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[ "Catching proteins in liquid helium droplets", "Catching proteins in liquid helium droplets" ]	[ "Frauke Bierau \nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany\n", "Peter Kupser \nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany\n", "Gerard Meijer \nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany\n", "Gert Von Helden \nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany\n" ]	[ "Fritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany", "Fritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany", "Fritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany", "Fritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany" ]	[]	An experimental approach is presented that allows for the incorporation of large mass/charge selected ions in liquid helium droplets. It is demonstrated that droplets can be efficiently doped with a mass/charge selected amino acid as well as with the much bigger m≈12 000 amu protein Cytochrome C in selected charge states. The sizes of the ion-doped droplets are determined via electrostatic deflection. Under the experimental conditions employed, the observed droplet sizes are very large and range, dependent on the incorporated ion, from 10 10 helium atoms for protonated Phenylalanine to 10 12 helium atoms for Cytochrome C. As a possible explanation, a simple model based on the size-and internal energy-dependence of the pickup efficiency is given.	10.1103/physrevlett.105.133402	[ "https://arxiv.org/pdf/1008.3816v2.pdf" ]	2,997,921	1008.3816	be0d244e8ba7cf2df501e51b01c877831d6b5d1d	Catching proteins in liquid helium droplets 29 Sep 2010 (Dated: September 30, 2010) Frauke Bierau Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4-6D-14195BerlinGermany Peter Kupser Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4-6D-14195BerlinGermany Gerard Meijer Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4-6D-14195BerlinGermany Gert Von Helden Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4-6D-14195BerlinGermany Catching proteins in liquid helium droplets 29 Sep 2010 (Dated: September 30, 2010)arXiv:1008.3816v2 [physics.atm-clus] An experimental approach is presented that allows for the incorporation of large mass/charge selected ions in liquid helium droplets. It is demonstrated that droplets can be efficiently doped with a mass/charge selected amino acid as well as with the much bigger m≈12 000 amu protein Cytochrome C in selected charge states. The sizes of the ion-doped droplets are determined via electrostatic deflection. Under the experimental conditions employed, the observed droplet sizes are very large and range, dependent on the incorporated ion, from 10 10 helium atoms for protonated Phenylalanine to 10 12 helium atoms for Cytochrome C. As a possible explanation, a simple model based on the size-and internal energy-dependence of the pickup efficiency is given. The isolation of foreign species in liquid helium nanodroplets is of fundamental interest and has found important applications in molecular spectroscopy [1][2][3][4][5]. Helium clusters have an internal equilibrium temperature of ∼ 0.37 K, which is maintained by evaporative cooling. Liquid helium is optically transparent from the deep UV to the far IR and superfluid helium droplets can serve as gentle matrices to provide an isothermal environment for embedded molecules at cryogenic temperatures. Further, due to the weak interactions of liquid helium, molecules embedded in helium nanodroplets can rotate freely and their optical spectra show narrow linewidths [6,7]. Helium droplets can be formed in free-jet expansions [8] of high pressure pre-cooled helium gas, and this technique is used by many groups to produce continuous or pulsed droplet beams. The choice of expansion parameters, in particular the orifice diameter d, source pressure P 0 and temperature T 0 , determines the resulting size distribution of the helium clusters and the average flux. Atoms or molecules can be incorporated in the droplets by pickup from a gas cell and by using this technique, small molecules [9] and biomolecules [10,11] as well as large species such as C 60 [12] have successfully been doped into helium droplets. Using laser vaporization, less volatile materials such as refractory metal atoms can be evaporated and clusters can grow inside helium droplets [13,14]. Those clusters, however, will occur in a distribution of sizes that is governed by Poisson statistics. For studying molecules in helium droplets, a prerequisite is the ability to bring the intact molecule into the gas phase. For many interesting species such as most larger biomolecules, this can not be done via evaporation. Pulsed laser desorption is one other possibility, doing so, however, yields low concentrations in the gas phase as well as frequently a mixture of species that additionally contains decomposed molecules and matrix molecules. More promising would be to use established techniques, as for example electrospray ionization followed by mass separation, and to incorporate those mass/charge selected ions into helium droplets. We here present an experimental approach in which mass/charge selected ions that are stored and accumulated in an ion trap are picked up by helium droplets traversing the trap. The approach is conceptually similar to pickup experiments of neutrals from gas cells, however a crucial difference is that in the case of the ion trap, use is made of the high kinetic energy of the heavy helium droplets, which allows ions only to leave the trap once they are inside a droplet. The so doped droplets can then be further investigated and here, an initial study on the droplet masses and beam velocities is presented. Having biomolecules embedded in helium droplets will allow many exciting experiments, such as spectroscopic investigations and possibly in the future single molecule diffraction experiments using x-ray free electron lasers [15][16][17] or electron diffraction experiments on aligned biomolecules [18]. For large molecules inside helium droplets, recent spectroscopic studies have shown to give spectra of much higher resolution, compared to those obtained for the bare gas-phase molecule in a molecular beam setup [19]. A scheme of the experimental setup is shown in fig.1. Ions are brought into the gas phase via electrospray ionization (ESI) and mass/charge selected in a quadrupole mass spectrometer. This part of the machine is a modified version of a QTof Ultima (Waters Corporation). The ion beam can then be bent 90 • by a quadrupole bender to be injected into a linear hexapole ion trap. Alternatively, when the bender is turned off, the ions are transferred into a time of flight mass spectrometer (not shown in fig.1). The ion trap consists of six 30 cm long, 5 mm diameter rods which are placed on a 14.1 mm diameter circle. The trap is operated with 200 V p−p , 1.7 MHz RF and two trapping electrodes that are kept 1-3 volts above the rod DC level are used for longitudinal trapping. The helium droplet source is pulsed and similar in design to that of A. Vilesov [20]. In brief, a commercial pulsed valve (Parker Series 99) with a Kel-F poppet is mounted to the low temperature stage of a closed-cycle He cryocooler (Sumitomo RDK 408D2). To improve the cooling of the nozzle, its faceplate is machined from copper with an inserted stainless-steel piece which provides the sealing to the main valve body. The valve opening is a 0.8 mm diameter hole followed by a 90 • , 1 cm long cone. The valve temperature can be adjusted between 4.7 K and 30 K, the helium pressure can be regulated up to 50 bar and the valve is typically operated with ≈ 200 µs long pulses at a 4 Hz repetition rate. Five cm downstream the nozzle, the beam is skimmed (Beam dynamics model 2, 2 mm opening) and enters the ion trap. The droplets move slowly (≈200 -300 ms −1 ), however their mass can be large and so will be their kinetic energy. When a droplet picks up an ion, the kinetic energy of the doped droplet will exceed the longitudinal trapping potential energy and, inside the droplets, the ions can leave the trap. The ion-doped helium droplets pass straight through the bender and can be analyzed and detected further downstream. The charged droplets can be deflected by applying voltages to a pair of parallel metal plates, and the ion current can be measured 50 cm after the plates on a movable copper plate. Due to the high droplet mass in the here presented experiments, it is not possible to use sensitive charged particle detectors that rely on electron release from a surface upon ion impact as the first step. Instead, the direct ion current is measured via a current to voltage amplifier. Two types of amplifiers are used. One (FEMTO DLPCA-200) has a calibrated maximal gain of 10 11 V/A and a bandwidth of 1.1 kHz. The other one (AMPTEK A250) provides a With increasing T0 the peak shifts to earlier arrival times. A considerable portion of the peak widths can be attributed to the time response of the current amplifier. Measured velocities (blue points and curve) of ion-doped helium droplets as a function of source temperature T0 (b) and pressure P0 (c). Also shown are predictions for an ideal He gas expansion (green solid curve) as well as a liquid expansion (red dashed curve) following the Bernoulli expression. better signal to noise, however is not calibrated and is the one used in most of the here presented experiments. In a typical experiment, a pulse of He buffer gas is injected into the ion trap. Incoming ions with a kinetic energy just barely above the trapping potential energy loose energy via collisions and are stored in the trap. After a few seconds, depending on the ion current, the number of ions in the trap increases no further as presumably the space-charge limit (10 6 − 10 7 e · cm −3 ) is reached. Then, the bender is turned off, helium droplets are generated for 15 seconds and the ion current of charged droplets is measured as a function of time after valve opening. After these 15 seconds, the number of ions in the trap is reduced by the helium droplets to about 50% of its initial value (measured in the time of flight mass spectrometer) [21] and the cycle is repeated after reloading the trap. At the ion densities in the trap, the pickup of multiple ions is unlikely. The experiments are performed on droplets doped with protonated singly charged Phenylalanine (Phe) as well as with the 104 amino acid protein Cytochrome C (CytC) in different charge states. This protein (Aldrich, horse heart) has a mass of 12327 amu and is known to be present in a broad range of charge states after electrospray, ranging from z=+9 up to z=+17. In the here presented experiments, the charge states z=+9, +14 and +17 are selected. In fig.2(a), time of flight profiles of helium droplets doped with CytC in the charge state +14 are shown as a function of source temperature T 0 . At early times, an interference caused by the trigger pulse of the valve can be observed. The ion signal appears at around 8 ms and has a width of ≈1 ms. Measuring the current with the calibrated amplifier gives peak currents of up to ≈ 20 pA, which implies about 10 4 ion-doped droplets per pulse. Such intensities should be sufficient to investigate the dopant ions via action spectroscopy (in which the wavelength dependent response of the chromophore upon light exposure is monitored) for which only few ions per pulse are needed [22]. All traces in fig.2 (a), are taken at P 0 = 30 bar. The lowest temperature investigated is 5 K. When increasing the temperature, the peaks shift to earlier times. At temperatures above 10 K, the intensity rapidly diminishes and no doped droplets can be observed above 11.5 K. The measured beam velocities as a function of source temperature T 0 are shown in fig.2(b) and range from 200 ms −1 at 5 K to 305 ms −1 at T 0 =11.5 K. The droplet velocities are also observed to vary with source pressure P 0 (see fig.2(c)). At a fixed temperature of 7 K, the velocity increases from 190 ms −1 at 20 bar to 240 ms −1 at 50 bar and below 20 bar, no signal is observed. For comparison, the expected velocities for a supersonic expansion of an ideal gas and the expected velocities from the Bernoulli equation v = c 2P 0 /ρ 0 , with the discharge coefficient c as 1.0 and with ρ 0 as the known values for the helium densities, are shown. Clearly, the ideal gas expression does not predict the observed pressure dependence, which is qualitatively reproduced by the Bernoulli expression. However, the ideal gas expression does better in predicting the temperature dependence of the beam velocities. Important parameters are the size and size distribution of the ion-doped droplets. For an experimental determination, a pair of oppositely charged metal plates with a spacing of d=7 mm and a length of L 1 =10 cm is installed behind the trap and the bender (see fig.1), about 120 cm downstream of the droplet source. Just in front of the plates, a 1.2 mm collimating slit is located. Applying a voltage ±U to the plates causes a deflection s y of the beam which depends on beam velocity v x , droplet mass m and droplet charge ze according to: s y = zeEL 1 (L 1 + 2L 2 ) 2mv 2 x(1) where E = 2U/d. The deflection is detected by measuring the ion current on a plate after a movable 1 mm slit at a distance L 2 =50 cm from the end of the deflection plates. The droplet source temperature and pressure are kept at 8 K and 30 bar, respectively. To measure the deflection for a particular electric field E, time of flight profiles are recorded as a function of detector position. In fig.3, the integrated ion signal for the three charge states of CytC as well as protonated Phe at a field of 5.7 kV/cm (diamonds) and with no deflection field applied (filled circles) are shown. It is apparent that applying an electric field causes the droplet beam profiles to shift and to broaden. At a constant field, the observed deflection should depend on the droplet mass and charge and higher charged droplets should give rise to a larger deflection. Strikingly, when comparing the highly charged CytC doped droplets to the singly charged Phe doped droplets, the opposite is observed. Comparing the differently charged CytC doped droplets among each other, an increased deflection with increasing charge is observed. The increased deflection of the Phe doped droplets must thus result from a smaller droplet size. In order to quantify the observations, the curves in fig.3 are fitted with a model. The curves with no field applied are fit with an instrumental function that contains the widths of the two slits as well as the absolute zero position. The model for the fits of the deflection curves contains the same instrumental function as well as a size distribution function of the droplets. For that, we take a log-normal distribution, which has previously been shown to be applicable for helium droplets [23]. Such a distribution is described by a mean size as well as by a width parameter. The width parameter σ, the standard deviation of the distribution of the logarithm of the droplet size, is kept constant at 1.0 for all CytC curves and at 0.58 for the Phe curve. The resulting (arithmetic) mean droplet masses are (2.3, 2.7 and 3.2) · 10 12 amu for CytC +9, +14 and +17, respectively and 2.2 · 10 10 amu for Phe doped droplets. It should be noted that in most He-droplet experiments reported to date, it is notoriously difficult to directly measure the droplet size distributions. Nonetheless, the here observed droplet sizes are considerably larger than those observed in this temperature and pressure range by others [23][24][25][26] and more similar to those obtained at much lower temperature from the breakup of a liquid helium beam [27]. The observation that the here observed droplet sizes are orders of magnitude larger could be explained by the larger nozzle orifice of 800 µm, compared to 500 µm used in the experiments of Vilesov [20]. In addition, the here employed nozzle is machined out of copper (compared to stainless steel used in other experiments [20]) and has a longer 90 degree conical section, which might further promote the formation of large droplets. An interesting and puzzling observation is the difference in droplet sizes in the CytC and Phe experiment. As the droplet size distribution emitted from the source is the same in both experiments, the pick up efficiency for a given droplet size must depend on the dopant molecule mass and/or charge. Apparently, a droplet size of ≈ 10 12 atoms is required to pick up CytC ions and the curves in fig.3 thus represent only the tail of the real size distribution. Whether the curve for Phe represents the droplet size distribution as emitted from the source is not yet clear. The question is which physical factors could give rise to such a size effect. The velocity of CytC in the trap can be neglected relative to the velocity of the helium droplets; the protein thus dives into liquid helium with about the droplet velocity (200-300 ms −1 ). A rough estimation based on an impulsive model gives that in a distance of about 0.02 µm, the particle is slowed down to the Landau velocity (60 ms −1 ), after which it could move without much friction through the droplet. When having a droplet of 4 µm diameter (≈ 10 12 atoms), the protein reaches in ≈ 70 ns the other edge of the droplet. Before the pickup by helium droplets, the proteins are thermalized in the ion trap at 300 K. Their internal vibrational energy is then ∼ 30 eV, estimated using typical values for the heat capacity of a protein [28]. After some time, this energy will be transferred to the helium, causing about 4 · 10 4 helium atoms to evaporate [1]. Recent observations have shown that vibrationally excited small ions can leave helium droplets, presumably assisted by a shell of "hot" helium surrounding them [29]. A possibility is thus, that times significantly smaller than 70 ns are insufficient to cool the protein and its surroundings, and when the still "warm" protein reaches the edge of the droplet, it will escape again. It should be noted that in the here presented CytC experiments, the internal energies deposited at once by a dopant molecule into a helium droplet are considerably larger than those in all previous helium droplet experiments. Protonated Phe has a much lower internal vibrational energy than that of CytC and therefore does not require long cooling times or large droplets to become trapped. In summary, we have presented a new technique that allows for the first time the pickup of large mass/charge selected biomolecular ions by helium droplets. The droplet sizes are determined via electrostatic deflection and are found to be very large, containing more than 10 10 helium atoms and being larger than 1 µm in size. The approach is general and allows the selective and clean incorporation of a wide range of other mass/charge selected species, such as for example cluster ions, into helium droplets and thus opens the helium droplet isolation technique to a wide range of new species. We gratefully thank Andrey Vilesov for help and advice with the pulsed droplet source and Waters for providing us the QTof mass spectrometer. FIG. 1 : 1Scheme of the experimental setup. Ions are brought into the gas phase via electrospray ionization, are mass/charge selected by a quadrupole MS and stored in an ion trap. Helium droplets can pick up those ions and after some distance downstream, the direct ion current is measured. Electric field deflection can be used to determine the droplet size distribution (see text). of flight profiles of CytC ions (z=+14) embedded in He droplets. The He droplets are produced at P0= 30 bar and varying source temperatures T0 from 5 K to 11 K. FIG. 3 : 3Deflection profiles for three different charge states of CytC z=+9, +14 and +17 and protonated singly charged Phe at an electric field of E=5.7 kV/cm (diamonds) and at zero field (filled circles). Also shown are the respective fitted curves (lines). * Electronic address: E-mail: helden@fhi-berlin. mpg.de* Electronic address: E-mail: [email protected] . J P Toennies, A F Vilesov, Angew. Chem. Int. Ed. 432622J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004). . J P Toennies, A F Vilesov, Ann. Rev. Phys. Chem. 491J. P. Toennies and A. F. Vilesov, Ann. Rev. Phys. Chem. 49, 1 (1998). . 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Slipchenko, S. Kuma, T. Momose, and A. F. Vilesov, Rev. Sci. Instrum. 73, 3600 (2002). Typical mass spectra of the ion trap contents are given as supplementary material. Typical mass spectra of the ion trap contents are given as supplementary material. . M A Duncan, Int. J. Mass Spectrom. 200545M. A. Duncan, Int. J. Mass Spectrom. 200, 545 (2000). . M Lewerenz, B Schilling, J P Toennies, Chem. Phys. Lett. 206381M. Lewerenz, B. Schilling, and J. P. Toennies, Chem. Phys. Lett. 206, 381 (1993). . H Buchenau, J. Chem. Phys. 926875H. Buchenau et al, J. Chem. Phys. 92, 6875 (1990). . T Jiang, J A Northby, Phys. Rev. Lett. 682620T. Jiang and J. A. Northby, Phys. Rev. Lett. 68, 2620 (1992). . D Pentlehner, Rev. Sci. Instrum. 8043302D. Pentlehner et al, Rev. Sci. Instrum. 80, 043302 (2009). . R E Grisenti, J P Toennies, Phys. Rev. Lett. 90234501R. E. Grisenti and J. P. Toennies, Phys. Rev. Lett. 90, 234501 (2003). . L Meinhold, F Merzel, J C Smith, Phys. Rev. Lett. 99138101L. Meinhold, F. Merzel, and J. C. Smith, Phys. Rev. Lett. 99, 138101 (2007). . S Smolarek, N B Brauer, W J Buma, M Drabbels, J. Am.Chem.Soc. in pressS. Smolarek, N. B. Brauer, W. J. Buma, and M. Drabbels, J. Am.Chem.Soc (in press) .	[]
[ "Fundamental asymmetries between spatial and temporal boundaries in electromagnetics Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX", "Fundamental asymmetries between spatial and temporal boundaries in electromagnetics Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX" ]	[ "Wending Mai [email protected] \nElectrical Engineering Department\nPennsylvania State University\n16802University ParkPAUSA\n", "ANDJingwei Xu \nElectrical Engineering Department\nPennsylvania State University\n16802University ParkPAUSA\n", "Douglas H Werner \nElectrical Engineering Department\nPennsylvania State University\n16802University ParkPAUSA\n" ]	[ "Electrical Engineering Department\nPennsylvania State University\n16802University ParkPAUSA", "Electrical Engineering Department\nPennsylvania State University\n16802University ParkPAUSA", "Electrical Engineering Department\nPennsylvania State University\n16802University ParkPAUSA" ]	[]	Time-varying materials bring an extra degree of design freedom compared to their conventional time-invariant counterparts. However, few discussions have focused on the underlying physical difference between spatial and temporal boundaries. In this letter, we thoroughly investigate those differences from the perspective of conservation laws. By doing so, the building blocks of optics and electromagnetics such as the reflection law, Snell's law, and Fresnel's equations can be analogously derived in a temporal context, but with completely different interpretations. Furthermore, we study the unique features of temporal boundaries, such as their nonconformance to energy conservation and causality.	10.3390/sym15040858	[ "https://arxiv.org/pdf/2207.04286v1.pdf" ]	250,426,125	2207.04286	36d18d2e1d97ff0c03cab6ce3cee3266bdfdb14f	Fundamental asymmetries between spatial and temporal boundaries in electromagnetics Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX Wending Mai [email protected] Electrical Engineering Department Pennsylvania State University 16802University ParkPAUSA ANDJingwei Xu Electrical Engineering Department Pennsylvania State University 16802University ParkPAUSA Douglas H Werner Electrical Engineering Department Pennsylvania State University 16802University ParkPAUSA Fundamental asymmetries between spatial and temporal boundaries in electromagnetics Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX *Corresponding author: Time-varying materials bring an extra degree of design freedom compared to their conventional time-invariant counterparts. However, few discussions have focused on the underlying physical difference between spatial and temporal boundaries. In this letter, we thoroughly investigate those differences from the perspective of conservation laws. By doing so, the building blocks of optics and electromagnetics such as the reflection law, Snell's law, and Fresnel's equations can be analogously derived in a temporal context, but with completely different interpretations. Furthermore, we study the unique features of temporal boundaries, such as their nonconformance to energy conservation and causality. INTRODUCTION There has been a long history of research interest in methodologies for artificially controlling electromagnetic (EM) wave propagation [1][2][3][4]. The rapid development of metamaterials in recent years has provided an unprecedented flexibility for manipulating EM waves. With carefully tailored electric and/or magnetic responses, such metamaterials may be engineered to exhibit extreme values of ε and/or μ [5][6][7]. The control over EM waves in the vast majority of metamaterials has been accomplished via the spatial distribution of the effective material properties. Reconfigurable and active metamaterials, however, are enabled by the time-variation in their material properties, which are unachievable by conventional materials [8][9][10][11][12]. This class of metamaterials with time-varying properties opens a new realm of possibilities for tailoring EM waves. To be more specific, in addition to considering the spatial distribution of the constituent materials (i.e., ε(x,y,z)), one could explore the desired functionalities with greater flexibility by introducing the additional dimension of "time", i.e., ε(x,y,z,t) [2]. In recent years, researchers have extensively explored these novel "4-D" material systems, with a particular focus on their time-varying aspect. It was found that when an EM wave travels through a medium whose material properties (i.e., ε or μ) change suddenly, it will split into two waves propagating in opposite directions [13]. This phenomenon is the temporal dual of reflection and transmission where a wave meets the boundary between two different materials. Since then, many wellknown concepts and applications such as homogenization, cloaks, aiming, photonic band gaps, polarization conversion, and quarter-wave transformers have been studied in the temporal domain [14][15][16][17][18]. While temporal boundaries have been researched widely, several fundamental questions still remain unanswered. For example, a temporal version of Fresnel's equation has been developed in [13]. Why are the reflection and transmission coefficients different from its spatial counterpart? As a building block in electromagnetics, does Snell law also have a temporal equivalent? In [17], the authors found that the wave energy of the proposed temporal system is not constant, and attribute it to external energy sources. Is there another perspective for interpreting this energy conservation enigma? Finally, some researchers have pointed out that causality plays an important role in time-varying systems. So, what role does causality play in an actual EM system? Moreover, how does causality shed light on the asymmetry between space and time? In this paper, we attempt to answer the above questions by comparing the differences between spatial and temporal boundaries. Specifically, we consider two cases: a conventional spatial boundary in a time-invariant system, and a temporal boundary in a spatially invariant system. From Noether's theorem, these two cases correspond to energy and momentum conservation, respectively. First, we compared the momentum conservation between the two cases, and derive the temporal version of the reflection law and Snell's law. Then we revisit Fresnel's equations in the spatial domain and Morgenthaler's equations [19] in the time domain. Furthermore, we discuss the energy conservation issue and causality, which are unique to temporal systems MOMENTUM CONSERVATION (REFLECTION LAW, SNELL'S LAW) Here we reexamine the momentum conservation law and establish a framework for how it leads to different conclusions in spatial and temporal systems. Specifically, we consider two scenarios: The first one considers a time-invariant 2D space, whose refractive index is n 1 and n 2 for the region z > z 0 and z < z 0 , respectively. The second one involves an unbounded and homogeneous system where the medium undergoes a sudden permittivity or permeability change at = 0 . The electric field of the incidence wave is defined as �⃗ = 0 − ( �⃗ • ⃗− ) , where = 2 / is the angular frequency, and �⃗ is the wave vector whose magnitude \| �⃗ \| = 2 /λ is the wavenumber (i.e., spatial frequency). Here we do not specify the polarization of the wave because the system is assumed to be isotropic. The angular frequency and wave vector are closely related to the wave energy and momentum through the Planck-Einstein equation and the de Broglie relation, respectively [20]. For the first scenario, the frequency remains constant because of photon energy conservation. Considering ⃗ = / �⃗ and \| ⃗\| = / , we have: � � � �⃗ � = � � �⃗ � � � �⃗ � = � � �⃗ � ,(1) where �⃗ , �⃗ , and �⃗ represent the wave vector of the incident, transmitted, and reflected wave of angle , , and , respectively. The magnitude of the wave vector is a function of the refractive index change. To further define the direction of the wave vector, its tangential components are taken into consideration: � � � �⃗ � ( ) = � � �⃗ � ( ) � � �⃗ � ( ) = � � �⃗ � ( ) .(2) Accordingly, Eqs. (1) and (2) together comprise the reflection law and Snell's law in the spatial domain. On the other hand, for the second scenario, the wave vector �⃗ is constant due to the momentum conservation. In this case it is the frequency that depends on the refractive index change: � = − = ,(3) where , , and represent the angular frequency of the incident, forward and backward wave. It is worth noting that the backward and forward waves have negative and positive frequencies, respectively. This is because the velocity of the forward wave is in the same direction as the incident wave, while that of the backward wave is in the opposite direction. The different expressions for Snell's law manifest the fundamental asymmetry in electromagnetics between space and time. In the first scenario, the incident and reflected waves appear in the same medium. Therefore, it is the reflection angle that is equal to the incidence angle. On the other hand, in the second scenario, the forward and backward waves exist in the same medium. Consequently, the magnitude of the backward wave velocity is equal to that of the forward wave, rather than the incident wave. Moreover, Eq. (1) in the first scenario is a vector equation. To determine the direction of the wave vector, its tangential components are considered in Eq. (2). On the contrary, Eq. (3) in the second scenario is a scalar equation. The direction of velocity is decided by the plus or minus sign of the frequency. The medium is spatially homogeneous, allowing the forward and backward waves to propagate in the same (or opposite) direction as the incident wave. In other words, the incident wave is always geometrically normal to the temporal boundary, which is a consequence of the fact that the temporal boundary is perpendicular to all three spatial directions. TRANSMISSION AND REFLECTION (FRESNEL EQUATIONS) Besides Snell's law, Fresnel's equations represent another important set of rules governing EM waves. In the case of normal incidence, and considering the spatial boundary conditions at z = z 0 ,the reflection and transmission coefficients can be expressed as: = \| ��⃗ \| \| ��⃗ \| = + ,(4)= \| ��⃗ \| \| ��⃗ \| = − + ,(5) where 1 and 2 are the wave impedances in medium 1 and 1 . This yields the following condition: + = (6) This expression not only reveals that the boundary condition is implicit in Fresnel's equations, but also indicates that the incidence and reflection appear on the same side of the spatial boundary, while the transmission is on the opposite side. For a temporal boundary (the second scenario), the field continuity conditions can be expressed as [13,19]: ��⃗ = � �⃗ = ( � �⃗ + � �⃗ ),(7)��⃗ = �� ⃗ = ( �� ⃗ − �� ⃗ ).(8) Notice that the existence of a temporal boundary does not break the spatial translational symmetry. Therefore, all three components of � �⃗ and �⃗ are conserved. Hence, it can be shown that [19]: = ��⃗ ��⃗ = � + � � �,(9)= ��⃗ ��⃗ = ( − � � ),(10) where τ and ρ are defined as transmission and reflection in the temporal scenario. Eqs. (7)- (10) were first derived by Morgenthaler [19], and have been widely used to analyze the transmission and reflection of EM waves passing through a temporal boundary. They can be viewed as the analog of Fresnel's equations in the temporal domain. Interestingly, Eq. (9) and (10) can be rewritten in a more suggestive form: = � + �,(11) = � − �. Notice that the first term on the right side of Eq. (11) and (12) is equal to \| � �⃗ \|/\| � �⃗ \| and \| � �⃗ \|/\| � �⃗ \| respectively. By defining the transmission and reflection coefficients in terms of � �⃗ instead of �⃗ , allows them to be rewritten in an expression very similar to Eq. (4) and (5): = ��⃗ ��⃗ = + ,(13)= ��⃗ ��⃗ = − .(14) These two equations lead to: + = .