior work in event-based vision has achieved outstanding detection performance but at the cost of substantial inference time, typically beyond 40 milliseconds.","By revisiting the high-level design of recurrent vision backbones, we reduce inference time by a factor of 5 while retaining similar performance.","To achieve this, we explore a multi-stage design that utilizes three key concepts in each stage: First, a convolutional prior that can be regarded as a conditional positional embedding.","Second, local- and dilated global self-attention for spatial feature interaction.","Third, recurrent temporal feature aggregation to minimize latency while retaining temporal information.","RVTs can be trained from scratch to reach state-of-the-art performance on event-based object detection - achieving an mAP of 47.5% on the Gen1 automotive dataset.","At the same time, RVTs offer fast inference (13 ms on a T4 GPU) and favorable parameter efficiency (5 times fewer than prior art).","Our study brings new insights into effective design choices that could be fruitful for research beyond event-based vision."],["Introduction","Time matters for object detection.","In 30 milliseconds, a human can run 0.3 meters, a car on public roads covers up to 1 meter, and a train can travel over 2 meters.","Yet, during this time, an ordinary camera captures only a single frame.","Frame-based sensors must strike a balance between latency and bandwidth.","Given a fixed bandwidth, a frame-based camera must trade-off camera resolution and frame rate.","However, in highly dynamic scenes, reducing the resolution or the frame rate may come at the cost of missing essential scene details, and, in safety-critical scenarios like automotive, this may even cause fatalities.","In recent years, event cameras have emerged as alternative sensor that offers a different trade-off.","Instead of counterbalancing bandwidth requirements and perceptual latency, they provide visual information at sub-millisecond latency but sacrifice absolute intensity information.","Instead of capturing intensity images, event cameras measure changes in intensity at the time they occur.","This results in a stream of events, which encode time, location, and polarity of brightness changes [14].","The main advantages of event cameras are their sub-millisecond latency, very high dynamic range ($>120$ dB), strong robustness to motion blur, and ability to provide events asynchronously in a continuous manner.","Figure: Detection performance vs inference time of our RVT models on the 1 Mpx detection dataset using a T4 GPU.","The circle areas are proportional to the model size.In this work, we aim to utilize these outstanding properties of event cameras for object detection in time-critical scenarios.","Therefore, our objective is to design an approach that reduces the processing latency as much as possible while maintaining high performance.","This is challenging because event cameras asynchronously trigger binary events that are spread of pixel space and time.","Hence, we need to develop detection algorithms that can continuously associate features in the spatio-temporal domain while simultaneously satisfying strict latency requirements.","Recent work has shown that dynamic graph neural networks (GNNs) [27], [40] and sparse neural networks [32], [54], [52], [10] can theoretically achieve low latency inference for event-based object detection.","Yet, to achieve this in practical scenarios they either require specialized hardware or their detection performance needs to be improved.","An alternative thread of research approaches the problem from the view of conventional, dense neural network designs [19], [7], [20], [35], [25].","These methods show impressive performance on event-based object detection, especially when using temporal recurrence in their architectures [35], [25].","Still, the processing latency of these approaches remains beyond 40 milliseconds such that the low-latency aspect of event cameras cannot be fully leveraged.","This raises the question: How can we achieve both high accuracy and efficiency without requiring specialized hardware?","We notice that common design choices yield a suboptimal trade-off between performance and compute.","For example, prior work uses expensive convolutional LSTM (Conv-LSTM) cells [41] extensively in their feature extraction stage [35], [25] or relies on heavy backbones such as the VGG architecture [25].","Sparse neural networks instead struggle to model global mixing of features which is crucial to correctly locate and classify large objects in the scene.","To achieve our main objective, we fundamentally revisit the design of vision backbones for event-based object detection.","In particular, we take inspiration from neural network design for conventional frame-based object detection and combine them with ideas that have proven successful in the event-based vision literature.","Our study deliberately focuses on macro design of the object detection backbone to identify key components for both high performance and fast inference on GPUs.","The resulting neural network is based on a single block that is repeated four times to form a multi-stage hierarchical backbone that can be used with off-the-self detection frameworks.","We identify three key components that enable an excellent trade-off between detection performance and inference time.","First, we find that interleaved local- and global self-attention [47] is ideally suited to mix both local and global features while offering linear complexity in the input resolution.","Second, this attention mechanism is most effective when preceded by a simple convolution that also downsamples the spatial resolution from the previous stage.","This convolution effectively provides a strong prior about the grid-structure of the pixel array and also acts as a conditional positional embedding for the transformer layers [9].","Third, temporal recurrence is paramount to achieve strong detection performance with events.","Differently from prior work, we find that Conv-LSTM cells can be replaced by plain LSTM cells [18] that operate on each feature separatelyequivalent to $1\\times 1$ kernel in a Conv-LSTM cell.","By doing so, we dramatically reduce the number of parameters and latency but also slightly improve the overall performance.","Our full framework achieves competitive performance and higher efficiency compared to state-of-the-art methods.","Specifically, we reduce parameter count (from 100M to 18.5 M) and inference time (from 72 ms to 13 ms) up to a factor of 5 compared to prior art [25].","At the same time, we train our networks from scratch, showing that these benefits do not originate from large-scale pretraining.","Our main contributions can be summarized as follows: (1) We re-examine predominant design choices in event-based object detection pipelines and reveal a set of key enablers for high performance in event-based object detection.","(2) We propose a simple, composable stage design that unifies the crucial building blocks in a compact way.","(3) We propose a hierarchical multi-stage backbone that is fast, lightweight and still offers performance comparable to the best reported so far.","(4) We present state-of-the-art object detection performance of 47.5% mAP on the Gen1 detection dataset [11] and highly competitive results on the 1 Mpx detection dataset [35] while training the proposed architecture from scratch.","In addition, we also provide insights into effective data augmentation techniques that contribute to reaching these results.","Figure: Overview of the unrolled computation graph of our multi-stage recurrent backbone.","Events are processed into a tensor representation before they are used as input to the first stage.","Each stage also reuses the LSTM states (c: cell, h: hidden) from the previous timestep.","Finally, the detection framework interfaces with the backbone from the second stage onwards.","Specifically, the hidden states of the LSTMs are used as features for the detection framework."],["Object Detection for Event Cameras","Object detection in the event camera literature can be broadly classified into three emerging research directions.","Recent work explores graph neural networks to dynamically construct a spatio-temporal graph [27], [40], [33].","New nodes and node edges are established by sub-sampling events and finding existing nodes that are close in space-time.","The main challenge is to design the architecture such that information can propagate over vast distances in the space-time volume.","This is relevant, for example, when large objects move slowly with respect to the camera.","Furthermore, aggressive sub-sampling of events can lead to the removal of potentially crucial information, but is often required to maintain low-latency inference.","A second line of work employs spiking neural networks (SNNs) that propagate information sparsely within the network [54], [52], [10].","SNNs are closely related to dense recurrent neural networks (RNNs) in that each spiking neuron has an internal state that is propagated in time.","Differently from RNNs, neurons in SNNs only emit spikes whenever a threshold is reached.","This spike generation mechanism is not differentiable, which leads to substantial difficulties in optimizing these networks [34], [42], [23], [53], [22], [4], [44].","One workaround is to avoid the aforementioned threshold and instead propagate features throughout the receptive field [32] .","The downside of this mechanism is that the sparse-processing property is lost within deeper layers of the network.","Overall, the design and training of SNNs still requires fundamental investigation before competitive performance can be reached.","A third research direction is concerned with exploring dense neural networks for object detection with event cameras.","The first step is the creation of a dense tensor (event representation) that enables compatibility with dense operations such as convolutions.","Early work directly uses a single event representation generated from a short temporal window of events to infer detections [19], [7], [20].","These approaches discard relevant information from beyond the considered temporal window such that detecting slowly moving objects becomes difficult or impossible.","Followup work addresses this issue by incorporating recurrent neural network layers [35], [25] which drastically improved the detection performance.","We follow this line of work but revamp dominant architecture choices to build a canonical framework that is fast, lightweight and highly performant."],["Vision Transformers for Spatio-Temporal Data","The success of attention-based models [49] in NLP has inspired the exploration of transformer-based architectures in computer vision [12].","Attention-based models have recently also been explored in video classification [45], [13], [1], [5] where the models are applied directly to a set of frames.","While these approaches have shown promising results in spatio-temporal modelling, they are optimized for offline processing of stored video data.","In event-based vision, attention-based components have found applications in classification [39], [50] and image reconstruction [51], but their use in event-based object detection has yet to be investigated.","Figure: RVT block structure.","The input is convolved with kernel size k×kk\\times k and stride ss.","Block-SA applies self-attention in local windows while Grid-SA is a global operation using dilated attention.","Finally, each block ends with an LSTM that reuses the (cell- and hidden) states from the previous timestep.","The LSTM is applied to each feature separately.","Normalization and activation layers are omitted for conciseness."],["Method","Our object detection approach is designed to process a stream of events sequentially as they arrive.","Incoming events are first processed into tensors that represent events in space and time.","In every timestep, our network takes a new event representation as input as well as the previous states of the recurrent neural network layers.","After each pass through the backbone, the output of the RNNs are used as input to the detection framework.","The following sections elaborate on each one of these steps.","Figure REF shows an overview of the RVT architecture."],["Event Processing","Each pixel of an event camera can independently trigger an event when a significant log brightness change occurs.","An event can be positive or negative depending on the sign of the brightness change.","We characterize an event with polarity $p_k\\in \\lbrace 0,1\\rbrace $ as a tuple $e_k=(x_k,y_k,t_k,p_k)$ that occurs at pixel $(x_k, y_k)$ at time $t_k$ .","Modern event cameras can produce 10s of millions of events per second which renders event-by-event processing out of reach on conventional processing units.","In this work we opt for a very simple preprocessing step to enable compatibility with convolutional neural network layers which, as we will show later in Sec.","REF , are an important contributor to the performance of our model.","Our preprocessing step starts with the creation of a 4-dimensional tensor $E$ .","The first dimension consists of two components and represents the polarity.","The second dimension has $T$ components and is associated with $T$ discretization steps of time.","The 3rd and 4th dimension represent height and width of the event camera.","We process set of events $\\mathcal {E}$ within a time duration $[t_a,t_b)$ the following way: $E(p, \\tau , x, y) &= \\sum _{e_k\\in \\mathcal {E}}\\delta (p - p_k)\\delta (x-x_k,y-y_k)\\delta (\\tau -\\tau _k),\\\\\\tau _k &= \\left\\lfloor \\frac{t_k-t_a}{t_b - t_a}\\cdot T\\right\\rfloor $ In words, we create $T$ 2-channel frames where each pixel contains the number of positive or negative events within one of the $T$ temporal frames.","As a final step, we flatten the polarity and time dimension to retrieve a 3-dimensional tensor with shape $(2T, H, W)$ to directly enable compatibility with 2D convolutions.","We implement the presented algorithm with byte tensors to save memory and bandwidth.","Other, more sophisticated representations are possible [6], [55], [16], [2], [25], [48], [3], but their thorough evaluation is not our focus."],["Mixing Spatial and Temporal Features","The main difficulty of object detection with event cameras is that at any given time, the neural network should be able to efficiently (1) extract local- and global task-relevant features in pixel space because objects can cover both very small regions or large portions of the field of view; (2) extract features from very recent events (e.g.","moving edges) as well as events from several seconds ago.","This is necessary because some objects or moving slowly with respect to the camera such that they generate very few events over time.","These observations motivate us to investigate transformer layers for spatial feature extraction and recurrent neural networks for efficient temporal feature extraction.","Figure REF illustrates the components of a single stage."],["Spatial Feature Extraction","The spatial feature extraction stage should incorporate a prior about the fact that pixels are arranged in a 2D grid as early as possible in the computation graph.","We enable this by using a convolution with overlapping kernels on the input features that at the same time spatially downsamples the input or features from the previous stage.","This convolution also endows our model with a conditional positional embedding [9] such that we do not require absolute [49], [12] or relative [30] positional embeddings.","Our ablation study in Sec.","REF shows that overlapping kernels lead to a substantial boost in detection performance.","In a subsequent step, the resulting features are transformed through multi-axis self-attention.","We quickly summarize the steps but refer to Tu et.","al [47] for an elaborate explanation.","Multi-axis attention consists of two stages using self-attention.","The first stage performs local feature interaction while the second stage enables dilated global feature mixing.","More specifically, the features are first grouped locally into non-overlapping windows: Let $X\\in \\mathbb {R}^{H\\times W\\times C}$ be the input feature tensor.","We reshape the tensor to a shape $(\\frac{H}{P}\\times \\frac{W}{P}, P\\times P, C)$ where $P\\times P$ is the window size in which multi-head self-attention [49] is applied.","This block attention (Block-SA in Fig.","REF ) is used to model local interactions.","As a next step, we would ideally be able to extract features globally.","One straightforward way to achieve this would be applying self-attention on the whole feature map.","Unfortunately, global self-attention has quadratic complexity in the number of features.","Instead, we use grid attention (Grid-SA in Fig.","REF ).","Grid attention partitions the feature maps into a grid of shape $(G\\times G, \\frac{H}{G}\\times \\frac{W}{G}, C)$ using a $G\\times G$ uniform grid.","The resulting windows are of size $\\frac{H}{G}\\times \\frac{W}{G}$ .","Self-attention is then applied to these windows which corresponds to global, dilated mixing of features.","We study alternative designs as part of our architecture in the ablation studies in Sec.","REF ."],["Temporal Feature Extraction","We opt for temporal feature aggregation with LSTM [18] cells at the end of the stage.","Differently from prior work [35], [25] we find that temporal and spatial feature aggregation can be completely separated.","This means that we use plain LSTM cells such that the states of the LSTMs do not interact with each other.","By avoiding Conv-LSTM units [41], we can drastically reduce the computational complexity and parameter count.","I.e.","a Conv-LSTM with kernel size $k\\times k$ and stride 1 demands $k^2$ the number of parameters and compute compared to the original LSTM cell.","We examine this aspect in the experimental Sec.","REF ."],["Model Details","We apply LayerNorm [24] before and LayerScale [46] after each attention and MLP module, and add a residual connection after each module.","We found that LayerScale enables a wider range of learning rates."],["Hierarchical Multi-Stage Design","We compose multiple RVT blocks together to form a multi-stage hierarchical backbone.","The overall architecture is shown in Fig.","REF .","At first, a local temporal slice of events is processed into a 2D tensor format as formulated in the beginning of this section.","Subsequently, each stage takes the previous features as input and optionally uses the LSTM state from the last timestep to compute features for the next stage.","By saving the LSTM states for the following timestep, each recurrent stage can retain temporal information for the whole feature map.","We follow prior work and use features from the second to the fourth stage for the object detection framework.","To do so, we reshape the hidden states of the LSTMs into 2D feature maps."],["Experiments","We conduct ablations and evaluation our model on the Gen1 [11] and 1 Mpx [35] event camera datasets.","We train two variants of our model: RVT-B the base model and a small model RVT-S on both datasets.","Parameter details for both models are shown in Tab.","REF ."],["Implementation Details","We initialize all layers randomly except LayerScale which is initialized to 1e-5 for each module.","We train our models for 400k iterations with the ADAM optimizer [21] with a OneCycle learning rate schedule [43] using a 2000 warmup iterations and linear decay from a maximum learning rate or both models on both datasets and all ablation studies.","For comparison with prior work, we use backpropagation through time (BPTT) over a sequence length of 21 for the Gen1 dataset and 11 for the 1 Mpx dataset.","We use a batch size of 8 and a maximum learning rate of 2e-4 for the Gen1 dataset and train the model on a single Tesla V100 GPU which takes approximately 2 days.","For the 1 Mpx dataset, we use an effective batch size of 12, a maximum learning rate of 2.45e-4 and train the model on four Tesla V100 GPUs which takes up to 3 days.","Our data augmention includes random horizontal flipping, zooming in and zooming out applied over the BPTT sequence.","More details on data augmentation are available in Sec.","REF and the supplementary material.","For evaluation, we feed the model full sequences and only reset the state of the recurrent layers at the beginning of a new sequence.","This is different from the training setting where we only train on chunks of randomly sampled shorter sequences for BPTT.","Finally, we use the YOLOX framework [15], which includes the IOU loss, class loss and regression loss.","These losses are averaged both over the batch and sequence length for each optimization step.","We discretize time into $T=10$ bins and summarize 50 milliseconds of events into an event representation."],["Datasets","The Gen1 Automotive Detection dataset [11] consists of 39 hours of event camera recordings at a resolution of $304 \\times 240$ .","In total, the Gen1 dataset contains 228k car and 28k pedestrian bounding boxes available at 1, 2 or 4 Hz.","We follow the evaluation protocol of prior work [35], [25] and remove bounding boxes with a side length of less than 10 pixels and a diagonal of less than 30 pixels.","The 1 MPx dataset [35] also features driving scenarios but provides recordings at a higher resolution of $720 \\times 1280$ over a period of several months at day and night.","It consists of approximately 15 hours of event data labeled at a frequency of 30 or 60 Hz with a total amount of 25 million bounding box labels for three classes (car, pedestrian, and two-wheeler).","We follow the evaluation protocol of prior work [35], [25].","That is, we downsample the input resolution by 2 to nHD resolution ($640\\times 360$ ) and remove bounding boxes with a side length of less than 20 pixels and a diagonal of less than 60 pixels (at the original resolution).","We provide qualitative examples of this dataset together with predictions of our base model in Fig.","REF .","For both datasets, mean average precision (mAP) is the main metric [28] that we consider."],["Ablation Studies","This section examines the two main contributors to the final performance of the proposed model.","First, we investigate key components and design choices of the proposed backbone.","Second, we study the influence of different data augmentation techniques that are compatible with our sequential problem setting.","The ablation studies are performed on the Gen1 dataset, unless stated otherwise, and the training sequence length for BPTT is set to 11 instead of 21 to reduce the training time.","We set the window size of window-based attention models to $8\\times 10$ at all scales in the backbone.","This means that the window at the fourth stage covers the whole feature map.","Our ablation studies are performed on the validation set where the best performing model is taken after 400k iterations.","Table: Spatial Aggregation.","Multi-axis attention leads to the best results on both the Gen1 and 1 Mpx dataset."],["Spatial Interaction","In Tab.","REF , we study different spatial aggregation techniques.","For a fair comparison, we keep the LSTM and convolutional downsampling layers identical and only exchange the attention and MLP modules.","We compare multi-axis attention with ConvNext blocks [31] and Swin transformer blocks [30].","ConvNext is a convolutional neural network architecture that has shown competitive performance with transformer-based models on a wide range of tasks, including object detection.","We use the default kernel size of $7 \\times 7$ as originally suggested and place three ConvNeXt blocks in each stage to approximately match the number of parameters of the reference model.","Swin, instead, is an attention-based model that applies local self-attention in windows that interact with each other through cyclic shifting.","We find that our Swin variant achieves better performance than the ConvNext variant, however, both are outperformed by multi-axis self-attention [47] on both the Gen1 and 1 Mpx dataset.","This experiment suggests that global interaction at every stage (multi-axis) is advantageous to purely local interaction (Swin, ConvNext).","Table: Downsampling Strategy.","The usage of overlapping kernels leads to higher performance at the expense of a slight increase in the number of parameters."],["Convolutional Downsampling","The original vision transformer [12] architecture does not perform local feature interaction with convolutional layers.","Some popular hierarchical counterparts also choose to apply downsample features without overlapping kernels [8], [30].","In Tab.","REF , we compare overlapping and non-overlapping convolutional kernels in both the input layer (patch embedding) and feature downsampling stage.","While non-overlapping convolutions reduce the number of parameters, they cause a substantial drop in performance.","Consequently, we choose overlapping kernels in all stages of the network.","Table: LSTM kernel size.","Conv-LSTM variants do not outperform the feature specific (1×11\\times 1) LSTM."],["LSTM with Convolutions","Prior state-of-the-art approaches on object detection with event cameras heavily rely on convolutional LSTM cells [35], [25].","We revisit this design choice and experiment with plain LSTM cells and a depthwise separable Conv-LSTM variant [36].","The depthwise separable Conv-LSTM first applies a depthwise separable convolution on both the input and hidden state before a point-wise ($1\\times 1$ ) convolution is applied.","Our results in Tab.","REF suggest that plain LSTM cells are sufficient in our model and even outperform both variations.","This is to some degree surprising because both variants are a strict superset of the plain LSTM.","We decide to use a plain LSTM cell based on these observations.","Table: LSTM placement.","LSTM cells contribute to the overall performance even in the early stages.Table: Comparisons on test sets of Gen1 and 1 Mpx datasets.","Best results in bold and second best underlined.","Brackets (·)(\\cdot ) in runtime indicate the inference time of the backbone without detection head.","A star * ^* suggests that this information was not directly available and estimated based on the publications.","Runtime is measured in milliseconds for a batch size of 1.","We used a T4 GPU for RVT to compare against indicated timings in prior work , on comparable GPUs (Titan Xp)."],["LSTM Placement","In this ablation we study the influence of using temporal recurrence only in a subset of stages or not at all.","For all comparisons, we leave the model exactly the same but reset the states of the LSTMs at selected stages in each timestep.","This way, we can simulate the absence of recurrent layers while keeping the number of parameters constant in the comparisons.","The results in Tab.","REF suggest that using no recurrence at all leads to a drastic decline of detection performance.","Enabling the LSTMs in each stage, starting from the fourth consistently leads to enhanced performance.","Surprisingly, we find that adding an LSTM to the first stage also leads to improvements, albeit the increase in mAP is not large.","In general, this experiment suggests that the detection framework benefits from features that have been augmented with temporal information.","Based on our observations, we decide to keep the LSTM also in the first stage.","Table: Data Augmentation.","Data augmentation consistently improves the results."],["Data Augmentation","While data augmentation is not directly related to the model itself, it greatly influences the final result as we will illustrate next.","Here, we investigate three data augmentation techniques that are suitable for object detection on spatio-temporal data: Random (1) horizontal flipping, (2) zoom-in, and (3) zoom-out.","Zoom-in augmentation randomly selects crops that contain at least one full bounding box at the final timestep of the BPTT sequence (i.e.","during training).","This crop is then applied to the rest of the sequence before the crops are rescaled to the default resolution.","This procedure ensures that we have at least a single label to compute the loss function while maintaining the same resolution during training.","Zoom-out augmentation resizes the full input to a lower resolution and randomly places the downscaled input in a zero-tensor initialized at the default resolution.","This procedure is then applied in an identical way to the remaining BPTT sequence.","Table REF shows that our model is performing poorly if no data augmentation is applied.","Overall, we find that data augmentation is important to combat overfittig not only on the Gen1 sequence but also on the 1 Mpx dataset.","The most effective augmentation is zoom-in, followed by zoom-out and horizontal flipping.","Based on these results, we decide to apply all data augmentation techniques.","We report the specific hyperparameters in the supplementary material.","Figure: Predictions on the 1 Mpx dataset.","All examples are thematically picked to illustrate the behaviour of the model in different scenarios.","(d) shows a scenario in which the model can still partially detect objects in absence of events due the temporal memory."],["Benchmark Comparisons","In this section, we compare our proposed neural network architecture against prior work on both the Gen1 [11] and 1 Mpx dataset [35] and summarize the results in Tab.","REF .","We train two models, a base model (RVT-B) with approximately 18.5 million parameters and a small variant (RVT-S) with 4.4 million parameters by reducing the channel dimensions by two in each stage.","Their architectural hyperparameters are outlined in Tab.","REF .","To compare with prior work, we choose the models based on their best performance on the validation set and evaluate them on the test set.","From Tab.","REF we can draw multiple conclusions.","First, we observe that models using recurrent layers consistently outperform other approaches, both sparse (GNNs, SNNs) and dense feed-forward models without recurrent layers (Inception+SSD, RRC-Events, YOLOv3 Events) by an mAP of more than 10 on both datasets.","One notable exception is MatrixLSTM [6] which applies LSTM cells directly at the input.","In contrast, RED [35] and ASTMNet [25] employ recurrent layers only in deeper layers.","Our base model achieves a new state-of-the-art performance of 47.5 mAP on the Gen1 dataset and 47.0 mAP on the 1 Mpx dataset.","ASTMNet claims comparable results on both datasets albeit at the cost of using a much larger backbone and increased inference time.","The RED model, also reports favorable results, but achieves 7.5 lower mAP on the Gen1 and 4 mAP lower mAP on the 1 Mpx dataset compared to our model.","Finally, our small model is amongst the smallest in our comparison.","Still, it achieves 4.3 higher mAP on the Gen1 dataset than the RED model while using 5 times fewer parameters."],["Inference Time","We also compute the inference time of our model in PyTorch eager mode on a T4 GPU with a batch size of 1.","Unfortunately, both RED and ASTMNet are not open source such that we cannot directly compare inference time on the same GPU model.","Instead, we use the timings provided by the authors that conducted their timing experiments on comparable GPUs (e.g.","Titan Xp).","We report the timing results of both our base and small model in Tab.","REF and also visualize them in Fig.","REF .","With an inference time of 10.4 ms on the Gen1 dataset ($304\\times 240$ ) our base model yields a latency reduction of approximately 6 milliseconds compared to RED and over 3 times lower inference time than ASTMNet.","On the the 1 Mpx dataset, using a higher resolution of $640\\times 360$ , our base model runs 3 times faster than RED and over 5 times faster than ASTMNet.","Our small model keeps the inference time on both datasets below 10 milliseconds.","We also report the inference time of the backbone alone, by excluding computation dedicated to the detection framework.","The numbers in brackets in the timing column indicate that the backbone typically occupies about half of the total inference time.","The inference times of both model variants are quite similar, although the small model uses approximately 4 times less compute.","This suggests that a proper deployment (e.g.","with TensorRT) could reduce the overhead of PyTorch eager execution and further reduce the overall latency."],["Discussion and Limitations","We use a very simple event representation which does not leverage the full potential of event-based data.","For example, we only have a weak prior on the order of events because we process the temporal dimension directly with fully connected layers.","Recent work has shown substantial gains by introducing temporal convolutions in early layers [25].","Efficient low-level processing of event data is still an open research problem that we have not addressed in this work.","Our approach currently only uses event streams to detect objects.","Frames yield complementary information that, when properly incorporated, will yield significantly enhanced detection performance.","For example, in Fig.","REF (d) we can see that our model can retain information over some period when no events are available.","Still, the memory of the network will fade and detection performance deteriorates.","High quality frames even at low frame-rate could provide the missing complementary information.","Hence, we believe that a multi-modal extension of our method on a suitable dataset is a promising next step."],["Conclusion","We introduced a novel backbone architecture for object detection with event cameras.","The architecture consists of a stage design that is repeatedly applied to create a multi-stage hierarchical neural network.","Each stage compactly incorporates convolutional priors, local- and sparse global attention and recurrent feature aggregation.","Our experiments highlight that recurrent vision transformers can be trained from scratch to reach state-of-the-art performance in object detection with event cameras.","The resulting canonical stage-design is directly compatible with existing detection frameworks, and paves the way to low-latency object detection with event cameras on conventional hardware.","Nonetheless, we hope that this work also inspires novel designs in future neuromorphic systems."],["Acknowledgment","This work was supported by Huawei Zurich Research Center; by the National Centre of Competence in Research (NCCR) Robotics (grant agreement No.","51NF40-185543) through the Swiss National Science Foundation (SNSF), and the European Research Council (ERC) under grant agreement No.","864042 (AGILEFLIGHT)."],["Overview","Here we supplement the main paper with details regarding data augmentation parameters in section A- (A- refers to the sections in this appendix) and additional experiments in section A-.","More specifically, section A-REF studies skip connections for the LSTM cells and section A-REF discusses the cross-dataset generalization capabilities with a deployment on the DSEC [17] dataset."],["Data Augmentation Details","We apply three data augmentation techniques to train our models from scratch.","Table REF summarizes the probability of each augmentation being used on an individual sample.","For each sample, we may apply horizontal flipping and also apply a zoom augmentation subsequently.","For the zoom augmentation we draw from a Bernoulli distribution to indicate whether we apply the augmentation at all.","If zoom augmentation shall be applied, we randomly choose between zoom-in or zoom-out augmentation based on the respective probability.","For the zoom augmentations, there is also the parameter that defines the magnitude with which the augmentation is applied.","A magnitude of 1 means that no zoom is applied while a magnitude greater than 1 indicates how much zooming in or zooming out is applied with respect to the original resolution.","The magnitude that we finally apply is drawn from a continuous uniform distribution with bounds min and max.","Table: Data Augmentation Parameters.","The probability defines the Bernoulli distribution from which we draw the decision whether to apply this augmentation on a given sample."],["Additional Experiments","This section provides two additional experiments that did not fit into the main paper.","First, we briefly discuss an ablation on the possibility of using residual LSTM layers.","Second, we follow with a qualitative study of cross-dataset generalization using our model trained on the 1 Mpx dataset."],["Residual LSTM Ablation","Our model employs LSTM cells [18] without skip/residual connections in the model.","We also experimented with adding skip connections to the LSTM cells on the Gen1 [11] dataset.","Table REF shows that adding a residual connection to the LSTM cells leads to worse results.","We hypothesize that this residual connection hampers the LSTM's ability to control the mixture of incoming (current timestep) and retained temporal features (previous timesteps).","For example, it would be difficult for the residual-LSTM combination to ignore the incoming feature because the output of the LSTM is simply added to this feature.","Without the residual connection, the LSTM could simply set the input gate to 0 to ignore the input.","Table: LSTM with and without residual connection.","Using a skip connection over the LSTM cells leads to worse results."],["Cross-Dataset Generalization: From 1 Mpx to DSEC","DSEC is a dataset that features event cameras and global shutter cameras close to each other.","Unlike the 1 Mpx dataset [35] which was recorded mostly urban scenarios in Paris, DSEC [17] provides recordings from urban and rural regions in Switzerland.","Furthermore, the event camera used in the DSEC dataset is a Gen 3 prophesee event camera instead of a Gen 4 camera as in the 1 Mpx dataset.","In this section, we qualitatively show that our model can be successfully deployed in a different environment and using different event cameras.","We deploy RVT-B, trained on the 1 Mpx dataset , on several sequences of the DSEC dataset to qualitatively assess the cross-dataset generalization.","While DSEC does not provide object detection labels, we can visually assess the quality of the detections by using the provided calibration files to map the frames of the global shutter camera to the event camera view.","Figure REF and REF show predictions of our model together with the images closest in time to the detections.","Figure REF shows our model can successfully detect cars in mountainous environments.","In particular, Figure REF (a) shows an HDR scene where the global shutter frame is overexposed such that the approaching car is barely visible.","Due to the high dynamic range of the event camera, our model can detect the approaching car without any difficulty.","Figure REF features more urban environments where our model also manages to detect objects correctly.","Figure: Prediction examples on the DSEC dataset featuring a mountainous environment.","Frames are shown only for visualization purposes and are not used by the model.","Column (a) shows a typical high-dynamic range (HDR) scenario where the vehicle is exiting a tunnel with a car approaching from outside the tunnel.","The HDR capabilities of the event cameras enables our model to detect the approaching car.","Column (b) shows a scenario with a wet road and challenging reflections.Figure: Prediction examples on the DSEC dataset featuring a (sub-)urban environments.","Frames are shown only for visualization purposes and are not used by the model.","Column (a) illustrates a typical urban situation where pedestrians, two-wheelers, cars and other road users occupy the street simultaneously.","In column (c), we show a failure case of our model where a street pillar is erroneously detected and classified as a two-wheeler."],["Discussion of Failure Cases","By and large, our model can successfully detect objects on DSEC even though it has only been trained on the 1 Mpx dataset.","Still, we found failure cases that might stem from distribution shift between the datasets.","For example, Figure REF (c) shows the erroneous detection of a two-wheeler that instead is a pillar on the street.","Overall, the model is good at detecting cars but is less confident and accurate at detecting two-wheelers and pedestrians.","This effect likely stems from the fact that the 1 Mpx dataset has almost twice as many car labels as pedestrian and two-wheeler labels combined.","The submission comes with two videos showing the predictions on two separate sequences on DSEC for a better qualitative assessment."],["Gen1 {{cite:1a57c7d3860ad2c81ecd07530cb1f14e26ad31f9}}","“Prophesee Gen1 Automotive Detection Dataset License Terms and Conditions”: https://www.prophesee.ai/2020/01/24/prophesee-gen1-automotive-detection-dataset/"],["1 Mpx {{cite:b7f6a351fa8f5aea53b2ab10daf2026d7fc7796f}}","“Prophesee 1MegaPixel Automotive Detection Dataset License Terms and Conditions”: https://www.prophesee.ai/2020/11/24/automotive-megapixel-event-based-dataset/"],["DSEC {{cite:eade248fcd80be136c742eef47239dceb3a1892f}}","“Creative Commons Attribution-ShareAlike 4.0 International public license (CC BY-SA 4.0)”: https://dsec.ifi.uzh.ch/"]],"string":"[\n [\n \"Recurrent Vision Transformers for Object Detection with Event Cameras\"\n ],\n [\n \"Abstract We present Recurrent Vision Transformers (RVTs), a novel backbone for object detection with event cameras.\",\n \"Event cameras provide visual information with sub-millisecond latency at a high-dynamic range and with strong robustness against motion blur.\",\n \"These unique properties offer great potential for low-latency object detection and tracking in time-critical scenarios.\",\n \"Prior work in event-based vision has achieved outstanding detection performance but at the cost of substantial inference time, typically beyond 40 milliseconds.\",\n \"By revisiting the high-level design of recurrent vision backbones, we reduce inference time by a factor of 5 while retaining similar performance.\",\n \"To achieve this, we explore a multi-stage design that utilizes three key concepts in each stage: First, a convolutional prior that can be regarded as a conditional positional embedding.\",\n \"Second, local- and dilated global self-attention for spatial feature interaction.\",\n \"Third, recurrent temporal feature aggregation to minimize latency while retaining temporal information.\",\n \"RVTs can be trained from scratch to reach state-of-the-art performance on event-based object detection - achieving an mAP of 47.5% on the Gen1 automotive dataset.\",\n \"At the same time, RVTs offer fast inference (13 ms on a T4 GPU) and favorable parameter efficiency (5 times fewer than prior art).\",\n \"Our study brings new insights into effective design choices that could be fruitful for research beyond event-based vision.\"\n ],\n [\n \"Introduction\",\n \"Time matters for object detection.\",\n \"In 30 milliseconds, a human can run 0.3 meters, a car on public roads covers up to 1 meter, and a train can travel over 2 meters.\",\n \"Yet, during this time, an ordinary camera captures only a single frame.\",\n \"Frame-based sensors must strike a balance between latency and bandwidth.\",\n \"Given a fixed bandwidth, a frame-based camera must trade-off camera resolution and frame rate.\",\n \"However, in highly dynamic scenes, reducing the resolution or the frame rate may come at the cost of missing essential scene details, and, in safety-critical scenarios like automotive, this may even cause fatalities.\",\n \"In recent years, event cameras have emerged as alternative sensor that offers a different trade-off.\",\n \"Instead of counterbalancing bandwidth requirements and perceptual latency, they provide visual information at sub-millisecond latency but sacrifice absolute intensity information.\",\n \"Instead of capturing intensity images, event cameras measure changes in intensity at the time they occur.\",\n \"This results in a stream of events, which encode time, location, and polarity of brightness changes [14].\",\n \"The main advantages of event cameras are their sub-millisecond latency, very high dynamic range ($>120$ dB), strong robustness to motion blur, and ability to provide events asynchronously in a continuous manner.\",\n \"Figure: Detection performance vs inference time of our RVT models on the 1 Mpx detection dataset using a T4 GPU.\",\n \"The circle areas are proportional to the model size.In this work, we aim to utilize these outstanding properties of event cameras for object detection in time-critical scenarios.\",\n \"Therefore, our objective is to design an approach that reduces the processing latency as much as possible while maintaining high performance.\",\n \"This is challenging because event cameras asynchronously trigger binary events that are spread of pixel space and time.\",\n \"Hence, we need to develop detection algorithms that can continuously associate features in the spatio-temporal domain while simultaneously satisfying strict latency requirements.\",\n \"Recent work has shown that dynamic graph neural networks (GNNs) [27], [40] and sparse neural networks [32], [54], [52], [10] can theoretically achieve low latency inference for event-based object detection.\",\n \"Yet, to achieve this in practical scenarios they either require specialized hardware or their detection performance needs to be improved.\",\n \"An alternative thread of research approaches the problem from the view of conventional, dense neural network designs [19], [7], [20], [35], [25].\",\n \"These methods show impressive performance on event-based object detection, especially when using temporal recurrence in their architectures [35], [25].\",\n \"Still, the processing latency of these approaches remains beyond 40 milliseconds such that the low-latency aspect of event cameras cannot be fully leveraged.\",\n \"This raises the question: How can we achieve both high accuracy and efficiency without requiring specialized hardware?\",\n \"We notice that common design choices yield a suboptimal trade-off between performance and compute.\",\n \"For example, prior work uses expensive convolutional LSTM (Conv-LSTM) cells [41] extensively in their feature extraction stage [35], [25] or relies on heavy backbones such as the VGG architecture [25].\",\n \"Sparse neural networks instead struggle to model global mixing of features which is crucial to correctly locate and classify large objects in the scene.\",\n \"To achieve our main objective, we fundamentally revisit the design of vision backbones for event-based object detection.\",\n \"In particular, we take inspiration from neural network design for conventional frame-based object detection and combine them with ideas that have proven successful in the event-based vision literature.\",\n \"Our study deliberately focuses on macro design of the object detection backbone to identify key components for both high performance and fast inference on GPUs.\",\n \"The resulting neural network is based on a single block that is repeated four times to form a multi-stage hierarchical backbone that can be used with off-the-self detection frameworks.\",\n \"We identify three key components that enable an excellent trade-off between detection performance and inference time.\",\n \"First, we find that interleaved local- and global self-attention [47] is ideally suited to mix both local and global features while offering linear complexity in the input resolution.\",\n \"Second, this attention mechanism is most effective when preceded by a simple convolution that also downsamples the spatial resolution from the previous stage.\",\n \"This convolution effectively provides a strong prior about the grid-structure of the pixel array and also acts as a conditional positional embedding for the transformer layers [9].\",\n \"Third, temporal recurrence is paramount to achieve strong detection performance with events.\",\n \"Differently from prior work, we find that Conv-LSTM cells can be replaced by plain LSTM cells [18] that operate on each feature separatelyequivalent to $1\\\\times 1$ kernel in a Conv-LSTM cell.\",\n \"By doing so, we dramatically reduce the number of parameters and latency but also slightly improve the overall performance.\",\n \"Our full framework achieves competitive performance and higher efficiency compared to state-of-the-art methods.\",\n \"Specifically, we reduce parameter count (from 100M to 18.5 M) and inference time (from 72 ms to 13 ms) up to a factor of 5 compared to prior art [25].\",\n \"At the same time, we train our networks from scratch, showing that these benefits do not originate from large-scale pretraining.\",\n \"Our main contributions can be summarized as follows: (1) We re-examine predominant design choices in event-based object detection pipelines and reveal a set of key enablers for high performance in event-based object detection.\",\n \"(2) We propose a simple, composable stage design that unifies the crucial building blocks in a compact way.\",\n \"(3) We propose a hierarchical multi-stage backbone that is fast, lightweight and still offers performance comparable to the best reported so far.\",\n \"(4) We present state-of-the-art object detection performance of 47.5% mAP on the Gen1 detection dataset [11] and highly competitive results on the 1 Mpx detection dataset [35] while training the proposed architecture from scratch.\",\n \"In addition, we also provide insights into effective data augmentation techniques that contribute to reaching these results.\",\n \"Figure: Overview of the unrolled computation graph of our multi-stage recurrent backbone.\",\n \"Events are processed into a tensor representation before they are used as input to the first stage.\",\n \"Each stage also reuses the LSTM states (c: cell, h: hidden) from the previous timestep.\",\n \"Finally, the detection framework interfaces with the backbone from the second stage onwards.\",\n \"Specifically, the hidden states of the LSTMs are used as features for the detection framework.\"\n ],\n [\n \"Object Detection for Event Cameras\",\n \"Object detection in the event camera literature can be broadly classified into three emerging research directions.\",\n \"Recent work explores graph neural networks to dynamically construct a spatio-temporal graph [27], [40], [33].\",\n \"New nodes and node edges are established by sub-sampling events and finding existing nodes that are close in space-time.\",\n \"The main challenge is to design the architecture such that information can propagate over vast distances in the space-time volume.\",\n \"This is relevant, for example, when large objects move slowly with respect to the camera.\",\n \"Furthermore, aggressive sub-sampling of events can lead to the removal of potentially crucial information, but is often required to maintain low-latency inference.\",\n \"A second line of work employs spiking neural networks (SNNs) that propagate information sparsely within the network [54], [52], [10].\",\n \"SNNs are closely related to dense recurrent neural networks (RNNs) in that each spiking neuron has an internal state that is propagated in time.\",\n \"Differently from RNNs, neurons in SNNs only emit spikes whenever a threshold is reached.\",\n \"This spike generation mechanism is not differentiable, which leads to substantial difficulties in optimizing these networks [34], [42], [23], [53], [22], [4], [44].\",\n \"One workaround is to avoid the aforementioned threshold and instead propagate features throughout the receptive field [32] .\",\n \"The downside of this mechanism is that the sparse-processing property is lost within deeper layers of the network.\",\n \"Overall, the design and training of SNNs still requires fundamental investigation before competitive performance can be reached.\",\n \"A third research direction is concerned with exploring dense neural networks for object detection with event cameras.\",\n \"The first step is the creation of a dense tensor (event representation) that enables compatibility with dense operations such as convolutions.\",\n \"Early work directly uses a single event representation generated from a short temporal window of events to infer detections [19], [7], [20].\",\n \"These approaches discard relevant information from beyond the considered temporal window such that detecting slowly moving objects becomes difficult or impossible.\",\n \"Followup work addresses this issue by incorporating recurrent neural network layers [35], [25] which drastically improved the detection performance.\",\n \"We follow this line of work but revamp dominant architecture choices to build a canonical framework that is fast, lightweight and highly performant.\"\n ],\n [\n \"Vision Transformers for Spatio-Temporal Data\",\n \"The success of attention-based models [49] in NLP has inspired the exploration of transformer-based architectures in computer vision [12].\",\n \"Attention-based models have recently also been explored in video classification [45], [13], [1], [5] where the models are applied directly to a set of frames.\",\n \"While these approaches have shown promising results in spatio-temporal modelling, they are optimized for offline processing of stored video data.\",\n \"In event-based vision, attention-based components have found applications in classification [39], [50] and image reconstruction [51], but their use in event-based object detection has yet to be investigated.\",\n \"Figure: RVT block structure.\",\n \"The input is convolved with kernel size k×kk\\\\times k and stride ss.\",\n \"Block-SA applies self-attention in local windows while Grid-SA is a global operation using dilated attention.\",\n \"Finally, each block ends with an LSTM that reuses the (cell- and hidden) states from the previous timestep.\",\n \"The LSTM is applied to each feature separately.\",\n \"Normalization and activation layers are omitted for conciseness.\"\n ],\n [\n \"Method\",\n \"Our object detection approach is designed to process a stream of events sequentially as they arrive.\",\n \"Incoming events are first processed into tensors that represent events in space and time.\",\n \"In every timestep, our network takes a new event representation as input as well as the previous states of the recurrent neural network layers.\",\n \"After each pass through the backbone, the output of the RNNs are used as input to the detection framework.\",\n \"The following sections elaborate on each one of these steps.\",\n \"Figure REF shows an overview of the RVT architecture.\"\n ],\n [\n \"Event Processing\",\n \"Each pixel of an event camera can independently trigger an event when a significant log brightness change occurs.\",\n \"An event can be positive or negative depending on the sign of the brightness change.\",\n \"We characterize an event with polarity $p_k\\\\in \\\\lbrace 0,1\\\\rbrace $ as a tuple $e_k=(x_k,y_k,t_k,p_k)$ that occurs at pixel $(x_k, y_k)$ at time $t_k$ .\",\n \"Modern event cameras can produce 10s of millions of events per second which renders event-by-event processing out of reach on conventional processing units.\",\n \"In this work we opt for a very simple preprocessing step to enable compatibility with convolutional neural network layers which, as we will show later in Sec.\",\n \"REF , are an important contributor to the performance of our model.\",\n \"Our preprocessing step starts with the creation of a 4-dimensional tensor $E$ .\",\n \"The first dimension consists of two components and represents the polarity.\",\n \"The second dimension has $T$ components and is associated with $T$ discretization steps of time.\",\n \"The 3rd and 4th dimension represent height and width of the event camera.\",\n \"We process set of events $\\\\mathcal {E}$ within a time duration $[t_a,t_b)$ the following way: $E(p, \\\\tau , x, y) &= \\\\sum _{e_k\\\\in \\\\mathcal {E}}\\\\delta (p - p_k)\\\\delta (x-x_k,y-y_k)\\\\delta (\\\\tau -\\\\tau _k),\\\\\\\\\\\\tau _k &= \\\\left\\\\lfloor \\\\frac{t_k-t_a}{t_b - t_a}\\\\cdot T\\\\right\\\\rfloor $ In words, we create $T$ 2-channel frames where each pixel contains the number of positive or negative events within one of the $T$ temporal frames.\",\n \"As a final step, we flatten the polarity and time dimension to retrieve a 3-dimensional tensor with shape $(2T, H, W)$ to directly enable compatibility with 2D convolutions.\",\n \"We implement the presented algorithm with byte tensors to save memory and bandwidth.\",\n \"Other, more sophisticated representations are possible [6], [55], [16], [2], [25], [48], [3], but their thorough evaluation is not our focus.\"\n ],\n [\n \"Mixing Spatial and Temporal Features\",\n \"The main difficulty of object detection with event cameras is that at any given time, the neural network should be able to efficiently (1) extract local- and global task-relevant features in pixel space because objects can cover both very small regions or large portions of the field of view; (2) extract features from very recent events (e.g.\",\n \"moving edges) as well as events from several seconds ago.\",\n \"This is necessary because some objects or moving slowly with respect to the camera such that they generate very few events over time.\",\n \"These observations motivate us to investigate transformer layers for spatial feature extraction and recurrent neural networks for efficient temporal feature extraction.\",\n \"Figure REF illustrates the components of a single stage.\"\n ],\n [\n \"Spatial Feature Extraction\",\n \"The spatial feature extraction stage should incorporate a prior about the fact that pixels are arranged in a 2D grid as early as possible in the computation graph.\",\n \"We enable this by using a convolution with overlapping kernels on the input features that at the same time spatially downsamples the input or features from the previous stage.\",\n \"This convolution also endows our model with a conditional positional embedding [9] such that we do not require absolute [49], [12] or relative [30] positional embeddings.\",\n \"Our ablation study in Sec.\",\n \"REF shows that overlapping kernels lead to a substantial boost in detection performance.\",\n \"In a subsequent step, the resulting features are transformed through multi-axis self-attention.\",\n \"We quickly summarize the steps but refer to Tu et.\",\n \"al [47] for an elaborate explanation.\",\n \"Multi-axis attention consists of two stages using self-attention.\",\n \"The first stage performs local feature interaction while the second stage enables dilated global feature mixing.\",\n \"More specifically, the features are first grouped locally into non-overlapping windows: Let $X\\\\in \\\\mathbb {R}^{H\\\\times W\\\\times C}$ be the input feature tensor.\",\n \"We reshape the tensor to a shape $(\\\\frac{H}{P}\\\\times \\\\frac{W}{P}, P\\\\times P, C)$ where $P\\\\times P$ is the window size in which multi-head self-attention [49] is applied.\",\n \"This block attention (Block-SA in Fig.\",\n \"REF ) is used to model local interactions.\",\n \"As a next step, we would ideally be able to extract features globally.\",\n \"One straightforward way to achieve this would be applying self-attention on the whole feature map.\",\n \"Unfortunately, global self-attention has quadratic complexity in the number of features.\",\n \"Instead, we use grid attention (Grid-SA in Fig.\",\n \"REF ).\",\n \"Grid attention partitions the feature maps into a grid of shape $(G\\\\times G, \\\\frac{H}{G}\\\\times \\\\frac{W}{G}, C)$ using a $G\\\\times G$ uniform grid.\",\n \"The resulting windows are of size $\\\\frac{H}{G}\\\\times \\\\frac{W}{G}$ .\",\n \"Self-attention is then applied to these windows which corresponds to global, dilated mixing of features.\",\n \"We study alternative designs as part of our architecture in the ablation studies in Sec.\",\n \"REF .\"\n ],\n [\n \"Temporal Feature Extraction\",\n \"We opt for temporal feature aggregation with LSTM [18] cells at the end of the stage.\",\n \"Differently from prior work [35], [25] we find that temporal and spatial feature aggregation can be completely separated.\",\n \"This means that we use plain LSTM cells such that the states of the LSTMs do not interact with each other.\",\n \"By avoiding Conv-LSTM units [41], we can drastically reduce the computational complexity and parameter count.\",\n \"I.e.\",\n \"a Conv-LSTM with kernel size $k\\\\times k$ and stride 1 demands $k^2$ the number of parameters and compute compared to the original LSTM cell.\",\n \"We examine this aspect in the experimental Sec.\",\n \"REF .\"\n ],\n [\n \"Model Details\",\n \"We apply LayerNorm [24] before and LayerScale [46] after each attention and MLP module, and add a residual connection after each module.\",\n \"We found that LayerScale enables a wider range of learning rates.\"\n ],\n [\n \"Hierarchical Multi-Stage Design\",\n \"We compose multiple RVT blocks together to form a multi-stage hierarchical backbone.\",\n \"The overall architecture is shown in Fig.\",\n \"REF .\",\n \"At first, a local temporal slice of events is processed into a 2D tensor format as formulated in the beginning of this section.\",\n \"Subsequently, each stage takes the previous features as input and optionally uses the LSTM state from the last timestep to compute features for the next stage.\",\n \"By saving the LSTM states for the following timestep, each recurrent stage can retain temporal information for the whole feature map.\",\n \"We follow prior work and use features from the second to the fourth stage for the object detection framework.\",\n \"To do so, we reshape the hidden states of the LSTMs into 2D feature maps.\"\n ],\n [\n \"Experiments\",\n \"We conduct ablations and evaluation our model on the Gen1 [11] and 1 Mpx [35] event camera datasets.\",\n \"We train two variants of our model: RVT-B the base model and a small model RVT-S on both datasets.\",\n \"Parameter details for both models are shown in Tab.\",\n \"REF .\"\n ],\n [\n \"Implementation Details\",\n \"We initialize all layers randomly except LayerScale which is initialized to 1e-5 for each module.\",\n \"We train our models for 400k iterations with the ADAM optimizer [21] with a OneCycle learning rate schedule [43] using a 2000 warmup iterations and linear decay from a maximum learning rate or both models on both datasets and all ablation studies.\",\n \"For comparison with prior work, we use backpropagation through time (BPTT) over a sequence length of 21 for the Gen1 dataset and 11 for the 1 Mpx dataset.\",\n \"We use a batch size of 8 and a maximum learning rate of 2e-4 for the Gen1 dataset and train the model on a single Tesla V100 GPU which takes approximately 2 days.\",\n \"For the 1 Mpx dataset, we use an effective batch size of 12, a maximum learning rate of 2.45e-4 and train the model on four Tesla V100 GPUs which takes up to 3 days.\",\n \"Our data augmention includes random horizontal flipping, zooming in and zooming out applied over the BPTT sequence.\",\n \"More details on data augmentation are available in Sec.\",\n \"REF and the supplementary material.\",\n \"For evaluation, we feed the model full sequences and only reset the state of the recurrent layers at the beginning of a new sequence.\",\n \"This is different from the training setting where we only train on chunks of randomly sampled shorter sequences for BPTT.\",\n \"Finally, we use the YOLOX framework [15], which includes the IOU loss, class loss and regression loss.\",\n \"These losses are averaged both over the batch and sequence length for each optimization step.\",\n \"We discretize time into $T=10$ bins and summarize 50 milliseconds of events into an event representation.\"\n ],\n [\n \"Datasets\",\n \"The Gen1 Automotive Detection dataset [11] consists of 39 hours of event camera recordings at a resolution of $304 \\\\times 240$ .\",\n \"In total, the Gen1 dataset contains 228k car and 28k pedestrian bounding boxes available at 1, 2 or 4 Hz.\",\n \"We follow the evaluation protocol of prior work [35], [25] and remove bounding boxes with a side length of less than 10 pixels and a diagonal of less than 30 pixels.\",\n \"The 1 MPx dataset [35] also features driving scenarios but provides recordings at a higher resolution of $720 \\\\times 1280$ over a period of several months at day and night.\",\n \"It consists of approximately 15 hours of event data labeled at a frequency of 30 or 60 Hz with a total amount of 25 million bounding box labels for three classes (car, pedestrian, and two-wheeler).\",\n \"We follow the evaluation protocol of prior work [35], [25].\",\n \"That is, we downsample the input resolution by 2 to nHD resolution ($640\\\\times 360$ ) and remove bounding boxes with a side length of less than 20 pixels and a diagonal of less than 60 pixels (at the original resolution).\",\n \"We provide qualitative examples of this dataset together with predictions of our base model in Fig.\",\n \"REF .\",\n \"For both datasets, mean average precision (mAP) is the main metric [28] that we consider.\"\n ],\n [\n \"Ablation Studies\",\n \"This section examines the two main contributors to the final performance of the proposed model.\",\n \"First, we investigate key components and design choices of the proposed backbone.\",\n \"Second, we study the influence of different data augmentation techniques that are compatible with our sequential problem setting.\",\n \"The ablation studies are performed on the Gen1 dataset, unless stated otherwise, and the training sequence length for BPTT is set to 11 instead of 21 to reduce the training time.\",\n \"We set the window size of window-based attention models to $8\\\\times 10$ at all scales in the backbone.\",\n \"This means that the window at the fourth stage covers the whole feature map.\",\n \"Our ablation studies are performed on the validation set where the best performing model is taken after 400k iterations.\",\n \"Table: Spatial Aggregation.\",\n \"Multi-axis attention leads to the best results on both the Gen1 and 1 Mpx dataset.\"\n ],\n [\n \"Spatial Interaction\",\n \"In Tab.\",\n \"REF , we study different spatial aggregation techniques.\",\n \"For a fair comparison, we keep the LSTM and convolutional downsampling layers identical and only exchange the attention and MLP modules.\",\n \"We compare multi-axis attention with ConvNext blocks [31] and Swin transformer blocks [30].\",\n \"ConvNext is a convolutional neural network architecture that has shown competitive performance with transformer-based models on a wide range of tasks, including object detection.\",\n \"We use the default kernel size of $7 \\\\times 7$ as originally suggested and place three ConvNeXt blocks in each stage to approximately match the number of parameters of the reference model.\",\n \"Swin, instead, is an attention-based model that applies local self-attention in windows that interact with each other through cyclic shifting.\",\n \"We find that our Swin variant achieves better performance than the ConvNext variant, however, both are outperformed by multi-axis self-attention [47] on both the Gen1 and 1 Mpx dataset.\",\n \"This experiment suggests that global interaction at every stage (multi-axis) is advantageous to purely local interaction (Swin, ConvNext).\",\n \"Table: Downsampling Strategy.\",\n \"The usage of overlapping kernels leads to higher performance at the expense of a slight increase in the number of parameters.\"\n ],\n [\n \"Convolutional Downsampling\",\n \"The original vision transformer [12] architecture does not perform local feature interaction with convolutional layers.\",\n \"Some popular hierarchical counterparts also choose to apply downsample features without overlapping kernels [8], [30].\",\n \"In Tab.\",\n \"REF , we compare overlapping and non-overlapping convolutional kernels in both the input layer (patch embedding) and feature downsampling stage.\",\n \"While non-overlapping convolutions reduce the number of parameters, they cause a substantial drop in performance.\",\n \"Consequently, we choose overlapping kernels in all stages of the network.\",\n \"Table: LSTM kernel size.\",\n \"Conv-LSTM variants do not outperform the feature specific (1×11\\\\times 1) LSTM.\"\n ],\n [\n \"LSTM with Convolutions\",\n \"Prior state-of-the-art approaches on object detection with event cameras heavily rely on convolutional LSTM cells [35], [25].\",\n \"We revisit this design choice and experiment with plain LSTM cells and a depthwise separable Conv-LSTM variant [36].\",\n \"The depthwise separable Conv-LSTM first applies a depthwise separable convolution on both the input and hidden state before a point-wise ($1\\\\times 1$ ) convolution is applied.\",\n \"Our results in Tab.\",\n \"REF suggest that plain LSTM cells are sufficient in our model and even outperform both variations.\",\n \"This is to some degree surprising because both variants are a strict superset of the plain LSTM.\",\n \"We decide to use a plain LSTM cell based on these observations.\",\n \"Table: LSTM placement.\",\n \"LSTM cells contribute to the overall performance even in the early stages.Table: Comparisons on test sets of Gen1 and 1 Mpx datasets.\",\n \"Best results in bold and second best underlined.\",\n \"Brackets (·)(\\\\cdot ) in runtime indicate the inference time of the backbone without detection head.\",\n \"A star * ^* suggests that this information was not directly available and estimated based on the publications.\",\n \"Runtime is measured in milliseconds for a batch size of 1.\",\n \"We used a T4 GPU for RVT to compare against indicated timings in prior work , on comparable GPUs (Titan Xp).\"\n ],\n [\n \"LSTM Placement\",\n \"In this ablation we study the influence of using temporal recurrence only in a subset of stages or not at all.\",\n \"For all comparisons, we leave the model exactly the same but reset the states of the LSTMs at selected stages in each timestep.\",\n \"This way, we can simulate the absence of recurrent layers while keeping the number of parameters constant in the comparisons.\",\n \"The results in Tab.\",\n \"REF suggest that using no recurrence at all leads to a drastic decline of detection performance.\",\n \"Enabling the LSTMs in each stage, starting from the fourth consistently leads to enhanced performance.\",\n \"Surprisingly, we find that adding an LSTM to the first stage also leads to improvements, albeit the increase in mAP is not large.\",\n \"In general, this experiment suggests that the detection framework benefits from features that have been augmented with temporal information.\",\n \"Based on our observations, we decide to keep the LSTM also in the first stage.\",\n \"Table: Data Augmentation.\",\n \"Data augmentation consistently improves the results.\"\n ],\n [\n \"Data Augmentation\",\n \"While data augmentation is not directly related to the model itself, it greatly influences the final result as we will illustrate next.\",\n \"Here, we investigate three data augmentation techniques that are suitable for object detection on spatio-temporal data: Random (1) horizontal flipping, (2) zoom-in, and (3) zoom-out.\",\n \"Zoom-in augmentation randomly selects crops that contain at least one full bounding box at the final timestep of the BPTT sequence (i.e.\",\n \"during training).\",\n \"This crop is then applied to the rest of the sequence before the crops are rescaled to the default resolution.\",\n \"This procedure ensures that we have at least a single label to compute the loss function while maintaining the same resolution during training.\",\n \"Zoom-out augmentation resizes the full input to a lower resolution and randomly places the downscaled input in a zero-tensor initialized at the default resolution.\",\n \"This procedure is then applied in an identical way to the remaining BPTT sequence.\",\n \"Table REF shows that our model is performing poorly if no data augmentation is applied.\",\n \"Overall, we find that data augmentation is important to combat overfittig not only on the Gen1 sequence but also on the 1 Mpx dataset.\",\n \"The most effective augmentation is zoom-in, followed by zoom-out and horizontal flipping.\",\n \"Based on these results, we decide to apply all data augmentation techniques.\",\n \"We report the specific hyperparameters in the supplementary material.\",\n \"Figure: Predictions on the 1 Mpx dataset.\",\n \"All examples are thematically picked to illustrate the behaviour of the model in different scenarios.\",\n \"(d) shows a scenario in which the model can still partially detect objects in absence of events due the temporal memory.\"\n ],\n [\n \"Benchmark Comparisons\",\n \"In this section, we compare our proposed neural network architecture against prior work on both the Gen1 [11] and 1 Mpx dataset [35] and summarize the results in Tab.\",\n \"REF .\",\n \"We train two models, a base model (RVT-B) with approximately 18.5 million parameters and a small variant (RVT-S) with 4.4 million parameters by reducing the channel dimensions by two in each stage.\",\n \"Their architectural hyperparameters are outlined in Tab.\",\n \"REF .\",\n \"To compare with prior work, we choose the models based on their best performance on the validation set and evaluate them on the test set.\",\n \"From Tab.\",\n \"REF we can draw multiple conclusions.\",\n \"First, we observe that models using recurrent layers consistently outperform other approaches, both sparse (GNNs, SNNs) and dense feed-forward models without recurrent layers (Inception+SSD, RRC-Events, YOLOv3 Events) by an mAP of more than 10 on both datasets.\",\n \"One notable exception is MatrixLSTM [6] which applies LSTM cells directly at the input.\",\n \"In contrast, RED [35] and ASTMNet [25] employ recurrent layers only in deeper layers.\",\n \"Our base model achieves a new state-of-the-art performance of 47.5 mAP on the Gen1 dataset and 47.0 mAP on the 1 Mpx dataset.\",\n \"ASTMNet claims comparable results on both datasets albeit at the cost of using a much larger backbone and increased inference time.\",\n \"The RED model, also reports favorable results, but achieves 7.5 lower mAP on the Gen1 and 4 mAP lower mAP on the 1 Mpx dataset compared to our model.\",\n \"Finally, our small model is amongst the smallest in our comparison.\",\n \"Still, it achieves 4.3 higher mAP on the Gen1 dataset than the RED model while using 5 times fewer parameters.\"\n ],\n [\n \"Inference Time\",\n \"We also compute the inference time of our model in PyTorch eager mode on a T4 GPU with a batch size of 1.\",\n \"Unfortunately, both RED and ASTMNet are not open source such that we cannot directly compare inference time on the same GPU model.\",\n \"Instead, we use the timings provided by the authors that conducted their timing experiments on comparable GPUs (e.g.\",\n \"Titan Xp).\",\n \"We report the timing results of both our base and small model in Tab.\",\n \"REF and also visualize them in Fig.\",\n \"REF .\",\n \"With an inference time of 10.4 ms on the Gen1 dataset ($304\\\\times 240$ ) our base model yields a latency reduction of approximately 6 milliseconds compared to RED and over 3 times lower inference time than ASTMNet.\",\n \"On the the 1 Mpx dataset, using a higher resolution of $640\\\\times 360$ , our base model runs 3 times faster than RED and over 5 times faster than ASTMNet.\",\n \"Our small model keeps the inference time on both datasets below 10 milliseconds.\",\n \"We also report the inference time of the backbone alone, by excluding computation dedicated to the detection framework.\",\n \"The numbers in brackets in the timing column indicate that the backbone typically occupies about half of the total inference time.\",\n \"The inference times of both model variants are quite similar, although the small model uses approximately 4 times less compute.\",\n \"This suggests that a proper deployment (e.g.\",\n \"with TensorRT) could reduce the overhead of PyTorch eager execution and further reduce the overall latency.\"\n ],\n [\n \"Discussion and Limitations\",\n \"We use a very simple event representation which does not leverage the full potential of event-based data.\",\n \"For example, we only have a weak prior on the order of events because we process the temporal dimension directly with fully connected layers.\",\n \"Recent work has shown substantial gains by introducing temporal convolutions in early layers [25].\",\n \"Efficient low-level processing of event data is still an open research problem that we have not addressed in this work.\",\n \"Our approach currently only uses event streams to detect objects.\",\n \"Frames yield complementary information that, when properly incorporated, will yield significantly enhanced detection performance.\",\n \"For example, in Fig.\",\n \"REF (d) we can see that our model can retain information over some period when no events are available.\",\n \"Still, the memory of the network will fade and detection performance deteriorates.\",\n \"High quality frames even at low frame-rate could provide the missing complementary information.\",\n \"Hence, we believe that a multi-modal extension of our method on a suitable dataset is a promising next step.\"\n ],\n [\n \"Conclusion\",\n \"We introduced a novel backbone architecture for object detection with event cameras.\",\n \"The architecture consists of a stage design that is repeatedly applied to create a multi-stage hierarchical neural network.\",\n \"Each stage compactly incorporates convolutional priors, local- and sparse global attention and recurrent feature aggregation.\",\n \"Our experiments highlight that recurrent vision transformers can be trained from scratch to reach state-of-the-art performance in object detection with event cameras.\",\n \"The resulting canonical stage-design is directly compatible with existing detection frameworks, and paves the way to low-latency object detection with event cameras on conventional hardware.\",\n \"Nonetheless, we hope that this work also inspires novel designs in future neuromorphic systems.\"\n ],\n [\n \"Acknowledgment\",\n \"This work was supported by Huawei Zurich Research Center; by the National Centre of Competence in Research (NCCR) Robotics (grant agreement No.\",\n \"51NF40-185543) through the Swiss National Science Foundation (SNSF), and the European Research Council (ERC) under grant agreement No.\",\n \"864042 (AGILEFLIGHT).\"\n ],\n [\n \"Overview\",\n \"Here we supplement the main paper with details regarding data augmentation parameters in section A- (A- refers to the sections in this appendix) and additional experiments in section A-.\",\n \"More specifically, section A-REF studies skip connections for the LSTM cells and section A-REF discusses the cross-dataset generalization capabilities with a deployment on the DSEC [17] dataset.\"\n ],\n [\n \"Data Augmentation Details\",\n \"We apply three data augmentation techniques to train our models from scratch.\",\n \"Table REF summarizes the probability of each augmentation being used on an individual sample.\",\n \"For each sample, we may apply horizontal flipping and also apply a zoom augmentation subsequently.\",\n \"For the zoom augmentation we draw from a Bernoulli distribution to indicate whether we apply the augmentation at all.\",\n \"If zoom augmentation shall be applied, we randomly choose between zoom-in or zoom-out augmentation based on the respective probability.\",\n \"For the zoom augmentations, there is also the parameter that defines the magnitude with which the augmentation is applied.\",\n \"A magnitude of 1 means that no zoom is applied while a magnitude greater than 1 indicates how much zooming in or zooming out is applied with respect to the original resolution.\",\n \"The magnitude that we finally apply is drawn from a continuous uniform distribution with bounds min and max.\",\n \"Table: Data Augmentation Parameters.\",\n \"The probability defines the Bernoulli distribution from which we draw the decision whether to apply this augmentation on a given sample.\"\n ],\n [\n \"Additional Experiments\",\n \"This section provides two additional experiments that did not fit into the main paper.\",\n \"First, we briefly discuss an ablation on the possibility of using residual LSTM layers.\",\n \"Second, we follow with a qualitative study of cross-dataset generalization using our model trained on the 1 Mpx dataset.\"\n ],\n [\n \"Residual LSTM Ablation\",\n \"Our model employs LSTM cells [18] without skip/residual connections in the model.\",\n \"We also experimented with adding skip connections to the LSTM cells on the Gen1 [11] dataset.\",\n \"Table REF shows that adding a residual connection to the LSTM cells leads to worse results.\",\n \"We hypothesize that this residual connection hampers the LSTM's ability to control the mixture of incoming (current timestep) and retained temporal features (previous timesteps).\",\n \"For example, it would be difficult for the residual-LSTM combination to ignore the incoming feature because the output of the LSTM is simply added to this feature.\",\n \"Without the residual connection, the LSTM could simply set the input gate to 0 to ignore the input.\",\n \"Table: LSTM with and without residual connection.\",\n \"Using a skip connection over the LSTM cells leads to worse results.\"\n ],\n [\n \"Cross-Dataset Generalization: From 1 Mpx to DSEC\",\n \"DSEC is a dataset that features event cameras and global shutter cameras close to each other.\",\n \"Unlike the 1 Mpx dataset [35] which was recorded mostly urban scenarios in Paris, DSEC [17] provides recordings from urban and rural regions in Switzerland.\",\n \"Furthermore, the event camera used in the DSEC dataset is a Gen 3 prophesee event camera instead of a Gen 4 camera as in the 1 Mpx dataset.\",\n \"In this section, we qualitatively show that our model can be successfully deployed in a different environment and using different event cameras.\",\n \"We deploy RVT-B, trained on the 1 Mpx dataset , on several sequences of the DSEC dataset to qualitatively assess the cross-dataset generalization.\",\n \"While DSEC does not provide object detection labels, we can visually assess the quality of the detections by using the provided calibration files to map the frames of the global shutter camera to the event camera view.\",\n \"Figure REF and REF show predictions of our model together with the images closest in time to the detections.\",\n \"Figure REF shows our model can successfully detect cars in mountainous environments.\",\n \"In particular, Figure REF (a) shows an HDR scene where the global shutter frame is overexposed such that the approaching car is barely visible.\",\n \"Due to the high dynamic range of the event camera, our model can detect the approaching car without any difficulty.\",\n \"Figure REF features more urban environments where our model also manages to detect objects correctly.\",\n \"Figure: Prediction examples on the DSEC dataset featuring a mountainous environment.\",\n \"Frames are shown only for visualization purposes and are not used by the model.\",\n \"Column (a) shows a typical high-dynamic range (HDR) scenario where the vehicle is exiting a tunnel with a car approaching from outside the tunnel.\",\n \"The HDR capabilities of the event cameras enables our model to detect the approaching car.\",\n \"Column (b) shows a scenario with a wet road and challenging reflections.Figure: Prediction examples on the DSEC dataset featuring a (sub-)urban environments.\",\n \"Frames are shown only for visualization purposes and are not used by the model.\",\n \"Column (a) illustrates a typical urban situation where pedestrians, two-wheelers, cars and other road users occupy the street simultaneously.\",\n \"In column (c), we show a failure case of our model where a street pillar is erroneously detected and classified as a two-wheeler.\"\n ],\n [\n \"Discussion of Failure Cases\",\n \"By and large, our model can successfully detect objects on DSEC even though it has only been trained on the 1 Mpx dataset.\",\n \"Still, we found failure cases that might stem from distribution shift between the datasets.\",\n \"For example, Figure REF (c) shows the erroneous detection of a two-wheeler that instead is a pillar on the street.\",\n \"Overall, the model is good at detecting cars but is less confident and accurate at detecting two-wheelers and pedestrians.\",\n \"This effect likely stems from the fact that the 1 Mpx dataset has almost twice as many car labels as pedestrian and two-wheeler labels combined.\",\n \"The submission comes with two videos showing the predictions on two separate sequences on DSEC for a better qualitative assessment.\"\n ],\n [\n \"Gen1 {{cite:1a57c7d3860ad2c81ecd07530cb1f14e26ad31f9}}\",\n \"“Prophesee Gen1 Automotive Detection Dataset License Terms and Conditions”: https://www.prophesee.ai/2020/01/24/prophesee-gen1-automotive-detection-dataset/\"\n ],\n [\n \"1 Mpx {{cite:b7f6a351fa8f5aea53b2ab10daf2026d7fc7796f}}\",\n \"“Prophesee 1MegaPixel Automotive Detection Dataset License Terms and Conditions”: https://www.prophesee.ai/2020/11/24/automotive-megapixel-event-based-dataset/\"\n ],\n [\n \"DSEC {{cite:eade248fcd80be136c742eef47239dceb3a1892f}}\",\n \"“Creative Commons Attribution-ShareAlike 4.0 International public license (CC BY-SA 4.0)”: https://dsec.ifi.uzh.ch/\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05598"}}},{"rowIdx":87,"cells":{"text":{"kind":"list like","value":[["UV-driven Chemistry as a Signpost for Late-stage Planet Formation"],["Abstract The chemical reservoir within protoplanetary disks has a direct impact on planetary compositions and the potential for life.","A long-lived carbon-and nitrogen-rich chemistry at cold temperatures (<=50K) is observed within cold and evolved planet-forming disks.","This is evidenced by bright emission from small organic radicals in 1-10 Myr aged systems that would otherwise have frozen out onto grains within 1 Myr.","We explain how the chemistry of a planet-forming disk evolves from a cosmic-ray/X-ray-dominated regime to an ultraviolet-dominated chemical equilibrium.","This, in turn, will bring about a temporal transition in the chemical reservoir from which planets will accrete.","This photochemical dominated gas phase chemistry develops as dust evolves via growth, settling and drift, and the small grain population is depleted from the disk atmosphere.","A higher gas-to-dust mass ratio allows for deeper penetration of ultraviolet photons is coupled with a carbon-rich gas (C/O > 1) to form carbon-bearing radicals and ions.","This further results in gas phase formation of organic molecules, which then would be accreted by any actively forming planets present in the evolved disk."],["Protoplanetary disks are the natal environments for planets.","Disks have three main components: a pebble-rich dusty midplane (dust grain radius $>\\sim $ 1mm), a gaseous atmosphere extending well above[73] and radially beyond (by a factor of $\\sim $ 2[6]) the pebble-rich midplane, and a small dust population (radius $<$ 10 $\\mu $ m) that is coupled to the gas.","Each component of the protoplanetary disk has an impact on shaping the chemistry of actively forming planets.","The solid cores of giant planets must form over a short timescale ($\\sim $ 1 Myr) for the eventual planet to obtain its full mass over the course of a typical lifetime of gas in a disk (3-10 Myr[48]) and to explain the widely observed gap and ring structures that are thought to be indicative of planet formation[5].","The compositions of pebbles and their icy mantles directly influence the final composition of a solid planetary core [78], [59].","After a core becomes sufficiently massive, planets start to accrete material from the gaseous reservoir surrounding the pebble-rich midplane of a protoplanetary disk to form their atmospheres [62].","It remains difficult to directly probe the gas within the planet-forming midplane due to the high dust densities leading to elevated dust optical depths that mask line emission, as well as cold temperatures which leads to the freezing out common gas tracers such as CO onto dust grains.","However, constraining the chemical environment of the planet-forming midplane is essential to connect sub-mm observations probing the warm intermediate regions above the disk midplane to the composition of actively forming planets.","The molecules $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ are two of many complex organic molecules (loosely defined as a molecules with at least four atoms, including multiple carbon atoms) that could act as basic precursors to prebiotic molecules[82], [83], [87].","$\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ have been observed and spatially resolved towards the protoplanetary disks around six young stars: GM Aur, AS 209, HD 163296, MWC 480, LkCa 15, and V4046 Sgr [13], [55], [60], [79].","$\\textrm {CH}_{3}\\textrm {CN}$ has also been spatially resolved observed toward the TW Hya disk[66], and a couple of small ($\\approx $ 4 source) surveys have detected unresolved $\\textrm {CH}_{3}\\textrm {CN}$ or $\\textrm {HC}_{3}\\textrm {N}$ emission from other young stellar objects[28].","These molecules exhibit bright emission signifying high gas phase abundances and column densities (N$_{\\rm {Total}}$ =10$^{12}$ -10$^{13} \\rm {cm}^{-2}$ [13], [55]).","$\\textrm {CH}_{3}\\textrm {CN}$ is an excellent probe of gas temperature due to multiple transitions tracing a range of energy states [lowest energy state E$_{L} \\approx $ 5 K, spanning lower energy states over $\\Delta $ E$_{L}$ =$\\sim $ 100 K] that can be observed simultaneously.","Each J-transition (where J is the rotational quanum number) has a series of K-ladder transitions (where K is the quantum number of angular momentum along the molecular axis) which are only sensitive to collisions, thus the ratio between K-transitions depends only on the gas density and temperature.","The K-transitions span a large range of temperatures and are sufficiently close in frequency to observe in one spectral setting with the Atacama Large Millimeter/submillimeter Array (ALMA).","Analyses of rotational diagrams[44] made from $\\textrm {CH}_{3}\\textrm {CN}$ observations demonstrate an origin in gas with a temperature between 25-50 K[55], well below the expected desorption temperature (100-124 K[30]).","Thus, the brightly observed flux from this species and similar nitriles and organics like $\\textrm {HC}_{3}\\textrm {N}$ , and CH$_{2}$ CN[26] present a chemical conundrum as they should not be present in the gaseous state; rather, they should be frozen on cold grain surfaces.","Where $\\textrm {CH}_{3}\\textrm {CN}$ and other carbon-rich molecules reside and how they are replenished in the gas phase has a substantial impact on our understanding of prebiotic enrichment into gaseous planet atmospheres.","The traditional solution to this apparent discrepancy has been to turn to grain-surface chemistry.","Simple carbon- and nitrogen-bearing molecules can undergo hydrogenation or other reactions to form complex organics on the surface of a grain that then non-thermally desorb the grain intact.","This solution has been used to explain the organic inventories locked onto pebbles and icy grains [78], [12].","This dust chemistry path requires both an intact photodesorption rate of $\\textrm {CH}_{3}\\textrm {CN}$ from the grains of 10$^{-3}$ mols/photon[93], [66] and a reactive desorption efficiency[92] of 1%.","Laboratory experiments find an intact photodesorption efficiency of $\\textrm {CH}_{3}\\textrm {CN}$ to be orders of magnitude less efficient (10$^{-5}$ mols/photon[10]) and reactive desorption has not been well studied in laboratory experiments for these species.","We posit here that there is a simpler alternative based solely on disk evolution processes that have been previously theorized and observed which link the disk gas chemistry to planet formation.","This involves two ingredients: (1) an elevated C-to-O ratio in the gas and (2) greater penetration of UV photons due to a reduction in the total surface area of small grain population due to pebble formation.","These combine to power a state of photo-chemistry equilibrium.","There has been mounting evidence for an elevated C-to-O gas phase ratio at large radial distances in evolved and cool gas-rich protoplanetary disks.","Brighter than expected emission from small hydrocarbons such as C$_{2}$ H [72], [21] and complex organics such as c-C$_{3}$ H$_{2}$ [29] in disks and $\\textrm {HC}_{3}\\textrm {N}$ and $\\textrm {CH}_{3}\\textrm {CN}$ in photon-dominated regions[65] have independently suggested a C-to-O ratio well above the solar abundance ratio (C-to-O$\\approx $ 0.55) [7].","As a radical, C$_{2}$ H has a short lifetime ($<$ 1000 years), but it is found to be abundant in the gas.","Bosman et al.","2021[21] find that the only way to reproduce the high column densities observed was to increase the C-to-O ratio throughout the full disk to between 1-2.","They found that it was also necessary for C$_{2}$ H to exist in high density gas while remaining above the disk midplane (height/radius$\\approx $ 0.1-0.2) where UV photons dominate the high-energy photon budget.","Together, this produces a long-lived carbon-rich gas phase cycle in photochemical equilibrium[21].","One critical factor is that C$_2$ H is predicted and is observed to exist above the disk midplane[64].","In contrast, $\\textrm {CH}_{3}\\textrm {CN}$ has been observed to emit closer to the midplane[55].","Extending the carbon-rich chemical environment that is used to explain C$_{2}$ H observations to the midplane could allow for gas phase formation of complex molecules in the planet-forming zone.","Elevated C-to-O ratios would be necessary to supply the materials to create hydrocarbons and nitriles.","However, this alone would not alleviate the issue of $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ existing in the gas at temperatures at which they should be frozen and locked onto grains.","A non-thermal desorption mechanism is needed to increase the number of molecules that are being desorbed from the grains, either intact or as fragments of larger molecules.","If the disk atmosphere is small-dust rich, with a gas-to-dust ratio close to the typical interstellar medium (ISM) mass ratio (gas-to-dust = 100) then UV radiation cannot penetrate deep into the disk.","In this case, only high energy radiation (X-rays and cosmic rays) can penetrate the dust and gas and drive the chemistry within and near the midplane.","Small dust grains are the main opacity source for UV photons which are readily produced by the young and active star.","As the protoplanetary disk evolves, small dust grains agglomerate and eventually settle downwards and drift inward radially as they grow in size[3], [17], decreasing the total surface area of the small grain population.","As the main opacity source begins to deplete, UV photons can penetrate deeper into the disk.","Here, complete desorption of $\\textrm {CH}_{3}\\textrm {CN}$ or $\\textrm {HC}_{3}\\textrm {N}$ is inhibited due to the dense and cold environment[65].","This mechanism would have an effect on all molecules residing on grains, and a brief discussion regarding this can be found in the Supplemental Materials, in the section `Implications on other molecules'.","The combined effect of a high C-to-O ratio and excess UV flux allows for complex molecules to exist within the gas phase at cold, midplane temperatures.","Thus bright hydrocarbon and nitrile (i.e.","C$_{2}$ H, CH$_{3}$ CN) emission from the cold midplane acts as a signpost for an evolved dust population, coincident with an advanced stage of planet formation including the accumulation of gas-giant atmospheres.","The chemical scenario we put forward, comparing early and late-stage disk environments is shown in Figure REF .","The disks observed in the Molecules with ALMA at Planet-forming Scales (MAPS) large program[80] provide plausible support for this theory.","The youngest disks in the MAPS sample reside around IM Lup and AS 209 which are on the order of 1-2 Myr old[80].","Towards the youngest source, IM Lup, there is no detection of $\\textrm {CH}_{3}\\textrm {CN}$ nor $\\textrm {HC}_{3}\\textrm {N}$ .","AS 209 has detections of $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ which suggest both molecules emit from z/r$>$ 0.1.","The disk systems surrounding stars $>$ 6 Myr have $\\textrm {CH}_{3}\\textrm {CN}$ detected at z/r$<$ 0.1 as determined by the modeled thermal structure and detected rotational temperature from the K-ladder transitions.","This complex organic-rich midplane will influence the atmospheres of planetary companions.","It is worth noting that the oldest disk systems in the MAPS program exist around Herbig stars while the youngest systems are T Tauri stars.","The UV-bright spectrum innate to Herbig stars may additionally influence the push of complex molecules towards the planet-forming midplane.","Currently, the MAPS sample contains the bulk of the resolved $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ data towards protoplanetary disks.","To explore the effect of the stellar spectrum on $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ emission more disks with a wide range of stellar host masses would need to be observed.","Figure: A schematic highlighting the physical evolution of a disk and how that physical environment can affect the chemistry.","At the top, we show a disk with a large amount of small dust that acts to block UV photons.","As the small dust settles, UV photons make their way deeper into the disk, allowing for photodesorption of complex species off grains.","Now, there is a cycle of carbon chemistry that can be observed in the gas phase.To test the validity of this proposed end-stage chemistry, we first explored single point models representative of the disk midplane with corresponding cold temperatures and high gas densities (approximately 35 K, 5$\\times $ 10$^{11}$ mol/cm$^{-3}$ ).","We varied the gas-to-dust mass ratio and initial carbon abundance among other physical and chemical variables including the nitrogen abundance, dust extinction, and ionization rate.","We found that the gas phase $\\textrm {CH}_{3}\\textrm {CN}$ abundance was the most sensitive to the gas-to-dust ratio and initial gas phase carbon abundance.","We then produced a thermo-chemical model representing the disk around the Herbig Ae star HD 163296.","This disk is old (approx.","7.6 Myr)[14] and is nearby at 101 pc[42] and has observed jets and winds [19], [96].","The HD 163296 disk has been widely observed in multiple gas and dust tracers with high spatial resolution ($\\sim $ 10 au) with clear gap and ring structures typically assumed to be associated with active planet formation[5], [80].","There is bright $\\textrm {CH}_{3}\\textrm {CN}$ emission coming from $\\sim $ 35 K gas as well as bright emission from $\\textrm {HC}_{3}\\textrm {N}$ and HCN[55], [15].","Our modeling efforts follow that of Zhang et al.","2021[97], and Calahan et al.","2021b[25] which set up a thermo-chemical model of the HD 163296 disk by reproducing the mm dust-continuum observations, the spectral-energy distribution (SED), the full line intensity and morphology of six CO isotopologue transitions, and the vertical distribution of the optically thick lines.","The model of HD 163296 started with a gas-to-small dust ratio of the disk equal to 500[97] which was needed to reproduce the disk's SED.","c|ccc|ccc 7 0pt Modeling Parameters: TW Hya & HD 163296 Parameters TW Hya HD 163296 Gas Small Dust Large Dust Gas Small Dust Large Dust$^{a}$ Mass (M$_{\\odot }$ ) 0.025 $1.0\\times 10^{-4}$ $4.0\\times 10^{-4}$ 0.14$2.6 \\times 10^{-5}$ 0.024 $\\Psi $ 1.1 1.2 1.2 1.08 1.081.08 $\\gamma $ 0.75 0.75 1.0 0.80.80.1 h$_{c}$ (au) 42 42 8.4 8.448.44 n/a r$_{c}$ (au) 400 400 400 165165 n/a r$_{in}$ (au) 0.1 0.5 1 0.45 0.450.45 r$_{out}$ (au) 200 200200 600 600240 Final values of the TW Hya and HD 163296 models that reproduce CO, HD, $\\textrm {CH}_{3}\\textrm {CN}$ , HCN, and $\\textrm {HC}_{3}\\textrm {N}$ observations when available.","r$_{in}$ and r$_{out}$ are the radial inner and outer limits of the disk, beyond these limits there is assumed to be no gas nor dust.","$^{a}$ The surface density of the large dust distribution in HD 163296 is empirically set by continuum observations, thus it is not smooth and it is not dictated by the parametric equations.","To produce this end-stage chemical environment, we deplete the small dust mass by a factor of 10 throughout the disk of HD 163296 making the new gas-to-dust mass ratio above the pebble disk midplane equal to 5,000.","We then enhanced the initial gas phase carbon abundance in the system in the form of C, CH$_{4}$ , or C$_{2}$ H, increasing the overall gas phase carbon-to-oxygen ratio in both disks to above unity.","There are two possible sources for excess carbon in protoplanetary disks, either through the destruction of refractory carbon grains[22] or CO depletion through mechanisms such as reactions with ionized molecules and atoms such as He$^{+}$ or H$_{3}^{+}$[86] and a series of chemical and freeze-out processes.","We note that while signatures of CO depletion are found in the HD 163296 disk[97], [25], CO destruction likely only occurs in environments with a high cosmic-ray flux [2$\\times $ 10$^{-17}$ s$^{-1}$ ][86].","Additionally, CO destruction would supply equal amounts of carbon and oxygen, while we seek to enhance carbon over oxygen.","Our thermo-chemical model is run for a full megayear (Myr), after which the chemistry reaches an equilibrium.","Notable chemical feedback due to dust evolution is predicted to occur over scales of $\\sim $ 1 Myr[61], [91], thus we use $\\sim $ 1 Myr as an approximate length of time for the chemical environment to transition from `early-stage' to `late-stage' dust evolution and subsequently exist in a state of photochemical equilibrium.","Gas phase chemical reactions and rates were taken from the chemical network derived in Bosman et al.","2018[20] which in turn relies on the UMIST Database for Astrochemistry Rate12 version[70] (see Methods section for more details).","We found that regardless of the carrier of carbon, using a C/O = 1-2 and a factor of 10 depletion of small dust roughly reproduces line ratios and fluxes of the radial intensity profiles of $\\textrm {CH}_{3}\\textrm {CN}$ , $\\textrm {HC}_{3}\\textrm {N}$ , and HCN (see Figure REF ).","In this proposed scenario, $\\sim $ 96% of the total mass of $\\textrm {CH}_{3}\\textrm {CN}$ continues to reside frozen out onto grains, but the increase in UV flux allows for sufficient $\\textrm {CH}_{3}\\textrm {CN}$ to exist and be formed in the gas to reproduce observed radial intensity profiles and column densities.","Figure REF shows the two-dimensional number densities of $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ .","The inner $\\sim $ 20 au exhibited brighter or more centrally peaked emission than was seen in observations if the C-to-O ratio = 2 throughout the full disk.","To counteract this, we set the C-to-O ratio = 0.47, or the ISM ratio, inside of ($\\sim $ 20 au) where Zhang et al.","2021[97] found an ISM ratio of $\\textrm {H}_{2}$ /CO (See Supplementary Figure 2).","The inner disk emission remains slightly brighter than observed in HCN and $\\textrm {CH}_{3}\\textrm {CN}$ (see Figure REF ) with this alteration.","This could be accounted for via a depletion in the nitrogen abundance within the N$_{2}$ ice line, a strong buildup of pebbles around the water ice line ($\\sim $ 5au), a lower HCN desorption energy, or a combination of these effects.","Additionally, the observations may also be affected by processes such as beam smearing and a higher dust opacity than modeled.","A comparison model without a depletion in small dust mass is shown in Supplementary Figure 5.","Figure: Radial intensity profiles of the observed complex organic molecules and HCN towards the disk around HD 163296.","The molecules shown are CH 3 CN \\textrm {CH}_{3}\\textrm {CN}, HCN, and HC 3 N\\textrm {HC}_{3}\\textrm {N} (left to right with two J transition of CH 3 CN \\textrm {CH}_{3}\\textrm {CN} shown in the first two panels).","Solid thick lines in the background correspond to the observations derived from Ilee et al 2021 and Guzmán et al.","2021 (which utilized Law et al.","2021) while dashed lines correspond to modeled radial profiles.","Our final model includes an increase in C/O ratio beyond 20 au and a depletion of small dust and represents 1 Myr of chemistry.","This model can simultaneously fit the flux, line ratios, and general morphology of the observed radial profiles of these complex organic molecules.Figure: The radial and vertical number density distributions of CH 3 CN \\textrm {CH}_{3}\\textrm {CN} and HC 3 N\\textrm {HC}_{3}\\textrm {N} The density distributions were determined by our final HD 163296 model with an increase in C/O ratio beyond 20 au and a depletion of small dust.","This distribution produced the radial profiles shown in Figure .","A z/r<<0.1 is good approximation for the midplane of the disk, and both CH 3 CN \\textrm {CH}_{3}\\textrm {CN} and HC 3 N\\textrm {HC}_{3}\\textrm {N} emit partially from the midplane.","A comparison can be made with Figure which lacks the inclusion of small dust depletion.$\\textrm {HC}_{3}\\textrm {N}$ and $\\textrm {CH}_{3}\\textrm {CN}$ are built up in the gas from simple carbon and nitrogen-based volatiles (see Methods section “Chemical Reactions\" for details).","A reservoir of these more simple molecules is maintained in the gas phase due to photodesorption from dust grain surfaces by an enhanced UV-field.","The main destruction products from $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ are simple volatiles that can cycle back to create larger nitrile molecules including their original parent molecule.","While there are enough complex organic molecules in the gas phase to observe bright emission, the majority of the $\\textrm {CH}_{3}\\textrm {CN}$ , $\\textrm {HC}_{3}\\textrm {N}$ , and HCN near the midplane still remains frozen out onto grains.","The carbon-rich gas reservoir and UV-dominated disk together allow for a cycle of carbon chemistry to remain active in the gas phase.","This implies that actively accreting gas giants will build their atmosphere out of this complex nitrile-enhanced material.","Gas giant planets with atmospheres containing a chemical make-up with a C-to-O ratio $>$ 1 can be explained by the natural evolution of dust and the observed high C-to-O ratios within protoplanetary disks.","This environment could be extended to the inner disk in some cases, as studies such as Najita et al.","2011[76] and Anderson et al.","2021[2] posit a higher than solar C-to-O ratio within the inner disk to account for Spitzer observations of HCN and C$_{2}$ H$_{2}$ around T Tauri stars.","The implications of this end-stage chemistry are far-reaching.","Primarily, we put forward a new chemically and physically coupled picture of planet formation and its direct impact on the chemistry of forming planet atmospheres.","Within the early stages of planet formation, pebble growth and accretion into solid planet cores has begun, and chemistry onto these cores are influenced primarily by the chemical make-up of large pebbles and their icy mantles.","The chemistry active in the midplane during this stage of formation is dominated by X-rays and cosmic rays[95].","After timescales of order 1 Myr, at the end-stage of pebble formation, small grains have grown and settled towards the midplane, allowing for UV photons to penetrate deeper in the disk than previously assumed.","This excess UV flux in concert with an above solar C-to-O ratio produces a carbon-rich gas phase cycle of production and destruction in the gas surrounding actively forming planets.","Bright emission from $\\textrm {CH}_{3}\\textrm {CN}$ , $\\textrm {HC}_{3}\\textrm {N}$ and other carbon-rich molecules emitting at cold temperatures are a sign-post of this evolved chemical environment.","These forming gas giants may accrete this surrounding gas and its chemical signature into their atmosphere, and may be responsible for high C-to-O measurements that have been observed in exo-planetary atmospheres as compared to their host star[23].","Secondly, the deeper penetration of UV photons pushes the CO and H$_{2}$ self-shielding layers deeper into the disk.","Due to this, there are more ions such as C$^{+}$ and H$^{+}$ throughout the disk than would have been present with a regular (ISM) level of small dust surface area.","As the ion-fraction increases so does the area of the disk that is subject to magneto-rotational instability (MRI).","In our final model, the UV field increases by a factor of 3-5 in the atmosphere and between 1-2 near the midplane, and thirteen times more gas mass is MRI-active, most notably the gas midplane within 10 au, and then at a z/r=0.1-0.2 within 30 au (see Figure 4).","This could be a source of accretion[43] for older disks.","Our proposed UV-dominated carbon-rich gas phase chemistry is a major shift in our understanding of the astrochemistry of planet formation.","This transition will have a strong effect on the composition of actively forming planets, disk MRI-activity, and subsequent disk accretion.","Figure: A comparison of the MRI active region within a non-elevated UV environment and an elevated UV environment.","The two left-most plots show regions within the HD 163296 disk which are MRI activated in an environment where small dust is depleted by a factor of 10 in total mass (gas/dust = 5,000 middle plot) and where small dust is not depleted (gas/dust=500 left).","The color corresponds to the gas density in mol/cm 3 ^{3}, highlighting the substantial increase in mass that is MRI-active.","The right-most plot shows a comparison of the UV field strength through a `regular' disk and a dust depleted disk.","The UV field increases upwards of a factor of 3-5 in the atmosphere of the disk and 1-2 below a z/r=0.2.","This has a notable effects on the chemistry of the midplane.For a thorough description of the observations and the techniques used to obtain the CLEAN-ed images and radial profiles of $\\textrm {CH}_{3}\\textrm {CN}$ , HCN, and $\\textrm {HC}_{3}\\textrm {N}$ towards HD 163296 see Czekala et al.","2021[31], Law et al.","2021[63], and Öberg et al.","2021[80].","A description of the observations of $\\textrm {CH}_{3}\\textrm {CN}$ towards TW Hya can be found in Loomis et al.","2018a[66].","A brief summary of these observations are as follows.","The data for HD 163296 is from the MAPS large program (Project ID 2018.1.01055.L)[80].","The image cubes were produced using the tclean task in the Common Astronomy Software Applications (CASA) package, version 6.1.0[71].","Keplerian masks based on the disk geometric parameters were used in the CLEANing process and the final images were then corrected for the Jorsater & van Moorsel effect to ensure that the image residuals are in units consistent with that of the CLEAN model.","For all lines, we used the beam-circularized and uv-tapered images, which had synthesised beam sizes of 0.\"3.","All radial intensity profiles were generated by deprojecting and azimuthally-averaging zeroth moment maps using the GoFish python package.","$\\textrm {CH}_{3}\\textrm {CN}$ was observed towards TW Hya as a part of ALMA project 2016.1.01046.S.","Each emission line was individually imaged using CLEAN and the synthesised beam for each transition were matched using small uv-tapers to 1$.\\!\\!^{\\prime \\prime }$ 05 x 0$.\\!\\!^{\\prime \\prime }$ 83.","A Keplerian mask was used to extract the flux for each transition.","See Loomis et al.","2018a[66] for more details.","Simple single point models of the disk environment were utilized to quickly and efficiently understand the chemical impacts of different physical and initial chemical conditions.","Our chemical network is derived from Bosman et al.","2018[20] which in turn is derived from the UMIST “RATE12” network[70], and for this study we disregarded most grain-surface chemical reactions in order to isolate gas-phase formation of $\\textrm {CH}_{3}\\textrm {CN}$ .","After our results from the modeling suggested a higher gas/dust ratio and carbon content, we turned to more comprehensive thermo-chemical codes which model the thermal physics and chemical evolution throughout the whole disk.","The code RAC2D (https://github.com/fjdu/rac-2d)[35], was used to create models of the disks around TW Hya and HD 163296.","The 2D temperature, density, and molecular abundance results were used to simulate observations of each disk with a raytracing code: RADMC-3D[37].","A brief description of the physical code of RAC2D is given below; a detailed description of the code can be found in Calahan et al 2021a[24].","RAC2D takes into account a gas and dust structure and stellar radiation field and computes the gas and dust temperature and chemical structure over time.","Our model consists of three mass components: gas, small dust, and large dust grains.","The spatial extent of each component is given by a global surface density distribution [68], which is widely used in protoplanetary disk modeling and corresponds to the self-similar solution of a viscously evolved disk.","$\\Sigma (r)=\\Sigma _{c}\\left(\\frac{r}{r_{c}}\\right)^{-\\gamma }\\exp {\\left[-\\left(\\frac{r}{r_{c}}\\right)^{2-\\gamma }\\right]},$ $r_{c}$ is the characteristic radius at which the surface density is $\\Sigma _{c}/\\rm {e}$ where $\\Sigma _{c}$ is the characteristic surface density, and $\\gamma $ is the power-law index that describes the radial behavior of the surface density.","A 2D density profile for the gas and dust populations can be derived from the surface density profile and a scale height: $\\rho (r,z) = \\frac{\\Sigma (r)}{\\sqrt{2\\pi }h(r)} \\exp {\\left[-\\frac{1}{2}\\left(\\frac{z}{h(r)}\\right)^{2}\\right]},$ $h=h_{c}\\left( \\frac{r}{r_{c}}\\right)^{\\Psi } ,$ where $h_{c}$ is the scale height at the characteristic radius, and $\\Psi $ is a power index that characterizes the flaring of the disk structure.","The modeling parameters used for TW Hya and HD 163296 are shown in Table 1.","For both TW Hya and HD 163296 models, each dust population follows an Mathis, Rumpl, and Nordsieck (MRN) grain distribution $n(a) \\propto a^{-3.5}$ [69], where `a' indicates the size of the grain.","The small dust grains have radii between $5 \\times 10^{-3}$ - $1 \\mu $ m, and the large grains have radii between $5 \\times 10^{-3}$ - $10^{3} \\mu $ m. The large dust population is settled in the midplane with a smaller vertical extent and radial extent (gas extends $\\sim $ 5 and 2.5 times above and beyond, respectively).","This settled large grain population is the result of dust evolution, namely growth in concert with vertical settling to the midplane and radial drift.","For the HD 163296 model, the large grain population has a unique, non-smooth, surface density profile that reproduces the millimeter continuum observations of the HD 163296 disk[58], [97].","Opacity values for the dust are calculated based on Birnstiel et al.","2018[18].","Large dust grains consist of water ice[94], silicates[32], troilites and refractory organics[53].","Small dust grains consist of 50% silicates and 50% refractory organics.","Further discussion on the modeling efforts for TW Hya can be found in Calahan et al.","2021a[24] and for HD 163296 in Calahan et al.","2021b[25].","The thermo-chemical model resulted in 2D distributions of the gas and dust temperature, density, and molecular abundances.","These were used as inputs for the raytracing code RADMC-3D[37].","The molecular properties of $\\textrm {CH}_{3}\\textrm {CN}$ , $\\textrm {HC}_{3}\\textrm {N}$ , and HCN were taken from the Leiden Atomic and Molecular Database (LAMDA[85]), with some molecular parameters updated according to data from the Cologne Database for Molecular Spectroscopy (CDMS[74], [75]).","The result from RADMC-3D was a 3D image cube of the molecular emission across velocity space.","We utilized GoFish[88] to compress these 3D images into zeroth moment maps and radial profiles which were then directly compared to the azimuthally-averaged observations.","The chemical reaction network utilized in this work comes from Bosman et al.","2018[20] which in turn relies on the Rate12 version of the UMIST Database for Astrochemistry[70]: a network that is widely used across astrochemical modeling efforts.","In our network there are 6,302 gas phase reactions and we have limited the grain-surface reactions to twelve.","We kept grain-surface reactions that involved the formation of H$_{2}$ , CH$_{2}$ OH, CH$_{3}$ OH, CO$_{2}$ , H$_{2}$ O and NH$_{3}$ due to each of these species being well-studied in the laboratory and come with strong evidence for active two-body chemistry on dust grains[1], [51], [45], [56], [57], [41], [77], [84] (see Table 2).","Both thermal adsorption and desorption are taken into account for every molecule in the network.","Our network contains molecules with at most eleven carbon atoms, and at most twelve total atoms.","We run our thermo-chemical model for 1 Myr as the $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ chemistry reaches an equilibrium after this time period (see Supplementary Figure 3).","In our evolved disk model, the main formation pathways for $\\textrm {CH}_{3}\\textrm {CN}$ once the chemistry has reached equilibrium is as follows: $\\rm {CH_{3}^{+} + HCN \\rightarrow CH_{3}CNH^{+} + e^{-} \\rightarrow CH_{3}CN + H}$ This reaction in the gas phase is the primary formation pathway for $\\textrm {CH}_{3}\\textrm {CN}$ , thus there is a strong reliance on HCN existing in the gas phase even at cold temperatures below its measured sublimation temperature ($\\sim $ 85-103 K[16]) and an ionization source (UV photons) to produce CH$_{3}^{+}$ .","$\\textrm {HC}_{3}\\textrm {N}$ can be readily produced by a number of different ways including: $\\rm {N + CH_{2}CCH \\rightarrow HC_{3}N + H_{2}}$ $\\rm {CN + C_{2}H_{2} \\rightarrow HC_{3}N + H}$ $\\rm {H + C_{3}N^{-} \\rightarrow HC_{3}N + e^{-}}$ The formation of $\\textrm {HC}_{3}\\textrm {N}$ strongly relies on the existence of carbon-rich molecules including radicals (CN) and ions (C$_{3}$ N$^{-}$ ).","c|c 7 0pt Dust Surface Reactions Reaction Reference gH + gH $\\rightarrow $ gH$_{2}$ Hasegawa, Herbst, & Leung 1992[51] gH + gOH $\\rightarrow $ gH$_{2}$ O Ioppolo et al.","2010[57] gH + gH$_{2}$ O$_{2}$ $\\rightarrow $ gH$_{2}$ O + gOH Ioppolo et al.","2008[56] gH + gCH$_{3}$ OH $\\rightarrow $ gH$_{2}$ + gCH$_{2}$ OH Extrapolated from Fuchs et al.","2009[41] gH + gCH$_{2}$ OH $\\rightarrow $ gCH$_{3}$ OH Extrapolated from Fuchs et al.","2009[41] gH$_{2}$ + gOH $\\rightarrow $ gH$_{2}$ O + gH Oba et al 2012[77] gOH + gCO $\\rightarrow $ gCO$_{2}$ + gH $^{a}$ Ruffle & Herbst 2001[84] gO + gCO $\\rightarrow $ gCO$_{2}$ $^{a}$ Goumans & Brown 2008[45] gO + gHCO $\\rightarrow $ gCO$_{2}$ + gH Goumans & Brown 2008[45] gH + gNH$_{2}$ $\\rightarrow $ gNH$_{3}$ Allen & Robinson 1977[1] The complete list dust surface reactions accounted for in this study.","$^{a}$ Have additional special treatment for three body reactions We additionally produced a thermo-chemical model representing the disk around T Tauri star TW Hya.","TW Hya is approximately 10 Myrs old[90] and hosts the closest Class II disk at 59.9 pc[42].","TW Hya has been widely observed in multiple gas and dust tracers with high spatial resolution ($\\sim $ 10 au)[4], [54].","It also exhibits bright $\\textrm {CH}_{3}\\textrm {CN}$ coming from $\\sim $ 33 K gas[66].","Our modeling efforts of TW Hya follow that of Calahan et al.","2021a[24] which sets up a thermo-chemical model of the disk by reproducing the spectral-energy distribution (SED), the full line intensity and morphology of seven CO isotopologue transitions, and an HD J=1-0 observation from the Hershel PACS instrument[11].","To reproduce all CO radial profiles as well as the HD flux, the small dust in the upper layers of the atmosphere of the TW Hya disk were effectively depleted slightly[24] due to having a slightly lower flaring angle than the gas population.","This was enough small dust depletion to reproduce the available $\\textrm {CH}_{3}\\textrm {CN}$ lines from Loomis et al.","2018a given a C/O ratio equal to 1.0 (see Figure REF ).","This disk is an additional piece of evidence supporting our evolved chemistry proposal.","This introduction of a late-stage photo-chemical equilibrium was motivated by observations of $\\textrm {CH}_{3}\\textrm {CN}$ , $\\textrm {HC}_{3}\\textrm {N}$ , and HCN.","However, the late-stage chemistry would have an effect on other molecules as well.","We predict in addition to these organic molecules, molecules such as C$_{X}$ H, C$_{X}$ H$_{2}$ , and HC$_{X}$ N will be abundant in the gas, enhanced by the cycle of carbon chemistry that reproduces observed $\\textrm {CH}_{3}\\textrm {CN}$ and $\\textrm {HC}_{3}\\textrm {N}$ .","The main carbon carrier, CO, is largely unaffected by the increase in photo-chemistry.","CO is largely in the gas phase in previous models that do not include this `late-stage' chemistry and in the region in which $\\textrm {CH}_{3}\\textrm {CN}$ was added into the gas, CO was already primarily in the gas phase.","We find a slight depletion of CO in the upper atmosphere of the disk due to the CO photodissociation layer being pushed down, and there is a slight enhancement of CO in the midplane due to the UV-enhancement.","However this accounts for less than 1% of the total abundance of CO thus did not have a strong impact on the modeled radial profiles.","Another key molecule is H$_{2}$ O.","In our models, we do not initialize our model with H$_{2}$ O, thus there is very little water to be affected by this `late-stage' chemistry.","Observational results of Du et al.","2017[36] support this as they found a low overall abundance in H$_{2}$ O in disks with a survey of 13 protoplanetary disks.","$\\textrm {CH}_{3}\\textrm {CN}$ nor $\\textrm {HC}_{3}\\textrm {N}$ would be seen to be at a high abundance if gas-phase H$_{2}$ O was in high abundance in the disk as it would disrupt the carbon-rich chemistry.","The depletion of small grains will affect the observed spectral energy distribution (SED) from each disk.","TW Hya’s dust population was not altered from that of Calahan et al.","2021a[24] and continues to match its observed SED.","A depletion of a factor of 10 in the small dust population around HD 163296 would cause a dimming in the mid-infrared part of the SED.","However, the small dust abundance, the distribution of dust grain size, and the assumed dust opacity sources are degenerate in the ways they may affect the disk SED.","We modeled protoplanetary disk SEDs using the code TORUS[49], [50].","TORUS is a Monte Carlo radiative transfer code utilizing radiative equilibrium[67] and silicate grains[33].","In Figure REF , we show a series of SEDs produced from models motivated by our thermo-chemical model of HD 163296 including the stellar parameters and dust distribution in Table 1 from Calahan et al.","2021b[25].","We find that by varying the minimum dust grain radius or the power law index, we can account for a factor of 10 in UV attenuation.","The total population mass, grain size, and how the grain sizes are distributed are strongly degenerate and can result in uncertainties of the dust mass by a factor of at least 10.","Thus our depletion of small dust continues to reproduce all previous observables including the SED.","With the increase in UV flux deeper into the disk, more ions are created.","Ions can be coupled with the magnetic fields that thread through the disk and interact with the bulk gas creating turbulence.","This mechanism is called the magneto-rotational instability or MRI[9].","Magneto-hydrodynamical processes are assumed to be present throughout protoplanetary disks due to young star’s active magnetic field, and are thought to be one of the main drivers of angular momentum transport [27].","An MRI-active zone may drive the bulk of the mass transportation in the disk and activate accretion onto the star, two vital processes that determine the future of a young solar system.","It is thought that planet formation may be aided within ‘dead-zones’ where MRI is non-active[46].","The magnetic Reynolds number ($Re$ ) and ambipolar diffusion term ($Am$ ) are two quantities that help quantify the presence of an MRI-active zone.","The Reynolds number quantifies the level of coupling between ionized gas and magnetic fields and is defined as $Re \\equiv \\frac{c_{s}h}{D} \\approx 1 \\left(\\frac{\\chi _{e}}{10^{-13}}\\right) \\left(\\frac{T}{100~\\rm {K}}\\right)^{1/2} \\left(\\frac{a}{\\rm {AU}}\\right)^{3/2} ,$ where $c_{s}$ is the sound speed, $h$ is the scale height of the disk, D is the magnetic diffusivity parameter, $\\chi _{e}$ is the electron abundance, $T$ temperature and $a$ radial location in the disk [81].","The ambipolar diffusion term describes the coupling of ionized molecules and their interaction with neutral gas particles: $Am \\equiv \\frac{\\chi _{i} n_{\\rm {H}_{2}} \\beta _{\\rm {in}}}{\\Omega } \\approx 1 \\left(\\frac{\\chi _{i}}{10^{-8}}\\right) \\left(\\frac{n_{\\rm {H}_{2}}}{10^{10}\\rm {cm^{-3}}}\\right) \\left(\\frac{a}{\\rm {AU}}\\right)^{3/2} ,$ where $\\chi _{i}$ is the ion abundance, $n_{\\rm {H}_{2}}$ is the number density of H$_{2}$ atoms, $\\Omega $ is the dynamical time, $\\beta _{\\rm {in}}$ is the collisional rate coefficient for singly charged species to share momentum with neutral species, and $a$ is the radial location in the disk [34], [81].","For MRI to act as a turbulent driver of neutral gas, both $Re$ and $Am$ must be sufficiently high.","Simulations show that values between 0.1-100 for $Am$ can trigger significant coupling between ions and neutrals [52], [8].","Models by Flock et al.","2012[40] suggest $Re \\approx $ 3,300-5,000 is required to sustain sufficient turbulence with a critical $Re$ = 3,000.","In our work, we assume a combined $Am >$ 100 and $Re >$ 3,000 to signify an MRI-active zone.","We calculate the regions in which the MRI is active in the HD 163296 disk in our models with an early-stage physical environment and a late-stage environment (signified by a depletion in the small dust population mass) in Figure REF .","Our solution allows for an increase of UV flux in the atmosphere by a factor of 3-5.","More of the disk becomes ion-rich due to the CO and H$_{2}$ self-shielding layers being located deeper into the disk.","The increase in the ion and electron abundance ($\\chi _{i}$ , $\\chi _{e}$ ) are the key factors that enhance the Re and Am values and thus produce additional MRI activity (see Figure REF ).","As a result, thirteen times more mass within the disk becomes MRI-active including within the midplane.","In the base model representing early stages of disk chemistry, the MRI activity at the midplane extends to 4 au.","By depleting the small dust population by an order of magnitude, the MRI activity at the midplane then extends out to $\\sim $ 10 au.","This increase in MRI activity can contribute to the reason behind why older disk systems, such as TW Hya, are actively accreting.","Meridional flows have been identified in the HD 163296 disk located at two of the largest dust gaps (approx.","45 and 86 AU)[89] and these vertical flows are coincident with the lower vertical limit of the MRI-active zone.","The MRI activation could have an effect on the height or continue to drive meridional flows.","Turbulence measurements of the HD 163296 disk have been derived in Flaherty et al.","2015 and 2017[38], [39] using molecular tracers, and they find turbulent velocities to be 5% or less of the sound speed between 30-300 au and at all measured heights.","It is not yet clear whether our MRI prediction is in tension with the observational evidence of low turbulence in HD 163296, more work needs to be done on the modeling of how MRI-active regions drive turbulence and more observational constraints are needed to constrain the inner 40 au where we find the strongest MRI-activity."],["Acknowledgements","Jenny Calahan is the corresponding author and can be contacted via jcalahan@umich.edu J.K.C.","acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No.","DGE 1256260 and the National Aeronautics and Space Administration FINESST grant, under Grant no.","80NSSC19K1534.","E.A.B.","acknowledges support from NSF AAG Grant #1907653.","A.D.B.","acknowledges support from NSF AAG Grant #1907653.","E.A.R.","acknowledges support from NSF AST 1830728.","Support for J. H. was provided by NASA through the NASA Hubble Fellowship grant #HST-HF2-51460.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.","This work was supported by a grant from the Simons Foundation 686302 and by an award from the Simons Foundation 321183FY19, KÖ.","This material is based upon work supported by the National Science Foundation under Grant No.","AST-1907832.","J.D.I.","acknowledges support from an STFC Ernest Rutherford Fellowship (ST/W004119/1) and a University Academic Fellowship from the University of Leeds.","C.W.","acknowledges financial support from the University of Leeds, the Science and Technology Facilities Council, and UK Research and Innovation (grant numbers ST/T000287/1 and MR/T040726/1).","V.V.G.","gratefully acknowledges support from FONDECYT Regular 1221352, ANID BASAL projects ACE210002 and FB210003, and ANID, – Millennium Science Initiative Program – NCN19_171.","We thank Tim Harries for help with and providing access to the TORUS modeling program."],["Data Availability","The data that support the findings of this study can be obtained as part of the MAPS program and are publicly available via alma-maps.info.","Data regarding TW Hya can be obtained via data-rich figures in Calahan et al 2021 published on the online publication of the Astrophysical Journal article.","This paper makes use of the following ALMA data: ADS/JAO.ALMA#2018.1.01055.L.","and 2016.1.01046.S ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.","The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.","The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.","This study relied on the following publicly available coding packages: rac2d: https://github.com/fjdu/rac-2d, RADMC-3D: https://www.ita.uni-heidelberg.de/ dullemond/software/radmc-3d/, and GoFish: https://github.com/richteague/gofish.","TORUS is a private code developed by Tim Harries and collaborators."]],"string":"[\n [\n \"UV-driven Chemistry as a Signpost for Late-stage Planet Formation\"\n ],\n [\n \"Abstract The chemical reservoir within protoplanetary disks has a direct impact on planetary compositions and the potential for life.\",\n \"A long-lived carbon-and nitrogen-rich chemistry at cold temperatures (<=50K) is observed within cold and evolved planet-forming disks.\",\n \"This is evidenced by bright emission from small organic radicals in 1-10 Myr aged systems that would otherwise have frozen out onto grains within 1 Myr.\",\n \"We explain how the chemistry of a planet-forming disk evolves from a cosmic-ray/X-ray-dominated regime to an ultraviolet-dominated chemical equilibrium.\",\n \"This, in turn, will bring about a temporal transition in the chemical reservoir from which planets will accrete.\",\n \"This photochemical dominated gas phase chemistry develops as dust evolves via growth, settling and drift, and the small grain population is depleted from the disk atmosphere.\",\n \"A higher gas-to-dust mass ratio allows for deeper penetration of ultraviolet photons is coupled with a carbon-rich gas (C/O > 1) to form carbon-bearing radicals and ions.\",\n \"This further results in gas phase formation of organic molecules, which then would be accreted by any actively forming planets present in the evolved disk.\"\n ],\n [\n \"Protoplanetary disks are the natal environments for planets.\",\n \"Disks have three main components: a pebble-rich dusty midplane (dust grain radius $>\\\\sim $ 1mm), a gaseous atmosphere extending well above[73] and radially beyond (by a factor of $\\\\sim $ 2[6]) the pebble-rich midplane, and a small dust population (radius $<$ 10 $\\\\mu $ m) that is coupled to the gas.\",\n \"Each component of the protoplanetary disk has an impact on shaping the chemistry of actively forming planets.\",\n \"The solid cores of giant planets must form over a short timescale ($\\\\sim $ 1 Myr) for the eventual planet to obtain its full mass over the course of a typical lifetime of gas in a disk (3-10 Myr[48]) and to explain the widely observed gap and ring structures that are thought to be indicative of planet formation[5].\",\n \"The compositions of pebbles and their icy mantles directly influence the final composition of a solid planetary core [78], [59].\",\n \"After a core becomes sufficiently massive, planets start to accrete material from the gaseous reservoir surrounding the pebble-rich midplane of a protoplanetary disk to form their atmospheres [62].\",\n \"It remains difficult to directly probe the gas within the planet-forming midplane due to the high dust densities leading to elevated dust optical depths that mask line emission, as well as cold temperatures which leads to the freezing out common gas tracers such as CO onto dust grains.\",\n \"However, constraining the chemical environment of the planet-forming midplane is essential to connect sub-mm observations probing the warm intermediate regions above the disk midplane to the composition of actively forming planets.\",\n \"The molecules $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ are two of many complex organic molecules (loosely defined as a molecules with at least four atoms, including multiple carbon atoms) that could act as basic precursors to prebiotic molecules[82], [83], [87].\",\n \"$\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ have been observed and spatially resolved towards the protoplanetary disks around six young stars: GM Aur, AS 209, HD 163296, MWC 480, LkCa 15, and V4046 Sgr [13], [55], [60], [79].\",\n \"$\\\\textrm {CH}_{3}\\\\textrm {CN}$ has also been spatially resolved observed toward the TW Hya disk[66], and a couple of small ($\\\\approx $ 4 source) surveys have detected unresolved $\\\\textrm {CH}_{3}\\\\textrm {CN}$ or $\\\\textrm {HC}_{3}\\\\textrm {N}$ emission from other young stellar objects[28].\",\n \"These molecules exhibit bright emission signifying high gas phase abundances and column densities (N$_{\\\\rm {Total}}$ =10$^{12}$ -10$^{13} \\\\rm {cm}^{-2}$ [13], [55]).\",\n \"$\\\\textrm {CH}_{3}\\\\textrm {CN}$ is an excellent probe of gas temperature due to multiple transitions tracing a range of energy states [lowest energy state E$_{L} \\\\approx $ 5 K, spanning lower energy states over $\\\\Delta $ E$_{L}$ =$\\\\sim $ 100 K] that can be observed simultaneously.\",\n \"Each J-transition (where J is the rotational quanum number) has a series of K-ladder transitions (where K is the quantum number of angular momentum along the molecular axis) which are only sensitive to collisions, thus the ratio between K-transitions depends only on the gas density and temperature.\",\n \"The K-transitions span a large range of temperatures and are sufficiently close in frequency to observe in one spectral setting with the Atacama Large Millimeter/submillimeter Array (ALMA).\",\n \"Analyses of rotational diagrams[44] made from $\\\\textrm {CH}_{3}\\\\textrm {CN}$ observations demonstrate an origin in gas with a temperature between 25-50 K[55], well below the expected desorption temperature (100-124 K[30]).\",\n \"Thus, the brightly observed flux from this species and similar nitriles and organics like $\\\\textrm {HC}_{3}\\\\textrm {N}$ , and CH$_{2}$ CN[26] present a chemical conundrum as they should not be present in the gaseous state; rather, they should be frozen on cold grain surfaces.\",\n \"Where $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and other carbon-rich molecules reside and how they are replenished in the gas phase has a substantial impact on our understanding of prebiotic enrichment into gaseous planet atmospheres.\",\n \"The traditional solution to this apparent discrepancy has been to turn to grain-surface chemistry.\",\n \"Simple carbon- and nitrogen-bearing molecules can undergo hydrogenation or other reactions to form complex organics on the surface of a grain that then non-thermally desorb the grain intact.\",\n \"This solution has been used to explain the organic inventories locked onto pebbles and icy grains [78], [12].\",\n \"This dust chemistry path requires both an intact photodesorption rate of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ from the grains of 10$^{-3}$ mols/photon[93], [66] and a reactive desorption efficiency[92] of 1%.\",\n \"Laboratory experiments find an intact photodesorption efficiency of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ to be orders of magnitude less efficient (10$^{-5}$ mols/photon[10]) and reactive desorption has not been well studied in laboratory experiments for these species.\",\n \"We posit here that there is a simpler alternative based solely on disk evolution processes that have been previously theorized and observed which link the disk gas chemistry to planet formation.\",\n \"This involves two ingredients: (1) an elevated C-to-O ratio in the gas and (2) greater penetration of UV photons due to a reduction in the total surface area of small grain population due to pebble formation.\",\n \"These combine to power a state of photo-chemistry equilibrium.\",\n \"There has been mounting evidence for an elevated C-to-O gas phase ratio at large radial distances in evolved and cool gas-rich protoplanetary disks.\",\n \"Brighter than expected emission from small hydrocarbons such as C$_{2}$ H [72], [21] and complex organics such as c-C$_{3}$ H$_{2}$ [29] in disks and $\\\\textrm {HC}_{3}\\\\textrm {N}$ and $\\\\textrm {CH}_{3}\\\\textrm {CN}$ in photon-dominated regions[65] have independently suggested a C-to-O ratio well above the solar abundance ratio (C-to-O$\\\\approx $ 0.55) [7].\",\n \"As a radical, C$_{2}$ H has a short lifetime ($<$ 1000 years), but it is found to be abundant in the gas.\",\n \"Bosman et al.\",\n \"2021[21] find that the only way to reproduce the high column densities observed was to increase the C-to-O ratio throughout the full disk to between 1-2.\",\n \"They found that it was also necessary for C$_{2}$ H to exist in high density gas while remaining above the disk midplane (height/radius$\\\\approx $ 0.1-0.2) where UV photons dominate the high-energy photon budget.\",\n \"Together, this produces a long-lived carbon-rich gas phase cycle in photochemical equilibrium[21].\",\n \"One critical factor is that C$_2$ H is predicted and is observed to exist above the disk midplane[64].\",\n \"In contrast, $\\\\textrm {CH}_{3}\\\\textrm {CN}$ has been observed to emit closer to the midplane[55].\",\n \"Extending the carbon-rich chemical environment that is used to explain C$_{2}$ H observations to the midplane could allow for gas phase formation of complex molecules in the planet-forming zone.\",\n \"Elevated C-to-O ratios would be necessary to supply the materials to create hydrocarbons and nitriles.\",\n \"However, this alone would not alleviate the issue of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ existing in the gas at temperatures at which they should be frozen and locked onto grains.\",\n \"A non-thermal desorption mechanism is needed to increase the number of molecules that are being desorbed from the grains, either intact or as fragments of larger molecules.\",\n \"If the disk atmosphere is small-dust rich, with a gas-to-dust ratio close to the typical interstellar medium (ISM) mass ratio (gas-to-dust = 100) then UV radiation cannot penetrate deep into the disk.\",\n \"In this case, only high energy radiation (X-rays and cosmic rays) can penetrate the dust and gas and drive the chemistry within and near the midplane.\",\n \"Small dust grains are the main opacity source for UV photons which are readily produced by the young and active star.\",\n \"As the protoplanetary disk evolves, small dust grains agglomerate and eventually settle downwards and drift inward radially as they grow in size[3], [17], decreasing the total surface area of the small grain population.\",\n \"As the main opacity source begins to deplete, UV photons can penetrate deeper into the disk.\",\n \"Here, complete desorption of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ or $\\\\textrm {HC}_{3}\\\\textrm {N}$ is inhibited due to the dense and cold environment[65].\",\n \"This mechanism would have an effect on all molecules residing on grains, and a brief discussion regarding this can be found in the Supplemental Materials, in the section `Implications on other molecules'.\",\n \"The combined effect of a high C-to-O ratio and excess UV flux allows for complex molecules to exist within the gas phase at cold, midplane temperatures.\",\n \"Thus bright hydrocarbon and nitrile (i.e.\",\n \"C$_{2}$ H, CH$_{3}$ CN) emission from the cold midplane acts as a signpost for an evolved dust population, coincident with an advanced stage of planet formation including the accumulation of gas-giant atmospheres.\",\n \"The chemical scenario we put forward, comparing early and late-stage disk environments is shown in Figure REF .\",\n \"The disks observed in the Molecules with ALMA at Planet-forming Scales (MAPS) large program[80] provide plausible support for this theory.\",\n \"The youngest disks in the MAPS sample reside around IM Lup and AS 209 which are on the order of 1-2 Myr old[80].\",\n \"Towards the youngest source, IM Lup, there is no detection of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ nor $\\\\textrm {HC}_{3}\\\\textrm {N}$ .\",\n \"AS 209 has detections of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ which suggest both molecules emit from z/r$>$ 0.1.\",\n \"The disk systems surrounding stars $>$ 6 Myr have $\\\\textrm {CH}_{3}\\\\textrm {CN}$ detected at z/r$<$ 0.1 as determined by the modeled thermal structure and detected rotational temperature from the K-ladder transitions.\",\n \"This complex organic-rich midplane will influence the atmospheres of planetary companions.\",\n \"It is worth noting that the oldest disk systems in the MAPS program exist around Herbig stars while the youngest systems are T Tauri stars.\",\n \"The UV-bright spectrum innate to Herbig stars may additionally influence the push of complex molecules towards the planet-forming midplane.\",\n \"Currently, the MAPS sample contains the bulk of the resolved $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ data towards protoplanetary disks.\",\n \"To explore the effect of the stellar spectrum on $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ emission more disks with a wide range of stellar host masses would need to be observed.\",\n \"Figure: A schematic highlighting the physical evolution of a disk and how that physical environment can affect the chemistry.\",\n \"At the top, we show a disk with a large amount of small dust that acts to block UV photons.\",\n \"As the small dust settles, UV photons make their way deeper into the disk, allowing for photodesorption of complex species off grains.\",\n \"Now, there is a cycle of carbon chemistry that can be observed in the gas phase.To test the validity of this proposed end-stage chemistry, we first explored single point models representative of the disk midplane with corresponding cold temperatures and high gas densities (approximately 35 K, 5$\\\\times $ 10$^{11}$ mol/cm$^{-3}$ ).\",\n \"We varied the gas-to-dust mass ratio and initial carbon abundance among other physical and chemical variables including the nitrogen abundance, dust extinction, and ionization rate.\",\n \"We found that the gas phase $\\\\textrm {CH}_{3}\\\\textrm {CN}$ abundance was the most sensitive to the gas-to-dust ratio and initial gas phase carbon abundance.\",\n \"We then produced a thermo-chemical model representing the disk around the Herbig Ae star HD 163296.\",\n \"This disk is old (approx.\",\n \"7.6 Myr)[14] and is nearby at 101 pc[42] and has observed jets and winds [19], [96].\",\n \"The HD 163296 disk has been widely observed in multiple gas and dust tracers with high spatial resolution ($\\\\sim $ 10 au) with clear gap and ring structures typically assumed to be associated with active planet formation[5], [80].\",\n \"There is bright $\\\\textrm {CH}_{3}\\\\textrm {CN}$ emission coming from $\\\\sim $ 35 K gas as well as bright emission from $\\\\textrm {HC}_{3}\\\\textrm {N}$ and HCN[55], [15].\",\n \"Our modeling efforts follow that of Zhang et al.\",\n \"2021[97], and Calahan et al.\",\n \"2021b[25] which set up a thermo-chemical model of the HD 163296 disk by reproducing the mm dust-continuum observations, the spectral-energy distribution (SED), the full line intensity and morphology of six CO isotopologue transitions, and the vertical distribution of the optically thick lines.\",\n \"The model of HD 163296 started with a gas-to-small dust ratio of the disk equal to 500[97] which was needed to reproduce the disk's SED.\",\n \"c|ccc|ccc 7 0pt Modeling Parameters: TW Hya & HD 163296 Parameters TW Hya HD 163296 Gas Small Dust Large Dust Gas Small Dust Large Dust$^{a}$ Mass (M$_{\\\\odot }$ ) 0.025 $1.0\\\\times 10^{-4}$ $4.0\\\\times 10^{-4}$ 0.14$2.6 \\\\times 10^{-5}$ 0.024 $\\\\Psi $ 1.1 1.2 1.2 1.08 1.081.08 $\\\\gamma $ 0.75 0.75 1.0 0.80.80.1 h$_{c}$ (au) 42 42 8.4 8.448.44 n/a r$_{c}$ (au) 400 400 400 165165 n/a r$_{in}$ (au) 0.1 0.5 1 0.45 0.450.45 r$_{out}$ (au) 200 200200 600 600240 Final values of the TW Hya and HD 163296 models that reproduce CO, HD, $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , HCN, and $\\\\textrm {HC}_{3}\\\\textrm {N}$ observations when available.\",\n \"r$_{in}$ and r$_{out}$ are the radial inner and outer limits of the disk, beyond these limits there is assumed to be no gas nor dust.\",\n \"$^{a}$ The surface density of the large dust distribution in HD 163296 is empirically set by continuum observations, thus it is not smooth and it is not dictated by the parametric equations.\",\n \"To produce this end-stage chemical environment, we deplete the small dust mass by a factor of 10 throughout the disk of HD 163296 making the new gas-to-dust mass ratio above the pebble disk midplane equal to 5,000.\",\n \"We then enhanced the initial gas phase carbon abundance in the system in the form of C, CH$_{4}$ , or C$_{2}$ H, increasing the overall gas phase carbon-to-oxygen ratio in both disks to above unity.\",\n \"There are two possible sources for excess carbon in protoplanetary disks, either through the destruction of refractory carbon grains[22] or CO depletion through mechanisms such as reactions with ionized molecules and atoms such as He$^{+}$ or H$_{3}^{+}$[86] and a series of chemical and freeze-out processes.\",\n \"We note that while signatures of CO depletion are found in the HD 163296 disk[97], [25], CO destruction likely only occurs in environments with a high cosmic-ray flux [2$\\\\times $ 10$^{-17}$ s$^{-1}$ ][86].\",\n \"Additionally, CO destruction would supply equal amounts of carbon and oxygen, while we seek to enhance carbon over oxygen.\",\n \"Our thermo-chemical model is run for a full megayear (Myr), after which the chemistry reaches an equilibrium.\",\n \"Notable chemical feedback due to dust evolution is predicted to occur over scales of $\\\\sim $ 1 Myr[61], [91], thus we use $\\\\sim $ 1 Myr as an approximate length of time for the chemical environment to transition from `early-stage' to `late-stage' dust evolution and subsequently exist in a state of photochemical equilibrium.\",\n \"Gas phase chemical reactions and rates were taken from the chemical network derived in Bosman et al.\",\n \"2018[20] which in turn relies on the UMIST Database for Astrochemistry Rate12 version[70] (see Methods section for more details).\",\n \"We found that regardless of the carrier of carbon, using a C/O = 1-2 and a factor of 10 depletion of small dust roughly reproduces line ratios and fluxes of the radial intensity profiles of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , $\\\\textrm {HC}_{3}\\\\textrm {N}$ , and HCN (see Figure REF ).