(15) It can be clearly observed that there is a similarity between Eq. (6) and Eq. (15). In short, the temporal form of Fresnel's equations is different from their spatial counterparts, which is partly due to the different boundary conditions employed for the two scenarios. But there are deeper reasons for this difference. First, the forward and backward waves exist after the temporal boundary, while the incident wave appears before it (notice that this difference also plays a key role in the previous section). Second, it is � �⃗ , rather than �⃗ , that is continuous, but the reflection and transmission coefficients are still expressed in terms of electric fields by convention. This difference provides a complete explanation for why in temporal scenarios things become more complicated. That is, we need two independent parameters (ε, μ) to determine the transmission and reflection behavior (see Eq. (9) and (10)). While in the spatial scenario, and are only a function of the impedance . ENERGY CONSERVATION Energy conservation is implicit in all time-invariant systems. When a wave propagates through a spatial boundary, the total energy is conserved ( . ., + = ⋅ / + = ), which can be derived from Eqs. (4) and (5). On the other hand, the existence of temporal boundaries breaks the time translation symmetry, and wave energy conservation can no longer be guaranteed. This raises the following questions: does the energy change after a wave meets a temporal boundary? If so, where does the energy change come from? Let us first analyze this problem from the wave perspective of light. The total power intensity of the forward and backward waves can be defined using the Poynting vector: = \| ��⃗ × ��⃗ \| \| ��⃗ × ��⃗ \| (16) = \| ��⃗ × ��⃗ \| \| ��⃗ × ��⃗ \| (17) Adding Eqs. (16) and (17) together, while taking into consideration Eqs. (9) and (10), we find that: = + = ( )(18) where and represent the total wave power intensities before and after the temporal boundary. The power intensity decreases if < , and increases if > . Then, from the particle perspective of light, we recognize that the power intensity of EM waves is also defined as the total amount of energy passing through a surface per unit area , and per unit time : = ħ / , where is the number of photons. It further goes as: = ħ \|� �⃗\| (19) where is the photon density (number of photons per unit volume). According to the Plank-Einstein relation, the energy of each photon is proportional to the angular frequency . Then by considering Eqs. (3) and (19), we can deduce that when a wave passes through a temporal boundary, the energy and speed of each photon change simultaneously, and each of them contributes / times to the total energy change. On the other hand, the photon density remains the same. CAUSALITY Causality is a unique property relative to the time domain. An interesting example demonstrating how causality plays a role in the time domain is to consider stacking two boundaries. As in the previous section, we consider two scenarios featuring spatial and temporal boundaries respectively. Fig. 1(a) demonstrates the multi-reflection process when a wave encounters a stacked boundary in the spatial domain. The EM wave bounces back and forth between the two boundaries, and will continue this process ad infinitum if a lossless system is considered [1]. On the contrary, as Fig. 1(b) shows, there are no "bouncing" waves between two temporal boundaries, because causality forbids the waves from propagating 'backwards' in time. To be more specific, the first temporal boundary causes the incident wave to split into a forward ( ) and backward ( ) term. After a period of time, these two waves propagating in opposite directions encounter the second temporal boundary, and each of them simultaneously splits into two waves (i.e., four waves in total). Comparing with the temporal case, the total transmission in the spatial case is composed of many extra reflection terms, as depicted in the red circle in Fig. 1(a). An additional 180° phase change is brought about by this extra reflection, which is not present in the temporal case due to causality. Therefore, the transmission coefficients in the spatial and temporal cases are different: It increases in the former case while decreasing in the latter. Now, let us validate the above physical interpretation in detail. First, we analyzed a specific system involving impedance matching. In practice, the two spatial boundaries can be separated by a distance: ∆ = / ( = , , … ) , as depicted in Fig. 1(a). The total transmission and reflection can be expressed as: = ∑ ( ) ∞ = (20) = + ∑ ( ) ∞ =(21) where = /( + ) and = ( − )/( + ) ( , = , , ) is the impedance of the medium with corresponding refractive index .The total transmission and reflection can be decomposed into an infinite sum of terms � � , for which all of them are positive. Therefore, all the partial transmission terms have the same phase, while the partial reflection terms possess a 180˚ phase difference relative to the first reflection term ( ). Consequently, one can maximize the transmission while minimizing the reflection. This is the mechanism behind the well-known quarter-wavelength impedance matching technique. The temporal analog of this technique, (i.e., antireflection temporal coatings (ATCs)) was studied in [15,21,22]. The incident wave will split into four components after encountering two temporal boundaries, as shown in Fig. 1 (b). For simplicity, here we assume that = = . By letting the time duration between the two temporal boundaries satisfy the relation ∆ = − = , the forward and backward waves can be expressed as: are backward. Hence, the phase difference of the total forward and backward waves is solely determined by the optical path difference . Consequently, with the antireflection temporal coating, both the reflection and transmission will be minimized. The total transmission with and without the ATCs can be derived as: / = � � / (24) / = � + � � (25) Obviously, the following condition / < / is met if ≠ . This result again indicates that the introduction of ATCs will reduce the total transmission. Causality forbids the waves from propagating in a time-reversed direction, and consequently will not play a role in time-invariant systems. This subtle distinction partly explains why, although the quarter-wave transformer and its temporal counterpart can both eliminate reflection, the transmission behavior is very different. The example also supports our claim in the energy conservation section. For the spatial case, the total wave energy is conserved (transmission increases and reflection decreases). However, for the temporal case, the total wave energy is not conserved (both the transmission and reflection are reduced). CONCLUSION Temporal boundaries in EM seem to be a perfect dual of spatial boundaries in many ways. However, there are some fundamental differences underlying the perceived similarities. Our work has investigated and summarized several important distinctions between spatial and temporal EM boundaries, from several different perspectives. First, it has been shown that spatial and temporal symmetry correspond to momentum and energy conservation, respectively. Second, there are some unique properties in temporal systems, such as causality, that differ from spatial systems. We found that many rules governing the spatial domain (i.e., time-invariant systems) cannot be simply employed directly in the time domain, which sometimes justifies that a more in-depth investigation be carried out. Our findings provide profound insight into the asymmetry between space and time in fundamental EM systems. This work also serves as a guide to design EM systems that operate solely in the time domain, which are analogous to but different from the traditional time-invariant systems. Funding. Penn State MRSEC, Center for Nanoscale Science (NSF DMR-1420620), and DARPA/DSO Extreme Optics and Imaging (EXTREME) Program (HR00111720032). Disclosures. The authors declare no conflicts of interest. Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. Fig. 1 . 1Schematics showing the transmission and reflection process when there are two boundaries. 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[ "Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion", "Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion", "Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion", "Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion" ]	[ "Arnab Sarkar \nDepartment of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India\n", "Amna Ali \nRsRL\nDubaiUnited Arab Emirates\n", "† K Rajesh Nayak \nIndian Institute of Science Education and Research (IISER)\nBengal-741246MohanpurWestIndia\n", "A S Majumdar \nDepartment of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India\n", "Arnab Sarkar \nDepartment of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India\n", "Amna Ali \nRsRL\nDubaiUnited Arab Emirates\n", "† K Rajesh Nayak \nIndian Institute of Science Education and Research (IISER)\nBengal-741246MohanpurWestIndia\n", "A S Majumdar \nDepartment of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India\n" ]	[ "Department of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India", "RsRL\nDubaiUnited Arab Emirates", "Indian Institute of Science Education and Research (IISER)\nBengal-741246MohanpurWestIndia", "Department of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India", "Department of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India", "RsRL\nDubaiUnited Arab Emirates", "Indian Institute of Science Education and Research (IISER)\nBengal-741246MohanpurWestIndia", "Department of Astrophysics and High Energy Physics\nS. N. Bose National Centre for Basic Sciences\nJD Block\nSector III, Salt lake cityKolkata-700106India" ]	[]	We consider the accretion of dark energy by constituent black holes in binary formations during the present epoch of the Universe. In the context of an observationally consistent dark energy model, we evaluate the growth of black holes' masses due to accretion. We show that accretion leads to faster circularization of the binary orbits. We compute the average power of the gravitational waves emitted from binaries, which exhibits a considerable enhancement due to the effect of growth of masses as a result of accretion. This in turn, leads to a significant reduction of the coalescence time of the binaries. We present examples pertaining to various choices of the initial masses of the black holes in the stellar mass range and above, in order to clearly establish a possible observational signature of dark energy in the emerging era of gravitational wave astronomy.	10.1103/physrevd.107.084038	[ "https://export.arxiv.org/pdf/2210.12502v1.pdf" ]	253,098,720	2210.12502	e0365a94006a0840a8c36ac18e189be48e04b048	Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion Arnab Sarkar Department of Astrophysics and High Energy Physics S. N. Bose National Centre for Basic Sciences JD Block Sector III, Salt lake cityKolkata-700106India Amna Ali RsRL DubaiUnited Arab Emirates † K Rajesh Nayak Indian Institute of Science Education and Research (IISER) Bengal-741246MohanpurWestIndia A S Majumdar Department of Astrophysics and High Energy Physics S. N. Bose National Centre for Basic Sciences JD Block Sector III, Salt lake cityKolkata-700106India Enhanced power of gravitational waves and rapid coalescence of black hole binaries through dark energy accretion (Dated: October 25, 2022) We consider the accretion of dark energy by constituent black holes in binary formations during the present epoch of the Universe. In the context of an observationally consistent dark energy model, we evaluate the growth of black holes' masses due to accretion. We show that accretion leads to faster circularization of the binary orbits. We compute the average power of the gravitational waves emitted from binaries, which exhibits a considerable enhancement due to the effect of growth of masses as a result of accretion. This in turn, leads to a significant reduction of the coalescence time of the binaries. We present examples pertaining to various choices of the initial masses of the black holes in the stellar mass range and above, in order to clearly establish a possible observational signature of dark energy in the emerging era of gravitational wave astronomy. I. INTRODUCTION After the first direct detection of gravitational waves from a merging binary of black holes by aLIGO [1], and subsequent series of detections from similar sources [2][3][4], a new era in observational astronomy has begun. Besides binaries of compact objects in bounded orbits, there are various other mechanisms of production of gravitational waves from a wide varieties of sources, such as nearby fly-pass of two compact objects in unbounded orbits [5], gravitational collapse of sufficiently massive stars [6], cosmological phase transitions [7,8], breaking of cosmic strings [9,10], inflation and pre-heating [11,12] etc.. However, till date the observations by the aLIGO and VIRGO detectors have been carried out from mainly one type of sources, which are the binaries of compact objects, viz., black holes and neutron stars. Gravitational wave observations have been used to estimate and constrain various astrophysical and cosmological parameters associated with the generation and propagation of these gravitational waves. Among these, some important ones worth mentioning are: (i) estimating the Hubble parameter [13,14], (ii) constraining a large class of cosmological scalar-tensor theories [15,16], (iii) constraining the mass of gravitons for bimetric-gravity theories [17], (iv) investigating the state of matter inside a neutron star [18], (v) constraining higher-dimensional theories [19], and there are several others. Attempts to constrain dark energy, responsible for the accelerated expansion of the late Universe [22], have been made indirectly using gravitational wave observations, either through the estimation of the Hubble parameter, or through constraining cosmological scalar-tensor theories. Late time acceleration of expansion of the Universe is one of the most intriguing discoveries of recent times, which was directly confirmed from supernovae Ia observations in 1998 [20,21] and was also supported by various indirect probes. Many theoretical approaches have been employed to explain the current cosmic acceleration. The component of the Universe providing the required negative pressure for driving this accelerated expansion is generically called 'dark energy' [23]. As normal matter (radiation, baryonic matter or cold dark matter) is gravitationally attractive, the standard lore is to assume the presence of a relativistic fluid which is repulsive in nature, as the dark energy candidate. The simplest candidate of dark energy is the cosmological constant Λ, which is mostly consistent with cosmological observations. However, it is plagued with conceptual problems, for example, fine-tuning and coincidence problems [24], which are theoretical in nature. With a hope to address these problems, cosmologists have proposed mechanisms where the role of dark energy is played by a completely different component of the Universe, which may have a variable equation-of-state parameter. Many varieties of dark energy models have been proposed, theoretically studied and observationally constrained till now. There exist a wide class of scalar field models coupled to gravity. Among these, minimally coupled ones, called quintessence, in which cosmic acceleration is driven by the potential energy [25,26], are known to alleviate some of the problems of the cosmological constant. Scalar fields, in which the cosmic acceleration is driven by the kinetic energy, called 'k-essence' [27][28][29], have also been studied, motivated from unification and quantum gravity scenarios. Such models may further yield a consistent picture of the complete evolution, starting from the early era inflation, the subsequent dark matter domination, and finally the late time acceleration [30,31]. Other alternatives include random barotropic fluids with predetermined forms of the equation-of-state parameter, such as the Chaplygin gas models [32], string theory motivated models [33][34][35] and braneworld models [36,37]. There also exist approaches without requiring additional fields [38][39][40][41]. A major difference between the scalar field and other fluid models of dark energy with the ΛCDM model (and other approaches not requiring additional fields) is that the former type of dark energy is subjected to accretion by the black holes present in the Universe. In fact, those back holes with surroundings containing insufficiently available other forms of matter-energy for accretion, would still accrete the scalar field dark energy, which is uniformly distributed throughout the Universe. Accretion of various types of dark energy by black holes has been a subject of theoretical interest for a considerable time [42][43][44][45][46]. On the basis of various works done till date, it is widely accepted that the mass of a black hole would increase due to steady spherical accretion if the equation-ofstate parameter of the dark energy w is > −1. On the other hand, accretion would result in mass loss of a black hole, for phantom type dark energy with w < −1. If dark energy exists in the Universe in a form which can be accreted by black holes, the result would not be limited to just the change of masses of the black holes. It is expected that other phenomena associated with the black holes would also be influenced. The evolution of binaries formed with the black holes, the gravitational wave emitted from those binaries and coalescence of those binaries are some of the physical processes which get directly affected if the masses of the concerned black holes are changing continuously instead of being constant. The efficacy of the above effects, in particular, whether the modified variation of gravitational energy of the binary system could be detectable via the rate of change of the orbital radius, has been a subject of debate [47,48] in the case of spherically symmetric accretion of dark energy [42]. In the present work our motivation is to explore the problem of associated modification of black hole binary parameters due to accretion in the context of a popular k-essence model of dark energy. Specifically, we consider a string theory [51] inspired low energy effective action framework containing a dilaton scalar field [52]. The resultant k-essence dark energy scenario [53] is compatible with cosmological observations [54]. Here we study spherical accretion of the k-essence dilatonic ghost condensate dark energy by black holes. This falls within a class of models known as 'ghost condensates' [55]. Considering binary formations of black holes in the early inspiral stage, we study main aspects of evolution of the orbits, due to continuous change of masses of the component black holes of such binaries, resulting from spherical accretion of the chosen model of k-essence dark energy. More specifically, we study the pace of shrinking of such an orbit and the average power of the gravitational wave emitted from the orbit in the course of its evolution, and perform quantitative comparisons of the differences with the case when the masses of component black holes are constant. We further investigate the modification in the time required to reach the coalescence stage of such a binary, in comparison with the constant mass case. The paper is organised as follows. In section II, we study the growth of black hole mass due to accretion of the chosen model of k-essence dark energy in the late Universe. In section III, we investigate the effect of growing masses of black holes on the evolution of binaries. We compute the rate of decrease of orbital radius after circularization of orbits, and the average power of the emitted gravitational waves. We compare these results with the case of binaries with black holes of constant masses without accretion. In section IV, we estimate the reduction in the time required for reaching coalescencestage by such binaries. We present our concluding remarks in section V. II. DARK ENERGY ACCRETION BY BLACK HOLES IN A K-ESSENCE MODEL K-essence scalar fields are the dynamical dark energy models where the acceleration is driven by kinetic term in the scalar field Lagrangian. Among many k-essence models, we choose a particular string-inspired ghost condensate model, called 'k-essence dilatonic ghost condensate', which can successfully describe the cosmological evolution, while simultaneously satisfies the necessary conditions of quantum stability and sound speed [53,56]. This model has also found to be observationally consistent [54]. The condition on sound speed for any scalar field dark energy model is simply that the sound speed can not exceed the speed of light in vacuum (c) i.e. it can not have super-luminal speed. In this regard, it is worthwhile to mention that the sound speed makes an important difference between quintessence and k-essence models. While for standard quintessence models, with canonical scalar fields, the sound speed is always equal to the speed of light ; for the k-essence models it is not so. This fact of having varying sound speeds through the cosmic evolution gives various ways of distinguishing different k-essence models from one-another and from the quintessence models [57]. In fact this difference of sound speed of k-essence models with the quintessence models is one of the main reasons, for which we have chosen a k-essence model for this study. The action of k-essence scalar field ϕ, along with nonrelativistic matter and radiation, can be generally written as [27] : S = d 4 x √ −g 1 2κ 2 R + L(ϕ, X) + Sm ,(1) where κ = (8πG/3) 1/2 , R is the Ricci-scalar and L is a function of the k-essence scalar field ϕ and its kinetic energy X = −(1/2)g µν ∂µϕ∂ν ϕ. Sm is the action contributed from the non-relativistic matter and radiation. In case of the specific model considered here, the Lagrangian density is given by [53,56]: L = −X + e κλϕ X 2 M 4 ,(2) where M is a constant having the dimension of mass and λ is a constant dimensionless parameter, which is set according to stability conditions. The set of equations governing the cosmological dynamics of this k-essence model can be conveniently written in terms of three dimensionless parameters: [53,56] : x1 = κφ √ 6H , x2 = ϕ 2 e κλϕ 2M 4 , x3 = κ √ ρr √ 3H ,(3) where H is the Hubble-parameter and ρr is the density of radiation in the Universe. With these dimentionless parameters x1, x2 and x3, the evolution equations can be cast in the following autonomous form: dx1 dN = − x1 2 6(2x2 − 1) + 3 √ 6λx1x2 (6x2 − 1) + x1 2 (3 − 3x 2 1 + 3x 2 1 x2 + x3) ,(4)dx2 dN = x2 3x2(4 − √ 6λx1) − √ 6( √ 6 − λx1) 1 − 6x2 ,(5)dx3 dN = − 3 2 (−x 2 1 + x 2 1 x2 + x 2 3 3 + 1) .(6) where N = ln(a) = −ln(1+z) ; while a and z are respectively the scale-factor and the redshift. N is generally called efoldings. The advantage of these autonomous equations and the dimensionless parameters is that, these are easier to solve numerically, and various important cosmological quantities can be given in terms of these dimensionless parameters x1, x2 and x3 viz. [53,56], w ef f = −1 − 2Ḣ 3H 2 = −x 2 1 + x 2 1 x2 + 1 3 x 2 3 ,(7)wϕ = 1 − x2 1 − 3x2 ,(8)c 2 s = 2x2 − 1 6x2 − 1 ,(9)Ωϕ = −x 2 1 + 3x 2 1 x2 ,(10)ΩR = x 2 3 ,(11)ΩM = 1 + x 2 1 − 3x 2 1 x2 − x 2 3 ,(12) where w ef f and wϕ are respectively the effective equation-ofstate parameter and the equation-of-state parameter of the kessence model. cs is the sound speed of the k-essence model. Ωϕ, ΩM and ΩR are respectively the fractional densities of the dark energy, non-relativistic matter and radiation in the Universe. In the Fig. 1, the evolutions of the fractional densities Ωϕ and ΩM have been depicted w.r.t. redshift z, for a certain period in the late Universe, with the initial conditions taken as x1 = 6.0 × 10 −11 , x2 = 0.5 + (1.0 × 10 −9 ), and x3 = 0.999 at redshift z ≈ 10 6.218 and the value of λ = 0.2 [56]. One can obtain simple equations for the Hubble-parameter H and the time t in terms of the e-foldings N , using equation 7 : 1 H 2 dH dt = − 3 2 (−x 2 1 + x 2 1 x2 + 1 3 x 2 3 + 1) .(13) As dN dt = H, the L.H.S. of the equation 13 can be expressed as : 1 H 2 dH dt = 1 H 2 dH dN dN dt = 1 H dH dN . Denoting h = ln H, we get : Using the equation 14, we can write the equation 13 as: dh dN = d dN (ln H) = 1 H dH dN .(14)dh dN = − 3 2 (−x 2 1 + x 2 1 x2 + 1 3 x 2 3 + 1) ,(15)dt dN = 1 H ,(16) to get the time t. Also, for the scale factor a, we have the equation: da dN = a .(17) Solving the above equation 17 one can obtain the scale factor a and corresponding redshift from the relation 1 + z = 1/a. We numerically solve the set of equations 4, 5, 6 along with the equations 15, 16 and 17, with appropriate initial conditions. We now consider accretion of dark energy by black holes in the context of the above dark energy model. It may be noted here that the rate of accretion is affected by the sound speed of the ambient fluid. The surface of accretion is defined by the black hole horizon if there is no critical point outside the horizon [42]. The fluid being accreted by a black hole has the critical point, if its speed increases from subsonic to transonic values. From the historical development of spherical accretion by black holes starting from the pioneering work by Bondi [49], it is evident that if a black hole moves through the ambient medium with a speed much lesser than speed of light in vacuum (c), and the medium, considered as a perfect fluid, has a sound speed less than c, then the accretion radius would be larger than the black hole horizon i.e. the Schwarzschild radius [50]. In the above scenario where the sound speed of the kessence model lies in the range 0 < cs/c < 1, the time-rate of change of mass of a black hole spherically accreting the k-essence dark energy is obtained by using the accretion radius ra = m/(v 2 rel + c 2 s ) which defines the relevant surface of accretion [50]. Hence, the rate of accretion is given by dm dt = 4πA G 2 m 2 (v 2 rel + c 2 s ) 3/2 (1 + wϕ)ρϕ ,(18) where cs = dP dρ 1/2 and v rel is the relative speed of the black hole with respect to the ambient cosmic fluid being accreted. ρϕ is the background density of the k-essence dark energy, wϕ is the equation-of-state parameter of the k-essence darkenergy. A is a proportionality-factor that can be taken to be of the order of ∼ 1 [50]. Note that for v rel << cs and cs ∼ c, the equation 18 leads to the rate of change of mass derived in Ref. [42]. For the present analysis we consider that the relative speed of the moving black hole with respect to the ambient cosmic fluid, viz., the k-essence dark energy, is negligible in comparison to the sound speed of the k-essence model, i.e., v rel << cs. This is valid for most of the black holes in late Universe, as it can be seen that in the dark energy dominated Universe, the sound speed of the chosen k-essence model is of the order of ∼ 0.1c (see Fig.2). So, in the denominator on the R.H.S. of equation 18, v 2 rel can be neglected in comparison to c 2 s . Thereby, the time-rate of change of mass of a black hole due to spherical accretion of the k-essence dark energy is given by dm dt = 4πA G 2 m 2 c 3 s (1 + wϕ)ρϕ ,(19) where 2GM/c 2 s is the effective accretion radius, sometimes referred as the 'Bondi radius'. We can determine the time-rate of change of mass of a black hole due to spherical accretion of the chosen k-essence dark energy using equation 19, where ρϕ = ρT Ωϕ, and ρT = (3/8πG)H 2 is obtained by solving the equation 15 for getting the Hubble-parameter H. We depict the evolution of masses of black holes, due to accretion of the chosen kessence dark energy, with four different initial masses taken as 10, 30, 50 and 70 times of the Solar-mass (M ) respectively, with respect to the redshift z in Fig.3. It can be seen from the Fig.3 that the amount of growth in masses of the black holes due to the dark energy accretion increases with the increase in their initial masses. It may be noted that ordinary stellar-mass black holes which are usually observed by electromagnetic signals emitted by various type of astrophysical mechanisms generally have masses in the range 5 − 20 M . However, aLIGO and VIGO have detected gravitational waves from mergers of binaries with component black holes having masses from 30 M to as large as 80 M [58,59]. It is quite evident from the Fig.3 that stellar-mass black holes, having mass in the range 5 − 20 M , can grow to heavier ones by means of continuous spherical accretion of similar type of dark energy. III. POWER OF GRAVITATIONAL WAVES EMITTED FROM BINARIES The instantaneous power of gravitational radiation due to the orbital motion of two black holes of masses m1 and m2 in the quadrupole-approximation is given by [60,61] : P(t) = 8 15 G 4 c 5 M (m1m2) 2 (rmin(1 + e)) 5 (1 + e Cosφ) 4 e 2 Sin 2 φ + 12(1 + e Cosφ) 2 ,(20) where M = m1 + m2, 'e' is the eccentricity of the orbit, φ is the angular position of the reduced mass µ on the plane of the orbit in a polar-coordinate system (r, φ) with origin at the center-of-mass, and rmin is the radial distance of closest approach. In the present case, the masses are continuously changing due to accretion of dark energy. Due to this continuous time-variation of the masses of the black holes, two extra terms (having single and double time-derivatives of the masses) arise along with the main term in the amplitude of gravitational radiation [62]. However, these terms are negligible in comparison to the main term in this case. Hence, the equation 20 needs to be considered here with time-dependent masses. When the orbit is bounded, the total energy carried away by the gravitational radiation due to the relative motion of the system of two black holes within one complete cycle or time-period, is given by ∆E = T 0 P(t)dt = 2π 0 P(t) dt dφ dφ .(21) It is known that energy of gravitational waves is well-defined when the average of the energy over several time-periods of the wave is taken. Also, a compact object in a Keplarian elliptical orbit emits gravitational waves with frequencies, which are integral multiples of the frequency ω0 = GM/a 3 1/2 , where a is the semi-major axis of the elliptical orbit. Hence, the period of the gravitational waves emitted due to this orbital-motion, is a fraction of the orbital-period. Therefore, a well-defined version of the power of the emitted gravitational waves is the average of the power taken over one period of the orbit. The average of the power Pavg over one period of the orbit can be written as [60] Pavg(t) = 1 T T 0 P(t)dt = 32G 4 (m1m2) 2 M 5c 5 a 5 f (e) ,(22) where the function f (e) of eccentricity e is given by : f (e) = 1 (1 − e 2 ) 7/2 1 + 73 24 e 2 + 37 96 e 4 .(23) For a circular orbit e = 0, thereby f (e) becomes 1 and a becomes the radius of the circular orbit. Note that in case of constant masses the eccentricity of the orbit changes only due to the emission of gravitational waves. However, in the present case since the masses of the black holes are continuously changing through accretion of dark energy, the change of eccentricity would be due to two different effects: (i) growth of the masses via accretion, and (ii) loss of energy and angular momentum carried away by gravitational waves. 1 Using the rate of change of energy and angular momentum of a binary of black holes in bounded orbit, the rate of change of the semi-major axis a and eccentricity e of the orbit can be obtained as [63], da dt = − 64 5 G 3 (m1m2)M c 5 a 3 1 (1 − e 2 ) 7/2 1 + 73 24 e 2 + 37 96 e 4 ,(24) and de dt = − 304 15 G 3 (m1m2)M c 5 a 4 e (1 − e 2 ) 5/2 1 + 121 304 e 2 .(25) It may be noted that the semi-major axis a and eccentricity e, governed by the above equations 24 and 25, are averages of these quantities over one period of the orbit, not their instantaneous values, as the corresponding equations of energy and angular momentum of the system, from which these are derived, govern their averages over one period. This is quite evident from the fact that these equations 24 and 25 do not contain the phase-angle φ. From the equation 25 it follows that, if the eccentricity e becomes zero(0), then de dt = 0 implying e = constant, i.e., e remains zero. Thereby, once the orbit becomes circular, it remains circular. We solve these equations 24 and 25 numerically for different initial masses of the black holes forming binaries and orbiting in elliptical orbits with initial eccentricity ei = 0.9. We choose the combinations of initial masses of the black holes forming the binaries to be 50, 60M ; 10, 60M and 10, 20M , respectively. 1 The angular momentum of the system of two black holes is not affected due to spherical accretion of dark energy because the scalar-field dark energy model considered here does not contain angular momentum, and hence, cannot impart any angular momentum to the system. The initial semi-major axis ai of the elliptical orbit has been taken as 10 6 times of the sum of their initial Schwarzschildradii, i.e., ai = 10 6 (2GMi/c 2 ), (Mi being the initial totalmass of the black holes) so that the Keplarian-approximation holds well. The time-period of the orbit is given by : T = 2π/ω0, where the angular-frequency ω0 is given by : ω0 = GM a 3 1/2 . The fall of the eccentricities of the elliptical orbits of the binaries, for three different combinations of initial masses of the constituent black holes, is depicted in the Fig.4 (Only certain portions of the full evolution-profiles have been shown here so that the differences can be visualized clearly). It can be seen from Fig.4 that the eccentricities for the orbits of binaries, where the masses of the black holes are growing due to accretion of the chosen model of k-essence dark energy, drop faster than those where the masses are constant. Moreover, the eccentricities of binaries with larger mass black holes drop faster. After the eccentricity vanishes, i.e., circularization of the orbit is achieved, the rate of change of radius r of the circular orbit is given by dr dt = − 64 5 G 3 (m1m2)M c 5 r 3 .(26) Correspondingly, the average power of the emitted gravitational wave for the circular orbit becomes, Pavg(t) = 32G 4 (m1m2) 2 M 5c 5 r 5 .(27) We first determine the patterns of shrinking of radius r, by solving the equation 26, for the circular orbits in which two black holes of masses m1 and m2 are in binary formations, for two different cases viz., (i) when the masses are changing due to spherical accretion of dark energy described by our chosen model and (ii) when the masses are constant, for three specified combinations of initial values of m1 and m2 for each of the cases. For this, we fix the initial radius for each of the cases to be 10 5 times of the sum of the initial Schwarzschild radii of the black holes, i.e., ri = 10 5 (2GMi/c 2 ) (where Mi stands for the initial total mass of the black holes). This choice for the initial radii of the circular orbits for each case is considered to study the comparative evolution with similar initial conditions. The radii of the circular orbits for three different combinations of initial values of masses m1 and m2, and for two different cases, as mentioned above, are plotted w.r.t. redshift z in Fig.5. It can be seen from the Fig.5 that, with the increasing difference in the masses and increasing total masses of the component black holes of the binaries, the difference in rate of shrinking of the radii of the circular orbits increases. We next study the variation of the average power Pavg(t) with the evolution of the circular orbits for each of the cases. We depict the variation of the average power w.r.t. the redshift for each of the cases in Fig.6. It can be seen from the plots in Fig.6 that within the same interval, the average power of the emitted gravitational wave grows significantly higher for the binaries of black holes with growing masses, in comparison with the case when the masses of the black holes are constant. The average power of the emitted gravitational wave in case of evolving masses of black holes in the binaries, grows faster in comparison to the case of constant masses of the black holes. A certain amount of increase of the masses of the black holes of binaries results in more amplification of the average power, because of the fact that the average power of the emitted gravitational waves is proportional to the quantity µ 2 M 3 (µ being the reduced-mass of the black holes forming the binary). So, a small increment in the masses of the black holes results in a comparatively greater increase in the average power of the emitted gravitational waves. Moreover, the faster shrinking of the radius of circular orbit for increasing masses of the black holes, in comparison to the case of constant masses, also contributes to the faster growth of the average power Pavg(t) in the former case, as it is proportionl to r −5 . IV. REDUCED COALESCENCE TIME From the previous analysis we have seen that as the masses of the black holes forming the binary increases due to accretion of dark energy, the average power of the emitted gravitational waves becomes significantly higher with the evolution of the orbit, in comparison to the case of constancy of masses when there is no accretion. Since the power of the emitted gravitational wave increases with time, the binary loses energy faster and shrinks more rapidly. As a result, the time taken by a binary to coalesce is shorter when the black holes' masses are growing, than for the case of constant masses of the black holes. Let us now estimate the decrease in coalescence time-interval of a binary, due to increasing masses of the component black holes that are spherically accreting the chosen model of k-essence dark energy. For a binary constituted with black holes of constant masses, the rate of loss of energy by the binary is equal to the power of the emitted gravitational waves, i.e., Pavg = −dEavg/dt, where Eavg is the average energy of the binary. Using the expression of average power from equation 27, the time-evolution of the frequency of gravitational waves (fgw) emitted from the binary is given by [63], dfgw dt = 96 5 π 8/3 GM c 3 5/3 f 11/3 gw ,(28) where M = (m1m2) 3/5 /M 1/5 is the chirp-mass of the binary. For the case of constant masses of the component black holes of the binary, the solution of the above equation 28 can be written as [63]: fgw = 1 π 5 256 1 τ 3/8 GM c 3 −5/8 ,(29) where tc is the time of coalescence of the binary and τ = tc −t is the time-interval required by the binary to reach coalescence, from any stage of its evolution at an arbitrary time t. Using the equation 29 evaluated at an initial time ti and the relation ω 2 s i = (GM/r 3 i ), where ωs i is the initial sourcefrequency, it can be shown that the time-interval τi = tc − ti required by the binary to reach the coalescence stage from initial instant, is related to the initial radius of the circular-orbit ri, as [63] τi = 5 256 c 5 r 4 i G 3 M m1m2 .(30) Now, for the case when the masses of the black holes are changing, the counterpart of the equation 29 valid in the present case is given by f −8/3 gw = 256 5 π 8/3 G c 3 5/3 tc t M 5/3 dt .(31) Making a change of variable from t to τ in the integral tc t M 5/3 dt, we can write: tc t M 5/3 dt = τ 0 M 5/3 dτ .(32) In following the suffix 'i' denotes the corresponding initial value of the quantity at initial time t i . Next, as was done for the case of constant masses, here also we evaluate the equation 31 at initial time t i and use the relation ω 2 s i = (GMi/r 3 i ). It follows that t i or alternatively τ i satisfies the equation : τ i 0 M 5/3 dτ M 5/3 i = 5 256 c 5 r 4 i G 3 Mim1im2i ,(33) as the counterpart of the equation 30 for the case of changing masses of black holes in binaries, due to dark energy accretion. The equations 30 and 33 provide the values of the coalescence time-intervals for the two different cases, viz., constant mass and varying mass of the black holes, respectively. Therefore, when the initial masses of the component black holes and initial radii of orbits are same for both the cases, the difference of the coalescence-time intervals corresponding to the two cases is given by, ∆τi = τ i − τi.