\",\n \"In this proposed scenario, $\\\\sim $ 96% of the total mass of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ continues to reside frozen out onto grains, but the increase in UV flux allows for sufficient $\\\\textrm {CH}_{3}\\\\textrm {CN}$ to exist and be formed in the gas to reproduce observed radial intensity profiles and column densities.\",\n \"Figure REF shows the two-dimensional number densities of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ .\",\n \"The inner $\\\\sim $ 20 au exhibited brighter or more centrally peaked emission than was seen in observations if the C-to-O ratio = 2 throughout the full disk.\",\n \"To counteract this, we set the C-to-O ratio = 0.47, or the ISM ratio, inside of ($\\\\sim $ 20 au) where Zhang et al.\",\n \"2021[97] found an ISM ratio of $\\\\textrm {H}_{2}$ /CO (See Supplementary Figure 2).\",\n \"The inner disk emission remains slightly brighter than observed in HCN and $\\\\textrm {CH}_{3}\\\\textrm {CN}$ (see Figure REF ) with this alteration.\",\n \"This could be accounted for via a depletion in the nitrogen abundance within the N$_{2}$ ice line, a strong buildup of pebbles around the water ice line ($\\\\sim $ 5au), a lower HCN desorption energy, or a combination of these effects.\",\n \"Additionally, the observations may also be affected by processes such as beam smearing and a higher dust opacity than modeled.\",\n \"A comparison model without a depletion in small dust mass is shown in Supplementary Figure 5.\",\n \"Figure: Radial intensity profiles of the observed complex organic molecules and HCN towards the disk around HD 163296.\",\n \"The molecules shown are CH 3 CN \\\\textrm {CH}_{3}\\\\textrm {CN}, HCN, and HC 3 N\\\\textrm {HC}_{3}\\\\textrm {N} (left to right with two J transition of CH 3 CN \\\\textrm {CH}_{3}\\\\textrm {CN} shown in the first two panels).\",\n \"Solid thick lines in the background correspond to the observations derived from Ilee et al 2021 and Guzmán et al.\",\n \"2021 (which utilized Law et al.\",\n \"2021) while dashed lines correspond to modeled radial profiles.\",\n \"Our final model includes an increase in C/O ratio beyond 20 au and a depletion of small dust and represents 1 Myr of chemistry.\",\n \"This model can simultaneously fit the flux, line ratios, and general morphology of the observed radial profiles of these complex organic molecules.Figure: The radial and vertical number density distributions of CH 3 CN \\\\textrm {CH}_{3}\\\\textrm {CN} and HC 3 N\\\\textrm {HC}_{3}\\\\textrm {N} The density distributions were determined by our final HD 163296 model with an increase in C/O ratio beyond 20 au and a depletion of small dust.\",\n \"This distribution produced the radial profiles shown in Figure .\",\n \"A z/r<<0.1 is good approximation for the midplane of the disk, and both CH 3 CN \\\\textrm {CH}_{3}\\\\textrm {CN} and HC 3 N\\\\textrm {HC}_{3}\\\\textrm {N} emit partially from the midplane.\",\n \"A comparison can be made with Figure which lacks the inclusion of small dust depletion.$\\\\textrm {HC}_{3}\\\\textrm {N}$ and $\\\\textrm {CH}_{3}\\\\textrm {CN}$ are built up in the gas from simple carbon and nitrogen-based volatiles (see Methods section “Chemical Reactions\\\" for details).\",\n \"A reservoir of these more simple molecules is maintained in the gas phase due to photodesorption from dust grain surfaces by an enhanced UV-field.\",\n \"The main destruction products from $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ are simple volatiles that can cycle back to create larger nitrile molecules including their original parent molecule.\",\n \"While there are enough complex organic molecules in the gas phase to observe bright emission, the majority of the $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , $\\\\textrm {HC}_{3}\\\\textrm {N}$ , and HCN near the midplane still remains frozen out onto grains.\",\n \"The carbon-rich gas reservoir and UV-dominated disk together allow for a cycle of carbon chemistry to remain active in the gas phase.\",\n \"This implies that actively accreting gas giants will build their atmosphere out of this complex nitrile-enhanced material.\",\n \"Gas giant planets with atmospheres containing a chemical make-up with a C-to-O ratio $>$ 1 can be explained by the natural evolution of dust and the observed high C-to-O ratios within protoplanetary disks.\",\n \"This environment could be extended to the inner disk in some cases, as studies such as Najita et al.\",\n \"2011[76] and Anderson et al.\",\n \"2021[2] posit a higher than solar C-to-O ratio within the inner disk to account for Spitzer observations of HCN and C$_{2}$ H$_{2}$ around T Tauri stars.\",\n \"The implications of this end-stage chemistry are far-reaching.\",\n \"Primarily, we put forward a new chemically and physically coupled picture of planet formation and its direct impact on the chemistry of forming planet atmospheres.\",\n \"Within the early stages of planet formation, pebble growth and accretion into solid planet cores has begun, and chemistry onto these cores are influenced primarily by the chemical make-up of large pebbles and their icy mantles.\",\n \"The chemistry active in the midplane during this stage of formation is dominated by X-rays and cosmic rays[95].\",\n \"After timescales of order 1 Myr, at the end-stage of pebble formation, small grains have grown and settled towards the midplane, allowing for UV photons to penetrate deeper in the disk than previously assumed.\",\n \"This excess UV flux in concert with an above solar C-to-O ratio produces a carbon-rich gas phase cycle of production and destruction in the gas surrounding actively forming planets.\",\n \"Bright emission from $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , $\\\\textrm {HC}_{3}\\\\textrm {N}$ and other carbon-rich molecules emitting at cold temperatures are a sign-post of this evolved chemical environment.\",\n \"These forming gas giants may accrete this surrounding gas and its chemical signature into their atmosphere, and may be responsible for high C-to-O measurements that have been observed in exo-planetary atmospheres as compared to their host star[23].\",\n \"Secondly, the deeper penetration of UV photons pushes the CO and H$_{2}$ self-shielding layers deeper into the disk.\",\n \"Due to this, there are more ions such as C$^{+}$ and H$^{+}$ throughout the disk than would have been present with a regular (ISM) level of small dust surface area.\",\n \"As the ion-fraction increases so does the area of the disk that is subject to magneto-rotational instability (MRI).\",\n \"In our final model, the UV field increases by a factor of 3-5 in the atmosphere and between 1-2 near the midplane, and thirteen times more gas mass is MRI-active, most notably the gas midplane within 10 au, and then at a z/r=0.1-0.2 within 30 au (see Figure 4).\",\n \"This could be a source of accretion[43] for older disks.\",\n \"Our proposed UV-dominated carbon-rich gas phase chemistry is a major shift in our understanding of the astrochemistry of planet formation.\",\n \"This transition will have a strong effect on the composition of actively forming planets, disk MRI-activity, and subsequent disk accretion.\",\n \"Figure: A comparison of the MRI active region within a non-elevated UV environment and an elevated UV environment.\",\n \"The two left-most plots show regions within the HD 163296 disk which are MRI activated in an environment where small dust is depleted by a factor of 10 in total mass (gas/dust = 5,000 middle plot) and where small dust is not depleted (gas/dust=500 left).\",\n \"The color corresponds to the gas density in mol/cm 3 ^{3}, highlighting the substantial increase in mass that is MRI-active.\",\n \"The right-most plot shows a comparison of the UV field strength through a `regular' disk and a dust depleted disk.\",\n \"The UV field increases upwards of a factor of 3-5 in the atmosphere of the disk and 1-2 below a z/r=0.2.\",\n \"This has a notable effects on the chemistry of the midplane.For a thorough description of the observations and the techniques used to obtain the CLEAN-ed images and radial profiles of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , HCN, and $\\\\textrm {HC}_{3}\\\\textrm {N}$ towards HD 163296 see Czekala et al.\",\n \"2021[31], Law et al.\",\n \"2021[63], and Öberg et al.\",\n \"2021[80].\",\n \"A description of the observations of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ towards TW Hya can be found in Loomis et al.\",\n \"2018a[66].\",\n \"A brief summary of these observations are as follows.\",\n \"The data for HD 163296 is from the MAPS large program (Project ID 2018.1.01055.L)[80].\",\n \"The image cubes were produced using the tclean task in the Common Astronomy Software Applications (CASA) package, version 6.1.0[71].\",\n \"Keplerian masks based on the disk geometric parameters were used in the CLEANing process and the final images were then corrected for the Jorsater & van Moorsel effect to ensure that the image residuals are in units consistent with that of the CLEAN model.\",\n \"For all lines, we used the beam-circularized and uv-tapered images, which had synthesised beam sizes of 0.\\\"3.\",\n \"All radial intensity profiles were generated by deprojecting and azimuthally-averaging zeroth moment maps using the GoFish python package.\",\n \"$\\\\textrm {CH}_{3}\\\\textrm {CN}$ was observed towards TW Hya as a part of ALMA project 2016.1.01046.S.\",\n \"Each emission line was individually imaged using CLEAN and the synthesised beam for each transition were matched using small uv-tapers to 1$.\\\\!\\\\!^{\\\\prime \\\\prime }$ 05 x 0$.\\\\!\\\\!^{\\\\prime \\\\prime }$ 83.\",\n \"A Keplerian mask was used to extract the flux for each transition.\",\n \"See Loomis et al.\",\n \"2018a[66] for more details.\",\n \"Simple single point models of the disk environment were utilized to quickly and efficiently understand the chemical impacts of different physical and initial chemical conditions.\",\n \"Our chemical network is derived from Bosman et al.\",\n \"2018[20] which in turn is derived from the UMIST “RATE12” network[70], and for this study we disregarded most grain-surface chemical reactions in order to isolate gas-phase formation of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ .\",\n \"After our results from the modeling suggested a higher gas/dust ratio and carbon content, we turned to more comprehensive thermo-chemical codes which model the thermal physics and chemical evolution throughout the whole disk.\",\n \"The code RAC2D (https://github.com/fjdu/rac-2d)[35], was used to create models of the disks around TW Hya and HD 163296.\",\n \"The 2D temperature, density, and molecular abundance results were used to simulate observations of each disk with a raytracing code: RADMC-3D[37].\",\n \"A brief description of the physical code of RAC2D is given below; a detailed description of the code can be found in Calahan et al 2021a[24].\",\n \"RAC2D takes into account a gas and dust structure and stellar radiation field and computes the gas and dust temperature and chemical structure over time.\",\n \"Our model consists of three mass components: gas, small dust, and large dust grains.\",\n \"The spatial extent of each component is given by a global surface density distribution [68], which is widely used in protoplanetary disk modeling and corresponds to the self-similar solution of a viscously evolved disk.\",\n \"$\\\\Sigma (r)=\\\\Sigma _{c}\\\\left(\\\\frac{r}{r_{c}}\\\\right)^{-\\\\gamma }\\\\exp {\\\\left[-\\\\left(\\\\frac{r}{r_{c}}\\\\right)^{2-\\\\gamma }\\\\right]},$ $r_{c}$ is the characteristic radius at which the surface density is $\\\\Sigma _{c}/\\\\rm {e}$ where $\\\\Sigma _{c}$ is the characteristic surface density, and $\\\\gamma $ is the power-law index that describes the radial behavior of the surface density.\",\n \"A 2D density profile for the gas and dust populations can be derived from the surface density profile and a scale height: $\\\\rho (r,z) = \\\\frac{\\\\Sigma (r)}{\\\\sqrt{2\\\\pi }h(r)} \\\\exp {\\\\left[-\\\\frac{1}{2}\\\\left(\\\\frac{z}{h(r)}\\\\right)^{2}\\\\right]},$ $h=h_{c}\\\\left( \\\\frac{r}{r_{c}}\\\\right)^{\\\\Psi } ,$ where $h_{c}$ is the scale height at the characteristic radius, and $\\\\Psi $ is a power index that characterizes the flaring of the disk structure.\",\n \"The modeling parameters used for TW Hya and HD 163296 are shown in Table 1.\",\n \"For both TW Hya and HD 163296 models, each dust population follows an Mathis, Rumpl, and Nordsieck (MRN) grain distribution $n(a) \\\\propto a^{-3.5}$ [69], where `a' indicates the size of the grain.\",\n \"The small dust grains have radii between $5 \\\\times 10^{-3}$ - $1 \\\\mu $ m, and the large grains have radii between $5 \\\\times 10^{-3}$ - $10^{3} \\\\mu $ m. The large dust population is settled in the midplane with a smaller vertical extent and radial extent (gas extends $\\\\sim $ 5 and 2.5 times above and beyond, respectively).\",\n \"This settled large grain population is the result of dust evolution, namely growth in concert with vertical settling to the midplane and radial drift.\",\n \"For the HD 163296 model, the large grain population has a unique, non-smooth, surface density profile that reproduces the millimeter continuum observations of the HD 163296 disk[58], [97].\",\n \"Opacity values for the dust are calculated based on Birnstiel et al.\",\n \"2018[18].\",\n \"Large dust grains consist of water ice[94], silicates[32], troilites and refractory organics[53].\",\n \"Small dust grains consist of 50% silicates and 50% refractory organics.\",\n \"Further discussion on the modeling efforts for TW Hya can be found in Calahan et al.\",\n \"2021a[24] and for HD 163296 in Calahan et al.\",\n \"2021b[25].\",\n \"The thermo-chemical model resulted in 2D distributions of the gas and dust temperature, density, and molecular abundances.\",\n \"These were used as inputs for the raytracing code RADMC-3D[37].\",\n \"The molecular properties of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , $\\\\textrm {HC}_{3}\\\\textrm {N}$ , and HCN were taken from the Leiden Atomic and Molecular Database (LAMDA[85]), with some molecular parameters updated according to data from the Cologne Database for Molecular Spectroscopy (CDMS[74], [75]).\",\n \"The result from RADMC-3D was a 3D image cube of the molecular emission across velocity space.\",\n \"We utilized GoFish[88] to compress these 3D images into zeroth moment maps and radial profiles which were then directly compared to the azimuthally-averaged observations.\",\n \"The chemical reaction network utilized in this work comes from Bosman et al.\",\n \"2018[20] which in turn relies on the Rate12 version of the UMIST Database for Astrochemistry[70]: a network that is widely used across astrochemical modeling efforts.\",\n \"In our network there are 6,302 gas phase reactions and we have limited the grain-surface reactions to twelve.\",\n \"We kept grain-surface reactions that involved the formation of H$_{2}$ , CH$_{2}$ OH, CH$_{3}$ OH, CO$_{2}$ , H$_{2}$ O and NH$_{3}$ due to each of these species being well-studied in the laboratory and come with strong evidence for active two-body chemistry on dust grains[1], [51], [45], [56], [57], [41], [77], [84] (see Table 2).\",\n \"Both thermal adsorption and desorption are taken into account for every molecule in the network.\",\n \"Our network contains molecules with at most eleven carbon atoms, and at most twelve total atoms.\",\n \"We run our thermo-chemical model for 1 Myr as the $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ chemistry reaches an equilibrium after this time period (see Supplementary Figure 3).\",\n \"In our evolved disk model, the main formation pathways for $\\\\textrm {CH}_{3}\\\\textrm {CN}$ once the chemistry has reached equilibrium is as follows: $\\\\rm {CH_{3}^{+} + HCN \\\\rightarrow CH_{3}CNH^{+} + e^{-} \\\\rightarrow CH_{3}CN + H}$ This reaction in the gas phase is the primary formation pathway for $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , thus there is a strong reliance on HCN existing in the gas phase even at cold temperatures below its measured sublimation temperature ($\\\\sim $ 85-103 K[16]) and an ionization source (UV photons) to produce CH$_{3}^{+}$ .\",\n \"$\\\\textrm {HC}_{3}\\\\textrm {N}$ can be readily produced by a number of different ways including: $\\\\rm {N + CH_{2}CCH \\\\rightarrow HC_{3}N + H_{2}}$ $\\\\rm {CN + C_{2}H_{2} \\\\rightarrow HC_{3}N + H}$ $\\\\rm {H + C_{3}N^{-} \\\\rightarrow HC_{3}N + e^{-}}$ The formation of $\\\\textrm {HC}_{3}\\\\textrm {N}$ strongly relies on the existence of carbon-rich molecules including radicals (CN) and ions (C$_{3}$ N$^{-}$ ).\",\n \"c|c 7 0pt Dust Surface Reactions Reaction Reference gH + gH $\\\\rightarrow $ gH$_{2}$ Hasegawa, Herbst, & Leung 1992[51] gH + gOH $\\\\rightarrow $ gH$_{2}$ O Ioppolo et al.\",\n \"2010[57] gH + gH$_{2}$ O$_{2}$ $\\\\rightarrow $ gH$_{2}$ O + gOH Ioppolo et al.\",\n \"2008[56] gH + gCH$_{3}$ OH $\\\\rightarrow $ gH$_{2}$ + gCH$_{2}$ OH Extrapolated from Fuchs et al.\",\n \"2009[41] gH + gCH$_{2}$ OH $\\\\rightarrow $ gCH$_{3}$ OH Extrapolated from Fuchs et al.\",\n \"2009[41] gH$_{2}$ + gOH $\\\\rightarrow $ gH$_{2}$ O + gH Oba et al 2012[77] gOH + gCO $\\\\rightarrow $ gCO$_{2}$ + gH $^{a}$ Ruffle & Herbst 2001[84] gO + gCO $\\\\rightarrow $ gCO$_{2}$ $^{a}$ Goumans & Brown 2008[45] gO + gHCO $\\\\rightarrow $ gCO$_{2}$ + gH Goumans & Brown 2008[45] gH + gNH$_{2}$ $\\\\rightarrow $ gNH$_{3}$ Allen & Robinson 1977[1] The complete list dust surface reactions accounted for in this study.\",\n \"$^{a}$ Have additional special treatment for three body reactions We additionally produced a thermo-chemical model representing the disk around T Tauri star TW Hya.\",\n \"TW Hya is approximately 10 Myrs old[90] and hosts the closest Class II disk at 59.9 pc[42].\",\n \"TW Hya has been widely observed in multiple gas and dust tracers with high spatial resolution ($\\\\sim $ 10 au)[4], [54].\",\n \"It also exhibits bright $\\\\textrm {CH}_{3}\\\\textrm {CN}$ coming from $\\\\sim $ 33 K gas[66].\",\n \"Our modeling efforts of TW Hya follow that of Calahan et al.\",\n \"2021a[24] which sets up a thermo-chemical model of the disk by reproducing the spectral-energy distribution (SED), the full line intensity and morphology of seven CO isotopologue transitions, and an HD J=1-0 observation from the Hershel PACS instrument[11].\",\n \"To reproduce all CO radial profiles as well as the HD flux, the small dust in the upper layers of the atmosphere of the TW Hya disk were effectively depleted slightly[24] due to having a slightly lower flaring angle than the gas population.\",\n \"This was enough small dust depletion to reproduce the available $\\\\textrm {CH}_{3}\\\\textrm {CN}$ lines from Loomis et al.\",\n \"2018a given a C/O ratio equal to 1.0 (see Figure REF ).\",\n \"This disk is an additional piece of evidence supporting our evolved chemistry proposal.\",\n \"This introduction of a late-stage photo-chemical equilibrium was motivated by observations of $\\\\textrm {CH}_{3}\\\\textrm {CN}$ , $\\\\textrm {HC}_{3}\\\\textrm {N}$ , and HCN.\",\n \"However, the late-stage chemistry would have an effect on other molecules as well.\",\n \"We predict in addition to these organic molecules, molecules such as C$_{X}$ H, C$_{X}$ H$_{2}$ , and HC$_{X}$ N will be abundant in the gas, enhanced by the cycle of carbon chemistry that reproduces observed $\\\\textrm {CH}_{3}\\\\textrm {CN}$ and $\\\\textrm {HC}_{3}\\\\textrm {N}$ .\",\n \"The main carbon carrier, CO, is largely unaffected by the increase in photo-chemistry.\",\n \"CO is largely in the gas phase in previous models that do not include this `late-stage' chemistry and in the region in which $\\\\textrm {CH}_{3}\\\\textrm {CN}$ was added into the gas, CO was already primarily in the gas phase.\",\n \"We find a slight depletion of CO in the upper atmosphere of the disk due to the CO photodissociation layer being pushed down, and there is a slight enhancement of CO in the midplane due to the UV-enhancement.\",\n \"However this accounts for less than 1% of the total abundance of CO thus did not have a strong impact on the modeled radial profiles.\",\n \"Another key molecule is H$_{2}$ O.\",\n \"In our models, we do not initialize our model with H$_{2}$ O, thus there is very little water to be affected by this `late-stage' chemistry.\",\n \"Observational results of Du et al.\",\n \"2017[36] support this as they found a low overall abundance in H$_{2}$ O in disks with a survey of 13 protoplanetary disks.\",\n \"$\\\\textrm {CH}_{3}\\\\textrm {CN}$ nor $\\\\textrm {HC}_{3}\\\\textrm {N}$ would be seen to be at a high abundance if gas-phase H$_{2}$ O was in high abundance in the disk as it would disrupt the carbon-rich chemistry.\",\n \"The depletion of small grains will affect the observed spectral energy distribution (SED) from each disk.\",\n \"TW Hya’s dust population was not altered from that of Calahan et al.\",\n \"2021a[24] and continues to match its observed SED.\",\n \"A depletion of a factor of 10 in the small dust population around HD 163296 would cause a dimming in the mid-infrared part of the SED.\",\n \"However, the small dust abundance, the distribution of dust grain size, and the assumed dust opacity sources are degenerate in the ways they may affect the disk SED.\",\n \"We modeled protoplanetary disk SEDs using the code TORUS[49], [50].\",\n \"TORUS is a Monte Carlo radiative transfer code utilizing radiative equilibrium[67] and silicate grains[33].\",\n \"In Figure REF , we show a series of SEDs produced from models motivated by our thermo-chemical model of HD 163296 including the stellar parameters and dust distribution in Table 1 from Calahan et al.\",\n \"2021b[25].\",\n \"We find that by varying the minimum dust grain radius or the power law index, we can account for a factor of 10 in UV attenuation.\",\n \"The total population mass, grain size, and how the grain sizes are distributed are strongly degenerate and can result in uncertainties of the dust mass by a factor of at least 10.\",\n \"Thus our depletion of small dust continues to reproduce all previous observables including the SED.\",\n \"With the increase in UV flux deeper into the disk, more ions are created.\",\n \"Ions can be coupled with the magnetic fields that thread through the disk and interact with the bulk gas creating turbulence.\",\n \"This mechanism is called the magneto-rotational instability or MRI[9].\",\n \"Magneto-hydrodynamical processes are assumed to be present throughout protoplanetary disks due to young star’s active magnetic field, and are thought to be one of the main drivers of angular momentum transport [27].\",\n \"An MRI-active zone may drive the bulk of the mass transportation in the disk and activate accretion onto the star, two vital processes that determine the future of a young solar system.\",\n \"It is thought that planet formation may be aided within ‘dead-zones’ where MRI is non-active[46].\",\n \"The magnetic Reynolds number ($Re$ ) and ambipolar diffusion term ($Am$ ) are two quantities that help quantify the presence of an MRI-active zone.\",\n \"The Reynolds number quantifies the level of coupling between ionized gas and magnetic fields and is defined as $Re \\\\equiv \\\\frac{c_{s}h}{D} \\\\approx 1 \\\\left(\\\\frac{\\\\chi _{e}}{10^{-13}}\\\\right) \\\\left(\\\\frac{T}{100~\\\\rm {K}}\\\\right)^{1/2} \\\\left(\\\\frac{a}{\\\\rm {AU}}\\\\right)^{3/2} ,$ where $c_{s}$ is the sound speed, $h$ is the scale height of the disk, D is the magnetic diffusivity parameter, $\\\\chi _{e}$ is the electron abundance, $T$ temperature and $a$ radial location in the disk [81].\",\n \"The ambipolar diffusion term describes the coupling of ionized molecules and their interaction with neutral gas particles: $Am \\\\equiv \\\\frac{\\\\chi _{i} n_{\\\\rm {H}_{2}} \\\\beta _{\\\\rm {in}}}{\\\\Omega } \\\\approx 1 \\\\left(\\\\frac{\\\\chi _{i}}{10^{-8}}\\\\right) \\\\left(\\\\frac{n_{\\\\rm {H}_{2}}}{10^{10}\\\\rm {cm^{-3}}}\\\\right) \\\\left(\\\\frac{a}{\\\\rm {AU}}\\\\right)^{3/2} ,$ where $\\\\chi _{i}$ is the ion abundance, $n_{\\\\rm {H}_{2}}$ is the number density of H$_{2}$ atoms, $\\\\Omega $ is the dynamical time, $\\\\beta _{\\\\rm {in}}$ is the collisional rate coefficient for singly charged species to share momentum with neutral species, and $a$ is the radial location in the disk [34], [81].\",\n \"For MRI to act as a turbulent driver of neutral gas, both $Re$ and $Am$ must be sufficiently high.\",\n \"Simulations show that values between 0.1-100 for $Am$ can trigger significant coupling between ions and neutrals [52], [8].\",\n \"Models by Flock et al.\",\n \"2012[40] suggest $Re \\\\approx $ 3,300-5,000 is required to sustain sufficient turbulence with a critical $Re$ = 3,000.\",\n \"In our work, we assume a combined $Am >$ 100 and $Re >$ 3,000 to signify an MRI-active zone.\",\n \"We calculate the regions in which the MRI is active in the HD 163296 disk in our models with an early-stage physical environment and a late-stage environment (signified by a depletion in the small dust population mass) in Figure REF .\",\n \"Our solution allows for an increase of UV flux in the atmosphere by a factor of 3-5.\",\n \"More of the disk becomes ion-rich due to the CO and H$_{2}$ self-shielding layers being located deeper into the disk.\",\n \"The increase in the ion and electron abundance ($\\\\chi _{i}$ , $\\\\chi _{e}$ ) are the key factors that enhance the Re and Am values and thus produce additional MRI activity (see Figure REF ).\",\n \"As a result, thirteen times more mass within the disk becomes MRI-active including within the midplane.\",\n \"In the base model representing early stages of disk chemistry, the MRI activity at the midplane extends to 4 au.\",\n \"By depleting the small dust population by an order of magnitude, the MRI activity at the midplane then extends out to $\\\\sim $ 10 au.\",\n \"This increase in MRI activity can contribute to the reason behind why older disk systems, such as TW Hya, are actively accreting.\",\n \"Meridional flows have been identified in the HD 163296 disk located at two of the largest dust gaps (approx.\",\n \"45 and 86 AU)[89] and these vertical flows are coincident with the lower vertical limit of the MRI-active zone.\",\n \"The MRI activation could have an effect on the height or continue to drive meridional flows.\",\n \"Turbulence measurements of the HD 163296 disk have been derived in Flaherty et al.\",\n \"2015 and 2017[38], [39] using molecular tracers, and they find turbulent velocities to be 5% or less of the sound speed between 30-300 au and at all measured heights.\",\n \"It is not yet clear whether our MRI prediction is in tension with the observational evidence of low turbulence in HD 163296, more work needs to be done on the modeling of how MRI-active regions drive turbulence and more observational constraints are needed to constrain the inner 40 au where we find the strongest MRI-activity.\"\n ],\n [\n \"Acknowledgements\",\n \"Jenny Calahan is the corresponding author and can be contacted via jcalahan@umich.edu J.K.C.\",\n \"acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No.\",\n \"DGE 1256260 and the National Aeronautics and Space Administration FINESST grant, under Grant no.\",\n \"80NSSC19K1534.\",\n \"E.A.B.\",\n \"acknowledges support from NSF AAG Grant #1907653.\",\n \"A.D.B.\",\n \"acknowledges support from NSF AAG Grant #1907653.\",\n \"E.A.R.\",\n \"acknowledges support from NSF AST 1830728.\",\n \"Support for J. H. was provided by NASA through the NASA Hubble Fellowship grant #HST-HF2-51460.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.\",\n \"This work was supported by a grant from the Simons Foundation 686302 and by an award from the Simons Foundation 321183FY19, KÖ.\",\n \"This material is based upon work supported by the National Science Foundation under Grant No.\",\n \"AST-1907832.\",\n \"J.D.I.\",\n \"acknowledges support from an STFC Ernest Rutherford Fellowship (ST/W004119/1) and a University Academic Fellowship from the University of Leeds.\",\n \"C.W.\",\n \"acknowledges financial support from the University of Leeds, the Science and Technology Facilities Council, and UK Research and Innovation (grant numbers ST/T000287/1 and MR/T040726/1).\",\n \"V.V.G.\",\n \"gratefully acknowledges support from FONDECYT Regular 1221352, ANID BASAL projects ACE210002 and FB210003, and ANID, – Millennium Science Initiative Program – NCN19_171.\",\n \"We thank Tim Harries for help with and providing access to the TORUS modeling program.\"\n ],\n [\n \"Data Availability\",\n \"The data that support the findings of this study can be obtained as part of the MAPS program and are publicly available via alma-maps.info.\",\n \"Data regarding TW Hya can be obtained via data-rich figures in Calahan et al 2021 published on the online publication of the Astrophysical Journal article.\",\n \"This paper makes use of the following ALMA data: ADS/JAO.ALMA#2018.1.01055.L.\",\n \"and 2016.1.01046.S ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.\",\n \"The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.\",\n \"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\",\n \"This study relied on the following publicly available coding packages: rac2d: https://github.com/fjdu/rac-2d, RADMC-3D: https://www.ita.uni-heidelberg.de/ dullemond/software/radmc-3d/, and GoFish: https://github.com/richteague/gofish.\",\n \"TORUS is a private code developed by Tim Harries and collaborators.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05539"}}},{"rowIdx":88,"cells":{"text":{"kind":"list like","value":[["Effects of spin orbit coupling on proximity induced superconductivity"],["Abstract We investigate the effect of spin orbit coupling on proximity induced superconductivity in a normal metal attached to a superconductor.","Specifically, we consider a heterostructure where the presence of interfaces gives rise to a Rashba spin orbit coupling.","The properties of the induced superconductivity in these systems are addressed within the tunneling Hamiltonian formalism.","We find that the spin orbit coupling induces a mixture of singlet and triplet pairing and, under specific circumstances, an odd frequency, even parity, spin triplet pairs can arise.","We also address the effect of impurity scattering on the induced pairs, and discuss our results in context of heterostructures consisting of materials with spin-momentum locking."],["Introduction","Hybrid nanostructures consisting of superconductors have been intensively studied, both experimentally and theoretically.","Such hybrid proximity structures provide a platform for the realization of novel superconducting states in the vicinity of interfaces that connect superconductors to non-superconducting materials.","The current heightened interest in these systems is also driven by the potential of these heterostructures to host Majorana fermions[1], [2], [3], [4].","As these Majorana fermions obey non-Abelian braiding statistics they may serve as building blocks for fault-tolerant quantum computation [5], [6], [7], [8], [9].","Superconductor and ferromagnetic (S/F) hybrid structures have been studied heavily[10], [11], [12], [13], [14], [15], with a recent focus on proximity structures with topological materials.","In such proximity structures, the spin orbit coupling (SOC) plays an important role.","It is usually induced by the breaking of inversion symmetry e.g.","through an underlying substrate or the presence of an interface.","In a superconductor, the SOC leads to a mixing of singlet and triplet pairing.","In this two-component superconductivity either singlet or triplet pairing can be dominating[16], [17].","Recent observation of triplet dominant two component superconductivity in CoSi$_2$ /TiSi$_2$ /Si heterostructures has confirmed the realization of dominant triplet pairing in these structures via a substrate induced SOC[18], [19].","The role of SOC in the S/F nanostructures has been studied extensively [20], [21], [22], [23], [24], where the junctions involved $s$ -wave superconductors.","These studies were carried out in the quasi-classical formalism in the diffusive limit[25].","In the diffusive limit, the most dominant energy scale in the problem is the elastic impurity scattering i. e. the impurity scattering rate $\\tau ^{-1} \\gg \\Delta , \\tau ^{-1}_{sf}, \\tau ^{-1}_{in} $ , where $\\Delta $ is the superconducting gap, $\\tau ^{-1}_{sf/in}$ is the scattering rate from spin-flip/inelastic scattering.","In such a regime, even frequency - even parity - spin singlet superconductor (EES) induces EES pairs in the diffusive normal metals, and even frequency - odd parity - spin triplet superconductors (EOT) leads to the formation of odd frequency - even parity - spin triplet pairs in the diffusive metal, as long as the interface in non-magnetic in nature[26].","A non-magnetic interface prevents triplet to singlet conversion.","Our main objective is to understand the properties of proximity-induced superconductivity in metals with sizable SOC.","We will examine the stability of the proximity-induced superconductivity against weak disorder and analyze the emergence of odd frequency pairs.","In this paper, we focus on the effect of SOC on induced superconductivity in a normal non-magnetic metal that is connected to an unconventional superconductor.","For concreteness, we consider a Rashba SOC that is induced by the underlying substrate beneath the normal metal component.","One reason for this particular choice of SOC interaction is that it can be generated and controlled by applying a gate voltage to the heterostructure.","Here, we focus on the structure of the induced superconductivity in the normal metal connected to a triplet superconductor.","In what follows, we adopt the tunneling Hamiltonian formalism[27], [28], [29], [30], [31], [32], [33].","In the next section, we introduce the basic model and the theoretical methods.","The subsequent section provides a discussion of our results.","The final section summarizes the key qualitative conclusions.","Figure: Schematic illustration of a superconductor - normal junction.","The interface is along the yzyz plane."],["Model & Formalism","Figure REF shows a schematic diagram of the superconductor - normal (SN) junction.","The interface is located at the $x=0$ plane.","We consider a SOC that is induced by the substrate and take the $\\hat{z}$ axis to be parallel the normal to the substrate.","The Hamiltonian for the normal component reads, $\\hat{H}_N &=& \\sum _{\\mathbf {k},\\sigma } c^\\dagger _{k\\sigma } \\left[ \\xi _\\mathbf {k} \\delta _{\\sigma \\sigma ^{\\prime }}+ \\hat{H} _{Rashba} \\right] c_{k\\sigma ^{\\prime }},$ where $c^\\dagger /c$ is the electron creation/annihilation operator, $\\xi _\\mathbf {k} $ is the electronic dispersion, $\\mathbf {k}$ denotes the momentum and $\\sigma $ represents the electron spin.","We denote a $4\\times 4$ matrix in Nambu-spin space with $\\check{\\square }$ while $\\hat{\\square }$ indicates a $2\\times 2$ matrix in the spin space.","The Rashba SOC term reads, $\\hat{\\mathcal {H}}_{Rashba} &=& -\\frac{\\alpha }{m} \\left( \\sigma \\times \\mathbf {k}\\right)\\cdot \\hat{z} = \\epsilon _N \\left( \\hat{z} \\times \\hat{\\mathbf {k}}\\right)\\cdot \\mathbf {\\sigma } , \\\\\\epsilon _N&=&\\frac{\\alpha |\\mathbf {k}|}{m}.$ Here, $\\alpha $ is the Rashba SOC coupling constant, $m$ is the effective mass and $\\sigma $ is the Pauli vector ($\\sigma _x,\\sigma _y,\\sigma _z$ ), where $\\sigma _{x/y/z}$ are the Pauli matrices in spin space.","The two helical bands generated by this term have energies $\\xi _\\mathbf {k}\\pm \\epsilon _N$ .","The Hamiltonian for the superconductor reads, $\\check{H}_{sc} &=& \\Psi ^\\dagger \\begin{pmatrix}\\xi _{\\mathbf {k}} & \\hat{\\Delta } \\\\\\hat{\\Delta }^\\dagger & -\\xi _{\\mathbf {k}}\\end{pmatrix} \\Psi $ where $\\xi _{\\mathbf {k}}$ is the dispersion in the superconductor, $\\Psi ^\\dagger =(a^\\dagger _{\\uparrow \\mathbf {k}},a^\\dagger _{\\downarrow \\mathbf {k}},a_{\\uparrow -\\mathbf {k}},a_{\\downarrow -\\mathbf {k}})$ , where $a^\\dagger /a$ is the creation/annihilation operator.","The gap $\\hat{\\Delta }=\\Delta i\\sigma _y$ for the singlet case while for the triplet case $\\hat{\\Delta }=\\Delta \\mathbf {d}\\cdot \\sigma i \\sigma _y$ .","Here, $\\mathbf {d}$ is the $d$ -vector of the triplet paring.","The tunneling Hamiltonian is, $\\check{H}_{tunneling} = \\gamma \\Psi ^\\dagger \\check{\\tau }_3 \\Phi + h. c.$ where $\\check{\\tau }_3=\\mathrm {diag}(1,1,-1,1)$ is a matrix in Nambu spin space, $\\Phi $ is the analog of $\\Psi $ in the normal metal and $\\gamma $ is the tunneling matrix element.","Here, we assume $\\gamma $ to be spin and momentum independent and take it to be real.","The mean-field expression for the Green's function of the superconducting component is $\\check{G}_{sc} = \\frac{1}{\\omega ^2-\\xi ^2-|\\Delta |^2} \\begin{pmatrix}(\\omega +\\xi ) \\sigma _0 & \\hat{\\Delta } \\\\\\hat{\\Delta }^\\dagger & (\\omega -\\xi ) \\sigma _0\\end{pmatrix}.$ Here, $\\sigma _0$ is the $2\\times 2$ identity matrix in spin space and for notational convenience, we abbreviate $\\xi _{\\mathbf {k}}$ as $\\xi $ .","The tunneling self energy for the normal side of the junction at the interface reads, $\\check{\\Sigma }_t(k_{||},\\omega ) &=& |\\gamma |^2 \\int \\frac{d k_\\perp }{2 \\pi } \\frac{1}{\\omega ^2-\\xi ^2-|\\Delta |^2} \\nonumber \\\\&\\times & \\begin{pmatrix}(\\omega +\\xi ) \\sigma _0 & -\\hat{\\Delta } \\\\-\\hat{\\Delta }^\\dagger & (\\omega -\\xi ) \\sigma _0\\end{pmatrix}.$ Here, $k_\\perp $ is the momentum component perpendicular to the SN interface and $k_{||}$ is the momentum parallel to it.","The self-energy has the same general structure as the structure of the Green's function in the superconductor.","The integration over the momentum component perpendicular to the interface may modify $\\mathbf {d}$ to $\\tilde{\\mathbf {d}}$ depending on the junction geometry.","To study the robustness of the induced pairs against disorder, we consider point-like impurities randomly distributed in the normal metal.","Within a self-consistent T-matrix approximation which incorporates all scattering processes from a single impurity site the impurity self energy contribution is, $\\check{\\Sigma }_{imp}( \\omega ) &=& n_{imp} \\check{\\tau }_3 {V}_{imp}\\left[ \\check{1}- \\check{\\mathbf {g}} \\check{\\tau }_3 {V}_{imp} \\right]^{-1},$ where $n_{imp}$ is the impurity concentration, $V_{imp}$ is the impurity potential, $\\check{1}$ is the $4\\times 4$ identity matrix and $\\check{\\mathbf {g}}$ is $\\check{\\mathbf {g}} &=& \\int _{\\mathbf {k}} \\check{\\mathbb {G}}.$ where $\\check{\\mathbb {G}}&=& \\frac{\\check{1}}{\\check{G}_0^{-1}-\\check{\\Sigma }_t( k_{||},\\omega )-\\check{\\Sigma }_{imp}(\\omega )}.$ Here $\\check{G}_0$ is the normal metal bare Green's function, and the impurity self-energy is calculated self-consistently."],["SN junction with a singlet superconductor","First, we consider a pure singlet superconductor attached to the normal metal component discussed above.","In that case, $\\hat{\\Delta }=\\Delta i \\sigma _y$ and the tunneling self-energy is given by, $\\check{\\Sigma }_t(k_{||},\\omega ) = |\\gamma |^2 \\begin{pmatrix}(\\Sigma _0 + \\Sigma _3)\\sigma _0 & i \\sigma _y {\\Sigma }_1 \\\\-i \\sigma _y {\\Sigma }_1 & (\\Sigma _0 - \\Sigma _3)\\sigma _0.\\end{pmatrix}$ In the case of a momentum independent $\\Delta $ and for a particle-hole symmetric system, the tunneling self-energy becomes, $\\check{\\Sigma }_t(k_{||},\\omega ) = -\\Gamma _t \\frac{1}{\\sqrt{\\Delta ^2-\\omega ^2}} \\begin{pmatrix}\\omega \\sigma _0 & i \\sigma _y {\\Delta } \\\\-i \\sigma _y \\Delta &\\omega \\sigma _0\\end{pmatrix}.$ Here $\\Gamma _t\\equiv \\pi |\\gamma |^2 \\nu _s$ is the energy scale associated with the tunneling process, where $\\nu _s$ is the normal state density of states (DOS) of the superconductor at the Fermi level.","Inclusion of particle-hole asymmetry gives finite $\\Sigma _3$ , that can be absorbed in the chemical potential.","As it turns out, the presence of particle-hole asymmetry does not lead to significant qualitative difference.","The normal metal Green's function in the clean limit is, $\\check{\\mathbb {G}} &=& \\begin{pmatrix}\\left[ \\bar{\\omega } - {\\xi } \\right] \\sigma _0 - \\epsilon _N (\\mathbf {w} \\cdot \\sigma ) & - i\\Sigma _1 \\sigma _y \\\\i \\Sigma _1 \\sigma _y & \\left[\\bar{\\omega } + {\\xi } \\right] \\sigma _0 - \\epsilon _N (\\mathbf {w} \\cdot \\sigma ^\\ast )\\end{pmatrix}^{-1}, \\nonumber \\\\\\check{\\mathbb {G}} &=& \\begin{pmatrix}\\hat{\\mathbb {G}}_{11} & \\hat{\\mathbb {G}}_{12} \\\\\\hat{\\mathbb {G}}_{21} & \\hat{\\mathbb {G}}_{22}.\\end{pmatrix}$ Here, $\\bar{\\omega }=\\omega -\\Sigma _0$ and $\\mathbf {w}=\\hat{z}\\times \\hat{\\mathbf {k}}$ .","The components of $\\check{\\mathbb {G}} $ are, $\\hat{\\mathbb {G}}_{11} &=& \\frac{1}{2} \\left[ \\mathcal {R}_+ + \\mathcal {R}_- \\right] + \\frac{1}{2} \\left[ \\mathcal {R}_+ - \\mathcal {R}_- \\right] (\\mathbf {w} \\cdot \\sigma ), \\\\\\mathcal {R}_\\pm &=& \\frac{\\bar{\\omega } + (\\bar{\\xi } \\pm \\epsilon _N)}{\\bar{\\omega }^2-(\\bar{\\xi } \\pm \\epsilon _N)^2 -\\Sigma _1^2}, \\\\\\hat{\\mathbb {G}}_{12} & =& \\left(\\frac{1}{2} \\left[ \\frac{1}{D_+} + \\frac{1}{D_-}\\right] + \\frac{1}{2} \\left[ \\frac{1}{D_+} - \\frac{1}{D_-}\\right] \\mathbf {w} \\cdot \\sigma \\right), \\nonumber \\\\& \\times & ( i \\sigma _y \\Sigma _1) \\\\D_\\pm &=& \\bar{\\omega }^2-(\\bar{\\xi } \\pm \\epsilon _N)^2 -\\Sigma _1^2.$ The structure of the induced pairs can be obtained from the anomalous Green's function $\\hat{\\mathbb {G}}_{12} $ .","The first term of $\\hat{\\mathbb {G}}_{12}$ in Eq.","() is the spin-singlet, even parity and even frequency component.","This is the conventional proximity effect for a singlet superconductor.","The second term of $\\hat{\\mathbb {G}}_{12} $ is a spin-triplet, odd parity and even frequency term.","This term is directly proportional to the strength of SOC.","Due to finite SOC, spin rotational symmetry is broken which allows for a mixing of singlet and triplet terms.","The $\\mathbf {d}$ -vector for triplet pairing is determined by the SOC vector $\\mathbf {w}$ .","At this point, we can include the effect of impurity scattering.","The impurity self-energy depends on the momentum integrated Green's function.","Since $\\mathbf {w}$ is an odd function of momentum, the terms linear in $\\mathbf {w}$ vanish in the momentum integrated Green's function for a singlet superconductor.","As a result, the momentum integrated Green's function is, $\\int _\\mathbf {\\mathbf {k}} \\check{\\mathbb {G}} &=& -\\pi N_0 \\frac{1}{\\sqrt{\\Sigma _1^2-\\bar{\\omega }^2}} \\begin{pmatrix}\\bar{\\omega } \\sigma _0 & \\Sigma _1 i \\sigma _y \\\\-i \\sigma _y \\Sigma _1 & \\bar{\\omega } \\sigma _0\\end{pmatrix},$ where $N_0$ is the normal metal DOS at the Fermi level.","The impurity self-energy can be expressed as, $\\check{\\Sigma }_{imp} &=&\\begin{pmatrix}{\\Sigma }_{imp0} \\sigma _0 & i \\sigma _y \\Sigma _{imp1} \\\\-i \\sigma _y \\Sigma _{imp1} & {\\Sigma }_{imp0} \\sigma _0,\\end{pmatrix}, \\\\\\Sigma _{imp0} &=& n_{imp}\\frac{ g_0 V^2}{1- V^2 (g_0^2 - g_1^2)}, \\\\\\Sigma _{imp1} &=& n_{imp}\\frac{-g_1 V^2}{1- V^2 (g_0^2 - g_1^2)},$ where we have neglected the $\\Sigma _3$ component of the impurity self energy, which vanishes for a particle hole symmetric system.","We can define impurity renormalized energy and induced off-diagonal self-energy as, $\\tilde{\\omega }&=& \\omega - \\Sigma _0(\\omega ) - \\Sigma _{imp0}(\\tilde{\\omega }), \\\\\\tilde{\\Sigma }_1 &=& \\Sigma _1(\\omega ) + \\Sigma _{imp1}(\\tilde{\\omega }).$ Here, the fully dressed Green's function is used to calculate $\\tilde{\\omega }$ , $\\tilde{\\Sigma }_1$ self-consistently.","The equations for renormalized $\\tilde{\\Sigma }_1$ and $\\tilde{\\omega }$ are identical as those for an $s$ -wave superconductors with nonmagnetic impurities.","Now, we rewrite the impurity and tunneling dressed Green's function $\\check{\\mathbb {G}}$ by replacing $\\bar{\\omega }$ and $\\Sigma _1$ with $\\tilde{\\omega }$ and $\\tilde{\\Sigma }_1$ , respectively.","After some straightforward algebra, we get $\\tilde{\\omega }/\\tilde{\\Sigma }_1 = \\bar{\\omega }/\\Sigma _1$ .","This ensures that the induced pairs remain robust against nonmagnetic disorder.","Next, we consider the interface DOS given by $N(\\omega )=N_0 \\mathrm {Im} \\frac{\\tilde{\\omega }}{\\sqrt{\\tilde{\\Sigma }_1^2-\\tilde{\\omega }^2}}.$ Fig.","REF (a) shows the interface DOS for several values of the tunneling energy scale $\\Gamma _t$ .","In the weak tunneling regime ($\\Gamma _t\\ll \\Delta $ ), the effective gap in the DOS is determined by $\\Gamma _t^2/\\Delta $ , and in the strong tunneling regime ($\\Gamma _t\\gg \\Delta $ ), the effective gap becomes identical to the size of the gap of the superconductor.","The sub-dominant triplet component does not induce any low energy sub-gap states, instead the low energy DOS is mainly controlled by the singlet order parameter, which is an isotropic s-wave in the present case.","The effect of impurity scattering is depicted in Fig.","REF (b), which shows no change in the DOS with increasing impurity scattering rate $\\Gamma _{imp}\\equiv n_{imp}\\pi N_0 V^2/(1+\\pi ^2 V^2 N_0^2) $ .","Figure: Proximity induced gap in the normal metal.","(a) Variation of normalized DOS with strength of tunneling energy scale Γ t \\Gamma _t.","(b) Normalized density of states at the interface as a function of energy for several values of impurity scattering rate."],["SN junction with a triplet superconductor","In a triplet superconductor the order parameter is $\\hat{\\Delta }=\\Delta \\mathbf {d}\\cdot \\sigma i\\sigma _y$ , where the $\\mathbf {d}$ -vector describes the pair structure in spin space.","Here, we restrict ourselves to unitary pairing ($\\mathbf {d}\\times \\mathbf {d}^\\ast =0$ ) in the superconducting component of the heterojunction.","In this case, we find for the tunneling self-energy, $\\check{\\Sigma }_t(k_{||},\\omega ) = |\\gamma |^2 \\begin{pmatrix}\\Sigma _0 \\sigma _0 & {\\Sigma }_1 \\tilde{\\mathbf {d}} \\cdot \\sigma i \\sigma _y \\\\-i \\sigma _y {\\Sigma }_1 \\tilde{\\mathbf {d}} ^\\ast \\cdot \\sigma & \\Sigma _0 \\sigma _0\\end{pmatrix},$ where the components are given by, $\\Sigma _0 & = & |\\gamma |^2 \\int \\frac{d k_\\perp }{2 \\pi } \\frac{\\omega }{\\omega ^2-\\xi ^2-\\Delta ^2 | \\mathbf {d}|^2}, \\\\\\mathbf {\\Sigma }_1 & = & |\\gamma |^2 \\int \\frac{d k_\\perp }{2 \\pi } \\frac{-\\Delta \\mathbf {d}}{\\omega ^2-\\xi ^2-\\Delta ^2 | \\mathbf {d}|^2 } = \\Sigma _1 \\tilde{\\mathbf {d}}.$ As before in the singlet case, we ignore the $\\Sigma _3$ self-energy.","The $\\mathbf {d}$ -vector is an odd function of momentum and $\\mathbf {\\Sigma }_1$ is also an odd function of the momentum parallel to the interface ($k_{||}$ ), and it does not depend on the momentum component normal to the interface ($k_\\perp $ ).","The Green's function in the normal metal can be written as, $\\check{\\mathbb {G}} &=& \\begin{pmatrix}\\left[ \\bar{\\omega } - {\\xi } \\right] \\sigma _0 - \\epsilon _N (\\mathbf {w} \\cdot \\sigma ) & - {\\Sigma }_1 \\tilde{\\mathbf {d}} \\cdot \\sigma i \\sigma _y \\\\i \\sigma _y {\\Sigma }_1 \\tilde{\\mathbf {d}} ^\\ast \\cdot \\sigma & \\left[\\bar{\\omega } + {\\xi } \\right] \\sigma _0 - \\epsilon _N (\\mathbf {w} \\cdot \\sigma ^\\ast )\\end{pmatrix}^{-1} \\nonumber \\\\&=& \\begin{pmatrix}\\hat{\\mathbb {G}}_{11} & \\hat{\\mathbb {G}}_{12} \\\\\\hat{\\mathbb {G}}_{21} & \\hat{\\mathbb {G}}_{22}.\\end{pmatrix}$ The general structure of the off-diagonal Green's function has the form, $\\hat{\\mathbb {G}}_{12} &\\propto & A_0 \\sigma _0 + A_1 \\mathbf {w}\\cdot \\sigma + A_2 \\tilde{\\mathbf {d}} \\cdot \\sigma + A_3 \\tilde{\\mathbf {d}} ^\\ast \\cdot \\sigma \\nonumber \\\\& & + A_4 \\left( \\tilde{\\mathbf {d}} \\times \\mathbf {w} \\right) \\cdot \\sigma ,$ where the specific values of the set $\\lbrace A_i\\rbrace $ (i=0,...4) will depend on geometry of the junction and the specifics of the $\\mathbf {d}$ -vector which in turn determine $\\tilde{\\mathbf {d}}$ .","In general, $\\tilde{\\mathbf {d}} \\times \\tilde{\\mathbf {d}} ^\\ast $ may not vanish despite unitary pairing in the superconductor ($\\mathbf {d}\\times \\mathbf {d}^\\ast =0$ ).","Eq.","(REF ) has singlet and triplet terms, but due to the anisotropic nature of the gap in the superconductor details of geometry and pairing structure is essential to understand the structure of induced pairs in the normal metal.","To proceed, we will consider a few pertinent cases."],["$\\mathbf {d}$ -vector {{formula:cd423bf1-efa1-4737-9b31-b95841a50102}}","Motivated by the recent experiments on CoSi$_2$ /TiSi$_2$ /Si heterostructures[18], [34], [19], we first consider the case of a $\\mathbf {d}$ -vector that is parallel to the SOC vector $\\mathbf {w}$ .","The experimental results for CoSi$_2$ /TiSi$_2$ on a Si substrate indicate the presence of a dominant triplet superconducting state in CoSi$_2$ with $\\mathbf {d}$ -vector along $\\mathbf {w}$ .","Due to the SOC which is induced by the Si substrate, there will also be a finite but weak singlet component in the superconductor which we ignore for now.","The general case of a mixed parity superconductor will be addressed in subsection REF .","For side-by-side coupled junctions as the one illustrated in Fig.","REF (a), the tunneling self-energies are, $\\Sigma _0 &=& \\Sigma _0(\\omega ,k_{||}), \\\\\\mathbf {\\Sigma }_1 &=& - k_y \\Sigma _{1}(\\omega ,k_{||}) \\hat{x} = \\Sigma _1 \\tilde{\\mathbf {d}} ,$ where $\\tilde{\\mathbf {d}} = -k_y \\hat{x}$ and the normal and anomalous part of the Green's functions are, $\\hat{G}_{11} &=& \\frac{1}{D} \\left[ \\left( a_0 b_+ b_- - b_0\\Sigma _1^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}} \\right) \\sigma _0 \\right.","\\nonumber \\\\& &- \\epsilon _N \\left( b_+ b_- +\\Sigma _1^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}} \\right)(\\mathbf {w} \\cdot \\sigma ) \\nonumber \\\\&& \\left.","-2\\epsilon _n \\Sigma _1^2 (\\tilde{\\mathbf {d}}\\cdot \\mathbf {w}) \\tilde{\\mathbf {d}} \\cdot \\sigma \\right] \\\\\\hat{G}_{12} &=& \\hat{\\mathtt {G}}_{12}\\frac{1}{D} i \\sigma _y \\\\\\hat{\\mathtt {G}} _{12}&=& (2 \\epsilon _N \\xi _\\mathbf {k} {\\Sigma }_1 \\tilde{\\mathbf {d}} \\cdot \\mathbf {w} ) \\sigma _0 \\nonumber \\\\&& -2 \\epsilon _N^2 {\\Sigma }_1 (\\tilde{\\mathbf {d}} \\cdot \\mathbf {w}) \\mathbf {w} \\cdot \\sigma \\nonumber \\\\&&+ ( \\bar{\\omega }^2 -\\xi _{\\mathbf {k}}^2 +\\epsilon _N^2 -{\\Sigma }_1^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}} ) {\\Sigma }_1 \\tilde{\\mathbf {d}} \\cdot \\sigma \\nonumber \\\\&&-2 i \\bar{\\omega } \\epsilon _N {\\Sigma }_1 (\\tilde{\\mathbf {d}} \\times \\mathbf {w} ) \\cdot \\sigma \\\\D&=& (\\bar{\\omega }^2 -\\xi _{\\mathbf {k}}^2+\\epsilon _N^2-{\\Sigma }_1^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}} )^2 \\nonumber \\\\&& + 4 \\epsilon _N^2\\left[ {\\Sigma }_1^2 (\\tilde{\\mathbf {d}} \\cdot \\mathbf {w})^2 -\\bar{\\omega }^2\\right].$ Here $a_0 = \\bar{\\omega }-\\xi _\\mathbf {k}$ , $b_0 = \\bar{\\omega }+\\xi _\\mathbf {k}$ and $b_\\pm $ is $\\bar{\\omega }+\\xi _\\mathbf {k}\\pm \\epsilon _N$ .","Eq.","() contains an even parity ($\\propto k_y^2$ ), even frequency singlet term that arises due to a non-zero SOC and disappears in the limit of vanishing SOC.","Apart from an expected triplet pairing with $\\tilde{\\mathbf {d}} \\cdot \\sigma $ , a non-vanishing SOC brings about another kind of triplet pairing with $\\mathbf {w}\\cdot \\sigma $ structure.","Both these triplet components are even in frequency and have odd parity.","The novel pairing that arises due to the SOC is described by the last term in Eq.","() which is $\\propto \\left( \\tilde{\\mathbf {d}} \\times \\mathbf {w} \\right)\\cdot \\sigma $ and possesses a momentum dependence $\\propto k_x k_y$ .","This term describes triplet pairs that are have even parity and are odd in frequency.","This term is absent when the SOC vanishes.","This term also vanishes for junctions possessing top-bottom geometry as the one shown in Fig.","REF (b) as $\\tilde{\\mathbf {d}} || \\mathbf {w}$ in this geometry.","Thus, there are only singlet and triplet components with even frequency structure in this junction geometry where the singlet pairs are generated by the SOC.","Figure: Two different SN junction geometries.Figure: The local density of states at the interface for several values of normal state scattering rate.","The parameter cc is the cotangent of ss-wave phase shift c≡cotθ s c\\equiv \\cot \\theta _s is 1 for panel (a) and 0.010.01 for panel(b).Next, we include the effect of impurity scattering as described in Sec.",".","We consider a two-dimensional electron gas and a side-by-side coupled geometry.","For a two dimensional superconductor with an order parameter characterized by $\\mathbf {d}=\\hat{z}\\times \\mathbf {k}$ , the Fermi surface is fully gapped.","In that case, the self-energy components are $\\Sigma _0 = -\\Gamma _t \\frac{\\omega }{\\sqrt{\\Delta ^2-\\omega ^2}}, \\\\\\mathbf {\\Sigma }_1 = \\Gamma _t \\frac{\\Delta }{\\sqrt{\\Delta ^2-\\omega ^2}} \\frac{k_y}{k_F} \\hat{x}.$ The momentum integrated Green's function is $g_0 \\check{{1}}$ .","The disorder renormalized $\\tilde{\\omega }$ is determined by, $\\tilde{\\omega }&=& \\omega + \\Gamma _t \\frac{\\omega }{\\sqrt{\\Delta ^2 - \\omega ^2}} +\\frac{n_{imp}}{\\pi N_0} \\frac{\\mathtt {g}_0}{\\cot ^2 \\theta _s- \\mathtt {g}_0^2}$ where $\\theta _s\\equiv \\tan ^{-1} (\\pi N_0 V) $ is the $s$ -wave scattering phase shift.","Fig.","REF (a) shows the local DOS at the interface for weak scattering ($c=1$ ) and panel (b) shows the local DOS for $c=0.01$ , i.e.","strong scattering.","In contrast to isotropic $s$ -wave, the impurity scattering rapidly suppresses the induced superconductivity in this case.","While the superconductor is fully gapped, the induced superconductivity has low-energy states and does not develop a gap in the DOS.","Figure: (a) The interface DOS at the Fermi level as a function of SOC energy in the clean limit.","The interface DOS as a function of energy for several values of SOC energy in the clean limit (b) and with a scattering rate Γ N =Δ\\Gamma _N=\\Delta for c=1c=1."],["$\\mathbf {d}$ -vector {{formula:cf57e828-e844-443c-b04a-0e912ff69a93}}","Next, we consider a chiral $p$ -wave state with $\\mathbf {d} =( p_x + i p_y)\\hat{z}$ which is a complex, but unitary order parameter ($\\mathbf {d}\\times \\mathbf {d}^\\ast =0$ ).","In this case, the tunneling self-energies read, $\\Sigma _0 &=& \\Sigma _0(\\omega ,k_{||}), \\\\\\mathbf {\\Sigma }_1 &=& i k_y \\Sigma _{1}(\\omega ,k_{||}) \\hat{z}= i\\Sigma _1 \\tilde{\\mathbf {d}} ,$ with ${\\tilde{\\mathbf {d}}} \\in \\mathcal {R}$ and $\\tilde{\\mathbf {d}}\\perp \\mathbf {w}$ .","The anomalous part of the Green's function is obtained as, $\\hat{G}_{11} &=&\\left[ \\left( a_0 b_+ b_- - b_0\\Sigma _1^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}} \\right) \\sigma _0 \\right.","\\nonumber \\\\& & \\left.","- \\epsilon _N \\left( b_+ b_- +\\Sigma _1^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}} \\right)(\\mathbf {w} \\cdot \\sigma ) \\right]\\frac{1}{D} , \\\\ \\hat{G}_{12} &=& \\frac{ \\hat{\\mathtt {G}}_{12} }{D} i \\sigma _y.","\\\\\\hat{\\mathtt {G}}_{12} &=& i\\Sigma _1 ( \\bar{\\omega }^2 -\\xi _{\\mathbf {k}}^2 +\\epsilon _N^2 -{{\\Sigma }}_1^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}} ) {\\tilde{\\mathbf {d}}} \\cdot {\\sigma } \\nonumber \\\\& & +2 \\bar{\\omega } \\epsilon _N {\\Sigma }_1( {\\tilde{\\mathbf {d}}}\\times \\mathbf {w} ) \\cdot \\sigma \\\\D&=& (\\bar{\\omega }^2 -\\xi _{\\mathbf {k}}^2+\\epsilon _N^2-{{\\Sigma }_1}^2 \\tilde{\\mathbf {d}}\\cdot \\tilde{\\mathbf {d}})^2- 4 \\epsilon _N^2 \\bar{\\omega }^2.$ As $\\tilde{\\mathbf {d}} \\perp \\mathbf {w}$ , any $\\tilde{\\mathbf {d}}\\cdot \\mathbf {w}$ component be it singlet or triplet, vanishes.","This also ensures the presence of an odd frequency component for both kinds of junction geometries.","The momentum structure of the odd frequency component turns out to be $\\propto k_x k_y \\hat{x} + k_y^2 \\hat{y}$ therefore, the $\\Sigma _{imp1}$ self-energy contribution becomes finite, in contrast to the previous case.","A non-vanishing $\\Sigma _{imp1}$ self-energy, as it turns out, converts the induced pairing into non-unitary pairing.","(See the general expression for the Green's function in the appendix ).","The impurity self-energies are given by Eq.","(REF ) and ().","However, the SOC itself kills the induced pairing as can be inferred from Fig.","REF (a), where it is shown that the DOS at the Fermi level reaches the normal state value as the SOC strength increases beyond the pairing energy scale of the superconductor.","In Fig.","REF (b), the DOS at the interface is shown for energies below the superconducting gap in the case of weak SOC.","We are, however, interested in the regime $\\Delta \\ll \\epsilon _N \\ll E_F$ , and in this regime the proximity induced superconductivity does not survive.","The inclusion of impurity scattering does not change this conclusion but further diminishes the proximity induced superconductivity even for weak SOC as be inferred from Fig.","REF (c).","Figure: The interface DOS for a mixed parity and normal junction without any impurity scattering.","(a) The interface DOS for r=0.8r=0.8 and r=1.2r=1.2 with no SOC in the normal metal.","The tunneling energy scale Γ t =0.5Δ 0 \\Gamma _t=0.5\\Delta _0.","(b) The interface DOS states for r=0.8r=0.8 and r=1.2r=1.2 with (dashed lines) and without (solid lines)."],["Mixed parity state","The last case that we consider is that of a superconductor with mixed parity order parameters.","Such a state is possible, if the superconductor itself is under the influence of a SOC.","The normal and anomalous mean field Green's functions in this case are, $\\hat{G}_{11} &=& \\frac{1}{2} \\left[ \\hat{G}_+ + \\hat{G}_- \\right] + \\frac{1}{2} \\left[ \\hat{G}_+ - \\hat{G}_- \\right] \\mathbf {d}\\cdot \\sigma , \\\\\\hat{G}_{\\pm } &=& \\frac{\\omega +\\xi _\\pm }{\\omega ^2 - \\xi _\\pm ^2 -\\Delta ^2_\\pm }, \\\\\\hat{G}_{12} &=& \\frac{1}{2}\\left[ (\\hat{F}_+ + \\hat{F}_-) + (\\hat{F}_+ - \\hat{F}_-) \\mathbf {d}\\cdot \\sigma \\right] i \\sigma _y, \\\\\\hat{F}_\\pm &=& \\frac{\\Delta _\\pm }{\\omega ^2 - \\xi _\\pm ^2 -\\Delta ^2_\\pm }.$ Here $\\Delta _\\pm = (\\Delta _s \\pm \\Delta _t)$ and $\\xi _\\pm =\\xi _{\\mathbf {k}}\\pm \\epsilon _S$ , where $\\epsilon _S$ is the SOC energy scale in the superconductor and $\\Delta _{s/t} $ is the singlet/triplet component of the gap.","The normal ($\\hat{\\Sigma }_{11}$ ) and anomalous ($\\hat{\\Sigma }_{12}$ ) tunneling self-energies are $\\Sigma _0 + \\Sigma _{soc} \\tilde{\\mathbf {d}}\\cdot \\sigma $ and $ \\left( \\Sigma _s + \\Sigma _t \\tilde{\\mathbf {d}}\\cdot \\sigma \\right) i\\sigma _y$ , where $\\Sigma _{soc}$ modifies the SOC in the normal metal and $\\Sigma _{s/t}$ is the singlet/triplet component.","They are given by $\\Sigma _{0/soc} &=& -\\Gamma _t \\left( \\frac{\\omega }{Q_+} \\pm \\frac{\\omega }{Q_-}\\right), \\\\\\Sigma _{s/t} &=& \\Gamma _t \\left( \\frac{\\Delta _+}{Q_+} \\pm \\frac{\\Delta _-}{Q_-}\\right).$ where $Q_\\pm = 2\\sqrt{\\Delta _\\pm ^2-\\omega ^2}$ and it was assumed that the SOC splitting energy in the superconductor is $\\epsilon _S \\ll E_F$ .","Consequently, we ignore the differences between the DOS of the two helical bands, which is of the order of $\\epsilon _S/E_F$ .","The renormalized SOC term in the normal metal can be rewritten as, $\\tilde{\\epsilon }_N \\widetilde{\\mathbf {w}}\\cdot \\sigma &=& \\epsilon _N \\mathbf {w}\\cdot \\sigma +\\Sigma _{soc} \\tilde{\\mathbf {d}}\\cdot \\sigma ,\\\\\\tilde{\\epsilon }_N &=& \\sqrt{(\\epsilon _N \\mathbf {w}+\\Sigma _{soc} \\tilde{\\mathbf {d}})\\cdot (\\epsilon _N \\mathbf {w}+\\Sigma _{soc} \\tilde{\\mathbf {d}})},\\\\\\widetilde{\\mathbf {w}}&=& \\frac{\\epsilon _N \\mathbf {w} + \\Sigma _{soc} \\tilde{\\mathbf {d}}}{\\tilde{\\epsilon }_N}.$ The anomalous Green's function in the normal metal is, $\\hat{G}_{12} &\\propto &\\left[ A_0 \\sigma _0 + A_1 \\widetilde{\\mathbf {w}}\\cdot \\sigma + A_2 \\tilde{\\mathbf {d}}\\cdot \\sigma \\right.","\\nonumber \\\\& & \\left.","+ A_3 ( \\tilde{\\mathbf {d}} \\times \\widetilde{\\mathbf {w}} )\\cdot \\sigma \\right]i\\sigma _y,$ where the set $\\lbrace A_i\\rbrace ~(i=0,\\dots ,4)$ of coefficients is provided in appendix .","The first term in Eq.","(REF ) is the even parity, spin singlet term and the second and third terms are odd parity, spin triplet terms.","There is an odd frequency, spin triplet, even parity (OTE) term, which only exists in the presence of both a finite triplet component in the superconductor and a finite SOC in the normal metal.","Figure: (a) The interface DOS for singlet dominant (r=0.8r=0.8) SN junction for several representative values of impurity scattering rate.","(b) Variation of interface DOS with disorder for triplet dominant (r=1.2r=1.2) case.We parameterize the singlet and the triplet gaps in the superconductor as $\\Delta _s = \\Delta _0/\\sqrt{1+r^2}$ and $\\Delta _t = \\Delta _0 r/\\sqrt{1+r^2}$ , respectively[34].","The parameter $r$ is the ratio of triplet and singlet order parameters, where $r>1$ describes a triplet dominant regime while $r<1$ presents a singlet dominant regime.","In the extreme singlet ($r\\rightarrow 0$ ) and triplet ($r\\rightarrow \\infty $ ) limit, we recover the results presented in Sec.","REF and REF , respectively.","In the intermediate regime, the DOS at the interface shows signatures of two energy scales $\\Delta _s$ and $\\Delta _t$ .","For a vanishing SOC in the normal metal, the singlet dominant regime ($r<1$ ) ensues, where the DOS shows a gap $~\\mathrm {min}\\lbrace \\Delta _+, \\Delta _-\\rbrace $ , as shown Fig.","REF (a) and for the triplet dominant case ($r>1$ ), the DOS is similar to the pure triplet case discussed above (see Sec.","REF ), but the relevant energy scale is again $\\mathrm {min}\\lbrace \\Delta _+, \\Delta _-\\rbrace $ .","This behavior remains qualitatively same in the presence of a SOC in the normal metal.","However, the SOC leads to a particle-hole asymmetric DOS as illustrated in Fig.","REF (b).","This happens due to the coupling of singlet and triplet pairs in the normal metal due to the SOC.","This coupling generates a term linear in $\\xi $ in the denominator of the Green's function, with a prefactor $\\propto \\tilde{\\epsilon }_N \\Sigma _s \\Sigma _t \\tilde{\\mathbf {d}}\\cdot \\widetilde{\\mathbf {w}}$ , which causes the breakdown of particle-hole symmetry.","The effect of impurity scattering for the singlet/triplet dominant case is qualitatively the same as for the pure singlet/triplet case.","The particle-hole asymmetry in the DOS also gets smeared by the impurity scattering as depicted in Fig.","REF (a) for the singlet dominant ($r<1$ ) case and in the Fig.","REF (b) for the triplet dominant case ($r>1$ )."],["Summary & Conclusion","In this paper, we have studied the effect of Rashba SOC on the structure of proximity induced Cooper pairs in a normal metal connected to a superconductor.","We considered several kinds of gap symmetries.","The SOC in the normal metal leads to a singlet-triplet mixed state if the SN junction involves a singlet superconductor.","The strength of the triplet component in this case depends on the strength of the SOC and the low energy quasiparticle spectrum remains gapped and robust against disorder.","The SOC driven triplet state does not lead to any low energy states.","This is reminiscent of the proximity induced mixed parity states that have been reported for topological insulator and $s$ -wave superconducting junctions where the origin of the singlet-triplet mixing is the spin-momentum locking [35], [36], [37], [38], , [30], [32].","In the case of an SN junction involving a triplet superconductor, the broken spin-rotational symmetry can lead to a non-zero singlet component, but its presence depends on the spin-structure of the gap in the superconductor and the junction geometry.","We find that a singlet component is present whenever the off-diagonal self-energy ($\\mathbf {\\Sigma }_1$ ) has a component parallel to the SOC vector ($\\mathbf {w}$ ).","On the other hand, an off-diagonal self-energy that is perpendicular to the SOC vector get suppressed by the SOC very quickly.","The induced triplet component may have a different spin-structure compared to the superconductor, depending on the junction geometry.","The induced triplet pairs have sub-gap low energy states, however, the induced triplet component turns out to be fragile against impurity scattering.","The effect of disorder is similar to the effect of disorder on proximity induced $p$ -wave superconductivity on the surface of topological insulators[31].","In the SN junctions with two component superconductors that have both singlet and triplet components, the low energy behavior is determined by the dominant component.","However, the SOC induced coupling between the singlet and triplet components leads to a particle-hole asymmetric DOS.","The disorder suppresses this particle hole asymmetry.","One of the key conclusions of this work is the formation of odd frequency, spin triplet, even parity pairs in the normal metal segment of an SN junction with a triplet superconductor.","Such odd frequency component only arises in the presence of SOC in the normal metal.","The induced $\\mathbf {d}$ -vector of the odd frequency term is $\\propto (\\mathbf {\\Sigma }_1 \\times \\mathbf {w})$ , which is an even function of momentum.","Similar OTE induced superconductivity has been reported in an $s$ -wave junction with a topological insulator with gap modulation near the interface[29] or in the presence of an exchange field[30].","In contrast, the formation of OTE pairs that we find for a triplet superconductor SN junction does not require gap modulation near the interface or any exchange field.","We have obtained the full momentum-spin-energy structure of the OTE pairs.","The OTE pairs have momentum dependence, which makes them vulnerable to impurity scattering.","OTE pairs have also been reported in the normal metal junctions with triplet superconductors where the normal metal did not have the SOC[26].","These studies were performed using the Usadel equations with Nazarov-Tanaka boundary conditions[40], [41] Here, the approach is different.","We calculate the normal state Green function right at the interface, thus avoiding ambiguity with respect to possible boundary conditions.","In summary, we have investigated the effect of Rashba SOC on proximity induced superconductivity in the SN junctions consisting of conventional and unconventional superconductor.","Eq.","(REF ) and () are very general, and the structure of the induced superconductivity is applicable to many other systems, such as surface states of topological insulators or systems with Dresselhaus SOC.","We examine the robustness of the induced superconductivity against disorder, and find that the induced triplet superconductivity gets suppressed by it.","In contrast, the fully gapped $s$ -wave superconductivity remains robust against disorder.","We find that the OTE state is induced in the SN junctions with triplet superconductors, but it does not show any low energy signature.","The OTE pairs may gets suppressed weakly or strongly by the disorder depending on their momentum structure.","We show that the formation of the OTE pairs requires a triplet superconductor in the SN junction, SOC and a favorable geometry.","OTE pairs are not induced in every triplet superconductor - normal metal junction."],["Acknowledgments","The authors are grateful to Shao-Pin Chiu and Juhn-Jong Lin for useful discussions.","VM, YL and FCZ are partially supported by NSFC grant 11674278 and by the priority program of the Chinese Academy of Sciences grant No.","XDB28000000, and YL is also supported by the China Postdoctoral Science Foundation under grant No.","2020M670422 and by the Fundamental Research Funds for the Central Universities under grant No.","E2E44305.","SK is supported by the Ministry of Science and Technology of Taiwan through grant numbers MOST 106-2112-M-009-007-MY4 and 110-2112-M-A49-015, and by the Ministry of Education of Taiwan through the Higher Education Sprout Project and acknowledges support by Yushan Fellowship Program of the Ministry of Education, Taiwan."],["General triplet case","This appendix provides the details of the normal metal Green's function for the general triplet case discussed in Sec.","REF .","The $\\hat{\\mathbf {G}}_{11}$ component of the normal state Green's function $\\check{\\mathbb {G}}$ is given by, $\\hat{\\mathbf {G}}_{11} &= &\\frac{b_+ b_-}{\\mathbf {D}} \\left[ M_0 \\sigma _0 - M_1 \\mathbf {w}\\cdot \\sigma - M_2 \\mathbf {q}\\cdot \\sigma \\right.","\\nonumber \\\\&& \\left.","- M_3 \\mathbf {\\Sigma }_1\\cdot \\sigma - M_4 \\mathbf {\\Sigma }_1^\\ast \\cdot \\sigma \\right] \\\\\\mathbf {D} &=& M_0^2 - M_1^2 (\\mathbf {w}\\cdot \\mathbf {w}) -M_2^2 (\\mathbf {q}\\cdot \\mathbf {q}) \\nonumber \\\\&& - M_3^2 (\\mathbf {\\Sigma }_1\\cdot \\mathbf {\\Sigma }_1) -M_4^2 (\\mathbf {\\Sigma }_1^\\ast \\cdot \\mathbf {\\Sigma }_1^\\ast ) \\nonumber \\\\&& -2 M_1 M_2 \\mathbf {q}\\cdot \\mathbf {w} - 4\\epsilon _N M_1 \\mathbf {\\Sigma }_1 \\cdot \\mathbf {w} \\mathbf {\\Sigma }_1^\\ast \\cdot \\mathbf {w} \\nonumber \\\\& &- 2\\epsilon _N^2|\\mathbf {\\Sigma }_1|^2 \\mathbf {\\Sigma }_1 \\cdot \\mathbf {w} \\mathbf {\\Sigma }_1^\\ast \\cdot \\mathbf {w}.$ Here $b_\\pm = \\bar{\\omega }+\\xi _{\\mathbf {k}}\\pm \\tilde{\\epsilon }_N$ , $\\mathbf {q}=i \\mathbf {d}\\times \\mathbf {d}^\\ast $ , and the coefficients $M_{i}$ ($i=0,..,4$ ) are, $M_0 &=& a_0 b_+ b_- -b_0 |\\mathbf {\\Sigma }_1|^2 - \\epsilon _N \\mathbf {q} \\cdot \\mathbf {w}, \\\\M_1 &=& -\\epsilon _N b_+ b_- - \\epsilon _N |\\mathbf {\\Sigma }_1|^2, \\\\M_2 &=& -b_0, \\\\M_3 &=& \\epsilon _N \\mathbf {\\Sigma }_1^\\ast \\cdot \\mathbf {w}, \\\\M_4 &=& \\epsilon _N \\mathbf {\\Sigma }_1 \\cdot \\mathbf {w}.$ The anomalous component $\\hat{\\mathbf {G}}_{12}$ of $\\check{\\mathbb {G}}$ reads, $\\hat{\\mathbf {G}}_{12} &= & \\hat{\\mathtt {G}}_{12} \\frac{i \\sigma _y}{\\mathbf {D}},$ where $\\hat{\\mathtt {G}}_{12} &=&\\left[ C_0 \\sigma _0 + C_1 \\mathbf {w}\\cdot \\sigma +C_2 \\mathbf {\\Sigma }_1\\cdot \\sigma \\right.\\nonumber \\\\&& \\left.","+ C_3 \\mathbf {\\Sigma }_1^\\ast \\cdot \\sigma + C_4 \\left( \\mathbf {\\Sigma } \\times \\mathbf {w}\\right) \\sigma \\right],$ $C_0 &=& 2\\epsilon _N \\xi b_+ b_- \\mathbf {\\Sigma }_1 \\cdot \\mathbf {w} \\nonumber \\\\& & + 2\\epsilon _N b_0 (\\mathbf {\\Sigma }_1 \\times (\\mathbf {\\Sigma }_1 \\times \\mathbf {\\Sigma }_1^\\ast )) \\cdot \\mathbf {w},\\\\C_1 &=& -2b_+ b_- \\epsilon _N^2 \\mathbf {\\Sigma }_1 \\cdot \\mathbf {w} \\nonumber \\\\& & - \\epsilon _N^2 (\\mathbf {\\Sigma }_1 \\times (\\mathbf {\\Sigma }_1 \\times \\mathbf {\\Sigma }_1^\\ast )) \\cdot \\mathbf {w}, \\\\C_2 &=& b_+ b_- \\left( a_0 b_0 + \\epsilon _N^2 \\right) \\nonumber \\\\& & + \\epsilon _N^2 \\left( \\mathbf {\\Sigma }_1 \\times \\mathbf {w} \\right)\\cdot \\left( \\mathbf {\\Sigma }_1^\\ast \\times \\mathbf {w} \\right), \\\\C_3 &=& -b_+ b_- \\mathbf {\\Sigma }_1 \\cdot \\mathbf {\\Sigma }_1 \\nonumber \\\\& & - \\epsilon _N^2 \\left( \\mathbf {\\Sigma }_1 \\times \\mathbf {w} \\right)\\cdot \\left( \\mathbf {\\Sigma }_1 \\times \\mathbf {w} \\right), \\\\C_4 &=& -2i\\epsilon _N \\bar{\\omega } b_+ b_- + i \\epsilon _N^2 (\\mathbf {q}\\cdot \\mathbf {w}).$ Here $a_0=\\bar{\\omega }-\\xi _{\\mathbf {k}}$ and $b_0=\\bar{\\omega }+\\xi _{\\mathbf {k}}$ ."],["Mixed parity","Details for the normal and anomalous Green's functions the mixed parity state of Sec.","REF are provided in this appendix.","The normal Green's function is, $\\hat{\\mathbf {G}}_{11} &= & \\frac{b_+ b_-}{\\mathtt {D}} \\left[ L_0 \\sigma _0 - L_1 \\widetilde{\\mathbf {w}}\\cdot \\sigma -L_2 \\tilde{\\mathbf {d}} \\cdot \\sigma \\right], \\\\L_0 &=& a_0 b_+ b_- -b_0 \\left( \\Sigma _s^2 + \\Sigma _t^2 \\tilde{\\mathbf {d}} \\cdot \\tilde{\\mathbf {d}} \\right) \\nonumber \\\\& & +2\\tilde{\\epsilon }_N \\Sigma _s \\Sigma _t \\tilde{\\mathbf {d}} \\cdot \\widetilde{\\mathbf {w}}, \\\\L_1 &=& -\\tilde{\\epsilon }_N \\left( b_+ b_- - \\Sigma _s^2 + \\Sigma _t^2 \\tilde{\\mathbf {d}} \\cdot \\tilde{\\mathbf {d}} \\right), \\\\L_2 &=& -2\\left( b_0 \\Sigma _s \\Sigma _t - \\tilde{\\epsilon }_N \\Sigma _t^2 \\tilde{\\mathbf {d}} \\cdot \\widetilde{\\mathbf {w}} \\right),\\\\\\mathtt {D}&=& L_0^2 -L_1^2 - L_2^2 \\tilde{\\mathbf {d}} \\cdot \\tilde{\\mathbf {d}} - 2L_1 L_2 \\tilde{\\mathbf {d}} \\cdot \\widetilde{\\mathbf {w}} .$ The anomalous Green's function can be cast into the form $\\hat{\\mathbf {G}}_{12} &= & \\hat{\\mathtt {G}}_{12} \\frac{i \\sigma _y}{\\mathtt {D}},$ with $\\hat{\\mathtt {G}}_{12} &=& \\left[ A_0 \\sigma _0 + A_1 \\widetilde{\\mathbf {w}}\\cdot \\sigma + A_2 \\tilde{\\mathbf {d}}\\cdot \\sigma \\right.","\\nonumber \\\\&& \\left.","+ A_3 ( \\tilde{\\mathbf {d}} \\times \\widetilde{\\mathbf {w}} )\\cdot \\sigma \\right], \\\\A_0 &=& (a_0 b_0-\\tilde{\\epsilon }_N^2) \\Sigma _s - \\Sigma _s^3 + \\Sigma _s \\Sigma _t^2 \\tilde{\\mathbf {d}} \\cdot \\tilde{\\mathbf {d}} \\nonumber \\\\&& +2\\xi _{\\mathbf {k}} \\tilde{\\epsilon }_N \\tilde{\\mathbf {d}} \\cdot \\widetilde{\\mathbf {w}} \\\\A_1 &=& 2\\xi _{\\mathbf {k}} \\tilde{\\epsilon }_N \\Sigma _s -2\\tilde{\\epsilon }_N^2\\Sigma _t \\widetilde{\\mathbf {w}} \\cdot \\tilde{\\mathbf {d}} , \\\\A_2 &=& \\Sigma _t \\left( a_0 b_0 +\\tilde{\\epsilon }_N^2 +\\Sigma _s^2 - \\Sigma _t^2 \\tilde{\\mathbf {d}} \\cdot \\tilde{\\mathbf {d}} \\right)\\ \\\\A_3 &=& -2i \\tilde{\\epsilon }_N \\bar{\\omega } \\Sigma _t.$"]],"string":"[\n [\n \"Effects of spin orbit coupling on proximity induced superconductivity\"\n ],\n [\n \"Abstract We investigate the effect of spin orbit coupling on proximity induced superconductivity in a normal metal attached to a superconductor.\",\n \"Specifically, we consider a heterostructure where the presence of interfaces gives rise to a Rashba spin orbit coupling.\",\n \"The properties of the induced superconductivity in these systems are addressed within the tunneling Hamiltonian formalism.\",\n \"We find that the spin orbit coupling induces a mixture of singlet and triplet pairing and, under specific circumstances, an odd frequency, even parity, spin triplet pairs can arise.\",\n \"We also address the effect of impurity scattering on the induced pairs, and discuss our results in context of heterostructures consisting of materials with spin-momentum locking.\"\n ],\n [\n \"Introduction\",\n \"Hybrid nanostructures consisting of superconductors have been intensively studied, both experimentally and theoretically.\",\n \"Such hybrid proximity structures provide a platform for the realization of novel superconducting states in the vicinity of interfaces that connect superconductors to non-superconducting materials.\",\n \"The current heightened interest in these systems is also driven by the potential of these heterostructures to host Majorana fermions[1], [2], [3], [4].\",\n \"As these Majorana fermions obey non-Abelian braiding statistics they may serve as building blocks for fault-tolerant quantum computation [5], [6], [7], [8], [9].\",\n \"Superconductor and ferromagnetic (S/F) hybrid structures have been studied heavily[10], [11], [12], [13], [14], [15], with a recent focus on proximity structures with topological materials.\",\n \"In such proximity structures, the spin orbit coupling (SOC) plays an important role.\",\n \"It is usually induced by the breaking of inversion symmetry e.g.\",\n \"through an underlying substrate or the presence of an interface.\",\n \"In a superconductor, the SOC leads to a mixing of singlet and triplet pairing.\",\n \"In this two-component superconductivity either singlet or triplet pairing can be dominating[16], [17].\",\n \"Recent observation of triplet dominant two component superconductivity in CoSi$_2$ /TiSi$_2$ /Si heterostructures has confirmed the realization of dominant triplet pairing in these structures via a substrate induced SOC[18], [19].\",\n \"The role of SOC in the S/F nanostructures has been studied extensively [20], [21], [22], [23], [24], where the junctions involved $s$ -wave superconductors.\",\n \"These studies were carried out in the quasi-classical formalism in the diffusive limit[25].\",\n \"In the diffusive limit, the most dominant energy scale in the problem is the elastic impurity scattering i. e. the impurity scattering rate $\\\\tau ^{-1} \\\\gg \\\\Delta , \\\\tau ^{-1}_{sf}, \\\\tau ^{-1}_{in} $ , where $\\\\Delta $ is the superconducting gap, $\\\\tau ^{-1}_{sf/in}$ is the scattering rate from spin-flip/inelastic scattering.\",\n \"In such a regime, even frequency - even parity - spin singlet superconductor (EES) induces EES pairs in the diffusive normal metals, and even frequency - odd parity - spin triplet superconductors (EOT) leads to the formation of odd frequency - even parity - spin triplet pairs in the diffusive metal, as long as the interface in non-magnetic in nature[26].\",\n \"A non-magnetic interface prevents triplet to singlet conversion.\",\n \"Our main objective is to understand the properties of proximity-induced superconductivity in metals with sizable SOC.\",\n \"We will examine the stability of the proximity-induced superconductivity against weak disorder and analyze the emergence of odd frequency pairs.\",\n \"In this paper, we focus on the effect of SOC on induced superconductivity in a normal non-magnetic metal that is connected to an unconventional superconductor.\",\n \"For concreteness, we consider a Rashba SOC that is induced by the underlying substrate beneath the normal metal component.\",\n \"One reason for this particular choice of SOC interaction is that it can be generated and controlled by applying a gate voltage to the heterostructure.\",\n \"Here, we focus on the structure of the induced superconductivity in the normal metal connected to a triplet superconductor.\",\n \"In what follows, we adopt the tunneling Hamiltonian formalism[27], [28], [29], [30], [31], [32], [33].\",\n \"In the next section, we introduce the basic model and the theoretical methods.\",\n \"The subsequent section provides a discussion of our results.\",\n \"The final section summarizes the key qualitative conclusions.\",\n \"Figure: Schematic illustration of a superconductor - normal junction.\",\n \"The interface is along the yzyz plane.\"\n ],\n [\n \"Model & Formalism\",\n \"Figure REF shows a schematic diagram of the superconductor - normal (SN) junction.\",\n \"The interface is located at the $x=0$ plane.\",\n \"We consider a SOC that is induced by the substrate and take the $\\\\hat{z}$ axis to be parallel the normal to the substrate.\",\n \"The Hamiltonian for the normal component reads, $\\\\hat{H}_N &=& \\\\sum _{\\\\mathbf {k},\\\\sigma } c^\\\\dagger _{k\\\\sigma } \\\\left[ \\\\xi _\\\\mathbf {k} \\\\delta _{\\\\sigma \\\\sigma ^{\\\\prime }}+ \\\\hat{H} _{Rashba} \\\\right] c_{k\\\\sigma ^{\\\\prime }},$ where $c^\\\\dagger /c$ is the electron creation/annihilation operator, $\\\\xi _\\\\mathbf {k} $ is the electronic dispersion, $\\\\mathbf {k}$ denotes the momentum and $\\\\sigma $ represents the electron spin.\",\n \"We denote a $4\\\\times 4$ matrix in Nambu-spin space with $\\\\check{\\\\square }$ while $\\\\hat{\\\\square }$ indicates a $2\\\\times 2$ matrix in the spin space.\",\n \"The Rashba SOC term reads, $\\\\hat{\\\\mathcal {H}}_{Rashba} &=& -\\\\frac{\\\\alpha }{m} \\\\left( \\\\sigma \\\\times \\\\mathbf {k}\\\\right)\\\\cdot \\\\hat{z} = \\\\epsilon _N \\\\left( \\\\hat{z} \\\\times \\\\hat{\\\\mathbf {k}}\\\\right)\\\\cdot \\\\mathbf {\\\\sigma } , \\\\\\\\\\\\epsilon _N&=&\\\\frac{\\\\alpha |\\\\mathbf {k}|}{m}.$ Here, $\\\\alpha $ is the Rashba SOC coupling constant, $m$ is the effective mass and $\\\\sigma $ is the Pauli vector ($\\\\sigma _x,\\\\sigma _y,\\\\sigma _z$ ), where $\\\\sigma _{x/y/z}$ are the Pauli matrices in spin space.\",\n \"The two helical bands generated by this term have energies $\\\\xi _\\\\mathbf {k}\\\\pm \\\\epsilon _N$ .\",\n \"The Hamiltonian for the superconductor reads, $\\\\check{H}_{sc} &=& \\\\Psi ^\\\\dagger \\\\begin{pmatrix}\\\\xi _{\\\\mathbf {k}} & \\\\hat{\\\\Delta } \\\\\\\\\\\\hat{\\\\Delta }^\\\\dagger & -\\\\xi _{\\\\mathbf {k}}\\\\end{pmatrix} \\\\Psi $ where $\\\\xi _{\\\\mathbf {k}}$ is the dispersion in the superconductor, $\\\\Psi ^\\\\dagger =(a^\\\\dagger _{\\\\uparrow \\\\mathbf {k}},a^\\\\dagger _{\\\\downarrow \\\\mathbf {k}},a_{\\\\uparrow -\\\\mathbf {k}},a_{\\\\downarrow -\\\\mathbf {k}})$ , where $a^\\\\dagger /a$ is the creation/annihilation operator.\",\n \"The gap $\\\\hat{\\\\Delta }=\\\\Delta i\\\\sigma _y$ for the singlet case while for the triplet case $\\\\hat{\\\\Delta }=\\\\Delta \\\\mathbf {d}\\\\cdot \\\\sigma i \\\\sigma _y$ .\",\n \"Here, $\\\\mathbf {d}$ is the $d$ -vector of the triplet paring.\",\n \"The tunneling Hamiltonian is, $\\\\check{H}_{tunneling} = \\\\gamma \\\\Psi ^\\\\dagger \\\\check{\\\\tau }_3 \\\\Phi + h. c.$ where $\\\\check{\\\\tau }_3=\\\\mathrm {diag}(1,1,-1,1)$ is a matrix in Nambu spin space, $\\\\Phi $ is the analog of $\\\\Psi $ in the normal metal and $\\\\gamma $ is the tunneling matrix element.\",\n \"Here, we assume $\\\\gamma $ to be spin and momentum independent and take it to be real.\",\n \"The mean-field expression for the Green's function of the superconducting component is $\\\\check{G}_{sc} = \\\\frac{1}{\\\\omega ^2-\\\\xi ^2-|\\\\Delta |^2} \\\\begin{pmatrix}(\\\\omega +\\\\xi ) \\\\sigma _0 & \\\\hat{\\\\Delta } \\\\\\\\\\\\hat{\\\\Delta }^\\\\dagger & (\\\\omega -\\\\xi ) \\\\sigma _0\\\\end{pmatrix}.$ Here, $\\\\sigma _0$ is the $2\\\\times 2$ identity matrix in spin space and for notational convenience, we abbreviate $\\\\xi _{\\\\mathbf {k}}$ as $\\\\xi $ .\",\n \"The tunneling self energy for the normal side of the junction at the interface reads, $\\\\check{\\\\Sigma }_t(k_{||},\\\\omega ) &=& |\\\\gamma |^2 \\\\int \\\\frac{d k_\\\\perp }{2 \\\\pi } \\\\frac{1}{\\\\omega ^2-\\\\xi ^2-|\\\\Delta |^2} \\\\nonumber \\\\\\\\&\\\\times & \\\\begin{pmatrix}(\\\\omega +\\\\xi ) \\\\sigma _0 & -\\\\hat{\\\\Delta } \\\\\\\\-\\\\hat{\\\\Delta }^\\\\dagger & (\\\\omega -\\\\xi ) \\\\sigma _0\\\\end{pmatrix}.$ Here, $k_\\\\perp $ is the momentum component perpendicular to the SN interface and $k_{||}$ is the momentum parallel to it.\",\n \"The self-energy has the same general structure as the structure of the Green's function in the superconductor.\",\n \"The integration over the momentum component perpendicular to the interface may modify $\\\\mathbf {d}$ to $\\\\tilde{\\\\mathbf {d}}$ depending on the junction geometry.\",\n \"To study the robustness of the induced pairs against disorder, we consider point-like impurities randomly distributed in the normal metal.\",\n \"Within a self-consistent T-matrix approximation which incorporates all scattering processes from a single impurity site the impurity self energy contribution is, $\\\\check{\\\\Sigma }_{imp}( \\\\omega ) &=& n_{imp} \\\\check{\\\\tau }_3 {V}_{imp}\\\\left[ \\\\check{1}- \\\\check{\\\\mathbf {g}} \\\\check{\\\\tau }_3 {V}_{imp} \\\\right]^{-1},$ where $n_{imp}$ is the impurity concentration, $V_{imp}$ is the impurity potential, $\\\\check{1}$ is the $4\\\\times 4$ identity matrix and $\\\\check{\\\\mathbf {g}}$ is $\\\\check{\\\\mathbf {g}} &=& \\\\int _{\\\\mathbf {k}} \\\\check{\\\\mathbb {G}}.$ where $\\\\check{\\\\mathbb {G}}&=& \\\\frac{\\\\check{1}}{\\\\check{G}_0^{-1}-\\\\check{\\\\Sigma }_t( k_{||},\\\\omega )-\\\\check{\\\\Sigma }_{imp}(\\\\omega )}.$ Here $\\\\check{G}_0$ is the normal metal bare Green's function, and the impurity self-energy is calculated self-consistently.\"\n ],\n [\n \"SN junction with a singlet superconductor\",\n \"First, we consider a pure singlet superconductor attached to the normal metal component discussed above.\",\n \"In that case, $\\\\hat{\\\\Delta }=\\\\Delta i \\\\sigma _y$ and the tunneling self-energy is given by, $\\\\check{\\\\Sigma }_t(k_{||},\\\\omega ) = |\\\\gamma |^2 \\\\begin{pmatrix}(\\\\Sigma _0 + \\\\Sigma _3)\\\\sigma _0 & i \\\\sigma _y {\\\\Sigma }_1 \\\\\\\\-i \\\\sigma _y {\\\\Sigma }_1 & (\\\\Sigma _0 - \\\\Sigma _3)\\\\sigma _0.\\\\end{pmatrix}$ In the case of a momentum independent $\\\\Delta $ and for a particle-hole symmetric system, the tunneling self-energy becomes, $\\\\check{\\\\Sigma }_t(k_{||},\\\\omega ) = -\\\\Gamma _t \\\\frac{1}{\\\\sqrt{\\\\Delta ^2-\\\\omega ^2}} \\\\begin{pmatrix}\\\\omega \\\\sigma _0 & i \\\\sigma _y {\\\\Delta } \\\\\\\\-i \\\\sigma _y \\\\Delta &\\\\omega \\\\sigma _0\\\\end{pmatrix}.$ Here $\\\\Gamma _t\\\\equiv \\\\pi |\\\\gamma |^2 \\\\nu _s$ is the energy scale associated with the tunneling process, where $\\\\nu _s$ is the normal state density of states (DOS) of the superconductor at the Fermi level.\",\n \"Inclusion of particle-hole asymmetry gives finite $\\\\Sigma _3$ , that can be absorbed in the chemical potential.\",\n \"As it turns out, the presence of particle-hole asymmetry does not lead to significant qualitative difference.\",\n \"The normal metal Green's function in the clean limit is, $\\\\check{\\\\mathbb {G}} &=& \\\\begin{pmatrix}\\\\left[ \\\\bar{\\\\omega } - {\\\\xi } \\\\right] \\\\sigma _0 - \\\\epsilon _N (\\\\mathbf {w} \\\\cdot \\\\sigma ) & - i\\\\Sigma _1 \\\\sigma _y \\\\\\\\i \\\\Sigma _1 \\\\sigma _y & \\\\left[\\\\bar{\\\\omega } + {\\\\xi } \\\\right] \\\\sigma _0 - \\\\epsilon _N (\\\\mathbf {w} \\\\cdot \\\\sigma ^\\\\ast )\\\\end{pmatrix}^{-1}, \\\\nonumber \\\\\\\\\\\\check{\\\\mathbb {G}} &=& \\\\begin{pmatrix}\\\\hat{\\\\mathbb {G}}_{11} & \\\\hat{\\\\mathbb {G}}_{12} \\\\\\\\\\\\hat{\\\\mathbb {G}}_{21} & \\\\hat{\\\\mathbb {G}}_{22}.\\\\end{pmatrix}$ Here, $\\\\bar{\\\\omega }=\\\\omega -\\\\Sigma _0$ and $\\\\mathbf {w}=\\\\hat{z}\\\\times \\\\hat{\\\\mathbf {k}}$ .\",\n \"The components of $\\\\check{\\\\mathbb {G}} $ are, $\\\\hat{\\\\mathbb {G}}_{11} &=& \\\\frac{1}{2} \\\\left[ \\\\mathcal {R}_+ + \\\\mathcal {R}_- \\\\right] + \\\\frac{1}{2} \\\\left[ \\\\mathcal {R}_+ - \\\\mathcal {R}_- \\\\right] (\\\\mathbf {w} \\\\cdot \\\\sigma ), \\\\\\\\\\\\mathcal {R}_\\\\pm &=& \\\\frac{\\\\bar{\\\\omega } + (\\\\bar{\\\\xi } \\\\pm \\\\epsilon _N)}{\\\\bar{\\\\omega }^2-(\\\\bar{\\\\xi } \\\\pm \\\\epsilon _N)^2 -\\\\Sigma _1^2}, \\\\\\\\\\\\hat{\\\\mathbb {G}}_{12} & =& \\\\left(\\\\frac{1}{2} \\\\left[ \\\\frac{1}{D_+} + \\\\frac{1}{D_-}\\\\right] + \\\\frac{1}{2} \\\\left[ \\\\frac{1}{D_+} - \\\\frac{1}{D_-}\\\\right] \\\\mathbf {w} \\\\cdot \\\\sigma \\\\right), \\\\nonumber \\\\\\\\& \\\\times & ( i \\\\sigma _y \\\\Sigma _1) \\\\\\\\D_\\\\pm &=& \\\\bar{\\\\omega }^2-(\\\\bar{\\\\xi } \\\\pm \\\\epsilon _N)^2 -\\\\Sigma _1^2.$ The structure of the induced pairs can be obtained from the anomalous Green's function $\\\\hat{\\\\mathbb {G}}_{12} $ .\",\n \"The first term of $\\\\hat{\\\\mathbb {G}}_{12}$ in Eq.\",\n \"() is the spin-singlet, even parity and even frequency component.\",\n \"This is the conventional proximity effect for a singlet superconductor.\",\n \"The second term of $\\\\hat{\\\\mathbb {G}}_{12} $ is a spin-triplet, odd parity and even frequency term.\",\n \"This term is directly proportional to the strength of SOC.\",\n \"Due to finite SOC, spin rotational symmetry is broken which allows for a mixing of singlet and triplet terms.\",\n \"The $\\\\mathbf {d}$ -vector for triplet pairing is determined by the SOC vector $\\\\mathbf {w}$ .\",\n \"At this point, we can include the effect of impurity scattering.\",\n \"The impurity self-energy depends on the momentum integrated Green's function.\",\n \"Since $\\\\mathbf {w}$ is an odd function of momentum, the terms linear in $\\\\mathbf {w}$ vanish in the momentum integrated Green's function for a singlet superconductor.\",\n \"As a result, the momentum integrated Green's function is, $\\\\int _\\\\mathbf {\\\\mathbf {k}} \\\\check{\\\\mathbb {G}} &=& -\\\\pi N_0 \\\\frac{1}{\\\\sqrt{\\\\Sigma _1^2-\\\\bar{\\\\omega }^2}} \\\\begin{pmatrix}\\\\bar{\\\\omega } \\\\sigma _0 & \\\\Sigma _1 i \\\\sigma _y \\\\\\\\-i \\\\sigma _y \\\\Sigma _1 & \\\\bar{\\\\omega } \\\\sigma _0\\\\end{pmatrix},$ where $N_0$ is the normal metal DOS at the Fermi level.\",\n \"The impurity self-energy can be expressed as, $\\\\check{\\\\Sigma }_{imp} &=&\\\\begin{pmatrix}{\\\\Sigma }_{imp0} \\\\sigma _0 & i \\\\sigma _y \\\\Sigma _{imp1} \\\\\\\\-i \\\\sigma _y \\\\Sigma _{imp1} & {\\\\Sigma }_{imp0} \\\\sigma _0,\\\\end{pmatrix}, \\\\\\\\\\\\Sigma _{imp0} &=& n_{imp}\\\\frac{ g_0 V^2}{1- V^2 (g_0^2 - g_1^2)}, \\\\\\\\\\\\Sigma _{imp1} &=& n_{imp}\\\\frac{-g_1 V^2}{1- V^2 (g_0^2 - g_1^2)},$ where we have neglected the $\\\\Sigma _3$ component of the impurity self energy, which vanishes for a particle hole symmetric system.\",\n \"We can define impurity renormalized energy and induced off-diagonal self-energy as, $\\\\tilde{\\\\omega }&=& \\\\omega - \\\\Sigma _0(\\\\omega ) - \\\\Sigma _{imp0}(\\\\tilde{\\\\omega }), \\\\\\\\\\\\tilde{\\\\Sigma }_1 &=& \\\\Sigma _1(\\\\omega ) + \\\\Sigma _{imp1}(\\\\tilde{\\\\omega }).$ Here, the fully dressed Green's function is used to calculate $\\\\tilde{\\\\omega }$ , $\\\\tilde{\\\\Sigma }_1$ self-consistently.\",\n \"The equations for renormalized $\\\\tilde{\\\\Sigma }_1$ and $\\\\tilde{\\\\omega }$ are identical as those for an $s$ -wave superconductors with nonmagnetic impurities.\",\n \"Now, we rewrite the impurity and tunneling dressed Green's function $\\\\check{\\\\mathbb {G}}$ by replacing $\\\\bar{\\\\omega }$ and $\\\\Sigma _1$ with $\\\\tilde{\\\\omega }$ and $\\\\tilde{\\\\Sigma }_1$ , respectively.\",\n \"After some straightforward algebra, we get $\\\\tilde{\\\\omega }/\\\\tilde{\\\\Sigma }_1 = \\\\bar{\\\\omega }/\\\\Sigma _1$ .\",\n \"This ensures that the induced pairs remain robust against nonmagnetic disorder.\",\n \"Next, we consider the interface DOS given by $N(\\\\omega )=N_0 \\\\mathrm {Im} \\\\frac{\\\\tilde{\\\\omega }}{\\\\sqrt{\\\\tilde{\\\\Sigma }_1^2-\\\\tilde{\\\\omega }^2}}.$ Fig.\",\n \"REF (a) shows the interface DOS for several values of the tunneling energy scale $\\\\Gamma _t$ .\",\n \"In the weak tunneling regime ($\\\\Gamma _t\\\\ll \\\\Delta $ ), the effective gap in the DOS is determined by $\\\\Gamma _t^2/\\\\Delta $ , and in the strong tunneling regime ($\\\\Gamma _t\\\\gg \\\\Delta $ ), the effective gap becomes identical to the size of the gap of the superconductor.\",\n \"The sub-dominant triplet component does not induce any low energy sub-gap states, instead the low energy DOS is mainly controlled by the singlet order parameter, which is an isotropic s-wave in the present case.\",\n \"The effect of impurity scattering is depicted in Fig.\",\n \"REF (b), which shows no change in the DOS with increasing impurity scattering rate $\\\\Gamma _{imp}\\\\equiv n_{imp}\\\\pi N_0 V^2/(1+\\\\pi ^2 V^2 N_0^2) $ .\",\n \"Figure: Proximity induced gap in the normal metal.\",\n \"(a) Variation of normalized DOS with strength of tunneling energy scale Γ t \\\\Gamma _t.\",\n \"(b) Normalized density of states at the interface as a function of energy for several values of impurity scattering rate.\"\n ],\n [\n \"SN junction with a triplet superconductor\",\n \"In a triplet superconductor the order parameter is $\\\\hat{\\\\Delta }=\\\\Delta \\\\mathbf {d}\\\\cdot \\\\sigma i\\\\sigma _y$ , where the $\\\\mathbf {d}$ -vector describes the pair structure in spin space.\",\n \"Here, we restrict ourselves to unitary pairing ($\\\\mathbf {d}\\\\times \\\\mathbf {d}^\\\\ast =0$ ) in the superconducting component of the heterojunction.\",\n \"In this case, we find for the tunneling self-energy, $\\\\check{\\\\Sigma }_t(k_{||},\\\\omega ) = |\\\\gamma |^2 \\\\begin{pmatrix}\\\\Sigma _0 \\\\sigma _0 & {\\\\Sigma }_1 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\sigma i \\\\sigma _y \\\\\\\\-i \\\\sigma _y {\\\\Sigma }_1 \\\\tilde{\\\\mathbf {d}} ^\\\\ast \\\\cdot \\\\sigma & \\\\Sigma _0 \\\\sigma _0\\\\end{pmatrix},$ where the components are given by, $\\\\Sigma _0 & = & |\\\\gamma |^2 \\\\int \\\\frac{d k_\\\\perp }{2 \\\\pi } \\\\frac{\\\\omega }{\\\\omega ^2-\\\\xi ^2-\\\\Delta ^2 | \\\\mathbf {d}|^2}, \\\\\\\\\\\\mathbf {\\\\Sigma }_1 & = & |\\\\gamma |^2 \\\\int \\\\frac{d k_\\\\perp }{2 \\\\pi } \\\\frac{-\\\\Delta \\\\mathbf {d}}{\\\\omega ^2-\\\\xi ^2-\\\\Delta ^2 | \\\\mathbf {d}|^2 } = \\\\Sigma _1 \\\\tilde{\\\\mathbf {d}}.$ As before in the singlet case, we ignore the $\\\\Sigma _3$ self-energy.\",\n \"The $\\\\mathbf {d}$ -vector is an odd function of momentum and $\\\\mathbf {\\\\Sigma }_1$ is also an odd function of the momentum parallel to the interface ($k_{||}$ ), and it does not depend on the momentum component normal to the interface ($k_\\\\perp $ ).\",\n \"The Green's function in the normal metal can be written as, $\\\\check{\\\\mathbb {G}} &=& \\\\begin{pmatrix}\\\\left[ \\\\bar{\\\\omega } - {\\\\xi } \\\\right] \\\\sigma _0 - \\\\epsilon _N (\\\\mathbf {w} \\\\cdot \\\\sigma ) & - {\\\\Sigma }_1 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\sigma i \\\\sigma _y \\\\\\\\i \\\\sigma _y {\\\\Sigma }_1 \\\\tilde{\\\\mathbf {d}} ^\\\\ast \\\\cdot \\\\sigma & \\\\left[\\\\bar{\\\\omega } + {\\\\xi } \\\\right] \\\\sigma _0 - \\\\epsilon _N (\\\\mathbf {w} \\\\cdot \\\\sigma ^\\\\ast )\\\\end{pmatrix}^{-1} \\\\nonumber \\\\\\\\&=& \\\\begin{pmatrix}\\\\hat{\\\\mathbb {G}}_{11} & \\\\hat{\\\\mathbb {G}}_{12} \\\\\\\\\\\\hat{\\\\mathbb {G}}_{21} & \\\\hat{\\\\mathbb {G}}_{22}.\\\\end{pmatrix}$ The general structure of the off-diagonal Green's function has the form, $\\\\hat{\\\\mathbb {G}}_{12} &\\\\propto & A_0 \\\\sigma _0 + A_1 \\\\mathbf {w}\\\\cdot \\\\sigma + A_2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\sigma + A_3 \\\\tilde{\\\\mathbf {d}} ^\\\\ast \\\\cdot \\\\sigma \\\\nonumber \\\\\\\\& & + A_4 \\\\left( \\\\tilde{\\\\mathbf {d}} \\\\times \\\\mathbf {w} \\\\right) \\\\cdot \\\\sigma ,$ where the specific values of the set $\\\\lbrace A_i\\\\rbrace $ (i=0,...4) will depend on geometry of the junction and the specifics of the $\\\\mathbf {d}$ -vector which in turn determine $\\\\tilde{\\\\mathbf {d}}$ .\",\n \"In general, $\\\\tilde{\\\\mathbf {d}} \\\\times \\\\tilde{\\\\mathbf {d}} ^\\\\ast $ may not vanish despite unitary pairing in the superconductor ($\\\\mathbf {d}\\\\times \\\\mathbf {d}^\\\\ast =0$ ).\",\n \"Eq.\",\n \"(REF ) has singlet and triplet terms, but due to the anisotropic nature of the gap in the superconductor details of geometry and pairing structure is essential to understand the structure of induced pairs in the normal metal.\",\n \"To proceed, we will consider a few pertinent cases.\"\n ],\n [\n \"$\\\\mathbf {d}$ -vector {{formula:cd423bf1-efa1-4737-9b31-b95841a50102}}\",\n \"Motivated by the recent experiments on CoSi$_2$ /TiSi$_2$ /Si heterostructures[18], [34], [19], we first consider the case of a $\\\\mathbf {d}$ -vector that is parallel to the SOC vector $\\\\mathbf {w}$ .\",\n \"The experimental results for CoSi$_2$ /TiSi$_2$ on a Si substrate indicate the presence of a dominant triplet superconducting state in CoSi$_2$ with $\\\\mathbf {d}$ -vector along $\\\\mathbf {w}$ .\",\n \"Due to the SOC which is induced by the Si substrate, there will also be a finite but weak singlet component in the superconductor which we ignore for now.\",\n \"The general case of a mixed parity superconductor will be addressed in subsection REF .\",\n \"For side-by-side coupled junctions as the one illustrated in Fig.\",\n \"REF (a), the tunneling self-energies are, $\\\\Sigma _0 &=& \\\\Sigma _0(\\\\omega ,k_{||}), \\\\\\\\\\\\mathbf {\\\\Sigma }_1 &=& - k_y \\\\Sigma _{1}(\\\\omega ,k_{||}) \\\\hat{x} = \\\\Sigma _1 \\\\tilde{\\\\mathbf {d}} ,$ where $\\\\tilde{\\\\mathbf {d}} = -k_y \\\\hat{x}$ and the normal and anomalous part of the Green's functions are, $\\\\hat{G}_{11} &=& \\\\frac{1}{D} \\\\left[ \\\\left( a_0 b_+ b_- - b_0\\\\Sigma _1^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\right) \\\\sigma _0 \\\\right.\",\n \"\\\\nonumber \\\\\\\\& &- \\\\epsilon _N \\\\left( b_+ b_- +\\\\Sigma _1^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\right)(\\\\mathbf {w} \\\\cdot \\\\sigma ) \\\\nonumber \\\\\\\\&& \\\\left.\",\n \"-2\\\\epsilon _n \\\\Sigma _1^2 (\\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\mathbf {w}) \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\sigma \\\\right] \\\\\\\\\\\\hat{G}_{12} &=& \\\\hat{\\\\mathtt {G}}_{12}\\\\frac{1}{D} i \\\\sigma _y \\\\\\\\\\\\hat{\\\\mathtt {G}} _{12}&=& (2 \\\\epsilon _N \\\\xi _\\\\mathbf {k} {\\\\Sigma }_1 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\mathbf {w} ) \\\\sigma _0 \\\\nonumber \\\\\\\\&& -2 \\\\epsilon _N^2 {\\\\Sigma }_1 (\\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\mathbf {w}) \\\\mathbf {w} \\\\cdot \\\\sigma \\\\nonumber \\\\\\\\&&+ ( \\\\bar{\\\\omega }^2 -\\\\xi _{\\\\mathbf {k}}^2 +\\\\epsilon _N^2 -{\\\\Sigma }_1^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}} ) {\\\\Sigma }_1 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\sigma \\\\nonumber \\\\\\\\&&-2 i \\\\bar{\\\\omega } \\\\epsilon _N {\\\\Sigma }_1 (\\\\tilde{\\\\mathbf {d}} \\\\times \\\\mathbf {w} ) \\\\cdot \\\\sigma \\\\\\\\D&=& (\\\\bar{\\\\omega }^2 -\\\\xi _{\\\\mathbf {k}}^2+\\\\epsilon _N^2-{\\\\Sigma }_1^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}} )^2 \\\\nonumber \\\\\\\\&& + 4 \\\\epsilon _N^2\\\\left[ {\\\\Sigma }_1^2 (\\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\mathbf {w})^2 -\\\\bar{\\\\omega }^2\\\\right].$ Here $a_0 = \\\\bar{\\\\omega }-\\\\xi _\\\\mathbf {k}$ , $b_0 = \\\\bar{\\\\omega }+\\\\xi _\\\\mathbf {k}$ and $b_\\\\pm $ is $\\\\bar{\\\\omega }+\\\\xi _\\\\mathbf {k}\\\\pm \\\\epsilon _N$ .\",\n \"Eq.\",\n \"() contains an even parity ($\\\\propto k_y^2$ ), even frequency singlet term that arises due to a non-zero SOC and disappears in the limit of vanishing SOC.\",\n \"Apart from an expected triplet pairing with $\\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\sigma $ , a non-vanishing SOC brings about another kind of triplet pairing with $\\\\mathbf {w}\\\\cdot \\\\sigma $ structure.\",\n \"Both these triplet components are even in frequency and have odd parity.\",\n \"The novel pairing that arises due to the SOC is described by the last term in Eq.\",\n \"() which is $\\\\propto \\\\left( \\\\tilde{\\\\mathbf {d}} \\\\times \\\\mathbf {w} \\\\right)\\\\cdot \\\\sigma $ and possesses a momentum dependence $\\\\propto k_x k_y$ .\",\n \"This term describes triplet pairs that are have even parity and are odd in frequency.\",\n \"This term is absent when the SOC vanishes.\",\n \"This term also vanishes for junctions possessing top-bottom geometry as the one shown in Fig.\",\n \"REF (b) as $\\\\tilde{\\\\mathbf {d}} || \\\\mathbf {w}$ in this geometry.\",\n \"Thus, there are only singlet and triplet components with even frequency structure in this junction geometry where the singlet pairs are generated by the SOC.\",\n \"Figure: Two different SN junction geometries.Figure: The local density of states at the interface for several values of normal state scattering rate.\",\n \"The parameter cc is the cotangent of ss-wave phase shift c≡cotθ s c\\\\equiv \\\\cot \\\\theta _s is 1 for panel (a) and 0.010.01 for panel(b).Next, we include the effect of impurity scattering as described in Sec.\",\n \".\",\n \"We consider a two-dimensional electron gas and a side-by-side coupled geometry.\",\n \"For a two dimensional superconductor with an order parameter characterized by $\\\\mathbf {d}=\\\\hat{z}\\\\times \\\\mathbf {k}$ , the Fermi surface is fully gapped.\",\n \"In that case, the self-energy components are $\\\\Sigma _0 = -\\\\Gamma _t \\\\frac{\\\\omega }{\\\\sqrt{\\\\Delta ^2-\\\\omega ^2}}, \\\\\\\\\\\\mathbf {\\\\Sigma }_1 = \\\\Gamma _t \\\\frac{\\\\Delta }{\\\\sqrt{\\\\Delta ^2-\\\\omega ^2}} \\\\frac{k_y}{k_F} \\\\hat{x}.$ The momentum integrated Green's function is $g_0 \\\\check{{1}}$ .\",\n \"The disorder renormalized $\\\\tilde{\\\\omega }$ is determined by, $\\\\tilde{\\\\omega }&=& \\\\omega + \\\\Gamma _t \\\\frac{\\\\omega }{\\\\sqrt{\\\\Delta ^2 - \\\\omega ^2}} +\\\\frac{n_{imp}}{\\\\pi N_0} \\\\frac{\\\\mathtt {g}_0}{\\\\cot ^2 \\\\theta _s- \\\\mathtt {g}_0^2}$ where $\\\\theta _s\\\\equiv \\\\tan ^{-1} (\\\\pi N_0 V) $ is the $s$ -wave scattering phase shift.\",\n \"Fig.\",\n \"REF (a) shows the local DOS at the interface for weak scattering ($c=1$ ) and panel (b) shows the local DOS for $c=0.01$ , i.e.\",\n \"strong scattering.\",\n \"In contrast to isotropic $s$ -wave, the impurity scattering rapidly suppresses the induced superconductivity in this case.\",\n \"While the superconductor is fully gapped, the induced superconductivity has low-energy states and does not develop a gap in the DOS.\",\n \"Figure: (a) The interface DOS at the Fermi level as a function of SOC energy in the clean limit.\",\n \"The interface DOS as a function of energy for several values of SOC energy in the clean limit (b) and with a scattering rate Γ N =Δ\\\\Gamma _N=\\\\Delta for c=1c=1.\"\n ],\n [\n \"$\\\\mathbf {d}$ -vector {{formula:cf57e828-e844-443c-b04a-0e912ff69a93}}\",\n \"Next, we consider a chiral $p$ -wave state with $\\\\mathbf {d} =( p_x + i p_y)\\\\hat{z}$ which is a complex, but unitary order parameter ($\\\\mathbf {d}\\\\times \\\\mathbf {d}^\\\\ast =0$ ).\",\n \"In this case, the tunneling self-energies read, $\\\\Sigma _0 &=& \\\\Sigma _0(\\\\omega ,k_{||}), \\\\\\\\\\\\mathbf {\\\\Sigma }_1 &=& i k_y \\\\Sigma _{1}(\\\\omega ,k_{||}) \\\\hat{z}= i\\\\Sigma _1 \\\\tilde{\\\\mathbf {d}} ,$ with ${\\\\tilde{\\\\mathbf {d}}} \\\\in \\\\mathcal {R}$ and $\\\\tilde{\\\\mathbf {d}}\\\\perp \\\\mathbf {w}$ .\",\n \"The anomalous part of the Green's function is obtained as, $\\\\hat{G}_{11} &=&\\\\left[ \\\\left( a_0 b_+ b_- - b_0\\\\Sigma _1^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\right) \\\\sigma _0 \\\\right.\",\n \"\\\\nonumber \\\\\\\\& & \\\\left.\",\n \"- \\\\epsilon _N \\\\left( b_+ b_- +\\\\Sigma _1^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\right)(\\\\mathbf {w} \\\\cdot \\\\sigma ) \\\\right]\\\\frac{1}{D} , \\\\\\\\ \\\\hat{G}_{12} &=& \\\\frac{ \\\\hat{\\\\mathtt {G}}_{12} }{D} i \\\\sigma _y.\",\n \"\\\\\\\\\\\\hat{\\\\mathtt {G}}_{12} &=& i\\\\Sigma _1 ( \\\\bar{\\\\omega }^2 -\\\\xi _{\\\\mathbf {k}}^2 +\\\\epsilon _N^2 -{{\\\\Sigma }}_1^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}} ) {\\\\tilde{\\\\mathbf {d}}} \\\\cdot {\\\\sigma } \\\\nonumber \\\\\\\\& & +2 \\\\bar{\\\\omega } \\\\epsilon _N {\\\\Sigma }_1( {\\\\tilde{\\\\mathbf {d}}}\\\\times \\\\mathbf {w} ) \\\\cdot \\\\sigma \\\\\\\\D&=& (\\\\bar{\\\\omega }^2 -\\\\xi _{\\\\mathbf {k}}^2+\\\\epsilon _N^2-{{\\\\Sigma }_1}^2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\tilde{\\\\mathbf {d}})^2- 4 \\\\epsilon _N^2 \\\\bar{\\\\omega }^2.$ As $\\\\tilde{\\\\mathbf {d}} \\\\perp \\\\mathbf {w}$ , any $\\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\mathbf {w}$ component be it singlet or triplet, vanishes.\",\n \"This also ensures the presence of an odd frequency component for both kinds of junction geometries.\",\n \"The momentum structure of the odd frequency component turns out to be $\\\\propto k_x k_y \\\\hat{x} + k_y^2 \\\\hat{y}$ therefore, the $\\\\Sigma _{imp1}$ self-energy contribution becomes finite, in contrast to the previous case.\",\n \"A non-vanishing $\\\\Sigma _{imp1}$ self-energy, as it turns out, converts the induced pairing into non-unitary pairing.\",\n \"(See the general expression for the Green's function in the appendix ).\",\n \"The impurity self-energies are given by Eq.\",\n \"(REF ) and ().\",\n \"However, the SOC itself kills the induced pairing as can be inferred from Fig.\",\n \"REF (a), where it is shown that the DOS at the Fermi level reaches the normal state value as the SOC strength increases beyond the pairing energy scale of the superconductor.\",\n \"In Fig.\",\n \"REF (b), the DOS at the interface is shown for energies below the superconducting gap in the case of weak SOC.\",\n \"We are, however, interested in the regime $\\\\Delta \\\\ll \\\\epsilon _N \\\\ll E_F$ , and in this regime the proximity induced superconductivity does not survive.\",\n \"The inclusion of impurity scattering does not change this conclusion but further diminishes the proximity induced superconductivity even for weak SOC as be inferred from Fig.\",\n \"REF (c).\",\n \"Figure: The interface DOS for a mixed parity and normal junction without any impurity scattering.\",\n \"(a) The interface DOS for r=0.8r=0.8 and r=1.2r=1.2 with no SOC in the normal metal.\",\n \"The tunneling energy scale Γ t =0.5Δ 0 \\\\Gamma _t=0.5\\\\Delta _0.\",\n \"(b) The interface DOS states for r=0.8r=0.8 and r=1.2r=1.2 with (dashed lines) and without (solid lines).\"\n ],\n [\n \"Mixed parity state\",\n \"The last case that we consider is that of a superconductor with mixed parity order parameters.\",\n \"Such a state is possible, if the superconductor itself is under the influence of a SOC.\",\n \"The normal and anomalous mean field Green's functions in this case are, $\\\\hat{G}_{11} &=& \\\\frac{1}{2} \\\\left[ \\\\hat{G}_+ + \\\\hat{G}_- \\\\right] + \\\\frac{1}{2} \\\\left[ \\\\hat{G}_+ - \\\\hat{G}_- \\\\right] \\\\mathbf {d}\\\\cdot \\\\sigma , \\\\\\\\\\\\hat{G}_{\\\\pm } &=& \\\\frac{\\\\omega +\\\\xi _\\\\pm }{\\\\omega ^2 - \\\\xi _\\\\pm ^2 -\\\\Delta ^2_\\\\pm }, \\\\\\\\\\\\hat{G}_{12} &=& \\\\frac{1}{2}\\\\left[ (\\\\hat{F}_+ + \\\\hat{F}_-) + (\\\\hat{F}_+ - \\\\hat{F}_-) \\\\mathbf {d}\\\\cdot \\\\sigma \\\\right] i \\\\sigma _y, \\\\\\\\\\\\hat{F}_\\\\pm &=& \\\\frac{\\\\Delta _\\\\pm }{\\\\omega ^2 - \\\\xi _\\\\pm ^2 -\\\\Delta ^2_\\\\pm }.$ Here $\\\\Delta _\\\\pm = (\\\\Delta _s \\\\pm \\\\Delta _t)$ and $\\\\xi _\\\\pm =\\\\xi _{\\\\mathbf {k}}\\\\pm \\\\epsilon _S$ , where $\\\\epsilon _S$ is the SOC energy scale in the superconductor and $\\\\Delta _{s/t} $ is the singlet/triplet component of the gap.\",\n \"The normal ($\\\\hat{\\\\Sigma }_{11}$ ) and anomalous ($\\\\hat{\\\\Sigma }_{12}$ ) tunneling self-energies are $\\\\Sigma _0 + \\\\Sigma _{soc} \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\sigma $ and $ \\\\left( \\\\Sigma _s + \\\\Sigma _t \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\sigma \\\\right) i\\\\sigma _y$ , where $\\\\Sigma _{soc}$ modifies the SOC in the normal metal and $\\\\Sigma _{s/t}$ is the singlet/triplet component.\",\n \"They are given by $\\\\Sigma _{0/soc} &=& -\\\\Gamma _t \\\\left( \\\\frac{\\\\omega }{Q_+} \\\\pm \\\\frac{\\\\omega }{Q_-}\\\\right), \\\\\\\\\\\\Sigma _{s/t} &=& \\\\Gamma _t \\\\left( \\\\frac{\\\\Delta _+}{Q_+} \\\\pm \\\\frac{\\\\Delta _-}{Q_-}\\\\right).$ where $Q_\\\\pm = 2\\\\sqrt{\\\\Delta _\\\\pm ^2-\\\\omega ^2}$ and it was assumed that the SOC splitting energy in the superconductor is $\\\\epsilon _S \\\\ll E_F$ .\",\n \"Consequently, we ignore the differences between the DOS of the two helical bands, which is of the order of $\\\\epsilon _S/E_F$ .\",\n \"The renormalized SOC term in the normal metal can be rewritten as, $\\\\tilde{\\\\epsilon }_N \\\\widetilde{\\\\mathbf {w}}\\\\cdot \\\\sigma &=& \\\\epsilon _N \\\\mathbf {w}\\\\cdot \\\\sigma +\\\\Sigma _{soc} \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\sigma ,\\\\\\\\\\\\tilde{\\\\epsilon }_N &=& \\\\sqrt{(\\\\epsilon _N \\\\mathbf {w}+\\\\Sigma _{soc} \\\\tilde{\\\\mathbf {d}})\\\\cdot (\\\\epsilon _N \\\\mathbf {w}+\\\\Sigma _{soc} \\\\tilde{\\\\mathbf {d}})},\\\\\\\\\\\\widetilde{\\\\mathbf {w}}&=& \\\\frac{\\\\epsilon _N \\\\mathbf {w} + \\\\Sigma _{soc} \\\\tilde{\\\\mathbf {d}}}{\\\\tilde{\\\\epsilon }_N}.$ The anomalous Green's function in the normal metal is, $\\\\hat{G}_{12} &\\\\propto &\\\\left[ A_0 \\\\sigma _0 + A_1 \\\\widetilde{\\\\mathbf {w}}\\\\cdot \\\\sigma + A_2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\sigma \\\\right.\",\n \"\\\\nonumber \\\\\\\\& & \\\\left.\",\n \"+ A_3 ( \\\\tilde{\\\\mathbf {d}} \\\\times \\\\widetilde{\\\\mathbf {w}} )\\\\cdot \\\\sigma \\\\right]i\\\\sigma _y,$ where the set $\\\\lbrace A_i\\\\rbrace ~(i=0,\\\\dots ,4)$ of coefficients is provided in appendix .\",\n \"The first term in Eq.\",\n \"(REF ) is the even parity, spin singlet term and the second and third terms are odd parity, spin triplet terms.\",\n \"There is an odd frequency, spin triplet, even parity (OTE) term, which only exists in the presence of both a finite triplet component in the superconductor and a finite SOC in the normal metal.\",\n \"Figure: (a) The interface DOS for singlet dominant (r=0.8r=0.8) SN junction for several representative values of impurity scattering rate.\",\n \"(b) Variation of interface DOS with disorder for triplet dominant (r=1.2r=1.2) case.We parameterize the singlet and the triplet gaps in the superconductor as $\\\\Delta _s = \\\\Delta _0/\\\\sqrt{1+r^2}$ and $\\\\Delta _t = \\\\Delta _0 r/\\\\sqrt{1+r^2}$ , respectively[34].\",\n \"The parameter $r$ is the ratio of triplet and singlet order parameters, where $r>1$ describes a triplet dominant regime while $r<1$ presents a singlet dominant regime.\",\n \"In the extreme singlet ($r\\\\rightarrow 0$ ) and triplet ($r\\\\rightarrow \\\\infty $ ) limit, we recover the results presented in Sec.\",\n \"REF and REF , respectively.\",\n \"In the intermediate regime, the DOS at the interface shows signatures of two energy scales $\\\\Delta _s$ and $\\\\Delta _t$ .\",\n \"For a vanishing SOC in the normal metal, the singlet dominant regime ($r<1$ ) ensues, where the DOS shows a gap $~\\\\mathrm {min}\\\\lbrace \\\\Delta _+, \\\\Delta _-\\\\rbrace $ , as shown Fig.\",\n \"REF (a) and for the triplet dominant case ($r>1$ ), the DOS is similar to the pure triplet case discussed above (see Sec.\",\n \"REF ), but the relevant energy scale is again $\\\\mathrm {min}\\\\lbrace \\\\Delta _+, \\\\Delta _-\\\\rbrace $ .\",\n \"This behavior remains qualitatively same in the presence of a SOC in the normal metal.\",\n \"However, the SOC leads to a particle-hole asymmetric DOS as illustrated in Fig.\",\n \"REF (b).\",\n \"This happens due to the coupling of singlet and triplet pairs in the normal metal due to the SOC.\",\n \"This coupling generates a term linear in $\\\\xi $ in the denominator of the Green's function, with a prefactor $\\\\propto \\\\tilde{\\\\epsilon }_N \\\\Sigma _s \\\\Sigma _t \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\widetilde{\\\\mathbf {w}}$ , which causes the breakdown of particle-hole symmetry.\",\n \"The effect of impurity scattering for the singlet/triplet dominant case is qualitatively the same as for the pure singlet/triplet case.\",\n \"The particle-hole asymmetry in the DOS also gets smeared by the impurity scattering as depicted in Fig.\",\n \"REF (a) for the singlet dominant ($r<1$ ) case and in the Fig.\",\n \"REF (b) for the triplet dominant case ($r>1$ ).\"\n ],\n [\n \"Summary & Conclusion\",\n \"In this paper, we have studied the effect of Rashba SOC on the structure of proximity induced Cooper pairs in a normal metal connected to a superconductor.\",\n \"We considered several kinds of gap symmetries.\",\n \"The SOC in the normal metal leads to a singlet-triplet mixed state if the SN junction involves a singlet superconductor.\",\n \"The strength of the triplet component in this case depends on the strength of the SOC and the low energy quasiparticle spectrum remains gapped and robust against disorder.\",\n \"The SOC driven triplet state does not lead to any low energy states.\",\n \"This is reminiscent of the proximity induced mixed parity states that have been reported for topological insulator and $s$ -wave superconducting junctions where the origin of the singlet-triplet mixing is the spin-momentum locking [35], [36], [37], [38], , [30], [32].\",\n \"In the case of an SN junction involving a triplet superconductor, the broken spin-rotational symmetry can lead to a non-zero singlet component, but its presence depends on the spin-structure of the gap in the superconductor and the junction geometry.\",\n \"We find that a singlet component is present whenever the off-diagonal self-energy ($\\\\mathbf {\\\\Sigma }_1$ ) has a component parallel to the SOC vector ($\\\\mathbf {w}$ ).\",\n \"On the other hand, an off-diagonal self-energy that is perpendicular to the SOC vector get suppressed by the SOC very quickly.\",\n \"The induced triplet component may have a different spin-structure compared to the superconductor, depending on the junction geometry.\",\n \"The induced triplet pairs have sub-gap low energy states, however, the induced triplet component turns out to be fragile against impurity scattering.\",\n \"The effect of disorder is similar to the effect of disorder on proximity induced $p$ -wave superconductivity on the surface of topological insulators[31].\",\n \"In the SN junctions with two component superconductors that have both singlet and triplet components, the low energy behavior is determined by the dominant component.\",\n \"However, the SOC induced coupling between the singlet and triplet components leads to a particle-hole asymmetric DOS.\",\n \"The disorder suppresses this particle hole asymmetry.\",\n \"One of the key conclusions of this work is the formation of odd frequency, spin triplet, even parity pairs in the normal metal segment of an SN junction with a triplet superconductor.\",\n \"Such odd frequency component only arises in the presence of SOC in the normal metal.\",\n \"The induced $\\\\mathbf {d}$ -vector of the odd frequency term is $\\\\propto (\\\\mathbf {\\\\Sigma }_1 \\\\times \\\\mathbf {w})$ , which is an even function of momentum.\",\n \"Similar OTE induced superconductivity has been reported in an $s$ -wave junction with a topological insulator with gap modulation near the interface[29] or in the presence of an exchange field[30].\",\n \"In contrast, the formation of OTE pairs that we find for a triplet superconductor SN junction does not require gap modulation near the interface or any exchange field.\",\n \"We have obtained the full momentum-spin-energy structure of the OTE pairs.\",\n \"The OTE pairs have momentum dependence, which makes them vulnerable to impurity scattering.\",\n \"OTE pairs have also been reported in the normal metal junctions with triplet superconductors where the normal metal did not have the SOC[26].\",\n \"These studies were performed using the Usadel equations with Nazarov-Tanaka boundary conditions[40], [41] Here, the approach is different.\",\n \"We calculate the normal state Green function right at the interface, thus avoiding ambiguity with respect to possible boundary conditions.\",\n \"In summary, we have investigated the effect of Rashba SOC on proximity induced superconductivity in the SN junctions consisting of conventional and unconventional superconductor.\",\n \"Eq.\",\n \"(REF ) and () are very general, and the structure of the induced superconductivity is applicable to many other systems, such as surface states of topological insulators or systems with Dresselhaus SOC.\",\n \"We examine the robustness of the induced superconductivity against disorder, and find that the induced triplet superconductivity gets suppressed by it.\",\n \"In contrast, the fully gapped $s$ -wave superconductivity remains robust against disorder.\",\n \"We find that the OTE state is induced in the SN junctions with triplet superconductors, but it does not show any low energy signature.\",\n \"The OTE pairs may gets suppressed weakly or strongly by the disorder depending on their momentum structure.\",\n \"We show that the formation of the OTE pairs requires a triplet superconductor in the SN junction, SOC and a favorable geometry.\",\n \"OTE pairs are not induced in every triplet superconductor - normal metal junction.\"\n ],\n [\n \"Acknowledgments\",\n \"The authors are grateful to Shao-Pin Chiu and Juhn-Jong Lin for useful discussions.\",\n \"VM, YL and FCZ are partially supported by NSFC grant 11674278 and by the priority program of the Chinese Academy of Sciences grant No.\",\n \"XDB28000000, and YL is also supported by the China Postdoctoral Science Foundation under grant No.\",\n \"2020M670422 and by the Fundamental Research Funds for the Central Universities under grant No.\",\n \"E2E44305.\",\n \"SK is supported by the Ministry of Science and Technology of Taiwan through grant numbers MOST 106-2112-M-009-007-MY4 and 110-2112-M-A49-015, and by the Ministry of Education of Taiwan through the Higher Education Sprout Project and acknowledges support by Yushan Fellowship Program of the Ministry of Education, Taiwan.\"\n ],\n [\n \"General triplet case\",\n \"This appendix provides the details of the normal metal Green's function for the general triplet case discussed in Sec.\",\n \"REF .\",\n \"The $\\\\hat{\\\\mathbf {G}}_{11}$ component of the normal state Green's function $\\\\check{\\\\mathbb {G}}$ is given by, $\\\\hat{\\\\mathbf {G}}_{11} &= &\\\\frac{b_+ b_-}{\\\\mathbf {D}} \\\\left[ M_0 \\\\sigma _0 - M_1 \\\\mathbf {w}\\\\cdot \\\\sigma - M_2 \\\\mathbf {q}\\\\cdot \\\\sigma \\\\right.\",\n \"\\\\nonumber \\\\\\\\&& \\\\left.\",\n \"- M_3 \\\\mathbf {\\\\Sigma }_1\\\\cdot \\\\sigma - M_4 \\\\mathbf {\\\\Sigma }_1^\\\\ast \\\\cdot \\\\sigma \\\\right] \\\\\\\\\\\\mathbf {D} &=& M_0^2 - M_1^2 (\\\\mathbf {w}\\\\cdot \\\\mathbf {w}) -M_2^2 (\\\\mathbf {q}\\\\cdot \\\\mathbf {q}) \\\\nonumber \\\\\\\\&& - M_3^2 (\\\\mathbf {\\\\Sigma }_1\\\\cdot \\\\mathbf {\\\\Sigma }_1) -M_4^2 (\\\\mathbf {\\\\Sigma }_1^\\\\ast \\\\cdot \\\\mathbf {\\\\Sigma }_1^\\\\ast ) \\\\nonumber \\\\\\\\&& -2 M_1 M_2 \\\\mathbf {q}\\\\cdot \\\\mathbf {w} - 4\\\\epsilon _N M_1 \\\\mathbf {\\\\Sigma }_1 \\\\cdot \\\\mathbf {w} \\\\mathbf {\\\\Sigma }_1^\\\\ast \\\\cdot \\\\mathbf {w} \\\\nonumber \\\\\\\\& &- 2\\\\epsilon _N^2|\\\\mathbf {\\\\Sigma }_1|^2 \\\\mathbf {\\\\Sigma }_1 \\\\cdot \\\\mathbf {w} \\\\mathbf {\\\\Sigma }_1^\\\\ast \\\\cdot \\\\mathbf {w}.$ Here $b_\\\\pm = \\\\bar{\\\\omega }+\\\\xi _{\\\\mathbf {k}}\\\\pm \\\\tilde{\\\\epsilon }_N$ , $\\\\mathbf {q}=i \\\\mathbf {d}\\\\times \\\\mathbf {d}^\\\\ast $ , and the coefficients $M_{i}$ ($i=0,..,4$ ) are, $M_0 &=& a_0 b_+ b_- -b_0 |\\\\mathbf {\\\\Sigma }_1|^2 - \\\\epsilon _N \\\\mathbf {q} \\\\cdot \\\\mathbf {w}, \\\\\\\\M_1 &=& -\\\\epsilon _N b_+ b_- - \\\\epsilon _N |\\\\mathbf {\\\\Sigma }_1|^2, \\\\\\\\M_2 &=& -b_0, \\\\\\\\M_3 &=& \\\\epsilon _N \\\\mathbf {\\\\Sigma }_1^\\\\ast \\\\cdot \\\\mathbf {w}, \\\\\\\\M_4 &=& \\\\epsilon _N \\\\mathbf {\\\\Sigma }_1 \\\\cdot \\\\mathbf {w}.$ The anomalous component $\\\\hat{\\\\mathbf {G}}_{12}$ of $\\\\check{\\\\mathbb {G}}$ reads, $\\\\hat{\\\\mathbf {G}}_{12} &= & \\\\hat{\\\\mathtt {G}}_{12} \\\\frac{i \\\\sigma _y}{\\\\mathbf {D}},$ where $\\\\hat{\\\\mathtt {G}}_{12} &=&\\\\left[ C_0 \\\\sigma _0 + C_1 \\\\mathbf {w}\\\\cdot \\\\sigma +C_2 \\\\mathbf {\\\\Sigma }_1\\\\cdot \\\\sigma \\\\right.\\\\nonumber \\\\\\\\&& \\\\left.\",\n \"+ C_3 \\\\mathbf {\\\\Sigma }_1^\\\\ast \\\\cdot \\\\sigma + C_4 \\\\left( \\\\mathbf {\\\\Sigma } \\\\times \\\\mathbf {w}\\\\right) \\\\sigma \\\\right],$ $C_0 &=& 2\\\\epsilon _N \\\\xi b_+ b_- \\\\mathbf {\\\\Sigma }_1 \\\\cdot \\\\mathbf {w} \\\\nonumber \\\\\\\\& & + 2\\\\epsilon _N b_0 (\\\\mathbf {\\\\Sigma }_1 \\\\times (\\\\mathbf {\\\\Sigma }_1 \\\\times \\\\mathbf {\\\\Sigma }_1^\\\\ast )) \\\\cdot \\\\mathbf {w},\\\\\\\\C_1 &=& -2b_+ b_- \\\\epsilon _N^2 \\\\mathbf {\\\\Sigma }_1 \\\\cdot \\\\mathbf {w} \\\\nonumber \\\\\\\\& & - \\\\epsilon _N^2 (\\\\mathbf {\\\\Sigma }_1 \\\\times (\\\\mathbf {\\\\Sigma }_1 \\\\times \\\\mathbf {\\\\Sigma }_1^\\\\ast )) \\\\cdot \\\\mathbf {w}, \\\\\\\\C_2 &=& b_+ b_- \\\\left( a_0 b_0 + \\\\epsilon _N^2 \\\\right) \\\\nonumber \\\\\\\\& & + \\\\epsilon _N^2 \\\\left( \\\\mathbf {\\\\Sigma }_1 \\\\times \\\\mathbf {w} \\\\right)\\\\cdot \\\\left( \\\\mathbf {\\\\Sigma }_1^\\\\ast \\\\times \\\\mathbf {w} \\\\right), \\\\\\\\C_3 &=& -b_+ b_- \\\\mathbf {\\\\Sigma }_1 \\\\cdot \\\\mathbf {\\\\Sigma }_1 \\\\nonumber \\\\\\\\& & - \\\\epsilon _N^2 \\\\left( \\\\mathbf {\\\\Sigma }_1 \\\\times \\\\mathbf {w} \\\\right)\\\\cdot \\\\left( \\\\mathbf {\\\\Sigma }_1 \\\\times \\\\mathbf {w} \\\\right), \\\\\\\\C_4 &=& -2i\\\\epsilon _N \\\\bar{\\\\omega } b_+ b_- + i \\\\epsilon _N^2 (\\\\mathbf {q}\\\\cdot \\\\mathbf {w}).$ Here $a_0=\\\\bar{\\\\omega }-\\\\xi _{\\\\mathbf {k}}$ and $b_0=\\\\bar{\\\\omega }+\\\\xi _{\\\\mathbf {k}}$ .\"\n ],\n [\n \"Mixed parity\",\n \"Details for the normal and anomalous Green's functions the mixed parity state of Sec.\",\n \"REF are provided in this appendix.\",\n \"The normal Green's function is, $\\\\hat{\\\\mathbf {G}}_{11} &= & \\\\frac{b_+ b_-}{\\\\mathtt {D}} \\\\left[ L_0 \\\\sigma _0 - L_1 \\\\widetilde{\\\\mathbf {w}}\\\\cdot \\\\sigma -L_2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\sigma \\\\right], \\\\\\\\L_0 &=& a_0 b_+ b_- -b_0 \\\\left( \\\\Sigma _s^2 + \\\\Sigma _t^2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\right) \\\\nonumber \\\\\\\\& & +2\\\\tilde{\\\\epsilon }_N \\\\Sigma _s \\\\Sigma _t \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\widetilde{\\\\mathbf {w}}, \\\\\\\\L_1 &=& -\\\\tilde{\\\\epsilon }_N \\\\left( b_+ b_- - \\\\Sigma _s^2 + \\\\Sigma _t^2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\right), \\\\\\\\L_2 &=& -2\\\\left( b_0 \\\\Sigma _s \\\\Sigma _t - \\\\tilde{\\\\epsilon }_N \\\\Sigma _t^2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\widetilde{\\\\mathbf {w}} \\\\right),\\\\\\\\\\\\mathtt {D}&=& L_0^2 -L_1^2 - L_2^2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\tilde{\\\\mathbf {d}} - 2L_1 L_2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\widetilde{\\\\mathbf {w}} .$ The anomalous Green's function can be cast into the form $\\\\hat{\\\\mathbf {G}}_{12} &= & \\\\hat{\\\\mathtt {G}}_{12} \\\\frac{i \\\\sigma _y}{\\\\mathtt {D}},$ with $\\\\hat{\\\\mathtt {G}}_{12} &=& \\\\left[ A_0 \\\\sigma _0 + A_1 \\\\widetilde{\\\\mathbf {w}}\\\\cdot \\\\sigma + A_2 \\\\tilde{\\\\mathbf {d}}\\\\cdot \\\\sigma \\\\right.\",\n \"\\\\nonumber \\\\\\\\&& \\\\left.\",\n \"+ A_3 ( \\\\tilde{\\\\mathbf {d}} \\\\times \\\\widetilde{\\\\mathbf {w}} )\\\\cdot \\\\sigma \\\\right], \\\\\\\\A_0 &=& (a_0 b_0-\\\\tilde{\\\\epsilon }_N^2) \\\\Sigma _s - \\\\Sigma _s^3 + \\\\Sigma _s \\\\Sigma _t^2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\nonumber \\\\\\\\&& +2\\\\xi _{\\\\mathbf {k}} \\\\tilde{\\\\epsilon }_N \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\widetilde{\\\\mathbf {w}} \\\\\\\\A_1 &=& 2\\\\xi _{\\\\mathbf {k}} \\\\tilde{\\\\epsilon }_N \\\\Sigma _s -2\\\\tilde{\\\\epsilon }_N^2\\\\Sigma _t \\\\widetilde{\\\\mathbf {w}} \\\\cdot \\\\tilde{\\\\mathbf {d}} , \\\\\\\\A_2 &=& \\\\Sigma _t \\\\left( a_0 b_0 +\\\\tilde{\\\\epsilon }_N^2 +\\\\Sigma _s^2 - \\\\Sigma _t^2 \\\\tilde{\\\\mathbf {d}} \\\\cdot \\\\tilde{\\\\mathbf {d}} \\\\right)\\\\ \\\\\\\\A_3 &=& -2i \\\\tilde{\\\\epsilon }_N \\\\bar{\\\\omega } \\\\Sigma _t.$\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05550"}}},{"rowIdx":89,"cells":{"text":{"kind":"list like","value":[["A Weyl's law for black holes"],["Abstract We discuss a Weyl's law for the quasi-normal modes of black holes that recovers the structural features of the standard Weyl's law for the eigenvalues of the Laplacian in compact regions.","Specifically, the asymptotics of the counting function $N(\\omega)$ of quasi-normal modes of $(d+1)$-dimensional black holes follows a power-law $N(\\omega)\\sim \\mathrm{Vol}_d^{\\mathrm{eff}}\\omega^d$, with $\\mathrm{Vol}_d^{\\mathrm{eff}}$ an effective volume determined by the light-trapping and decay properties of the black hole geometry.","Closed forms are presented for the Schwarzschild black hole and a quasi-normal mode Weyl's law is proposed for generic black holes.","As an application, such Weyl's law could provide a probe into the effective dimensionality of spacetime and the relevant resonant scales of actual astrophysical black holes, upon the counting of sufficiently many overtones in the observed ringdown signal of binary black hole mergers."],["Spectrum asymptotics, degrees of freedom counting and geometry","The goal of the present note is to present the basic elements of a Weyl's law proposed for black hole (BH) quasi-normal modes (QNMs).","Such a law provides an estimate of the number of QNMs, inside a circle of a given radius in the frequency-complex plane, in terms of the asymptotics of high-frequency QNM overtones and a particular scale defined by the geometry of the black hole spacetime.","It directly extends to the BH QNM setting the standard Weyl's law for the eigenvalues of the Laplacian of a compact Riemannian manifold.","Weyl's law [1], [2] relates three distinct notions: the number of states of a physical system contained in a given spatial domain $D$ , the high-frequency asymptotics of the spectrum of the operator controlling its dynamics, and the geometry of the region $D$ .","The study of this triple relation has proved to be a far-reaching catalyst for key developments both in physics and mathematics.","On the physics side, it lays e.g.","at the core of the developments in radiation theory aiming at understanding the black body radiation and, therefore, it is ultimately interwoven with the birth of quantum theory (see e.g.","[3], [4]).","On the mathematical side, it dwells in the more general setting of inverse problems in spectral geometry that have unveiled a rich set of correspondences between the spectral properties of geometric differential operators defined on a (Riemanninan) manifold, namely the Laplacian, and the properties of families of geodesics on such manifolds [3], [5], [6].","Specifically, Weyl's law for the Laplacian stands as an archetype of such results in spectral geometry with direct implications in physics.","For concreteness, let us consider the asymptotics of the homogenous scalar Helmholtz operator on a compact domain $D$ $(\\Delta + k_n^2)\\phi _n = 0 \\ ,$ subject to homogeneous (Dirichlet, Neumann or Robin) boundary conditions on its boundary $\\partial D$ .","From the compacity of $D$ the eigenvalues $k_n^2$ form a discrete set and, denoting by $N(k)$ the number of eigenvalues $k_n^2$ below a certain wave-number $k$ , the following large-$k_n$ asymptotic relation holds $N(k) \\sim \\mathrm {Vol}_d(D) k^d + o(k^{d-1}) \\ \\ , \\ \\ (k\\rightarrow \\infty )$ where $\\mathrm {Vol}_d(D)$ is the volume of the compact domain $D$ and $d$ is the dimension of $D$ , namely $d=\\mathrm {dim}(D)$ .","This is the Weyl's law.","Crucially this `universal' asymptotic leading term does not depend on the shape of $D$ , the boundary $\\partial D$ only entering at the subleading term in the asymptotic expansion (see e.g.","[7]).","Similar Weyl's law relations hold for other fields (e.g.","the electromagnetic field) and differential operators [6].","In ref.","[8] a Weyl's law was proposed for BH QNMs.","QNMs encode the linear response of a resonator under an external perturbation, corresponding to exponentially decaying oscillating solutions to an appropriate wave equation under outgoing boundary conditions in a (generally) non-compact domain $D$ (cf.","[9], [10], [11], [12] for reviews of QNMs in the gravitational seeting and [13] for optical nanoresonators).","Specifically in the BH case, QNM complex frequencies $\\omega _n$ (the real part providing the QNM oscillation frequency and the imaginary part the inverse of its time-decay scale) encode invariant information on the BH stationary background.","A crucial feature of QNM frequencies, in spite of the non-compactness of $D$ , is that they form a discrete set.","The number of frequencies $\\omega _n$ in a given circle of the complex plane can therefore be counted and the question of a QNM version of Weyl's law can meaningfully be posed.","Consequently, QNM Weyl's laws have been the subject of extensive research in the literature (see e.g.","[14], [15], [16].","However, the situation for QNMs is more subtle than in the standard self-adjoint case (REF ) and a Weyl's law expression analogous to (REF ) is not generic beyond the one-dimensional case and for compact support (or very fastly decaying) scattering potentials [17], [18].","In this setting it seems quite remarkable that asymptotically flat (spherically symmetric) BH QNMs do admit [8] a Weyl-like's law of the standard functional form in (REF ) $N(\\omega ) \\sim \\frac{1}{c^d}\\mathrm {Vol}_d^{\\mathrm {eff}} \\omega ^d + o(^{d-1}) \\ \\ , \\ \\ (\\omega \\rightarrow \\infty )$ where $N(\\omega )$ amounts for the number of (complex) QNM frequencies $\\omega _n$ such that $|\\omega _n|\\le \\omega $ , where $\\omega >0$ , and $\\mathrm {Vol}_d^{\\mathrm {eff}}$ is an $d$ -dimension effective volume determined by the BH geometry (the appropriate power of the light speed $c$ has been introduced for dimensional reasons).","Note that, given the non-compact support of the BH geometry, the factor $\\mathrm {Vol}_d^{\\mathrm {eff}}$ does not admit the direct interpretation of a compact “cavity” volume in (REF ).","It is tempting, however, to seek interpreting $\\mathrm {Vol}_d^{\\mathrm {eff}}$ as providing a length/time (cubic) scale where the resonance phenomenon takes place.","Upon this assumption, one can invert the logic in reading BH QNM Weyl's law (REF ), as compared to the standard Weyl's law (REF ).","Indeed, whereas in the latter the dimension $d$ and the cavity's volume $\\mathrm {Vol}_d(D)$ are known Or the interest lies only on the density of states $N(k)/\\mathrm {Vol}_d(D)$ , as in the black body radiation problem.","In either case, the cavity's volume $\\mathrm {Vol}_d(D)$ is not the quantity of interest in itself, just playing the (key) role of being the only trace of the cavity's geometry and therefore guaranteeing the universality of the Weyl's law (actually the original motivation for Weyl's proof).","a priori, so the interest is focused on determining the number of states $N(k)$ , the former can be used to probe: i) The (effective) spacetime dimension $(d+1)$ .","ii) The dynamical scale for the BH resonant phenomenon, under the assumption that $N(\\omega )$ can be independently (for instance, observationally) accessed.","This possibility of using the proposed BH QNM Weyl's law as a probe into spacetime features defines our interest in this QNM asymptotic problem.","The article is organized as follows.","In section we review the Weyl's law in the classical setting of the eigenvalues of the Laplacian in a compact region.","In section we discuss the Weyl's law for QNMs associated with the scattering (in non-compact domains) by compact support potentials, separating the special one-dimensional case from the general discussion in higher dimensions.","In section we discuss the application to the BH case, focusing on the Schwarzschild BH case and then proposing a QNM Weyl's law for generic BHs.","Finally, section presents the conclusions and some perspectives."],["Weyl's law: space(-time) dimension and length scales","We present here a brief account of the elements of Weyl's law relevant in our later discussion on BH QNMs, with a focus on the volume factor and the power-law dependence on the dimension.","In a first step we review the Weyl's in the classical self-adjoint case and, in a second stage, we comment on the known results in the QNM (scattering resonances) setting."],["Weyl's law in closed manifolds","For concreteness, we focus here on the case of a compact manifold without boundaries [5].","This case contains all the specific elements we aim at addressing in the QNM case, providing a simplified setting but adequate setting.","The same leading-order result is found for compact manifolds with boundary, the differences related to the boundary conditions only appearing at the subleading-order, that we will not address here.","Given a closed (compact without boundary) Riemannian manifold $(M, g_{ab})$ of dimension $d$ with Levi-Civita connection $\\nabla _a$ , we consider the eigenvalue problem associated with scalar Laplace-Beltrami operator $\\Delta _g = \\nabla ^a\\nabla _a$ , namely $-\\Delta _g\\phi _n = \\lambda _n\\phi _n \\ ,$ eigenvalues $\\lambda _n$ are real and non-negative.","Given $\\lambda \\in \\mathbb {R}^+$ we define the eigenvalue counting function $N(\\lambda )$ as $N(\\lambda ) = \\hbox{number of } \\lambda _n \\hbox{ such that } \\lambda _n\\le \\lambda \\ .$ The Weyl's law states that, in the large-$\\lambda $ asymptotic limit $N(\\lambda ) \\sim C_d \\mathrm {Vol}_d(M) \\lambda ^{\\frac{d}{2}} \\ \\ , \\ \\ (\\lambda \\rightarrow \\infty )$ where $\\mathrm {Vol}_d(M)$ is the ($d-$ dimensional) volume of $M$ and $C_d$ are universal constants only depending on the dimension $d$ $C_d = \\frac{\\mathrm {Vol}_d(B^d_1)}{(2\\pi )^d} \\ ,$ with $\\mathrm {Vol}_d(B^d_1)$ the Euclidean volume of the unit $d$ -dimensional ball $B^d_1$ .","To get a taste and for later comparison, we give some examples for the lowest dimensions: i) Case $d=1$: for the ball $B^1_1$ , namely the $[-1,1]$ interval $\\mathrm {Vol}_1(B^1_1) = 2 \\ \\ ,$ and therefore, denoting $\\mathrm {Vol}_1 = L$ , we get $N(\\lambda ) \\sim \\frac{2}{2\\pi }L \\lambda ^{\\frac{1}{2}} =\\Big (\\frac{L}{\\pi }\\Big ) \\lambda ^{\\frac{1}{2}}\\ \\ , \\ \\ (\\lambda \\rightarrow \\infty ) \\ .$ ii) Case $d=2$: for the ball $B^2_1$ , the unit circle in the plane $\\mathrm {Vol}_2(B^2_1) = \\pi \\ \\ ,$ and denoting $\\mathrm {Vol}_2(M) = A$ , namely the area, we get $N(\\lambda ) \\sim \\frac{\\pi }{(2\\pi )^2} A \\; \\lambda = \\Big (\\frac{A}{4\\pi }\\Big )\\lambda \\ \\ , \\ \\ (\\lambda \\rightarrow \\infty ) \\ .$ iii) Case $d=3$: the ball $B^3_1$ is the unit Euclidean ball, so $V_3(B^2_1) = \\frac{4}{3}\\pi \\ \\ ,$ and denoting $\\mathrm {Vol}_3(M) = V$ we get $N(\\lambda ) \\sim \\frac{4\\pi /3}{(2\\pi )^3} V \\; \\lambda ^{\\frac{3}{2}}= \\Big (\\frac{V}{6\\pi ^2}\\Big )\\lambda ^{\\frac{3}{2}} \\ \\ , \\ \\ (\\lambda \\rightarrow \\infty ) \\ .$ iv) Case general $d$: for dimension $d$ , the unit Euclidean $d$ -ball has volume (with $\\Gamma (x)$ Euler's gamma function) $\\mathrm {Vol}_d(B^d_1) = \\frac{\\pi ^{\\frac{d}{2}}}{\\Gamma (\\frac{d}{2}+1)} \\ \\ ,$ and denoting $\\mathrm {Vol}_d(M) = V_d$ we get $N(\\lambda ) &\\sim & \\frac{\\pi ^{\\frac{d}{2}}/\\Gamma (\\frac{d}{2}+1)}{(2\\pi )^d} V_d \\lambda ^{\\frac{d}{2}} \\nonumber \\\\&=& \\bigg (\\frac{V_d}{(4\\pi )^{\\frac{d}{2}}\\Gamma (\\frac{d}{2}+1)}\\bigg )\\lambda ^{\\frac{d}{2}}\\ \\ , \\ \\ (\\lambda \\rightarrow \\infty ) \\ .$ The important point we would like to stress here is that, from the leading order of Weyl's law, one can extract both the dimension of the manifold and an averaged (volumetric) typical length scale $L_{\\mathrm {vol}}:=(V_d)^{\\frac{1}{d}}$ ."],["Weyl's law in compact manifolds with boundaries","As commented above, the Weyl's law asymptotics for the leading term of the eigenvalue counting function $N(\\lambda )$ holds also for the spectral problem of the Laplacian in a compact domain $D\\subset \\mathbb {R}^d$ .","The differences in $N(\\lambda )$ following from distinct boundary conditions show up only at the subleading order.","Specifically, given the spectral problem $\\left\\lbrace \\begin{array}{lcl}\\Delta \\phi _n = \\lambda _n \\phi _n \\ \\ &,& \\ \\ (D\\subset \\mathbb {R}^d, D \\hbox{ compact}) \\\\\\left.\\phi _n\\right|_{\\partial D} = 0 \\ \\ &,& \\ \\ \\hbox{(Dirichlet boundary conditions)}\\\\\\hbox{or} &&\\\\\\left.\\partial _s\\phi _n\\right|_{\\partial D} = 0 \\ \\ &,& \\ \\ \\hbox{(Neumann boundary conditions)}\\end{array}\\right.","\\ ,$ with $\\partial _s\\phi _n$ the normal derivative at the boundary $\\partial D$ .","In this case, under appropriate conditions [6], `Weyl's conjecture' $N(\\lambda ) \\sim C_d \\mathrm {Vol}_d(D) \\lambda ^{\\frac{d}{2}} \\mp \\frac{1}{4}C_{d-1} \\mathrm {Vol}_{d-1}(\\partial D) \\lambda ^{\\frac{d-1}{2}} \\ \\ ,$ holds when $\\lambda \\rightarrow \\infty $ , with the “minus” sign corresponding to the Dirichlet case and the “plus” to the Neumann one."],["QNM Weyl's law","For the sake of concreteness, in order to introduce QNMs we consider the scattering problem corresponding to the wave equation of a scalar field propagating in the (non-compact) space $\\mathbb {R}^d$ (namely scalar wave in $(d+1)$ -dimensional Minkowski spacetime) with scattering potential $V(x)$ , where $x\\in \\mathbb {R}^d$ and $V(x)$ is time-independent More general time-dependent situations can be considered (see e.g.","[53]), but in the present Weyl's law setting of stay in the time-independent case.","$\\left(\\partial ^2_t - \\Delta + V(x)\\right) \\phi = 0 \\ ,$ subject to outgoing boundary conditions and with appropriate initial data.","By taking the Fourier transform in time, and denoting the frequency spectral parameter as $\\omega $ , we can write the following spectral problem $\\left(- \\Delta + V(x)\\right) \\phi = \\omega ^2 \\phi \\ ,$ complemented to the asymptotic outgoing boundary conditions.","Introducing the Schrödinger operator $P = -\\Delta + V(x)$ QNMs can be formally understood as corresponding to discrete The continuous part of the spectrum would correspond to the so-called `tails'.","part of the spectrum of this operator, crucially, defined under outgoing boundary conditions $\\left\\lbrace \\begin{array}{l}P\\phi _n = \\left(-\\Delta + V(x)\\right)\\phi _n = \\omega ^2_n \\phi _n \\ \\ , \\ \\ \\hbox{(in } \\mathbb {R}^d)\\\\\\hbox{Outgoing boundary conditions at infinity}\\end{array}\\right.$ The dissipative character of the outgoing boundary conditions entails the non-selfadjoint character of the operator, so the QNM frequencies $\\omega _n$ 's are generically complex numbers.","From a more formal perspective, QNMs can be introduced in terms of the resolvent $R_P(\\omega ) = (P- \\omega \\mathrm {I})^{-1}$ of $P$ .","Under the appropriate conditions on $V(x)$ (and with the appropriate convention for $\\omega $ , namely $\\phi (t,x)\\sim e^{i\\omega t}\\phi (x)$ ), the resolvent $R_P(\\omega )$ is analytic in the lower-half $\\omega $ -complex plane.","QNMs frequencies $\\omega _n$ correspond then to the poles of the meromorphic extension of the resolvent $R_P(\\omega )$ to the upper-half $\\omega $ -complex plane (see, for instance [15], [16]).","Alternatively, a characterization of QNMs can be given as proper eigenvalues of an non-selfadjoint operator defined in terms of hyperboloidal spacetime foliations [22], [23], [24], [25], [26].","The latter is actually the approach we adopt in the calculations presented below.","A crucial feature in the present Weyl's law setting is the discrete nature of QNM frequencies $\\omega _n$ 's, in spite of the non-compact nature of the integration domain.","This allows to introduce a counting function for such $\\omega _n$ 's, as in the case of eigenvalues $\\lambda _n$ 's of the Laplace operator in compact domains, namely Eqs.","(REF ) and (REF ).","However, given that the problem is now being defined in the complex plane (not an ordered set), different natural strategies for the counting can be considered.","In this setting, given $\\omega \\in \\mathbb {R}^+$ , here we consider the QNM counting function $N(\\omega )$ defined as $N(\\omega ) = \\hbox{number of } \\omega _n \\hbox{ such that } |\\omega _n|\\le \\omega \\ ,$ that is, $N(\\omega )$ counts the number of complex QNM frequencies $\\omega _n$ 's contained in a circle in the complex plane of radius $\\omega $ and centered in the origin.","Having introduced the counting function $N(\\omega )$ the study of its large-$\\omega $ asymptotics is naturally posed, in particular the question about a possible Weyl's law.","The assessment of the Weyl's law in the case of QNM frequencies is more delicate than in the classical selfadjoint case.","A definite positive answer can be given in the one-dimensional case for compact support or fast decreasing potentials, but the situation becomes more complicate in higher dimensions.","In contrast, a powerful approach in terms of a semi-classical treatment emerges, providing rich information on the distribution of QNMs in such semiclassical limit.","In the following we briefly review the main situation in these respective cases."],["QNM Weyl's law: one-dimensional case","Let us consider a compact support, bounded potential $V(x)$ in the one-dimensional case $d=1$ .","Considering the support of $V(x)$ , namely $\\mathrm {supp(V)}$ , the so-called “convex hull” of $\\mathrm {supp(V)}$ , denoted as $\\mathrm {chsupp(V)}$ , is the minimal convex set containing $\\mathrm {supp(V)}$ .","In other words, $\\mathrm {chsupp(V)}$ is the smallest interval containing the support of $V(x)$ (cf.","Fig.","REF for an illustration of this concept).","Denoting by $L$ such $\\mathrm {chsupp(V)}$ , the following asymptotic Weyl's law in one dimensions hold [27], [17], [15], [16] $N(\\omega ) \\sim 2 \\left(\\frac{L}{c\\pi }\\right) \\omega \\ \\ , \\ \\ (\\omega \\rightarrow \\infty ) \\ .$ We note that this expression recovers, up to a factor 2, Weyl's law (REF ), with the $\\omega /c = k = \\lambda ^{\\frac{1}{2}}$ (where $k$ is the “wave number” in expression (REF ) and $c$ is the “light speed”) and with the identification of $L$ as the “effective volume” associated with the potential $V(x)$ .","This follows from the existence of two QNM branches, symmetric with respect to the imaginary axis.","We discuss further this in section REF , reinterpreting $L$ .","For later convenience, we note that this one-dimensional Weyl's law is intimately related to the following result [27], [17] concerning the asymptotic distribution of the QNMs frequencies $\\omega _n$ 's, with $n\\gg 1$ , of a one-dimensional potential $V(x)$ of low regularity (class $C^p$ , $p<\\infty )$ $\\!\\!\\!\\!\\!\\omega ^{\\rm R}_n \\sim \\pm \\frac{\\pi c}{L}\\left( n + \\tilde{\\gamma }\\right), \\,\\,\\omega ^{\\rm I}_n \\sim \\frac{c}{L} \\bigg [\\gamma \\ln \\left( \\left|\\omega _n^{\\rm R}\\right| + \\gamma ^{\\prime }\\right) - \\ln S\\bigg ],$ where $\\gamma $ , $\\tilde{\\gamma }$ , $\\gamma ^{\\prime }$ and $S$ are constants depending on the potential $V(x)$ .","This asymptotic distribution of $\\omega _n^{\\prime }$ defines the so-called “Regge QNM branches” [27], [17].","Note the asymptotic regular distribution of frequencies $\\omega _n$ 's along the logarithmic QNM branches, with constant $\\Delta \\omega ^{\\rm R} := \\omega ^{\\rm R}_{n+1}-\\omega ^{\\rm R}_{n}$ , so that we can estimate in this case, for each QNM branch, $\\displaystyle N(\\omega ) \\sim \\frac{\\omega }{\\Delta \\omega ^{\\rm R}} = \\frac{L}{c\\pi }\\omega $ .","Multiplying by the 2 branches we recover (REF ).","Figure: Illustration of the “convex hull” chsupp (V)\\mathrm {chsupp(V)} of the support of a potentialV(x)V(x).","This notion, namely the “diameter” of the region where the potentialdoes not vanish, provides the relevant multiplicative scale in QNM Weyl's law for compact support potentials.The one-dimensional result (REF ) is extended in [18] (see also [28]) beyond the case of compact support potentials $V(x)$ to a Weyl-like law valid for potentials with a very fast decay, namely with a so-called `super-exponentially decreasing' behaviour.","Specifically it holds (cf.","expression in “Conjecture 1.2” in [18]) $N(\\omega ) \\sim \\frac{s_V}{2\\pi \\rho } \\omega ^\\rho \\ \\ , \\ \\ (\\omega \\rightarrow \\infty ) \\ ,$ where $s_V$ is a quantity determined by $V(x)$ , in particular its decay rate at infinity, and $\\rho $ is the so-called `order $\\rho $ ' of $V(x)$ (e.g.","[18]).","In particular, for functions referred to as of `exponential type' (see again [18]), it holds $\\rho =1$ .","Unfortunately, the theorem in [18] requires the potential $V(x)$ to be super-exponentially decreasing, with $\\rho >1$ .","This shows in particular that for generic potentials, even those satisfying a Weyl-like power law in $\\omega $ , the connection between the power of $\\omega $ and space dimension $d$ is generically lost.","This is illustrated by a class of potentials including Gaussians $V(x)=e^{-ax^2}$ , for which Weyl's law (REF ) is shown to correspond to $N(\\omega ) \\sim 2\\frac{a}{c^2\\pi } \\omega ^2 \\ \\ , \\ \\ (\\omega \\rightarrow \\infty )$ (we have introduced the factor $c^2$ , as compared to the expression in [18] to keep track of the physical dimensions).","The situation for more generic potentials is therefore quite open and depending on the details of $V(x)$ and it does not get better when increasing the space dimension $d$ , as we see below.","As commented above, the QNM Weyl's law (REF ) presents a factor 2 with respect to the standard Weyl's law (REF ) for bounded states.","This factor 2 is naturally understood in terms of the existence of two QNM branches symmetric with respect the imaginary axis in the $\\omega $ -complex plane: QNMs frequencies $\\omega _n = \\omega _n^{\\rm R} + i \\omega ^{\\rm I}_n$ indeed come in pairs, $\\omega _n^\\mp = \\pm |\\omega _n^{\\rm R}| + i \\omega ^{\\rm I}_n$ , corresponding to the two modes propagating respectively towards the left ($\\omega _n^-$ ) and towards the right ($\\omega _n^+$ ).","From a physical perspective, and in the context of a mode propagating to the left and another propagating to the right, it is natural to interpret the quantity $2 L/c$ as the time required for the thermalization at the scale $L$ , namely the time to travel forth and back the distance $L$ $T_{\\mathrm {therm}} = 2\\left(\\frac{L}{c}\\right) \\ .$ (see [29] for an interpretation along these lines).","Such time scale seems also natural in a setting employing a frequency spectral parameter $\\omega $ , rather than a wavenumber $k$ .","We can then rewrite (REF ) as $N(\\omega ) \\sim \\left(\\frac{T_{\\mathrm {therm}}}{\\pi }\\right) \\omega \\ \\ , \\ \\ (\\omega \\rightarrow \\infty ) \\ .$"],["QNM Weyl's law: higher dimensions","Considering QNMs of a compact support, bounded potential $V(x)$ in odd-dimensions $d\\ge 3$ , the following upper bound for the QNM counting function holds [16] $N(\\omega ) \\le C_V \\omega ^d \\ .$ This provides a bound for $N(\\omega )$ but, in contrast with the one-dimensional case where the power in Weyl's law (REF ) is indeed given by the dimension $d=1$ , in the $d\\ge 3$ case this is not enough to establish Weyl's law in the large-$\\omega $ asymptotics of $N(\\omega )$ for generic compact support potentials.","An important exception to this is the radial case $V(x)=V(||x||)$ .","Specifically, given a radially-symmetric bounded, compact support potential $V(||x||)$ in odd dimension $d\\ge 3$ , with support inside a ball $B^d_R$ of radius $R$ (namely $\\mathrm {chsupp}(V)=B^d_R$ ) and with a jump at its boundary $V(R)\\ne 0$ ), it holds [30] $N(\\omega ) \\sim C_R \\omega ^d + o(\\omega ^{d-1}) \\ \\ , \\ \\ (\\omega \\rightarrow \\infty ) \\ .$ This is a fine result.","Remarkably, an (almost) explicit form can be given for the constant $C_R$ .","Specifically it holds [31] $C_R &=& \\left(2 \\frac{(\\mathrm {Vol}_d(B^d_1))^2}{(2\\pi )^d} + A_{S^{d-1}}\\right) R^d \\nonumber \\\\&=& \\left(2 \\frac{(\\mathrm {Vol}_d(B^d_1))}{(2\\pi )^d} \\mathrm {Vol}_d(B^d_R) + A_{S^{d-1}} R^d\\right) \\nonumber \\\\&=& \\left(2 C_d \\mathrm {Vol}_d(\\mathrm {chsupp}(V)) + A_{S^{d-1}} R^d\\right)$ where we have used the expression $\\mathrm {Vol}_d(B^d_R)= \\mathrm {Vol}_d(B^d_1)R^d$ for the Euclidean volume of the ball of radius $R$ .","Here $A_{S^{d-1}}$ is the constant in the QNM Weyl's law of the associated “obstacle problem”.","By the later we mean the $d$ -dimensional scattering problem in the exterior of a $(d-1)$ -sphere of radius $R$ , namely $S_R^{d-1}$ .","That is, the QNM problem defined as $\\left\\lbrace \\begin{array}{l}-\\Delta \\phi _n = \\omega ^2_n \\phi _n \\ \\ , \\ \\ \\hbox{(in } \\mathbb {R}^d\\setminus B^d_R)\\\\\\hbox{Homogeneous boundary conditions at } S_R^{d-1} \\\\\\hbox{Outgoing boundary conditions at infinity}\\end{array}\\right.$ for which a Weyl's law holds as [31] $N(\\omega ) \\sim A_{S^{d-1}} \\omega ^d + o(\\omega ^{d-1}) \\ \\ , \\ \\ (\\omega \\rightarrow \\infty )$ independently An explicit (but formal) expression can indeed be found for $A_{S^{d-1}}$ .","See Eq.","(6) in [31].","of the Dirichlet, Neumann or Robin (homogenous) boundary conditions at $S_R^{d-1}$ .","Note the additive form of $C_R$ with two terms $C_R = C_R^V + C_R^{\\mathrm {ext}} \\ .$ where: i) $C_R^V= 2 C_d \\mathrm {Vol}_d(\\mathrm {chsupp}(V))$ .","This is a contribution associated with the potential $V(x)$ .","This is exactly twice the expression for the selfadjoint Weyl's law in a compact domain $D$ , with the volume of $D$ substituted by the volume of the convex hull of the support of $V(x)$ .","ii) $C_R^{\\mathrm {ext}}= A_{S^{d-1}} R^d$ .","This is contribution associated with the “exterior problem”, in particular with $V(x)=0$ .","We will come back to this additive character of $C_R$ , later in the discussion of the BH QNM Weyl's law.","If we relax the compact support condition on the potential $V(x)$ , we already lose control on the validity of the upper bound (REF ) and a general Weyl's law is also out of reach, as dramatically illustrated in the one-dimensional case We have not discussed the even dimension $d$ case.","References to bounds on $N(\\omega )$ can be found in [31]..","The actual fact is that the problem of the asymptotics of the QNM counting function $N(\\omega )$ for QNMs is a much more involved than in the self-adjoint case.","As stated in Ref.","[16]: “Weyl laws for counting resonant states are more complicated and richer [than for bound states] as they involve both energy and rates of decay.","Even the leading term can be affected by dynamical properties of the system”.","Therefore in the most generic case, in particular for non-compact potentials in high-dimensions and with slow decay, a QNM Weyl's law sharing the universal features of the self-adjoint one is not expected.","Again, citing [15] in reference to the QNM counting function: “Except in dimension one [...] the issue of asymptotics or even optimal lower bounds remains unclear”."],["QNM Weyl's law for black holes","As discussed at the end section REF , the status of the QNM Weyl's law in a general setting remains an open problem.","This is in particular the case for non-compactly supported potentials and for the discussion in higher-dimensions.","In this setting, it seems a quite remarkable feature that QNM of $(3+1)$ -dimensional BHs do indeed satisfy the standard Weyl's law [8] of compact manifolds discussed in section .","Stationary vacuum BHs are very particular solutions in Einstein theory, as a consequence of the uniqueness theorem [34], showing very special structural features [35].","One might ask then if satisfying Weyl's law is a consequence of this special character.","This might be true regarding the large scale (long-wavenumber) aspects of these spacetimes, but it seems not to be the case for the small scales (high-wavenumber).","Indeed, as discussed in [8], the Weyl's law is robust under high-frequency (actually high-wavenumber) perturbations, in spite of the ultraviolet instability of QNM overtones [26], [8], [36], [26].","In this section we discuss QNM Weyl's law for asymptotically flat BHs.","We do not present a derivation of such results (that seems quite a formidable challenge, given the commented difficulties for QNM Weyl's law), but present a verification by straightforward evaluation.","We dwell in the spherically symmetric case that, under the light of the discussion in section REF , is indeed a strong restriction.","However, this case contains already genuine non-trivial features and offers key insight of what can be expected in the general case."],["Effective one-dimensional BH potentials","We focus our Weyl's law discussion in the $(3+1)$ -Schwarzschild BH, proceeding in several stages from the reduction to a one-dimensional effective problem to the full three dimensional Weyl's law.","Specifically, we consider here QNM frequencies associated to gravitational perturbations.","Making use of its spherical symmetry, gravitational perturbations in Schwarzschild can be reduced to two independent scalar one-dimensional wave equations of the form (REF ), with $x$ the so-called tortoise coordinate $r^*\\in ]-\\infty , \\infty [$ $\\left(-\\frac{d}{dr^*} + V(r(r^*))\\right) \\phi _{n\\ell m} = \\omega _{n\\ell m}\\phi _{n\\ell m} \\ ,$ where the angular modes $(\\ell , m)$ correspond to the spherical mode decomposition.","More explicitly, the “axial” and “polar” Schwarzschild gravitational parities have respectively associated the Regger-Wheeler $V^{\\mathrm {RW},s}_{\\ell }(r)$ and Zerilli $V^{\\mathrm {Z}}_{\\ell }(r)$ potentials [37], [38], [39], [9], [40], with $r$ the radial Schwarzschild coordinate $V^{\\mathrm {RW},s}_{\\ell }(r) = \\left(1-\\frac{2M}{r}\\right) \\left(\\frac{\\ell (\\ell + 1)}{r^2} +(1-s^2) \\frac{2M}{r^3} \\right)\\ ,$ for the axial case ($s=0,1,2$ correspond to the scalar, electromagnetic and (linearized) gravitational cases), and $&&V^{\\mathrm {Z}}_{\\ell }(r) = \\left(1-\\frac{2M}{r}\\right) \\nonumber \\\\&&\\left(\\frac{2n^2(n+1)r^3 + 6n^2Mr^2+18nM^2r+18M^3}{r^3(nr+3M)^2} \\right) \\ ,$ with $n = \\dfrac{(\\ell -1)(\\ell +2)}{2} \\ .$ for the polar case (the expression $r^* = r + 2M\\ln (r/2M-1)$ gives the relation between the $r^*$ and the standard radial Schwarzschild coordinate $r$ in the potentials' expressions).","A key feature of Schwarzschild is its so-called QNM isospectrality, namely, QNMs associated with the $V^{\\mathrm {RW},s}_{\\ell }(r)$ and $V^{\\mathrm {Z}}_{\\ell }(r)$ coincide (see [41], [42] for a fine discussion of this issue in terms of underlying integrability properties; see also references therein).","We can therefore choose either of the two potentials above in our discussion.","For simplicity one can think of $V^{\\mathrm {RW},s}$ with $s=2$ (gravitational case).","The key point in our discussion is that the relevant one-dimensional potential is of non-compact support.","Therefore it is not in the class of potentials under the hypothesis of the theorem in [17] leading to the Weyl's asymptotics (REF ).","Even more, when expressed in terms of the $r^*$ coordinate entering in the QNM equation (REF ), the decay is only exponential at $r^*\\rightarrow -\\infty $ and, even worse, a power-law at $r^*\\rightarrow \\infty $ .","Such decays are therefore too slow and we do not fall in the hypotheses of the theorem in [18] leading to (REF ).","As a consequence, there is no reason a priori to expect that the asymptotics of the QNM counting function $N(\\omega )$ for Schwarzschild satisfies a Weyl's law.","And, in spite of this, it does.","To show it, let us consider the large-$n$ QNM asymptotics of $\\omega _{n\\ell m}$ for a fixed mode $(\\ell m)$ (QNM frequencies are actually degenerated in the azimuthal mode $m$ , but we keep it in the present QNM counting context).","Following [43] it holds $\\!\\!\\!\\!","2M\\omega _{n\\ell m} = \\pm 0.0874247 + \\frac{1}{2}(n - \\frac{1}{2}) + \\ldots \\ \\ (n \\gg 1) ,$ that it is not only independent of $m$ , but also of $\\ell $ .","As sketched in Fig.","REF , QNMs are placed asymptotically parallel to the imaginary axis.","Therefore, for sufficiently large $n$ , the asymptotics of $N_{\\ell m}$ are controlled by (REF ) in a simple manner.","Concretely, the number $N_{\\ell m}$ of QNMs $\\omega _{n\\ell m}$ with fixed $(\\ell m)$ are homogeneously distributed parallel to the imaginary axis, with a constant vertical gap $\\Delta \\omega _{n\\ell m} =\\omega _{(n+1)\\ell m}-\\omega _{n \\ell m}=\\frac{1}{4M} \\ .$ Therefore, for each $(\\ell , m)$ and each parallel of the two QNM branches parallel to the imaginary axis, the number $N_{\\ell m}(\\omega )$ of QNMs contained in that circle of radius $\\omega $ is approximately given by the simple quotient $N_{\\ell m}(\\omega ) \\sim \\frac{2M \\omega }{1/2} = 4 M \\omega \\ \\ , \\ \\ (\\omega \\rightarrow \\infty ) \\ .$ Counting the two branches, we finally have $N_{\\ell m}(\\omega ) \\sim 8M \\omega \\ \\ , \\ \\ (\\omega \\rightarrow \\infty ) \\ .$ This satisfies indeed the one-dimensional Weyl's law (REF ) with the proper $d=1$ dimensionality in the power law and a length scale (independent of $(\\ell m)$ angular numbers) fixed by the relation $2 L/\\pi = 8 M$ , that is $L=4\\pi M \\ \\ .$ We do not have a justification for this verification of the Weyl's law in a “strong sense”, apart from the direct verification of large $n$ QNM asymptotics found by Nollert.","In particular, the length $L$ is an effective one fixed by the potential.","However, the latter not being compact-supported, it is not clear how to interpret it.","Some heuristics, lying at a physical speculative level without providing a proper geometric understanding, are presented in [8] and will be expanded elsewhere.","Figure: Sketch of the figure used in the “counting argument”of QNMs in the Schwarzschild BHs.","The key features arethe asymptotic distribution with a constant gap of QNMs along a line parallelto the imaginary axis, as well as thebound on the real part of QNM frequencies ω n \\omega _n'sin terms of the real part of the fundamental QNM.As commented above and presented in [8], the Weyl's law (REF ) is robust under `ultraviolet' perturbations of the effective potentials (REF ) and (REF ).","More specifically, under high-wavenumber (small scales) tiny perturbations of $V(r)$ , the large-$n$ asymptotics of $N(\\omega $ ) keeps a power law with the same power $d=1$ , though the multiplicative constant depends on the perturbation scale (fixed by its wave number), transitioning from the Schwarzschild value to approximately three times that value (cf.","behaviour of $L^{\\mathrm {W}}$ in Fig.","2 of [8]).","It is a remarkable feature that such robustness of the Weyl's law under ultraviolet perturbations happens in spite of the completely different distribution in the complex plane of unperturbed and perturbed QNM overtones: whereas in the unperturbed case QNM overtones distribute asymptotically in parallel to the imaginary axis, in the perturbed case they migrate and 'open' to new branches in the complex plane, with unbounded real part, distributing themselves along (Nollert-Price-like) asymptotically open logarithmic branches Actually, together with this logarithmic QNM branches, another family of “inner QNMs” appear.","The latter seem to play a key role in the change of the multiplicative length scale in the Weyl's law [8].","This will be systematically studied elsewhere..","The rationale for the preservation of Weyl's law in the perturbed case seems to be intimately related to the structure of Regge QNMs in (REF ) that, as discussed above, encodes the Weyl's law in dimension one.","Specifically, Weyl's law in the ultraviolet perturbed case would be a consequence of the conjecture proposed in [8] and refined in [45], according to which the ultraviolet QNM overtone instability is a general relativity low-regularity phenomenon in which perturbed Nollert-Price overtone branches tend towards Regge QNM branches in the infinite wavenumber limit of the perturbations.","The key point is that, although the branches dramatically migrate to different regions in the complex plane, the distribution of QNM overtones along the branches remains homogeneous with an approximately constant separation along the branch, so that the simple counting argument presented above and illustrated in Fig.","REF remains essentially unchanged.","The detailed analysis of this limit and its consequences on Weyl's law, in particular, the dependence of the length scale in the Weyl's law with the wavenumber of the perturbation, will be presented elsewhere."],["Three dimensional case: analytic estimation","We assess now the Weyl's law in the full ($(3+1)$ -dimensional, spherically symmetric) Schwarzschild BH, in particular recovering the power $d=3$ .","The argument is not fine enough to correctly estimate the multiplicative volume scale in Weyl's law, see below in section REF , but it offers non-trivial insight.","Expression (REF ) provides the QNM counting function for a fixed $(\\ell , m)$ mode.","Let us consider now the sum over $(\\ell , m)$ $N(\\omega ) &=& \\sum _{\\ell =2}^{\\ell _{\\mathrm {max}}} \\sum _{m=-\\ell }^{\\ell }N_{\\ell m}= \\sum _{\\ell =2}^{\\ell _{\\mathrm {max}}} \\sum _{m=-\\ell }^{\\ell } 8 M \\omega \\nonumber \\\\&=& 8 M \\omega \\sum _{\\ell =2}^{\\ell _{\\mathrm {max}}} (2\\ell + 1) \\ ,$ where we have made use of the degeneracy in the azimuthal number $m$ .","The key point is to estimate $\\ell _{\\mathrm {max}}$ (in particular, it is a finite number).","This depends critically on the specific distribution of QNMs in the complex plane for Schwarzschild.","Such structure is illustrated, for instance, in the upper panel of Fig.","REF : The real part of QNM branches of fixed $\\ell $ is bounded by its value for the fundamental mode $\\omega _{0,\\ell }$ , then the two branches cross at the algebraicly special QNM on the imaginary axis and the asymptotic a line parallel to the imaginary axis.","The key point is that for a given circle of radius $\\omega $ , all QNM branches with $\\mathrm {Re}(\\omega _{0,\\ell })>\\omega $ lie outside that circle (note that horizontal and vertical scales in the figure are the same, so the depicted circle is faithful to the present argument), so they do not contribute to $N(\\omega )$ .","For a given $\\ell $ , we can estimate the value of $\\mathrm {Re}(\\omega _{0,\\ell })$ from the large-$\\ell $ asymptotics (cf.","e.g.","Eq.","(32) in [9]) $3\\sqrt{3}M\\omega _{n,\\ell } \\sim \\ell + \\frac{1}{2} + i\\left(n +\\frac{1}{2}\\right) \\ \\ , \\ \\ (\\ell \\rightarrow \\infty )$ that are intimately related to the light ring radius, $r=3M$ .","Therefore, keeping the real part (for $n=0$ ) $3\\sqrt{3}M \\mathrm {Re}(\\omega _{0,\\ell }) \\sim \\ell + \\frac{1}{2} \\ \\ , \\ \\ (\\ell \\rightarrow \\infty )$ As argued above, branches with $\\mathrm {Re}(\\omega _{0,\\ell })>\\omega $ do not enter in the QNM counting, so QNMs with $\\ell > 3\\sqrt{3}M \\omega $ do not contribute to $N(\\omega )$ , this providing the estimate $\\ell _{\\mathrm {max}} \\sim 3\\sqrt{3}M \\omega \\ .$ As seen in Fig.","REF , this overestimates the counting QNMs for large $\\ell \\le \\ell _{\\mathrm {max}}$ , but has the virtue of relating the counting to the light ring structure.","We can then write (REF ) as $N(\\omega ) \\sim 8M \\omega \\sum _{\\ell =2}^{3\\sqrt{3}M\\omega } (2\\ell + 1)\\sim 8M \\omega \\cdot 2 \\sum _{\\ell =2}^{3\\sqrt{3}M\\omega } \\ell $ Using $\\displaystyle \\sum _{i=2}^p i \\sim \\sum _{i=1}^p i = \\frac{p(p+1)}{2}\\sim \\frac{p^2}{2}$ (for $p\\gg 1$ ), we get $N(\\omega ) \\sim 8M \\omega \\cdot 2 \\frac{27M^2 \\omega ^2}{2} = (6M)^3 \\omega ^3$ As commented above, the factor $(6M)^3$ overestimates $N(\\omega )$ , so we should rather keep as a proper result of this argument $N \\sim C_{\\mathrm {Schwarz}}\\;\\omega ^3 \\ ,$ with $C_{\\mathrm {Schwarz}}$ a constant.","We proceed now to estimate it from direct counting in numerically calculated QMNs."],["Three dimensional case: numerical computation","Using the hyperboloidal, compactified scheme in [24], [46], [47], [26], [8] for the numerical calculation of QNMs in a spherically symmetric setting, the following result follows from a straightforward computation (with $2\\le \\ell \\le 20$ in the gravitational case, $s=2$ , cf.","top panel in Fig.","REF ) $N(s) = \\alpha \\cdot (4M)^3 \\omega ^{3.0} \\ ,$ with $\\alpha \\sim 2.98$ .","On the one hand, this is consistent with the power law $d=3$ , that we recovered with the analytic estimation in (REF ).","Regarding the multiplicative factor, it is tempting to approach $\\alpha \\sim 3$ , although this requires the use of more $\\ell $ 's in the calculation and this becomes rapidly challenging from a numerical point of view.","Under this assumption, the analytical estimation $N^\\mathrm {analytic}(\\omega )$ counting function (REF ) overestimates the numerical estimation $N^\\mathrm {numerical}(\\omega )$ in (REF ) by a factor $\\frac{N^\\mathrm {analytic}(\\omega )}{N^\\mathrm {numerical}(\\omega )}\\sim \\frac{(6M)^3 \\omega ^3}{3(4M)^3 \\omega ^3} = \\frac{9}{8} \\ .$ Given the approximations in the argument leading to (REF ), this seems quite a reasonable estimation, thus supporting the role of the light-ring scale, $L^{\\mathrm {LR}} = 3M$ , as a relevant scale in the QNM counting problem.","This is consistent, of course, with the key role of the light ring for QNMs (cf.","e.g. [11]).","As a curious remark, the factor $9/8$ is the one appearing in the so-called Buchdahl bound requiring $R > (9/8)R_{\\mathrm {Schwarz}}$ (where $R$ is the areal radius) for the stability of static, spherically symmetric matter configurations.","If this Buchdahl bound plays a role in our QNM discussion, it is for now intriguing and unclear.","Figure: Numerical study of the QNM Weyl's law for Schwarzschild.Top panel: Numerically calculated BH QNMs for a gravitational perturbation (ss=||spin|| = 2) on the (3+1)(3+1)-dimensionalSchwarzschild spacetime.","Within the circle |r h ω|=8.1|r_{\\rm h} \\omega | = 8.1, with r h =2Mr_{\\rm h}=2M,one counts the number of QNMs for the angular modes ℓ=2,⋯,20\\ell =2,\\cdots , 20.","Bottom panel: Weyl's law N(ω)N(\\omega ) in (3+1)(3+1)-dimensionalSchwarzschild BHs for scalar (ss=||spin|| = 0), electromagnetic (ss=||spin|| = 1) and gravitational (ss=||spin|| = 2) perturbations.The asymptotic N(ω)∼|ω| 3 N(\\omega ) \\sim |\\omega |^3 is recovered, while we observe a length scale L W ∼3r h L^{\\rm W}\\sim 3\\,r_h.Finally, the bottom panel of Fig.","REF shows the results of the Weyl's law for all types of perturbations in the Regge-Wheeler potential (REF ), namely scalar ($s=0$ ), electromagnetic ($=1$ ) and gravitational ($s = 2$ ) perturbations.","The obtained Weyl's law seem to be independent of the type of perturbation, in spite of the differences in the associated potentials, supporting the idea that the multiplicative scale is a geometric property intrinsic to the underlying Schwarzschild BH spacetime."],["BH QNM Weyl's law: a proposal","We have just seen, both using an analytic estimation or a straightforward numerical calculation, that the asymptotics of the counting function $N(\\omega )$ for the Schwarzschild BH is consistent with the classical Weyl's law, with a power law given in terms of the dimension $d$ of spatial sections of the spacetime.","Given the spherical symmetry of the Schwarzschild BH, this is certainly suggested by the result for spherically symmetric potentials in odd dimensions $d$ discussed in section REF , namely the asymptotics (REF ) for $N(\\omega )$ .","However there is a key difference Another minor point to highlight is that the Schwarzschild QNM problem is not of the form $(-\\Delta + V(r))\\phi _n = \\omega ^2\\phi _n$ for the Euclidean Laplacian $-\\Delta $ in [30], [31].",": in contrast with the (key) assumption in [30], [31] on the compact support of the potential, the potentials $V(r)$ in (REF ) and (REF ) are not only of non-compact support, but they present a slow decay.","It is therefore remarkable that the standard Weyl's law holds.","We conjecture that this is indeed the case for generic asymptotically flat BH $(d+1)$ -dimensional spacetimes, with odd $d$ , so the asymptotics (REF ) hold.","This is a bold statement, given the special nature of the spherically symmetric case and in the absence of a study specifically devoted to a non-spherically symmetric case, namely Kerr.","But it is also in the spirit of the conjecture of Weyl's law by Sommerfeld [49] and Lorentz [50] preceding (and leading) to Weyl's proof one year later [1], [2] (see also the discussion in [4]).","In the specific case of BH spacetimes, such a proposal is actually encouraged by the genericity and universality properties of stationary BH vacuum solutions.","Such a universality is akin to the universality of Weyl's law.","Essentially motivated by structural consistency with the standard Weyl's law (REF ) for bound states of selfadjoint operators in compact manifolds, we can refine a bit further our proposal (REF ) in what refers to the volume factor.","Specifically, we start from (REF ) and (REF ) and consider the additive splitting of $C_R$ in (REF ).","As discussed after Eq.","(REF ), there is a contribution $C_R^V$ from the (convex hull of the) compact-support potential and a contribution $C_R^{\\mathrm {ext}}$ from the “exterior” scattering problem without potential (an “obstacle”).","We can consider now pushing further and further away the boundary of the support of $V(||x||)$ .","Then, in a formal sense, non-compact support potentials (as it is the case for Schwarzschild) can be seen as the (singular) limit in which the support of $V(||x||)$ is pushed to infinity and no “exterior region” remains.","In particular, no contribution of $C_R^{\\mathrm {ext}}$ would enter into $C_R$ , the latter being then given solely by $C_R^V= 2 C_d \\mathrm {Vol}_d(\\mathrm {chsupp}(V))$ .","Of course, this is just a formal expression, since $\\mathrm {Vol}_d(\\mathrm {chsupp}(V))$ diverges and it should be substituted by an effective volume term $\\mathrm {Vol}_d^{\\mathrm {eff}}$ completely fixed by the BH geometry.","Upon these considerations, our proposal for the QNM Weyl law for $(d+1)$ -dimensional BH spacetimes, with $d$ odd, is $N(\\omega ) \\sim \\frac{2}{c^d}C_d\\mathrm {Vol}_d^{\\mathrm {eff}} \\omega ^d + o(^{d-1}) \\ \\ , \\ \\ (\\omega \\rightarrow \\infty )$ We stress that $\\mathrm {Vol}_d^{\\mathrm {eff}}$ is a spacetime quantity, not linked with a particular spatial foliation of the spacetime: it should be related to the intrinsic properties of the spacetime curvature."],["QNM Weyl's law for the Schwarzschild BH","We can revisit now the discussion in section under the light of the proposal (REF ).","We can determine $\\mathrm {Vol}_d^{\\mathrm {eff}}$ for the Schwarzschild BH from (REF ), from the expression for $N(\\omega )$ in (REF ) and from the value of of $C_d$ for $d=3$ , namely $\\displaystyle C_3 = \\frac{1}{6\\pi ^2}$ (cf.","(REF ).","Adopting $\\alpha =3$ in (REF ), we have the Weyl's law $N(\\omega )\\sim 3\\cdot (4 M)^3\\omega ^3 + o(\\omega ^2)$ .","From this it follows $\\mathrm {Vol}_d^{\\mathrm {eff}} = (3\\pi )^2 (4M)^3 \\ .$ This expression is not very illuminating.","We take an alternative perspective by making use of the thermalization time $T_{\\mathrm {therm}}$ introduced in (REF ).","From a physical perspective (and in the spirit of the reasoning in the black body problem [4]) , instead of thinking in terms of a density of states $\\rho (\\omega )$ $\\rho (\\omega ) \\sim \\frac{N(\\omega )}{\\mathrm {Volume}} \\ ,$ we can consider the related notion of radiation flux per unit of time and area $F(\\omega )$ , namely $F(\\omega ) \\sim \\frac{N(\\omega )}{\\mathrm {Time}\\cdot \\mathrm {Area}} \\ .$ Under this perspective, the multiplicative factor in the Weyl's law has naturally the dimensions $[\\mathrm {Time}]\\cdot [\\mathrm {Area}]$ .","We can rewrite the factor in (REF ), for $d=3$ , in terms of the product of a thermalization time $T_{\\mathrm {therm}}$ and an effective area $\\mathrm {Area}_{2}^{\\mathrm {eff}}$ , as $\\frac{2}{c^3}C_3\\mathrm {Vol}_3^{\\mathrm {eff}} = \\frac{1}{c^{2}}C_3 T_{\\mathrm {therm}} \\mathrm {Area}^{\\mathrm {eff}} \\ .$ This poses question about the natural sphere for evaluating $\\mathrm {Area}^{\\mathrm {eff}}$ .","Invoking the role of the light-ring radius $R = 3M$ as a relevant (key) scale in the Schwarzschild QNM counting problem, cf.","discussion after Eqs.","(REF ) and (REF ), we adopt $\\mathrm {Area}^{\\mathrm {eff}} = \\mathrm {Area}^{\\mathrm {LR}} = 4\\pi \\cdot (3M)^2 \\ .$ This expression for $\\mathrm {Area}^{\\mathrm {eff}}$ , together with Eq.","(REF ) fixes the value of the thermalization time, resulting in $T_{\\mathrm {therm}} = 32\\pi M \\ .$ This is a suggestive result, both for its simplicity and its natural connection to BH `thermal issues'.","Indeed, if we regard the “radial scale” in $\\mathrm {Vol}_3^{\\mathrm {eff}}$ as a “linear length”, rather than as a time, it is not natural to explain the presence of the “$\\pi $ factor”, that appears naturally related to “angular lengths”.","On the contrary when looking at it as a `time', and moreover as a `thermalization time' (precisely because of the factor 2 in QNM Weyl's laws), the factor $\\pi $ appears naturally as connected to the thermal aspects of BH physics.","Indeed, the factor $\\displaystyle \\frac{\\kappa }{8\\pi }$ in BH first law (with $\\kappa =(4M)^{-1}$ the BH surface gravity) $\\delta M = \\left(\\frac{\\kappa }{8\\pi }\\right)\\cdot \\delta \\mathrm {Area}^{\\mathrm {Hor}} \\ ,$ is precisely the inverse of the thermalization time $T_{\\mathrm {therm}}$ (REF ) $T_{\\mathrm {therm}} = \\frac{8\\pi }{\\kappa } \\ .$ This connection with BH thermodynamics further supports the interpretation of $T_{\\mathrm {therm}}$ as a thermalization time Heuristic speculations on the possible connections between $T_{\\mathrm {therm}}$ and the BH evaporation process and Hawking radiation will be discussed elsewhere.","and the change of perspective implicit in the passage from the “density of states” perspective in (REF ) to the “flux” one in (REF ).","From a purely BH dynamical perspective, we can rewrite the first law (REF ), in a dynamical rather than a variational version $\\frac{d\\mathrm {Area}^{\\mathrm {Hor}}}{dt} = (32\\pi M) \\frac{dM}{dt} = T_{\\mathrm {therm}} \\frac{dM}{dt} \\ ,$ that relates this purely QNM thermalization notion, $T_{\\mathrm {therm}}$ , with the rate of change of the BH physical quantities.","In this “thermal setting”, the QNM Weyl's law for the Schwarzschild BH adopts the following form that highlights the relevant mechanisms underlying the description of the BH resonances $N(\\omega ) \\sim \\frac{1}{c^2}\\left(\\frac{1}{6\\pi ^2}\\right) T_{\\mathrm {therm}} \\cdot \\mathrm {Area}^{\\mathrm {LR}} \\omega ^3 + o(^{2})\\ ,$ when $\\omega \\rightarrow \\infty $ .","Remarkably, inspired by this thermal setting (but fully agnostic and independent of thermal issues), a proper intrinsic geometric expression can be formulated for Schwarzschild QNM Weyl's law.","Making use of Smarr formula [52] (valid for general vacuum stationary BHs), we can rewrite the first law (REF ) as an integrated expression, namely $M = 2 \\left(\\frac{\\kappa }{8\\pi } \\right)\\cdot \\mathrm {Area}^{\\mathrm {Hor}} \\ ,$ The factor 2 being the consequence of the application of Euler's theorem for homogeneous functions (Kerr's expression for $M$ is homogeneous of degree $1/2$ in the $A$ variable).","Then $T_{\\mathrm {therm}} = 2 \\cdot \\frac{\\mathrm {Area}^{\\mathrm {Hor}}}{M} \\ ,$ and we can write (REF ) as $N(\\omega ) \\sim \\frac{2}{c^3}\\left(\\frac{1}{6\\pi ^2}\\right) \\left(\\frac{\\mathrm {Area}^{\\mathrm {Hor}}\\cdot \\mathrm {Area}^{\\mathrm {LR}}}{M}\\right)\\omega ^3 + o(^{2})\\ ,$ This is a genuine geometric expression, depending solely on BH intrinsic quantities.","In particular, it is oblivious to any thermal reasoning (though the latter has proved a key catalyst leading to expression (REF )).","We note the fine intertwining between Euler's theorem and BH quantities, leading to the reintroduction of the factor 2 in BH QNM Weyl's law, when re-expressed in purely geometric terms.","This leads in particular to the identification of the effective volume $\\mathrm {Vol}_d^{\\mathrm {eff}}$ in (REF ) as $\\mathrm {Vol}_d^{\\mathrm {eff}} = \\left(\\frac{\\mathrm {Area}^{\\mathrm {Hor}}\\cdot \\mathrm {Area}^{\\mathrm {LR}}}{M}\\right) \\ ,$ a much more transparent expression than (REF )."],["Generic BHs: a heuristic (formal) proposal ","Unfortunately, expressions (REF ) and (REF ) do not extend straightforwardly to the generic case BH, due to the generic absence of light-rings.","Indeed, in the generic case, the region of trapping for light geodesics is no longer a surface but a volume.","The intuition gained from (REF ) and (REF ) suggests that the effective volume $\\mathrm {Vol}_d^{\\mathrm {eff}}$ in Weyl's law (REF ) should know about an (effective) measure of the light trapping region.","Formally, for a general ($d+1$ )-dimensional BH, one can write the following version of QNM Weyl's law $N(\\omega ) \\sim \\frac{1}{c^{d-1}}C_d\\left(T_{\\mathrm {therm}}\\cdot \\mathrm {Area}_{d-1}^{\\mathrm {eff-LT}}\\right)\\omega ^d + o(^{d-1}) \\ ,$ (for $\\omega \\rightarrow \\infty $ ) as an analogue for (REF ) and $N(\\omega ) \\sim \\frac{2}{c^{d}}C_d\\mathrm {Vol}_d^{\\mathrm {eff-LT}} \\omega ^d + o(^{d-1}) \\ \\ , \\ \\ (\\omega \\rightarrow \\infty )$ with $T_{\\mathrm {therm}}$ given by the well-defined BH geometric expression (REF ) and where $\\mathrm {Area}_{d-1}^{\\mathrm {eff-LT}}$ and $\\mathrm {Vol}_d^{\\mathrm {eff-LT}}$ are, respectively, effective area and volume associated with the light-trapped region, related as $2\\cdot \\mathrm {Vol}_d^{\\mathrm {eff-LT}} = c T_{\\mathrm {therm}}\\cdot \\mathrm {Area}_{d-1}^{\\mathrm {eff-LT}}$ .","How such an effective $\\mathrm {Vol}_d^{\\mathrm {eff-LT}}$ is related to the actual volume of the light-trapping region is unclear, since an appropriate notion of “convex hull” for such a region (taking into account the presence of the BH horizon) could be needed."],["Conclusions","We have discussed the Weyl's law for BH QNMs introduced in ref.","([8]), showing its consistency with the standard Weyl's law for bound states in self-adjoint problems.","In particular, for $(d+1)$ -dimensional BHs it recovers the power-law dependence on the (space) dimension $d$ .","Regarding the multiplicative factor in the Weyl's law, we have proposed that the effective volume playing the role of the finite volume associated with compact support potentials in the standard Weyl's law, is controlled by the size of BH light-trapping region.","Such a BH QNM Weyl's law can be used in two directions.","On the one hand, given the BH spacetime dimension and having a candidate for the BH light-trapping region, one can estimate the asymptotics of the number of QNMs and from there infer expressions for a notions of “QNM density states” or “QNM radiation flux”.","Perhaps more interestingly, one could proceed in the reverse direction.","Assuming a (hypothetical future) direct observational measurement of sufficiently many high-overtones in gravitational wave signals detected by interferometric gravitational antennae, an observational QNM counting function $N^{\\mathrm {obs}}(\\omega )$ could be constructed.","Using the proposed BH QNM Weyl's law one could then infer: i) an observational estimation of spacetime dimension, $(d^{\\mathrm {obs}}+1)$ , and ii) an estimation of the relevant (length/time) scale in the BH QNM resonant phenomenon.","These quantities thus provide observational probes into the geometry of BH spacetimes.","Regarding future perspectives, we can comment on the following lines: i) Assessment of the QNM BH Weyl's law in the non-spherically symmetric case, namely in the Kerr BH.","This is crucial, since part of our inferences (e.g.","the dimension $d$ in QNM Weyl's law) heavily rely in results that could be specific of the spherically symmetric case.","ii) Related to point i), assessment of the QNM Weyl's law expression (REF ) in the spherically Reissner-Nordström case.","This is a non-trivial test of our discussion.","iii) Systematic study of the impact of the ultraviolet BH QNM overtone instability on the BH QNM Weyl's law, in particular assessing the “phase transition” in [8], associated with the change of Weyl's law (light-trapping) scale as the wavenumber of the perturbations increases.","iv) Systematic assessment of the proposed BH QNM Weyl's law in more general scenarios: BHs in higher dimensions, different spacetime asymptotics (Anti-de Sitter, de Sitter...), dependence on the type of perturbation (scalar, electromagnetic, gravitational...), et cetera.","Acknowledgments.","We would like to thank Rémi M. Mokdad and Johannes Sjöstrand.","This work was supported by the French “Investissements d'Avenir” program through project ISITE-BFC (ANR-15-IDEX-03), the ANR “Quantum Fields interacting with Geometry” (QFG) project (ANR-20-CE40-0018-02), the EIPHI Graduate School (ANR-17-EURE-0002), the Spanish FIS2017-86497-C2-1 project (with FEDER contribution), the European Research Council Grant ERC-2014-StG 639022-NewNGR “New frontiers in numerical general relativity\" and the European Commission Marie Sklodowska-Curie grant No 843152 (Horizon 2020 programme).","The project used Queen Mary's Apocrita HPC facility, supported by QMUL Research-IT, and CCuB computational resources (université de Bourgogne)."]],"string":"[\n [\n \"A Weyl's law for black holes\"\n ],\n [\n \"Abstract We discuss a Weyl's law for the quasi-normal modes of black holes that recovers the structural features of the standard Weyl's law for the eigenvalues of the Laplacian in compact regions.\",\n \"Specifically, the asymptotics of the counting function $N(\\\\omega)$ of quasi-normal modes of $(d+1)$-dimensional black holes follows a power-law $N(\\\\omega)\\\\sim \\\\mathrm{Vol}_d^{\\\\mathrm{eff}}\\\\omega^d$, with $\\\\mathrm{Vol}_d^{\\\\mathrm{eff}}$ an effective volume determined by the light-trapping and decay properties of the black hole geometry.\",\n \"Closed forms are presented for the Schwarzschild black hole and a quasi-normal mode Weyl's law is proposed for generic black holes.\",\n \"As an application, such Weyl's law could provide a probe into the effective dimensionality of spacetime and the relevant resonant scales of actual astrophysical black holes, upon the counting of sufficiently many overtones in the observed ringdown signal of binary black hole mergers.\"\n ],\n [\n \"Spectrum asymptotics, degrees of freedom counting and geometry\",\n \"The goal of the present note is to present the basic elements of a Weyl's law proposed for black hole (BH) quasi-normal modes (QNMs).\",\n \"Such a law provides an estimate of the number of QNMs, inside a circle of a given radius in the frequency-complex plane, in terms of the asymptotics of high-frequency QNM overtones and a particular scale defined by the geometry of the black hole spacetime.\",\n \"It directly extends to the BH QNM setting the standard Weyl's law for the eigenvalues of the Laplacian of a compact Riemannian manifold.\",\n \"Weyl's law [1], [2] relates three distinct notions: the number of states of a physical system contained in a given spatial domain $D$ , the high-frequency asymptotics of the spectrum of the operator controlling its dynamics, and the geometry of the region $D$ .\",\n \"The study of this triple relation has proved to be a far-reaching catalyst for key developments both in physics and mathematics.\",\n \"On the physics side, it lays e.g.\",\n \"at the core of the developments in radiation theory aiming at understanding the black body radiation and, therefore, it is ultimately interwoven with the birth of quantum theory (see e.g.\",\n \"[3], [4]).\",\n \"On the mathematical side, it dwells in the more general setting of inverse problems in spectral geometry that have unveiled a rich set of correspondences between the spectral properties of geometric differential operators defined on a (Riemanninan) manifold, namely the Laplacian, and the properties of families of geodesics on such manifolds [3], [5], [6].\",\n \"Specifically, Weyl's law for the Laplacian stands as an archetype of such results in spectral geometry with direct implications in physics.\",\n \"For concreteness, let us consider the asymptotics of the homogenous scalar Helmholtz operator on a compact domain $D$ $(\\\\Delta + k_n^2)\\\\phi _n = 0 \\\\ ,$ subject to homogeneous (Dirichlet, Neumann or Robin) boundary conditions on its boundary $\\\\partial D$ .\",\n \"From the compacity of $D$ the eigenvalues $k_n^2$ form a discrete set and, denoting by $N(k)$ the number of eigenvalues $k_n^2$ below a certain wave-number $k$ , the following large-$k_n$ asymptotic relation holds $N(k) \\\\sim \\\\mathrm {Vol}_d(D) k^d + o(k^{d-1}) \\\\ \\\\ , \\\\ \\\\ (k\\\\rightarrow \\\\infty )$ where $\\\\mathrm {Vol}_d(D)$ is the volume of the compact domain $D$ and $d$ is the dimension of $D$ , namely $d=\\\\mathrm {dim}(D)$ .\",\n \"This is the Weyl's law.\",\n \"Crucially this `universal' asymptotic leading term does not depend on the shape of $D$ , the boundary $\\\\partial D$ only entering at the subleading term in the asymptotic expansion (see e.g.\",\n \"[7]).\",\n \"Similar Weyl's law relations hold for other fields (e.g.\",\n \"the electromagnetic field) and differential operators [6].\",\n \"In ref.\",\n \"[8] a Weyl's law was proposed for BH QNMs.\",\n \"QNMs encode the linear response of a resonator under an external perturbation, corresponding to exponentially decaying oscillating solutions to an appropriate wave equation under outgoing boundary conditions in a (generally) non-compact domain $D$ (cf.\",\n \"[9], [10], [11], [12] for reviews of QNMs in the gravitational seeting and [13] for optical nanoresonators).\",\n \"Specifically in the BH case, QNM complex frequencies $\\\\omega _n$ (the real part providing the QNM oscillation frequency and the imaginary part the inverse of its time-decay scale) encode invariant information on the BH stationary background.\",\n \"A crucial feature of QNM frequencies, in spite of the non-compactness of $D$ , is that they form a discrete set.\",\n \"The number of frequencies $\\\\omega _n$ in a given circle of the complex plane can therefore be counted and the question of a QNM version of Weyl's law can meaningfully be posed.\",\n \"Consequently, QNM Weyl's laws have been the subject of extensive research in the literature (see e.g.\",\n \"[14], [15], [16].\",\n \"However, the situation for QNMs is more subtle than in the standard self-adjoint case (REF ) and a Weyl's law expression analogous to (REF ) is not generic beyond the one-dimensional case and for compact support (or very fastly decaying) scattering potentials [17], [18].\",\n \"In this setting it seems quite remarkable that asymptotically flat (spherically symmetric) BH QNMs do admit [8] a Weyl-like's law of the standard functional form in (REF ) $N(\\\\omega ) \\\\sim \\\\frac{1}{c^d}\\\\mathrm {Vol}_d^{\\\\mathrm {eff}} \\\\omega ^d + o(^{d-1}) \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty )$ where $N(\\\\omega )$ amounts for the number of (complex) QNM frequencies $\\\\omega _n$ such that $|\\\\omega _n|\\\\le \\\\omega $ , where $\\\\omega >0$ , and $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ is an $d$ -dimension effective volume determined by the BH geometry (the appropriate power of the light speed $c$ has been introduced for dimensional reasons).\",\n \"Note that, given the non-compact support of the BH geometry, the factor $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ does not admit the direct interpretation of a compact “cavity” volume in (REF ).\",\n \"It is tempting, however, to seek interpreting $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ as providing a length/time (cubic) scale where the resonance phenomenon takes place.\",\n \"Upon this assumption, one can invert the logic in reading BH QNM Weyl's law (REF ), as compared to the standard Weyl's law (REF ).\",\n \"Indeed, whereas in the latter the dimension $d$ and the cavity's volume $\\\\mathrm {Vol}_d(D)$ are known Or the interest lies only on the density of states $N(k)/\\\\mathrm {Vol}_d(D)$ , as in the black body radiation problem.\",\n \"In either case, the cavity's volume $\\\\mathrm {Vol}_d(D)$ is not the quantity of interest in itself, just playing the (key) role of being the only trace of the cavity's geometry and therefore guaranteeing the universality of the Weyl's law (actually the original motivation for Weyl's proof).\",\n \"a priori, so the interest is focused on determining the number of states $N(k)$ , the former can be used to probe: i) The (effective) spacetime dimension $(d+1)$ .\",\n \"ii) The dynamical scale for the BH resonant phenomenon, under the assumption that $N(\\\\omega )$ can be independently (for instance, observationally) accessed.\",\n \"This possibility of using the proposed BH QNM Weyl's law as a probe into spacetime features defines our interest in this QNM asymptotic problem.\",\n \"The article is organized as follows.\",\n \"In section we review the Weyl's law in the classical setting of the eigenvalues of the Laplacian in a compact region.\",\n \"In section we discuss the Weyl's law for QNMs associated with the scattering (in non-compact domains) by compact support potentials, separating the special one-dimensional case from the general discussion in higher dimensions.\",\n \"In section we discuss the application to the BH case, focusing on the Schwarzschild BH case and then proposing a QNM Weyl's law for generic BHs.\",\n \"Finally, section presents the conclusions and some perspectives.\"\n ],\n [\n \"Weyl's law: space(-time) dimension and length scales\",\n \"We present here a brief account of the elements of Weyl's law relevant in our later discussion on BH QNMs, with a focus on the volume factor and the power-law dependence on the dimension.\",\n \"In a first step we review the Weyl's in the classical self-adjoint case and, in a second stage, we comment on the known results in the QNM (scattering resonances) setting.\"\n ],\n [\n \"Weyl's law in closed manifolds\",\n \"For concreteness, we focus here on the case of a compact manifold without boundaries [5].\",\n \"This case contains all the specific elements we aim at addressing in the QNM case, providing a simplified setting but adequate setting.\",\n \"The same leading-order result is found for compact manifolds with boundary, the differences related to the boundary conditions only appearing at the subleading-order, that we will not address here.\",\n \"Given a closed (compact without boundary) Riemannian manifold $(M, g_{ab})$ of dimension $d$ with Levi-Civita connection $\\\\nabla _a$ , we consider the eigenvalue problem associated with scalar Laplace-Beltrami operator $\\\\Delta _g = \\\\nabla ^a\\\\nabla _a$ , namely $-\\\\Delta _g\\\\phi _n = \\\\lambda _n\\\\phi _n \\\\ ,$ eigenvalues $\\\\lambda _n$ are real and non-negative.\",\n \"Given $\\\\lambda \\\\in \\\\mathbb {R}^+$ we define the eigenvalue counting function $N(\\\\lambda )$ as $N(\\\\lambda ) = \\\\hbox{number of } \\\\lambda _n \\\\hbox{ such that } \\\\lambda _n\\\\le \\\\lambda \\\\ .$ The Weyl's law states that, in the large-$\\\\lambda $ asymptotic limit $N(\\\\lambda ) \\\\sim C_d \\\\mathrm {Vol}_d(M) \\\\lambda ^{\\\\frac{d}{2}} \\\\ \\\\ , \\\\ \\\\ (\\\\lambda \\\\rightarrow \\\\infty )$ where $\\\\mathrm {Vol}_d(M)$ is the ($d-$ dimensional) volume of $M$ and $C_d$ are universal constants only depending on the dimension $d$ $C_d = \\\\frac{\\\\mathrm {Vol}_d(B^d_1)}{(2\\\\pi )^d} \\\\ ,$ with $\\\\mathrm {Vol}_d(B^d_1)$ the Euclidean volume of the unit $d$ -dimensional ball $B^d_1$ .\",\n \"To get a taste and for later comparison, we give some examples for the lowest dimensions: i) Case $d=1$: for the ball $B^1_1$ , namely the $[-1,1]$ interval $\\\\mathrm {Vol}_1(B^1_1) = 2 \\\\ \\\\ ,$ and therefore, denoting $\\\\mathrm {Vol}_1 = L$ , we get $N(\\\\lambda ) \\\\sim \\\\frac{2}{2\\\\pi }L \\\\lambda ^{\\\\frac{1}{2}} =\\\\Big (\\\\frac{L}{\\\\pi }\\\\Big ) \\\\lambda ^{\\\\frac{1}{2}}\\\\ \\\\ , \\\\ \\\\ (\\\\lambda \\\\rightarrow \\\\infty ) \\\\ .$ ii) Case $d=2$: for the ball $B^2_1$ , the unit circle in the plane $\\\\mathrm {Vol}_2(B^2_1) = \\\\pi \\\\ \\\\ ,$ and denoting $\\\\mathrm {Vol}_2(M) = A$ , namely the area, we get $N(\\\\lambda ) \\\\sim \\\\frac{\\\\pi }{(2\\\\pi )^2} A \\\\; \\\\lambda = \\\\Big (\\\\frac{A}{4\\\\pi }\\\\Big )\\\\lambda \\\\ \\\\ , \\\\ \\\\ (\\\\lambda \\\\rightarrow \\\\infty ) \\\\ .$ iii) Case $d=3$: the ball $B^3_1$ is the unit Euclidean ball, so $V_3(B^2_1) = \\\\frac{4}{3}\\\\pi \\\\ \\\\ ,$ and denoting $\\\\mathrm {Vol}_3(M) = V$ we get $N(\\\\lambda ) \\\\sim \\\\frac{4\\\\pi /3}{(2\\\\pi )^3} V \\\\; \\\\lambda ^{\\\\frac{3}{2}}= \\\\Big (\\\\frac{V}{6\\\\pi ^2}\\\\Big )\\\\lambda ^{\\\\frac{3}{2}} \\\\ \\\\ , \\\\ \\\\ (\\\\lambda \\\\rightarrow \\\\infty ) \\\\ .$ iv) Case general $d$: for dimension $d$ , the unit Euclidean $d$ -ball has volume (with $\\\\Gamma (x)$ Euler's gamma function) $\\\\mathrm {Vol}_d(B^d_1) = \\\\frac{\\\\pi ^{\\\\frac{d}{2}}}{\\\\Gamma (\\\\frac{d}{2}+1)} \\\\ \\\\ ,$ and denoting $\\\\mathrm {Vol}_d(M) = V_d$ we get $N(\\\\lambda ) &\\\\sim & \\\\frac{\\\\pi ^{\\\\frac{d}{2}}/\\\\Gamma (\\\\frac{d}{2}+1)}{(2\\\\pi )^d} V_d \\\\lambda ^{\\\\frac{d}{2}} \\\\nonumber \\\\\\\\&=& \\\\bigg (\\\\frac{V_d}{(4\\\\pi )^{\\\\frac{d}{2}}\\\\Gamma (\\\\frac{d}{2}+1)}\\\\bigg )\\\\lambda ^{\\\\frac{d}{2}}\\\\ \\\\ , \\\\ \\\\ (\\\\lambda \\\\rightarrow \\\\infty ) \\\\ .$ The important point we would like to stress here is that, from the leading order of Weyl's law, one can extract both the dimension of the manifold and an averaged (volumetric) typical length scale $L_{\\\\mathrm {vol}}:=(V_d)^{\\\\frac{1}{d}}$ .\"\n ],\n [\n \"Weyl's law in compact manifolds with boundaries\",\n \"As commented above, the Weyl's law asymptotics for the leading term of the eigenvalue counting function $N(\\\\lambda )$ holds also for the spectral problem of the Laplacian in a compact domain $D\\\\subset \\\\mathbb {R}^d$ .\",\n \"The differences in $N(\\\\lambda )$ following from distinct boundary conditions show up only at the subleading order.\",\n \"Specifically, given the spectral problem $\\\\left\\\\lbrace \\\\begin{array}{lcl}\\\\Delta \\\\phi _n = \\\\lambda _n \\\\phi _n \\\\ \\\\ &,& \\\\ \\\\ (D\\\\subset \\\\mathbb {R}^d, D \\\\hbox{ compact}) \\\\\\\\\\\\left.\\\\phi _n\\\\right|_{\\\\partial D} = 0 \\\\ \\\\ &,& \\\\ \\\\ \\\\hbox{(Dirichlet boundary conditions)}\\\\\\\\\\\\hbox{or} &&\\\\\\\\\\\\left.\\\\partial _s\\\\phi _n\\\\right|_{\\\\partial D} = 0 \\\\ \\\\ &,& \\\\ \\\\ \\\\hbox{(Neumann boundary conditions)}\\\\end{array}\\\\right.\",\n \"\\\\ ,$ with $\\\\partial _s\\\\phi _n$ the normal derivative at the boundary $\\\\partial D$ .\",\n \"In this case, under appropriate conditions [6], `Weyl's conjecture' $N(\\\\lambda ) \\\\sim C_d \\\\mathrm {Vol}_d(D) \\\\lambda ^{\\\\frac{d}{2}} \\\\mp \\\\frac{1}{4}C_{d-1} \\\\mathrm {Vol}_{d-1}(\\\\partial D) \\\\lambda ^{\\\\frac{d-1}{2}} \\\\ \\\\ ,$ holds when $\\\\lambda \\\\rightarrow \\\\infty $ , with the “minus” sign corresponding to the Dirichlet case and the “plus” to the Neumann one.\"\n ],\n [\n \"QNM Weyl's law\",\n \"For the sake of concreteness, in order to introduce QNMs we consider the scattering problem corresponding to the wave equation of a scalar field propagating in the (non-compact) space $\\\\mathbb {R}^d$ (namely scalar wave in $(d+1)$ -dimensional Minkowski spacetime) with scattering potential $V(x)$ , where $x\\\\in \\\\mathbb {R}^d$ and $V(x)$ is time-independent More general time-dependent situations can be considered (see e.g.\",\n \"[53]), but in the present Weyl's law setting of stay in the time-independent case.\",\n \"$\\\\left(\\\\partial ^2_t - \\\\Delta + V(x)\\\\right) \\\\phi = 0 \\\\ ,$ subject to outgoing boundary conditions and with appropriate initial data.\",\n \"By taking the Fourier transform in time, and denoting the frequency spectral parameter as $\\\\omega $ , we can write the following spectral problem $\\\\left(- \\\\Delta + V(x)\\\\right) \\\\phi = \\\\omega ^2 \\\\phi \\\\ ,$ complemented to the asymptotic outgoing boundary conditions.\",\n \"Introducing the Schrödinger operator $P = -\\\\Delta + V(x)$ QNMs can be formally understood as corresponding to discrete The continuous part of the spectrum would correspond to the so-called `tails'.\",\n \"part of the spectrum of this operator, crucially, defined under outgoing boundary conditions $\\\\left\\\\lbrace \\\\begin{array}{l}P\\\\phi _n = \\\\left(-\\\\Delta + V(x)\\\\right)\\\\phi _n = \\\\omega ^2_n \\\\phi _n \\\\ \\\\ , \\\\ \\\\ \\\\hbox{(in } \\\\mathbb {R}^d)\\\\\\\\\\\\hbox{Outgoing boundary conditions at infinity}\\\\end{array}\\\\right.$ The dissipative character of the outgoing boundary conditions entails the non-selfadjoint character of the operator, so the QNM frequencies $\\\\omega _n$ 's are generically complex numbers.\",\n \"From a more formal perspective, QNMs can be introduced in terms of the resolvent $R_P(\\\\omega ) = (P- \\\\omega \\\\mathrm {I})^{-1}$ of $P$ .\",\n \"Under the appropriate conditions on $V(x)$ (and with the appropriate convention for $\\\\omega $ , namely $\\\\phi (t,x)\\\\sim e^{i\\\\omega t}\\\\phi (x)$ ), the resolvent $R_P(\\\\omega )$ is analytic in the lower-half $\\\\omega $ -complex plane.\",\n \"QNMs frequencies $\\\\omega _n$ correspond then to the poles of the meromorphic extension of the resolvent $R_P(\\\\omega )$ to the upper-half $\\\\omega $ -complex plane (see, for instance [15], [16]).\",\n \"Alternatively, a characterization of QNMs can be given as proper eigenvalues of an non-selfadjoint operator defined in terms of hyperboloidal spacetime foliations [22], [23], [24], [25], [26].\",\n \"The latter is actually the approach we adopt in the calculations presented below.\",\n \"A crucial feature in the present Weyl's law setting is the discrete nature of QNM frequencies $\\\\omega _n$ 's, in spite of the non-compact nature of the integration domain.\",\n \"This allows to introduce a counting function for such $\\\\omega _n$ 's, as in the case of eigenvalues $\\\\lambda _n$ 's of the Laplace operator in compact domains, namely Eqs.\",\n \"(REF ) and (REF ).\",\n \"However, given that the problem is now being defined in the complex plane (not an ordered set), different natural strategies for the counting can be considered.\",\n \"In this setting, given $\\\\omega \\\\in \\\\mathbb {R}^+$ , here we consider the QNM counting function $N(\\\\omega )$ defined as $N(\\\\omega ) = \\\\hbox{number of } \\\\omega _n \\\\hbox{ such that } |\\\\omega _n|\\\\le \\\\omega \\\\ ,$ that is, $N(\\\\omega )$ counts the number of complex QNM frequencies $\\\\omega _n$ 's contained in a circle in the complex plane of radius $\\\\omega $ and centered in the origin.\",\n \"Having introduced the counting function $N(\\\\omega )$ the study of its large-$\\\\omega $ asymptotics is naturally posed, in particular the question about a possible Weyl's law.\",\n \"The assessment of the Weyl's law in the case of QNM frequencies is more delicate than in the classical selfadjoint case.\",\n \"A definite positive answer can be given in the one-dimensional case for compact support or fast decreasing potentials, but the situation becomes more complicate in higher dimensions.\",\n \"In contrast, a powerful approach in terms of a semi-classical treatment emerges, providing rich information on the distribution of QNMs in such semiclassical limit.\",\n \"In the following we briefly review the main situation in these respective cases.\"\n ],\n [\n \"QNM Weyl's law: one-dimensional case\",\n \"Let us consider a compact support, bounded potential $V(x)$ in the one-dimensional case $d=1$ .\",\n \"Considering the support of $V(x)$ , namely $\\\\mathrm {supp(V)}$ , the so-called “convex hull” of $\\\\mathrm {supp(V)}$ , denoted as $\\\\mathrm {chsupp(V)}$ , is the minimal convex set containing $\\\\mathrm {supp(V)}$ .\",\n \"In other words, $\\\\mathrm {chsupp(V)}$ is the smallest interval containing the support of $V(x)$ (cf.\",\n \"Fig.\",\n \"REF for an illustration of this concept).\",\n \"Denoting by $L$ such $\\\\mathrm {chsupp(V)}$ , the following asymptotic Weyl's law in one dimensions hold [27], [17], [15], [16] $N(\\\\omega ) \\\\sim 2 \\\\left(\\\\frac{L}{c\\\\pi }\\\\right) \\\\omega \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty ) \\\\ .$ We note that this expression recovers, up to a factor 2, Weyl's law (REF ), with the $\\\\omega /c = k = \\\\lambda ^{\\\\frac{1}{2}}$ (where $k$ is the “wave number” in expression (REF ) and $c$ is the “light speed”) and with the identification of $L$ as the “effective volume” associated with the potential $V(x)$ .\",\n \"This follows from the existence of two QNM branches, symmetric with respect to the imaginary axis.\",\n \"We discuss further this in section REF , reinterpreting $L$ .\",\n \"For later convenience, we note that this one-dimensional Weyl's law is intimately related to the following result [27], [17] concerning the asymptotic distribution of the QNMs frequencies $\\\\omega _n$ 's, with $n\\\\gg 1$ , of a one-dimensional potential $V(x)$ of low regularity (class $C^p$ , $p<\\\\infty )$ $\\\\!\\\\!\\\\!\\\\!\\\\!\\\\omega ^{\\\\rm R}_n \\\\sim \\\\pm \\\\frac{\\\\pi c}{L}\\\\left( n + \\\\tilde{\\\\gamma }\\\\right), \\\\,\\\\,\\\\omega ^{\\\\rm I}_n \\\\sim \\\\frac{c}{L} \\\\bigg [\\\\gamma \\\\ln \\\\left( \\\\left|\\\\omega _n^{\\\\rm R}\\\\right| + \\\\gamma ^{\\\\prime }\\\\right) - \\\\ln S\\\\bigg ],$ where $\\\\gamma $ , $\\\\tilde{\\\\gamma }$ , $\\\\gamma ^{\\\\prime }$ and $S$ are constants depending on the potential $V(x)$ .\",\n \"This asymptotic distribution of $\\\\omega _n^{\\\\prime }$ defines the so-called “Regge QNM branches” [27], [17].\",\n \"Note the asymptotic regular distribution of frequencies $\\\\omega _n$ 's along the logarithmic QNM branches, with constant $\\\\Delta \\\\omega ^{\\\\rm R} := \\\\omega ^{\\\\rm R}_{n+1}-\\\\omega ^{\\\\rm R}_{n}$ , so that we can estimate in this case, for each QNM branch, $\\\\displaystyle N(\\\\omega ) \\\\sim \\\\frac{\\\\omega }{\\\\Delta \\\\omega ^{\\\\rm R}} = \\\\frac{L}{c\\\\pi }\\\\omega $ .\",\n \"Multiplying by the 2 branches we recover (REF ).\",\n \"Figure: Illustration of the “convex hull” chsupp (V)\\\\mathrm {chsupp(V)} of the support of a potentialV(x)V(x).\",\n \"This notion, namely the “diameter” of the region where the potentialdoes not vanish, provides the relevant multiplicative scale in QNM Weyl's law for compact support potentials.The one-dimensional result (REF ) is extended in [18] (see also [28]) beyond the case of compact support potentials $V(x)$ to a Weyl-like law valid for potentials with a very fast decay, namely with a so-called `super-exponentially decreasing' behaviour.\",\n \"Specifically it holds (cf.\",\n \"expression in “Conjecture 1.2” in [18]) $N(\\\\omega ) \\\\sim \\\\frac{s_V}{2\\\\pi \\\\rho } \\\\omega ^\\\\rho \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty ) \\\\ ,$ where $s_V$ is a quantity determined by $V(x)$ , in particular its decay rate at infinity, and $\\\\rho $ is the so-called `order $\\\\rho $ ' of $V(x)$ (e.g.\",\n \"[18]).\",\n \"In particular, for functions referred to as of `exponential type' (see again [18]), it holds $\\\\rho =1$ .\",\n \"Unfortunately, the theorem in [18] requires the potential $V(x)$ to be super-exponentially decreasing, with $\\\\rho >1$ .\",\n \"This shows in particular that for generic potentials, even those satisfying a Weyl-like power law in $\\\\omega $ , the connection between the power of $\\\\omega $ and space dimension $d$ is generically lost.\",\n \"This is illustrated by a class of potentials including Gaussians $V(x)=e^{-ax^2}$ , for which Weyl's law (REF ) is shown to correspond to $N(\\\\omega ) \\\\sim 2\\\\frac{a}{c^2\\\\pi } \\\\omega ^2 \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty )$ (we have introduced the factor $c^2$ , as compared to the expression in [18] to keep track of the physical dimensions).\",\n \"The situation for more generic potentials is therefore quite open and depending on the details of $V(x)$ and it does not get better when increasing the space dimension $d$ , as we see below.\",\n \"As commented above, the QNM Weyl's law (REF ) presents a factor 2 with respect to the standard Weyl's law (REF ) for bounded states.\",\n \"This factor 2 is naturally understood in terms of the existence of two QNM branches symmetric with respect the imaginary axis in the $\\\\omega $ -complex plane: QNMs frequencies $\\\\omega _n = \\\\omega _n^{\\\\rm R} + i \\\\omega ^{\\\\rm I}_n$ indeed come in pairs, $\\\\omega _n^\\\\mp = \\\\pm |\\\\omega _n^{\\\\rm R}| + i \\\\omega ^{\\\\rm I}_n$ , corresponding to the two modes propagating respectively towards the left ($\\\\omega _n^-$ ) and towards the right ($\\\\omega _n^+$ ).\",\n \"From a physical perspective, and in the context of a mode propagating to the left and another propagating to the right, it is natural to interpret the quantity $2 L/c$ as the time required for the thermalization at the scale $L$ , namely the time to travel forth and back the distance $L$ $T_{\\\\mathrm {therm}} = 2\\\\left(\\\\frac{L}{c}\\\\right) \\\\ .$ (see [29] for an interpretation along these lines).\",\n \"Such time scale seems also natural in a setting employing a frequency spectral parameter $\\\\omega $ , rather than a wavenumber $k$ .\",\n \"We can then rewrite (REF ) as $N(\\\\omega ) \\\\sim \\\\left(\\\\frac{T_{\\\\mathrm {therm}}}{\\\\pi }\\\\right) \\\\omega \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty ) \\\\ .$\"\n ],\n [\n \"QNM Weyl's law: higher dimensions\",\n \"Considering QNMs of a compact support, bounded potential $V(x)$ in odd-dimensions $d\\\\ge 3$ , the following upper bound for the QNM counting function holds [16] $N(\\\\omega ) \\\\le C_V \\\\omega ^d \\\\ .$ This provides a bound for $N(\\\\omega )$ but, in contrast with the one-dimensional case where the power in Weyl's law (REF ) is indeed given by the dimension $d=1$ , in the $d\\\\ge 3$ case this is not enough to establish Weyl's law in the large-$\\\\omega $ asymptotics of $N(\\\\omega )$ for generic compact support potentials.\",\n \"An important exception to this is the radial case $V(x)=V(||x||)$ .\",\n \"Specifically, given a radially-symmetric bounded, compact support potential $V(||x||)$ in odd dimension $d\\\\ge 3$ , with support inside a ball $B^d_R$ of radius $R$ (namely $\\\\mathrm {chsupp}(V)=B^d_R$ ) and with a jump at its boundary $V(R)\\\\ne 0$ ), it holds [30] $N(\\\\omega ) \\\\sim C_R \\\\omega ^d + o(\\\\omega ^{d-1}) \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty ) \\\\ .$ This is a fine result.\",\n \"Remarkably, an (almost) explicit form can be given for the constant $C_R$ .\",\n \"Specifically it holds [31] $C_R &=& \\\\left(2 \\\\frac{(\\\\mathrm {Vol}_d(B^d_1))^2}{(2\\\\pi )^d} + A_{S^{d-1}}\\\\right) R^d \\\\nonumber \\\\\\\\&=& \\\\left(2 \\\\frac{(\\\\mathrm {Vol}_d(B^d_1))}{(2\\\\pi )^d} \\\\mathrm {Vol}_d(B^d_R) + A_{S^{d-1}} R^d\\\\right) \\\\nonumber \\\\\\\\&=& \\\\left(2 C_d \\\\mathrm {Vol}_d(\\\\mathrm {chsupp}(V)) + A_{S^{d-1}} R^d\\\\right)$ where we have used the expression $\\\\mathrm {Vol}_d(B^d_R)= \\\\mathrm {Vol}_d(B^d_1)R^d$ for the Euclidean volume of the ball of radius $R$ .\",\n \"Here $A_{S^{d-1}}$ is the constant in the QNM Weyl's law of the associated “obstacle problem”.\",\n \"By the later we mean the $d$ -dimensional scattering problem in the exterior of a $(d-1)$ -sphere of radius $R$ , namely $S_R^{d-1}$ .\",\n \"That is, the QNM problem defined as $\\\\left\\\\lbrace \\\\begin{array}{l}-\\\\Delta \\\\phi _n = \\\\omega ^2_n \\\\phi _n \\\\ \\\\ , \\\\ \\\\ \\\\hbox{(in } \\\\mathbb {R}^d\\\\setminus B^d_R)\\\\\\\\\\\\hbox{Homogeneous boundary conditions at } S_R^{d-1} \\\\\\\\\\\\hbox{Outgoing boundary conditions at infinity}\\\\end{array}\\\\right.$ for which a Weyl's law holds as [31] $N(\\\\omega ) \\\\sim A_{S^{d-1}} \\\\omega ^d + o(\\\\omega ^{d-1}) \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty )$ independently An explicit (but formal) expression can indeed be found for $A_{S^{d-1}}$ .\",\n \"See Eq.\",\n \"(6) in [31].\",\n \"of the Dirichlet, Neumann or Robin (homogenous) boundary conditions at $S_R^{d-1}$ .\",\n \"Note the additive form of $C_R$ with two terms $C_R = C_R^V + C_R^{\\\\mathrm {ext}} \\\\ .$ where: i) $C_R^V= 2 C_d \\\\mathrm {Vol}_d(\\\\mathrm {chsupp}(V))$ .\",\n \"This is a contribution associated with the potential $V(x)$ .\",\n \"This is exactly twice the expression for the selfadjoint Weyl's law in a compact domain $D$ , with the volume of $D$ substituted by the volume of the convex hull of the support of $V(x)$ .\",\n \"ii) $C_R^{\\\\mathrm {ext}}= A_{S^{d-1}} R^d$ .\",\n \"This is contribution associated with the “exterior problem”, in particular with $V(x)=0$ .\",\n \"We will come back to this additive character of $C_R$ , later in the discussion of the BH QNM Weyl's law.\",\n \"If we relax the compact support condition on the potential $V(x)$ , we already lose control on the validity of the upper bound (REF ) and a general Weyl's law is also out of reach, as dramatically illustrated in the one-dimensional case We have not discussed the even dimension $d$ case.\",\n \"References to bounds on $N(\\\\omega )$ can be found in [31]..\",\n \"The actual fact is that the problem of the asymptotics of the QNM counting function $N(\\\\omega )$ for QNMs is a much more involved than in the self-adjoint case.\",\n \"As stated in Ref.\",\n \"[16]: “Weyl laws for counting resonant states are more complicated and richer [than for bound states] as they involve both energy and rates of decay.\",\n \"Even the leading term can be affected by dynamical properties of the system”.\",\n \"Therefore in the most generic case, in particular for non-compact potentials in high-dimensions and with slow decay, a QNM Weyl's law sharing the universal features of the self-adjoint one is not expected.\",\n \"Again, citing [15] in reference to the QNM counting function: “Except in dimension one [...] the issue of asymptotics or even optimal lower bounds remains unclear”.\"\n ],\n [\n \"QNM Weyl's law for black holes\",\n \"As discussed at the end section REF , the status of the QNM Weyl's law in a general setting remains an open problem.\",\n \"This is in particular the case for non-compactly supported potentials and for the discussion in higher-dimensions.\",\n \"In this setting, it seems a quite remarkable feature that QNM of $(3+1)$ -dimensional BHs do indeed satisfy the standard Weyl's law [8] of compact manifolds discussed in section .\",\n \"Stationary vacuum BHs are very particular solutions in Einstein theory, as a consequence of the uniqueness theorem [34], showing very special structural features [35].\",\n \"One might ask then if satisfying Weyl's law is a consequence of this special character.\",\n \"This might be true regarding the large scale (long-wavenumber) aspects of these spacetimes, but it seems not to be the case for the small scales (high-wavenumber).\",\n \"Indeed, as discussed in [8], the Weyl's law is robust under high-frequency (actually high-wavenumber) perturbations, in spite of the ultraviolet instability of QNM overtones [26], [8], [36], [26].\",\n \"In this section we discuss QNM Weyl's law for asymptotically flat BHs.\",\n \"We do not present a derivation of such results (that seems quite a formidable challenge, given the commented difficulties for QNM Weyl's law), but present a verification by straightforward evaluation.\",\n \"We dwell in the spherically symmetric case that, under the light of the discussion in section REF , is indeed a strong restriction.\",\n \"However, this case contains already genuine non-trivial features and offers key insight of what can be expected in the general case.\"\n ],\n [\n \"Effective one-dimensional BH potentials\",\n \"We focus our Weyl's law discussion in the $(3+1)$ -Schwarzschild BH, proceeding in several stages from the reduction to a one-dimensional effective problem to the full three dimensional Weyl's law.\",\n \"Specifically, we consider here QNM frequencies associated to gravitational perturbations.\",\n \"Making use of its spherical symmetry, gravitational perturbations in Schwarzschild can be reduced to two independent scalar one-dimensional wave equations of the form (REF ), with $x$ the so-called tortoise coordinate $r^*\\\\in ]-\\\\infty , \\\\infty [$ $\\\\left(-\\\\frac{d}{dr^*} + V(r(r^*))\\\\right) \\\\phi _{n\\\\ell m} = \\\\omega _{n\\\\ell m}\\\\phi _{n\\\\ell m} \\\\ ,$ where the angular modes $(\\\\ell , m)$ correspond to the spherical mode decomposition.\",\n \"More explicitly, the “axial” and “polar” Schwarzschild gravitational parities have respectively associated the Regger-Wheeler $V^{\\\\mathrm {RW},s}_{\\\\ell }(r)$ and Zerilli $V^{\\\\mathrm {Z}}_{\\\\ell }(r)$ potentials [37], [38], [39], [9], [40], with $r$ the radial Schwarzschild coordinate $V^{\\\\mathrm {RW},s}_{\\\\ell }(r) = \\\\left(1-\\\\frac{2M}{r}\\\\right) \\\\left(\\\\frac{\\\\ell (\\\\ell + 1)}{r^2} +(1-s^2) \\\\frac{2M}{r^3} \\\\right)\\\\ ,$ for the axial case ($s=0,1,2$ correspond to the scalar, electromagnetic and (linearized) gravitational cases), and $&&V^{\\\\mathrm {Z}}_{\\\\ell }(r) = \\\\left(1-\\\\frac{2M}{r}\\\\right) \\\\nonumber \\\\\\\\&&\\\\left(\\\\frac{2n^2(n+1)r^3 + 6n^2Mr^2+18nM^2r+18M^3}{r^3(nr+3M)^2} \\\\right) \\\\ ,$ with $n = \\\\dfrac{(\\\\ell -1)(\\\\ell +2)}{2} \\\\ .$ for the polar case (the expression $r^* = r + 2M\\\\ln (r/2M-1)$ gives the relation between the $r^*$ and the standard radial Schwarzschild coordinate $r$ in the potentials' expressions).\",\n \"A key feature of Schwarzschild is its so-called QNM isospectrality, namely, QNMs associated with the $V^{\\\\mathrm {RW},s}_{\\\\ell }(r)$ and $V^{\\\\mathrm {Z}}_{\\\\ell }(r)$ coincide (see [41], [42] for a fine discussion of this issue in terms of underlying integrability properties; see also references therein).\",\n \"We can therefore choose either of the two potentials above in our discussion.\",\n \"For simplicity one can think of $V^{\\\\mathrm {RW},s}$ with $s=2$ (gravitational case).\",\n \"The key point in our discussion is that the relevant one-dimensional potential is of non-compact support.\",\n \"Therefore it is not in the class of potentials under the hypothesis of the theorem in [17] leading to the Weyl's asymptotics (REF ).\",\n \"Even more, when expressed in terms of the $r^*$ coordinate entering in the QNM equation (REF ), the decay is only exponential at $r^*\\\\rightarrow -\\\\infty $ and, even worse, a power-law at $r^*\\\\rightarrow \\\\infty $ .\",\n \"Such decays are therefore too slow and we do not fall in the hypotheses of the theorem in [18] leading to (REF ).\",\n \"As a consequence, there is no reason a priori to expect that the asymptotics of the QNM counting function $N(\\\\omega )$ for Schwarzschild satisfies a Weyl's law.\",\n \"And, in spite of this, it does.\",\n \"To show it, let us consider the large-$n$ QNM asymptotics of $\\\\omega _{n\\\\ell m}$ for a fixed mode $(\\\\ell m)$ (QNM frequencies are actually degenerated in the azimuthal mode $m$ , but we keep it in the present QNM counting context).\",\n \"Following [43] it holds $\\\\!\\\\!\\\\!\\\\!\",\n \"2M\\\\omega _{n\\\\ell m} = \\\\pm 0.0874247 + \\\\frac{1}{2}(n - \\\\frac{1}{2}) + \\\\ldots \\\\ \\\\ (n \\\\gg 1) ,$ that it is not only independent of $m$ , but also of $\\\\ell $ .\",\n \"As sketched in Fig.\",\n \"REF , QNMs are placed asymptotically parallel to the imaginary axis.\",\n \"Therefore, for sufficiently large $n$ , the asymptotics of $N_{\\\\ell m}$ are controlled by (REF ) in a simple manner.\",\n \"Concretely, the number $N_{\\\\ell m}$ of QNMs $\\\\omega _{n\\\\ell m}$ with fixed $(\\\\ell m)$ are homogeneously distributed parallel to the imaginary axis, with a constant vertical gap $\\\\Delta \\\\omega _{n\\\\ell m} =\\\\omega _{(n+1)\\\\ell m}-\\\\omega _{n \\\\ell m}=\\\\frac{1}{4M} \\\\ .$ Therefore, for each $(\\\\ell , m)$ and each parallel of the two QNM branches parallel to the imaginary axis, the number $N_{\\\\ell m}(\\\\omega )$ of QNMs contained in that circle of radius $\\\\omega $ is approximately given by the simple quotient $N_{\\\\ell m}(\\\\omega ) \\\\sim \\\\frac{2M \\\\omega }{1/2} = 4 M \\\\omega \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty ) \\\\ .$ Counting the two branches, we finally have $N_{\\\\ell m}(\\\\omega ) \\\\sim 8M \\\\omega \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty ) \\\\ .$ This satisfies indeed the one-dimensional Weyl's law (REF ) with the proper $d=1$ dimensionality in the power law and a length scale (independent of $(\\\\ell m)$ angular numbers) fixed by the relation $2 L/\\\\pi = 8 M$ , that is $L=4\\\\pi M \\\\ \\\\ .$ We do not have a justification for this verification of the Weyl's law in a “strong sense”, apart from the direct verification of large $n$ QNM asymptotics found by Nollert.\",\n \"In particular, the length $L$ is an effective one fixed by the potential.\",\n \"However, the latter not being compact-supported, it is not clear how to interpret it.\",\n \"Some heuristics, lying at a physical speculative level without providing a proper geometric understanding, are presented in [8] and will be expanded elsewhere.\",\n \"Figure: Sketch of the figure used in the “counting argument”of QNMs in the Schwarzschild BHs.\",\n \"The key features arethe asymptotic distribution with a constant gap of QNMs along a line parallelto the imaginary axis, as well as thebound on the real part of QNM frequencies ω n \\\\omega _n'sin terms of the real part of the fundamental QNM.As commented above and presented in [8], the Weyl's law (REF ) is robust under `ultraviolet' perturbations of the effective potentials (REF ) and (REF ).\",\n \"More specifically, under high-wavenumber (small scales) tiny perturbations of $V(r)$ , the large-$n$ asymptotics of $N(\\\\omega $ ) keeps a power law with the same power $d=1$ , though the multiplicative constant depends on the perturbation scale (fixed by its wave number), transitioning from the Schwarzschild value to approximately three times that value (cf.\",\n \"behaviour of $L^{\\\\mathrm {W}}$ in Fig.\",\n \"2 of [8]).\",\n \"It is a remarkable feature that such robustness of the Weyl's law under ultraviolet perturbations happens in spite of the completely different distribution in the complex plane of unperturbed and perturbed QNM overtones: whereas in the unperturbed case QNM overtones distribute asymptotically in parallel to the imaginary axis, in the perturbed case they migrate and 'open' to new branches in the complex plane, with unbounded real part, distributing themselves along (Nollert-Price-like) asymptotically open logarithmic branches Actually, together with this logarithmic QNM branches, another family of “inner QNMs” appear.\",\n \"The latter seem to play a key role in the change of the multiplicative length scale in the Weyl's law [8].\",\n \"This will be systematically studied elsewhere..\",\n \"The rationale for the preservation of Weyl's law in the perturbed case seems to be intimately related to the structure of Regge QNMs in (REF ) that, as discussed above, encodes the Weyl's law in dimension one.\",\n \"Specifically, Weyl's law in the ultraviolet perturbed case would be a consequence of the conjecture proposed in [8] and refined in [45], according to which the ultraviolet QNM overtone instability is a general relativity low-regularity phenomenon in which perturbed Nollert-Price overtone branches tend towards Regge QNM branches in the infinite wavenumber limit of the perturbations.\",\n \"The key point is that, although the branches dramatically migrate to different regions in the complex plane, the distribution of QNM overtones along the branches remains homogeneous with an approximately constant separation along the branch, so that the simple counting argument presented above and illustrated in Fig.\",\n \"REF remains essentially unchanged.\",\n \"The detailed analysis of this limit and its consequences on Weyl's law, in particular, the dependence of the length scale in the Weyl's law with the wavenumber of the perturbation, will be presented elsewhere.\"\n ],\n [\n \"Three dimensional case: analytic estimation\",\n \"We assess now the Weyl's law in the full ($(3+1)$ -dimensional, spherically symmetric) Schwarzschild BH, in particular recovering the power $d=3$ .\",\n \"The argument is not fine enough to correctly estimate the multiplicative volume scale in Weyl's law, see below in section REF , but it offers non-trivial insight.\",\n \"Expression (REF ) provides the QNM counting function for a fixed $(\\\\ell , m)$ mode.\",\n \"Let us consider now the sum over $(\\\\ell , m)$ $N(\\\\omega ) &=& \\\\sum _{\\\\ell =2}^{\\\\ell _{\\\\mathrm {max}}} \\\\sum _{m=-\\\\ell }^{\\\\ell }N_{\\\\ell m}= \\\\sum _{\\\\ell =2}^{\\\\ell _{\\\\mathrm {max}}} \\\\sum _{m=-\\\\ell }^{\\\\ell } 8 M \\\\omega \\\\nonumber \\\\\\\\&=& 8 M \\\\omega \\\\sum _{\\\\ell =2}^{\\\\ell _{\\\\mathrm {max}}} (2\\\\ell + 1) \\\\ ,$ where we have made use of the degeneracy in the azimuthal number $m$ .\",\n \"The key point is to estimate $\\\\ell _{\\\\mathrm {max}}$ (in particular, it is a finite number).\",\n \"This depends critically on the specific distribution of QNMs in the complex plane for Schwarzschild.\",\n \"Such structure is illustrated, for instance, in the upper panel of Fig.\",\n \"REF : The real part of QNM branches of fixed $\\\\ell $ is bounded by its value for the fundamental mode $\\\\omega _{0,\\\\ell }$ , then the two branches cross at the algebraicly special QNM on the imaginary axis and the asymptotic a line parallel to the imaginary axis.\",\n \"The key point is that for a given circle of radius $\\\\omega $ , all QNM branches with $\\\\mathrm {Re}(\\\\omega _{0,\\\\ell })>\\\\omega $ lie outside that circle (note that horizontal and vertical scales in the figure are the same, so the depicted circle is faithful to the present argument), so they do not contribute to $N(\\\\omega )$ .\",\n \"For a given $\\\\ell $ , we can estimate the value of $\\\\mathrm {Re}(\\\\omega _{0,\\\\ell })$ from the large-$\\\\ell $ asymptotics (cf.\",\n \"e.g.\",\n \"Eq.\",\n \"(32) in [9]) $3\\\\sqrt{3}M\\\\omega _{n,\\\\ell } \\\\sim \\\\ell + \\\\frac{1}{2} + i\\\\left(n +\\\\frac{1}{2}\\\\right) \\\\ \\\\ , \\\\ \\\\ (\\\\ell \\\\rightarrow \\\\infty )$ that are intimately related to the light ring radius, $r=3M$ .\",\n \"Therefore, keeping the real part (for $n=0$ ) $3\\\\sqrt{3}M \\\\mathrm {Re}(\\\\omega _{0,\\\\ell }) \\\\sim \\\\ell + \\\\frac{1}{2} \\\\ \\\\ , \\\\ \\\\ (\\\\ell \\\\rightarrow \\\\infty )$ As argued above, branches with $\\\\mathrm {Re}(\\\\omega _{0,\\\\ell })>\\\\omega $ do not enter in the QNM counting, so QNMs with $\\\\ell > 3\\\\sqrt{3}M \\\\omega $ do not contribute to $N(\\\\omega )$ , this providing the estimate $\\\\ell _{\\\\mathrm {max}} \\\\sim 3\\\\sqrt{3}M \\\\omega \\\\ .$ As seen in Fig.\",\n \"REF , this overestimates the counting QNMs for large $\\\\ell \\\\le \\\\ell _{\\\\mathrm {max}}$ , but has the virtue of relating the counting to the light ring structure.\",\n \"We can then write (REF ) as $N(\\\\omega ) \\\\sim 8M \\\\omega \\\\sum _{\\\\ell =2}^{3\\\\sqrt{3}M\\\\omega } (2\\\\ell + 1)\\\\sim 8M \\\\omega \\\\cdot 2 \\\\sum _{\\\\ell =2}^{3\\\\sqrt{3}M\\\\omega } \\\\ell $ Using $\\\\displaystyle \\\\sum _{i=2}^p i \\\\sim \\\\sum _{i=1}^p i = \\\\frac{p(p+1)}{2}\\\\sim \\\\frac{p^2}{2}$ (for $p\\\\gg 1$ ), we get $N(\\\\omega ) \\\\sim 8M \\\\omega \\\\cdot 2 \\\\frac{27M^2 \\\\omega ^2}{2} = (6M)^3 \\\\omega ^3$ As commented above, the factor $(6M)^3$ overestimates $N(\\\\omega )$ , so we should rather keep as a proper result of this argument $N \\\\sim C_{\\\\mathrm {Schwarz}}\\\\;\\\\omega ^3 \\\\ ,$ with $C_{\\\\mathrm {Schwarz}}$ a constant.\",\n \"We proceed now to estimate it from direct counting in numerically calculated QMNs.\"\n ],\n [\n \"Three dimensional case: numerical computation\",\n \"Using the hyperboloidal, compactified scheme in [24], [46], [47], [26], [8] for the numerical calculation of QNMs in a spherically symmetric setting, the following result follows from a straightforward computation (with $2\\\\le \\\\ell \\\\le 20$ in the gravitational case, $s=2$ , cf.\",\n \"top panel in Fig.\",\n \"REF ) $N(s) = \\\\alpha \\\\cdot (4M)^3 \\\\omega ^{3.0} \\\\ ,$ with $\\\\alpha \\\\sim 2.98$ .\",\n \"On the one hand, this is consistent with the power law $d=3$ , that we recovered with the analytic estimation in (REF ).\",\n \"Regarding the multiplicative factor, it is tempting to approach $\\\\alpha \\\\sim 3$ , although this requires the use of more $\\\\ell $ 's in the calculation and this becomes rapidly challenging from a numerical point of view.\",\n \"Under this assumption, the analytical estimation $N^\\\\mathrm {analytic}(\\\\omega )$ counting function (REF ) overestimates the numerical estimation $N^\\\\mathrm {numerical}(\\\\omega )$ in (REF ) by a factor $\\\\frac{N^\\\\mathrm {analytic}(\\\\omega )}{N^\\\\mathrm {numerical}(\\\\omega )}\\\\sim \\\\frac{(6M)^3 \\\\omega ^3}{3(4M)^3 \\\\omega ^3} = \\\\frac{9}{8} \\\\ .$ Given the approximations in the argument leading to (REF ), this seems quite a reasonable estimation, thus supporting the role of the light-ring scale, $L^{\\\\mathrm {LR}} = 3M$ , as a relevant scale in the QNM counting problem.\",\n \"This is consistent, of course, with the key role of the light ring for QNMs (cf.\",\n \"e.g. [11]).\",\n \"As a curious remark, the factor $9/8$ is the one appearing in the so-called Buchdahl bound requiring $R > (9/8)R_{\\\\mathrm {Schwarz}}$ (where $R$ is the areal radius) for the stability of static, spherically symmetric matter configurations.\",\n \"If this Buchdahl bound plays a role in our QNM discussion, it is for now intriguing and unclear.\",\n \"Figure: Numerical study of the QNM Weyl's law for Schwarzschild.Top panel: Numerically calculated BH QNMs for a gravitational perturbation (ss=||spin|| = 2) on the (3+1)(3+1)-dimensionalSchwarzschild spacetime.\",\n \"Within the circle |r h ω|=8.1|r_{\\\\rm h} \\\\omega | = 8.1, with r h =2Mr_{\\\\rm h}=2M,one counts the number of QNMs for the angular modes ℓ=2,⋯,20\\\\ell =2,\\\\cdots , 20.\",\n \"Bottom panel: Weyl's law N(ω)N(\\\\omega ) in (3+1)(3+1)-dimensionalSchwarzschild BHs for scalar (ss=||spin|| = 0), electromagnetic (ss=||spin|| = 1) and gravitational (ss=||spin|| = 2) perturbations.The asymptotic N(ω)∼|ω| 3 N(\\\\omega ) \\\\sim |\\\\omega |^3 is recovered, while we observe a length scale L W ∼3r h L^{\\\\rm W}\\\\sim 3\\\\,r_h.Finally, the bottom panel of Fig.\",\n \"REF shows the results of the Weyl's law for all types of perturbations in the Regge-Wheeler potential (REF ), namely scalar ($s=0$ ), electromagnetic ($=1$ ) and gravitational ($s = 2$ ) perturbations.\",\n \"The obtained Weyl's law seem to be independent of the type of perturbation, in spite of the differences in the associated potentials, supporting the idea that the multiplicative scale is a geometric property intrinsic to the underlying Schwarzschild BH spacetime.\"\n ],\n [\n \"BH QNM Weyl's law: a proposal\",\n \"We have just seen, both using an analytic estimation or a straightforward numerical calculation, that the asymptotics of the counting function $N(\\\\omega )$ for the Schwarzschild BH is consistent with the classical Weyl's law, with a power law given in terms of the dimension $d$ of spatial sections of the spacetime.\",\n \"Given the spherical symmetry of the Schwarzschild BH, this is certainly suggested by the result for spherically symmetric potentials in odd dimensions $d$ discussed in section REF , namely the asymptotics (REF ) for $N(\\\\omega )$ .\",\n \"However there is a key difference Another minor point to highlight is that the Schwarzschild QNM problem is not of the form $(-\\\\Delta + V(r))\\\\phi _n = \\\\omega ^2\\\\phi _n$ for the Euclidean Laplacian $-\\\\Delta $ in [30], [31].\",\n \": in contrast with the (key) assumption in [30], [31] on the compact support of the potential, the potentials $V(r)$ in (REF ) and (REF ) are not only of non-compact support, but they present a slow decay.\",\n \"It is therefore remarkable that the standard Weyl's law holds.\",\n \"We conjecture that this is indeed the case for generic asymptotically flat BH $(d+1)$ -dimensional spacetimes, with odd $d$ , so the asymptotics (REF ) hold.\",\n \"This is a bold statement, given the special nature of the spherically symmetric case and in the absence of a study specifically devoted to a non-spherically symmetric case, namely Kerr.\",\n \"But it is also in the spirit of the conjecture of Weyl's law by Sommerfeld [49] and Lorentz [50] preceding (and leading) to Weyl's proof one year later [1], [2] (see also the discussion in [4]).\",\n \"In the specific case of BH spacetimes, such a proposal is actually encouraged by the genericity and universality properties of stationary BH vacuum solutions.\",\n \"Such a universality is akin to the universality of Weyl's law.\",\n \"Essentially motivated by structural consistency with the standard Weyl's law (REF ) for bound states of selfadjoint operators in compact manifolds, we can refine a bit further our proposal (REF ) in what refers to the volume factor.\",\n \"Specifically, we start from (REF ) and (REF ) and consider the additive splitting of $C_R$ in (REF ).\",\n \"As discussed after Eq.\",\n \"(REF ), there is a contribution $C_R^V$ from the (convex hull of the) compact-support potential and a contribution $C_R^{\\\\mathrm {ext}}$ from the “exterior” scattering problem without potential (an “obstacle”).\",\n \"We can consider now pushing further and further away the boundary of the support of $V(||x||)$ .\",\n \"Then, in a formal sense, non-compact support potentials (as it is the case for Schwarzschild) can be seen as the (singular) limit in which the support of $V(||x||)$ is pushed to infinity and no “exterior region” remains.\",\n \"In particular, no contribution of $C_R^{\\\\mathrm {ext}}$ would enter into $C_R$ , the latter being then given solely by $C_R^V= 2 C_d \\\\mathrm {Vol}_d(\\\\mathrm {chsupp}(V))$ .\",\n \"Of course, this is just a formal expression, since $\\\\mathrm {Vol}_d(\\\\mathrm {chsupp}(V))$ diverges and it should be substituted by an effective volume term $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ completely fixed by the BH geometry.\",\n \"Upon these considerations, our proposal for the QNM Weyl law for $(d+1)$ -dimensional BH spacetimes, with $d$ odd, is $N(\\\\omega ) \\\\sim \\\\frac{2}{c^d}C_d\\\\mathrm {Vol}_d^{\\\\mathrm {eff}} \\\\omega ^d + o(^{d-1}) \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty )$ We stress that $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ is a spacetime quantity, not linked with a particular spatial foliation of the spacetime: it should be related to the intrinsic properties of the spacetime curvature.\"\n ],\n [\n \"QNM Weyl's law for the Schwarzschild BH\",\n \"We can revisit now the discussion in section under the light of the proposal (REF ).\",\n \"We can determine $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ for the Schwarzschild BH from (REF ), from the expression for $N(\\\\omega )$ in (REF ) and from the value of of $C_d$ for $d=3$ , namely $\\\\displaystyle C_3 = \\\\frac{1}{6\\\\pi ^2}$ (cf.\",\n \"(REF ).\",\n \"Adopting $\\\\alpha =3$ in (REF ), we have the Weyl's law $N(\\\\omega )\\\\sim 3\\\\cdot (4 M)^3\\\\omega ^3 + o(\\\\omega ^2)$ .\",\n \"From this it follows $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}} = (3\\\\pi )^2 (4M)^3 \\\\ .$ This expression is not very illuminating.\",\n \"We take an alternative perspective by making use of the thermalization time $T_{\\\\mathrm {therm}}$ introduced in (REF ).\",\n \"From a physical perspective (and in the spirit of the reasoning in the black body problem [4]) , instead of thinking in terms of a density of states $\\\\rho (\\\\omega )$ $\\\\rho (\\\\omega ) \\\\sim \\\\frac{N(\\\\omega )}{\\\\mathrm {Volume}} \\\\ ,$ we can consider the related notion of radiation flux per unit of time and area $F(\\\\omega )$ , namely $F(\\\\omega ) \\\\sim \\\\frac{N(\\\\omega )}{\\\\mathrm {Time}\\\\cdot \\\\mathrm {Area}} \\\\ .$ Under this perspective, the multiplicative factor in the Weyl's law has naturally the dimensions $[\\\\mathrm {Time}]\\\\cdot [\\\\mathrm {Area}]$ .\",\n \"We can rewrite the factor in (REF ), for $d=3$ , in terms of the product of a thermalization time $T_{\\\\mathrm {therm}}$ and an effective area $\\\\mathrm {Area}_{2}^{\\\\mathrm {eff}}$ , as $\\\\frac{2}{c^3}C_3\\\\mathrm {Vol}_3^{\\\\mathrm {eff}} = \\\\frac{1}{c^{2}}C_3 T_{\\\\mathrm {therm}} \\\\mathrm {Area}^{\\\\mathrm {eff}} \\\\ .$ This poses question about the natural sphere for evaluating $\\\\mathrm {Area}^{\\\\mathrm {eff}}$ .\",\n \"Invoking the role of the light-ring radius $R = 3M$ as a relevant (key) scale in the Schwarzschild QNM counting problem, cf.\",\n \"discussion after Eqs.\",\n \"(REF ) and (REF ), we adopt $\\\\mathrm {Area}^{\\\\mathrm {eff}} = \\\\mathrm {Area}^{\\\\mathrm {LR}} = 4\\\\pi \\\\cdot (3M)^2 \\\\ .$ This expression for $\\\\mathrm {Area}^{\\\\mathrm {eff}}$ , together with Eq.\",\n \"(REF ) fixes the value of the thermalization time, resulting in $T_{\\\\mathrm {therm}} = 32\\\\pi M \\\\ .$ This is a suggestive result, both for its simplicity and its natural connection to BH `thermal issues'.\",\n \"Indeed, if we regard the “radial scale” in $\\\\mathrm {Vol}_3^{\\\\mathrm {eff}}$ as a “linear length”, rather than as a time, it is not natural to explain the presence of the “$\\\\pi $ factor”, that appears naturally related to “angular lengths”.\",\n \"On the contrary when looking at it as a `time', and moreover as a `thermalization time' (precisely because of the factor 2 in QNM Weyl's laws), the factor $\\\\pi $ appears naturally as connected to the thermal aspects of BH physics.\",\n \"Indeed, the factor $\\\\displaystyle \\\\frac{\\\\kappa }{8\\\\pi }$ in BH first law (with $\\\\kappa =(4M)^{-1}$ the BH surface gravity) $\\\\delta M = \\\\left(\\\\frac{\\\\kappa }{8\\\\pi }\\\\right)\\\\cdot \\\\delta \\\\mathrm {Area}^{\\\\mathrm {Hor}} \\\\ ,$ is precisely the inverse of the thermalization time $T_{\\\\mathrm {therm}}$ (REF ) $T_{\\\\mathrm {therm}} = \\\\frac{8\\\\pi }{\\\\kappa } \\\\ .$ This connection with BH thermodynamics further supports the interpretation of $T_{\\\\mathrm {therm}}$ as a thermalization time Heuristic speculations on the possible connections between $T_{\\\\mathrm {therm}}$ and the BH evaporation process and Hawking radiation will be discussed elsewhere.\",\n \"and the change of perspective implicit in the passage from the “density of states” perspective in (REF ) to the “flux” one in (REF ).\",\n \"From a purely BH dynamical perspective, we can rewrite the first law (REF ), in a dynamical rather than a variational version $\\\\frac{d\\\\mathrm {Area}^{\\\\mathrm {Hor}}}{dt} = (32\\\\pi M) \\\\frac{dM}{dt} = T_{\\\\mathrm {therm}} \\\\frac{dM}{dt} \\\\ ,$ that relates this purely QNM thermalization notion, $T_{\\\\mathrm {therm}}$ , with the rate of change of the BH physical quantities.\",\n \"In this “thermal setting”, the QNM Weyl's law for the Schwarzschild BH adopts the following form that highlights the relevant mechanisms underlying the description of the BH resonances $N(\\\\omega ) \\\\sim \\\\frac{1}{c^2}\\\\left(\\\\frac{1}{6\\\\pi ^2}\\\\right) T_{\\\\mathrm {therm}} \\\\cdot \\\\mathrm {Area}^{\\\\mathrm {LR}} \\\\omega ^3 + o(^{2})\\\\ ,$ when $\\\\omega \\\\rightarrow \\\\infty $ .\",\n \"Remarkably, inspired by this thermal setting (but fully agnostic and independent of thermal issues), a proper intrinsic geometric expression can be formulated for Schwarzschild QNM Weyl's law.\",\n \"Making use of Smarr formula [52] (valid for general vacuum stationary BHs), we can rewrite the first law (REF ) as an integrated expression, namely $M = 2 \\\\left(\\\\frac{\\\\kappa }{8\\\\pi } \\\\right)\\\\cdot \\\\mathrm {Area}^{\\\\mathrm {Hor}} \\\\ ,$ The factor 2 being the consequence of the application of Euler's theorem for homogeneous functions (Kerr's expression for $M$ is homogeneous of degree $1/2$ in the $A$ variable).\",\n \"Then $T_{\\\\mathrm {therm}} = 2 \\\\cdot \\\\frac{\\\\mathrm {Area}^{\\\\mathrm {Hor}}}{M} \\\\ ,$ and we can write (REF ) as $N(\\\\omega ) \\\\sim \\\\frac{2}{c^3}\\\\left(\\\\frac{1}{6\\\\pi ^2}\\\\right) \\\\left(\\\\frac{\\\\mathrm {Area}^{\\\\mathrm {Hor}}\\\\cdot \\\\mathrm {Area}^{\\\\mathrm {LR}}}{M}\\\\right)\\\\omega ^3 + o(^{2})\\\\ ,$ This is a genuine geometric expression, depending solely on BH intrinsic quantities.\",\n \"In particular, it is oblivious to any thermal reasoning (though the latter has proved a key catalyst leading to expression (REF )).\",\n \"We note the fine intertwining between Euler's theorem and BH quantities, leading to the reintroduction of the factor 2 in BH QNM Weyl's law, when re-expressed in purely geometric terms.\",\n \"This leads in particular to the identification of the effective volume $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ in (REF ) as $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}} = \\\\left(\\\\frac{\\\\mathrm {Area}^{\\\\mathrm {Hor}}\\\\cdot \\\\mathrm {Area}^{\\\\mathrm {LR}}}{M}\\\\right) \\\\ ,$ a much more transparent expression than (REF ).\"\n ],\n [\n \"Generic BHs: a heuristic (formal) proposal \",\n \"Unfortunately, expressions (REF ) and (REF ) do not extend straightforwardly to the generic case BH, due to the generic absence of light-rings.\",\n \"Indeed, in the generic case, the region of trapping for light geodesics is no longer a surface but a volume.\",\n \"The intuition gained from (REF ) and (REF ) suggests that the effective volume $\\\\mathrm {Vol}_d^{\\\\mathrm {eff}}$ in Weyl's law (REF ) should know about an (effective) measure of the light trapping region.\",\n \"Formally, for a general ($d+1$ )-dimensional BH, one can write the following version of QNM Weyl's law $N(\\\\omega ) \\\\sim \\\\frac{1}{c^{d-1}}C_d\\\\left(T_{\\\\mathrm {therm}}\\\\cdot \\\\mathrm {Area}_{d-1}^{\\\\mathrm {eff-LT}}\\\\right)\\\\omega ^d + o(^{d-1}) \\\\ ,$ (for $\\\\omega \\\\rightarrow \\\\infty $ ) as an analogue for (REF ) and $N(\\\\omega ) \\\\sim \\\\frac{2}{c^{d}}C_d\\\\mathrm {Vol}_d^{\\\\mathrm {eff-LT}} \\\\omega ^d + o(^{d-1}) \\\\ \\\\ , \\\\ \\\\ (\\\\omega \\\\rightarrow \\\\infty )$ with $T_{\\\\mathrm {therm}}$ given by the well-defined BH geometric expression (REF ) and where $\\\\mathrm {Area}_{d-1}^{\\\\mathrm {eff-LT}}$ and $\\\\mathrm {Vol}_d^{\\\\mathrm {eff-LT}}$ are, respectively, effective area and volume associated with the light-trapped region, related as $2\\\\cdot \\\\mathrm {Vol}_d^{\\\\mathrm {eff-LT}} = c T_{\\\\mathrm {therm}}\\\\cdot \\\\mathrm {Area}_{d-1}^{\\\\mathrm {eff-LT}}$ .\",\n \"How such an effective $\\\\mathrm {Vol}_d^{\\\\mathrm {eff-LT}}$ is related to the actual volume of the light-trapping region is unclear, since an appropriate notion of “convex hull” for such a region (taking into account the presence of the BH horizon) could be needed.\"\n ],\n [\n \"Conclusions\",\n \"We have discussed the Weyl's law for BH QNMs introduced in ref.\",\n \"([8]), showing its consistency with the standard Weyl's law for bound states in self-adjoint problems.\",\n \"In particular, for $(d+1)$ -dimensional BHs it recovers the power-law dependence on the (space) dimension $d$ .\",\n \"Regarding the multiplicative factor in the Weyl's law, we have proposed that the effective volume playing the role of the finite volume associated with compact support potentials in the standard Weyl's law, is controlled by the size of BH light-trapping region.\",\n \"Such a BH QNM Weyl's law can be used in two directions.\",\n \"On the one hand, given the BH spacetime dimension and having a candidate for the BH light-trapping region, one can estimate the asymptotics of the number of QNMs and from there infer expressions for a notions of “QNM density states” or “QNM radiation flux”.\",\n \"Perhaps more interestingly, one could proceed in the reverse direction.\",\n \"Assuming a (hypothetical future) direct observational measurement of sufficiently many high-overtones in gravitational wave signals detected by interferometric gravitational antennae, an observational QNM counting function $N^{\\\\mathrm {obs}}(\\\\omega )$ could be constructed.\",\n \"Using the proposed BH QNM Weyl's law one could then infer: i) an observational estimation of spacetime dimension, $(d^{\\\\mathrm {obs}}+1)$ , and ii) an estimation of the relevant (length/time) scale in the BH QNM resonant phenomenon.\",\n \"These quantities thus provide observational probes into the geometry of BH spacetimes.\",\n \"Regarding future perspectives, we can comment on the following lines: i) Assessment of the QNM BH Weyl's law in the non-spherically symmetric case, namely in the Kerr BH.\",\n \"This is crucial, since part of our inferences (e.g.\",\n \"the dimension $d$ in QNM Weyl's law) heavily rely in results that could be specific of the spherically symmetric case.\",\n \"ii) Related to point i), assessment of the QNM Weyl's law expression (REF ) in the spherically Reissner-Nordström case.\",\n \"This is a non-trivial test of our discussion.\",\n \"iii) Systematic study of the impact of the ultraviolet BH QNM overtone instability on the BH QNM Weyl's law, in particular assessing the “phase transition” in [8], associated with the change of Weyl's law (light-trapping) scale as the wavenumber of the perturbations increases.\",\n \"iv) Systematic assessment of the proposed BH QNM Weyl's law in more general scenarios: BHs in higher dimensions, different spacetime asymptotics (Anti-de Sitter, de Sitter...), dependence on the type of perturbation (scalar, electromagnetic, gravitational...), et cetera.\",\n \"Acknowledgments.\",\n \"We would like to thank Rémi M. Mokdad and Johannes Sjöstrand.\",\n \"This work was supported by the French “Investissements d'Avenir” program through project ISITE-BFC (ANR-15-IDEX-03), the ANR “Quantum Fields interacting with Geometry” (QFG) project (ANR-20-CE40-0018-02), the EIPHI Graduate School (ANR-17-EURE-0002), the Spanish FIS2017-86497-C2-1 project (with FEDER contribution), the European Research Council Grant ERC-2014-StG 639022-NewNGR “New frontiers in numerical general relativity\\\" and the European Commission Marie Sklodowska-Curie grant No 843152 (Horizon 2020 programme).\",\n \"The project used Queen Mary's Apocrita HPC facility, supported by QMUL Research-IT, and CCuB computational resources (université de Bourgogne).\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05570"}}},{"rowIdx":90,"cells":{"text":{"kind":"list like","value":[["DISCO: Adversarial Defense with Local Implicit Functions"],["Abstract The problem of adversarial defenses for image classification, where the goal is to robustify a classifier against adversarial examples, is considered.","Inspired by the hypothesis that these examples lie beyond the natural image manifold, a novel aDversarIal defenSe with local impliCit functiOns (DISCO) is proposed to remove adversarial perturbations by localized manifold projections.","DISCO consumes an adversarial image and a query pixel location and outputs a clean RGB value at the location.","It is implemented with an encoder and a local implicit module, where the former produces per-pixel deep features and the latter uses the features in the neighborhood of query pixel for predicting the clean RGB value.","Extensive experiments demonstrate that both DISCO and its cascade version outperform prior defenses, regardless of whether the defense is known to the attacker.","DISCO is also shown to be data and parameter efficient and to mount defenses that transfers across datasets, classifiers and attacks."],["Introduction","It has long been hypothesized that vision is only possible because the natural world contains substantial statistical regularities, which are exploited by the vision system to overcome the difficulty of scene understanding [12], [49], [119], [42], [146], [43], [58], [130], [137], [101], [13], [142].","Under this hypothesis, natural images form a low-dimension manifold in image space, denoted as the image manifold, to which human vision is highly tuned.","While deep neural networks (DNNs) [140], [52], [167], [141], [131] aim to classify natural images with human-like accuracy, they have been shown prone to adversarial attacks that, although imperceptible to humans, significantly decrease their performance.","As shown in Fig.","REF (a) and (b), these attacks typically consist of adding an imperceptible perturbation, which can be generated in various manners [46], [88], [17], [79], to the image.","Over the past few years, the introduction of more sophisticated attacks has exposed a significant vulnerability of DNNs to this problem [138], [38], [37], [29], [29], [125].","In fact, it has been shown that adversarial examples crafted with different classifiers and optimization procedures even transfer across networks [34], [117], [159], [61], [108].","A potential justification for the success of adversarial attacks and their transferability is that they create images that lie just barely outside the image manifold [135], [124], [81], [45], [71], [11].","We refer to these images as barely outliers.","While humans have the ability to project these images into the manifold, probably due to an history of training under adverse conditions, such as environments with low-light or ridden with occlusions, this is not the case for current classifiers.","A key factor to the success of the human projection is likely the accurate modeling of the image manifold.","Hence, several defenses against adversarial attacks are based on models of natural image statistics.","These are usually global image representations [92], [139], [124], [166] or conditional models of image pixel statistics [135], [11], [145], [122], [74].","For example, PixelDefend [135] and HPD [11] project malicious images into the natural image manifold using the PixelCNN model [74], which predicts a pixel value conditioned on the remainder of the image.","However, these strategies [135], [11], [92], [139], [124] can be easily defeated by existing attacks.","We hypothesize that this is due to the difficulty of learning generative image models, which require global image modelling, a highly complex task.","It is well known that the synthesis of realistic natural images requires very large model sizes and training datasets [69], [14], [70].","Even with these, it is not clear that the manifold is modeled in enough detail to defend adversarial attacks.","In this work, we argue that, unlike image synthesis, the manifold projection required for adversarial defense is a conditional operation: the synthesis of a natural image given the perturbed one.","Assuming that the attack does not alter the global structure of the image (which would likely not be imperceptible to humans) it should suffice for this function to be a conditional model of local image (i.e.","patch) statistics.","We argue that this conditional modeling can be implemented with an implicit function [24], [134], [94], [121], [100], [73], [65], [25], where the network learns a conditional representation of the image appearance in the neighborhood of each pixel, given a feature extracted at that pixel.","This strategy is denoted aDversarIal defenSe with local impliCit functiOns (DISCO).","Local implicit models have recently been shown to provide impressive results for 3D modeling [134], [94], [121], [100], [160], [73], [65], [25], [93], [109], [44], [156] and image interpolation [24].","We show that such models can be trained to project barely outliers into the patch manifold, with much smaller parameter and dataset sizes than generative models, while enabling much more precise control of the manifold projection operation.","This is illustrated in Fig.","REF , which presents an image, its adversarial attack, and the output of the DISCO defense.","To train DISCO, a dataset of adversarial-clean pairs is first curated.","During training, DISCO inputs an adversarial image and a query pixel location, for which it predicts a new RGB value.","This is implemented with a feature encoder and a local implicit function.","The former is composed by a set of residual blocks with stacked convolution layers and produces a deep feature per pixel.","The latter consumes the query location and the features in a small neighborhood of the query location.","The implicit function is learned to minimize the $L_1$ loss between the predicted RGB value and that of the clean image.","The restriction of the manifold modeling to small image neighborhoods is a critical difference with respect to previous defenses based on the modeling of the natural image manifold.","Note that, as shown in Fig.","REF , DISCO does not project the entire image into the manifold, only each pixel neighborhood.","This considerably simplifies the modeling and allows a much more robust defense in a parameter and data efficient manner.","This is demonstrated by evaluating the performance of DISCO under both the oblivious and adaptive settings [163], [115], [161].","Under the oblivious setting, the popular RobustBench [27] benchmark is considered, for both $L_{\\infty }$ and $L_{2}$ attacks with Autoattack [29].","DISCO achieves SOTA robust accuracy (RA) performance, e.g.","outperforming the prior art on Cifar10, Cifar100 and ImageNet by 17%, 19% and 19% on $L_{\\infty }$ Autoattack.","A comparison to recent test-time defenses [164], [4], [120], [28], [90], [99] also shows that DISCO is a more effective defensive strategy across various datasets and attacks.","Furthermore, a study of the defense transferability across datasets, classifiers and attacks shows that the DISCO defense maintains much of its robustness even when deployed in a setting that differs from that used for training by any combination of these three factors.","Finally, the importance of the local manifold modeling is illustrated by experiments on ImageNet [35], where it is shown that even when trained with only 0.5% of the dataset DISCO outperforms all prior defenses.","Under the adaptive setting, DISCO is evaluated using the BPDA [8] attack, known to circumvent most defenses based on image transformation [135], [124], [139], [161], [40].","While DISCO is more vulnerable under this setting, where the defense is known to the attacker, it still outperforms existing approaches by 46.76%.","More importantly, we show that the defense can be substantially strengthened by cascading DISCO stages, which magnifies the gains of DISCO to 57.77%.","This again leverages the parameter efficiency of the local modeling of image statistics, which allows the implementation of DISCO cascades with low complexity.","The ability to cascade DISCO stages also allows a new type of defense, where the number of DISCO stages is randomized on a per image basis.","This introduces some degree of uncertainty about the defense even under the adaptive setting and further improves robustness.","Overall, this work makes four contributions.","First, it proposes the use of defenses based purely on the conditional modeling of local image statistics.","Second, it introduces a novel defense of this type, DISCO, based on local implicit functions.","Third, it leverages the parameter efficiency of the local modeling to propose a cascaded version of DISCO that is shown robust even to adaptive attacks.","Finally, DISCO is shown to outperform prior defenses on RobustBench [27] and other 11 attacks, as well as test-time defenses under various experimental settings.","Extensive ablations demonstrate that DISCO has unmatched defense transferrability in the literature, across datasets, attacks and classifiers."],["Related Work","Adversarial Attack and Defense.","We briefly review adversarial attacks and defenses for classification and prior art related to our work.","Please refer to [19], [103], [2] for more complete reviews.","Adversarial Attacks aim to fool the classifier by generating an imperceptible perturbation (under $L_p$ norm constraint) that is applied to the clean image.","Attack methods have evolved from simple addition of sign gradient, as in FGSM [46], to more sophisticated approaches [79], [17], [138], [144], [88], [29], [38], [37], [29].","While most white-box attacks assume access to the classifier gradient, BPDA [8] proposed a gradient approximation attack that can circumvent defenses built on obfuscated gradients.","In general, these attacks fall into two settings, oblivious or adaptive, depending on whether the attacker is aware of the defense strategy [163], [115], [161].","DISCO is evaluated under both settings.","Adversarial Defenses can be categorized into adversarially trained and transformation based.","The former are trained against adversarial examples generated on-the-fly during training [114], [48], [47], [113], [126], [104], [126], allowing the resulting robust classifier to defend against the adversarial examples.","While adversarially trained defenses dominate the literature, they are bound together with the classifier.","Hence, re-training is required if the classifier changes and the cost of adversarial training increases for larger classifiers.","Transformation based defenses [50], [161], [64], [135], [139] instead introduce an additional defense module, which can be applied to many classifiers.","This module preprocesses the input image before passing it to the classifier.","The proposed preprocessing steps include JPEG compression [33], [40], [87], bit reduction [50], [161], [64], pixel deflection [111] or applications of random transformations [158], [50] and median filters [102].","Beyond pixel space defenses, malicious images can also be reconstructed to better match natural image statistics using autoencoders [92], [139], GANs [124], [7], [166], or other generative models, such as the PixelCNN [122].","The latter is used to project the malicious image into the natural image manifold by methods like PixelDefend [135] or HPD [11].","These methods can only produce images of fixed size [92], [124], [139], [166] and model pixel likelihoods over the entire image [135], [11].","This is unlike DISCO, which models conditional local statistics and can produce outputs of various size.","The idea of performing adversarial purification before feeding the image into the classifier is central to a recent wave of test-time defenses [164], [4], [90], [99].","[164] addresses the impracticality of previous Monte-Carlo purification models by introducing a Denoising Score-Matching and a random noise injection mechanism.","[4] prepends an anti-adversary layer to the classifier, with the goal of maximizing the classifier confidence of the predicted label.","[90] reverses the adversarial examples using self-supervised contrastive loss.","[99] proposed a diffusion model for adversarial removal.","Unlike these prior works, DISCO purifies the adversarial image by modeling the local patch statistics.","Such characteristics results in data and parameter efficiency, which have not been demonstrated for [164], [4], [90], [99].","Furthermore, DISCO outperforms all prior works in terms of robust accuracy, under the various settings they proposed.","Implicit Function.","refers to the use of a neural network to model a continuous function [134].","This has been widely used in applications involving audio [176], [51], [134], 2D images [24], [39] and 3D shapes [134], [94], [121], [100], [160], [73], [65], [25], [93], [82], [109], [44], [156].","In the 3D literature, local implicit functions have become popular models of object shape [100], [121], [109] or complex 3D scenes [94].","This also inspired 2D applications to super-resolution [24], image [68] and video generation [165].","In the adversarial attack literature, implicit functions have recently been proposed to restore adversarial point clouds of 3D shape, through the IF-Defense [156].","To the best of our knowledge, ours is the first paper to propose local implicit functions for 2D adversarial defense.","Table: NO_CAPTION"],["Method","In this section, we introduce the architecture of DISCO and its training and testing procedure."],["Motivation","Under the hypothesis that natural images lie on a low-dimension image manifold, classification networks can be robustified by learning to project barely outliers (i.e.","adversarial images) into the manifold, a process that can be seen as manifold thickening.","Images in a shell around the manifold are projected into it, leaving a larger margin to images that should be recognized as outliers.","While this idea has been studied [45], [71], [85], [63], its success hinges on the ability of classification models to capture the complexities of the image manifold.","This is a very hard problem, as evidenced by the difficulty of model inversion algorithms that aim to synthesize images with a classifier [148], [89], [98], [162].","These algorithms fail to produce images comparable to the state of the art in generative modeling, such as GANs [70], [69], [15].","Recently, however, it has been shown that it is possible to synthesize realistic images and 3D shapes with implicit functions, through the use of deep networks [44], [24], [134], [94], [121], [100], [65], [25] that basically memorize images or objects as continuous functions.","The assumption behind DISCO is that these implicit functions can capture the local statistics of images or 3D shapes, and can be trained for manifold thickening, that is to learn how to projecting barely outliers into the image manifold."],["Model Architecture and Training","Data Preparation To train DISCO, a dataset $D=\\lbrace (x^i_{cln}, x^i_{adv})\\rbrace _{i=1}^N$ containing a set of paired clean $x^i_{cln}$ and adversarial $x^i_{adv}$ images is curated.","For this, a classifier $P_{trn}$ and an attack procedure $A_{trn}$ are first selected.","For each image $x^i_{cln}$ , the adversarial image $x^i_{adv}$ is generated by attacking the predictions $P_{trn}(x^i_{cln})$ using $A_{trn}$ , as shown in Fig.","REF (a).","Training As shown in Fig.","REF (b), the DISCO defense is trained with pairs of random patches cropped at the same location of the images $x_{cln}$ and $x_{adv}$ .","For example, random patches of size $48\\times 48$ are sampled from training pairs of the ImageNet dataset [35].","The network is then trained to minimize the L1 loss between the RGB values of the clean $x_{cln}$ and defense output $x_{def}$ .","Architecture DISCO performs manifold thickening by leveraging the LIIF [25] architecture to purify adversarial patches.","It takes an adversarial image $x_{adv} \\in \\mathbb {R}^{H\\times W \\times 3}$ and a query pixel location $p=[i,j] \\in \\mathbb {R}^2$ as input and predicts a clean RGB value $v \\in \\mathbb {R}^3$ at the query pixel location, ideally identical to the pixel value of the clean image $x_{cln} \\in \\mathbb {R}^{H\\times W \\times 3}$ at that location.","The defense output $x_{def} \\in \\mathbb {R}^{H^{\\prime }\\times W^{\\prime } \\times 3}$ can then be synthesized by predicting a RGB value for each pixel location in a grid of size ${H^{\\prime }\\times W^{\\prime } \\times 3}$ .","Note that it is not a requirement that the size of $x_{def}$ be the same as that of $x_{cln}$ .","In fact, the size of $x_{def}$ could be changed during inference.","Figure: The DISCO architecture includes an encoder and a local implicit module.","The network is trained to map Adversarial into Defense images, using an L 1 L_1 loss to Clean images.To implement this, DISCO is composed of two modules, illustrated in Fig.","REF .","The first is an encoder $E$ that extracts the per-pixel feature of an input image $x$ .","The encoder architecture resembles the design of EDSR [84], originally proposed for super-resolution.","It contains a sequence of 15 residual blocks, each composed of a convolution layer, a ReLu layer and a second convolution layer.","The encoder output is a feature map $f=E(x) \\in \\mathbb {R}^{H\\times W \\times C}$ , with $C=64$ channels.","The feature at location $p=[i,j]$ is denoted as $f_{ij}$ .","The second module is the the local implicit module $L$ , which is implemented by a MLP.","Given query pixel location $p$ , $L$ first finds the nearest pixel value $p^*=[i^*,j^*]$ in the input image $x$ and corresponding feature $f_{i^*j^*}$ .","$L$ then takes the features in the neighborhood of $p^*$ into consideration to predict the clean RGB value $v$ .","More specifically, let $\\hat{f}_{i^*j^*}$ denote a concatenation of the features whose location is within the kernel size $s$ of $p^*$ .","The local implicit module $L$ takes the concatenated feature, the relative position $r=p-p^*$ between $p$ and $p^*$ , and the pixel shape as input, to predict a RGB value $v$ .","By default, the kernel size $s$ is set to be 3.","Since the network implements a continuous function, based only on neighboring pixels, the original grid size $H\\times W$ is not important.","The image coordinates can be normalized so that $(i,j) \\in [-1,1]^2$ and the pixel shape is the height and width of the output pixel in the normalized coordinates.","This makes DISCO independent of the original image size or resolution."],["Inference","For inference, DISCO takes either a clean or an adversarial image as input.","Given a specified output size for $x_{def}$ , DISCO loops over all the output pixel locations, predicting an RGB value per location.","Note that this is not computationally intensive because the encoder feature map $f=E(x)$ is computed once and used to the predict the RGB values of all query pixel locations.","Furthermore, while the training pairs are generated with classifier $P_{trn}$ and attack $A_{trn}$ , the inference time classifier $P_{tst}$ and attack $A_{tst}$ could be different.","In the experimental section we show that DISCO is quite flexible, performing well when (1) $P_{trn}$ and $P_{tst}$ consume images of different input size and (2) the attack, classifier and dataset used for inference are different than those used for training.","In fact, DISCO is shown to be more robust to these configuration changes than previous methods."],["DISCO Cascades","DISCO is computationally very appealing because it disentangles the training of the defense from that of the classifier.","This can be a big practical advantage, since classifier retraining is needed whenever training settings, such as architecture, hyper-parameters, or number of classes, change.","Adversarial defenses require retraining on the entire dataset when this is the case, which is particularly expensive for large models (like SENet [55] or EfficientNet [141]) trained on large datasets (like ImageNet [35] and OpenImages [75]).","Unsurprisingly, RobustBench [27], one of the largest adversarial learning benchmarks, reports more than 70 baselines for Cifar10, but less than 5 on ImageNet.","DISCO does not have this defense complexity, since it is trained independently of the classifier.","Furthermore, because DISCO is a model of local statistics, it is particularly parameter efficient.","As shown in Fig.","REF , DISCO has a lightweight design with only 1.6M parameters, which is significantly less than most recent classifier [141], [55], [131], [52] and GAN [70], [14] models with good performance for ImageNet-like images.","This also leads to a computationally efficient defense.","Our experiments show that DISCO can be trained with only 50,000 training pairs.","In fact, we show that it can beat the prior SOTA using less than 0.5% of ImageNet as training data (Table REF ).","One major benefit of this efficiency is that it creates a large unbalance between the costs of defense and attack.","Consider memory usage, which is dominated by the computation of gradients needed for either the attack or the backpropagation of training.","Let $N_d$ and $N_c$ be the number of parameters of the DISCO network and classifier, respectively.","The per image memory cost of training the DISCO defense is $O(N_d)$ .","On the other hand, the attack cost depends on the information available to the attacker.","We consider two settings, commonly considered in the literature.","In the oblivious setting, only the classifier is known to that attacker and the attack has cost $O(N_c)$ .","In the adaptive setting, both the classifier and the DISCO are exposed and backpropagation has memory cost $O(N_c + N_d)$ .","In experiments, we show that DISCO is quite effective against oblivious attacks.","Adaptive attacks are more challenging.","However, as shown in Fig REF , it is usually the case that $N_c > N_d$ , making the complexity of the attack larger than that of the defense.","This is unlike adversarial training, where attack and defense require backpropagation on the same model and thus have the same per-image cost.","This asymmetry between the memory cost of the attack and defense under DISCO can be magnified by cascading DISCO networks.","If $K$ identical stages of DISCO are cascaded, the defense complexity remains $O(N_d)$ but that of the attack raises to $O(N_c + K N_d)$ .","Hence the ratio of attack-to-defense cost raises to $O(K + N_c/N_d)$ .","Interestingly, our experiments (see Section REF ) show that when $K$ is increased the defense performance of the DISCO cascade increases as well.","Hence, DISCO cascades combine high robust accuracy with a large ratio of attack-to-defense cost."],["Experiments","In this section, we discuss experiments performed to evaluate the robustness of DISCO.","Results are discussed for both the oblivious and adaptive settings [163], [115], [161] and each result is averaged over 3 trials.","$\\epsilon _{p}$ denotes the perturbation magnitude under the $L_p$ norm.","All experiments are conducted on a single Nvidia Titan Xp GPU with Intel Xeon CPU E5-2630 using Pytorch [110].","Please see appendix for more training details, quantitative results and visualizations.","We adopt the code from LIIF [24] for implementation.","Training Dataset: The following training configuration is used unless noted.","Three datasets are considered: Cifar10 [76], Cifar100 [77] and Imagenet [35].","For each, 50,000 adversarial-clean training pairs are curated.","For Cifar10 and Cifar100, these are the images in the training set, while for ImageNet, 50 images are randomly sampled per class.","Following RobustBench [27], the evaluation is then conducted on the test set of each dataset.","To create training pairs, PGD [88] ($\\epsilon _{\\infty }=8/255$ with step size is 2/255 and the number of steps 100) is used to attack a ResNet18 and a WideResNet28 on Cifar10/ImageNet and Cifar100, respectively.","Attack and Benchmark: DISCO is evaluated on RobustBench [27], which contains more than 100 baselines evaluated using Autoattack [29].","This is an ensemble of four sequential attacks, including the PGD [88] attack with two different optimization losses, the FAB attack [26] and the black-box Square Attack [5].","DISCO is compared to defense baselines under both $L_{\\infty }$ and $L_{2}$ norms.","To study defense generalization, 11 additional attacks are considered, including FGSM [46], BIM [79], BPDA [8] and EotPgd [86].","Note that DISCO is not trained specifically for these attacks.","Metric: Standard (SA) and robust (RA) accuracy are considered.","The former measures classification accuracy on clean examples, the latter on adversarial.","The average of SA and RA is also used.","Table: NO_CAPTIONFigure: Comparison of DISCO to No Defense, Adversarially Trained, and Transformation based baselines.","(a) Cifar10, (b) Cifar100, and (c) ImageNet.","Top-row: trade-off between SA and RA.","Bottom row: average accuracy of each of theRobustBench baselines and DISCO."],["Oblivious Adversary","SOTA on RobustBench: DISCO achieves SOTA performance on RobustBench.","Table REF and REF compare DISCO to the RobustBench baselines on Cifar10 under $L_{\\infty }$ and $L_2$ Autoattack, respectively.","Baselines are categorized into (1) no defense (first block), (2) adversarially trained (second block) and (3) transformation based (third block).","The methods presented in each table are those of highest RA performance in each category.","The full table is given in the supplemental, together with those of Cifar100 and ImageNet.","Note that the RobustBench comparison slightly favours the adversarially trained methods, which use a larger classifier.","A detailed comparison to all RobustBench baselines is given in Fig.","REF , for the three datasets.","The upper row visualizes the trade-off between SA and RA.","The bottom row plots the averaged SA/RA across baselines.","Blue, green, cyan and red indicate no defense, adversarially trained baselines, transformation based baselines and DISCO, respectively.","These results show that, without a defense, the attack fools the classifier on nearly all examples.","Adversarially trained baselines improve RA by training against the adversarial examples.","Some of these [114], [48], [47], [67], [113], [57], [153], [136], [155] also leverage additional training data.","Transformation based defenses require no modification of the pre-trained classifier and can generalize across attack strategies [124], [139].","While early methods (like Jpeg Compression [40] and Bit Reduction [161]) are not competitive, recent defenses [139] outperform adversarially trained baselines on Cifar100 and ImageNet.","DISCO is an even more powerful transformation-based defense, which clearly outperforms the prior SOTA RA by a large margin (17 % on Cifar10, 19 % on Cifar100 and 19 % on ImageNet).","In the upper row of Fig.","REF , it clearly beats the prior Pareto front for the SA vs. RA trade-off.","Table REF and Table REF also show that previous transformation based methods tend to perform better for $L_2$ than $L_{\\infty }$ Autoattack.","DISCO is more robust, achieving similar RA for $L_2$ and $L_{\\infty }$ Autoattacks.","Table: Defense Transfer of L ∞ L_{\\infty } trained defenses to L 2 L_{2} attacks on Cifar10.","Top block: adversarially trained, Bottom block: transformation based.Improving SOTA Methods: While DISCO outperforms the SOTA methods on RobustBench, it can also be combined with the latter.","Table REF shows that adding DISCO improves the performance of top three ResNet50 robust baselines for ImageNet [123], [41], [152] by 16.77 (for RA) and 8.12 (for averaged SA/RA) on average.","This demonstrates the plug-and-play ability of DISCO.","Comparison to Test-Time Defenses First, we compare DISCO to four recent test-time defenses.","Following the setup of [164], DISCO is evaluated on Cifar10 using a WRN28-10 network under the PGD40 attack ($\\epsilon =8/255$ ).","While [164] reported an RA of 80.24 for the default setting, DISCO achieves 80.80, even though it is not optimized for this experiment and has much fewer parameters (1.6M vs 29.7M).","Second, a comparison to [4], [90] under Autoattack, shows that [4] achieves RAs of 79.21/40.68 and [90] of 67.79/33.16 on the Cifar10/Cifar100 datasets.","These numbers are much lower than those reported for DISCO (85.56/67.93) on Table 1 & Appendix Table C. Third, under the APgd [29] attack, [4] achieves 80.65/47.63 RA on Cifar10/Cifar100 dataset, while DISCO achieves 85.79/77.33 (Appendix Table E & Table 3).","This shows that DISCO clearly outperforms [4] on two different attacks and datasets.","Finally, like DISCO, [99] compares to defenses in RobustBench.","For Cifar10 and a WRN28-10 classifier, [99] achieves 70.64/78.58 RA under $\\epsilon _\\infty =8/255$ and $\\epsilon _2=0.5$ respectively, while DISCO achieves 85.56/88.47 (Table 1 & Table 2).","On ImageNet, [99] achieves 40.93/44.39 RA with ResNet50/WRN50, while DISCO achieves 68.2/69.5 (Appendix Table D).","In summary, DISCO outperforms all these approaches in the various settings they considered, frequently achieving large gains in RA.","Dataset Size: Table REF shows the SA and RA performance of DISCO when training pairs are sampled from a random subset of the ImageNet classes (100 and 500).","Compared to the ImageNet SOTA [123] RA of 38.14% (See Appendix), DISCO outperforms the prior art by 21.7% (59.84 vs 38.14) when trained on about 0.5% of the ImageNet training data.","Defense Transferability The transferability of the DISCO defense is investigated across attacks, classifiers and datasets.","Transfer across Attacks.","RobustBench evaluates the model on Autoattack, which includes the PGD attack used to train DISCO.","Table REF summarizes the transfer performance of DISCO, trained with PGD attacks, when subject to ten different $L_{\\infty }$ attacks at inference.","This is compared to the transfer peformance of the two top Cifar100 baselines on RobustBench.","DISCO outperforms these baselines with an average RA gain greater than 24.49%.","When compared to the baseline that uses the same classifier (WRN28-10), this gain increases to 29.5%.","Fig.","REF visualizes the gains of DISCO (red bar) on Cifar10 and ImageNet.","Among 3 datasets and 10 attacks, DISCO outperforms the baselines on 24 results.","The average gains are largest on Cifar100 and ImageNet, where the RA of the prior approaches is lower.","Note that the defense is more challenging on ImageNet, due to the higher dimensionality of its images [128].","The full table can be found in the supplemental.","We next evaluate the transfer ability of DISCO trained with the $L_{\\infty }$ PGD attacks to four $L_2$ norm inference attacks: FGSM [46], BIM [79], CW [17] and DeepFool [95].","Table REF compares the defense transferability of DISCO to both adversarially trained (top block) and transformation baselines (lower block).","DISCO generalizes well to $L_2$ attacks.","It can defend more attacks than adversarially trained baselines (top block) and is more robust than the prior SOTA transformation based defenses.","Beyond different test attacks $A_{tst}$ on a PGD-trained DISCO, we also evaluated the effect of changing the training attack $A_{trn}$ used to generate the adversarial-clean pairs on which DISCO is trained.","In rows 3-5 of Table REF , PGD [88], BIM [79] and FGSM [46] are used to generate training pairs, while Autoattack is used as testing attack.","BIM and PGD have comparable results, which are stronger than those of FGSM.","Nevertheless all methods outperform the SOTA RobustBench defense [114] for Autoattack on Cifar10, shown in the first row.","These results suggests that DISCO is robust to many combinations of training and inference attacks.","Table: NO_CAPTIONTable: NO_CAPTIONTransfer across Classifiers.","The first section of Table REF shows the results when the testing classifier is different from the training classifier.","While the ResNet18 is always used to curate the training pairs of DISCO, the testing classifier varies between ResNet18, WideResNet28 and VGG16.","The small impact of the classifier used for inference on the overall RA shows that DISCO is classifier agnostic and can be applied to multiple testing classifiers once it is trained.","Transfer across Datasets.","The evidence that adversarial attacks push images away from the natural image manifold [175], [83], [139], [40], [63], [85] and that attacks can be transferred across classifiers [34], [150], [154], [60], [149], suggest that it may be possible to transfer defenses across datasets.","This, however, has not been studied in detail in the literature, partly because adversarially trained baselines entangle the defense and the classifier training.","This is unlike DISCO and other transformation based baselines, which can be transferred across datasets.","The bottom section of Table REF shows the test performance on Cifar10 of DISCO trained on Cifar100 and ImageNet.","Since Cifar100 images are more similar to those of Cifar10 than ImageNet ones, the Cifar100 trained DISCO transfers better than that trained on ImageNet.","However, even the RA of the latter is 7.72% higher than the best RA reported on RobustBench [114].","Note that the DISCO trained on Cifar100 and ImageNet never see images from Cifar10 and the transfer is feasible because no limitation is imposed on the output size of DISCO."],["Adaptive Adversary","The adaptive adversary assumes both the classifier and defense strategy are exposed to the attacker.","As noted by [8], [139], [143], this setting is more challenging, especially for transformation based defenses.","We adopt the BPDA [8] attack, which is known as an effective attack for transformation based defenses, such as DISCO.","Fig.","REF compares the RA of DISCO trained with PGD attack to the results published for other methods in [139].","For fair comparison, DISCO is combined with a VGG16 classifier.","The figure confirms that both prior transformation defenses and a single stage DISCO ($K=1$ ) are vulnerable to an adaptive adversary.","However, without training against BPDA, DISCO is 46.76% better than prior methods.","More importantly, this gain increases substantially with $K$ , saturating at RA of 57.77% for $K=5$ stages, which significantly outperforms the SOTA by 57.35%."],["Cascade DISCO","So far, we have considered the setting where the structure of the cascade DISCO is known to the attacker.","DISCO supports a more sophisticated and practical setting, where the number $K$ of DISCO stages used by the defense is randomized on a per-image basis.","In this case, even if the use of DISCO is exposed to the attacker, there is still uncertainty about how many stages to use in the attack.","We investigated the consequences of this uncertainty by measuring the defense performance when different values of $K$ are use for attack and defense, denoted as $K_{adv}$ and $K_{def}$ , respectively.","The oblivious setting has $K_{adv}=0$ and $K_{def}\\ge 1$ , while $K_{adv}=K_{def}$ in the adaptive setting.","We now consider the case where $K_{adv} \\ne K_{def}$ .","Fig.","REF investigates the effectiveness of cascade DISCO trained with PGD attack when faced with the BPDA [8] attack in this setting, where RA($K_{adv}$ ,$K_{def}$ ) is the RA when $K_{adv}$ and $K_{def}$ are used, and $K_{adv} \\in \\lbrace i\\rbrace _{i=0}^5$ , $K_{def} \\in \\lbrace i\\rbrace _{i=1}^3$ .","Under the setting of $K_{adv} \\ne K_{def}$ , the RA is higher than that of the adaptive setting.","Take $K_{adv}=2$ for example.","Both RA(2,1)=55.3 and RA(2,3)=59.8 outperform RA(2,2)=52.","In addition, Fig.","REF compares the time to generate a single adversarial example on Cifar10 and defend against it using DISCO.","Clearly, the computational resources needed to generate an attack are significantly higher than those of the defense and the ratio of attack-to-defense cost raises with $K$ .","Both this and the good defense performance for mismatched $K$ s give the defender a strong advantage.","It appears that the defense is more vulnerable when the attacker knows $K$ (adaptive setting) and even there, as seen in the previous section, the defense can obtain the upper hand by casacading several DISCO stages.","Table: NO_CAPTION"],["Discussion, Societal Impact and Limitations","In this work, we have proposed the use of local implicit functions for adversarial defense.","Given an input adversarial image and a query location, the DISCO model is proposed to project the RGB value of each image pixel into the image manifold, conditional on deep features centered in the pixel neighborhood.","By training this projection with adversarial and clean images, DISCO learns to remove the adversarial perturbation.","Experiments demonstrate DISCO's computational efficiency, its outstanding defense performance and transfer ability across attacks, datasets and classifiers.","The cascaded version of DISCO further strengthens the defense with minor additional cost.","Limitations: While DISCO shows superior performance on the attacks studied in this work (mainly norm-bounded attacks), it remains to be tested whether it is robust to other type of attacks [138], [16], [56], [54], [80], such as one pixel attack [138], patch attacks [16], [56] or functional adversarial attack [80].","In addition, more evaluation configurations across attacks, datasets and classifiers will be investigated in the future.","Societal Impact: We hope the idea of using local implicit functions can inspire better defenses and prevent the nefarious effects of deep learning attacks.","Obviously, better defenses can also be leveraged by bad actors to improve resistance to the efforts of law enforcement, for example."],["Acknowledgments","This work was partially funded by NSF awards IIS1924937 and IIS-2041009, a gift from Amazon, a gift from Qualcomm, and NVIDIA GPU donations.","We also acknowledge and thank the use of the Nautilus platform for some of the experiments discussed above.","Appendix"],["Compare to SOTA in RobustBench","In this section, we list the quantitative result of the baselines in RobustBench [27] .","Table REF , REF and REF correspond to Fig.6(a), (b) and (c) of the main paper, respectively.","Table REF shows the baselines under Autoattack with $\\epsilon _{2}=0.5$ .","The index displayed in each table corresponds to the index shown in Fig.6 in the main paper.","The baselines of each table are grouped into No defense (first block), Adversarially trained defense in RobustBench (second block), Transformation based defense (third block) and DISCO (last block).","The results of adversarially trained baselines are copied from RobustBench, while the results of transformation-based defenses are obtained with our implementation.","For STL [139], models with different sparse constraints $\\lambda $ are used from the publicly available STL githubhttps://github.com/GitBoSun/AdvDefense_CSC.","DISCO is also combined with various classifiers for evaluation.","More discussion can be found in Sec.","4.1 of the paper.","Table: Cifar10 baselines and DISCO under Autoattack (ϵ ∞ =8/255\\epsilon _{\\infty }=8/255).","This table corresponds to Fig.","6(a) in the main paper.Table: Cifar10 baselines and DISCO under Autoattack (ϵ 2 =0.5\\epsilon _{2}=0.5).Table: Cifar100 baselines and DISCO under Autoattack (ϵ ∞ =8/255\\epsilon _{\\infty }=8/255).","This table corresponds to Fig.","6(b) in the main paper.Table: ImageNet baselines and DISCO under Autoattack (ϵ ∞ =4/255\\epsilon _{\\infty }=4/255).","This table corresponds to Fig.","6(c) in the main paper."],["Defense Transfer","In this section, we discuss the qualitative results of DISCO transferability across attacks.","Table REF , REF and REF represents the results for Cifar10, Cifar100 and ImageNet, respectively.","The corresponding plots are illustrated in Fig.","REF , REF and REF .","More discussion can be found in Sec.","4.1 of the paper.","Table: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTION"],["Improving Cifar10 and Cifar100 SOTA on RobustBench","Sec.","4.1 in the main paper shows that DISCO can improve the prior SOTA defenses on the ImageNet dataset.","In Table REF , we further investigate the gain of applying DISCO on SOTA Cifar10 and Cifar100 defenses.","The first and second block of Table REF show the gains of applying DISCO on [114], which is the prior SOTA defense against $L_2$ and $L_{\\infty }$ Autoattack on Cifar10.","DISCO also improves the prior SOTA defense [47] on Cifar100 by 2.89%.","These results indicate that, beyond being a robust defense by itself, DISCO can also be applied to existing defenses to improve their robustness."],["Kernel Size s","In this section, we ablate the kernel size used to train DISCO on ImageNet.","The kernel size $s$ controls the feature neighborhood forwarded to the local implicit module.","Table REF shows that $s=3$ achieves the best performance, which degrades for $s=5$ by a significant margin (3.26%).","This shows that while tasks like classification require large and global receptive fields, the projection of adversarial images into the natural image manifold can be done on small neighborhoods.","Given that the complexity of modeling the manifold increases with the neighborhood size, it is not surprising that larger $s$ lead to weaker performance.","This is consistent with the well known complexity of synthesizing images with global models, such as GANs.","What is somewhat surprising is that even $s=1$ is sufficient to enable a robust defense.","By default, we use $s=3$ in all our experiments."],["Computation Time for STL and DISCO","Table REF compares the inference time of STL [139], DISCO and cascade DISCO (from $K=$ 2 to 5) on Cifar10 and ImageNet.","For a single image Cifar10 of size 32x32, STL requires an Cifar10 5.9$\\times $ (0.65 vs 0.011) larger than that of DISCO ($K$ =1).","When cascade DISCO is used, inference time increases approximately linearly with $K$ .","For a single ImageNet image of size 224, STL requires 23.71 seconds while DISCO (K=1) only requires 0.027.","The inference time difference increases to $878.15\\times $ (23.71 vs 0.027) on ImageNet , which is significantly larger than that of Cifar10 (5.9$\\times $ ).","This shows that DISCO is a better defense in the sense that it can handle widely varying input image sizes with minor variations of computing cost."],["Training Details","On Cifar10 and Cifar100, we train the DISCO for 40 epochs.","On ImageNet, DISCO is only trained for 3 epochs because ImageNet images are larger and produce more random crops.","The learning rate is set to 0.0001 and the Adam optimizer is used in all experiments.","All experiments are conducted using Pytorch [110].","All time measurements, for both baselines and DISCO, are made on a single Nvidia Titan Xp GPU with Intel Xeon CPU E5-2630, with batch size 1 and averaged over 100 images."],["Adopted Code and Benchmark","In this section, we list the url links that are used for training and evaluating DISCO.","To create the adversarial-clean training pairs, we adopt the code from TorchAttackhttps://adversarial-attacks-pytorch.readthedocs.io/en/latest/ and Areshttps://github.com/thu-ml/ares, which support the multiple attack methods.","These attack methods are then used to attack pretrained classifiers on Cifar10, Cifar100 and ImageNet.","We use the ResNet18 classifiers from Ares for Cifar10, the WideResNet Cifar100 classifiers from this repository https://github.com/xternalz/WideResNet-pytorch and the ResNet18 ImageNet classifiers of Pytorch [110].","To evaluate DISCO, we adopt Autoattack from RobustBench [27]https://github.com/RobustBench/robustbench and compare to the pretrained defenses on the RobustBench leaderboard.","In addition to Autoattack, we use the AdverTorchhttps://github.com/BorealisAI/advertorch library to implement the BPDA attack [8] and the TorchAttackhttps://adversarial-attacks-pytorch.readthedocs.io/en/latest/ library for other attacks, like FGSM [46] and BIM [79].","For the adversarially trained defense baselines, we adopt the pretrained weights from RobustBench [27]https://github.com/RobustBench/robustbench, while the codes for transformation based baselines are adopted from Ares, Cifar autoencoder https://github.com/chenjie/PyTorch-CIFAR-10-autoencoder and STL [139].","To implement DISCO, we use code from LIIFhttps://github.com/yinboc/liif [24]."],["Visualizations","DISCO defense outputs against FGSM [46] and BIM [79] and PGD [88] attacks are visualized in Fig.","REF , REF and REF , respectively.","Take Fig.","REF for example.","The first and second rows show the clean and adversarial images, while rows 3-5 show the output of DISCO and cascade DISCO ($K=2$ and $K=3$ ).","Clearly, both DISCO and its cascade version can effectively remove the adversarial perturbation.","Note that these images are produced from the same DISCO model without retraining for any attack.","Figure: Comparison of Clean image, Adversarial image and DISCO output from K=K= 1 to 3 under FGSM attack.Figure: Comparison of Clean image, Adversarial image and DISCO output from K=K= 1 to 3 under BIM attack.Figure: Comparison of Clean image, Adversarial image and DISCO output from K=K= 1 to 3 under PGD attack."]],"string":"[\n [\n \"DISCO: Adversarial Defense with Local Implicit Functions\"\n ],\n [\n \"Abstract The problem of adversarial defenses for image classification, where the goal is to robustify a classifier against adversarial examples, is considered.\",\n \"Inspired by the hypothesis that these examples lie beyond the natural image manifold, a novel aDversarIal defenSe with local impliCit functiOns (DISCO) is proposed to remove adversarial perturbations by localized manifold projections.\",\n \"DISCO consumes an adversarial image and a query pixel location and outputs a clean RGB value at the location.\",\n \"It is implemented with an encoder and a local implicit module, where the former produces per-pixel deep features and the latter uses the features in the neighborhood of query pixel for predicting the clean RGB value.\",\n \"Extensive experiments demonstrate that both DISCO and its cascade version outperform prior defenses, regardless of whether the defense is known to the attacker.\",\n \"DISCO is also shown to be data and parameter efficient and to mount defenses that transfers across datasets, classifiers and attacks.\"\n ],\n [\n \"Introduction\",\n \"It has long been hypothesized that vision is only possible because the natural world contains substantial statistical regularities, which are exploited by the vision system to overcome the difficulty of scene understanding [12], [49], [119], [42], [146], [43], [58], [130], [137], [101], [13], [142].\",\n \"Under this hypothesis, natural images form a low-dimension manifold in image space, denoted as the image manifold, to which human vision is highly tuned.\",\n \"While deep neural networks (DNNs) [140], [52], [167], [141], [131] aim to classify natural images with human-like accuracy, they have been shown prone to adversarial attacks that, although imperceptible to humans, significantly decrease their performance.\",\n \"As shown in Fig.\",\n \"REF (a) and (b), these attacks typically consist of adding an imperceptible perturbation, which can be generated in various manners [46], [88], [17], [79], to the image.\",\n \"Over the past few years, the introduction of more sophisticated attacks has exposed a significant vulnerability of DNNs to this problem [138], [38], [37], [29], [29], [125].\",\n \"In fact, it has been shown that adversarial examples crafted with different classifiers and optimization procedures even transfer across networks [34], [117], [159], [61], [108].\",\n \"A potential justification for the success of adversarial attacks and their transferability is that they create images that lie just barely outside the image manifold [135], [124], [81], [45], [71], [11].\",\n \"We refer to these images as barely outliers.\",\n \"While humans have the ability to project these images into the manifold, probably due to an history of training under adverse conditions, such as environments with low-light or ridden with occlusions, this is not the case for current classifiers.\",\n \"A key factor to the success of the human projection is likely the accurate modeling of the image manifold.\",\n \"Hence, several defenses against adversarial attacks are based on models of natural image statistics.\",\n \"These are usually global image representations [92], [139], [124], [166] or conditional models of image pixel statistics [135], [11], [145], [122], [74].\",\n \"For example, PixelDefend [135] and HPD [11] project malicious images into the natural image manifold using the PixelCNN model [74], which predicts a pixel value conditioned on the remainder of the image.\",\n \"However, these strategies [135], [11], [92], [139], [124] can be easily defeated by existing attacks.\",\n \"We hypothesize that this is due to the difficulty of learning generative image models, which require global image modelling, a highly complex task.\",\n \"It is well known that the synthesis of realistic natural images requires very large model sizes and training datasets [69], [14], [70].\",\n \"Even with these, it is not clear that the manifold is modeled in enough detail to defend adversarial attacks.\",\n \"In this work, we argue that, unlike image synthesis, the manifold projection required for adversarial defense is a conditional operation: the synthesis of a natural image given the perturbed one.\",\n \"Assuming that the attack does not alter the global structure of the image (which would likely not be imperceptible to humans) it should suffice for this function to be a conditional model of local image (i.e.\",\n \"patch) statistics.\",\n \"We argue that this conditional modeling can be implemented with an implicit function [24], [134], [94], [121], [100], [73], [65], [25], where the network learns a conditional representation of the image appearance in the neighborhood of each pixel, given a feature extracted at that pixel.\",\n \"This strategy is denoted aDversarIal defenSe with local impliCit functiOns (DISCO).\",\n \"Local implicit models have recently been shown to provide impressive results for 3D modeling [134], [94], [121], [100], [160], [73], [65], [25], [93], [109], [44], [156] and image interpolation [24].\",\n \"We show that such models can be trained to project barely outliers into the patch manifold, with much smaller parameter and dataset sizes than generative models, while enabling much more precise control of the manifold projection operation.\",\n \"This is illustrated in Fig.\",\n \"REF , which presents an image, its adversarial attack, and the output of the DISCO defense.\",\n \"To train DISCO, a dataset of adversarial-clean pairs is first curated.\",\n \"During training, DISCO inputs an adversarial image and a query pixel location, for which it predicts a new RGB value.\",\n \"This is implemented with a feature encoder and a local implicit function.\",\n \"The former is composed by a set of residual blocks with stacked convolution layers and produces a deep feature per pixel.\",\n \"The latter consumes the query location and the features in a small neighborhood of the query location.\",\n \"The implicit function is learned to minimize the $L_1$ loss between the predicted RGB value and that of the clean image.\",\n \"The restriction of the manifold modeling to small image neighborhoods is a critical difference with respect to previous defenses based on the modeling of the natural image manifold.\",\n \"Note that, as shown in Fig.\",\n \"REF , DISCO does not project the entire image into the manifold, only each pixel neighborhood.\",\n \"This considerably simplifies the modeling and allows a much more robust defense in a parameter and data efficient manner.\",\n \"This is demonstrated by evaluating the performance of DISCO under both the oblivious and adaptive settings [163], [115], [161].\",\n \"Under the oblivious setting, the popular RobustBench [27] benchmark is considered, for both $L_{\\\\infty }$ and $L_{2}$ attacks with Autoattack [29].\",\n \"DISCO achieves SOTA robust accuracy (RA) performance, e.g.\",\n \"outperforming the prior art on Cifar10, Cifar100 and ImageNet by 17%, 19% and 19% on $L_{\\\\infty }$ Autoattack.\",\n \"A comparison to recent test-time defenses [164], [4], [120], [28], [90], [99] also shows that DISCO is a more effective defensive strategy across various datasets and attacks.\",\n \"Furthermore, a study of the defense transferability across datasets, classifiers and attacks shows that the DISCO defense maintains much of its robustness even when deployed in a setting that differs from that used for training by any combination of these three factors.\",\n \"Finally, the importance of the local manifold modeling is illustrated by experiments on ImageNet [35], where it is shown that even when trained with only 0.5% of the dataset DISCO outperforms all prior defenses.\",\n \"Under the adaptive setting, DISCO is evaluated using the BPDA [8] attack, known to circumvent most defenses based on image transformation [135], [124], [139], [161], [40].\",\n \"While DISCO is more vulnerable under this setting, where the defense is known to the attacker, it still outperforms existing approaches by 46.76%.