(34) In order to perform a comparative estimate of the reduction in coalescence time-intervals due to dark energy accretion for the examples of binaries studied in the present work, we fix the initial time to be same for both the cases (pertaining to constant and changing masses), and evaluate the corresponding times of coalescence. We obtain the times of coalescence for binaries of the three different combinations of initial black hole masses considered by us in the previous section, for studying the evolution of eccentricities of the elliptical orbits, shrinking of radii of the circular orbits and the average power of emitted gravitational waves. We choose the initial time at the e-folding value N = −1, or the corresponding redshift z ≈ For each of the cases, we set the initial radii ri of the circular orbits to be 10 5 times of the sum of the initial Schwarzschild radii of the black holes. We display the decrease in coalescence time-intervals for these three different examples in the Table I. From the above analysis it is evident that for binaries consisting of black holes, having masses in the stellar-mass range and few-times greater than the stellar-mass black holes, specifically those from which several merging events have been detected by the aLIGO and VIRGO detectors, the time required for coalescence gets significantly reduced due to the increase in masses of the black holes caused by accretion of the chosen model of dark energy. The magnitude of reduction in the coalescence time-interval is ∼ 10 8 years, when both the component black holes are stellar-mass black holes (note the column for the combination of 10 and 20 M in Table I). However, for larger mass black holes (having masses few times larger than stellar-mass ones) the effect of accretion of the dark energy is greater. For example, the coalescence timeinterval gets reduced by ∼ 10 9 years (see the third column in Table I), and even more if there is a significant difference in the initial masses of the constituent black holes (see the second column in Table I). Note though, that the magnitude of decrease in coalescence time-interval also depends on the initial radius of the circular orbit. V. CONCLUSIONS A variety of cosmological observations have revealed that the present Universe is undergoing a phase of accelerated expansion, and such observations lend support to dynamical dark energy models responsible for the present acceleration. The string theory inspired dilatonic scalar field model, chosen in this work, in which acceleration is driven by the scalar field kinetic energy [53], seems to be observationally consistent [54]. If dark energy exists in an accretable form, it is inevitable that the black holes existing in the present Universe would evolve by accreting it. This in turn, would have a natural imprint on the evolution of binaries constituted by the black holes, as we have shown in this work. Specifically, we have studied the effect of growth of masses of black holes due to the spherical accretion of the chosen model of k-essence dark energy on several important parameters of binaries constituted by those black holes. We have investigated the effect of changing masses of the black holes on the evolution of the binaries and the average power of the emitted gravitational waves. We have found that accretion of the chosen model of k-essence dark energy leads to rapid circularization of binary orbits in comparison to the case of constancy of masses i.e. without accretion. Further, in comparison with the constant-mass case, the average power of gravitational waves increases significantly faster due to the increase in masses of the black holes. Since the average power grows as Pavg ∝ µ 2 M 3 , a comparatively small amount of increase in masses due to accretion leads to a much larger increment of the average power of emitted gravitational waves within the concerned time-scale. Finally, we have analysed how the effect of the increase in masses of the black holes leads to the reduction in the coalescence-time intervals of black hole binaries in the stellar mass range and above. Our work establishes the fact that if dark energy is similar to scalar field models like the k-essence model considered here, then it would result in reduced coalescence-time intervals of the binaries of black holes present in the current era of the Universe. The reduction in coalescence-time intervals means increased rate of coalescences. A possible upshot of the effect of accretion of dark energy by black holes in binary formations is that if this effect is observationally detectable in the new era of gravitational wave astronomy, it can lead to independent constraints on the equation-of-state parameter w of the dark energy model. Such observations on local candidates are associated with much less noise compared to certain other dark energy observations involving all-sky surveys such as cosmic microwave background and baryon acoustic oscillations. Observations with aLIGO and VIRGO detectors should be useful in this regard, as we have demonstrated the significance of the effect for the binaries of black holes within mass-ranges, from which several merging events have been detected by these detectors. Moreover, upcoming observations using the planned futuristic detector LISA, may also be able to investigate the imprint of dark energy accretion on coalescence time-intervals for binaries of supermassive black holes formed during galaxy mergers or even extreme mass-ratio inspirals (EMRIs) also. FIG. 1 : 1Evolution of the fractional densities of k-essence dark energy denoted by Ω DE ≡ Ωϕ and non-relativistic matter denoted by Ω M , w.r.t. redshift z. The fractional density of radiation Ω R is negligible in this era of the Universe. FIG. 2 : 2The variation of the sound speed cs of the k-essence model w.r.t. redshift z. FIG. 3 : 3= 10 M ⊙ m i = 30 M ⊙ m i = 50 M ⊙ m i = 70 M ⊙ Growth of mass of black holes with various initial masses due to accretion of the k-essence dark energy w.r.t. redshift z. = 50M ⊙ , m 2 = 60M ⊙ ; changing m 1 = 50M ⊙ , m 2 = 60M ⊙ ; constant m 1 = 10M ⊙ , m 2 = 60M ⊙ ; changing m 1 = 10M ⊙ , m 2 = 60M ⊙ ; constant m 1 = 10M ⊙ , m 2 = 20M ⊙ ; changing m 1 = 10M ⊙ , m 2 = 20M ⊙ ; constant FIG. 4: Evolution of eccentricities of elliptical orbits, from the initial value 0.9, w.r.t. redshift z, for three different combinations of the initial masses of black holes, for two different cases, (i) growing masses and (ii) constant masses. = 50M ⊙ , m 2 = 60M ⊙ ; changing m 1 = 50M ⊙ , m 2 = 60M ⊙ ; constant m 1 = 10M ⊙ , m 2 = 60M ⊙ ; changingm 1 = 10M ⊙ , m 2 = 60M ⊙ ; constant m 1 = 10M ⊙ , m 2 = 20M ⊙ ; changing m 1 = 10M ⊙ , m 2 = 20M ⊙ ; constantFIG. 5: Variation of the radius r of circular orbit of two black holes in binary formation w.r.t. redshift z, in three different combinations of initial masses and two cases viz., (i) growing masses, and (ii) constant masses. ==FIG. 6 : 610M ⊙ , m 2 = 60M ⊙ ; changing m 1 = 10M ⊙ , m 2 = 60M ⊙ ; constant 10M ⊙ , m 2 = 20M ⊙ ; changing m 1 = 10M ⊙ , m 2 = 20M ⊙ ; constant Evolution of average power Pavg of the gravitational wave emitted due to the orbital-motion of two black holes in binary formation, w.r.t. redshift z, for three different combinations of initial masses and for two different (constant and changing mass) cases. A range of the full evolution profiles have been shown for visual clarity. by solving which we can get h and consequently H = e h . After solving equation 15 for the Hubble-parameter, we can simply solve the equation: 1.72. Gy Decrease in Coalescence-time ∆τi = τ i − τi 15.14 × 10 7 y 2.146 Gy 1.637 GyInitial masses 10 and 20 M 10 and 60 M 50 and 60 M Initial radius of orbit ri 8.899 × 10 9 m 20.764 × 10 9 m 32.628 × 10 9 m tc for constant masses 4.817 Gy 6.956 Gy 6.329 Gy t c for varying masses 4.665 Gy 4.81 Gy 4.693 Gy τi for constant masses 66.135 × 10 7 y 2.8 Gy 2.173 Gy τ i for varying masses 50.99 × 10 7 y 0.655 Gy 0.537 TABLE I : IReduction in coalescence time-intervals due to accretion of the chosen model of dark energy. Acknowledgements: Arnab Sarkar thanks S. N. Bose National Centre for Basic Sciences, Kol-106, under Dept. of Science and Technology, Govt. of India, for funding through institute-fellowship. . B P Abbott, LIGO-VIRGO CollaborationPhys. Rev. Lett. 11661102B. P. 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[ "A Double Machine Learning Trend Model for Citizen Science Data", "A Double Machine Learning Trend Model for Citizen Science Data" ]	[ "Daniel Fink \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Alison Johnston \nSchool of Maths and Statistics\nCentre for Research into Ecological and Environmental Modelling\nUniversity of St Andrews\nSt AndrewsUK\n", "Matt Strimas-Mackey \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Tom Auer \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Wesley M Hochachka \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Shawn Ligocki \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Lauren Oldham Jaromczyk \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Orin Robinson \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Chris Wood \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Steve Kelling \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n", "Amanda D Rodewald \nCornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA\n" ]	[ "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "School of Maths and Statistics\nCentre for Research into Ecological and Environmental Modelling\nUniversity of St Andrews\nSt AndrewsUK", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA", "Cornell Lab of Ornithology\nCornell University\n14850IthacaNYUSA" ]	[]	1. Citizen and community-science (CS) datasets are typically collected using flexible protocols. These protocols enable large volumes of data to be collected globally every year, however, the consequence is that these data typically lack the structure necessary to maintain consistent sampling across years. Thus, it is not possible to estimate accurate population trends with simple methods, because population changes over time are confounded with changes in the observation process. 2. Here we describe a novel modeling approach designed to estimate species population trends while controlling for the interannual confounding common in citizen science data. The approach is based on Double Machine Learning, a statistical framework that uses machine learning methods to estimate population change and the propensity scores used to adjust for confounding discovered in the data. Additionally, we present a simulation method to identify and adjust for residual confounding missed by the propensity scores. Machine learning makes it possible to use large feature sets to control for confounding and model heterogeneity in trends. 3. To illustrate the approach, we estimated species trends using data from the CS project eBird. We used a simulation study to assess the ability of the method to estimate spatially varying trends in the face of real-world confounding. Results showed that the trend estimates distinguished between spatially constant and spatially varying trends.There were low error rates on the estimated direction of population change (increasing/decreasing) at each location and high correlations on the estimated magnitude of population change. 4. The ability to estimate spatially explicit trends while accounting for confounding inherent in citizen science data has the potential to fill important information gaps, helping to estimate population trends for species and/or regions lacking rigorous monitoring data.	null	[ "https://export.arxiv.org/pdf/2210.15524v2.pdf" ]	253,157,684	2210.15524	58c43834bb186caa1d37fb7ec0805c7ff82dc4e4	A Double Machine Learning Trend Model for Citizen Science Data Daniel Fink Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Alison Johnston School of Maths and Statistics Centre for Research into Ecological and Environmental Modelling University of St Andrews St AndrewsUK Matt Strimas-Mackey Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Tom Auer Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Wesley M Hochachka Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Shawn Ligocki Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Lauren Oldham Jaromczyk Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Orin Robinson Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Chris Wood Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Steve Kelling Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA Amanda D Rodewald Cornell Lab of Ornithology Cornell University 14850IthacaNYUSA A Double Machine Learning Trend Model for Citizen Science Data 1 Running headline: Trend Estimation with Double Machine Learningcausal forestscausal inferencecitizen scienceconfoundingdouble machine learningmachine learningpropensity scoretrends 1. Citizen and community-science (CS) datasets are typically collected using flexible protocols. These protocols enable large volumes of data to be collected globally every year, however, the consequence is that these data typically lack the structure necessary to maintain consistent sampling across years. Thus, it is not possible to estimate accurate population trends with simple methods, because population changes over time are confounded with changes in the observation process. 2. Here we describe a novel modeling approach designed to estimate species population trends while controlling for the interannual confounding common in citizen science data. The approach is based on Double Machine Learning, a statistical framework that uses machine learning methods to estimate population change and the propensity scores used to adjust for confounding discovered in the data. Additionally, we present a simulation method to identify and adjust for residual confounding missed by the propensity scores. Machine learning makes it possible to use large feature sets to control for confounding and model heterogeneity in trends. 3. To illustrate the approach, we estimated species trends using data from the CS project eBird. We used a simulation study to assess the ability of the method to estimate spatially varying trends in the face of real-world confounding. Results showed that the trend estimates distinguished between spatially constant and spatially varying trends.There were low error rates on the estimated direction of population change (increasing/decreasing) at each location and high correlations on the estimated magnitude of population change. 4. The ability to estimate spatially explicit trends while accounting for confounding inherent in citizen science data has the potential to fill important information gaps, helping to estimate population trends for species and/or regions lacking rigorous monitoring data. structured surveys where, ideally, the same observers follow the same survey protocols at the same locations, dates and times each year. This controlled survey structure is used to minimize the interannual variation in the observation process that can lead to confounding. This same structure enables estimation using regression models that do not explicitly account for confounding (e.g. Kery & Royle, 2020;Link et al., 2020). However, these survey requirements also make it difficult to collect species-observation data at the scales necessary to monitor large groups of species across broad spatial extents, and at arbitrary times of year. Citizen science projects are collecting increasingly large volumes of data on a variety of taxa (Pocock et al., 2017); however, the opportunistic approach typically taken towards data collection makes these data susceptible to interannual changes in the observation process. For example, several studies have documented interannual variation in spatial site selection (August et al. 2020, Shirey et al. 2021, Zhang et al. 2021. Participant populations change as new participants join projects and continuing participants improve the way they conduct surveys (Johnston et al., 2022). Data collection protocols change, either through deliberate choice or for uncontrollable reasons. Examples of deliberate changes are those caused by the use of shortterm incentives or games (Xue et al., 2016), and in the long-term promotion of 'best practice' protocols (e.g. submission of complete checklists; . Uncontrollable changes include improvements to equipment, such as binoculars, the development of species identification apps, and external forces shaping observers' behaviour, such as the COVID pandemic (e.g. Hochachka et al., 2021). Studies of citizen science data have also shown how interannual changes in the observation process bias trend estimates. Bowler et al. showed how changes in the spatial site selection produce species-specific biases in trend estimates. Zhang et al. 2021 showed how bias can arise despite interannual survey structure, documenting how unexpected changes in survey censoring due to urbanization biased trends in species richness. Thus, a key challenge for using citizen science data for trend estimation is controlling for the plethora of potentially confounding sources of interannual variation. Recent analytical developments in other fields have created an opportunity to use citizen science data for trend estimation whilst controlling for interannual confounding. Double Machine Learning (DML) is a statistical framework developed to utilize generic Machine Learning (ML) methods (e.g., random forests, lasso, penalized-regression, boosted models, deep neural networks) for causal inference (Chernozhukov et al., 2018b). The DML framework has been increasingly used to estimate heterogenous treatment effects within large, feature-rich observational datasets, an important problem in many disciplines where confounding is a central concern, from economics (Athey, 2017) to personalized medicine (Obermeyer & Emanuel, 2016). We consider DML for estimating spatially explicit species population trends from citizen science data. Conceptually, DML divides trend estimation into three separate modeling tasks. The goal of the first task is to predict local population sizes averaged across the study period. To do this, a species distribution model is trained to learn how observations of species vary with a set of features (e.g. climate, landcover, search effort). The goal of the second task is to identify confounding sources of variation in the data. To do this, a propensity score model is trained, which describes how the features vary systematically over time ). In the third task, the expected population size and observation year are used as benchmark values to help isolate the trend so it can be estimated without the influence of confounding features. The inclusion of spatial features in the third prediction task creates the opportunity to generate spatially explicit trend estimates. The ability to capture trends with high resolution (e.g. landscape scale) is valuable for studying the processes affecting populations at these scales (e.g. agriculture, energy development, urbanization) (Rose et al., 2017). Developing the propensity score model, plays a critical role in identifying the patterns of interannual variation in sampling that can lead to confounding bias. While the ability to include large feature sets in the propensity score model can provide broad, detailed control for confounding, it is nevertheless important to be able to quantify the degree to which this model adequately captures the interannual variation in sampling. However, we are not aware of any standard diagnostics for assessing the effectiveness of the propensity score model. We propose a novel simulation-based diagnostic tool to help identify residual confounding, i.e. confounding sources of variation in the available feature data missed by the propensity score model. Information gained from these diagnostic simulations can be used to adjust the trend estimates. We use the DML trend model and residual confounding for a real-world application estimating population trends with data from eBird, a popular citizen science project that has been collecting bird observation data since 2002 . eBird engages large numbers of participants who each decide where, when, and how to participate. As with many other citizen science projects, the limited structure in eBird has given rise to an evolving, heterogenous observation process where interannual confounding is a central concern when estimating population trends. The goal is to estimate the average annual rate of change in breeding season abundance 2007-2021 at a 27km resolution across North America for three species of birds with different distributions and habitat preferences: wood thrush (Hylocichla mustelina), Canada warbler (Cardellina canadensis), and long-billed curlew (Numenius americanus). Speciesspecific simulation studies were used to assess overall performance and the ability of the method to estimate spatially varying trends. SECTION 2: The DML Trend Model In this section, we introduce the DML trend model and our proposed simulation-based residual confounding analysis. Then we present a simple synthetic example to demonstrate how interannual confounding can be controlled using this approach. SECTION 2.1 Double Machine Learning To estimate population trends, we begin with the model that describes variation in species abundance , the response variable (also called the outcome or label variable), as = + ( ) + . (1) The objective is to estimate parameter , the rate of change in abundance per unit time . For convenience we assume is a real-valued measure or index of species abundance, but integer counts, or binary indicators of a species' occurrence can be accommodated without loss of generality. We also assume that measures time in units of years and that is the interannual trend, but other units can be accommodated to estimate trends over different time scales without loss of generality. The function is a non-parametric function of the vector = ( ! , … , " ) consisting of the features (also called covariates or predictor variables) that capture effects that are constant across years. Features in can include both ecological process variables (e.g. habitat or climatic conditions) and observation process variables (e.g. search effort or survey time of day). The number of features, , can be large. The variable is a stochastic error term. To understand how confounding can affect trend estimation in (1) it is useful to consider an idealized data set where the features include all important sources of variation in abundance and the observations are collected as a random sample across , independently drawn each year of the study period. Under these conditions, [ \| , ] = 0 and the trend can be estimated without bias. In practice, confounding can arise when there are systematic year-to-year changes in the observation process that affect the distribution of . For example, surveys could be conducted in sites with better habitat quality over time. In such situations, biased estimates of would result from employing estimation methods that do not account for the confounding, because all inter-annual changes are ascribed to in equation (1) (Imbens & Rubin, 2015). A common strategy to account for confounding when analyzing non-experimental data uses propensity scores to adjust estimates (Rosenbaum & Rubin, 1983). In this approach a propensity score model is introduced to keep track of confounding, which can be framed here as the dependence of on features . The propensity score model is written as, = ( ) + ,(2) where is a non-parametric function of the features and is a stochastic error term where [ \| ] = 0. DML solves equations (1) and (2) using the plug-in estimator of . The plug-in estimator is constructed by substituting the conditional mean response averaged across T, (4) The residual on the left side of (4) isolates the change in abundance by regressing out yearinvariant effects of features on and the residual on the right side removes the effects of confounding by regressing out the effects of features on . This formulation motivates the plug-in estimator where ( ) and ( ) are separately predicted and then plugged into equation 4 to estimate the trend, . Chernozhukov et al. (2018) showed how the plug-in estimator can accurately estimate even when the predictions of ( ) and ( ) are noisy and suffer from regularization bias. This makes it possible to take advantage of large feature sets using generic statistical and machine learning methods (e.g. penalized regressions, lasso, random forests, boosted models, deep neural networks and ensembles of these methods). The ability to include large, rich feature sets is practically important because it can improve inference by strengthening the conditional mean model while providing for strong confounding control through the propensity score model. ( ) = [ \| ] = ( ) + ( ),(3) In this paper we used Causal Forests (Athey et al., 2019), an implementation of the DML framework that uses Random Forests (Breiman, 2001) as the machine learning model for each of the prediction tasks. Moreover, by considering the trend to be a non-parametric function of the feature vector = ( ! , … , # ), where the number of features can be large, Causal Forests extend the response model to = ( ) + ( ) + . (5) The ability to estimate trends conditional on a set of features can be used to identify and study heterogeneity in population change. For example, by including spatial features in , spatially explicit trends can be estimated. In the causal inference literature, equation 5 is known as a heterogenous treatment effect or conditional average treatment effect estimator. In statistics it is equivalent to a varying coefficient model for (Hastie & Tibshirani, 1993). (Please see SI Section S1 for a brief review of DML literature) The goal in this paper is to use DML to strengthen inference about population trends by controlling for interannual confounding, but with additional assumptions it can also be used for causal inference. The models in equations 1 and 5 are closely connected to the potential outcomes framework ) that describe the conditions necessary for casual inference. Within this framework, DML and Causal Forest estimators are unbiased and normally distributed (Athey et al., 2019;Chernozhukov et al., 2018a). Thus, unlike many machine learning methods, DML can be used for statistical and causal inference. (See SI Section S2 for more information about the potential outcomes framework.) SECTION 2.2 Residual Confounding Propensity scores provide a theoretically justified strategy to control for confounding in but in practice good performance requires that the model ( ) accurately describes the confounding. Model and data limitations can make this challenging. When ( ) fails to fully capture the confounding then there will be what we call residual confounding with respect to . Residual confounding can arise if are well suited to explain patterns of year-invariant variation in abundance but are not well suited to reveal interannual variation. For example, search duration, the amount of time spent searching for species, is often an important predictor of abundance. If the amount of intra-annual variation in search duration is large compared to the inter-annual changes, then the latter may be more difficult to predict. However, we are not aware of any standard diagnostics for assessing the effectiveness of the propensity score model. Here we propose the use of a simulation-based diagnostic for residual confounding in the DML model. First, we simulate datasets { * , * , * , * } with a known species population trend %&# while maintaining all the interannual confounding in the original features. Then to diagnose residual confounding, we look for systematic differences between %&# and * , the DML trend estimate based on the simulated data. The differences 8τ '() − τ * : can be used to adjust the estimate. Simulating data from a null model, i.e., zero-trend, while maintaining all the interannual confounding in the original features is straightforward and such data can be computed for any DML estimate in two steps: 1. Generate a synthetic feature set { * , * , * } by resampling with replacement from { , , } stratified by , and, then, 2. Compute synthetic responses * = ( * ) based on the conditional mean model (3). The first step is to generates realistic feature sets while maintaining the interannual variation in . Using the conditional mean model (3) in the second step ensures that the synthetic data have zero trend while maintaining year-invariant patterns of variation in abundance associated with . Moreover, because these synthetic data are based on a zero or null-trend model, additional assumptions (beyond those of the DML) about the unknown trend are avoided. We illustrate the residual confounding diagnostic and adjustment in the synthetic example presented in the next Section. In Section 3.4 we extend the use of residual confounding simulation to also assess DML trend estimates across a suite of simulated non-zero trends. SECTION 2.3 Synthetic Example We present a simple example to illustrate how the DML trend model uses propensity scores and the residual confounding adjustment to control for interannual confounding. To do this, we generated fully synthetic data sets { , , , } with known structure and then compared DML trend estimates based on the synthetic data sets with and without adjustments. To generate the synthetic data, the log abundance & reported on the i th survey was specified using the following response model, & = * + @ + , +-! ℎ +& + @ + , +-! +& + & D * + @ + , +-! +& F + & ,(6) where ℎ +& was the j th habitat feature, +& was the j th observation process feature, &+ was the j th trend feature, & was the year of the i th survey, and = 1, … , . Additionally, the synthetic data were constructed with two confounding sources of variation, . J = ℎ !& + !& + & ,(7) where participants 1) selected sites with more habitat type ℎ ! , and 2) spent increasing amounts of effort ! searching for species at later times . The continuous time variable K was transformed into discrete years = 0, … , 10 using empirical quantiles. Terms & and & were independent stochastic error terms.) To mimic problems where the availability of large, potentially informative feature sets motivates the use of machine learning, we set = 25 but with only three non-zero parameters, ( ! = 10, ! = 10, * = −0.05). These parameters were chosen to generate data sets with moderate confounding and where the inter-annual variation in the species population size * was relatively small compared to the intra-annual variation (Fig. 1A). These are common challenges in many real applications. For example, the spatial variation in species abundance across its range may be greater than the interannual variation of the trend itself, especially for trends over short time periods. Similarly, the heterogeneity of the citizen science observation processes may generate high levels of confounding variation. Here, the interannual increases in survey coverage and search effort (equation 7) both generate positive biases because they are associated with reporting of higher species abundance (Equation 6). The difference in slopes between and the trend fit with a simple linear regression show the effects of these biases ( Fig 1A). We generated 100 synthetic datasets and estimated trends using Causal Forests with the package grf (Tibshirani, et al. 2021). Each of the synthetic datasets had a sample size of 1100 consisting of 100 surveys per year. The 50 features = {ℎ, } were used to train the random forests used to predict the conditional means and propensity scores. The 25 trend features were used to train the random forest trend model. To assess the impact of the propensity score and residual confounding adjustments we computed four trend estimates for each synthetic data set: 1) ! : The Causal Forest trend without adjusting for the propensity scores. This was computed by using ( ) as a constant value for the propensity scores, 2) /01 : The Causal Forests trend with propensity score adjustment, 3) τ * : The Causal Forests trend with propensity score adjustment based on data simulated from the conditional mean null model (Section 2.2), 4) 23 = /01 -τ * : The Causal Forests trend with propensity score and residual confounding adjustments. Without any adjustment for confounding, trend estimates ! were strongly biased (Fig. 1B, purple distribution). Including the propensity score adjustment reduced but did not completely remove the confounding bias (Fig. 1B, dark blue distribution). The residual confounding adjustment provided additional control for confounding (light blue distribution). Together, these adjustments have debiased 23 enough for the 95% confidence interval to cover the true trend in 94% of the simulations. The differences between /01 and 23 reflect the difficulty of modeling the propensity scores and conditional means using Random Forests to identify sparse, noisy, linear signals with only moderate sample sizes. In general, the residual confounding adjustments function to ensure control of confounding when the propensity score model suffers from deficiencies. The residual confounding adjustments will be strongest when the propensity score model is weak and will tend to zero the stronger the propensity scores model is. Thus, we expect residual confounding will be most valuable when the set of features is limited, signal to noise ratios are low, or when confounding is not well understood. SECTION 3: Trend Analysis of North American Breeding Birds In addition to the synthetic example in Section 2.3 we also estimate trends based on data from eBird, a popular citizen science project that has been collecting bird observation data since 2002 . This application presents the challenges of estimating spatially explicit trends in abundance across large geographic extents in the face of confounding and temporally correlated observations. In this section we describe the eBird data, the species abundance model underlying trend estimation, the Causal Forest implementation, the residual confounding analysis, and the species-specific simulation study used assess the performance of the method. All computing was done in the R statistical computing language (R Core Team, 2019). SECTION 3.1: Data eBird is a semi-structured survey because its flexibility allows participants to collect data in the ways they choose, but ancillary data are collected that describe the variation in data collection methods. To help control for variation in observation process, we analyzed the subset of the data where participants report all bird species detected and identified during the survey period, resulting in complete checklists of bird species. This limits variation in preferential reporting rates across species and provides a basis to infer species non-detections. We also required all checklists to include key ancillary variables describing characteristics of each birdwatching event, for example the time of day and distance travelled. These variables and others can be used to adjust for variation in detection rates . We calculated trends for three species that represent a range of different breeding niches, observation processes, and processes driving population change. Wood thrush is a commonly reported bird of the deciduous forest in eastern North America. Canada warblers are a less commonly reported forest bird that breed in the boreal forests of North America. Long-billed curlew is an infrequently reported shorebird that breeds in the grasslands of the arid interior of North America. We analyzed eBird data from 2007 to 2021 within each species' previously identified breeding range and season (Fink, Auer, Johnston, Strimas-Mackey, et al., 2020). To prepare the data for the trend analysis we aggregated data using a (27km × 27km x 1week) grid based on checklist latitudes, longitudes, and dates. We computed grid cell averages for four classes of information: (1) The number of individuals of the given species reported in each grid cell was used as the response variable ( ); (2) Five observation-effort features describing how participants conducted surveys were used as features to account for variation in detection rates ; (3) 14 features describing short-term temporal variation -date, time of day, and hourly weather -were used as features to account for variation in availability for detection; (4) A suite of 57 spatial features describing the composition and configuration of landscapes in each grid cell were used to capture associations between species and elevation, topography, land & water cover, land use, hydrology, and road density. Please see the SI Section S3 for details about data and data processing. SECTION 3.2 Species Abundance Model Species expected abundance can be defined as the product of the species' occurrence rate and the expected count of the species given occurrence, within a given area and time window (Zuur et al., 2009). Based on this definition and the chain rule, the rate of change in species abundance is the sum of