\",\n \"More importantly, we show that the defense can be substantially strengthened by cascading DISCO stages, which magnifies the gains of DISCO to 57.77%.\",\n \"This again leverages the parameter efficiency of the local modeling of image statistics, which allows the implementation of DISCO cascades with low complexity.\",\n \"The ability to cascade DISCO stages also allows a new type of defense, where the number of DISCO stages is randomized on a per image basis.\",\n \"This introduces some degree of uncertainty about the defense even under the adaptive setting and further improves robustness.\",\n \"Overall, this work makes four contributions.\",\n \"First, it proposes the use of defenses based purely on the conditional modeling of local image statistics.\",\n \"Second, it introduces a novel defense of this type, DISCO, based on local implicit functions.\",\n \"Third, it leverages the parameter efficiency of the local modeling to propose a cascaded version of DISCO that is shown robust even to adaptive attacks.\",\n \"Finally, DISCO is shown to outperform prior defenses on RobustBench [27] and other 11 attacks, as well as test-time defenses under various experimental settings.\",\n \"Extensive ablations demonstrate that DISCO has unmatched defense transferrability in the literature, across datasets, attacks and classifiers.\"\n ],\n [\n \"Related Work\",\n \"Adversarial Attack and Defense.\",\n \"We briefly review adversarial attacks and defenses for classification and prior art related to our work.\",\n \"Please refer to [19], [103], [2] for more complete reviews.\",\n \"Adversarial Attacks aim to fool the classifier by generating an imperceptible perturbation (under $L_p$ norm constraint) that is applied to the clean image.\",\n \"Attack methods have evolved from simple addition of sign gradient, as in FGSM [46], to more sophisticated approaches [79], [17], [138], [144], [88], [29], [38], [37], [29].\",\n \"While most white-box attacks assume access to the classifier gradient, BPDA [8] proposed a gradient approximation attack that can circumvent defenses built on obfuscated gradients.\",\n \"In general, these attacks fall into two settings, oblivious or adaptive, depending on whether the attacker is aware of the defense strategy [163], [115], [161].\",\n \"DISCO is evaluated under both settings.\",\n \"Adversarial Defenses can be categorized into adversarially trained and transformation based.\",\n \"The former are trained against adversarial examples generated on-the-fly during training [114], [48], [47], [113], [126], [104], [126], allowing the resulting robust classifier to defend against the adversarial examples.\",\n \"While adversarially trained defenses dominate the literature, they are bound together with the classifier.\",\n \"Hence, re-training is required if the classifier changes and the cost of adversarial training increases for larger classifiers.\",\n \"Transformation based defenses [50], [161], [64], [135], [139] instead introduce an additional defense module, which can be applied to many classifiers.\",\n \"This module preprocesses the input image before passing it to the classifier.\",\n \"The proposed preprocessing steps include JPEG compression [33], [40], [87], bit reduction [50], [161], [64], pixel deflection [111] or applications of random transformations [158], [50] and median filters [102].\",\n \"Beyond pixel space defenses, malicious images can also be reconstructed to better match natural image statistics using autoencoders [92], [139], GANs [124], [7], [166], or other generative models, such as the PixelCNN [122].\",\n \"The latter is used to project the malicious image into the natural image manifold by methods like PixelDefend [135] or HPD [11].\",\n \"These methods can only produce images of fixed size [92], [124], [139], [166] and model pixel likelihoods over the entire image [135], [11].\",\n \"This is unlike DISCO, which models conditional local statistics and can produce outputs of various size.\",\n \"The idea of performing adversarial purification before feeding the image into the classifier is central to a recent wave of test-time defenses [164], [4], [90], [99].\",\n \"[164] addresses the impracticality of previous Monte-Carlo purification models by introducing a Denoising Score-Matching and a random noise injection mechanism.\",\n \"[4] prepends an anti-adversary layer to the classifier, with the goal of maximizing the classifier confidence of the predicted label.\",\n \"[90] reverses the adversarial examples using self-supervised contrastive loss.\",\n \"[99] proposed a diffusion model for adversarial removal.\",\n \"Unlike these prior works, DISCO purifies the adversarial image by modeling the local patch statistics.\",\n \"Such characteristics results in data and parameter efficiency, which have not been demonstrated for [164], [4], [90], [99].\",\n \"Furthermore, DISCO outperforms all prior works in terms of robust accuracy, under the various settings they proposed.\",\n \"Implicit Function.\",\n \"refers to the use of a neural network to model a continuous function [134].\",\n \"This has been widely used in applications involving audio [176], [51], [134], 2D images [24], [39] and 3D shapes [134], [94], [121], [100], [160], [73], [65], [25], [93], [82], [109], [44], [156].\",\n \"In the 3D literature, local implicit functions have become popular models of object shape [100], [121], [109] or complex 3D scenes [94].\",\n \"This also inspired 2D applications to super-resolution [24], image [68] and video generation [165].\",\n \"In the adversarial attack literature, implicit functions have recently been proposed to restore adversarial point clouds of 3D shape, through the IF-Defense [156].\",\n \"To the best of our knowledge, ours is the first paper to propose local implicit functions for 2D adversarial defense.\",\n \"Table: NO_CAPTION\"\n ],\n [\n \"Method\",\n \"In this section, we introduce the architecture of DISCO and its training and testing procedure.\"\n ],\n [\n \"Motivation\",\n \"Under the hypothesis that natural images lie on a low-dimension image manifold, classification networks can be robustified by learning to project barely outliers (i.e.\",\n \"adversarial images) into the manifold, a process that can be seen as manifold thickening.\",\n \"Images in a shell around the manifold are projected into it, leaving a larger margin to images that should be recognized as outliers.\",\n \"While this idea has been studied [45], [71], [85], [63], its success hinges on the ability of classification models to capture the complexities of the image manifold.\",\n \"This is a very hard problem, as evidenced by the difficulty of model inversion algorithms that aim to synthesize images with a classifier [148], [89], [98], [162].\",\n \"These algorithms fail to produce images comparable to the state of the art in generative modeling, such as GANs [70], [69], [15].\",\n \"Recently, however, it has been shown that it is possible to synthesize realistic images and 3D shapes with implicit functions, through the use of deep networks [44], [24], [134], [94], [121], [100], [65], [25] that basically memorize images or objects as continuous functions.\",\n \"The assumption behind DISCO is that these implicit functions can capture the local statistics of images or 3D shapes, and can be trained for manifold thickening, that is to learn how to projecting barely outliers into the image manifold.\"\n ],\n [\n \"Model Architecture and Training\",\n \"Data Preparation To train DISCO, a dataset $D=\\\\lbrace (x^i_{cln}, x^i_{adv})\\\\rbrace _{i=1}^N$ containing a set of paired clean $x^i_{cln}$ and adversarial $x^i_{adv}$ images is curated.\",\n \"For this, a classifier $P_{trn}$ and an attack procedure $A_{trn}$ are first selected.\",\n \"For each image $x^i_{cln}$ , the adversarial image $x^i_{adv}$ is generated by attacking the predictions $P_{trn}(x^i_{cln})$ using $A_{trn}$ , as shown in Fig.\",\n \"REF (a).\",\n \"Training As shown in Fig.\",\n \"REF (b), the DISCO defense is trained with pairs of random patches cropped at the same location of the images $x_{cln}$ and $x_{adv}$ .\",\n \"For example, random patches of size $48\\\\times 48$ are sampled from training pairs of the ImageNet dataset [35].\",\n \"The network is then trained to minimize the L1 loss between the RGB values of the clean $x_{cln}$ and defense output $x_{def}$ .\",\n \"Architecture DISCO performs manifold thickening by leveraging the LIIF [25] architecture to purify adversarial patches.\",\n \"It takes an adversarial image $x_{adv} \\\\in \\\\mathbb {R}^{H\\\\times W \\\\times 3}$ and a query pixel location $p=[i,j] \\\\in \\\\mathbb {R}^2$ as input and predicts a clean RGB value $v \\\\in \\\\mathbb {R}^3$ at the query pixel location, ideally identical to the pixel value of the clean image $x_{cln} \\\\in \\\\mathbb {R}^{H\\\\times W \\\\times 3}$ at that location.\",\n \"The defense output $x_{def} \\\\in \\\\mathbb {R}^{H^{\\\\prime }\\\\times W^{\\\\prime } \\\\times 3}$ can then be synthesized by predicting a RGB value for each pixel location in a grid of size ${H^{\\\\prime }\\\\times W^{\\\\prime } \\\\times 3}$ .\",\n \"Note that it is not a requirement that the size of $x_{def}$ be the same as that of $x_{cln}$ .\",\n \"In fact, the size of $x_{def}$ could be changed during inference.\",\n \"Figure: The DISCO architecture includes an encoder and a local implicit module.\",\n \"The network is trained to map Adversarial into Defense images, using an L 1 L_1 loss to Clean images.To implement this, DISCO is composed of two modules, illustrated in Fig.\",\n \"REF .\",\n \"The first is an encoder $E$ that extracts the per-pixel feature of an input image $x$ .\",\n \"The encoder architecture resembles the design of EDSR [84], originally proposed for super-resolution.\",\n \"It contains a sequence of 15 residual blocks, each composed of a convolution layer, a ReLu layer and a second convolution layer.\",\n \"The encoder output is a feature map $f=E(x) \\\\in \\\\mathbb {R}^{H\\\\times W \\\\times C}$ , with $C=64$ channels.\",\n \"The feature at location $p=[i,j]$ is denoted as $f_{ij}$ .\",\n \"The second module is the the local implicit module $L$ , which is implemented by a MLP.\",\n \"Given query pixel location $p$ , $L$ first finds the nearest pixel value $p^*=[i^*,j^*]$ in the input image $x$ and corresponding feature $f_{i^*j^*}$ .\",\n \"$L$ then takes the features in the neighborhood of $p^*$ into consideration to predict the clean RGB value $v$ .\",\n \"More specifically, let $\\\\hat{f}_{i^*j^*}$ denote a concatenation of the features whose location is within the kernel size $s$ of $p^*$ .\",\n \"The local implicit module $L$ takes the concatenated feature, the relative position $r=p-p^*$ between $p$ and $p^*$ , and the pixel shape as input, to predict a RGB value $v$ .\",\n \"By default, the kernel size $s$ is set to be 3.\",\n \"Since the network implements a continuous function, based only on neighboring pixels, the original grid size $H\\\\times W$ is not important.\",\n \"The image coordinates can be normalized so that $(i,j) \\\\in [-1,1]^2$ and the pixel shape is the height and width of the output pixel in the normalized coordinates.\",\n \"This makes DISCO independent of the original image size or resolution.\"\n ],\n [\n \"Inference\",\n \"For inference, DISCO takes either a clean or an adversarial image as input.\",\n \"Given a specified output size for $x_{def}$ , DISCO loops over all the output pixel locations, predicting an RGB value per location.\",\n \"Note that this is not computationally intensive because the encoder feature map $f=E(x)$ is computed once and used to the predict the RGB values of all query pixel locations.\",\n \"Furthermore, while the training pairs are generated with classifier $P_{trn}$ and attack $A_{trn}$ , the inference time classifier $P_{tst}$ and attack $A_{tst}$ could be different.\",\n \"In the experimental section we show that DISCO is quite flexible, performing well when (1) $P_{trn}$ and $P_{tst}$ consume images of different input size and (2) the attack, classifier and dataset used for inference are different than those used for training.\",\n \"In fact, DISCO is shown to be more robust to these configuration changes than previous methods.\"\n ],\n [\n \"DISCO Cascades\",\n \"DISCO is computationally very appealing because it disentangles the training of the defense from that of the classifier.\",\n \"This can be a big practical advantage, since classifier retraining is needed whenever training settings, such as architecture, hyper-parameters, or number of classes, change.\",\n \"Adversarial defenses require retraining on the entire dataset when this is the case, which is particularly expensive for large models (like SENet [55] or EfficientNet [141]) trained on large datasets (like ImageNet [35] and OpenImages [75]).\",\n \"Unsurprisingly, RobustBench [27], one of the largest adversarial learning benchmarks, reports more than 70 baselines for Cifar10, but less than 5 on ImageNet.\",\n \"DISCO does not have this defense complexity, since it is trained independently of the classifier.\",\n \"Furthermore, because DISCO is a model of local statistics, it is particularly parameter efficient.\",\n \"As shown in Fig.\",\n \"REF , DISCO has a lightweight design with only 1.6M parameters, which is significantly less than most recent classifier [141], [55], [131], [52] and GAN [70], [14] models with good performance for ImageNet-like images.\",\n \"This also leads to a computationally efficient defense.\",\n \"Our experiments show that DISCO can be trained with only 50,000 training pairs.\",\n \"In fact, we show that it can beat the prior SOTA using less than 0.5% of ImageNet as training data (Table REF ).\",\n \"One major benefit of this efficiency is that it creates a large unbalance between the costs of defense and attack.\",\n \"Consider memory usage, which is dominated by the computation of gradients needed for either the attack or the backpropagation of training.\",\n \"Let $N_d$ and $N_c$ be the number of parameters of the DISCO network and classifier, respectively.\",\n \"The per image memory cost of training the DISCO defense is $O(N_d)$ .\",\n \"On the other hand, the attack cost depends on the information available to the attacker.\",\n \"We consider two settings, commonly considered in the literature.\",\n \"In the oblivious setting, only the classifier is known to that attacker and the attack has cost $O(N_c)$ .\",\n \"In the adaptive setting, both the classifier and the DISCO are exposed and backpropagation has memory cost $O(N_c + N_d)$ .\",\n \"In experiments, we show that DISCO is quite effective against oblivious attacks.\",\n \"Adaptive attacks are more challenging.\",\n \"However, as shown in Fig REF , it is usually the case that $N_c > N_d$ , making the complexity of the attack larger than that of the defense.\",\n \"This is unlike adversarial training, where attack and defense require backpropagation on the same model and thus have the same per-image cost.\",\n \"This asymmetry between the memory cost of the attack and defense under DISCO can be magnified by cascading DISCO networks.\",\n \"If $K$ identical stages of DISCO are cascaded, the defense complexity remains $O(N_d)$ but that of the attack raises to $O(N_c + K N_d)$ .\",\n \"Hence the ratio of attack-to-defense cost raises to $O(K + N_c/N_d)$ .\",\n \"Interestingly, our experiments (see Section REF ) show that when $K$ is increased the defense performance of the DISCO cascade increases as well.\",\n \"Hence, DISCO cascades combine high robust accuracy with a large ratio of attack-to-defense cost.\"\n ],\n [\n \"Experiments\",\n \"In this section, we discuss experiments performed to evaluate the robustness of DISCO.\",\n \"Results are discussed for both the oblivious and adaptive settings [163], [115], [161] and each result is averaged over 3 trials.\",\n \"$\\\\epsilon _{p}$ denotes the perturbation magnitude under the $L_p$ norm.\",\n \"All experiments are conducted on a single Nvidia Titan Xp GPU with Intel Xeon CPU E5-2630 using Pytorch [110].\",\n \"Please see appendix for more training details, quantitative results and visualizations.\",\n \"We adopt the code from LIIF [24] for implementation.\",\n \"Training Dataset: The following training configuration is used unless noted.\",\n \"Three datasets are considered: Cifar10 [76], Cifar100 [77] and Imagenet [35].\",\n \"For each, 50,000 adversarial-clean training pairs are curated.\",\n \"For Cifar10 and Cifar100, these are the images in the training set, while for ImageNet, 50 images are randomly sampled per class.\",\n \"Following RobustBench [27], the evaluation is then conducted on the test set of each dataset.\",\n \"To create training pairs, PGD [88] ($\\\\epsilon _{\\\\infty }=8/255$ with step size is 2/255 and the number of steps 100) is used to attack a ResNet18 and a WideResNet28 on Cifar10/ImageNet and Cifar100, respectively.\",\n \"Attack and Benchmark: DISCO is evaluated on RobustBench [27], which contains more than 100 baselines evaluated using Autoattack [29].\",\n \"This is an ensemble of four sequential attacks, including the PGD [88] attack with two different optimization losses, the FAB attack [26] and the black-box Square Attack [5].\",\n \"DISCO is compared to defense baselines under both $L_{\\\\infty }$ and $L_{2}$ norms.\",\n \"To study defense generalization, 11 additional attacks are considered, including FGSM [46], BIM [79], BPDA [8] and EotPgd [86].\",\n \"Note that DISCO is not trained specifically for these attacks.\",\n \"Metric: Standard (SA) and robust (RA) accuracy are considered.\",\n \"The former measures classification accuracy on clean examples, the latter on adversarial.\",\n \"The average of SA and RA is also used.\",\n \"Table: NO_CAPTIONFigure: Comparison of DISCO to No Defense, Adversarially Trained, and Transformation based baselines.\",\n \"(a) Cifar10, (b) Cifar100, and (c) ImageNet.\",\n \"Top-row: trade-off between SA and RA.\",\n \"Bottom row: average accuracy of each of theRobustBench baselines and DISCO.\"\n ],\n [\n \"Oblivious Adversary\",\n \"SOTA on RobustBench: DISCO achieves SOTA performance on RobustBench.\",\n \"Table REF and REF compare DISCO to the RobustBench baselines on Cifar10 under $L_{\\\\infty }$ and $L_2$ Autoattack, respectively.\",\n \"Baselines are categorized into (1) no defense (first block), (2) adversarially trained (second block) and (3) transformation based (third block).\",\n \"The methods presented in each table are those of highest RA performance in each category.\",\n \"The full table is given in the supplemental, together with those of Cifar100 and ImageNet.\",\n \"Note that the RobustBench comparison slightly favours the adversarially trained methods, which use a larger classifier.\",\n \"A detailed comparison to all RobustBench baselines is given in Fig.\",\n \"REF , for the three datasets.\",\n \"The upper row visualizes the trade-off between SA and RA.\",\n \"The bottom row plots the averaged SA/RA across baselines.\",\n \"Blue, green, cyan and red indicate no defense, adversarially trained baselines, transformation based baselines and DISCO, respectively.\",\n \"These results show that, without a defense, the attack fools the classifier on nearly all examples.\",\n \"Adversarially trained baselines improve RA by training against the adversarial examples.\",\n \"Some of these [114], [48], [47], [67], [113], [57], [153], [136], [155] also leverage additional training data.\",\n \"Transformation based defenses require no modification of the pre-trained classifier and can generalize across attack strategies [124], [139].\",\n \"While early methods (like Jpeg Compression [40] and Bit Reduction [161]) are not competitive, recent defenses [139] outperform adversarially trained baselines on Cifar100 and ImageNet.\",\n \"DISCO is an even more powerful transformation-based defense, which clearly outperforms the prior SOTA RA by a large margin (17 % on Cifar10, 19 % on Cifar100 and 19 % on ImageNet).\",\n \"In the upper row of Fig.\",\n \"REF , it clearly beats the prior Pareto front for the SA vs. RA trade-off.\",\n \"Table REF and Table REF also show that previous transformation based methods tend to perform better for $L_2$ than $L_{\\\\infty }$ Autoattack.\",\n \"DISCO is more robust, achieving similar RA for $L_2$ and $L_{\\\\infty }$ Autoattacks.\",\n \"Table: Defense Transfer of L ∞ L_{\\\\infty } trained defenses to L 2 L_{2} attacks on Cifar10.\",\n \"Top block: adversarially trained, Bottom block: transformation based.Improving SOTA Methods: While DISCO outperforms the SOTA methods on RobustBench, it can also be combined with the latter.\",\n \"Table REF shows that adding DISCO improves the performance of top three ResNet50 robust baselines for ImageNet [123], [41], [152] by 16.77 (for RA) and 8.12 (for averaged SA/RA) on average.\",\n \"This demonstrates the plug-and-play ability of DISCO.\",\n \"Comparison to Test-Time Defenses First, we compare DISCO to four recent test-time defenses.\",\n \"Following the setup of [164], DISCO is evaluated on Cifar10 using a WRN28-10 network under the PGD40 attack ($\\\\epsilon =8/255$ ).\",\n \"While [164] reported an RA of 80.24 for the default setting, DISCO achieves 80.80, even though it is not optimized for this experiment and has much fewer parameters (1.6M vs 29.7M).\",\n \"Second, a comparison to [4], [90] under Autoattack, shows that [4] achieves RAs of 79.21/40.68 and [90] of 67.79/33.16 on the Cifar10/Cifar100 datasets.\",\n \"These numbers are much lower than those reported for DISCO (85.56/67.93) on Table 1 & Appendix Table C. Third, under the APgd [29] attack, [4] achieves 80.65/47.63 RA on Cifar10/Cifar100 dataset, while DISCO achieves 85.79/77.33 (Appendix Table E & Table 3).\",\n \"This shows that DISCO clearly outperforms [4] on two different attacks and datasets.\",\n \"Finally, like DISCO, [99] compares to defenses in RobustBench.\",\n \"For Cifar10 and a WRN28-10 classifier, [99] achieves 70.64/78.58 RA under $\\\\epsilon _\\\\infty =8/255$ and $\\\\epsilon _2=0.5$ respectively, while DISCO achieves 85.56/88.47 (Table 1 & Table 2).\",\n \"On ImageNet, [99] achieves 40.93/44.39 RA with ResNet50/WRN50, while DISCO achieves 68.2/69.5 (Appendix Table D).\",\n \"In summary, DISCO outperforms all these approaches in the various settings they considered, frequently achieving large gains in RA.\",\n \"Dataset Size: Table REF shows the SA and RA performance of DISCO when training pairs are sampled from a random subset of the ImageNet classes (100 and 500).\",\n \"Compared to the ImageNet SOTA [123] RA of 38.14% (See Appendix), DISCO outperforms the prior art by 21.7% (59.84 vs 38.14) when trained on about 0.5% of the ImageNet training data.\",\n \"Defense Transferability The transferability of the DISCO defense is investigated across attacks, classifiers and datasets.\",\n \"Transfer across Attacks.\",\n \"RobustBench evaluates the model on Autoattack, which includes the PGD attack used to train DISCO.\",\n \"Table REF summarizes the transfer performance of DISCO, trained with PGD attacks, when subject to ten different $L_{\\\\infty }$ attacks at inference.\",\n \"This is compared to the transfer peformance of the two top Cifar100 baselines on RobustBench.\",\n \"DISCO outperforms these baselines with an average RA gain greater than 24.49%.\",\n \"When compared to the baseline that uses the same classifier (WRN28-10), this gain increases to 29.5%.\",\n \"Fig.\",\n \"REF visualizes the gains of DISCO (red bar) on Cifar10 and ImageNet.\",\n \"Among 3 datasets and 10 attacks, DISCO outperforms the baselines on 24 results.\",\n \"The average gains are largest on Cifar100 and ImageNet, where the RA of the prior approaches is lower.\",\n \"Note that the defense is more challenging on ImageNet, due to the higher dimensionality of its images [128].\",\n \"The full table can be found in the supplemental.\",\n \"We next evaluate the transfer ability of DISCO trained with the $L_{\\\\infty }$ PGD attacks to four $L_2$ norm inference attacks: FGSM [46], BIM [79], CW [17] and DeepFool [95].\",\n \"Table REF compares the defense transferability of DISCO to both adversarially trained (top block) and transformation baselines (lower block).\",\n \"DISCO generalizes well to $L_2$ attacks.\",\n \"It can defend more attacks than adversarially trained baselines (top block) and is more robust than the prior SOTA transformation based defenses.\",\n \"Beyond different test attacks $A_{tst}$ on a PGD-trained DISCO, we also evaluated the effect of changing the training attack $A_{trn}$ used to generate the adversarial-clean pairs on which DISCO is trained.\",\n \"In rows 3-5 of Table REF , PGD [88], BIM [79] and FGSM [46] are used to generate training pairs, while Autoattack is used as testing attack.\",\n \"BIM and PGD have comparable results, which are stronger than those of FGSM.\",\n \"Nevertheless all methods outperform the SOTA RobustBench defense [114] for Autoattack on Cifar10, shown in the first row.\",\n \"These results suggests that DISCO is robust to many combinations of training and inference attacks.\",\n \"Table: NO_CAPTIONTable: NO_CAPTIONTransfer across Classifiers.\",\n \"The first section of Table REF shows the results when the testing classifier is different from the training classifier.\",\n \"While the ResNet18 is always used to curate the training pairs of DISCO, the testing classifier varies between ResNet18, WideResNet28 and VGG16.\",\n \"The small impact of the classifier used for inference on the overall RA shows that DISCO is classifier agnostic and can be applied to multiple testing classifiers once it is trained.\",\n \"Transfer across Datasets.\",\n \"The evidence that adversarial attacks push images away from the natural image manifold [175], [83], [139], [40], [63], [85] and that attacks can be transferred across classifiers [34], [150], [154], [60], [149], suggest that it may be possible to transfer defenses across datasets.\",\n \"This, however, has not been studied in detail in the literature, partly because adversarially trained baselines entangle the defense and the classifier training.\",\n \"This is unlike DISCO and other transformation based baselines, which can be transferred across datasets.\",\n \"The bottom section of Table REF shows the test performance on Cifar10 of DISCO trained on Cifar100 and ImageNet.\",\n \"Since Cifar100 images are more similar to those of Cifar10 than ImageNet ones, the Cifar100 trained DISCO transfers better than that trained on ImageNet.\",\n \"However, even the RA of the latter is 7.72% higher than the best RA reported on RobustBench [114].\",\n \"Note that the DISCO trained on Cifar100 and ImageNet never see images from Cifar10 and the transfer is feasible because no limitation is imposed on the output size of DISCO.\"\n ],\n [\n \"Adaptive Adversary\",\n \"The adaptive adversary assumes both the classifier and defense strategy are exposed to the attacker.\",\n \"As noted by [8], [139], [143], this setting is more challenging, especially for transformation based defenses.\",\n \"We adopt the BPDA [8] attack, which is known as an effective attack for transformation based defenses, such as DISCO.\",\n \"Fig.\",\n \"REF compares the RA of DISCO trained with PGD attack to the results published for other methods in [139].\",\n \"For fair comparison, DISCO is combined with a VGG16 classifier.\",\n \"The figure confirms that both prior transformation defenses and a single stage DISCO ($K=1$ ) are vulnerable to an adaptive adversary.\",\n \"However, without training against BPDA, DISCO is 46.76% better than prior methods.\",\n \"More importantly, this gain increases substantially with $K$ , saturating at RA of 57.77% for $K=5$ stages, which significantly outperforms the SOTA by 57.35%.\"\n ],\n [\n \"Cascade DISCO\",\n \"So far, we have considered the setting where the structure of the cascade DISCO is known to the attacker.\",\n \"DISCO supports a more sophisticated and practical setting, where the number $K$ of DISCO stages used by the defense is randomized on a per-image basis.\",\n \"In this case, even if the use of DISCO is exposed to the attacker, there is still uncertainty about how many stages to use in the attack.\",\n \"We investigated the consequences of this uncertainty by measuring the defense performance when different values of $K$ are use for attack and defense, denoted as $K_{adv}$ and $K_{def}$ , respectively.\",\n \"The oblivious setting has $K_{adv}=0$ and $K_{def}\\\\ge 1$ , while $K_{adv}=K_{def}$ in the adaptive setting.\",\n \"We now consider the case where $K_{adv} \\\\ne K_{def}$ .\",\n \"Fig.\",\n \"REF investigates the effectiveness of cascade DISCO trained with PGD attack when faced with the BPDA [8] attack in this setting, where RA($K_{adv}$ ,$K_{def}$ ) is the RA when $K_{adv}$ and $K_{def}$ are used, and $K_{adv} \\\\in \\\\lbrace i\\\\rbrace _{i=0}^5$ , $K_{def} \\\\in \\\\lbrace i\\\\rbrace _{i=1}^3$ .\",\n \"Under the setting of $K_{adv} \\\\ne K_{def}$ , the RA is higher than that of the adaptive setting.\",\n \"Take $K_{adv}=2$ for example.\",\n \"Both RA(2,1)=55.3 and RA(2,3)=59.8 outperform RA(2,2)=52.\",\n \"In addition, Fig.\",\n \"REF compares the time to generate a single adversarial example on Cifar10 and defend against it using DISCO.\",\n \"Clearly, the computational resources needed to generate an attack are significantly higher than those of the defense and the ratio of attack-to-defense cost raises with $K$ .\",\n \"Both this and the good defense performance for mismatched $K$ s give the defender a strong advantage.\",\n \"It appears that the defense is more vulnerable when the attacker knows $K$ (adaptive setting) and even there, as seen in the previous section, the defense can obtain the upper hand by casacading several DISCO stages.\",\n \"Table: NO_CAPTION\"\n ],\n [\n \"Discussion, Societal Impact and Limitations\",\n \"In this work, we have proposed the use of local implicit functions for adversarial defense.\",\n \"Given an input adversarial image and a query location, the DISCO model is proposed to project the RGB value of each image pixel into the image manifold, conditional on deep features centered in the pixel neighborhood.\",\n \"By training this projection with adversarial and clean images, DISCO learns to remove the adversarial perturbation.\",\n \"Experiments demonstrate DISCO's computational efficiency, its outstanding defense performance and transfer ability across attacks, datasets and classifiers.\",\n \"The cascaded version of DISCO further strengthens the defense with minor additional cost.\",\n \"Limitations: While DISCO shows superior performance on the attacks studied in this work (mainly norm-bounded attacks), it remains to be tested whether it is robust to other type of attacks [138], [16], [56], [54], [80], such as one pixel attack [138], patch attacks [16], [56] or functional adversarial attack [80].\",\n \"In addition, more evaluation configurations across attacks, datasets and classifiers will be investigated in the future.\",\n \"Societal Impact: We hope the idea of using local implicit functions can inspire better defenses and prevent the nefarious effects of deep learning attacks.\",\n \"Obviously, better defenses can also be leveraged by bad actors to improve resistance to the efforts of law enforcement, for example.\"\n ],\n [\n \"Acknowledgments\",\n \"This work was partially funded by NSF awards IIS1924937 and IIS-2041009, a gift from Amazon, a gift from Qualcomm, and NVIDIA GPU donations.\",\n \"We also acknowledge and thank the use of the Nautilus platform for some of the experiments discussed above.\",\n \"Appendix\"\n ],\n [\n \"Compare to SOTA in RobustBench\",\n \"In this section, we list the quantitative result of the baselines in RobustBench [27] .\",\n \"Table REF , REF and REF correspond to Fig.6(a), (b) and (c) of the main paper, respectively.\",\n \"Table REF shows the baselines under Autoattack with $\\\\epsilon _{2}=0.5$ .\",\n \"The index displayed in each table corresponds to the index shown in Fig.6 in the main paper.\",\n \"The baselines of each table are grouped into No defense (first block), Adversarially trained defense in RobustBench (second block), Transformation based defense (third block) and DISCO (last block).\",\n \"The results of adversarially trained baselines are copied from RobustBench, while the results of transformation-based defenses are obtained with our implementation.\",\n \"For STL [139], models with different sparse constraints $\\\\lambda $ are used from the publicly available STL githubhttps://github.com/GitBoSun/AdvDefense_CSC.\",\n \"DISCO is also combined with various classifiers for evaluation.\",\n \"More discussion can be found in Sec.\",\n \"4.1 of the paper.\",\n \"Table: Cifar10 baselines and DISCO under Autoattack (ϵ ∞ =8/255\\\\epsilon _{\\\\infty }=8/255).\",\n \"This table corresponds to Fig.\",\n \"6(a) in the main paper.Table: Cifar10 baselines and DISCO under Autoattack (ϵ 2 =0.5\\\\epsilon _{2}=0.5).Table: Cifar100 baselines and DISCO under Autoattack (ϵ ∞ =8/255\\\\epsilon _{\\\\infty }=8/255).\",\n \"This table corresponds to Fig.\",\n \"6(b) in the main paper.Table: ImageNet baselines and DISCO under Autoattack (ϵ ∞ =4/255\\\\epsilon _{\\\\infty }=4/255).\",\n \"This table corresponds to Fig.\",\n \"6(c) in the main paper.\"\n ],\n [\n \"Defense Transfer\",\n \"In this section, we discuss the qualitative results of DISCO transferability across attacks.\",\n \"Table REF , REF and REF represents the results for Cifar10, Cifar100 and ImageNet, respectively.\",\n \"The corresponding plots are illustrated in Fig.\",\n \"REF , REF and REF .\",\n \"More discussion can be found in Sec.\",\n \"4.1 of the paper.\",\n \"Table: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTION\"\n ],\n [\n \"Improving Cifar10 and Cifar100 SOTA on RobustBench\",\n \"Sec.\",\n \"4.1 in the main paper shows that DISCO can improve the prior SOTA defenses on the ImageNet dataset.\",\n \"In Table REF , we further investigate the gain of applying DISCO on SOTA Cifar10 and Cifar100 defenses.\",\n \"The first and second block of Table REF show the gains of applying DISCO on [114], which is the prior SOTA defense against $L_2$ and $L_{\\\\infty }$ Autoattack on Cifar10.\",\n \"DISCO also improves the prior SOTA defense [47] on Cifar100 by 2.89%.\",\n \"These results indicate that, beyond being a robust defense by itself, DISCO can also be applied to existing defenses to improve their robustness.\"\n ],\n [\n \"Kernel Size s\",\n \"In this section, we ablate the kernel size used to train DISCO on ImageNet.\",\n \"The kernel size $s$ controls the feature neighborhood forwarded to the local implicit module.\",\n \"Table REF shows that $s=3$ achieves the best performance, which degrades for $s=5$ by a significant margin (3.26%).\",\n \"This shows that while tasks like classification require large and global receptive fields, the projection of adversarial images into the natural image manifold can be done on small neighborhoods.\",\n \"Given that the complexity of modeling the manifold increases with the neighborhood size, it is not surprising that larger $s$ lead to weaker performance.\",\n \"This is consistent with the well known complexity of synthesizing images with global models, such as GANs.\",\n \"What is somewhat surprising is that even $s=1$ is sufficient to enable a robust defense.\",\n \"By default, we use $s=3$ in all our experiments.\"\n ],\n [\n \"Computation Time for STL and DISCO\",\n \"Table REF compares the inference time of STL [139], DISCO and cascade DISCO (from $K=$ 2 to 5) on Cifar10 and ImageNet.\",\n \"For a single image Cifar10 of size 32x32, STL requires an Cifar10 5.9$\\\\times $ (0.65 vs 0.011) larger than that of DISCO ($K$ =1).\",\n \"When cascade DISCO is used, inference time increases approximately linearly with $K$ .\",\n \"For a single ImageNet image of size 224, STL requires 23.71 seconds while DISCO (K=1) only requires 0.027.\",\n \"The inference time difference increases to $878.15\\\\times $ (23.71 vs 0.027) on ImageNet , which is significantly larger than that of Cifar10 (5.9$\\\\times $ ).\",\n \"This shows that DISCO is a better defense in the sense that it can handle widely varying input image sizes with minor variations of computing cost.\"\n ],\n [\n \"Training Details\",\n \"On Cifar10 and Cifar100, we train the DISCO for 40 epochs.\",\n \"On ImageNet, DISCO is only trained for 3 epochs because ImageNet images are larger and produce more random crops.\",\n \"The learning rate is set to 0.0001 and the Adam optimizer is used in all experiments.\",\n \"All experiments are conducted using Pytorch [110].\",\n \"All time measurements, for both baselines and DISCO, are made on a single Nvidia Titan Xp GPU with Intel Xeon CPU E5-2630, with batch size 1 and averaged over 100 images.\"\n ],\n [\n \"Adopted Code and Benchmark\",\n \"In this section, we list the url links that are used for training and evaluating DISCO.\",\n \"To create the adversarial-clean training pairs, we adopt the code from TorchAttackhttps://adversarial-attacks-pytorch.readthedocs.io/en/latest/ and Areshttps://github.com/thu-ml/ares, which support the multiple attack methods.\",\n \"These attack methods are then used to attack pretrained classifiers on Cifar10, Cifar100 and ImageNet.\",\n \"We use the ResNet18 classifiers from Ares for Cifar10, the WideResNet Cifar100 classifiers from this repository https://github.com/xternalz/WideResNet-pytorch and the ResNet18 ImageNet classifiers of Pytorch [110].\",\n \"To evaluate DISCO, we adopt Autoattack from RobustBench [27]https://github.com/RobustBench/robustbench and compare to the pretrained defenses on the RobustBench leaderboard.\",\n \"In addition to Autoattack, we use the AdverTorchhttps://github.com/BorealisAI/advertorch library to implement the BPDA attack [8] and the TorchAttackhttps://adversarial-attacks-pytorch.readthedocs.io/en/latest/ library for other attacks, like FGSM [46] and BIM [79].\",\n \"For the adversarially trained defense baselines, we adopt the pretrained weights from RobustBench [27]https://github.com/RobustBench/robustbench, while the codes for transformation based baselines are adopted from Ares, Cifar autoencoder https://github.com/chenjie/PyTorch-CIFAR-10-autoencoder and STL [139].\",\n \"To implement DISCO, we use code from LIIFhttps://github.com/yinboc/liif [24].\"\n ],\n [\n \"Visualizations\",\n \"DISCO defense outputs against FGSM [46] and BIM [79] and PGD [88] attacks are visualized in Fig.\",\n \"REF , REF and REF , respectively.\",\n \"Take Fig.\",\n \"REF for example.\",\n \"The first and second rows show the clean and adversarial images, while rows 3-5 show the output of DISCO and cascade DISCO ($K=2$ and $K=3$ ).\",\n \"Clearly, both DISCO and its cascade version can effectively remove the adversarial perturbation.\",\n \"Note that these images are produced from the same DISCO model without retraining for any attack.\",\n \"Figure: Comparison of Clean image, Adversarial image and DISCO output from K=K= 1 to 3 under FGSM attack.Figure: Comparison of Clean image, Adversarial image and DISCO output from K=K= 1 to 3 under BIM attack.Figure: Comparison of Clean image, Adversarial image and DISCO output from K=K= 1 to 3 under PGD attack.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05630"}}},{"rowIdx":91,"cells":{"text":{"kind":"list like","value":[["Nearby SNR: a possible common origin to multi-messenger anomalies in\n spectra, ratios and anisotropy of cosmic rays"],["Abstract The multi-messenger anomalies, including spectral hardening or excess for nuclei, leptons, ratios of $\\bar p/p$ and B/C, and anisotropic reversal, were observed in past years.","AMS-02 experiment also revealed different spectral break for positron and electron at 284 GeV and beyond TeV respectively.","It is natural to ask whether all those anomalies originate from one unified physical scenario.","In this work, the spatially-dependent propagation (SDP) with a nearby SNR source is adopted to reproduce above mentioned anomalies.","There possibly exists dense molecular cloud(DMC) around SNRs and the secondary particles can be produced by pp-collision or fragmentation between the accelerated primary cosmic rays and DMC.","As a result, the spectral hardening for primary, secondary particles and ratios of $B/C$ and $\\bar p/p$ can be well reproduced.","Due to the energy loss at source age of 330 kyrs, the characteristic spectral break-off for primary electron is at about 1 TeV hinted from the measurements.","The secondary positron and electron from charged pion take up $5\\%$ energy from their mother particles, so the positron spectrum has a cut-off at $\\sim$250 GeV.","Therefore, the different spectral break for positron and electron together with other anomalies can be fulfilled in this unified physical scenario.","More interesting is that we also obtain the featured structures as spectral break-off at 5 TV for secondary particles of Li, Be, B, which can be served to verify our model.","We hope that those tagged structures can be observed by the new generation of space-borne experiment HERD in future."],["Introduction","The origin of cosmic rays (CRs) has been a centurial mystery since its discovery.","The scientists have been always devoted to resolve this problem.","With new generation space-borne and ground-based experiments, CRs measurements are stepping into an era of high precision and a series of new phenomena in spectra, ratio of secondary-to-primary and anisotropy, as multi-messenger anomalies, are revealed now.","It may be an effective way to pinpoint the origin problem by joint study of multi-messenger information.","Firstly, the nuclei spectra have been measured with unprecedent precision and a fine structure of spectral hardening at 200 GV has been discovered by ATIC-2, CREAM and PAMELA experiments [85], [86], [35], [103], [14].","Lately, AMS-02 experiment also confirmed it [18], [19] and further revealed that other heavy nuclei including secondary particles have similar anomaly [23], [22], [24], [27], [28], [29].","More interesting is that the spectral break-off around $\\sim $ 14 TeV was observed by CREAM, NUCLEON and DAMPE experiments [103], [48], [49], [46].","Furthermore, recent spectral measurement of Helium showed that the drop-off starts from $\\sim $ 34 TV, which supported the rigidity dependent cut-off [37].","Several kinds of models have been proposed to explain the origin of spectral hardening, including the nearby source [94], [75], [76], [87], [101], the combined effects from different group sources and the spatially-dependent propagation(SDP)[69], [70], [77], [79].","Considering the break-off at the rigidity of 14 TV in spectrum, it seems that the nearby source model becomes natural and accessible.","However, other observational clues are required to support this point of view.","Secondly, the spectra of positron and electron is another good choice to shed new light on this topic.","The famous spectral excess of positron above 20 GeV was discovered by PAMELA experiment [13].","Then the AMS-02 experiment confirmed this remarkable result [11], [25].","Just recently, a sharp drop-off at 284 GeV was observed by AMS-02 experiment with above 4$\\sigma $ confidence level [25].","As for electrons, the measurement by the AMS-02 experiment showed that the energy spectrum could not be described by a single power-law form, the power index changes at about 42 GeV.","Contrary to the positron flux, which has an exponential energy cutoff of about 810 GeV, at the 5$\\sigma $ level the electron flux does not have an energy cutoff below 1.9 TeV [26].","However, the drop-off around 1 TeV in the total spectrum of positron and electron was first reported by HESS collaboration [31], [33] and validated by MAGIC [54], VERITAS [92] experiments.","The DAMPE experiment also performed the direct measurement to this feature and announced that the break-off was at $\\sim $ 0.9 TeV [59].","It is obvious that the spectra of positron and electron have extra-components at high energy similar to nuclei one.","The nearby source is also an alternative for interpreting the above feasures.","One of the natural advantages of positron and electron is their fast cooling in the interstellar radiation field (ISRF), which can roughly decide the source distance and age.","For example, the spectral cut-off at $\\sim $ TeV requires that the age of nearby source is about 330 kyrs [64], [72], [108], [78], which will set a much more strict constrain on its place.","Where is the nearby source and its rough direction may point out a new bright road.","Lastly, the anisotropy of CRs is one of the best choices to fulfill this role.","Thanks to unremitting efforts of ground-based experiments, the measurements of large-scale anisotropy has made great progress from hundreds of GeV to several PeV [41], [44], [2], [1], [51].","It is obvious that the phase has reversed at 100 TeV and the direction roughly point to local magnetic field and Galactic Center (GC) below and above 100 TeV respectively.","Coincidentally, the amplitude has a dip structure at $\\sim $ 100 TeV, starting from $\\sim $ 10 TeV.","The most importance thing is that there exists a common transition energy scale between the structures of the energy spectra and the anisotropies.","The local source possibly plays a very important role to resolve the conjunct problems of spectra and anisotropies.","Furthermore, the direction of anisotropy can roughly outline the position of such local source.","In our recent work, we proposed a local source under the SDP model to reproduce the co-evolution of the spectra and anisotropies and found that the optimal candidate of local source is possible a SNR at Geminga’s birth place [76], [87].","Based on above discussions, the local source is necessary to understand the multi-messenger anomalies.","However, the latest observations bring new challenges into this model, such as the different spectral break-off for positron and electron and a series of results of nuclei spectra and ratios that were published by AMS-02 experiment recently.","Therefore, a systematic study is useful to understand those new observations.","More important is that the featured structure is required and necessary to verify this model.","In this work, a unified physical scenario as the SDP with a nearby source, Geminga SNR, is adopted to reproduce all the above anomalies.","Simultaneously, we obtain the tagged structure to examine our model.","The paper is organized as follows.","Section 2 describes the model and method briefly, Section 3 presents all the calculated results and Section 4 gives the conclusion."],["Model and Method Description","The CRs in solar system come from two parts as the global one from the galactic background sources (bkg) and the local one from nearby SNRs (loc SNR).","For the background sources, it is viable to assume that the spatial distribution of CRs from them arrives at steady state.","Nevertheless for the nearby single SNR, the time-dependent transport of CRs after injection is requisite.","Furthermore, the dense molecular cloud (DMC) plays a key role in the star formation, which means that there possibly exists DMC around SNRs [32], [99], [107].","In our model, we assume that the DMC or dense interstellar medium(DISM) exists around the nearby SNR.","Therefore, the general physical picture can be sketched as three steps.","Firstly, the nuclei and electrons can be accelerated to very high energy (VHE) together during the SNR explosion.","Then the VHE CR nuclei and electrons undergo the interaction with DMC and ISRF and then produce secondary particles.","Lastly, the primary and secondary particles will go through the interstellar space and experience a long travel in the galaxy.","Certainly, a limited part will arrive at the earth and be observed by the various experiments.","The results can be imaged as the cartoon illustration of Figure REF .","Here, the spectral break-off of primary nuclei and electrons with exponential form is adopted to be 5 TV for the sake of reproducing the observed bump structure of proton at $\\sim $ 14 TV by DAMPE satellite experiment [46].","The primary electrons have to suffer the energy loss cooling by scattering off the ISRF with the time around 330 kyrs [91], [80], [61], which leads to the sharp dropping of electron spectrum in the energy region of $\\sim $ TeV as observed by DAMPE, HESS, MAGIC and VERITAS [31], [33], [54], [92], [59].","The anti-proton and positron will be produced in the pp-collision between VHE proton and DMC.","Due to the spectral break-off of primary nuclei at the rigidity of 5 TV, this causes the cut-off of positron, the secondary particle, at about $\\sim $ 300 GeV.","Simultaneously, the $\\gamma $ -rays from $\\pi ^0$ decay in the pp-collision is produced and has a spectral cut-off around 500 GeV.","In addition, the heavy nuclei will undergo the fragmentation with the DISM or DMC and then the secondary nuclei such as $Li, Be, B$ are produced.","The secondary nuclei from the fragmentation inherit the property of their mother particle and have the same morphology of energy break-off around 5 TV, which can be served to differentiate with other models [109], [77].","The detailed descriptions of cosmic ray propagation, Galactic background sources, and nearby SNRs are displayed in the following appendix A, B and C respectively.","Table: The parameters of transport spectrum with SDP model.Table: Injection parameters of the background and local source.Figure: The cartoon illustration to describe the production mechanism of the multi-messenger anomalies: the spectral hardening and break-off at rigidity of 200 GV and 14 TV for both primary and secondary nuclei, the energy cut-off for positron and electron at around 300 GeV and 1 TeV respectively."],["Results","Based on above discussions, the spectra of primary, secondary and all particles are calculated to reproduce the measurements.","Simultaneously, the ratios between primary to primary, secondary to primary and secondary to secondary are presented.","For the sake of completeness of this work, we also give the anisotropy for CR nuclei and electrons.","In the model calculations, the parameters of propagation and inject spectra both for local source and galactic ones are listed in Table REF and REF respectively."],["Spectra","The CR spectra are the most important effects to understand their propagation in the galaxy.","Thanks to the new generations of spaced-borne experiments, the spectral measurements of CRs are stepping into precise era and revealed a series of new phenomena, such as the spectral hardening at 200 GV and break-off at $\\sim $ 14 TV, the famous excess of positron and extra-component of electron at high energy.","In this section, the unified model calculations are described to reproduce and understand the measurements.","Figure REF and REF show the measurements and model calculations for most of nuclei species individually.","In the model calculations, the red solid line with shadow is the contribution from local SNR, the blue solid line is from galactic background sources and the black solid line is the sum of them.","It is obvious that the spectral hardening can be reproduced well for all the species and its origin dominantly originates from the contribution of local source.","To satisfy the energy break-off of proton at the rigidity of 14 TV observed by DAMPE experiment [46], the injection spectrum of local source is parameterized as a cutoff power-law form, $q_{\\rm inj}({\\cal R})=q_0{\\cal R}^{-\\nu ^{\\prime }}\\exp (-{\\cal R}/{\\cal R}^{\\prime }_{\\rm c})$ , where the normalization $q_0$ and spectral index $\\nu ^{\\prime }$ are determined through fitting to the CR energy spectra.","The parameter ${\\cal R}^{\\prime }_{\\rm c}$ is adopted to be 15 TV, which leads to the bend of CR spectra starting around the rigidity of 5 TV.","For detailed parameter information about the spectra, please refer to Table REF .","Figure REF shows that the spectral break-off of proton is consistent well between model expection and data points from DAMPE and CREAM measurements [103], [46], but the Helium species has a little difference with DAMPE measurement and is roughly consistent with CREAM observations at several highest energy points.","The reason is that the measured spectral break-off from DAMPE for Helium species is at the rigidity of $\\sim $ 34 TV, which is a little higher than proton under the Z-dependent energy cut-off frame [37].","To keep the uniformity of all the nuclei, we choose the cut-off rigidity at 15 TV, which doesn't affect the results of other heavier primary nuclei and the conclusions.","Under this physical scenario, our model expects that all the heavier primary nuclei have the same rigidity break-off at 5 TV as shown in Figure REF , which can be observed by the HERD experiment in future [73].","Figure: The comparison between model calculations and experimental measurements for proton and Helium spectra as (a) and (b) panel.","Here the red shadow shape is from the contribution of local SNR (e.g.","Geminga), the blue solid line comes from the background components and the black solid line is the total contribution.","The data points are measured by AMS-02, CREAM and DAMPE experiments, , , , .Figure: Similar to Figure , the individual species of C,O,Ne,Mg,SiC, O, Ne, Mg, Si and FeFe from panel (a) to (f).","The data points are from the AMS-02 experiment, , ."],["Secondary Particles","Following the primary species, the secondary particles spectra also have two components as the local and the global sources.","Figure REF gives the spectra of $\\bar{p}$ , $Li, Be$ and $B$ from panel (a) to (d).","The model calculations are well consistent with the observations from AMS-02 [20], [23].","The hardening of $\\bar{p}$ spectrum starts from tens of GeV owing to 200 GV hardening of its mother proton.","For the species of $Li, Be$ and $B$ , they are produced through fragmentation of heavier nuclei, such as $C, O$ and so on.","They keep the same behaviour as their mother particles with the hardening at 200 GV and break-off at around $\\sim $ 10 TV.","The typical energy break-off for secondary particles is pivotal to validate the dominant interactions around source regions for the nearby source, which are probe to unveil the origin of positron and $\\bar{p}$ excess at high energy.","We hope that the spectral bend around $\\sim $ TeV can be observed by HERD experiment in near future.","Figure: Similar to Figure , the secondary particles of p ¯,Li,Be\\bar{p}, Li, Be and BB from panel (a) to panel (d).","The data points are from the AMS-02 experiment., ."],["Two components particles","There are also some special particles, such as $N, Na, Al$ , including both the primary and secondary components.","They are thought to be produced both in galactic background sources, and by the collisions of heavier nuclei with the interstellar medium (ISM) ($\\rm O\\rightarrow \\rm N+ \\rm X$ , $\\rm Mg\\rightarrow \\rm Na+ \\rm X$ , $\\rm Si\\rightarrow \\rm Na+ \\rm X$ , $\\rm Si\\rightarrow \\rm Al+ \\rm X$ )[67], [53], [93].","As shown in Figure REF , the red solid line with shadow is the contribution from primary component accelerated directly by local SNR, the green solid line with shadow indicates the secondary component produced by the collisions of heavier nuclei with ISM, the blue solid line is from galactic background sources and the black solid line is the sum of them.","It can be seen that the model calculations are agreement with the observations, and their spectra also have a break-off structure similar to that of primary particles at $\\sim 5$ TV.","Figure: Similar to Figure , the individual species of N,Na,AlN, Na, Al, including both primary and secondary components, from panel (a) to (c).","The data points are from the AMS-02 experiment, ."],["All particles","The space-borne experiments have decisive advantages in the seperation ability for different species.","However they have limited effective detector area, which lead to the lower statistical event numbers in high energy.","On the contrary, the ground-based experiments have opposite properties.","This makes that the spectral measurements for individual species step into dilemma above tens of TeV.","The all-particle spectra can make up this shortcoming to constrain the high energy contributions.","Figure REF shows the model calculations and measurements for all-particle spectrum.","In the calculations, four groups as $H+He, C+N+O, Ne+Mg+Si$ and $Fe$ are demonstrated in blue, orange, green and red solid lines.","It is clear that the knee structure of the all-particle spectrum can be properly reproduced by the background component assuming a Z-dependent cutoff with $R_c \\sim 7$ PV.","In this case, the light species of protons and He nuclei dominantly composes the knee structure.","This is because we try to fit the KASCADE spectra of proton and Helium, which was also favoured by the diffuse $\\gamma $ -ray measurement at ultra-high energy by AS$\\gamma $ experiment [45].","Figure: The observed and calculated all-particle spectra.","In the model calculations, the solid blue, orange, green, red and black line represent the P+He, C+N+O, Ne+Mg+Si, Fe species and all-particle spectra respectively.","The data points are measured by TALE , Tibet-ASγ\\gamma , ICETOP , experiments and from weighted by Horandel ."],["Positron and Electron","It is a hot topic for the positron excess and electron spectral hardening at high energy.","Similar to nuclei secondary particles, the positron is also composed of two parts as around Geminga SNR and global background component from pp-collision.","Figure REF shows the spectra of positron, electron, their sum and the ratio of positron to the sum of positron and electron.","The model calculations are good reproduction of experimental data.","Particularly, the positron will take $5\\%$ energy from its mother species proton in pp-collision.","Owing to the spectral break-off around 5 TeV of proton, the positron spectrum has cut-off around 250 GeV and successfully reproduce the AMS-02 measurements as shown in panel (a).","As the discussion in section 2, the age of Geminga SNR is $3.3\\times 10^5$ yrs, which leads to the energy break of electron spectrum about TeV as shown in panel (b) [50].","Then the difference of energy break-off between positron and electron can be naturally understood.","For the ratio of positron to the sum of positron and electron, it increases with increasing energy above about TeV, which results from the cut-off of the total electron spectrum at $\\sim $ TeV.","Figure: Similar to the Figure , the spectra of positron, electron and their sum from panel (a) to (c) and the ratio of positron to the sum of positron and electron in panel (d).","The data points are adopted from AMS-02 experiments , ."],["Ratios","The ratios are important to understand the acceleration, propagation and interaction properties of CRs.","Thanks to the unprecedented precise measurements from AMS-02, the ratios of primary to primary, secondary to secondary and secondary to secondary species can be well measured and clearly shown the difference [22], [24], [28], [27], [29].","In this section, the corresponding model calculations are obtained to reproduce those observations."],["Primary to primary species","The ratios of primary to primary species carry the acceleration information in the source region.","The model of diffusive shock acceleration predicts that the individual species should have identical power law spectrum [90], [30].","Figure REF shows the comparison between model calculations and observations for the ratios of primary to primary species for $He/O$ , $C/O$ ,$N/O$ , $Ne/O$ , $Mg/O$ , $Si/O$ , $Ne/Mg$ , $Si/Mg$ , $Na/Si$ , $Al/Si$ , $Fe/He$ , $Fe/O$ and $Fe/Si$ .","The red solid lines with shadow indicate the contributions from the nearby SNR.","The black solid lines represent the ratio results of model expectation, taking into account the contributions from the nearby source.","In fact, the individual spectrum has well reproduced the observations as shown in Figure REF , REF and REF , so the ratios should be also consistent between data and model calculations.","Figure REF lists the ratios of primary to primary species, which has good consistency with observations."],["secondary to primary species","It is believed that most of the secondary nuclei originate from collisions of CRs with ISM in propagation.","Therefore the information about CR propagation can be extracted from comparison between the spectra of secondary particles and those of primary CRs [105], [104], [106].","Figure REF shows the ratios of secondary to primary species for $\\bar{p}/p$ , $Li/C$ , $Be/C$ , $B/C$ , $Li/O$ , $Be/O$ and $B/O$ .","The blue solid lines represent the ratios of background component of secondary particles to the total amount of primary particles, the red solid lines show the ratios of nearby SNR component of secondary particles to the total amount of primary particles and the black solid line is the ratio of total secondary to total primary.","The model calculations work well to reproduce the observations.","Here except $\\bar{p}/p$ with energy independent (constant) distribution above 10 GeV, all other heavier nuclei have hardening above 200 GeV for model calculations.","Just recently, the similar hardening has also been discovered in $B/C$ and $B/O$ above 100 GeV/n by DAMPE experiment [58].","It is obvious that our model calculations work well to reproduce this hardening.","Similar to the spectra of secondary species, the ratios have a break-off around TeV for $\\bar{p}/p$ and 5 TeV for heavier nuclei, which can offer a crucial and definitive identification with other kinds of models.","Figure: Comparison between model calculations and observations for the ratios of secondary to primary species.","The data points are adopted from AMS-02 measurements , , ."],["Secondary to secondary species","The ratios of secondary to secondary species take important information of interaction in CR propagation.","If they originate from the same mother particle, the spectral behavior reflects the common pecularity considering the known interaction cross-section, the same ISM and the same interaction time.","This property can be served to understand the CR origin puzzle, such as positron [13].","Figure REF shows the ratio comparison between model calculations and measurements for $\\bar{p}/e^+$ , $Li/B$ and $Be/B$ .","Our model calculations are consistent with the measurements.","More interesting is that the energy independent distribution is clear shown from them above 10 GeV.","The model calculation of $\\bar{p}/e^+$ rises up sharply above 300 GeV.","This is because the energy cut-off of positron around 300 GeV.","Figure: Similar to Figure .","Comparison between model calculations and observations for the ratios of secondary to secondary species.","The data points are adopted from AMS-02 measurements , .Similar to the results of spectra and ratios, the anisotropy is also the joint contributions from the background and local sources of CRs.","Due to more abundant sources in the inner disk, the phase of anisotropy directs to the galactic center.","However, the observations of phase roughly point to the direction of anti-galactic center from 100 GeV to 100 TeV.","The local source, located at the outer of galactic disk, plays the dominant roles in this energy region.","So the anisotropy demonstrates mutually repressive competition between local source and background.","The dip structure at 100 TeV is the transition energy point for the two kinds of sources.","Figure REF shows the amplitude and phase of anisotropy for CRs.","The CR anistropy can be fitted well under this physical scenario.","Figure REF presents amplitude and phase for CREs.","The expection of amplitude for CREs is lower than the up-limit of Fermi-LAT experiment [9].","Figure: Comparison between model calculations and observations of the amplitude and phase of anisotropy for CRs.","The data points are taken from underground muon detectors:Norikura (),Ottawa (),London (),Bolivia (),Budapest (),Hobart (),London (),Misato (),Socorro (),Yakutsk (),Banksan (),Hong Kong (),Sakashita (),Utah (),Liapootah (),Matsushiro (),Poatina (),Kamiokande (),Marco (),SuperKamiokande ();and air shower array experiments:PeakMusala (),Baksan (),Norikura (),EAS-TOP (, , ),Baksan (),Milagro (),IceCube (, ),Ice-Top (),ARGO-YBJ (),Tibet (, , )K-Grande().Figure: Comparison between model calculations and up-limit of the amplitude and phase of anisotropy for CREs.","The data points are adopted from"],["Summary","The new generation of space-borne and ground-based experiments took unprecedented precise measurements for CR spectra and anisotropy and revealed multi-messenger anomalies for them.","In this work, we propose that the local source, Geminga SNR, is the common origin for all those anomalies.","The Geminga SNR has three critical advantages: perfect age of 330 kyrs, suitable position with a distance of 330 pc and assumed DMC around it.","The physical figure can be summarized as following.","Firstly, the diffusive shock around Geminga SNR can accelerate the CR nuclei and electrons to very high energy together.","Here we adopt the break-off rigidity to be $\\sim $ 5 TV for the sake of 14 TV energy cut-off observation by DAMPE experiment [46].","The electron will suffer the energy loss for 330 kyrs, which lead to the spectral break around TeV observed by DAMPE, HESS [31], [33], [59].","Due to the contributions of the local source, the spectral hardening and drop-off for all nuclei and electron respective can be well reproduced.","Secondarily, the interaction of pp-collision and fragmentation with DMC happen to produce secondary particles, such as $\\bar{p}, e^+, \\gamma , Li,Be,B$ and so on.","Inheriting the properties from their mother particles, the $\\bar{p}, e^+, \\gamma $ as products of pp-collision will take $10\\%, 5\\%, 10\\%$ of proton energy, which result in the corresponding energy break-off at 250, 500, 500 GeV respectively for them.","So the hardening and drop-off for secondary particles can also be well consistent with the measurements.","Simultaneously, the heavier secondary nuclei from fragmentation process, such as $Li,Be,B$ , will have the same energy break-off with their mothers around $\\sim $ 5 TeV.","This property is typical for this kind model, which can be served to test the model in future.","Lastly, thanks to the appropriate location of Geminga SNR, the special evolution with energy of anisotropy in amplitude and phase can roughly reproduce the observations.","Summarily, the multi-messenger anomaly of spectra, ratios and anisotropy can be well reproduced in one unified physical mechanism.","Particularly, the difference of spectral break-off between positron and electron can also be explained in the same scenario.","More interesting is that we expect the typical energy break-off feature for secondary nuclei $Li,Be,B$ and their ratios with primary nuclei.","This feature will also play important role to resolve the origin puzzle of positron and $\\bar{p}$ excesses.","We hope that this feature can be observed by HERD experiment in future.","This work is supported by the National Key Research and Development Program (No.","2018YFA0404202), the National Natural Science Foundation of China (Nos.","12275279, U2031110, 12175248)."],["The propagation of CRs","It has been recognized in recent years that the propagation of CRs in the Milky Way should depend on the spatial locations, as inferred by the HAWC and LHAASO observations of extended $\\gamma $ -ray halos around pulsars [10], [34] and the spatial variations of the CR intensities and spectral indices from Fermi-LAT observations [102], [12].","The spatially-dependent propagation (SDP) model was also proposed to explain the observed hardening of CRs [97], [98], [62], [69], [77], [70], and also the large-scale anisotropies by means of a nearby source[76], [87].","In the SDP model, the diffusive halo is divided into two parts, the inner halo (disk) and the outer halo.","In the inner halo, the diffusion coefficient is much smaller than that in the outer halo, as indicated by the HAWC observations.","The propagation equation of CRs in the magnetic halo and the descriptions of specific iterms in it can be refered to [69].","The spatial diffusion coefficient $D_{xx}$ can be parameterized as $D_{xx}(r,z, {\\cal R} )= D_{0}F(r,z)\\beta ^{\\eta } \\left(\\frac{\\cal R}{{\\cal R}_{0}} \\right)^{\\delta _{0}F(r,z)},$ where $r$ and $z$ are cylindrical coordinate, ${\\cal R}$ is the particle's rigidity, $\\beta $ is the particle's velocity in unit of light speed, $D_0$ and $\\delta _0$ are constants representing the diffusion coefficient and its high-energy rigidity dependence in the outer halo, $\\eta $ is a phenomenological constant in order to fit the low-energy data.","The spatial dependent function $F(r, z)$ is given as $F(r,z) = \\left\\lbrace \\begin{array}{ll}{\\frac{N_m}{1+f(r,z)}+\\left[1-\\frac{N_{m}}{1+f(r,z)}\\right]}\\left(\\frac{{z}}{\\xi {z}_{\\rm h}} \\right)^{n}, & {{|z|} \\le \\xi {z}_{\\rm h}} \\\\\\\\1, & { {|z|} > \\xi {z}_{\\rm h}} \\\\\\end{array},\\right.$ where the total half-thickness of the propagation halo is $z_h$ , and the half-thickness of the inner halo is $\\xi z_{h}$ .","The constant $n$ describes the smoothness of the parameters at the transition between the two halos.","The expression $f(r, z)$ is the source density distribution.","In this work, we adopt the diffusion re-acceleration model, with the diffusive re-acceleration coefficient $D_{pp}$ , which correlated with $D_{xx}$ via $D_{pp}D_{xx} = \\frac{4p^{2}v_{A}^{2}}{3\\delta (4-\\delta ^{2})(4-\\delta )}$ , where $v_A$ is the Alfvén velocity, $p$ is the momentum, and $\\delta $ is the rigidity dependence slope of the diffusion coefficient [89].","The parameters of SDP model used here are listed in Table REF ."],["Background sources","All of the SNRs other than the nearby one are labeled as background sources.","The source density distribution is approximated as an axisymmetric form parametrized as $f(r, z) = \\left(\\frac{r}{r_\\odot } \\right)^\\alpha \\exp \\left[-\\frac{\\beta (r-r_\\odot )}{r_\\odot } \\right] \\exp \\left(-\\frac{|z|}{z_s} \\right) ~,$ where $r_\\odot \\equiv 8.5$ kpc represents the distance from the Galactic center to the solar system.","Parameters $\\alpha $ and $\\beta $ are taken to be 1.69 and 3.33 [55].","The density of the source distribution decreases exponentially along the vertical height from the Galactic plane, with $z_s$ = 200 pc.","The parameters of SDP model used here are listed in Table REF .","The injection spectrum of nuclei and primary electrons are assumed to be an exponentially cutoff broken power-law function of particle rigidity ${\\cal R}$ , i.e.","$q({\\cal R}) = Q_{0} \\left\\lbrace \\begin{array}{lll}\\left(\\frac{{\\cal R}}{{\\cal R}_{\\rm br}} \\right)^{\\nu _{1}}, & {{\\cal R} \\le {\\cal R}_{\\rm br}} \\\\\\\\\\left(\\frac{{\\cal R}}{{\\cal R}_{\\rm br}} \\right)^{\\nu _{2}} \\exp \\left[-\\frac{\\cal R}{{\\cal R}_{\\rm c}} \\right], & { {\\cal R} > {\\cal R}_{\\rm br}} \\\\\\end{array}\\right.,$ where $Q_{0}$ is the normalization factor, ${\\cal R}_{\\rm br}$ is break rigidity, $\\nu _{1,2}$ are the spectral incides before and after the break rigidity, ${\\cal R}_{\\rm c}$ is the cutoff rigidity.","Table REF shows the injection spectrum parameters of different CR nuclei species.","The numerical package DRAGON is used to solve the propagation equation of CRs [60].","For energies smaller than tens of GeV, the fluxes of CRs are suppressed by the solar modulation effect.","We use the force-field approximation [65] to account for the solar modulation.","The secondary CR nuclei, such as Li, Be, B, can be brought forth from the fragmentation of heavier parent nuclei throughout the transport.","The production rate is expressed as follows $Q_j = \\sum _{i = \\rm C, N, O} (n_{\\rm H} \\sigma _{i+{\\rm H}\\rightarrow j} +n_{\\rm He} \\sigma _{i+{\\rm He} \\rightarrow j} ) v \\psi _i ~,$ where $n_{{\\rm H}/{\\rm He}}$ is the number density of hydrogen/helium in the ISM and $\\sigma _{i+{\\rm H/He}\\rightarrow j}$ is the total cross section of the corresponding hadronic interaction.","Unlike above secondary CR nuclei, secondary $\\rm e^+$ and $\\rm \\bar{p}$ are produced through the pp collisions between the primary CR nuclei from background sources and ISM.","Therefore the source term of both $\\rm e^+$ and $\\rm \\bar{p}$ is the convolution of the energy spectra of primary nuclei $\\Phi _i(E)$ and the relevant differential cross section $d \\sigma _{i + {\\rm H/He} \\rightarrow j}/d E_j$ , i.e.","$\\nonumber Q_j &=& \\sum _{i = \\rm p, He} \\int dp_i v \\left\\lbrace n_{\\rm H} \\frac{ \\sigma _{i+{\\rm H}\\rightarrow j}(p_i, p_j)}{ dp_j} \\right.","\\\\& & \\left.","+n_{\\rm He} \\frac{\\sigma _{i+{\\rm He}\\rightarrow j}(p_i, p_j)}{dp_j} \\right\\rbrace \\psi _i(p_i) ~,$ Furthermore, antiprotons may still undergo non-annihilated inelastic scattering with ISM protons during propagation, which can generate the tertiary production."],["Nearby supernova remnant","We assume that one supernova explosion in the vicinity of the solar neighborhood occurred in a giant MC about $10^5-10^6$ years ago.","The CR charged particles were continually accelerated by passing back and forth across the shock front with the expansion of supernova ejecta.","Under the SDP scenario of GCR propagation, [78] demonstrated that, among the observed local SNRs, only Geminga SNR is able to explain both the spectra and anisotropies observations of CRs simultaneously.","Since the location of Monogem is similar to that of Geminga, their impacts on CR flux may degenerate with each other.","Here, for simplicity, we take Geminga SNR as the major contributor of the local source in this work.","The Geminga SNR locates at its birth place with the distance and age of r = 330 pc and $t_{inj} = 3.3\\times 10^5$ yrs.","The direction in the galactic coordinate is $l=194^{\\circ }$ (galactic longitude) and $b=-13^{\\circ }$ (galactic latitude) [91].","Its distance, age and direction jointly decide the important role as an optimal candidate of nearby source to CRs [76], [87], [110].","The injection process of SNR is approximated as burst-like.","So the primary and secondary particles have experienced 300 kyrs and a tiny part of them enter into solar system in the end.","In this work, the propagated spectrum from Geminga SNR is thus a convolution of the Green's function and the time-dependent injection rate $Q_0(t)$[50], i.e.","$\\varphi (\\vec{r}, {\\cal R}, t) = \\int _{t_i}^{t} G(\\vec{r}-\\vec{r}^\\prime , t-t^\\prime , {\\cal R}) Q_0(t^\\prime ) d t^\\prime .$ The normalization is determined through fitting Galactic cosmic rays energy spectra and the detailed parameters is listed in Table REF .","Besides, the CR nuclei generated by the local SNR also collide with the molecular gas around them and give birth to prolific daughter particles, like B, $\\rm e^{\\pm }$ , $\\rm \\bar{p}$ , and so forth.","The yields of B and $\\rm e^{\\pm }$ , $\\rm \\bar{p}$ inside the MC are respectively $Q_j = \\sum _{i = \\rm C, N, O} (n_{\\rm H} \\sigma _{i+{\\rm H}\\rightarrow j} +n_{\\rm He} \\sigma _{i+{\\rm He}\\rightarrow j} )v Q_i(E) t_{\\rm col}$ and $Q_j \\; = \\;\\sum _{i = \\rm p, He} {\\displaystyle \\int \\limits _{E_{\\rm th}}^{+ \\infty }} \\; d E_i \\;v \\; \\left\\lbrace n_{\\, \\rm H}{\\displaystyle \\frac{d \\sigma _{i + {\\rm H} \\rightarrow j}}{d E_j }} \\right.","\\\\\\left.","+n_{\\, \\rm He} {\\displaystyle \\frac{d \\sigma _{i + {\\rm He} \\rightarrow j}}{d E_j }} \\right\\rbrace Q_i(E_i) t_{\\rm col} ~,$ where $n_{\\rm H/He}$ is the number density of hydrogen/helium in MC.","In this work, we assume that it is 1000 times greater than the mean value of ISM.","$t_{\\rm col}$ is the duration of collision of 420 yrs.","$Q_i(E)$ is the accelerated spectrum of primary nuclei inside local SNR.","Summarily, the local source, Geminga SNR, is responsible for the spectral hardening at 200 GV and cut-off at 14 TV for all the nuclei species.","The primary electron can be accelerated to $\\sim $ 5 TeV, similar to nuclei, but will undergo the energy loss in ISRF and have a spectral cut-off around TeV due to its age of $3.3\\times 10^5$ yrs.","For the secondary particles, the spectral cut-off depend on their mother ones and the interaction modes.","For positron, the energy break-off will located at $\\sim $ 300 GeV.","The different energy break-off between positron and electron can be natural understood in this scenario.","The secondary heavier nuclei will have the same spectral cut-off as their mothers at $\\sim $ 5 TV, which can be tested in future experiments like HERD [73]."]],"string":"[\n [\n \"Nearby SNR: a possible common origin to multi-messenger anomalies in\\n spectra, ratios and anisotropy of cosmic rays\"\n ],\n [\n \"Abstract The multi-messenger anomalies, including spectral hardening or excess for nuclei, leptons, ratios of $\\\\bar p/p$ and B/C, and anisotropic reversal, were observed in past years.\",\n \"AMS-02 experiment also revealed different spectral break for positron and electron at 284 GeV and beyond TeV respectively.\",\n \"It is natural to ask whether all those anomalies originate from one unified physical scenario.\",\n \"In this work, the spatially-dependent propagation (SDP) with a nearby SNR source is adopted to reproduce above mentioned anomalies.\",\n \"There possibly exists dense molecular cloud(DMC) around SNRs and the secondary particles can be produced by pp-collision or fragmentation between the accelerated primary cosmic rays and DMC.\",\n \"As a result, the spectral hardening for primary, secondary particles and ratios of $B/C$ and $\\\\bar p/p$ can be well reproduced.\",\n \"Due to the energy loss at source age of 330 kyrs, the characteristic spectral break-off for primary electron is at about 1 TeV hinted from the measurements.\",\n \"The secondary positron and electron from charged pion take up $5\\\\%$ energy from their mother particles, so the positron spectrum has a cut-off at $\\\\sim$250 GeV.\",\n \"Therefore, the different spectral break for positron and electron together with other anomalies can be fulfilled in this unified physical scenario.\",\n \"More interesting is that we also obtain the featured structures as spectral break-off at 5 TV for secondary particles of Li, Be, B, which can be served to verify our model.\",\n \"We hope that those tagged structures can be observed by the new generation of space-borne experiment HERD in future.\"\n ],\n [\n \"Introduction\",\n \"The origin of cosmic rays (CRs) has been a centurial mystery since its discovery.\",\n \"The scientists have been always devoted to resolve this problem.\",\n \"With new generation space-borne and ground-based experiments, CRs measurements are stepping into an era of high precision and a series of new phenomena in spectra, ratio of secondary-to-primary and anisotropy, as multi-messenger anomalies, are revealed now.\",\n \"It may be an effective way to pinpoint the origin problem by joint study of multi-messenger information.\",\n \"Firstly, the nuclei spectra have been measured with unprecedent precision and a fine structure of spectral hardening at 200 GV has been discovered by ATIC-2, CREAM and PAMELA experiments [85], [86], [35], [103], [14].\",\n \"Lately, AMS-02 experiment also confirmed it [18], [19] and further revealed that other heavy nuclei including secondary particles have similar anomaly [23], [22], [24], [27], [28], [29].\",\n \"More interesting is that the spectral break-off around $\\\\sim $ 14 TeV was observed by CREAM, NUCLEON and DAMPE experiments [103], [48], [49], [46].\",\n \"Furthermore, recent spectral measurement of Helium showed that the drop-off starts from $\\\\sim $ 34 TV, which supported the rigidity dependent cut-off [37].\",\n \"Several kinds of models have been proposed to explain the origin of spectral hardening, including the nearby source [94], [75], [76], [87], [101], the combined effects from different group sources and the spatially-dependent propagation(SDP)[69], [70], [77], [79].\",\n \"Considering the break-off at the rigidity of 14 TV in spectrum, it seems that the nearby source model becomes natural and accessible.\",\n \"However, other observational clues are required to support this point of view.\",\n \"Secondly, the spectra of positron and electron is another good choice to shed new light on this topic.\",\n \"The famous spectral excess of positron above 20 GeV was discovered by PAMELA experiment [13].\",\n \"Then the AMS-02 experiment confirmed this remarkable result [11], [25].\",\n \"Just recently, a sharp drop-off at 284 GeV was observed by AMS-02 experiment with above 4$\\\\sigma $ confidence level [25].\",\n \"As for electrons, the measurement by the AMS-02 experiment showed that the energy spectrum could not be described by a single power-law form, the power index changes at about 42 GeV.\",\n \"Contrary to the positron flux, which has an exponential energy cutoff of about 810 GeV, at the 5$\\\\sigma $ level the electron flux does not have an energy cutoff below 1.9 TeV [26].\",\n \"However, the drop-off around 1 TeV in the total spectrum of positron and electron was first reported by HESS collaboration [31], [33] and validated by MAGIC [54], VERITAS [92] experiments.\",\n \"The DAMPE experiment also performed the direct measurement to this feature and announced that the break-off was at $\\\\sim $ 0.9 TeV [59].\",\n \"It is obvious that the spectra of positron and electron have extra-components at high energy similar to nuclei one.\",\n \"The nearby source is also an alternative for interpreting the above feasures.\",\n \"One of the natural advantages of positron and electron is their fast cooling in the interstellar radiation field (ISRF), which can roughly decide the source distance and age.\",\n \"For example, the spectral cut-off at $\\\\sim $ TeV requires that the age of nearby source is about 330 kyrs [64], [72], [108], [78], which will set a much more strict constrain on its place.\",\n \"Where is the nearby source and its rough direction may point out a new bright road.\",\n \"Lastly, the anisotropy of CRs is one of the best choices to fulfill this role.\",\n \"Thanks to unremitting efforts of ground-based experiments, the measurements of large-scale anisotropy has made great progress from hundreds of GeV to several PeV [41], [44], [2], [1], [51].\",\n \"It is obvious that the phase has reversed at 100 TeV and the direction roughly point to local magnetic field and Galactic Center (GC) below and above 100 TeV respectively.\",\n \"Coincidentally, the amplitude has a dip structure at $\\\\sim $ 100 TeV, starting from $\\\\sim $ 10 TeV.\",\n \"The most importance thing is that there exists a common transition energy scale between the structures of the energy spectra and the anisotropies.\",\n \"The local source possibly plays a very important role to resolve the conjunct problems of spectra and anisotropies.\",\n \"Furthermore, the direction of anisotropy can roughly outline the position of such local source.\",\n \"In our recent work, we proposed a local source under the SDP model to reproduce the co-evolution of the spectra and anisotropies and found that the optimal candidate of local source is possible a SNR at Geminga’s birth place [76], [87].\",\n \"Based on above discussions, the local source is necessary to understand the multi-messenger anomalies.\",\n \"However, the latest observations bring new challenges into this model, such as the different spectral break-off for positron and electron and a series of results of nuclei spectra and ratios that were published by AMS-02 experiment recently.\",\n \"Therefore, a systematic study is useful to understand those new observations.\",\n \"More important is that the featured structure is required and necessary to verify this model.\",\n \"In this work, a unified physical scenario as the SDP with a nearby source, Geminga SNR, is adopted to reproduce all the above anomalies.\",\n \"Simultaneously, we obtain the tagged structure to examine our model.\",\n \"The paper is organized as follows.\",\n \"Section 2 describes the model and method briefly, Section 3 presents all the calculated results and Section 4 gives the conclusion.\"\n ],\n [\n \"Model and Method Description\",\n \"The CRs in solar system come from two parts as the global one from the galactic background sources (bkg) and the local one from nearby SNRs (loc SNR).\",\n \"For the background sources, it is viable to assume that the spatial distribution of CRs from them arrives at steady state.\",\n \"Nevertheless for the nearby single SNR, the time-dependent transport of CRs after injection is requisite.\",\n \"Furthermore, the dense molecular cloud (DMC) plays a key role in the star formation, which means that there possibly exists DMC around SNRs [32], [99], [107].\",\n \"In our model, we assume that the DMC or dense interstellar medium(DISM) exists around the nearby SNR.\",\n \"Therefore, the general physical picture can be sketched as three steps.\",\n \"Firstly, the nuclei and electrons can be accelerated to very high energy (VHE) together during the SNR explosion.\",\n \"Then the VHE CR nuclei and electrons undergo the interaction with DMC and ISRF and then produce secondary particles.\",\n \"Lastly, the primary and secondary particles will go through the interstellar space and experience a long travel in the galaxy.\",\n \"Certainly, a limited part will arrive at the earth and be observed by the various experiments.\",\n \"The results can be imaged as the cartoon illustration of Figure REF .\",\n \"Here, the spectral break-off of primary nuclei and electrons with exponential form is adopted to be 5 TV for the sake of reproducing the observed bump structure of proton at $\\\\sim $ 14 TV by DAMPE satellite experiment [46].\",\n \"The primary electrons have to suffer the energy loss cooling by scattering off the ISRF with the time around 330 kyrs [91], [80], [61], which leads to the sharp dropping of electron spectrum in the energy region of $\\\\sim $ TeV as observed by DAMPE, HESS, MAGIC and VERITAS [31], [33], [54], [92], [59].\",\n \"The anti-proton and positron will be produced in the pp-collision between VHE proton and DMC.\",\n \"Due to the spectral break-off of primary nuclei at the rigidity of 5 TV, this causes the cut-off of positron, the secondary particle, at about $\\\\sim $ 300 GeV.\",\n \"Simultaneously, the $\\\\gamma $ -rays from $\\\\pi ^0$ decay in the pp-collision is produced and has a spectral cut-off around 500 GeV.\",\n \"In addition, the heavy nuclei will undergo the fragmentation with the DISM or DMC and then the secondary nuclei such as $Li, Be, B$ are produced.\",\n \"The secondary nuclei from the fragmentation inherit the property of their mother particle and have the same morphology of energy break-off around 5 TV, which can be served to differentiate with other models [109], [77].\",\n \"The detailed descriptions of cosmic ray propagation, Galactic background sources, and nearby SNRs are displayed in the following appendix A, B and C respectively.\",\n \"Table: The parameters of transport spectrum with SDP model.Table: Injection parameters of the background and local source.Figure: The cartoon illustration to describe the production mechanism of the multi-messenger anomalies: the spectral hardening and break-off at rigidity of 200 GV and 14 TV for both primary and secondary nuclei, the energy cut-off for positron and electron at around 300 GeV and 1 TeV respectively.\"\n ],\n [\n \"Results\",\n \"Based on above discussions, the spectra of primary, secondary and all particles are calculated to reproduce the measurements.\",\n \"Simultaneously, the ratios between primary to primary, secondary to primary and secondary to secondary are presented.\",\n \"For the sake of completeness of this work, we also give the anisotropy for CR nuclei and electrons.\",\n \"In the model calculations, the parameters of propagation and inject spectra both for local source and galactic ones are listed in Table REF and REF respectively.\"\n ],\n [\n \"Spectra\",\n \"The CR spectra are the most important effects to understand their propagation in the galaxy.\",\n \"Thanks to the new generations of spaced-borne experiments, the spectral measurements of CRs are stepping into precise era and revealed a series of new phenomena, such as the spectral hardening at 200 GV and break-off at $\\\\sim $ 14 TV, the famous excess of positron and extra-component of electron at high energy.\",\n \"In this section, the unified model calculations are described to reproduce and understand the measurements.\",\n \"Figure REF and REF show the measurements and model calculations for most of nuclei species individually.\",\n \"In the model calculations, the red solid line with shadow is the contribution from local SNR, the blue solid line is from galactic background sources and the black solid line is the sum of them.\",\n \"It is obvious that the spectral hardening can be reproduced well for all the species and its origin dominantly originates from the contribution of local source.\",\n \"To satisfy the energy break-off of proton at the rigidity of 14 TV observed by DAMPE experiment [46], the injection spectrum of local source is parameterized as a cutoff power-law form, $q_{\\\\rm inj}({\\\\cal R})=q_0{\\\\cal R}^{-\\\\nu ^{\\\\prime }}\\\\exp (-{\\\\cal R}/{\\\\cal R}^{\\\\prime }_{\\\\rm c})$ , where the normalization $q_0$ and spectral index $\\\\nu ^{\\\\prime }$ are determined through fitting to the CR energy spectra.\",\n \"The parameter ${\\\\cal R}^{\\\\prime }_{\\\\rm c}$ is adopted to be 15 TV, which leads to the bend of CR spectra starting around the rigidity of 5 TV.\",\n \"For detailed parameter information about the spectra, please refer to Table REF .\",\n \"Figure REF shows that the spectral break-off of proton is consistent well between model expection and data points from DAMPE and CREAM measurements [103], [46], but the Helium species has a little difference with DAMPE measurement and is roughly consistent with CREAM observations at several highest energy points.\",\n \"The reason is that the measured spectral break-off from DAMPE for Helium species is at the rigidity of $\\\\sim $ 34 TV, which is a little higher than proton under the Z-dependent energy cut-off frame [37].\",\n \"To keep the uniformity of all the nuclei, we choose the cut-off rigidity at 15 TV, which doesn't affect the results of other heavier primary nuclei and the conclusions.\",\n \"Under this physical scenario, our model expects that all the heavier primary nuclei have the same rigidity break-off at 5 TV as shown in Figure REF , which can be observed by the HERD experiment in future [73].\",\n \"Figure: The comparison between model calculations and experimental measurements for proton and Helium spectra as (a) and (b) panel.\",\n \"Here the red shadow shape is from the contribution of local SNR (e.g.\",\n \"Geminga), the blue solid line comes from the background components and the black solid line is the total contribution.\",\n \"The data points are measured by AMS-02, CREAM and DAMPE experiments, , , , .Figure: Similar to Figure , the individual species of C,O,Ne,Mg,SiC, O, Ne, Mg, Si and FeFe from panel (a) to (f).\",\n \"The data points are from the AMS-02 experiment, , .\"\n ],\n [\n \"Secondary Particles\",\n \"Following the primary species, the secondary particles spectra also have two components as the local and the global sources.\",\n \"Figure REF gives the spectra of $\\\\bar{p}$ , $Li, Be$ and $B$ from panel (a) to (d).\",\n \"The model calculations are well consistent with the observations from AMS-02 [20], [23].\",\n \"The hardening of $\\\\bar{p}$ spectrum starts from tens of GeV owing to 200 GV hardening of its mother proton.\",\n \"For the species of $Li, Be$ and $B$ , they are produced through fragmentation of heavier nuclei, such as $C, O$ and so on.\",\n \"They keep the same behaviour as their mother particles with the hardening at 200 GV and break-off at around $\\\\sim $ 10 TV.\",\n \"The typical energy break-off for secondary particles is pivotal to validate the dominant interactions around source regions for the nearby source, which are probe to unveil the origin of positron and $\\\\bar{p}$ excess at high energy.\",\n \"We hope that the spectral bend around $\\\\sim $ TeV can be observed by HERD experiment in near future.\",\n \"Figure: Similar to Figure , the secondary particles of p ¯,Li,Be\\\\bar{p}, Li, Be and BB from panel (a) to panel (d).\",\n \"The data points are from the AMS-02 experiment., .\"\n ],\n [\n \"Two components particles\",\n \"There are also some special particles, such as $N, Na, Al$ , including both the primary and secondary components.\",\n \"They are thought to be produced both in galactic background sources, and by the collisions of heavier nuclei with the interstellar medium (ISM) ($\\\\rm O\\\\rightarrow \\\\rm N+ \\\\rm X$ , $\\\\rm Mg\\\\rightarrow \\\\rm Na+ \\\\rm X$ , $\\\\rm Si\\\\rightarrow \\\\rm Na+ \\\\rm X$ , $\\\\rm Si\\\\rightarrow \\\\rm Al+ \\\\rm X$ )[67], [53], [93].\",\n \"As shown in Figure REF , the red solid line with shadow is the contribution from primary component accelerated directly by local SNR, the green solid line with shadow indicates the secondary component produced by the collisions of heavier nuclei with ISM, the blue solid line is from galactic background sources and the black solid line is the sum of them.\",\n \"It can be seen that the model calculations are agreement with the observations, and their spectra also have a break-off structure similar to that of primary particles at $\\\\sim 5$ TV.\",\n \"Figure: Similar to Figure , the individual species of N,Na,AlN, Na, Al, including both primary and secondary components, from panel (a) to (c).\",\n \"The data points are from the AMS-02 experiment, .\"\n ],\n [\n \"All particles\",\n \"The space-borne experiments have decisive advantages in the seperation ability for different species.\",\n \"However they have limited effective detector area, which lead to the lower statistical event numbers in high energy.\",\n \"On the contrary, the ground-based experiments have opposite properties.\",\n \"This makes that the spectral measurements for individual species step into dilemma above tens of TeV.\",\n \"The all-particle spectra can make up this shortcoming to constrain the high energy contributions.\",\n \"Figure REF shows the model calculations and measurements for all-particle spectrum.\",\n \"In the calculations, four groups as $H+He, C+N+O, Ne+Mg+Si$ and $Fe$ are demonstrated in blue, orange, green and red solid lines.\",\n \"It is clear that the knee structure of the all-particle spectrum can be properly reproduced by the background component assuming a Z-dependent cutoff with $R_c \\\\sim 7$ PV.\",\n \"In this case, the light species of protons and He nuclei dominantly composes the knee structure.\",\n \"This is because we try to fit the KASCADE spectra of proton and Helium, which was also favoured by the diffuse $\\\\gamma $ -ray measurement at ultra-high energy by AS$\\\\gamma $ experiment [45].\",\n \"Figure: The observed and calculated all-particle spectra.\",\n \"In the model calculations, the solid blue, orange, green, red and black line represent the P+He, C+N+O, Ne+Mg+Si, Fe species and all-particle spectra respectively.\",\n \"The data points are measured by TALE , Tibet-ASγ\\\\gamma , ICETOP , experiments and from weighted by Horandel .\"\n ],\n [\n \"Positron and Electron\",\n \"It is a hot topic for the positron excess and electron spectral hardening at high energy.\",\n \"Similar to nuclei secondary particles, the positron is also composed of two parts as around Geminga SNR and global background component from pp-collision.\",\n \"Figure REF shows the spectra of positron, electron, their sum and the ratio of positron to the sum of positron and electron.\",\n \"The model calculations are good reproduction of experimental data.\",\n \"Particularly, the positron will take $5\\\\%$ energy from its mother species proton in pp-collision.\",\n \"Owing to the spectral break-off around 5 TeV of proton, the positron spectrum has cut-off around 250 GeV and successfully reproduce the AMS-02 measurements as shown in panel (a).\",\n \"As the discussion in section 2, the age of Geminga SNR is $3.3\\\\times 10^5$ yrs, which leads to the energy break of electron spectrum about TeV as shown in panel (b) [50].\",\n \"Then the difference of energy break-off between positron and electron can be naturally understood.\",\n \"For the ratio of positron to the sum of positron and electron, it increases with increasing energy above about TeV, which results from the cut-off of the total electron spectrum at $\\\\sim $ TeV.\",\n \"Figure: Similar to the Figure , the spectra of positron, electron and their sum from panel (a) to (c) and the ratio of positron to the sum of positron and electron in panel (d).\",\n \"The data points are adopted from AMS-02 experiments , .\"\n ],\n [\n \"Ratios\",\n \"The ratios are important to understand the acceleration, propagation and interaction properties of CRs.\",\n \"Thanks to the unprecedented precise measurements from AMS-02, the ratios of primary to primary, secondary to secondary and secondary to secondary species can be well measured and clearly shown the difference [22], [24], [28], [27], [29].\",\n \"In this section, the corresponding model calculations are obtained to reproduce those observations.\"\n ],\n [\n \"Primary to primary species\",\n \"The ratios of primary to primary species carry the acceleration information in the source region.\",\n \"The model of diffusive shock acceleration predicts that the individual species should have identical power law spectrum [90], [30].\",\n \"Figure REF shows the comparison between model calculations and observations for the ratios of primary to primary species for $He/O$ , $C/O$ ,$N/O$ , $Ne/O$ , $Mg/O$ , $Si/O$ , $Ne/Mg$ , $Si/Mg$ , $Na/Si$ , $Al/Si$ , $Fe/He$ , $Fe/O$ and $Fe/Si$ .\",\n \"The red solid lines with shadow indicate the contributions from the nearby SNR.\",\n \"The black solid lines represent the ratio results of model expectation, taking into account the contributions from the nearby source.\",\n \"In fact, the individual spectrum has well reproduced the observations as shown in Figure REF , REF and REF , so the ratios should be also consistent between data and model calculations.\",\n \"Figure REF lists the ratios of primary to primary species, which has good consistency with observations.\"\n ],\n [\n \"secondary to primary species\",\n \"It is believed that most of the secondary nuclei originate from collisions of CRs with ISM in propagation.\",\n \"Therefore the information about CR propagation can be extracted from comparison between the spectra of secondary particles and those of primary CRs [105], [104], [106].\",\n \"Figure REF shows the ratios of secondary to primary species for $\\\\bar{p}/p$ , $Li/C$ , $Be/C$ , $B/C$ , $Li/O$ , $Be/O$ and $B/O$ .\",\n \"The blue solid lines represent the ratios of background component of secondary particles to the total amount of primary particles, the red solid lines show the ratios of nearby SNR component of secondary particles to the total amount of primary particles and the black solid line is the ratio of total secondary to total primary.\",\n \"The model calculations work well to reproduce the observations.\",\n \"Here except $\\\\bar{p}/p$ with energy independent (constant) distribution above 10 GeV, all other heavier nuclei have hardening above 200 GeV for model calculations.\",\n \"Just recently, the similar hardening has also been discovered in $B/C$ and $B/O$ above 100 GeV/n by DAMPE experiment [58].\",\n \"It is obvious that our model calculations work well to reproduce this hardening.\",\n \"Similar to the spectra of secondary species, the ratios have a break-off around TeV for $\\\\bar{p}/p$ and 5 TeV for heavier nuclei, which can offer a crucial and definitive identification with other kinds of models.\",\n \"Figure: Comparison between model calculations and observations for the ratios of secondary to primary species.\",\n \"The data points are adopted from AMS-02 measurements , , .\"\n ],\n [\n \"Secondary to secondary species\",\n \"The ratios of secondary to secondary species take important information of interaction in CR propagation.\",\n \"If they originate from the same mother particle, the spectral behavior reflects the common pecularity considering the known interaction cross-section, the same ISM and the same interaction time.\",\n \"This property can be served to understand the CR origin puzzle, such as positron [13].\",\n \"Figure REF shows the ratio comparison between model calculations and measurements for $\\\\bar{p}/e^+$ , $Li/B$ and $Be/B$ .\",\n \"Our model calculations are consistent with the measurements.\",\n \"More interesting is that the energy independent distribution is clear shown from them above 10 GeV.\",\n \"The model calculation of $\\\\bar{p}/e^+$ rises up sharply above 300 GeV.\",\n \"This is because the energy cut-off of positron around 300 GeV.\",\n \"Figure: Similar to Figure .\",\n \"Comparison between model calculations and observations for the ratios of secondary to secondary species.\",\n \"The data points are adopted from AMS-02 measurements , .Similar to the results of spectra and ratios, the anisotropy is also the joint contributions from the background and local sources of CRs.\",\n \"Due to more abundant sources in the inner disk, the phase of anisotropy directs to the galactic center.\",\n \"However, the observations of phase roughly point to the direction of anti-galactic center from 100 GeV to 100 TeV.\",\n \"The local source, located at the outer of galactic disk, plays the dominant roles in this energy region.\",\n \"So the anisotropy demonstrates mutually repressive competition between local source and background.\",\n \"The dip structure at 100 TeV is the transition energy point for the two kinds of sources.\",\n \"Figure REF shows the amplitude and phase of anisotropy for CRs.\",\n \"The CR anistropy can be fitted well under this physical scenario.\",\n \"Figure REF presents amplitude and phase for CREs.\",\n \"The expection of amplitude for CREs is lower than the up-limit of Fermi-LAT experiment [9].\",\n \"Figure: Comparison between model calculations and observations of the amplitude and phase of anisotropy for CRs.\",\n \"The data points are taken from underground muon detectors:Norikura (),Ottawa (),London (),Bolivia (),Budapest (),Hobart (),London (),Misato (),Socorro (),Yakutsk (),Banksan (),Hong Kong (),Sakashita (),Utah (),Liapootah (),Matsushiro (),Poatina (),Kamiokande (),Marco (),SuperKamiokande ();and air shower array experiments:PeakMusala (),Baksan (),Norikura (),EAS-TOP (, , ),Baksan (),Milagro (),IceCube (, ),Ice-Top (),ARGO-YBJ (),Tibet (, , )K-Grande().Figure: Comparison between model calculations and up-limit of the amplitude and phase of anisotropy for CREs.\",\n \"The data points are adopted from\"\n ],\n [\n \"Summary\",\n \"The new generation of space-borne and ground-based experiments took unprecedented precise measurements for CR spectra and anisotropy and revealed multi-messenger anomalies for them.\",\n \"In this work, we propose that the local source, Geminga SNR, is the common origin for all those anomalies.\",\n \"The Geminga SNR has three critical advantages: perfect age of 330 kyrs, suitable position with a distance of 330 pc and assumed DMC around it.\",\n \"The physical figure can be summarized as following.\",\n \"Firstly, the diffusive shock around Geminga SNR can accelerate the CR nuclei and electrons to very high energy together.\",\n \"Here we adopt the break-off rigidity to be $\\\\sim $ 5 TV for the sake of 14 TV energy cut-off observation by DAMPE experiment [46].\",\n \"The electron will suffer the energy loss for 330 kyrs, which lead to the spectral break around TeV observed by DAMPE, HESS [31], [33], [59].\",\n \"Due to the contributions of the local source, the spectral hardening and drop-off for all nuclei and electron respective can be well reproduced.\",\n \"Secondarily, the interaction of pp-collision and fragmentation with DMC happen to produce secondary particles, such as $\\\\bar{p}, e^+, \\\\gamma , Li,Be,B$ and so on.\",\n \"Inheriting the properties from their mother particles, the $\\\\bar{p}, e^+, \\\\gamma $ as products of pp-collision will take $10\\\\%, 5\\\\%, 10\\\\%$ of proton energy, which result in the corresponding energy break-off at 250, 500, 500 GeV respectively for them.\",\n \"So the hardening and drop-off for secondary particles can also be well consistent with the measurements.\",\n \"Simultaneously, the heavier secondary nuclei from fragmentation process, such as $Li,Be,B$ , will have the same energy break-off with their mothers around $\\\\sim $ 5 TeV.\",\n \"This property is typical for this kind model, which can be served to test the model in future.\",\n \"Lastly, thanks to the appropriate location of Geminga SNR, the special evolution with energy of anisotropy in amplitude and phase can roughly reproduce the observations.\",\n \"Summarily, the multi-messenger anomaly of spectra, ratios and anisotropy can be well reproduced in one unified physical mechanism.\",\n \"Particularly, the difference of spectral break-off between positron and electron can also be explained in the same scenario.\",\n \"More interesting is that we expect the typical energy break-off feature for secondary nuclei $Li,Be,B$ and their ratios with primary nuclei.\",\n \"This feature will also play important role to resolve the origin puzzle of positron and $\\\\bar{p}$ excesses.\",\n \"We hope that this feature can be observed by HERD experiment in future.\",\n \"This work is supported by the National Key Research and Development Program (No.\",\n \"2018YFA0404202), the National Natural Science Foundation of China (Nos.\",\n \"12275279, U2031110, 12175248).\"\n ],\n [\n \"The propagation of CRs\",\n \"It has been recognized in recent years that the propagation of CRs in the Milky Way should depend on the spatial locations, as inferred by the HAWC and LHAASO observations of extended $\\\\gamma $ -ray halos around pulsars [10], [34] and the spatial variations of the CR intensities and spectral indices from Fermi-LAT observations [102], [12].\",\n \"The spatially-dependent propagation (SDP) model was also proposed to explain the observed hardening of CRs [97], [98], [62], [69], [77], [70], and also the large-scale anisotropies by means of a nearby source[76], [87].\",\n \"In the SDP model, the diffusive halo is divided into two parts, the inner halo (disk) and the outer halo.\",\n \"In the inner halo, the diffusion coefficient is much smaller than that in the outer halo, as indicated by the HAWC observations.\",\n \"The propagation equation of CRs in the magnetic halo and the descriptions of specific iterms in it can be refered to [69].\",\n \"The spatial diffusion coefficient $D_{xx}$ can be parameterized as $D_{xx}(r,z, {\\\\cal R} )= D_{0}F(r,z)\\\\beta ^{\\\\eta } \\\\left(\\\\frac{\\\\cal R}{{\\\\cal R}_{0}} \\\\right)^{\\\\delta _{0}F(r,z)},$ where $r$ and $z$ are cylindrical coordinate, ${\\\\cal R}$ is the particle's rigidity, $\\\\beta $ is the particle's velocity in unit of light speed, $D_0$ and $\\\\delta _0$ are constants representing the diffusion coefficient and its high-energy rigidity dependence in the outer halo, $\\\\eta $ is a phenomenological constant in order to fit the low-energy data.\",\n \"The spatial dependent function $F(r, z)$ is given as $F(r,z) = \\\\left\\\\lbrace \\\\begin{array}{ll}{\\\\frac{N_m}{1+f(r,z)}+\\\\left[1-\\\\frac{N_{m}}{1+f(r,z)}\\\\right]}\\\\left(\\\\frac{{z}}{\\\\xi {z}_{\\\\rm h}} \\\\right)^{n}, & {{|z|} \\\\le \\\\xi {z}_{\\\\rm h}} \\\\\\\\\\\\\\\\1, & { {|z|} > \\\\xi {z}_{\\\\rm h}} \\\\\\\\\\\\end{array},\\\\right.$ where the total half-thickness of the propagation halo is $z_h$ , and the half-thickness of the inner halo is $\\\\xi z_{h}$ .\",\n \"The constant $n$ describes the smoothness of the parameters at the transition between the two halos.\",\n \"The expression $f(r, z)$ is the source density distribution.\",\n \"In this work, we adopt the diffusion re-acceleration model, with the diffusive re-acceleration coefficient $D_{pp}$ , which correlated with $D_{xx}$ via $D_{pp}D_{xx} = \\\\frac{4p^{2}v_{A}^{2}}{3\\\\delta (4-\\\\delta ^{2})(4-\\\\delta )}$ , where $v_A$ is the Alfvén velocity, $p$ is the momentum, and $\\\\delta $ is the rigidity dependence slope of the diffusion coefficient [89].\",\n \"The parameters of SDP model used here are listed in Table REF .\"\n ],\n [\n \"Background sources\",\n \"All of the SNRs other than the nearby one are labeled as background sources.\",\n \"The source density distribution is approximated as an axisymmetric form parametrized as $f(r, z) = \\\\left(\\\\frac{r}{r_\\\\odot } \\\\right)^\\\\alpha \\\\exp \\\\left[-\\\\frac{\\\\beta (r-r_\\\\odot )}{r_\\\\odot } \\\\right] \\\\exp \\\\left(-\\\\frac{|z|}{z_s} \\\\right) ~,$ where $r_\\\\odot \\\\equiv 8.5$ kpc represents the distance from the Galactic center to the solar system.\",\n \"Parameters $\\\\alpha $ and $\\\\beta $ are taken to be 1.69 and 3.33 [55].\",\n \"The density of the source distribution decreases exponentially along the vertical height from the Galactic plane, with $z_s$ = 200 pc.\",\n \"The parameters of SDP model used here are listed in Table REF .\",\n \"The injection spectrum of nuclei and primary electrons are assumed to be an exponentially cutoff broken power-law function of particle rigidity ${\\\\cal R}$ , i.e.\",\n \"$q({\\\\cal R}) = Q_{0} \\\\left\\\\lbrace \\\\begin{array}{lll}\\\\left(\\\\frac{{\\\\cal R}}{{\\\\cal R}_{\\\\rm br}} \\\\right)^{\\\\nu _{1}}, & {{\\\\cal R} \\\\le {\\\\cal R}_{\\\\rm br}} \\\\\\\\\\\\\\\\\\\\left(\\\\frac{{\\\\cal R}}{{\\\\cal R}_{\\\\rm br}} \\\\right)^{\\\\nu _{2}} \\\\exp \\\\left[-\\\\frac{\\\\cal R}{{\\\\cal R}_{\\\\rm c}} \\\\right], & { {\\\\cal R} > {\\\\cal R}_{\\\\rm br}} \\\\\\\\\\\\end{array}\\\\right.,$ where $Q_{0}$ is the normalization factor, ${\\\\cal R}_{\\\\rm br}$ is break rigidity, $\\\\nu _{1,2}$ are the spectral incides before and after the break rigidity, ${\\\\cal R}_{\\\\rm c}$ is the cutoff rigidity.\",\n \"Table REF shows the injection spectrum parameters of different CR nuclei species.\",\n \"The numerical package DRAGON is used to solve the propagation equation of CRs [60].\",\n \"For energies smaller than tens of GeV, the fluxes of CRs are suppressed by the solar modulation effect.\",\n \"We use the force-field approximation [65] to account for the solar modulation.\",\n \"The secondary CR nuclei, such as Li, Be, B, can be brought forth from the fragmentation of heavier parent nuclei throughout the transport.\",\n \"The production rate is expressed as follows $Q_j = \\\\sum _{i = \\\\rm C, N, O} (n_{\\\\rm H} \\\\sigma _{i+{\\\\rm H}\\\\rightarrow j} +n_{\\\\rm He} \\\\sigma _{i+{\\\\rm He} \\\\rightarrow j} ) v \\\\psi _i ~,$ where $n_{{\\\\rm H}/{\\\\rm He}}$ is the number density of hydrogen/helium in the ISM and $\\\\sigma _{i+{\\\\rm H/He}\\\\rightarrow j}$ is the total cross section of the corresponding hadronic interaction.\",\n \"Unlike above secondary CR nuclei, secondary $\\\\rm e^+$ and $\\\\rm \\\\bar{p}$ are produced through the pp collisions between the primary CR nuclei from background sources and ISM.\",\n \"Therefore the source term of both $\\\\rm e^+$ and $\\\\rm \\\\bar{p}$ is the convolution of the energy spectra of primary nuclei $\\\\Phi _i(E)$ and the relevant differential cross section $d \\\\sigma _{i + {\\\\rm H/He} \\\\rightarrow j}/d E_j$ , i.e.\",\n \"$\\\\nonumber Q_j &=& \\\\sum _{i = \\\\rm p, He} \\\\int dp_i v \\\\left\\\\lbrace n_{\\\\rm H} \\\\frac{ \\\\sigma _{i+{\\\\rm H}\\\\rightarrow j}(p_i, p_j)}{ dp_j} \\\\right.\",\n \"\\\\\\\\& & \\\\left.\",\n \"+n_{\\\\rm He} \\\\frac{\\\\sigma _{i+{\\\\rm He}\\\\rightarrow j}(p_i, p_j)}{dp_j} \\\\right\\\\rbrace \\\\psi _i(p_i) ~,$ Furthermore, antiprotons may still undergo non-annihilated inelastic scattering with ISM protons during propagation, which can generate the tertiary production.\"\n ],\n [\n \"Nearby supernova remnant\",\n \"We assume that one supernova explosion in the vicinity of the solar neighborhood occurred in a giant MC about $10^5-10^6$ years ago.\",\n \"The CR charged particles were continually accelerated by passing back and forth across the shock front with the expansion of supernova ejecta.\",\n \"Under the SDP scenario of GCR propagation, [78] demonstrated that, among the observed local SNRs, only Geminga SNR is able to explain both the spectra and anisotropies observations of CRs simultaneously.\",\n \"Since the location of Monogem is similar to that of Geminga, their impacts on CR flux may degenerate with each other.\",\n \"Here, for simplicity, we take Geminga SNR as the major contributor of the local source in this work.\",\n \"The Geminga SNR locates at its birth place with the distance and age of r = 330 pc and $t_{inj} = 3.3\\\\times 10^5$ yrs.\",\n \"The direction in the galactic coordinate is $l=194^{\\\\circ }$ (galactic longitude) and $b=-13^{\\\\circ }$ (galactic latitude) [91].\",\n \"Its distance, age and direction jointly decide the important role as an optimal candidate of nearby source to CRs [76], [87], [110].\",\n \"The injection process of SNR is approximated as burst-like.\",\n \"So the primary and secondary particles have experienced 300 kyrs and a tiny part of them enter into solar system in the end.\",\n \"In this work, the propagated spectrum from Geminga SNR is thus a convolution of the Green's function and the time-dependent injection rate $Q_0(t)$[50], i.e.\",\n \"$\\\\varphi (\\\\vec{r}, {\\\\cal R}, t) = \\\\int _{t_i}^{t} G(\\\\vec{r}-\\\\vec{r}^\\\\prime , t-t^\\\\prime , {\\\\cal R}) Q_0(t^\\\\prime ) d t^\\\\prime .$ The normalization is determined through fitting Galactic cosmic rays energy spectra and the detailed parameters is listed in Table REF .\",\n \"Besides, the CR nuclei generated by the local SNR also collide with the molecular gas around them and give birth to prolific daughter particles, like B, $\\\\rm e^{\\\\pm }$ , $\\\\rm \\\\bar{p}$ , and so forth.\",\n \"The yields of B and $\\\\rm e^{\\\\pm }$ , $\\\\rm \\\\bar{p}$ inside the MC are respectively $Q_j = \\\\sum _{i = \\\\rm C, N, O} (n_{\\\\rm H} \\\\sigma _{i+{\\\\rm H}\\\\rightarrow j} +n_{\\\\rm He} \\\\sigma _{i+{\\\\rm He}\\\\rightarrow j} )v Q_i(E) t_{\\\\rm col}$ and $Q_j \\\\; = \\\\;\\\\sum _{i = \\\\rm p, He} {\\\\displaystyle \\\\int \\\\limits _{E_{\\\\rm th}}^{+ \\\\infty }} \\\\; d E_i \\\\;v \\\\; \\\\left\\\\lbrace n_{\\\\, \\\\rm H}{\\\\displaystyle \\\\frac{d \\\\sigma _{i + {\\\\rm H} \\\\rightarrow j}}{d E_j }} \\\\right.\",\n \"\\\\\\\\\\\\left.\",\n \"+n_{\\\\, \\\\rm He} {\\\\displaystyle \\\\frac{d \\\\sigma _{i + {\\\\rm He} \\\\rightarrow j}}{d E_j }} \\\\right\\\\rbrace Q_i(E_i) t_{\\\\rm col} ~,$ where $n_{\\\\rm H/He}$ is the number density of hydrogen/helium in MC.\",\n \"In this work, we assume that it is 1000 times greater than the mean value of ISM.\",\n \"$t_{\\\\rm col}$ is the duration of collision of 420 yrs.\",\n \"$Q_i(E)$ is the accelerated spectrum of primary nuclei inside local SNR.\",\n \"Summarily, the local source, Geminga SNR, is responsible for the spectral hardening at 200 GV and cut-off at 14 TV for all the nuclei species.\",\n \"The primary electron can be accelerated to $\\\\sim $ 5 TeV, similar to nuclei, but will undergo the energy loss in ISRF and have a spectral cut-off around TeV due to its age of $3.3\\\\times 10^5$ yrs.\",\n \"For the secondary particles, the spectral cut-off depend on their mother ones and the interaction modes.\",\n \"For positron, the energy break-off will located at $\\\\sim $ 300 GeV.\",\n \"The different energy break-off between positron and electron can be natural understood in this scenario.\",\n \"The secondary heavier nuclei will have the same spectral cut-off as their mothers at $\\\\sim $ 5 TV, which can be tested in future experiments like HERD [73].\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05641"}}},{"rowIdx":92,"cells":{"text":{"kind":"list like","value":[["Generic Tagging for RISC-V Binaries"],["Abstract With the widespread popularity of RISC-V -- an open-source ISA -- custom hardware security solutions targeting specific defense needs are gaining popularity.","These solutions often require specialized compilers that can insert metadata (called tags) into the generated binaries, and/or extend the RISC-V ISA with new instructions.","Developing such compilers can be a tedious and time-consuming process.","In this paper, we present COGENT, a generic instruction tag generator for RISC-V architecture.","COGENT is capable of associating a tag of configurable and varying widths (1 to 20 bits) to each instruction.","It is also capable of emitting labels that are central to the implementation of control-flow integrity (CFI) solutions.","COGENT encodes all tags and labels as nop instructions thereby providing full backward compatibility.","We evaluate COGENT on a subset of programs from the SPEC CPU2017 benchmark suite and report the binary size increase to be 29.3% and 18.27% for the lowest and highest tag coverage levels respectively.","Additionally, we executed tagged programs on COTS RISC-V unmodified hardware and found the execution time overhead (with respect to backward compatibility) to be 13.4% and 5.72% for the lowest and highest coverage levels respectively.","Finally, using a case study, we present possible use case scenarios where COGENT can be applied."],["Introduction","Over the years, the advent of Internet of Things (IoT) along with an increasing number of mobile, desktop and server systems has led to a steady increase in cybersecurity attacks.","Most of the attacks–even those targeted at the hardware (e.g.","Spectre[1], and Meltdown[2]) are initiated through software.","The defense community has focused efforts at different layers of deployments.","For instance, the software community has explored software-only defenses such as StackGuard[3], and CFI via program instrumentation (e.g., [4], [5], [6]).","Meanwhile, the hardware community has pursued defenses that leverage hardware features in order to detect and flag exploits (e.g., Intel CET[7], W⌃X) or prevent information leakage (e.g., Intel SGX[8], ARM TrustZone[9]).","While software-only security measures are highly desirable and can be deployed with little effort, they typically suffer higher performance penalties compared to hardware solutions.","Furthermore, the security mechanisms themselves may be a part of the attack surface, which brings about transparency issues.","On the other hand, hardware-only security mechanisms must bridge the semantic gap problem and recover high-level security-relevant program semantics in order to enforce rich and expressive policy abstractions.","Further, the tolerance for errors is low and “fixing\" design errors is very expensive (e.g.","Intel MPX[10], [11] had problems that resulted in eventual deprecation).","With the gaining popularity of RISC-V, an open-source ISA, hardware - software cohesive defenses have gained center stage and are desirable.","Typically, defenders take advantage of (a) fine-grained control over the RISC-V hardware including (but not limited to) changes to ISA in combination with (b) curated compilers capable of enriching binaries with security metadata/stubs, deploying highly effective and fast performing defenses.","For example, recent defenses such as Cheri [12], Zero [13], RetTag [14], Framer [15], etc., all rely on a dedicated compiler that can generate instrumented binaries for their modified hardware.","The modified binaries contain metadata called “tags” that provides the hardware with specific information regarding the code or data unit in the program.","Typically, there is a one-to-one mapping between a tag and corresponding code/data unit.","These defenses strike a balance between security and performance requirements and are often tailor-made to deployment needs.","They rely on performant compilers and/or hardware design and engineering, yet compiler development remains an expensive process often requiring several hundred developer-hours for development and testing.","More importantly, availability of a suitable compiler would attract more hardware defenses that are currently impeded by the lack of a compiler.","Solutions that shorten the development effort and allow quick incorporation of metadata into binaries are missing and highly sought after.","In this paper, we address the above need and present COGENT: COmpiler for GENeric Tagging, a generic LLVM-based compiler that incorporates instruction tags (or code metadata) into RISC-V binaries.","In essence, COGENT removes the burden of compiler development from RISC-V hardware defenses that rely on embedding instruction metadata into binaries.","Motivated by the needs of Control-Flow Integrity (CFI) defenses that embed CFI labels into code, COGENT interleaves tag metadata into the instruction stream.","COGENT has several key features that make it highly favorable for hardware-software cohesive defenses.","First, it is highly flexible.","It can accommodate various tag widths (1 to 20 bits per instruction).","This flexibility is essential in order to achieve rich expressiveness in tags.","Second, it is functionally correct.","That is, it ensures that the inserted tags do not interfere with program logic.","Functional correctness is non-trivial due to several challenges (see Section REF ).","This is important since a tag system is not usable without such guarantees.","Third, COGENT preserves backward compatibility.","That is, tagged binaries will seamlessly run on generic RISC-V processors that are tag-unaware.","We accomplish backward compatibility by encoding tag information into NOP instructions that are simply ignored by unsupported hardware.","Finally, our changes are segregated into LLVM IR-level and target-specific changes.","Other architectures (e.g., ARM) can be supported by reusing the IR-specific changes and rewriting ARM LLVM backend.","We make the following contributions: We implement a flexible and defense-scheme agnostic implementation codenamed COGENT, a LLVM-based compiler for RISC-V target.","COGENT provides configurable instruction-level tagging support to several different tag widths and coverage.","We also make changes to the LLD linker to support COGENT.","We demonstrate the efficacy of COGENT via a CFI implementation.","We evaluate the example CFI implementation for correctness and backwards compatibility.","We demonstrate moderate to low performance overhead and binary bloat.","Through case studies, we demonstrate potential real-world applicability of COGENT.","Summary of Results: We evaluated the correctness of COGENT by compiling and running a subset of the SPEC CPU2017 benchmark suite on Commercial Off The Shelf (COTS) hardware.","Our subset has small function sizes, heavy memory overhead, and high loop counts, all of which stress instruction caches.","For our example CFI implementation, we found that COGENT increased the size of the .text section of these programs by an average of 29.32% for the lowest tag coverage, and 18.27% for the highest.","We found that backward-compatibility performance (i.e., overhead imposed by tagged binaries running on tag-unaware hardware) was impacted by an average of 13.40% for the low coverage and 5.72% for the highest.","We also present several possible use cases for the system, including specific recent CVEs that could be addressed.","The rest of the paper is organized as follows: sec:background provides the technical background necessary to understand the remainder of the paper.","In sec:overview and sec:technicaldetails, we present the overview and technical details of COGENT.","We evaluate COGENT in sec:eval and provide case studies to highlight benefits of using COGENT.","Finally, we present related work in sec:relatedwork and conclude in sec:conclusions.","Tagging is a technique that very broadly associates some amount of metadata (the tag), with a memory location and registers.","Specially designed architectures then consume that tag and enforce a security policy.","Tags can be associated with a) memory objects (data tagging) allowing policies to track and protect data in memory, and/or on b) code (instruction tagging), allowing policies to change or extend the behavior of the tagged instructions.","Of particular importance to this paper is instruction tagging.","Instruction Tagging: allows a tag-aware architecture to execute additional code or checks on any instruction associated with a particular tag.","Instruction tags are useful in implementing Control-Flow Integrity (CFI) policies.","Identical tags are used at control flow sources (call/jump sites) and control flow targets in order to signify legal control flow.","Tags can be placed on load/stores to protect sensitive data, forbidding access to a memory region if the tag is incorrect.","These kinds of tags often require corresponding metadata placed with the data they are meant to access.","The tag policies and enforcement are implementation specific and form the essence of a tag-based defense.","The specific instruction tags are a function of the intended security policies, and therefore vary greatly between designs [16].","Differences in tag requirements can even result in different designs which use the same hardware for similar purposes [17].","This implementation-specific design requirement is inherent to the nature of tagging.","Tags are simply a way of providing more information to the program at run time, but the information needed is dependent on the design of the system itself.","Broadly speaking, there is a trade-off made in the design of tag systems between how much information it provides to the hardware, and how much overhead the tags introduce into the system.","This overhead can include power consumption, hardware complexity, performance or any other costs paid by the design.","Depending on how much information one's defense needs, tags can be arbitrarily large, or as small as a single bit.","Designers often use techniques such as encoding the metadata information into the unused upper bits of a pointer, but this is often insufficient on its own [14], requiring additional tagging or ISA(instruction set architecture) changes.","The choices of where to store metadata, how much metadata to store, and how to access it are defining characteristics of tagging systems."],["LLVM Compiler Infrastructure","The LLVM compiler infrastructure [18] is a fully featured compiler designed with modularity in mind.","It is broken down into components, which operate on different parts of the compilation process.","Important to COGENT is the idea of the language-specific front end, the target architecture-specific back end, and the language- and target-agnostic intermediate representation (IR).","The LLVM design requires the language-specific front end to convert source code into LLVM IR, the optimizer schedules an analysis and transformation pass pipeline on that IR, and its output is IR for consumption by the back end.","The back end might also perform additional transformations on the code via its own set of passes and built-in functionality.","The final product of the back end is then output to an object file, which is fed into a linker to create the final binary.","Modifying the LLVM compiler can thus take two main forms.","The simplest scenario is to create and modify transformation and analysis passes within the existing structure of the pipeline.","These transformation passes can be scheduled to run before or after any specific pass or combination of passes, allowing one to place new transformations and analysis at any point in the pipeline.","The more complex scenario is to modify the internal components of LLVM, changing or extending the functionality of pre-existing code in the LLVM infrastructure."],["Relocations and Symbols","As part of the creation of an object file, the back end will emit target specific information called a relocation into object files.","These relocations are designed to inform the linker of how to handle information that cannot be finalized before the linking phase of compilation.","Examples of this for RISC-V include encoding addresses, which are often emitted as a pair of relocations (named HI and LOW), covering two instructions (i.e.","auipc, jalr).","These relocations tell the linker to place the final address of the target symbol across two instructions when it is required.","Symbols themselves can be kept as offsets from an unknown starting address, allowing the linker to know the final address during the linking process, even though the absolute position cannot be determined by the compiler back end.","Tagging schemes are complex and varied, and different schemes require different configurations.","There are some that require more space (i.e., metadata) to be accessible for any given instruction, while some might require only a single bit for some defenses [13].","Additionally tagging schemes must be able to provide the metadata to the hardware.","A generic instruction tagging scheme must be able to meet this requirement under as many different configurations as possible.","To that end we set out to create COGENT.","The goal of COGENT is to enable embedding configurable metadata tags into instruction stream, and not implementation details for different defense schemes such as forward or backward edge CFI, heap or stack, stack protection, or any other particular defense scheme.","We did not set out to create a specific defense with this paper, or provide analysis that can be used to create one.","Instead we focused on three major tasks.","Flexibility: COGENT must be able to generate binaries for RISC-V architecture that require varying tag widths, with minimal changes.","Additionally, we aim to modularize our implementation in order to minimize the developer effort in modifying COGENT to support a new architecture.","COGENT must also be able conducive to supporting different types of defense schemes including those that require large amounts of metadata.","Functional Correctness: Insertion of tags and metadata must not interfere with program logic.","That is, all necessary tags must be appropriately inserted, and inserted tags must not alter the control flow or corrupt data.","Backwards Compatibility: The tool must allow created code to be backwards compatible.","That is, tagged binaries must be able to seamlessly execute on tag-unaware hardware.","While backwards compatibility is a desirable feature for many systems, we recognize that it may not be necessary in every case.","If turning it off could provide benefits to a scheme COGENT should allow that as well.","A tag or metadata lookup action in the hardware could be expensive and inefficient.","Therefore it is best to reduce the number of abstractions required to retrieve any tag for a given instruction.","Likewise having separate (but mapped) memory adds complexity to the process of bringing the tags into the hardware."],["Tag Semantic Gap","During the compilation process, programs pass through different stages of compilation.","While the tags are eventually assigned on the raw instruction stream, the actual assignment of tags may occur at a higher level of abstraction (i.e., source code or intermediate representation).","Depending on the stage of compilation where tags are assigned, percolating them down to the instructions is important and requires bridging of the semantic gap between the higher and lower levels of program representation.","Further, during compilation the compiler will often expand IR instructions/source statements into multiple assembly instructions.","This manifests as a Many-to-One relationship where many tags map to a single IR instruction/source statement.","A very common example in RISC-V is the lowering of the LLVM-IR pseudo instruction PSEUDOCALL from MachineInstr level into the MCinst level.","During this process the single PSEUDOCALL is expanded into an auipc,jalr pair.","Likewise there are One-to-Many relationships where multiple instructions at a higher-level of abstraction are lowered into a single assembly instruction (e.g.","bitcast instructions).","These one-to-many and many-to-one relationships can occur during any point of the lowering process, and where possible, COGENT must provide ways to handle them correctly.","Unfortunately there are many cases without a generic solution i.e.","in the lowering from IR level instructions to machine Instructions."],["Compiler-inserted Code","During compilation the compiler may introduce instructions that did not exist at the previous (i.e., higher) level of representation.","This is typically done to meet back end specifications or optimizations.","Examples include in-lining and function prologue/epilogue insertion.","COGENT must provide a way to tag compiler-inserted code, and must ensure that it does not disrupt the tag schemes."],["Pre-calculated Offsets, Symbols, and Relocations","Features such as C++’s try/catch block implementation often pre-calculate the distance between two symbols instead of leaving it as a relocation wherever possible.","COGENT must be sensitive to these pre-calculated values.","More generally, wherever possible the compiler calculates the offset to a relocation or a symbol from the beginning of a compilation unit.","COGENT must preserve the relative locations in the code section though any changes it makes."],["ISA Changes","Prior efforts have altered the ISA to enrich code with metadata.","Any ISA changes would by their very nature break backwards compatibility.","COGENT must be able to encode tags without any modifications to the ISA."],["Our Approach","When examining both the challenges and the goals, we converged on the following design choices for COGENT.","These solutions address both the challenges and goals."],["Flexibility","To address flexibility and the many back-end specific challenges we encountered, we designed COGENT to be as modular as possible, with front end and back end components in order to allow for adaptability.","Unfortunately many (but not all) of the back end changes are target specific in the best case, and design specific in the worst.","To solve this we clearly separate the COGENT process into five major pieces.","IR Level Tagging API.","This component allows the creation of IR level passes that can take advantage of the front and back end agnostic nature of the IR to perform analysis and select tags for IR level instructions.","These IR level tagging passes can be shared between any source and any target, and so are the most portable method of tag selection.","This component consists of an analysis pass that provides propagation of assigned tags into any back-end passes where possible.","Backend Level Tagging API.","This component allows the creation of LLVM MachineInstr and MCinst level passes that can tag code after it has been lowered into a target specific back end.","This allows tagging of code inserted after IR level optimizations or back end insertions.","Backend and Defense Specific Changes.","These changes are both back end and defense specific.","Representing choices that must be made in how the target back-end and the scheme must interact.","For the RISC-V prototype we developed (see sec:example), One example would be the insertion of a metadata label between an auipc jalr pair to include a CFI label.","A second would be the specific behavior of tags in the many to-one and one-to-many lowering relationships.","These changes cannot be handled by back end passes alone and must be hard coded into the back end itself on a case-by-case basis.","Defense Scheme Agnostic Back End Changes.","These are changes to the back end that must be done to allow for the insertion of the tag instructions as late as possible into the code stream.","This includes turning off the pre-calculated offsets, ensuring correct tag placement, and adjusting symbol and relocation offsets before object file emissions.","These do not interact with the logic of tag selection, only with the emission of the tags themselves into an object file.","Configuration File.","This file allows a user to specify simply what emission options they would like for the final tag instructions.","Details of the provided options are presented in sec:select, but the configuration file allows developers to switch between tag widths for their own instruction tagging schemes.","To address the flexibility requirement of providing additional metadata where needed, we introduce metadata labels as a tool for a developer to use.","These labels can be repurposed for any use, and are encoded as nops by the compiler in order to maintain backwards compatibility.","In the RISC-V back end we chose lui instruction which provides 20 bits of metadata per inserted label."],["Functional Correctness",".","Addressing correct tag placement becomes challenging when inserting new tag instructions at specified alignments.","Any changes made to the code after a tag has been inserted will displace the tags and must be adjusted in order to maintain consistent tag placement.","To this end, all functions are forced to be aligned.","Additionally, we assume that relaxation/compressed instruction generation is turned off (see sec:compressed for a discussion on compressed instructions).","Additionally optimizations such as pre-calculating offsets based on the MCInst’s as described in challenges must be turned off.","This adds additional one time overhead to the linking process, but no extra run time cost beyond the insertion of the tags themselves.","Optimizations introduced by IR passes can be left in place however, by insuring tag insertion is completed at the last point possible in the pipeline.","Finally tag instructions can be placed in locations that disrupt relocation pairs, meaning the linker must be made aware of this possibility."],["Backward Compatibility","To allow for backwards compatibility, at least one of the possible tag instruction configurations must be encoded as a nop instruction.","For the RISC-V prototype, the instruction lui x0, <imm> was selected, as it is functionally a nop and gives the maximum number of bytes to encode tag values.","Other target architectures would have to select different nop-equivalent instructions with differing amounts of bits to ensure backwards compatibility.","Table: Tag instructions used by COGENT."],["Details","With the solutions outlined in sec:overview we created a prototype version of COGENT.","This prototype comes in two parts, The underlying changes to LLVM itself, and an example implementation that provides metadata labels on computed jumps and computed callsites."],["LLVM Changes","We make all of the changes described in this section to LLVM version 10.0.0.","The first and the most important step in COGENT implementation is to provide a way for the developer to select how they wish to represent the tags in the instruction stream.","This representation is referred to as the tag instruction.","The selection of this tag instruction determines How many bits a user will have available to store tags How many instructions any single tag can cover Backwards compatibility fig:taginstructions shows the different choices we provide via a configuration file.","We detail their bit layout, how many bits they provide, and the number of bits available for each instruction based on the coverage desired.","If backwards compatibility is not required, users can select a custom opcode for the tag instruction.","The tag enumeration itself is defined in a separate header file that must be included to the relevant passes.","Figure: Demonstration of COGENT using the simple CFI described in sec:example, contains compiler inserted code (0x58-0x68), many to one relationships (0x94-0x9C), relocation and symbol adjustments (0x6c-0x74), a tag inserted between a relocation (0x70).","Additionally it includes data that must be persisted from the IR (the function labels) to the backend inserted labels at (0x50) and (0x98)."],["Tagging API","Once the tag instruction has been selected, the next COGENT process is to assign tags to LLVM’s IR level instructions or machineInstr.","In the backend, this process is done via a simple API we expose to the user, consisting of standard .set(<tag>) and .get(<tag>) extensions on top of LLVM’s base machineInst classes.","These are employed by LLVM backend passes that developers insert into the compiler pipeline.","In general tag instrumentation passes should be inserted as late as possible in the pipeline to prevent optimizations from interfering with tag lowering.","We also provide label emission, via an emission of a single lui nop at any point in the compilation process.","This lui must be tagged appropriately to indicate to the hardware to process this label as a particular type of metadata.","For tagging in the IR level, we provide a pass that IR level passes can push a mapping of IRInstruction : tag into, and that backend passes can register with to access that tag mapping to assign those tags to machineInstr.","The lowering of these IR tags is complicated by the needs of the target specific backend, as described in the next section."],["Lowering Behavior","For IR instruction tagging, we provide a default behavior in a modified RISCV backend.","However depending on the specific deployment needs, developers must investigate the lowering process and make changes where appropriate to their planned tagging behavior.","By default one-to-many transformations apply the tag on the IR instruction on every machine instruction created by that instruction during the lowering process.","For many-to-one, the default behavior is the first tag.","Any instructions not directly created from another instruction (an optimization or specification specific to the back end) will be assigned the default tag value from the tag configuration file.","This process may be insufficient for specific defense needs, but there is no one-size-fits-all approach to the lowering problem and necessitates developer effort.","For machine instruction inserted tags, we modify the lowering process from machineInstr to MCInst allowing the tag field to be copied over.","In cases where this is a one to one lowering, we simply copy the field from the machineInstr to the MCInst.","For one to many relationships, such as PSEUDOCALL, we assign multiple tags to the machineInstr which are assigned to each of the MCInst’s during this expansion process.","Specific tag values for a given expansion will depend on the tagging scheme itself.","Similar to IR level lowering there is no one size fits all solution to expansion.","Fortunately at the machineInstr level there are only a few of these cases to handle in the RISCV back-end."],["Emitting the Tags","The actual tag emission into the instruction stream occurs in the finalize function during code emission in the LLVM backend.","At this point the byte code for instructions has been placed into containers called fragments, and their layout relative to each other is known.","However they have not been emitted into the final binary.","We modify LLVM such that as instructions are encoded into their fragment, their ordered list of tags are also added into a new “tag” array.","With this tag array in place, and the knowledge of the fragments offset (provided by llvm) we create a new data array for the fragment, that has the tag instructions inserted into the instructions.","We do this for every fragment that contains instructions.","In this manner we can be sure that any MCinst that has received a tag, will have that tag emitted into the final binary Any non-zero fragment offset must be recalculated to account for tags that will have been inserted into previous fragments.","However because this is a simple one tag instruction per-number of encountered instructions, it is always possible to calculate this correctly with just the original offset.","(I.E.","If the offset is 12 bytes, and the tag coverage is 3, there would be a single tag instruction inserted before that offset, and the new offset would then be 16) Like the fragment offsets, during tag emission symbols and relocation must be adjusted to account for the tag insertions.","As they contain offsets that are byte distances from the beginning of a fragments, these values must be adjusted by the same one per n calculation as the fragment offsets.","For both symbols and fragments we create a new array, with the updated values, and then overwrite the unmodified arrays using a ConstCast.","There are some corner cases in symbol resolution that must be addressed.","Specifically the compiler will attempt to calculate the distance between two symbols before the finalize layout step if it is able to determine the distance early (i.e., if they are in the same fragment), this is a compiler optimization to increase symbol resolution speed in the linker.","To deal with this, we disable this optimization and allow the linker to resolve the difference after tag insertion.","Additionally, the calculation of if a jump should be PC-relative or indirect is done before the final emission based on the maximum distance allowed in a single PC-relative jump instruction.","This calculation must be adjusted to assume an additional one tag instruction per number of instructions insertion.","An example of the final result of these changes, and the final emission of tags is presented in fig:cfiexample"],["Linker Changes","To support both the insertion of tags and possibly metadata labels, the linker must be made aware that there can be instructions inserted between a HI and LO relocation pair.","Normally HI-LO relocation's must appear as a pair, directly next to each other.","We modify the linker to allow for up to three instructions inserted between the pair, and adjust the final value calculations to account for the added distance.","Three additional instructions are selected as a default to allow HI, tag, METADATA_LABEL, METADATA_LABEL, LO configurations, however this number can be adjusted if more metadata is desired."],["Example CFI Implementation","We provide an example CFI implementation that demonstrates the common design challenges and changes of a simple COGENT implementation.","CFI is a very well studied problem [6][19][20] and is a common use case for tagging implementations [16][13][14].","This example CFI implementation places metadata labels at the beginning of exported or address taken functions, and before the function call at computed call sites.","It consists of the following components.","A tag configuration with 2 tags, using a lui-nop to provide backwards compatibility.","The tags are simply N (A normal instruction) or CFL (control flow label) this allows any of the coverage's supported by lui fig:taginstructions to be selected.","An IR level pass that assigns CFI labels to functions and callsites based on their function signature.","Function-signature-based CFI policy is consistent with the CFI approach in LLVM.","A callsite and a destination must have matching CFL labels.","A Machine instruction level pass that inserts and tags the CFI labels with CFL at the beginning of every non-internal function.","A back-end and scheme specific change to insert the CFL label between the AUIPC-JALR pair that RISCV generates for call sites.","An example of this CFI implementation on a from source to object file is provided in fig:cfiexample.","This example also includes many of the common challenges faced by COGENT.","CFI is such a well studied problem, and our example solution is based on previous work [18].","As such we do not claim this implementation as a novel contribution."],["Tag Expansion","Although we do not claim it as a contribution, there is no fundamental reason that placing two tag instructions next to each other would not work.","Such a configuration could allow for higher coverage at the cost of higher space overhead.","This may be useful when space is a nonissue, or for inserting complex metadata for testing purposes."],["Compressed Instruction Support","Likewise there is no fundamental barrier to support compressed instructions.","The instruction opcode in RISCV indicates if an instruction is compressed or non-compressed.","With that knowledge and the current PC, it is possible to infer what bits in the tag instruction contain the corresponding tag.","For compressed instruction support, care must be taken when designing the tags and the tag policy, as the tag bits must be small enough such that a single tag instruction can cover every instruction (compressed or not), even if they are all tagged.","This can be done by using tags small enough to cover all possible instructions (i.e., 3 bit tags for an lui with an uncompressed coverage of 3) or a two-tiered tag system, where uncompressed instructions receive a larger share of the tag space.","The compiler must additionally be modified to insert alignment NOPs in the case where a non-compressed instruction would cross a tag instruction, pushing it under the coverage of the next tag."],["Dynamic Linking","Currently COGENT only supports statically linked binaries.","This is to ensure that the tags for any externally linked libraries are always located in the correct place for interpretation.","We plan to support dynamic linking in future versions of COGENT."],["Handwritten Assembly Support","Handwritten Assembly sections are loaded into the compiler backend for parsing before being emitted to the final object file.","This allows COGENT to insert the tag instructions in hand written assembly segments.","However its ability to correctly determine tags on these sections is limited.","The correct tagging of these sections would be left to the implementation of a specific tagging scheme, in much the same way as backend and scheme specific tagging would be and may only be possible with manual tagging."],["Additional Configurations","Other tag layouts beyond the single one we chose are desirable in a generic tagging system.","Possibilities include Placing all the tags for a function's instructions at the beginning of that function.","This would consist of a series of tag instructions sufficient to cover every instruction in the function.","On non tag aware systems this would simply be the equivalent of a series of NOPs that run before a function starts.","Placing the tags for a basic block at the beginning of that basic block.","Again, this would consist of a series of tag instructions large enough to cover every instruction in that basic block.","Placing the tag a separate .text.tag section in the ELF executable file.","It would be up to the hardware to determine how to use this section, but it can be desirable, and non tag aware hardware can simply ignore the section completely.","While COGENT currently does not support these additional configurations, we have plans to extend support to them in the near future.","Figure: Backward compatibility results for Lui NOP, coverage 3, 7, and 15, with and without labels in percentage overheadCOGENT is a generic system with many different possible use cases.","As such we evaluate it Four primary ways.","Overhead on COTS hardware (i.e.","the overhead of backwards compatibility.","), and binary size overhead, are presented to demonstrate the cost of cogent in its most common configurations and uses.","The last two, are a comparison of related works, and case studies demonstrating how the system can easily be adapted to different use-cases.","The off the shell performance tests were done on a SiFive U540 SoC, which contains four 64-bit RISC-V cores and 8GB of memory, running Gentoo Linux with the kernel version 5.2.9.","For the backwards compatibility evaluation, we use the lui nop described in sec:select.","This is chosen to allow the maximum amount of space per tag and still allow the binaries to run on unmodified hardware.","Coverage 3 provides 6 tag bits per instructions covered, 7 provides 2, and 15 provides one.","While the labels were inserted as described in the example CFI configuration (see sec:example), evaluating them requires special hardware capable of interpreting the tags.","This is explored in a separate hardware paper [21].","We chose a subset of the SPEC CPU2017 rate benchmarks for our tests.","This subset has a wide range of characteristics, including heavy memory operations in mcf_r, and rapid arithmetic operations in deepsjeng_r [22].","Testing against these characteristics reveals what kind of performance impacts can be expected on unmodified hardware.","SPEC CPU2017 provides three different types of workloads.","Test, training, and reference.","The test workload is a simple test of the executable, the training is designed to give feedback for optimizations, and the reference workload is the fully timed data set.","For these tests we used the full reference workload.","The size overhead includes the overhead of label insertion on the coverage 3 configuration.","This is done comparing the size of a binary compiled with and without labels for each benchmark program.","A simple count of the number of inserted labels would be insufficient when including tags, as the position and number of labels can add to the number of tags inserted into the program."],["Backwards Compatible Results","Overall the average backward-compatibility performance overhead of inserting a tagged lui nop every three instructions and labels was 13.40%.","Every 7 instructions and labels was 6.98%, and every 15 instructions and labels was 5.72%.","This is not a linear reduction due to two major factors.","The inclusion of the labels, whose overhead is the same over every coverage, and varied properties of benchmarks themselves.","Additionally, for small loops and functions in binaries the number of nop’s encountered in the code stream can be greater than simply 1 per 3 normal instructions.","loop_start: <TAG> addi addi addi <TAG> j loop_start Consider the simplified toy example in asm-ex.","In this set of instructions, we encounter 2 tags every 4 normal instructions every time we iterate over the loop.","Examples can be found for every coverage size, using larger loops or functions."],["Binary Properties","The properties of the benchmarks have a major impact on the results of our evaluation.","For programs that are memory operation intensive, even in backwards compatibility mode, the overhead of additional nops added to the instruction stream is quickly overwhelmed by the cost of memory operations.","mcf_r has this property, and has an overhead of 3.79% for coverage 3, and 1.09% for coverage 15 in backwards compatibility mode.","Conversely, small looping in-register operations can cause the nops to have an oversized impact on program performance.","Deepsjeng fits this profile, and the resulting overhead is 21.79% for C3, 12.45% for C7, and 8.39% for C15.","Programs that do not exhibit such extremes in behavior tend to have overhead results close to the average results.","Full results can be found in fig:overhead."],["Size Overhead","Overall we expect the text section to increase in proportion to the number of tag instructions we insert.","However complicating this number is the function alignment requirements.","In the text section of the binary, the space between functions is filled in with zeros that extend from the end of one function to the beginning of the next.","This effect is more pronounced in C++ programs with many small functions such as omnetpp_r and xlancbmk_r.","Labels have a minimal impact on binary size, averaging 0.78%.","Overall the text section size increase is on average 29.32% for coverage three with labels, 20.49% for coverage seven, and 18.27% for coverage 15.","The full text section size results are available in tab:size.","Table: Text section size in bytes for baseline and 4 different configurations.","Luic3 is coverage 3 using the lui nop.","luic7 is coverage 7 using the lui nop.","luic15 is coverage 15 using the lui nop.Table: State of the art comparison.","*PUMP's tags are sizeof(int *) per memory word, some applications will not use the full space.","**PUMP allows arbitrary metadata by allowing a pointer to be encoded into the tag"],["Feature Comparison Against State of the Art","COGENT was created to enable flexibility in its deployment.","To demonstrate this against the current state of the art, we compared COGENT against recent tag-based architectures using several criteria.","How easy is it to employ the tool in a different manner then one of the creators designs?","A high rating means the tool was designed for that purpose, while a low indicates it was created for one specific task.","Does this tool have a Variable tag size?","Can this tool provide an arbitrary amount of metadata to a user above and beyond the size of the tag, if that user desires it.","Does this tool change the underlying ISA, removing backwards compatibility.","Does this tool use the unused pointer bits?","What kind of additional memory protection does this tool employ to protect its tags?","The results of this investigation are available in tab:comp.","In general we find that unless a tool was designed for reuse (i.e.","PUMP), it would be difficult to re-target the hardware for other tasks.","State of the art solutions use ISA changes to provide the compiler with the wrappers and tag aware instructions they need to manipulate their metadata.","These ISA changes break backwards compatibility.","The common use of unused pointer bits, or custom pointers, render solutions incompatible with each other.","Table: Sample CVE's and tag-based solutions"],["Case Studies and Use Cases","COGENT is a generic tool designed to allow for the easy implementation of other tagging and labeling schemes.","We present a case study, offering a detailed look at systems COGENT could be used to achieve.","We selected CVE-2021-3330 to explore with detailed solutions presenting and describing Tag Aware Arithmetic sec:2021-3330. tab:CWECVE shows a selection of recent CVEs to which our findings could be applied, including the one we selected for our case study.","Additionally, we present an alternate usage of COGENT for fuzzing and testing, where we use COGENT to keep track of coverage and branching behavior of a program at runtime.","Table: Overflow check tagging schemeCVE-2021-3330 is a vulnerability that affects the Zephyr RTOS, and comes from a series of errors that includes an improperly constructed cyclical list, where a singularly linked list is expected.","Iterating over this now cyclical list causes an unsigned value to be repeatedly subtracted from another unsigned value, leading to an integer underflow.","This value is then fed as the length argument into [language=C]|memmove()|, leading to a massive memory move that can overwrite kernel structures.","This CVE consists of several points of failure.","Improper list construction, integer underflow, and the overwrite itself.","In this case study, we focus on a potential solution for the integer underflow, which would prevent the memory overwrite by triggering an exception when the underflow occurs."],["Type Aware Arithmetic","To do this we will introduce type aware arithmetic, requiring a tag with a width of a single bit, giving two possible states for every instruction as shown in tab:overflowcheck.","These two tags are N and UN_ARTH.","The N tag is given to any instruction in the stream that will not be checked for integer overflows.","This can be because the instruction does not perform any arithmetic on an unsigned integer, or because the programmer or the compiler has chosen to disable the check on that instruction.","While the UN_ARTH would be automatically placed on any instruction that performs arithmetic operations on unsigned types.","To insert this tag, an llvm compiler pass would be created that examines the left hand side of any arithmetic expression, determining if the type of the result is unsigned.","If it is, the compiler will add the tag to any and all arithmetic instructions that result from the IR level instruction, which the COGENT system would then propagate down into the final binary.","There are two cases where the compiler would not place the UN_ARTH tag on an instruction that creates an unsigned value.","The first is when the compiler can determine that values being subtracted are hard coded immediate values, in this case we would have to assume the programmer is attempting to exploit the defined behavior of unsigned (IE unsigned MAX = 0 - 1).","The second would be a provided annotation that allows the programmer to opt out of the checks if they had a reason to.","This is required as in the C standard integer over/underflow are defined behavior.","Forcing the user to opt out of the behavior is done in an attempt to minimize developer errors when using this defense."],["Required Hardware Support","The required hardware support for this tag scheme is very simple, if an instruction is tagged with UN_ARTH, an exception should be thrown if the results from that operation over or underflow.","Hardware would require this additional tag support as without metadata, it would be unable to differentiate between an operations on signed and unsigned integers.","This can either be a hard stop to the program, or be call an exception handler if the policy calls for it.","In this case, for CVE-2021-3330, we would halt the program execution, as the corruption that results from this bug can overwrite kernel data structures."],["Code and Tag Layout","For CVE 2021-3330, the original integer overflow occurs in the following two lines of code.","memmove(frag->data, frag->data + hdr_len, frag->len - hdr_len); frag->len -= hdr_len; These lines of code operate only on unsigned integers, and do not have the opt-out annotation.","After compilation, this code produces the following relevant lines of assembly, loading the required values into registers, and then subtracting them.","linerange=1-10, breaklines=true, frame=single, caption=CVE-2021-3330 underflow ASM ]listings/taggedunarth There are two places in this code that will need to be tagged, sub a2, a0, a2 and sub a1, a3, a1.","This information can be determined by the compiler, and so they will receive the UN_ARTH tags.","Resulting int he following final code layout.","Note the tag instructions themselves are not included in this listing, as there exact position would depend on compiler optimizations levels, and overall layout of the code, however they would be inserted into the instruction stream at the appropriate alignments.","linerange=13-22, breaklines=true, frame=single, tabsize=2, caption=CVE-2021-3330 underflow ASM ]listings/taggedunarth With all this in place, when this tagged code is run and the bug is triggered, operation would proceed normally until the subtraction that causes the arithmetic underflow.","Once that happens, the hardware would trigger an exception based on the UN_ARTH tag, causing an extra check as part of the instruction processing phase.","A final strength of this solution is that because of the simple nature of the two tags employed, it would be a trivial exercise for COGENT (although not necessarily the hardware) to layer this tag with another single bit tag that would not be placed on arithmetic instructions.","An example could be the single bit return address protection provided by [16] and [13].","These two protections could be layered without the need for extra tag sizes."],["Description","Inline tagging could be used during testing and fuzzing to allow for quick tracking of program coverage, counts of reached instructions, and paths taken through the CFG.","In this use case, run time security is not considered as part of the design.","Instead COGENT inserts metadata informing testing hardware what information it needs to record, and where it needs to record it.","Hardware changes are limited to what is required to facilitate these tasks.","Table: Coverage tags summary"],["Tag and Label Design","This use case requires the coverage 7, 2 bit LUI labeling scheme.","Giving 4 possible tags, each indicating a different behavior to the testing hardware.","Additionally we introduce counting labels, with the initialized value of 0 as the primary fast coverage tracking method.","An in depth description of each tag and the labels are as follows, with a summary provided in tab:coverage.","CL (Counting label) - Indicates that this instruction is a Counting label.","When this tag, and the associated label are encountered in the instruction stream.","First the value contained in them is incremented by one, Then written back into the label.","This requires that the system be allowed to write values back into the code section.","These will be placed at the beginning of every basic block.","CI (Counting Instruction) - Indicates that this instruction should be counted.","When the instruction stream reaches this tag, it will check the current program counter against a mapping of PC : Int to see if this instruction has been encountered before.","If it has, it will increment the associated Int by one.","If not, it will create a new entry in the map, with a value of one.","This counting instruction is explicitly slower than the counting label, due to the need for the mapping lookup, but could be used for specific purposes.","BCF (Branching Control Flow) - Indicates a branching point in the instruction stream that needs to be tracked.","This tells the hardware to check if the current PC is in the BCF mappings.","If it is not, it creates a new entry in a {PC : {Next_PC : Count}} mapping.","After it has either found, or created, the entry for the current PC, the Hardware then checks if the next program counter is in the {Next_PC: Count} mapping, adding it if it does not find it, or increment the counter if it does.","In this way the BCF tag builds a map of each branching instruction, the targets it has reached, and how many times it has reached those targets.","N (Normal) - This tells the hardware that no special hardware actions are needed to enforce the overall security policies within this instruction."],["Compiler Support","For coverage tracking, the compiler would be configured to place counting labels and their associated tags at the beginning of every basic block at the machine instruction level.","Additionally, a pass would be added to enable two levels of branching control flow tracking depending on the desires of the tester.","BCF labels could be placed on every control flow instruction or, just on computed control flow instructions."],["Hardware Support","This coverage system requires two major changes to the supporting hardware beyond the ability to read tags.","First is the ability to write the incremented counters back into the counting labels.","Second is the ability to record the PC of instructions tagged with CI or BCF into mappings.","To do this we would reserve a known memory for the CI and BCF mappings, and ensure that the size of each entry is known to the hardware.","This means for BCF mappings there would be a configurable maximum number of targets in order to keep lookup times reasonable."],["Example Code","The RISC-V assembly code below is an example of how the coverage tagging would look like on a simple basic block, followed by a branching jump.","linerange=1-19, breaklines=true, frame=single, tabsize=2, caption=Coverage Example ASM ]listings/coverage Assuming the above was entered three times, jumping once and falling through twice, the counting label at 0x4 would be |lui x0, 3| and at 0x1C it would be |lui x0, 2|.","LBB0_2 is always reached at 0x34, making its value |lui x0, 3|.","The branching table would contain 0x18 : 0x1C: 2, 0x30: 1, showing the exact behavior of the branch over three runs."],["Related Work","PUMP [16] provides a flexible system to associate metadata with memory locations, including on instructions.","Its large fixed-width tag sizeof(*) provides unlimited metadata size per memory location via allowing the tag to simply be a pointer to additional memory.","However schemes that actually use this amount of space are not common, and even then, the common case for any given memory location is a small amount of metadata.","To help alleviate the large overhead of having so much unused space in there tags, they propose and implement tag compression schemes to speed up retrieval.","Compared to PUMP, COGENT's instruction tagging allows simpler protection schemes to use smaller tag sizes when needed, while at the same time allowing that same arbitrarily sized metadata though the use of any number of inline labels.","Importantly though, PUMP has an actual hardware design backing it, while COGENT does not have specialized hardware, so the only comparison can be between the location and storage size of metadata, not on performance, hardware complexity, or overall overhead.","CHERI [12] is a well known capability based architectural enhancement for RISC instruction sets.","Work on it is current and on-going [24] [25] [26].","CHERI provides its security though architectural capabilities, and tagged memory.","Additionally it is a full stack solution, and includes compiler, OS, and hardware support.","Cheri itself requires that compiler to support on all levels.","Language specific front end, IR level passes, and backend emissions.","RetTag [14] uses compiler modifications to insert new instructions in the function prologue and epilogue to generate a new instruction (thus modifying the ISA and preventing backward compatibility) called pac.","This instruction generates a unique PAC using the a unique function ID, the virtual address of the functions RA the SP, and a 128-bit RAA key and stores it in the 16 unused bits of a function pointer, and it is used to protect the return address from corruption.","ZERØ [27], provides a set of new instructions via a modification of the ISA and a method of encoding metadata that keeps the overhead to eight bytes per 4kb page.","This work discards backwards compatibility by including new ISA instructions for memory and metadata management.","Additionally it stores ten bits of information in the unused bits of pointers to enforce its own version of CFI using unique-per-function signature labels.","It modifies the compiler to insert its new instructions at the appropriate places.","Timber-V [28] allows the creation of up to four protected enclaves on a RISC-V system.","It likewise uses a new set of instructions in order to tell the hardware when it can check and manipulate tags.","Like ZERØ this removes its ability to provide backwards compatible code.","While Timber-V does not require compiler support, they specify that compiler support would allow for better integration.","Several works including Framer [15], RetTag [14] and ZERØ [13] use the upper unused bits of a pointer as place to store there metadata.","Doing this gives these works strong backwards compatibility and low overhead.","However by their nature it is difficult to layer any works that use the unused pointer bits.","Additional space is limited, leading framer to use a novel encoding scheme to provide access to the pointers metadata, relying as much as possible on relative offsets from known points (i.e.","frames).","Such works could be combined with COGENT as we do not manipulate the pointers themselves, leaving the unused parts of the pointer available for any other scheme, or to be used for a CFI label.","FlexFit [29] is a generalized solution presented as a method to allow users to configure a set of instruction filters for use at run time.","They implement this on a RISC-V FPGA board to demonstrate the practicality, feasibility, and usability of there generic filtering scheme.","They store the protection information in four unused bits of the RISC-V PTE, giving up to 16 protected domains at a page granularity.","Using the unused bits in the PTE, has been explored by other works as well [30] [31], which is why FlexFit limits its protection domains to 16, using only 4 of the available bits.","While FlexFit does not use a compiler, the authors suggest a specialized compiler and loader for future work."],["Conclusion","This paper presents COGENT, a modified LLVM compiler capable of generating RISCV binaries with configurable width instruction tags.","These tags can be used to design security solutions that convey rich semantic information to a tag-aware architecture.","COGENT tackles several challenges including one-to-many and many-to-one instruction-level tag associations.","We evaluate COGENT on a subset of SPEC 2017 benchmark programs for performance and binary bloat.","Further, we present case studies that highlight potential use cases of COGENT."]],"string":"[\n [\n \"Generic Tagging for RISC-V Binaries\"\n ],\n [\n \"Abstract With the widespread popularity of RISC-V -- an open-source ISA -- custom hardware security solutions targeting specific defense needs are gaining popularity.\",\n \"These solutions often require specialized compilers that can insert metadata (called tags) into the generated binaries, and/or extend the RISC-V ISA with new instructions.\",\n \"Developing such compilers can be a tedious and time-consuming process.\",\n \"In this paper, we present COGENT, a generic instruction tag generator for RISC-V architecture.\",\n \"COGENT is capable of associating a tag of configurable and varying widths (1 to 20 bits) to each instruction.\",\n \"It is also capable of emitting labels that are central to the implementation of control-flow integrity (CFI) solutions.\",\n \"COGENT encodes all tags and labels as nop instructions thereby providing full backward compatibility.\",\n \"We evaluate COGENT on a subset of programs from the SPEC CPU2017 benchmark suite and report the binary size increase to be 29.3% and 18.27% for the lowest and highest tag coverage levels respectively.\",\n \"Additionally, we executed tagged programs on COTS RISC-V unmodified hardware and found the execution time overhead (with respect to backward compatibility) to be 13.4% and 5.72% for the lowest and highest coverage levels respectively.\",\n \"Finally, using a case study, we present possible use case scenarios where COGENT can be applied.\"\n ],\n [\n \"Introduction\",\n \"Over the years, the advent of Internet of Things (IoT) along with an increasing number of mobile, desktop and server systems has led to a steady increase in cybersecurity attacks.\",\n \"Most of the attacks–even those targeted at the hardware (e.g.\",\n \"Spectre[1], and Meltdown[2]) are initiated through software.\",\n \"The defense community has focused efforts at different layers of deployments.\",\n \"For instance, the software community has explored software-only defenses such as StackGuard[3], and CFI via program instrumentation (e.g., [4], [5], [6]).\",\n \"Meanwhile, the hardware community has pursued defenses that leverage hardware features in order to detect and flag exploits (e.g., Intel CET[7], W⌃X) or prevent information leakage (e.g., Intel SGX[8], ARM TrustZone[9]).\",\n \"While software-only security measures are highly desirable and can be deployed with little effort, they typically suffer higher performance penalties compared to hardware solutions.\",\n \"Furthermore, the security mechanisms themselves may be a part of the attack surface, which brings about transparency issues.\",\n \"On the other hand, hardware-only security mechanisms must bridge the semantic gap problem and recover high-level security-relevant program semantics in order to enforce rich and expressive policy abstractions.\",\n \"Further, the tolerance for errors is low and “fixing\\\" design errors is very expensive (e.g.\",\n \"Intel MPX[10], [11] had problems that resulted in eventual deprecation).\",\n \"With the gaining popularity of RISC-V, an open-source ISA, hardware - software cohesive defenses have gained center stage and are desirable.\",\n \"Typically, defenders take advantage of (a) fine-grained control over the RISC-V hardware including (but not limited to) changes to ISA in combination with (b) curated compilers capable of enriching binaries with security metadata/stubs, deploying highly effective and fast performing defenses.\",\n \"For example, recent defenses such as Cheri [12], Zero [13], RetTag [14], Framer [15], etc., all rely on a dedicated compiler that can generate instrumented binaries for their modified hardware.\",\n \"The modified binaries contain metadata called “tags” that provides the hardware with specific information regarding the code or data unit in the program.\",\n \"Typically, there is a one-to-one mapping between a tag and corresponding code/data unit.\",\n \"These defenses strike a balance between security and performance requirements and are often tailor-made to deployment needs.\",\n \"They rely on performant compilers and/or hardware design and engineering, yet compiler development remains an expensive process often requiring several hundred developer-hours for development and testing.\",\n \"More importantly, availability of a suitable compiler would attract more hardware defenses that are currently impeded by the lack of a compiler.\",\n \"Solutions that shorten the development effort and allow quick incorporation of metadata into binaries are missing and highly sought after.\",\n \"In this paper, we address the above need and present COGENT: COmpiler for GENeric Tagging, a generic LLVM-based compiler that incorporates instruction tags (or code metadata) into RISC-V binaries.\",\n \"In essence, COGENT removes the burden of compiler development from RISC-V hardware defenses that rely on embedding instruction metadata into binaries.\",\n \"Motivated by the needs of Control-Flow Integrity (CFI) defenses that embed CFI labels into code, COGENT interleaves tag metadata into the instruction stream.\",\n \"COGENT has several key features that make it highly favorable for hardware-software cohesive defenses.\",\n \"First, it is highly flexible.\",\n \"It can accommodate various tag widths (1 to 20 bits per instruction).\",\n \"This flexibility is essential in order to achieve rich expressiveness in tags.\",\n \"Second, it is functionally correct.\",\n \"That is, it ensures that the inserted tags do not interfere with program logic.\",\n \"Functional correctness is non-trivial due to several challenges (see Section REF ).\",\n \"This is important since a tag system is not usable without such guarantees.\",\n \"Third, COGENT preserves backward compatibility.\",\n \"That is, tagged binaries will seamlessly run on generic RISC-V processors that are tag-unaware.\",\n \"We accomplish backward compatibility by encoding tag information into NOP instructions that are simply ignored by unsupported hardware.\",\n \"Finally, our changes are segregated into LLVM IR-level and target-specific changes.\",\n \"Other architectures (e.g., ARM) can be supported by reusing the IR-specific changes and rewriting ARM LLVM backend.\",\n \"We make the following contributions: We implement a flexible and defense-scheme agnostic implementation codenamed COGENT, a LLVM-based compiler for RISC-V target.\",\n \"COGENT provides configurable instruction-level tagging support to several different tag widths and coverage.\",\n \"We also make changes to the LLD linker to support COGENT.\",\n \"We demonstrate the efficacy of COGENT via a CFI implementation.\",\n \"We evaluate the example CFI implementation for correctness and backwards compatibility.\",\n \"We demonstrate moderate to low performance overhead and binary bloat.\",\n \"Through case studies, we demonstrate potential real-world applicability of COGENT.\",\n \"Summary of Results: We evaluated the correctness of COGENT by compiling and running a subset of the SPEC CPU2017 benchmark suite on Commercial Off The Shelf (COTS) hardware.\",\n \"Our subset has small function sizes, heavy memory overhead, and high loop counts, all of which stress instruction caches.\",\n \"For our example CFI implementation, we found that COGENT increased the size of the .text section of these programs by an average of 29.32% for the lowest tag coverage, and 18.27% for the highest.\",\n \"We found that backward-compatibility performance (i.e., overhead imposed by tagged binaries running on tag-unaware hardware) was impacted by an average of 13.40% for the low coverage and 5.72% for the highest.\",\n \"We also present several possible use cases for the system, including specific recent CVEs that could be addressed.\",\n \"The rest of the paper is organized as follows: sec:background provides the technical background necessary to understand the remainder of the paper.\",\n \"In sec:overview and sec:technicaldetails, we present the overview and technical details of COGENT.\",\n \"We evaluate COGENT in sec:eval and provide case studies to highlight benefits of using COGENT.\",\n \"Finally, we present related work in sec:relatedwork and conclude in sec:conclusions.\",\n \"Tagging is a technique that very broadly associates some amount of metadata (the tag), with a memory location and registers.\",\n \"Specially designed architectures then consume that tag and enforce a security policy.\",\n \"Tags can be associated with a) memory objects (data tagging) allowing policies to track and protect data in memory, and/or on b) code (instruction tagging), allowing policies to change or extend the behavior of the tagged instructions.\",\n \"Of particular importance to this paper is instruction tagging.\",\n \"Instruction Tagging: allows a tag-aware architecture to execute additional code or checks on any instruction associated with a particular tag.\",\n \"Instruction tags are useful in implementing Control-Flow Integrity (CFI) policies.\",\n \"Identical tags are used at control flow sources (call/jump sites) and control flow targets in order to signify legal control flow.\",\n \"Tags can be placed on load/stores to protect sensitive data, forbidding access to a memory region if the tag is incorrect.\",\n \"These kinds of tags often require corresponding metadata placed with the data they are meant to access.\",\n \"The tag policies and enforcement are implementation specific and form the essence of a tag-based defense.\",\n \"The specific instruction tags are a function of the intended security policies, and therefore vary greatly between designs [16].\",\n \"Differences in tag requirements can even result in different designs which use the same hardware for similar purposes [17].\",\n \"This implementation-specific design requirement is inherent to the nature of tagging.\",\n \"Tags are simply a way of providing more information to the program at run time, but the information needed is dependent on the design of the system itself.\",\n \"Broadly speaking, there is a trade-off made in the design of tag systems between how much information it provides to the hardware, and how much overhead the tags introduce into the system.\",\n \"This overhead can include power consumption, hardware complexity, performance or any other costs paid by the design.\",\n \"Depending on how much information one's defense needs, tags can be arbitrarily large, or as small as a single bit.\",\n \"Designers often use techniques such as encoding the metadata information into the unused upper bits of a pointer, but this is often insufficient on its own [14], requiring additional tagging or ISA(instruction set architecture) changes.\",\n \"The choices of where to store metadata, how much metadata to store, and how to access it are defining characteristics of tagging systems.\"\n ],\n [\n \"LLVM Compiler Infrastructure\",\n \"The LLVM compiler infrastructure [18] is a fully featured compiler designed with modularity in mind.\",\n \"It is broken down into components, which operate on different parts of the compilation process.\",\n \"Important to COGENT is the idea of the language-specific front end, the target architecture-specific back end, and the language- and target-agnostic intermediate representation (IR).\",\n \"The LLVM design requires the language-specific front end to convert source code into LLVM IR, the optimizer schedules an analysis and transformation pass pipeline on that IR, and its output is IR for consumption by the back end.\",\n \"The back end might also perform additional transformations on the code via its own set of passes and built-in functionality.\",\n \"The final product of the back end is then output to an object file, which is fed into a linker to create the final binary.\",\n \"Modifying the LLVM compiler can thus take two main forms.\",\n \"The simplest scenario is to create and modify transformation and analysis passes within the existing structure of the pipeline.\",\n \"These transformation passes can be scheduled to run before or after any specific pass or combination of passes, allowing one to place new transformations and analysis at any point in the pipeline.\",\n \"The more complex scenario is to modify the internal components of LLVM, changing or extending the functionality of pre-existing code in the LLVM infrastructure.\"\n ],\n [\n \"Relocations and Symbols\",\n \"As part of the creation of an object file, the back end will emit target specific information called a relocation into object files.\",\n \"These relocations are designed to inform the linker of how to handle information that cannot be finalized before the linking phase of compilation.\",\n \"Examples of this for RISC-V include encoding addresses, which are often emitted as a pair of relocations (named HI and LOW), covering two instructions (i.e.\",\n \"auipc, jalr).\",\n \"These relocations tell the linker to place the final address of the target symbol across two instructions when it is required.\",\n \"Symbols themselves can be kept as offsets from an unknown starting address, allowing the linker to know the final address during the linking process, even though the absolute position cannot be determined by the compiler back end.\",\n \"Tagging schemes are complex and varied, and different schemes require different configurations.\",\n \"There are some that require more space (i.e., metadata) to be accessible for any given instruction, while some might require only a single bit for some defenses [13].\",\n \"Additionally tagging schemes must be able to provide the metadata to the hardware.\",\n \"A generic instruction tagging scheme must be able to meet this requirement under as many different configurations as possible.\",\n \"To that end we set out to create COGENT.\",\n \"The goal of COGENT is to enable embedding configurable metadata tags into instruction stream, and not implementation details for different defense schemes such as forward or backward edge CFI, heap or stack, stack protection, or any other particular defense scheme.\",\n \"We did not set out to create a specific defense with this paper, or provide analysis that can be used to create one.\",\n \"Instead we focused on three major tasks.\",\n \"Flexibility: COGENT must be able to generate binaries for RISC-V architecture that require varying tag widths, with minimal changes.\",\n \"Additionally, we aim to modularize our implementation in order to minimize the developer effort in modifying COGENT to support a new architecture.\",\n \"COGENT must also be able conducive to supporting different types of defense schemes including those that require large amounts of metadata.\",\n \"Functional Correctness: Insertion of tags and metadata must not interfere with program logic.\",\n \"That is, all necessary tags must be appropriately inserted, and inserted tags must not alter the control flow or corrupt data.\",\n \"Backwards Compatibility: The tool must allow created code to be backwards compatible.\",\n \"That is, tagged binaries must be able to seamlessly execute on tag-unaware hardware.\",\n \"While backwards compatibility is a desirable feature for many systems, we recognize that it may not be necessary in every case.\",\n \"If turning it off could provide benefits to a scheme COGENT should allow that as well.\",\n \"A tag or metadata lookup action in the hardware could be expensive and inefficient.\",\n \"Therefore it is best to reduce the number of abstractions required to retrieve any tag for a given instruction.\",\n \"Likewise having separate (but mapped) memory adds complexity to the process of bringing the tags into the hardware.\"\n ],\n [\n \"Tag Semantic Gap\",\n \"During the compilation process, programs pass through different stages of compilation.\",\n \"While the tags are eventually assigned on the raw instruction stream, the actual assignment of tags may occur at a higher level of abstraction (i.e., source code or intermediate representation).\",\n \"Depending on the stage of compilation where tags are assigned, percolating them down to the instructions is important and requires bridging of the semantic gap between the higher and lower levels of program representation.\",\n \"Further, during compilation the compiler will often expand IR instructions/source statements into multiple assembly instructions.\",\n \"This manifests as a Many-to-One relationship where many tags map to a single IR instruction/source statement.\",\n \"A very common example in RISC-V is the lowering of the LLVM-IR pseudo instruction PSEUDOCALL from MachineInstr level into the MCinst level.\",\n \"During this process the single PSEUDOCALL is expanded into an auipc,jalr pair.\",\n \"Likewise there are One-to-Many relationships where multiple instructions at a higher-level of abstraction are lowered into a single assembly instruction (e.g.\",\n \"bitcast instructions).\",\n \"These one-to-many and many-to-one relationships can occur during any point of the lowering process, and where possible, COGENT must provide ways to handle them correctly.\",\n \"Unfortunately there are many cases without a generic solution i.e.\",\n \"in the lowering from IR level instructions to machine Instructions.\"\n ],\n [\n \"Compiler-inserted Code\",\n \"During compilation the compiler may introduce instructions that did not exist at the previous (i.e., higher) level of representation.\",\n \"This is typically done to meet back end specifications or optimizations.\",\n \"Examples include in-lining and function prologue/epilogue insertion.\",\n \"COGENT must provide a way to tag compiler-inserted code, and must ensure that it does not disrupt the tag schemes.\"\n ],\n [\n \"Pre-calculated Offsets, Symbols, and Relocations\",\n \"Features such as C++’s try/catch block implementation often pre-calculate the distance between two symbols instead of leaving it as a relocation wherever possible.\",\n \"COGENT must be sensitive to these pre-calculated values.\",\n \"More generally, wherever possible the compiler calculates the offset to a relocation or a symbol from the beginning of a compilation unit.\",\n \"COGENT must preserve the relative locations in the code section though any changes it makes.\"\n ],\n [\n \"ISA Changes\",\n \"Prior efforts have altered the ISA to enrich code with metadata.\",\n \"Any ISA changes would by their very nature break backwards compatibility.\",\n \"COGENT must be able to encode tags without any modifications to the ISA.\"\n ],\n [\n \"Our Approach\",\n \"When examining both the challenges and the goals, we converged on the following design choices for COGENT.\",\n \"These solutions address both the challenges and goals.\"\n ],\n [\n \"Flexibility\",\n \"To address flexibility and the many back-end specific challenges we encountered, we designed COGENT to be as modular as possible, with front end and back end components in order to allow for adaptability.\",\n \"Unfortunately many (but not all) of the back end changes are target specific in the best case, and design specific in the worst.\",\n \"To solve this we clearly separate the COGENT process into five major pieces.\",\n \"IR Level Tagging API.\",\n \"This component allows the creation of IR level passes that can take advantage of the front and back end agnostic nature of the IR to perform analysis and select tags for IR level instructions.\",\n \"These IR level tagging passes can be shared between any source and any target, and so are the most portable method of tag selection.\",\n \"This component consists of an analysis pass that provides propagation of assigned tags into any back-end passes where possible.\",\n \"Backend Level Tagging API.\",\n \"This component allows the creation of LLVM MachineInstr and MCinst level passes that can tag code after it has been lowered into a target specific back end.\",\n \"This allows tagging of code inserted after IR level optimizations or back end insertions.\",\n \"Backend and Defense Specific Changes.\",\n \"These changes are both back end and defense specific.\",\n \"Representing choices that must be made in how the target back-end and the scheme must interact.\",\n \"For the RISC-V prototype we developed (see sec:example), One example would be the insertion of a metadata label between an auipc jalr pair to include a CFI label.\",\n \"A second would be the specific behavior of tags in the many to-one and one-to-many lowering relationships.\",\n \"These changes cannot be handled by back end passes alone and must be hard coded into the back end itself on a case-by-case basis.\",\n \"Defense Scheme Agnostic Back End Changes.\",\n \"These are changes to the back end that must be done to allow for the insertion of the tag instructions as late as possible into the code stream.\",\n \"This includes turning off the pre-calculated offsets, ensuring correct tag placement, and adjusting symbol and relocation offsets before object file emissions.\",\n \"These do not interact with the logic of tag selection, only with the emission of the tags themselves into an object file.\",\n \"Configuration File.\",\n \"This file allows a user to specify simply what emission options they would like for the final tag instructions.\",\n \"Details of the provided options are presented in sec:select, but the configuration file allows developers to switch between tag widths for their own instruction tagging schemes.\",\n \"To address the flexibility requirement of providing additional metadata where needed, we introduce metadata labels as a tool for a developer to use.\",\n \"These labels can be repurposed for any use, and are encoded as nops by the compiler in order to maintain backwards compatibility.\",\n \"In the RISC-V back end we chose lui instruction which provides 20 bits of metadata per inserted label.\"\n ],\n [\n \"Functional Correctness\",\n \".\",\n \"Addressing correct tag placement becomes challenging when inserting new tag instructions at specified alignments.\",\n \"Any changes made to the code after a tag has been inserted will displace the tags and must be adjusted in order to maintain consistent tag placement.\",\n \"To this end, all functions are forced to be aligned.\",\n \"Additionally, we assume that relaxation/compressed instruction generation is turned off (see sec:compressed for a discussion on compressed instructions).\",\n \"Additionally optimizations such as pre-calculating offsets based on the MCInst’s as described in challenges must be turned off.\",\n \"This adds additional one time overhead to the linking process, but no extra run time cost beyond the insertion of the tags themselves.\",\n \"Optimizations introduced by IR passes can be left in place however, by insuring tag insertion is completed at the last point possible in the pipeline.\",\n \"Finally tag instructions can be placed in locations that disrupt relocation pairs, meaning the linker must be made aware of this possibility.\"\n ],\n [\n \"Backward Compatibility\",\n \"To allow for backwards compatibility, at least one of the possible tag instruction configurations must be encoded as a nop instruction.\",\n \"For the RISC-V prototype, the instruction lui x0, <imm> was selected, as it is functionally a nop and gives the maximum number of bytes to encode tag values.\",\n \"Other target architectures would have to select different nop-equivalent instructions with differing amounts of bits to ensure backwards compatibility.\",\n \"Table: Tag instructions used by COGENT.\"\n ],\n [\n \"Details\",\n \"With the solutions outlined in sec:overview we created a prototype version of COGENT.\",\n \"This prototype comes in two parts, The underlying changes to LLVM itself, and an example implementation that provides metadata labels on computed jumps and computed callsites.\"\n ],\n [\n \"LLVM Changes\",\n \"We make all of the changes described in this section to LLVM version 10.0.0.\",\n \"The first and the most important step in COGENT implementation is to provide a way for the developer to select how they wish to represent the tags in the instruction stream.\",\n \"This representation is referred to as the tag instruction.\",\n \"The selection of this tag instruction determines How many bits a user will have available to store tags How many instructions any single tag can cover Backwards compatibility fig:taginstructions shows the different choices we provide via a configuration file.\",\n \"We detail their bit layout, how many bits they provide, and the number of bits available for each instruction based on the coverage desired.\",\n \"If backwards compatibility is not required, users can select a custom opcode for the tag instruction.\",\n \"The tag enumeration itself is defined in a separate header file that must be included to the relevant passes.\",\n \"Figure: Demonstration of COGENT using the simple CFI described in sec:example, contains compiler inserted code (0x58-0x68), many to one relationships (0x94-0x9C), relocation and symbol adjustments (0x6c-0x74), a tag inserted between a relocation (0x70).\",\n \"Additionally it includes data that must be persisted from the IR (the function labels) to the backend inserted labels at (0x50) and (0x98).\"\n ],\n [\n \"Tagging API\",\n \"Once the tag instruction has been selected, the next COGENT process is to assign tags to LLVM’s IR level instructions or machineInstr.\",\n \"In the backend, this process is done via a simple API we expose to the user, consisting of standard .set(<tag>) and .get(<tag>) extensions on top of LLVM’s base machineInst classes.\",\n \"These are employed by LLVM backend passes that developers insert into the compiler pipeline.\",\n \"In general tag instrumentation passes should be inserted as late as possible in the pipeline to prevent optimizations from interfering with tag lowering.\",\n \"We also provide label emission, via an emission of a single lui nop at any point in the compilation process.\",\n \"This lui must be tagged appropriately to indicate to the hardware to process this label as a particular type of metadata.\",\n \"For tagging in the IR level, we provide a pass that IR level passes can push a mapping of IRInstruction : tag into, and that backend passes can register with to access that tag mapping to assign those tags to machineInstr.\",\n \"The lowering of these IR tags is complicated by the needs of the target specific backend, as described in the next section.\"\n ],\n [\n \"Lowering Behavior\",\n \"For IR instruction tagging, we provide a default behavior in a modified RISCV backend.\",\n \"However depending on the specific deployment needs, developers must investigate the lowering process and make changes where appropriate to their planned tagging behavior.\",\n \"By default one-to-many transformations apply the tag on the IR instruction on every machine instruction created by that instruction during the lowering process.\",\n \"For many-to-one, the default behavior is the first tag.\",\n \"Any instructions not directly created from another instruction (an optimization or specification specific to the back end) will be assigned the default tag value from the tag configuration file.\",\n \"This process may be insufficient for specific defense needs, but there is no one-size-fits-all approach to the lowering problem and necessitates developer effort.\",\n \"For machine instruction inserted tags, we modify the lowering process from machineInstr to MCInst allowing the tag field to be copied over.\",\n \"In cases where this is a one to one lowering, we simply copy the field from the machineInstr to the MCInst.\",\n \"For one to many relationships, such as PSEUDOCALL, we assign multiple tags to the machineInstr which are assigned to each of the MCInst’s during this expansion process.\",\n \"Specific tag values for a given expansion will depend on the tagging scheme itself.\",\n \"Similar to IR level lowering there is no one size fits all solution to expansion.\",\n \"Fortunately at the machineInstr level there are only a few of these cases to handle in the RISCV back-end.\"\n ],\n [\n \"Emitting the Tags\",\n \"The actual tag emission into the instruction stream occurs in the finalize function during code emission in the LLVM backend.\",\n \"At this point the byte code for instructions has been placed into containers called fragments, and their layout relative to each other is known.\",\n \"However they have not been emitted into the final binary.\",\n \"We modify LLVM such that as instructions are encoded into their fragment, their ordered list of tags are also added into a new “tag” array.\",\n \"With this tag array in place, and the knowledge of the fragments offset (provided by llvm) we create a new data array for the fragment, that has the tag instructions inserted into the instructions.\",\n \"We do this for every fragment that contains instructions.\",\n \"In this manner we can be sure that any MCinst that has received a tag, will have that tag emitted into the final binary Any non-zero fragment offset must be recalculated to account for tags that will have been inserted into previous fragments.\",\n \"However because this is a simple one tag instruction per-number of encountered instructions, it is always possible to calculate this correctly with just the original offset.\",\n \"(I.E.\",\n \"If the offset is 12 bytes, and the tag coverage is 3, there would be a single tag instruction inserted before that offset, and the new offset would then be 16) Like the fragment offsets, during tag emission symbols and relocation must be adjusted to account for the tag insertions.\",\n \"As they contain offsets that are byte distances from the beginning of a fragments, these values must be adjusted by the same one per n calculation as the fragment offsets.\",\n \"For both symbols and fragments we create a new array, with the updated values, and then overwrite the unmodified arrays using a ConstCast.\",\n \"There are some corner cases in symbol resolution that must be addressed.\",\n \"Specifically the compiler will attempt to calculate the distance between two symbols before the finalize layout step if it is able to determine the distance early (i.e., if they are in the same fragment), this is a compiler optimization to increase symbol resolution speed in the linker.\",\n \"To deal with this, we disable this optimization and allow the linker to resolve the difference after tag insertion.\",\n \"Additionally, the calculation of if a jump should be PC-relative or indirect is done before the final emission based on the maximum distance allowed in a single PC-relative jump instruction.\",\n \"This calculation must be adjusted to assume an additional one tag instruction per number of instructions insertion.\",\n \"An example of the final result of these changes, and the final emission of tags is presented in fig:cfiexample\"\n ],\n [\n \"Linker Changes\",\n \"To support both the insertion of tags and possibly metadata labels, the linker must be made aware that there can be instructions inserted between a HI and LO relocation pair.\",\n \"Normally HI-LO relocation's must appear as a pair, directly next to each other.\",\n \"We modify the linker to allow for up to three instructions inserted between the pair, and adjust the final value calculations to account for the added distance.\",\n \"Three additional instructions are selected as a default to allow HI, tag, METADATA_LABEL, METADATA_LABEL, LO configurations, however this number can be adjusted if more metadata is desired.\"\n ],\n [\n \"Example CFI Implementation\",\n \"We provide an example CFI implementation that demonstrates the common design challenges and changes of a simple COGENT implementation.\",\n \"CFI is a very well studied problem [6][19][20] and is a common use case for tagging implementations [16][13][14].\",\n \"This example CFI implementation places metadata labels at the beginning of exported or address taken functions, and before the function call at computed call sites.\",\n \"It consists of the following components.\",\n \"A tag configuration with 2 tags, using a lui-nop to provide backwards compatibility.\",\n \"The tags are simply N (A normal instruction) or CFL (control flow label) this allows any of the coverage's supported by lui fig:taginstructions to be selected.\",\n \"An IR level pass that assigns CFI labels to functions and callsites based on their function signature.\",\n \"Function-signature-based CFI policy is consistent with the CFI approach in LLVM.\",\n \"A callsite and a destination must have matching CFL labels.\",\n \"A Machine instruction level pass that inserts and tags the CFI labels with CFL at the beginning of every non-internal function.\",\n \"A back-end and scheme specific change to insert the CFL label between the AUIPC-JALR pair that RISCV generates for call sites.\",\n \"An example of this CFI implementation on a from source to object file is provided in fig:cfiexample.\",\n \"This example also includes many of the common challenges faced by COGENT.\",\n \"CFI is such a well studied problem, and our example solution is based on previous work [18].\",\n \"As such we do not claim this implementation as a novel contribution.\"\n ],\n [\n \"Tag Expansion\",\n \"Although we do not claim it as a contribution, there is no fundamental reason that placing two tag instructions next to each other would not work.\",\n \"Such a configuration could allow for higher coverage at the cost of higher space overhead.\",\n \"This may be useful when space is a nonissue, or for inserting complex metadata for testing purposes.\"\n ],\n [\n \"Compressed Instruction Support\",\n \"Likewise there is no fundamental barrier to support compressed instructions.\",\n \"The instruction opcode in RISCV indicates if an instruction is compressed or non-compressed.\",\n \"With that knowledge and the current PC, it is possible to infer what bits in the tag instruction contain the corresponding tag.\",\n \"For compressed instruction support, care must be taken when designing the tags and the tag policy, as the tag bits must be small enough such that a single tag instruction can cover every instruction (compressed or not), even if they are all tagged.\",\n \"This can be done by using tags small enough to cover all possible instructions (i.e., 3 bit tags for an lui with an uncompressed coverage of 3) or a two-tiered tag system, where uncompressed instructions receive a larger share of the tag space.\",\n \"The compiler must additionally be modified to insert alignment NOPs in the case where a non-compressed instruction would cross a tag instruction, pushing it under the coverage of the next tag.\"\n ],\n [\n \"Dynamic Linking\",\n \"Currently COGENT only supports statically linked binaries.\",\n \"This is to ensure that the tags for any externally linked libraries are always located in the correct place for interpretation.\",\n \"We plan to support dynamic linking in future versions of COGENT.\"\n ],\n [\n \"Handwritten Assembly Support\",\n \"Handwritten Assembly sections are loaded into the compiler backend for parsing before being emitted to the final object file.\",\n \"This allows COGENT to insert the tag instructions in hand written assembly segments.\",\n \"However its ability to correctly determine tags on these sections is limited.\",\n \"The correct tagging of these sections would be left to the implementation of a specific tagging scheme, in much the same way as backend and scheme specific tagging would be and may only be possible with manual tagging.\"\n ],\n [\n \"Additional Configurations\",\n \"Other tag layouts beyond the single one we chose are desirable in a generic tagging system.\",\n \"Possibilities include Placing all the tags for a function's instructions at the beginning of that function.\",\n \"This would consist of a series of tag instructions sufficient to cover every instruction in the function.\",\n \"On non tag aware systems this would simply be the equivalent of a series of NOPs that run before a function starts.\",\n \"Placing the tags for a basic block at the beginning of that basic block.\",\n \"Again, this would consist of a series of tag instructions large enough to cover every instruction in that basic block.\",\n \"Placing the tag a separate .text.tag section in the ELF executable file.\",\n \"It would be up to the hardware to determine how to use this section, but it can be desirable, and non tag aware hardware can simply ignore the section completely.\",\n \"While COGENT currently does not support these additional configurations, we have plans to extend support to them in the near future.\",\n \"Figure: Backward compatibility results for Lui NOP, coverage 3, 7, and 15, with and without labels in percentage overheadCOGENT is a generic system with many different possible use cases.\",\n \"As such we evaluate it Four primary ways.\",\n \"Overhead on COTS hardware (i.e.\",\n \"the overhead of backwards compatibility.\",\n \"), and binary size overhead, are presented to demonstrate the cost of cogent in its most common configurations and uses.\",\n \"The last two, are a comparison of related works, and case studies demonstrating how the system can easily be adapted to different use-cases.\",\n \"The off the shell performance tests were done on a SiFive U540 SoC, which contains four 64-bit RISC-V cores and 8GB of memory, running Gentoo Linux with the kernel version 5.2.9.\",\n \"For the backwards compatibility evaluation, we use the lui nop described in sec:select.\",\n \"This is chosen to allow the maximum amount of space per tag and still allow the binaries to run on unmodified hardware.\",\n \"Coverage 3 provides 6 tag bits per instructions covered, 7 provides 2, and 15 provides one.\",\n \"While the labels were inserted as described in the example CFI configuration (see sec:example), evaluating them requires special hardware capable of interpreting the tags.\",\n \"This is explored in a separate hardware paper [21].\",\n \"We chose a subset of the SPEC CPU2017 rate benchmarks for our tests.\",\n \"This subset has a wide range of characteristics, including heavy memory operations in mcf_r, and rapid arithmetic operations in deepsjeng_r [22].\",\n \"Testing against these characteristics reveals what kind of performance impacts can be expected on unmodified hardware.\",\n \"SPEC CPU2017 provides three different types of workloads.\",\n \"Test, training, and reference.\",\n \"The test workload is a simple test of the executable, the training is designed to give feedback for optimizations, and the reference workload is the fully timed data set.\",\n \"For these tests we used the full reference workload.\",\n \"The size overhead includes the overhead of label insertion on the coverage 3 configuration.\",\n \"This is done comparing the size of a binary compiled with and without labels for each benchmark program.\",\n \"A simple count of the number of inserted labels would be insufficient when including tags, as the position and number of labels can add to the number of tags inserted into the program.\"\n ],\n [\n \"Backwards Compatible Results\",\n \"Overall the average backward-compatibility performance overhead of inserting a tagged lui nop every three instructions and labels was 13.40%.\",\n \"Every 7 instructions and labels was 6.98%, and every 15 instructions and labels was 5.72%.\",\n \"This is not a linear reduction due to two major factors.\",\n \"The inclusion of the labels, whose overhead is the same over every coverage, and varied properties of benchmarks themselves.\",\n \"Additionally, for small loops and functions in binaries the number of nop’s encountered in the code stream can be greater than simply 1 per 3 normal instructions.\",\n \"loop_start: <TAG> addi addi addi <TAG> j loop_start Consider the simplified toy example in asm-ex.\",\n \"In this set of instructions, we encounter 2 tags every 4 normal instructions every time we iterate over the loop.\",\n \"Examples can be found for every coverage size, using larger loops or functions.\"\n ],\n [\n \"Binary Properties\",\n \"The properties of the benchmarks have a major impact on the results of our evaluation.\",\n \"For programs that are memory operation intensive, even in backwards compatibility mode, the overhead of additional nops added to the instruction stream is quickly overwhelmed by the cost of memory operations.\",\n \"mcf_r has this property, and has an overhead of 3.79% for coverage 3, and 1.09% for coverage 15 in backwards compatibility mode.\",\n \"Conversely, small looping in-register operations can cause the nops to have an oversized impact on program performance.\",\n \"Deepsjeng fits this profile, and the resulting overhead is 21.79% for C3, 12.45% for C7, and 8.39% for C15.\",\n \"Programs that do not exhibit such extremes in behavior tend to have overhead results close to the average results.\",\n \"Full results can be found in fig:overhead.\"\n ],\n [\n \"Size Overhead\",\n \"Overall we expect the text section to increase in proportion to the number of tag instructions we insert.\",\n \"However complicating this number is the function alignment requirements.\",\n \"In the text section of the binary, the space between functions is filled in with zeros that extend from the end of one function to the beginning of the next.\",\n \"This effect is more pronounced in C++ programs with many small functions such as omnetpp_r and xlancbmk_r.\",\n \"Labels have a minimal impact on binary size, averaging 0.78%.\",\n \"Overall the text section size increase is on average 29.32% for coverage three with labels, 20.49% for coverage seven, and 18.27% for coverage 15.\",\n \"The full text section size results are available in tab:size.\",\n \"Table: Text section size in bytes for baseline and 4 different configurations.\",\n \"Luic3 is coverage 3 using the lui nop.\",\n \"luic7 is coverage 7 using the lui nop.\",\n \"luic15 is coverage 15 using the lui nop.Table: State of the art comparison.\",\n \"*PUMP's tags are sizeof(int *) per memory word, some applications will not use the full space.\",\n \"**PUMP allows arbitrary metadata by allowing a pointer to be encoded into the tag\"\n ],\n [\n \"Feature Comparison Against State of the Art\",\n \"COGENT was created to enable flexibility in its deployment.\",\n \"To demonstrate this against the current state of the art, we compared COGENT against recent tag-based architectures using several criteria.\",\n \"How easy is it to employ the tool in a different manner then one of the creators designs?\",\n \"A high rating means the tool was designed for that purpose, while a low indicates it was created for one specific task.\",\n \"Does this tool have a Variable tag size?\",\n \"Can this tool provide an arbitrary amount of metadata to a user above and beyond the size of the tag, if that user desires it.\",\n \"Does this tool change the underlying ISA, removing backwards compatibility.\",\n \"Does this tool use the unused pointer bits?\",\n \"What kind of additional memory protection does this tool employ to protect its tags?\",\n \"The results of this investigation are available in tab:comp.\",\n \"In general we find that unless a tool was designed for reuse (i.e.\",\n \"PUMP), it would be difficult to re-target the hardware for other tasks.\",\n \"State of the art solutions use ISA changes to provide the compiler with the wrappers and tag aware instructions they need to manipulate their metadata.\",\n \"These ISA changes break backwards compatibility.\",\n \"The common use of unused pointer bits, or custom pointers, render solutions incompatible with each other.\",\n \"Table: Sample CVE's and tag-based solutions\"\n ],\n [\n \"Case Studies and Use Cases\",\n \"COGENT is a generic tool designed to allow for the easy implementation of other tagging and labeling schemes.\",\n \"We present a case study, offering a detailed look at systems COGENT could be used to achieve.\",\n \"We selected CVE-2021-3330 to explore with detailed solutions presenting and describing Tag Aware Arithmetic sec:2021-3330. tab:CWECVE shows a selection of recent CVEs to which our findings could be applied, including the one we selected for our case study.\",\n \"Additionally, we present an alternate usage of COGENT for fuzzing and testing, where we use COGENT to keep track of coverage and branching behavior of a program at runtime.\",\n \"Table: Overflow check tagging schemeCVE-2021-3330 is a vulnerability that affects the Zephyr RTOS, and comes from a series of errors that includes an improperly constructed cyclical list, where a singularly linked list is expected.\",\n \"Iterating over this now cyclical list causes an unsigned value to be repeatedly subtracted from another unsigned value, leading to an integer underflow.\",\n \"This value is then fed as the length argument into [language=C]|memmove()|, leading to a massive memory move that can overwrite kernel structures.\",\n \"This CVE consists of several points of failure.\",\n \"Improper list construction, integer underflow, and the overwrite itself.\",\n \"In this case study, we focus on a potential solution for the integer underflow, which would prevent the memory overwrite by triggering an exception when the underflow occurs.\"\n ],\n [\n \"Type Aware Arithmetic\",\n \"To do this we will introduce type aware arithmetic, requiring a tag with a width of a single bit, giving two possible states for every instruction as shown in tab:overflowcheck.\",\n \"These two tags are N and UN_ARTH.\",\n \"The N tag is given to any instruction in the stream that will not be checked for integer overflows.\",\n \"This can be because the instruction does not perform any arithmetic on an unsigned integer, or because the programmer or the compiler has chosen to disable the check on that instruction.\",\n \"While the UN_ARTH would be automatically placed on any instruction that performs arithmetic operations on unsigned types.\",\n \"To insert this tag, an llvm compiler pass would be created that examines the left hand side of any arithmetic expression, determining if the type of the result is unsigned.\",\n \"If it is, the compiler will add the tag to any and all arithmetic instructions that result from the IR level instruction, which the COGENT system would then propagate down into the final binary.\",\n \"There are two cases where the compiler would not place the UN_ARTH tag on an instruction that creates an unsigned value.\",\n \"The first is when the compiler can determine that values being subtracted are hard coded immediate values, in this case we would have to assume the programmer is attempting to exploit the defined behavior of unsigned (IE unsigned MAX = 0 - 1).\",\n \"The second would be a provided annotation that allows the programmer to opt out of the checks if they had a reason to.\",\n \"This is required as in the C standard integer over/underflow are defined behavior.\",\n \"Forcing the user to opt out of the behavior is done in an attempt to minimize developer errors when using this defense.\"\n ],\n [\n \"Required Hardware Support\",\n \"The required hardware support for this tag scheme is very simple, if an instruction is tagged with UN_ARTH, an exception should be thrown if the results from that operation over or underflow.\",\n \"Hardware would require this additional tag support as without metadata, it would be unable to differentiate between an operations on signed and unsigned integers.\",\n \"This can either be a hard stop to the program, or be call an exception handler if the policy calls for it.\",\n \"In this case, for CVE-2021-3330, we would halt the program execution, as the corruption that results from this bug can overwrite kernel data structures.\"\n ],\n [\n \"Code and Tag Layout\",\n \"For CVE 2021-3330, the original integer overflow occurs in the following two lines of code.\",\n \"memmove(frag->data, frag->data + hdr_len, frag->len - hdr_len); frag->len -= hdr_len; These lines of code operate only on unsigned integers, and do not have the opt-out annotation.\",\n \"After compilation, this code produces the following relevant lines of assembly, loading the required values into registers, and then subtracting them.\",\n \"linerange=1-10, breaklines=true, frame=single, caption=CVE-2021-3330 underflow ASM ]listings/taggedunarth There are two places in this code that will need to be tagged, sub a2, a0, a2 and sub a1, a3, a1.\",\n \"This information can be determined by the compiler, and so they will receive the UN_ARTH tags.\",\n \"Resulting int he following final code layout.\",\n \"Note the tag instructions themselves are not included in this listing, as there exact position would depend on compiler optimizations levels, and overall layout of the code, however they would be inserted into the instruction stream at the appropriate alignments.\",\n \"linerange=13-22, breaklines=true, frame=single, tabsize=2, caption=CVE-2021-3330 underflow ASM ]listings/taggedunarth With all this in place, when this tagged code is run and the bug is triggered, operation would proceed normally until the subtraction that causes the arithmetic underflow.\",\n \"Once that happens, the hardware would trigger an exception based on the UN_ARTH tag, causing an extra check as part of the instruction processing phase.\",\n \"A final strength of this solution is that because of the simple nature of the two tags employed, it would be a trivial exercise for COGENT (although not necessarily the hardware) to layer this tag with another single bit tag that would not be placed on arithmetic instructions.\",\n \"An example could be the single bit return address protection provided by [16] and [13].\",\n \"These two protections could be layered without the need for extra tag sizes.\"\n ],\n [\n \"Description\",\n \"Inline tagging could be used during testing and fuzzing to allow for quick tracking of program coverage, counts of reached instructions, and paths taken through the CFG.\",\n \"In this use case, run time security is not considered as part of the design.\",\n \"Instead COGENT inserts metadata informing testing hardware what information it needs to record, and where it needs to record it.\",\n \"Hardware changes are limited to what is required to facilitate these tasks.\",\n \"Table: Coverage tags summary\"\n ],\n [\n \"Tag and Label Design\",\n \"This use case requires the coverage 7, 2 bit LUI labeling scheme.\",\n \"Giving 4 possible tags, each indicating a different behavior to the testing hardware.\",\n \"Additionally we introduce counting labels, with the initialized value of 0 as the primary fast coverage tracking method.\",\n \"An in depth description of each tag and the labels are as follows, with a summary provided in tab:coverage.\",\n \"CL (Counting label) - Indicates that this instruction is a Counting label.\",\n \"When this tag, and the associated label are encountered in the instruction stream.\",\n \"First the value contained in them is incremented by one, Then written back into the label.\",\n \"This requires that the system be allowed to write values back into the code section.\",\n \"These will be placed at the beginning of every basic block.\",\n \"CI (Counting Instruction) - Indicates that this instruction should be counted.\",\n \"When the instruction stream reaches this tag, it will check the current program counter against a mapping of PC : Int to see if this instruction has been encountered before.\",\n \"If it has, it will increment the associated Int by one.\",\n \"If not, it will create a new entry in the map, with a value of one.\",\n \"This counting instruction is explicitly slower than the counting label, due to the need for the mapping lookup, but could be used for specific purposes.\",\n \"BCF (Branching Control Flow) - Indicates a branching point in the instruction stream that needs to be tracked.\",\n \"This tells the hardware to check if the current PC is in the BCF mappings.\",\n \"If it is not, it creates a new entry in a {PC : {Next_PC : Count}} mapping.\",\n \"After it has either found, or created, the entry for the current PC, the Hardware then checks if the next program counter is in the {Next_PC: Count} mapping, adding it if it does not find it, or increment the counter if it does.\",\n \"In this way the BCF tag builds a map of each branching instruction, the targets it has reached, and how many times it has reached those targets.\",\n \"N (Normal) - This tells the hardware that no special hardware actions are needed to enforce the overall security policies within this instruction.\"\n ],\n [\n \"Compiler Support\",\n \"For coverage tracking, the compiler would be configured to place counting labels and their associated tags at the beginning of every basic block at the machine instruction level.\",\n \"Additionally, a pass would be added to enable two levels of branching control flow tracking depending on the desires of the tester.\",\n \"BCF labels could be placed on every control flow instruction or, just on computed control flow instructions.\"\n ],\n [\n \"Hardware Support\",\n \"This coverage system requires two major changes to the supporting hardware beyond the ability to read tags.\",\n \"First is the ability to write the incremented counters back into the counting labels.\",\n \"Second is the ability to record the PC of instructions tagged with CI or BCF into mappings.\",\n \"To do this we would reserve a known memory for the CI and BCF mappings, and ensure that the size of each entry is known to the hardware.\",\n \"This means for BCF mappings there would be a configurable maximum number of targets in order to keep lookup times reasonable.\"\n ],\n [\n \"Example Code\",\n \"The RISC-V assembly code below is an example of how the coverage tagging would look like on a simple basic block, followed by a branching jump.\",\n \"linerange=1-19, breaklines=true, frame=single, tabsize=2, caption=Coverage Example ASM ]listings/coverage Assuming the above was entered three times, jumping once and falling through twice, the counting label at 0x4 would be |lui x0, 3| and at 0x1C it would be |lui x0, 2|.\",\n \"LBB0_2 is always reached at 0x34, making its value |lui x0, 3|.\",\n \"The branching table would contain 0x18 : 0x1C: 2, 0x30: 1, showing the exact behavior of the branch over three runs.\"\n ],\n [\n \"Related Work\",\n \"PUMP [16] provides a flexible system to associate metadata with memory locations, including on instructions.\",\n \"Its large fixed-width tag sizeof(*) provides unlimited metadata size per memory location via allowing the tag to simply be a pointer to additional memory.\",\n \"However schemes that actually use this amount of space are not common, and even then, the common case for any given memory location is a small amount of metadata.\",\n \"To help alleviate the large overhead of having so much unused space in there tags, they propose and implement tag compression schemes to speed up retrieval.\",\n \"Compared to PUMP, COGENT's instruction tagging allows simpler protection schemes to use smaller tag sizes when needed, while at the same time allowing that same arbitrarily sized metadata though the use of any number of inline labels.\",\n \"Importantly though, PUMP has an actual hardware design backing it, while COGENT does not have specialized hardware, so the only comparison can be between the location and storage size of metadata, not on performance, hardware complexity, or overall overhead.\",\n \"CHERI [12] is a well known capability based architectural enhancement for RISC instruction sets.\",\n \"Work on it is current and on-going [24] [25] [26].\",\n \"CHERI provides its security though architectural capabilities, and tagged memory.\",\n \"Additionally it is a full stack solution, and includes compiler, OS, and hardware support.\",\n \"Cheri itself requires that compiler to support on all levels.\",\n \"Language specific front end, IR level passes, and backend emissions.\",\n \"RetTag [14] uses compiler modifications to insert new instructions in the function prologue and epilogue to generate a new instruction (thus modifying the ISA and preventing backward compatibility) called pac.\",\n \"This instruction generates a unique PAC using the a unique function ID, the virtual address of the functions RA the SP, and a 128-bit RAA key and stores it in the 16 unused bits of a function pointer, and it is used to protect the return address from corruption.\",\n \"ZERØ [27], provides a set of new instructions via a modification of the ISA and a method of encoding metadata that keeps the overhead to eight bytes per 4kb page.\",\n \"This work discards backwards compatibility by including new ISA instructions for memory and metadata management.\",\n \"Additionally it stores ten bits of information in the unused bits of pointers to enforce its own version of CFI using unique-per-function signature labels.\",\n \"It modifies the compiler to insert its new instructions at the appropriate places.\",\n \"Timber-V [28] allows the creation of up to four protected enclaves on a RISC-V system.\",\n \"It likewise uses a new set of instructions in order to tell the hardware when it can check and manipulate tags.\",\n \"Like ZERØ this removes its ability to provide backwards compatible code.\",\n \"While Timber-V does not require compiler support, they specify that compiler support would allow for better integration.\",\n \"Several works including Framer [15], RetTag [14] and ZERØ [13] use the upper unused bits of a pointer as place to store there metadata.\",\n \"Doing this gives these works strong backwards compatibility and low overhead.\",\n \"However by their nature it is difficult to layer any works that use the unused pointer bits.\",\n \"Additional space is limited, leading framer to use a novel encoding scheme to provide access to the pointers metadata, relying as much as possible on relative offsets from known points (i.e.\",\n \"frames).\",\n \"Such works could be combined with COGENT as we do not manipulate the pointers themselves, leaving the unused parts of the pointer available for any other scheme, or to be used for a CFI label.\",\n \"FlexFit [29] is a generalized solution presented as a method to allow users to configure a set of instruction filters for use at run time.\",\n \"They implement this on a RISC-V FPGA board to demonstrate the practicality, feasibility, and usability of there generic filtering scheme.\",\n \"They store the protection information in four unused bits of the RISC-V PTE, giving up to 16 protected domains at a page granularity.\",\n \"Using the unused bits in the PTE, has been explored by other works as well [30] [31], which is why FlexFit limits its protection domains to 16, using only 4 of the available bits.\",\n \"While FlexFit does not use a compiler, the authors suggest a specialized compiler and loader for future work.\"\n ],\n [\n \"Conclusion\",\n \"This paper presents COGENT, a modified LLVM compiler capable of generating RISCV binaries with configurable width instruction tags.\",\n \"These tags can be used to design security solutions that convey rich semantic information to a tag-aware architecture.\",\n \"COGENT tackles several challenges including one-to-many and many-to-one instruction-level tag associations.\",\n \"We evaluate COGENT on a subset of SPEC 2017 benchmark programs for performance and binary bloat.\",\n \"Further, we present case studies that highlight potential use cases of COGENT.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05614"}}},{"rowIdx":93,"cells":{"text":{"kind":"list like","value":[["YoloCurvSeg: You Only Label One Noisy Skeleton for Vessel-style\n Curvilinear Structure Segmentation"],["Abstract Weakly-supervised learning (WSL) has been proposed to alleviate the conflict between data annotation cost and model performance through employing sparsely-grained (i.e., point-, box-, scribble-wise) supervision and has shown promising performance, particularly in the image segmentation field.","However, it is still a very challenging problem due to the limited supervision, especially when only a small number of labeled samples are available.","Additionally, almost all existing WSL segmentation methods are designed for star-convex structures which are very different from curvilinear structures such as vessels and nerves.","In this paper, we propose a novel sparsely annotated segmentation framework for curvilinear structures, named YoloCurvSeg, based on image synthesis.","A background generator delivers image backgrounds that closely match real distributions through inpainting dilated skeletons.","The extracted backgrounds are then combined with randomly emulated curves generated by a Space Colonization Algorithm-based foreground generator and through a multilayer patch-wise contrastive learning synthesizer.","In this way, a synthetic dataset with both images and curve segmentation labels is obtained, at the cost of only one or a few noisy skeleton annotations.","Finally, a segmenter is trained with the generated dataset and possibly an unlabeled dataset.","The proposed YoloCurvSeg is evaluated on four publicly available datasets (OCTA500, CORN, DRIVE and CHASEDB1) and the results show that YoloCurvSeg outperforms state-of-the-art WSL segmentation methods by large margins.","With only one noisy skeleton annotation (respectively 0.14%, 0.03%, 1.40%, and 0.65% of the full annotation), YoloCurvSeg achieves more than 97% of the fully-supervised performance on each dataset.","Code and datasets will be released at https://github.com/llmir/YoloCurvSeg."],["Introduction","Curvilinear structures are elongated, curved, multi-scale structures that often appear tree-like and are commonly found in natural images (e.g., cracks and aerial road maps) and biomedical images (e.g., vessels, nerves and cell membranes).","Automatic and precise segmentation of these curvilinear structures plays a significant role in both computer vision and biomedical image analysis.","For example, road mapping serves as a prerequisite in both autonomous driving and urban planning.","In the biomedical field, studies [1], [2], [3], [4] have suggested that the morphology and topology of specific curvilinear anatomy (e.g., retinal vessels and corneal nerve fibers) are highly relevant to the presence or severity of various diseases such as hypertension, arteriolosclerosis, keratitis, age-related macular degeneration, diabetic retinopathy, and so on.","Retinal vessels are observable in retinal fundus images and optical coherence tomography angiography (OCTA) images, while corneal nerve fibers are identifiable in confocal corneal microscopy (CCM) images.","It has been suggested that early signs of many ophthalmic diseases are reflected by microvascular and capillary abnormalities [5], [6].","Collectively, accurate segmentation of various curvilinear structures is of great importance for computer-aided diagnosis, quantitative analysis and early screening, especially in ophthalmology.","In recent years, benefiting from the development of deep learning (DL), many DL-based segmentation algorithms for curvilinear structures have been proposed and have shown overwhelming performance compared to traditional (e.g., matched filter-based and morphological processing-based [7], [8]) methods.","Most existing works are dedicated to designing sophisticated network architectures [9], [10], [11] or deploying strategies to preserve curvilinear structures' topology by employing generative adversarial networks (GANs) [2], [12] or topology-preserving loss functions [13], [14].","These methods are typically fully-supervised, wherein large-scale well-annotated datasets are required.","However, collecting and labeling a large-scale dataset with full annotation is very costly and time-consuming, particularly for medical images since their annotation requires expert knowledge and clinical experience.","Furthermore, annotating curvilinear structures is even more challenging, given that curvilinear structures are slender, multi-scale, and complex in shape with fine details.","More recently, many efforts have been made to reduce the annotation cost for DL model training.","For example, semi-supervised learning (SSL) trains models by combining limited amounts of annotated data with massive unlabeled data [15], [16], [17].","While effective, most state-of-the-art (SOTA) SSL methods still require about 5%-30% of the accurately and precisely labeled data to achieve about 85%-95% of the fully-supervised performance, which is still not sufficiently cost-effective and still time-consuming when it comes to labeling curvilinear structures.","Weakly supervised learning (WSL) attempts to alleviate the annotation issue from another perspective by performing sparsely-grained (i.e., point-, scribble-, bounding box-wise) supervision and attains promising performance [18], [19], [20], [21], [22].","Compared with either point or bounding box, scribble is a relatively more flexible and generalizable form of sparse annotation that can be used to annotate complex structures [23].","Existing scribble-supervised segmentation methods mainly fall into two categories.","The first line of research exploits structural or volumetric priors to expand scribbles into more accurate pseudo proposals; for example, grouping pixels with similar grayscale intensities or locations into the same class [18], [19], [24].","However, the expansion process may introduce noisy proposals, which may induce error accumulation and deteriorate the performance of the segmentation model.","Some work [25] also points out the inherent weakness of these methods, namely models retain their own predictions and thus resist updating.","The second line learns adversarial shape priors utilizing extra unpaired but fully-annotated masks.","Such approaches somewhat contradict the motivation of saving annotation costs, especially for complex curvilinear structures [26], [27], [28].","Moreover, most WSL methods still require sparsely labeling the entire dataset (or a large portion), and they are mainly designed and validated on relatively simple structures (e.g., cardiac structures or abdominal organs) with assumptions and priors that may not apply to complex structures (e.g., curvilinear structures).","To address these aforementioned challenges, we here present a novel WSL segmentation framework for vessel-style curvilinear structures, namely You Only Label One Noisy Skeleton for Curvilinear Structure Segmentation (YoloCurvSeg).","For curvilinear structures, label noises/errors are inevitable, and a good segmentation approach should be noise tolerant.","Therefore, instead of utilizing only the annotated pixels for supervision, YoloCurvSeg ingeniously converts the weakly-supervised problem into a fully- or semi-supervised one via image synthesis.","It employs a trained inpainting network as a background generator, which takes one (or multiple depending on availability) noisy skeleton (as shown in Fig.","REF ) and dilates it to serve as an inpainting mask to obtain a background that closely matches the real distribution.","The extracted background is then augmented and combined with randomly emulated curves generated by a Space Colonization Algorithm-based foreground generator, from which a synthetic dataset is obtained through a multilayer patch-wise contrastive learning synthesizer.","Finally, a segmenter performs coarse-to-fine two-stage training using the synthetic dataset and an unlabeled dataset (if available).","Our main contributions are summarized as follows: We propose a novel weakly-supervised framework for one-shot skeleton/scribble-supervised curvilinear structure segmentation, namely YoloCurvSeg.","To the best of our knowledge, YoloCurvSeg is the first sparsely-annotated and weakly-supervised segmentation method for curvilinear structures.","YoloCurvSeg novelly converts a WSL problem into a fully supervised one through four steps: curve generation, image inpainting, image translation and coarse-to-fine segmentation.","The proposed framework is noise-robust, sample-insensitive and easily extensible to various curvilinear structures.","We evaluate YoloCurvSeg on four challenging curvilinear structure segmentation datasets, namely OCTA500 [29], CORN [30], DRIVE [31], CHASEDB1 [32].","Experimental results show that YoloCurvSeg outperforms SOTA WSL and noisy label learning methods by large margins.","Meanwhile, we demonstrate that $>$ 97% of the fully-supervised performance can be achieved with only one noisy skeleton label (approximately 0.1% or 1% of the full annotation), which shall also inspire subsequent works on WSL and curvilinear dataset construction.","Existing automatic curvilinear structure segmentation algorithms can be roughly divided into two categories.","The first category is traditional unsupervised methods, mainly including mathematical morphology methods and various filtering methods [10].","For instance, Zana et al.","[33] segment vascular-like patterns using a hybrid framework of morphological filtering and cross-curvature analysis.","Passat et al.","[34] present a preliminary approach to strengthen the segmentation of cerebral vessels by incorporating high-level anatomical knowledge into the segmentation process.","Filtering methods include Hessian matrix-based filters [35], matched filters [8], [36], multi-oriented filters [37], symmetry filter [38], etc.","The other category is supervised methods, wherein data with ground truth labels are used to train segmenters based on predefined or model-extracted features.","Traditional machine-learning-based approaches are dedicated to pixel-level classification using handcrafted features [39], [40].","Recently, DL-based approaches have made significant progress in various segmentation tasks.","For example, Ronneberger et al.","[41] propose U-Net, which has been widely used in numerous medical image segmentation tasks.","Existing curvilinear structure segmentation works focus on well-designed network architectures by introducing multi-scale [11], [42], multi-task [6], [9], [43], or various attention mechanisms [10], [44] and well-playing morphological and topological properties by introducing GANs or morphology-/topology-preserving loss functions [13], [14].","Still, data availability and annotation quality are the main limitations of these methods."],["Weakly-supervised Segmentation","Weakly-supervised segmentation aims to reduce the labeling costs by training segmentation models on data annotated with coarse granularity [18].","Among various formats of sparse annotations, scribble is recognized as the most flexible and versatile one that can be used to annotate complex structures [23], [27].","Existing scribble-supervised segmentation methods fall into two main categories.","The first one exploits structural or volumetric priors to expand scribble annotations by assigning a same class to pixels with similar intensities or nearby locations [18], [19], [24].","The main limitation of such approaches is that they heavily rely on pseudo proposals and often contain multiple stages, which can be time-consuming and prone to errors that may be propagated during model training.","The second category learns adversarial shape priors utilizing extra unpaired but fully-annotated masks.","Such approaches somewhat contradict the motivation of saving annotation costs, especially for complex curvilinear structures [26], [27], [28].","Additionally, these methods still require sparsely labeling the entire dataset or a large portion, and they are mainly designed and validated on relatively simple structures like cardiac structures or abdominal organs with assumptions and priors that may not apply to complex structures (e.g., curvilinear ones).","In this paper, we make use of noisy skeletons that differ from scribbles in two ways: (1) skeletons are more label demanding since all branches are supposed to be covered; (2) they are more likely to contain errors or noises, which are inevitable when quickly labeling slender structures.","We convert sparse and noisy skeleton annotations to accurate ones via an image synthesis pipeline, thus requiring only one noisy skeleton label.","This significantly reduces the annotation cost."],["Medical Image Synthesis","GAN [45] has become the mainstay of medical image synthesis, with common applications in intra-modality augmentation [46], cross-domain image-to-image translation [47], quality enhancement [48], missing modality generation [49], etc.","Below we briefly review previous works on retinal image synthesis, the topic of which is relevant to our work.","Costa et al.","[50] employ a U-Net trained with paired fundus images and vessel masks.","It employs a conditional GAN (Pix2pix [52]) to learn a mapping from vessel masks to the corresponding fundus images.","To simplify the framework, they propose an adversarial autoencoder (AAE) for retinal vascularity synthesis and a GAN for generating retinal images [51].","Similarly, Guibas et al.","[53] present a two-stage approach that consists of a DCGAN for generating vasculature from noise and a cGAN (Pix2pix) to synthesize the corresponding fundus image.","These methods require an extra set of vessel annotations to train AAE or DCGAN and may sometimes generate vessels with unrealistic morphology.","The generated images also lack diversity.","Zhao et al.","[54] develop Tub-sGAN, which incorporates style transfer into the GAN framework to generate more diverse outputs.","Note that cGAN requires paired images and vessel masks for training, which is a strict condition to some extent.","In another work, SkrGAN [55] is proposed to introduce a sketch prior related constraint to guide the image generation process.","Yet, the sketches utilized are extracted by the Sobel edge operator, and cannot be used as segmentation masks.","In this paper, we employ a multilayer patch-wise contrastive foreground-background fusion GAN for several considerations.","According to previous research, training a GAN to learn a direct mapping from a curvilinear structure mask to the corresponding image is difficult, especially under few-shot conditions [56].","Therefore, we provide GAN with extracted real backgrounds, enabling implicit skip-connection that allows the GAN to focus more on mapping the foreground regions.","Such a design not only enhances performance but also accelerates convergence.","Multilayer patch-wise contrastive learning allows the provided mask and the foreground region of the generated image to be spatially aligned (via unpaired training), which further benefits the subsequent segmenter.","YoloCurvSeg comprises four main components: (1) a Curve Generator that produces binary curve masks that well accommodate the corresponding image modality of interest; (2) an Inpainter for extracting backgrounds from labeled samples; (3) a Synthesizer that synthesizes images from the generated curve masks and the image backgrounds; and (4) a two-stage Segmenter trained with the synthetic dataset and an unlabeled dataset.","The overall framework is shown in Fig.","REF .","Figure: Overview of our proposed YoloCurvSeg, which comprises four main components: a space colonization algorithm-based curve generator, a background inpainter, a multilayer patch-wise contrastive foreground-background fusion synthesizer, and a two-stage coarse-to-fine segmenter."],["Curvilinear Structure Generation","Space colonization is a procedural modeling algorithm in computer graphics that simulates the growth of branching networks or tree-like structures [57], [58], including vasculature, leaf venations, root systems, etc.","It is employed in YoloCurvSeg for modeling the iterative growth of curvilinear structures with two fundamental elements: attractors and nodes.","Its core steps are described in the top left panel of Fig.","REF , wherein blue dots denote attractors and black ones denote nodes: a) place a set of attractors randomly or following a predefined pattern, and then associate nodes with nearby attractors (if their distance is within an attraction distance $D_a$ ); b) for each node, calculate its average direction from all attractors affecting it; c) calculate the position of new nodes via normalizing the average direction to a unit vector and scaling it by a pre-defined segment length $L_s$ ; d) place nodes at the calculated positions and check if any nodes are within an attractor's kill zone; e) prune an attractor if there are nodes staying within its kill distance $D_k$ ; f) repeat steps b)-e) until the maximum number of nodes is reached.","Through observing the pattern of the foreground/curve in a single image or a few images that are accessible, including the curves' starting point, boundary, and degree of curvature, etc., it is relatively straightforward to set the corresponding hyperparameters, such as the root node coordinates $C_r$ (e.g., the starting point of the vessels in fundus lies in the optic disc region), as well as the bounds and obstacles.","For $D_a$ , $D_k$ and $L_s$ , the commonly used values of 5, 30 and 5 can be re-tuned as needed.","Table REF summarizes the parameters and post-processing operations we employ for generating the four types of curves and representative examples are demonstrated in the top panel of Fig.","REF .","Please note that our adopted settings and post-processing operations only represent our empirical choices and are not necessarily the best-performing ones; users can make further adjustments based on their observations and experiences.","In addition to the curvilinear shape, we also need to simulate the thickness of each branch $R^n = R_1^n + R_2^n,$ where $R$ , $R_1$ and $R_2$ respectively denote the radii of a father branch and its two child branches.","$n$ is set to be 3 according to Murray's law [59].","The calculation is performed recursively from the branch tips (whose radii are set to be 1) towards the tree base.","Several intuitive demos can be accessed at linkhttps://jasonwebb.github.io/2d-space-colonization-experiments/.","By setting random grid attractors and root nodes via predefined parameters, we construct four curve banks for our four datasets of interest, which are then utilized for training the synthesizers and the segmenters.","Table: parameters for generating different types of curves.","D a D_a, D k D_k, L s L_s and C r C_r respectively denote the attraction distance, kill distance, segment length and root node coordinates.","rr and ll denote radius and length.","⨁\\bigoplus and ⨀\\bigodot represent the union and element-wise multiplication operations.","there are two rows for OCTA500 since two components are needed."],["Inpainting for Background Extraction","Inpainting is the task of reconstructing missing or masked regions in an image.","Similar to removing watermarks or extraneous pedestrians from images, we employ an inpainting model here to remove foregrounds (e.g., vessels and nerve fibers) from the images of interest, under the hypothesis that the dilated noisy skeletons can fully cover the foregrounds.","In inpainting, common concerns are the network's ability to grasp local and global context and to generalize to a different (especially higher) resolution."],["Architecture","Inspired by [60], we adopt an inpainting network based on the recently proposed fast Fourier convolutions (FFCs) [61] with image-wide receptive fields, strong generalizability and relatively few parameters.","Given a masked image $I \\odot (1-m)$ , where $I$ and $m$ respectively denote the original image and the binary mask of inpainting regions, the feed-forward inpainting network $f_\\theta (\\cdot )$ aims to output an inpainted image $\\hat{I}$ = $f_\\theta (I^{\\prime })$ taking a four-channel input $I^{\\prime }=\\operatorname{concat}(I\\odot (1-m),m)$ .","FFC builds its basis on channel-wise fast Fourier transform (FFT) and has a receptive field covering the whole image.","It splits channels into two parallel branches: a local branch uses conventional convolutions and a global branch uses real FFT to capture global context, as shown in Fig.","REF .","real FFT is only applicable to real-valued signals, and inverse real FFT ensures the output is real-valued.","Compared to FFT, real FFT uses only half of the spectrum.","In FFC, real FFT is first applied to the input tensor and a ComplexToReal operation is performed by concatenating the real and imaginary parts.","Then, it applies convolutions in the frequency domain.","Inverse real FFT is performed to transform features from the frequency domain to the spatial domain through the RealToComplex operation.","Finally, the local and global branches are fused.","For the upsampling and downsampling of the Inpainter and the discriminator architecture in adversarial training, we follow the ResNet settings respectively employed in [62] and [60].","The training is performed on [image, randomly synthesized mask] pairs.","We adopt the mask generation strategy in [60], containing multiple rectangles with arbitrary aspect ratios and wide polygonal chains."],["Objective","Compared with naive supervised losses which may result in blurry predictions, perceptual loss [63] evaluates the distance between feature maps of the inpainted image and the original image via a pre-trained network $\\phi (\\cdot )$ .","It does not require exact reconstruction and allows for variations in the reconstructed image.","Given that inpainting focuses on understanding the global structure, we introduce a perceptual loss of a large receptive field $\\mathcal {L}_{HRP}$ through a pre-trained ResNet50 $\\phi _{HRF}(\\cdot )$ with dilated convolutions provided in [60] $\\mathcal {L}_{HRP}(I, \\hat{I})=\\mathcal {M}([\\phi _{HRF}(I)-\\phi _{HRF}(\\hat{I})]^2),$ where $\\mathcal {M}$ is a sequential two-stage mean operation, i.e., obtaining the inter-layer mean of intra-layer means.","Additionally, an adversarial loss $\\mathcal {L}_{adv}$ is utilized to encourage the inpainted image to be realistic.","Specifically, we use a PatchGAN [52] discriminator $\\mathcal {D}_{\\xi }(\\cdot )$ and label patches that overlap with the mask as fake and the others as real.","The non-saturating adversarial loss is defined as ${\\begin{array}{c}\\mathcal {L}_D=-E_I[\\log D_{\\xi }(I)]-E_{I, m}[\\log D_{\\xi }(\\hat{I}) \\odot (1-m)] \\\\-E_{I, m}[\\log (1-D_{\\xi }(\\hat{I})) \\odot m],\\end{array}}$ $\\mathcal {L}_G=-E_{I, m}[\\log D_{\\xi }(\\hat{I})],$ $\\mathcal {L}_{adv}=\\operatorname{sg}_\\theta (\\mathcal {L}_D)+\\operatorname{sg}_{\\xi }(\\mathcal {L}_G) \\rightarrow \\min _{\\theta , \\xi },\\vspace{-2.84544pt}$ where $\\hat{I} = f_\\theta (I^\\prime )$ is the output of the inpainting network and $\\operatorname{sg}_{var}$ represents stop gradient w.r.t.","$var$ .","To further stabilize the training, we use a gradient penalty $\\mathcal {L}_{GP} = E_I\\Vert \\nabla D_{\\xi }(I)\\Vert _2^2$ [64] and a perceptual loss defined on features of the discriminator $\\mathcal {L}_{DP}$ [65].","The final objective of the Inpainter is $\\mathcal {L}_{inpaint}=\\mathcal {L}_{HRP}+\\lambda _{adv}\\mathcal {L}_{adv}+\\lambda _{DP}\\mathcal {L}_{DP}+\\lambda _{GP}\\mathcal {L}_{GP},$ where $\\lambda _{adv}$ , $\\lambda _{DP}$ and $\\lambda _{GP}$ are hyper-parameters balancing the contributions of different losses.","$\\mathcal {L}_{HRP}$ is responsible for supervised signals and global structure consistency while $\\mathcal {L}_{adv}$ and $\\mathcal {L}_{DP}$ are responsible for local details and realism.","Figure: The architecture of the Inpainter.","The input is a four-channel image with the first three channels being the original image and the last channel being the binary mask of the inpainting regions.","The output is the inpainted image.","The dimensional change of the feature map in FFC is shown in the lower right panel."],["Training","Given that the training of the Inpainter does not require annotation and it learns a general ability to recover missing regions through contextual understanding, we initialize the model with pre-trained parameters from the Places-Challenge dataset [66] and fine-tune it on images accessible within each training set.","The validation set consists of both accessible training set images and validation set images (paired with predefined masks obtained using a specific generation strategy [60]).","The training is conducted with a batch size of 8 and an Adam optimizer is adopted with a learning rate of $10^{-3}$ for 50 epochs.","The training data are augmented through random flipping, rotation and color jittering.","We empirically set $\\lambda _{adv}=3$ , $\\lambda _{DP}=10$ and $\\lambda _{GP}=10^{-4}$ .","Once trained, the Inpainter is used to remove the foregrounds from the skeleton-labeled samples taking the dilated noisy annotations as the masks.","Then we construct a background bank for each dataset by augmenting the extracted backgrounds through random flipping, rotation, etc.","Now we have a curve (foreground) bank $B_{curv} = \\lbrace c^{1}, \\cdots , c^{N}\\rbrace $ and a background bank $B_{bg} = \\lbrace b^{1}, \\cdots , b^{N}\\rbrace $ respectively from the Curve Generator and the Inpainter, for each given dataset.","We construct an intermediate dataset $X_{inter} = \\lbrace x^{1}, \\cdots , x^{N}\\rbrace $ through randomly sampling a curve $c^{i}$ from $B_{curv}$ and a background $b^{i}$ from $B_{bg}$ , and then concatenating them to form a temporary sample $x^{i} = \\operatorname{concat}(b^{i},c^{i})$ .","The problem now turns into an unpaired image-to-image translation task, i.e., designing a synthesizer to learn a mapping from $X_{inter}$ to the corresponding real dataset $Y$ .","It is desirable that the local context especially the foreground of the synthetic image $\\hat{y}^{i}$ is spatially aligned with that of the corresponding intermediate image $x^{i}$ (especially $c^{i}$ ) as much as possible.","Previously, for unpaired image translation, most existing methods apply GANs with a cycle structure, relying on cycle-consistency to ensure high-level correspondence [67].","While effective, the underlying bijective assumption behind cycle-consistency is sometimes too restrictive, which may reduce the diversity of the generated samples.","More importantly, the cycle-consistency is not suitable for our task since it does not guarantee any explicit or implicit spatial constraints.","In such context, we introduce a multilayer patch-wise contrastive learning based synthesizer to learn a mapping from $X_{inter}$ to $Y$ inspired by [68], [69], which is illustrated in the lower middle panel of Fig.","REF .","It is trained in a generative adversarial manner with an internal contrastive learning pretext task.","The generator (i.e., synthesizer) $G$ is a U-shape network, which firstly down-samples the input image into high-level features via an encoder $E$ with three residual blocks equipped with instance normalization and ReLU activation.","As such, each pixel in the high-level feature map represents the embedding feature vector of a patch in the original image.","Several layers of interest $E_{l \\in L}(x)$ in $E$ are selected to extract multi-scale features of patches and each passes through a two-layer multilayer perceptron (MLP) $H_l$ ($l$ indexes a layer), obtaining a feature stack $\\lbrace v_{l \\in L}=H_{l \\in L}[E_{l \\in L}(x)]\\rbrace $ .","Given patchwise features $v_{l}$ and the corresponding pair $\\lbrace H_l(E_l(x))^{s_1}, H_l(E_l(G(x)))^{s_2}\\rbrace $ with $s_1$ and $s_2$ denoting the spatial locations of the patches of interest, we set $v^+$ to represent a patch at the same location as $v$ and $v^-_n$ to denote the $n^{th}$ among $N$ patches at different locations.","The objective of the contrastive learning task is to maintain the local information at the same spatial location.","Similar to the noise contrastive estimation loss [70], our objective function can be written as $\\hspace{-5.406pt}\\mathcal {L}_{c}=-\\sum _{l \\in L} \\log \\frac{\\exp (v_l \\cdot v_l^{+} / \\tau )}{\\exp (v_l \\cdot v_l^{+} / \\tau )+\\sum _{n=1}^N \\exp (v_l \\cdot v_{l n}^{-} / \\tau )},$ where $\\tau $ is a temperature hyper-parameter.","Besides, we employ the identity loss, which was first proposed in [67] for regularizing the generator $G$ .","We pass each real sample $y \\in Y$ through the encoder $E$ and obtain the patchwise features $v^*$ , the negative samples $v^{*-}$ and the positive samples $v_n^{*+}$ .","The identity loss is formulated as $\\hspace{-2.84526pt}\\mathcal {L}_{id}=-\\sum _{l \\in L} \\log \\frac{\\exp (v_l^* \\cdot v_l^{*+} / \\tau )}{\\exp (v_l^* \\cdot v_l^{*+} / \\tau )+\\sum _{n=1}^N \\exp (v_l^* \\cdot v_{l n}^{*-} / \\tau )}.$ We use the LSGAN loss as our adversarial loss $\\mathcal {L}_{adv}$ [71] to make the synthetic images as realistic as possible.","Therefore, with trade-off parameters $\\lambda _{adv}$ , $\\lambda _{c}$ and $\\lambda _{id}$ , the overall loss of the synthesizer is defined as $\\mathcal {L}_{syn} = \\lambda _{adv}\\mathcal {L}_{adv} + \\lambda _{c}\\mathcal {L}_{c} + \\lambda _{id}\\mathcal {L}_{id}.$ The training of the synthesizer is conducted employing an Adam optimizer with a learning rate of $10^{-4}$ and a cosine decay strategy, together with a batch size of 1.","We utilize images that are accessible within each training set as the corresponding real dataset $Y$ for training the synthesizer.","The hyperparameters and model weights are selected based on the Fréchet Inception Distances (FIDs) [76] between the synthesized images and $Y$ .","We set $\\lambda _{adv}$ as 1, $\\lambda _{c}$ as 1, $\\lambda _{id}$ as 0.5 and $\\tau $ as 0.07.","The training process lasted for 300 epochs."],["Two-stage Coarse-to-Fine Segmentation","A synthetic dataset $\\mathcal {D}_{syn} = \\lbrace (\\hat{y}^{1},c^1),\\cdots ,(\\hat{y}^{N},c^N)\\rbrace $ , with $\\hat{y}^{i}$ being a synthetic image and $c^i$ being the corresponding curve ground truth, is created by the Synthesizer.","The weakly-supervised task is then transformed into a fully- or semi- one when making use of solely the synthetic dataset or a combination of an unlabeled dataset $\\mathcal {D}_{ori}$ and the synthetic dataset $\\mathcal {D}_{syn}$ .","In this section, we introduce a two-stage coarse-to-fine segmentation pipeline to tackle the task.","A specific segmentation network is first trained on $\\mathcal {D}_{syn}$ to obtain a coarse model $S_{coarse}$ with a segmentation loss $\\mathcal {L}_{seg}$ $\\hspace{-5.406pt}\\mathcal {L}_{seg} = 0.5\\times \\mathcal {L}_{ce} + 0.5\\times \\mathcal {L}_{dice},$ where $\\mathcal {L}_{ce}$ and $\\mathcal {L}_{dice}$ respectively denote the cross-entropy loss and the Dice loss.","We observe and conclude that the performance of $S_{coarse}$ is mainly limited by two issues; one is that the curve generated by the Curve Generator still has a certain morphological gap with the foreground of the real image, and the other one is that there is also a slight but inevitable intensity gap between the Synthesizer-generated image and the real image.","We target at relieving the latter issue by making use of $\\mathcal {D}_{ori}$ to further boost the segmentation performance.","We employ predictions on $\\mathcal {D}_{ori}$ from $S_{coarse}$ as pseudo-labels and train a fine model $S_{fine}$ (initialized on $S_{coarse}$ ) on the combination of $\\mathcal {D}_{ori}$ and $\\mathcal {D}_{syn}$ via a final loss $\\mathcal {L}_{final}$ which is formulated as $\\mathcal {L}_{final} = \\mathcal {L}_{seg} + \\lambda _{psd}\\mathcal {L}_{psd},$ where $\\mathcal {L}_{psd}$ denotes the loss on $\\mathcal {D}_{ori}$ sharing the same formulation as $\\mathcal {L}_{seg}$ in Eq.","(REF ) and $\\lambda _{psd}$ is a trade-off parameter.","We employ vanilla U-Net with feature channels of 16, 32, 64, 128 and 256 as our $S_{coarse}$ 's and $S_{fine}$ 's architecture.","We use an SGD optimizer (weight decay = $10^{-4}$ , momentum = 0.9) for training both $S_{coarse}$ and $S_{fine}$ with a batch size of 12 and an initial learning rate of $10^{-2}$ .","The total iterations and $\\lambda _{psd}$ are respectively set to be 30$k$ and 1.","In this section, we extensively evaluate the effectiveness of our YoloCurvSeg framework on four representative curvilinear structure segmentation datasets."],["Datasets and Preprocessing","We comprehensively evaluate YoloCurvSeg on four ophthalmic datasets: OCTA500, CORN, DRIVE and CHASEDB1.","OCTA500 is used for retinal microvascular segmentation, and only the subset that contains 300 samples with a $6\\times 6$ $mm^2$ field of view (FOV) and a $400\\times 400$ resolution is utilized.","We only make use of the $en$ -$face$ images generated by maximum projection between the internal limiting membrane layer and the outer plexiform layer.","CORN consists of 1578 CCM images for nerve fiber segmentation.","It also provides two subsets respectively consisting of 340 low-quality and 288 high-quality images.","All CCM images have a resolution of $384\\times 384$ and an FOV of $400\\times 400$ $\\mu m^2$ .","Instead of following the dataset's original division, we use 1532 images (samples that overlap with the test set are removed and the validation split ratio is 0.2) for training and validation, and test on 60 relatively accurately labeled samples provided in its subset.","DRIVE and CHASEDB1 are used for retinal vessel segmentation and respectively have resolutions of $565\\times 584$ and $999\\times 960$ .","These two fundus datasets are cropped via the provided FOV masks and respectively resized to $576\\times 576$ and $960\\times 960$ .","For DRIVE, we utilize the original division of 20 training samples and 20 testing samples.","For CHASEDB1, we follow the division in [2], [11], with the first 20 images serving as the training set and the remaining 8 used for testing.","For OCTA500, we respectively utilize 200, 10 and 90 samples as the training, validation and testing sets.","Images are first normalized and data augmentation consists of random rotation, flipping and Bézier Curve transformation [72]."],["Implementation Details","We implement YoloCurvSeg and other compared methods by PyTorch on a workstation equipped with 8 RTX 3090Ti GPUs.","In the Synthesizer, the indices of the layers selected to calculate $\\mathcal {L}_c$ include $\\lbrace 0,4,8,12,16\\rbrace $ .","For training the Segmenters $S_{coarse}$ and $S_{fine}$ , the polynomial policy with $power=0.9$ is used to adjust the learning rate online [73].","Other hyperparameters, training details and model architecture are already provided in previous sections.","It is worth noting that manually-delineated vessel segmentation labels are provided for OCTA500, DRIVE and CHASEDB1.","To generate noisy skeleton annotations for those three datasets, we perform the skeletonize operation in scikit-image [74] to obtain the skeletons of the original ground truth masks and then employ elastic transformation to simulate jitter noises that may be introduced during fast manual labeling.","For CORN, only noisy skeleton labels are provided, and thus are directly used in all our experiments.","For this dataset, we dilate each skeleton to a 3-pixel width to serve as the full mask in its fully-supervised learning setting, and the same operation is also applied to the testing set annotations.","For sparse labels used in other comparative WSL methods, skeletons of the backgrounds are also generated via skeletonization.","The synthesis process in YoloCurvSeg can be online or offline.","For better reproducibility and fair comparison, we use the offline version in our experiments, i.e., we first generate the synthetic dataset and then train the Segmenters.","We generate 1276, 5005, 1240, and 1604 synthetic samples respectively for OCTA500, CORN, DRIVE and CHASEDB1 if all training samples are labeled.","If only one sample is labeled, we respectively generate 100, 100, 60, and 80 synthetic samples."],["Synthesis Performance","Before comparing with SOTA WSL methods, we first qualitatively and quantitatively evaluate the synthesis performance of YoloCurvSeg.","We visualize representative examples, in terms of the noisy skeleton labels, dilated masks for inpainting, extracted backgrounds, generated curves and synthesized images, in Fig.","REF .","It can be observed from the last column that the generated curves well match the synthetic images.","Figure: Visualization of synthetic data from YoloCurvSeg.","From left to right are examples of the noisy skeleton label, the inflated inpainting mask, the extracted background, the generated foreground, the synthesized image and the generated foreground superimposed on the synthesized image.","Zoom in for details.We also compare the intensity distributions of the synthetic datasets with the real ones in Fig.","REF , exhibiting high intensity similarities between the synthetic and real images in terms of both background and foreground.","From the t-SNE [75] visualization in Fig.","REF , the synthetic datasets are generally in line and well mixed with the real ones.","In most cases, the synthetic data are even more uniformly and widely distributed, having a similar effect as data augmentation.","The FIDs between the synthetic components and the corresponding real ones are tabulated in Table REF to measure the difficulty of learning a mapping between the two distributions.","Column A shows the morphological gap between the synthetic and real curvilinear structures.","Columns B and C illustrate there is great difficulty in translating directly to the image distribution from foreground masks, while the background image with the foreground removed is less distant from the original distribution.","Since YoloCurvSeg incorporates background information, which implicitly acts as skip connections, it reduces the difficulty of image translation and thus can synthesize images close to the real ones even under few-shot settings.","Our method achieves competitive FID scores on all four datasets, two of which are even smaller than those between the real training and test sets, as shown in column E of Table REF .","Figure: Histograms of the four datasets in terms of the real data (top) and the corresponding synthetic data (bottom).Table: FID scores between various synthetic components and the real ones.","A: synthetic mask vs. real mask, B: synthetic mask vs. real image, C: synthetic background vs. real image, D: synthetic image vs. real image, E: real training image vs. real test image."],["Comparisons with SOTA","Since noisy skeletons can be considered as sparse annotations with a certain degree of noise or directly utilized as noisy labels, we compare YoloCurvSeg with two categories of methods: 1) WSL methods and 2) noisy label learning (NLL) methods.","The Dice similarity coefficient (DSC) and the average symmetric surface distance (ASSD) are used as the evaluation metrics."],["Comparison with WSL methods","Firstly, we compare YoloCurvSeg with 11 scribble-supervised segmentation methods employing the same skeleton set that we generate: pCE (baseline); random walker pseudo labeling (RW); uncertainty aware self-ensembling and transformation-consistent model (USTM); Scribble2Label (S2L); Mumford–shah Loss (MLoss); entropy minimization (EM); dense CRF loss and gated CRF loss; active contour loss (AC); dual-branch network with dynamically mixed pseudo labels supervision (DBDM) and tree energy loss, the results of which are shown in Table REF .","For fair comparisons, YoloCurvSeg does not go through the fine stage and is denoted as Ours (coarse) (i.e., the performance of $S_{coarse}$ ).","The upper part of the table indicates that all training data are sparsely labeled, while the lower part indicates that only one representative sample is labeled and all other data are unlabeled and not utilized.","YoloCurvSeg achieves the best performance on all datasets under both settings, outperforming other WSL methods by large margins.","Comparing \"All\" versus \"One\", apparently, YoloCurvSeg is not sensitive to the sample size of the labeled data and achieves 96.1%, 106.3%, 95.3% and 96.2% fully-supervised performance on the four datasets with only 0.14%, 0.03%, 1.40% and 0.65% labeled pixels in terms of DSC.","Representative visualization results are shown in Fig.","REF .","Figure: t-SNE visualization of the four real and synthetic datasets.","CORN good and CORN poor respectively denote the high-quality and low-quality subsets in CORN."],["Comparison with NLL methods","In Table REF , we also compare YoloCurvSeg with several NLL methods, including generalized cross-entropy loss (GCE), co-teaching (COT), TriNet, confident learning with spatial label smoothing (CLS LS) and divergence-aware selective training (DAST) on OCTA500 and DRIVE.","Despite utilizing one full mask and multiple or even all skeleton samples, these methods are inferior to YoloCurvSeg which employs only one skeleton sample.","We also find that all NLL methods perform worse than the fully-supervised model (FS in Table REF ) trained solely with the same single fully labeled sample, illustrating that additional noisily labeled samples are not beneficial to model performance under such noisy conditions.","Table: Comparison with NLL methods on OCTA500 and DRIVE.","M and S respectively indicate full mask and noisy skeleton.","FS denotes fully-supervised learning."],["Robustness Analysis and Ablation Study","To verify the robustness of YoloCurvSeg for the selected one-shot sparsely labeled sample, we randomly select 10 samples from each dataset and compare the performance with the fully-supervised model trained on the same sample.","As demonstrated in Fig.","REF , YoloCurvSeg exceeds full supervision in almost all cases and delivers highly stable performance decoupled from image/annotation quality which nevertheless induces great fluctuations in the performance of the fully-supervised models.","In addition to robustness, the predictions from YoloCurvSeg also have smaller variances.","Both aspects indicate that YoloCurvSeg is sample-insensitive and can reduce the risk of selecting a wrong sample to label.","As for the ablation study, we first remove the background bank extracted by the Inpainter and perform direct curve-to-image translation.","As shown in panel (a) of Fig.","REF , the synthetic images present unrealistic background texture due to the large gap between pre- and post-translation distributions (which is also shown in column B of Table REF ).","For high-resolution image datasets, the foregrounds of the synthetic images are distorted and fail to spatially align with the corresponding curve masks, which also occurs when we remove $\\mathcal {L}_c$ of the Synthesizer (and use CycleGAN [67] for substitution), as shown in Fig.","REF (b).","This indicates that the contrastive Synthesizer (especially $\\mathcal {L}_c$ ) is crucial for maintaining the corresponding local context at the same spatial location.","Figure: Performance of YoloCurvSeg given different labeled samples.Figure: Qualitative visualization of representative results from our S coarse S_{coarse} and other SOTA WSL methods under one-shot setting.It is worth pointing out that the results we have reported in the previous section represent the performance of $S_{coarse}$ .","We also explore the performance of $S_{fine}$ with and without utilizing $\\mathcal {D}_{syn}$ .","It can be observed in Fig.","REF that the performance of $S_{fine}$ is further improved by utilizing $\\mathcal {D}_{syn}$ , which may be attributed to the fact that the synthetic curves have high degrees of continuity and can reduce the model's outlier predictions.","We also conduct a performance comparison of the fully-supervised model with and without pretraining on $\\mathcal {D}_{syn}$ .","Results show that the synthetic images from YoloCurvSeg also have great potential in serving as pretraining images; pretraining on $\\mathcal {D}_{syn}$ increases DSC by 0.55, 0.63, 0.81 and decreases ASSD by 0.061, 0.041, 0.148, respectively on OCTA500, DRIVE and CHASEDB1.","Ultimately, via further utilizing an additional unlabeled dataset $\\mathcal {D}_{ori}$ , YoloCurvSeg achieves 97.00%, 97.49% and 97.63% of the fully-supervised performance (with the full masks of all samples employed) with only one noisy skeleton annotation on those three datasets.","Figure: Visualization of representative synthetic results from the ablation study.","Arrows mark the unrealistic background regions, structures, or misalignments between the masks and the regions of interest.Figure: Performance of YoloCurvSeg under different training paradigms.","FS denotes fully-supervised learning.","This paper presents a novel sparsely annotated segmentation framework for curvilinear structures, named YoloCurvSeg.","Extensive experiments are conducted on four publicly accessible datasets, with superiority of our proposed framework being successfully established.","Potential future directions are transferring YoloCurvSeg to 3D scenarios and exploring a better pipeline to further reduce the domain gap between synthetic images and real images."]],"string":"[\n [\n \"YoloCurvSeg: You Only Label One Noisy Skeleton for Vessel-style\\n Curvilinear Structure Segmentation\"\n ],\n [\n \"Abstract Weakly-supervised learning (WSL) has been proposed to alleviate the conflict between data annotation cost and model performance through employing sparsely-grained (i.e., point-, box-, scribble-wise) supervision and has shown promising performance, particularly in the image segmentation field.\",\n \"However, it is still a very challenging problem due to the limited supervision, especially when only a small number of labeled samples are available.\",\n \"Additionally, almost all existing WSL segmentation methods are designed for star-convex structures which are very different from curvilinear structures such as vessels and nerves.\",\n \"In this paper, we propose a novel sparsely annotated segmentation framework for curvilinear structures, named YoloCurvSeg, based on image synthesis.\",\n \"A background generator delivers image backgrounds that closely match real distributions through inpainting dilated skeletons.\",\n \"The extracted backgrounds are then combined with randomly emulated curves generated by a Space Colonization Algorithm-based foreground generator and through a multilayer patch-wise contrastive learning synthesizer.\",\n \"In this way, a synthetic dataset with both images and curve segmentation labels is obtained, at the cost of only one or a few noisy skeleton annotations.\",\n \"Finally, a segmenter is trained with the generated dataset and possibly an unlabeled dataset.\",\n \"The proposed YoloCurvSeg is evaluated on four publicly available datasets (OCTA500, CORN, DRIVE and CHASEDB1) and the results show that YoloCurvSeg outperforms state-of-the-art WSL segmentation methods by large margins.\",\n \"With only one noisy skeleton annotation (respectively 0.14%, 0.03%, 1.40%, and 0.65% of the full annotation), YoloCurvSeg achieves more than 97% of the fully-supervised performance on each dataset.\",\n \"Code and datasets will be released at https://github.com/llmir/YoloCurvSeg.\"\n ],\n [\n \"Introduction\",\n \"Curvilinear structures are elongated, curved, multi-scale structures that often appear tree-like and are commonly found in natural images (e.g., cracks and aerial road maps) and biomedical images (e.g., vessels, nerves and cell membranes).\",\n \"Automatic and precise segmentation of these curvilinear structures plays a significant role in both computer vision and biomedical image analysis.\",\n \"For example, road mapping serves as a prerequisite in both autonomous driving and urban planning.\",\n \"In the biomedical field, studies [1], [2], [3], [4] have suggested that the morphology and topology of specific curvilinear anatomy (e.g., retinal vessels and corneal nerve fibers) are highly relevant to the presence or severity of various diseases such as hypertension, arteriolosclerosis, keratitis, age-related macular degeneration, diabetic retinopathy, and so on.\",\n \"Retinal vessels are observable in retinal fundus images and optical coherence tomography angiography (OCTA) images, while corneal nerve fibers are identifiable in confocal corneal microscopy (CCM) images.\",\n \"It has been suggested that early signs of many ophthalmic diseases are reflected by microvascular and capillary abnormalities [5], [6].\",\n \"Collectively, accurate segmentation of various curvilinear structures is of great importance for computer-aided diagnosis, quantitative analysis and early screening, especially in ophthalmology.\",\n \"In recent years, benefiting from the development of deep learning (DL), many DL-based segmentation algorithms for curvilinear structures have been proposed and have shown overwhelming performance compared to traditional (e.g., matched filter-based and morphological processing-based [7], [8]) methods.\",\n \"Most existing works are dedicated to designing sophisticated network architectures [9], [10], [11] or deploying strategies to preserve curvilinear structures' topology by employing generative adversarial networks (GANs) [2], [12] or topology-preserving loss functions [13], [14].\",\n \"These methods are typically fully-supervised, wherein large-scale well-annotated datasets are required.\",\n \"However, collecting and labeling a large-scale dataset with full annotation is very costly and time-consuming, particularly for medical images since their annotation requires expert knowledge and clinical experience.\",\n \"Furthermore, annotating curvilinear structures is even more challenging, given that curvilinear structures are slender, multi-scale, and complex in shape with fine details.\",\n \"More recently, many efforts have been made to reduce the annotation cost for DL model training.\",\n \"For example, semi-supervised learning (SSL) trains models by combining limited amounts of annotated data with massive unlabeled data [15], [16], [17].\",\n \"While effective, most state-of-the-art (SOTA) SSL methods still require about 5%-30% of the accurately and precisely labeled data to achieve about 85%-95% of the fully-supervised performance, which is still not sufficiently cost-effective and still time-consuming when it comes to labeling curvilinear structures.\",\n \"Weakly supervised learning (WSL) attempts to alleviate the annotation issue from another perspective by performing sparsely-grained (i.e., point-, scribble-, bounding box-wise) supervision and attains promising performance [18], [19], [20], [21], [22].\",\n \"Compared with either point or bounding box, scribble is a relatively more flexible and generalizable form of sparse annotation that can be used to annotate complex structures [23].\",\n \"Existing scribble-supervised segmentation methods mainly fall into two categories.\",\n \"The first line of research exploits structural or volumetric priors to expand scribbles into more accurate pseudo proposals; for example, grouping pixels with similar grayscale intensities or locations into the same class [18], [19], [24].\",\n \"However, the expansion process may introduce noisy proposals, which may induce error accumulation and deteriorate the performance of the segmentation model.\",\n \"Some work [25] also points out the inherent weakness of these methods, namely models retain their own predictions and thus resist updating.\",\n \"The second line learns adversarial shape priors utilizing extra unpaired but fully-annotated masks.\",\n \"Such approaches somewhat contradict the motivation of saving annotation costs, especially for complex curvilinear structures [26], [27], [28].\",\n \"Moreover, most WSL methods still require sparsely labeling the entire dataset (or a large portion), and they are mainly designed and validated on relatively simple structures (e.g., cardiac structures or abdominal organs) with assumptions and priors that may not apply to complex structures (e.g., curvilinear structures).\",\n \"To address these aforementioned challenges, we here present a novel WSL segmentation framework for vessel-style curvilinear structures, namely You Only Label One Noisy Skeleton for Curvilinear Structure Segmentation (YoloCurvSeg).\",\n \"For curvilinear structures, label noises/errors are inevitable, and a good segmentation approach should be noise tolerant.\",\n \"Therefore, instead of utilizing only the annotated pixels for supervision, YoloCurvSeg ingeniously converts the weakly-supervised problem into a fully- or semi-supervised one via image synthesis.\",\n \"It employs a trained inpainting network as a background generator, which takes one (or multiple depending on availability) noisy skeleton (as shown in Fig.\",\n \"REF ) and dilates it to serve as an inpainting mask to obtain a background that closely matches the real distribution.\",\n \"The extracted background is then augmented and combined with randomly emulated curves generated by a Space Colonization Algorithm-based foreground generator, from which a synthetic dataset is obtained through a multilayer patch-wise contrastive learning synthesizer.\",\n \"Finally, a segmenter performs coarse-to-fine two-stage training using the synthetic dataset and an unlabeled dataset (if available).\",\n \"Our main contributions are summarized as follows: We propose a novel weakly-supervised framework for one-shot skeleton/scribble-supervised curvilinear structure segmentation, namely YoloCurvSeg.\",\n \"To the best of our knowledge, YoloCurvSeg is the first sparsely-annotated and weakly-supervised segmentation method for curvilinear structures.\",\n \"YoloCurvSeg novelly converts a WSL problem into a fully supervised one through four steps: curve generation, image inpainting, image translation and coarse-to-fine segmentation.\",\n \"The proposed framework is noise-robust, sample-insensitive and easily extensible to various curvilinear structures.\",\n \"We evaluate YoloCurvSeg on four challenging curvilinear structure segmentation datasets, namely OCTA500 [29], CORN [30], DRIVE [31], CHASEDB1 [32].\",\n \"Experimental results show that YoloCurvSeg outperforms SOTA WSL and noisy label learning methods by large margins.\",\n \"Meanwhile, we demonstrate that $>$ 97% of the fully-supervised performance can be achieved with only one noisy skeleton label (approximately 0.1% or 1% of the full annotation), which shall also inspire subsequent works on WSL and curvilinear dataset construction.\",\n \"Existing automatic curvilinear structure segmentation algorithms can be roughly divided into two categories.\",\n \"The first category is traditional unsupervised methods, mainly including mathematical morphology methods and various filtering methods [10].\",\n \"For instance, Zana et al.\",\n \"[33] segment vascular-like patterns using a hybrid framework of morphological filtering and cross-curvature analysis.\",\n \"Passat et al.\",\n \"[34] present a preliminary approach to strengthen the segmentation of cerebral vessels by incorporating high-level anatomical knowledge into the segmentation process.\",\n \"Filtering methods include Hessian matrix-based filters [35], matched filters [8], [36], multi-oriented filters [37], symmetry filter [38], etc.\",\n \"The other category is supervised methods, wherein data with ground truth labels are used to train segmenters based on predefined or model-extracted features.\",\n \"Traditional machine-learning-based approaches are dedicated to pixel-level classification using handcrafted features [39], [40].\",\n \"Recently, DL-based approaches have made significant progress in various segmentation tasks.\",\n \"For example, Ronneberger et al.\",\n \"[41] propose U-Net, which has been widely used in numerous medical image segmentation tasks.\",\n \"Existing curvilinear structure segmentation works focus on well-designed network architectures by introducing multi-scale [11], [42], multi-task [6], [9], [43], or various attention mechanisms [10], [44] and well-playing morphological and topological properties by introducing GANs or morphology-/topology-preserving loss functions [13], [14].\",\n \"Still, data availability and annotation quality are the main limitations of these methods.\"\n ],\n [\n \"Weakly-supervised Segmentation\",\n \"Weakly-supervised segmentation aims to reduce the labeling costs by training segmentation models on data annotated with coarse granularity [18].\",\n \"Among various formats of sparse annotations, scribble is recognized as the most flexible and versatile one that can be used to annotate complex structures [23], [27].\",\n \"Existing scribble-supervised segmentation methods fall into two main categories.\",\n \"The first one exploits structural or volumetric priors to expand scribble annotations by assigning a same class to pixels with similar intensities or nearby locations [18], [19], [24].\",\n \"The main limitation of such approaches is that they heavily rely on pseudo proposals and often contain multiple stages, which can be time-consuming and prone to errors that may be propagated during model training.\",\n \"The second category learns adversarial shape priors utilizing extra unpaired but fully-annotated masks.\",\n \"Such approaches somewhat contradict the motivation of saving annotation costs, especially for complex curvilinear structures [26], [27], [28].\",\n \"Additionally, these methods still require sparsely labeling the entire dataset or a large portion, and they are mainly designed and validated on relatively simple structures like cardiac structures or abdominal organs with assumptions and priors that may not apply to complex structures (e.g., curvilinear ones).\",\n \"In this paper, we make use of noisy skeletons that differ from scribbles in two ways: (1) skeletons are more label demanding since all branches are supposed to be covered; (2) they are more likely to contain errors or noises, which are inevitable when quickly labeling slender structures.\",\n \"We convert sparse and noisy skeleton annotations to accurate ones via an image synthesis pipeline, thus requiring only one noisy skeleton label.\",\n \"This significantly reduces the annotation cost.\"\n ],\n [\n \"Medical Image Synthesis\",\n \"GAN [45] has become the mainstay of medical image synthesis, with common applications in intra-modality augmentation [46], cross-domain image-to-image translation [47], quality enhancement [48], missing modality generation [49], etc.\",\n \"Below we briefly review previous works on retinal image synthesis, the topic of which is relevant to our work.\",\n \"Costa et al.\",\n \"[50] employ a U-Net trained with paired fundus images and vessel masks.\",\n \"It employs a conditional GAN (Pix2pix [52]) to learn a mapping from vessel masks to the corresponding fundus images.\",\n \"To simplify the framework, they propose an adversarial autoencoder (AAE) for retinal vascularity synthesis and a GAN for generating retinal images [51].\",\n \"Similarly, Guibas et al.\",\n \"[53] present a two-stage approach that consists of a DCGAN for generating vasculature from noise and a cGAN (Pix2pix) to synthesize the corresponding fundus image.\",\n \"These methods require an extra set of vessel annotations to train AAE or DCGAN and may sometimes generate vessels with unrealistic morphology.\",\n \"The generated images also lack diversity.\",\n \"Zhao et al.\",\n \"[54] develop Tub-sGAN, which incorporates style transfer into the GAN framework to generate more diverse outputs.\",\n \"Note that cGAN requires paired images and vessel masks for training, which is a strict condition to some extent.\",\n \"In another work, SkrGAN [55] is proposed to introduce a sketch prior related constraint to guide the image generation process.\",\n \"Yet, the sketches utilized are extracted by the Sobel edge operator, and cannot be used as segmentation masks.\",\n \"In this paper, we employ a multilayer patch-wise contrastive foreground-background fusion GAN for several considerations.\",\n \"According to previous research, training a GAN to learn a direct mapping from a curvilinear structure mask to the corresponding image is difficult, especially under few-shot conditions [56].\",\n \"Therefore, we provide GAN with extracted real backgrounds, enabling implicit skip-connection that allows the GAN to focus more on mapping the foreground regions.\",\n \"Such a design not only enhances performance but also accelerates convergence.\",\n \"Multilayer patch-wise contrastive learning allows the provided mask and the foreground region of the generated image to be spatially aligned (via unpaired training), which further benefits the subsequent segmenter.\",\n \"YoloCurvSeg comprises four main components: (1) a Curve Generator that produces binary curve masks that well accommodate the corresponding image modality of interest; (2) an Inpainter for extracting backgrounds from labeled samples; (3) a Synthesizer that synthesizes images from the generated curve masks and the image backgrounds; and (4) a two-stage Segmenter trained with the synthetic dataset and an unlabeled dataset.\",\n \"The overall framework is shown in Fig.\",\n \"REF .\",\n \"Figure: Overview of our proposed YoloCurvSeg, which comprises four main components: a space colonization algorithm-based curve generator, a background inpainter, a multilayer patch-wise contrastive foreground-background fusion synthesizer, and a two-stage coarse-to-fine segmenter.\"\n ],\n [\n \"Curvilinear Structure Generation\",\n \"Space colonization is a procedural modeling algorithm in computer graphics that simulates the growth of branching networks or tree-like structures [57], [58], including vasculature, leaf venations, root systems, etc.\",\n \"It is employed in YoloCurvSeg for modeling the iterative growth of curvilinear structures with two fundamental elements: attractors and nodes.\",\n \"Its core steps are described in the top left panel of Fig.\",\n \"REF , wherein blue dots denote attractors and black ones denote nodes: a) place a set of attractors randomly or following a predefined pattern, and then associate nodes with nearby attractors (if their distance is within an attraction distance $D_a$ ); b) for each node, calculate its average direction from all attractors affecting it; c) calculate the position of new nodes via normalizing the average direction to a unit vector and scaling it by a pre-defined segment length $L_s$ ; d) place nodes at the calculated positions and check if any nodes are within an attractor's kill zone; e) prune an attractor if there are nodes staying within its kill distance $D_k$ ; f) repeat steps b)-e) until the maximum number of nodes is reached.\",\n \"Through observing the pattern of the foreground/curve in a single image or a few images that are accessible, including the curves' starting point, boundary, and degree of curvature, etc., it is relatively straightforward to set the corresponding hyperparameters, such as the root node coordinates $C_r$ (e.g., the starting point of the vessels in fundus lies in the optic disc region), as well as the bounds and obstacles.\",\n \"For $D_a$ , $D_k$ and $L_s$ , the commonly used values of 5, 30 and 5 can be re-tuned as needed.\",\n \"Table REF summarizes the parameters and post-processing operations we employ for generating the four types of curves and representative examples are demonstrated in the top panel of Fig.\",\n \"REF .\",\n \"Please note that our adopted settings and post-processing operations only represent our empirical choices and are not necessarily the best-performing ones; users can make further adjustments based on their observations and experiences.