two terms: 1) the rate of change in the species occurrence, and 2) the rate of change in species count given occurrence. Intuitively, trends in species abundance can arise from trends in the occurrence rate (e.g., as a function of whether the habitat is even suitable for a species) and/or trends in expected counts given occurrence. We estimated each trend component with its own DML. To estimate the interannual rate of change in occurrence rate a Causal Forest was trained based on the binary response variable indicating the detection/non-detection of the species and the features. Then to estimate the interannual rate of change in the log transformed species count, a separate Causal Forest was trained based on the continuous response variable (log transformed count) and features, using the subset of surveys where the species was detected (i.e., all counts were positive). To quantify the sampling variation in abundance trend estimates arising jointly from the estimated trend in species occurrence rates and the estimated trend in species counts, given occurrence, we adopted a data resampling approach and computed an ensemble of 100 estimates. We calculated 80% confidence intervals using the lower 10th and upper 90th percentiles across the ensemble. Additionally, averaging estimates across the ensemble provides a simple way to control for overfitting (Efron, 2014). Please see the SI Section S4 for additional information about constructing the ensemble. SECTION 3.3 Causal Forest Implementation Causal Forests were fit using the grf package and were grown with 2000 trees using automatic parameter tuning for all parameters. The feature sets for the conditional mean and propensity score models included (1) observation effort, (2) short-term temporal, and (3) spatial features. We also included latitude and longitude as features in the marginal model to account for residual spatial patterns of abundance. To account for spatial variation in trends we included all spatial features in W along with latitude and longitude to account for residual patterns. Section 3.4: Residual confounding The simulation-based residual confounding assessment presented in Sections 2.2 and 2.3 was based on a simulated null model. For the eBird analysis we extended the simulations to also assess confounding bias under a range of important nonzero trend scenarios. (See Section 3.5 for a description of the scenarios.) We implemented the residual confounding diagnostics and adjustments at the species level because interannual variation in how participants conduct surveys can generate distinct biases for each species. The goal was to find a set of parameters to describe and correct for residual confounding that would generalize well for all locations in the species range, regardless of the direction, magnitude, or spatial pattern of the unknown trend. To do this we fit a linear regression model for each species, estimated using all locations within the species' range for all simulation scenarios. Predictions from this regression model were used to make residual confounding adjustments (See Figure 2). SECTION 3.5: Simulation Study The species-specific simulations were constructed to create data meeting four objectives for each species: 1) realistic patterns of year-invariant patterns of occurrence and counts on eBird checklists; 2) with specified trends in abundance; 3) including temporal correlation that arises from environmental stochasticity in population growth rates; while 4) maintaining the interannual confounding in the original eBird data. To assess the overall performance in detecting and describing spatial trend patterns, trends were simulated at a 27km × 27km spatial scale across each species range and across 10 scenarios with zero and non-zero trends varying in direction, magnitude, and spatial pattern. Magnitudes were set to <1% (weak), 3.3% (moderate) and 6.7% per year (strong) based on IUCN Red List criteria (IUCN, 2019). The spatially varying trends were constructed to vary in direction and magnitude along a gradient from the core to the edge of the species' population (Figures 3, SI-3, and SI-4). All scenarios also included temporally correlated stochasticity as expected from interannual environmental factors, an important characteristic of species abundance data. The simulations were used for two tasks: 1) The training task to compute the residual confounding estimates, and 2) The testing task to evaluate the trend estimates after accounting for residual confounding. To maintain independence among these tasks, we independently generated two sets of simulated data sets for training and testing. Each simulated training data sets contained data from one of five different simulation scenarios (one null and 4 with varying magnitude, spatially constant and variable trends). Ten datasets were independently generated for each of the 5 scenarios. Thus, the simulated training data included 50 data sets generated under 5 different trend scenarios. Fifty simulated test data were generated independently of the training data, from 5 comparable but different scenarios. See the SI S5 for details about the simulations. Figure 2: A schematic workflow for the DML abundance trend model with residual confounding adjustment. This schematic is based on the eBird analysis with numbers corresponding to the steps in Section 3.5. The residual confounding adjustment (LEFT) is based on the systematic differences between simulated trends (Step 2; expanded in Figure SI-1) and the DML estimates of these trends (Step 3, red boxes). Step 4 estimates the residual confounding coefficients by formally linking Step 2 and Step 3. The real data analysis (RIGHT) begins by calculating the DML trend estimate (Step 1) and then adjusts for residual confounding (Step 5). An ensemble of trend estimates is generated at each location, from which the point estimates and confidence intervals are calculated. Analysis workflow To compute each species' DML trend estimate with the residual confounding adjustment we followed the steps in the schematic workflow shown in Figure 2: 1) Estimate /01 using Causal Forests with propensity score adjustments based on the original data { , , , } (Sections 3.2 and 3.3), 2) Simulate data { * , * , * , * } with specified trends τ '() (Section 3.5), 3) Estimate τ * using Causal Forests with propensity score adjustment based on the simulated training data { * , * , * , * }, 4) Fit %&# = * + ! * , the residual confounding regression (Section 3.4), 5) Apply the residual confounding adjustment to the original DML estimate: 23 = * + ! /01 using simulation-based parameters ( * , ! ) to adjust /01 based on the original data. Section 3.5.1: Confounding bias in eBird To measure the strength of the confounding bias and the performance of the propensity score and residual confounding adjustments we performed a simulation analysis comparing the trend estimates, ! , the Causal Forest estimate without adjusting for the propensity scores to /01 and 23 , both computed using the workflow above. To measure the performance of the trend estimates we used the simulated test data with the specified trends 45%4 %&# as the original data. This allowed us to assess the species-level confounding bias by fitting the regression 45%4 %&# = * + ! separately for each of the estimates = ( ! , /01 and 23 ). The intercept parameter * measured the distance between a zero-trend estimate and the corresponding expected value of the simulated trend. Thus, the intercept described the bias when estimating the trend direction (increasing or decreasing), with a value of zero indicating no directional bias. The slope parameter ! measured how simulated trends scaled with the direction and magnitude of the estimated trends, with a value of 1 indicating no scaling bias. The impact of the propensity score on directional bias is assessed by comparing bias coefficients between ! and /01 , where an improvement is * moving towards 0 and ! moving towards 1. Estimates show that the propensity score adjustments reduced, though did not eliminate, directional bias for all three species (Table 1). The propensity score adjustment also reduced the directional bias for Wood thrush, with a smaller reduction for Canada warbler and a small increase for long-billed curlew (Table 1) ( Table 1). Species The impact of the residual confounding adjustment on directional bias is assessed by comparing bias coefficients between /01 and 23 . The residual confounding adjustment strongly reduced both directional and scaling bias for long billed curlew but had smaller effects on the other two species (Table 1). For Canada Warbler the residual confounding adjustment increased the magnitude of the directional bias and slightly decreased the magnitude of the scaling bias. For wood thrush the residual confounding adjustment led to a slight increase in the magnitude of the directional bias and it decreased the magnitude of the scaling bias. SECTION 3.5.2: Estimate performance Next, we assessed the performance of 23 , the DML trend estimates computed with both the propensity score and the residual confounding adjustments, using the simulated test data as the original data in the workflow and then comparing estimates with the specified trends 45%4 %&# . We evaluated the quality of the estimated trend magnitude (the average percent-per-year (PPY) rate of change in abundance 2007-21) and the trend direction (increasing/decreasing), two important inferential objectives for population monitoring. Directional errors were defined to occur when trends were estimated to be significantly different from zero but were in the opposite direction to the simulated trend. We considered estimates to be non-zero if the 80% confidence interval did not contain zero. Because directional errors varied strongly with trend magnitude ( Figure SI-1), we reported the mean directional error rate, binned into categories of trend magnitude (see Supplemental Information for more details about the directional error). We also computed Pearson's correlation between simulated and estimated trends for non-zero trend estimates. Finally, we assessed the coverage of the resampling-based uncertainty estimates as the percentage of all 27km locations where the estimated intervals contained the simulated trend value. All assessments were based on independent test set data. The mean directional error rate among non-zero trends was low for all species (Table 2). The correlations among non-zero estimates and simulated true values 45%4 %&# was high. As expected, the directional error rates increased and the correlations decreased with the volume of species' data that were non-zero counts; from wood thrush (a commonly reported species in a region with high data density), to Canada warbler (less commonly reported in regions with lower data density), to long-billed curlew (infrequently reported compared to Wood Thrush and Canada Warbler within a relatively low-data density region of the continent). Interval coverage increased with decreasing amounts of species data (note, both sample sizes and detection rates decrease among species), though it was markedly less than the nominal confidence 80% level for all species. Species Trend Scenarios SECTION 3.5.3: Identifying spatial trends Finally, we assessed model performance identifying spatial heterogeneity in trends. We compared model performance between spatially constant and spatially varying scenarios for each species (Table 2). The similarity in performance between constant and varying trends highlights the ability of the model to adapt to heterogenous trends. Figure 4 shows maps of the estimated average annual percent-per-year change in abundance from 2007-2021 for all three species based on the real data. The Wood Thrush population shows steep declines in the northeast and increases in the southwest of its breeding season population, a pattern similar to other published studies (e.g. Fink, Auer, Johnston, Ruiz-Gutierrez, et al., 2020) The estimated population change for Canada Warbler also shows spatial patterning, though the uncertainty is relatively high. Long-billed Curlew shows strong, significant range-wide declines consistent with previous analysis (Rosenberg et al., 2019). SECTION 3.6: Species trend estimates SECTION 4: DISCUSSSION Our work shows how Double Machine Learning can be used to estimate the interannual rate of population change from citizen science data that contain confounding variation through time. Simulation results showed that the propensity score adjustment reduced, though did not eliminate, confounding bias in eBird data. The simulation-based residual confounding adjustment provided further bias reductions. The resulting trend estimates accurately estimated trend direction and magnitude in most cases. These estimates were sufficiently accurate to distinguish between spatially constant and spatially varying patterns at a 27km×27km resolution, across multiple simulation scenarios. Our study also highlighted several challenges with the DLM trend model including the task of modeling propensity scores, estimating uncertainty, and accounting for temporal and spatial correlation. In this section we discuss these challenges and how DML trend models may be used for other citizen science datasets and applications. SECTION 4.1: Confounding in Citizen Science data We found that confounding bias is a challenge when estimating interannual trends with data from the citizen science project eBird. Without any correction for confounding, the simulation results showed that bias can be strong (4.7% directional bias for long-billed curlew), though it varied among species (0.6% directional bias for wood thrush). The ablation study showed that the propensity score adjustment performed by DML reduced bias, though did not completely remove it. The simulation results also demonstrated that the residual confounding adjustment made further reductions in bias. SECTION 4.1.1: Propensity Scores These results highlight the important role of the propensity scores in the DML model. Adjustments based on propensity scores have the potential to provide strong bias control, but in practice, model and data challenges can limit their effectiveness. Even with eBird, a relatively well studied citizen science data set (e.g. Johnston et al., 2019), the residual confounding detected gaps in control. This highlights the importance of future work to improve our understanding of citizen science observation processes and how they evolve over time. Improvements in models and data will be valuable to improve the application of the DML trend model. The goal of the propensity score model is to capture sources of interannual variation in the observation process that also impact the reported abundance of the species. This suggests a strategy for selecting features to include in the propensity score model where all important sources of variation in year-invariant abundance are included as the set of potential confounders. This is the strategy implicit our presentation where the features used for the conditional mean model are also used in the propensity score model. The propensity score model is not limited to this set of features, and we expect there are situations where it will be advantageous to consider others. However, we caution against indiscriminately including features in the propensity score models to avoid introducing or amplifying bias (Hernán & Robins, 2020). Again, this highlights the importance of future work to improve our understanding of citizen science observation processes and how they evolve over time. SECTION 4.1.2: Residual Confounding In general, residual confounding adjustments function to ensure control of confounding when the propensity score model suffers from deficiencies. The residual confounding adjustments will be strongest when the propensity score model is weak and will tend to zero the stronger the propensity score model is. Thus, we expect residual confounding will be most valuable when the set of features is limited, signal to noise ratios are low, or when confounding is not well understood. Critical to the success of any residual confounding analysis is the construction of the underlying simulated data. In general, the simulated data needs to have a known population trend while maintaining all the interannual confounding in the original features. The zero-trend, or null trend simulation in Section 2.2 is a convenient, general-purpose residual confounding analysis based on the conditional mean model. Thus, the applicability and performance of this approach will depend strongly on the quality of the conditional mean model. For the eBird data analysis, it was also important to assess residual confounding for non-zero and spatially structured trends. To do this we extended the residual confounding simulations to include trends that varied in direction, magnitude, and spatial patterns. Our strategy was to inform the simulation data generating process by using real data as much as possible while reducing the synthetic components (and assumptions therein) to a minimum (see for other examples of empirically driven simulations.) The synthetic components in our simulation were based on two key assumptions, 1) that populations change at the same rate across the study period, and, 2) that spatial patterning aligned with edge-core population structure. Testing these assumptions is an area for further research. SECTION 4.1.2: Inferential Scope An important goal of this paper was to investigate the use of DML to control for interannual confounding when estimating trends based on citizen science data. The results show that DML can be used to reduce confounding bias leading to more accurate estimates and stronger associative inferences. These results are in line with other efforts that seek to improve associative inferences by harnessing approaches and ideas originally developed for causal inference (Bühlmann, 2020; Cui & Athey, 2022). With additional assumptions DML can also be used for causal inference. The key assumption to make causal inference is to assert the absence of confounders that are missing from the analysis and independent of the original features (sometimes called missing, hidden, or unmeasured confounders). Practically, asserting the absence of missing confounders requires assumptions that go beyond the data in hand (Hernán & Robins, 2020). Neither the propensity score model nor the simulation-based residual confounding analysis can detect or control for missing confounders. Thus, end-users need to carefully consider the strength of their domain knowledge and the limits of inference. Section 4.2 Estimating Uncertainty Estimating uncertainty for the modelling of trends using eBird data presented two challenges that were not present in the synthetic example. The first challenge was to capture the sampling variation from both steps of the abundance hurdle model. To do this we used a resampling approach to propagate uncertainty from both estimation steps when estimating the 80% intervals. The second challenge was to assess the DML performance at estimating uncertainty based on temporally correlated counts. Temporal correlation in observed species count data can be induced by several important ecological processes, so it is important to include such correlations in the simulation study. This presents an analytical challenge the DML outcome model does not include structural components or features necessary to account for temporal correlation. The simulation results showed that confidence interval coverage for the eBird analysis was below the nominal 80% level. We believe this is caused, at least partially, by the outcome model not accounting for the temporal correlation. Nevertheless, these same simulation results demonstrated that there was strong directional error control when we used the interval estimates to identify non-zero trends, i.e., trend estimates whose intervals did not contain zero. We interpret these two results to indicate that the uncertainty may not be scaling appropriately with the magnitude of the trend. Accounting for temporal correlation in the outcome is an interesting direction for further research into use of the DML framework. Residual spatial structure is another common feature of large-scale geographic studies that is absent from the DML. Given that the processes driving population change are more likely to vary locally when data come from large geographic extents (Rose et al., 2017), an important avenue of additional research is into accounting for nonstationarity of the drivers and confounders of trends. Incorporating more powerful rules to identify non-zero trends that control for multiple comparisons (e.g. false detection rate thresholding) and spatial correlation could also serve to improve the power of the approach and the scope of inference for species with weaker trend signals like Canada warbler. Section 4.3: Other Applications In this study we showed how spatial features can be used to estimate spatial patterns of variation in the trends (Fig.3, 4, SI-3, SI-4). The ability to associate features and trends can also be used to study other patterns of variation. For example, including indicators in the trend feature set can be used to estimate the effects of different survey protocols, management actions, or policies on population change. This could be useful for conservation planning, assessment, to inform integrated analysis, or for future survey design. Moreover, by including other features in the trend model that capture other potential trend effects (e.g. changes in landcover) it is possible to study systematic management differences after accounting for changes in landcover. This may be useful for Before-After-Control-Impact studies with citizen science data where accounting for the simultaneous impacts of other management and environmental changes is a challenge (Kerr et al., 2019). Finally, learned associations between trends and features can also be used to forecast expected population changes, conditional on a given set of features values and the assumption that the underlying processes driving population change during the study period will persist into the future. Section 4.4: Other Sources of Citizen Science Data The DML trend model and the simulation-based adjustment presented here can be used with other data types and applications. The Causal Forest implementation can accommodate binary and real valued outcome variables, making it possible to estimate trends in species occurrence rates, expected trends, and other indices of abundance. For the eBird analysis presented here we used the fact that species counts were collected as part of a complete checklist of birds, which allowed us to infer the zero counts associated with non-detection. The same approach can be used with other checklist-based citizen science projects to analyse counts as well as binary response presence-absence data (e.g. birds BTO 2017; Swiss bird project; and butterflies van Swaay et al., 2008). Even when observations are not collected in the form of complete lists, for some taxa observations can be assembled into pseudo-checklists (Henckel et al., 2020;van Strien et al., 2013) making them amenable to DML trend analysis. It may also be possible to analyse presence-only data (e.g. iNaturalist.org) by carefully selecting (Valavi et al., 2021) or weighting (Fithian & Hastie, 2013) background data. However, more research will be needed to carefully consider biases and confounding associated with presence-only data (e.g. Stoudt et al., 2022). The DML trend model may even be useful for the analysis of data collected from structured surveys where confounding can arise despite survey structure (e.g. Zhang et al. 2021). Section S1: DML Literature Review Causal machine learning is a broad, active field of research (Kaddour et al., 2022) where machine learning methods are employed to reason about the factors that affect data generating processes (interventions or exposures) and what would have happened in hindsight (counterfactuals). This includes work adapting standard machine learning methods to flexibly estimate heterogenous treatment effects along a potentially large number of covariates. Recent advances for estimating heterogenous treatment effects include methods based on random forests (Athey et al., 2019;Wager & Athey, 2018) the LASSO (Chen et al., 2017), Bayesian additive regression trees (Hahn et al., 2020), boosting (Powers et al., 2018), neural networks (Shalit et al., 2017), and metalearners based on combinations of these methods (Künzel et al., 2019). DLM approaches include (Chen et al., 2017;Chernozhukov et al., 2018b;Colangelo & Lee, 2022;Foster & Syrgkanis, 2020;Jung et al., 2021;Kennedy, 2022;. Carvalho et al. (2019) and have compared the empirical performance of several of these methods. Several packages are available including the grf R package for Causal Forests the CausalML package for DML is available in R and Python (Chen et al., 2020) and the EconML package is available in Python (Battocchi, 2019). Supplemental Information for Section S2: The Potential Outcomes Framework The potential outcomes framework provides a theoretical basis for causal inference using counterfactuals (outcomes that did not occur but would likely have occurred if the cause had occurred differently) when analyzing the relationship between cause and effect under different exposure or treatment conditions. The outcome model in Equation 1 (and equation S1) is closely connected to the potential outcomes framework where we posit potential outcomes 8 & ( = 1), & ( = 2), … , & ( = ): corresponding to the species abundance & on the i th survey that would have been reported had it been conducted on different years of the study period, = (1, … , ). In this setup, the average per year rate of change in population size is estimated as the exposure or treatment effect. Under the following assumptions the trend, , can be estimated without bias in the potential outcomes framework . First, potential outcomes are independent of the observation year conditional on features , the set of potential confounders. This assumption has two important implications. The first implication is that there are no hidden or missing confounders. For this reason, this assumption is often termed the no unmeasured confounding, conditional ignorability, or unconfoundedness assumption (Rosenbaum & Rubin, 1983). Ultimately, justifying this assumption requires subject-matter knowledge that extends beyond the data in hand and can be particularly challenging to justify in applications where subjectmatter knowledge may be insufficient to identify all important confounders (Hernán & Robins, 2020). The second implication of the unconfoundedness assumption is that the propensity score model adequately captures the interannual variation necessary to achieve the conditional independence of the observation year given features . Unlike the first implication of the unconfoundedness assumption, this is a modeling assumption that can be checked with data and is the motivation the residual confounding diagnostic and adjustment proposed in Section 2.2 of the main paper. The second assumption states that each survey must have nonzero probability of being surveyed in any year during the study period = (1, … , ). This follows from the logic that if a given survey could only be conducted on a single year, then the trend, defined as a difference of potential outcomes, would be undefined. Practically, this assumption can be checked by investigating the degree of overlap or common support in the distribution of features among years in the propensity score model ). This assumption is often termed the common support assumption. The third assumption states that the outcome reported for a given survey is not affected by the year of observation assigned to other checklists. This assumption is known as the stable unit treatment value assumption (Rubin, 1980) and generally requires that treatments or exposures at one unit do not affect the outcomes for other units. Section S3: Data description In this supplement we provide a detailed description of the data and data processing, including the aggregation, used for the eBird analysis. We estimated species trends using data from eBird, a popular citizen science project that has been collecting bird observation data since 2002 . eBird is a semistructured survey because its flexibility allows participants to collect data in the ways they choose, but auxillary data are collected that describe the variation in data collection methods. There are two important components of data collection that are followed by many eBird participants. First, participants are encouraged to report all bird species detected and identified during the survey period, resulting in a complete checklist of bird species. This limits variation in preferential reporting rates across species and provides a basis to infer species non-detections. Second, participants are also encouraged to report characteristics of their birdwatching, for example the time of day and distance travelled. These variables and others can be used to adjust for variation in detection rates . The bird observation data were obtained from the citizen science project, eBird . We used a subset of data in which the time, date, and location of each survey were reported, and observers recorded the number of individuals of all bird species detected and identified during the survey period, resulting in a complete checklist of species on the survey . Only the first participant's checklist was considered for each group of linked ("shared") checklists; shared checklists are produced by the duplication of an original checklist followed by reassigning the copy to a different observer who was part of the same group of observers during the period of observation. We further restricted checklists to those collected with 'stationary' or 'traveling' protocols from January 1, 2007 to December 31, 2021. Traveling surveys were restricted to those ≤ 10km. The resultant dataset consisted of 43,814,030 checklists. Each species' trend analysis and simulations were based on breeding season subsets of this dataset. We used the species-specific breeding season range and dates published in (Fink, Auer, Johnston, Strimas-Mackey, et al., 2020). The observation effort variables were: (a) the duration spent searching for birds, (b) whether the observer was stationary or traveling, (c) the distance traveled during the search, (d) the number of people in the search party, and (e) the Checklist Calibration Index (CCI), a standardized measure indexing differences in the rate at which observers accumulate new species, controlling for geographic region, habitat and other factors that determine the number of species that potentially could be observed (Kelling et al., 2015). Note, only the first observer's CCI was associated with checklists that contain more than one observer. The temporal variables included observation time of the day, which was standardized across time zones and daylight savings times as the difference from solar noon, was used to model variation in availability for detection, e.g. variation in behavior such as participation in the dawn chorus (Diefenbach et al., 2007). The day of the year (1-366) on which the search was conducted was used to capture intra-annual variation and the year of the observation was included to account for inter-annual variation. Spatial and spatiotemporal descriptors of the local environment were variables describing elevation, topography, shorelines, islands, land cover, land use, hydrology, and road density (Meijer et al., 2018). To account for the effects of elevation and topography, each checklist location was associated with elevation (Tozer et al., 2019), eastness, and northness. These latter two topographic variables combine slope and aspect to provide a continuous measure describing geographic orientation in combination with slope at both 90m 2 and 1km 2 resolutions (Amatulli et al., 2018). Each checklist was also linked to a series of covariates derived from the NASA MODIS land cover, land use, and hydrology data; MCD12Q1 (Friedl & Sulla-Menashe, 2019). We selected this data product for its moderately high spatial resolution and annual temporal resolution to capture spatial patterns of change in land cover, land use, and hydrology. We used the FAO-Land Cover Classification System which classifies each 500m pixel into land cover one of 21 vegetative cover classes, along with additional classifications describing the land use and hydrology of each pixel. Checklists were linked to the MODIS data by year from 2001-2020, capturing inter-annual changes in land cover. The checklist data for 2021 were matched to the 2020 data, as MODIS data from after 2020 were unavailable at the time of analysis. Additionally, to delineate the interface between terrestrial, aquatic, and marine environments we used NASA MODIS land water classification MOD44W (Carroll et al., 2017, p. 44) in conjunction with 30m shoreline and island data (Sayre et al., 2019) and the elevation data described above to classify each pixel into land, ocean, inland water, and coastal areas. To identify habitat for coastal species, tidal mudflats were classified in three-year windows . Finally, hourly weather variables were assigned from the Copernicus ERA5 reanalysis product at 30km resolution (Hersbach et al., 2020). To prepare these data for the trend analysis we aggregated all checklists, separately for each year, using a spatio-temporal grid whose dimensions were (27km × 27km x 1week) based on the latitude, longitude, date, and year of each checklist. The response variable for each grid cell was the mean count of the given species averaged for all checklists within each grid cell. Grid cells without any detections of this species during any weeks during any years were removed. Averaging counts in this way strengthens abundance signals making it easier to detect trends among species that are detected less frequently and species with relatively low abundance. The feature sets were based on the variables in Table SI-1. We aggregated all observation effort variables by summing to represent the total effort within each grid cell. We also calculated the mean search duration and distance within each grid cell. Grid cell CCI values were computed as the average CCI within each grid cell weighted by search duration. The grid cell mean values were calculated for the time of day when surveys were initiated and for the day of the year when surveys were conducted. To keep track of the survey year, we also created a new variable for each grid cell indicating the unique year when surveys were conducted within the cell. All spatial and spatiotemporal variables were summarized within grid cells using two metrics. The first metric describes the mean or composition of the variable across the grid cell landscape. The second metric describes the variation or spatial configuration of the variable across the grid cell landscape. For the categorical class variables, we computed the composition as the proportion of each class within each grid cell (PLAND) and we computed the spatial configuration using an index of its edge density within each grid cell (ED) using the R package landscapemetrics (Hesselbarth et al., 2019;McGarigal et al., 2012) . For the continuous variables (elevation, eastness, and northness) we computed the median and standard deviations for the composition and configuration, respectively. SECTION S4: The Abundance Ensemble The ensemble was generated by refitting the causal forests with training datasets randomized to capture sampling variation and variation arising from the randomized spatiotemporal aggregation procedure. First, 75% of the available checklists we randomly sampled. Second, we aggregate the sampled checklists using a randomly located grid. To generate additional independence between the occurrence and count stages of the hurdle model, the training data were independently sampled and aggregated for the occurrence and count model training. We computed 10 ensemble estimates each for the occurrence and count models. To estimate trends in population abundance and their uncertainty we considered the ensemble consisting of all 100 unique combinations of occurrence and count estimates. We did not choose to compute more ensemble members because of the high computational cost of computing ensemble estimates for the simulation analyses. Section S5: eBird Simulation In this supplement we provide additional details about eBird simulation used for the residual confounding analysis and model assessment. This section includes an overview of the simulation construction and subsections on the generative model, the simulation scenarios, and the discrete-time stochastic growth rate model. Section S5.1 Simulation Overview These simulations were constructed to meet four objectives: 1) To generate realistic patterns of each species' occurrence and counts on eBird checklists; 2) with specified trends in abundance; 3) including temporal correlation that arises from environmental stochasticity in population growth rates; and 4) replicating the interannual confounding in the eBird data. The following steps were used to generate the simulated datasets for each species (Figure S1-1): 1) Train generative model to learn species' occurrence and abundance. We trained a predictive model (Fink, Auer, Johnston, Ruiz-Gutierrez, et al., 2020) to learn and then generate realistic patterns of variation in species' occurrence and counts on eBird checklists that were constant across years. To ensure that the generative model learned patterns constant across years it was trained using a random subset of the species' data with a permuted version of the observation year feature: 8 . This model learned patterns for both the eBird observation process (e.g., variation in detection rates associated with varying amounts of search effort) and species ecological processes (e.g., variation in occurrence and counts associated with environmental features). 