\",\n \"In addition to the curvilinear shape, we also need to simulate the thickness of each branch $R^n = R_1^n + R_2^n,$ where $R$ , $R_1$ and $R_2$ respectively denote the radii of a father branch and its two child branches.\",\n \"$n$ is set to be 3 according to Murray's law [59].\",\n \"The calculation is performed recursively from the branch tips (whose radii are set to be 1) towards the tree base.\",\n \"Several intuitive demos can be accessed at linkhttps://jasonwebb.github.io/2d-space-colonization-experiments/.\",\n \"By setting random grid attractors and root nodes via predefined parameters, we construct four curve banks for our four datasets of interest, which are then utilized for training the synthesizers and the segmenters.\",\n \"Table: parameters for generating different types of curves.\",\n \"D a D_a, D k D_k, L s L_s and C r C_r respectively denote the attraction distance, kill distance, segment length and root node coordinates.\",\n \"rr and ll denote radius and length.\",\n \"⨁\\\\bigoplus and ⨀\\\\bigodot represent the union and element-wise multiplication operations.\",\n \"there are two rows for OCTA500 since two components are needed.\"\n ],\n [\n \"Inpainting for Background Extraction\",\n \"Inpainting is the task of reconstructing missing or masked regions in an image.\",\n \"Similar to removing watermarks or extraneous pedestrians from images, we employ an inpainting model here to remove foregrounds (e.g., vessels and nerve fibers) from the images of interest, under the hypothesis that the dilated noisy skeletons can fully cover the foregrounds.\",\n \"In inpainting, common concerns are the network's ability to grasp local and global context and to generalize to a different (especially higher) resolution.\"\n ],\n [\n \"Architecture\",\n \"Inspired by [60], we adopt an inpainting network based on the recently proposed fast Fourier convolutions (FFCs) [61] with image-wide receptive fields, strong generalizability and relatively few parameters.\",\n \"Given a masked image $I \\\\odot (1-m)$ , where $I$ and $m$ respectively denote the original image and the binary mask of inpainting regions, the feed-forward inpainting network $f_\\\\theta (\\\\cdot )$ aims to output an inpainted image $\\\\hat{I}$ = $f_\\\\theta (I^{\\\\prime })$ taking a four-channel input $I^{\\\\prime }=\\\\operatorname{concat}(I\\\\odot (1-m),m)$ .\",\n \"FFC builds its basis on channel-wise fast Fourier transform (FFT) and has a receptive field covering the whole image.\",\n \"It splits channels into two parallel branches: a local branch uses conventional convolutions and a global branch uses real FFT to capture global context, as shown in Fig.\",\n \"REF .\",\n \"real FFT is only applicable to real-valued signals, and inverse real FFT ensures the output is real-valued.\",\n \"Compared to FFT, real FFT uses only half of the spectrum.\",\n \"In FFC, real FFT is first applied to the input tensor and a ComplexToReal operation is performed by concatenating the real and imaginary parts.\",\n \"Then, it applies convolutions in the frequency domain.\",\n \"Inverse real FFT is performed to transform features from the frequency domain to the spatial domain through the RealToComplex operation.\",\n \"Finally, the local and global branches are fused.\",\n \"For the upsampling and downsampling of the Inpainter and the discriminator architecture in adversarial training, we follow the ResNet settings respectively employed in [62] and [60].\",\n \"The training is performed on [image, randomly synthesized mask] pairs.\",\n \"We adopt the mask generation strategy in [60], containing multiple rectangles with arbitrary aspect ratios and wide polygonal chains.\"\n ],\n [\n \"Objective\",\n \"Compared with naive supervised losses which may result in blurry predictions, perceptual loss [63] evaluates the distance between feature maps of the inpainted image and the original image via a pre-trained network $\\\\phi (\\\\cdot )$ .\",\n \"It does not require exact reconstruction and allows for variations in the reconstructed image.\",\n \"Given that inpainting focuses on understanding the global structure, we introduce a perceptual loss of a large receptive field $\\\\mathcal {L}_{HRP}$ through a pre-trained ResNet50 $\\\\phi _{HRF}(\\\\cdot )$ with dilated convolutions provided in [60] $\\\\mathcal {L}_{HRP}(I, \\\\hat{I})=\\\\mathcal {M}([\\\\phi _{HRF}(I)-\\\\phi _{HRF}(\\\\hat{I})]^2),$ where $\\\\mathcal {M}$ is a sequential two-stage mean operation, i.e., obtaining the inter-layer mean of intra-layer means.\",\n \"Additionally, an adversarial loss $\\\\mathcal {L}_{adv}$ is utilized to encourage the inpainted image to be realistic.\",\n \"Specifically, we use a PatchGAN [52] discriminator $\\\\mathcal {D}_{\\\\xi }(\\\\cdot )$ and label patches that overlap with the mask as fake and the others as real.\",\n \"The non-saturating adversarial loss is defined as ${\\\\begin{array}{c}\\\\mathcal {L}_D=-E_I[\\\\log D_{\\\\xi }(I)]-E_{I, m}[\\\\log D_{\\\\xi }(\\\\hat{I}) \\\\odot (1-m)] \\\\\\\\-E_{I, m}[\\\\log (1-D_{\\\\xi }(\\\\hat{I})) \\\\odot m],\\\\end{array}}$ $\\\\mathcal {L}_G=-E_{I, m}[\\\\log D_{\\\\xi }(\\\\hat{I})],$ $\\\\mathcal {L}_{adv}=\\\\operatorname{sg}_\\\\theta (\\\\mathcal {L}_D)+\\\\operatorname{sg}_{\\\\xi }(\\\\mathcal {L}_G) \\\\rightarrow \\\\min _{\\\\theta , \\\\xi },\\\\vspace{-2.84544pt}$ where $\\\\hat{I} = f_\\\\theta (I^\\\\prime )$ is the output of the inpainting network and $\\\\operatorname{sg}_{var}$ represents stop gradient w.r.t.\",\n \"$var$ .\",\n \"To further stabilize the training, we use a gradient penalty $\\\\mathcal {L}_{GP} = E_I\\\\Vert \\\\nabla D_{\\\\xi }(I)\\\\Vert _2^2$ [64] and a perceptual loss defined on features of the discriminator $\\\\mathcal {L}_{DP}$ [65].\",\n \"The final objective of the Inpainter is $\\\\mathcal {L}_{inpaint}=\\\\mathcal {L}_{HRP}+\\\\lambda _{adv}\\\\mathcal {L}_{adv}+\\\\lambda _{DP}\\\\mathcal {L}_{DP}+\\\\lambda _{GP}\\\\mathcal {L}_{GP},$ where $\\\\lambda _{adv}$ , $\\\\lambda _{DP}$ and $\\\\lambda _{GP}$ are hyper-parameters balancing the contributions of different losses.\",\n \"$\\\\mathcal {L}_{HRP}$ is responsible for supervised signals and global structure consistency while $\\\\mathcal {L}_{adv}$ and $\\\\mathcal {L}_{DP}$ are responsible for local details and realism.\",\n \"Figure: The architecture of the Inpainter.\",\n \"The input is a four-channel image with the first three channels being the original image and the last channel being the binary mask of the inpainting regions.\",\n \"The output is the inpainted image.\",\n \"The dimensional change of the feature map in FFC is shown in the lower right panel.\"\n ],\n [\n \"Training\",\n \"Given that the training of the Inpainter does not require annotation and it learns a general ability to recover missing regions through contextual understanding, we initialize the model with pre-trained parameters from the Places-Challenge dataset [66] and fine-tune it on images accessible within each training set.\",\n \"The validation set consists of both accessible training set images and validation set images (paired with predefined masks obtained using a specific generation strategy [60]).\",\n \"The training is conducted with a batch size of 8 and an Adam optimizer is adopted with a learning rate of $10^{-3}$ for 50 epochs.\",\n \"The training data are augmented through random flipping, rotation and color jittering.\",\n \"We empirically set $\\\\lambda _{adv}=3$ , $\\\\lambda _{DP}=10$ and $\\\\lambda _{GP}=10^{-4}$ .\",\n \"Once trained, the Inpainter is used to remove the foregrounds from the skeleton-labeled samples taking the dilated noisy annotations as the masks.\",\n \"Then we construct a background bank for each dataset by augmenting the extracted backgrounds through random flipping, rotation, etc.\",\n \"Now we have a curve (foreground) bank $B_{curv} = \\\\lbrace c^{1}, \\\\cdots , c^{N}\\\\rbrace $ and a background bank $B_{bg} = \\\\lbrace b^{1}, \\\\cdots , b^{N}\\\\rbrace $ respectively from the Curve Generator and the Inpainter, for each given dataset.\",\n \"We construct an intermediate dataset $X_{inter} = \\\\lbrace x^{1}, \\\\cdots , x^{N}\\\\rbrace $ through randomly sampling a curve $c^{i}$ from $B_{curv}$ and a background $b^{i}$ from $B_{bg}$ , and then concatenating them to form a temporary sample $x^{i} = \\\\operatorname{concat}(b^{i},c^{i})$ .\",\n \"The problem now turns into an unpaired image-to-image translation task, i.e., designing a synthesizer to learn a mapping from $X_{inter}$ to the corresponding real dataset $Y$ .\",\n \"It is desirable that the local context especially the foreground of the synthetic image $\\\\hat{y}^{i}$ is spatially aligned with that of the corresponding intermediate image $x^{i}$ (especially $c^{i}$ ) as much as possible.\",\n \"Previously, for unpaired image translation, most existing methods apply GANs with a cycle structure, relying on cycle-consistency to ensure high-level correspondence [67].\",\n \"While effective, the underlying bijective assumption behind cycle-consistency is sometimes too restrictive, which may reduce the diversity of the generated samples.\",\n \"More importantly, the cycle-consistency is not suitable for our task since it does not guarantee any explicit or implicit spatial constraints.\",\n \"In such context, we introduce a multilayer patch-wise contrastive learning based synthesizer to learn a mapping from $X_{inter}$ to $Y$ inspired by [68], [69], which is illustrated in the lower middle panel of Fig.\",\n \"REF .\",\n \"It is trained in a generative adversarial manner with an internal contrastive learning pretext task.\",\n \"The generator (i.e., synthesizer) $G$ is a U-shape network, which firstly down-samples the input image into high-level features via an encoder $E$ with three residual blocks equipped with instance normalization and ReLU activation.\",\n \"As such, each pixel in the high-level feature map represents the embedding feature vector of a patch in the original image.\",\n \"Several layers of interest $E_{l \\\\in L}(x)$ in $E$ are selected to extract multi-scale features of patches and each passes through a two-layer multilayer perceptron (MLP) $H_l$ ($l$ indexes a layer), obtaining a feature stack $\\\\lbrace v_{l \\\\in L}=H_{l \\\\in L}[E_{l \\\\in L}(x)]\\\\rbrace $ .\",\n \"Given patchwise features $v_{l}$ and the corresponding pair $\\\\lbrace H_l(E_l(x))^{s_1}, H_l(E_l(G(x)))^{s_2}\\\\rbrace $ with $s_1$ and $s_2$ denoting the spatial locations of the patches of interest, we set $v^+$ to represent a patch at the same location as $v$ and $v^-_n$ to denote the $n^{th}$ among $N$ patches at different locations.\",\n \"The objective of the contrastive learning task is to maintain the local information at the same spatial location.\",\n \"Similar to the noise contrastive estimation loss [70], our objective function can be written as $\\\\hspace{-5.406pt}\\\\mathcal {L}_{c}=-\\\\sum _{l \\\\in L} \\\\log \\\\frac{\\\\exp (v_l \\\\cdot v_l^{+} / \\\\tau )}{\\\\exp (v_l \\\\cdot v_l^{+} / \\\\tau )+\\\\sum _{n=1}^N \\\\exp (v_l \\\\cdot v_{l n}^{-} / \\\\tau )},$ where $\\\\tau $ is a temperature hyper-parameter.\",\n \"Besides, we employ the identity loss, which was first proposed in [67] for regularizing the generator $G$ .\",\n \"We pass each real sample $y \\\\in Y$ through the encoder $E$ and obtain the patchwise features $v^*$ , the negative samples $v^{*-}$ and the positive samples $v_n^{*+}$ .\",\n \"The identity loss is formulated as $\\\\hspace{-2.84526pt}\\\\mathcal {L}_{id}=-\\\\sum _{l \\\\in L} \\\\log \\\\frac{\\\\exp (v_l^* \\\\cdot v_l^{*+} / \\\\tau )}{\\\\exp (v_l^* \\\\cdot v_l^{*+} / \\\\tau )+\\\\sum _{n=1}^N \\\\exp (v_l^* \\\\cdot v_{l n}^{*-} / \\\\tau )}.$ We use the LSGAN loss as our adversarial loss $\\\\mathcal {L}_{adv}$ [71] to make the synthetic images as realistic as possible.\",\n \"Therefore, with trade-off parameters $\\\\lambda _{adv}$ , $\\\\lambda _{c}$ and $\\\\lambda _{id}$ , the overall loss of the synthesizer is defined as $\\\\mathcal {L}_{syn} = \\\\lambda _{adv}\\\\mathcal {L}_{adv} + \\\\lambda _{c}\\\\mathcal {L}_{c} + \\\\lambda _{id}\\\\mathcal {L}_{id}.$ The training of the synthesizer is conducted employing an Adam optimizer with a learning rate of $10^{-4}$ and a cosine decay strategy, together with a batch size of 1.\",\n \"We utilize images that are accessible within each training set as the corresponding real dataset $Y$ for training the synthesizer.\",\n \"The hyperparameters and model weights are selected based on the Fréchet Inception Distances (FIDs) [76] between the synthesized images and $Y$ .\",\n \"We set $\\\\lambda _{adv}$ as 1, $\\\\lambda _{c}$ as 1, $\\\\lambda _{id}$ as 0.5 and $\\\\tau $ as 0.07.\",\n \"The training process lasted for 300 epochs.\"\n ],\n [\n \"Two-stage Coarse-to-Fine Segmentation\",\n \"A synthetic dataset $\\\\mathcal {D}_{syn} = \\\\lbrace (\\\\hat{y}^{1},c^1),\\\\cdots ,(\\\\hat{y}^{N},c^N)\\\\rbrace $ , with $\\\\hat{y}^{i}$ being a synthetic image and $c^i$ being the corresponding curve ground truth, is created by the Synthesizer.\",\n \"The weakly-supervised task is then transformed into a fully- or semi- one when making use of solely the synthetic dataset or a combination of an unlabeled dataset $\\\\mathcal {D}_{ori}$ and the synthetic dataset $\\\\mathcal {D}_{syn}$ .\",\n \"In this section, we introduce a two-stage coarse-to-fine segmentation pipeline to tackle the task.\",\n \"A specific segmentation network is first trained on $\\\\mathcal {D}_{syn}$ to obtain a coarse model $S_{coarse}$ with a segmentation loss $\\\\mathcal {L}_{seg}$ $\\\\hspace{-5.406pt}\\\\mathcal {L}_{seg} = 0.5\\\\times \\\\mathcal {L}_{ce} + 0.5\\\\times \\\\mathcal {L}_{dice},$ where $\\\\mathcal {L}_{ce}$ and $\\\\mathcal {L}_{dice}$ respectively denote the cross-entropy loss and the Dice loss.\",\n \"We observe and conclude that the performance of $S_{coarse}$ is mainly limited by two issues; one is that the curve generated by the Curve Generator still has a certain morphological gap with the foreground of the real image, and the other one is that there is also a slight but inevitable intensity gap between the Synthesizer-generated image and the real image.\",\n \"We target at relieving the latter issue by making use of $\\\\mathcal {D}_{ori}$ to further boost the segmentation performance.\",\n \"We employ predictions on $\\\\mathcal {D}_{ori}$ from $S_{coarse}$ as pseudo-labels and train a fine model $S_{fine}$ (initialized on $S_{coarse}$ ) on the combination of $\\\\mathcal {D}_{ori}$ and $\\\\mathcal {D}_{syn}$ via a final loss $\\\\mathcal {L}_{final}$ which is formulated as $\\\\mathcal {L}_{final} = \\\\mathcal {L}_{seg} + \\\\lambda _{psd}\\\\mathcal {L}_{psd},$ where $\\\\mathcal {L}_{psd}$ denotes the loss on $\\\\mathcal {D}_{ori}$ sharing the same formulation as $\\\\mathcal {L}_{seg}$ in Eq.\",\n \"(REF ) and $\\\\lambda _{psd}$ is a trade-off parameter.\",\n \"We employ vanilla U-Net with feature channels of 16, 32, 64, 128 and 256 as our $S_{coarse}$ 's and $S_{fine}$ 's architecture.\",\n \"We use an SGD optimizer (weight decay = $10^{-4}$ , momentum = 0.9) for training both $S_{coarse}$ and $S_{fine}$ with a batch size of 12 and an initial learning rate of $10^{-2}$ .\",\n \"The total iterations and $\\\\lambda _{psd}$ are respectively set to be 30$k$ and 1.\",\n \"In this section, we extensively evaluate the effectiveness of our YoloCurvSeg framework on four representative curvilinear structure segmentation datasets.\"\n ],\n [\n \"Datasets and Preprocessing\",\n \"We comprehensively evaluate YoloCurvSeg on four ophthalmic datasets: OCTA500, CORN, DRIVE and CHASEDB1.\",\n \"OCTA500 is used for retinal microvascular segmentation, and only the subset that contains 300 samples with a $6\\\\times 6$ $mm^2$ field of view (FOV) and a $400\\\\times 400$ resolution is utilized.\",\n \"We only make use of the $en$ -$face$ images generated by maximum projection between the internal limiting membrane layer and the outer plexiform layer.\",\n \"CORN consists of 1578 CCM images for nerve fiber segmentation.\",\n \"It also provides two subsets respectively consisting of 340 low-quality and 288 high-quality images.\",\n \"All CCM images have a resolution of $384\\\\times 384$ and an FOV of $400\\\\times 400$ $\\\\mu m^2$ .\",\n \"Instead of following the dataset's original division, we use 1532 images (samples that overlap with the test set are removed and the validation split ratio is 0.2) for training and validation, and test on 60 relatively accurately labeled samples provided in its subset.\",\n \"DRIVE and CHASEDB1 are used for retinal vessel segmentation and respectively have resolutions of $565\\\\times 584$ and $999\\\\times 960$ .\",\n \"These two fundus datasets are cropped via the provided FOV masks and respectively resized to $576\\\\times 576$ and $960\\\\times 960$ .\",\n \"For DRIVE, we utilize the original division of 20 training samples and 20 testing samples.\",\n \"For CHASEDB1, we follow the division in [2], [11], with the first 20 images serving as the training set and the remaining 8 used for testing.\",\n \"For OCTA500, we respectively utilize 200, 10 and 90 samples as the training, validation and testing sets.\",\n \"Images are first normalized and data augmentation consists of random rotation, flipping and Bézier Curve transformation [72].\"\n ],\n [\n \"Implementation Details\",\n \"We implement YoloCurvSeg and other compared methods by PyTorch on a workstation equipped with 8 RTX 3090Ti GPUs.\",\n \"In the Synthesizer, the indices of the layers selected to calculate $\\\\mathcal {L}_c$ include $\\\\lbrace 0,4,8,12,16\\\\rbrace $ .\",\n \"For training the Segmenters $S_{coarse}$ and $S_{fine}$ , the polynomial policy with $power=0.9$ is used to adjust the learning rate online [73].\",\n \"Other hyperparameters, training details and model architecture are already provided in previous sections.\",\n \"It is worth noting that manually-delineated vessel segmentation labels are provided for OCTA500, DRIVE and CHASEDB1.\",\n \"To generate noisy skeleton annotations for those three datasets, we perform the skeletonize operation in scikit-image [74] to obtain the skeletons of the original ground truth masks and then employ elastic transformation to simulate jitter noises that may be introduced during fast manual labeling.\",\n \"For CORN, only noisy skeleton labels are provided, and thus are directly used in all our experiments.\",\n \"For this dataset, we dilate each skeleton to a 3-pixel width to serve as the full mask in its fully-supervised learning setting, and the same operation is also applied to the testing set annotations.\",\n \"For sparse labels used in other comparative WSL methods, skeletons of the backgrounds are also generated via skeletonization.\",\n \"The synthesis process in YoloCurvSeg can be online or offline.\",\n \"For better reproducibility and fair comparison, we use the offline version in our experiments, i.e., we first generate the synthetic dataset and then train the Segmenters.\",\n \"We generate 1276, 5005, 1240, and 1604 synthetic samples respectively for OCTA500, CORN, DRIVE and CHASEDB1 if all training samples are labeled.\",\n \"If only one sample is labeled, we respectively generate 100, 100, 60, and 80 synthetic samples.\"\n ],\n [\n \"Synthesis Performance\",\n \"Before comparing with SOTA WSL methods, we first qualitatively and quantitatively evaluate the synthesis performance of YoloCurvSeg.\",\n \"We visualize representative examples, in terms of the noisy skeleton labels, dilated masks for inpainting, extracted backgrounds, generated curves and synthesized images, in Fig.\",\n \"REF .\",\n \"It can be observed from the last column that the generated curves well match the synthetic images.\",\n \"Figure: Visualization of synthetic data from YoloCurvSeg.\",\n \"From left to right are examples of the noisy skeleton label, the inflated inpainting mask, the extracted background, the generated foreground, the synthesized image and the generated foreground superimposed on the synthesized image.\",\n \"Zoom in for details.We also compare the intensity distributions of the synthetic datasets with the real ones in Fig.\",\n \"REF , exhibiting high intensity similarities between the synthetic and real images in terms of both background and foreground.\",\n \"From the t-SNE [75] visualization in Fig.\",\n \"REF , the synthetic datasets are generally in line and well mixed with the real ones.\",\n \"In most cases, the synthetic data are even more uniformly and widely distributed, having a similar effect as data augmentation.\",\n \"The FIDs between the synthetic components and the corresponding real ones are tabulated in Table REF to measure the difficulty of learning a mapping between the two distributions.\",\n \"Column A shows the morphological gap between the synthetic and real curvilinear structures.\",\n \"Columns B and C illustrate there is great difficulty in translating directly to the image distribution from foreground masks, while the background image with the foreground removed is less distant from the original distribution.\",\n \"Since YoloCurvSeg incorporates background information, which implicitly acts as skip connections, it reduces the difficulty of image translation and thus can synthesize images close to the real ones even under few-shot settings.\",\n \"Our method achieves competitive FID scores on all four datasets, two of which are even smaller than those between the real training and test sets, as shown in column E of Table REF .\",\n \"Figure: Histograms of the four datasets in terms of the real data (top) and the corresponding synthetic data (bottom).Table: FID scores between various synthetic components and the real ones.\",\n \"A: synthetic mask vs. real mask, B: synthetic mask vs. real image, C: synthetic background vs. real image, D: synthetic image vs. real image, E: real training image vs. real test image.\"\n ],\n [\n \"Comparisons with SOTA\",\n \"Since noisy skeletons can be considered as sparse annotations with a certain degree of noise or directly utilized as noisy labels, we compare YoloCurvSeg with two categories of methods: 1) WSL methods and 2) noisy label learning (NLL) methods.\",\n \"The Dice similarity coefficient (DSC) and the average symmetric surface distance (ASSD) are used as the evaluation metrics.\"\n ],\n [\n \"Comparison with WSL methods\",\n \"Firstly, we compare YoloCurvSeg with 11 scribble-supervised segmentation methods employing the same skeleton set that we generate: pCE (baseline); random walker pseudo labeling (RW); uncertainty aware self-ensembling and transformation-consistent model (USTM); Scribble2Label (S2L); Mumford–shah Loss (MLoss); entropy minimization (EM); dense CRF loss and gated CRF loss; active contour loss (AC); dual-branch network with dynamically mixed pseudo labels supervision (DBDM) and tree energy loss, the results of which are shown in Table REF .\",\n \"For fair comparisons, YoloCurvSeg does not go through the fine stage and is denoted as Ours (coarse) (i.e., the performance of $S_{coarse}$ ).\",\n \"The upper part of the table indicates that all training data are sparsely labeled, while the lower part indicates that only one representative sample is labeled and all other data are unlabeled and not utilized.\",\n \"YoloCurvSeg achieves the best performance on all datasets under both settings, outperforming other WSL methods by large margins.\",\n \"Comparing \\\"All\\\" versus \\\"One\\\", apparently, YoloCurvSeg is not sensitive to the sample size of the labeled data and achieves 96.1%, 106.3%, 95.3% and 96.2% fully-supervised performance on the four datasets with only 0.14%, 0.03%, 1.40% and 0.65% labeled pixels in terms of DSC.\",\n \"Representative visualization results are shown in Fig.\",\n \"REF .\",\n \"Figure: t-SNE visualization of the four real and synthetic datasets.\",\n \"CORN good and CORN poor respectively denote the high-quality and low-quality subsets in CORN.\"\n ],\n [\n \"Comparison with NLL methods\",\n \"In Table REF , we also compare YoloCurvSeg with several NLL methods, including generalized cross-entropy loss (GCE), co-teaching (COT), TriNet, confident learning with spatial label smoothing (CLS LS) and divergence-aware selective training (DAST) on OCTA500 and DRIVE.\",\n \"Despite utilizing one full mask and multiple or even all skeleton samples, these methods are inferior to YoloCurvSeg which employs only one skeleton sample.\",\n \"We also find that all NLL methods perform worse than the fully-supervised model (FS in Table REF ) trained solely with the same single fully labeled sample, illustrating that additional noisily labeled samples are not beneficial to model performance under such noisy conditions.\",\n \"Table: Comparison with NLL methods on OCTA500 and DRIVE.\",\n \"M and S respectively indicate full mask and noisy skeleton.\",\n \"FS denotes fully-supervised learning.\"\n ],\n [\n \"Robustness Analysis and Ablation Study\",\n \"To verify the robustness of YoloCurvSeg for the selected one-shot sparsely labeled sample, we randomly select 10 samples from each dataset and compare the performance with the fully-supervised model trained on the same sample.\",\n \"As demonstrated in Fig.\",\n \"REF , YoloCurvSeg exceeds full supervision in almost all cases and delivers highly stable performance decoupled from image/annotation quality which nevertheless induces great fluctuations in the performance of the fully-supervised models.\",\n \"In addition to robustness, the predictions from YoloCurvSeg also have smaller variances.\",\n \"Both aspects indicate that YoloCurvSeg is sample-insensitive and can reduce the risk of selecting a wrong sample to label.\",\n \"As for the ablation study, we first remove the background bank extracted by the Inpainter and perform direct curve-to-image translation.\",\n \"As shown in panel (a) of Fig.\",\n \"REF , the synthetic images present unrealistic background texture due to the large gap between pre- and post-translation distributions (which is also shown in column B of Table REF ).\",\n \"For high-resolution image datasets, the foregrounds of the synthetic images are distorted and fail to spatially align with the corresponding curve masks, which also occurs when we remove $\\\\mathcal {L}_c$ of the Synthesizer (and use CycleGAN [67] for substitution), as shown in Fig.\",\n \"REF (b).\",\n \"This indicates that the contrastive Synthesizer (especially $\\\\mathcal {L}_c$ ) is crucial for maintaining the corresponding local context at the same spatial location.\",\n \"Figure: Performance of YoloCurvSeg given different labeled samples.Figure: Qualitative visualization of representative results from our S coarse S_{coarse} and other SOTA WSL methods under one-shot setting.It is worth pointing out that the results we have reported in the previous section represent the performance of $S_{coarse}$ .\",\n \"We also explore the performance of $S_{fine}$ with and without utilizing $\\\\mathcal {D}_{syn}$ .\",\n \"It can be observed in Fig.\",\n \"REF that the performance of $S_{fine}$ is further improved by utilizing $\\\\mathcal {D}_{syn}$ , which may be attributed to the fact that the synthetic curves have high degrees of continuity and can reduce the model's outlier predictions.\",\n \"We also conduct a performance comparison of the fully-supervised model with and without pretraining on $\\\\mathcal {D}_{syn}$ .\",\n \"Results show that the synthetic images from YoloCurvSeg also have great potential in serving as pretraining images; pretraining on $\\\\mathcal {D}_{syn}$ increases DSC by 0.55, 0.63, 0.81 and decreases ASSD by 0.061, 0.041, 0.148, respectively on OCTA500, DRIVE and CHASEDB1.\",\n \"Ultimately, via further utilizing an additional unlabeled dataset $\\\\mathcal {D}_{ori}$ , YoloCurvSeg achieves 97.00%, 97.49% and 97.63% of the fully-supervised performance (with the full masks of all samples employed) with only one noisy skeleton annotation on those three datasets.\",\n \"Figure: Visualization of representative synthetic results from the ablation study.\",\n \"Arrows mark the unrealistic background regions, structures, or misalignments between the masks and the regions of interest.Figure: Performance of YoloCurvSeg under different training paradigms.\",\n \"FS denotes fully-supervised learning.\",\n \"This paper presents a novel sparsely annotated segmentation framework for curvilinear structures, named YoloCurvSeg.\",\n \"Extensive experiments are conducted on four publicly accessible datasets, with superiority of our proposed framework being successfully established.\",\n \"Potential future directions are transferring YoloCurvSeg to 3D scenarios and exploring a better pipeline to further reduce the domain gap between synthetic images and real images.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05566"}}},{"rowIdx":94,"cells":{"text":{"kind":"list like","value":[["New examples of twisted Brill-Noether loci I"],["Abstract Our purpose in this paper is to construct new examples of twisted Brill-Noether loci on curves of genus $g\\ge2$.","Many of these examples have negative expected dimension.","We deduce also the existence of a new region in the Brill-Noether map, whose points support non-empty standard Brill-Noether loci."],["Introduction","Our object in this paper is to construct twisted Brill-Noether loci different from those constructed in [12].","In particular, the relevance of many of these examples is that they have negative Brill-Noether numbers, whereas the examples constructed in [12] all have positive Brill-Noether numbers.","We deduce also the existence of a new region in the Brill-Noether map (see below for the definition), the points of which support non-empty Brill-Noether loci, whose existence was not previously known.","This implies that our new examples also differ from those constructed in [12] (see Remark REF (iii)).","In what follows, we shall usually abbreviate Brill-Noether to BN.","Let $C$ be a smooth irreducible projective curve of genus $g\\ge 2$ defined over the complex numbers.","For $i=1,2$ , let $M_i:=M(n_i,d_i)$ denote the moduli space of stable bundles of rank $n_i$ and degree $d_i$ .","Similarly, we write $\\widetilde{M}_i$ for the corresponding moduli space of S-equivalence classes of semistable bundles.","Let $E_2$ be any vector bundle on $C$ of rank $n_2$ and degree $d_2$ .","For any integers $n_1$ , $d_1$ , $k$ with $n_1\\ge 1$ , one can define the twisted BN locus $B(n_1,d_1,k)(E_2):=\\lbrace E_1\\in M_1\\,|\\,h^0(E_1\\otimes E_2)\\ge k\\rbrace .$ The locus $B(n_1,d_1,k)(E_2)$ is a closed subscheme of $M_1$ .","In fact, it is a determinantal locus; it follows that, if $\\beta (n_1,d_1,k)(E_2)\\le n_1^2(g-1)+1$ , then every irreducible component of $B(n_1,d_1,k)(E_2)$ has dimension greater than or equal to the BN number $\\beta (n_1,d_1,k)(E_2):=n_1^2(g-1)+1-k(k-\\chi ),$ where $\\chi :=n_2d_1+n_1d_2-n_1n_2(g-1).$ For this reason, the BN number is often referred to as the expected dimension.","We define also $\\widetilde{B}(n_1,d_1,k)(E_2):=\\lbrace [E_1]\\in \\widetilde{M}_1\\,|\\,h^0(\\operatorname{gr}E_1 \\otimes E_2)\\ge k\\rbrace ,$ where $[E_1]$ denotes the S-equivalence class of $E_1$ and $\\text{gr}E_1$ is the corresponding graded bundle.","This is a closed subscheme of $\\widetilde{M}_1$ , but can have components of dimension less than the BN number.","In the special case $E_2=\\mathcal {O}_C$ , we obtain the standard (untwisted) higher rank BN locus, denoted by $B(n_1,d_1,k)$ , with expected dimension $\\beta (n_1,d_1,k):=n_1^2(g-1)+1-k(k-d_1+n_1(g-1)).$ In particular, $B(1,d_1,k)$ is the classical BN locus $W^{k-1}_{d_1}$ .","In the case $n_1=1$ , twisted BN loci have been studied for some time and include the special case of maximal line subbundles.","A systematic study for arbitrary $n_1$ was started in [12], where, in particular, examples of twisted BN loci were constructed [12] using the technique of [18].","As we shall see in Theorem REF , these have the important feature that the BN number (REF ) is positive.","We can now introduce the loci which we are going to study in this paper.","This is a special case of a more general construction introduced in [12], where it was used primarily to obtain results for $B(n_1,d_1,k)(E_2)$ .","If $\\gcd (n_i,d_i)=1$ for $i=1,2$ , then there exist universal bundles $\\mathcal {U}_i$ on $M_i\\times C$ and, for any integer $k$ , we define the universal twisted BN locus as follows: $B^k(\\mathcal {U}_1,\\mathcal {U}_2):=\\lbrace (E_1,E_2)\\in M_1\\times M_2\\, |\\,h^0(E_1\\otimes E_2)\\ge k\\rbrace .$ In the non-coprime case, we need to lift to étale coverings of $M_1$ and $M_2$ in order to construct “universal” bundles [20].","However (REF ) still makes sense and we shall use the same notation.","The locus $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ is a closed subscheme of $M_1\\times M_2$ ; if $k\\le 0$ , then $B^k(\\mathcal {U}_1,\\mathcal {U}_2)=M_1\\times M_2$ .","The structure of $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ as a determinantal locus allows us to define a BN number (see [12] $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2):=\\dim M_1+\\dim M_2-k(k-\\chi ),$ which may again be referred to as the expected dimension.","Note that $\\chi =\\chi (E_1\\otimes E_2)$ for all $(E_1,E_2)\\in M_1\\times M_2$ .","If $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)\\le \\dim M_1+\\dim M_2$ , then every irreducible component of $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ has dimension greater than or equal to $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)$ .","We define similarly the corresponding semistable locus $\\widetilde{B}(\\mathcal {U}_1,\\mathcal {U}_2):=\\lbrace ([E_1],[E_2])\\in \\widetilde{M}_1\\times \\widetilde{M}_2\\,|\\,h^0(\\text{gr}E_1\\otimes \\text{gr}E_2)\\ge k\\rbrace .$ Note that the definition is symmetric and $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\cong B^k(\\mathcal {U}_2,\\mathcal {U}_1)$ .","Moreover, by Serre duality, $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\cong B^{k_1}(\\mathcal {U}_1^*,\\mathcal {U}_2^*\\otimes p_C^*K_C)$ , where $k_1=k-\\chi $ and $K_C$ denotes the canonical line bundle on $C$ .","Note also that $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\cong B^k(\\mathcal {U}_1\\otimes p_C^*L^*,\\mathcal {U}_2\\otimes p_C^*L)$ for any line bundle $L$ .","We take account of these possibilities in Theorem REF (ii) and Remark REF (iii) (see also Remark REF (i)).","Remark 1.1 The following conditions are equivalent: $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ ; $B(n_1,d_1,k)(E_2)\\ne \\emptyset $ for some $E_2\\in M_2$ ; $B(n_2,d_2,k)(E_1)\\ne \\emptyset $ for some $E_1\\in M_1$ .","Many of our results are valid for any smooth curve.","For some, however, we require the concept of a Petri curve.","The curve $C$ is said to be Petri if the classical Petri map $H^0(L)\\otimes H^0(K_C\\otimes L^*)\\longrightarrow H^0(K_C)$ is injective for every line bundle $L$ on $C$ .","The general curve is Petri in the sense that Petri curves of genus $g$ form a non-empty open subset of the moduli space of curves of genus $g$ [11], [16].","By classical BN theory, if $C$ is Petri, the BN locus $B(1,d_1,k)$ has dimension precisely $\\beta (1,d_1,k)$ whenever $0\\le \\beta (1,d_1,k)\\le g$ and is empty when $\\beta (1,d_1,k)<0$ .","Sometimes we will not be able to obtain results for arbitrary Petri curves but only for curves which are general in the sense that they live in an unspecified open subset of the moduli space of curves of genus $g$ .","The BN loci $B(1,d_1,k)(E_2)$ , with $E_2$ general, behave in a similar way to the classical BN loci [12].","By Remark REF , this implies non-emptiness results for $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ when $n_1=1$ or $n_2=1$ .","Moreover, when $n_2=1$ , we have an isomorphism $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\longrightarrow B(n_1,d_1+n_1d_2,k)\\times \\mbox{Pic}^{d_2}(C):\\ (E_1,L)\\mapsto (E_1\\otimes L,L),$ giving a description of $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ in terms of an untwisted BN locus.","We know many cases in which such loci are non-empty; see, in particular, Section .","In this paper, we shall assume that $n_i\\ge 2$ for $i=1,2$ .","We can now state our first main theorem.","Theorem 1.2 Suppose that $C$ is a smooth curve of genus $g\\ge 2$ and that $\\mu _i$ , $\\lambda _i$ $(i=1,2)$ are positive rational numbers.","Let $B(n_i,d_i,k_i)$ be non-empty BN loci with $n_i\\ge 2$ , $d_i/n_i=\\mu _i$ and $k_i/n_i=\\lambda _i$ and suppose that $k=k_1k_2$ .","If $\\mu _1<2$ and $\\mu _2\\le 2g$ , then $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ .","If, in addition, $\\mu _1+\\mu _2<\\lambda _1\\lambda _2+g-1$ , then, for sufficiently large $n_1$ , $n_2$ , $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ .","The major part of this theorem will be stated and proved in a more detailed version as Theorem REF , the last part being covered by Lemma REF .","Using the results of Section , it is easy to find many cases where the hypotheses of the theorem are satisfied, thus obtaining the examples we are seeking (see Example REF ).","There exist examples even when $n_1=n_2=2$ (see Example REF ).","In Theorem REF , we show that this construction gives a new region in the $(\\mu ,\\lambda )$ -plane, whose points support non-empty (untwisted) BN loci.","Our second main theorem is of a rather different nature, in that $d_2$ is negative and so $h^0(E_2)=0$ for all $E_2\\in M_2$ .","Theorem 1.3 Suppose that $C$ is a smooth curve of genus $g\\ge 2$ , $n_1\\ge 2$ , $k_1>n_1$ , $n$ a positive integer and either $d>2ng$ or $d=2ng$ and $C$ is non-hyperelliptic.","Suppose further that $B(n_1,d_1,k_1)\\ne \\emptyset $ and that $k\\le (d-n(g-1))(k_1-n_1)-nd_1.$ Let $(n_2,d_2)=(d-ng,-d)$ .","Then $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ .","If, in addition, $k=d(k_1-n_1)-e$ , where $e\\ge n(g-1)(k_1-n_1)+nd_1$ , and $d_1<k_1+n_1(g-1)-\\frac{g-1}{k_1-n_1},$ then, for any fixed values of $n_1$ , $d_1$ , $k_1$ , $n$ and $e$ , and $d\\gg 0$ , $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ .","A more detailed version of this theorem will be stated and proved as Theorem REF .","In Corollary REF and the related Example REF , we show that the requirement that $B(n_1,d_1,k_1)\\ne \\emptyset $ is compatible with the condition $d_1<k_1+n_1(g-1)-\\frac{g-1}{k_1-n_1}$ .","Hence Theorem REF does lead to examples of twisted BN loci with negative expected dimension.","Section is concerned with the non-emptiness of $B(n_1,d_1,k_1)$ .","In subsection REF , we introduce the BN map and describe some sufficient conditions for non-emptiness.","Subsection REF ) contains a description of kernel bundles and the dual span construction.","We include also a description of BN loci for bundles of slope $\\le 2$ (subsection REF ).","In section , we obtain an extended version of the main existence theorem of [12].","Sections and contain our main results.","The dual span construction works also when we consider pairs $(E,V)$ where $V$ is a subspace of $H^0(E)$ which generates $E$ .","In the case where $E$ is a line bundle, there are then further stability results, which have consequences for the non-emptiness of $B(n_1,d_1,n_1+1)$ .","We shall investigate applications of this to twisted BN loci in a future paper [7].","Throughout the paper, $C$ will denote a smooth irreducible projective curve of genus $g\\ge 2$ defined over ${\\mathbb {C}}$ .","For a vector bundle $E$ on $C$ of rank $n$ and degree $d$ , we denote by $\\mu (E):=\\frac{d}{n}$ the slope of $E$ .","We write also $h^0(E)$ for the dimension of the space of sections $H^0(E)$ .","The canonical line bundle on $C$ is denoted by $K_C$ .","For any real number $a$ , we write $\\lfloor a\\rfloor $ and $\\lceil a\\rceil $ for the largest integer $\\le a$ and the smallest integer $\\ge a$ , respectively.","We thank Lilia Alanis Lopez for help with the figures."],["Non-emptiness of BN loci","In this section, we will recall and extend some of the principal known results on the non-emptiness of the BN loci $B(n_1,d_1,k_1)$ .","The answer is completely known for $k_1\\le n_1$ , but the problem is much more complicated (and still not completely solved) for $k_1>n_1$ (except for $g\\le 3$ and (almost) for hyperelliptic curves) (see [6]).","For Petri curves, the naïve conjecture is that $B(n_1,d_1,k_1)$ is non-empty if and only if $\\beta (n_1,d_1,k_1)\\ge 0$ .","From (REF ), this is equivalent to $d_1\\ge k_1+n_1(g-1)-\\frac{n_1(g-1)}{k_1/n_1}-\\frac{1}{k_1}.$ In fact, numerous examples show that (REF ) is neither necessary nor sufficient for non-emptiness of $B(n_1,d_1,k_1)$ (see, for example,[4], [2], [17], [15], [1])."],["The BN map","Following [6], the non-emptiness of $B(n_1,d_1,k_1)$ can be understood in terms of the BN map.","For this, we map any non-empty BN locus $B(n_1,d_1,k_1)$ (or $\\widetilde{B}(n_1,d_1,k_1)$ ) with $0\\le d_1\\le n_1(2g-2)$ to the point in the plane $(\\mu ,\\lambda )=(d_1/n_1,k_1/n_1).$ We shall say that $(\\mu ,\\lambda )$ supports $B(n_1,d_1,k_1)$ (resp., $\\widetilde{B}(n_1,d_1,k_1)$ ) if (REF ) holds and $B(n_1,d_1,k_1)\\ne \\emptyset $ (resp., $\\widetilde{B}(n_1,d_1,k_1)\\ne \\emptyset $ ).","We are interested here in regions in the $(\\mu ,\\lambda )$ -plane for which it is known that all points with rational coordinates support infinitely many BN loci.","The first examples of such regions were determined by Teixidor i Bigas [21] and Mercat [18].","To describe these in terms of the BN map, we follow [6].","For $s\\ge 1$ , write $\\widehat{\\eta }^{\\prime }(s):=s+g-2-\\left\\lfloor \\frac{g-1}{s}\\right\\rfloor $ ; note that $d\\ge \\widehat{\\eta }^{\\prime }(s)$ if and only if $\\beta (1,d+1,s)\\ge 1$ , and that $\\widehat{\\eta }^{\\prime }(1)=0$ , $\\widehat{\\eta }^{\\prime }(g)=2g-2$ .","Now define a function $t_g$ for $0\\le \\mu \\le 2g-2$ as follows: $t_g(\\mu ):={\\left\\lbrace \\begin{array}{ll}0,&\\mu =0;\\\\\\mu -\\lceil \\mu \\rceil +s,&\\mu \\in ]\\widehat{\\eta }^{\\prime }(s),\\widehat{\\eta }^{\\prime }(s)+1];\\\\s,&\\mu \\in ]\\widehat{\\eta }^{\\prime }(s)+1,\\widehat{\\eta }^{\\prime }(s+1)].\\end{array}\\right.","}$ We define T to be the region in the $(\\mu ,\\lambda )$ -plane given by $\\mu ,\\lambda \\in {\\mathbb {Q}},\\ 0\\le \\mu \\le 2g-2,\\ 0<\\lambda \\le t_g(\\mu ).$ The region T is shown in [6] in the case $g=7$ .","Note that Clifford's Theorem can be stated in the form $\\lambda \\le \\frac{\\mu }{2}+1$ and the boundary line for this inequality (the Clifford line) is shown in the figure.","Moreover, the condition $\\beta (n_1,d_1,k_1)=1$ becomes $\\lambda (\\lambda -\\mu +g-1)=g-1$ (see (REF )), which defines a branch of a hyperbola with asymptotes $\\lambda =0$ and $\\lambda =\\mu -g+1$ (the BN curve), and is also shown in the figure.","The following lemma summarises the results of Teixidor [21] and Mercat [18].","Teixidor's proof is for general $C$ ; for $\\widetilde{B}(n_1,d_1,k_1)$ , this is sufficient to prove it for any smooth curve.","Mercat's result [18] is valid for any smooth curve and implies our lemma.","For the version stated below, see [6].","Lemma 2.1 Let $C$ be a smooth curve of genus $g\\ge 2$ and suppose that $n_1\\ge 2$ .","If $(\\mu ,\\lambda )\\in \\operatorname{T}$ , then $(\\mu ,\\lambda )$ supports $\\widetilde{B}(n_1,d_1,k_1)$ for all values of $n_1$ for which $n_1\\mu $ and $n_1\\lambda $ are both integers.","The same is true for $B(n_1,d_1,k_1)$ , provided that the points $(\\mu ,\\lambda )=(\\widehat{\\eta }^{\\prime }(s)+1,\\lambda ),\\ s-1<\\lambda \\le s,\\ \\widehat{\\eta }^{\\prime }(s)+1\\ne \\widehat{\\eta }^{\\prime }(s+1).$ are excluded.","Remark 2.2 Lemma REF is equivalent to the statement that $\\widetilde{B}(n_1,d_1,k_1)\\ne \\emptyset $ if $d_1\\ge k_1+n_1(g-1)-n_1\\left\\lfloor \\frac{g-1}{\\lceil k_1/n_1\\rceil }\\right\\rfloor .$ This can also be stated as $\\mu \\ge \\lambda +g-1-\\left\\lfloor \\frac{g-1}{\\left\\lceil \\lambda \\right\\rceil }\\right\\rfloor .$ For $B(n_1,d_1,k_1)$ , the case $d_1= n_1\\left(\\left\\lceil \\frac{k_1}{n_1}\\right\\rceil +g-1- \\left\\lfloor \\frac{g-1}{\\lceil k_1/n_1\\rceil }\\right\\rfloor \\right)$ must be excluded when $\\beta (1,d_1/n_1+1,\\left\\lceil k_1/n_1\\right\\rceil +1)\\le 0$ .","(REF ) and (REF ) are essentially a restatement of Teixidor's conditions [21] and relate closely to Mercat's construction [18].","Another method of constructing non-empty BN loci on a smooth curve $C$ is described in [6].","This starts from the results of [4], [17], [19], which give precise conditions for non-emptiness when $d_1\\le 2n_1$ , and proceeds by tensoring by line bundles $L$ with $h^0(L)=s\\ge 1$ .","Serre duality extends the region further and we denote the result by BMNO.","To describe BMNO in the BN map, we write $\\hat{\\eta }(s):=s+g-1-\\left\\lfloor \\frac{g}{s}\\right\\rfloor $ for $s\\ge 1$ .","The condition $d\\ge \\hat{\\eta }(s)$ is then equivalent to $\\beta (1,d,s)\\ge 0$ .","On a Petri curve, this is equivalent by classical BN theory to the assertion that $B(1,d,s)\\ne \\emptyset $ .","Note that $\\hat{\\eta }(1)=0$ and, on a Petri curve, $\\hat{\\eta }(2)$ is the gonality of $C$ .","We now define a function $f_g(\\mu )$ for $0\\le \\mu \\le g-1$ as follows: $f_g(\\mu ):={\\left\\lbrace \\begin{array}{ll}\\frac{s}{g}(\\mu -\\left\\lceil \\mu \\right\\rceil )+s,&\\mu \\in [\\hat{\\eta }(s),\\hat{\\eta }(s)+1];\\\\\\frac{s}{g}(\\mu -\\left\\lceil \\mu \\right\\rceil +1)+s,&\\mu \\in ]\\hat{\\eta }(s)+1,\\hat{\\eta }(s+1)-1];\\\\\\frac{\\hat{\\eta }(s+1)-s}{g}(\\mu -\\left\\lceil \\mu \\right\\rceil +1)+s,&\\mu \\in ]\\hat{\\eta }(s+1)-1,\\hat{\\eta }(s+1)[.\\end{array}\\right.","}$ We extend the definition of $f_g$ , which is modified from that in [6] to allow $\\mu =0$ , to the whole interval $[0,2g-2]$ by Serre duality.","The region BMNO in the $(\\mu ,\\lambda )$ -plane is then defined by the conditions $0\\le \\mu \\le 2g-2,\\ 0\\le \\lambda \\le f_g(\\mu ).$ For $g=10$ , this region is illustrated in [6], which also includes the top boundary of T, the Clifford line and the BN curve.","In Figure 1, we recall [6], including also an indication of the excluded points from Lemmas REF and REF and Remark REF (ii).","The following lemma summarises the principal result of [6], taking account of the results of [19] (see [6]).","Lemma 2.3 Let $C$ be a smooth curve of genus $g\\ge 2$ and $n_1\\ge 2$ .","If $(\\mu ,\\lambda )\\in \\operatorname{BMNO}$ , then $(\\mu ,\\lambda )$ supports $\\widetilde{B}(n_1,d_1,k_1)$ for infinitely many values of $n_1$ .","If $C$ is non-hyperelliptic, the same holds for $B(n_1,d_1,k_1)$ , provided that points of the form $(\\mu ,\\lambda )$ such that $\\mu =\\hat{\\eta }(s)$ with $\\mu \\le g-1$ and $\\lambda >\\frac{s-1}{g}+s-1$ , together with their Serre duals and the points $((\\widehat{\\eta }(s)+1,s)$ , are excluded.","Remark 2.4 (i) [6] provide information about which values of $n_1$ are included in the above result for any given $(\\mu ,\\lambda )$ .","In particular, there is a dense set of points in BMNO for which all values of $n_1$ with $n_1\\mu , n_1\\lambda \\in {\\mathbb {Z}}$ are allowed.","(ii) Some points excluded in Lemma REF are included in Lemma REF .","In fact, if $g$ is a multiple of $s$ , then $\\widehat{\\eta }^{\\prime }(s)=\\widehat{\\eta }(s)$ , and the points $(\\widehat{\\eta }(s)+1,\\lambda )$ for $s-1<\\lambda <s$ are excluded by (REF ), but included by Lemma REF .","(iii) For $g\\ge 4$ , there are many non-empty $B(n_1,d_1,k_1)$ corresponding to points outside the union of T and BMNO; see, for example, the figures in [13], [14], [15].","We shall construct later a new region of the BN map, giving new points for all $g\\ge 5$ (see Theorem REF ).","(iv)[6] also contains comprehensive results for hyperelliptic curves.","Figure: NO_CAPTION"],["Kernel and dual span bundles","Let $E$ be a generated bundle of rank $n$ and degree $d$ .","Consider the exact sequence $0\\longrightarrow D_E^*\\longrightarrow H^0(E)\\otimes \\mathcal {O}\\longrightarrow E\\longrightarrow 0.$ The bundle $D_E^*$ may be referred to as a kernel bundle (or syzygy bundle) and has rank $d-ng$ and degree $-d$ .","Dualising (REF ), we have $0\\longrightarrow E^*\\longrightarrow H^0(E)^*\\otimes \\mathcal {O}\\longrightarrow D_E\\longrightarrow 0.$ Following [10], we call $D_E$ a dual span bundle.","The following theorem addresses the stability of $D_E$ and summarises results of Butler and Mercat (see also [5]).","We write $S:=\\lbrace E\\in M(n,d)|D_E\\text{ is stable}\\rbrace .$ Theorem 2.5 Let $C$ be a smooth curve of genus $g\\ge 2$ .","Suppose that either $d>2ng$ and $E\\in M(n,d)$ or $d=2ng$ , $C$ is non-hyperelliptic and $E$ is a general element of $M(n,d)$ .","Then $E$ is generated and $D_E$ is stable of slope $\\le 2$ .","In particular, $S$ is a non-empty open subset of $M(n,d)$ and $S=M(n,d)$ when $d>2ng$ .","Moreover, the morphism $S\\longrightarrow B(d-ng,d,d-n(g-1)): E\\mapsto D_E$ is an isomorphism.","Any stable bundle of slope $>2g-1$ is generated.","Moreover, under the given hypotheses, $D_E$ is stable by [9].","The fact that (REF ) is an isomorphism follows from [17] and [19]."],["Small slope","When $d_1<2n_1$ , Lemma REF includes a complete answer to the question of the non-emptiness of $B(n_1,d_1,k_1)$ .","The references [4], [17] (see also [3]) contain much further information about these BN loci, in particular that there exists $E_1\\in B(n_1,d_1,k_1)$ , taking one of a number of special forms.","Theorem 2.6 Let $C$ be a curve of genus $g\\ge 2$ .","If $d_1<2n_1$ and $B(n_1,d_1,k_1)\\ne \\emptyset $ , then there exists $E_1\\in B(n_1,d_1,k_1)$ taking one of the following forms.","I $d_1<n_1+g$ .","$E_1$ fits into an exact sequence $0\\longrightarrow \\mathcal {O}^{k_1}\\longrightarrow E_1\\longrightarrow F\\longrightarrow 0$ for some sheaf $F$ .","II $d_1=n_1+g\\ell <2n_1$ , $\\ell \\ge 1$ .","$E_1$ is the dual span of a stable bundle $E$ of slope $>2g$ and is therefore given by an exact sequence $0\\longrightarrow E^*\\longrightarrow H^0(E)^*\\otimes \\mathcal {O}\\longrightarrow E_1\\longrightarrow 0.$ Moreover, $h^0(E)=h^0(E_1)= n_1+\\ell $ .","III $d_1=n_1+g\\ell +\\ell ^{\\prime }<2n_1$ with $\\ell >0$ and $0<\\ell ^{\\prime }<g$ .","There exist exact sequences $0\\longrightarrow D_{E^{\\prime }}^*\\longrightarrow H^0(E^{\\prime })\\otimes \\mathcal {O}\\longrightarrow E^{\\prime }\\longrightarrow 0$ with $E^{\\prime }$ stable of rank $n_1+\\ell ^{\\prime }$ , degree $d_1$ and $h^0(E^{\\prime })=n_1+\\ell +\\ell ^{\\prime }$ , and $0\\longrightarrow \\mathcal {O}^{\\ell ^{\\prime }}\\longrightarrow E^{\\prime }\\longrightarrow E_1\\longrightarrow 0.$ I.","By [4] (for $d_1\\le n_1$ ) and [17] (for $n_1<d_1<n_1+g$ ), the sections of $E_1$ are all carried by a subsheaf $\\mathcal {O}^{k_1}$ .","This gives (REF ) II.","We know that $B(n_1,d_1,k_1)\\ne \\emptyset $ if and only if $d_1\\ge n_1+g(k_1-n_1)$ .","We can therefore choose $E_1\\in B(n_1,d_1,n_1+\\ell )$ and then $h^0(E_1)=n_1+\\ell $ .","By [17], $E_1=D_E$ for some stable bundle $E$ of rank $\\ell $ and degree $d_1$ , hence of slope $>2g$ .","So $h^0(E)=n_1+\\ell $ and we have the exact sequence (REF ).","III.","By [17] (see Case II), there exists a stable bundle $E^{\\prime }$ of rank $n_1+\\ell ^{\\prime }$ and degree $d_1$ with $h^0(E^{\\prime })=n_1+\\ell +\\ell ^{\\prime }$ , fitting in an exact sequence REF .","Now, as in the proof of [17], we have an exact sequence (REF ), with $E_1\\in B(n_1,d_1,n_1+\\ell )$ .","Since $B(n_1,d_1,k_1)\\ne \\emptyset $ , we have (see [17]) $n_1+g\\ell +\\ell ^{\\prime }=d_1\\ge n_1+g(k_1-n_1),$ hence $\\ell \\ge k_1-n_1$ and $E_1\\in B(n_1,d_1,k_1)$ .","Remark 2.7 Theorem REF can be extended to the case $d_1=2n_1$ (see [19]).","We have two cases.","IV $C$ non-hyperelliptic.","By the results of [19], we are in one of Cases I-III, except that in Case II, we now have $\\deg E=2n_1g$ .","There is one further case, namely when $E_1=D_{K_C}\\in B(g-1,2g-2,g)$ .","V $C$ hyperelliptic.","In this case $B(n_1,2n_1,k_1)\\ne \\emptyset $ if and only if $k_1\\le n_1$ (see [19]).","When this holds, there exists $E_1\\in B(n_1,2n_1,k_1)$ which admits $\\mathcal {O}^{k_1}$ as a subsheaf."],["A non-emptiness result for twisted BN loci","In this section we give an extended version of [12], which we need for comparison purposes.","Theorem 3.1 Let $C$ be a smooth curve of genus $g\\ge 2$ and $E_2$ any vector bundle of rank $n_2$ and degree $d_2$ on $C$ .","Let $d_0$ and $k_0$ be integers satisfying $\\beta (1,d_0,k_0)(E_2)\\ge 1$ , and suppose that $n_1\\ge 2$ .","Then, (i) if $k\\le n_1k_0$ and $d_1\\ge n_1d_0+1\\ (\\text{resp.","}, d_1\\ge n_1d_0)$ , the twisted BN locus $B(n_1,d_1,k)(E_2) \\ (\\text{resp.","}, \\widetilde{B}(n_1,d_1,k)(E_2))$ is non-empty and $\\beta (n_1,d_1,k)(E_2)>(\\text{resp.","}, \\ge )\\,1$ ; (ii) if $k_1:=k-n_2d_1+n_1d_2-n_1n_2(g-1)\\le n_1k_0$ and $-d_1\\ge n_1d_0+1\\ (\\text{resp.","}, -d_1\\ge n_1d_0)$ , the twisted BN locus $B(n_1,d_1,k)(K_C\\otimes E_2^*)\\ (\\text{resp.","}, \\widetilde{B}(n_1,d_1,k)(K_C\\otimes E_2^*))$ is non-empty and $\\beta (n_1,d_1,k)(K_C\\otimes E_2^*)>(\\text{resp.","}, \\ge )\\,1.$ (i) The non-emptiness of $B(n_1,d_1,k)(E_2)$ (resp., $\\widetilde{B}(n_1,d_1,k)(E_2)$ ) is [12].","The inequality $\\beta (n_1,d_1,k)(E_2)>(\\ge )1$ follows from the observation that $\\beta (n_1,n_1d_0,n_1k_0)(E_2)\\ge 1 \\Longleftrightarrow \\beta (1,d_0,k_0)(E_2)\\ge 1$ and the fact that $\\beta (n_1,d_1,k)(E_2)$ is a strictly increasing function of $d_1$ and a decreasing function of $k$ .","(ii) Under the stated hypotheses, in the case $-d_1\\ge n_1d_0+1$ , (i) implies that $B(n_1,-d_1,k_1)(E_2)\\ne \\emptyset $ .","By Serre duality, $B(n_1,d_1,k)(K_C\\otimes E_2^*)\\cong B(n_1,-d_1,k_1)(E_2).$ So $B(n_1,d_1,k)(K_C\\otimes E_2^*)\\ne \\emptyset $ .","Moreover, as in the proof of (i), $\\beta (n_1,-d_1,k_1)(E_2)>1$ and $\\beta (n_1,d_1,k)(K_C\\otimes E_2^*)=n_1^2(g-1)+1-kk_1=\\beta (n_1,-d_1,k_1)(E_2).$ The same proof applies in the case $-d_1\\ge n_1d_0$ if we replace $>$ in (REF ) by $\\ge $ and $B$ with $\\widetilde{B}$ throughout.","Corollary 3.2 Under the hypotheses of Theorem REF , the twisted BN locus $B^k(\\mathcal {U}_!,\\mathcal {U}_2)\\ (\\text{resp.","}, \\widetilde{B}(\\mathcal {U}_1,\\mathcal {U}_2))$ is non-empty.","Moreover $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)>n_2^2(g-1)+2\\ >0\\ (\\mbox{resp.,}\\ge n_2^2(g-1)+2).$ In the theorem, we can take $E_2$ to be stable; the required non-emptiness then follows (see Remark REF ).","The inequality for $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)$ follows from the fact that $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)= \\dim M_2+\\beta (n_1,d_1,k)(E_2)$ (see (REF ) and (REF )).","Remark 3.3 (i) As indicated in the introduction, the fact that the BN numbers are positive will be crucial for us in establishing that many of the examples constructed in this paper are new.","(ii) (REF ) makes Theorem REF (i) the precise analogue of the theorem of Teixidor i Bigas [21] for twisted BN loci $B(n_1,d_1,k)(E_2)$ , at least on a general curve; the proof in [12] depends on Mercat's construction [18] and holds for any curve.","(iii) If we replace $E_2$ by $E_2\\otimes L$ , where $L$ is a line bundle of degree $\\ell $ , and simultaneously replace $d_0$ by $d_0-\\ell $ and $d_1$ by $d_1-n_1\\ell $ in Theorem REF (i), the BN numbers and the isomorphism classes of the BN loci are unchanged.","This applies also to Theorem REF (ii) except that now $d_1$ must be replaced by $d_1+n_1\\ell $ ."],["Examples with $k=k_1k_2$","In this section, we consider the following situation.","Suppose that $B(n_i,d_i,k_i)\\ne \\emptyset $ for $i=1,2$ and that there exist $E_i\\in B(n_i,d_i,k_i)$ such that $h^0(E_1\\otimes E_2)\\ge k_1k_2$ .","When $n_2=1$ , this is exactly the situation considered in [6].","Here, however, we assume $n_i\\ge 2$ .","One scenario to which this applies is when $E_1=D_E$ ."],["The construction","Proposition 4.1 Let $C$ be a smooth curve of genus $g$ .","Suppose that $E$ is a generated stable (resp., semistable) bundle of rank $n$ and degree $d_1$ and that $h^0(E)=k_1=n+n_1$ .","Suppose further that $D_E$ is stable (resp., semistable) and that $E_2\\in B(n_2,d_2,k_2)\\ (\\text{resp.","}, [E_2]\\in \\widetilde{B}(n_2,d_2,k_2))$ with $\\mu (E_2)<\\mu (E)$ .","Then $h^0(D_E\\otimes E_2)\\ge k_1k_2\\ (\\text{resp.","}, h^0(D_E\\otimes \\text{gr}E_2)\\ge k_1k_2)$ .","In particular, $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset \\ (\\text{resp.","}, \\widetilde{B}^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset )$ .","Consider the sequence (REF ) and suppose first that we are in the stable case.","Note that $D_E\\in B(n_1,d_1,k_1)$ , since $D_E$ is stable by hypothesis and $h^0(E^*)=0$ .","Note also that $h^0(E^*\\otimes E_2)=0$ since $E^*\\otimes E_2$ is semistable of negative slope.","Tensoring (REF ) by $E_2$ and taking global sections, we obtain $h^0(D_E\\otimes E_2)\\ge h^0(E)\\cdot h^0(E_2)\\ge k_1k_2.$ The semistable case proceeds similarly.","Note that, by (REF ), we have $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)=(n_1^2+n_2^2)(g-1)+2-k(k-\\chi ).$ We have also $\\beta (n_1n_2,n_2d_1+n_1d_2,k)=n_1^2n_2^2(g-1)+1-k(k-\\chi ).$ Lemma 4.2 Let $\\mu _i$ , $\\lambda _i$ be positive rational numbers for $i=1,2$ and suppose that $d_i=n_i\\mu _i$ , $k_i=n_i\\lambda _i$ , $k=k_1k_2$ .","If $\\mu _1+\\mu _2<\\lambda _1\\lambda _2+g-1,$ then, for fixed $\\mu _i$ , $\\lambda _i$ and sufficiently large $n_1$ , $n_2$ , $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ .","Since $k=k_1k_2$ , (REF ) is equivalent to $\\frac{\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)}{n_1^2n_2^2}=\\left(\\frac{1}{n_2^2}+\\frac{1}{n_1^2}\\right)(g-1)+\\frac{2}{n_1^2n_2^2}-\\lambda _1\\lambda _2(\\lambda _1\\lambda _2-(\\mu _1+\\mu _2)+g-1).$ The result follows.","Theorem 4.3 Suppose that $C$ is a smooth curve of genus $g\\ge 2$ and that $\\mu _i$ , $\\lambda _i$ ($i=1,2$ ) are positive rational numbers, $\\mu _1<2$ and $\\mu _2\\le 2g$ .","(i) If $B(n_i,d_i,k_i)$ are non-empty BN loci with $n_i\\ge 2$ supported by $(\\mu _i,\\lambda _i)$ and $k=k_1k_2$ , then $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ .","(ii) If $\\widetilde{B}(n_i,d_i,k_i)$ are non-empty BN loci with $n_i\\ge 2$ supported by $(\\mu _i,\\lambda _i)$ and $k=k_1k_2$ , then $\\widetilde{B}^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ and $\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)\\ne \\emptyset .$ (i) Let $E_2\\in B(n_2,d_2,k_2)$ with $d_2\\le 2n_2g$ .","We prove that there exists $E_1\\in B(n_1,d_1,k_1)$ such that $h^0(E_1\\otimes E_2)\\ge k_1k_2$ .","The proof is the same as in [6] with $L$ replaced by $E_2$ .","We need to consider cases I-III of Theorem REF .","I.","$\\mathcal {O}^{k_1}\\otimes E_2$ is a subsheaf of $E_1\\otimes E_2$ .","It follows at once that $k_1k_2\\le h^0(E_1\\otimes E_2$ ).","II.","in (REF ), $E_1\\cong D_E$ and $\\mu (E_2)\\le 2g<\\mu (E)$ .","The result follows from Proposition REF .","III.","Tensoring (REF ) and (REF ) by $E_2$ and noting that $D_{E^{\\prime }}^*\\otimes E_2$ is semistable of negative slope, we obtain $h^0(E^{\\prime }\\otimes E_2)\\ge h^0(E^{\\prime }) h^0(E_2)$ and hence $h^0(E_1\\otimes E_2)\\ge ( h^0(E^{\\prime })-\\ell ^{\\prime })h^0(E_2)=(n_1+\\ell )h^0(E_2)\\ge k_1k_2.$ (ii) This follows in the same way as (i), together with the fact that the tensor product of semistable bundles is semistable.","Remark 4.4 If $C$ is non-hyperelliptic, the theorem holds under the hypotheses $\\mu _1\\le 2$ , $\\mu _2<2g$ (see Case IV in Remark REF )."],["Examples","Using Lemmas REF and REF , it is easy to find many cases as in Theorem REF where (REF ) is also satisfied, thus yielding examples of non-empty BN loci $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ with negative BN number.","These cannot be obtained from Theorem REF .","Example 4.5 Suppose that $\\mu _i\\le \\frac{1}{2}$ for $i=1,2$ .","From [4] (see Lemma REF and Figure 1), we can take $\\lambda _i$ to be any positive rational numbers with $\\lambda _i\\le 1-\\frac{1}{g}(1-\\mu _i)$ .","Then, Lemma REF and Theorem REF apply, giving $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ and $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ for sufficiently large $n_1$ , $n_2$ , even when $g=2$ .","More generally, we can take $\\mu _1\\le 1$ and $\\mu _2\\le g-2$ for $g\\ge 3$ , or $\\mu _1<2$ and $\\mu _2\\le g-3$ for $g\\ge 4$ , with $\\lambda _i$ any positive rational numbers for which there exist non-empty BN loci $B(n_i,d_i,k_i)$ supported by $(\\mu _i,\\lambda _i)$ .","It is even possible to find examples with $n_1=n_2=2$ .","Example 4.6 Suppose that $n_1=n_2=2$ .","Then, under the hypotheses of Theorem REF , $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ if and only if $\\left(\\frac{1}{2}-\\lambda _1\\lambda _2\\right)(g-1)+\\frac{1}{8}-\\lambda _1\\lambda _2(\\lambda _1\\lambda _2-(\\mu _1+\\mu _2))<0$ (see (REF )).","We can take $B(n_i,d_i,k_i)=B(2,3,2)$ for $i=1,2$ ; this is non-empty by either Lemma REF or Lemma REF .","(REF ) now becomes $-\\frac{1}{2}(g-1)+\\frac{1}{8}+2<0,$ which is true for $g\\ge 6$ .","Here, $B(2,3,2)$ is the only candidate for $B(2,d_1,k_1)$ for applying Theorem REF , but, for $i=2$ , we could take any non-empty $B(2,d_2,k_2)$ instead of $B(2,3,2)$ ; once we have fixed $d_2$ with $d_2\\le 4g$ and $k_2\\ge 2$ , (REF ) will hold for sufficiently large $g$ ."],["A new region in the BN map","Substituting $k=k_1k_2$ in (REF ) and dividing by $n_1^2n_2^2$ , we have $\\beta :=\\frac{\\beta (n_1n_2,n_2d_1+n_1d_2,k)}{n_1^2n_2^2}=\\\\=g-1+\\frac{1}{n_1^2n_2^2}-\\lambda _1\\lambda _2(\\lambda _1\\lambda _2-(\\mu _1+\\mu _2)+g-1).$ It is an interesting question as to whether $\\beta $ can ever be negative under the hypotheses of Theorem REF and the assumption that $\\beta (n_2,d_2,k_2)\\ge 1$ .","One can see easily that $\\beta >0$ if $\\lambda _1\\le 1$ , but other cases are not so straightforward.","Independently of this, one can ask whether the theorem can give rise to new examples of non-empty BN loci in $M(n_1n_2,n_2d_1+n_1d_2,k_1k_2)$ .","Our next result shows that this is indeed the case.","We define a new region of the BN map as follows.","Let $R:=\\left\\lbrace (\\mu _1,\\mu _2,\\lambda _1,\\lambda _2)\\in {\\mathbb {Q}}^4\\,\\left|\\begin{array}{c}\\,0<\\mu _1<2,0<\\mu _2\\le 2g-2-\\mu _1,\\\\\\lambda _i\\le \\max \\lbrace f_g(\\mu _i),t_g(\\mu _i)\\rbrace \\end{array}\\right.\\right\\rbrace $ and let $h:R\\rightarrow {\\mathbb {Q}}^2$ be defined by $h(\\mu _1,\\mu _2,\\lambda _1,\\lambda _2)=(\\mu _1+\\mu _2,\\lambda _1\\lambda _2).$ Definition 4.7 The region BPN in the BN map is defined to be the union of $\\operatorname{Im}h$ and its Serre dual.","Theorem 4.8 Let $C$ be a smooth curve of genus $g\\ge 2$ and $n_i\\ge 2$ .","(i) If $(\\mu ,\\lambda )=h(\\mu _1,\\mu _2,\\lambda _1,\\lambda _2)\\in \\operatorname{BPN}$ , then $(\\mu _1+\\mu _2,\\lambda _1\\lambda _2)$ supports a BN locus $\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)$ with $k=k_1k_2$ for infinitely many values of $n_1$ , $n_2$ ; (ii) If $g\\ge 5$ , $\\operatorname{BPN} $ is not contained in $\\operatorname{T\\cup BMNO}$ .","(i) By Lemmas REF and REF , $(\\mu _i,\\lambda _i)$ supports the BN locus $\\widetilde{B}(n_i,d_i,k_i)$ ($i=1,2$ ) for infinitely many values of $n_i$ .","The result follows from Theorem REF (ii).","(ii) Suppose that $2<\\mu <4$ .","We can write $\\mu =\\mu _1+\\mu _2$ with $1<\\mu _i<2$ for $i=1,2$ .","Now, if $\\lambda _i=1+\\frac{1}{g}(\\mu _i-1)$ , then, by [17] (see also Lemma REF ), $(\\mu _i,\\lambda _i$ ) is in the region BMNO and $(\\mu _i,\\lambda _i)$ supports $B(n_i,d_i,k_i)$ for any allowable $n_i$ .","So, by Theorem REF (ii), $\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)\\ne \\emptyset $ when $k=k_1k_2$ .","On the other hand, $\\lambda _1\\lambda _2>1+\\frac{1}{g}(\\mu _1+\\mu _2-2)\\ge 1+\\frac{1}{g}(\\mu _1+\\mu _2-\\left\\lceil \\mu _1+\\mu _2\\right\\rceil +1).$ It follows from Lemma REF and REF that $(\\mu _1+\\mu _2,\\lambda _1\\lambda _2)$ is not in the region BMNO if $\\widehat{\\eta }(2)\\ge 5$ , in other words, if $g\\ge 7$ .","It is clear also that $(\\mu _1+\\mu _2,\\lambda _1\\lambda _2)$ is not in the region T, whose upper boundary is given by $\\lambda =1$ in this range.","In fact, our construction also gives new points for $g=5$ (compare [14]) and $g=6$ (compare [15]).","(When $g=6$ , there is a missing grey area in [15]; the upper boundary for $\\operatorname{BMNO}$ should be given by $\\lambda =1+\\frac{1}{2}(\\mu -3)$ for $3<\\mu <4$ .","This does not affect our assertion.)","Remark 4.9 The construction in the proof of Theorem REF shows that, for $g\\ge 7$ , the top boundary of BPN in the range $2<\\mu <4$ is given by $\\lambda =1+\\frac{1}{g}(\\mu -2)+\\frac{1}{g^2}\\left(\\frac{\\mu -2}{2}\\right)^2$ , whereas the top boundary of BMNO is given by $\\lambda =1+\\frac{1}{g}(\\mu -2)$ for $2<\\mu <3$ and by $\\lambda =1+\\frac{1}{g}(\\mu -3)$ for $3<\\mu <4$ .","At the point $(2,1)$ , the line $\\lambda =1+\\frac{1}{g}(\\mu -2)$ is the tangent to the parabola $\\lambda =1+\\frac{1}{g}(\\mu -2)+\\frac{1}{g^2}\\left(\\frac{\\mu -2}{2}\\right)^2$ .","For $g=10$ , BPN is illustrated in Figure 2, which extends to the range $4<\\mu <5$ .","Figure 2 also shows an expanded version of the BN map close to $\\mu =3$ .","Figure: NO_CAPTION Remark 4.10 (i) BPN is a new region in the BN map (see Figure 2), whereas the papers [14], [15] give examples for isolated values of $\\mu $ , some of which extend to higher values of $g$ .","When $g=4$ , on the other hand, there is an additional grey area of the BN map (see [13]), which contains the new part of the region BPN in the range $2<\\mu <3$ .","(ii) At least for Petri curves, some of the non-empty BN loci $\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)\\subset \\widetilde{M}(n_1n_2,n_2d_1+n_1d_2)$ constructed in Theorem REF are new.","It is possible to choose the ranks and degrees so that $\\gcd (n_1n_2,n_2d_1+n_1d_2)=1$ ; for example, if $g$ is odd, one can take $(n_1,d_1,k_1)=(g+1,2g+1,g+2),\\ (n_2,d_2,k_2)=(g+2,2g+2,g+3).$ Then $B(n_1n_2,n_2d_1+n_1d_2,k)\\ne \\emptyset $ .","(iii) One can check that, in the framework of Theorem REF , the examples of twisted BN loci with negative expected dimension constructed in [12] give rise to points of T. It follows that our examples are different."],["The construction","The essential ingredient in proving Theorem REF is Mercat's construction [18], while Theorem REF depends on the constructions of [4], [17].","Another way of constructing non-empty twisted BN loci is described in [8] and uses kernel bundles.","The following proposition is contained in [8] and plays an essential rôle in the proof of [8].","We include a proof for the convenience of the reader.","Proposition 5.1 Suppose that $E_1$ has rank $n_1$ and degree $d_1$ with $h^0(E_1)\\ge k_1$ and that $E$ is as in (REF ).","If $h^1(E_1\\otimes E)=0$ and $k\\le (d-n(g-1))(k_1-n_1)-nd_1,$ then $h^0(E_1\\otimes D_E^*)\\ge k$ .","We have $h^0(E_1\\otimes E)=nd_1+n_1d-nn_1(g-1)$ and $h^0(E)\\ge d-n(g-1)$ .","So, tensoring (REF ) by $E_1$ and taking global sections, $h^0(E_1\\otimes D_E^*)\\ge (d-n(g-1))k_1-nd_1-n_1d+nn_1(g-1).$ Simplifying and comparing with (REF ), this gives the result.","We have the following immediate corollary (see Remark REF ).","Corollary 5.2 Suppose that the hypotheses of Proposition REF are satisfied.","(i) If $E_1$ is stable $(\\text{resp., semistable})$ , then $B(n_1,d_1,k)(D_E^*)\\ne \\emptyset $ $(\\text{resp.","}, \\widetilde{B}(n_1,d_1,k)(D_E^*)\\ne \\emptyset )$ .","(ii) If $D_E^*$ is stable $(\\text{resp., semistable})$ , then $B(h^0(E)-n,-d,k)(E_1)\\ne \\emptyset \\ (\\text{resp.","},\\widetilde{B}(h^0(E)-n,-d,k)(E_1)\\ne \\emptyset )$ .","(iii) If both $E_1$ and $D_E^*$ are stable $(\\text{resp., semistable})$ and $(n_2,d_2)=(h^0(E)-n,-d)$ , then $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset \\ (\\text{resp.","}, \\widetilde{B}^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset )$ .","Our object here is to use Corollary REF (iii) to obtain new non-empty examples of twisted BN loci.","We assume that $n_1\\ge 2$ .","In order for (REF ) to yield positive values of $k$ when $d>n(g-1)$ , which will always be the case, we clearly require $k_1>n_1$ .","The key point we require in order to apply Corollary REF (iii) is that $D_E^*$ and $E_1$ are stable.","We have already discussed the existence of stable $E_1$ in subsection REF and the stability of $D_E$ in subsection REF .","We can now state and prove a general theorem, which includes Theorem REF .","Theorem 5.3 Suppose that $C$ is a smooth curve of genus $g\\ge 2$ , $n_1\\ge 2$ , $k_1>n_1$ and either $d>2ng$ or $d=2ng$ and $C$ is non-hyperelliptic.","Suppose further that $B(n_1,d_1,k_1)\\ne \\emptyset $ and that (REF ) holds.","Let $(n_2,d_2)=(d-ng,-d)$ and suppose that $S$ is defined as in (REF ).","Then $S\\ne \\emptyset $ and the morphism $B(n_1,d_1,k_1)\\times S\\longrightarrow M_1\\times M_2:(E_1,E)\\mapsto (E_1,D_E^*)$ is injective and has image contained in $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ ; in particular, $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset .$ If, in addition, $k=d(k_1-n_1)-e$ , where $e\\ge n(g-1)(k_1-n_1)+nd_1,$ and $d_1<k_1+n_1(g-1)-\\frac{g-1}{k_1-n_1},$ then, for any fixed values of $n_1$ , $d_1$ , $k_1$ , $n$ and $e$ satisfying (REF ), and $d\\gg 0$ , $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0.$ The first part follows from Theorem REF , Proposition REF and Corollary REF (iii).","We need to note that $h^1(E_1\\otimes E)=0$ since $E_1\\otimes E$ is semistable of slope greater than $2g$ .","For the second part, we have $\\nonumber \\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)&=&n_1^2(g-1)+1+(d-ng)^2(g-1)+1\\\\&&-(d(k_1-n_1)-e)(d(k_1-n_1)-e-\\chi ),$ where, by (REF ), $\\nonumber \\chi &=&(d-ng)d_1-n_1d-n_1(d-ng)(g-1)\\\\&=&(d-ng)(d_1-n_1g)-nn_1g.$ The right-hand side of (REF ) is a quadratic in $d$ , with leading coefficient $g-1-(k_1-n_1)(k_1-n_1-d_1+n_1g).$ So $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ for $d\\gg 0$ provided that $d_1<k_1+n_1(g-1)-\\frac{g-1}{k_1-n_1}.$ Remark 5.4 (i) Under the hypotheses of Theorem REF , including the inequalities (REF ) and REF , we deduce that the non-emptiness of $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ cannot be obtained from Theorem REF .","In fact, there are several choices we can make in attempting to apply Theorem REF to this situation.","We can take $E_2$ to be any of $E_1\\otimes L$ , $D_E^*\\otimes L$ , $K_C\\otimes E_1^*\\otimes L$ and $K_C\\otimes D_E\\otimes L$ , where $L$ is any line bundle, with the appropriate choice of $(n_1,d_1)$ in each case.","In all cases, we obtain $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ge 1$ .","In the absence of (REF ), we would have to work through all these cases separately.","(ii) If we assume only that $\\widetilde{B}(n_1,d_1,k_1)\\ne \\emptyset $ and $d\\ge 2ng$ with $C$ any smooth curve of genus $g\\ge 2$ , we obtain $\\widetilde{B}(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ .","Also, if (REF ) holds and $d\\gg 0$ , then the non-emptiness of $\\widetilde{B}(\\mathcal {U}_1,\\mathcal {U}_2)$ cannot be obtained from Theorem REF ."],["Examples","To show that Theorem REF provides the examples we are seeking, we need to show that the conditions $B(n_1,d_1,k_1)\\ne \\emptyset $ and (REF ) are compatible.","For this, we need more precise statements concerning the possible values of $d_1$ .","Corollary 5.5 Suppose that $C$ is a smooth curve of genus $g\\ge 3$ , $n_1\\ge 2$ , $k_1>n_1$ and $(n_2,d_2)=(d-ng,-d)$ with either $d>2ng$ or $d=2ng$ and $C$ non-hyperelliptic.","Suppose further that $k_1+n_1(g-1)-n_1\\left\\lfloor \\frac{g-1}{\\lceil k_1/n_1\\rceil }\\right\\rfloor \\le d_1<k_1+n_1(g-1)-\\frac{g-1}{k_1-n_1}$ and that $d_1$ is not divisible by $n_1$ .","Then, $B(n_1,d_1,k_1)\\ne \\emptyset $ .","Moreover, for any fixed value of $e$ satisfying (REF ) and $d\\gg 0$ , $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ , but $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ .","In particular, $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ is of negative expected dimension and the non-emptiness of $B^k(\\mathcal {U}_1,\\mathcal {U}_2)$ cannot be obtained from Theorem REF .","The fact that $B(n_1,d_1,k_1)\\ne \\emptyset $ follows from Remark REF .","Theorem REF then implies that $B^k(\\mathcal {U}_1,\\mathcal {U}_2)\\ne \\emptyset $ and that $\\beta ^k(\\mathcal {U}_1,\\mathcal {U}_2)<0$ when (REF ) holds and $d\\gg 0$ .","Hence, the non-emptiness cannot be obtained from Theorem REF (see Remark REF (i)).","To obtain examples from this corollary, we need, in particular, to prove the existence of $d_1$ satisfying (REF ).","This is equivalent to showing that $n_1\\left\\lfloor \\frac{g-1}{\\lceil k_1/n_1\\rceil }\\right\\rfloor >\\frac{g-1}{k_1-n_1}$ In fact (REF ) cannot hold if $g=2$ (which is why we have assumed $g\\ge 3$ in Corollary REF ) or, more generally, if $k_1\\ge n_1(g-1)+1$ , since, in that case, the left-hand side of (REF ) is 0.","We therefore assume that $n_1<k_1\\le n_1(g-1)$ Example 5.6 Suppose that $g=3$ and (REF ) holds.","Then (REF ) holds if and only $n_1(k_1-n_1)>2$ , in other words, either $n_1\\ge 3$ or $n_1=2$ and $k_1=4$ .","In the latter case, the only possible value for $d_1$ is $d_1=6$ .","This is not permitted by the hypotheses of Corollary REF and, in fact $B(2,6,4)=\\emptyset $ .","For $n_1\\ge 3$ , there do exist solutions of (REF ) for which $d_1$ is not divisible by $n_1$ .","In fact, if $k_1=n_1+2$ , (REF ) gives $2n_1+2\\le d_1<3n_1+1$ , giving $n_1-2$ values of $d_1$ not divisible by $n_1$ .","If $n_1+3\\le k_1\\le 2n_1$ , there are $n_1-1$ permitted values of $d_1$ in the range $k_1+n_1\\le d_1<k_1+2n_1$ .","A similar analysis can be carried out for $g\\ge 4$ ; in this case, in particular, $B(2,4g-5,2g-2)\\ne \\emptyset $ and (REF ) is satisfied, so examples exist for $n_1=2$ , as well as for larger $n_1$ .","Remark 5.7 When $k_1=n_1+1$ , there are additional cases in which the non-emptiness of $B(n_1,d_1,k_1)$ is known.","This is related to the extended dual span construction, which involves the stability of bundles $D_{L,V}$ , where $L$ is a line bundle and $V$ is a subspace of $H^0(L)$ which generates $L$ .","Applications of this will be considered in a paper currently in preparation [7]."]],"string":"[\n [\n \"New examples of twisted Brill-Noether loci I\"\n ],\n [\n \"Abstract Our purpose in this paper is to construct new examples of twisted Brill-Noether loci on curves of genus $g\\\\ge2$.\",\n \"Many of these examples have negative expected dimension.\",\n \"We deduce also the existence of a new region in the Brill-Noether map, whose points support non-empty standard Brill-Noether loci.\"\n ],\n [\n \"Introduction\",\n \"Our object in this paper is to construct twisted Brill-Noether loci different from those constructed in [12].\",\n \"In particular, the relevance of many of these examples is that they have negative Brill-Noether numbers, whereas the examples constructed in [12] all have positive Brill-Noether numbers.\",\n \"We deduce also the existence of a new region in the Brill-Noether map (see below for the definition), the points of which support non-empty Brill-Noether loci, whose existence was not previously known.\",\n \"This implies that our new examples also differ from those constructed in [12] (see Remark REF (iii)).\",\n \"In what follows, we shall usually abbreviate Brill-Noether to BN.\",\n \"Let $C$ be a smooth irreducible projective curve of genus $g\\\\ge 2$ defined over the complex numbers.\",\n \"For $i=1,2$ , let $M_i:=M(n_i,d_i)$ denote the moduli space of stable bundles of rank $n_i$ and degree $d_i$ .\",\n \"Similarly, we write $\\\\widetilde{M}_i$ for the corresponding moduli space of S-equivalence classes of semistable bundles.\",\n \"Let $E_2$ be any vector bundle on $C$ of rank $n_2$ and degree $d_2$ .\",\n \"For any integers $n_1$ , $d_1$ , $k$ with $n_1\\\\ge 1$ , one can define the twisted BN locus $B(n_1,d_1,k)(E_2):=\\\\lbrace E_1\\\\in M_1\\\\,|\\\\,h^0(E_1\\\\otimes E_2)\\\\ge k\\\\rbrace .$ The locus $B(n_1,d_1,k)(E_2)$ is a closed subscheme of $M_1$ .\",\n \"In fact, it is a determinantal locus; it follows that, if $\\\\beta (n_1,d_1,k)(E_2)\\\\le n_1^2(g-1)+1$ , then every irreducible component of $B(n_1,d_1,k)(E_2)$ has dimension greater than or equal to the BN number $\\\\beta (n_1,d_1,k)(E_2):=n_1^2(g-1)+1-k(k-\\\\chi ),$ where $\\\\chi :=n_2d_1+n_1d_2-n_1n_2(g-1).$ For this reason, the BN number is often referred to as the expected dimension.\",\n \"We define also $\\\\widetilde{B}(n_1,d_1,k)(E_2):=\\\\lbrace [E_1]\\\\in \\\\widetilde{M}_1\\\\,|\\\\,h^0(\\\\operatorname{gr}E_1 \\\\otimes E_2)\\\\ge k\\\\rbrace ,$ where $[E_1]$ denotes the S-equivalence class of $E_1$ and $\\\\text{gr}E_1$ is the corresponding graded bundle.\",\n \"This is a closed subscheme of $\\\\widetilde{M}_1$ , but can have components of dimension less than the BN number.\",\n \"In the special case $E_2=\\\\mathcal {O}_C$ , we obtain the standard (untwisted) higher rank BN locus, denoted by $B(n_1,d_1,k)$ , with expected dimension $\\\\beta (n_1,d_1,k):=n_1^2(g-1)+1-k(k-d_1+n_1(g-1)).$ In particular, $B(1,d_1,k)$ is the classical BN locus $W^{k-1}_{d_1}$ .\",\n \"In the case $n_1=1$ , twisted BN loci have been studied for some time and include the special case of maximal line subbundles.\",\n \"A systematic study for arbitrary $n_1$ was started in [12], where, in particular, examples of twisted BN loci were constructed [12] using the technique of [18].\",\n \"As we shall see in Theorem REF , these have the important feature that the BN number (REF ) is positive.\",\n \"We can now introduce the loci which we are going to study in this paper.\",\n \"This is a special case of a more general construction introduced in [12], where it was used primarily to obtain results for $B(n_1,d_1,k)(E_2)$ .\",\n \"If $\\\\gcd (n_i,d_i)=1$ for $i=1,2$ , then there exist universal bundles $\\\\mathcal {U}_i$ on $M_i\\\\times C$ and, for any integer $k$ , we define the universal twisted BN locus as follows: $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2):=\\\\lbrace (E_1,E_2)\\\\in M_1\\\\times M_2\\\\, |\\\\,h^0(E_1\\\\otimes E_2)\\\\ge k\\\\rbrace .$ In the non-coprime case, we need to lift to étale coverings of $M_1$ and $M_2$ in order to construct “universal” bundles [20].\",\n \"However (REF ) still makes sense and we shall use the same notation.\",\n \"The locus $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ is a closed subscheme of $M_1\\\\times M_2$ ; if $k\\\\le 0$ , then $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)=M_1\\\\times M_2$ .\",\n \"The structure of $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ as a determinantal locus allows us to define a BN number (see [12] $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2):=\\\\dim M_1+\\\\dim M_2-k(k-\\\\chi ),$ which may again be referred to as the expected dimension.\",\n \"Note that $\\\\chi =\\\\chi (E_1\\\\otimes E_2)$ for all $(E_1,E_2)\\\\in M_1\\\\times M_2$ .\",\n \"If $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\le \\\\dim M_1+\\\\dim M_2$ , then every irreducible component of $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ has dimension greater than or equal to $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ .\",\n \"We define similarly the corresponding semistable locus $\\\\widetilde{B}(\\\\mathcal {U}_1,\\\\mathcal {U}_2):=\\\\lbrace ([E_1],[E_2])\\\\in \\\\widetilde{M}_1\\\\times \\\\widetilde{M}_2\\\\,|\\\\,h^0(\\\\text{gr}E_1\\\\otimes \\\\text{gr}E_2)\\\\ge k\\\\rbrace .$ Note that the definition is symmetric and $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\cong B^k(\\\\mathcal {U}_2,\\\\mathcal {U}_1)$ .\",\n \"Moreover, by Serre duality, $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\cong B^{k_1}(\\\\mathcal {U}_1^*,\\\\mathcal {U}_2^*\\\\otimes p_C^*K_C)$ , where $k_1=k-\\\\chi $ and $K_C$ denotes the canonical line bundle on $C$ .\",\n \"Note also that $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\cong B^k(\\\\mathcal {U}_1\\\\otimes p_C^*L^*,\\\\mathcal {U}_2\\\\otimes p_C^*L)$ for any line bundle $L$ .\",\n \"We take account of these possibilities in Theorem REF (ii) and Remark REF (iii) (see also Remark REF (i)).\",\n \"Remark 1.1 The following conditions are equivalent: $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ ; $B(n_1,d_1,k)(E_2)\\\\ne \\\\emptyset $ for some $E_2\\\\in M_2$ ; $B(n_2,d_2,k)(E_1)\\\\ne \\\\emptyset $ for some $E_1\\\\in M_1$ .\",\n \"Many of our results are valid for any smooth curve.\",\n \"For some, however, we require the concept of a Petri curve.\",\n \"The curve $C$ is said to be Petri if the classical Petri map $H^0(L)\\\\otimes H^0(K_C\\\\otimes L^*)\\\\longrightarrow H^0(K_C)$ is injective for every line bundle $L$ on $C$ .\",\n \"The general curve is Petri in the sense that Petri curves of genus $g$ form a non-empty open subset of the moduli space of curves of genus $g$ [11], [16].\",\n \"By classical BN theory, if $C$ is Petri, the BN locus $B(1,d_1,k)$ has dimension precisely $\\\\beta (1,d_1,k)$ whenever $0\\\\le \\\\beta (1,d_1,k)\\\\le g$ and is empty when $\\\\beta (1,d_1,k)<0$ .\",\n \"Sometimes we will not be able to obtain results for arbitrary Petri curves but only for curves which are general in the sense that they live in an unspecified open subset of the moduli space of curves of genus $g$ .\",\n \"The BN loci $B(1,d_1,k)(E_2)$ , with $E_2$ general, behave in a similar way to the classical BN loci [12].\",\n \"By Remark REF , this implies non-emptiness results for $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ when $n_1=1$ or $n_2=1$ .\",\n \"Moreover, when $n_2=1$ , we have an isomorphism $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\longrightarrow B(n_1,d_1+n_1d_2,k)\\\\times \\\\mbox{Pic}^{d_2}(C):\\\\ (E_1,L)\\\\mapsto (E_1\\\\otimes L,L),$ giving a description of $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ in terms of an untwisted BN locus.\",\n \"We know many cases in which such loci are non-empty; see, in particular, Section .\",\n \"In this paper, we shall assume that $n_i\\\\ge 2$ for $i=1,2$ .\",\n \"We can now state our first main theorem.\",\n \"Theorem 1.2 Suppose that $C$ is a smooth curve of genus $g\\\\ge 2$ and that $\\\\mu _i$ , $\\\\lambda _i$ $(i=1,2)$ are positive rational numbers.\",\n \"Let $B(n_i,d_i,k_i)$ be non-empty BN loci with $n_i\\\\ge 2$ , $d_i/n_i=\\\\mu _i$ and $k_i/n_i=\\\\lambda _i$ and suppose that $k=k_1k_2$ .\",\n \"If $\\\\mu _1<2$ and $\\\\mu _2\\\\le 2g$ , then $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ .\",\n \"If, in addition, $\\\\mu _1+\\\\mu _2<\\\\lambda _1\\\\lambda _2+g-1$ , then, for sufficiently large $n_1$ , $n_2$ , $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ .\",\n \"The major part of this theorem will be stated and proved in a more detailed version as Theorem REF , the last part being covered by Lemma REF .\",\n \"Using the results of Section , it is easy to find many cases where the hypotheses of the theorem are satisfied, thus obtaining the examples we are seeking (see Example REF ).\",\n \"There exist examples even when $n_1=n_2=2$ (see Example REF ).\",\n \"In Theorem REF , we show that this construction gives a new region in the $(\\\\mu ,\\\\lambda )$ -plane, whose points support non-empty (untwisted) BN loci.\",\n \"Our second main theorem is of a rather different nature, in that $d_2$ is negative and so $h^0(E_2)=0$ for all $E_2\\\\in M_2$ .\",\n \"Theorem 1.3 Suppose that $C$ is a smooth curve of genus $g\\\\ge 2$ , $n_1\\\\ge 2$ , $k_1>n_1$ , $n$ a positive integer and either $d>2ng$ or $d=2ng$ and $C$ is non-hyperelliptic.\",\n \"Suppose further that $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ and that $k\\\\le (d-n(g-1))(k_1-n_1)-nd_1.$ Let $(n_2,d_2)=(d-ng,-d)$ .\",\n \"Then $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ .\",\n \"If, in addition, $k=d(k_1-n_1)-e$ , where $e\\\\ge n(g-1)(k_1-n_1)+nd_1$ , and $d_1<k_1+n_1(g-1)-\\\\frac{g-1}{k_1-n_1},$ then, for any fixed values of $n_1$ , $d_1$ , $k_1$ , $n$ and $e$ , and $d\\\\gg 0$ , $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ .\",\n \"A more detailed version of this theorem will be stated and proved as Theorem REF .\",\n \"In Corollary REF and the related Example REF , we show that the requirement that $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ is compatible with the condition $d_1<k_1+n_1(g-1)-\\\\frac{g-1}{k_1-n_1}$ .\",\n \"Hence Theorem REF does lead to examples of twisted BN loci with negative expected dimension.\",\n \"Section is concerned with the non-emptiness of $B(n_1,d_1,k_1)$ .\",\n \"In subsection REF , we introduce the BN map and describe some sufficient conditions for non-emptiness.\",\n \"Subsection REF ) contains a description of kernel bundles and the dual span construction.\",\n \"We include also a description of BN loci for bundles of slope $\\\\le 2$ (subsection REF ).\",\n \"In section , we obtain an extended version of the main existence theorem of [12].\",\n \"Sections and contain our main results.\",\n \"The dual span construction works also when we consider pairs $(E,V)$ where $V$ is a subspace of $H^0(E)$ which generates $E$ .\",\n \"In the case where $E$ is a line bundle, there are then further stability results, which have consequences for the non-emptiness of $B(n_1,d_1,n_1+1)$ .\",\n \"We shall investigate applications of this to twisted BN loci in a future paper [7].\",\n \"Throughout the paper, $C$ will denote a smooth irreducible projective curve of genus $g\\\\ge 2$ defined over ${\\\\mathbb {C}}$ .\",\n \"For a vector bundle $E$ on $C$ of rank $n$ and degree $d$ , we denote by $\\\\mu (E):=\\\\frac{d}{n}$ the slope of $E$ .\",\n \"We write also $h^0(E)$ for the dimension of the space of sections $H^0(E)$ .\",\n \"The canonical line bundle on $C$ is denoted by $K_C$ .\",\n \"For any real number $a$ , we write $\\\\lfloor a\\\\rfloor $ and $\\\\lceil a\\\\rceil $ for the largest integer $\\\\le a$ and the smallest integer $\\\\ge a$ , respectively.\",\n \"We thank Lilia Alanis Lopez for help with the figures.\"\n ],\n [\n \"Non-emptiness of BN loci\",\n \"In this section, we will recall and extend some of the principal known results on the non-emptiness of the BN loci $B(n_1,d_1,k_1)$ .\",\n \"The answer is completely known for $k_1\\\\le n_1$ , but the problem is much more complicated (and still not completely solved) for $k_1>n_1$ (except for $g\\\\le 3$ and (almost) for hyperelliptic curves) (see [6]).\",\n \"For Petri curves, the naïve conjecture is that $B(n_1,d_1,k_1)$ is non-empty if and only if $\\\\beta (n_1,d_1,k_1)\\\\ge 0$ .\",\n \"From (REF ), this is equivalent to $d_1\\\\ge k_1+n_1(g-1)-\\\\frac{n_1(g-1)}{k_1/n_1}-\\\\frac{1}{k_1}.$ In fact, numerous examples show that (REF ) is neither necessary nor sufficient for non-emptiness of $B(n_1,d_1,k_1)$ (see, for example,[4], [2], [17], [15], [1]).\"\n ],\n [\n \"The BN map\",\n \"Following [6], the non-emptiness of $B(n_1,d_1,k_1)$ can be understood in terms of the BN map.\",\n \"For this, we map any non-empty BN locus $B(n_1,d_1,k_1)$ (or $\\\\widetilde{B}(n_1,d_1,k_1)$ ) with $0\\\\le d_1\\\\le n_1(2g-2)$ to the point in the plane $(\\\\mu ,\\\\lambda )=(d_1/n_1,k_1/n_1).$ We shall say that $(\\\\mu ,\\\\lambda )$ supports $B(n_1,d_1,k_1)$ (resp., $\\\\widetilde{B}(n_1,d_1,k_1)$ ) if (REF ) holds and $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ (resp., $\\\\widetilde{B}(n_1,d_1,k_1)\\\\ne \\\\emptyset $ ).\",\n \"We are interested here in regions in the $(\\\\mu ,\\\\lambda )$ -plane for which it is known that all points with rational coordinates support infinitely many BN loci.\",\n \"The first examples of such regions were determined by Teixidor i Bigas [21] and Mercat [18].\",\n \"To describe these in terms of the BN map, we follow [6].\",\n \"For $s\\\\ge 1$ , write $\\\\widehat{\\\\eta }^{\\\\prime }(s):=s+g-2-\\\\left\\\\lfloor \\\\frac{g-1}{s}\\\\right\\\\rfloor $ ; note that $d\\\\ge \\\\widehat{\\\\eta }^{\\\\prime }(s)$ if and only if $\\\\beta (1,d+1,s)\\\\ge 1$ , and that $\\\\widehat{\\\\eta }^{\\\\prime }(1)=0$ , $\\\\widehat{\\\\eta }^{\\\\prime }(g)=2g-2$ .\",\n \"Now define a function $t_g$ for $0\\\\le \\\\mu \\\\le 2g-2$ as follows: $t_g(\\\\mu ):={\\\\left\\\\lbrace \\\\begin{array}{ll}0,&\\\\mu =0;\\\\\\\\\\\\mu -\\\\lceil \\\\mu \\\\rceil +s,&\\\\mu \\\\in ]\\\\widehat{\\\\eta }^{\\\\prime }(s),\\\\widehat{\\\\eta }^{\\\\prime }(s)+1];\\\\\\\\s,&\\\\mu \\\\in ]\\\\widehat{\\\\eta }^{\\\\prime }(s)+1,\\\\widehat{\\\\eta }^{\\\\prime }(s+1)].\\\\end{array}\\\\right.\",\n \"}$ We define T to be the region in the $(\\\\mu ,\\\\lambda )$ -plane given by $\\\\mu ,\\\\lambda \\\\in {\\\\mathbb {Q}},\\\\ 0\\\\le \\\\mu \\\\le 2g-2,\\\\ 0<\\\\lambda \\\\le t_g(\\\\mu ).$ The region T is shown in [6] in the case $g=7$ .\",\n \"Note that Clifford's Theorem can be stated in the form $\\\\lambda \\\\le \\\\frac{\\\\mu }{2}+1$ and the boundary line for this inequality (the Clifford line) is shown in the figure.\",\n \"Moreover, the condition $\\\\beta (n_1,d_1,k_1)=1$ becomes $\\\\lambda (\\\\lambda -\\\\mu +g-1)=g-1$ (see (REF )), which defines a branch of a hyperbola with asymptotes $\\\\lambda =0$ and $\\\\lambda =\\\\mu -g+1$ (the BN curve), and is also shown in the figure.\",\n \"The following lemma summarises the results of Teixidor [21] and Mercat [18].\",\n \"Teixidor's proof is for general $C$ ; for $\\\\widetilde{B}(n_1,d_1,k_1)$ , this is sufficient to prove it for any smooth curve.\",\n \"Mercat's result [18] is valid for any smooth curve and implies our lemma.\",\n \"For the version stated below, see [6].\",\n \"Lemma 2.1 Let $C$ be a smooth curve of genus $g\\\\ge 2$ and suppose that $n_1\\\\ge 2$ .\",\n \"If $(\\\\mu ,\\\\lambda )\\\\in \\\\operatorname{T}$ , then $(\\\\mu ,\\\\lambda )$ supports $\\\\widetilde{B}(n_1,d_1,k_1)$ for all values of $n_1$ for which $n_1\\\\mu $ and $n_1\\\\lambda $ are both integers.\",\n \"The same is true for $B(n_1,d_1,k_1)$ , provided that the points $(\\\\mu ,\\\\lambda )=(\\\\widehat{\\\\eta }^{\\\\prime }(s)+1,\\\\lambda ),\\\\ s-1<\\\\lambda \\\\le s,\\\\ \\\\widehat{\\\\eta }^{\\\\prime }(s)+1\\\\ne \\\\widehat{\\\\eta }^{\\\\prime }(s+1).$ are excluded.\",\n \"Remark 2.2 Lemma REF is equivalent to the statement that $\\\\widetilde{B}(n_1,d_1,k_1)\\\\ne \\\\emptyset $ if $d_1\\\\ge k_1+n_1(g-1)-n_1\\\\left\\\\lfloor \\\\frac{g-1}{\\\\lceil k_1/n_1\\\\rceil }\\\\right\\\\rfloor .$ This can also be stated as $\\\\mu \\\\ge \\\\lambda +g-1-\\\\left\\\\lfloor \\\\frac{g-1}{\\\\left\\\\lceil \\\\lambda \\\\right\\\\rceil }\\\\right\\\\rfloor .$ For $B(n_1,d_1,k_1)$ , the case $d_1= n_1\\\\left(\\\\left\\\\lceil \\\\frac{k_1}{n_1}\\\\right\\\\rceil +g-1- \\\\left\\\\lfloor \\\\frac{g-1}{\\\\lceil k_1/n_1\\\\rceil }\\\\right\\\\rfloor \\\\right)$ must be excluded when $\\\\beta (1,d_1/n_1+1,\\\\left\\\\lceil k_1/n_1\\\\right\\\\rceil +1)\\\\le 0$ .\",\n \"(REF ) and (REF ) are essentially a restatement of Teixidor's conditions [21] and relate closely to Mercat's construction [18].\",\n \"Another method of constructing non-empty BN loci on a smooth curve $C$ is described in [6].\",\n \"This starts from the results of [4], [17], [19], which give precise conditions for non-emptiness when $d_1\\\\le 2n_1$ , and proceeds by tensoring by line bundles $L$ with $h^0(L)=s\\\\ge 1$ .\",\n \"Serre duality extends the region further and we denote the result by BMNO.\",\n \"To describe BMNO in the BN map, we write $\\\\hat{\\\\eta }(s):=s+g-1-\\\\left\\\\lfloor \\\\frac{g}{s}\\\\right\\\\rfloor $ for $s\\\\ge 1$ .\",\n \"The condition $d\\\\ge \\\\hat{\\\\eta }(s)$ is then equivalent to $\\\\beta (1,d,s)\\\\ge 0$ .\",\n \"On a Petri curve, this is equivalent by classical BN theory to the assertion that $B(1,d,s)\\\\ne \\\\emptyset $ .\",\n \"Note that $\\\\hat{\\\\eta }(1)=0$ and, on a Petri curve, $\\\\hat{\\\\eta }(2)$ is the gonality of $C$ .\",\n \"We now define a function $f_g(\\\\mu )$ for $0\\\\le \\\\mu \\\\le g-1$ as follows: $f_g(\\\\mu ):={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{s}{g}(\\\\mu -\\\\left\\\\lceil \\\\mu \\\\right\\\\rceil )+s,&\\\\mu \\\\in [\\\\hat{\\\\eta }(s),\\\\hat{\\\\eta }(s)+1];\\\\\\\\\\\\frac{s}{g}(\\\\mu -\\\\left\\\\lceil \\\\mu \\\\right\\\\rceil +1)+s,&\\\\mu \\\\in ]\\\\hat{\\\\eta }(s)+1,\\\\hat{\\\\eta }(s+1)-1];\\\\\\\\\\\\frac{\\\\hat{\\\\eta }(s+1)-s}{g}(\\\\mu -\\\\left\\\\lceil \\\\mu \\\\right\\\\rceil +1)+s,&\\\\mu \\\\in ]\\\\hat{\\\\eta }(s+1)-1,\\\\hat{\\\\eta }(s+1)[.\\\\end{array}\\\\right.\",\n \"}$ We extend the definition of $f_g$ , which is modified from that in [6] to allow $\\\\mu =0$ , to the whole interval $[0,2g-2]$ by Serre duality.\",\n \"The region BMNO in the $(\\\\mu ,\\\\lambda )$ -plane is then defined by the conditions $0\\\\le \\\\mu \\\\le 2g-2,\\\\ 0\\\\le \\\\lambda \\\\le f_g(\\\\mu ).$ For $g=10$ , this region is illustrated in [6], which also includes the top boundary of T, the Clifford line and the BN curve.\",\n \"In Figure 1, we recall [6], including also an indication of the excluded points from Lemmas REF and REF and Remark REF (ii).\",\n \"The following lemma summarises the principal result of [6], taking account of the results of [19] (see [6]).\",\n \"Lemma 2.3 Let $C$ be a smooth curve of genus $g\\\\ge 2$ and $n_1\\\\ge 2$ .\",\n \"If $(\\\\mu ,\\\\lambda )\\\\in \\\\operatorname{BMNO}$ , then $(\\\\mu ,\\\\lambda )$ supports $\\\\widetilde{B}(n_1,d_1,k_1)$ for infinitely many values of $n_1$ .\",\n \"If $C$ is non-hyperelliptic, the same holds for $B(n_1,d_1,k_1)$ , provided that points of the form $(\\\\mu ,\\\\lambda )$ such that $\\\\mu =\\\\hat{\\\\eta }(s)$ with $\\\\mu \\\\le g-1$ and $\\\\lambda >\\\\frac{s-1}{g}+s-1$ , together with their Serre duals and the points $((\\\\widehat{\\\\eta }(s)+1,s)$ , are excluded.\",\n \"Remark 2.4 (i) [6] provide information about which values of $n_1$ are included in the above result for any given $(\\\\mu ,\\\\lambda )$ .\",\n \"In particular, there is a dense set of points in BMNO for which all values of $n_1$ with $n_1\\\\mu , n_1\\\\lambda \\\\in {\\\\mathbb {Z}}$ are allowed.\",\n \"(ii) Some points excluded in Lemma REF are included in Lemma REF .\",\n \"In fact, if $g$ is a multiple of $s$ , then $\\\\widehat{\\\\eta }^{\\\\prime }(s)=\\\\widehat{\\\\eta }(s)$ , and the points $(\\\\widehat{\\\\eta }(s)+1,\\\\lambda )$ for $s-1<\\\\lambda <s$ are excluded by (REF ), but included by Lemma REF .\",\n \"(iii) For $g\\\\ge 4$ , there are many non-empty $B(n_1,d_1,k_1)$ corresponding to points outside the union of T and BMNO; see, for example, the figures in [13], [14], [15].\",\n \"We shall construct later a new region of the BN map, giving new points for all $g\\\\ge 5$ (see Theorem REF ).\",\n \"(iv)[6] also contains comprehensive results for hyperelliptic curves.\",\n \"Figure: NO_CAPTION\"\n ],\n [\n \"Kernel and dual span bundles\",\n \"Let $E$ be a generated bundle of rank $n$ and degree $d$ .\",\n \"Consider the exact sequence $0\\\\longrightarrow D_E^*\\\\longrightarrow H^0(E)\\\\otimes \\\\mathcal {O}\\\\longrightarrow E\\\\longrightarrow 0.$ The bundle $D_E^*$ may be referred to as a kernel bundle (or syzygy bundle) and has rank $d-ng$ and degree $-d$ .\",\n \"Dualising (REF ), we have $0\\\\longrightarrow E^*\\\\longrightarrow H^0(E)^*\\\\otimes \\\\mathcal {O}\\\\longrightarrow D_E\\\\longrightarrow 0.$ Following [10], we call $D_E$ a dual span bundle.\",\n \"The following theorem addresses the stability of $D_E$ and summarises results of Butler and Mercat (see also [5]).\",\n \"We write $S:=\\\\lbrace E\\\\in M(n,d)|D_E\\\\text{ is stable}\\\\rbrace .$ Theorem 2.5 Let $C$ be a smooth curve of genus $g\\\\ge 2$ .\",\n \"Suppose that either $d>2ng$ and $E\\\\in M(n,d)$ or $d=2ng$ , $C$ is non-hyperelliptic and $E$ is a general element of $M(n,d)$ .\",\n \"Then $E$ is generated and $D_E$ is stable of slope $\\\\le 2$ .\",\n \"In particular, $S$ is a non-empty open subset of $M(n,d)$ and $S=M(n,d)$ when $d>2ng$ .\",\n \"Moreover, the morphism $S\\\\longrightarrow B(d-ng,d,d-n(g-1)): E\\\\mapsto D_E$ is an isomorphism.\",\n \"Any stable bundle of slope $>2g-1$ is generated.\",\n \"Moreover, under the given hypotheses, $D_E$ is stable by [9].\",\n \"The fact that (REF ) is an isomorphism follows from [17] and [19].\"\n ],\n [\n \"Small slope\",\n \"When $d_1<2n_1$ , Lemma REF includes a complete answer to the question of the non-emptiness of $B(n_1,d_1,k_1)$ .\",\n \"The references [4], [17] (see also [3]) contain much further information about these BN loci, in particular that there exists $E_1\\\\in B(n_1,d_1,k_1)$ , taking one of a number of special forms.\",\n \"Theorem 2.6 Let $C$ be a curve of genus $g\\\\ge 2$ .\",\n \"If $d_1<2n_1$ and $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ , then there exists $E_1\\\\in B(n_1,d_1,k_1)$ taking one of the following forms.\",\n \"I $d_1<n_1+g$ .\",\n \"$E_1$ fits into an exact sequence $0\\\\longrightarrow \\\\mathcal {O}^{k_1}\\\\longrightarrow E_1\\\\longrightarrow F\\\\longrightarrow 0$ for some sheaf $F$ .\",\n \"II $d_1=n_1+g\\\\ell <2n_1$ , $\\\\ell \\\\ge 1$ .\",\n \"$E_1$ is the dual span of a stable bundle $E$ of slope $>2g$ and is therefore given by an exact sequence $0\\\\longrightarrow E^*\\\\longrightarrow H^0(E)^*\\\\otimes \\\\mathcal {O}\\\\longrightarrow E_1\\\\longrightarrow 0.$ Moreover, $h^0(E)=h^0(E_1)= n_1+\\\\ell $ .\",\n \"III $d_1=n_1+g\\\\ell +\\\\ell ^{\\\\prime }<2n_1$ with $\\\\ell >0$ and $0<\\\\ell ^{\\\\prime }<g$ .\",\n \"There exist exact sequences $0\\\\longrightarrow D_{E^{\\\\prime }}^*\\\\longrightarrow H^0(E^{\\\\prime })\\\\otimes \\\\mathcal {O}\\\\longrightarrow E^{\\\\prime }\\\\longrightarrow 0$ with $E^{\\\\prime }$ stable of rank $n_1+\\\\ell ^{\\\\prime }$ , degree $d_1$ and $h^0(E^{\\\\prime })=n_1+\\\\ell +\\\\ell ^{\\\\prime }$ , and $0\\\\longrightarrow \\\\mathcal {O}^{\\\\ell ^{\\\\prime }}\\\\longrightarrow E^{\\\\prime }\\\\longrightarrow E_1\\\\longrightarrow 0.$ I.\",\n \"By [4] (for $d_1\\\\le n_1$ ) and [17] (for $n_1<d_1<n_1+g$ ), the sections of $E_1$ are all carried by a subsheaf $\\\\mathcal {O}^{k_1}$ .\",\n \"This gives (REF ) II.\",\n \"We know that $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ if and only if $d_1\\\\ge n_1+g(k_1-n_1)$ .\",\n \"We can therefore choose $E_1\\\\in B(n_1,d_1,n_1+\\\\ell )$ and then $h^0(E_1)=n_1+\\\\ell $ .\",\n \"By [17], $E_1=D_E$ for some stable bundle $E$ of rank $\\\\ell $ and degree $d_1$ , hence of slope $>2g$ .\",\n \"So $h^0(E)=n_1+\\\\ell $ and we have the exact sequence (REF ).\",\n \"III.\",\n \"By [17] (see Case II), there exists a stable bundle $E^{\\\\prime }$ of rank $n_1+\\\\ell ^{\\\\prime }$ and degree $d_1$ with $h^0(E^{\\\\prime })=n_1+\\\\ell +\\\\ell ^{\\\\prime }$ , fitting in an exact sequence REF .\",\n \"Now, as in the proof of [17], we have an exact sequence (REF ), with $E_1\\\\in B(n_1,d_1,n_1+\\\\ell )$ .\",\n \"Since $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ , we have (see [17]) $n_1+g\\\\ell +\\\\ell ^{\\\\prime }=d_1\\\\ge n_1+g(k_1-n_1),$ hence $\\\\ell \\\\ge k_1-n_1$ and $E_1\\\\in B(n_1,d_1,k_1)$ .\",\n \"Remark 2.7 Theorem REF can be extended to the case $d_1=2n_1$ (see [19]).\",\n \"We have two cases.\",\n \"IV $C$ non-hyperelliptic.\",\n \"By the results of [19], we are in one of Cases I-III, except that in Case II, we now have $\\\\deg E=2n_1g$ .\",\n \"There is one further case, namely when $E_1=D_{K_C}\\\\in B(g-1,2g-2,g)$ .\",\n \"V $C$ hyperelliptic.\",\n \"In this case $B(n_1,2n_1,k_1)\\\\ne \\\\emptyset $ if and only if $k_1\\\\le n_1$ (see [19]).\",\n \"When this holds, there exists $E_1\\\\in B(n_1,2n_1,k_1)$ which admits $\\\\mathcal {O}^{k_1}$ as a subsheaf.\"\n ],\n [\n \"A non-emptiness result for twisted BN loci\",\n \"In this section we give an extended version of [12], which we need for comparison purposes.\",\n \"Theorem 3.1 Let $C$ be a smooth curve of genus $g\\\\ge 2$ and $E_2$ any vector bundle of rank $n_2$ and degree $d_2$ on $C$ .\",\n \"Let $d_0$ and $k_0$ be integers satisfying $\\\\beta (1,d_0,k_0)(E_2)\\\\ge 1$ , and suppose that $n_1\\\\ge 2$ .\",\n \"Then, (i) if $k\\\\le n_1k_0$ and $d_1\\\\ge n_1d_0+1\\\\ (\\\\text{resp.\",\n \"}, d_1\\\\ge n_1d_0)$ , the twisted BN locus $B(n_1,d_1,k)(E_2) \\\\ (\\\\text{resp.\",\n \"}, \\\\widetilde{B}(n_1,d_1,k)(E_2))$ is non-empty and $\\\\beta (n_1,d_1,k)(E_2)>(\\\\text{resp.\",\n \"}, \\\\ge )\\\\,1$ ; (ii) if $k_1:=k-n_2d_1+n_1d_2-n_1n_2(g-1)\\\\le n_1k_0$ and $-d_1\\\\ge n_1d_0+1\\\\ (\\\\text{resp.\",\n \"}, -d_1\\\\ge n_1d_0)$ , the twisted BN locus $B(n_1,d_1,k)(K_C\\\\otimes E_2^*)\\\\ (\\\\text{resp.\",\n \"}, \\\\widetilde{B}(n_1,d_1,k)(K_C\\\\otimes E_2^*))$ is non-empty and $\\\\beta (n_1,d_1,k)(K_C\\\\otimes E_2^*)>(\\\\text{resp.\",\n \"}, \\\\ge )\\\\,1.$ (i) The non-emptiness of $B(n_1,d_1,k)(E_2)$ (resp., $\\\\widetilde{B}(n_1,d_1,k)(E_2)$ ) is [12].\",\n \"The inequality $\\\\beta (n_1,d_1,k)(E_2)>(\\\\ge )1$ follows from the observation that $\\\\beta (n_1,n_1d_0,n_1k_0)(E_2)\\\\ge 1 \\\\Longleftrightarrow \\\\beta (1,d_0,k_0)(E_2)\\\\ge 1$ and the fact that $\\\\beta (n_1,d_1,k)(E_2)$ is a strictly increasing function of $d_1$ and a decreasing function of $k$ .\",\n \"(ii) Under the stated hypotheses, in the case $-d_1\\\\ge n_1d_0+1$ , (i) implies that $B(n_1,-d_1,k_1)(E_2)\\\\ne \\\\emptyset $ .\",\n \"By Serre duality, $B(n_1,d_1,k)(K_C\\\\otimes E_2^*)\\\\cong B(n_1,-d_1,k_1)(E_2).$ So $B(n_1,d_1,k)(K_C\\\\otimes E_2^*)\\\\ne \\\\emptyset $ .\",\n \"Moreover, as in the proof of (i), $\\\\beta (n_1,-d_1,k_1)(E_2)>1$ and $\\\\beta (n_1,d_1,k)(K_C\\\\otimes E_2^*)=n_1^2(g-1)+1-kk_1=\\\\beta (n_1,-d_1,k_1)(E_2).$ The same proof applies in the case $-d_1\\\\ge n_1d_0$ if we replace $>$ in (REF ) by $\\\\ge $ and $B$ with $\\\\widetilde{B}$ throughout.\",\n \"Corollary 3.2 Under the hypotheses of Theorem REF , the twisted BN locus $B^k(\\\\mathcal {U}_!,\\\\mathcal {U}_2)\\\\ (\\\\text{resp.\",\n \"}, \\\\widetilde{B}(\\\\mathcal {U}_1,\\\\mathcal {U}_2))$ is non-empty.\",\n \"Moreover $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)>n_2^2(g-1)+2\\\\ >0\\\\ (\\\\mbox{resp.,}\\\\ge n_2^2(g-1)+2).$ In the theorem, we can take $E_2$ to be stable; the required non-emptiness then follows (see Remark REF ).\",\n \"The inequality for $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ follows from the fact that $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)= \\\\dim M_2+\\\\beta (n_1,d_1,k)(E_2)$ (see (REF ) and (REF )).\",\n \"Remark 3.3 (i) As indicated in the introduction, the fact that the BN numbers are positive will be crucial for us in establishing that many of the examples constructed in this paper are new.\",\n \"(ii) (REF ) makes Theorem REF (i) the precise analogue of the theorem of Teixidor i Bigas [21] for twisted BN loci $B(n_1,d_1,k)(E_2)$ , at least on a general curve; the proof in [12] depends on Mercat's construction [18] and holds for any curve.\",\n \"(iii) If we replace $E_2$ by $E_2\\\\otimes L$ , where $L$ is a line bundle of degree $\\\\ell $ , and simultaneously replace $d_0$ by $d_0-\\\\ell $ and $d_1$ by $d_1-n_1\\\\ell $ in Theorem REF (i), the BN numbers and the isomorphism classes of the BN loci are unchanged.\",\n \"This applies also to Theorem REF (ii) except that now $d_1$ must be replaced by $d_1+n_1\\\\ell $ .\"\n ],\n [\n \"Examples with $k=k_1k_2$\",\n \"In this section, we consider the following situation.\",\n \"Suppose that $B(n_i,d_i,k_i)\\\\ne \\\\emptyset $ for $i=1,2$ and that there exist $E_i\\\\in B(n_i,d_i,k_i)$ such that $h^0(E_1\\\\otimes E_2)\\\\ge k_1k_2$ .\",\n \"When $n_2=1$ , this is exactly the situation considered in [6].\",\n \"Here, however, we assume $n_i\\\\ge 2$ .\",\n \"One scenario to which this applies is when $E_1=D_E$ .\"\n ],\n [\n \"The construction\",\n \"Proposition 4.1 Let $C$ be a smooth curve of genus $g$ .\",\n \"Suppose that $E$ is a generated stable (resp., semistable) bundle of rank $n$ and degree $d_1$ and that $h^0(E)=k_1=n+n_1$ .\",\n \"Suppose further that $D_E$ is stable (resp., semistable) and that $E_2\\\\in B(n_2,d_2,k_2)\\\\ (\\\\text{resp.\",\n \"}, [E_2]\\\\in \\\\widetilde{B}(n_2,d_2,k_2))$ with $\\\\mu (E_2)<\\\\mu (E)$ .\",\n \"Then $h^0(D_E\\\\otimes E_2)\\\\ge k_1k_2\\\\ (\\\\text{resp.\",\n \"}, h^0(D_E\\\\otimes \\\\text{gr}E_2)\\\\ge k_1k_2)$ .\",\n \"In particular, $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset \\\\ (\\\\text{resp.\",\n \"}, \\\\widetilde{B}^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset )$ .\",\n \"Consider the sequence (REF ) and suppose first that we are in the stable case.\",\n \"Note that $D_E\\\\in B(n_1,d_1,k_1)$ , since $D_E$ is stable by hypothesis and $h^0(E^*)=0$ .\",\n \"Note also that $h^0(E^*\\\\otimes E_2)=0$ since $E^*\\\\otimes E_2$ is semistable of negative slope.\",\n \"Tensoring (REF ) by $E_2$ and taking global sections, we obtain $h^0(D_E\\\\otimes E_2)\\\\ge h^0(E)\\\\cdot h^0(E_2)\\\\ge k_1k_2.$ The semistable case proceeds similarly.\",\n \"Note that, by (REF ), we have $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)=(n_1^2+n_2^2)(g-1)+2-k(k-\\\\chi ).$ We have also $\\\\beta (n_1n_2,n_2d_1+n_1d_2,k)=n_1^2n_2^2(g-1)+1-k(k-\\\\chi ).$ Lemma 4.2 Let $\\\\mu _i$ , $\\\\lambda _i$ be positive rational numbers for $i=1,2$ and suppose that $d_i=n_i\\\\mu _i$ , $k_i=n_i\\\\lambda _i$ , $k=k_1k_2$ .\",\n \"If $\\\\mu _1+\\\\mu _2<\\\\lambda _1\\\\lambda _2+g-1,$ then, for fixed $\\\\mu _i$ , $\\\\lambda _i$ and sufficiently large $n_1$ , $n_2$ , $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ .\",\n \"Since $k=k_1k_2$ , (REF ) is equivalent to $\\\\frac{\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)}{n_1^2n_2^2}=\\\\left(\\\\frac{1}{n_2^2}+\\\\frac{1}{n_1^2}\\\\right)(g-1)+\\\\frac{2}{n_1^2n_2^2}-\\\\lambda _1\\\\lambda _2(\\\\lambda _1\\\\lambda _2-(\\\\mu _1+\\\\mu _2)+g-1).$ The result follows.\",\n \"Theorem 4.3 Suppose that $C$ is a smooth curve of genus $g\\\\ge 2$ and that $\\\\mu _i$ , $\\\\lambda _i$ ($i=1,2$ ) are positive rational numbers, $\\\\mu _1<2$ and $\\\\mu _2\\\\le 2g$ .\",\n \"(i) If $B(n_i,d_i,k_i)$ are non-empty BN loci with $n_i\\\\ge 2$ supported by $(\\\\mu _i,\\\\lambda _i)$ and $k=k_1k_2$ , then $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ .\",\n \"(ii) If $\\\\widetilde{B}(n_i,d_i,k_i)$ are non-empty BN loci with $n_i\\\\ge 2$ supported by $(\\\\mu _i,\\\\lambda _i)$ and $k=k_1k_2$ , then $\\\\widetilde{B}^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ and $\\\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)\\\\ne \\\\emptyset .$ (i) Let $E_2\\\\in B(n_2,d_2,k_2)$ with $d_2\\\\le 2n_2g$ .\",\n \"We prove that there exists $E_1\\\\in B(n_1,d_1,k_1)$ such that $h^0(E_1\\\\otimes E_2)\\\\ge k_1k_2$ .\",\n \"The proof is the same as in [6] with $L$ replaced by $E_2$ .\",\n \"We need to consider cases I-III of Theorem REF .\",\n \"I.\",\n \"$\\\\mathcal {O}^{k_1}\\\\otimes E_2$ is a subsheaf of $E_1\\\\otimes E_2$ .\",\n \"It follows at once that $k_1k_2\\\\le h^0(E_1\\\\otimes E_2$ ).\",\n \"II.\",\n \"in (REF ), $E_1\\\\cong D_E$ and $\\\\mu (E_2)\\\\le 2g<\\\\mu (E)$ .\",\n \"The result follows from Proposition REF .\",\n \"III.\",\n \"Tensoring (REF ) and (REF ) by $E_2$ and noting that $D_{E^{\\\\prime }}^*\\\\otimes E_2$ is semistable of negative slope, we obtain $h^0(E^{\\\\prime }\\\\otimes E_2)\\\\ge h^0(E^{\\\\prime }) h^0(E_2)$ and hence $h^0(E_1\\\\otimes E_2)\\\\ge ( h^0(E^{\\\\prime })-\\\\ell ^{\\\\prime })h^0(E_2)=(n_1+\\\\ell )h^0(E_2)\\\\ge k_1k_2.$ (ii) This follows in the same way as (i), together with the fact that the tensor product of semistable bundles is semistable.\",\n \"Remark 4.4 If $C$ is non-hyperelliptic, the theorem holds under the hypotheses $\\\\mu _1\\\\le 2$ , $\\\\mu _2<2g$ (see Case IV in Remark REF ).\"\n ],\n [\n \"Examples\",\n \"Using Lemmas REF and REF , it is easy to find many cases as in Theorem REF where (REF ) is also satisfied, thus yielding examples of non-empty BN loci $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ with negative BN number.\",\n \"These cannot be obtained from Theorem REF .\",\n \"Example 4.5 Suppose that $\\\\mu _i\\\\le \\\\frac{1}{2}$ for $i=1,2$ .\",\n \"From [4] (see Lemma REF and Figure 1), we can take $\\\\lambda _i$ to be any positive rational numbers with $\\\\lambda _i\\\\le 1-\\\\frac{1}{g}(1-\\\\mu _i)$ .\",\n \"Then, Lemma REF and Theorem REF apply, giving $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ and $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ for sufficiently large $n_1$ , $n_2$ , even when $g=2$ .\",\n \"More generally, we can take $\\\\mu _1\\\\le 1$ and $\\\\mu _2\\\\le g-2$ for $g\\\\ge 3$ , or $\\\\mu _1<2$ and $\\\\mu _2\\\\le g-3$ for $g\\\\ge 4$ , with $\\\\lambda _i$ any positive rational numbers for which there exist non-empty BN loci $B(n_i,d_i,k_i)$ supported by $(\\\\mu _i,\\\\lambda _i)$ .\",\n \"It is even possible to find examples with $n_1=n_2=2$ .\",\n \"Example 4.6 Suppose that $n_1=n_2=2$ .\",\n \"Then, under the hypotheses of Theorem REF , $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ if and only if $\\\\left(\\\\frac{1}{2}-\\\\lambda _1\\\\lambda _2\\\\right)(g-1)+\\\\frac{1}{8}-\\\\lambda _1\\\\lambda _2(\\\\lambda _1\\\\lambda _2-(\\\\mu _1+\\\\mu _2))<0$ (see (REF )).\",\n \"We can take $B(n_i,d_i,k_i)=B(2,3,2)$ for $i=1,2$ ; this is non-empty by either Lemma REF or Lemma REF .\",\n \"(REF ) now becomes $-\\\\frac{1}{2}(g-1)+\\\\frac{1}{8}+2<0,$ which is true for $g\\\\ge 6$ .\",\n \"Here, $B(2,3,2)$ is the only candidate for $B(2,d_1,k_1)$ for applying Theorem REF , but, for $i=2$ , we could take any non-empty $B(2,d_2,k_2)$ instead of $B(2,3,2)$ ; once we have fixed $d_2$ with $d_2\\\\le 4g$ and $k_2\\\\ge 2$ , (REF ) will hold for sufficiently large $g$ .\"\n ],\n [\n \"A new region in the BN map\",\n \"Substituting $k=k_1k_2$ in (REF ) and dividing by $n_1^2n_2^2$ , we have $\\\\beta :=\\\\frac{\\\\beta (n_1n_2,n_2d_1+n_1d_2,k)}{n_1^2n_2^2}=\\\\\\\\=g-1+\\\\frac{1}{n_1^2n_2^2}-\\\\lambda _1\\\\lambda _2(\\\\lambda _1\\\\lambda _2-(\\\\mu _1+\\\\mu _2)+g-1).$ It is an interesting question as to whether $\\\\beta $ can ever be negative under the hypotheses of Theorem REF and the assumption that $\\\\beta (n_2,d_2,k_2)\\\\ge 1$ .\",\n \"One can see easily that $\\\\beta >0$ if $\\\\lambda _1\\\\le 1$ , but other cases are not so straightforward.\",\n \"Independently of this, one can ask whether the theorem can give rise to new examples of non-empty BN loci in $M(n_1n_2,n_2d_1+n_1d_2,k_1k_2)$ .\",\n \"Our next result shows that this is indeed the case.\",\n \"We define a new region of the BN map as follows.\",\n \"Let $R:=\\\\left\\\\lbrace (\\\\mu _1,\\\\mu _2,\\\\lambda _1,\\\\lambda _2)\\\\in {\\\\mathbb {Q}}^4\\\\,\\\\left|\\\\begin{array}{c}\\\\,0<\\\\mu _1<2,0<\\\\mu _2\\\\le 2g-2-\\\\mu _1,\\\\\\\\\\\\lambda _i\\\\le \\\\max \\\\lbrace f_g(\\\\mu _i),t_g(\\\\mu _i)\\\\rbrace \\\\end{array}\\\\right.\\\\right\\\\rbrace $ and let $h:R\\\\rightarrow {\\\\mathbb {Q}}^2$ be defined by $h(\\\\mu _1,\\\\mu _2,\\\\lambda _1,\\\\lambda _2)=(\\\\mu _1+\\\\mu _2,\\\\lambda _1\\\\lambda _2).$ Definition 4.7 The region BPN in the BN map is defined to be the union of $\\\\operatorname{Im}h$ and its Serre dual.\",\n \"Theorem 4.8 Let $C$ be a smooth curve of genus $g\\\\ge 2$ and $n_i\\\\ge 2$ .\",\n \"(i) If $(\\\\mu ,\\\\lambda )=h(\\\\mu _1,\\\\mu _2,\\\\lambda _1,\\\\lambda _2)\\\\in \\\\operatorname{BPN}$ , then $(\\\\mu _1+\\\\mu _2,\\\\lambda _1\\\\lambda _2)$ supports a BN locus $\\\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)$ with $k=k_1k_2$ for infinitely many values of $n_1$ , $n_2$ ; (ii) If $g\\\\ge 5$ , $\\\\operatorname{BPN} $ is not contained in $\\\\operatorname{T\\\\cup BMNO}$ .\",\n \"(i) By Lemmas REF and REF , $(\\\\mu _i,\\\\lambda _i)$ supports the BN locus $\\\\widetilde{B}(n_i,d_i,k_i)$ ($i=1,2$ ) for infinitely many values of $n_i$ .\",\n \"The result follows from Theorem REF (ii).\",\n \"(ii) Suppose that $2<\\\\mu <4$ .\",\n \"We can write $\\\\mu =\\\\mu _1+\\\\mu _2$ with $1<\\\\mu _i<2$ for $i=1,2$ .\",\n \"Now, if $\\\\lambda _i=1+\\\\frac{1}{g}(\\\\mu _i-1)$ , then, by [17] (see also Lemma REF ), $(\\\\mu _i,\\\\lambda _i$ ) is in the region BMNO and $(\\\\mu _i,\\\\lambda _i)$ supports $B(n_i,d_i,k_i)$ for any allowable $n_i$ .\",\n \"So, by Theorem REF (ii), $\\\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)\\\\ne \\\\emptyset $ when $k=k_1k_2$ .\",\n \"On the other hand, $\\\\lambda _1\\\\lambda _2>1+\\\\frac{1}{g}(\\\\mu _1+\\\\mu _2-2)\\\\ge 1+\\\\frac{1}{g}(\\\\mu _1+\\\\mu _2-\\\\left\\\\lceil \\\\mu _1+\\\\mu _2\\\\right\\\\rceil +1).$ It follows from Lemma REF and REF that $(\\\\mu _1+\\\\mu _2,\\\\lambda _1\\\\lambda _2)$ is not in the region BMNO if $\\\\widehat{\\\\eta }(2)\\\\ge 5$ , in other words, if $g\\\\ge 7$ .\",\n \"It is clear also that $(\\\\mu _1+\\\\mu _2,\\\\lambda _1\\\\lambda _2)$ is not in the region T, whose upper boundary is given by $\\\\lambda =1$ in this range.\",\n \"In fact, our construction also gives new points for $g=5$ (compare [14]) and $g=6$ (compare [15]).\",\n \"(When $g=6$ , there is a missing grey area in [15]; the upper boundary for $\\\\operatorname{BMNO}$ should be given by $\\\\lambda =1+\\\\frac{1}{2}(\\\\mu -3)$ for $3<\\\\mu <4$ .\",\n \"This does not affect our assertion.)\",\n \"Remark 4.9 The construction in the proof of Theorem REF shows that, for $g\\\\ge 7$ , the top boundary of BPN in the range $2<\\\\mu <4$ is given by $\\\\lambda =1+\\\\frac{1}{g}(\\\\mu -2)+\\\\frac{1}{g^2}\\\\left(\\\\frac{\\\\mu -2}{2}\\\\right)^2$ , whereas the top boundary of BMNO is given by $\\\\lambda =1+\\\\frac{1}{g}(\\\\mu -2)$ for $2<\\\\mu <3$ and by $\\\\lambda =1+\\\\frac{1}{g}(\\\\mu -3)$ for $3<\\\\mu <4$ .\",\n \"At the point $(2,1)$ , the line $\\\\lambda =1+\\\\frac{1}{g}(\\\\mu -2)$ is the tangent to the parabola $\\\\lambda =1+\\\\frac{1}{g}(\\\\mu -2)+\\\\frac{1}{g^2}\\\\left(\\\\frac{\\\\mu -2}{2}\\\\right)^2$ .\",\n \"For $g=10$ , BPN is illustrated in Figure 2, which extends to the range $4<\\\\mu <5$ .\",\n \"Figure 2 also shows an expanded version of the BN map close to $\\\\mu =3$ .\",\n \"Figure: NO_CAPTION Remark 4.10 (i) BPN is a new region in the BN map (see Figure 2), whereas the papers [14], [15] give examples for isolated values of $\\\\mu $ , some of which extend to higher values of $g$ .\",\n \"When $g=4$ , on the other hand, there is an additional grey area of the BN map (see [13]), which contains the new part of the region BPN in the range $2<\\\\mu <3$ .\",\n \"(ii) At least for Petri curves, some of the non-empty BN loci $\\\\widetilde{B}(n_1n_2,n_2d_1+n_1d_2,k)\\\\subset \\\\widetilde{M}(n_1n_2,n_2d_1+n_1d_2)$ constructed in Theorem REF are new.\",\n \"It is possible to choose the ranks and degrees so that $\\\\gcd (n_1n_2,n_2d_1+n_1d_2)=1$ ; for example, if $g$ is odd, one can take $(n_1,d_1,k_1)=(g+1,2g+1,g+2),\\\\ (n_2,d_2,k_2)=(g+2,2g+2,g+3).$ Then $B(n_1n_2,n_2d_1+n_1d_2,k)\\\\ne \\\\emptyset $ .\",\n \"(iii) One can check that, in the framework of Theorem REF , the examples of twisted BN loci with negative expected dimension constructed in [12] give rise to points of T. It follows that our examples are different.\"\n ],\n [\n \"The construction\",\n \"The essential ingredient in proving Theorem REF is Mercat's construction [18], while Theorem REF depends on the constructions of [4], [17].\",\n \"Another way of constructing non-empty twisted BN loci is described in [8] and uses kernel bundles.\",\n \"The following proposition is contained in [8] and plays an essential rôle in the proof of [8].\",\n \"We include a proof for the convenience of the reader.\",\n \"Proposition 5.1 Suppose that $E_1$ has rank $n_1$ and degree $d_1$ with $h^0(E_1)\\\\ge k_1$ and that $E$ is as in (REF ).\",\n \"If $h^1(E_1\\\\otimes E)=0$ and $k\\\\le (d-n(g-1))(k_1-n_1)-nd_1,$ then $h^0(E_1\\\\otimes D_E^*)\\\\ge k$ .\",\n \"We have $h^0(E_1\\\\otimes E)=nd_1+n_1d-nn_1(g-1)$ and $h^0(E)\\\\ge d-n(g-1)$ .\",\n \"So, tensoring (REF ) by $E_1$ and taking global sections, $h^0(E_1\\\\otimes D_E^*)\\\\ge (d-n(g-1))k_1-nd_1-n_1d+nn_1(g-1).$ Simplifying and comparing with (REF ), this gives the result.\",\n \"We have the following immediate corollary (see Remark REF ).\",\n \"Corollary 5.2 Suppose that the hypotheses of Proposition REF are satisfied.\",\n \"(i) If $E_1$ is stable $(\\\\text{resp., semistable})$ , then $B(n_1,d_1,k)(D_E^*)\\\\ne \\\\emptyset $ $(\\\\text{resp.\",\n \"}, \\\\widetilde{B}(n_1,d_1,k)(D_E^*)\\\\ne \\\\emptyset )$ .\",\n \"(ii) If $D_E^*$ is stable $(\\\\text{resp., semistable})$ , then $B(h^0(E)-n,-d,k)(E_1)\\\\ne \\\\emptyset \\\\ (\\\\text{resp.\",\n \"},\\\\widetilde{B}(h^0(E)-n,-d,k)(E_1)\\\\ne \\\\emptyset )$ .\",\n \"(iii) If both $E_1$ and $D_E^*$ are stable $(\\\\text{resp., semistable})$ and $(n_2,d_2)=(h^0(E)-n,-d)$ , then $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset \\\\ (\\\\text{resp.\",\n \"}, \\\\widetilde{B}^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset )$ .\",\n \"Our object here is to use Corollary REF (iii) to obtain new non-empty examples of twisted BN loci.\",\n \"We assume that $n_1\\\\ge 2$ .\",\n \"In order for (REF ) to yield positive values of $k$ when $d>n(g-1)$ , which will always be the case, we clearly require $k_1>n_1$ .\",\n \"The key point we require in order to apply Corollary REF (iii) is that $D_E^*$ and $E_1$ are stable.\",\n \"We have already discussed the existence of stable $E_1$ in subsection REF and the stability of $D_E$ in subsection REF .\",\n \"We can now state and prove a general theorem, which includes Theorem REF .\",\n \"Theorem 5.3 Suppose that $C$ is a smooth curve of genus $g\\\\ge 2$ , $n_1\\\\ge 2$ , $k_1>n_1$ and either $d>2ng$ or $d=2ng$ and $C$ is non-hyperelliptic.\",\n \"Suppose further that $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ and that (REF ) holds.\",\n \"Let $(n_2,d_2)=(d-ng,-d)$ and suppose that $S$ is defined as in (REF ).\",\n \"Then $S\\\\ne \\\\emptyset $ and the morphism $B(n_1,d_1,k_1)\\\\times S\\\\longrightarrow M_1\\\\times M_2:(E_1,E)\\\\mapsto (E_1,D_E^*)$ is injective and has image contained in $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ ; in particular, $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset .$ If, in addition, $k=d(k_1-n_1)-e$ , where $e\\\\ge n(g-1)(k_1-n_1)+nd_1,$ and $d_1<k_1+n_1(g-1)-\\\\frac{g-1}{k_1-n_1},$ then, for any fixed values of $n_1$ , $d_1$ , $k_1$ , $n$ and $e$ satisfying (REF ), and $d\\\\gg 0$ , $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0.$ The first part follows from Theorem REF , Proposition REF and Corollary REF (iii).\",\n \"We need to note that $h^1(E_1\\\\otimes E)=0$ since $E_1\\\\otimes E$ is semistable of slope greater than $2g$ .\",\n \"For the second part, we have $\\\\nonumber \\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)&=&n_1^2(g-1)+1+(d-ng)^2(g-1)+1\\\\\\\\&&-(d(k_1-n_1)-e)(d(k_1-n_1)-e-\\\\chi ),$ where, by (REF ), $\\\\nonumber \\\\chi &=&(d-ng)d_1-n_1d-n_1(d-ng)(g-1)\\\\\\\\&=&(d-ng)(d_1-n_1g)-nn_1g.$ The right-hand side of (REF ) is a quadratic in $d$ , with leading coefficient $g-1-(k_1-n_1)(k_1-n_1-d_1+n_1g).$ So $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ for $d\\\\gg 0$ provided that $d_1<k_1+n_1(g-1)-\\\\frac{g-1}{k_1-n_1}.$ Remark 5.4 (i) Under the hypotheses of Theorem REF , including the inequalities (REF ) and REF , we deduce that the non-emptiness of $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ cannot be obtained from Theorem REF .\",\n \"In fact, there are several choices we can make in attempting to apply Theorem REF to this situation.\",\n \"We can take $E_2$ to be any of $E_1\\\\otimes L$ , $D_E^*\\\\otimes L$ , $K_C\\\\otimes E_1^*\\\\otimes L$ and $K_C\\\\otimes D_E\\\\otimes L$ , where $L$ is any line bundle, with the appropriate choice of $(n_1,d_1)$ in each case.\",\n \"In all cases, we obtain $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ge 1$ .\",\n \"In the absence of (REF ), we would have to work through all these cases separately.\",\n \"(ii) If we assume only that $\\\\widetilde{B}(n_1,d_1,k_1)\\\\ne \\\\emptyset $ and $d\\\\ge 2ng$ with $C$ any smooth curve of genus $g\\\\ge 2$ , we obtain $\\\\widetilde{B}(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ .\",\n \"Also, if (REF ) holds and $d\\\\gg 0$ , then the non-emptiness of $\\\\widetilde{B}(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ cannot be obtained from Theorem REF .\"\n ],\n [\n \"Examples\",\n \"To show that Theorem REF provides the examples we are seeking, we need to show that the conditions $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ and (REF ) are compatible.\",\n \"For this, we need more precise statements concerning the possible values of $d_1$ .\",\n \"Corollary 5.5 Suppose that $C$ is a smooth curve of genus $g\\\\ge 3$ , $n_1\\\\ge 2$ , $k_1>n_1$ and $(n_2,d_2)=(d-ng,-d)$ with either $d>2ng$ or $d=2ng$ and $C$ non-hyperelliptic.\",\n \"Suppose further that $k_1+n_1(g-1)-n_1\\\\left\\\\lfloor \\\\frac{g-1}{\\\\lceil k_1/n_1\\\\rceil }\\\\right\\\\rfloor \\\\le d_1<k_1+n_1(g-1)-\\\\frac{g-1}{k_1-n_1}$ and that $d_1$ is not divisible by $n_1$ .\",\n \"Then, $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ .\",\n \"Moreover, for any fixed value of $e$ satisfying (REF ) and $d\\\\gg 0$ , $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ , but $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ .\",\n \"In particular, $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ is of negative expected dimension and the non-emptiness of $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)$ cannot be obtained from Theorem REF .\",\n \"The fact that $B(n_1,d_1,k_1)\\\\ne \\\\emptyset $ follows from Remark REF .\",\n \"Theorem REF then implies that $B^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)\\\\ne \\\\emptyset $ and that $\\\\beta ^k(\\\\mathcal {U}_1,\\\\mathcal {U}_2)<0$ when (REF ) holds and $d\\\\gg 0$ .\",\n \"Hence, the non-emptiness cannot be obtained from Theorem REF (see Remark REF (i)).\",\n \"To obtain examples from this corollary, we need, in particular, to prove the existence of $d_1$ satisfying (REF ).\",\n \"This is equivalent to showing that $n_1\\\\left\\\\lfloor \\\\frac{g-1}{\\\\lceil k_1/n_1\\\\rceil }\\\\right\\\\rfloor >\\\\frac{g-1}{k_1-n_1}$ In fact (REF ) cannot hold if $g=2$ (which is why we have assumed $g\\\\ge 3$ in Corollary REF ) or, more generally, if $k_1\\\\ge n_1(g-1)+1$ , since, in that case, the left-hand side of (REF ) is 0.\",\n \"We therefore assume that $n_1<k_1\\\\le n_1(g-1)$ Example 5.6 Suppose that $g=3$ and (REF ) holds.\",\n \"Then (REF ) holds if and only $n_1(k_1-n_1)>2$ , in other words, either $n_1\\\\ge 3$ or $n_1=2$ and $k_1=4$ .\",\n \"In the latter case, the only possible value for $d_1$ is $d_1=6$ .\",\n \"This is not permitted by the hypotheses of Corollary REF and, in fact $B(2,6,4)=\\\\emptyset $ .\",\n \"For $n_1\\\\ge 3$ , there do exist solutions of (REF ) for which $d_1$ is not divisible by $n_1$ .\",\n \"In fact, if $k_1=n_1+2$ , (REF ) gives $2n_1+2\\\\le d_1<3n_1+1$ , giving $n_1-2$ values of $d_1$ not divisible by $n_1$ .\",\n \"If $n_1+3\\\\le k_1\\\\le 2n_1$ , there are $n_1-1$ permitted values of $d_1$ in the range $k_1+n_1\\\\le d_1<k_1+2n_1$ .\",\n \"A similar analysis can be carried out for $g\\\\ge 4$ ; in this case, in particular, $B(2,4g-5,2g-2)\\\\ne \\\\emptyset $ and (REF ) is satisfied, so examples exist for $n_1=2$ , as well as for larger $n_1$ .\",\n \"Remark 5.7 When $k_1=n_1+1$ , there are additional cases in which the non-emptiness of $B(n_1,d_1,k_1)$ is known.\",\n \"This is related to the extended dual span construction, which involves the stability of bundles $D_{L,V}$ , where $L$ is a line bundle and $V$ is a subspace of $H^0(L)$ which generates $L$ .\",\n \"Applications of this will be considered in a paper currently in preparation [7].\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05573"}}},{"rowIdx":95,"cells":{"text":{"kind":"list like","value":[["Multimodal and Explainable Internet Meme Classification"],["Abstract Warning: this paper contains content that may be offensive or upsetting.","In the current context where online platforms have been effectively weaponized in a variety of geo-political events and social issues, Internet memes make fair content moderation at scale even more difficult.","Existing work on meme classification and tracking has focused on black-box methods that do not explicitly consider the semantics of the memes or the context of their creation.","In this paper, we pursue a modular and explainable architecture for Internet meme understanding.","We design and implement multimodal classification methods that perform example- and prototype-based reasoning over training cases, while leveraging both textual and visual SOTA models to represent the individual cases.","We study the relevance of our modular and explainable models in detecting harmful memes on two existing tasks: Hate Speech Detection and Misogyny Classification.","We compare the performance between example- and prototype-based methods, and between text, vision, and multimodal models, across different categories of harmfulness (e.g., stereotype and objectification).","We devise a user-friendly interface that facilitates the comparative analysis of examples retrieved by all of our models for any given meme, informing the community about the strengths and limitations of these explainable methods."],["Introduction","The moderation of content on social media is becoming one of the main societal challenges as online platforms have been effectively weaponized in a variety of geo-political events and social issues [37], [36], [10], [38].","While lowering barriers to information sharing can guarantee freedom of expression, research showed that it also facilitates the diffusion of harmful narratives, including violent content and misinformation [5], [6], [50].","The detection of harmful content is challenging, given that content can be easily created in different modalities, ranging from text to multimedia content, and spread very quickly, sometimes amplified by coordinated accounts involved in influence operations (e.g., bots and trolls) [30], [53], [29], [45], and often across platforms with different degrees and strategies for moderation [48], [52].","Meanwhile, determining toxicity, or inappropriateness broadly is non-obvious even for humans, as social media interactions are integrated into both the virtual and the real-world context.","Content moderation policies, or the lack thereof, can have serious implications on individuals, groups, and society as a whole.","On the one hand, content moderators may react late, inconsistently, or unfairly, thus angering users [19], as well as contributing to reinforcing and exacerbating conspiratorial narratives [10], [28].","On the other hand, minimal content moderation may permit coordinated influence operations [16] or enable the spontaneous formation of toxic and dangerous communities, e.g., the study by [31] demonstrates how “the Manosphere”, a conglomerate of men-centered online communities, may serve as a gateway to far-right movements.","A recent study [14] revealed worrying patterns of online abuse, estimating 1.1 million toxic tweets sent to women over one year.","Their study also reveals that black women were 84% more likely than white women to experience abuse on the platform.","These studies collectively show that sexism and misogyny are still prevalent all over the globe [21], despite initiatives such as the UN Sustainable Development Goals [7] that emphasize the importance of gender equality, peace, and justice.","The recent explosion of multimedia content, in the form of Internet memes (IMs), makes content moderation even more difficult, especially when the context is not taken into account.","An Internet meme can be roughly defined as “a piece of culture, typically a joke, which gains influence through online transmission” [13].","An Internet meme is based on a medium, typically an image representing a well-understood reference to a prototypical situation within a certain community.","Given that IMs are potential vectors for misinformation, political propaganda, and hate speech, enabling their scalable analysis is essential.","Nonetheless, the automated analysis of IMs is challenging because of their nature: IMs are multimodal, i.e., they combine visual and language information creatively.","Notably, IMs are not just funny; they are relatable and, thus, they are community-dependent.","Therefore, their correct interpretation passes from the identification of the right virtual context.","Moreover, IMs are succinct, i.e., they spread complex messages with a minimal information unit that connects the virtual circumstances to the real ones.","Finally, IMs are fluid, i.e., they are subject to variations and alterations.","In one study by Meta [1], 121,605 different variants of one particular meme were posted across 1.14 million status updates.","The inaccurate classification of memes can lead to inadequate moderation interventions (removal, flagging, demotion, etc.)","that, combined with the lack of tracing mechanisms across platforms, has the potential to further decrease public trust in social media platforms, and related moderation policies.","Existing work on meme tracking and classification has focused on their temporal spread over time (i.e., virality) [32], [49], [25] and high-level categorization tasks like hate speech detection that focus on perceptual features [23], [17].","Little work has focused on the aspects of semantics and pragmatics of a meme, which require precise feature extraction from images and from the text.","Moreover, memes assume rich background knowledge about the spatio-temporal and cultural context in which they came into existence.","Combining text, vision, and extra knowledge is an AI-complete problem.","In this paper, we explore explainable multimodal methods for IMs classification.","We rely on the general idea of case-based reasoning, where a method prediction can be traced back to similar memes that the method has observed at training time.","Considering the complex nature of Internet memes, we opt for case-based reasoning because it can provide transparent insights into the model reasoning, while still leveraging the representation learning ability of state-of-the-art (SOTA) models.","Our contributions are: We build on prior work to employ explainable reasoning methods for meme understanding.","These methods perform example- and prototype-based reasoning over training cases, while leveraging both textual and visual SOTA models to extract features for the individual cases.","We study the relevance of our modular and explainable models in detecting harmful memes on two tasks: Hate Speech Detection and Misogyny Classification.","We compare the performance between example- and prototype-based methods and between text, vision, and multimodal models, across different categories of harmfulness (e.g., stereotype and objectification).","We devise a user-friendly interface that facilitates the comparative analysis of examples retrieved by all of our models for any given meme.","We leverage the user interface to understand the ability of different explainable models to retrieve useful instances for case-based reasoning and inform future work about these methods' strengths and limitations.","We make our code available to facilitate future research on explainable IM classification for social good.https://github.com/usc-isi-i2/meme-understanding Figure: Classification and feature extraction model within the Example-based explanation model."],["Method","A meme classification model that detects offensive or inappropriate memes can be easily trained.","However, the black-box nature of ML models makes it difficult to interpret why a meme is flagged [2], especially when flagged wrongly.","We adopt example-based and prototype-based approaches to make explainable predictions for internet meme classification tasks.","Both approaches utilize a frozen pre-trained model to extract features from a meme in a transfer learning setup with a separate downstream classification model, which leverages the features to make a final decision.","The modularity of the approach enables an easy comparison over the combination of the pre-trained model and the explanation method used.","We further develop a web-based visualization tool to study these explanation methodologies.","Figure: Example-based explanation based on similarity-based meme search.","The Train meme-features database contains pre-computed features using the Feature Extractor module."],["Example-based Meme Classification","We adopt an example-based method [41] to make predictions and explain them by displaying similar memes to end-users.","Example-based explanation works by showing training examples that have a similar representation to the test example from the model's point of view to act as a proxy to understand the model's behavior.","We use example-based classification because it helps users to understand how the classification model represents a meme compared to the training dataset supporting the model prediction.","Although Internet memes involve text, image, and often need commonsense or cultural-specific knowledge to be well understood, and that might limit the efficacy of example-based explanation, still even as a heuristic, similar examples can help the end-users understand the reasoning that is done by the model [42].","This approach further helps analyze misclassifications and detect latent biases in the dataset [47].","Figure REF shows the meme classification model, which applies a classification head (L1, L2, L3 and Predictions layer) over the frozen pre-trained model for prediction (see the last Subsection about Pretrained models).","The last hidden state (output of L3) of the trained classifier is used as the extracted features for calculating the similarity between memes using cosine similarity.","Then, for an unlabelled meme, we predict the labels using this classification model.","The features can be fed into a query engine to select similar images (Figure REF ) from a database that stores pre-computed features corresponding to the training memes.","To display the retrieved similar memes in a user-friendly way, we develop a visualization tool to display the model-wise predictions and similar memes from the training dataset, thus supporting the predictions with example-based explanations.","Figure: Our architecture for prototype-based explainable classification called Explainable Deep Neural Networks (xDNN).","Figure reused from ."],["Prototype-based Meme Classification","Unlike an example-based explanation, a prototype-based explanation is not a post-hoc approach.","Instead, prototype-based classification relies on learning label-wise prototypes from the training dataset followed by a rule-based decision algorithm for the classification, which makes these models inherently interpretable.","Prototype-based explanation is based on prototype theory [43], which is a theory of categorization in psychology and cognitive linguistics, in which there is a graded degree of belonging to a conceptual category, and some members are more central than others.","In prototype theory, any given concept in any given language has a real-world example that best represents this concept, i.e., its prototype.","Like example-based explanation, prototype theory is also an instance of case-based reasoning, and there has been some controversy over the superiority of one over the other.","There are both claims about the superiority of prototypical examples over normal examples [20], as well as their counterparts [33] who state that a context theory of classification, which derives concepts purely from exemplars works better than a class of theories that included prototype theory.","We reuse the implementation of Explainable Deep Neural Networks (xDNN) [3].","It uses the training data features extracted using the pre-trained model to create class-wise prototypes (local peaks for class distribution).","As shown in Figure REF , the new unlabelled sample can be evaluated against these prototypes and then classified using rule-based local and global decision-making stages."],["Pretrained Models for Feature Extraction","We chose the following pre-trained models for feature extraction to analyze the information captured by models trained over different modalities and pretraining strategies.","Textual Models: We use the BERT$_{base}$ model [15], trained on BooksCorpus (800M words) and English Wikipedia (2,500M words) using two unsupervised tasks of Masked LM and Next Sentence Prediction.","We expect that BERT would help analyze explainability for general-purpose formal language.","Also, we used the BERTweet model [35] having the same architecture as BERT$_{base}$ and trained using the RoBERTa [26] pretraining procedure over 80 GB corpus of 850M English tweets.","BERTweet is supposed to be more contextually related to a meme text as tweets have short text lengths and generally contain informal grammar with irregular vocabulary, similar to IMs.","Vision Models: To capture visual information, we used the CLIP (Contrastive Language-Image Pre-training) model [39].","It is trained with Natural Language Supervision over 400 million (image, text) pairs collected from the Internet with the contrastive objective of creating similar features for an image and text pair.","Because of the variety of training data and unrestricted text supervision, CLIP reaches SOTA-comparable zero-shot performance over various tasks like fine-grained object classification and geo-localization, action recognition in videos, and OCR.","CLIP is robust toward distribution shift between various datasets and shows better domain generalization over various datasets.","Mixed Models: To capture both graphical and textual information simultaneously, we concatenate features from both BERTweet and CLIP together and use them for downstream predictions."],["Experimental Setup","This section discusses the setup for evaluating our approaches over the explainable meme classification tasks."],["Tasks and Datasets","We experimented with meme classification tasks over two existing datasets: MAMI and Hateful Memes.","SemEval-2022 Task 5: Multimedia Automatic Misogyny Identification (MAMI) dataset [17] consists of two sub-tasks of misogyny detection and its type classification.","Sub-task A: Misogyny Detection Task focuses on detecting whether a meme is misogynous.","The inter-annotator Fleiss-k Agreement for sub-task A is $0.5767$ .","Sub-task B: Misogyny Type Classification Task is a multi-class task that categorizes a meme into one or more misogyny types, namely, shaming, stereotype, objectification, and violence.","A more formal description of these categories can be found in [17].","The inter-annotator Fleiss-k Agreement for sub-task B is $0.3373$ .","Data statistics for both subtasks of the MAMI dataset are presented in Table REF .","The inter-annotator Fleiss-k Agreement clearly shows that sub-task B is comparatively more difficult than sub-task A. Hateful Memes dataset [22] consists of a single task of meme hate detection.","The dataset consists of 10K memes equally divided into hateful and not-hateful classes; the dev and test set consist of $5\\%$ and $10\\%$ of the dataset, respectively.","The human accuracy for the classification was $84.70\\%$ , ranging from $83\\%$ to $87\\%$ ."],["Evaluation","We keep our evaluation of classification performance consistent with the original paper about the MAMI dataset.","Sub-task A is evaluated using macro-average F1 measure for each class label (misogynous and not misogynous).","Likewise, Sub-task B is evaluated using weighted-average F1 measure, weighted by the true label count for each label.","For the Hateful Meme dataset, we compare the models based on the macro-average F1-score between the hateful and not-hateful classes.","Table REF shows performance statistics for all the participants in the MAMI task within the SemEval-2022 competition [17].","Table: Classification Head parameters for the Example-based method.In addition, we manually evaluate the example-based explanation approach using the visualization tool by analyzing the prediction and similar memes from the training dataset.","We evaluate the prototype-based explanation method (xDNN) by its classification performance and manually investigating the prototypes identified from the training dataset.","Table: MAMI dataset characteristics.Table: Basic statistics of the results for the participating systems in Sub-task A and Sub-task B, expressed in terms of macro-averaged and weighted-average F1-score respectively.Table: Classification results for MAMI dataset and Hateful memes dataset."],["Model Training Details","The classification model (Figure REF ) used in the example-based explanation setup applies a trainable neural head over frozen pre-trained models, which is trained with the Binary Cross Entropy Loss using the Adam optimizer with a learning rate of $10^{-4}$ .","Table REF describes each layer of the classification head, and the hidden state of L3 is used for feature extraction for similar example searches over the training dataset.","xDNN [3] is a generative model, i.e., it learns prototypes and respective distributions automatically from the training data with no user/problem specific parameters.","We reuse the publicly available xDNN implementationhttps://github.com/Plamen-Eduardo/xDNN---Python and experiment with different pre-trained models described in the subsection on Pretrained Models.","Figure: Explanatory interface for our Example-based classification method."],["Analysis","In this section, we analyze and compare different methods and feature extraction models for meme classification.","As our methods focus on both accuracy and explainability, we present an analysis on both of them in turn."],["Results","Table REF shows the performance of individual models on the MAMI and Hateful memes datasets.","The table compares the explainability strategies' performance over different pre-trained models' choices.","MAMI v/s Hateful Meme: Between the subtask A (misogyny detection) of the MAMI dataset and Hate detection over the Hateful Meme dataset, all models have better performance for the misogyny detection task.","This can be because of the fact that the presence of misogyny directly relates to the mention of women (or related terminology); in contrast, hate is a more open-ended problem.","For both tasks and methods, we also observe that the BERTweet model, which is trained on Twitter data, performs better than the BERT-based models for the MAMI dataset, though the difference between the two models is within one point difference.","This shows that exposure to social media (Twitter) data has a positive, yet limited, impact on models for meme content classification.","However, for the Hateful meme task, BERT performs better than the BERTweet model suggesting the shift in distribution between the two datasets.","Prototype-based (xDNN) v/s Example-based Explanation (Deep learning): For both datasets and different pre-training models, the Example-based method, which uses a neural classification head, performs better than the Prototype-based (xDNN) on the same pre-trained model.","This indicates that the prototype-based models rely entirely on the pre-trained features and might lose performance on learning complex patterns, which the deep learning model can learn.","However, in terms of training time, xDNN is much faster than training the neural classifier head, as it needs just a single pass over the training dataset.","Modality performance analysis (Text v/s Image v/s Mixed): For each meme dataset and method combination, the CLIP-based image model performs comparatively better than BERT-based text models.","The combined model using CLIP and BERTweet features outperforms all models, including those using CLIP alone.","However, the improvement of the joint model over CLIP is relatively low (0.5-1.5 absolute points) compared to the improvement over BERT models (10-10 absolute points) for both explanation strategies.","This means that either visual information is more important than text or that the CLIP model can also capture the textual information in the meme or a combination of both reasons.","For the task of misogyny classification, the combination of CLIP and BERTweet also performs the best, with the CLIP-only model performing very closely as a second best."],["Explanability Analysis","Example-based Classification Figure REF shows the visualization tool for the Example-based classification prediction over one test image.","The tool displays the model-wise predictions for BERTTweet, CLIP, and their combination, together with similar memes from the training dataset for explainability.","The test image is misogynous, portraying shaming and objectification of women.","The predictions from each model are correct with high confidence about the misogyny detection and shaming type classification, which is easily explainable from the similar examples from the training dataset.","Looking at the most similar images per model, we observe that the combined model retrieves images that also depict misogyny and either shaming, objectification, or both.","This shows that the examples retrieved by this model are most reliable, which also correlates with its best performance.","The retrieved examples by CLIP and BERTTweet are partially useful, with CLIP retrieving more relevant images than BERTTweet in most cases.","Prototype-based Classification xDNN model is inherently explainable as it predicts the label for a meme based on the closest prototype as shown in REF .","To our surprise, xDNN for both the MAMI and the Hateful Meme datasets creates prototypes equal to the training supports for the class.","This can be because of the fact that the memes, even though belonging to the same category, can have very different textual/visual information content and representation.","Nevertheless, we are further investigating this behavior in depth to find the exact reason."],["Discussion","Our experiments revealed that methods for meme classification can balance the goals of explainability and good accuracy.","While the example-based method is simpler, it achieved higher performance and its explanatory power was more intuitive, as demonstrated through our tool-supported analysis.","Among the feature extractors, we observed that vision models were more effective than language models, and the combination of the two achieved the best performance.","We next elevate these analyses to high-level lessons and discussion points that may benefit future work on explainable meme classification for tasks like hate speech detection and misogyny classification.","Lack of virtual and real-world context During our analysis, we found that memes often rely on real-world, day-to-day context to be understood.","As the pre-trained models lack this context, they may wrongly classify a meme that is very contextual to real-world common sense, especially in social media and misogyny.","Our models have no access to this additional context.","For instance, the usage of the meme can be occurring in an exchange between parties on social media / messaging platforms, which is not provided as a context in the task.","A hollistic method for meme classification and explaination needs to account for the cultural and folkloric nature of memes [4]: memes often start from a seed of a concept or an idea (this real-world common sense or common cultural background / reference), which then gets derived to achieve an intent.","The way this idea is adapted or degenerated depends on the intent and can happen by using figures of speech such as an oxymoron, in which case the image and the meme's text will be antonymous.","Integration of background knowledge and figures of speech The example in Figure REF includes a test image where the canvas image depicts Nicki Minaj.","The shaming, objectification, and misogyny in this case are probably linked to assumed background knowledge, such as Nicky's public image, a referenced song with a music video, and the lyrics that belong to the song in that music video.","While the meme caption includes slang terms like “ass” and “std”, the cadence and the formulation of the meme sentence rely on the aforementioned song.","As for the other images on the right-hand side where some context can be detected, we have the center image, which is from Victoria's Secret fashion show (which has been criticized for objectifying women).","While some of these cues might be implicitly captured by CLIP, such examples demonstrate the need for background commonsense and factual knowledge, as well as Internet folklore, to build robust and explainable meme classification methods in the future.","This example also motivates the need for the integration of an analysis of figures of speech like the center image being somewhat paradoxical.","Subjectivity of Ground Truth Labels Another observation that came out of our analysis is that the problem statement of misogyny detection and type classification is inherently subjective, and the labeler's background and familiarity with social norms affect it.","This argument is also clear from the inter-annotator agreement score for the MAMI dataset.","Hence, for subjective problem statements like these, the ground truth labels are always questionable and are biased by the labelers.","This observation relates to recent work that highlights issues with crowdsourced labeling for hate speech detection or sentiment analysis on social media, as examples [34], [51] or [12] to address the bias in labeling hate or abuse datasets.","It also connects to discussions of evaluating toxicity [9].","Concerns about the consistency of labeling aspects such as sexism are also prominent in computational social sciences like psychology [44]."],["Related Work","Most prior works on Internet memes in AI have focused on understanding their virality and spread on social media over time [32], [49], [25].","Another popular direction has been detecting forms of hate speech in memes.","The Hateful Memes Challenge and Dataset [23] is a competition and open source dataset with over 10 thousand examples, where the goal is to leverage vision and language understanding to identify memes that employ hate speech.","Kirk et al.","[24] compare memes in this challenge to memes in the `wild', observing that extraction of captions is an open challenge, and that open-world memes are more diverse than traditional memes.","The Multimedia Automatic Misogyny Identification (MAMI) [17] challenge asks systems to identify misogynous memes, based on both text and images in the input memes.","Methods for these challenges typically employ Transformer-based models that incorporate vision and language, like ViLBERT [27], UNITER [11], and CLIP [40].","The work by Sheratt [46] aims to organize memes into a genealogy, with the goal of building a comprehensive knowledge base going forward.","The combination of efforts to explain IMs with explicit knowledge and the generalization power of large visual, textual, and multimodal models holds a promise to advance the SOTA of meme understanding and classification.","However, to our knowledge, no prior work has focused on such multi-faceted and explainable methods for understanding IMs.","To bridge this gap, we design a modular architecture that integrates visual and textual models with prototype- and example-based reasoning methods.","Our framework thus balances the goals of obtaining SOTA performance and providing transparent access to the model reasoning.","There has been a surge in using example-based explanations to enhance people's comprehension of black-box deep learning models' behavior and acquired knowledge.","[8] propose and evaluate two kinds of example-based explanations in the visual domain.","The extracted similar training data points help the end-users understand and recognize the capabilities of the model better.","Although [18] have the same conclusion as [8] confirming the effect of examples to boost the comprehension of the model by end-users, they do not see any evidence supporting the same effect about the trust of end-users when presented with example-based explanations.","Similarly, methods for prototype-based classification have been developed for visual tasks in the past, such as xDNN [3].","However, to our knowledge, we are the first work to employ example-based and prototype-based methods for downstream tasks of IM classification."],["Conclusions","In this work, we implemented and analyzed example- and prototype-based approaches for explainable Misogyny Identification and Hate Speech Detection in IMs.","Our experiments revealed that methods for IMs classification can balance the goals of explainability and good accuracy.","While the example-based method is simpler, it achieved higher performance and its explanatory power was more intuitive, as demonstrated through our tool-supported analysis.","Among the feature extractors, we observed that vision models were more effective than language models, and the combination of the two achieved the best performance.","We connected these findings to thorny challenges about including background knowledge and real-world context in complex multimodal tasks, as well as concerns about the subjective nature of tasks that revolve around these tasks.","We make our code available in hope that subsequent research can help us in pursuing these challenges together."],["Acknowledgements","The first two authors have been supported by armasuisse Science and Technology, Switzerland under contract No.","8003532866.","The experiments were run on a cluster provided by armasuisse Science and Technology."]],"string":"[\n [\n \"Multimodal and Explainable Internet Meme Classification\"\n ],\n [\n \"Abstract Warning: this paper contains content that may be offensive or upsetting.