2) Generate simulated checklists, * . To replicate the interannual variation in the original features we generated the simulated checklists * by resampling the original features with replacement, stratified by year. The stratification by year ensures that any interannual changes in sampling are reflected in the simulated checklists * . 3) Simulate species trends. Using checklists * we initialized the simulated population for the first year of the study using the expected species occurrence rates and counts predicted from the generative model. Then population dynamics were simulated based on the permuted observation year feature, 8 , using a discrete-time stochastic exponential growth model with a spatially explicit growth rate. Ten different simulation scenarios were created for each species with different patterns of population increase and decrease. The expected growth rate at each location was constant across years and varied according to the given simulation scenario. The ten simulation scenarios were generated, varying in direction, magnitude, and spatial patterns. Magnitudes were set to <1% (weak), 3.3% (moderate) and 6.7% per year (strong) based on IUCN Red List criteria . The spatially varying trends were constructed to vary in direction and magnitude along a gradient from the core to the edge of the species' population (See Figure 4 and Figures SI-2 and SI-3). Finally, we included a stochastic component of the simulated growth rate to replicate the temporal correlation expected from interannual environmental stochasticity, an important characteristic of real species abundance data. The stochastic component within each cell was drawn from a random distribution with the same cell-specific mean, with distributions varying independently among 27km grid cells and years with a standard deviation of approximately 6% per year. 4) Simulate observations. The observed checklist counts were simulated from the trajectories of expected population abundance created in step 3. The reported species detection was generated as a realization from a Bernoulli distribution based on the expected occurrence rate. Conditional on detection, we generated the reported checklist count as a realization from a Poisson distribution with rate set to be the logarithm of the expected count at each site in each year. Figure S1-1: A schematic illustration of the data generating process used to simulate species-specific data with specified trends and data-driven confounding. Each species simulation begins with a subset of data and progresses through the four steps described in Section SI-5. The data generating process was informed as much as possible by real data to capture the full range of confounding while reducing the scope of simplifying assumptions necessary to generate synthetic trends. Section S5.2: The Generative model The goal of the generative model was to train a model to predict expected patterns of species abundance based on user-specified population trends along with realistic ecological and observational patterns learned from data. To do this we used the Boosted Regression Trees (BRT) hurdle model described in Fink, Auer, Johnston, Ruiz-Gutierrez, et al., (2020). Let ( , , 5 , 9 , ) be the set of training data for a given region, season, and species where: • is the 1 vector of observed counts on the n surveys in the training data, • is the 1 vector that indicates the checklists with count greater than zero, • 5 is the matrix of k predictors that describe the ecological process, • 9 is the matrix of j predictors that describe the observation process, and • is the 1 vector of the year each survey was conducted. We begin with the BRT hurdle model for species abundance and then explain how it was modified for generative modeling. In the first step of the unmodified BRT hurdle model, a Bernoulli response BRT is trained to predict the probability of occurrence: ~ ( ) ( ) = ( 5 , 9 , ) where is the probability of occurrence and the function () is fit using boosted decision trees. In the second step, the Poisson response BRT, ~ ( ) ( ) = ( 5 , 9 , ) is trained to predict the expected counts , using the subset of the training data observed to be present, i.e. > 0. To generate simulated counts of reported birds we modify the BRT as follows. The first modification permutes the predictor variable. This ensures that the BRTs cannot learn year-to-year variation from the training data. The second modification trains the hurdle BRT with an offset constructed to impart a user-specified trend when training the Poisson response model. The modified fitting procedure begins with the Bernoulli response BRT, trained with the permuted year, ~ ( ) ( ) = ( 5 , 9 , 8 ), The Poisson response BRT is trained with the permuted year and the offset, , ~ ( ) ( ) + = ( 5 , 9 , 8 ). The offset can be considered as an adjustment to the expected observed counts, on the log-link scale. We construct the offset = ( 8 ) where () is a function of the permuted year value, 8 . Because the offset is the only source of interannual variation with respect to 8 , it forces the boosting procedure to learn the user specified population trend. Section S5.3: The Simulation Scenarios The three types of spatial trend offsets constructed were: 1) spatially constant trends, 2) spatially varying trends and 3) no trend. We used the following linear model to construct the trend offsets, = 8 + : 8 : , where controls the strength and direction of the overall year-to-year changes in the expected log count and : controls the strength of the interaction between and : , the interacting variable. To avoid scaling issues, we assume that the interacting variable has been transformed to vary between -1 and 1. Note that because an intercept is fit as part of (), we do not include an intercept term in the offset. Spatially uniform trends were generated by setting : = 0. Trends that affect a population uniformly over a region may indicate the indirect effects of broad-spatial scale processes like climate change. Spatially varying trends can be generated by setting = 0 and specifying a spatially patterned variable : to interact with 8 . To assess if spatial patterns associated with density dependent population processes can be detected, the spatially interacting variable was selected to be the land cover composition feature with highest importance score from the occurrence model and that had sufficient detections of the focal species across a range of land cover values. In this way the interacting variable functions as an index of population density. Processes like habitat loss, disease, and dispersal can interact with population density to generate spatially varying trend patterns, e.g. (Massimino et al., 2015). The spatially constant trend values were generated by using annual changes in the count part of the model on the log scale defined as values =(-0.08, -0.04, 0.04, 0.08). The largest absolute values for the spatially constant trend (-0.08 and 0.08) were selected to generate spatially constant trends of approximately 6.7% per year averaged across species' ranges. With the spatially variable trends, positive and negative directions generated inversely related trend patterns, but both varying in association with the same landcover covariate (See Figure SECTION S5.4: The Discrete-time stochastic exponential growth rate model In this supplemental section we describe the discrete-time stochastic exponential growth rate model used to incorporate population dynamics and temporal correlation into the simulated data sets. To set notation, we begin with the deterministic discrete-time exponential growth model, 4;!,% = (1 + % ) 4,% where • 4,% is an index of population size at time t and location s. In this study, 4,% is the count of the species on the checklist indexed by year t and location s, • % is the growth rate at location s. In our simulations % is constant across years. We describe the rate of population change in terms of the Percent Per Year ( ) change, so that % = /100, and • indexes years 1…T of the study period. The stochastic growth rate model can be written as 4;!,% = 4,% 4,% , where 4,% is a random variable with 8 4,% : = (1 + % ), the deterministic component of population growth, and 8 4,% :, that captures the stochastic component of population growth. To parameterize the model, it is convenient to take logarithms, 4;!,% = 4,% + 4,% . We let 4,%~ o 2,% , 2 = ( )q and write 4;!,% = 4,% + 2,% + 4 ( ), Where 2,% , = r 4,% s and 4 ( )~ o0, 2 = ( )q. Assuming that locations are indexed according to a spatial grid, the stochastic component of 4,% varies independently among spatial grid cells and among years. This process can represent environmental stochasticity like the year-to-year effects of extreme weather on reproductive success. Populations within the same grid cell experience the same environmental conditions affecting population growth. Despite the year-toyear independence of the stochastic effects, the population trajectories generated by this process will be temporally correlated within each grid cell due to the compounded effects of the stochasticity across years. To generate realizations from the discrete-time stochastic exponential growth rate model two parameters need to be specified: % describing the deterministic component of population growth and 2 ( ) describing the stochastic component of population growth. For each simulation scenario, the values of % are constructed from the generative model. To specify 2 ( ) consider the following relationship between parameters of the normal and lognormal distributions, Because % = >>? !** , the range of values in which we are interested is relatively small, i.e. closer to 0.0 than to 1.0 in magnitude. For example, consider the 6.7 and 3.2 percent per year values used as IUCN RedList criteria (IUCN, 2019) and in the simulations. It follows that the standard deviation of % will likely be small, i.e. closer to 0.0 than to 1.0. Using the assumption that these values should be small, and that 4,% = 1 + % , we know that 8 4,% : ≈ 1, t8exp( 2 = ( ): − 1 ≈ 2 ( ) and that 8 4,% : = ( % ). Using these facts and substituting into the expression for ( 4,% ) yields ( 4,% ) = 2 ( ). Thus, 100 2 ( ) is approximately the standard deviation in units of percent-per-year population growth. For all simulation scenarios we specified 100 2 ( ) to be six percent per year. We chose this value to achieve a coefficient of variation for the growth rate of approximately 1 for the strong magnitude simulation scenarios and a coefficient of variation less than one for moderate and weak scenarios. SECTION S6 Directional Error In this supplemental section we describe how we assessed estimates of the trend direction (increasing or decreasing). Directional errors were defined to occur when non-zero trends estimated the direction incorrectly. Non-zero trends were defined to occur when the 80% confidence interval did not contain zero. Filtering out non-zero locations was an important step because estimating trends with sparse, noisy data is a difficult estimation task. This can be seen from the size of the uncertainty estimates ( Figure 3 in the main text). Like many classification tasks, we expected that the ability to detect non-zero trends (a measure of power) and the ability to avoid directional errors (false positive errors) would vary with the magnitude of the trend, reflecting the underlying signal-to-noise ratio. Understanding the associations between power and error rates and trend magnitude is useful for interpreting simulation performance and generalizing to the analysis of real data. To do this we summarized the proportion of interval estimates overlapping zero and the proportion of directional errors as a function of estimated trend magnitude, binned into half percent increments. This summarization was carried out across all 27km locations for 10 realizations from each of the 5 simulation scenarios held aside for model assessment. To summarize the species-level directional error rate across simulations we chose to report the mean directional error rate, averaged by magnitude binned into half-degree increments ( Table 2 in main text). Implicitly, this measure gives each magnitude bin equal importance. This is a useful property when comparing analyses because it avoids confounding differences in the distribution of simulated trend magnitudes with the difference in error rates we are interested in. SECTION SI-7 Simulation Results for Canada warbler and long-billed curlew This section includes figures showing trend maps for each of the ten scenarios for Canada warbler and long-billed curlew. Figure SI-3: Canada Warbler Trend Simulations. All trend maps show the average annual percent-per-year change in abundance from 2007-2021 within 27km pixels (red=decline, blue=increase, white=non-significantly different from 0 at alpha =0.2), intensity (darker colors indicate stronger trends). Simulated trends show scenarios varying by direction and magnitude along rows: weak (includes trends ~\|1%/ \|), moderate (includes regions with trends ~\|3.5%/ \|), and strong trends (includes regions with trends ~\|6.7%/ \|). The columns show simulated and estimated trends for spatially constant and varying simulation scenarios. Figure SI-4: Long-Billed Curlew Trend Simulations. All trend maps show the average annual percent-per-year change in abundance from 2007-2021 within 27km pixels (red=decline, blue=increase, white=non-significantly different from 0 at alpha =0.2), intensity (darker colors indicate stronger trends). Simulated trends show scenarios varying by direction and magnitude along rows: weak (includes trends ~\|1%/ \|), moderate (includes regions with trends ~\|3.5%/ \|), and strong trends (includes regions with trends ~\|6.7%/ \|). The columns show simulated and estimated trends for spatially constant and varying simulation scenarios. into equation (1) to yield the residual-on-residual regression, 8 − ( ): = 8 − ( ): + . Figure 1 : 1Synthetic example. (A) Scatterplot of a single realization of simulated log abundances for years 0-10 with site selection and search effort confounding. The simulated trend with slope −0.05 (orange line) reflects a small portion of the overall variation in abundance. The difference in slopes between the simulated trend and the trend fit with a simple linear regression (purple line) reflects the effects of confounding. (B) Density plots across 100 realizations for trend estimates ! without any correction for confounding (purple), trend estimates /01 with the propensity score adjustment (dark blue), trend estimates 23 with the propensity score and the residual confounding adjustment (light blue) with the (orange) vertical line showing the simulated trend at −0.05. Figure 4 4shows trend maps for Wood Thrush for a single realization of each simulation scenario (See Supplemental Information Figures SI3 & SI4 for Canada warbler and long-billed curlew). These maps show how the model adapted to simulations with different directions, magnitudes, and spatial patterns. Figure 3 Wood 3Thrush Trend Simulations. All trend maps show the average annual percentper-year change in abundance from 2007-2021 within 27km pixels (red=decline, blue=increase, white=80% confidence interval contained zero), intensity (darker colors indicate stronger trends). Simulated trends show scenarios varying by direction and magnitude along rows: weak (includes trends ~\|1%/ \|), moderate (includes regions with trends ~\|3.5%/ \|), and strong trends (includes regions with trends ~\|6.7%/ \|). The columns show simulated and estimated trends for spatially constant and varying simulation scenarios. Figure 4 4Trend estimates Wood Thrush, Canada Warbler and Long-billed Curlew. All trend maps show the average annual percent-per-year change in abundance from 2007-2021 within 27km pixels (red=decline, blue=increase), intensity (darker colors indicate stronger trends). The top row shows the estimated trends, middle row shows confidence interval length, and the bottom panel shows the non-zero trends in red and blue with white in locations where 80% confidence interval contained zero. A Double Machine Learning Trend Model for Citizen Science Data Daniel Fink 1* , Alison Johnston 2 , Matt Strimas-Mackey 1 , Tom Auer 1 , Wesley M. Hochachka 1 , Shawn Ligocki 1 , Lauren Oldham Jaromczyk 1 , Orin Robinson 1 , Chris Wood 1 , Steve Kelling 1 , and Amanda D. Rodewald 1 1 Cornell Lab of Ornithology, Cornell University, Ithaca, NY 14850, USA. 2 Centre for Research into Ecological and Environmental Modelling, School of Maths and Statistics, University of St Andrews, St Andrews, UK. *Corresponding author. [email protected] Contents This supplemental Information document contains the following sections: d. The Discrete-time stochastic exponential growth rate model 6) Directional Error 7) Simulation figures for Canada warbler and long-billed curlew 8) References for Supplemental Information 2 in the main text and Figures SI-2 and SI-3). The spatially varying trend values were generated with the parameter interacting with landcover varying across values : = (-0.40, -0.20, 0.20, 0.40), with the largest values selected to generate relatively large regions within the species' range experiencing changes in population size of at least 6.7% per year. Figure SI-1 shows how the power to detect non-zero trends increases rapidly with trend magnitude and the directional error rate decreases with trend magnitude across all simulations for all three species. Figure SI- 2 2Power detecting non-zero trends and directional error rates as functions of trend magnitude. Blue lines show the power to detect non-zero trends as a function of trend magnitude binned into half Percent Per Year (PPY) increments for wood thrush (squares), Canada warbler (triangles), and long-billed curlew (circles). Red lines show the directional error rate as a function of trend magnitude binned into half Percent Per Year (PPY). Table 1: Species-level estimates of confounding bias. Slope and intercept estimates and standard errors (SE) are presented for each species for trend estimates ! without any correction for confounding, trend estimates /01 with the Propensity Score (PS) adjustment, and trend estimates 23 with the propensity score and the Residual Confounding (RC) adjustment.Estimator PS RC Intercept Intercept SE Slope Slope SE Wood thrush 23 Yes Yes 0.295 0.005 1.045 0.001 /01 Yes No 0.254 0.005 1.342 0.001 ! No No -0.610 0.005 1.480 0.001 Canada warbler 23 Yes Yes 0.641 0.011 0.931 0.002 /01 Yes No -0.212 0.010 1.159 0.002 ! 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[ "LOG-SUBHARMONICITY", "LOG-SUBHARMONICITY" ]	[ "Fiberwise Bergman Kernels ", "Vector Bundles " ]	[]	[]	In this article, we consider Bergman kernels related to modules at boundary points for singular hermitian metrics on holomorphic vector bundles, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a lower estimate of weighted L 2 integrals on sublevel sets of plurisubharmonic functions, and reprove an effectiveness result of the strong openness property of the modules.	null	[ "https://export.arxiv.org/pdf/2210.16601v2.pdf" ]	253,237,378	2210.16601	65ca59fbda841bf33c4dbf9ff29e33335176ffb5	LOG-SUBHARMONICITY 10 May 2023 Fiberwise Bergman Kernels Vector Bundles LOG-SUBHARMONICITY 10 May 2023 In this article, we consider Bergman kernels related to modules at boundary points for singular hermitian metrics on holomorphic vector bundles, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a lower estimate of weighted L 2 integrals on sublevel sets of plurisubharmonic functions, and reprove an effectiveness result of the strong openness property of the modules. Introduction It is well-known that the strong openness property of multiplier ideal sheaves (see e.g. [36,32,33,13,14,11,15,31,34,35,12,26]) has a great influence in the study of several complex variables, complex geometry and complex algebraic geometry (see e.g. [23,29,5,6,17,7,37,25,4,38,39,18,30,8]). Demailly [11,12] conjectured the strong openness property and Guan-Zhou [23] gave the proof (Jonsson-Mustaţȃ [27] proved the 2-dimensional case). In order to prove the strong openness property, Jonsson and Mustaţȃ (see [28], see also [27]) posed the following conjecture, which played an important role in their proof of 2-dimensional strong openness property: Conjecture J-M: If c F o (ψ) < +∞, 1 r 2 µ({c F o (ψ)ψ − log \|F \| < log r}) has a uniform positive lower bound independent of r ∈ (0, 1), where c F o (ψ) := sup{c ≥ 0 : \|F \| 2 e −2cψ is locally L 1 near o}, and µ is the Lebesgue measure. Guan-Zhou [24] proved Conjecture J-M by using the strong openness property. Bao-Guan-Yuan [3] (see also [19] by Guan-Mi-Yuan) gave an approach to Conjecture J-M independent of the strong openness property by establishing a concavity property of the minimal L 2 integrals with respect to a module at a boundary point of the sub-level sets, and obtained a sharp effectiveness result of Conjecture J-M meanwhile. In [1] (see also [2]), we considered Bergman kernels related to modules at boundary points of the sub-level sets, and obtained the log-subharmonicity property of the Bergman kernels. We applied the log-subharmonicity to get a lower estimate of weighted L 2 integrals on sublevel sets, and reproved the effectiveness result of strong openness property of modules at boundary points. Recently, for singular hermitian metrics on holomorphic vector bundles, Guan-Mi-Yuan ( [20]) established a concavity property of minimal L 2 integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points of the sublevel sets, inducing the strong openness property and its effectiveness result of the modules. It is natural to ask: Question 1.1. Is there an approach from optimal L 2 extension theorem to the strong openness property and its effectiveness result related to modules at boundary points for singular hermitian metrics on holomorphic vector bundles? In this article, we give an affirmative answer to Question 1.1. We recall some definitions. Let M be an n−dimensional complex manifold. Let E be a rank r holomorphic vector bundle over M and E be the conjugate of E, E * be the dual bundle of E. Recall that a section h of the vector bundle E * ⊗ E * with measurable coefficients, such that h is an almost everywhere positive definite hermitian form on E, is a measurable metric on E. And recall that we call a measurable metricĥ on E has a positive locally lower bound if for any compact subset K of M , there exists a constant C K > 0 such thatĥ > C K h 1 on K, where h 1 is a smooth metric on E. Then we recall the following definition of singular hermitian metrics on vector bundles. Definition 1.2 (see [20]). Let M , E and h be as above and Σ ⊂ M be a closed set of measure zero. Let {M j } +∞ j=1 be a sequence of relatively compact subsets of M such that M 1 ⊂⊂ M 2 ⊂⊂ . . . ⊂⊂ M j ⊂⊂ M j+1 ⊂⊂ . . . and +∞ j=1 M j = M . Assume that for each M j , there exists a sequence of hermitian metrics {h j,s } +∞ s=1 on M j of class C 2 such that lim s→+∞ h j,s = h point-wisely on M j \ Σ. Then the collection of data (M, E, Σ, M j , h, h j,s ) is called a singular hermitian metric on E. Next we recall the following singular version of Nakano positivity. Let D be a hermitian metric on M , θ be a hermitian form on T M with continuous coefficients, and (M, E, Σ, M j , h, h j,s ) be a singular hermitian metric on E. Definition 1.3 (see [20]). We write: Θ h (E) ≥ s N ak θ ⊗ Id E if the following requirements are met. For each M j , there exists a sequence of continuous functions λ j,s on M j and a continuous function λ j on M j subject to the following requirements: (1) for any x ∈ M j : \|e x \| hj,s ≤ \|e x \| hj,s+1 for any s ∈ N and any e x ∈ E x ; (2) Θ hj,s (E) ≥ N ak θ − λ j,s ω ⊗ Id E on M j ; (3) λ j,s → 0 a.e. on M j ; (4) 0 ≤ λ j,s ≤ λ j on M j for any s. If F (z) = 0 for some z ∈ M , set Ψ(z) = −T . Let E be a holomorphic vector bundle on M with rank r. Let (V, z) be a local coordinate near a point z 0 of M and E\| V is trivial. Then for any g ∈ H 0 (V, O(K M ⊗ E)), there exists a holomorphic (n, 0) formĝ on V such that g =ĝ ⊗ e locally, where e is a local section of E on V . Denote that \|g\| 2 h0 \| V := √ −1 n 2 g ∧ḡ e, e h0 , where h 0 is any (smooth or singular) metric on E. It can be checked that \|g\| 2 h0 \| V is invariant under the coordinate change and \|g\| 2 h0 is a globally defined (n, n) form on V . Note that for any t ≥ T , M t = {ψ + 2 log \|1/F \| < −t} on M \ {F = 0}. Hence M t is a Stein submanifold of M for any t ≥ T (see [16]) , and Ψ = ψ + 2 log \|1/F \| is a plurisubharmonic function on M t . Letĥ be a smooth metric on E. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Assume (M, E, Σ, M j , h, h j,s ) is a singular metric on E, and Θ h (E) ≥ s N ak 0. For any t ≥ T , denote that A 2 (M t , h) := {f ∈ H 0 (M t , O(K M ⊗ E)) : Mt \|f \| 2 h < +∞}. For any t ∈ [T, +∞) and λ > 0, denote that Ψ λ,t := λ max{Ψ + t, 0}. And for any f ∈ A 2 (M T , h), denote that f λ,t := MT \|f \| 2 h e −Ψ λ,t 1/2 . Note that f 2 T := f 2 λ,T = MT \|f \| 2 h for any λ > 0, and e λ(T −t) f 2 T ≤ f 2 λ,t ≤ f 2 T < +∞ for any t ≥ T . We will state that A 2 (M T , h) is a Hilbert space in Section 2. Denote the dual space of A 2 (M T , h) by A 2 (M T , h) * . For any ξ ∈ A 2 (M T , h) * , denote that the Bergman kernel with respect to ξ is K h ξ,Ψ,λ (t) := sup f ∈A 2 (MT ,h) \|ξ · f \| 2 f 2 λ,t for any t ∈ [T, +∞), where K h ξ,Ψ,λ (t) = 0 if A 2 (M T , h) = {0}. Denote U T := (T, +∞) + √ −1R := {w ∈ C : Re w > T } ⊂ C. We obtain the following log-subharmonicity property of the Bergman kernel K h ξ,Ψ,λ . Theorem 1.4. Assume that A 2 (M T , h) = {0}. Then log K h ξ,Ψ,λ (Re w) is subhar- monic with respect to w ∈ U T . When F ≡ 1, we have Ψ ≡ ψ on {ψ < −T }, and Theorem 1.4 induces the following corollary related to fiberwise Bergman kernels with respect to plurisubharmonic functions. Corollary 1.5. Assume that K h ξ,ψ,λ (t 0 ) ∈ (0, +∞) for some t 0 ≥ T . Then log K h ξ,ψ,λ (Re w) is subharmonic with respect to w ∈ U T . We recall some notations in [20]. Let z 0 be a point in M . Denote that J(E, Ψ) z0 := {f ∈ H 0 ({Ψ < −t}∩V, O(E)) : t ∈ R, V is a neighborhood of z 0 }, and J(E, Ψ) z0 :=J(E, Ψ) z0 / ∼, where the equivalence relation '∼' is as follows: f ∼ g ⇔ f = g on {Ψ < −t} ∩ V, where t ≫ T, V is a neighborhood of z 0 . For any f ∈J(E, Ψ) z0 , denote the equivalence class of f in J(E, Ψ) z0 by f z0 . And for any f z0 , g z0 ∈ J(E, Ψ) z0 , and (q, z 0 ) ∈ O M,z0 , define f z0 + g z0 := (f + g) z0 , (q, z 0 ) · f z0 := (qf ) z0 . It is clear that J(E, Ψ) z0 is an O M,z0 −module. For any a ≥ 0, denote that I(h, aΨ) z0 := f z0 ∈ J(E, Ψ) z0 : ∃t ≫ T, V is a neighborhood of z 0 , s.t. {Ψ<−t}∩V \|f \| 2 h e −aΨ dV M < +∞ , where dV M is a contin- uous volume form on M . Then it is clear that I(h, aΨ) z0 is an O M,z0 −submodule of J(E, Ψ) z0 . Especially, we denote that I z0 := I(ĥ, 0Ψ) z0 , whereĥ is the smooth metric on E. If z 0 ∈ t>T {Ψ < −t}, then I z0 = O(E) z0 . Let Z 0 be a subset of t>T {Ψ < −t}. Let J z0 be an O M,z0 −submodule of J(E, Ψ) z0 for any z 0 ∈ Z 0 . For any t ≥ T , denote that A 2 (M t , h) ∩ J := f ∈ A 2 (M t , h) : f z0 ∈ O(K M ) z0 ⊗ J z0 , for any z 0 ∈ Z 0 . Assume that A 2 (M T , h) ∩ J is a proper subspace of A 2 (M T , h). Using Theorem 1.4, we obtain the following concavity and monotonicity property related to K h ξ,Ψ,λ . Theorem 1.6. Assume that A 2 (M T , h) = {0}, I(h, Ψ) z0 ⊂ J z0 for any z 0 ∈ Z 0 , and ξ ∈ A 2 (M T , h) * such that ξ\| A 2 (MT ,h)∩J ≡ 0. Then − log K h ξ,Ψ,λ (t)+t is concave and increasing with respect to t ∈ [T, +∞). Let F ≡ 1, and let Z 0 ⊂ {ψ = −∞}. Additionally, we let modules I(h, ψ) z0 and J z0 be ideals of O M,z0 for any z 0 ∈ Z 0 . Then Theorem 1.6 induces the following corollary related to Bergman kernels with respect to interior points. Corollary 1.7. Assume that A 2 (M T , h) = {0}, I(h, ψ) z0 ⊂ J z0 for any z 0 ∈ Z 0 , and ξ ∈ A 2 (M T , h) * such that ξ\| A 2 (MT ,h)∩J ≡ 0. Then − log K h ξ,ψ,λ (t) +t is concave and increasing with respect to t ∈ [T, +∞). If F (z) = 0 for some z ∈ M , set Ψ(z) = −T . Let E be a holomorphic vector bundle on M with rank r. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Assume (M, E, Σ, M j , h, h j,s ) is a singular metric on E. We give the following lower estimate of L 2 integrals on sublevel sets {Ψ < −t} by Theorem 1.4 and Theorem 1.6. Corollary 1.8 (see [20]). Let f be an E−valued holomorphic (n, 0) form on {Ψ < −t 0 } for some t 0 ≥ T such that f ∈ A 2 (M t0 , h). Let z 0 ∈ M t0 , and assume that a f z0 (Ψ; h) < +∞ and Θ h (E) ≥ s N ak 0, where a f z0 (Ψ; h) := sup{a ≥ 0 : f z0 ∈ O(K M ) z0 ⊗ I(h, 2aΨ) z0 }. Then for any r ∈ (0, e −a f z 0 (Ψ;h)t0 ], we have 1 r 2 {a f z 0 (Ψ;h)Ψ<log r} \|f \| 2 h ≥ e 2a f z 0 (Ψ;h)t0 C, where C :=C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ) := inf Mt 0 \|f \| 2 h :f ∈ A 2 (M t0 , h) & (f − f ) z0 ∈ O(K M ) z0 ⊗ I + (h, 2a f z0 (Ψ; h)Ψ) z0 , and I + (h, pΨ) z0 = p ′ >p I(h, p ′ Ψ) z0 for any p > 0. Remark 1.9. In Corollary 1.8, for any z 0 ∈ M , the proof of a f z0 (Ψ; h) > 0 can be referred to [20]. When F ≡ 1, Corollary 1.8 gives a lower estimate of L 2 integrals on sublevel sets of plurisubharmonic function. Corollary 1.10. Let f be an E−valued holomorphic (n, 0) form on {ψ < −t 0 } for some t 0 ≥ T such that f ∈ A 2 (M t0 , h). Let z 0 ∈ M t0 , and assume that a f z0 (ψ; h) < +∞ and Θ h (E) ≥ s N ak 0, where a f z0 (ψ; h) := sup{a ≥ 0 : f z0 ∈ O(K M ) z0 ⊗ I(h, 2aψ) z0 }. Then for any r ∈ (0, e −a f z 0 (ψ;h)t0 ], we have 1 r 2 {a f z 0 (ψ;h)ψ<log r} \|f \| 2 h ≥ e 2a f z 0 (ψ;h)t0 C, where C :=C(ψ, h, I + (h, 2a f z0 (ψ; h)ψ) z0 , f, M t0 ) := inf Mt 0 \|f \| 2 h :f ∈ A 2 (M t0 , h) & (f − f ) z0 ∈ O(K M ) z0 ⊗ I + (h, 2a f z0 (ψ; h)ψ) z0 , and I + (h, pψ) z0 = p ′ >p I(h, p ′ ψ) z0 for any p > 0. Theorem 1.4 and Theorem 1.6 also deduce a reproof of the following effectiveness result of strong openness property of the module I(h, aΨ) z0 on vector bundles. Corollary 1.11 (see [20]). Let f be a holomorphic (n, 0) form on M t0 = {Ψ < −t 0 } for some t 0 ≥ T such that f ∈ A 2 (M t0 , h). Let z 0 ∈ M . Assume that a f z0 (Ψ; h) < +∞ and Θ h (E) ≥ s N ak 0. Let C 1 and C 2 be two positive constants. If (1) Mt 0 \|f \| 2 h e −Ψ ≤ C 1 ; (2) C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ) ≥ C 2 , then for any q > 1 satisfying θ(q) > C 1 C 2 , we have f z0 ∈ O(K M ) z0 ⊗ I(h, qΨ) z0 , where θ(q) = q q−1 e t0 . For F ≡ 1, Corollary 1.11 degenerates to the effectiveness result of strong openness property with respect to interior points. Corollary 1.12. Let f be a holomorphic (n, 0) form on M t0 = {ψ < −t 0 } for some t 0 ≥ T such that f ∈ A 2 (M t0 , h). Let z 0 ∈ M . Assume that a f z0 (ψ; h) < +∞ and Θ h (E) ≥ s N ak 0. Let C 1 and C 2 be two positive constants. If (1) Mt 0 \|f \| 2 h e −ψ ≤ C 1 ; (2) C(ψ, h, I + (h, 2a f z0 (ψ; h)ψ) z0 , f, M t0 ) ≥ C 2 , then for any q > 1 satisfying θ(q) > C 1 C 2 , we have f z0 ∈ O(K M ) z0 ⊗ I(h, qψ) z0 , where θ(q) = q q−1 e t0 . Preparations L 2 methods. We need the following optimal L 2 extension theorem, which can be referred to [21]. And for the convenience of readers, we give a proof in appendix. Let M be an n−dimensional Stein manifold. Let D = ∆ w0,r = {w ∈ C : \|w − w 0 \| < r} ⊂ U T , where w 0 ∈ U T , r > 0, and w is the coordinate on D. Let Ω := M × D be an (n + 1)−dimensional complex manifold, and p 1 , p 2 be the natural projections from Ω to D and M . Let E be a holomorphic vector bundle on M with rank r. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Assume (M, E, Σ, M j , h, h j,s ) is a singular metric on E, and Θ h (E) ≥ s N ak 0. Let E ′ := p * 2 (E) be a vector bundle over Ω. Then p * 2 (h) is a measurable metric on E ′ induced by the construction of E ′ . It can be checked that p * 2 (h) has a positive locally lower bound on E ′ , (Ω, E ′ , Σ×D, M j ×D, p * 2 (h), p * 2 (h j,s )) is a singular metric on E ′ , and Θ p * 2 (h) (E ′ ) ≥ s N ak 0. LetΨ be a bounded plurisubharmonic function on Ω. Denote thatΨ w := Ψ\| M×{w} . Let M be an n−dimensional Stein manifold. Let F ≡ 0 be a holomorphic function on M , and ψ be a plurisubharmonic function on M . Let E be a holomorphic vector bundle on M with rank r. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Denote thath := he −ψ . Let (M, E, Σ, M j ,h,h j,s ) be a singular metric on E, and assume that Θh(E) ≥ s N ak 0. Let T be a real number. Denote that ϕ := max{ψ + T, 2 log \|F \|}, and Ψ := min{ψ − 2 log \|F \|, −T }. If F (z) = 0 for some z ∈ M , set Ψ(z) = −T . The following lemma will be used to prove Theorem 1.6. Lemma 2.2 ([20]). Let t 0 ∈ (T, +∞) be arbitrary given. Let f be an E−valued holomorphic (n, 0) form on {Ψ < −t 0 } such that {Ψ<−t0} \|f \| 2 h < +∞. Then there exists an E−valued holomorphic (n, 0) formF on M such that M \|F − (1 − b t0 (Ψ))f F 2 \| 2 h e vt 0 (Ψ)−φ ≤ C M I {−t0−1<Ψ<−t0} \|f F \| 2 h , where b t0 (t) = t −∞ I {−t0−1<s<−t0} ds, v t0 (t) = t −t0 b t0 (s)ds− t 0 and C is a positive constant independent of t 0 and f . 2.2. Some lemmas about submodules of J(E, Ψ). Let F be a holomorphic function on a pseudoconvex domain D ⊂ C n containing the origin o ∈ C n . Let ψ be a plurisubharmonic function on D. Let f = (f 1 , . . . , f r ) be a holomorphic section of E := D × C r . Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Assume (D, E, Σ, D j , h, h j,s ) is a singular metric on E, and Θ h (E) ≥ s N ak 0. Let T be a real number. Denote that Ψ := min{ψ − 2 log \|F \|, −T }. If F (z) = 0 for some z ∈ D, we set Ψ(z) = −T . We recall the following lemma. Lemma 2.3 ([20]). Let J o be an O C n ,o −submodule of I(h, 0Ψ) o such that I(h, Ψ) o ⊂ J o . Assume that f o ∈ J(E, Ψ) o . Let U 0 be a Stein open neighborhood of o. Let {f j } j≥1 be a sequence of E−valued holomorphic (n, 0) forms on U 0 ∩ {Ψ < −t j } for any j ≥ 1, where t j ∈ (T, +∞). Assume that t 0 = lim j→+∞ t j ∈ [T, +∞), lim sup j→+∞ U0∩{Ψ<−tj } \|f j \| 2 h ≤ C < +∞, and (f j −f ) o ∈ J o . Then there exists a subsequence of {f j } j≥1 compactly convergent to an E−valued holomorphic (n, 0) form f 0 on {Ψ < −t 0 } ∩ U 0 which satisfies U0∩{Ψ<−t0} \|f 0 \| 2 h ≤ C, and (f 0 − f ) o ∈ J o . Let M be an n−dimensional complex manifold. Let K M be the canonical line bundle on M . Let ψ be a plurisubharmonic function on M . Let F ≡ 0 be a holomorphic function on M , and let T ∈ [−∞, +∞). Denote that Ψ := min{ψ − 2 log \|F \|, −T }. If F (z) = 0 for some z ∈ M , set Ψ(z) = −T . Let E be a holomorphic vector bundle on M with rank r. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Assume (M, E, Σ, M j , h, h j,s ) is a singular metric on E, and Θ h (E) ≥ s N ak 0. Recall that A 2 (M t , h) := {f ∈ H 0 (M t , O(K M ⊗ E)) : Mt \|f \| 2 h < +∞} for any t ≥ T . Let Z 0 be a subset of M . Let J z0 be an O M,z0 −submodule of J(E, Ψ) z0 for any z 0 ∈ Z 0 . For any t ≥ T , denote that A 2 (M t , h) ∩ J := f ∈ A 2 (M t , h) : f z0 ∈ O(K M ) z0 ⊗ J z0 , for any z 0 ∈ Z 0 . We state that A 2 (M T , h) ∩ J is a closed subspace of A 2 (M T , h) if J z0 ⊃ I(h, Ψ) z0 for any z 0 ∈ Z 0 . Lemma 2.4. Assume that J z0 ⊃ I(h, Ψ) z0 for any z 0 ∈ Z 0 . Then A 2 (M T , h) ∩ J is closed in A 2 (M T , h). Proof. Let {f j } be a sequence of E−valued holomorphic (n, 0) forms in A 2 (M T , h)∩ J, such that {f j } isMT \|f 0 \| 2 h ≤ lim inf j→+∞ MT \|f kj \| 2 h < +∞, which means that f 0 ∈ A 2 (M T , h). For any ǫ > 0, there exists N > 0 such that for any m, n > N , we have MT \|f m − f n \| 2 h < ǫ. Then for any m > N , it follows from Fatou's Lemma that MT \|f m − f 0 \| 2 h ≤ lim inf j→+∞ MT \|f m − f kj \| 2 h ≤ ǫ. This shows that {f j } converges to f 0 under the topology of A 2 (M T , h). Note that (f j ) z0 ∈ O(K M ) z0 ⊗ J z0 for any j and z 0 ∈ Z 0 . According to Lemma 2.3, we can get that (f 0 ) z0 ∈ O(K M ) z0 ⊗ J z0 for any z 0 ∈ Z 0 , which means that f 0 ∈ A 2 (M T , h) ∩ J. The we know that A 2 (M T , h) ∩ J is closed in A 2 (M T , h). Note that when Z 0 = ∅ (or J z0 = J(E, Ψ) z0 for any z 0 ∈ Z 0 ), Lemma 2.4 implies that A 2 (M T , h) is a Hilbert space. Corollary 2.5. A 2 (M T , h) is a Hilbert space. Some lemmas about functionals on A 2 (M T , h). The following two lemmas will be used in the proof of Theorem 1.4. Let M be an n−dimensional complex manifold. Let E be a holomorphic vector bundle on M with rank r. Letĥ be a smooth metric on E. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Assume (M, E, Σ, M j , h, h j,s ) is a singular metric on E, and Θ h (E) ≥ s N ak 0. Lemma 2.6. Let {f j } be a sequence in A 2 (M, h), such that M \|f j \| 2 h is uniformly bounded for any j ∈ N + . Assume that f j compactly converges to f 0 ∈ A 2 (M, h). Then for any ξ ∈ A 2 (M, h) * , lim j→+∞ ξ · f j = ξ · f 0 . Proof. For any f ∈ A 2 (M, h), denote that f 2 := M \|f \| 2 h . Let {f kj } be any subsequence of {f j }. Since A 2 (M, h) is a Hilbert space, and f kj 2 is uniformly bounded, there exists a subsequence of {f kj } (denoted by {f k l j }) weakly convergent to somef ∈ A 2 (M, h). Let {U l } be an open cover of the complex manifold M , such that E\| U l is trivial. Let (U l , w l ) be the local coordinate on each U l , and e l = (e l,1 , . . . , e l,r ) is a local section of E on U l . Then we may denote that f j = r k=1 f j,l,k dw l ⊗ e l,k , f 0 = r k=1 g 0,l,k dw l ⊗ e l,k , andf = r k=1g l,k dw l ⊗ e l,k on each U l , where f j,l,k , g 0,l,k and g l,k are holomorphic functions on U l . For any z ∈ M , denote that S z := {l : z ∈ U l }. For any l ∈ S z and k ∈ {1, . . . , r}, let ξ z,l,k be the functional defined as follows: ξ z,l,k : A 2 (M, h) −→ C f −→ f l,k (z), where f = r k=1 f l,k dw l ⊗ e l,k on U l , and f l,k is a holomorphic function on U l . It is clear that the functional ξ z,l,k ∈ A 2 (M, h) * for any z ∈ M , l ∈ S z and k ∈ {1, . . . , r}, since h has a positive locally lower bound. Then we have g 0,l,k (z) = lim j→+∞ ξ z,l,k ·f j = lim j→+∞ ξ z,l,k ·f k l j = ξ z,l ·f =g l,k (z), ∀z ∈ M, l ∈ S z , 1 ≤ k ≤ r, thus f 0 =f . It means that {f kj } has a subsequence weakly convergent to f 0 . Since {f kj } is an arbitrary subsequence of {f j }, we get that {f j } weakly converges to f 0 . In other words, for any ξ ∈ A 2 (M, h) * , lim j→+∞ ξ · f j = ξ · f 0 . Let Ω := M × D, where M is an n−dimensional complex manifold, and D is a domain in C. Let E be a holomorphic vector bundle on M with rank r. Let E ′ := E ⊠ (D × C) be a holomorphic vector bundle on Ω, here D × C is the trivial line bundle on D. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Assume (M, E, Σ, M j , h, h j,s ) is a singular metric on E, and Θ h (E) ≥ s N ak 0. Let f be an E ′ −valued holomorphic (n + 1, 0) form on Ω. For any τ ∈ D, denote that f τ := f dτ \| Mτ is an E ′ −valued holomorphic (n, 0) form on M τ , where M τ := π −1 2 (τ ) , and π 2 is the natural projection from Ω to D. Assume that D Mτ \|f τ \| 2 h dλ D < +∞, where λ D is the Lebesgue measure on D. Lemma 2.7. For any ξ ∈ A 2 (M, h) * , ξ · f τ is holomorphic with respect to τ ∈ D. Proof. We only need to prove that h(τ ) := ξ · f τ is holomorphic near any τ 0 ∈ D. Since τ 0 ∈ D, there exists r > 0 such that ∆(τ 0 , 2r) ⊂⊂ D. Then for any τ ∈ ∆(τ 0 , r), according to sub-mean value inequality of subharmonic functions, we have M \|f τ \| 2 h ≤ 1 πr 2 ∆(τ,r) M \|f \| 2 h dλ D < +∞, which implies that f τ ∈ A 2 (M, h) and there exists C > 0 such that M \|f τ \| 2 h ≤ C for any τ ∈ ∆(τ 0 , r). Let {U l } be an open cover of the complex manifold M , and (U l , w l ) be the local coordinate on each U l . For any z ∈ M , Denote that S z := {l : z ∈ U l }. And for any l ∈ S z , k ∈ {1, . . . , r}, let ξ z,l,k be the functional in the proof of Lemma 2.6. In the Hilbert space A 2 (M, h), by Riesz representation theorem, there exists φ z,l,k ∈ A 2 (M, h) such that ξ z,l,k · g = √ −1 n 2 M g, φ z,l,k h for any z ∈ M , l ∈ S z , k ∈ {1, . . . , r}. Denote that H := span{φ z,l,k : z ∈ M, l ∈ S z , 1 ≤ k ≤ r} is a closed subspace of A 2 (M, h). If H = A 2 (M, h), then the closed subspace H ⊥ = {0}. Choosing some g 0 ∈ H ⊥ with g 0 = 0, we have that for any z ∈ M , l ∈ S z , and k ∈ {1, . . . , r}, ξ z,l,k · g 0 = 0 holds. Then it is clear that g 0 = 0, which is a contradiction. Thus H = A 2 (M, h). Denote that L := span{ξ z,l,k : z ∈ M, l ∈ S z , 1 ≤ k ≤ r} ⊂ A 2 (M, h) * . Since H = A 2 (M, h), we can find a sequence {ξ j } ⊂ L ⊂ A 2 (M, h) * , such that lim k→+∞ ξ j − ξ A 2 (M,h) * = 0. It is clear that for any z ∈ M , l ∈ S z and k ∈ {1, . . . , r}, ξ z,l,k · f τ is holomorphic with respect to τ ∈ D. Then for any k, h j (τ ) := ξ j · f τ is holomorphic with respect to τ ∈ D. Besides, for any τ ∈ ∆(τ 0 , r), we have \|h j (τ ) − h(τ )\| 2 =\|(ξ j − ξ) · f τ \| 2 ≤ ξ j − ξ 2 A 2 (M,h) * M \|f τ \| 2 h ≤C ξ j − ξ 2 A 2 (M,h) * ,which means that h j uniformly converges to h on ∆(τ 0 , r). According to Weierstrass theorem, we know that h is holomorphic on ∆(τ 0 , r), i.e. near τ 0 . Then we get that ξ · f τ is holomorphic with respect to τ ∈ D. 2.4. Some properties of K h ξ,Ψ,λ (t). In this section, we prove some properties of the Bergman kernel K h ξ,Ψ,λ (t). Let ξ ∈ A 2 (M T , h) * \ {0}. We need the following lemma. Lemma 2.8. For any t ∈ [T, +∞), if K h ξ,Ψ,λ (t) ∈ (0, +∞), then there existsf ∈ A 2 (M T , h), such that K h ξ,Ψ,λ (t) = \|ξ ·f \| 2 f 2 λ,t . Proof. By the definition of K h ξ,Ψ,λ (t), there exists a sequence {f j } of E−valued holomorphic (n, 0) forms in A 2 (M T , h), such that f j λ,t = 1, and lim j→+∞ \|ξ · f j \| 2 = K h ξ,Ψ,λ (t). Then MT \|f j \| 2 h is uniformly bounded. Following from Montel's theorem, we can get a subsequence of {f j } compactly convergent to an E−valued holomorphic (n, 0) formf on M T . According to Fatou's lemma, we have f λ,t ≤ 1, and according to Lemma 2.6, we have \|ξ ·f \| 2 = K h ξ,Ψ,λ (t), thus K h ξ,Ψ,λ (t) ≤ \|ξ·f\| 2 f 2 λ,t . Note that f λ,t ≤ 1 impliesf ∈ A 2 (M T , h), which means K h ξ,Ψ,λ (t) ≥ \|ξ·f \| 2 f 2 λ,t . Then we get that K h ξ,Ψ,λ (t) = \|ξ·f\| 2 f 2 λ,t . Recall that Z 0 is a subset of M , and J z0 is an O M,z0 −submodule of J(E, Ψ) z0 such that I(h, Ψ) z0 ⊂ J z0 for any z 0 ∈ Z 0 . For any t ≥ T , recall that A 2 (M t , h) ∩ J := f ∈ A 2 (M t , h) : f z0 ∈ O(K M ) z0 ⊗ J z0 , for any z 0 ∈ Z 0 . Following from Lemma 2.4, we know that A 2 (M T , h) ∩ J is a closed subspace of A 2 (M T , h). Let f ∈ A 2 (M T , h), such that f / ∈ A 2 (M T , h) ∩ J. Recall the minimal L 2 integral ( [20]) related to J as follows: C(Ψ, h, J, f, M T ) := inf MT \|f \| 2 h : (f − f ) z0 ∈ (O(K M )) z0 ⊗ J z0 for any z 0 ∈ Z 0 &f ∈ H 0 (M T , O(K M ⊗ E)) . Then the following lemma holds. ξ∈A 2 (MT ,h) * \{0} ξ\| A 2 (M T ,h)∩J ≡0 \|ξ · f \| 2 K h ξ,Ψ,λ (T ) . (2.1) Proof. Denote that (f − f ) ∈ J if (f − f ) z0 ∈ (O(K M )) z0 ⊗ J z0 for any z 0 ∈ Z 0 . Note that ξ ·f = ξ · f for anyf ∈ A 2 (M T , h) with (f − f ) ∈ J, and ξ ∈ A 2 (M T , h) * satisfying ξ\| A 2 (MT ,h)∩J ≡ 0. Then we have K h ξ,Ψ,λ (T ) = sup g∈A 2 (MT ,h) \|ξ · g\| 2 MT \|g\| 2 h ≥ sup f ∈A 2 (MT ,h) (f −f )∈J \|ξ ·f \| 2 MT \|f \| 2 h = sup f ∈A 2 (MT ,h) (f −f )∈J \|ξ · f \| 2 MT \|f \| 2 h . Thus we get that sup ξ∈A 2 (MT ,h) * \{0} ξ\| A 2 (M T ,h)∩J ≡0 \|ξ · f \| 2 K h ξ,Ψ,λ (T ) ≤ inf f ∈A 2 (MT ,h) (f −f )∈J MT \|f \| 2 h =C(Ψ, h, J, f, M T ). Since A 2 (M T , h) is a Hilbert space, and A 2 (M T , h) ∩ J is a closed proper sub- space of A 2 (M T , h), there exists a closed subspace H of A 2 (M T , h) such that H = (A 2 (M T , h) ∩ J) ⊥ = {0} . Then for f ∈ A 2 (M T , h), we can make the decomposition f = f J + f H , such that f J ∈ A 2 (M T , h) ∩ J, and f H ∈ H. Note that the linear functional ξ f defined as follows: ξ f · g := MT g, f H h , ∀g ∈ A 2 (M T , h), satisfies that ξ f ∈ A 2 (M T , h) * \ {0} and ξ f \| A 2 (MT ,h)∩J ≡ 0. Then we have sup ξ∈A 2 (MT ,h) * \{0} ξ\| A 2 (M T ,h)∩J ≡0 \|ξ · f \| 2 K h ξ,Ψ,λ (T ) ≥ \|ξ f · f \| 2 K h ξ f ,Ψ,λ (T ) . Besides, we can know that K h ξ f ,Ψ,λ (T ) = sup u∈A 2 (MT ,h) \| MT u, f H h \| 2 MT \|u\| 2 h ≤ MT \|f H \| 2 h , and ξ f · f = ξ f · (f J + f H ) = ξ f · f H = MT \|f H \| 2 h . Then we have \|ξ f · f \| 2 K h ξ f ,Ψ,λ (T ) ≥ MT \|f H \| 2 h ≥ C(Ψ, h, J, f, M T ), which implies that sup ξ∈A 2 (MT ,h) * \{0} ξ\| A 2 (M T ,h)∩J ≡0 \|ξ · f \| 2 K h ξ,Ψ,λ (T ) ≥ C(Ψ, h, J, f, M T ). Lemma 2.9 is proved. Let ξ ∈ A 2 (M T , h) * , and recall that the Bergman kernel related to ξ is K h ξ,Ψ,λ (t) := sup f ∈A 2 (MT ,h) \|ξ · f \| 2 f 2 λ,t for any t ∈ [T, +∞) and λ > 0. We state the following Lemma. Lemma 2.10. K h ξ,Ψ,λ (t) is upper-semicontinuous with respect to t ∈ [T, +∞), i.e., for any sequence {t j } ∞ j=1 in [T, +∞) such that lim j→+∞ t j = t 0 ∈ [T, +∞), we have lim sup j→+∞ K h ξ,Ψ,λ (t j ) ≤ K h ξ,Ψ,λ (t 0 ). Proof. Denote that K(t) := K h ξ,Ψ,λ (t) for any t ∈ [T, +∞). It can be seen that e λ(s−t) f 2 λ,s ≤ f 2 λ,t ≤ f 2 λ,s for any t > s ≥ T and f ∈ A 2 (M T , h). Note that K(s) = 0 for some s ≥ T induces K(t) = 0 for any t ≥ T . Then it suffices to prove Lemma 2.10 for K(t 0 ) ∈ (0, +∞) and K(t j ) ∈ (0, +∞), ∀j ∈ N + . We assume that {t kj } is the subsequence of {t j } such that lim j→+∞ K(t kj ) = lim sup j→+∞ K(t j ). By Lemma 2.8, there exists a sequence of E−valued holomorphic (n, 0) forms {f j } on M T such that f j ∈ A 2 (M T , h), f j λ,tj = 1, and \|ξ · f j \| 2 = K(t j ), for any j ∈ N + . Since {t j } is bounded in C, there exists some s 0 < +∞, such that t j < s 0 for any j, which implies that MT \|f j \| 2 h ≤ e λ(s0−T ) f j 2 λ,tj = e λ(s0−T ) , ∀j ∈ N + . Then following from Montel's theorem, we can get a subsequence of {f kj } (denoted by {f kj } itself) compactly convergent to an E−valued holomorphic (n, 0) form f 0 on M T . According to Fatou's lemma, we have f 0 λ,t0 = MT \|f 0 \| 2 h e −λ max{Ψ+t0,0} = MT lim j→+∞ \|f kj \| 2 h e −λ max{Ψ+t k j ,0} ≤ lim inf j→+∞ MT \|f kj \| 2 h e −λ max{Ψ+t k j ,0} = lim inf j→+∞ f kj λ,tj = 1. h). Lemma 2.6 shows that \|ξ·f 0 \| 2 = lim j→+∞ \|ξ·f kj \| 2 = lim sup j→+∞ K(t j ). Thus Then MT \|f 0 \| 2 h ≤ e λ(t0−T ) f 0 2 λ,t0 ≤ e λ(s0−T ) < +∞, which implies that f 0 ∈ A 2 (M T ,K(t 0 ) ≥ \|ξ · f 0 \| 2 f 0 2 λ,t0 ≥ lim sup j→+∞ K(t j ), which means that K(t) is upper semi-continuous with respect to t ∈ [T, +∞). Proof of Theorem 1.4 We prove Theorem 1.4 by using Lemma 2.1. Proof of Theorem 1.4. Denote that Ω := M T × U T . Denote that π 1 , π 2 are the natural projections from Ω to M T and U T . Let E ′ := π * 1 (M T ) be a vector bundle on Ω. LetΨ := λ max{Ψ(z) + Re w, 0} for any (z, w) ∈ Ω with z ∈ M T and w ∈ U T . ThenΨ is a plurisubharmonic function on Ω T := M T × U T , where it can be seen that Ω is a Stein manifold. Denote that K(w) := K h ξ,Ψ,λ (Re w) for any w ∈ U T . We prove that log K(w) is a subharmonic function with respect to w ∈ U T . Firstly we prove that log K(w) is upper semicontinuous. Let w j ∈ U T such that lim λ→+∞ w j = w 0 ∈ U T . Then lim j→+∞ Re w j = Re w 0 ∈ [T, +∞). Following from Lemma 2.10, we get that lim sup j→+∞ log K(w j ) ≤ log K(w 0 ). Thus log K(w) is upper semicontinuous with respect to w ∈ U T . Secondly we prove that log K(w) satisfies the sub-mean value inequality on U T . Let w 0 ∈ U T , and ∆(w 0 , r) ⊂ U T be the disc centered at w 0 with radius r. Let Ω ′ := M T × ∆(w 0 , r) ⊂ Ω be a submanifold of Ω. Let f 0 ∈ A 2 (M T , h) such that K(w 0 ) = \|ξ · f 0 \| 2 f 0 2 λ,Re w0 by Lemma 2.8. Note that M T is a Stein manifold, andΨ(z, w) = Ψ λ,Re w = λ max{Ψ(z) + Re w, 0} is a bounded plurisubharmonic function on Ω ′ . Using Lemma 2.1, we can get an E ′ −valued holomorphic (n + 1, 0) formf on Ω ′ such thatf dw \| MT ×{w0} = f 0 , and 1 πr 2 Ω ′ \|f \| 2 π * 1 (h) e −Ψ ≤ MT \|f 0 \| 2 h e −Ψ λ,Re w 0 . (3.1) Denote thatf w =f dw \| MT ×{w} . Since the function y = log x is concave, according to Jensen's inequality and inequality (3.1), we have log f 0 2 λ,Re w0 = log MT \|f 0 \| 2 h e −Ψ λ,Re w 0 ≥ log 1 πr 2 Ω ′ \|f \| 2 h e −Ψ = log 1 πr 2 ∆(w0,r) MT ×{w} \|f w \| 2 h e −Ψ λ,Re w dµ ∆w 0 ,r (w) ≥ 1 πr 2 ∆(w0,r) log f w 2 λ,Re w dµ ∆w 0 ,r (w) ≥ 1 πr 2 ∆(w0,r) log \|ξ ·f w \| 2 − log K(w)\| dµ ∆w 0 ,r (w). (3.2) Where µ ∆w 0 ,r is the Lebesgue measure on ∆ w0,r . It follows from Lemma 2.7 that ξ·f w is holomorphic with respect to w, which implies that log \|ξ·f w \| 2 is subharmonic with respect to w. Then we have log \|ξ · f 0 \| 2 ≤ 1 πr 2 ∆(w0,r) log \|ξ ·f w \| 2 dµ ∆w 0 ,r (w). Combining with inequlity (3.2), we get log f 0 2 λ,Re w0 ≥ log \|ξ · f 0 \| 2 − 1 πr 2 ∆(w0,r) log K(w)dµ ∆w 0 ,r (w), which means log K(w 0 ) ≤ 1 πr 2 ∆(w0,r) log K(w)dµ ∆w 0 ,r (w). Since log K(w) is upper semicontinuous and satisfies the sub-mean value inequality on U T , we know that log K(w) is a subharmonic function on U T . Proof of Theorem 1.6 In this section, we give the proof of Theorem 1.6. We need the following lemma. Proof of Theorem 1.6. It follows from Theorem 1.4 that log K h ξ,Ψ,λ (Re w) is subharmonic with respect to w ∈ (T, +∞)+ √ −1R. Note that log K h ξ,Ψ,λ (Re w) is only dependent on Re w, then following from Lemma 4.1, we get that log K h ξ,Ψ,λ (t) = log K h ξ,Ψ,λ (t + √ −1R) is convex with respect to t ∈ (T, +∞). Combining with Lemma 2.10, we get that log K h ξ,Ψ,λ (t) is convex with respect to t ∈ [T, +∞), which implies that − log K h ξ,Ψ,λ (t) + t is concave with respect to t ∈ [T, +∞). Then for any ξ ∈ A 2 (M T , h) * with ξ\| A 2 (MT ,h)∩J ≡ 0, to prove that log −K h ξ,Ψ,λ (t) + t is increasing, we only need to prove that log −K h ξ,Ψ,λ (t) + t has a lower bound on [T, +∞). Using Lemma 2.8, we obtain that there exists f t ∈ A 2 (M T , h) for any t ∈ [T, +∞), such that ξ · f t = 1 and K h ξ,Ψ,λ (t) = 1 f t 2 λ,t . (4.1) In addition, according to Lemma 2.2, there exists an E−valued holomorphic (n, 0) formF on M T such that MT \|F − (1 − b t (Ψ))f t F 2 \| 2 h e −φ+vt(Ψ) ≤ C MT I {−t−1<Ψ<−t} \|f t F \| 2 h ,(4.2) whereφ = max{ψ + T, 2 log \|F \|},h = he −ψ , and C is a positive constant independent of F and t. Then it follows from inequality (4.2) that MT \|F − (1 − b t (Ψ))f t F 2 \| 2 h e −φ+vt(Ψ) ≤C MT I {−t−1<Ψ<−t} \|f t F \| 2 h ≤Ce t+1 {Ψ<−t} \|f t \| 2 h .MT \|F t − (1 − b t (Ψ))f t \| 2 h e vt(Ψ)−Ψ ≤ Ce t+1 {Ψ<−t} \|f t \| 2 h < +∞. (4.4) According to inequality (4.4), we can get that (F t − f t ) z0 ∈ O(K M ) z0 ⊗ I(h, Ψ) z0 ⊂ O(K M ) z0 ⊗ J z0 , which means that ξ ·F t = ξ · f t = 1. Besides, since v t (Ψ) ≥ Ψ, we have MT \|F t − (1 − b t (Ψ))f t \| 2 h e vt(Ψ)−Ψ 1/2 ≥ MT \|F t − (1 − b t (Ψ))f t \| 2 h 1/2 ≥ MT \|F t \| 2 h 1/2 − MT \|(1 − b t (Ψ))f t \| 2 h 1/2 ≥ MT \|F t \| 2 h 1/2 − {Ψ<−t} \|f t \| 2 h 1/2 . Combining with inequality (4.4), we have MT \|F t \| 2 h ≤2 MT \|F t − (1 − b t (Ψ))f t \| 2 h e vt(Ψ)−Ψ + 2 {Ψ<−t} \|f t \| 2 h ≤2(Ce t+1 + 1) {Ψ<−t} \|f t \| 2 h . Note that f t 2 λ,t = MT \|f t \| 2 h e −Ψ λ,t = {Ψ<−t} \|f t \| 2 h + {T >Ψ≥−t} \|f t \| 2 h e −λ(Ψ+t) ≥ {Ψ<−t} \|f t \| 2 h . FIBERWISE BERGMAN KERNELS, VECTOR BUNDLES, AND LOG-SUBHARMONICITY 17 Then we have MT \|F t \| 2 h ≤ 2(Ce t+1 + 1) f t 2 λ,t ≤ C 1 e t K h ξ,Ψ,λ (t) , where C 1 is a positive constant independent on t. In addition, ξ ·F t = 1 implies that MT \|F t \| 2 h = F t 2 λ,T ≥ (K h ξ,Ψ,λ (T )) −1 . Then we get that − log K h ξ,Ψ,λ (t) + t ≥ C 2 , ∀t ∈ [T, +∞), where C 2 := log(C −1 1 K h ξ,Ψ,λ (T )) is a finite constant. Since − log K h ξ,Ψ,λ (t) + t is concave, we get that − log K h ξ,Ψ,λ (t)+t is increasing with respect to t ∈ [T, +∞). Proofs of Corollary 1.8 and Corollary 1.11 In this section, we give the proofs of Corollary 1.8 and Corollary 1.11. Before the proofs, we do some preparations. Let h be a measurable metric on E satisfying that h has a positive locally lower bound. Let (M, E, Σ, M j , h, h j,s ) be a singular metric on E. Assume that Θ h ≥ s N ak , and t ∈ [pt 0 , +∞). Note that , h), and K ξ,p,λ (pt 0 ) ∈ (0, +∞). Then Theorem 1.6 tells us that − log K ξ,p,λ (t) + t is increasing with respect to t ∈ [pt 0 , +∞), which implies that 0. Let f be a holomorphic (n, 0) form on M t0 = {Ψ < −t 0 } for some t 0 ≥ T such that f ∈ A 2 (M t0 , h). Let z 0 ∈ M ,pΨ = min{pψ + (2⌈p⌉ − 2p) log \|F \| − 2 log \|F ⌈p⌉ \|, −pt 0 } on {pΨ < −pt 0 } for any p > 0, where ⌈p⌉ := min{m ∈ Z : m ≥ p}. Then definition of J p shows that f ∈ A 2 (M t0 , h) \ (A 2 (M t0 , h) ∩ J p ), which implies that A 2 (M t0 , h) ∩ J p is a proper subspace of A 2 (M t0− log K ξ,p,λ (t) + t ≥ − log K ξ,p,λ (pt 0 ) + pt 0 , ∀t ∈ [pt 0 , +∞). (5.1) Since f ∈ A 2 (M t0 , h), following from inequality (5.1), we get that f 2 p,λ,t ≥ \|ξ · f \| 2 K ξ,p,λ (t) ≥ e −t+pt0 \|ξ · f \| 2 K ξ,p,λ (pt 0 ) , ∀t ∈ [pt 0 , +∞). In addition, since f / ∈ A 2 (M t0 , h) ∩ J p , according to Lemma 2.9, we have f 2 p,λ,t ≥ sup ξ∈A 2 (Mt 0 ,h) * \{0} ξ\| A 2 (M t 0 ,h)∩Jp ≡0 e −t+pt0 \|ξ · f \| 2 K ξ,p,λ (pt 0 ) = e −t+pt0 C(pΨ, h, J p , f, M t0 ), ∀t ∈ [pt 0 , +∞). (5.2) Note that for any t ∈ [pt 0 , +∞), f 2 p,λ,t = {pΨ<−t} \|f \| 2 h + {−pt0>pΨ≥−t} \|f \| 2 h e −λ(pΨ+t) . (5.3) Since for any λ > 0, {−pt0>pΨ≥−t} \|f \| 2 h e −λ(pΨ+t) ≤ {−pt0>pΨ≥−t} \|f \| 2 h < +∞,C(pΨ, h, J p , f, M t0 ) ≥ C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ), ∀p > 2a f z0 (Ψ; h). Since Mt 0 \|f \| 2 h < +∞, it follows from Lebesgue's dominated convergence theorem and inequality (5.5) that {2a f z 0 (Ψ;h)Ψ<−t} \|f \| 2 h = lim p→2a f z 0 (Ψ;h)+0 {pΨ<−t} \|f \| 2 h ≥ lim sup p→2a f z 0 (Ψ;h)+0 e −t+pt0 C(pΨ, h, J p , f, M t0 ) ≥e −t+2a f z 0 (Ψ;h)t0 C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ),(5.6) for any t ∈ (2a f z0 (Ψ; h)t 0 , +∞). For t = 2a f z0 (Ψ; h)t 0 , it is clear that the above inequality also holds by the definition of C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ). Let r := e −t/2 , and we get that Corollary 1.8 holds. In the following we give the proof of Corollary 1.11. Proof of Corollary 1.11. For any q > 2a f z0 (Ψ; h), according to inequality (5.5), we get that for any t ∈ [qt 0 , +∞), {qΨ<−t} \|f \| 2 h ≥ e −t+qt0 C(qΨ, h, J q , f, M t0 ). (5.7) It follows from Fubini's Theorem that {Ψ<−t0} \|f \| 2 h e −Ψ = {Ψ<−t0} \|f \| 2 h e −Ψ 0 ds = +∞ 0 {Ψ<−t0}∩{s<e −Ψ } \|f \| 2 h ds = +∞ −∞ {qΨ<−qt}∩{Ψ<−t0} \|f \| 2 h e t dt. Inequality (5.7) implies that for any q > 2a f o (Ψ; ϕ), +∞ t0 {qΨ<−qt}∩{Ψ<−t0} \|f \| 2 h e t dt ≥ +∞ t0 e −qt+qt0 C(qΨ, h, J q , f, M t0 ) · e t dt = 1 q − 1 e t0 C(qΨ, h, J q , f, M t0 ), and t0 −∞ {qΨ<−qt}∩{Ψ<−t0} \|f \| 2 h e t dt ≥ t0 −∞ C(qΨ, h, J q , f, M t0 ) · e t dt =e t0 C(qΨ, h, J q , f, M t0 ). Then we have Mt 0 \|f \| 2 h e −Ψ ≥ q q − 1 e t0 C(qΨ, h, J q , f, M t0 ). (5.8) for any q > 2a f z0 (Ψ; h). Note that J q ⊂ I + (h, 2a f z0 (Ψ; h)Ψ) z0 for any q > 2a f z0 (Ψ; h), which implies C(qΨ, h, J q , f, M t0 ) ≥ C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ), ∀q > 2a f z0 (Ψ; h). Then inequality (5.8) induces Mt 0 \|f \| 2 h e −Ψ ≥ q q − 1 e t0 C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ). (5.9) Let q → 2a f z0 (Ψ; h) + 0, then inequality (5.9) also holds for q ≥ 2a f z0 (Ψ; h). Thus if q > 1 satisfying Mt 0 \|f \| 2 h e −Ψ < q q − 1 e t0 C(Ψ, h, I + (h, 2a f z0 (Ψ; h)Ψ) z0 , f, M t0 ),(5.10) we have q < 2a f z0 (Ψ; h), which means that f z0 ∈ O(K M ) z0 ⊗ I(h, qΨ) z0 . Proof of Corollary 1.11 is done. Appendix In this section, we give the proof of Lemma 2.1. We firstly recall some notations and lemmas. Let M be a complex manifold. Let ω be a continuous hermitian metric on M . Let dV M be a continuous volume form on M . We denote by L 2 p,q (M, ω, dV M ) the spaces of L 2 integrable (p, q) forms over M with respect to ω and dV M . It is known that L 2 p,q (M, ω, dV M ) is a Hilbert space. Lemma 6.1 (see [20]). Let {u n } +∞ n=1 be a sequence of (p, q) forms in L 2 p,q (M, ω, dV M ) which is weakly convergent to u. Let {v n } +∞ n=1 be a sequence of Lebesgue measurable real functions on M which converges point-wisely to v. We assume that there exists a constant C > 0 such that \|v n \| ≤ C for any n. Then {v n u n } +∞ n=1 weakly converges to vu in L 2 p,q (M, ω, dV M ). Lemma 6.2 (see [22]). Let Q be a Hermitian vector bundle on a Kähler manifold M of dimension n with a Kähler metric ω. Let θ be a continuous (1, 0) form on M . Then we have [ √ −1θ ∧θ, Λ ω ]α =θ ∧ (α (θ) ♯ ), for any (n, 1) form α with value in Q. Moreover, for any positive (1, 1) form β, we have [β, Λ ω ] is semipositive. Let X be an n−dimensional complex manifold and ω be a hermitian metric on X. Let Q be a vector bundle on X with rank r. Let D ′′ : L 2 (M, ∧ n,q T * M ⊗ Q) → L 2 (M, ∧ n,q+1 T * M ⊗ Q) be the extension of∂−operator in the sense of distribution. Let {h i } +∞ i=1 be a family of C 2 smooth hermitian metric on Q and h be a measurable metric on Q such that lim i→+∞ h i = h almost everywhere on X. We assume that {h i } +∞ i=1 and h satisfy one of the following conditions, (A) h i is increasingly convergent to h as i → +∞; (B) there exists a continuous metricĥ on Q and a constant C > 0 such that for any i ≥ 0, 1 Cĥ ≤ h i ≤ Cĥ and 1 Cĥ ≤ h ≤ Cĥ. Denote H i := L 2 (X, K X ⊗ Q, h i , dV ω ) and H := L 2 (X, K X ⊗ Q, h, dV ω ). Note that H ⊂ H i ⊂ H 1 for any i ∈ Z >0 . Denote P i := H i → KerD ′′ and P := H → KerD ′′ be the orthogonal projections with respect to h i and h respectively. {f i } +∞ i=1 which sat- isfies f i ∈ H i and \|\|f i \|\| hi ≤ C 1 for some constant C 1 > 0, there exists a Q-valued (n, 0)-form f 0 ∈ H such that there exists a subsequence of {f i } +∞ i=1 (also denoted by {f i } +∞ i=1 ) weakly converges to f 0 in H 1 and P i (f i ) weakly converges to P (f 0 ) in H 1 . We need the following result in Hilbert spaces. Lemma 6.4. Let ·, · 1 , ·, · 2 be two inner products on a vector space H such that both (H, ·, · 1 ) and (H, ·, · 2 ) are Hilbert spaces. Assume that there exists some C > 0 such that · 2 ≤ C · 1 , where · 1 , · 2 are the norms induced by ·, · 1 and ·, · 2 respectively. Then for any sequence {x j } ⊂ H weakly convergent to x ∈ H in (H, ·, · 1 ), then {x j } also weakly converges to x in (H, ·, · 2 ). Proof. For any w ∈ H, we denote a functional L w over (H, ·, · 1 ) as follows: L w : H −→ C z −→ z, w 2 . It is clear that L w is linear. In addition, for any z ∈ H, we have \|L w (z)\| = \| z, y 2 \| ≤ z 2 w 2 ≤ C w 2 z 1 . Then L w is a continuous functional over (H, ·, · 1 ), which implies that there exists some T w ∈ H such that z, w 2 = L w (z) = z, T w 1 for any z ∈ H by Riesz representation theorem. We have that T : H → H is a continuous linear operator. It follows that any weakly convergent sequence in (H, ·, · 1 ) is also a weakly convergent sequence in (H, ·, · 2 ). Lemma 6.5 (see [20]). Let M be a complex manifold admitting a complete Kähler metric, and ω is a Kähler metric on M (not necessarily complete). Let (Q, h) be a hermitian vector bundle over M . Assume that η and g are smooth bounded positive functions on M such that η + g −1 are smooth bounded positive functions on M such that η + g −1 is a smooth bounded positive functions on M and let B := [η √ −1Θ Q − √ −1∂∂η − √ −1g∂η ∧∂η, Λ ω ]. Assume thatλ ≥ 0 is a bounded continuous function on M such that B +λI is positive definite everywhere on ∧ n,q T * M ⊗Q for some q ≥ 1. Then given a form v ∈ L 2 (M, ∧ n,q T * M ⊗Q) such that D ′′ v = 0 and M (B +λI) −1 v, v Q,ω dV ω < +∞, there exists an approximate solution u ∈ L 2 (M, ∧ n,q−1 T * M ⊗ Q) and a correcting term τ ∈ L 2 (M, ∧ n,q T * M ⊗ Q) such that D ′′ u + P h ( λ τ ) = v, where P h : L 2 (M, ∧ n,q T * M ⊗ Q) → Ker D ′′ is the orthogonal projection and M (η + g −1 ) −1 \|u\| 2 Q,ω dV ω + M \|τ \| 2 Q,ω dV ω ≤ M (B +λI) −1 v, v Q,ω dV ω . (6.1) Lemma 6.6 (see [16]). Let X be a Stein manifold and ϕ a plurisubharmonic function on X. Then there exists a sequence {ϕ n } n=1,··· of smooth strongly plurisubharmonic functions such that ϕ n ↓ ϕ. Lemma 6.7 (Lemma 6.9 in [9]). Let Ω be an open subset of C n and Z be a complex analytic subset of Ω. Assume that v is a (p, q − 1)−form with L 2 loc coefficients and h is an L 1 loc (p, q)−form coefficients such that∂v = h on Ω \ Z (in the sense of distribution theory). Then∂v = h on Ω. In the following, we give the proof of Lemma 2.1. Note that M is a Stein manifold, there exists a smooth plurisubharmonic exhaustion function P on M . Let M j := {P < j} (k = 1, 2, ..., ). We choose P such that M 1 = ∅. Then M 1 ⋐ M 2 ⋐ ... ⋐ M j ⋐ M j+1 ⋐ ... and ∪ +∞ j=1 M j = M . Each M j is a Stein manifold. For any smooth metricĥ on M , since h has a positive locally lower bound, we can find some C K > 0 such that \|e\| h ≥ C K \|e\|ĥ on K for any compact subset K of M and any local holomorphic section e of E. Then it follows from M \|u\| 2 h e −Ψw 0 < +∞ that K \|u\| 2 h < +∞ for any compact subset K of M . Step 1: Regularization ofΨ. According to Lemma 6.6, we can find a sequence of smooth strongly plurisubharmonic functions {Ψ m } +∞ m=1 on Ω such thatΨ m ↓Ψ on Ω. Additionally, Let r i ∈ (0, r) be a sequence of real numbers such that r i → r as i → +∞, and D i := {\|w − w 0 \| < r i } ⊂ D, Ω i := M × D i . If for any i, there exists some extensionũ i of u such that Ωi \|ũ i \| 2 p * 2 (h) e −Ψ ≤ 1 πr 2 i M \|u\| 2 h e −Ψw 0 , then by Montel's theorem and the diagonal method, we can find an extensionũ of u on Ω such that Ω \|ũ\| 2 p * 2 (h) e −Ψ ≤ 1 πr 2 M \|u\| 2 h e −Ψw 0 . SinceΨ is bounded, and M j × D i is relatively compact in Ω, combining with the above discussion, we can assumeΨ m is uniformly bounded in M j × D with respect to m for any fixed j (see [22]). Step 2: Recall some constructions. Let t 0 ∈ (0, +∞), B > 0. In the following, to simplify our notations, we denote b t0,B (t) by b(t) and v t0,B (t) by v(t). Let ǫ ∈ (0, 1 8 B). Let {v ǫ } ǫ∈(0, 1 8 B) be a family of smooth increasing convex functions on R, such that: (1) v ǫ (t) = t for t ≥ −t 0 − ǫ, v ǫ (t) = constant for t < −t 0 − B + ǫ; (2) v ǫ ′′ (t) are convergence pointwisely to 1 B I (−t0−B,−t0) ,when ǫ → 0, and 0 ≤ v ǫ ′′ (t) ≤ 2 B I (−t0−B+ǫ,−t0−ǫ) for ant t ∈ R; (3) v ǫ ′ (t) are convergence pointwisely to b(t) which is a continuous function on R when ǫ → 0 and 0 ≤ v ǫ ′ (t) ≤ 1 for any t ∈ R. One can construct the family {v ǫ } ǫ∈(0, 1 8 B) by setting v ǫ (t) := t −∞ ( t1 −∞ ( 1 B − 4ǫ I (−t0−B+2ǫ,−t0−2ǫ) * ρ 1 4 ǫ )(s)ds)dt 1 − −t0 −∞ ( t1 −∞ ( 1 B − 4ǫ I (−t0−B+2ǫ,−t0−2ǫ) * ρ 1 4 ǫ )(s)ds)dt 1 − t 0 , where ρ 1 4 ǫ is the kernel of convolution satisfying supp(ρ 1 4 ǫ ) ⊂ (− 1 4 ǫ, 1 4 ǫ). Then it follows that v ǫ ′′ (t) = 1 B − 4ǫ I (−t0−B+2ǫ,−t0−2ǫ) * ρ 1 4 ǫ (t), and v ǫ ′ (t) = t −∞ ( 1 B − 4ǫ I (−t0−B+2ǫ,−t0−2ǫ) * ρ 1 4 ǫ )(s)ds. Let ψ 0 := p * 1 (2 log \|w − w 0 \| − 2 log r) be a plurisubharmonic function on Ω. Let η = s(−v ǫ (ψ 0 )) and φ = u(−v ǫ (ψ 0 )), where s ∈ C ∞ ((0, +∞)) satisfies s > 0 and u ∈ C ∞ ((0, +∞)), such that s ′ (t) = 0 for any t, u ′′ s − s ′′ > 0 and s ′ − u ′ s = 1. Recall that (M, E, Σ, M j , h, h j,m ′ ) is a singular hermitian metric on E. Then there exists a sequence of hermitian metrics {h j,m ′ } +∞ m ′ =1 on M j+1 of class C 2 such that lim m ′ →+∞ h j,m ′ = h almost everywhere on M j+1 and {h j,m ′ } +∞ m ′ =1 satisfies the conditions of Definition 1.3. We will fix j until the last step (Step 9), thus we simply denote h j,m ′ by h m ′ . Denote thath := p * 2 (h m ′ )e −Φm , where Φ m :=Ψ m + φ + ψ 0 . Step 3: Solving∂-equation with error term. Set B = [η √ −1Θh − √ −1∂∂η ⊗ Id E ′ − √ −1g∂η ∧∂η ⊗ Id E ′ , Λ ω∧ω0 ], where ω 0 = √ −1 2 dw ∧ dw is the standard Kähler form on C, and g is a positive function. We will determine g by calculations. On M j × (D \ {w 0 }), direct calculation shows that ∂∂η = − s ′ (−v ǫ (ψ 0 ))∂∂(v ǫ (ψ 0 )) + s ′′ (−v ǫ (ψ 0 ))∂(v ǫ (ψ 0 )) ∧∂(v ǫ (ψ 0 )), ∂∂φ = − u ′ (−v ǫ (ψ 0 ))∂∂(v ǫ (ψ 0 )) + u ′′ (−v ǫ (ψ 0 ))∂(v ǫ (ψ 0 )) ∧∂(v ǫ (ψ 0 )), ηΘh =η∂∂φ ⊗ Id E ′ + ηΘ h m ′ ⊠h0 + η∂∂(Ψ m ) ⊗ Id E ′ + η∂∂ψ 0 ⊗ Id E ′ =su ′′ (−v ǫ (ψ 0 ))∂(v ǫ (ψ 0 )) ∧∂(v ǫ (ψ 0 )) ⊗ Id E ′ − su ′ (−v ǫ (ψ 0 ))∂∂(v ǫ (ψ 0 )) ⊗ Id E ′ + sΘ p * 2 (h m ′ ) + s∂∂(Ψ m ) ⊗ Id E ′ . Therefore, η √ −1Θh − √ −1∂∂η ⊗ Id E − √ −1g∂η ∧∂η ⊗ Id E =sΘ p * 2 (h m ′ ) + s∂∂(Ψ m ) ⊗ Id E ′ + (s ′ − su ′ )(v ′ ǫ (ψ 0 ) √ −1∂∂ψ 0 + v ′′ ǫ (ψ 0 ) √ −1∂ψ 0 ∧∂ψ 0 ) ⊗ Id E ′ + ((u ′′ s − s ′′ ) − gs ′2 ) √ −1∂(v ǫ (ψ 0 )) ∧∂(v ǫ (ψ 0 )) ⊗ Id E ′ . We omit the composition item (−v ǫ (ψ 0 )) after s ′ − su ′ and (u ′′ s − s ′′ ) − gs ′2 in the above equalities. Note that u ′′ s − s ′′ > 0. Let g = u ′′ s−s ′′ s ′2 (−v ǫ (ψ 0 )). We have η + g −1 = (s + s ′2 u ′′ s−s ′′ )(−v ǫ (ψ 0 )). Note that s ′ − su ′ = 1, 0 ≤ v ′ ǫ (ψ 0 ) ≤ 1. Then η √ −1Θh − √ −1∂∂η ⊗ Id E ′ − √ −1∂η ∧∂η ⊗ Id E ′ =sΘ p * 2 (h m ′ ) + s∂∂(Ψ m ) ⊗ Id E ′ + v ′ ǫ (ψ 0 ) √ −1∂∂ψ 0 ⊗ Id E ′ + v ′′ ǫ (ψ 0 ) √ −1∂ψ 0 ∧∂ψ 0 ⊗ Id E ′ =v ′ ǫ (ψ 0 ) √ −1∂∂ψ 0 ⊗ Id E ′ + v ′′ ǫ (ψ 0 ) √ −1∂ψ 0 ∧∂ψ 0 ⊗ Id E ′ + s(Θ h m ′ +λ m ′ ω ⊗ Id E ) ∧ ω 0 ⊗ Id D×C + s∂∂Ψ m ⊗ Id E ′ −λ m ′ ω ∧ ω 0 ⊗ Id E ′ ≥v ′′ ǫ (ψ 0 ) √ −1∂ψ 0 ∧∂ψ 0 ⊗ Id E ′ − sλ m ′ ω ∧ ω 0 ⊗ Id E ′ . Here from Definition 1. 3,λ m ′ satisfies Θ h m ′ (E) ≥ N ak −λ m ′ ω ⊗ Id E on M j . It can be seen that s(−v ǫ (ψ 0 )) is uniformly upper bounded on M j × D with respect to j, m, m ′ , ǫ. Let N 1 be the uniformly upper bound of s(−v ǫ (ψ 0 )) on M j × D. Then on M j × (D \ {w 0 }), we have η √ −1Θh − √ −1∂∂η ⊗ Id E ′ − √ −1∂η ∧∂η ⊗ Id E ′ ≥v ′′ ǫ (ψ 0 ) √ −1∂ψ 0 ∧∂ψ 0 ⊗ Id E ′ − N 1λm ′ ω ∧ ω 0 ⊗ Id E ′ . Then for any E ′ −valued (n + 1, 1) form α, we have (B + N 1λm ′ Id E ′ )α, α h ≥ [v ′′ ǫ (ψ 0 )∂(ψ 0 ) ∧∂(ψ 0 ) ⊗ Id E ′ , Λ ω∧ω0 ]α, α h = (v ′′ ǫ (ψ 0 )∂(ψ 0 ) ∧ (α (∂ψ 0 ) ♯ ))α, α h .(6. 2) It follows from Lemma 6.2 that B + N 1λm ′ Id E ′ is semipositive. Denoteλ m ′ := λ m ′ + 1 m ′ , thenB := B + N 1λm ′ Id E ′ is positive. According to inequality (6.2), we have \| v ′′ ǫ (ψ 0 )∂ψ 0 ∧ γ,α h \| 2 =\| v ′′ ǫ (ψ 0 )γ,α (∂ψ 0 ) ♯ h \| 2 ≤ (v ′′ ǫ (ψ 0 )γ, γ) h (v ′′ ǫ (ψ 0 ))\|α (∂ψ 0 ) ♯ \| 2 h = (v ′′ ǫ (ψ 0 )γ, γ) h (v ′′ ǫ (ψ 0 ))∂ψ 0 ∧ (α (∂ψ 0 ) ♯ ),α h ≤ (v ′′ ǫ (ψ 0 )γ, γ) h Bα ,α) h (6.3) for any E ′ −valued (n + 1, 0) form γ and E ′ −valued (n + 1, 1) formα. Let f := u ∧ dw be the trivial extension of u from M j × {w 0 } to Ω. Then µ :=∂ (1 − v ′ ǫ (ψ 0 ))f is well defined and smooth on M j × D. Note that µ = −∂v ′ ǫ (ψ 0 ) ∧ f. Take γ = f ,α =B −1 µ. Then it follows from inequality (6.3) that B −1 µ, µ h ≤ v ′′ ǫ (ψ 0 )\|f \| 2 h . Thus we have Mj ×(D\{w0}) B −1 µ, µ h ≤ Mj ×(D\{w0}) v ′′ ǫ (ψ 0 )\|f \| 2 h (6.4) Recall thath = p * 2 (h m ′ )e −Φm and Φ m = φ +Ψ m + ψ 0 . Note that 0 ≤ v ′′ ǫ (t) ≤ 2 B I (−t0−B+ǫ,−t0−ǫ) , e −φ is bounded function on M j × D, h m ′ ≤ h, andΨ m is lower bounded on Ω. Then Mj ×(D\{w0}) v ′′ ǫ (ψ 0 )\|f \| 2 h ≤e t0+B−ǫ sup Mj ×D (e −φ−Ψm ) Mj ×D 2 B I (−t0−B+ǫ,−t0−ǫ) \|f \| 2 h⊠h0 < +∞. It is clear that M j × (D \ {w 0 }) carries a complete Kähler metric since M j is Stein. Then it follows from Lemma 6.5 that there exists u m,m ′ ,ǫ,j ∈ L 2 (M j × (D \ {w 0 }), K Ω ⊗ E ′ , p * 2 (h m ′ )e −Φm ), h m,m ′ ,ǫ,j ∈ L 2 (M j × (D \ {w 0 }), ∧ n+1,1 T * Ω ⊗ E ′ , p * 2 (h m ′ )e −Φm ), such that∂u m,m ′ ,ǫ,j + P m,m ′ N 1λm ′ h m,(B + N 1λm ′ Id E ′ ) −1 µ, µ h ≤ Mj ×(D\{w0}) v ′′ ǫ (ψ 0 )\|f \| 2 p * 2 (h m ′ ) e −Φm < + ∞. Assume that we can choose η and φ such that (η + g −1 ) −1 = e vǫ(ψ0) e φ . Then we have As \|e x \| h m ′ ≤ \|e x \| h m ′ +1 for any m ′ ∈ Z ≥0 , for any fixed i, we have sup m ′ Mj ×D \|u m,m ′ ,ǫ,j \| 2 p * 2 (hi) e −ψ0 < +∞. Especially letting h i = h 1 , since the closed unit ball of the Hilbert space is weakly compact, we can extract a subsequence u m,m ′′ ,ǫ,j weakly convergent to u m,ǫ,j in L 2 (M j × D, K Ω ⊗ E ′ , p * 2 (h 1 )e −ψ0 ) as m ′′ → +∞. It follows from Lemma 6.1 that u m,m ′′ ,ǫ,j e vǫ(ψ0)−Ψ weakly converges to u m,ǫ,j e vǫ(ψ0)−Ψ in L 2 (M j × D, K Ω ⊗ E ′ , p * 2 (h 1 )e −ψ0 ) as m ′′ → +∞. For fixed i ∈ Z ≥0 , as h 1 and h i are both C 2 smooth hermitian metrics on M j ⊂⊂ X, we know that the two norms in L 2 (M j × D, K Ω ⊗ E ′ , p * 2 (h 1 )e −ψ0 ) and L 2 (M j × D, K Ω ⊗E ′ , p * 2 (h i )e −ψ0 ) are equivalent. Note that sup m ′′ Mj ×D \|u m,m ′′ ,ǫ,j \| 2 p * 2 (hi) e −ψ0 < +∞. Hence we know that u m,m ′′ ,ǫ,j e vǫ(ψ0)−Ψ also weakly converges to u m,ǫ,j e vǫ(ψ0)−Ψ in L 2 (M j × D, K Ω ⊗ E ′ , p * 2 (h i )e −ψ0 ) as m ′′ → +∞ by Lemma 6.4. Then we have As \|e x \| h m ′ ≤ \|e x \| h m ′ +1 for any m ′ ∈ Z ≥0 , we have sup m ′′ Mj ×D \|h m,m ′′ ,ǫ,j \| 2 p * 2 (h1) e −ψ0 < +∞. Since the closed unit ball of the Hilbert space is weakly compact, we can extract a subsequence of h m,m ′′ ,ǫ,j (also denoted by h m,m ′′ ,ǫ,j ) weakly convergent to h m,ǫ,j in L 2 (M j × D, K Ω ⊗ E ′ , p * 2 (h 1 )e −ψ0 ) as m ′′ → +∞. As 0 ≤λ m ′′ ≤λ 1 + 1 and M j is relatively compact in X, we have sup m ′′ Mj ×D N 1λm ′′ \|h m,m ′′ ,ǫ,j \| 2 p * 2 (h m ′′ ) e −ψ0 < +∞. It follows from Lemma 6.3 that there exists a subsequence of {m ′′ } (also denoted by {m ′′ }, such that N 1λm ′′ h m,m ′′ ,ǫ,j is weakly convergent to someh m,ǫ,j and Especially, we know sup ǫ Mj ×D \|F m,ǫ,j \| 2 p * 2 (h1) < +∞. Note that h 1 is a C 2 hermitian metric on E, M j ⊂⊂ M and F m,ǫ,j is an E ′ -valued holomorphic (n+ 1, 0) form on M j × D. Then there exists a subsequence of {F m,ǫ,j } ǫ (also denoted by {F m,ǫ,j } ǫ ) compactly convergent to an E ′ -valued holomorphic (n + 1, 0) form F m,j on M j × D. Then it follows from Fatou's lemma that we have Mj ×D \|F m,j − (1 − b(ψ 0 ))f \| 2 p * 2 (h) e v(ψ0)−ψ0−Ψm ≤ lim inf ǫ→0 Mj ×D \|F m,ǫ,j − (1 − v ′ ǫ (ψ 0 ))f \| 2 p * 2 (h) e vǫ(ψ0)−Ψm−ψ0 ≤ lim sup ǫ→0 Mj ×D \|F m,ǫ,j − (1 − v ′ ǫ (ψ 0 ))f \| 2 p * 2 (h) e vǫ(ψ0)−Ψm−ψ0 ≤ lim sup ǫ→0 Mj ×D v ′′ ǫ (ψ 0 )\|f \| 2 p * 2 (h) e −u(−vǫ(ψ0))−Ψm−ψ0 ≤ sup Mj ×D e −u(−v(ψ0)) Mj ×D 1 B I {−t0−B<ψ0<−t0} \|f \| 2 p * 2 (h) e −Ψm−ψ0 , (6.14) where b(t) = b t0 (t) = 1 B t −∞ I {−t0−B<s<−t0} ds, v(t) = v t0 (t) = t −t0 b t0 (s)ds − t 0 . Step 6: ODE System. Now we want to find η and φ such that (η + g −1 ) = e −vǫ(ψ0) e −φ . As η = s(−v ǫ (ψ 0 )) and φ = u(−v ǫ (ψ 0 )), we have (η+g −1 )e vǫ(ψ0) e φ = (s + s ′2 u ′′ s−s ′′ )e −t e u • (−v ǫ (ψ 0 )). Summarizing the above discussion about s and u, we are naturally led to a system of ODEs: 1). s + s ′2 u ′′ s − s ′′ e u−t = 1, 2). s ′ − su ′ = 1, (6.15) where t ∈ (0, +∞). We solve the ODE system (6.15) and get u(t) = − log(1 − e −t ), s(t) = t 1 − e −t − 1. It follows that s ∈ C ∞ (0, +∞) satisfies s > 0 and u ∈ C ∞ (0, +∞) satisfies u ′′ s − s ′′ > 0. As u(t) = − log(1 − e −t ) is decreasing with respect to t, then it follows from 0 ≥ v(t) ≥ max{t, −t 0 − B} ≥ −t 0 − B, for any t < 0 that Combining with inequality (6.14), we have 1. 1 . 1Main result. Let M be an n−dimensional Stein manifold. Let K M be the canonical line bundle on M . Let dV M be a continuous volume form on M . Let ψ be a plurisubharmonic function on M . Let F ≡ 0 be a holomorphic function on M , and let T ∈ [−∞, +∞). Denote that Ψ := min{ψ − 2 log \|F \|, −T }. 1. 2 . 2Applications. Let M be an n−dimensional Stein manifold. Let K M be the canonical line bundle on M . Let ψ be a plurisubharmonic function on M . Let F ≡ 0 be a holomorphic function on M , and let T ∈ [−∞, +∞). Denote that Ψ := min{ψ − 2 log \|F \|, −T }. Lemma 2 . 1 . 21For any E−valued holomorphic (n, 0) form u on M such that M \|u\| 2 h e −Ψw 0 < +∞, there exists an E ′ −valued holomorphic (n + 1, 0) formũ on Ω, such that u = u ∧ dw on M × {w 0 }, and 1 πr 2 Ω \|ũ\| 2 p * 2 (h) e −Ψ ≤ M \|u\| 2 h e −Ψw 0 . a Cauchy sequence under the topology of A 2 (M T , h). Then MT \|f j \| 2 h is uniformly bounded. Using Lemma 2.3 and diagonal method, for any subsequence {f kj } of {f j }, we can find a further subsequence compactly convergent to an E−valued holomorphic (n, 0) form f 0 on M T . With Fatou's Lemma, we have Lemma 2. 9 . 9Assume that C(Ψ, h, J, f, M T ) ∈ (0, +∞), then C(Ψ, h, J, f, M T ) = sup Lemma 4 . 1 41(see[10]). Let D = I + √ −1R := {z = x + √ −1y ∈ C : x ∈ I, y ∈ R} be a subset of C, where I is an interval in R. Let φ(z) be a subharmonic function on D which is only dependent on x = Re z. Then φ(x) := φ(x + √ −1R) is a convex function with respect to x ∈ I. t :=F /F 2 on M T , thenF t is an E−valued holomorphic (n, 0) form on M T . Note thatφ = 2 log \|F \| and Ψ = ψ − 2 log \|F \| on M T . Then inequality (4.3) implies that and assume that a f z0 (Ψ; h) < +∞. According to Remark 1.9, we know that a f z0 (Ψ; h) ∈ (0, +∞). Let p > 2a f z0 (Ψ; h) and λ > 0. Let ξ ∈ A 2 (M t0 , h) * \{0} satisfying ξ\| A 2 (Mt 0 ,h)∩Jp ≡ 0, where J p := I(h, pΨ) z0 . Denote that K ξ,p,λ (t) := sup f ∈A 2 (Mt 0 ,h) Lemma 6.3 ([20]). For any sequence of Q-valued (n, 0)-forms m ′ ,ǫ,j = µ holds on M j × (D \ {w 0 }) where P m,m ′ : L 2 (M j × (D \ {w 0 }), ∧ n+1,1 T * Ω ⊗ E ′ , p * 2 (h m ′ )e −Φm ) → KerD ′′ is the orthogonal projection, and Mj ×(D\{w0}) 1 η + g −1 \|u m,m ′ ,ǫ, 22 hi) e vǫ(ψ0)−Ψm−ψ0 ≤ lim inf m ′′ →+∞ Mj ×D \|u m,m ′′ ,ǫ,j \| 2 p * 2 (hi) e vǫ(ψ0)−Ψm−ψ0 ≤ lim inf m ′′ →+∞ Mj ×D v ′′ ǫ (ψ 0 )\|f \| 2 p * 2 (h m ′′ ) e u(vǫ(ψ0))(h) e u(vǫ(ψ0))−Ψm−ψ0 < +∞.Letting i → +∞, by monotone convergence theorem, we have (h) e −u(−vǫ(ψ0))−Ψm−ψ0 < +∞. (6.9)It follows from inf Mj ×D e −u(−vǫ(ψ0))−Ψm > 0, inequalities (6.6), (6.8) that sup m ′′ Mj ×D \|h m,m ′′ ,ǫ,j \| 2 p * 2 (h m ′′ ) e −ψ0 < +∞. e −u(t) = 1 − e −t0−B . (6.16) Letting λ → +∞ in inequality (5.2), we get that for any t ∈ [pt 0 , +∞),and lim λ→+∞ e −λ(pΨ+t) = 0 on {−pt 0 > pΨ ≥ −t}, according to Lebesgue's domi- nated convergence theorem, we have lim λ→+∞ {−pt0>pΨ≥−t} \|f \| 2 h e −λ(pΨ+t) = 0. Then equality (5.3) implies lim λ→+∞ f 2 p,λ,t = {pΨ<−t} \|f \| 2 h , ∀t ∈ [pt 0 , +∞). (5.4) {pΨ<−t} \|f \| 2 h ≥ e −t+pt0 C(pΨ, h, J p , f, M t0 ). (5.5) Now we give the proof the Corollary 1.8. Proof of Corollary 1.8. Note that J p ⊂ I + (h, 2a f z0 (Ψ; h)Ψ) z0 for any p > 2a f z0 (Ψ; h). Then we have (h m ′ ) e −ψ0 < +∞. (h) < +∞, which implies that sup ǫ Mj ×D \|F m,ǫ,j \| 2 p * 2 (h) < +∞. (h) e −Ψm = 0.