\",\n \"In the current context where online platforms have been effectively weaponized in a variety of geo-political events and social issues, Internet memes make fair content moderation at scale even more difficult.\",\n \"Existing work on meme classification and tracking has focused on black-box methods that do not explicitly consider the semantics of the memes or the context of their creation.\",\n \"In this paper, we pursue a modular and explainable architecture for Internet meme understanding.\",\n \"We design and implement multimodal classification methods that perform example- and prototype-based reasoning over training cases, while leveraging both textual and visual SOTA models to represent the individual cases.\",\n \"We study the relevance of our modular and explainable models in detecting harmful memes on two existing tasks: Hate Speech Detection and Misogyny Classification.\",\n \"We compare the performance between example- and prototype-based methods, and between text, vision, and multimodal models, across different categories of harmfulness (e.g., stereotype and objectification).\",\n \"We devise a user-friendly interface that facilitates the comparative analysis of examples retrieved by all of our models for any given meme, informing the community about the strengths and limitations of these explainable methods.\"\n ],\n [\n \"Introduction\",\n \"The moderation of content on social media is becoming one of the main societal challenges as online platforms have been effectively weaponized in a variety of geo-political events and social issues [37], [36], [10], [38].\",\n \"While lowering barriers to information sharing can guarantee freedom of expression, research showed that it also facilitates the diffusion of harmful narratives, including violent content and misinformation [5], [6], [50].\",\n \"The detection of harmful content is challenging, given that content can be easily created in different modalities, ranging from text to multimedia content, and spread very quickly, sometimes amplified by coordinated accounts involved in influence operations (e.g., bots and trolls) [30], [53], [29], [45], and often across platforms with different degrees and strategies for moderation [48], [52].\",\n \"Meanwhile, determining toxicity, or inappropriateness broadly is non-obvious even for humans, as social media interactions are integrated into both the virtual and the real-world context.\",\n \"Content moderation policies, or the lack thereof, can have serious implications on individuals, groups, and society as a whole.\",\n \"On the one hand, content moderators may react late, inconsistently, or unfairly, thus angering users [19], as well as contributing to reinforcing and exacerbating conspiratorial narratives [10], [28].\",\n \"On the other hand, minimal content moderation may permit coordinated influence operations [16] or enable the spontaneous formation of toxic and dangerous communities, e.g., the study by [31] demonstrates how “the Manosphere”, a conglomerate of men-centered online communities, may serve as a gateway to far-right movements.\",\n \"A recent study [14] revealed worrying patterns of online abuse, estimating 1.1 million toxic tweets sent to women over one year.\",\n \"Their study also reveals that black women were 84% more likely than white women to experience abuse on the platform.\",\n \"These studies collectively show that sexism and misogyny are still prevalent all over the globe [21], despite initiatives such as the UN Sustainable Development Goals [7] that emphasize the importance of gender equality, peace, and justice.\",\n \"The recent explosion of multimedia content, in the form of Internet memes (IMs), makes content moderation even more difficult, especially when the context is not taken into account.\",\n \"An Internet meme can be roughly defined as “a piece of culture, typically a joke, which gains influence through online transmission” [13].\",\n \"An Internet meme is based on a medium, typically an image representing a well-understood reference to a prototypical situation within a certain community.\",\n \"Given that IMs are potential vectors for misinformation, political propaganda, and hate speech, enabling their scalable analysis is essential.\",\n \"Nonetheless, the automated analysis of IMs is challenging because of their nature: IMs are multimodal, i.e., they combine visual and language information creatively.\",\n \"Notably, IMs are not just funny; they are relatable and, thus, they are community-dependent.\",\n \"Therefore, their correct interpretation passes from the identification of the right virtual context.\",\n \"Moreover, IMs are succinct, i.e., they spread complex messages with a minimal information unit that connects the virtual circumstances to the real ones.\",\n \"Finally, IMs are fluid, i.e., they are subject to variations and alterations.\",\n \"In one study by Meta [1], 121,605 different variants of one particular meme were posted across 1.14 million status updates.\",\n \"The inaccurate classification of memes can lead to inadequate moderation interventions (removal, flagging, demotion, etc.)\",\n \"that, combined with the lack of tracing mechanisms across platforms, has the potential to further decrease public trust in social media platforms, and related moderation policies.\",\n \"Existing work on meme tracking and classification has focused on their temporal spread over time (i.e., virality) [32], [49], [25] and high-level categorization tasks like hate speech detection that focus on perceptual features [23], [17].\",\n \"Little work has focused on the aspects of semantics and pragmatics of a meme, which require precise feature extraction from images and from the text.\",\n \"Moreover, memes assume rich background knowledge about the spatio-temporal and cultural context in which they came into existence.\",\n \"Combining text, vision, and extra knowledge is an AI-complete problem.\",\n \"In this paper, we explore explainable multimodal methods for IMs classification.\",\n \"We rely on the general idea of case-based reasoning, where a method prediction can be traced back to similar memes that the method has observed at training time.\",\n \"Considering the complex nature of Internet memes, we opt for case-based reasoning because it can provide transparent insights into the model reasoning, while still leveraging the representation learning ability of state-of-the-art (SOTA) models.\",\n \"Our contributions are: We build on prior work to employ explainable reasoning methods for meme understanding.\",\n \"These methods perform example- and prototype-based reasoning over training cases, while leveraging both textual and visual SOTA models to extract features for the individual cases.\",\n \"We study the relevance of our modular and explainable models in detecting harmful memes on two tasks: Hate Speech Detection and Misogyny Classification.\",\n \"We compare the performance between example- and prototype-based methods and between text, vision, and multimodal models, across different categories of harmfulness (e.g., stereotype and objectification).\",\n \"We devise a user-friendly interface that facilitates the comparative analysis of examples retrieved by all of our models for any given meme.\",\n \"We leverage the user interface to understand the ability of different explainable models to retrieve useful instances for case-based reasoning and inform future work about these methods' strengths and limitations.\",\n \"We make our code available to facilitate future research on explainable IM classification for social good.https://github.com/usc-isi-i2/meme-understanding Figure: Classification and feature extraction model within the Example-based explanation model.\"\n ],\n [\n \"Method\",\n \"A meme classification model that detects offensive or inappropriate memes can be easily trained.\",\n \"However, the black-box nature of ML models makes it difficult to interpret why a meme is flagged [2], especially when flagged wrongly.\",\n \"We adopt example-based and prototype-based approaches to make explainable predictions for internet meme classification tasks.\",\n \"Both approaches utilize a frozen pre-trained model to extract features from a meme in a transfer learning setup with a separate downstream classification model, which leverages the features to make a final decision.\",\n \"The modularity of the approach enables an easy comparison over the combination of the pre-trained model and the explanation method used.\",\n \"We further develop a web-based visualization tool to study these explanation methodologies.\",\n \"Figure: Example-based explanation based on similarity-based meme search.\",\n \"The Train meme-features database contains pre-computed features using the Feature Extractor module.\"\n ],\n [\n \"Example-based Meme Classification\",\n \"We adopt an example-based method [41] to make predictions and explain them by displaying similar memes to end-users.\",\n \"Example-based explanation works by showing training examples that have a similar representation to the test example from the model's point of view to act as a proxy to understand the model's behavior.\",\n \"We use example-based classification because it helps users to understand how the classification model represents a meme compared to the training dataset supporting the model prediction.\",\n \"Although Internet memes involve text, image, and often need commonsense or cultural-specific knowledge to be well understood, and that might limit the efficacy of example-based explanation, still even as a heuristic, similar examples can help the end-users understand the reasoning that is done by the model [42].\",\n \"This approach further helps analyze misclassifications and detect latent biases in the dataset [47].\",\n \"Figure REF shows the meme classification model, which applies a classification head (L1, L2, L3 and Predictions layer) over the frozen pre-trained model for prediction (see the last Subsection about Pretrained models).\",\n \"The last hidden state (output of L3) of the trained classifier is used as the extracted features for calculating the similarity between memes using cosine similarity.\",\n \"Then, for an unlabelled meme, we predict the labels using this classification model.\",\n \"The features can be fed into a query engine to select similar images (Figure REF ) from a database that stores pre-computed features corresponding to the training memes.\",\n \"To display the retrieved similar memes in a user-friendly way, we develop a visualization tool to display the model-wise predictions and similar memes from the training dataset, thus supporting the predictions with example-based explanations.\",\n \"Figure: Our architecture for prototype-based explainable classification called Explainable Deep Neural Networks (xDNN).\",\n \"Figure reused from .\"\n ],\n [\n \"Prototype-based Meme Classification\",\n \"Unlike an example-based explanation, a prototype-based explanation is not a post-hoc approach.\",\n \"Instead, prototype-based classification relies on learning label-wise prototypes from the training dataset followed by a rule-based decision algorithm for the classification, which makes these models inherently interpretable.\",\n \"Prototype-based explanation is based on prototype theory [43], which is a theory of categorization in psychology and cognitive linguistics, in which there is a graded degree of belonging to a conceptual category, and some members are more central than others.\",\n \"In prototype theory, any given concept in any given language has a real-world example that best represents this concept, i.e., its prototype.\",\n \"Like example-based explanation, prototype theory is also an instance of case-based reasoning, and there has been some controversy over the superiority of one over the other.\",\n \"There are both claims about the superiority of prototypical examples over normal examples [20], as well as their counterparts [33] who state that a context theory of classification, which derives concepts purely from exemplars works better than a class of theories that included prototype theory.\",\n \"We reuse the implementation of Explainable Deep Neural Networks (xDNN) [3].\",\n \"It uses the training data features extracted using the pre-trained model to create class-wise prototypes (local peaks for class distribution).\",\n \"As shown in Figure REF , the new unlabelled sample can be evaluated against these prototypes and then classified using rule-based local and global decision-making stages.\"\n ],\n [\n \"Pretrained Models for Feature Extraction\",\n \"We chose the following pre-trained models for feature extraction to analyze the information captured by models trained over different modalities and pretraining strategies.\",\n \"Textual Models: We use the BERT$_{base}$ model [15], trained on BooksCorpus (800M words) and English Wikipedia (2,500M words) using two unsupervised tasks of Masked LM and Next Sentence Prediction.\",\n \"We expect that BERT would help analyze explainability for general-purpose formal language.\",\n \"Also, we used the BERTweet model [35] having the same architecture as BERT$_{base}$ and trained using the RoBERTa [26] pretraining procedure over 80 GB corpus of 850M English tweets.\",\n \"BERTweet is supposed to be more contextually related to a meme text as tweets have short text lengths and generally contain informal grammar with irregular vocabulary, similar to IMs.\",\n \"Vision Models: To capture visual information, we used the CLIP (Contrastive Language-Image Pre-training) model [39].\",\n \"It is trained with Natural Language Supervision over 400 million (image, text) pairs collected from the Internet with the contrastive objective of creating similar features for an image and text pair.\",\n \"Because of the variety of training data and unrestricted text supervision, CLIP reaches SOTA-comparable zero-shot performance over various tasks like fine-grained object classification and geo-localization, action recognition in videos, and OCR.\",\n \"CLIP is robust toward distribution shift between various datasets and shows better domain generalization over various datasets.\",\n \"Mixed Models: To capture both graphical and textual information simultaneously, we concatenate features from both BERTweet and CLIP together and use them for downstream predictions.\"\n ],\n [\n \"Experimental Setup\",\n \"This section discusses the setup for evaluating our approaches over the explainable meme classification tasks.\"\n ],\n [\n \"Tasks and Datasets\",\n \"We experimented with meme classification tasks over two existing datasets: MAMI and Hateful Memes.\",\n \"SemEval-2022 Task 5: Multimedia Automatic Misogyny Identification (MAMI) dataset [17] consists of two sub-tasks of misogyny detection and its type classification.\",\n \"Sub-task A: Misogyny Detection Task focuses on detecting whether a meme is misogynous.\",\n \"The inter-annotator Fleiss-k Agreement for sub-task A is $0.5767$ .\",\n \"Sub-task B: Misogyny Type Classification Task is a multi-class task that categorizes a meme into one or more misogyny types, namely, shaming, stereotype, objectification, and violence.\",\n \"A more formal description of these categories can be found in [17].\",\n \"The inter-annotator Fleiss-k Agreement for sub-task B is $0.3373$ .\",\n \"Data statistics for both subtasks of the MAMI dataset are presented in Table REF .\",\n \"The inter-annotator Fleiss-k Agreement clearly shows that sub-task B is comparatively more difficult than sub-task A. Hateful Memes dataset [22] consists of a single task of meme hate detection.\",\n \"The dataset consists of 10K memes equally divided into hateful and not-hateful classes; the dev and test set consist of $5\\\\%$ and $10\\\\%$ of the dataset, respectively.\",\n \"The human accuracy for the classification was $84.70\\\\%$ , ranging from $83\\\\%$ to $87\\\\%$ .\"\n ],\n [\n \"Evaluation\",\n \"We keep our evaluation of classification performance consistent with the original paper about the MAMI dataset.\",\n \"Sub-task A is evaluated using macro-average F1 measure for each class label (misogynous and not misogynous).\",\n \"Likewise, Sub-task B is evaluated using weighted-average F1 measure, weighted by the true label count for each label.\",\n \"For the Hateful Meme dataset, we compare the models based on the macro-average F1-score between the hateful and not-hateful classes.\",\n \"Table REF shows performance statistics for all the participants in the MAMI task within the SemEval-2022 competition [17].\",\n \"Table: Classification Head parameters for the Example-based method.In addition, we manually evaluate the example-based explanation approach using the visualization tool by analyzing the prediction and similar memes from the training dataset.\",\n \"We evaluate the prototype-based explanation method (xDNN) by its classification performance and manually investigating the prototypes identified from the training dataset.\",\n \"Table: MAMI dataset characteristics.Table: Basic statistics of the results for the participating systems in Sub-task A and Sub-task B, expressed in terms of macro-averaged and weighted-average F1-score respectively.Table: Classification results for MAMI dataset and Hateful memes dataset.\"\n ],\n [\n \"Model Training Details\",\n \"The classification model (Figure REF ) used in the example-based explanation setup applies a trainable neural head over frozen pre-trained models, which is trained with the Binary Cross Entropy Loss using the Adam optimizer with a learning rate of $10^{-4}$ .\",\n \"Table REF describes each layer of the classification head, and the hidden state of L3 is used for feature extraction for similar example searches over the training dataset.\",\n \"xDNN [3] is a generative model, i.e., it learns prototypes and respective distributions automatically from the training data with no user/problem specific parameters.\",\n \"We reuse the publicly available xDNN implementationhttps://github.com/Plamen-Eduardo/xDNN---Python and experiment with different pre-trained models described in the subsection on Pretrained Models.\",\n \"Figure: Explanatory interface for our Example-based classification method.\"\n ],\n [\n \"Analysis\",\n \"In this section, we analyze and compare different methods and feature extraction models for meme classification.\",\n \"As our methods focus on both accuracy and explainability, we present an analysis on both of them in turn.\"\n ],\n [\n \"Results\",\n \"Table REF shows the performance of individual models on the MAMI and Hateful memes datasets.\",\n \"The table compares the explainability strategies' performance over different pre-trained models' choices.\",\n \"MAMI v/s Hateful Meme: Between the subtask A (misogyny detection) of the MAMI dataset and Hate detection over the Hateful Meme dataset, all models have better performance for the misogyny detection task.\",\n \"This can be because of the fact that the presence of misogyny directly relates to the mention of women (or related terminology); in contrast, hate is a more open-ended problem.\",\n \"For both tasks and methods, we also observe that the BERTweet model, which is trained on Twitter data, performs better than the BERT-based models for the MAMI dataset, though the difference between the two models is within one point difference.\",\n \"This shows that exposure to social media (Twitter) data has a positive, yet limited, impact on models for meme content classification.\",\n \"However, for the Hateful meme task, BERT performs better than the BERTweet model suggesting the shift in distribution between the two datasets.\",\n \"Prototype-based (xDNN) v/s Example-based Explanation (Deep learning): For both datasets and different pre-training models, the Example-based method, which uses a neural classification head, performs better than the Prototype-based (xDNN) on the same pre-trained model.\",\n \"This indicates that the prototype-based models rely entirely on the pre-trained features and might lose performance on learning complex patterns, which the deep learning model can learn.\",\n \"However, in terms of training time, xDNN is much faster than training the neural classifier head, as it needs just a single pass over the training dataset.\",\n \"Modality performance analysis (Text v/s Image v/s Mixed): For each meme dataset and method combination, the CLIP-based image model performs comparatively better than BERT-based text models.\",\n \"The combined model using CLIP and BERTweet features outperforms all models, including those using CLIP alone.\",\n \"However, the improvement of the joint model over CLIP is relatively low (0.5-1.5 absolute points) compared to the improvement over BERT models (10-10 absolute points) for both explanation strategies.\",\n \"This means that either visual information is more important than text or that the CLIP model can also capture the textual information in the meme or a combination of both reasons.\",\n \"For the task of misogyny classification, the combination of CLIP and BERTweet also performs the best, with the CLIP-only model performing very closely as a second best.\"\n ],\n [\n \"Explanability Analysis\",\n \"Example-based Classification Figure REF shows the visualization tool for the Example-based classification prediction over one test image.\",\n \"The tool displays the model-wise predictions for BERTTweet, CLIP, and their combination, together with similar memes from the training dataset for explainability.\",\n \"The test image is misogynous, portraying shaming and objectification of women.\",\n \"The predictions from each model are correct with high confidence about the misogyny detection and shaming type classification, which is easily explainable from the similar examples from the training dataset.\",\n \"Looking at the most similar images per model, we observe that the combined model retrieves images that also depict misogyny and either shaming, objectification, or both.\",\n \"This shows that the examples retrieved by this model are most reliable, which also correlates with its best performance.\",\n \"The retrieved examples by CLIP and BERTTweet are partially useful, with CLIP retrieving more relevant images than BERTTweet in most cases.\",\n \"Prototype-based Classification xDNN model is inherently explainable as it predicts the label for a meme based on the closest prototype as shown in REF .\",\n \"To our surprise, xDNN for both the MAMI and the Hateful Meme datasets creates prototypes equal to the training supports for the class.\",\n \"This can be because of the fact that the memes, even though belonging to the same category, can have very different textual/visual information content and representation.\",\n \"Nevertheless, we are further investigating this behavior in depth to find the exact reason.\"\n ],\n [\n \"Discussion\",\n \"Our experiments revealed that methods for meme classification can balance the goals of explainability and good accuracy.\",\n \"While the example-based method is simpler, it achieved higher performance and its explanatory power was more intuitive, as demonstrated through our tool-supported analysis.\",\n \"Among the feature extractors, we observed that vision models were more effective than language models, and the combination of the two achieved the best performance.\",\n \"We next elevate these analyses to high-level lessons and discussion points that may benefit future work on explainable meme classification for tasks like hate speech detection and misogyny classification.\",\n \"Lack of virtual and real-world context During our analysis, we found that memes often rely on real-world, day-to-day context to be understood.\",\n \"As the pre-trained models lack this context, they may wrongly classify a meme that is very contextual to real-world common sense, especially in social media and misogyny.\",\n \"Our models have no access to this additional context.\",\n \"For instance, the usage of the meme can be occurring in an exchange between parties on social media / messaging platforms, which is not provided as a context in the task.\",\n \"A hollistic method for meme classification and explaination needs to account for the cultural and folkloric nature of memes [4]: memes often start from a seed of a concept or an idea (this real-world common sense or common cultural background / reference), which then gets derived to achieve an intent.\",\n \"The way this idea is adapted or degenerated depends on the intent and can happen by using figures of speech such as an oxymoron, in which case the image and the meme's text will be antonymous.\",\n \"Integration of background knowledge and figures of speech The example in Figure REF includes a test image where the canvas image depicts Nicki Minaj.\",\n \"The shaming, objectification, and misogyny in this case are probably linked to assumed background knowledge, such as Nicky's public image, a referenced song with a music video, and the lyrics that belong to the song in that music video.\",\n \"While the meme caption includes slang terms like “ass” and “std”, the cadence and the formulation of the meme sentence rely on the aforementioned song.\",\n \"As for the other images on the right-hand side where some context can be detected, we have the center image, which is from Victoria's Secret fashion show (which has been criticized for objectifying women).\",\n \"While some of these cues might be implicitly captured by CLIP, such examples demonstrate the need for background commonsense and factual knowledge, as well as Internet folklore, to build robust and explainable meme classification methods in the future.\",\n \"This example also motivates the need for the integration of an analysis of figures of speech like the center image being somewhat paradoxical.\",\n \"Subjectivity of Ground Truth Labels Another observation that came out of our analysis is that the problem statement of misogyny detection and type classification is inherently subjective, and the labeler's background and familiarity with social norms affect it.\",\n \"This argument is also clear from the inter-annotator agreement score for the MAMI dataset.\",\n \"Hence, for subjective problem statements like these, the ground truth labels are always questionable and are biased by the labelers.\",\n \"This observation relates to recent work that highlights issues with crowdsourced labeling for hate speech detection or sentiment analysis on social media, as examples [34], [51] or [12] to address the bias in labeling hate or abuse datasets.\",\n \"It also connects to discussions of evaluating toxicity [9].\",\n \"Concerns about the consistency of labeling aspects such as sexism are also prominent in computational social sciences like psychology [44].\"\n ],\n [\n \"Related Work\",\n \"Most prior works on Internet memes in AI have focused on understanding their virality and spread on social media over time [32], [49], [25].\",\n \"Another popular direction has been detecting forms of hate speech in memes.\",\n \"The Hateful Memes Challenge and Dataset [23] is a competition and open source dataset with over 10 thousand examples, where the goal is to leverage vision and language understanding to identify memes that employ hate speech.\",\n \"Kirk et al.\",\n \"[24] compare memes in this challenge to memes in the `wild', observing that extraction of captions is an open challenge, and that open-world memes are more diverse than traditional memes.\",\n \"The Multimedia Automatic Misogyny Identification (MAMI) [17] challenge asks systems to identify misogynous memes, based on both text and images in the input memes.\",\n \"Methods for these challenges typically employ Transformer-based models that incorporate vision and language, like ViLBERT [27], UNITER [11], and CLIP [40].\",\n \"The work by Sheratt [46] aims to organize memes into a genealogy, with the goal of building a comprehensive knowledge base going forward.\",\n \"The combination of efforts to explain IMs with explicit knowledge and the generalization power of large visual, textual, and multimodal models holds a promise to advance the SOTA of meme understanding and classification.\",\n \"However, to our knowledge, no prior work has focused on such multi-faceted and explainable methods for understanding IMs.\",\n \"To bridge this gap, we design a modular architecture that integrates visual and textual models with prototype- and example-based reasoning methods.\",\n \"Our framework thus balances the goals of obtaining SOTA performance and providing transparent access to the model reasoning.\",\n \"There has been a surge in using example-based explanations to enhance people's comprehension of black-box deep learning models' behavior and acquired knowledge.\",\n \"[8] propose and evaluate two kinds of example-based explanations in the visual domain.\",\n \"The extracted similar training data points help the end-users understand and recognize the capabilities of the model better.\",\n \"Although [18] have the same conclusion as [8] confirming the effect of examples to boost the comprehension of the model by end-users, they do not see any evidence supporting the same effect about the trust of end-users when presented with example-based explanations.\",\n \"Similarly, methods for prototype-based classification have been developed for visual tasks in the past, such as xDNN [3].\",\n \"However, to our knowledge, we are the first work to employ example-based and prototype-based methods for downstream tasks of IM classification.\"\n ],\n [\n \"Conclusions\",\n \"In this work, we implemented and analyzed example- and prototype-based approaches for explainable Misogyny Identification and Hate Speech Detection in IMs.\",\n \"Our experiments revealed that methods for IMs classification can balance the goals of explainability and good accuracy.\",\n \"While the example-based method is simpler, it achieved higher performance and its explanatory power was more intuitive, as demonstrated through our tool-supported analysis.\",\n \"Among the feature extractors, we observed that vision models were more effective than language models, and the combination of the two achieved the best performance.\",\n \"We connected these findings to thorny challenges about including background knowledge and real-world context in complex multimodal tasks, as well as concerns about the subjective nature of tasks that revolve around these tasks.\",\n \"We make our code available in hope that subsequent research can help us in pursuing these challenges together.\"\n ],\n [\n \"Acknowledgements\",\n \"The first two authors have been supported by armasuisse Science and Technology, Switzerland under contract No.\",\n \"8003532866.\",\n \"The experiments were run on a cluster provided by armasuisse Science and Technology.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05612"}}},{"rowIdx":96,"cells":{"text":{"kind":"list like","value":[["Using Multiple Instance Learning to Build Multimodal Representations"],["Abstract Image-text multimodal representation learning aligns data across modalities and enables important medical applications, e.g., image classification, visual grounding, and cross-modal retrieval.","In this work, we establish a connection between multimodal representation learning and multiple instance learning.","Based on this connection, we propose a generic framework for constructing permutation-invariant score functions with many existing multimodal representation learning approaches as special cases.","Furthermore, we use the framework to derive a novel contrastive learning approach and demonstrate that our method achieves state-of-the-art results on a number of downstream tasks."],["Introduction","In this paper, we propose a framework for designing multimodal representation learning methods that encompasses previous approaches as special cases and implies a new algorithm for multimodal learning that advances the state of the art.","Specifically, we establish a connection between self-supervised representation learning based on contrastive learning and multiple instance learning [3] and show that they share similar assumptions and goals.","We bring insights from multiple instance learning to offer a fresh perspective on self-supervised representation learning and ideas for performance improvements.","With this connection in mind, we derive a novel algorithm for learning image-text representations that capture the structure shared between the two modalities and generalize well in a variety of downstream tasks.","We aim to establish alignment between images and associated text to improve clinical workflow.","For example, an image model that mimics the radiologists' interpretation could retroactively label images to select relevant patients for a clinical trial.","Further, local alignment between image regions and text fragments (e.g., sentences) promises to benefit many downstream tasks.","For example, cross-modal retrieval can provide description of an image region for automated documentation or enable comparisons with similar previously imaged patients for better interpretation based on local anatomy or pathology.","Similarly, radiologists documenting findings can verify the accuracy of the report by noting if the referred location (i.e., visual grounding of the text) is consistent with their impression of the image.","Self-supervised representation learning is a useful tool for reducing annotation burden for machine learning models in medical imaging.","Despite the need and opportunities for automation, development of robust machine learning methods is held back by the lack of annotations that serve as the supervision signal for learning.","Self-supervised representation learning on paired image-text data offers two advantages: (i) learning requires no further annotations and (ii) treating text as “labels” enables us to use natural language to reference visual concepts and vice versa [27].","Thus, we focus on learning image-text multimodal representations but the proposed framework is broadly applicable to representation learning on other multimodal data.","Learning joint representations involves training image and text encoders to perform self-supervised tasks on paired image-text data [5], [23] and evaluating on relevant downstream tasks.","We focus on contrastive learning, i.e., classifying image-text pairs as matched (i.e., corresponding to the same imaging event), or mismatched.","Contrastive learning has been applied to medical domain, demonstrating impressive transfer capabilities on a diverse set of tasks [2], [4], [14], [33].","The biggest improvements come from addressing challenges unique to this domain, e.g., the use of cross attention to deal with the lack of effective pathology detectors [14] and adaptation of language models to address linguistic challenges in clinical notes [2].","Training the models has involved increasingly complex contrastive loss functions that treat image and text symmetrically [2], [4], [14], [33] and on multiple scales [2], [14].","In contrast to previous work that relies on many loss terms, our proposed contrastive loss is simple to implement and yields superior performance.","Borrowing ideas from multiple instance learning, we treat local image region features as “data” and sentence features as (complex) “labels”.","Multiple instance learning is a type of weakly supervised learning that is effective for problems that lack fine-grain annotations [3].","For example, it can localize tumor cells in whole slide images with just image-level labels [22].","Central to multiple instance learning is the construction of permutation-invariant score functions [15], and the choice of how the instance scores or features are aggregated to be evaluated against an image-level label.","Effective instance aggregators take advantage of domain knowledge [8], e.g., the Noisy-OR aggregator for drug activity prediction [24], the Noisy-AND aggregator for cellular phenotype classification [20].","In our work, we extend multiple instance classification to contrastive learning by constructing permutation-invariant image-text score functions.","Building on insights from multiple instance classification with correlated instances [22], our proposed instance aggregator exploits correlation across instances to build representations that perform well in downstream tasks.","Many prior multiple instance learning methods focused on one particular task of interest, e.g., detection [32], region classification [7], or retrieval [18].","Some investigated the choices of instance aggregators for more than one downstream task [11], [25] but are limited in generality (i.e., not meant to be applicable to other applications) and scope (i.e., explored a few simple instance aggregators).","In contrast, the proposed framework for constructing permutation-invariant score functions can be readily applied to other applications.","We systematically investigate instance aggregators and their effect on representation learning, which yields a novel approach to learning joint representations.","We evaluate the resulting image-text representations on a diverse set of downstream tasks and demonstrate state-of-the-art performance across all tasks in the context of a large set of chest X-ray images and associated radiological reports."],["Method","We first introduce notation and discuss the local and global approaches for constructing permutation-invariant image-document score functions at the core of the learning procedure.","We then instantiate the framework for a specific choice of aggregators for contrastive learning."],["Problem Setup","A local $D$ -dimensional representation of an image with $N$ proposed regions is a collection of $N$ features vectors $x_n\\in \\mathcal {X}\\subset \\mathbb {R}^D$ , $n\\in \\left\\lbrace 1,\\cdots ,N \\right\\rbrace $ .","In our experiments, we use regular tiling to generate image regions and leave more sophisticated proposal methods (e.g., [28]) for future work.","A local representation of a $M$ -sentence document (e.g., a radiology report) is a collection of sentence feature vectors $y_m \\in \\mathcal {Y}\\subset \\mathbb {R}^D$ , $m\\in \\left\\lbrace 1,\\cdots ,M \\right\\rbrace $ .","Function $h: \\mathcal {X}\\times \\mathcal {Y}\\rightarrow \\mathbb {R}$ measures the similarity between representations, e.g., $h(x_n,y_m)$ is the similarity between a region and a sentence.","In our experiments, we use cosine similarity $h(x, y) = \\langle x, y \\rangle / \\left( \\left\\Vert x\\right\\Vert \\left\\Vert y\\right\\Vert \\right)$ , though the formulation accepts any differentiable similarity function.","For any vector space $\\mathcal {U}$ , aggregator function $\\pi : \\mathcal {P}(\\mathcal {U}) \\rightarrow \\mathcal {U}$ aggregates elements in the input set into a “representative”.","$\\mathcal {P}(\\mathcal {U})$ is the set of all finite subsets of $\\mathcal {U}$ .","For example, $\\pi (\\left\\lbrace x_n \\right\\rbrace ) = \\frac{1}{N}\\sum _n x_n$ aggregates $N$ region features $x_n\\in \\mathcal {X}$ by averaging them, while $\\pi (\\left\\lbrace h_n \\right\\rbrace ) = \\max _n h_n$ aggregates $N$ similarity scores into a single score by computing the maximum score.","We restrict our attention to aggregators that are permutation-invariant, i.e., they treat their input as an unordered set rather than an ordered vector.","Permutation-invariant image-document score function $S: \\mathcal {P}(\\mathcal {X}) \\times \\mathcal {P}(\\mathcal {Y}) \\rightarrow \\mathbb {R}$ measures the similarity between an image and a document based on region features $\\left\\lbrace x_n \\right\\rbrace $ and sentence features $\\left\\lbrace y_m \\right\\rbrace $ .","Figure: Local (top) and global (bottom) image-document score functions."],["Local & Global Permutation-Invariant Score Functions","Contrastive representation learning can be seen as maximizing the likelihood of correctly classifying image-text pairs as matched or mismatched.","Since supervision is provided at the image-document level, we define a framework to build permutation-invariant image-document score functions.","The local approach aggregates region-sentence scores into an image-sentence score.","The image-sentence score $g_m$ for sentence $m$ in the document is obtained by applying a local aggregator function $\\pi _l$ to region-sentence scores, i.e., $g_m = \\pi _l ( \\left\\lbrace h(x_n, y_m) \\right\\rbrace _n ) \\triangleq \\pi _l ( \\left\\lbrace h(x_1, y_m), \\cdots , h(x_N, y_m) \\right\\rbrace ) $ .","The global approach first aggregates local region features $\\left\\lbrace x_n \\right\\rbrace $ into a single image feature vector $\\pi _g(\\left\\lbrace x_n \\right\\rbrace )$ using a global aggregator function $\\pi _g$ .","The image-sentence score $g_m$ is computed using the similarity function $h$ on the image feature vector $\\pi _g(\\left\\lbrace x_n \\right\\rbrace )$ and sentence feature vector $y_m$ , i.e., $g_m = h(\\pi _g(\\left\\lbrace x_n \\right\\rbrace ), y_m)$ .","In both approaches, the image-document score $S$ is obtained by aggregating image-sentence scores with another aggregator function $\\pi _s$ , i.e., $S(\\left\\lbrace x_n \\right\\rbrace , \\left\\lbrace y_m \\right\\rbrace ) = \\pi _s(\\left\\lbrace g_m \\right\\rbrace )$ .","Figure REF illustrates the framework for constructing $S$ .","To summarize, the local and global image-document scores $S_l$ and $S_g$ are computed as follows: $S_l(\\left\\lbrace x_n \\right\\rbrace , \\left\\lbrace y_m \\right\\rbrace )&= \\pi _s(\\left\\lbrace \\pi _l(\\left\\lbrace h(x_n, y_m) \\right\\rbrace _n) \\right\\rbrace _m), \\\\S_g(\\left\\lbrace x_n \\right\\rbrace , \\left\\lbrace y_m \\right\\rbrace )&= \\pi _s(\\left\\lbrace h( \\pi _g(\\left\\lbrace x_n \\right\\rbrace ), y_m ) \\right\\rbrace _m).$ As the aggregator functions are permutation-invariant, the image-document score function $S$ is naturally permutation-invariant as well.","We emphasize that $S$ treats image features and text features differently, and that the order of application of similarity evaluation $h(\\cdot )$ and aggregators $\\pi (\\cdot )$ is empirically relevant.","This design decision is motivated by the fact that each sentence in a radiology report represent a concept and its location in the image, i.e., it is akin to a label for some region in the image.","The converse is not necessarily true as some parts of the image are not described in the report."],["Representation Learning with LSE$+$ NL Aggregators","In this section, we introduce our method LSE$+$ NL for learning multimodal representations that relies on a combination of local and global image-document score functions and an asymmetric text-to-image contrastive loss.","Inspired by [22], we use a soft maximum function to identify the most relevant region for a sentence, i.e., the critical region, and attend more to regions that are similar to the critical region.","Specifically, the local aggregator $\\pi _l$ is the log-sum-exp (LSE) function $\\pi _l(\\left\\lbrace h_n \\right\\rbrace )= \\frac{1}{\\gamma _l} \\log \\sum _{n=1}^N \\exp (\\gamma _l\\,h_n),$ where $\\gamma _l$ is a scale parameter that controls how well the LSE function approximates the max function.","The global aggregator $\\pi _g$ linearly combine the region features using the distance to the critical region as weights, i.e., $\\pi _g(\\left\\lbrace x_n \\right\\rbrace )= \\sum _{n=1}^N \\dfrac{ \\exp ( \\gamma _g\\, \\langle Ax_n, Ax_k \\rangle ) }{ \\sum _{n^{\\prime }=1}^N \\exp (\\gamma _g\\, \\langle Ax_{n^{\\prime }}, Ax_k \\rangle ) } \\, x_n,$ where $k$ is the index of the critical region, i.e., $k=\\operatornamewithlimits{\\arg \\,\\max }_n h(x_n,y_m)$ , $A$ is a learned weight matrix, and $\\gamma _g$ is the scale parameter for the softmax function.","We can interpret $\\pi _g$ as a form of attention where regions that are more similar to the critical region are given a higher attention weight.","In effect, $\\pi _g$ exploits the correlation between each region and the critical region using attention.","In addition, $\\pi _g$ can be seen as a form of non-local (NL) network [31].","Both $\\pi _l$ and $\\pi _g$ are permutation-invariant functions.","We choose $\\pi _s$ to be the average function.","We use the local and global image-document scores in (REF ) and () computed with our choice of $\\pi _l$ and $\\pi _g$ for contrastive learning.","Given a document, we form an image-document score vector $s \\triangleq (s^+,s^-_1,\\cdots ,s^-_K)$ where $s^+\\in \\mathbb {R}$ is the image-document score with its matched image and $s^-_k\\in \\mathbb {R}$ for $k=1,\\cdots ,K$ is the image-document score with $K$ mismatched images.","We use $s_l$ and $s_g$ to denote $(K+1)$ -length score vectors defined above computed using the local and the global score functions respectively.","The image and text encoders are trained to minimize $\\mathcal {L}(s_l)+\\mathcal {L}(s_g)$ over documents in the training set where $\\mathcal {L}$ is the text-to-image contrastive loss [26], [33] $\\mathcal {L}(s)\\triangleq -\\log \\frac{ \\exp (\\gamma \\,s^+) }{ \\exp (\\gamma \\,s^+) + \\sum _{k=1}^K \\exp (\\gamma \\,s^-_k) }$ with scale parameter $\\gamma $ .","In the equation above, $s$ is either vector $s_l$ computed using (REF ) with $\\pi _l$ defined in (REF ) or vector $s_g$ computed using () with $\\pi _g$ defined in (REF ).","The image-to-text contrastive loss where the negative scores are computed for an image with $K$ different mismatched documents is often used alongside $\\mathcal {L}$ in prior work [2], [4], [14], [33].","We choose to treat images and text asymmetrically and show that the simple text-to-image contrastive loss is sufficient to induce representations that generalize well.","In multiple instance learning [3], a set that contains many instances $\\left\\lbrace x_1,\\cdots ,x_N \\right\\rbrace $ is referred to as a bag.","The training set consists of bags and their associated bag labels $y$ while the instance labels are not provided.","For binary bag labels, a positive bag is guaranteed to include at least one positive instance, while a negative bag includes no positive instances.","The bag-level labels are used to train classifier to assign instance-level and bag-level labels in new, unseen bags.","Existing image-text representation learning algorithms that are either predictive [6] or contrastive [27] can be seen as a form of multiple instance learning.","Specifically, we can view an image as a bag of region features and the corresponding sentence that describes the image as the bag label.","Instead of taking on binary values, the bag labels can represent arbitrary categories via natural language.","Although the exact region that corresponds to the sentence is unknown, the matched image contains at least one region that corresponds to the text while a randomly sampled image most likely does not.","Similar to multiple instance learning, self-supervised representation learning methods use these assumptions for learning.","More generally, we consider the text label as a bag of sentences.","For example, sentences describing findings within a chest X-ray image most likely can be permuted without changing the overall meaning.","Therefore, representation learning can be interpreted as predicting the label bag $\\left\\lbrace y_m \\right\\rbrace $ given the input bag $\\left\\lbrace x_m \\right\\rbrace $ .","This setup corresponds to multi-instance multi-label learning [34].","Moreover, multiple instance learning and multimodal representation learning share comparable goals.","Multiple instance learning aims to align instances and bags with labels such that the pre-trained model performs well in classification tasks.","Multimodal representation learning aims to align images and their subregions with text such that the pre-trained model perform well on tasks that rely on such alignment, e.g., image classification relies on image-sentence alignment, visual grounding and cross-modal retrieval rely on region-sentence alignment.","There are two main multiple instance learning approaches, instance-level and embedding-level approaches [1].","Instance-level approach computes the bag score by aggregating the instance scores.","Embedding-level approach computes the bag score based on a bag feature that is aggregated from the instance features.","The local and global approaches in Section REF are extensions of the instance and embedding approaches to contrastive learning.","This parallel enables us to analyze prior methods as instances of the framework defined in Section REF that is inspired by multiple instance learning (Table REF ).","We make one generalization to the formulation in Section REF to accommodate cross attention [21]: the local aggregator function $\\pi _l$ can potentially rely on label features $y_m$ to multiplex its behavior, i.e., $\\pi _l: \\mathcal {P}(\\mathcal {X})\\times \\mathcal {Y}\\rightarrow \\mathcal {X}$ .","In summary, a diverse set of aggregators $\\pi _l,\\pi _g,\\pi _s$ have been demonstrated on multimodal representation learning at varying scales, implying there may not be a single set of aggregators that works well for every problem.","More realistically, the best aggregator functions are the ones that fit application-specific assumptions well."],["Experiments","We illustrate the proposed approach by building a representation of frontal chest X-ray images and associated radiology reports and using it in downstream tasks.","In all the experiments, data used for representation learning is disjoint from the test sets used to evaluate the downstream tasks.","We normalize the images and resize them to 512x512 resolution.","We apply random image augmentations, i.e., 480x480 random crops, brightness and contrast variations, and random affine transforms (only for image model fine-tuning during evaluation).","We use PySBD [29] for sentence tokenization.","We employ ResNet-50 [13] as the image region encoder and CXR-BERT [2] as the sentence encoder.","Each encoder is followed by a linear projection to a 128 dimension embedding space.","In particular, the projected ResNet-50 conv-5 activations act as the region features $\\left\\lbrace x_n \\right\\rbrace $ and the projected mean-pooled contextualized word embeddings acts as the sentence features $\\left\\lbrace y_m \\right\\rbrace $ ."],["Representation learning","We use a subset of 234,073 chest X-ray images and report from MIMIC-CXR [16] for representation learning.","We randomly initialize the image encoder and use the CXR-BERT model [2] pretrained on a biomedical corpus (i.e., the stage II model) as the sentence encoder.","We use the AdamW optimizer and decay the initial learning rate of 5e-5 using a cosine schedule with 2k warmup steps.","we initialize $\\gamma =14$ and optimize this hyperparameter alongside the encoder parameters.","We set other scale parameters as follows: $\\gamma _l=0.1, \\gamma _g=e$ .","We use a batch size of 64.","For every image in the batch, we sample 5 sentences, with replacement if needed, to make up the label bag.","Here, $N=225$ and $M=5$ ."],["Downstream Tasks","Image Classification To evaluate zero-shot (ZS) and fine-tuned (FT) classification performance, we use the same split of RSNA Pneumonia (RSNA) [30] as in [14], specifically, 18,678/4,003/4,003 for training/validation/testing.","To evaluate in-distribution fine-tuned classification performance in the ablation study, we use 5 CheXpert labels (Atelectasis, Cardiomegaly, Edema, Pleural Effusion, Pneumothorax) on the MIMIC-CXR data set [16] that we denote MIMIC-CheXpert (CheX).","There are roughly 1k images in the test set associated with each CheXpert label.","To evaluate the data efficiency of representation learning approaches, we use different amounts of training data (1% and 100%).","For zero-shot image classification, we first tokenize and encode the class-specific text prompts (e.g., “Findings suggesting pneumonia.” and “No evidence of pneumonia.”).","For each image, we assign a binary label that corresponds to the prompt with the higher image-sentence score.","We find it important to normalize the scores to $[0,1]$ for each class before applying the softmax.","For fine-tuned image classification, we use the Adam optimizer [19] with a learning rate of 3e-3 to optimize the randomly initialized weights and a bias over the mean-pooled region features while keeping the encoder weights fixed.","For RSNA Pneumonia, we report accuracy and AUC.","For MIMIC-CheXpert, we report the average AUC over five binary classification tasks.","Visual Grounding We evaluate visual grounding performance using the MS-CXR region-sentence annotations [2].","This data set consists of 1,448 bounding boxes over 1,162 images, where each bounding box is associated with a sentence that describes its dominant radiological feature.","We compute region-sentence scores to quantify how well the sentence is localized in the image.","We report a measure of discrepancy between region-sentence scores inside and outside the bounding box, i.e., contrast-to-noise ratio (CNR) [2], and how well the thresholded region-sentence scores overlap with the bounding box on average, i.e., mean intersection over union (mIoU).","In contrast to [2], we pick thresholds that span $[-1,1]$ in $0.05$ increments to compute the mIoU for a fair comparison.","Cross-Modal Retrieval We evaluate cross-modal retrieval performance using the MS-CXR data set as well.","We compute the bounding box features from the region features using RoIAlign [12].","We compute box-sentence scores for ranking.","Specifically, we retrieve items in one modality given a query from the other modality by sorting based on the box-sentence scores.","The correctly retrieved item is the one that is paired with the query item.","We report the fraction of times the correct item was found in the top K results (R@K) and the median rank of the correct item in the ranked list (MedR)."],["Results","Comparison with State-of-the-art Methods We compare the proposed approach LSE$+$ NL with the state-of-the-art methods GLoRIA [14] and BioViL [2].","GLoRIA is a representation learning method that learns based on image-sentence and region-word pairs.","BioViL improves upon GLoRIA by using a better text encoder, relying on a symmetric contrastive loss and masked language modeling for representation learning.","We show that our simple model provides consistently better performance than these state-of-the-art algorithms.","We omit reporting GLoRIA's classification and visual grounding performance for GLoRIA as [2] showed that BioViL is consistently better than GLoRIA on these tasks.","Table REF reports image classification accuracy based on the learned representations for different amounts of data used to fine-tune the representation for the downstream task (zero-shot, 1%, and 100%).","Our method is competitive or better than the baseline, especially in the zero-shot setup, underscoring its promise for limited annotation scenarios.","Table REF and Table REF report the methods' performance on visual grounding and cross-modal retrieval respectively.","Our method is significantly better than the baseline.","Figure REF illustrates examples of visual grounding.","Unlike [2], we do not smooth the region-sentence scores produced by our model.","Our method yield qualitatively better region-sentence scores than BioViL on a few challenging failure cases discussed in [2].","In particular, our pre-trained model captures location specifications more effectively, e.g., recognizing “at both lung bases” in the first image and “right” in the third image.","Both our method and BioViL are prone to false positives, i.e., regions outside the ground-truth bounding box with high region-sentence scores, which highlights the need for further improvements.","Figure: Example visual grounding results for several challenging cases for BioVil (top row) and our method (bottom row).","Text queries and the corresponding ground truth bounding boxes are shown for each image.","Colormap overlay visualizes region-sentence scores (blue corresponds to low scores, red highlights regions with high scores).","Our method provides maps that align better with the ground truth bounding boxes.Table: Ablation study results.","For each variant of the method, performance statistics are reported for each downstream task consistently with Tables , , and .","RSNA is RSNA Pneumonia.","CheX is MIMIC-CheXpert.","FT is fine-tuned classification using 100% of the labels.","ZS is zero-shot classification.","We report AUC for image classification.","Local representations perform well for image classification, while visual grounding and cross-modal retrieval benefit from integration of local and global representations.Figure: Effects of aggregator choice on the performance.","Performance of models trained with local aggregators (shades of blue), global aggregators (shades of orange) and combinations of local and global aggregators (shades of green) is shown for image classification (AUC), visual grounding (CNR) and cross-modality retrieval (MedR averaged for both directions).","The metrics are normalized to unit interval for easier comparisons across tasks.","The choice of aggregators effects image classification performance much less than that of visual grounding and cross-modality retrieval.","There is high performance variations within each group.","Combination approaches do well on all tasks.Ablation In the ablation study (Table REF ), we compare our method LSE$+$ NL with using either the local LSE or the global NL approach only, as well as replacing the NL with average as the region aggregator, i.e., LSE$+$ Average.","To enable extensive experimentation, we use ResNet-18 as the image encoder.","LSE$+$ NL provides good trade-off between region-sentence and image-sentence alignment.","LSE$+$ NL has comparable performance to LSE for image classification tasks while significantly outperforming all alternatives in visual grounding and cross-modal retrieval.","Using a larger image encoder model ResNet-50 provides only a modest improvement in visual grounding.","Aggregator Choices Figure REF compares a few instance aggregators' performance on downstream tasks.","We compare the local approach (e.g., LSE, NOR [24], NAND [20]) the global approach (e.g., Max, Average, Att [15]) and a combination of local and global approaches (e.g., LSE$+$ Att, LSE$+$ NL).","Aggregators within each approach exhibits high performance variations.","The best local aggregator is superior to the best global aggregators we explored on all downstream tasks; While a combination of local and global approaches yields the best performing method."],["Conclusions","In this paper, we propose a framework to construct permutation-invariant image-document score functions for multimodal contrastive learning.","Taking inspiration from multiple instance learning, we introduce LSE$+$ NL for learning multimodal representations that rely on both local and global score functions and exploit correlation between image regions.","Our method outperforms the state-of-the-art approaches on image classification, visual grounding, and cross-modal retrieval.","In addition, we show that contrastive representation learning is a form of multiple instance learning, providing us with valuable insights from a related field for solving shared challenges to learn representations that generalized well."]],"string":"[\n [\n \"Using Multiple Instance Learning to Build Multimodal Representations\"\n ],\n [\n \"Abstract Image-text multimodal representation learning aligns data across modalities and enables important medical applications, e.g., image classification, visual grounding, and cross-modal retrieval.\",\n \"In this work, we establish a connection between multimodal representation learning and multiple instance learning.\",\n \"Based on this connection, we propose a generic framework for constructing permutation-invariant score functions with many existing multimodal representation learning approaches as special cases.\",\n \"Furthermore, we use the framework to derive a novel contrastive learning approach and demonstrate that our method achieves state-of-the-art results on a number of downstream tasks.\"\n ],\n [\n \"Introduction\",\n \"In this paper, we propose a framework for designing multimodal representation learning methods that encompasses previous approaches as special cases and implies a new algorithm for multimodal learning that advances the state of the art.\",\n \"Specifically, we establish a connection between self-supervised representation learning based on contrastive learning and multiple instance learning [3] and show that they share similar assumptions and goals.\",\n \"We bring insights from multiple instance learning to offer a fresh perspective on self-supervised representation learning and ideas for performance improvements.\",\n \"With this connection in mind, we derive a novel algorithm for learning image-text representations that capture the structure shared between the two modalities and generalize well in a variety of downstream tasks.\",\n \"We aim to establish alignment between images and associated text to improve clinical workflow.\",\n \"For example, an image model that mimics the radiologists' interpretation could retroactively label images to select relevant patients for a clinical trial.\",\n \"Further, local alignment between image regions and text fragments (e.g., sentences) promises to benefit many downstream tasks.\",\n \"For example, cross-modal retrieval can provide description of an image region for automated documentation or enable comparisons with similar previously imaged patients for better interpretation based on local anatomy or pathology.\",\n \"Similarly, radiologists documenting findings can verify the accuracy of the report by noting if the referred location (i.e., visual grounding of the text) is consistent with their impression of the image.\",\n \"Self-supervised representation learning is a useful tool for reducing annotation burden for machine learning models in medical imaging.\",\n \"Despite the need and opportunities for automation, development of robust machine learning methods is held back by the lack of annotations that serve as the supervision signal for learning.\",\n \"Self-supervised representation learning on paired image-text data offers two advantages: (i) learning requires no further annotations and (ii) treating text as “labels” enables us to use natural language to reference visual concepts and vice versa [27].\",\n \"Thus, we focus on learning image-text multimodal representations but the proposed framework is broadly applicable to representation learning on other multimodal data.\",\n \"Learning joint representations involves training image and text encoders to perform self-supervised tasks on paired image-text data [5], [23] and evaluating on relevant downstream tasks.\",\n \"We focus on contrastive learning, i.e., classifying image-text pairs as matched (i.e., corresponding to the same imaging event), or mismatched.\",\n \"Contrastive learning has been applied to medical domain, demonstrating impressive transfer capabilities on a diverse set of tasks [2], [4], [14], [33].\",\n \"The biggest improvements come from addressing challenges unique to this domain, e.g., the use of cross attention to deal with the lack of effective pathology detectors [14] and adaptation of language models to address linguistic challenges in clinical notes [2].\",\n \"Training the models has involved increasingly complex contrastive loss functions that treat image and text symmetrically [2], [4], [14], [33] and on multiple scales [2], [14].\",\n \"In contrast to previous work that relies on many loss terms, our proposed contrastive loss is simple to implement and yields superior performance.\",\n \"Borrowing ideas from multiple instance learning, we treat local image region features as “data” and sentence features as (complex) “labels”.\",\n \"Multiple instance learning is a type of weakly supervised learning that is effective for problems that lack fine-grain annotations [3].\",\n \"For example, it can localize tumor cells in whole slide images with just image-level labels [22].\",\n \"Central to multiple instance learning is the construction of permutation-invariant score functions [15], and the choice of how the instance scores or features are aggregated to be evaluated against an image-level label.\",\n \"Effective instance aggregators take advantage of domain knowledge [8], e.g., the Noisy-OR aggregator for drug activity prediction [24], the Noisy-AND aggregator for cellular phenotype classification [20].\",\n \"In our work, we extend multiple instance classification to contrastive learning by constructing permutation-invariant image-text score functions.\",\n \"Building on insights from multiple instance classification with correlated instances [22], our proposed instance aggregator exploits correlation across instances to build representations that perform well in downstream tasks.\",\n \"Many prior multiple instance learning methods focused on one particular task of interest, e.g., detection [32], region classification [7], or retrieval [18].\",\n \"Some investigated the choices of instance aggregators for more than one downstream task [11], [25] but are limited in generality (i.e., not meant to be applicable to other applications) and scope (i.e., explored a few simple instance aggregators).\",\n \"In contrast, the proposed framework for constructing permutation-invariant score functions can be readily applied to other applications.\",\n \"We systematically investigate instance aggregators and their effect on representation learning, which yields a novel approach to learning joint representations.\",\n \"We evaluate the resulting image-text representations on a diverse set of downstream tasks and demonstrate state-of-the-art performance across all tasks in the context of a large set of chest X-ray images and associated radiological reports.\"\n ],\n [\n \"Method\",\n \"We first introduce notation and discuss the local and global approaches for constructing permutation-invariant image-document score functions at the core of the learning procedure.\",\n \"We then instantiate the framework for a specific choice of aggregators for contrastive learning.\"\n ],\n [\n \"Problem Setup\",\n \"A local $D$ -dimensional representation of an image with $N$ proposed regions is a collection of $N$ features vectors $x_n\\\\in \\\\mathcal {X}\\\\subset \\\\mathbb {R}^D$ , $n\\\\in \\\\left\\\\lbrace 1,\\\\cdots ,N \\\\right\\\\rbrace $ .\",\n \"In our experiments, we use regular tiling to generate image regions and leave more sophisticated proposal methods (e.g., [28]) for future work.\",\n \"A local representation of a $M$ -sentence document (e.g., a radiology report) is a collection of sentence feature vectors $y_m \\\\in \\\\mathcal {Y}\\\\subset \\\\mathbb {R}^D$ , $m\\\\in \\\\left\\\\lbrace 1,\\\\cdots ,M \\\\right\\\\rbrace $ .\",\n \"Function $h: \\\\mathcal {X}\\\\times \\\\mathcal {Y}\\\\rightarrow \\\\mathbb {R}$ measures the similarity between representations, e.g., $h(x_n,y_m)$ is the similarity between a region and a sentence.\",\n \"In our experiments, we use cosine similarity $h(x, y) = \\\\langle x, y \\\\rangle / \\\\left( \\\\left\\\\Vert x\\\\right\\\\Vert \\\\left\\\\Vert y\\\\right\\\\Vert \\\\right)$ , though the formulation accepts any differentiable similarity function.\",\n \"For any vector space $\\\\mathcal {U}$ , aggregator function $\\\\pi : \\\\mathcal {P}(\\\\mathcal {U}) \\\\rightarrow \\\\mathcal {U}$ aggregates elements in the input set into a “representative”.\",\n \"$\\\\mathcal {P}(\\\\mathcal {U})$ is the set of all finite subsets of $\\\\mathcal {U}$ .\",\n \"For example, $\\\\pi (\\\\left\\\\lbrace x_n \\\\right\\\\rbrace ) = \\\\frac{1}{N}\\\\sum _n x_n$ aggregates $N$ region features $x_n\\\\in \\\\mathcal {X}$ by averaging them, while $\\\\pi (\\\\left\\\\lbrace h_n \\\\right\\\\rbrace ) = \\\\max _n h_n$ aggregates $N$ similarity scores into a single score by computing the maximum score.\",\n \"We restrict our attention to aggregators that are permutation-invariant, i.e., they treat their input as an unordered set rather than an ordered vector.\",\n \"Permutation-invariant image-document score function $S: \\\\mathcal {P}(\\\\mathcal {X}) \\\\times \\\\mathcal {P}(\\\\mathcal {Y}) \\\\rightarrow \\\\mathbb {R}$ measures the similarity between an image and a document based on region features $\\\\left\\\\lbrace x_n \\\\right\\\\rbrace $ and sentence features $\\\\left\\\\lbrace y_m \\\\right\\\\rbrace $ .\",\n \"Figure: Local (top) and global (bottom) image-document score functions.\"\n ],\n [\n \"Local & Global Permutation-Invariant Score Functions\",\n \"Contrastive representation learning can be seen as maximizing the likelihood of correctly classifying image-text pairs as matched or mismatched.\",\n \"Since supervision is provided at the image-document level, we define a framework to build permutation-invariant image-document score functions.\",\n \"The local approach aggregates region-sentence scores into an image-sentence score.\",\n \"The image-sentence score $g_m$ for sentence $m$ in the document is obtained by applying a local aggregator function $\\\\pi _l$ to region-sentence scores, i.e., $g_m = \\\\pi _l ( \\\\left\\\\lbrace h(x_n, y_m) \\\\right\\\\rbrace _n ) \\\\triangleq \\\\pi _l ( \\\\left\\\\lbrace h(x_1, y_m), \\\\cdots , h(x_N, y_m) \\\\right\\\\rbrace ) $ .\",\n \"The global approach first aggregates local region features $\\\\left\\\\lbrace x_n \\\\right\\\\rbrace $ into a single image feature vector $\\\\pi _g(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace )$ using a global aggregator function $\\\\pi _g$ .\",\n \"The image-sentence score $g_m$ is computed using the similarity function $h$ on the image feature vector $\\\\pi _g(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace )$ and sentence feature vector $y_m$ , i.e., $g_m = h(\\\\pi _g(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace ), y_m)$ .\",\n \"In both approaches, the image-document score $S$ is obtained by aggregating image-sentence scores with another aggregator function $\\\\pi _s$ , i.e., $S(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace , \\\\left\\\\lbrace y_m \\\\right\\\\rbrace ) = \\\\pi _s(\\\\left\\\\lbrace g_m \\\\right\\\\rbrace )$ .\",\n \"Figure REF illustrates the framework for constructing $S$ .\",\n \"To summarize, the local and global image-document scores $S_l$ and $S_g$ are computed as follows: $S_l(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace , \\\\left\\\\lbrace y_m \\\\right\\\\rbrace )&= \\\\pi _s(\\\\left\\\\lbrace \\\\pi _l(\\\\left\\\\lbrace h(x_n, y_m) \\\\right\\\\rbrace _n) \\\\right\\\\rbrace _m), \\\\\\\\S_g(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace , \\\\left\\\\lbrace y_m \\\\right\\\\rbrace )&= \\\\pi _s(\\\\left\\\\lbrace h( \\\\pi _g(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace ), y_m ) \\\\right\\\\rbrace _m).$ As the aggregator functions are permutation-invariant, the image-document score function $S$ is naturally permutation-invariant as well.\",\n \"We emphasize that $S$ treats image features and text features differently, and that the order of application of similarity evaluation $h(\\\\cdot )$ and aggregators $\\\\pi (\\\\cdot )$ is empirically relevant.\",\n \"This design decision is motivated by the fact that each sentence in a radiology report represent a concept and its location in the image, i.e., it is akin to a label for some region in the image.\",\n \"The converse is not necessarily true as some parts of the image are not described in the report.\"\n ],\n [\n \"Representation Learning with LSE$+$ NL Aggregators\",\n \"In this section, we introduce our method LSE$+$ NL for learning multimodal representations that relies on a combination of local and global image-document score functions and an asymmetric text-to-image contrastive loss.\",\n \"Inspired by [22], we use a soft maximum function to identify the most relevant region for a sentence, i.e., the critical region, and attend more to regions that are similar to the critical region.\",\n \"Specifically, the local aggregator $\\\\pi _l$ is the log-sum-exp (LSE) function $\\\\pi _l(\\\\left\\\\lbrace h_n \\\\right\\\\rbrace )= \\\\frac{1}{\\\\gamma _l} \\\\log \\\\sum _{n=1}^N \\\\exp (\\\\gamma _l\\\\,h_n),$ where $\\\\gamma _l$ is a scale parameter that controls how well the LSE function approximates the max function.\",\n \"The global aggregator $\\\\pi _g$ linearly combine the region features using the distance to the critical region as weights, i.e., $\\\\pi _g(\\\\left\\\\lbrace x_n \\\\right\\\\rbrace )= \\\\sum _{n=1}^N \\\\dfrac{ \\\\exp ( \\\\gamma _g\\\\, \\\\langle Ax_n, Ax_k \\\\rangle ) }{ \\\\sum _{n^{\\\\prime }=1}^N \\\\exp (\\\\gamma _g\\\\, \\\\langle Ax_{n^{\\\\prime }}, Ax_k \\\\rangle ) } \\\\, x_n,$ where $k$ is the index of the critical region, i.e., $k=\\\\operatornamewithlimits{\\\\arg \\\\,\\\\max }_n h(x_n,y_m)$ , $A$ is a learned weight matrix, and $\\\\gamma _g$ is the scale parameter for the softmax function.\",\n \"We can interpret $\\\\pi _g$ as a form of attention where regions that are more similar to the critical region are given a higher attention weight.\",\n \"In effect, $\\\\pi _g$ exploits the correlation between each region and the critical region using attention.\",\n \"In addition, $\\\\pi _g$ can be seen as a form of non-local (NL) network [31].\",\n \"Both $\\\\pi _l$ and $\\\\pi _g$ are permutation-invariant functions.\",\n \"We choose $\\\\pi _s$ to be the average function.\",\n \"We use the local and global image-document scores in (REF ) and () computed with our choice of $\\\\pi _l$ and $\\\\pi _g$ for contrastive learning.\",\n \"Given a document, we form an image-document score vector $s \\\\triangleq (s^+,s^-_1,\\\\cdots ,s^-_K)$ where $s^+\\\\in \\\\mathbb {R}$ is the image-document score with its matched image and $s^-_k\\\\in \\\\mathbb {R}$ for $k=1,\\\\cdots ,K$ is the image-document score with $K$ mismatched images.\",\n \"We use $s_l$ and $s_g$ to denote $(K+1)$ -length score vectors defined above computed using the local and the global score functions respectively.\",\n \"The image and text encoders are trained to minimize $\\\\mathcal {L}(s_l)+\\\\mathcal {L}(s_g)$ over documents in the training set where $\\\\mathcal {L}$ is the text-to-image contrastive loss [26], [33] $\\\\mathcal {L}(s)\\\\triangleq -\\\\log \\\\frac{ \\\\exp (\\\\gamma \\\\,s^+) }{ \\\\exp (\\\\gamma \\\\,s^+) + \\\\sum _{k=1}^K \\\\exp (\\\\gamma \\\\,s^-_k) }$ with scale parameter $\\\\gamma $ .\",\n \"In the equation above, $s$ is either vector $s_l$ computed using (REF ) with $\\\\pi _l$ defined in (REF ) or vector $s_g$ computed using () with $\\\\pi _g$ defined in (REF ).\",\n \"The image-to-text contrastive loss where the negative scores are computed for an image with $K$ different mismatched documents is often used alongside $\\\\mathcal {L}$ in prior work [2], [4], [14], [33].\",\n \"We choose to treat images and text asymmetrically and show that the simple text-to-image contrastive loss is sufficient to induce representations that generalize well.\",\n \"In multiple instance learning [3], a set that contains many instances $\\\\left\\\\lbrace x_1,\\\\cdots ,x_N \\\\right\\\\rbrace $ is referred to as a bag.\",\n \"The training set consists of bags and their associated bag labels $y$ while the instance labels are not provided.\",\n \"For binary bag labels, a positive bag is guaranteed to include at least one positive instance, while a negative bag includes no positive instances.\",\n \"The bag-level labels are used to train classifier to assign instance-level and bag-level labels in new, unseen bags.\",\n \"Existing image-text representation learning algorithms that are either predictive [6] or contrastive [27] can be seen as a form of multiple instance learning.\",\n \"Specifically, we can view an image as a bag of region features and the corresponding sentence that describes the image as the bag label.\",\n \"Instead of taking on binary values, the bag labels can represent arbitrary categories via natural language.\",\n \"Although the exact region that corresponds to the sentence is unknown, the matched image contains at least one region that corresponds to the text while a randomly sampled image most likely does not.\",\n \"Similar to multiple instance learning, self-supervised representation learning methods use these assumptions for learning.\",\n \"More generally, we consider the text label as a bag of sentences.\",\n \"For example, sentences describing findings within a chest X-ray image most likely can be permuted without changing the overall meaning.\",\n \"Therefore, representation learning can be interpreted as predicting the label bag $\\\\left\\\\lbrace y_m \\\\right\\\\rbrace $ given the input bag $\\\\left\\\\lbrace x_m \\\\right\\\\rbrace $ .\",\n \"This setup corresponds to multi-instance multi-label learning [34].\",\n \"Moreover, multiple instance learning and multimodal representation learning share comparable goals.\",\n \"Multiple instance learning aims to align instances and bags with labels such that the pre-trained model performs well in classification tasks.\",\n \"Multimodal representation learning aims to align images and their subregions with text such that the pre-trained model perform well on tasks that rely on such alignment, e.g., image classification relies on image-sentence alignment, visual grounding and cross-modal retrieval rely on region-sentence alignment.\",\n \"There are two main multiple instance learning approaches, instance-level and embedding-level approaches [1].\",\n \"Instance-level approach computes the bag score by aggregating the instance scores.\",\n \"Embedding-level approach computes the bag score based on a bag feature that is aggregated from the instance features.\",\n \"The local and global approaches in Section REF are extensions of the instance and embedding approaches to contrastive learning.\",\n \"This parallel enables us to analyze prior methods as instances of the framework defined in Section REF that is inspired by multiple instance learning (Table REF ).\",\n \"We make one generalization to the formulation in Section REF to accommodate cross attention [21]: the local aggregator function $\\\\pi _l$ can potentially rely on label features $y_m$ to multiplex its behavior, i.e., $\\\\pi _l: \\\\mathcal {P}(\\\\mathcal {X})\\\\times \\\\mathcal {Y}\\\\rightarrow \\\\mathcal {X}$ .\",\n \"In summary, a diverse set of aggregators $\\\\pi _l,\\\\pi _g,\\\\pi _s$ have been demonstrated on multimodal representation learning at varying scales, implying there may not be a single set of aggregators that works well for every problem.\",\n \"More realistically, the best aggregator functions are the ones that fit application-specific assumptions well.\"\n ],\n [\n \"Experiments\",\n \"We illustrate the proposed approach by building a representation of frontal chest X-ray images and associated radiology reports and using it in downstream tasks.\",\n \"In all the experiments, data used for representation learning is disjoint from the test sets used to evaluate the downstream tasks.\",\n \"We normalize the images and resize them to 512x512 resolution.\",\n \"We apply random image augmentations, i.e., 480x480 random crops, brightness and contrast variations, and random affine transforms (only for image model fine-tuning during evaluation).\",\n \"We use PySBD [29] for sentence tokenization.\",\n \"We employ ResNet-50 [13] as the image region encoder and CXR-BERT [2] as the sentence encoder.\",\n \"Each encoder is followed by a linear projection to a 128 dimension embedding space.\",\n \"In particular, the projected ResNet-50 conv-5 activations act as the region features $\\\\left\\\\lbrace x_n \\\\right\\\\rbrace $ and the projected mean-pooled contextualized word embeddings acts as the sentence features $\\\\left\\\\lbrace y_m \\\\right\\\\rbrace $ .\"\n ],\n [\n \"Representation learning\",\n \"We use a subset of 234,073 chest X-ray images and report from MIMIC-CXR [16] for representation learning.\",\n \"We randomly initialize the image encoder and use the CXR-BERT model [2] pretrained on a biomedical corpus (i.e., the stage II model) as the sentence encoder.\",\n \"We use the AdamW optimizer and decay the initial learning rate of 5e-5 using a cosine schedule with 2k warmup steps.\",\n \"we initialize $\\\\gamma =14$ and optimize this hyperparameter alongside the encoder parameters.\",\n \"We set other scale parameters as follows: $\\\\gamma _l=0.1, \\\\gamma _g=e$ .\",\n \"We use a batch size of 64.\",\n \"For every image in the batch, we sample 5 sentences, with replacement if needed, to make up the label bag.\",\n \"Here, $N=225$ and $M=5$ .\"\n ],\n [\n \"Downstream Tasks\",\n \"Image Classification To evaluate zero-shot (ZS) and fine-tuned (FT) classification performance, we use the same split of RSNA Pneumonia (RSNA) [30] as in [14], specifically, 18,678/4,003/4,003 for training/validation/testing.\",\n \"To evaluate in-distribution fine-tuned classification performance in the ablation study, we use 5 CheXpert labels (Atelectasis, Cardiomegaly, Edema, Pleural Effusion, Pneumothorax) on the MIMIC-CXR data set [16] that we denote MIMIC-CheXpert (CheX).\",\n \"There are roughly 1k images in the test set associated with each CheXpert label.\",\n \"To evaluate the data efficiency of representation learning approaches, we use different amounts of training data (1% and 100%).\",\n \"For zero-shot image classification, we first tokenize and encode the class-specific text prompts (e.g., “Findings suggesting pneumonia.” and “No evidence of pneumonia.”).\",\n \"For each image, we assign a binary label that corresponds to the prompt with the higher image-sentence score.\",\n \"We find it important to normalize the scores to $[0,1]$ for each class before applying the softmax.\",\n \"For fine-tuned image classification, we use the Adam optimizer [19] with a learning rate of 3e-3 to optimize the randomly initialized weights and a bias over the mean-pooled region features while keeping the encoder weights fixed.\",\n \"For RSNA Pneumonia, we report accuracy and AUC.\",\n \"For MIMIC-CheXpert, we report the average AUC over five binary classification tasks.\",\n \"Visual Grounding We evaluate visual grounding performance using the MS-CXR region-sentence annotations [2].\",\n \"This data set consists of 1,448 bounding boxes over 1,162 images, where each bounding box is associated with a sentence that describes its dominant radiological feature.\",\n \"We compute region-sentence scores to quantify how well the sentence is localized in the image.\",\n \"We report a measure of discrepancy between region-sentence scores inside and outside the bounding box, i.e., contrast-to-noise ratio (CNR) [2], and how well the thresholded region-sentence scores overlap with the bounding box on average, i.e., mean intersection over union (mIoU).\",\n \"In contrast to [2], we pick thresholds that span $[-1,1]$ in $0.05$ increments to compute the mIoU for a fair comparison.\",\n \"Cross-Modal Retrieval We evaluate cross-modal retrieval performance using the MS-CXR data set as well.\",\n \"We compute the bounding box features from the region features using RoIAlign [12].\",\n \"We compute box-sentence scores for ranking.\",\n \"Specifically, we retrieve items in one modality given a query from the other modality by sorting based on the box-sentence scores.\",\n \"The correctly retrieved item is the one that is paired with the query item.\",\n \"We report the fraction of times the correct item was found in the top K results (R@K) and the median rank of the correct item in the ranked list (MedR).\"\n ],\n [\n \"Results\",\n \"Comparison with State-of-the-art Methods We compare the proposed approach LSE$+$ NL with the state-of-the-art methods GLoRIA [14] and BioViL [2].\",\n \"GLoRIA is a representation learning method that learns based on image-sentence and region-word pairs.\",\n \"BioViL improves upon GLoRIA by using a better text encoder, relying on a symmetric contrastive loss and masked language modeling for representation learning.\",\n \"We show that our simple model provides consistently better performance than these state-of-the-art algorithms.\",\n \"We omit reporting GLoRIA's classification and visual grounding performance for GLoRIA as [2] showed that BioViL is consistently better than GLoRIA on these tasks.\",\n \"Table REF reports image classification accuracy based on the learned representations for different amounts of data used to fine-tune the representation for the downstream task (zero-shot, 1%, and 100%).\",\n \"Our method is competitive or better than the baseline, especially in the zero-shot setup, underscoring its promise for limited annotation scenarios.\",\n \"Table REF and Table REF report the methods' performance on visual grounding and cross-modal retrieval respectively.\",\n \"Our method is significantly better than the baseline.\",\n \"Figure REF illustrates examples of visual grounding.\",\n \"Unlike [2], we do not smooth the region-sentence scores produced by our model.\",\n \"Our method yield qualitatively better region-sentence scores than BioViL on a few challenging failure cases discussed in [2].\",\n \"In particular, our pre-trained model captures location specifications more effectively, e.g., recognizing “at both lung bases” in the first image and “right” in the third image.\",\n \"Both our method and BioViL are prone to false positives, i.e., regions outside the ground-truth bounding box with high region-sentence scores, which highlights the need for further improvements.\",\n \"Figure: Example visual grounding results for several challenging cases for BioVil (top row) and our method (bottom row).\",\n \"Text queries and the corresponding ground truth bounding boxes are shown for each image.\",\n \"Colormap overlay visualizes region-sentence scores (blue corresponds to low scores, red highlights regions with high scores).\",\n \"Our method provides maps that align better with the ground truth bounding boxes.Table: Ablation study results.\",\n \"For each variant of the method, performance statistics are reported for each downstream task consistently with Tables , , and .\",\n \"RSNA is RSNA Pneumonia.\",\n \"CheX is MIMIC-CheXpert.\",\n \"FT is fine-tuned classification using 100% of the labels.\",\n \"ZS is zero-shot classification.\",\n \"We report AUC for image classification.\",\n \"Local representations perform well for image classification, while visual grounding and cross-modal retrieval benefit from integration of local and global representations.Figure: Effects of aggregator choice on the performance.\",\n \"Performance of models trained with local aggregators (shades of blue), global aggregators (shades of orange) and combinations of local and global aggregators (shades of green) is shown for image classification (AUC), visual grounding (CNR) and cross-modality retrieval (MedR averaged for both directions).\",\n \"The metrics are normalized to unit interval for easier comparisons across tasks.\",\n \"The choice of aggregators effects image classification performance much less than that of visual grounding and cross-modality retrieval.\",\n \"There is high performance variations within each group.\",\n \"Combination approaches do well on all tasks.Ablation In the ablation study (Table REF ), we compare our method LSE$+$ NL with using either the local LSE or the global NL approach only, as well as replacing the NL with average as the region aggregator, i.e., LSE$+$ Average.\",\n \"To enable extensive experimentation, we use ResNet-18 as the image encoder.\",\n \"LSE$+$ NL provides good trade-off between region-sentence and image-sentence alignment.\",\n \"LSE$+$ NL has comparable performance to LSE for image classification tasks while significantly outperforming all alternatives in visual grounding and cross-modal retrieval.\",\n \"Using a larger image encoder model ResNet-50 provides only a modest improvement in visual grounding.\",\n \"Aggregator Choices Figure REF compares a few instance aggregators' performance on downstream tasks.\",\n \"We compare the local approach (e.g., LSE, NOR [24], NAND [20]) the global approach (e.g., Max, Average, Att [15]) and a combination of local and global approaches (e.g., LSE$+$ Att, LSE$+$ NL).\",\n \"Aggregators within each approach exhibits high performance variations.\",\n \"The best local aggregator is superior to the best global aggregators we explored on all downstream tasks; While a combination of local and global approaches yields the best performing method.\"\n ],\n [\n \"Conclusions\",\n \"In this paper, we propose a framework to construct permutation-invariant image-document score functions for multimodal contrastive learning.\",\n \"Taking inspiration from multiple instance learning, we introduce LSE$+$ NL for learning multimodal representations that rely on both local and global score functions and exploit correlation between image regions.\",\n \"Our method outperforms the state-of-the-art approaches on image classification, visual grounding, and cross-modal retrieval.\",\n \"In addition, we show that contrastive representation learning is a form of multiple instance learning, providing us with valuable insights from a related field for solving shared challenges to learn representations that generalized well.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05561"}}},{"rowIdx":97,"cells":{"text":{"kind":"list like","value":[["ResFed: Communication Efficient Federated Learning by Transmitting Deep\n Compressed Residuals"],["Abstract Federated learning enables cooperative training among massively distributed clients by sharing their learned local model parameters.","However, with increasing model size, deploying federated learning requires a large communication bandwidth, which limits its deployment in wireless networks.","To address this bottleneck, we introduce a residual-based federated learning framework (ResFed), where residuals rather than model parameters are transmitted in communication networks for training.","In particular, we integrate two pairs of shared predictors for the model prediction in both server-to-client and client-to-server communication.","By employing a common prediction rule, both locally and globally updated models are always fully recoverable in clients and the server.","We highlight that the residuals only indicate the quasi-update of a model in a single inter-round, and hence contain more dense information and have a lower entropy than the model, comparing to model weights and gradients.","Based on this property, we further conduct lossy compression of the residuals by sparsification and quantization and encode them for efficient communication.","The experimental evaluation shows that our ResFed needs remarkably less communication costs and achieves better accuracy by leveraging less sensitive residuals, compared to standard federated learning.","For instance, to train a 4.08 MB CNN model on CIFAR-10 with 10 clients under non-independent and identically distributed (Non-IID) setting, our approach achieves a compression ratio over 700X in each communication round with minimum impact on the accuracy.","To reach an accuracy of 70%, it saves around 99% of the total communication volume from 587.61 Mb to 6.79 Mb in up-streaming and to 4.61 Mb in down-streaming on average for all clients."],["Introduction","Federated learning has become an emerged machine learning paradigm, which enables distributed training on broad data sources without disclosing their original data [18].","Instead of transmitting raw data, only parameters (mostly model weights or gradients) in federated learning are iteratively shared between clients and a server via heterogeneous networks.","Federated learning has been successfully applied to various applications [17], such as mobile keyboard prediction, speech recognition, image object detection, etc, However, with the increasing size of machine learning models, the existing mobile communication infrastructure cannot always meet the requirement in terms of bandwidth and latency in federated learning, which constraints the wide deployment of federated learning.","For instance, to train a transformer model with billions of parameters usually 32-bit float parameters), the size of a message in a single federated learning round can be several 10 or 100 Gigabytes, e.g.","a CTRL model ([10]) with 1.6 billions parameters or a T5 model ([21]) with up to 11 billions parameters.","That can cause an enormous and extremely costly data traffic, even in 5G NR networks, where the throughput can be from 5 Gbps to 18 Gbps.","Another application scenario is to improve machine learning models for road traffic object recognition and detection in V2X (Vehicle-to-Everything) communication networks, where the bandwidth for V2X is also occupied for other traffic services at the same time, e.g.","collective perception service, and obviously the safety-related services should have higher priority.","Therefore, communication efficiency is a pivotal component for deploying federated learning, especially in wireless networks.","In an attempt to tackle the communication bottleneck, the parameter compression is considered as one of the most effective approaches, which allows for updating the models by transmitting much smaller size of messages in networks, and thereby reduces the required time per communication round in federated learning.","The approaches proposed by [28], [22], [7] can effectively reduce the communication volume in each round by various quantization techniques, however they only consider the communication efficiency for uploading (client-to-server) but not for downloading (server-to-client).","[16] compress the gradients instead of model parameters for distributed learning, which can not well fit federated learning, where clients can train multiple epochs in each round.","[26] use knowledge distillation ([6]) to learn and transmit a smaller model, where the original model structure is affected.","Furthermore, all of those works attempt to compress the model parameters or gradients based on the model in a specific round, without consideration of inter-round model update similarity, which contains additional redundancy sequentially.","Inspired by residuals in video compression protocols from [13], we introduce a residual-based federated learning framework, termed as ResFed.","It allows the server and clients to share and update models by sharing model parameter residuals rather than model parameters or gradients.","Particularly, by observing training trajectory in each local client and the aggregation trajectory in the server, we believe model updates in both clients and the server can be predictable.","Those predictive models – in analogy with predictive frames in video transmission – can foresee model updates in the federated learning.","After each communication round, we use the deviations between the predictive and the actual updated model parameters, which we call the model residuals, for the communication in networks.","Note that the actual updated models can be always recoverable by acquiring the residuals, as the predictors in the senders are shared to the receivers in ResFed.","More details are provided in Sec.","and .","Unlike transmitting model weights, ResFed can wring out the potential redundancy by removing the predictive information from history updates and only keep the residuals for communication.","Compared to transmitting model gradients after each training epoch, ResFed allows the models to be trained locally multiple times.","Compared to transmitting residual accumulation for multiple epochs, ResFed further minimizes the information by predicting the model updates from history.","As shown in Fig.REF , the values of residuals are overall smaller than weights and gradients during the entire training process.","To further shrink the size of messages for communication, we then compress only residuals using sparisification and quantization, and encode the messages for information sharing in client-to-server and server-to-client.","Our main contributions are summarized as follows: We introduce and formulate the model residuals for the communication efficiency in federated learning and indicate the residuals contain more dense information than model weights and gradients.","We propose a novel federated learning framework (ResFed) based on deep residual compression, which consists of the following steps: predictor sharing, model prediction, residual generation, residual compression, residual communicating, model recovering and model trajectory synchronization.","We provide the experimental evaluation of our framework with various communication cost budgets in both up- and down-streaming, which gives an insight in deploying it in resource-constrained communication environments.","The open source implementation of ResFed will be publicly available."],["System Setup","We first introduce the related concepts and techniques that will be used in our framework.","Given $N$ clients and a server in a federated learning system, we only focus on the information sharing between one single client $k$ and the server from communication round $t$ to $t+1$ , as shown in Fig REF .","The information sharing for other clients is the same."],["Model Update","Client.","Given a client $k$ with a local dataset $\\mathcal {D}_k$ , the initial local model in a new round $t$ is $w_k^{t-1}$ .","Before the local training starts, the client initially receives the global model $w^t$ from the server and updates the local model to $\\hat{w}_{k}^t$ .","After that, local model $\\hat{w}_{k}^t$ is trained on $\\mathcal {D}_k$ and transited to $w_{k}^t$ .","We mark the first update as $w_{k}^{t-1} \\rightarrow \\hat{w}_{k}^t$ and the second one as $\\hat{w}_{k}^t \\rightarrow w_{k}^t$ .","Note that the first updated model is equal or similar to the global updated model $w^t$ , i.e.","$\\hat{w}_{k}^t \\simeq w^t$ .","If lossy compression is used for communication and the loss due to compression can not be repaired, then $\\hat{w}_{k}^t \\sim w^t$ .","Server.","Similarly, the global model $w_{t-1}$ in the server is also updated twice after one round of communication $t$ .","The first update happens when it receives models from the clients, i.e.","$w^{t-1} \\rightarrow \\lbrace \\hat{w}_i^t|i=1,2,...,N\\rbrace $ .","Then the aggregation leads to the second update, $\\lbrace \\hat{w}_i^t|i=1,2,...,N\\rbrace \\rightarrow w^{t}$ ."],["Model Trajectory","Client.","Given a client $k$ at time point $t$ , we cache the updated models with a sliding time window $[t-T, t]$ in two different queues, that distinguish by two model updates.","We refer the time sequence of local model updates $\\mathcal {L}_{k}^t=\\lbrace w_{k}^{t-T}, ...,w_{k}^t\\rbrace $ from $w_{k}^t \\rightarrow \\hat{w}_{k}^t$ as a local model trajectory, and $\\mathcal {\\hat{G}}_{k}^t=\\lbrace \\hat{w}_{k}^{t-T}, ...,\\hat{w}_{k}^t\\rbrace $ from $\\hat{w}_{k}^t \\rightarrow w_{k}^{t+1}$ as a global model trajectory.","Server.","Correspondingly, we cache the local and global model updates in the local and global model trajectories for all client at the server, i.e.","$\\lbrace \\mathcal {\\hat{L}}_i|i=1,2,...,N\\rbrace $ and $\\lbrace \\mathcal {G}_i|i=1,2,...,N\\rbrace $ .","Note that if the server can always send the lossless global model update to all clients, the global trajectories at time $t$ are the same for all clients."],["Model Prediction","Client.","Given a client $k$ at time point $t$ , we predict $\\hat{w}_{k}^t \\rightarrow w_{k}^t$ from the local and global training trajectories, $\\mathcal {L}_{k}^{t-1}$ and $\\mathcal {\\hat{G}}_{k}^{t-1}$ as follows: $\\tilde{w}_{k}^{t} = f_{predict,k}(\\mathcal {L}_{k}^{t-1}, \\mathcal {\\hat{G}}_{k}^{t-1}, \\hat{w}_{k}^{t}) = \\operatornamewithlimits{arg\\,max}_{w_{k}^t} p(w_{k}^t|\\underbrace{w_{k}^{t-T}, ...,w_{k}^{t-1}}_{\\text{$\\mathcal {L}_{k}^{t-1}$}},\\underbrace{\\hat{w}_{k}^{t-T}, ...,\\hat{w}_{k}^{t-1}}_{\\text{$\\mathcal {\\hat{G}}_{k}^{t-1}$}}, \\hat{w}_{k}^{t})$ where $f_{predict,k}$ is the used predictor for model prediction in the client $k$ .","Server.","For the server, we predict model updates $\\hat{w}_i^t\\rightarrow w^t$ for each client $i$ from local and global trajectories $\\mathcal {\\hat{L}}_{i}^{t-1}$ and $\\mathcal {G}_{i}^{t-1}$ as follows: $\\tilde{w}_{i}^{t} = h_{predict,i}(\\mathcal {\\hat{L}}_{i}^{t-1}, \\mathcal {G}_{i}^{t-1}, \\hat{w}_{i}^{t}) = \\operatornamewithlimits{arg\\,max}_{w_{i}^t} p(w_{i}^t|\\underbrace{w_{i}^{t-T}, ...,w_{i}^{t-1}}_{\\text{$\\mathcal {\\hat{L}}_{k}^{t-1}$}},\\underbrace{\\hat{w}_{i}^{t-T}, ...,\\hat{w}_{i}^{t-1}}_{\\text{$\\mathcal {G}_{k}^{t-1}$}}, \\hat{w}_{i}^{t}), \\forall i\\in {1,...,N}$ where $h_{predict,i}$ is the used predictor for model prediction in the server for each client $i$ ."],["Model Residual","Given a model update $\\hat{w}_{i}^t \\rightarrow w_{i}^t$ for the client $i$ at time $t$ in the server or in the corresponding client, if we can compute the model prediction $\\tilde{w}_{i}^{t}$ based on Eq.","REF , we define the model residual as follows: $r_{i}^t = w_{i}^t - \\tilde{w}_{i}^t$ Note that $r_{i, ul}^t$ and $r_{i, dl}^t$ are the residuals of clients for uploading and the residuals of the server for downloading, respectively.More understanding of residuals is provided in Sec.","."],["Deep Compression","To reduce the model size for more efficient communication, we shrink the model size before sending it out.","We define a compressed model in the client k: $\\bar{w}_{k}^t = f_{compress}(w_{k}^t)$ and in the server: $\\bar{w}_{i}^t = h_{compress}(w_{i}^t), \\forall i\\in \\lbrace 1,...,N\\rbrace $ where $f_{compress}$ and $h_{compress}$ is the used compressor for model compression in clients and server respectively.","In our system, we consider to compress and communicate model residuals instead of model itself in the client k: $\\bar{r}_{k,ul}^t = f_{compress}(r_{k,ul}^t) = f_{compress}( w_{k}^t - f_{predict, i}(w_{k}^{t-T}, ...,w_{i}^{t-1},\\hat{w}_{k}^{t-T}, ...,\\hat{w}_{k}^{t-1}, \\hat{w}_{k}^{t}))$ and in the server: $\\bar{r}_{i,dl}^t = h_{compress}(r_{i,dl}^t) = h_{compress}( w_{i}^t - f_{predict, i}(w_{i}^{t-T}, ...,w_{i}^{t-1},\\hat{w}_{i}^{t-T}, ...,\\hat{w}_{i}^{t-1}, \\hat{w}_{i}^{t})), \\forall i\\in \\lbrace 1,...,N\\rbrace $"],["ResFed: Residual-based Federated Learning Framework","[t]: ResFed: Residual-based federated learning framework [1] Server runs: initialize the global model $w$ initialize the empty local model trajectories $\\mathcal {\\hat{L}}_{1},...,\\mathcal {\\hat{L}}_{N}$ initialize the global model trajectories $\\mathcal {G}_{1},...,\\mathcal {G}_{N}$ initialize the predictor $h_{predict}$ $i \\in \\lbrace 1,2,...,N\\rbrace $ initialize an empty local model trajectories $\\mathcal {L}_{i}$ @client $i$ initialize an empty global model trajectories $\\mathcal {\\hat{G}}_{i}$ @client $i$ $h^{\\prime }_{predict,i} = h_{predict}$ sharing predictors to client $i$ $f^{\\prime }_{predict,i} = f_{predict, i}$ get the shared predictors from client $i$ $t \\in \\lbrace 1,2,...,M\\rbrace $ $i \\in \\lbrace 1,2,...,N\\rbrace $ in parallel $t < T$ $\\mathcal {\\hat{G}}_i \\leftarrow cache(w)$ cache the global model in $\\mathcal {\\hat{G}}_i $ @client $i$ Server communicates $w$ to the client $i$ $\\hat{w}_i \\leftarrow \\textbf {LocalTrain}(w)$ @client $i$ Client $i$ communicates $\\hat{w}_i$ to the server $\\mathcal {L}_i \\leftarrow cache(\\hat{w}_i)$ cache the local model in $\\mathcal {L}_i$ @client $i$ Server communicates $\\bar{r}_{i,dl}$ to the client $i$ $\\bar{r}_{i,ul} \\leftarrow $ ResFedClientUpdate $(i, \\bar{r}_{i,dl})$ @client $i$ Client $i$ communicates $\\bar{r}_{i,ul}$ to the server $\\tilde{w}_i \\leftarrow f^{\\prime }_{predict,i}(\\mathcal {G}_i, \\mathcal {\\hat{L}}_i, \\hat{w})$ predict updated model $\\hat{w}_i \\leftarrow \\tilde{w}_i + \\bar{r}_{i,ul}$ recover models $\\mathcal {\\hat{L}}_i \\leftarrow cache(\\hat{w}_i)$ update local trajectory $w \\leftarrow \\textbf {Aggregate}(\\hat{w}_1,...,\\hat{w}_N)$ $i \\in \\lbrace 1,2,...,N\\rbrace $ $\\tilde{w}_i \\leftarrow h_{predict}(\\mathcal {G}_i, \\mathcal {\\hat{L}}_i, w)$ predict updated model $r_{i,dl} \\leftarrow w - \\tilde{w}_i$ compute model residuals $\\bar{r}_{i,dl} \\leftarrow h_{compress}(r_{i,dl})$ compress model residuals $\\mathcal {G}_i \\leftarrow cache(\\tilde{w}_i + \\bar{r}_{i,dl})$ synchronize global trajectory return $w$ ResFedClientUpdate $(k, \\bar{r})$ $\\tilde{w} \\leftarrow h^{\\prime }_{predict, i}(\\mathcal {\\hat{G}}_k, \\mathcal {L}_k, w_k)$ predict updated model $\\hat{w} \\leftarrow \\tilde{w} + \\bar{r}$ recover models $\\mathcal {\\hat{G}}_k \\leftarrow cache(\\hat{w})$ update global trajectory $w_k \\leftarrow \\textbf {LocalTrain}(\\hat{w})$ $\\tilde{w}_k \\leftarrow f_{predict,i}(\\mathcal {\\hat{G}}_k, \\mathcal {L}_k, \\hat{w})$ predict updated model $r_k \\leftarrow w_k - \\tilde{w}_k$ compute model residuals $\\bar{r}_k \\leftarrow f_{compress}(r_k)$ compress model residuals $\\mathcal {L}_k \\leftarrow cache(\\tilde{w}_k + \\bar{r}_k)$ synchronize local trajectory return $\\bar{r}_k$ The overview of the ResFed is shown in Fig.","REF and the detailed steps with lossy compression in one communication round is given in Fig.","REF .","In particular, we introduce (a) predictor sharing, (b) model prediction, and (c) residual generation in Sec REF .","In Sec.","REF , (d) residual compression is formulated in Eq.","REF and Eq.","REF .","Then, after (e) communicating residual bits, we provide details on (f) model recovering in Sec.","REF .","Finally in Sec.","REF , we describe (g) trajectory synchronization."],["Predictor Sharing","Then, we consider to deploy a pair of predictors in both clients and the server, which can execute the model predictions based on the local and global model trajectories in the time series.","In our framework, the server caches the local model trajectories in all clients once the local model update is received.","Given a client $k$ , if the received models in the server is exactly the same as the local model, and the models in $\\mathcal {\\hat{G}}_k$ also the same, we say that both trajectories in client $k$ are fully observable in the server.","Give a predictor $f_{predict, k}$ in the client $k$ , we share it to make the predictor in the server $f^{\\prime }_{predict, k}=f_{predict, k}$ in ResFed.","If the trajectories in client are fully observable in the server, we can get the same model predictions in both server and clients from Eq.","REF .","Then, by communicating the model residuals, the new model update at time $t+1$ can be recovered by the model residual and the model at last time point.","Also, we share the set of predictors in the server $\\lbrace h_{predict, i}|i=1,...,N\\rbrace $ to the corresponding client, i.e.","$h^{\\prime }_{predict, i}=h_{predict, i}, \\forall i\\in \\lbrace 1,...,N\\rbrace $ .","Then we can get the same model predictions based on Eq.","REF .","For the predictor, various design choices exist.","In this work, we employ the predictor with respect to the model transition dynamics in a sliding history time window.","The predictor is formulated as follows: $\\tilde{w}_{k}^{t} = f_{predict}(\\mathcal {L}_{k}^{t-1}, \\mathcal {\\hat{G}}_{k}^{t-1}, \\hat{w}_{k}^t) ={\\left\\lbrace \\begin{array}{ll}\\hat{w}_{k}^t, & \\text{$T=0$}.\\\\\\hat{w}_{k}^t + \\sum _{\\tau =1}^T (-1)^{T-\\tau }(T-\\tau +1)(w_{k}^{t-\\tau }-\\hat{w}_{k}^{t-\\tau }), & \\text{$T>0$}.\\end{array}\\right.","}$ To reduce the used memory for caching trajectories in the client, we apply a short time window $[t-T, t]$ in the prediction process.","We term (i) stationary predictor when $T=0$ ; and (ii) linear predictor when $T=1$ .","Note that we consider the model updates in [1], [19], [8] as special residuals, which calculated by stationary predictor.","Specifically, the stationary predictor uses the current model for the prediction of the next model, $\\tilde{w}_{k}^{t} = \\hat{w}_{k}^t$ , where the model residual is always $r_{k}^{t} = w_{k}^{t} - \\hat{w}_{k}^{t}$ .","Note that when the number of local training epochs is fixed to 1, the stationary residuals is proportional to gradients, i.e.","$r = \\eta g$ , where $\\eta $ is the learning rate and $g$ represents the gradients.","In the linear predictor, $\\tilde{w}_{k}^{t} = \\hat{w}_{k}^{t} + w_{k}^{t-1} - \\hat{w}_{k}^{t-1}$ , the model transition in the last local training step is always considered, where $r_{k}^{t} = w_{k}^{t} - \\hat{w}_{k}^{t} - w_{k}^{t-1} + \\hat{w}_{k}^{t-1}$ .","The predictor for the client $k$ in the server $h_{predict,k}$ is similar to $f_{predict,k}$ ."],["Model Recovery","We cache the trajectories in both server and clients.","Each client has two model trajectories for local and global model updates in the history.","In the server, it caches the global trajectory and the local model trajectories of all connecting clients in the history.","In this case, the two trajectories in each client are fully observable in the server.","Through sharing predictors, given a client $k$ at round $t$ , the server is able to get the same model prediction $\\tilde{w}_{k}^t$ as the client $k$ .","If uncompressed model residuals ($\\hat{w}_{k}^t = w^t$ ) are received from client $k$ , the model update after local training $w_{k}^t$ can be recovered in the server as follows: $w_{k}^t = r_{k}^{t} + \\hat{w}_{k}^t = r_{k}^{t} + w^t$ where $\\hat{w}_{k}^t$ is the global model in the last round.","Similarly, if we predict the global model update in the client, through sharing predictors and uncompressed residuals, the aggregated model can also be recovered in the client."],["Model Trajectory Synchronization","Since for the model residuals lossy compression is applied, i.e.","$ \\bar{r} \\ne r$ , the updated model of a sender $w$ can be recovered in receivers as $\\hat{w} = \\bar{r} + \\tilde{w} $ .","Therefore, we say that the $w$ cannot be fully recovered in receivers, as $\\hat{w} \\ne w $ .","If we cache the original models $w$ in the sender and $\\hat{w}$ in receivers, the trajectories in the sender and receiver are different, which leads to drift in results of shared predictors.","To avoid the drift effect, we synchronize the model trajectories in the sender by simulating the recovering process: The originally updated models are not cached in the trajectories; instead, we recover the model from compressed model residuals locally, in order to enforce the trajectories in the sender in the same way as the trajectories in the receiver.","The ResFed pseudocode is given in Algorithm ."],["Experimental Settings","Guided by the previous work by [2][14], we process and distribute the datasets MNIST ([12]), Fashion-MNIST ([27]), SVHN ([20]), CIFAR-10 ([11]) and CIFAR-100 ([20]) on a set of clients, and train LeNet-5, CNNIt consists of 5 convolutional and 3 fully connected layers., ResNet-18 on those federated dataset distributively, as shown in Tab.","REF .","We provide details on the experimental settings in Sec.","."],["Residuals vs Gradients and Weights","On top of the basic evaluation in Fig.","REF , we believe deep residual compression can save more communication volume in federated learning with minimum impact on the accuracy.","Thus, we demonstrate the federated learning integrating compressing weights, gradients and two different residuals, i.e.","stationary and linear residuals in Eq.","REF .","As shown in Fig.","REF , the testing accuracy on both IID and Non-IID datasets from deep residual compression always outperforms weight and gradient compression.","Also, the linear residuals can achieve a higher accuracy and faster convergence than stationary residuals.","The results indicate communicating residuals in federated learning can enable larger compression ratio per communication round, compared to communicating other parameters.","Figure: Comparison of compressing weights, gradients, residuals with stationary (Res-0) and linear (Res-1) predictors.","The sparsities are 0.2 and 0.01 for training on MNIST and CIFAR-100 distributed in 10 clients, respectively.","We quantize the non-zero parameters and use 1 bit to represent each of them.","For Non-IID setting, each client owns only the data with half classes."],["Communication Efficiency Improvement","Next, we evaluate the required communication volume for training three sizes of models on different datasets in both IID and Non-IID settings.","Tab.","REF shows that to reach a promising target accuracy, ResFed with lossy compression (compression ratio is set on $350\\times -375\\times $ ) can save on average around 99% of the total communication volume for all clients in only up- or down-streaming.","Furthermore, the bitsaving ratios of ResFed on IID and Non-IID settings are similar, which indicates the compression performance of ResFed is robust to data heterogeneity in federated learning.","We show testing accuracy and training loss change with increasing required communication volume in ResFed in Fig.","REF .","The results indicate communicating residuals in federated learning can remarkably save overall communication volume."],["Scalability for Resource-constrained Communication Environments","Finally, we explore the scalability of ResFed by tuning compression ratios for client-to-server and server-to-client, as in real application scenarios.","The available network resources for up- and down-streaming can be heterogeneous.","Fig REF shows the test accuracy effected for different values of sparsity, which leads to various compression ratios for each communication round for up- and down-streaming.","From Fig.","REF , we can observe the testing accuracy reduces with higher compression ratio per communication round, when the number of communication rounds is always the same, i.e.","set to 300.","However, when we consider the dedicated budget for communication costs in up- or down-streaming, a large compression ratio in ResFed can achieve better accuracy, as shown in Fig.","REF .","By adapting the compression ratio in resource-constrained communication environments, ResFed can effectively enhance the federated learning using deep residual compression."],["Related Work","Deep compression.","Deep compression was originally proposed by [5] and aims at compressing deep learning models by a pipeline including sparsification (pruning), quantization and encoding for more efficient deployment.","Based on deep compression, [16] have proposed deep gradient compression to reduce the communication costs in distributed learning by compressing gradients rather than model weights, which can also be used in federated learning.","However, models are usually trained more than one epoch locally in federated learning ([18], [15], [25], [9]), which results in gradient accumulation instead of gradients in other distributed learning scenarios ([23]).","Our conducted experiments also show compressing residuals can achieve better communication efficiency comparing to compress gradients due to the additional prediction step.","In ResFed, we especially consider residuals, which eliminate the model similarity in a single inter-round of federated learning communication and achieve a better compression performance by leveraging the deep residual compression.","Federated learning and communication efficiency.","Communication efficiency is the key for deploying federated learning in real application scenarios, especially to train a large model.","Previous research by [29], [9], [4] attempted to reduce the number of needed communication rounds for a better communication efficiency.","Meanwhile, the proposed approaches by [28], [22], [7] are built upon deep compression and focus on improving communication efficiency by decreasing the communication volume.","However, unlike compressing residuals in ResFed, they compressed model weights without consideration of any potential redundancy in sequential updating of federated learning.","The recent work by [30] has also mentioned the predictive model update in federated learning, which is concurrent to our work, but the information in the history of model updating is not considered for reducing the parameter redundancy there.","Additionally, all those algorithms above can only be used to improve the communication efficiency for up-streaming, while ResFed can handle with up- and/or down-streaming for heterogeneous resource-constrained environments.","Residuals in video encoding.","The residuals have been widely and successfully utilized in video encoding since H.261 ([3], [13]).","By considering inter-frame correlations, the pixel values in the current frame are predicted from history frames and then only residuals, i.e.","the deviations between predicted and the actual pixel values in the current frame, are encoded and streamed to the receivers.","Inspired by the residuals in video encoding, we integrate the model residuals into federated learning in ResFed, where the inter-round similarity of a model update is analogous to inter-frame correlation in video encoding."],["Conclusion","In this work, we introduce a residual-based federated learning framework, which allows clients and the server to share residuals instead of weights or gradients.","It achieves more efficient communication for both up- and down-streaming in federated learning by leveraging deep residual compression, and hence can be flexibly deployed in heterogeneous network environments.","Our conducted experimental evaluation shows that the framework remarkably reduces overall communication volume to reach the same prediction accuracy in standard federated learning.","Compared to compressing model weights or gradients, ResFed achieves higher accuracy and faster convergence speed.","Limitations.","We cache the recovered models as local and global trajectories for continual model prediction and residual computing in all clients and server.","Assuming that we perform ResFed with $N$ clients for training a model with $V$ 32-bit float parameters and we set the trajectory length on $T$ , each client should use $2*32*V*T$ bits memory to cache the 2 trajectories.","Thus, the additional required memory size in each client is proportional to $T*V$ .","For the server, it needs $2*32*V*T*N$ bits memory to cache the local and global trajectories for all clients.","In order to reduce the required memory, a potential solution is to cache the compressed models in the trajectories for both sender and receiver symmetrically, after model recovery.","However, the accuracy of model prediction based on compressed trajectories is reduced, then the memory-accuracy trade-off needs to be investigated in future work."],["Reproducibility statement","We provide the source-code for the implementation and evaluation of our proposed framework ResFed in the code appendix.","The user guidelines for installation and execution is given in the file README.md.","The details on our conducted experiments are provided in Sec. .","The source-code will be made publicly available after double-blind review.","Tables of Notations We provide an overview of the most relevant notations in Tab.","REF .","Table: Summary of mainly used notations in this paper.","Experimental Details and Further Results In this section, we provide the details on our conducted experiment in .","We run on a computer cluster with 4$\\times $ NVIDIA-A100-PCIE-40GB GPUs and 4$\\times $ 32-Core-AMD-EPYC-7513 CPUs.","The environment is a Linux system with Pytorch 1.8.1 and Cuda 11.1.","We demonstrate the learning task on 5 different datasets: MNIST [12]: 60000 data points in the training set and 10000 data points in the test set.","Each data point is a 28x28 gray-scale digit image, associated with a label from 10 classes.","CIFAR-10 [11]: 50000 data points in the training set and 10000 data points in the test set.","Each data point is a 32x32 RGB image, associated with a label from 10 classes.","Fashion-MNIST [27]: 60000 data points in the training set and 10000 data points in the test set.","Each data point is a 28x28 gray-scale image, associated with a label from 10 classes.","SVHN [20]: 73257 data points in the training set and 26032 data points in the test set.","Each data point is a 32x32 RGB digit image, associated with a label from 10 classes.","CIFAR-100 [20]: 50000 data points in the training set and 10000 data points in the test set.","Each data point is a 32x32 RGB image, associated with a label from 100 classes.","The models trained on those dataset are shown in Tab.","REF .","Table: Dataset and models in experimentsExperiments for Sec.","REF We divide the dataset, i.e.","MNIST and CIFAR-100, into 10 clients and run the FedAvg with local optimizer of stochastic gradient descent (SGD) (momentum is 0.9) to train LeNet-5 and CNN (with 5 convolutional and 3 fully connected layers), respectively.","The learning rate is fixed on 0.01 and the batch size is 64 for all tests.","In Non-IID data setting, each client owns only 2 out of 10 classes in MNIST and 50 out of 100 classes in CIFAR-100.","In clients, we consider 5 different approaches as follows: No Compression: The standard federated learning without any compression methods is used as the baseline, which provides the best results among all of the approaches, when number of communication rounds is the same.","Compress Weights: Before communication in standard federated learning, the model weights are first compressed.","Compress Gradients: The gradients in each epoch are compressed and communicated to the server.","Compress Res-0: The residuals are computed by stationary predictor, i.e.","Eq.","REF with $T=0$ .","Compress Res-1: The residuals are computed by linear predictor, i.e.","Eq.","REF with $T=1$ .","For each approach on each dataset, we run 10 tests with different seeds and show the mean value and standard variance in Fig.","REF .","We compress the those model parameters using deep compression pipeline ([5]) only for client-to-server.","In particular, we set sparsity on 80% and 99% for residuals in LetNet and CNN, respectively.","We use SGD optimizer momentum of 0.9.","The number of local epoch is 1.","Those sparsified parameters are zero-parameters and the number of the continually appearing zero-parameters are encoded as a 16-bit float parameters ([16]).","After that, we quantize the non-zero parameters in 1 bit with median value of positive and negative parameters ([28]).","Finally, Huffman encoding ([24]) is used.","Experiments for Sec.","REF We train LeNet-5, CNN and ResNet18 (size from small to large) on Fashion-MNIST, CIFAR-10 and SVHN with 10 clients in both IID and Non-IID settings.","We demonstrate the ResFed and lossy compress the residuals either only for uploading (UL) or downloading (DL) to study the effects on each direction separately.","The learning rate is fixed on 0.01 and the batch size is 64 for all tests.","In Non-IID data setting, each client owns 50% classes (5 out of 10).","We use mean values from 5 tests in each experiment for the evaluation in Tab.","REF and show the results with standard variance in Fig.","REF , which indicate the overall communication volume can be remarkably reduced in ResFed.","Figure: Overall communication efficiency enhanced by ResFed with Lossy Compression (LC) for only Uploading (UL) or Downloading (DL)For all experiments, we set the sparsity on 99% and use 1 bit for each non-zero parameters.","Consequently, the compression ratio per communication round for LetNet-5 is about $350\\times $ , CNN is about $375\\times $ and ResNet-18 is about $356\\times $ .","Experiments for Sec.","REF To show the correlation between deep residual compression in up- und down-streaming, we train the CNN model on IID CIFAR-10 with 10 clients and tune the sparsity for realizing different compression ratios per communication round in ResFed.","The learning rate is fixed on 0.01 and the batch size is 64.","We use SGD optimizer momentum of 0.9.","The number of local epoch is 1.","Specifically, the value of sparsity is $\\lbrace 0\\%, 90\\%, 95\\%, 99\\%, 99.5\\%\\rbrace $ for both up- and down-streaming and then set 1 bit for all non-zero parameters in quantization.","Understanding Residuals We provide an illustration of model transitions during federated learning in Fig.","REF .","Given a sender and a receiver (both can be a client or a server), the communication and operation result in model transition.","Note that for a client, the operation is local training; for a server, the operation is aggregation.","We consider the model transition caused by an operation as a internal model transition, and by communication as a external model transition.","Then, as explained in Sec.","REF , the model is updated twice between two communication rounds, which is shown in Fig.","REF as dual model transition.","Consequently, we can have an internal and an external model transition trajectory in both sender and receiver.","Note that for a client, the internal model transition trajectory is a local model trajectory; for a server, the internal model transition trajectory is a global model trajectory, as described in Sec.","REF .","Figure: Residual generation in ResFed during model transitions.","For a better overview, we simplify the system by disregarding the trajectory synchronization step in Sec.",".Figure: Value comparison of model parameters, gradients and residuals in federated learning.","We train a LeNet with 61706 weights of 32-bits float on MNIST distributed among 10 clients, with fixed learning rate 0.0010.001 and batch size 64.","For fairly comparing gradients and residuals, the number of local epoch in each client is set as 1.","We set 6 checkpoints when the number of communication rounds is {1,4,8,16,32,128}\\lbrace 1, 4, 8, 16, 32, 128\\rbrace .","The results show that most values of residuals are smaller than weights and gradients during the training.","It indicates that lossly compressing residuals naturally lose less information and have a smaller affect on the accuracy.Finally, ResFed allows the sender to predict the model for the next internal model transition, which is shown in orange.","Meanwhile the sender does the operation to execute the internal model transition and residuals (in purple) are deduced from the difference of both model transition results.","We believe the predicted model can be closer to the updated model than the previous model, which leads to smaller values of residuals.","To evaluate it, we set 6 checkpoints when the number of communication rounds is $\\lbrace 1, 4, 8, 16, 32, 128\\rbrace $ , and show the values of model weights, gradients and residuals in Fig.","REF .","Based on this, the residuals can be compressed smaller than weights and gradients."],["Tables of Notations","We provide an overview of the most relevant notations in Tab.","REF .","Table: Summary of mainly used notations in this paper."],["Experimental Details and Further Results","In this section, we provide the details on our conducted experiment in .","We run on a computer cluster with 4$\\times $ NVIDIA-A100-PCIE-40GB GPUs and 4$\\times $ 32-Core-AMD-EPYC-7513 CPUs.","The environment is a Linux system with Pytorch 1.8.1 and Cuda 11.1.","We demonstrate the learning task on 5 different datasets: MNIST [12]: 60000 data points in the training set and 10000 data points in the test set.","Each data point is a 28x28 gray-scale digit image, associated with a label from 10 classes.","CIFAR-10 [11]: 50000 data points in the training set and 10000 data points in the test set.","Each data point is a 32x32 RGB image, associated with a label from 10 classes.","Fashion-MNIST [27]: 60000 data points in the training set and 10000 data points in the test set.","Each data point is a 28x28 gray-scale image, associated with a label from 10 classes.","SVHN [20]: 73257 data points in the training set and 26032 data points in the test set.","Each data point is a 32x32 RGB digit image, associated with a label from 10 classes.","CIFAR-100 [20]: 50000 data points in the training set and 10000 data points in the test set.","Each data point is a 32x32 RGB image, associated with a label from 100 classes.","The models trained on those dataset are shown in Tab.","REF .","Table: Dataset and models in experiments"],["Experiments for Sec. ","We divide the dataset, i.e.","MNIST and CIFAR-100, into 10 clients and run the FedAvg with local optimizer of stochastic gradient descent (SGD) (momentum is 0.9) to train LeNet-5 and CNN (with 5 convolutional and 3 fully connected layers), respectively.","The learning rate is fixed on 0.01 and the batch size is 64 for all tests.","In Non-IID data setting, each client owns only 2 out of 10 classes in MNIST and 50 out of 100 classes in CIFAR-100.","In clients, we consider 5 different approaches as follows: No Compression: The standard federated learning without any compression methods is used as the baseline, which provides the best results among all of the approaches, when number of communication rounds is the same.","Compress Weights: Before communication in standard federated learning, the model weights are first compressed.","Compress Gradients: The gradients in each epoch are compressed and communicated to the server.","Compress Res-0: The residuals are computed by stationary predictor, i.e.","Eq.","REF with $T=0$ .","Compress Res-1: The residuals are computed by linear predictor, i.e.","Eq.","REF with $T=1$ .","For each approach on each dataset, we run 10 tests with different seeds and show the mean value and standard variance in Fig.","REF .","We compress the those model parameters using deep compression pipeline ([5]) only for client-to-server.","In particular, we set sparsity on 80% and 99% for residuals in LetNet and CNN, respectively.","We use SGD optimizer momentum of 0.9.","The number of local epoch is 1.","Those sparsified parameters are zero-parameters and the number of the continually appearing zero-parameters are encoded as a 16-bit float parameters ([16]).","After that, we quantize the non-zero parameters in 1 bit with median value of positive and negative parameters ([28]).","Finally, Huffman encoding ([24]) is used."],["Experiments for Sec. ","We train LeNet-5, CNN and ResNet18 (size from small to large) on Fashion-MNIST, CIFAR-10 and SVHN with 10 clients in both IID and Non-IID settings.","We demonstrate the ResFed and lossy compress the residuals either only for uploading (UL) or downloading (DL) to study the effects on each direction separately.","The learning rate is fixed on 0.01 and the batch size is 64 for all tests.","In Non-IID data setting, each client owns 50% classes (5 out of 10).","We use mean values from 5 tests in each experiment for the evaluation in Tab.","REF and show the results with standard variance in Fig.","REF , which indicate the overall communication volume can be remarkably reduced in ResFed.","Figure: Overall communication efficiency enhanced by ResFed with Lossy Compression (LC) for only Uploading (UL) or Downloading (DL)For all experiments, we set the sparsity on 99% and use 1 bit for each non-zero parameters.","Consequently, the compression ratio per communication round for LetNet-5 is about $350\\times $ , CNN is about $375\\times $ and ResNet-18 is about $356\\times $ ."],["Experiments for Sec. ","To show the correlation between deep residual compression in up- und down-streaming, we train the CNN model on IID CIFAR-10 with 10 clients and tune the sparsity for realizing different compression ratios per communication round in ResFed.","The learning rate is fixed on 0.01 and the batch size is 64.","We use SGD optimizer momentum of 0.9.","The number of local epoch is 1.","Specifically, the value of sparsity is $\\lbrace 0\\%, 90\\%, 95\\%, 99\\%, 99.5\\%\\rbrace $ for both up- and down-streaming and then set 1 bit for all non-zero parameters in quantization."],["Understanding Residuals","We provide an illustration of model transitions during federated learning in Fig.","REF .","Given a sender and a receiver (both can be a client or a server), the communication and operation result in model transition.","Note that for a client, the operation is local training; for a server, the operation is aggregation.","We consider the model transition caused by an operation as a internal model transition, and by communication as a external model transition.","Then, as explained in Sec.","REF , the model is updated twice between two communication rounds, which is shown in Fig.","REF as dual model transition.","Consequently, we can have an internal and an external model transition trajectory in both sender and receiver.","Note that for a client, the internal model transition trajectory is a local model trajectory; for a server, the internal model transition trajectory is a global model trajectory, as described in Sec.","REF .","Figure: Residual generation in ResFed during model transitions.","For a better overview, we simplify the system by disregarding the trajectory synchronization step in Sec.",".Figure: Value comparison of model parameters, gradients and residuals in federated learning.","We train a LeNet with 61706 weights of 32-bits float on MNIST distributed among 10 clients, with fixed learning rate 0.0010.001 and batch size 64.","For fairly comparing gradients and residuals, the number of local epoch in each client is set as 1.","We set 6 checkpoints when the number of communication rounds is {1,4,8,16,32,128}\\lbrace 1, 4, 8, 16, 32, 128\\rbrace .","The results show that most values of residuals are smaller than weights and gradients during the training.","It indicates that lossly compressing residuals naturally lose less information and have a smaller affect on the accuracy.Finally, ResFed allows the sender to predict the model for the next internal model transition, which is shown in orange.","Meanwhile the sender does the operation to execute the internal model transition and residuals (in purple) are deduced from the difference of both model transition results.","We believe the predicted model can be closer to the updated model than the previous model, which leads to smaller values of residuals.","To evaluate it, we set 6 checkpoints when the number of communication rounds is $\\lbrace 1, 4, 8, 16, 32, 128\\rbrace $ , and show the values of model weights, gradients and residuals in Fig.","REF .","Based on this, the residuals can be compressed smaller than weights and gradients."]],"string":"[\n [\n \"ResFed: Communication Efficient Federated Learning by Transmitting Deep\\n Compressed Residuals\"\n ],\n [\n \"Abstract Federated learning enables cooperative training among massively distributed clients by sharing their learned local model parameters.\",\n \"However, with increasing model size, deploying federated learning requires a large communication bandwidth, which limits its deployment in wireless networks.\",\n \"To address this bottleneck, we introduce a residual-based federated learning framework (ResFed), where residuals rather than model parameters are transmitted in communication networks for training.\",\n \"In particular, we integrate two pairs of shared predictors for the model prediction in both server-to-client and client-to-server communication.\",\n \"By employing a common prediction rule, both locally and globally updated models are always fully recoverable in clients and the server.\",\n \"We highlight that the residuals only indicate the quasi-update of a model in a single inter-round, and hence contain more dense information and have a lower entropy than the model, comparing to model weights and gradients.\",\n \"Based on this property, we further conduct lossy compression of the residuals by sparsification and quantization and encode them for efficient communication.\",\n \"The experimental evaluation shows that our ResFed needs remarkably less communication costs and achieves better accuracy by leveraging less sensitive residuals, compared to standard federated learning.\",\n \"For instance, to train a 4.08 MB CNN model on CIFAR-10 with 10 clients under non-independent and identically distributed (Non-IID) setting, our approach achieves a compression ratio over 700X in each communication round with minimum impact on the accuracy.\",\n \"To reach an accuracy of 70%, it saves around 99% of the total communication volume from 587.61 Mb to 6.79 Mb in up-streaming and to 4.61 Mb in down-streaming on average for all clients.\"\n ],\n [\n \"Introduction\",\n \"Federated learning has become an emerged machine learning paradigm, which enables distributed training on broad data sources without disclosing their original data [18].\",\n \"Instead of transmitting raw data, only parameters (mostly model weights or gradients) in federated learning are iteratively shared between clients and a server via heterogeneous networks.\",\n \"Federated learning has been successfully applied to various applications [17], such as mobile keyboard prediction, speech recognition, image object detection, etc, However, with the increasing size of machine learning models, the existing mobile communication infrastructure cannot always meet the requirement in terms of bandwidth and latency in federated learning, which constraints the wide deployment of federated learning.\",\n \"For instance, to train a transformer model with billions of parameters usually 32-bit float parameters), the size of a message in a single federated learning round can be several 10 or 100 Gigabytes, e.g.\",\n \"a CTRL model ([10]) with 1.6 billions parameters or a T5 model ([21]) with up to 11 billions parameters.\",\n \"That can cause an enormous and extremely costly data traffic, even in 5G NR networks, where the throughput can be from 5 Gbps to 18 Gbps.\",\n \"Another application scenario is to improve machine learning models for road traffic object recognition and detection in V2X (Vehicle-to-Everything) communication networks, where the bandwidth for V2X is also occupied for other traffic services at the same time, e.g.\",\n \"collective perception service, and obviously the safety-related services should have higher priority.\",\n \"Therefore, communication efficiency is a pivotal component for deploying federated learning, especially in wireless networks.\",\n \"In an attempt to tackle the communication bottleneck, the parameter compression is considered as one of the most effective approaches, which allows for updating the models by transmitting much smaller size of messages in networks, and thereby reduces the required time per communication round in federated learning.\",\n \"The approaches proposed by [28], [22], [7] can effectively reduce the communication volume in each round by various quantization techniques, however they only consider the communication efficiency for uploading (client-to-server) but not for downloading (server-to-client).\",\n \"[16] compress the gradients instead of model parameters for distributed learning, which can not well fit federated learning, where clients can train multiple epochs in each round.\",\n \"[26] use knowledge distillation ([6]) to learn and transmit a smaller model, where the original model structure is affected.\",\n \"Furthermore, all of those works attempt to compress the model parameters or gradients based on the model in a specific round, without consideration of inter-round model update similarity, which contains additional redundancy sequentially.\",\n \"Inspired by residuals in video compression protocols from [13], we introduce a residual-based federated learning framework, termed as ResFed.\",\n \"It allows the server and clients to share and update models by sharing model parameter residuals rather than model parameters or gradients.\",\n \"Particularly, by observing training trajectory in each local client and the aggregation trajectory in the server, we believe model updates in both clients and the server can be predictable.\",\n \"Those predictive models – in analogy with predictive frames in video transmission – can foresee model updates in the federated learning.\",\n \"After each communication round, we use the deviations between the predictive and the actual updated model parameters, which we call the model residuals, for the communication in networks.\",\n \"Note that the actual updated models can be always recoverable by acquiring the residuals, as the predictors in the senders are shared to the receivers in ResFed.\",\n \"More details are provided in Sec.\",\n \"and .\",\n \"Unlike transmitting model weights, ResFed can wring out the potential redundancy by removing the predictive information from history updates and only keep the residuals for communication.\",\n \"Compared to transmitting model gradients after each training epoch, ResFed allows the models to be trained locally multiple times.\",\n \"Compared to transmitting residual accumulation for multiple epochs, ResFed further minimizes the information by predicting the model updates from history.\",\n \"As shown in Fig.REF , the values of residuals are overall smaller than weights and gradients during the entire training process.\",\n \"To further shrink the size of messages for communication, we then compress only residuals using sparisification and quantization, and encode the messages for information sharing in client-to-server and server-to-client.\",\n \"Our main contributions are summarized as follows: We introduce and formulate the model residuals for the communication efficiency in federated learning and indicate the residuals contain more dense information than model weights and gradients.\",\n \"We propose a novel federated learning framework (ResFed) based on deep residual compression, which consists of the following steps: predictor sharing, model prediction, residual generation, residual compression, residual communicating, model recovering and model trajectory synchronization.