(6.22) Acknowledgements. We would like to thank Dr. Zhitong Mi and Zheng Yuan for checking this paper. The second author was supported by National Key R&D Program of China 2021YFA1003100, NSFC-11825101, NSFC-11522101 and NSFC-11431013.It is clear that, h m,m ′ ,ǫ,j ∈ L 2 (M j × D, ∧ n+1,1 T * Ω ⊗ E ′ , p * 2 (h m ′ )e −Φm ). In addition, it follows from inequality(6.5) thatMj ×D \|u m,m ′ ,ǫ,j \| 2By the construction of v ǫ (t), we know e vǫ(ψ0) has a positive lower bound on M j × D.By the constructions of v ǫ (t) and u, we know e −φ = e −u(−vǫ(ψ0)) has a positive lower bound on M j × D. Also we know e −Ψm has a positive lower bound on M j × D.Note that h m ′ is C 2 smooth on M j ⋐ M . Hence by Lemma 6.7 we haveon M j × D.Step 4: Letting m ′ → +∞.It follows from Lebesgue's dominated convergence theorem thatsinceΨ m is bounded and Mj ×D \|f \| 2 p * 2 (h) < +∞. It follows from inf Mj ×D e −vǫ(ψ0)−Ψm > 0, inequalities (6.6), (6.8) that. It follows from 0 ≤λ m ′′ ≤λ 1 + 1,λ m ′ → 0, a.e., M j is relatively compact in X and Lemma 6.1 that. It follows from the uniqueness of weak limit thath m,ǫ,j = 0. Then we have. Replacing m ′ by m ′′ in equality (6.7) and letting m ′′ go to +∞, we haveIt follows from equality (6.11) and inequality (6.9) that we know F m,ǫ,j is an E ′ -valued holomorphic (n + 1, 0) form on M j × D and(6.12)Step 5:(6.13)It follows from dominated convergence theorem thatCombining with inf(6.17)Step 7: Letting t 0 → +∞. According to inequality (6.17), for any t 0 > 0, there exists some F m,j,t0 such thatsinceΨ m is smooth andΨ m ≥Ψ, where dλ D is the Lebesgue measure on D.Note that ψ 0 = p * 1 (2 log \|w−w 0 \|−2 log r),Ψ m is bounded on any M j ×D, h ≥ h j,1 where h j,1 is smooth on M j , and F m,j,t0 , f are holomorphic on M j ×D. Considering that e −ψ0 is not integrable near M ×{w 0 }, we can find that F m,j,t0 \| Mj ×{w0} = u∧dw by inequality (6.18). Inequality (6.18) and inequality(6.19) also imply that lim supThen according to Lebesgue's dominated convergence theorem, we get lim t0→+∞ Mj ×DCombining inequalities (6.21) and (6.22), we know Mj ×D \|F m,j,t0 \| 2 p * 2 (h) is uniformly upper bounded with respect to t 0 ∈ (0, +∞). Note that h is locally lower bounded, thus we can find a subsequence of {F m,j,t0 } (also denoted by {F m,j,t0 } itself) compactly convergent toF m,j , whereF m,j is an E ′ −valued holomorphic (n + 1, 0) form on M j × D. Then following from Fatou's lemma, inequalities (6.21) and (6.22), we haveIn addition, we haveF m,j \| Mj ×{w0} = u ∧ dw, since F m,j,t0 \| Mj ×{w0} = u ∧ dw for any t 0 .Step 8: Letting m → +∞. SinceΨ m is uniformly bounded on any M j × D for any fixed j, we know thatis uniformly bounded with respect to m by inequality (6.23). Note that h is locally lower bounded, thus we can find a subsequence of {F m,j } (also denoted by {F m,j } itself) compactly convergent toF j , whereF j is an E ′ −valued holomorphic (n + 1, 0) form on M j × D. According to Fatou's Lemma, we have Additionally, we haveF j \| Mj ×{w0} = u ∧ dw.Step 9: Letting j → +∞. SinceΨ is bounded on Ω = M ×D, we have Mj ×D \|F j \| 2 p * 2 (h) is uniformly bounded with respect to j by inequality(6.24). Note that h is locally lower bounded, thus by diagonal method we can find a subsequence of {F j } (also denoted by {F j } itself) convergent toũ on any M j × D, whereũ is an E ′ −valued holomorphic (n + 1, 0) form on Ω. Then it follows from Fatou's Lemma that According toF j \| Mj ×{w0} = u ∧ dw, we also haveũ\| Mj ×{w0} = u ∧ dw, whichũ is actually the extension of u what we need. Then the proof of Lemma 2.1 is done. 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[ "Diffusion Probabilistic Models for Graph-Structured Prediction", "Diffusion Probabilistic Models for Graph-Structured Prediction" ]	[ "Hyosoon Jang ", "Sangwoo Mo ", "Sungsoo Ahn " ]	[]	[]	This paper studies graph-structured prediction for supervised learning on graphs with node-wise or edge-wise target dependencies. To solve this problem, recent works investigated combining graph neural networks (GNNs) with conventional structured prediction algorithms like conditional random fields. However, in this work, we pursue an alternative direction building on the recent successes of diffusion probabilistic models (DPMs). That is, we propose a new framework using DPMs to make graph-structured predictions. In the fully supervised setting, our DPM captures the target dependencies by iteratively updating each target estimate based on the estimates of nearby targets. We also propose a variational expectation maximization algorithm to train our DPM in the semisupervised setting. Extensive experiments verify that our framework consistently outperforms existing neural structured prediction models on inductive and transductive node classification. We also demonstrate the competitive performance of our framework for algorithmic reasoning tasks. 1	10.48550/arxiv.2302.10506	[ "https://export.arxiv.org/pdf/2302.10506v3.pdf" ]	257,050,271	2302.10506	e6c1b2de9a895c39fdb44505948d1e3963e43e2e	Diffusion Probabilistic Models for Graph-Structured Prediction Hyosoon Jang Sangwoo Mo Sungsoo Ahn Diffusion Probabilistic Models for Graph-Structured Prediction This paper studies graph-structured prediction for supervised learning on graphs with node-wise or edge-wise target dependencies. To solve this problem, recent works investigated combining graph neural networks (GNNs) with conventional structured prediction algorithms like conditional random fields. However, in this work, we pursue an alternative direction building on the recent successes of diffusion probabilistic models (DPMs). That is, we propose a new framework using DPMs to make graph-structured predictions. In the fully supervised setting, our DPM captures the target dependencies by iteratively updating each target estimate based on the estimates of nearby targets. We also propose a variational expectation maximization algorithm to train our DPM in the semisupervised setting. Extensive experiments verify that our framework consistently outperforms existing neural structured prediction models on inductive and transductive node classification. We also demonstrate the competitive performance of our framework for algorithmic reasoning tasks. 1 Introduction Supervised learning often involves predicting a set of targets that are related to one another through a graph structure. One example is classifying documents in citation networks (Sen et al., 2008). In this task, documents closely connected in the network, such as through citations, are more likely to belong to the same class. To solve this type of problem, structured prediction algorithms output a joint predictive distribution that considers the element-wise dependency between targets, as described by the given graph structure. To this end, researchers have developed structured prediction frameworks such as conditional random fields (Lafferty et al., 2001, CRFs) or iterative classification algorithms (Chakrabarti et al., 1998, ICAs). CRFs model the output dependencies via Markov networks conditioned on inputs. ICAs recursively update predictions on each target using the past predictions of neighboring targets until convergence. With the recent advent of graph neural networks (Kipf & Welling, 2016a;Xu et al., 2018, GNNs), researchers investigated combining GNNs with structured prediction algorithms (Ma et al., 2019b;Qu et al., 2019;Graber & Schwing, 2019;Qu et al., 2022). For example, Ma et al. (2019b); Qu et al. (2019; proposed to parameterize a CRF using GNNs. They differ by training the CRF using pseudolikelihood or a GNN-based proxy. Next, Hang et al. (2021) proposed to aggregate multiple GNN predictions as an ICA. In this work, we pursue an alternative research direction for GNN-based structured prediction. In particular, we propose to use the diffusion probabilistic model (Ho et al., 2020, DPM), which makes a prediction via iterative denoising of a diffusion process. We also propose a new variational expectation-maximization (EM) algorithm to train DPMs for semi-supervised settings. See Figure 1 for an illustration of our DPM for node classification. Our DPM expresses the target-wise dependencies via recursively updating a nodewise prediction using past predictions of neighboring nodes. arXiv:2302.10506v3 [cs.LG] 23 Feb 2023 We advocate that using DPMs for the graph-structured prediction can exploit their extraordinary capability in learning highly structured data (Li et al., 2022;Chen et al., 2022b;Hoogeboom et al., 2022b). In particular, our DPM makes a prediction using a GNN-based reverse diffusion process. This allows structured prediction with a range of dependencies proportional to the number of diffusion steps. Such a dependency requires an impractically large number of layers to learn using other GNN-based deep generative models, e.g., graph variational autoencoder (Kipf & Welling, 2016b). We design our variational EM to train DPMs in semisupervised settings, e.g., transductive node classification. Primarily, we suggest two strategies for adapting variational EM to our problem. First, we use a buffer to stabilize the optimization. We also use manifold-constrained sampling (Chung et al., 2022) to sample from DPM conditioned on ground truth labels during the expectation step. We extensively validate our method on eight inductive and eight transductive node classification problems constructed from ten datasets: DBLP (Tang et al., 2008), PPI (Zitnik & Leskovec, 2017), Pubmed, Cora and Citeseer (Yang et al., 2016), Photo and Computer (Shchur et al., 2018), and Cornell, Wisconsin, and Texas (Zhu et al., 2020). Our framework consistently outperforms the existing graph-structured prediction algorithms. We also show the superiority of our algorithm for graph algorithmic reasoning tasks that aim to predict the algorithmic outputs for edge copy, shortest path, and connected components (Du et al., 2022). Related Work Graph neural networks (GNNs). In recent years, GNNs have shown great success for graph representation learning (Kipf & Welling, 2016a;Hamilton et al., 2017;Veličković et al., 2017;Chamberlain et al., 2021). However, without additional modification, GNNs make unstructured predictions ignoring the dependencies between targets. For example, when predicting node-wise labels y = {y i : i ∈ V}, GNNs output a predictive distribution p θ (y) where the node-wise predictions are independent, i.e., p θ (y) = i∈V p θ (y i ). Structured prediction algorithms. Structured prediction algorithms aim to model the dependencies within the output space. One representative framework is conditional random fields (Sutton et al., 2012, CRFs), which output Markov random fields conditioned on the input. CRFs describe the dependencies between predictions using pair-wise potential functions. They have shown success in many applications, e.g., part-of-speech tagging (PVS & Karthik, 2007) and entity recognition (Patil et al., 2020). In another line of research, iterative classification algorithms (Neville & Jensen, 2000, ICAs) incorporate the target-wise dependencies via iterative message-passing updates between predictions. Graph-structured prediction with GNNs. Recently, several works focus on combining deep neural networks with structured prediction algorithms (Belanger & McCallum, 2016;Graber et al., 2018;Garcia Satorras et al., 2019;Qu et al., 2019;Hang et al., 2021;Du et al., 2022;Qu et al., 2022). In particular, researchers have considered GNNbased graph-structured prediction for node classification (Qu et al., 2019;Hang et al., 2021;Qu et al., 2022). First, Qu et al. (2019) parameterize the potential functions of a CRF using a GNN and train the CRF using pseudo-likelihood maximization (Besag, 1975). As an alternative way to train the CRF, Qu et al. (2022) proposed a training scheme based on a GNN-based proxy for the potential functions. Both methods achieve superior performance by combining the expressive power of GNNs and the ability of CRFs to make graph-structured predictions. Hang et al. (2021) propose an ICA-based learning framework of GNNs that updates each node-wise prediction by aggregating previous predictions on neighboring nodes. Finally, Ma et al. (2019a) solves the problem with a generative framework that learn the joint distribution of node features, labels, and graph structure. Diffusion probabilistic models (DPMs). Inspired by nonequilibrium thermodynamics, DPMs are deep generative models (Sohl-Dickstein et al., 2015) that learn to reverse a diffusion process and construct data from a noise distribution. Recent works demonstrated great success of DPMs in various domains, e.g., generating images (Dhariwal & Nichol, 2021), 3D shapes (Zhou et al., 2021), text (Li et al., 2022), and molecules (Hoogeboom et al., 2022b). In particular, researchers showed that DPMs even make predictions with complex structures, e.g., image segmentation (Chen et al., 2022b) and object detection (Chen et al., 2022a). Diffusion Probabilistic Model for Graph-Structured Prediction (DPM-GSP) In this section, we introduce our framework to use a diffusion probabilistic model for graph-structured prediction (DPM-GSP). We consider supervised learning on graphs where the task is to predict node-wise or edge-wise targets. We aim to make a graph-structured prediction that captures the target-wise dependencies characterized by a graph, e.g., in homogeneous graphs, neighboring nodes are more likely to be associated with similar targets (Wang & Zhang, 2006). In the following, we describe our DPM-GSP for fullysupervised and semi-supervised node classification. Our algorithm is applicable to both transductive and inductive settings without modification, i.e., when the test input is seen or unseen during training. We remark that our DPM-GSP is not only capable of predicting node-wise targets, but can also be extended to predict edge-wise targets. Fully supervised node classification Here, we consider supervised learning on a graph G with attributes x to predict targets y. To be specific, the graph G = (V, E) consists of a node set V and an edge set E. We assume that the attributes and the targets are associated with nodes, i.e., x = {x i : i ∈ V} and y = {y i : i ∈ V}. 2 If the targets y are discrete, we relax them into an one-hot vector to yield continuous values. Our DPM consists of forward and reverse diffusion processes, following the design of Ho et al. (2020). The fixed forward diffusion process iteratively injects noise into the targets until converging to a standard Gaussian distribution. 3 Conversely, the GNN-based reverse diffusion process aims to reconstruct the targets from the standard Gaussian distribution, conditioned on a given graph. By construction, our GNN-based reverse diffusion process can express longrange dependencies without training very deep GNNs. 4 We train the DPM to maximize the lower bound of the loglikelihood for the reverse diffusion process, with the ultimate goal of generating the correct targets from a given graph. Forward diffusion process. Given a target y (0) = y and the number of diffusion steps T our forward diffusion process constructs a sequence of noisy targets y (1) , . . . , y (T ) using a fixed distribution q(·) defined as follows: q(y (1) , . . . , y (T ) \|y (0) ) = T t=1 q(y (t) \|y (t−1) ), q(y (t) \|y (t−1) ) = N (y (t) ; √ α t y (t−1) , β t I), where I is an identity matrix, β 1 , . . . , β T are fixed variance schedules, and α t = 1 − β t . We set the variance schedule to promote q(y (T ) \|y (0) ) ≈ N (y (T ) ; 0, I) by setting β t < β t+1 for t = 0, . . . , T − 1 and β T = 1. We provide a detailed variance scheduling method in Appendix A. Reverse diffusion process. Given an initial sample y (T ) sampled from p(y (T ) ) = N (y (T ) ; 0, I), the reverse dif-2 One can easily extend DPM-GSP to learn on graphs associated with edge-wise attributes and targets (see Appendix C.3). 3 We also compare with other diffusion strategies in Table 5. 4 We highlight this point in Figure 4. fusion process p θ (y (0) , . . . , y (T −1) \|x, y (T ) , G) learns to reconstruct the forward diffusion process given by q(·). To be specific, it is formulated as follows: p θ (y (0) , . . . , y (T −1) \|y (T ) ) = T t=1 p θ (y (t−1) \|y (t) ), p θ (y (t−1) \|y (t) ) = N (y (t−1) ; µ θ (x, y (t) , G, t), σ 2 t I), where we omit the conditional dependency of p θ on graph G and attributes x for simplicity of notation. We parameterize the denoised mean value µ θ (·) of y (t−1) using a graph neural network (GNN). We provide a detailed description of the GNNs in Appendix A. Training objective. We train our DPM to maximize the variational lower bound of the marginal log-likelihood log p θ (y (0) ). To this end, we minimize the simplified training objective defined as follows: L = T t=1 1 σ 2 t E q(y (t) \|y (0) ) [ µ θ (x, y (t) , G, t) −μ 2 2 ]. (1) Here,μ denotes average of the conditional distribution q(y (t−1) \|y (t) , y (0 ) that can be computed in a closed form. In practice, we optimize the residual from the mean value; the detailed description is in Appendix A. Deterministic inference. We propose a deterministic inference algorithm to create a unique prediction of labels required for evaluation. Recall that the reverse diffusion process of DPM can produce a diverse set of predictions, but the classification problem needs a single best guess. To this end, when evaluating our DPM-GSP, we set y (T ) to be a zero vector 0, which is the average of p(y (T ) ) = N (y (T ) ; 0, I). Then for t = T, . . . , 1, the reverse diffusion process deterministically reconstructs y (t−1) from y (t) as the average of p θ (y (t−1) \|y (t) ). If the target is one-hot relaxation of discrete labels, we discretize the final prediction by choosing a dimension with maximum value. 5 Figure 3. Illustration of variational expectation-maximization algorithm for DPM-GSP in the semi-supervised scenario. We alternatively (a) estimate the unlabeled nodes and (b) maximize log-likelihood using the estimated unlabeled nodes. Algorithm 1 Semi-supervised training of DPM-GSP 1: Input: Graph G, attributes x, and targets y L . 2: Train a mean-field GNN p φ (y\|x, G) on y L and G. 3: Initialize the buffer B with samples from p φ (y U \|x, G). Get y U ∼ p θ (y U \|y L , x, G) using manifold-constrained sampling. Appendix B 7: Update B ← B ∪ {y U }. 8: If \|B\| > K, remove oldest entity from B. Sample y U ∼ B. 12: Update θ to minimize L in Equation (1) with G, x, and y = y L ∪ y U . 13: end for 14: until converged Semi-supervised node classification In the semi-supervised setting, we consider training on a graph G with attributes x where the targets y are partially labeled. We denote the labeled and the unlabeled targets by y L and y U , respectively. To this end, we train a model to maximize the marginal log-likelihood log p θ (y L \|G, x) = log y U p θ (y L , y U \|G, x). However, since the marginalization over y U is intractable, we apply the variational expectation-maximization (Dempster et al., 1977;Wainwright et al., 2008, EM) algorithm to maximize a variational lower bound: log p θ (y L ) ≥ E q * (y U ) [log p θ (y L , y U ) − log q * (y U )], where q * is a variational distribution and the equality holds when q * (y U ) = p θ (y U \|y L ). We omit the dependency of probabilities on a graph G and attributes x for simplicity. The variational EM algorithm maximizes the lower bound by iterating between an "expectation" step and a "maximization" step. The expectation step updates the variational distribution q * to tighten the lower bound, i.e., update q * (y U ) to match the true distribution p θ (y U \|y L ). Then, the maximization step trains the distribution p θ to maximize the lower bound using the current q * . We describe the overall procedure in Algorithm 1. Expectation step. In the expectation step, we update the variational distribution q * (y U ) to match the distribution of unobserved targets, i.e., q * (y U ) = p θ (y U \|y L ). To this end, we introduce a buffer B to store the last K samples from p θ (y U \|y L ) and set q * (y U ) as the empirical distribution of the buffer. Then one can infer that q * (y U ) ≈ p θ (y U \|y L ) if the buffer is sufficiently large and the parameter θ does not change much after each training step. However, sampling from p θ (y U \|y L ) is non-trivial since the reverse diffusion process of DPM only allows unconditional prediction p θ (y). To this end, we apply manifoldconstrained sampling (Chung et al., 2022), which constrains the reverse diffusion process to obtain a sample y ∼ p θ (y) with maintaining the ground truth labels for y L . We describe the detailed sampling procedure in Appendix B. We empirically observe a performance improvement from initializing the variational distribution q * (y U ) as a good approximation to the true node label distribution p(y U \|y L ). To this end, we initialize the buffer B using samples from a pre-trained "mean-field" GNN p φ (y\|x, G) which outputs an independent joint distribution over node-wise labels, i.e., p φ (y\|x, G) = i∈V p φ (y i \|x, G). Maximization step. In the maximization step, we update the distribution p θ (y) to maximize the lower bound with respect to the current variational distribution q * (y U ), i.e., samples from the buffer B. This is equivalent to minimizing Equation (1) with targets y set to the union of true targets y L and predicted targets y U ∼ B, i.e., y = y L ∪ y U . Experiments Inductive node classification We first evaluate our DPM-GSP in an inductive setting. Here, we consider training a model on a set of fully labeled graphs and then testing the model on unseen graphs. As the backbone of DPM-GSP, we consider graph convolutional neural network (Kipf & Welling, 2016a, GCN), graph sample and aggregation (Hamilton et al., 2017, GraphSAGE), and graph attention network (Veličković et al., 2017, GAT). Baselines. We compare DPM-GSP with a base GNN that we train to make unstructured prediction p θ (y\|G, x) = i∈V p θ (y i \|G, x). We also compare with the recent GNNbased graph-structured prediction algorithms: graph Markov Table 1. The inductive node classification accuracy. N-Acc and G-Acc denote the node-level and graph-level accuracy, respectively. Bold numbers indicate the best performance among the graph-structured prediction methods using the same GNN. Underlined numbers indicate the best performance. † We use the numbers reported by Qu et al. (2022). Pubmed Cora Citeseer Table 2. The inductive node classification performance. Bold numbers indicate the best performance among the graph-structured prediction methods using the same GNN. Underlined numbers indicate the best performance. † We use the numbers reported by Qu et al. (2022). neural network (Qu et al., 2019, GMNN), structured proxy network (Qu et al., 2022, SPN), and collective learning graph neural network (Hang et al., 2021, CLGNN). We describe the hyper-parameter settings in Appendix C.1. Method N-Acc G-Acc N-Acc G-Acc N-Acc G-Acc GMNN † 78.00 ±1DBLP PPI-1 PPI-2 PPI-10 PPI-20 Method N-Acc N-Acc F1 N-Acc F1 N-Acc F1 N-Acc F1 GMNN † 76.54 ±2 Datasets. We follow Qu et al. (2022) and experiment on both small-scale and large-scale graphs. We construct smallscale graphs from Cora, Citeseer, and Pubmed (Yang et al., 2016). We also construct large-scale graphs from PPI (Zitnik & Leskovec, 2017) and DBLP datasets (Tang et al., 2008). In addition, to make the task challenging, we consider limiting the number of training graphs in the PPI dataset. We let PPI-k denote a dataset with k graphs for training. Evaluation metrics. For all datasets, we evaluate nodelevel accuracy, i.e., the ratio of nodes associated with a correct prediction. For small-scale graphs, i.e., Pubmed, Cora, and Citeseer, we also report the graph-level accuracy, which measures the ratio of graphs where all the node-wise predictions are correct. For the PPI datasets, we also report the micro-F1 score that is robust against the imbalance in the distribution of labels. We report the performance measured using ten and five different random seeds for small-scale and large-scale graphs, respectively. Results. We report the results in Table 1 and Table 2 for small-scale and large-scale graphs, respectively. Here, one may observe that our DPM-GSP provides consistent gains to the GNNs that make unstructured predictions. Furthermore, our DPM-GSP outperforms all the baselines consistently except for DBLP. Our algorithm also achieves the best score for graph-level accuracy, which highlights the ability of our algorithm to incorporate structure into the prediction. Table 3. The transductive node classification accuracy. Bold numbers indicate the best performance among the graph-structured prediction methods using the same GNN. Underlined numbers indicate the best performance. † ‡ * We use the numbers reported by Bo et al. (2022), Ma et al. (2019b), and Wang et al. (2021) Transductive node classification Next, we evaluate the performance of our algorithm for transductive node classification. We apply Algorithm 1 to train our DPM-GSP in this setting. Baselines. We first compare our method with label propagation (Wang & Zhang, 2006, LP), and LP-based methods such as generalized label propagation (Li et al., 2019, GLP), and propagation then training adaptively (Dong et al., 2021, PTA). Then we consider learning the coupling matrix (Wang et al., 2021, LCM), which is a CRF-based method. We also compare with base GNN, GMNN, and CLGNN, introduced in Section 4.1. Finally, we also compare with generative graph models with graph neural networks (Ma et al., 2019b, G3NN). We describe the hyper-parameter settings in Appendix C.2. Datasets. We use eight datasets: Pubmed, Cora, Citeseer, Amazon Photo and Computer (Shchur et al., 2018), Cornell, Wisconsin, and Texas (Pei et al., 2019). The data partition criterion used for Pubmed, Cora, and Citeseer is the same as in Yang et al. (2016), where 20 nodes per class are used for training. For Photo and Computer, the criterion is the same as in Shchur et al. (2018), where 20 nodes per class are used for training. For Cornell, Wisconsin, and Texas, the criterion is the same as in Zhu et al. (2020), where 48% of nodes are used for training. We report the accuracy measured using ten random seeds for all datasets. Results. We report the results in Table 3. One can again observe how our algorithm consistently improves the base GNNs. Furthermore, our outperforms all the existing al-gorithms, even when sharing the GNN backbone with some structured prediction algorithms. Along with the experiments for inductive node classification, these results highlight the superiority of DPM-GSP in making graphstructured predictions. Algorithmic reasoning We also evaluate our DPM-GSP to predict the outcomes of graph algorithms, e.g., computing the shortest between two nodes. Solving such a task using GNNs has gained much attention since it builds connections between deep learning and classical computer science algorithms. To this end, researchers have developed tailored algorithms like iterative energy minimization (Du et al., 2022, IREM) and PonderNet (Banino et al., 2021). Here, we show that the capability of DPM-GSP to make a structured prediction even allows good reasoning ability. We evaluate the performance of our DPM-GSP on three graph algorithmic reasoning benchmarks: edge copy, shortest path, and connected component, proposed by Du et al. (2022). The edge copy task aims to replicate the edge attributes in the output. The shortest path task computes the shortest path between all pairs of nodes. The connected component task identifies which nodes belong to the same connected component. As these tasks are defined on edgewise targets, we modify DPM-GSP to make edge-wise predictions. The detailed diffusion scheme and the GNN architectures are described in Appendix C.3. Datasets. Following Du et al. (2022), we use a training dataset composed of graphs of varying sizes, ranging from two to ten nodes. Then, we evaluate performance on graphs with ten nodes. Furthermore, we also use graphs with 15 nodes to evaluate generalization capabilities. We report the performance using element-wise mean square error. Baselines. We compare our method with five methods reported by Du et al. (2022), including a feedforward neural network, recurrent neural network, Pondernet, iterative feedforward, and IREM. Results. We report the results in Table 4. Our DPM-GSP outperforms the considered baselines for five out of the six tasks. In particular, DPM-GSP significantly improves all the baselines for shortest path and connected components tasks. These results suggest that the diffusion model can easily solve algorithmic reasoning tasks thanks to its superior ability to make structured predictions. Ablation studies Here, we conduct ablation studies to analyze our framework. All the results are averaged over ten different random seeds. Number of GNN layers vs. diffusion steps. We first verify that our DPM-GSP method does not solely rely on increasing the number of GNN layers to improve node classification performance. To this end, we measure performance by varying the number of GNN layers and the number of diffusion steps. We report the corresponding results in the Figure 4. One can observe that just increasing the number of GNN layers does not improve performance, whereas increasing the number of diffusion steps results in better performance. Accuracy over diffusion steps. We investigate whether the iteration in the reverse diffusion process progressively improves the quality of predictions. In Figure 5, we plot the changes in node-level and graph-level accuracy in the reverse diffusion process as the number of iterations increases. The results confirm that the iterative inference process gradually increases accuracy, eventually reaching convergence. Deterministic vs. stochastic inference. We compare the original stochastic reverse diffusion process with our deterministic inference. In Figure 6, we plot the changes in accuracy for various temperatures λ, where the reverse diffusion process uses λ 2 σ 2 t as the variance for each step t = 1, . . . , T . Note that setting λ = 0 reduces the stochastic inference scheme to our deterministic one. One can see that reducing the randomness of DPM gives a better prediction. We also consider sampling various numbers of predictions for node-wise aggregation to improve performance. In Figure 7, when the sample size is increased to 1000, the deterministic and the stochastic inference schemes show similar node-level accuracy, whereas deterministic inference still yields a higher graph-level accuracy. This is because nodewise aggregation ignores the graph structure by aggregating the labels independently for each node. Table 5. The inductive node classification accuracy. N-Acc and G-Acc denote the node-level and graph-level accuracy, respectively. The backbone network is GAT. The Gaussian diffusion process outperforms the considered baselines specialized to discrete label. Pubmed Cora Citeseer Method N-Acc G-Acc N-Acc G-Acc N-Acc G-Acc Table 6. The inductive node classification accuracy and inference time (ms). N-Acc and G-Acc denote the node-level and graph-level accuracy, respectively. The backbone network is GCN. We let SPN (T ) and DPM-GSP (T ) denote the algorithms with T belief propagation steps and T diffusion steps, respectively. DPM-GSP shows a competitive trade-off between running time and accuracy. Gaussian vs. discrete diffusion. We validate our Gaussian diffusion process that relaxes the discrete label space by comparing with diffusion processes explicitly designed for discrete target space. We compare with absorbing and uniform transition proposed by Austin et al. (2021). We also adapt the bit upscaling transition proposed by Hoogeboom et al. (2022a) to our setting. We report the results in Table 5. Surprisingly, one can observe that the Gaussian diffusion process outperforms the considered baselines. We hypothesize that continuous relaxation allows better propagation of "uncertainty" in predictions, similar to how algorithm-like belief propagation (Murphy et al., 2013) works. We leave the investigation of a diffusion process specialized to our setting for future work. Running time vs. performance. Finally, we investigate whether the DPM-GSP can make a good trade-off between running time and performance. In Table 6, we report the performances of DPM-GSP with varying numbers of diffusion steps and running times. One can observe that our DPM-GSP with only three diffusion steps shows competitive performances compared to the SPN and CLGNN while running in similar or less time. Also, while increasing the inference times of the SPN does not enhance performance, DPM-GSP shows further performance improvement. Conclusion We propose diffusion probabilistic models for solving graphstructured prediction (DPM-GSP) in both fully supervised and semi-supervised settings. Extensive experiments show that DPM-GSP outperforms existing methods. We believe accelerating our framework with faster diffusion-based models, e.g., denoising diffusion implicit models (Song et al., DDIM), or combining with other diffusion schemes, e.g., simplex diffusion (Richemond et al., 2022), are important future research directions. Chen, S., Sun, P., Song, Y., and Luo, P. Diffusiondet: Diffusion model for object detection. arXiv preprint arXiv:2211.09788, 2022a. 2 Chen, T., Li, L., Saxena, S., Hinton, G., and Fleet, D. J. A generalist framework for panoptic segmentation of images and videos. arXiv preprint arXiv:2210.06366, 2022b. Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 2020. 1, 3, 12 Hoogeboom, E., Gritsenko, A. A., Bastings, J., Poole, B., van den Berg, R., and Salimans, T. Autoregressive diffusion models. In International Conference on Learning Representations, 2022a. URL https:// openreview.net/forum?id=Lm8T39vLDTE. A. Detailed descriptions of DPM-GSP In this section, we describe the detailed components of DPM-GSP. Variances schedule. To determine the variances schedule β 1 , . . . , β T , we use the cosine beta schedule suggested by Nichol & Dhariwal (2021). Here, we set the offset parameter to 0.008. Detailed reverse diffusion process. In Section 3.1, we parameterize probabilistic distribution of the reverse diffusion step p θ (y (t−1) \|y (t) ) as N (y (t−1) ; µ θ (x, y (t) , G, t), σ 2 t ). Here, we set σ 2 t to β t . Following Ho et al. (2020), we also define µ θ (x, y (t) , G, t) as follows: µ θ (x, y (t) , G, t) = 1 √ α t y (t) − β t √ 1 −ᾱ t θ (x, y (t) , G, t) ,(2) whereᾱ t is t i=1 α i . Here, parameterization of the residual function θ (x, y (t) , G, t) enables residual-matching training objective (Ho et al., 2020), which is further simplification of Equation (1). Specifically, we parameterize θ (x, y (t) , G, t) using a L-layer GNN as follows: θ (x, y (t) , G, t) = g(h (L) ) h ( ) i = (COMBINE ( ) (h ( −1) i , a ( ) i ) + f (t)) y (t) i , a ( ) i = AGGREGATE ( ) ({h ( −1) j \|(i, j) ∈ E}), where g(h (L) ) is an MLP that estimates the residual using the final node representation. AGGREGATE(·) and COMBINE(·) functions are identical to the backbone GNN, and · · indicates the concatenation. Here, h 0 i is initialized by concatenating the node features and noisy label x i y (t) i . The f (·) is a time embedding function that consists of a sinusoidal positional embedding function (Vaswani et al., 2017) and two-layer MLP. In this paper, we fix the dimension of sinusoidal positional embedding to 128. Detailed training objective. In Section 3.1, we define the training objective of DPM-GSP in terms of estimated average µ θ (x, y (t) , G, t). Here, Equation (2) enables defining the further simplified objective as follows: L = T t=1 E ∼N (0,I) β 2 t 2σ 2 t α t (1 −ᾱ t ) − θ (x, √ᾱ t y (0) + √ 1 −ᾱ t , G, t) 2 2 where the detailed derivation follows Ho et al. (2020). An additional suggestion from Ho et al. (2020) is to set all weights of the mean squared error to one instead of β 2 t 2σ 2 t αt(1−ᾱt) . We employ this idea by introducing a hyper-parameter called "unweighted mean squared error", with a True value indicating that all weights of the mean square error are set to one. B. Manifold-constrained sampling To sample y U from p θ (y U \|y L ), we introduce a manifold-constrained sampling method proposed by Chung et al. (2022). Here, the update rule of the reverse process for t = 0, . . . , T − 1 is modified as follows: y (t) = 1 √ α t y (t+1) − β t+1 √ 1 −ᾱ t+1 θ (x, y (t+1) , G, t + 1) + σ t+1 z, z ∼ N (z; 0, λ 2 I), y (t) U =P y (t) − γ ∂ ∂y (t+1) y L − Pŷ 2 2 y (t) L = √ᾱ t y L + √ 1 −ᾱ t z, z ∼ N (z; 0, λ 2 I). The first equation is a temporal reverse diffusion step before applying the manifold-constrained samplings. The second equation applies the manifold-constrained gradient γ ∂ ∂y (t+1) y L − Pŷ 2 2 , where P andP is a masking matrix that selects the labeled nodes and unlabeled nodes, respectively. Here,ŷ is y (t+1) − 1−ᾱt+1 √ 1−ᾱt+1 θ (x, y (t+1) , G, t + 1) / √ᾱ t+1 , and γ is a hyper-parameter. Following Chung et al. (2022), we set γ to 1/ y L − Pŷ 2 2 . The third equation is a consistency step. Additionally, we introduce a parameter λ to control the randomness; when we set λ to zero, the modified reverse step becomes deterministic. We describe the detailed sampling algorithm in Algorithm 2. Algorithm 2 Manifold-constrained sampling 1: Input: Graph G, node attributes x, labels y L , and temperature of randomness λ 2 . 