\",\n \"We provide the experimental evaluation of our framework with various communication cost budgets in both up- and down-streaming, which gives an insight in deploying it in resource-constrained communication environments.\",\n \"The open source implementation of ResFed will be publicly available.\"\n ],\n [\n \"System Setup\",\n \"We first introduce the related concepts and techniques that will be used in our framework.\",\n \"Given $N$ clients and a server in a federated learning system, we only focus on the information sharing between one single client $k$ and the server from communication round $t$ to $t+1$ , as shown in Fig REF .\",\n \"The information sharing for other clients is the same.\"\n ],\n [\n \"Model Update\",\n \"Client.\",\n \"Given a client $k$ with a local dataset $\\\\mathcal {D}_k$ , the initial local model in a new round $t$ is $w_k^{t-1}$ .\",\n \"Before the local training starts, the client initially receives the global model $w^t$ from the server and updates the local model to $\\\\hat{w}_{k}^t$ .\",\n \"After that, local model $\\\\hat{w}_{k}^t$ is trained on $\\\\mathcal {D}_k$ and transited to $w_{k}^t$ .\",\n \"We mark the first update as $w_{k}^{t-1} \\\\rightarrow \\\\hat{w}_{k}^t$ and the second one as $\\\\hat{w}_{k}^t \\\\rightarrow w_{k}^t$ .\",\n \"Note that the first updated model is equal or similar to the global updated model $w^t$ , i.e.\",\n \"$\\\\hat{w}_{k}^t \\\\simeq w^t$ .\",\n \"If lossy compression is used for communication and the loss due to compression can not be repaired, then $\\\\hat{w}_{k}^t \\\\sim w^t$ .\",\n \"Server.\",\n \"Similarly, the global model $w_{t-1}$ in the server is also updated twice after one round of communication $t$ .\",\n \"The first update happens when it receives models from the clients, i.e.\",\n \"$w^{t-1} \\\\rightarrow \\\\lbrace \\\\hat{w}_i^t|i=1,2,...,N\\\\rbrace $ .\",\n \"Then the aggregation leads to the second update, $\\\\lbrace \\\\hat{w}_i^t|i=1,2,...,N\\\\rbrace \\\\rightarrow w^{t}$ .\"\n ],\n [\n \"Model Trajectory\",\n \"Client.\",\n \"Given a client $k$ at time point $t$ , we cache the updated models with a sliding time window $[t-T, t]$ in two different queues, that distinguish by two model updates.\",\n \"We refer the time sequence of local model updates $\\\\mathcal {L}_{k}^t=\\\\lbrace w_{k}^{t-T}, ...,w_{k}^t\\\\rbrace $ from $w_{k}^t \\\\rightarrow \\\\hat{w}_{k}^t$ as a local model trajectory, and $\\\\mathcal {\\\\hat{G}}_{k}^t=\\\\lbrace \\\\hat{w}_{k}^{t-T}, ...,\\\\hat{w}_{k}^t\\\\rbrace $ from $\\\\hat{w}_{k}^t \\\\rightarrow w_{k}^{t+1}$ as a global model trajectory.\",\n \"Server.\",\n \"Correspondingly, we cache the local and global model updates in the local and global model trajectories for all client at the server, i.e.\",\n \"$\\\\lbrace \\\\mathcal {\\\\hat{L}}_i|i=1,2,...,N\\\\rbrace $ and $\\\\lbrace \\\\mathcal {G}_i|i=1,2,...,N\\\\rbrace $ .\",\n \"Note that if the server can always send the lossless global model update to all clients, the global trajectories at time $t$ are the same for all clients.\"\n ],\n [\n \"Model Prediction\",\n \"Client.\",\n \"Given a client $k$ at time point $t$ , we predict $\\\\hat{w}_{k}^t \\\\rightarrow w_{k}^t$ from the local and global training trajectories, $\\\\mathcal {L}_{k}^{t-1}$ and $\\\\mathcal {\\\\hat{G}}_{k}^{t-1}$ as follows: $\\\\tilde{w}_{k}^{t} = f_{predict,k}(\\\\mathcal {L}_{k}^{t-1}, \\\\mathcal {\\\\hat{G}}_{k}^{t-1}, \\\\hat{w}_{k}^{t}) = \\\\operatornamewithlimits{arg\\\\,max}_{w_{k}^t} p(w_{k}^t|\\\\underbrace{w_{k}^{t-T}, ...,w_{k}^{t-1}}_{\\\\text{$\\\\mathcal {L}_{k}^{t-1}$}},\\\\underbrace{\\\\hat{w}_{k}^{t-T}, ...,\\\\hat{w}_{k}^{t-1}}_{\\\\text{$\\\\mathcal {\\\\hat{G}}_{k}^{t-1}$}}, \\\\hat{w}_{k}^{t})$ where $f_{predict,k}$ is the used predictor for model prediction in the client $k$ .\",\n \"Server.\",\n \"For the server, we predict model updates $\\\\hat{w}_i^t\\\\rightarrow w^t$ for each client $i$ from local and global trajectories $\\\\mathcal {\\\\hat{L}}_{i}^{t-1}$ and $\\\\mathcal {G}_{i}^{t-1}$ as follows: $\\\\tilde{w}_{i}^{t} = h_{predict,i}(\\\\mathcal {\\\\hat{L}}_{i}^{t-1}, \\\\mathcal {G}_{i}^{t-1}, \\\\hat{w}_{i}^{t}) = \\\\operatornamewithlimits{arg\\\\,max}_{w_{i}^t} p(w_{i}^t|\\\\underbrace{w_{i}^{t-T}, ...,w_{i}^{t-1}}_{\\\\text{$\\\\mathcal {\\\\hat{L}}_{k}^{t-1}$}},\\\\underbrace{\\\\hat{w}_{i}^{t-T}, ...,\\\\hat{w}_{i}^{t-1}}_{\\\\text{$\\\\mathcal {G}_{k}^{t-1}$}}, \\\\hat{w}_{i}^{t}), \\\\forall i\\\\in {1,...,N}$ where $h_{predict,i}$ is the used predictor for model prediction in the server for each client $i$ .\"\n ],\n [\n \"Model Residual\",\n \"Given a model update $\\\\hat{w}_{i}^t \\\\rightarrow w_{i}^t$ for the client $i$ at time $t$ in the server or in the corresponding client, if we can compute the model prediction $\\\\tilde{w}_{i}^{t}$ based on Eq.\",\n \"REF , we define the model residual as follows: $r_{i}^t = w_{i}^t - \\\\tilde{w}_{i}^t$ Note that $r_{i, ul}^t$ and $r_{i, dl}^t$ are the residuals of clients for uploading and the residuals of the server for downloading, respectively.More understanding of residuals is provided in Sec.\",\n \".\"\n ],\n [\n \"Deep Compression\",\n \"To reduce the model size for more efficient communication, we shrink the model size before sending it out.\",\n \"We define a compressed model in the client k: $\\\\bar{w}_{k}^t = f_{compress}(w_{k}^t)$ and in the server: $\\\\bar{w}_{i}^t = h_{compress}(w_{i}^t), \\\\forall i\\\\in \\\\lbrace 1,...,N\\\\rbrace $ where $f_{compress}$ and $h_{compress}$ is the used compressor for model compression in clients and server respectively.\",\n \"In our system, we consider to compress and communicate model residuals instead of model itself in the client k: $\\\\bar{r}_{k,ul}^t = f_{compress}(r_{k,ul}^t) = f_{compress}( w_{k}^t - f_{predict, i}(w_{k}^{t-T}, ...,w_{i}^{t-1},\\\\hat{w}_{k}^{t-T}, ...,\\\\hat{w}_{k}^{t-1}, \\\\hat{w}_{k}^{t}))$ and in the server: $\\\\bar{r}_{i,dl}^t = h_{compress}(r_{i,dl}^t) = h_{compress}( w_{i}^t - f_{predict, i}(w_{i}^{t-T}, ...,w_{i}^{t-1},\\\\hat{w}_{i}^{t-T}, ...,\\\\hat{w}_{i}^{t-1}, \\\\hat{w}_{i}^{t})), \\\\forall i\\\\in \\\\lbrace 1,...,N\\\\rbrace $\"\n ],\n [\n \"ResFed: Residual-based Federated Learning Framework\",\n \"[t]: ResFed: Residual-based federated learning framework [1] Server runs: initialize the global model $w$ initialize the empty local model trajectories $\\\\mathcal {\\\\hat{L}}_{1},...,\\\\mathcal {\\\\hat{L}}_{N}$ initialize the global model trajectories $\\\\mathcal {G}_{1},...,\\\\mathcal {G}_{N}$ initialize the predictor $h_{predict}$ $i \\\\in \\\\lbrace 1,2,...,N\\\\rbrace $ initialize an empty local model trajectories $\\\\mathcal {L}_{i}$ @client $i$ initialize an empty global model trajectories $\\\\mathcal {\\\\hat{G}}_{i}$ @client $i$ $h^{\\\\prime }_{predict,i} = h_{predict}$ sharing predictors to client $i$ $f^{\\\\prime }_{predict,i} = f_{predict, i}$ get the shared predictors from client $i$ $t \\\\in \\\\lbrace 1,2,...,M\\\\rbrace $ $i \\\\in \\\\lbrace 1,2,...,N\\\\rbrace $ in parallel $t < T$ $\\\\mathcal {\\\\hat{G}}_i \\\\leftarrow cache(w)$ cache the global model in $\\\\mathcal {\\\\hat{G}}_i $ @client $i$ Server communicates $w$ to the client $i$ $\\\\hat{w}_i \\\\leftarrow \\\\textbf {LocalTrain}(w)$ @client $i$ Client $i$ communicates $\\\\hat{w}_i$ to the server $\\\\mathcal {L}_i \\\\leftarrow cache(\\\\hat{w}_i)$ cache the local model in $\\\\mathcal {L}_i$ @client $i$ Server communicates $\\\\bar{r}_{i,dl}$ to the client $i$ $\\\\bar{r}_{i,ul} \\\\leftarrow $ ResFedClientUpdate $(i, \\\\bar{r}_{i,dl})$ @client $i$ Client $i$ communicates $\\\\bar{r}_{i,ul}$ to the server $\\\\tilde{w}_i \\\\leftarrow f^{\\\\prime }_{predict,i}(\\\\mathcal {G}_i, \\\\mathcal {\\\\hat{L}}_i, \\\\hat{w})$ predict updated model $\\\\hat{w}_i \\\\leftarrow \\\\tilde{w}_i + \\\\bar{r}_{i,ul}$ recover models $\\\\mathcal {\\\\hat{L}}_i \\\\leftarrow cache(\\\\hat{w}_i)$ update local trajectory $w \\\\leftarrow \\\\textbf {Aggregate}(\\\\hat{w}_1,...,\\\\hat{w}_N)$ $i \\\\in \\\\lbrace 1,2,...,N\\\\rbrace $ $\\\\tilde{w}_i \\\\leftarrow h_{predict}(\\\\mathcal {G}_i, \\\\mathcal {\\\\hat{L}}_i, w)$ predict updated model $r_{i,dl} \\\\leftarrow w - \\\\tilde{w}_i$ compute model residuals $\\\\bar{r}_{i,dl} \\\\leftarrow h_{compress}(r_{i,dl})$ compress model residuals $\\\\mathcal {G}_i \\\\leftarrow cache(\\\\tilde{w}_i + \\\\bar{r}_{i,dl})$ synchronize global trajectory return $w$ ResFedClientUpdate $(k, \\\\bar{r})$ $\\\\tilde{w} \\\\leftarrow h^{\\\\prime }_{predict, i}(\\\\mathcal {\\\\hat{G}}_k, \\\\mathcal {L}_k, w_k)$ predict updated model $\\\\hat{w} \\\\leftarrow \\\\tilde{w} + \\\\bar{r}$ recover models $\\\\mathcal {\\\\hat{G}}_k \\\\leftarrow cache(\\\\hat{w})$ update global trajectory $w_k \\\\leftarrow \\\\textbf {LocalTrain}(\\\\hat{w})$ $\\\\tilde{w}_k \\\\leftarrow f_{predict,i}(\\\\mathcal {\\\\hat{G}}_k, \\\\mathcal {L}_k, \\\\hat{w})$ predict updated model $r_k \\\\leftarrow w_k - \\\\tilde{w}_k$ compute model residuals $\\\\bar{r}_k \\\\leftarrow f_{compress}(r_k)$ compress model residuals $\\\\mathcal {L}_k \\\\leftarrow cache(\\\\tilde{w}_k + \\\\bar{r}_k)$ synchronize local trajectory return $\\\\bar{r}_k$ The overview of the ResFed is shown in Fig.\",\n \"REF and the detailed steps with lossy compression in one communication round is given in Fig.\",\n \"REF .\",\n \"In particular, we introduce (a) predictor sharing, (b) model prediction, and (c) residual generation in Sec REF .\",\n \"In Sec.\",\n \"REF , (d) residual compression is formulated in Eq.\",\n \"REF and Eq.\",\n \"REF .\",\n \"Then, after (e) communicating residual bits, we provide details on (f) model recovering in Sec.\",\n \"REF .\",\n \"Finally in Sec.\",\n \"REF , we describe (g) trajectory synchronization.\"\n ],\n [\n \"Predictor Sharing\",\n \"Then, we consider to deploy a pair of predictors in both clients and the server, which can execute the model predictions based on the local and global model trajectories in the time series.\",\n \"In our framework, the server caches the local model trajectories in all clients once the local model update is received.\",\n \"Given a client $k$ , if the received models in the server is exactly the same as the local model, and the models in $\\\\mathcal {\\\\hat{G}}_k$ also the same, we say that both trajectories in client $k$ are fully observable in the server.\",\n \"Give a predictor $f_{predict, k}$ in the client $k$ , we share it to make the predictor in the server $f^{\\\\prime }_{predict, k}=f_{predict, k}$ in ResFed.\",\n \"If the trajectories in client are fully observable in the server, we can get the same model predictions in both server and clients from Eq.\",\n \"REF .\",\n \"Then, by communicating the model residuals, the new model update at time $t+1$ can be recovered by the model residual and the model at last time point.\",\n \"Also, we share the set of predictors in the server $\\\\lbrace h_{predict, i}|i=1,...,N\\\\rbrace $ to the corresponding client, i.e.\",\n \"$h^{\\\\prime }_{predict, i}=h_{predict, i}, \\\\forall i\\\\in \\\\lbrace 1,...,N\\\\rbrace $ .\",\n \"Then we can get the same model predictions based on Eq.\",\n \"REF .\",\n \"For the predictor, various design choices exist.\",\n \"In this work, we employ the predictor with respect to the model transition dynamics in a sliding history time window.\",\n \"The predictor is formulated as follows: $\\\\tilde{w}_{k}^{t} = f_{predict}(\\\\mathcal {L}_{k}^{t-1}, \\\\mathcal {\\\\hat{G}}_{k}^{t-1}, \\\\hat{w}_{k}^t) ={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\hat{w}_{k}^t, & \\\\text{$T=0$}.\\\\\\\\\\\\hat{w}_{k}^t + \\\\sum _{\\\\tau =1}^T (-1)^{T-\\\\tau }(T-\\\\tau +1)(w_{k}^{t-\\\\tau }-\\\\hat{w}_{k}^{t-\\\\tau }), & \\\\text{$T>0$}.\\\\end{array}\\\\right.\",\n \"}$ To reduce the used memory for caching trajectories in the client, we apply a short time window $[t-T, t]$ in the prediction process.\",\n \"We term (i) stationary predictor when $T=0$ ; and (ii) linear predictor when $T=1$ .\",\n \"Note that we consider the model updates in [1], [19], [8] as special residuals, which calculated by stationary predictor.\",\n \"Specifically, the stationary predictor uses the current model for the prediction of the next model, $\\\\tilde{w}_{k}^{t} = \\\\hat{w}_{k}^t$ , where the model residual is always $r_{k}^{t} = w_{k}^{t} - \\\\hat{w}_{k}^{t}$ .\",\n \"Note that when the number of local training epochs is fixed to 1, the stationary residuals is proportional to gradients, i.e.\",\n \"$r = \\\\eta g$ , where $\\\\eta $ is the learning rate and $g$ represents the gradients.\",\n \"In the linear predictor, $\\\\tilde{w}_{k}^{t} = \\\\hat{w}_{k}^{t} + w_{k}^{t-1} - \\\\hat{w}_{k}^{t-1}$ , the model transition in the last local training step is always considered, where $r_{k}^{t} = w_{k}^{t} - \\\\hat{w}_{k}^{t} - w_{k}^{t-1} + \\\\hat{w}_{k}^{t-1}$ .\",\n \"The predictor for the client $k$ in the server $h_{predict,k}$ is similar to $f_{predict,k}$ .\"\n ],\n [\n \"Model Recovery\",\n \"We cache the trajectories in both server and clients.\",\n \"Each client has two model trajectories for local and global model updates in the history.\",\n \"In the server, it caches the global trajectory and the local model trajectories of all connecting clients in the history.\",\n \"In this case, the two trajectories in each client are fully observable in the server.\",\n \"Through sharing predictors, given a client $k$ at round $t$ , the server is able to get the same model prediction $\\\\tilde{w}_{k}^t$ as the client $k$ .\",\n \"If uncompressed model residuals ($\\\\hat{w}_{k}^t = w^t$ ) are received from client $k$ , the model update after local training $w_{k}^t$ can be recovered in the server as follows: $w_{k}^t = r_{k}^{t} + \\\\hat{w}_{k}^t = r_{k}^{t} + w^t$ where $\\\\hat{w}_{k}^t$ is the global model in the last round.\",\n \"Similarly, if we predict the global model update in the client, through sharing predictors and uncompressed residuals, the aggregated model can also be recovered in the client.\"\n ],\n [\n \"Model Trajectory Synchronization\",\n \"Since for the model residuals lossy compression is applied, i.e.\",\n \"$ \\\\bar{r} \\\\ne r$ , the updated model of a sender $w$ can be recovered in receivers as $\\\\hat{w} = \\\\bar{r} + \\\\tilde{w} $ .\",\n \"Therefore, we say that the $w$ cannot be fully recovered in receivers, as $\\\\hat{w} \\\\ne w $ .\",\n \"If we cache the original models $w$ in the sender and $\\\\hat{w}$ in receivers, the trajectories in the sender and receiver are different, which leads to drift in results of shared predictors.\",\n \"To avoid the drift effect, we synchronize the model trajectories in the sender by simulating the recovering process: The originally updated models are not cached in the trajectories; instead, we recover the model from compressed model residuals locally, in order to enforce the trajectories in the sender in the same way as the trajectories in the receiver.\",\n \"The ResFed pseudocode is given in Algorithm .\"\n ],\n [\n \"Experimental Settings\",\n \"Guided by the previous work by [2][14], we process and distribute the datasets MNIST ([12]), Fashion-MNIST ([27]), SVHN ([20]), CIFAR-10 ([11]) and CIFAR-100 ([20]) on a set of clients, and train LeNet-5, CNNIt consists of 5 convolutional and 3 fully connected layers., ResNet-18 on those federated dataset distributively, as shown in Tab.\",\n \"REF .\",\n \"We provide details on the experimental settings in Sec.\",\n \".\"\n ],\n [\n \"Residuals vs Gradients and Weights\",\n \"On top of the basic evaluation in Fig.\",\n \"REF , we believe deep residual compression can save more communication volume in federated learning with minimum impact on the accuracy.\",\n \"Thus, we demonstrate the federated learning integrating compressing weights, gradients and two different residuals, i.e.\",\n \"stationary and linear residuals in Eq.\",\n \"REF .\",\n \"As shown in Fig.\",\n \"REF , the testing accuracy on both IID and Non-IID datasets from deep residual compression always outperforms weight and gradient compression.\",\n \"Also, the linear residuals can achieve a higher accuracy and faster convergence than stationary residuals.\",\n \"The results indicate communicating residuals in federated learning can enable larger compression ratio per communication round, compared to communicating other parameters.\",\n \"Figure: Comparison of compressing weights, gradients, residuals with stationary (Res-0) and linear (Res-1) predictors.\",\n \"The sparsities are 0.2 and 0.01 for training on MNIST and CIFAR-100 distributed in 10 clients, respectively.\",\n \"We quantize the non-zero parameters and use 1 bit to represent each of them.\",\n \"For Non-IID setting, each client owns only the data with half classes.\"\n ],\n [\n \"Communication Efficiency Improvement\",\n \"Next, we evaluate the required communication volume for training three sizes of models on different datasets in both IID and Non-IID settings.\",\n \"Tab.\",\n \"REF shows that to reach a promising target accuracy, ResFed with lossy compression (compression ratio is set on $350\\\\times -375\\\\times $ ) can save on average around 99% of the total communication volume for all clients in only up- or down-streaming.\",\n \"Furthermore, the bitsaving ratios of ResFed on IID and Non-IID settings are similar, which indicates the compression performance of ResFed is robust to data heterogeneity in federated learning.\",\n \"We show testing accuracy and training loss change with increasing required communication volume in ResFed in Fig.\",\n \"REF .\",\n \"The results indicate communicating residuals in federated learning can remarkably save overall communication volume.\"\n ],\n [\n \"Scalability for Resource-constrained Communication Environments\",\n \"Finally, we explore the scalability of ResFed by tuning compression ratios for client-to-server and server-to-client, as in real application scenarios.\",\n \"The available network resources for up- and down-streaming can be heterogeneous.\",\n \"Fig REF shows the test accuracy effected for different values of sparsity, which leads to various compression ratios for each communication round for up- and down-streaming.\",\n \"From Fig.\",\n \"REF , we can observe the testing accuracy reduces with higher compression ratio per communication round, when the number of communication rounds is always the same, i.e.\",\n \"set to 300.\",\n \"However, when we consider the dedicated budget for communication costs in up- or down-streaming, a large compression ratio in ResFed can achieve better accuracy, as shown in Fig.\",\n \"REF .\",\n \"By adapting the compression ratio in resource-constrained communication environments, ResFed can effectively enhance the federated learning using deep residual compression.\"\n ],\n [\n \"Related Work\",\n \"Deep compression.\",\n \"Deep compression was originally proposed by [5] and aims at compressing deep learning models by a pipeline including sparsification (pruning), quantization and encoding for more efficient deployment.\",\n \"Based on deep compression, [16] have proposed deep gradient compression to reduce the communication costs in distributed learning by compressing gradients rather than model weights, which can also be used in federated learning.\",\n \"However, models are usually trained more than one epoch locally in federated learning ([18], [15], [25], [9]), which results in gradient accumulation instead of gradients in other distributed learning scenarios ([23]).\",\n \"Our conducted experiments also show compressing residuals can achieve better communication efficiency comparing to compress gradients due to the additional prediction step.\",\n \"In ResFed, we especially consider residuals, which eliminate the model similarity in a single inter-round of federated learning communication and achieve a better compression performance by leveraging the deep residual compression.\",\n \"Federated learning and communication efficiency.\",\n \"Communication efficiency is the key for deploying federated learning in real application scenarios, especially to train a large model.\",\n \"Previous research by [29], [9], [4] attempted to reduce the number of needed communication rounds for a better communication efficiency.\",\n \"Meanwhile, the proposed approaches by [28], [22], [7] are built upon deep compression and focus on improving communication efficiency by decreasing the communication volume.\",\n \"However, unlike compressing residuals in ResFed, they compressed model weights without consideration of any potential redundancy in sequential updating of federated learning.\",\n \"The recent work by [30] has also mentioned the predictive model update in federated learning, which is concurrent to our work, but the information in the history of model updating is not considered for reducing the parameter redundancy there.\",\n \"Additionally, all those algorithms above can only be used to improve the communication efficiency for up-streaming, while ResFed can handle with up- and/or down-streaming for heterogeneous resource-constrained environments.\",\n \"Residuals in video encoding.\",\n \"The residuals have been widely and successfully utilized in video encoding since H.261 ([3], [13]).\",\n \"By considering inter-frame correlations, the pixel values in the current frame are predicted from history frames and then only residuals, i.e.\",\n \"the deviations between predicted and the actual pixel values in the current frame, are encoded and streamed to the receivers.\",\n \"Inspired by the residuals in video encoding, we integrate the model residuals into federated learning in ResFed, where the inter-round similarity of a model update is analogous to inter-frame correlation in video encoding.\"\n ],\n [\n \"Conclusion\",\n \"In this work, we introduce a residual-based federated learning framework, which allows clients and the server to share residuals instead of weights or gradients.\",\n \"It achieves more efficient communication for both up- and down-streaming in federated learning by leveraging deep residual compression, and hence can be flexibly deployed in heterogeneous network environments.\",\n \"Our conducted experimental evaluation shows that the framework remarkably reduces overall communication volume to reach the same prediction accuracy in standard federated learning.\",\n \"Compared to compressing model weights or gradients, ResFed achieves higher accuracy and faster convergence speed.\",\n \"Limitations.\",\n \"We cache the recovered models as local and global trajectories for continual model prediction and residual computing in all clients and server.\",\n \"Assuming that we perform ResFed with $N$ clients for training a model with $V$ 32-bit float parameters and we set the trajectory length on $T$ , each client should use $2*32*V*T$ bits memory to cache the 2 trajectories.\",\n \"Thus, the additional required memory size in each client is proportional to $T*V$ .\",\n \"For the server, it needs $2*32*V*T*N$ bits memory to cache the local and global trajectories for all clients.\",\n \"In order to reduce the required memory, a potential solution is to cache the compressed models in the trajectories for both sender and receiver symmetrically, after model recovery.\",\n \"However, the accuracy of model prediction based on compressed trajectories is reduced, then the memory-accuracy trade-off needs to be investigated in future work.\"\n ],\n [\n \"Reproducibility statement\",\n \"We provide the source-code for the implementation and evaluation of our proposed framework ResFed in the code appendix.\",\n \"The user guidelines for installation and execution is given in the file README.md.\",\n \"The details on our conducted experiments are provided in Sec. .\",\n \"The source-code will be made publicly available after double-blind review.\",\n \"Tables of Notations We provide an overview of the most relevant notations in Tab.\",\n \"REF .\",\n \"Table: Summary of mainly used notations in this paper.\",\n \"Experimental Details and Further Results In this section, we provide the details on our conducted experiment in .\",\n \"We run on a computer cluster with 4$\\\\times $ NVIDIA-A100-PCIE-40GB GPUs and 4$\\\\times $ 32-Core-AMD-EPYC-7513 CPUs.\",\n \"The environment is a Linux system with Pytorch 1.8.1 and Cuda 11.1.\",\n \"We demonstrate the learning task on 5 different datasets: MNIST [12]: 60000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 28x28 gray-scale digit image, associated with a label from 10 classes.\",\n \"CIFAR-10 [11]: 50000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 32x32 RGB image, associated with a label from 10 classes.\",\n \"Fashion-MNIST [27]: 60000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 28x28 gray-scale image, associated with a label from 10 classes.\",\n \"SVHN [20]: 73257 data points in the training set and 26032 data points in the test set.\",\n \"Each data point is a 32x32 RGB digit image, associated with a label from 10 classes.\",\n \"CIFAR-100 [20]: 50000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 32x32 RGB image, associated with a label from 100 classes.\",\n \"The models trained on those dataset are shown in Tab.\",\n \"REF .\",\n \"Table: Dataset and models in experimentsExperiments for Sec.\",\n \"REF We divide the dataset, i.e.\",\n \"MNIST and CIFAR-100, into 10 clients and run the FedAvg with local optimizer of stochastic gradient descent (SGD) (momentum is 0.9) to train LeNet-5 and CNN (with 5 convolutional and 3 fully connected layers), respectively.\",\n \"The learning rate is fixed on 0.01 and the batch size is 64 for all tests.\",\n \"In Non-IID data setting, each client owns only 2 out of 10 classes in MNIST and 50 out of 100 classes in CIFAR-100.\",\n \"In clients, we consider 5 different approaches as follows: No Compression: The standard federated learning without any compression methods is used as the baseline, which provides the best results among all of the approaches, when number of communication rounds is the same.\",\n \"Compress Weights: Before communication in standard federated learning, the model weights are first compressed.\",\n \"Compress Gradients: The gradients in each epoch are compressed and communicated to the server.\",\n \"Compress Res-0: The residuals are computed by stationary predictor, i.e.\",\n \"Eq.\",\n \"REF with $T=0$ .\",\n \"Compress Res-1: The residuals are computed by linear predictor, i.e.\",\n \"Eq.\",\n \"REF with $T=1$ .\",\n \"For each approach on each dataset, we run 10 tests with different seeds and show the mean value and standard variance in Fig.\",\n \"REF .\",\n \"We compress the those model parameters using deep compression pipeline ([5]) only for client-to-server.\",\n \"In particular, we set sparsity on 80% and 99% for residuals in LetNet and CNN, respectively.\",\n \"We use SGD optimizer momentum of 0.9.\",\n \"The number of local epoch is 1.\",\n \"Those sparsified parameters are zero-parameters and the number of the continually appearing zero-parameters are encoded as a 16-bit float parameters ([16]).\",\n \"After that, we quantize the non-zero parameters in 1 bit with median value of positive and negative parameters ([28]).\",\n \"Finally, Huffman encoding ([24]) is used.\",\n \"Experiments for Sec.\",\n \"REF We train LeNet-5, CNN and ResNet18 (size from small to large) on Fashion-MNIST, CIFAR-10 and SVHN with 10 clients in both IID and Non-IID settings.\",\n \"We demonstrate the ResFed and lossy compress the residuals either only for uploading (UL) or downloading (DL) to study the effects on each direction separately.\",\n \"The learning rate is fixed on 0.01 and the batch size is 64 for all tests.\",\n \"In Non-IID data setting, each client owns 50% classes (5 out of 10).\",\n \"We use mean values from 5 tests in each experiment for the evaluation in Tab.\",\n \"REF and show the results with standard variance in Fig.\",\n \"REF , which indicate the overall communication volume can be remarkably reduced in ResFed.\",\n \"Figure: Overall communication efficiency enhanced by ResFed with Lossy Compression (LC) for only Uploading (UL) or Downloading (DL)For all experiments, we set the sparsity on 99% and use 1 bit for each non-zero parameters.\",\n \"Consequently, the compression ratio per communication round for LetNet-5 is about $350\\\\times $ , CNN is about $375\\\\times $ and ResNet-18 is about $356\\\\times $ .\",\n \"Experiments for Sec.\",\n \"REF To show the correlation between deep residual compression in up- und down-streaming, we train the CNN model on IID CIFAR-10 with 10 clients and tune the sparsity for realizing different compression ratios per communication round in ResFed.\",\n \"The learning rate is fixed on 0.01 and the batch size is 64.\",\n \"We use SGD optimizer momentum of 0.9.\",\n \"The number of local epoch is 1.\",\n \"Specifically, the value of sparsity is $\\\\lbrace 0\\\\%, 90\\\\%, 95\\\\%, 99\\\\%, 99.5\\\\%\\\\rbrace $ for both up- and down-streaming and then set 1 bit for all non-zero parameters in quantization.\",\n \"Understanding Residuals We provide an illustration of model transitions during federated learning in Fig.\",\n \"REF .\",\n \"Given a sender and a receiver (both can be a client or a server), the communication and operation result in model transition.\",\n \"Note that for a client, the operation is local training; for a server, the operation is aggregation.\",\n \"We consider the model transition caused by an operation as a internal model transition, and by communication as a external model transition.\",\n \"Then, as explained in Sec.\",\n \"REF , the model is updated twice between two communication rounds, which is shown in Fig.\",\n \"REF as dual model transition.\",\n \"Consequently, we can have an internal and an external model transition trajectory in both sender and receiver.\",\n \"Note that for a client, the internal model transition trajectory is a local model trajectory; for a server, the internal model transition trajectory is a global model trajectory, as described in Sec.\",\n \"REF .\",\n \"Figure: Residual generation in ResFed during model transitions.\",\n \"For a better overview, we simplify the system by disregarding the trajectory synchronization step in Sec.\",\n \".Figure: Value comparison of model parameters, gradients and residuals in federated learning.\",\n \"We train a LeNet with 61706 weights of 32-bits float on MNIST distributed among 10 clients, with fixed learning rate 0.0010.001 and batch size 64.\",\n \"For fairly comparing gradients and residuals, the number of local epoch in each client is set as 1.\",\n \"We set 6 checkpoints when the number of communication rounds is {1,4,8,16,32,128}\\\\lbrace 1, 4, 8, 16, 32, 128\\\\rbrace .\",\n \"The results show that most values of residuals are smaller than weights and gradients during the training.\",\n \"It indicates that lossly compressing residuals naturally lose less information and have a smaller affect on the accuracy.Finally, ResFed allows the sender to predict the model for the next internal model transition, which is shown in orange.\",\n \"Meanwhile the sender does the operation to execute the internal model transition and residuals (in purple) are deduced from the difference of both model transition results.\",\n \"We believe the predicted model can be closer to the updated model than the previous model, which leads to smaller values of residuals.\",\n \"To evaluate it, we set 6 checkpoints when the number of communication rounds is $\\\\lbrace 1, 4, 8, 16, 32, 128\\\\rbrace $ , and show the values of model weights, gradients and residuals in Fig.\",\n \"REF .\",\n \"Based on this, the residuals can be compressed smaller than weights and gradients.\"\n ],\n [\n \"Tables of Notations\",\n \"We provide an overview of the most relevant notations in Tab.\",\n \"REF .\",\n \"Table: Summary of mainly used notations in this paper.\"\n ],\n [\n \"Experimental Details and Further Results\",\n \"In this section, we provide the details on our conducted experiment in .\",\n \"We run on a computer cluster with 4$\\\\times $ NVIDIA-A100-PCIE-40GB GPUs and 4$\\\\times $ 32-Core-AMD-EPYC-7513 CPUs.\",\n \"The environment is a Linux system with Pytorch 1.8.1 and Cuda 11.1.\",\n \"We demonstrate the learning task on 5 different datasets: MNIST [12]: 60000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 28x28 gray-scale digit image, associated with a label from 10 classes.\",\n \"CIFAR-10 [11]: 50000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 32x32 RGB image, associated with a label from 10 classes.\",\n \"Fashion-MNIST [27]: 60000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 28x28 gray-scale image, associated with a label from 10 classes.\",\n \"SVHN [20]: 73257 data points in the training set and 26032 data points in the test set.\",\n \"Each data point is a 32x32 RGB digit image, associated with a label from 10 classes.\",\n \"CIFAR-100 [20]: 50000 data points in the training set and 10000 data points in the test set.\",\n \"Each data point is a 32x32 RGB image, associated with a label from 100 classes.\",\n \"The models trained on those dataset are shown in Tab.\",\n \"REF .\",\n \"Table: Dataset and models in experiments\"\n ],\n [\n \"Experiments for Sec. \",\n \"We divide the dataset, i.e.\",\n \"MNIST and CIFAR-100, into 10 clients and run the FedAvg with local optimizer of stochastic gradient descent (SGD) (momentum is 0.9) to train LeNet-5 and CNN (with 5 convolutional and 3 fully connected layers), respectively.\",\n \"The learning rate is fixed on 0.01 and the batch size is 64 for all tests.\",\n \"In Non-IID data setting, each client owns only 2 out of 10 classes in MNIST and 50 out of 100 classes in CIFAR-100.\",\n \"In clients, we consider 5 different approaches as follows: No Compression: The standard federated learning without any compression methods is used as the baseline, which provides the best results among all of the approaches, when number of communication rounds is the same.\",\n \"Compress Weights: Before communication in standard federated learning, the model weights are first compressed.\",\n \"Compress Gradients: The gradients in each epoch are compressed and communicated to the server.\",\n \"Compress Res-0: The residuals are computed by stationary predictor, i.e.\",\n \"Eq.\",\n \"REF with $T=0$ .\",\n \"Compress Res-1: The residuals are computed by linear predictor, i.e.\",\n \"Eq.\",\n \"REF with $T=1$ .\",\n \"For each approach on each dataset, we run 10 tests with different seeds and show the mean value and standard variance in Fig.\",\n \"REF .\",\n \"We compress the those model parameters using deep compression pipeline ([5]) only for client-to-server.\",\n \"In particular, we set sparsity on 80% and 99% for residuals in LetNet and CNN, respectively.\",\n \"We use SGD optimizer momentum of 0.9.\",\n \"The number of local epoch is 1.\",\n \"Those sparsified parameters are zero-parameters and the number of the continually appearing zero-parameters are encoded as a 16-bit float parameters ([16]).\",\n \"After that, we quantize the non-zero parameters in 1 bit with median value of positive and negative parameters ([28]).\",\n \"Finally, Huffman encoding ([24]) is used.\"\n ],\n [\n \"Experiments for Sec. \",\n \"We train LeNet-5, CNN and ResNet18 (size from small to large) on Fashion-MNIST, CIFAR-10 and SVHN with 10 clients in both IID and Non-IID settings.\",\n \"We demonstrate the ResFed and lossy compress the residuals either only for uploading (UL) or downloading (DL) to study the effects on each direction separately.\",\n \"The learning rate is fixed on 0.01 and the batch size is 64 for all tests.\",\n \"In Non-IID data setting, each client owns 50% classes (5 out of 10).\",\n \"We use mean values from 5 tests in each experiment for the evaluation in Tab.\",\n \"REF and show the results with standard variance in Fig.\",\n \"REF , which indicate the overall communication volume can be remarkably reduced in ResFed.\",\n \"Figure: Overall communication efficiency enhanced by ResFed with Lossy Compression (LC) for only Uploading (UL) or Downloading (DL)For all experiments, we set the sparsity on 99% and use 1 bit for each non-zero parameters.\",\n \"Consequently, the compression ratio per communication round for LetNet-5 is about $350\\\\times $ , CNN is about $375\\\\times $ and ResNet-18 is about $356\\\\times $ .\"\n ],\n [\n \"Experiments for Sec. \",\n \"To show the correlation between deep residual compression in up- und down-streaming, we train the CNN model on IID CIFAR-10 with 10 clients and tune the sparsity for realizing different compression ratios per communication round in ResFed.\",\n \"The learning rate is fixed on 0.01 and the batch size is 64.\",\n \"We use SGD optimizer momentum of 0.9.\",\n \"The number of local epoch is 1.\",\n \"Specifically, the value of sparsity is $\\\\lbrace 0\\\\%, 90\\\\%, 95\\\\%, 99\\\\%, 99.5\\\\%\\\\rbrace $ for both up- and down-streaming and then set 1 bit for all non-zero parameters in quantization.\"\n ],\n [\n \"Understanding Residuals\",\n \"We provide an illustration of model transitions during federated learning in Fig.\",\n \"REF .\",\n \"Given a sender and a receiver (both can be a client or a server), the communication and operation result in model transition.\",\n \"Note that for a client, the operation is local training; for a server, the operation is aggregation.\",\n \"We consider the model transition caused by an operation as a internal model transition, and by communication as a external model transition.\",\n \"Then, as explained in Sec.\",\n \"REF , the model is updated twice between two communication rounds, which is shown in Fig.\",\n \"REF as dual model transition.\",\n \"Consequently, we can have an internal and an external model transition trajectory in both sender and receiver.\",\n \"Note that for a client, the internal model transition trajectory is a local model trajectory; for a server, the internal model transition trajectory is a global model trajectory, as described in Sec.\",\n \"REF .\",\n \"Figure: Residual generation in ResFed during model transitions.\",\n \"For a better overview, we simplify the system by disregarding the trajectory synchronization step in Sec.\",\n \".Figure: Value comparison of model parameters, gradients and residuals in federated learning.\",\n \"We train a LeNet with 61706 weights of 32-bits float on MNIST distributed among 10 clients, with fixed learning rate 0.0010.001 and batch size 64.\",\n \"For fairly comparing gradients and residuals, the number of local epoch in each client is set as 1.\",\n \"We set 6 checkpoints when the number of communication rounds is {1,4,8,16,32,128}\\\\lbrace 1, 4, 8, 16, 32, 128\\\\rbrace .\",\n \"The results show that most values of residuals are smaller than weights and gradients during the training.\",\n \"It indicates that lossly compressing residuals naturally lose less information and have a smaller affect on the accuracy.Finally, ResFed allows the sender to predict the model for the next internal model transition, which is shown in orange.\",\n \"Meanwhile the sender does the operation to execute the internal model transition and residuals (in purple) are deduced from the difference of both model transition results.\",\n \"We believe the predicted model can be closer to the updated model than the previous model, which leads to smaller values of residuals.\",\n \"To evaluate it, we set 6 checkpoints when the number of communication rounds is $\\\\lbrace 1, 4, 8, 16, 32, 128\\\\rbrace $ , and show the values of model weights, gradients and residuals in Fig.\",\n \"REF .\",\n \"Based on this, the residuals can be compressed smaller than weights and gradients.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05602"}}},{"rowIdx":98,"cells":{"text":{"kind":"list like","value":[["Universality of the local limit of preferential attachment models"],["Abstract We study preferential attachment models where vertices enter the network with i.i.d.","random numbers of edges that we call the out-degree.","We identify the local limit of such models, substantially extending the work of Berger et al.(2014).","The degree distribution of this limiting random graph, which we call the random P\\'{o}lya point tree, has a surprising size-biasing phenomenon.","Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d.","out-degrees.","Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees.","Our proof of local convergence consists of two main steps: a P\\'olya urn description of our graphs, and an explicit identification of the neighbourhoods in them.","We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the P\\'{o}lya urn graph.","Our result proves a density convergence result, for fixed ages of vertices in the local limit."],["[enumerate]itemsep = -0.2em figurename=figure"]],"string":"[\n [\n \"Universality of the local limit of preferential attachment models\"\n ],\n [\n \"Abstract We study preferential attachment models where vertices enter the network with i.i.d.\",\n \"random numbers of edges that we call the out-degree.\",\n \"We identify the local limit of such models, substantially extending the work of Berger et al.(2014).\",\n \"The degree distribution of this limiting random graph, which we call the random P\\\\'{o}lya point tree, has a surprising size-biasing phenomenon.\",\n \"Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d.\",\n \"out-degrees.\",\n \"Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees.\",\n \"Our proof of local convergence consists of two main steps: a P\\\\'olya urn description of our graphs, and an explicit identification of the neighbourhoods in them.\",\n \"We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the P\\\\'{o}lya urn graph.\",\n \"Our result proves a density convergence result, for fixed ages of vertices in the local limit.\"\n ],\n [\n \"[enumerate]itemsep = -0.2em figurename=figure\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05551"}}},{"rowIdx":99,"cells":{"text":{"kind":"list like","value":[["Morphology of Shocked Lateral Outflows in Colliding Hydrodynamic Flows"],["Abstract Supersonic interacting flows occurring in phenomena such as protostellar jets give rise to strong shocks, and have been demonstrated in several laboratory experiments.","To study such colliding flows, we use the AstroBEAR AMR code to conduct hydrodynamic simulations in three dimensions.","We introduce variations in the flow parameters of density, velocity, and cross sectional radius of the colliding flows %radius in order to study the propagation and conical shape of the bow shock formed by collisions between two, not necessarily symmetric, hypersonic flows.","We find that the motion of the interaction region is driven by imbalances in ram pressure between the two flows, while the conical structure of the bow shock is a result of shocked lateral outflows (SLOs) being deflected from the horizontal when the flows are of differing cross-section."],["Introduction","Radiative shocks occur in a variety of settings, such as High Energy Density Plasma (HEDP), protostellar jets [1], supernova explosions [2], gamma ray bursts [3], and active galactic nuclei [4].","Over the years several efforts have been made to study these phenomena in the laboratory.","Despite large differences in physical scale, results obtained in the laboratory can be used to understand phenomena in the astrophysical setting, thanks to the use of dimensionless parameters [5], [6], [7].","Experiments at Imperial College London have produced single flows using a pulsed-power generator to drive radial wire arrays [8] and radial foils [9], [10]; similar experiments have also been performed at Cornell University [11].","Collimation of these flows was driven by toroidal magnetic fields in a manner consistent with the “Magnetic Tower\" model [12].","Extending this work, [13] found that the presence of an ambient medium results in the formation of a shock, driven by ablation of halo plasma.","While astronomical shocks are generally produced via time-variations in velocity, a similar effect can be obtained in a laboratory setting by creating a collision between two supersonic flows.","[14] conducted such an experiment, images of which are shown in figure REF .","Among the results of that study are the emergence of small scale structures behind a conical bow shock, which was observed to propagate in a downward direction.","When two flows collide, an interaction region forms, with the jet shocks becoming the boundaries.","This region usually consists of a cooling region, in which gas cools from post-shock temperature $T_s$ immediately behind the jet shocks to a lower temperature further behind the shock.","After passing through the cooling region, gas reaches its final post-cooling temperature and is deposited onto the cold slab, where densities are highest (see figure REF ), some shocked material gets ejected laterally by the high pressures throughout the interaction region, producing shocked lateral outflows (SLOs) [15].","Figure: (left) Images of a colliding flow experiment.","Adapted from figure 2 of .","(right) a diagram showing the structure of colliding flows; adapted from In our previous paper [16] we began a series of simulations in an effort to study colliding radiative flows like those of [14].","In that paper we used hydrodynamic simulations with an analytic form of radiative cooling, focusing our attention on instabilities in an effort to explain the origin of small scale structures.","While [14] had suggested that the [17] instability was a source of structure, we did not see any evidence of this mode present in the case of analytic cooling.","Instead, long-term evolution was found to be dominated by the bending modes characteristic of the nonlinear thin shell instability (NTSI) [18], which could be triggered either by sufficiently short cooling lengths or by oscillations resulting from the radiative shock instability [19].","In this paper, we continue to investigate the results of [14] by extending the results of [16].","Here we shift our focus away from instabilities (which were discussed in significant depth in [16]) to a study of flow parameter variation, allowing us to better understand the conical shape and downward propagation of the bow shock and lateral outflows.","We will continue to focus solely on hydrodynamic simulations, but the analytic cooling curve previously used will be replaced with a more realistic cooling function; this will be described in further detail in section .","Our approach of continuing to remain in the radiative hydrodynamic case for now will eventually allow us to have a better understanding of the magnetic case by providing a reference for comparison.","This paper is organized as follows: In Section we discuss the model system, the selected cooling function, and simulation parameters.","In section , we present the results of the simulations.","Section will include a discussion of motion of the interaction region, properties of the cold slab, and deflection of the SLOs."],["Methods and Model","The simulations in this study were conducted using AstroBEARhttps://astrobear.pas.rochester.edu/${ }^{,}$ [21], [22], which is a massively parallelized adaptive mesh refinement (AMR) code that includes a variety of multiphysics solvers, such as magnetic resistivity, radiative transport, self-gravity, heat conduction, and ionization dynamics.","Our study uses only the hydro solvers with an energy source term associated with the radiative cooling.","Thus our governing equations are; $\\frac{\\partial \\rho }{\\partial t} + {\\nabla } \\cdot \\rho {v} = 0$ $\\frac{\\partial \\rho {v}}{\\partial t} + {\\nabla } \\cdot \\left( \\rho {v} \\otimes {v} \\right)= - \\nabla p$ $\\frac{\\partial E}{\\partial t} + {\\nabla } \\cdot ((E + p) {v}) =-n^2\\Lambda (n,T)$ where $\\rho $ is the mass density, $n$ is the number density of nuclei, ${v}$ is the fluid velocity, $p$ is the thermal pressure, and $E = \\frac{1}{\\gamma - 1} p + \\frac{1}{2}\\rho v^2$ is the combined internal and kinetic energies.","$n^2\\Lambda (n,T)$ is a cooling function, which gives the energy loss per unit volume.","While in [16] we used an analytic cooling function of the form $\\Lambda (T) = \\alpha \\left(\\frac{T}{T_0}\\right)^\\beta $ , in this paper we use a more realistic cooling function.","Such a cooling function introduces variation in the slope $\\frac{\\operatorname{d} \\log \\Lambda }{\\operatorname{d}\\log T}$ , which is important to both the radiative shock instability and to the Field instability.","The specific function $\\Lambda (n,T)$ we used was chosen in attempt to provide correspondence with the experiments of [14] and is plotted in figure REF .","This function was calculated using ABAKO/RAPCAL [23], [24].","The required atomic data were obtained using FAC [25], while the resulting function was parameterized in terms of density and temperature using PARPRA [26].","Cooling is strongest at high temperatures and lower densities; as a fluid cools, $\\Lambda (n,T)$ decreases until a local minimum is encountered at a temperature on the order of 10 eV.","Cooling then increases again as temperature continues to decrease, but this effect is reduced at higher densities.","Figure: The cooling function plotted (top) as a function of temperature, for selected densities, and (bottom) as a colour-map in terms of density and temperature.Similarly to [16], we drove two cylindrical jets, one from the top and bottom z-boundaries respectively.","For the first run, both jets had densities of $3.0\\times 10^{18}$ particles per cm$^3$ , speeds of 70.0 km s$^{-1}$ , temperature of 720 K, and cross sectional radii of 1.5 mm.","The ambient medium density was $1\\times 10^{18}$ particles per cm$^3$ at a temperature of 4320 K. Extrapolated boundary conditions were used in all directions.","Runs were conducted in a 6.4mm cubic space, and consisted of the reference run described above and nine other runs in which jet parameters were varied as follows (summarised in table REF ).","For the first two runs, we changed the density of the bottom jet to $2.0\\times 10^{18}$ and $1.0\\times 10^{18}$ particles per cm$^3$ .","For third and fourth runs, the radius of the bottom jet was varied to 2.0 and 1.0 mm, and the density was varied to $1.69\\times 10^{18}$ and $6.75\\times 10^{18}$ particles per cm$^3$ such that the mass of the jets remained unaltered.","For the fifth and sixth runs, the radius of the bottom jet was varied to 2.0 and 1.0 mm, but this time the density of the jet was reverted to the original value.","Finally, the seventh and eighth runs varied velocities of the top and bottom jets to 80 and 60 km s$^{-1}$ respectively; these values were chosen to match the experiments in [14].","The seventh run retained the densities of the reference run, while the eighth run used densities of $2.63\\times 10^{18}$ and $3.50\\times 10^{18}$ so that the jets were of equal momentum density, and the ninth run used densities of $2.30\\times 10^{18}$ and $4.08\\times 10^{18}$ so that the jets were of equal momentum flux.","Table: Densities, velocities, and radii of jets in the first set of runs.","For each pair of numbers, the first describes the top jet while the second describes the bottom jet."],["Results","We will first examine the reference run (run 0 in table REF ), shown in figure REF .","This run shows a collision between two identical jets, similar to those studied in [16] but now using the complex cooling curve described in section .","As gas crosses the jet shock and enters the interaction region, the density increases from $3\\times 10^{18}$ cm$^{-3}$ to $1.2\\times 10^{19}$ cm$^{-3}$ .","The density varies as the gas cools but remains at the same order of magnitude throughout the cooling region, but is seen to be as high as $2.5\\times 10^{21}$ cm$^{-3}$ in the cold slab.","As gas exits the interaction region through the sides and into the shocked lateral outflows, densities drop to a value of order $1\\times 10^{18}$ cm$^{-3}$ and lower as gas begins to mix into the ambient medium.","At later times clumps are observed to form in the cold slab, though the origin of these is uncertain given that the cold slab is not well resolved at just a few cells ($10^{-3}$ cm) in width.","Since the jets are identical, the interaction region remains stationary and shows two identical jet shocks; while the shocks exhibit some curvature, no conical structure is observed.","The shocked lateral outflows flow primarily horizontally even though some material leaks from the top and bottom of the SLOs.","Figure: From left to right, runs 0, 1, and 2 from table .","For these runs, the density of the top jet remains fixed while the density of the bottom jet decreases with each successive run.Next, we examine cases in which the density of the bottom jet is varied, while the density of the top jet is fixed at $1.0\\times 10^{18}$ particles per cm$^3$ .","Figure REF shows jets with densities $3.0\\times 10^{18}, 2.0\\times 10^{18}$ , and $1.0\\times 10^{18}$ particles per cm$^3$ .","As the density of the jets becomes imbalanced, the interaction region no longer remains in place, with the velocity of the interaction region in this case being proportional to the density imbalance.","Lateral outflows are seen to be nearly horizontal.","Clump formation in the cold slab begins much sooner in collisions with a density imbalance; while the details of smaller structures are beyond the scope of this paper, we conjecture that this is the result of asymmetry exacerbating the formation of perturbations within the cold slab.","Increasing the imbalance further does not significantly change the onset of clump formation.","These clumps merge with each other as time progresses, though the regions of cold slab between these clumps is once again not well resolved.","Study of the the long term behaviour of these clumps is also limited by the interaction region hitting the wall after a relatively short amount of time.","Figure: From left to right, runs 3, 0, and 4 from table .","For these runs, the density and radius of the top jet remains fixed while the bottom jet both increases in density and decreases in radius with each successive run.Next, we examine cases in which the density of the bottom jet is again varied, this time varying the radius of the bottom jet as $r \\propto \\rho ^{-\\frac{1}{2}}$ so that the total mass flux through a horizontal slice remains fixed.","Figure REF shows jets with densities $1.69\\times 10^{18}, 3.0\\times 10^{18}$ , and $6.75\\times 10^{18}$ particles per cm$^3$ and radii 2.0, 1.5, and 1.0 mm.","The jet with the smaller radius, having a higher density, drives the motion of the interaction region.","As with the previous set, an imbalance of density also results in formation of clumps within the cold slab at earlier times, with these clumps again merging into fewer but larger clumps as time progresses.","Since the jets are of differing radii, the SLOs are no longer horizontal, instead bending away from the jet of larger radius.","Figure: From left to right, runs 5, 0, and 6 from table .","For these runs, the radius of the top jet remains fixed while the radius of the bottom jet decreases with each successive run.We next examine cases of equal speed jets for which the radius of the bottom jet is varied at fixed density.","Figure REF shows jets with radii 2.0, 1.5, and 1.0 mm.","For equal velocities and equal opposing jet densities, ram pressure is balanced and the interaction region remains at a fixed location.","As with density imbalances, an imbalance of radius alone appears to be sufficient to promote earlier clump formation, though in the case with a 1.5 mm jet colliding with a 1.0 mm jet the merging of clumps appears to be inhibited.","Since the speeds and densities are balanced and the interaction region never approaches a wall, these cases can be observed for longer times.","After material ejected in the initial collision flies away, a lateral outflow remains.","When the jets differ in size, the direction of the SLO occurs at an angle to the horizontal plane and points away from the larger jet.","The exact angle increases with the ratio $\\frac{(r_+^2 - r_-^2)}{r_-}$ , where $r_+$ and $r_-$ are the radii of the larger and smaller jet respectively, and is explored in section REF .","The interaction region also exhibits curvature, bending away from the larger flow.","Figure: From left to right, runs 7, 8, 9 from table .","While the previous runs use v=70v = 70 km s -1 ^{-1} for both jets, these runs use jet velocities of 80 and 60 km s -1 ^{-1} for the top and bottom jets respectively.","The density of the top jet decreases (and the bottom jet increases) with each successive run.Figure: Run 9 from table , at intervals of 200 ns.","Density is plotted in the left column, while the right column shows a plot of temperature.","At later times the Nonlinear Thin Shell Instability can be observed.Lastly, we examine effects of changing the jet velocity.","While previous scenarios used a velocity of 70 km s$^{-1}$ , the final set of runs uses velocities of 80 km s$^{-1}$ for the top jet and 60 km s$^{-1}$ for the bottom jet.","Figure REF shows three simulations with these velocities.","The first case shows two jets of equal density; here the imbalance of velocity results in the interaction region moving with a velocity of 0 km s$^{-1}$ after collision.","The second case shows two jets of equal momentum density (and thus equal mass flux); the velocity of the interaction region is significantly reduced, but motion still occurs as a result of ram pressure imbalance.","Both of these runs exhibit clumping behaviour similar to that of the runs with a density imbalance.","The final case shows two jets of equal ram pressure.","The interaction region now remains stationary.","While clumping behaviour is initially similar to the other two cases, clump mergers in this case produce a full cold slab spanning the entire radius of the jet.","At later times (figure REF ), this cold slab is broken apart by the nonlinear thin shell instability, similar to the cases examined in [16]."],["Motion of the Interaction Region","As seen in figures REF , REF , and REF , the interaction region has a nonzero velocity unless the ram pressure $\\rho v^2$ is equal for both the top and bottom jet.","Otherwise, it moves away from the jet with higher ram pressure with velocity $V$ .","In a reference frame comoving with $V$ , the momentum flux entering the interaction region from one jet must balance the momentum flux entering from the other jet.","This constraint can be expressed as $ \\rho _1 \\left( v_1 - V \\right)^2 + p_1= \\rho _2 \\left( v_2 - V \\right)^2 + p_2$ where positive velocities indicate the downward direction; $v_2$ is therefore negative.","We shall also neglect the thermal pressure difference $p_1-p_2$ .","The requirement that both jets flow into the interaction region produces the constraint $v_2 < V < v_1$ .","Taking the square root of equation REF , subject to this constraint gives us $\\sqrt{\\rho _1}\\left( v_1 - V \\right) = \\sqrt{\\rho _2} \\left(V - v_2 \\right)$ which can be solved for $V$ : $ V = \\frac{ v_1 \\sqrt{\\rho _1} + v_2 \\sqrt{\\rho _2} }{\\sqrt{\\rho _1}+\\sqrt{\\rho _2}}$ We will now examine this result in the context of figures REF -REF .","Velocities can be measured as follows: the location of the cold slab, lower jet shock, and upper jet shock are measured at $t=130$ ns and $t=230$ ns.","The positions at each time step are averaged; taking the difference of these averages between the two time steps gives us the displacement (positive is down) over the 100 ns time interval.","For runs 7-9, a similar procedure is used except that the measurements are taken at $t=140$ ns and $t=280$ ns.","Results are given in table REF .","Table: Densities and velocities of the jets in each run, along with the estimated (equation ) and observed value for the interaction region velocity VV.","All velocities are given in km s -1 ^{-1} and densities are given in 10 18 10^{18} cm -3 ^{-3}We can connect this prediction to the experiments of [14], in which the jet velocities were measured to be $v_1 = 80\\pm 10$ km s${ }^{-1}$ and $v_2 = -60\\pm 10$ km s${ }^{-1}$ .","Substantial uncertainties ($\\sim 60\\%$ ) were noted for measurements of density, but in any case the ratio $\\frac{\\rho _1}{\\rho _2}$ was within an order of magnitude of unity.","If equation REF is instead solved for $\\frac{\\rho _1}{\\rho _2}$ we find $\\frac{\\rho _1}{\\rho _2} = \\left(\\frac{v_2-V}{v_1-V}\\right)^2$ The bow shock velocity is stated to be $40\\pm 10$ km s${ }^{-1}$ , so the density ratio evaluates to $\\frac{\\rho _1}{\\rho _2} = 6.25 \\pm 4.38$ ; this on the higher end of what is in agreement with the measurements presented in [14].","Another useful quantity to compute for collisions between flows of unequal ram pressure is the Mach number in the centre-of-mass frame.","In this paper we define Mach number to be $M^2 = \\frac{\\rho v^2}{\\gamma p}$ , and $M_1$ and $M_2$ to be the Mach numbers for top and bottom flows in the laboratory frame.","When combined with equation REF this definition gives us $M^2 = \\frac{\\rho _1(v_1-V)^2}{\\gamma p_1} = \\frac{\\rho _1}{\\gamma p_1}\\left(v_1-\\frac{v_1\\sqrt{\\rho _1}+v_2\\sqrt{\\rho _2}}{\\sqrt{\\rho _1}+\\sqrt{\\rho _2}}\\right)^2$ for the centre-of-mass frame (note that Mach numbers must be equal when ram pressures balance).","This simplifies to $M^2 = \\frac{\\rho _1\\rho _2}{\\gamma p_i}\\left(\\frac{v_1-v_2}{\\sqrt{\\rho _1}+\\sqrt{\\rho _2}}\\right)^2$ which, assuming $v_2 < 0 < v_1$ , can be further simplified to $ M = \\frac{M_1\\sqrt{\\rho _2}+M_2\\sqrt{\\rho _1}}{\\sqrt{\\rho _2}+\\sqrt{\\rho _1}}$ Note that while $v_2$ is assumed to be negative, we always define the Mach number to be positive.","Note that while the interaction region velocity is an average weighed by the square roots of densities, the Mach number is an average weighted by the square roots of inverse densities."],["Cold Slab in One Dimension","We now begin our examination of shocked lateral outflow by considering a scenario with no outflow: the growth of the cold slab for a flow colliding with a wall in one dimension.","In subsequent sections, these results will be useful in the development of a three-dimensional model.","Define $\\rho _i, v_i$ , and $T_i$ to be the initial pre-shock density, velocity, and temperature; also define $\\rho _f, v_f,$ and $T_f$ to be the same values for material in the cold slab.","Velocities are specified in a “shock\" frame (see figure REF ), in which the front (i.e.","the side closer to the shock) of the cold slab remains stationarywe assume that the radiative shock instability can be ignored, so that the distance between the shock and the front of the cold slab remains fixed.","Any growth of the slab appears in this frame as the far side of the slab moving further away from the shock.","Figure: Temperature (red) and Density (blue) for a one dimensional shock in a) the shock frame and b) the lab frame.In the shock frame, the shock remains stationary while material in the cold slab flows away from the shock.","In the lab frame, the shock moves to the left (dashed curve) while material already in the cold slab remains stationary.Let $h$ be the height of the cold slab and $m = \\rho _f hA$ be the mass of the cold slab within area $A$ .","Mass continuity tells us that $\\frac{\\operatorname{d} m}{\\operatorname{d} t}$ is equal to the incoming mass flux $\\rho _f v_f$ times the area $A$ , so $ \\rho _f v_f A = \\rho _f A \\frac{\\operatorname{d} h}{\\operatorname{d} t} + Ah\\frac{\\operatorname{d} \\rho _f}{\\operatorname{d} t}$ which can be solved for $\\frac{\\operatorname{d} h}{\\operatorname{d} t}$ $ \\frac{\\operatorname{d} h}{\\operatorname{d} t} =v_f - \\frac{h}{\\rho _f}\\frac{\\operatorname{d} \\rho _f}{\\operatorname{d} t}$ .","Going forward we shall assume that $\\rho _f$ reaches a constant value once $h$ reaches an appreciable length, allowing us to neglect the $\\frac{h}{\\rho _f}\\frac{\\operatorname{d} \\rho _f}{\\operatorname{d} t} $ term; we therefore conclude that the rate of growth of the slab is equal to the velocity of fluid flow within the slab.","With no way for material to exit the slab, the slab will grow indefinitely.","We now establish a relation between the preshock and cold slab states.","Conservation of momentum requires that $\\rho _i v_i^2 + p_i = \\rho _f v_f^2 + p_f$ Using mass continuity $\\rho _i v_i = \\rho _f v_f$ gives $v_f\\left(v_i^2 + \\frac{p_i}{\\rho _i}\\right) = v_i\\left(v_f^2 + \\frac{p_f}{\\rho _f}\\right)$ Under the assumption of an isothermal shock ($\\frac{p_f}{\\rho _f} = \\frac{p_i}{\\rho _i}$ ), this has two solutions: $v_f = v_i$ (preshock), and $v_f = \\frac{p_i}{\\rho _i v_i}$ (cold slab).","For the latter solution mass continuity then gives $\\rho _f = \\frac{\\rho _i v_i}{v_f} = \\frac{\\rho _i^2 v_i^2}{p_i}.$ Since the Mach number in the shock frame is defined as $M^2 = \\frac{\\rho v_i^2}{\\gamma p_i}$ , we can express the state of the cold slab asNote that the pressure in the cold slab exceeds the immediate postshock pressure by a factor of $\\frac{\\gamma +1}{2}$ .","$ \\frac{\\rho _f}{\\rho _i} = \\frac{p_f}{p_i} = \\frac{v_i}{v_f} = \\gamma M^2.","$ We can apply this result to estimate the density of the cold slab in our simulations; we will do this for run 0.","The density of the incoming flow is $3\\times 10^{18}$ cm$^{-3}$ while the incoming pressure is $2.98\\times 10^{5}$ erg cm$^{-3}$ .","Using a mass of 1 amu, we find a sound speed of 3.15 km s$^{-1}$ , which for a jet velocity of 70 km s$^{-1}$ gives us a Mach number of 22.","This gives us a cold slab density of $2.4\\times 10^{21}$ cm$^{-3}$ , which is in good agreement with what is observed in our simulations.","Before moving on to three dimensions, we briefly discuss this problem in the “lab\" frame, which is useful to consider for a flow running into a stationary wall.","If we return to figure REF , we see that in this frame the far side of the cold slab remains stationary while the shock moves away from the wall.","We therefore can define $v_i^{\\prime } = v_i - v_f$ and $M^{\\prime } = \\frac{v_i^{\\prime }}{v_i}M$ as the preshock velocity and Mach number in this frame.","Using this, we find that equation REF becomes $ \\frac{\\rho _f}{\\rho _i} = \\frac{p_f}{p_i} = \\frac{v_i^{\\prime }}{v_f}+1 = \\gamma M^{\\prime 2}\\left(1+\\frac{v_f}{v_i^{\\prime }}\\right)^2$ the solution of which is given by $ \\frac{v_f}{v_i^{\\prime }}= \\sqrt{\\frac{1}{4}+\\frac{1}{\\gamma M^{\\prime 2}} }-\\frac{1}{2}, $ and $ \\frac{\\rho _f}{\\rho _i}= \\gamma M^{\\prime 2} \\left[\\frac{1}{2}+ \\sqrt{\\frac{1}{4}+\\frac{1}{\\gamma M^{\\prime 2}} }\\right] +1.","$ These expressions can also be derived by considering conservation laws in the lab frame, noting that the mass of any region enclosing the shock increases over time."],["Cold Slab for Cylindrical flows","We now wish to extend the model developed in the previous section to a cylindrical model in which gas is permitted to escape laterally.","Our primary aim in this section is to use mass conservation to estimate the size of the cold slab.","Although some gas also escapes from the sides of the cooling regions without ever reaching the cold slab, the cooling region accounts for just 10% of the mass of the interaction despite being around 90% of the volume.","We therefore make the approximation that the mass of the SLO originates entirely from the cold slab material.","A more accurate calculation could be done by dividing the cooling region into layers and computing the SLO for each layer.","Let $r_-$ be the radius of the incoming flow (the reason for labelling as minus will become apparent in the next section).","Assume $h$ is the height of our cylinder, and that mass flows into the cylinder from both the top and bottom and out the sides.","The outgoing mass flux is equal to $\\rho _h (2\\pi r_0 h)v_\\perp $ , so equations REF and REF become $ \\rho _f v_f (2\\pi r_-^2) -\\rho _hv_\\perp (2\\pi r_- h) = \\rho _f (\\pi r_-^2) \\frac{\\operatorname{d} h}{\\operatorname{d} t} + \\pi r_-^2h\\frac{\\operatorname{d} \\rho _f}{\\operatorname{d} t}$ $ \\frac{\\operatorname{d} h}{\\operatorname{d} t} =2v_f - \\frac{h}{\\rho _f}\\frac{\\operatorname{d} \\rho _f}{\\operatorname{d} t} - \\frac{\\rho _h (2\\pi r_- h)}{\\rho _f(\\pi r_-^2)} v_\\perp .$ In equilibrium this gives an expression for the SLO velocity $ v_\\perp =\\frac{\\rho _fv_f r_-}{\\rho _h h}.$ Figure: A plot of Mach number near the edge of the interaction region, for run 0 at time 660 ns.","A black contour is drawn at M=1M=1.Let us begin by guessing that $\\rho _h = \\rho _f$ .","We will also assume that $v_\\perp $ is equal to the sound speed (which must equal that of the incoming flows).","This is justifiable both theoretically [15] and by observation of our simulations (see figure REF ).","We therefore find $\\frac{v_i}{M} = v_f \\frac{r_-}{h}$ Since the slab has no net growth we can use equation REF for $\\frac{v_f}{v_i}$ .","Solving for $\\frac{h}{r_-}$ therefore gives us $ \\frac{h}{r_-} = \\frac{1}{\\gamma M}$ The actual ratio may be slightly smaller as a result of aforementioned small amount of gas leaking laterally out of the cooling region.","The jets in our simulations have $\\gamma = \\frac{5}{3}$ and $M\\sim 20$ so this factor should be approximately 0.03.","For a jet of radius 1.5 mm the size of the cold slab would be just 0.045 mm, which is comparable to what we observe.","We must note however that such a length corresponds to less than 4 cells on the grid at the highest level of refinement, so numerical errors are likely to be significant."],["Deflection Angle for Jets with Different Radii","Finally, we shall examine the deflection angle for SLOs produced by jets with different radii.","We will assume that the jets are of equal density and velocity.","Under these assumptions, the deflection angle can be approximated as the collision between the SLO of a system of identical jets with radius $r_-$ (the radius of the smaller jet), and an incoming flow from the larger jet between $r_-$ and $r_+$ (the radius of the larger jet).","Define angle $\\theta $ to be the angle of deflection, relative to the surface normal to the incoming flows of density $\\rho _i$ , speed, $v_i$ , and pressure $p_i$ (see figure REF ).","Define area $A_\\text{SLO}\\operatorname{d}\\varphi $ as the area of some surface which is normal to the SLO and covers the extent of the SLO through angle $\\operatorname{d}\\varphi $ about the jet axis.","Figure: A diagram discussing the geometry of the problem for determining properties of the SLO for multiple jets.Let the SLO be characterized by $\\rho _\\text{SLO}$ , $v_\\text{SLO}$ , and $p_\\text{SLO}$ ; note that the $\\text{SLO}$ subscript denotes material in the deflected SLO, while an $h$ subscript describes a horizontal SLO.","The outgoing horizontal and vertical momentum fluxes must therefore be $F_x = (\\rho _\\text{SLO}v_\\text{SLO}^2+p_\\text{SLO})A_\\text{SLO}\\cos \\theta \\operatorname{d}\\varphi $ $F_y = (\\rho _\\text{SLO}v_\\text{SLO}^2+p_\\text{SLO})A_\\text{SLO}\\sin \\theta \\operatorname{d}\\varphi $ We now express the horizontal momentum flux leaving the interaction region in terns if the incoming quantities.","The SLO from the collision within the inner radius can be characterized as having density $\\rho _h = \\rho _f = \\gamma M^2 \\rho _i$ and velocity $v_\\perp = \\sqrt{\\frac{\\gamma p_i}{\\rho _i}}$ (see equation REF and the discussion in section REF ).","The pressure $p_h$ can be estimated using conservation of energy within the cold slab (which features no cooling).","$\\left[\\frac{\\gamma p_f v_f}{\\gamma -1}+\\frac{\\rho _fv_f^3}{2}\\right](2\\pi r_-^2) = \\left[\\frac{\\gamma p_h v_\\perp }{\\gamma -1}+\\frac{\\rho _hv_\\perp ^3}{2}\\right](2\\pi r_-h)$ Note that we use $v_f$ (the state of material entering the cold slab) instead of $v_i$ as we are only considering energy flux in and out of the cold slab rather than the interaction.","Expressing $\\rho _h$ and $v_\\perp $ in terms of $\\rho _f, p_f$ , and $M$ allows us to solve for $p_h$ $p_h = \\left[\\frac{3-\\gamma }{2} +\\frac{\\gamma -1}{2\\gamma ^2M^2}\\right] p_f$ Therefore in the limit of high Mach number, the horizontal momentum flux produced by the cold slab is equal to $ F_x^h = (\\rho _h v_\\perp ^2 + p_h) (r_- h \\operatorname{d}\\varphi ) \\approx Mp_i \\left(\\frac{3+\\gamma }{2}\\right) (r_-^2 \\operatorname{d}\\varphi )$ While neglecting the cooling region is sufficient when considering mass flux, the high pressures found in the cooling region also contribute significantly to the momentum flux.","We can however continue to ignore the kinetic term $\\rho _\\text{cr} v_\\perp ^2$ as this term approaches a constant in the limit of high Mach number and thus accounts for less than one percent of the momentum flux for collisions with $M\\sim 20$ .","We shall approxmiate the pressure within the cooling region to be equal to the immediate postshock pressure; this pressure within the cooling region should exceed this, but by a factor of less than $\\frac{\\gamma +1}{2}$ (assuming pressure in the cooling region never exceeds that of the cold slab).","We therefore conclude that a lower bound for the contribution of the cooling region to the momentum flux is given by $ F_x^L = p_\\text{ps} (2r_- L \\operatorname{d}\\varphi ) = \\frac{4\\gamma M^2 \\xi }{\\gamma +1} p_i(r_-^2 \\operatorname{d}\\varphi )$ where $L$ is the (vertical) size of the cooling region and $\\xi = \\frac{L}{r_-}$ .","In [16] we showed how to calculate $L$ for a power-law cooling function as an improvement of the approximation given in [29], and a similar approach can be used for a more complex function.","While performing such a calculation is necessary if one wishes to predict the outflow angle without running a simulation, when comparing analytical predictions to simulation results we will instead take the approach of measuring $L$ from the simulation results.","In the vertical direction, the incoming jet flux is $ F_y = (\\rho _iv_i^2+p_i)\\left(\\frac{r_+^2 - r_-^2}{2} \\operatorname{d}\\varphi \\right) = (\\gamma M^2+1)p_i\\left(\\frac{r_+^2 - r_-^2}{2}\\right) \\operatorname{d}\\varphi $ Using equations REF we can deduce $\\tan \\theta = \\frac{F_y}{F_x^h+F_x^L}$ ; combining this with equations REF - REF allows us to estimate the deflection angle: $\\tan \\theta = \\left[\\frac{(\\gamma +1)(\\gamma M^2 +1)}{8\\gamma \\xi M^2 + (\\gamma +1)(\\gamma +3)M}\\right] \\frac{r_+^2 - r_-^2}{r_-^2}$ We now compare equation REF to our simulations.","Using $\\gamma =\\frac{5}{3}$ , $M=20$ , $L = 0.4$ mm, $r_- = 1.0$ mm and $r_+ = 1.5$ mm, we find $\\tan \\theta = 0.93$ or $\\theta = 43^\\circ $ ; this is confirmed by the 600 ns frame of figure REF which shows a deflection angle of $\\theta = 41^\\circ $ .","Meanwhile if we use $r_- = 1.5$ mm and $r_+ = 2.0$ mm we find $\\tan \\theta = 0.73$ or $\\theta = 39^\\circ $ , with the measured angle being $36^\\circ $ .","To further test equation REF , we ran an additional set of simulations.","These simulations were similar to run 6 (see figure REF ), but the radius of the top jet varied between values of 1.0 mm, 1.5 mm, 2.0 mm, and 2.5 mm; these runs were also conducted at lower resolution than the other runs.","Using the 1.0 mm case we estimate $\\xi = 0.3$ , and for each subsequent run we took four measurements of deflection angle, shown in magenta in figure REF .","We find that equation REF was very accurate for the 1.5 mm case, but sorely overestimates for $\\frac{r_+}{r_-} \\gtrsim 2$ .","An improved result may be obtained by considering the curvature in the region between $r_-$ and $r_+$ , as equation REF is best suited to the limit where that region is small.","Figure: (top) A set of simulations used to further test equation .","(bottom) Deflection angles vs the ratio of radii.Each red cross corresponds to a magenta line in the simulations, while the black curve is the estimated deflection."],["Conclusions","In this paper we have continued our studies of colliding flows.","While in [16] we used identical flows and an analytic cooling model to isolate the effects of instabilities, in this paper we used a more realistic cooling function and variation of flow parameters; this was done in part to bring our simulations closer to the experiments of [14] while still remaining the hydrodynamic regime.","Our simulations suggest explanations for two important features of the bow shock.","First, for flows of differing density or velocity, the interaction region (and thus a shock which bounds it) moves at a speed given by equation REF .","Second, we find that the conical shape of the bow shock arises from a mismatch in the cross-sectional radii of the flows, with the shocked lateral outflows being deflected away from the jet of larger radius.","All of the simulations in these first two papers have been conducted in the hydrodynamic limit.","While this is useful for isolating various aspects of the underlying physics, magnetic fields are likely to be significant in both laboratory and astrophysical flows, and are thus a worthwhile inclusion in future papers on the subject.","Effects relating to optical depth may also be worth exploring, particularly in connection to laboratory contexts.","This work used the computational and visualization resources in the Center for Integrated Research Computing (CIRC) at the University of Rochester.","Financial support for this project was provided in part by the Department of Energy grants GR523126, DE-SC0001063, and DE-SC0020432, the National Science Foundation grant GR506177, and the Space Telescope Science Institute grant GR528562.","Additional funding for this research was provided by the Center for Matter at Atomic Pressures (CMAP), a National Science Foundation (NSF) Physics Frontiers Center, under Award PHY-2020249.","Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation."],["Data Availability Statement","The data that support the findings of this study are available from the corresponding author upon reasonable request.","The authors have no conflicts to disclose."]],"string":"[\n [\n \"Morphology of Shocked Lateral Outflows in Colliding Hydrodynamic Flows\"\n ],\n [\n \"Abstract Supersonic interacting flows occurring in phenomena such as protostellar jets give rise to strong shocks, and have been demonstrated in several laboratory experiments.\",\n \"To study such colliding flows, we use the AstroBEAR AMR code to conduct hydrodynamic simulations in three dimensions.\",\n \"We introduce variations in the flow parameters of density, velocity, and cross sectional radius of the colliding flows %radius in order to study the propagation and conical shape of the bow shock formed by collisions between two, not necessarily symmetric, hypersonic flows.\",\n \"We find that the motion of the interaction region is driven by imbalances in ram pressure between the two flows, while the conical structure of the bow shock is a result of shocked lateral outflows (SLOs) being deflected from the horizontal when the flows are of differing cross-section.\"\n ],\n [\n \"Introduction\",\n \"Radiative shocks occur in a variety of settings, such as High Energy Density Plasma (HEDP), protostellar jets [1], supernova explosions [2], gamma ray bursts [3], and active galactic nuclei [4].\",\n \"Over the years several efforts have been made to study these phenomena in the laboratory.\",\n \"Despite large differences in physical scale, results obtained in the laboratory can be used to understand phenomena in the astrophysical setting, thanks to the use of dimensionless parameters [5], [6], [7].\",\n \"Experiments at Imperial College London have produced single flows using a pulsed-power generator to drive radial wire arrays [8] and radial foils [9], [10]; similar experiments have also been performed at Cornell University [11].\",\n \"Collimation of these flows was driven by toroidal magnetic fields in a manner consistent with the “Magnetic Tower\\\" model [12].\",\n \"Extending this work, [13] found that the presence of an ambient medium results in the formation of a shock, driven by ablation of halo plasma.\",\n \"While astronomical shocks are generally produced via time-variations in velocity, a similar effect can be obtained in a laboratory setting by creating a collision between two supersonic flows.\",\n \"[14] conducted such an experiment, images of which are shown in figure REF .\",\n \"Among the results of that study are the emergence of small scale structures behind a conical bow shock, which was observed to propagate in a downward direction.\",\n \"When two flows collide, an interaction region forms, with the jet shocks becoming the boundaries.\",\n \"This region usually consists of a cooling region, in which gas cools from post-shock temperature $T_s$ immediately behind the jet shocks to a lower temperature further behind the shock.\",\n \"After passing through the cooling region, gas reaches its final post-cooling temperature and is deposited onto the cold slab, where densities are highest (see figure REF ), some shocked material gets ejected laterally by the high pressures throughout the interaction region, producing shocked lateral outflows (SLOs) [15].\",\n \"Figure: (left) Images of a colliding flow experiment.\",\n \"Adapted from figure 2 of .\",\n \"(right) a diagram showing the structure of colliding flows; adapted from In our previous paper [16] we began a series of simulations in an effort to study colliding radiative flows like those of [14].\",\n \"In that paper we used hydrodynamic simulations with an analytic form of radiative cooling, focusing our attention on instabilities in an effort to explain the origin of small scale structures.\",\n \"While [14] had suggested that the [17] instability was a source of structure, we did not see any evidence of this mode present in the case of analytic cooling.\",\n \"Instead, long-term evolution was found to be dominated by the bending modes characteristic of the nonlinear thin shell instability (NTSI) [18], which could be triggered either by sufficiently short cooling lengths or by oscillations resulting from the radiative shock instability [19].\",\n \"In this paper, we continue to investigate the results of [14] by extending the results of [16].\",\n \"Here we shift our focus away from instabilities (which were discussed in significant depth in [16]) to a study of flow parameter variation, allowing us to better understand the conical shape and downward propagation of the bow shock and lateral outflows.\",\n \"We will continue to focus solely on hydrodynamic simulations, but the analytic cooling curve previously used will be replaced with a more realistic cooling function; this will be described in further detail in section .\",\n \"Our approach of continuing to remain in the radiative hydrodynamic case for now will eventually allow us to have a better understanding of the magnetic case by providing a reference for comparison.\",\n \"This paper is organized as follows: In Section we discuss the model system, the selected cooling function, and simulation parameters.\",\n \"In section , we present the results of the simulations.\",\n \"Section will include a discussion of motion of the interaction region, properties of the cold slab, and deflection of the SLOs.\"\n ],\n [\n \"Methods and Model\",\n \"The simulations in this study were conducted using AstroBEARhttps://astrobear.pas.rochester.edu/${ }^{,}$ [21], [22], which is a massively parallelized adaptive mesh refinement (AMR) code that includes a variety of multiphysics solvers, such as magnetic resistivity, radiative transport, self-gravity, heat conduction, and ionization dynamics.\",\n \"Our study uses only the hydro solvers with an energy source term associated with the radiative cooling.\",\n \"Thus our governing equations are; $\\\\frac{\\\\partial \\\\rho }{\\\\partial t} + {\\\\nabla } \\\\cdot \\\\rho {v} = 0$ $\\\\frac{\\\\partial \\\\rho {v}}{\\\\partial t} + {\\\\nabla } \\\\cdot \\\\left( \\\\rho {v} \\\\otimes {v} \\\\right)= - \\\\nabla p$ $\\\\frac{\\\\partial E}{\\\\partial t} + {\\\\nabla } \\\\cdot ((E + p) {v}) =-n^2\\\\Lambda (n,T)$ where $\\\\rho $ is the mass density, $n$ is the number density of nuclei, ${v}$ is the fluid velocity, $p$ is the thermal pressure, and $E = \\\\frac{1}{\\\\gamma - 1} p + \\\\frac{1}{2}\\\\rho v^2$ is the combined internal and kinetic energies.\",\n \"$n^2\\\\Lambda (n,T)$ is a cooling function, which gives the energy loss per unit volume.\",\n \"While in [16] we used an analytic cooling function of the form $\\\\Lambda (T) = \\\\alpha \\\\left(\\\\frac{T}{T_0}\\\\right)^\\\\beta $ , in this paper we use a more realistic cooling function.\",\n \"Such a cooling function introduces variation in the slope $\\\\frac{\\\\operatorname{d} \\\\log \\\\Lambda }{\\\\operatorname{d}\\\\log T}$ , which is important to both the radiative shock instability and to the Field instability.\",\n \"The specific function $\\\\Lambda (n,T)$ we used was chosen in attempt to provide correspondence with the experiments of [14] and is plotted in figure REF .\",\n \"This function was calculated using ABAKO/RAPCAL [23], [24].\",\n \"The required atomic data were obtained using FAC [25], while the resulting function was parameterized in terms of density and temperature using PARPRA [26].\",\n \"Cooling is strongest at high temperatures and lower densities; as a fluid cools, $\\\\Lambda (n,T)$ decreases until a local minimum is encountered at a temperature on the order of 10 eV.\",\n \"Cooling then increases again as temperature continues to decrease, but this effect is reduced at higher densities.\",\n \"Figure: The cooling function plotted (top) as a function of temperature, for selected densities, and (bottom) as a colour-map in terms of density and temperature.Similarly to [16], we drove two cylindrical jets, one from the top and bottom z-boundaries respectively.\",\n \"For the first run, both jets had densities of $3.0\\\\times 10^{18}$ particles per cm$^3$ , speeds of 70.0 km s$^{-1}$ , temperature of 720 K, and cross sectional radii of 1.5 mm.\",\n \"The ambient medium density was $1\\\\times 10^{18}$ particles per cm$^3$ at a temperature of 4320 K. Extrapolated boundary conditions were used in all directions.\",\n \"Runs were conducted in a 6.4mm cubic space, and consisted of the reference run described above and nine other runs in which jet parameters were varied as follows (summarised in table REF ).\",\n \"For the first two runs, we changed the density of the bottom jet to $2.0\\\\times 10^{18}$ and $1.0\\\\times 10^{18}$ particles per cm$^3$ .\",\n \"For third and fourth runs, the radius of the bottom jet was varied to 2.0 and 1.0 mm, and the density was varied to $1.69\\\\times 10^{18}$ and $6.75\\\\times 10^{18}$ particles per cm$^3$ such that the mass of the jets remained unaltered.\",\n \"For the fifth and sixth runs, the radius of the bottom jet was varied to 2.0 and 1.0 mm, but this time the density of the jet was reverted to the original value.\",\n \"Finally, the seventh and eighth runs varied velocities of the top and bottom jets to 80 and 60 km s$^{-1}$ respectively; these values were chosen to match the experiments in [14].\",\n \"The seventh run retained the densities of the reference run, while the eighth run used densities of $2.63\\\\times 10^{18}$ and $3.50\\\\times 10^{18}$ so that the jets were of equal momentum density, and the ninth run used densities of $2.30\\\\times 10^{18}$ and $4.08\\\\times 10^{18}$ so that the jets were of equal momentum flux.\",\n \"Table: Densities, velocities, and radii of jets in the first set of runs.\",\n \"For each pair of numbers, the first describes the top jet while the second describes the bottom jet.\"\n ],\n [\n \"Results\",\n \"We will first examine the reference run (run 0 in table REF ), shown in figure REF .\",\n \"This run shows a collision between two identical jets, similar to those studied in [16] but now using the complex cooling curve described in section .\",\n \"As gas crosses the jet shock and enters the interaction region, the density increases from $3\\\\times 10^{18}$ cm$^{-3}$ to $1.2\\\\times 10^{19}$ cm$^{-3}$ .\",\n \"The density varies as the gas cools but remains at the same order of magnitude throughout the cooling region, but is seen to be as high as $2.5\\\\times 10^{21}$ cm$^{-3}$ in the cold slab.\",\n \"As gas exits the interaction region through the sides and into the shocked lateral outflows, densities drop to a value of order $1\\\\times 10^{18}$ cm$^{-3}$ and lower as gas begins to mix into the ambient medium.\",\n \"At later times clumps are observed to form in the cold slab, though the origin of these is uncertain given that the cold slab is not well resolved at just a few cells ($10^{-3}$ cm) in width.\",\n \"Since the jets are identical, the interaction region remains stationary and shows two identical jet shocks; while the shocks exhibit some curvature, no conical structure is observed.\",\n \"The shocked lateral outflows flow primarily horizontally even though some material leaks from the top and bottom of the SLOs.\",\n \"Figure: From left to right, runs 0, 1, and 2 from table .\",\n \"For these runs, the density of the top jet remains fixed while the density of the bottom jet decreases with each successive run.Next, we examine cases in which the density of the bottom jet is varied, while the density of the top jet is fixed at $1.0\\\\times 10^{18}$ particles per cm$^3$ .\",\n \"Figure REF shows jets with densities $3.0\\\\times 10^{18}, 2.0\\\\times 10^{18}$ , and $1.0\\\\times 10^{18}$ particles per cm$^3$ .\",\n \"As the density of the jets becomes imbalanced, the interaction region no longer remains in place, with the velocity of the interaction region in this case being proportional to the density imbalance.\",\n \"Lateral outflows are seen to be nearly horizontal.\",\n \"Clump formation in the cold slab begins much sooner in collisions with a density imbalance; while the details of smaller structures are beyond the scope of this paper, we conjecture that this is the result of asymmetry exacerbating the formation of perturbations within the cold slab.\",\n \"Increasing the imbalance further does not significantly change the onset of clump formation.\",\n \"These clumps merge with each other as time progresses, though the regions of cold slab between these clumps is once again not well resolved.\",\n \"Study of the the long term behaviour of these clumps is also limited by the interaction region hitting the wall after a relatively short amount of time.\",\n \"Figure: From left to right, runs 3, 0, and 4 from table .\",\n \"For these runs, the density and radius of the top jet remains fixed while the bottom jet both increases in density and decreases in radius with each successive run.Next, we examine cases in which the density of the bottom jet is again varied, this time varying the radius of the bottom jet as $r \\\\propto \\\\rho ^{-\\\\frac{1}{2}}$ so that the total mass flux through a horizontal slice remains fixed.\",\n \"Figure REF shows jets with densities $1.69\\\\times 10^{18}, 3.0\\\\times 10^{18}$ , and $6.75\\\\times 10^{18}$ particles per cm$^3$ and radii 2.0, 1.5, and 1.0 mm.\",\n \"The jet with the smaller radius, having a higher density, drives the motion of the interaction region.\",\n \"As with the previous set, an imbalance of density also results in formation of clumps within the cold slab at earlier times, with these clumps again merging into fewer but larger clumps as time progresses.\",\n \"Since the jets are of differing radii, the SLOs are no longer horizontal, instead bending away from the jet of larger radius.\",\n \"Figure: From left to right, runs 5, 0, and 6 from table .\",\n \"For these runs, the radius of the top jet remains fixed while the radius of the bottom jet decreases with each successive run.We next examine cases of equal speed jets for which the radius of the bottom jet is varied at fixed density.\",\n \"Figure REF shows jets with radii 2.0, 1.5, and 1.0 mm.\",\n \"For equal velocities and equal opposing jet densities, ram pressure is balanced and the interaction region remains at a fixed location.\",\n \"As with density imbalances, an imbalance of radius alone appears to be sufficient to promote earlier clump formation, though in the case with a 1.5 mm jet colliding with a 1.0 mm jet the merging of clumps appears to be inhibited.\",\n \"Since the speeds and densities are balanced and the interaction region never approaches a wall, these cases can be observed for longer times.\",\n \"After material ejected in the initial collision flies away, a lateral outflow remains.\",\n \"When the jets differ in size, the direction of the SLO occurs at an angle to the horizontal plane and points away from the larger jet.\",\n \"The exact angle increases with the ratio $\\\\frac{(r_+^2 - r_-^2)}{r_-}$ , where $r_+$ and $r_-$ are the radii of the larger and smaller jet respectively, and is explored in section REF .\",\n \"The interaction region also exhibits curvature, bending away from the larger flow.\",\n \"Figure: From left to right, runs 7, 8, 9 from table .\",\n \"While the previous runs use v=70v = 70 km s -1 ^{-1} for both jets, these runs use jet velocities of 80 and 60 km s -1 ^{-1} for the top and bottom jets respectively.\",\n \"The density of the top jet decreases (and the bottom jet increases) with each successive run.Figure: Run 9 from table , at intervals of 200 ns.\",\n \"Density is plotted in the left column, while the right column shows a plot of temperature.\",\n \"At later times the Nonlinear Thin Shell Instability can be observed.Lastly, we examine effects of changing the jet velocity.\",\n \"While previous scenarios used a velocity of 70 km s$^{-1}$ , the final set of runs uses velocities of 80 km s$^{-1}$ for the top jet and 60 km s$^{-1}$ for the bottom jet.\",\n \"Figure REF shows three simulations with these velocities.\",\n \"The first case shows two jets of equal density; here the imbalance of velocity results in the interaction region moving with a velocity of 0 km s$^{-1}$ after collision.\",\n \"The second case shows two jets of equal momentum density (and thus equal mass flux); the velocity of the interaction region is significantly reduced, but motion still occurs as a result of ram pressure imbalance.\",\n \"Both of these runs exhibit clumping behaviour similar to that of the runs with a density imbalance.\",\n \"The final case shows two jets of equal ram pressure.\",\n \"The interaction region now remains stationary.\",\n \"While clumping behaviour is initially similar to the other two cases, clump mergers in this case produce a full cold slab spanning the entire radius of the jet.\",\n \"At later times (figure REF ), this cold slab is broken apart by the nonlinear thin shell instability, similar to the cases examined in [16].\"\n ],\n [\n \"Motion of the Interaction Region\",\n \"As seen in figures REF , REF , and REF , the interaction region has a nonzero velocity unless the ram pressure $\\\\rho v^2$ is equal for both the top and bottom jet.\",\n \"Otherwise, it moves away from the jet with higher ram pressure with velocity $V$ .\",\n \"In a reference frame comoving with $V$ , the momentum flux entering the interaction region from one jet must balance the momentum flux entering from the other jet.\",\n \"This constraint can be expressed as $ \\\\rho _1 \\\\left( v_1 - V \\\\right)^2 + p_1= \\\\rho _2 \\\\left( v_2 - V \\\\right)^2 + p_2$ where positive velocities indicate the downward direction; $v_2$ is therefore negative.\",\n \"We shall also neglect the thermal pressure difference $p_1-p_2$ .\",\n \"The requirement that both jets flow into the interaction region produces the constraint $v_2 < V < v_1$ .\",\n \"Taking the square root of equation REF , subject to this constraint gives us $\\\\sqrt{\\\\rho _1}\\\\left( v_1 - V \\\\right) = \\\\sqrt{\\\\rho _2} \\\\left(V - v_2 \\\\right)$ which can be solved for $V$ : $ V = \\\\frac{ v_1 \\\\sqrt{\\\\rho _1} + v_2 \\\\sqrt{\\\\rho _2} }{\\\\sqrt{\\\\rho _1}+\\\\sqrt{\\\\rho _2}}$ We will now examine this result in the context of figures REF -REF .\",\n \"Velocities can be measured as follows: the location of the cold slab, lower jet shock, and upper jet shock are measured at $t=130$ ns and $t=230$ ns.\",\n \"The positions at each time step are averaged; taking the difference of these averages between the two time steps gives us the displacement (positive is down) over the 100 ns time interval.\",\n \"For runs 7-9, a similar procedure is used except that the measurements are taken at $t=140$ ns and $t=280$ ns.\",\n \"Results are given in table REF .\",\n \"Table: Densities and velocities of the jets in each run, along with the estimated (equation ) and observed value for the interaction region velocity VV.\",\n \"All velocities are given in km s -1 ^{-1} and densities are given in 10 18 10^{18} cm -3 ^{-3}We can connect this prediction to the experiments of [14], in which the jet velocities were measured to be $v_1 = 80\\\\pm 10$ km s${ }^{-1}$ and $v_2 = -60\\\\pm 10$ km s${ }^{-1}$ .\",\n \"Substantial uncertainties ($\\\\sim 60\\\\%$ ) were noted for measurements of density, but in any case the ratio $\\\\frac{\\\\rho _1}{\\\\rho _2}$ was within an order of magnitude of unity.\",\n \"If equation REF is instead solved for $\\\\frac{\\\\rho _1}{\\\\rho _2}$ we find $\\\\frac{\\\\rho _1}{\\\\rho _2} = \\\\left(\\\\frac{v_2-V}{v_1-V}\\\\right)^2$ The bow shock velocity is stated to be $40\\\\pm 10$ km s${ }^{-1}$ , so the density ratio evaluates to $\\\\frac{\\\\rho _1}{\\\\rho _2} = 6.25 \\\\pm 4.38$ ; this on the higher end of what is in agreement with the measurements presented in [14].\",\n \"Another useful quantity to compute for collisions between flows of unequal ram pressure is the Mach number in the centre-of-mass frame.\",\n \"In this paper we define Mach number to be $M^2 = \\\\frac{\\\\rho v^2}{\\\\gamma p}$ , and $M_1$ and $M_2$ to be the Mach numbers for top and bottom flows in the laboratory frame.\",\n \"When combined with equation REF this definition gives us $M^2 = \\\\frac{\\\\rho _1(v_1-V)^2}{\\\\gamma p_1} = \\\\frac{\\\\rho _1}{\\\\gamma p_1}\\\\left(v_1-\\\\frac{v_1\\\\sqrt{\\\\rho _1}+v_2\\\\sqrt{\\\\rho _2}}{\\\\sqrt{\\\\rho _1}+\\\\sqrt{\\\\rho _2}}\\\\right)^2$ for the centre-of-mass frame (note that Mach numbers must be equal when ram pressures balance).\",\n \"This simplifies to $M^2 = \\\\frac{\\\\rho _1\\\\rho _2}{\\\\gamma p_i}\\\\left(\\\\frac{v_1-v_2}{\\\\sqrt{\\\\rho _1}+\\\\sqrt{\\\\rho _2}}\\\\right)^2$ which, assuming $v_2 < 0 < v_1$ , can be further simplified to $ M = \\\\frac{M_1\\\\sqrt{\\\\rho _2}+M_2\\\\sqrt{\\\\rho _1}}{\\\\sqrt{\\\\rho _2}+\\\\sqrt{\\\\rho _1}}$ Note that while $v_2$ is assumed to be negative, we always define the Mach number to be positive.\",\n \"Note that while the interaction region velocity is an average weighed by the square roots of densities, the Mach number is an average weighted by the square roots of inverse densities.\"\n ],\n [\n \"Cold Slab in One Dimension\",\n \"We now begin our examination of shocked lateral outflow by considering a scenario with no outflow: the growth of the cold slab for a flow colliding with a wall in one dimension.\",\n \"In subsequent sections, these results will be useful in the development of a three-dimensional model.\",\n \"Define $\\\\rho _i, v_i$ , and $T_i$ to be the initial pre-shock density, velocity, and temperature; also define $\\\\rho _f, v_f,$ and $T_f$ to be the same values for material in the cold slab.\",\n \"Velocities are specified in a “shock\\\" frame (see figure REF ), in which the front (i.e.\",\n \"the side closer to the shock) of the cold slab remains stationarywe assume that the radiative shock instability can be ignored, so that the distance between the shock and the front of the cold slab remains fixed.\",\n \"Any growth of the slab appears in this frame as the far side of the slab moving further away from the shock.\",\n \"Figure: Temperature (red) and Density (blue) for a one dimensional shock in a) the shock frame and b) the lab frame.In the shock frame, the shock remains stationary while material in the cold slab flows away from the shock.\",\n \"In the lab frame, the shock moves to the left (dashed curve) while material already in the cold slab remains stationary.Let $h$ be the height of the cold slab and $m = \\\\rho _f hA$ be the mass of the cold slab within area $A$ .\",\n \"Mass continuity tells us that $\\\\frac{\\\\operatorname{d} m}{\\\\operatorname{d} t}$ is equal to the incoming mass flux $\\\\rho _f v_f$ times the area $A$ , so $ \\\\rho _f v_f A = \\\\rho _f A \\\\frac{\\\\operatorname{d} h}{\\\\operatorname{d} t} + Ah\\\\frac{\\\\operatorname{d} \\\\rho _f}{\\\\operatorname{d} t}$ which can be solved for $\\\\frac{\\\\operatorname{d} h}{\\\\operatorname{d} t}$ $ \\\\frac{\\\\operatorname{d} h}{\\\\operatorname{d} t} =v_f - \\\\frac{h}{\\\\rho _f}\\\\frac{\\\\operatorname{d} \\\\rho _f}{\\\\operatorname{d} t}$ .\",\n \"Going forward we shall assume that $\\\\rho _f$ reaches a constant value once $h$ reaches an appreciable length, allowing us to neglect the $\\\\frac{h}{\\\\rho _f}\\\\frac{\\\\operatorname{d} \\\\rho _f}{\\\\operatorname{d} t} $ term; we therefore conclude that the rate of growth of the slab is equal to the velocity of fluid flow within the slab.\",\n \"With no way for material to exit the slab, the slab will grow indefinitely.\",\n \"We now establish a relation between the preshock and cold slab states.\",\n \"Conservation of momentum requires that $\\\\rho _i v_i^2 + p_i = \\\\rho _f v_f^2 + p_f$ Using mass continuity $\\\\rho _i v_i = \\\\rho _f v_f$ gives $v_f\\\\left(v_i^2 + \\\\frac{p_i}{\\\\rho _i}\\\\right) = v_i\\\\left(v_f^2 + \\\\frac{p_f}{\\\\rho _f}\\\\right)$ Under the assumption of an isothermal shock ($\\\\frac{p_f}{\\\\rho _f} = \\\\frac{p_i}{\\\\rho _i}$ ), this has two solutions: $v_f = v_i$ (preshock), and $v_f = \\\\frac{p_i}{\\\\rho _i v_i}$ (cold slab).\",\n \"For the latter solution mass continuity then gives $\\\\rho _f = \\\\frac{\\\\rho _i v_i}{v_f} = \\\\frac{\\\\rho _i^2 v_i^2}{p_i}.$ Since the Mach number in the shock frame is defined as $M^2 = \\\\frac{\\\\rho v_i^2}{\\\\gamma p_i}$ , we can express the state of the cold slab asNote that the pressure in the cold slab exceeds the immediate postshock pressure by a factor of $\\\\frac{\\\\gamma +1}{2}$ .\",\n \"$ \\\\frac{\\\\rho _f}{\\\\rho _i} = \\\\frac{p_f}{p_i} = \\\\frac{v_i}{v_f} = \\\\gamma M^2.\",\n \"$ We can apply this result to estimate the density of the cold slab in our simulations; we will do this for run 0.\",\n \"The density of the incoming flow is $3\\\\times 10^{18}$ cm$^{-3}$ while the incoming pressure is $2.98\\\\times 10^{5}$ erg cm$^{-3}$ .\",\n \"Using a mass of 1 amu, we find a sound speed of 3.15 km s$^{-1}$ , which for a jet velocity of 70 km s$^{-1}$ gives us a Mach number of 22.\",\n \"This gives us a cold slab density of $2.4\\\\times 10^{21}$ cm$^{-3}$ , which is in good agreement with what is observed in our simulations.\",\n \"Before moving on to three dimensions, we briefly discuss this problem in the “lab\\\" frame, which is useful to consider for a flow running into a stationary wall.\",\n \"If we return to figure REF , we see that in this frame the far side of the cold slab remains stationary while the shock moves away from the wall.\",\n \"We therefore can define $v_i^{\\\\prime } = v_i - v_f$ and $M^{\\\\prime } = \\\\frac{v_i^{\\\\prime }}{v_i}M$ as the preshock velocity and Mach number in this frame.\",\n \"Using this, we find that equation REF becomes $ \\\\frac{\\\\rho _f}{\\\\rho _i} = \\\\frac{p_f}{p_i} = \\\\frac{v_i^{\\\\prime }}{v_f}+1 = \\\\gamma M^{\\\\prime 2}\\\\left(1+\\\\frac{v_f}{v_i^{\\\\prime }}\\\\right)^2$ the solution of which is given by $ \\\\frac{v_f}{v_i^{\\\\prime }}= \\\\sqrt{\\\\frac{1}{4}+\\\\frac{1}{\\\\gamma M^{\\\\prime 2}} }-\\\\frac{1}{2}, $ and $ \\\\frac{\\\\rho _f}{\\\\rho _i}= \\\\gamma M^{\\\\prime 2} \\\\left[\\\\frac{1}{2}+ \\\\sqrt{\\\\frac{1}{4}+\\\\frac{1}{\\\\gamma M^{\\\\prime 2}} }\\\\right] +1.\",\n \"$ These expressions can also be derived by considering conservation laws in the lab frame, noting that the mass of any region enclosing the shock increases over time.\"\n ],\n [\n \"Cold Slab for Cylindrical flows\",\n \"We now wish to extend the model developed in the previous section to a cylindrical model in which gas is permitted to escape laterally.\",\n \"Our primary aim in this section is to use mass conservation to estimate the size of the cold slab.\",\n \"Although some gas also escapes from the sides of the cooling regions without ever reaching the cold slab, the cooling region accounts for just 10% of the mass of the interaction despite being around 90% of the volume.\",\n \"We therefore make the approximation that the mass of the SLO originates entirely from the cold slab material.\",\n \"A more accurate calculation could be done by dividing the cooling region into layers and computing the SLO for each layer.\",\n \"Let $r_-$ be the radius of the incoming flow (the reason for labelling as minus will become apparent in the next section).\",\n \"Assume $h$ is the height of our cylinder, and that mass flows into the cylinder from both the top and bottom and out the sides.\",\n \"The outgoing mass flux is equal to $\\\\rho _h (2\\\\pi r_0 h)v_\\\\perp $ , so equations REF and REF become $ \\\\rho _f v_f (2\\\\pi r_-^2) -\\\\rho _hv_\\\\perp (2\\\\pi r_- h) = \\\\rho _f (\\\\pi r_-^2) \\\\frac{\\\\operatorname{d} h}{\\\\operatorname{d} t} + \\\\pi r_-^2h\\\\frac{\\\\operatorname{d} \\\\rho _f}{\\\\operatorname{d} t}$ $ \\\\frac{\\\\operatorname{d} h}{\\\\operatorname{d} t} =2v_f - \\\\frac{h}{\\\\rho _f}\\\\frac{\\\\operatorname{d} \\\\rho _f}{\\\\operatorname{d} t} - \\\\frac{\\\\rho _h (2\\\\pi r_- h)}{\\\\rho _f(\\\\pi r_-^2)} v_\\\\perp .$ In equilibrium this gives an expression for the SLO velocity $ v_\\\\perp =\\\\frac{\\\\rho _fv_f r_-}{\\\\rho _h h}.$ Figure: A plot of Mach number near the edge of the interaction region, for run 0 at time 660 ns.\",\n \"A black contour is drawn at M=1M=1.Let us begin by guessing that $\\\\rho _h = \\\\rho _f$ .\",\n \"We will also assume that $v_\\\\perp $ is equal to the sound speed (which must equal that of the incoming flows).\",\n \"This is justifiable both theoretically [15] and by observation of our simulations (see figure REF ).\",\n \"We therefore find $\\\\frac{v_i}{M} = v_f \\\\frac{r_-}{h}$ Since the slab has no net growth we can use equation REF for $\\\\frac{v_f}{v_i}$ .\",\n \"Solving for $\\\\frac{h}{r_-}$ therefore gives us $ \\\\frac{h}{r_-} = \\\\frac{1}{\\\\gamma M}$ The actual ratio may be slightly smaller as a result of aforementioned small amount of gas leaking laterally out of the cooling region.\",\n \"The jets in our simulations have $\\\\gamma = \\\\frac{5}{3}$ and $M\\\\sim 20$ so this factor should be approximately 0.03.\",\n \"For a jet of radius 1.5 mm the size of the cold slab would be just 0.045 mm, which is comparable to what we observe.\",\n \"We must note however that such a length corresponds to less than 4 cells on the grid at the highest level of refinement, so numerical errors are likely to be significant.\"\n ],\n [\n \"Deflection Angle for Jets with Different Radii\",\n \"Finally, we shall examine the deflection angle for SLOs produced by jets with different radii.\",\n \"We will assume that the jets are of equal density and velocity.\",\n \"Under these assumptions, the deflection angle can be approximated as the collision between the SLO of a system of identical jets with radius $r_-$ (the radius of the smaller jet), and an incoming flow from the larger jet between $r_-$ and $r_+$ (the radius of the larger jet).\",\n \"Define angle $\\\\theta $ to be the angle of deflection, relative to the surface normal to the incoming flows of density $\\\\rho _i$ , speed, $v_i$ , and pressure $p_i$ (see figure REF ).\",\n \"Define area $A_\\\\text{SLO}\\\\operatorname{d}\\\\varphi $ as the area of some surface which is normal to the SLO and covers the extent of the SLO through angle $\\\\operatorname{d}\\\\varphi $ about the jet axis.\",\n \"Figure: A diagram discussing the geometry of the problem for determining properties of the SLO for multiple jets.Let the SLO be characterized by $\\\\rho _\\\\text{SLO}$ , $v_\\\\text{SLO}$ , and $p_\\\\text{SLO}$ ; note that the $\\\\text{SLO}$ subscript denotes material in the deflected SLO, while an $h$ subscript describes a horizontal SLO.\",\n \"The outgoing horizontal and vertical momentum fluxes must therefore be $F_x = (\\\\rho _\\\\text{SLO}v_\\\\text{SLO}^2+p_\\\\text{SLO})A_\\\\text{SLO}\\\\cos \\\\theta \\\\operatorname{d}\\\\varphi $ $F_y = (\\\\rho _\\\\text{SLO}v_\\\\text{SLO}^2+p_\\\\text{SLO})A_\\\\text{SLO}\\\\sin \\\\theta \\\\operatorname{d}\\\\varphi $ We now express the horizontal momentum flux leaving the interaction region in terns if the incoming quantities.\",\n \"The SLO from the collision within the inner radius can be characterized as having density $\\\\rho _h = \\\\rho _f = \\\\gamma M^2 \\\\rho _i$ and velocity $v_\\\\perp = \\\\sqrt{\\\\frac{\\\\gamma p_i}{\\\\rho _i}}$ (see equation REF and the discussion in section REF ).\",\n \"The pressure $p_h$ can be estimated using conservation of energy within the cold slab (which features no cooling).\",\n \"$\\\\left[\\\\frac{\\\\gamma p_f v_f}{\\\\gamma -1}+\\\\frac{\\\\rho _fv_f^3}{2}\\\\right](2\\\\pi r_-^2) = \\\\left[\\\\frac{\\\\gamma p_h v_\\\\perp }{\\\\gamma -1}+\\\\frac{\\\\rho _hv_\\\\perp ^3}{2}\\\\right](2\\\\pi r_-h)$ Note that we use $v_f$ (the state of material entering the cold slab) instead of $v_i$ as we are only considering energy flux in and out of the cold slab rather than the interaction.\",\n \"Expressing $\\\\rho _h$ and $v_\\\\perp $ in terms of $\\\\rho _f, p_f$ , and $M$ allows us to solve for $p_h$ $p_h = \\\\left[\\\\frac{3-\\\\gamma }{2} +\\\\frac{\\\\gamma -1}{2\\\\gamma ^2M^2}\\\\right] p_f$ Therefore in the limit of high Mach number, the horizontal momentum flux produced by the cold slab is equal to $ F_x^h = (\\\\rho _h v_\\\\perp ^2 + p_h) (r_- h \\\\operatorname{d}\\\\varphi ) \\\\approx Mp_i \\\\left(\\\\frac{3+\\\\gamma }{2}\\\\right) (r_-^2 \\\\operatorname{d}\\\\varphi )$ While neglecting the cooling region is sufficient when considering mass flux, the high pressures found in the cooling region also contribute significantly to the momentum flux.\",\n \"We can however continue to ignore the kinetic term $\\\\rho _\\\\text{cr} v_\\\\perp ^2$ as this term approaches a constant in the limit of high Mach number and thus accounts for less than one percent of the momentum flux for collisions with $M\\\\sim 20$ .\",\n \"We shall approxmiate the pressure within the cooling region to be equal to the immediate postshock pressure; this pressure within the cooling region should exceed this, but by a factor of less than $\\\\frac{\\\\gamma +1}{2}$ (assuming pressure in the cooling region never exceeds that of the cold slab).\",\n \"We therefore conclude that a lower bound for the contribution of the cooling region to the momentum flux is given by $ F_x^L = p_\\\\text{ps} (2r_- L \\\\operatorname{d}\\\\varphi ) = \\\\frac{4\\\\gamma M^2 \\\\xi }{\\\\gamma +1} p_i(r_-^2 \\\\operatorname{d}\\\\varphi )$ where $L$ is the (vertical) size of the cooling region and $\\\\xi = \\\\frac{L}{r_-}$ .\",\n \"In [16] we showed how to calculate $L$ for a power-law cooling function as an improvement of the approximation given in [29], and a similar approach can be used for a more complex function.\",\n \"While performing such a calculation is necessary if one wishes to predict the outflow angle without running a simulation, when comparing analytical predictions to simulation results we will instead take the approach of measuring $L$ from the simulation results.\",\n \"In the vertical direction, the incoming jet flux is $ F_y = (\\\\rho _iv_i^2+p_i)\\\\left(\\\\frac{r_+^2 - r_-^2}{2} \\\\operatorname{d}\\\\varphi \\\\right) = (\\\\gamma M^2+1)p_i\\\\left(\\\\frac{r_+^2 - r_-^2}{2}\\\\right) \\\\operatorname{d}\\\\varphi $ Using equations REF we can deduce $\\\\tan \\\\theta = \\\\frac{F_y}{F_x^h+F_x^L}$ ; combining this with equations REF - REF allows us to estimate the deflection angle: $\\\\tan \\\\theta = \\\\left[\\\\frac{(\\\\gamma +1)(\\\\gamma M^2 +1)}{8\\\\gamma \\\\xi M^2 + (\\\\gamma +1)(\\\\gamma +3)M}\\\\right] \\\\frac{r_+^2 - r_-^2}{r_-^2}$ We now compare equation REF to our simulations.\",\n \"Using $\\\\gamma =\\\\frac{5}{3}$ , $M=20$ , $L = 0.4$ mm, $r_- = 1.0$ mm and $r_+ = 1.5$ mm, we find $\\\\tan \\\\theta = 0.93$ or $\\\\theta = 43^\\\\circ $ ; this is confirmed by the 600 ns frame of figure REF which shows a deflection angle of $\\\\theta = 41^\\\\circ $ .\",\n \"Meanwhile if we use $r_- = 1.5$ mm and $r_+ = 2.0$ mm we find $\\\\tan \\\\theta = 0.73$ or $\\\\theta = 39^\\\\circ $ , with the measured angle being $36^\\\\circ $ .\",\n \"To further test equation REF , we ran an additional set of simulations.\",\n \"These simulations were similar to run 6 (see figure REF ), but the radius of the top jet varied between values of 1.0 mm, 1.5 mm, 2.0 mm, and 2.5 mm; these runs were also conducted at lower resolution than the other runs.\",\n \"Using the 1.0 mm case we estimate $\\\\xi = 0.3$ , and for each subsequent run we took four measurements of deflection angle, shown in magenta in figure REF .\",\n \"We find that equation REF was very accurate for the 1.5 mm case, but sorely overestimates for $\\\\frac{r_+}{r_-} \\\\gtrsim 2$ .\",\n \"An improved result may be obtained by considering the curvature in the region between $r_-$ and $r_+$ , as equation REF is best suited to the limit where that region is small.\",\n \"Figure: (top) A set of simulations used to further test equation .\",\n \"(bottom) Deflection angles vs the ratio of radii.Each red cross corresponds to a magenta line in the simulations, while the black curve is the estimated deflection.\"\n ],\n [\n \"Conclusions\",\n \"In this paper we have continued our studies of colliding flows.\",\n \"While in [16] we used identical flows and an analytic cooling model to isolate the effects of instabilities, in this paper we used a more realistic cooling function and variation of flow parameters; this was done in part to bring our simulations closer to the experiments of [14] while still remaining the hydrodynamic regime.\",\n \"Our simulations suggest explanations for two important features of the bow shock.\",\n \"First, for flows of differing density or velocity, the interaction region (and thus a shock which bounds it) moves at a speed given by equation REF .\",\n \"Second, we find that the conical shape of the bow shock arises from a mismatch in the cross-sectional radii of the flows, with the shocked lateral outflows being deflected away from the jet of larger radius.\",\n \"All of the simulations in these first two papers have been conducted in the hydrodynamic limit.\",\n \"While this is useful for isolating various aspects of the underlying physics, magnetic fields are likely to be significant in both laboratory and astrophysical flows, and are thus a worthwhile inclusion in future papers on the subject.\",\n \"Effects relating to optical depth may also be worth exploring, particularly in connection to laboratory contexts.\",\n \"This work used the computational and visualization resources in the Center for Integrated Research Computing (CIRC) at the University of Rochester.\",\n \"Financial support for this project was provided in part by the Department of Energy grants GR523126, DE-SC0001063, and DE-SC0020432, the National Science Foundation grant GR506177, and the Space Telescope Science Institute grant GR528562.\",\n \"Additional funding for this research was provided by the Center for Matter at Atomic Pressures (CMAP), a National Science Foundation (NSF) Physics Frontiers Center, under Award PHY-2020249.\",\n \"Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation.\"\n ],\n [\n \"Data Availability Statement\",\n \"The data that support the findings of this study are available from the corresponding author upon reasonable request.\",\n \"The authors have no conflicts to disclose.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"2212.05631"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":1867602,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3MjczNTM3OSwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzcyNzM4OTc5LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.-5hTXB7Yuuef_kzVODUGc8Pr8NbHlcbSrGH-IVRHUwhIvbpA-6OCpqzrFqKIOoaKxhNNyd-ALhnmumWhdQifDA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":1867602,"numBytesParquet":44989201432}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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[
"The overtone level spacing of a black hole quasinormal frequencies: a\n fingerprint of a local $SL(2,\\mathbb{R})$ symmetry"
],
[
"Abstract The imaginary part of the quasinormal frequencies spectrum for a static and spherically symmetric black hole is analytically known to be equally spaced, both f... | 2212.05538 |
[
[
"Extending TrOCR for Text Localization-Free OCR of Full-Page Scanned\n Receipt Images"
],
[
"Abstract Digitization of scanned receipts aims to extract text from receipt images and save it into structured documents.",
"This is usually split into two sub-tasks: text localization and optical charac... | 2212.05525 |
[
[
"Crossover from exciton polarons to trions in doped two-dimensional\n semiconductors at finite temperature"
],
[
"Abstract We study systematically the role of temperature in the optical response of doped two-dimensional semiconductors.",
"By making use of a finite-temperature Fermi-polaron theor... | 2212.05635 |
[
[
"Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature\n Equation Outside Symmetry"
],
[
"Abstract We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hype... | 2212.05620 |
[
[
"The Critical and Cocritical Degrees of a Totally Acyclic Complex over a\n Complete Intersection"
],
[
"Abstract It is widely known that the minimal free resolution of a module over a complete intersection ring has nice patterns eventually arising in its Betti sequence.",
"In 1997, Avramov, Gash... | 2212.05567 |
[
[
"Human-Robot Team Performance Compared to Full Robot Autonomy in 16\n Real-World Search and Rescue Missions: Adaptation of the DARPA Subterranean\n Challenge"
],
[
"Abstract Human operators in human-robot teams are commonly perceived to be critical for mission success.",
"To explore the direct ... | 2212.05626 |
[
[
"Energy-based General Sequential Episodic Memory Networks at the\n Adiabatic Limit"
],
[
"Abstract The General Associative Memory Model (GAMM) has a constant state-dependant energy surface that leads the output dynamics to fixed points, retrieving single memories from a collection of memories that c... | 2212.05563 |
[
[
"Bayesian inversion with {\\alpha}-stable priors"
],
[
"Abstract We propose to use L\\'evy {\\alpha}-stable distributions for constructing priors for Bayesian inverse problems.",
"The construction is based on Markov fields with stable-distributed increments.",
"Special cases include the Cauch... | 2212.05555 |
[
[
"A Study of Slang Representation Methods"
],
[
"Abstract Warning: this paper contains content that may be offensive or upsetting.",
"Considering the large amount of content created online by the minute, slang-aware automatic tools are critically needed to promote social good, and assist policymak... | 2212.05613 |
[
[
"Finding two-level systems in glasses through machine learning"
],
[
"Abstract Two-level systems (TLS) are rare quantum tunneling defects which govern the physics of glasses at very low temperature.",
"Because of their extremely low density, it is very hard to directly identify them in computer s... | 2212.05582 |
[
[
"A Euclidean comparison theory for the size of sets"
],
[
"Abstract We discuss two main ways in comparing and evaluating the size of sets: the \"Cantorian\" way, grounded on the so called Hume principle (two sets have equal size if they are equipotent), and the \"Euclidean\" way, maintaining Euclid's... | 2212.05527 |
[
[
"Gaussian random projections of convex cones: approximate kinematic\n formulae and applications"
],
[
"Abstract Understanding the stochastic behavior of random projections of geometric sets constitutes a fundamental problem in high dimension probability that finds wide applications in diverse fields... | 2212.05545 |
[
[
"PromptCAL: Contrastive Affinity Learning via Auxiliary Prompts for\n Generalized Novel Category Discovery"
],
[
"Abstract Although existing semi-supervised learning models achieve remarkable success in learning with unannotated in-distribution data, they mostly fail to learn on unlabeled data sampl... | 2212.05590 |
[
[
"Dynamical fluctuations in the Riesz gas"
],
[
"Abstract We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the $|x-y|^{-s}$ Riesz potential, causing the over-damped motion of particles.",
"We investigate fluctuations of the integra... | 2212.05583 |
[
[
"Detecting Code Injections in Noisy Environments Through EM Signal\n Analysis and SVD Denoising"
],
[
"Abstract The penetration of embedded devices in networks that support critical applications has rendered them a lucrative target for attackers and evildoers.",
"However, traditional protection ... | 2212.05643 |
[
[
"On a sum of a multiplicative function linked to the divisor function\n over the set of integers B-multiple of 5"
],
[
"Abstract Let $d(n)$ and $d^{\\ast}(n)$ be the numbers of divisors and the numbers of unitary divisors of the integer $n\\geq1$.",
"In this paper, we prove that \\[ \\underset{n... | 2212.05549 |
[
[
"Cross-Modal Learning with 3D Deformable Attention for Action Recognition"
],
[
"Abstract An important challenge in vision-based action recognition is the embedding of spatiotemporal features with two or more heterogeneous modalities into a single feature.",
"In this study, we propose a new 3D de... | 2212.05638 |
[
[
"A field redefinition invariant Lagrange multiplier formalism"
],
[
"Abstract In this paper, we propose a field redefinition invariant Lagrange multiplier (LM) formalism in which new ghost-like fields, analogous to Lee-Yang ghosts, are introduced.",
"These ghost fields are required to restore the... | 2212.05629 |
[
[
"Elliptically symmetric distributions for directional data of arbitrary\n dimension"
],
[
"Abstract We formulate a class of angular Gaussian distributions that allows different degrees of isotropy for directional random variables of arbitrary dimension.",
"Through a series of novel reparameteriz... | 2212.05634 |
[
[
"Transductive Linear Probing: A Novel Framework for Few-Shot Node\n Classification"
],
[
"Abstract Few-shot node classification is tasked to provide accurate predictions for nodes from novel classes with only few representative labeled nodes.",
"This problem has drawn tremendous attention for it... | 2212.05606 |
[
[
"Physics-informed data-driven prediction of Jet A-1 spray characteristics\n using time-resolved flame chemiluminescence and sparse Mie scattering"
],
[
"Abstract A time-lag and linear regression-based framework is developed and its performance is assessed for predicting temporally resolved spray num... | 2212.05543 |
[
[
"Maximal first Betti number rigidity for open manifolds of nonnegative\n Ricci curvature"
],
[
"Abstract Let $M$ be an open Riemannian $n$-manifold with nonnegative Ricci curvature.",
"We prove that if the first Betti number of $M$ equals $n-1$, then $M$ is flat."
],
[
"Introduction",
... | 2212.05530 |
[
[
"Maximum spread of $K_{2,t}$-minor-free graphs"
],
[
"Abstract The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$.",
"In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor.",
"We show that for any ... | 2212.05540 |
[
[
"von Neumann algebra description of inflationary cosmology"
],
[
"Abstract We study the von Neumann algebra description of the inflationary quasi-de Sitter (dS) space.",
"Unlike perfect dS space, quasi-dS space allows the nonzero energy flux across the horizon, which can be identified with the ex... | 2212.05637 |
[
[
"Information causality in multipartite scenarios"
],
[
"Abstract Bell nonlocality is one of the most intriguing and counter-intuitive phenomena displayed by quantum systems.",
"Interestingly, such stronger-than-classical quantum correlations are somehow constrained, and one important question to ... | 2212.05601 |
[
[
"Decomposition of the Leinster-Cobbold Diversity Index"
],
[
"Abstract The Leinster and Cobbold diversity index possesses a number of merits; in particular, it generalises many existing indices and defines an effective number.",
"We present a scheme to quantify the contribution of richness, evenn... | 2212.05617 |
[
[
"Local Volume dwarf KK242: radial velocity, SF region, and metallicity"
],
[
"Abstract KK242 is a LV dwarf of transition type residing in the void environment.",
"Koda et al.",
"present clear indications on its connection with Scd galaxy NGC6503.",
"This implies the distance to KK242 of ~... | 2212.05624 |
[
[
"Error-aware Quantization through Noise Tempering"
],
[
"Abstract Quantization has become a predominant approach for model compression, enabling deployment of large models trained on GPUs onto smaller form-factor devices for inference.",
"Quantization-aware training (QAT) optimizes model paramete... | 2212.05603 |
[
[
"Approximating a flexible beam model in the Loewner framework"
],
[
"Abstract The paper develops the Loewner approach for data-based modeling of a linear distributed-parameter system.",
"This approach is applied to a controlled flexible beam model coupled with a spring-mass system.",
"The ori... | 2212.05600 |
[
[
"An overview of maximal distance minimizers problem"
],
[
"Abstract Consider a compact $M \\subset \\mathbb{R}^d$ and $l > 0$.",
"A maximal distance minimizer problem is to find a connected compact set $\\Sigma$ of the length (one-dimensional Hausdorff measure $\\H$) at most $l$ that minimizes \\... | 2212.05607 |
[
[
"DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator\n Splitting"
],
[
"Abstract Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications.",
"Alternative to tradit... | 2212.05571 |
[
[
"Current-voltage characteristics of superconductor-normal\n metal-superconductor junctions"
],
[
"Abstract We develop a theory of current-voltage (I-U) characteristics for superconductor-normal metal-superconductor (SNS) junctions.",
"At small voltages and sufficiently low temperatures the I-U c... | 2212.05625 |
[
[
"Learning Neural Volumetric Field for Point Cloud Geometry Compression"
],
[
"Abstract Due to the diverse sparsity, high dimensionality, and large temporal variation of dynamic point clouds, it remains a challenge to design an efficient point cloud compression method.",
"We propose to code the ge... | 2212.05589 |
[
[
"Robust Relationship Between Mid-latitudes CAPE and Moist Static Energy\n in Present and Future Simulations"
],
[
"Abstract Convective available potential energy (CAPE), a metric associated with severe weather, is expected to increase with warming.",
"Under the most widely-accepted theory, devel... | 2212.05548 |
[
[
"Technical Debt Management in OSS Projects: An Empirical Study on GitHub"
],
[
"Abstract Technical debt (TD) refers to delayed tasks and immature artifacts that may bring short-term benefits but incur extra costs of change during maintenance and evolution in the long term.",
"TD has been extensiv... | 2212.05537 |
[
[
"Retire: Robust Expectile Regression in High Dimensions"
],
[
"Abstract High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects.",
"Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity ... | 2212.05562 |
[
[
"A subperiodic tree whose intermediate branching number is strictly less\n than the intermediate growth rate"
],
[
"Abstract We construct an example of a subperiodic tree whose intermediate branching number is strictly less than the intermediate growth rate.",
"This answers a question of Amir an... | 2212.05553 |
[
[
"Algorithms approaching the threshold for semi-random planted clique"
],
[
"Abstract We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian~\\cite{FK01}.",
"The previous best algorithms for this model succeed if the pl... | 2212.05619 |
[
[
"Dark Energy Nature in Logarithmic $f(R,T)$ Cosmology"
],
[
"Abstract The present research paper is an investigation of dark energy nature of logarithmic $f(R, T)$-gravity cosmology in a flat FLRW space-time universe.",
"We have derived modified Einstein's field equations for the function $f(R, T... | 2212.05605 |
[
[
"Global existence for coupled reaction-diffusion equations with a balance\n law and nonlinearities with non-constant sign"
],
[
"Abstract This paper aims to prove the global existence of solutions for coupled reaction diffusion equations with a balance Law and nonlinearities with a non constant sign... | 2212.05623 |
[
[
"Electrodynamics under Action of Null Cosmic Strings"
],
[
"Abstract A method to study electromagnetic (EM) effects generated by a straight null cosmic string moving in classical EM fields is suggested.",
"The string is shown to induce an additional EM field which can be described as a solution t... | 2212.05564 |
[
[
"Robust Estimation and Inference for Expected Shortfall Regression with\n Many Regressors"
],
[
"Abstract Expected Shortfall (ES), also known as superquantile or Conditional Value-at-Risk, has been recognized as an important measure in risk analysis and stochastic optimization, and is also finding a... | 2212.05565 |
[
[
"SIPGI: an interactive pipeline for spectroscopic data reduction"
],
[
"Abstract SIPGI is a spectroscopic pipeline for the data reduction of optical/near-infrared data acquired by slit-based spectrographs.",
"SIPGI is a complete spectroscopic data reduction environment retaining the high level of... | 2212.05580 |
[
[
"Estimation of the photon production rate using imaginary momentum\n correlators"
],
[
"Abstract The thermal photon emission rate is determined by the spatially transverse, in-medium spectral function of the electromagnetic current.",
"Accessing the spectral function using Euclidean data is, how... | 2212.05622 |
[
[
"Clustered Formation of Massive Stars within an Ionized Rotating Disk"
],
[
"Abstract We present ALMA observations with a 800 au resolution and radiative-transfer modelling of the inner part ($r\\approx6000$ au) of the ionized accretion flow around a compact star cluster in formation at the center of... | 2212.05633 |
[
[
"Intrinsic control of interlayer exciton generation rate in van der Waals\n materials via Janus layers"
],
[
"Abstract We demonstrate the possibility of engineering the optical properties of transition metal dichalcogenide heterobilayers when one of the constitutive layers has a Janus structure.",
... | 2212.05615 |
[
[
"Reconciling a decelerating Universe with cosmological observations"
],
[
"Abstract Can modern cosmological observations be reconciled with a general-relativistic Universe without an anti-gravitating energy source?",
"Usually, the answer to this question by cosmologists is in the negative, and it... | 2212.05568 |
[
[
"Authorship Identification of Source Code Segments Written by Multiple\n Authors Using Stacking Ensemble Method"
],
[
"Abstract Source code segment authorship identification is the task of identifying the author of a source code segment through supervised learning.",
"It has vast importance in p... | 2212.05610 |
[
[
"Black hole solutions in the quadratic Weyl conformal geometric theory of\n gravity"
],
[
"Abstract We consider numerical black hole solutions in the Weyl conformal geometry, and its associated conformally invariant Weyl quadratic gravity.",
"In this model Einstein gravity (with a positive cosmo... | 2212.05542 |
[
[
"Gravitational waves from the early universe"
],
[
"Abstract These lecture notes are based on the course \"Gravitational waves from the early universe\" given at the 27th W.E.",
"Heraeus \"Saalburg\" Summer School 2021 by Valerie Domcke.",
"Ongoing and future collaborations will probe differe... | 2212.05594 |
[
[
"How and How Much is ${g_A}$ {\\it Fundamentally} Quenched in Nuclei?"
],
[
"Abstract The superallowed Gamow-Teller transition in the doubly-magic-shell nucleus $^{100}$Sn and the high resolution spectral shape analysis in the fourth-forbidden nonunique transition in $^{115}$In indicate as much as $\... | 2212.05558 |
[
[
"Hadronic contributions to the muon $g-2$ in holographic QCD"
],
[
"Abstract We discuss the recent progress made in using bottom-up holographic QCD models in calculating hadronic contributions to the anomalous magnetic moment of the muon, in particular the hadronic light-by-light scattering contribut... | 2212.05547 |
[
[
"A Compact TPC for the sPHENIX Experiment"
],
[
"Abstract The sPHENIX detector to be installed at RHIC in 2022 is designed to precisely measure jets, jet correlations, and dilepton pairs in heavy-ion collisions.",
"With these measurements in mind, sPHENIX will employ a compact TPC covering 20cm <... | 2212.05541 |
[
[
"Robust Inference in High Dimensional Linear Model with Cluster\n Dependence"
],
[
"Abstract Cluster standard error (Liang and Zeger, 1986) is widely used by empirical researchers to account for cluster dependence in linear model.",
"It is well known that this standard error is biased.",
"We... | 2212.05554 |
[
[
"A systematic literature review on Robotic Process Automation security"
],
[
"Abstract The technocrat epoch is overflowing with new technologies and such cutting-edge facilities accompany the risks and pitfalls.",
"Robotic process automation is another innovation that empowers the computerization... | 2212.05544 |
[
[
"Context-aware 6D Pose Estimation of Known Objects using RGB-D data"
],
[
"Abstract 6D object pose estimation has been a research topic in the field of computer vision and robotics.",
"Many modern world applications like robot grasping, manipulation, autonomous navigation etc, require the correct... | 2212.05560 |
[
[
"Doubly and triply extended MSRD codes"
],
[
"Abstract In this work, doubly extended linearized Reed--Solomon codes and triply extended Reed--Solomon codes are generalized.",
"We obtain a general result in which we characterize when a multiply extended code for a general metric attains the Single... | 2212.05528 |
[
[
"From Bloch Oscillations to a Steady-State Current in Strongly Biased\n Mesoscopic Devices"
],
[
"Abstract It has long been known that quantum particles in a periodic lattice exhibit an oscillatory motion that is solely driven by a constant and uniform force field.",
"In a strongly biased mesosc... | 2212.05574 |
[
[
"Nonequilibrium polariton condensation in biannular optically induced\n traps"
],
[
"Abstract We report the mean field model of nonequilibrium polariton condensation in annular effective non-Hermitian potential traps, stemming from incoherent optically induced excitonic reservoirs of annular shape."... | 2212.05533 |
[
[
"Local Limit of Nonlocal Gravity: A Teleparallel Extension of General\n Relativity"
],
[
"Abstract We describe a general constitutive framework for a teleparallel extension of the general theory of relativity.",
"This approach goes beyond the teleparallel equivalent of general relativity (TEGR) ... | 2212.05536 |
[
[
"Modeling halo and central galaxy orientations on the SO(3) manifold with\n score-based generative models"
],
[
"Abstract Upcoming cosmological weak lensing surveys are expected to constrain cosmological parameters with unprecedented precision.",
"In preparation for these surveys, large simulati... | 2212.05592 |
[
[
"Random Feature Models for Learning Interacting Dynamical Systems"
],
[
"Abstract Particle dynamics and multi-agent systems provide accurate dynamical models for studying and forecasting the behavior of complex interacting systems.",
"They often take the form of a high-dimensional system of diffe... | 2212.05591 |
[
[
"Periodic and compacton travelling wave solutions of discrete nonlinear\n Klein-Gordon lattices"
],
[
"Abstract We prove the existence of periodic travelling wave solutions for general discrete nonlinear Klein-Gordon systems, considering both cases of hard and soft on-site potentials.",
"In the ... | 2212.05575 |
[
[
"Nearby voids and their galaxies: recent progress and prospects"
],
[
"Abstract Voids occupy about 3/4 of the volume of the Universe and contain about 15% of its mass.",
"Due to various observational selection effects, these structure elements and galaxies populating voids, are highly under-explo... | 2212.05640 |
[
[
"Gravity from the Determinant of the Energy-Momentum: Astrophysical\n Implications"
],
[
"Abstract Determinants of the second-rank tensors stand useful in forming generally invariant terms as in the case of the volume element of the gravitational actions.",
"Here, we extend the action of the mat... | 2212.05585 |
[
[
"Accelerating Self-Supervised Learning via Efficient Training Strategies"
],
[
"Abstract Recently the focus of the computer vision community has shifted from expensive supervised learning towards self-supervised learning of visual representations.",
"While the performance gap between supervised a... | 2212.05611 |
[
[
"Galaxies on graph neural networks: towards robust synthetic galaxy\n catalogs with deep generative models"
],
[
"Abstract The future astronomical imaging surveys are set to provide precise constraints on cosmological parameters, such as dark energy.",
"However, production of synthetic data for ... | 2212.05596 |
[
[
"Virial expansion for the optical response of doped two-dimensional\n semiconductors"
],
[
"Abstract We present a quantum virial expansion for the optical response of a doped two-dimensional semiconductor.",
"As we show, this constitutes a perturbatively exact theory in the high-temperature or l... | 2212.05627 |
[
[
"A non-singular generalized entropy and its implications on bounce\n cosmology"
],
[
"Abstract We propose a new five-parameter entropy function that proves to be singular-free during the entire cosmic evolution of the universe, and at the same time, also generalizes the Tsallis, Barrow, R\\'{e}nyi, ... | 2212.05531 |
[
[
"Energy-recurrence Breakdown and Chaos in Disordered\n Fermi-Pasta-Ulam-Tsingou Lattices"
],
[
"Abstract In this paper, we consider the classic Fermi-Pasta-Ulam-Tsingou system as a model of interacting particles connected by harmonic springs with a quadratic nonlinear term (first system) and a set o... | 2212.05644 |
[
[
"Normal forms of $\\mathbb Z$-graded $Q$-manifolds"
],
[
"Abstract Following recent results of A.K.",
"and V.S.",
"on $\\mathbb Z$-graded manifolds, we give several local and global normal forms results for $Q$-structures on those, i.e.",
"for differential graded manifolds.",
"In part... | 2212.05579 |
[
[
"A Bombieri-Vinogradov-type theorem for moduli with small radical"
],
[
"Abstract In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\\le x^{1/4-\\varepsilon}$ with $p\\le (\\log x)^C$ to sparse sets of moduli $s\\le x^{1/4-\\varepsilo... | 2212.05576 |
[
[
"Dirac gauge theory for topological spinors in 3+1 dimensional networks"
],
[
"Abstract Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation.",
"We propose a Dirac gauge theory for... | 2212.05621 |
[
[
"Exact results for sheared polar active suspensions with variable liquid\n crystalline order"
],
[
"Abstract We consider a confined sheared active polar liquid crystal with a uniform orientation and study the effect of variations in the magnitude of polarization.",
"Restricting our analysis to o... | 2212.05534 |
[
[
"A Bit Stream Feature-Based Energy Estimator for HEVC Software Encoding"
],
[
"Abstract The total energy consumption of today's video coding systems is globally significant and emphasizes the need for sustainable video coder applications.",
"To develop such sustainable video coders, the knowledge... | 2212.05609 |
[
[
"Generation of nearly pure and highly directional magnetic light in\n fluorescence of rare earth ions"
],
[
"Abstract We discover regimes for promoting decay rates of magnetic light due to magnetic dipole (MD) transitions of trivalent rare earth ions located inside or near dielectric homogeneous sph... | 2212.05552 |
[
[
"Efficient algorithms to solve atom reconfiguration problems. II. The\n assignment-rerouting-ordering (aro) algorithm"
],
[
"Abstract Programmable arrays of optical traps enable the assembly of configurations of single atoms to perform controlled experiments on quantum many-body systems.",
"Find... | 2212.05586 |
[
[
"Schauder type theorems for mild solutions to non autonomous\n Ornstein-Uhlenbeck equations"
],
[
"Abstract We prove smoothing properties along suitable directions of the Ornstein-Uhlenbeck evolution operator, namely the evolution operator formally associated to the non autonomous Ornstein-Uhlenbeck... | 2212.05559 |
[
[
"Orthogonal SVD Covariance Conditioning and Latent Disentanglement"
],
[
"Abstract Inserting an SVD meta-layer into neural networks is prone to make the covariance ill-conditioned, which could harm the model in the training stability and generalization abilities.",
"In this paper, we systematical... | 2212.05599 |
[
[
"Towards a Learner-Centered Explainable AI: Lessons from the learning\n sciences"
],
[
"Abstract In this short paper, we argue for a refocusing of XAI around human learning goals.",
"Drawing upon approaches and theories from the learning sciences, we propose a framework for the learner-centered ... | 2212.05588 |
[
[
"Studying Postmerger Outflows from Magnetized Neutrino-cooled Accretion\n Disks"
],
[
"Abstract Neutrino-cooled accretion flow around a spinning black hole, produced by a compact binary merger is a promising scenario for jet formation and launching magnetically-driven outflows.",
"Based on GW170... | 2212.05628 |
[
[
"FactorJoin: A New Cardinality Estimation Framework for Join Queries"
],
[
"Abstract Cardinality estimation is one of the most fundamental and challenging problems in query optimization.",
"Neither classical nor learning-based methods yield satisfactory performance when estimating the cardinality... | 2212.05526 |
[
[
"Deep Inelastic Scattering with Application to Nuclear Targets: Lectures\n at the 1985 Los Alamos School on Relativistic Dynamics and Quark Nuclear\n Physics"
],
[
"Abstract This paper is essentially a verbatim reconstruction of lectures that I gave at the Los Alamos School on Relativistic Dynamics... | 2212.05616 |
[
[
"Mediation analysis with the mediator and outcome missing not at random"
],
[
"Abstract Mediation analysis is widely used for investigating direct and indirect causal pathways through which an effect arises.",
"However, many mediation analysis studies are often challenged by missingness in the me... | 2212.05577 |
[
[
"A Formalization of Doob's Martingale Convergence Theorems in mathlib"
],
[
"Abstract We present the formalization of Doob's martingale convergence theorems in the mathlib library for the Lean theorem prover.",
"These theorems give conditions under which (sub)martingales converge, almost everywhe... | 2212.05578 |
[
[
"Bayesian inference for partial orders from random linear extensions:\n power relations from 12th Century Royal Acta"
],
[
"Abstract We give a new class of models for time series data in which actors are listed in order of precedence.",
"We model the lists as a realisation of a queue in which qu... | 2212.05524 |
[
[
"Recurrent Vision Transformers for Object Detection with Event Cameras"
],
[
"Abstract We present Recurrent Vision Transformers (RVTs), a novel backbone for object detection with event cameras.",
"Event cameras provide visual information with sub-millisecond latency at a high-dynamic range and wi... | 2212.05598 |
[
[
"UV-driven Chemistry as a Signpost for Late-stage Planet Formation"
],
[
"Abstract The chemical reservoir within protoplanetary disks has a direct impact on planetary compositions and the potential for life.",
"A long-lived carbon-and nitrogen-rich chemistry at cold temperatures (<=50K) is observ... | 2212.05539 |
[
[
"Effects of spin orbit coupling on proximity induced superconductivity"
],
[
"Abstract We investigate the effect of spin orbit coupling on proximity induced superconductivity in a normal metal attached to a superconductor.",
"Specifically, we consider a heterostructure where the presence of inter... | 2212.05550 |
[
[
"A Weyl's law for black holes"
],
[
"Abstract We discuss a Weyl's law for the quasi-normal modes of black holes that recovers the structural features of the standard Weyl's law for the eigenvalues of the Laplacian in compact regions.",
"Specifically, the asymptotics of the counting function $N(\\... | 2212.05570 |
[
[
"DISCO: Adversarial Defense with Local Implicit Functions"
],
[
"Abstract The problem of adversarial defenses for image classification, where the goal is to robustify a classifier against adversarial examples, is considered.",
"Inspired by the hypothesis that these examples lie beyond the natural... | 2212.05630 |
[
[
"Nearby SNR: a possible common origin to multi-messenger anomalies in\n spectra, ratios and anisotropy of cosmic rays"
],
[
"Abstract The multi-messenger anomalies, including spectral hardening or excess for nuclei, leptons, ratios of $\\bar p/p$ and B/C, and anisotropic reversal, were observed in p... | 2212.05641 |
[
[
"Generic Tagging for RISC-V Binaries"
],
[
"Abstract With the widespread popularity of RISC-V -- an open-source ISA -- custom hardware security solutions targeting specific defense needs are gaining popularity.",
"These solutions often require specialized compilers that can insert metadata (calle... | 2212.05614 |
[
[
"YoloCurvSeg: You Only Label One Noisy Skeleton for Vessel-style\n Curvilinear Structure Segmentation"
],
[
"Abstract Weakly-supervised learning (WSL) has been proposed to alleviate the conflict between data annotation cost and model performance through employing sparsely-grained (i.e., point-, box-... | 2212.05566 |
[
[
"New examples of twisted Brill-Noether loci I"
],
[
"Abstract Our purpose in this paper is to construct new examples of twisted Brill-Noether loci on curves of genus $g\\ge2$.",
"Many of these examples have negative expected dimension.",
"We deduce also the existence of a new region in the Br... | 2212.05573 |
[
[
"Multimodal and Explainable Internet Meme Classification"
],
[
"Abstract Warning: this paper contains content that may be offensive or upsetting.",
"In the current context where online platforms have been effectively weaponized in a variety of geo-political events and social issues, Internet meme... | 2212.05612 |
[
[
"Using Multiple Instance Learning to Build Multimodal Representations"
],
[
"Abstract Image-text multimodal representation learning aligns data across modalities and enables important medical applications, e.g., image classification, visual grounding, and cross-modal retrieval.",
"In this work, w... | 2212.05561 |
[
[
"ResFed: Communication Efficient Federated Learning by Transmitting Deep\n Compressed Residuals"
],
[
"Abstract Federated learning enables cooperative training among massively distributed clients by sharing their learned local model parameters.",
"However, with increasing model size, deploying f... | 2212.05602 |
[
[
"Universality of the local limit of preferential attachment models"
],
[
"Abstract We study preferential attachment models where vertices enter the network with i.i.d.",
"random numbers of edges that we call the out-degree.",
"We identify the local limit of such models, substantially extendin... | 2212.05551 |
[
[
"Morphology of Shocked Lateral Outflows in Colliding Hydrodynamic Flows"
],
[
"Abstract Supersonic interacting flows occurring in phenomena such as protostellar jets give rise to strong shocks, and have been demonstrated in several laboratory experiments.",
"To study such colliding flows, we use ... | 2212.05631 |