2: Get y (T ) ∼ N (y (T ) ; 0, λ 2 I) Initial sampling 3: for t = T − 1, . . . , 0 do 4: Get z ∼ N (z; 0, λ 2 I) 5: y (t) ← 1 √ αt y (t+1) − βt+1 √ 1−ᾱt+1 θ (x, y (t+1) , G, t + 1) + σ t+1 z Temporal reverse diffusion step 6:ŷ ← y (t+1) − 1−ᾱt+1 √ 1−ᾱt+1 θ (x, y (t+1) , G, t + 1) / √ᾱ t+1 7: y (t) U ←P y (t) − γ ∂ ∂y (t+1) y L − Pŷ 2 2 Manifold-constrained gradient 8: Get z ∼ N (z; 0, λ 2 I) 9: y (t) L ← √ᾱ t y L + √ 1 −ᾱ t z Consistency step 10: y (t) ← y (t) L ∪ y (t) U 11: end for 12: return y (0) U C. Experimental details In this section, we describe detailed experiment settings. The unique hyper-parameters of DPM-GSP are listed as follows. • Number of diffusion steps (T ). • Unweighted mean squared error (unweighted MSE). C.1. Inductive node classification Inductive node classification (Pubmed, Cora, Citeseer, and DBLP). In this experiment, we set the hidden dimension and the number of layers in the GNN similarly to the experimental settings in Qu et al. (2022). • GCN: We set the hidden dimensions and the number of layers to 16 and two, respectively. Here, we use ReLU (Nair & Hinton, 2010) as the activation function. • GraphSAGE: We set the hidden dimensions to 80. The number of layers is determined through a grid search within the range {2, 4}. Here, we use ReLU (Nair & Hinton, 2010) as the activation function. • GAT: We set the hidden dimensions and the number of heads to 256 and four, respectively. The number of layers is determined through a grid search within the range {2, 4}. Here, we use ELU (Clevert et al., 2015) as the activation function. We include an optional linear skip connection between each layer. The obtained node representations are then passed through the MLP, with the number of layers determined through a grid search within the range of {1, 2}. Additional hyperparameter search ranges for DPM-GSP, CLGNN, and SPN can be found in Table 7, Table 8, and Table 9, respectively. For edge temperatures in the SPN, we use the values reported by Qu et al. (2022). In the case of the base GNN, the grid search ranges for the learning rate and weight decay are the same as the search ranges of CLGNN. We use Adam (Kingma & Ba, 2014) to optimize DPM-GSP, SPN, CLGNN, and GNN. In the case of GMNN, we use the results reported by Qu et al. (2022). Inductive node classification (PPI). In this experiment, we set the number of layers in the GNN similarly to the experimental settings in Qu et al. (2022). • GCN: We set the hidden dimensions and the number of layers to 1024 and two, respectively. Here, we use ReLU (Nair & Hinton, 2010) as the activation function. We also include the linear skip connection between each layer. • GraphSAGE: We set the hidden dimensions and the number of layers to 1024 and four, respectively. Here, we use ReLU (Nair & Hinton, 2010) as the activation function. We also include the linear skip connection between each layer. • GAT: We set the hidden dimensions, the number of heads, and the number of layers to 256, four, and four, respectively. Here, we use ELU (Clevert et al., 2015) as the activation function. We also include the linear skip connection between each layer. The obtained node representations are then passed through the three-layer MLP. Additional hyperparameter search ranges for DPM-GSP, and CLGNN can be found in Table 10 and Table 11, respectively. Since this task involves the loss for 121 labels, we set the learning rate low. In the case of the base GNN, the grid search ranges for the learning rate and weight decay are the same as the search ranges of CLGNN. We use Adam (Kingma & Ba, 2014) to optimize DPM-GSP, CLGNN, and base GNN. In the case of GMNN and SPN, we use the results reported by Qu et al. (2022). C.2. Transductive node classification The unique hyper-parameters of semi-supervised DPM-GSP are listed as follows. • Size of the buffer (K). • Number of iterations in expectation step (N 1 ). • Number of iterations in maximization step (N 2 ). • Number of initial iterations (N initial 2 ): During this iteration, the maximization step utilizes the buffer initialized by the mean-field GNN. • Temperature of manifold-constrained sampling in constructing buffer (λ buffer ): Following the temperature annealing approach of Qu et al. (2019), we also control the sampling temperature λ buffer in obtaining y U for buffer construction. Transductive node classification (Pubmed, Cora, and Citeseer). In this experiment, we set the hidden dimensions, and the number of layers as follows: • GCN and GraphSAGE: We set the hidden dimensions and the number of layers to 64 and two, respectively. Here, we use ReLU (Nair & Hinton, 2010) as the activation function. We apply dropout with a probability of 0.5. • GAT: We set the hidden dimensions, the number of heads, and the number of layers to eight, eight, and two, respectively. Here, we use ELU (Clevert et al., 2015) as the activation function. We include an optional linear skip connection between each layer. We apply dropout with a probability of 0.5. The obtained node representations are then passed through the one-layer MLP. Additional hyperparameter search ranges for DPM-GSP, and CLGNN can be found in Table 12 and Table 13, respectively. In the case of the base GNN, the grid search ranges for the learning rate and weight decay are the same as the search ranges of CLGNN. We use Adam (Kingma & Ba, 2014) to optimize DPM-GSP, CLGNN, and base GNN. We also use hyper-parameters reported by Qu et al. (2019) in optimizing the GMNN. In the case of G3NN, LCM, and LP-baSed methods, we use the results reported by Ma et al. (2019a), Wang et al. (2021), and Bo et al. (2022), respectively. Transductive node classification (Amazon Photo and Computer). In this experiment, we set the hidden dimensions, and the number of layers as follows: • GCN and GraphSAGE: We set the hidden dimensions and the number of layers to 128 and two, respectively. Here, we use ReLU (Nair & Hinton, 2010) as the activation function. We apply dropout with a probability of 0.5. • GAT: We set the hidden dimensions, the number of heads, and the number of layers to 16, eight, and two, respectively. Here, we use ELU (Clevert et al., 2015) as the activation function. We apply dropout with a probability of 0.5. The obtained node representations are then passed through the one-layer MLP. Additional hyperparameter search ranges for DPM-GSP, CLGNN, and GMNN can be found in Table 14 and Table 15, and Table 16, respectively. In the case of the base GNN, the grid search ranges for the learning rate and weight decay are the same as the search ranges of CLGNN. We use Adam (Kingma & Ba, 2014) to optimize DPM-GSP, CLGNN, GMNN, and base GNN. In the case of LP-based methods, we use the results reported by Bo et al. (2022). Transductive node classification (Cornell, Wisconsin, and Texas). In this experiment, we set the hidden dimensions, and the number of layers as follows: • GCN and GraphSAGE: We set the hidden dimensions and the number of layers to 64 and one, respectively. Here, we use ReLU (Nair & Hinton, 2010) as the activation function. We apply dropout with a probability of 0.5. • GAT: We set the hidden dimensions, the number of heads, and the number of layers to eight, eight, and one, respectively. Here, we use ELU (Clevert et al., 2015) as the activation function. We apply dropout with a probability of 0.5. The obtained node representations are then passed through the one-layer MLP. We provide the additional hyper-parameters search ranges of DPM-GSP, CLGNN, and GMNN in Table 17, Table 18 and Table 19, respectively. We use Adam (Kingma & Ba, 2014) to optimize DPM-GSP, CLGNN, GMNN, and base GNN. In the case of the base GNN, the grid search ranges for the learning rate and weight decay are the same as the search ranges of CLGNN. C.3. Graph algorithmic reasoning In this section, we consider solving graph algorithmic reasoning tasks with DPM-GSP. Here, the attributes and the targets are associated with edges, i.e., x = {x i : i ∈ E} and y = {y i : i ∈ E}. If the targets y are discrete, we relax them into a one-hot vector to yield continuous values. The forward and reverse diffusion scheme is the same as that in the Section 3.1, i.e., injects and eliminates Gaussian noise on the edge targets. The parameterization of the reverse process is similar to the parameterization of the energy-based model of IREM (Du et al., 2022). Specifically, the noisy edge target y (t) is updated as follows. First, the noisy edge targets y (t) and edge features x are concatenated and passed through to the GINEConv layer, which aggregates the neighboring node features and edge features to output the node representation (Hu et al., 2020). Here, the number of GINEConv layers and hidden dimensions are three and 128, respectively. The node features are initialized to zero. After applying each GINEConv layer, we add the time embedding vector to the node representation. Then, we concatenate a pair of final node representations and noisy targets for the given edges and then apply a two-layer MLP to obtain y (t−1) . In contrast to the node classification, we introduce randomness in the reverse process, i.e., we use the stochastic reverse process for obtaining y (0) . This approach is consistent with the IREM, which also includes randomness. The detailed hyper-parameters are reported in Table 20 Figure 1 . 1Preprint. Under review.1 Code: https://github.com/hsjang0/DPM-GSP. Illustration of our DPM solving a node classification problem. DPM makes a prediction by denoising a diffusion process on an input graph. Node color indicates the label type, and opacity is proportional to the likelihood of the true label in a prediction. Figure 2 . 2Overview of our DPM-GSP in the fully supervised setting. DPM-GSP predicts the noisy targets by iteratively updating the previous predictions with a GNN-based reverse diffusion process. Figure 4 . 4Performance with varying number of GNN layers and number of diffusion steps. Increasing the number of GNN layers alone does not lead to improved accuracy, but increasing the number of diffusion steps results in better accuracy. Figure 5 . 5Intermediate accuracy with steps in the reverse diffusion process. The solid line and shaded region represent the mean and standard deviation of the accuracy, respectively. The iteration in the reverse diffusion process gradually increases accuracy. Figure 6 . 6Accuracy with varying temperature of stochastic prediction. The solid line and shaded regions represent the mean and standard deviation of the accuracy, respectively. Reducing the randomness of DPM increases the accuracy. Figure 7 . 7Accuracy with varying number of samples for stochastic inference. The solid line and shaded regions represent the mean and standard deviation of the accuracy, respectively. The dashed line represents the accuracy of deterministic inference scheme. Deterministic inference performs at least as well as stochastic inference for the considered number of samples. 8 Hoogeboom 8, E., Satorras, V. G., Vignac, C., and Welling, M. Equivariant diffusion for molecule generation in 3d. In International Conference on Machine Learning, pp. 8867-8887. PMLR, 2022b. 2 Hu, W., Liu, B., Gomes, J., Zitnik, M., Liang, P., Pande, V., and Leskovec, J. Strategies for pre-training graph neural networks. In International Conference on Learning Representations (ICLR), 2020. 17 Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. 14, 15, 16, 17 Sutton, C., McCallum, A., et al. An introduction to conditional random fields. Foundations and Trends® in Machine Learning, 4(4):267-373, 2012. 2 Tang, J., Zhang, J., Yao, L., Li, J., Zhang, L., and Su, Z. Arnetminer: extraction and mining of academic social networks. In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 990-998, 2008. 2, 5 Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., and Polosukhin, I. Attention is all you need. Advances in neural information processing systems, 30, 2017. 12 Veličković, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., and Bengio, Y. Graph attention networks. arXiv preprint arXiv:1710.10903, 2017. 2, 4 Wainwright, M. J., Jordan, M. I., et al. Graphical models, exponential families, and variational inference. Foundations and Trends® in Machine Learning, 1(1-2):1-305, 2008. 4 . We use Adam (Kingma & Ba, 2014) to optimize DPM-GSP. The reported results of other baselines are taken from Du et al. (2022). +DPM-GSP 79.89 ±0.26 54.60 ±0.48 84.36 ±0.59 60.15 ±0.51 76.20 ±0.65 52.03 ±0.90.04 51.70 ±1.23 79.90 ±0.93 54.30 ±1.15 72.18 ±0.48 48.46 ±1.06 GCN 78.52 ±0.72 51.04 ±0.46 81.06 ±0.64 56.62 ±0.74 72.96 ±0.72 47.62 ±0.58 +SPN 79.20 ±0.52 52.90 ±0.41 82.23 ±0.48 59.16 ±0.30 73.96 ±0.84 48.66 ±1.04 +CLGNN 79.22 ±0.46 52.64 ±0.68 82.09 ±0.43 58.15 ±0.58 74.00 ±0.76 48.24 ±0.88 +DPM-GSP 80.01 ±0.47 54.49 ±0.33 83.90 ±0.69 59.72 ±0.66 74.96 ±0.89 49.86 ±0.85 GraphSAGE 77.42 ±0.32 49.24 ±1.36 81.68 ±0.22 58.15 ±0.66 73.59 ±0.72 45.12 ±0.94 +SPN 78.21 ±0.65 52.68 ±1.01 81.88 ±0.70 58.85 ±0.65 74.18 ±1.36 45.98 ±1.22 +CLGNN 77.97 ±1.24 51.26 ±2.71 82.28 ±0.44 59.56 ±1.27 74.57 ±0.44 46.69 ±1.13 +DPM-GSP 78.91 ±0.68 53.07 ±0.65 83.23 ±0.37 60.04 ±0.97 76.33 ±0.87 50.33 ±1.10 GAT 78.56 ±0.42 51.68 ±0.62 81.78 ±0.88 55.98 ±1.32 74.16 ±0.18 47.86 ±0.34 +SPN 79.16 ±0.21 52.65 ±0.66 83.42 ±0.52 58.51 ±0.64 74.78 ±0.66 48.76 ±0.87 +CLGNN 78.88 ±0.36 53.16 ±0.70 83.18 ±0.38 56.92 ±0.90 74.59 ±0.36 48.64 ±0.69 ±0.10 77.07 ±0.05 54.15 ±0.17 78.02 ±0.05 55.73 ±0.15 80.59 ±0.04 61.36 ±0.11 82.56 ±0.20 66.70 ±0.77 +CLGNN 81.41 ±1.17 80.09 ±0.33 64.76 ±0.20 84.20 ±0.12 73.34 ±0.64 97.45 ±0.10 95.64 ±0.20 99.31 ±0.01 98.66 ±0.01 +DPM-GSP 83.93 ±1.07 83.00 ±0.30 71.20 ±0.38 87.24 ±0.24 77.35 ±0.45 98.17 ±0.08 96.87 ±0.15 99.45 ±0.02 99.10 ±0.02 GraphSAGE 75.24 ±3.26 80.04 ±0.07 66.83 ±0.37 83.65 ±0.35 73.21 ±0.26 97.67 ±0.05 96.05 ±0.09 99.47 ±0.02 99.10 ±0.02 +SPN † 78.86 ±3.70 82.11 ±0.03 68.56 ±0.07 85.40 ±0.05 74.45 ±0.07 95.28 ±0.02 91.99 ±0.04 98.55 ±0.02 97.56 ±0.03 +CLGNN 76.43 ±2.43 81.50 ±0.11 67.74 ±0.22 84.05 ±0.40 73.74 ±0.38 97.68 ±0.03 96.08 ±0.04 99.49 ±0.02 99.14 ±0.04 +DPM-GSP 81.04 ±2.57 83.50 ±0.27 72.09 ±0.22 87.80 ±0.11 78.88 ±0.23 98.27 ±0.07 97.05 ±0.09 99.60 ±0.03 99.31 ±0.04 GAT 78.64 ±3.12 77.87 ±0.94 63.64 ±0.25 81.70 ±0.58 69.86 ±0.87 97.18 ±0.23 95.19 ±0.41 99.45 ±0.01 99.07 ±0.03 +SPN † 84.84 ±0.73 78.13 ±0.21 62.25 ±0.32 83.55 ±0.12 72.37 ±0.18 96.68 ±0.13 94.41 ±0.21 99.04 ±0.06 98.38 ±0.10 +CLGNN 82.79 ±1.45 79.01 ±0.17 64.02 ±0.40 82.78 ±0.18 70.39 ±0.79 97.24 ±0.24 95.26 ±0.04 99.46 ±0.02 99.09 ±0.03 +DPM-GSP 83.89 ±2.53 81.68 ±0.73 69.89 ±0.49 86.21 ±0.20 75.58 ±0.23 98.24 ±0.08 97.00 ±0.15 99.62 ±0.01 99.34 ±0.02.93 77.55 ±0.53 57.20 ±2.63 81.21 ±0.87 67.46 ±2.92 94.67 ±2.77 90.72 ±5.28 97.00 ±2.98 94.69 ±5.60 GCN 77.20 ±3.42 79.61 ±0.24 63.92 ±0.29 83.96 ±0.16 73.47 ±0.46 97.21 ±0.13 95.24 ±0.23 99.24 ±0.00 98.34 ±0.01 +SPN † 79.50 respectively. ±0.52 81.5 ±0.68 71.6 ±0.72 90.4 ±0.70 82.3 ±1.51 57.6 ±6.12 58.7 ±5.58 57.2 ±6.01 + G3NN-LSM ‡ 77.9 ±0.40 82.5 ±0.20 74.4 ±0.30 ±0.36 82.4 ±0.66 72.1 ±0.68 90.6 ±1.11 83.3 ±1.53 58.6 ±4.26 57.8 ±6.71 58.1 ±3.48 + G3NN-LSM ‡ 77.6 ±0.70 82.9 ±0.30 73.1 ±0.50Method Pubmed Cora Citeseer Photo Computer Cornell Wisconsin Texas LP † 70.4 ±0.00 50.6 ±0.00 71.8 ±0.00 79.8 ±3.40 79.0 ±4.80 - - - GLP † 78.8 ±0.40 80.3 ±0.20 71.7 ±0.60 89.6 ±0.70 81.9 ±1.10 - - - PTA † 80.1 ±0.10 83.0 ±0.50 71.6 ±0.40 90.7 ±2.10 82.3 ±0.90 - - - LCM * 77.7 ±1.90 83.3 ±0.70 72.2 ±0.50 - - - - - GMNN 81.7 ±0.56 83.8 ±0.86 73.0 ±0.72 91.3 ±1.56 82.8 ±1.52 53.5 ±7.02 61.8 ±8.24 68.1 ±7.72 GCN 78.6 - - - - - + G3NN-SBM ‡ 78.4 ±0.60 82.2 ±0.20 74.5 ±0.40 - - - - - + CLGNN 80.2 ±0.49 81.6 ±0.32 71.4 ±0.68 90.8 ±1.11 82.9 ±1.75 55.6 ±7.06 62.3 ±2.74 65.9 ±5.43 + DPM-GSP 82.6 ±0.90 82.9 ±0.41 74.0 ±0.62 91.5 ±1.01 83.4 ±1.88 63.6 ±5.72 67.5 ±2.76 68.4 ±3.19 GraphSAGE 77.2 ±0.88 80.0 ±0.87 71.2 ±0.66 91.0 ±0.84 82.1 ±0.97 74.3 ±2.16 81.1 ±5.56 82.1 ±5.43 + CLGNN 77.4 ±0.47 81.0 ±0.56 71.9 ±0.82 91.2 ±0.83 83.1 ±1.78 74.0 ±2.75 83.1 ±3.94 81.6 ±6.59 + DPM-GSP 78.4 ±1.38 83.2 ±0.43 72.6 ±0.96 91.9 ±1.16 84.0 ±1.43 75.1 ±2.91 84.1 ±3.44 83.0 ±4.68 GAT 79.1 - - - - - + G3NN-SBM ‡ 77.4 ±0.40 82.9 ±0.30 74.0 ±0.30 - - - - - + CLGNN 80.8 ±0.42 83.3 ±0.51 73.2 ±0.66 90.9 ±1.15 83.6 ±1.38 57.3 ±4.32 65.7 ±3.64 61.9 ±6.33 + DPM-GSP 82.0 ±0.29 84.2 ±0.70 74.6 ±0.62 92.2 ±1.05 85.2 ±1.31 63.0 ±5.27 67.1 ±4.36 64.0 ±5.58 Table 4 . 4Performance on graph algorithmic reasoning tasks. Bold numbers indicate the best performance for each task. The same size and large size indicate the performance on graphs with ten and 15 nodes, respectively. † We use the numbers reported byDu et al. (2022).Edge copy Connected components Shortest path Method Same size Large size Same size Large size Same size Large size Feedforward † 0.3016 0.3124 0.1796 0.3460 0.1233 1.4089 Recurrent † 0.3015 0.3113 0.1794 0.2766 0.1259 0.1083 Programmatic † 0.3053 0.4409 0.2338 3.1381 0.1375 0.1375 Iterative feedforward † 0.6163 0.6163 0.4908 0.4908 0.4908 0.7688 IREM † 0.0019 0.0019 0.1424 0.2171 0.0274 0.0464 DPM-GSP 0.0011 0.0038 0.0724 0.1884 0.0138 0.0286 5 10 20 40 80 Diffusion steps ±0.47 54.49 ±0.33 123. 83.90 ±0.69 59.72 ±0.66 129. 74.96 ±0.89 49.86 ±0.85 120.Pubmed Cora Citeseer Methods N-Acc G-Acc Time N-Acc G-Acc Time N-Acc G-Acc Time SPN (20) 79.12 ±0.60 52.88 ±0.44 4.85 82.23 ±0.48 59.16 ±0.30 4.95 73.96 ±0.84 48.66 ±1.04 4.52 SPN (100) 79.20 ±0.52 52.90 ±0.41 18.3 82.18 ±0.50 59.14 ±0.34 19.8 73.98 ±0.76 48.65 ±0.94 17.6 CLGNN 79.22 ±0.46 52.64 ±0.68 24.8 82.09 ±0.43 58.15 ±0.58 25.6 74.06 ±0.72 48.24 ±0.88 23.2 DPM-GSP (3) 79.19 ±0.69 52.27 ±1.01 5.05 83.06 ±0.56 59.19 ±0.78 5.58 74.12 ±0.69 48.60 ±0.88 5.21 DPM-GSP (80) 80.01 Table 7 . 7Hyper-parameters search ranges of DPM-GSP for Pubmed, Cora, and Citeseer.Network learning rate weight decay T unweighted MSE GCN 0 for DBLP and {0.005, 0.001, 0.0005} for others {0.003, 0.001, 0.0005} {40, 80} {True, False} GraphSAGE 0 for DBLP and {0.001, 0.0005} for others {0.001, 0.0005} {40, 80} {True, False} GAT 0 for DBLP and {0.001, 0.0005, 0.0001} for others {0.001, 0.0005, 0.0001} {40, 80} {True, False} Table 8 . 8Hyper-parameters search ranges of CLGNN for Pubmed, Cora, and Citeseer.Models learning rate weight decay number of Monte Carlo samples GCN {0.01, 0.005, 0.001} {0.003, 0.001, 0.0005} {5, 10} GraphSAGE {0.01, 0.005, 0.001} {0.001, 0.0005} {5, 10} GAT {0.01, 0.005, 0.001} {0.001, 0.0005, 0.0001} {5, 10} Table 9. Hyper-parameters search ranges of SPN for Pubmed, Cora, and Citeseer. Network learning rate (node GNN) learning rate (edge GNN) weight decay GCN {0.01, 0.005} {0.01, 0.005} {0.003, 0.001, 0.0005} GraphSAGE {0.005, 0.001} {0.005, 0.001} {0.001, 0.0005} GAT {0.005, 0.001} {0.005, 0.001, 0.0005} {0.001, 0.0005, 0.0001} Table 10 . 10Hyper-parameters search ranges of DPM-GSP for PPI.Table 11. Hyper-parameters search ranges of CLGNN for PPI. Network learning rate weight decay number of Monte Carlo samples GCN, GraphSAGE, and GAT {0.0003, 0.0001} 0 5Network learning rate weight decay T unweighted MSE GCN, GraphSAGE, and GAT {0.0003, 0.0001} 0 {80, 200} True Table 12 . 12Hyper-parameters search ranges of DPM-GSP for Pubmed, Cora, and Citeseer (semi-supervised learning).Table 13. Hyper-parameters search ranges of CLGNN for Pubmed, Cora, and Citeseer (semi-supervised learning).Models lr weight decay T unweighted MSE K N 1 N 2 N initial 2 λ buffer GCN and GraphSAGE {0.01, 0.005, 0.001} {0.01, 0.005, 0.001} 80 True 50 5 100 1000 {0.3, 0.5, 1} GAT {0.01, 0.005, 0.001, 0.0005} {0.01, 0.005, 0.001} 80 True 50 5 100 2000 {0.3, 0.5, 1} Models learning rate weight decay number of Monte Carlo samples GCN and GraphSAGE {0.05, 0.01, 0.005} {0.01, 0.005, 0.001} {5, 10} GAT {0.01, 0.005, 0.001} {0.01, 0.005, 0.001} {5, 10} Table 14 .Table 16 . 1416Hyper-parameters search ranges of DPM-GSP for Amazon Photo and Computer (semi-supervised learning). {0.1, 0.3, 0.5, 1} GraphSAGE and GAT {0.001, 0.0005} {0.001, 0.0005, 0.0001} 80Table 15. Hyper-parameters search ranges of CLGNN for Amazon Photo and Computer (semi-supervised learning). GraphSAGE and GAT {0.005, 0.001} {0.001, 0.0005, 0.0001} 5 Hyper-parameter search ranges of GMNN for Amazon Photo and Computer (semi-supervised learning). {0.005, 0.001} {0.005, 0.001, 0.0005} 128 {0.1, 0.3, 0.5, 1}Models lr weight decay T unweighted MSE K N 1 N 2 N initial 2 λ buffer GCN {0.005, 0.001} {0.005, 0.001, 0.0005} 80 True 50 5 100 2000 True 50 5 100 2000 {0.1, 0.3, 0.5, 1} Models learning rate weight decay number of Monte Carlo samples GCN {0.005, 0.001} {0.005, 0.001, 0.0005} 5 learning rate weight decay hidden dimension annealing parameter λ Table 17 . 17Hyper-parameters search ranges of DPM-GSP for Cornell, Wisconsin, and Texas (semi-supervised learning). GraphSAGE, and GAT {0.01, 0.005, 0.001} {0.01, 0.005, 0.001} 80Models lr weight decay T unweighted MSE K N 1 N 2 N initial 2 λ buffer GCN, True 50 5 50 1000 {0.1, 0.3, 0.5, 1} Table 18 . 18Hyper-parameters search ranges of CLGNN for Cornell, Wisconsin, and Texas (semi-supervised learning). GraphSAGE, and GAT {0.01, 0.005, 0.001} {0.01, 0.005, 0.001} {5, 10} Table 19. Hyper-parameter search ranges of GMNN for Cornell, Wisconsin, and Texas (semi-supervised learning). {0.01, 0.005, 001} {0.01, 0.005, 0.001} 64 {0.1, 0.3, 0.5, 1}Models learning rate weight decay number of Monte Carlo samples GCN, learning rate weight decay hidden dimension annealing parameter λ Table 20 . 20Hyper-parameters of DPM-GSP for graph algorithmic reasoning. unweighted MSE temperature of reverse process {0.0004, 0.0001} {1e-6, 1e-5} {50,100} True {0.3, 0.5, 1}learning rate weight decay T POSTECH 2 KAIST. Correspondence to: Sungsoo Ahn <[email protected]>. We verify that our deterministic inference strategy indeed improves over the typical stochastic inference strategy inFigure 7. Structured denoising diffusion models in discrete state-spaces. J Austin, D D Johnson, J Ho, D Tarlow, Van Den, R Berg, Advances in Neural Information Processing Systems. 34Austin, J., Johnson, D. 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[ "Hesitancy, Awareness and Vaccination: A Computational Analysis on Complex Networks", "Hesitancy, Awareness and Vaccination: A Computational Analysis on Complex Networks" ]	[ "Dibyajyoti Mallick \nDepartment of Physics\nNational Institute of Technology\nDurgapur\n", "Aniruddha Ray \nDepartment of Physics\nNational Institute of Technology\nDurgapur\n", "Ankita Das \nDepartment of Physics\nNational Institute of Technology\nDurgapur\n", "Sayantari Ghosh \nDepartment of Physics\nNational Institute of Technology\nDurgapur\n" ]	[ "Department of Physics\nNational Institute of Technology\nDurgapur", "Department of Physics\nNational Institute of Technology\nDurgapur", "Department of Physics\nNational Institute of Technology\nDurgapur", "Department of Physics\nNational Institute of Technology\nDurgapur" ]	[]	Considering the global pandemic of coronavirus disease 2019 , around the world several vaccines are being developed. Till now, these vaccines are the most effective way to reduce the high burden on the global health infrastructure. However, the public acceptance towards vaccination is a crucial and pressing problem for health authorities. This study has been designed to determine the parameters affecting the decisions of common individuals towards COVID-19 vaccine. In our study, using the platforms of compartmental model and network simulation, we categorize people and observe their motivation towards vaccination in a mathematical social contagion process. In our model, we consider peer influence as an important factor in this dynamics, and study how individuals are influencing each other for vaccination. The efficiency of the vaccination process is estimated by the period of time required to vaccinate a substantial fraction of total population. We discovered the major barriers and drivers of this dynamics, and concluded that it is required to formulate specific strategies by the healthcare workers which could be more effective for the undecided and vaccine hesitant group of people.	null	[ "https://export.arxiv.org/pdf/2302.10474v1.pdf" ]	257,050,842	2302.10474	0654037145110295faa0f34bf1023b1419738309	Hesitancy, Awareness and Vaccination: A Computational Analysis on Complex Networks Dibyajyoti Mallick Department of Physics National Institute of Technology Durgapur Aniruddha Ray Department of Physics National Institute of Technology Durgapur Ankita Das Department of Physics National Institute of Technology Durgapur Sayantari Ghosh Department of Physics National Institute of Technology Durgapur Hesitancy, Awareness and Vaccination: A Computational Analysis on Complex Networks * These authors share equal credit in this work. Considering the global pandemic of coronavirus disease 2019 , around the world several vaccines are being developed. Till now, these vaccines are the most effective way to reduce the high burden on the global health infrastructure. However, the public acceptance towards vaccination is a crucial and pressing problem for health authorities. This study has been designed to determine the parameters affecting the decisions of common individuals towards COVID-19 vaccine. In our study, using the platforms of compartmental model and network simulation, we categorize people and observe their motivation towards vaccination in a mathematical social contagion process. In our model, we consider peer influence as an important factor in this dynamics, and study how individuals are influencing each other for vaccination. The efficiency of the vaccination process is estimated by the period of time required to vaccinate a substantial fraction of total population. We discovered the major barriers and drivers of this dynamics, and concluded that it is required to formulate specific strategies by the healthcare workers which could be more effective for the undecided and vaccine hesitant group of people. Introduction The novel coronavirus disease 2019 , caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was first reported in Wuhan, Hubei Province of China. This disease is now a global pandemic which is spreading rapidly from person to person causing major public health concerns and economic crisis [14] [1]. A variety of active intervention policies have been introduced to suppress the spreading of this disease, such as hand sanitizing, social distancing, travel restrictions, partial or complete lockdown, wearing mask, quarantining, etc. After the declaration of the pandemic by WHO (World Health Organization) in March 2020, pharmaceutical companies and scientists have encountered a race against time to develop vaccines [6]. The recent availability of multiple vaccines against coronavirus has brought hope for prevention of the spreading and a rapid recovery of our badly affected economy, with a promise of sooner resumption of normal life. However, the widespread hesitancy about vaccines are becoming a major obstacle for global health. Many people have strong hesitation towards vaccination which is defined as the confusion about safety and effectiveness of the vaccine. The origin of this usually lies in some rumors regarding vaccine, concerning about side effects. This is has become a huge challenge for governments and public health authorities for reaching the expected and required vaccination coverage [3] [9], [19]. The key priority now is ensuring the vaccine acceptance because the lag in vaccination may provide a window for spread the new variants and can also be a major obstacle in developing society-wide herd immunity. Till date many researchers, scientists have tried to conduct different surveys to understand the behaviour towards vaccine acceptance and hesitancy [8,10,2]. Some studies [7], have applied different theoretical models to explain the vaccine acceptance, hesitancy, willingness of individuals towards vaccine as well as refusal to vaccinate which may vary depending upon personal decisions and epidemiological conditions [18]. Many advance countries are trying to conduct country specific surveys [11], while several reports are being prepared by different countries such as United Kingdom(UK), United States of America(USA), China, India, Saudi Arab to understand this vaccine acceptance behaviour. [5] [16] [12]. However, most of this studies are based on heuristic arguments rather than mathematical analysis. A computational framework having mathematical foundation helps to draw quantitative conclusions, and much more effective in predictive modeling purposes. This paper represents a new mathematical model regarding vaccination process which incorporates behavioural changes of every individuals in a society, driven by the global and local factors. The main assumption is here that the vaccination dynamics can be considered as social contagion process. This study aimed to identify the acceptability of covid-19 vaccine, information in support or opposing vaccination are flowing in a society like a viral infection. We take into account of refusal and hesitancy factors as well as the positive attitude of people towards vaccination [17]. In our study we focused on to identify the effects of the factors that could increase the vaccination coverage through numerical simulations on an artificial society. Moreover, we could justify several results found by survey based studies where this kind of questions have already been explored using survey results [4,15]. Our results address the vaccination acceptance problem using a comprehensive mathematical and computational framework, that would help us figuring out correct strategies, to encourage community for vaccine uptake and to stop further spreading of this pandemic. Model Formulation To understand this kind of problem, we have considered a set of differential equations to depict the possible transitions. To describe the vaccine dynamics we have our compartmental model as shown in figure 1 , at any time t, the total population N (t) is subdivided into four states: Ignorant (I), Hesitant (H), Unwilling(U ) , Vaccinated(V ). • Ignorant(I) − As the name suggests, in this group, people are ignorant about vaccine. They have no idea about vaccination, and have no clear opinion. In general, these people will not involve themselves in vaccination process. But they may be influenced by hesitant and vaccinated groups as well. Each people in this group are represented by I(t). • Hesitant(H) − In this group people know about vaccine nevertheless, they feel hesitation while taking decision regarding vaccination, being affected by the rumors. This group may be influenced by the global advertisement campaigns in favour of vaccination, and take vaccine eventually. On the other hand, they could also be influenced further by rumors and transit to unwilling state. Each people in this group are represented by H(t). • Unwilling(U ) − In this group people believe on negative rumors regarding vaccine and they push themselves for vaccination. Again some Unwilling people may return to Hesitant class as far they can not decided whether to get vaccine or not, by getting influenced through vaccinated subpopulation. Each people in this group are represented by U (t). • Vaccinated(V ) − In this group people are completely vaccinated and they are trying to influence other groups for joining them. Each people in this group are represented by V (t). In this model the key variable we have is the vaccinated population, who are influencing other populations to get vaccinated. This is also the target population which have to maximised over a certain period of time. Here, this group of people is playing an effective role to control the epidemic, and spread the infodemic. Considering the total population = 1 (normalized form), I + H + U + V = 1 Now, we will discuss the following possibilities of transition of people from one subpopulation to another over a given time period 't'. Let us discuss all the possible transitions considered in model, categorically. • Possible transitions related to I subpopulation: In this population µ is the rate at which people are entering into the ignorant population. This is added to consider the demographic variations. Now the members of I may be influenced by H and V population. H and V people might spread words on their inclination and decision, and encourage others for vaccination. Let us consider α be the effective contact rate. If each member of I group is going to be influenced by hesitant and vaccinated group with that rate, then α(H + V )I amount will be subtracted from I group and must be added to hesitant group. In the normal course, there must be some people who can die from that population, hence the term µ should also be deducted from group I in the rate equation. • Possible transitions related to H subpopulation: As mentioned in the previous case, due to the effect of influence from group H and V , the term α(H + V ) should be added in this rate equation. As usual people from this group can also die hence the term µH also be deducted from H group in its rate equation. Due to some reasons (influencing from other groups or due to development of self awareness among people), they are going for vaccination at a rate λ and some are entering into Unwilling group at a rate β. Hence the term βH and λH should be deducted from the rate equation. Again from Unwilling group some people are coming back into Hesitant group by believing on some rumors, hence the term γU V should be added to the rate equation of H. • Possible transitions related to U subpopulation: From the previous case discussed above, it should be noted that the term γU V and µU should be deducted from its rate equation, due to their hesitant behaviour and natural death from this group respectively. • Possible transitions related to V subpopulation: From the discussion of case 2, it is evident that the term λH must be added to its rate equation due to direct wishing for vaccination from group H and µV must be deducted due to natural death of people from this group. Hence all the rate equations of this model are compiled as follows: dI dt = µ − α(H + V )I − µI (1) dH dt = α(H + V )I − βH − λH + γ U V − µH (2) dU dt = βH − γ U V − µU (3) dV dt = λH − µV(4) Model Analysis on Complex Network In ODE-based models one of the major issues are homogeneous mixing, which indicates every individuals in a population have same probability of having contact with each other [20]. Our society in highly heterogeneous, and to accommodate that fact into our findings, we study the model on the heterogeneous setting of a complex network. The simulations are performed on a random network having 10000 nodes with an average degree 5. Here we choose EoN module in Networkx from python [13] and run these following simulations in Google Colaboratory. Varying the transition rates of each parameter we see the effects of that particular parameter on the time evolution of the dynamics through our model. The general dynamics has been depicted in Figure 2. It shows that eventually most of the people in population get vaccinated, however the time of coverage might be different, depending on the parameter values. Thus, to quantify this growth curve, we define, vaccination coverage time, τ V : τ V = {t\|V (t) = 0.9 V max } This means that we consider a timescale, τ V within which 90% of maximum vaccination, V max has been achieved. This will give us an estimate of the vaccination coverage speed. Effect of Vaccination Rate Vaccination rate is a major parameter of this dynamics. If you carefully observe different subpopulations, the effect could be prominently observed, as reported in Figure 3. • Effect on vaccinated people: As we can see from the results in Figure 3(a), that increasing the rate does not change the maximum number of vaccinated people but it shifts the saturation point leftwards, which means the system is reaching the saturation point faster. • Effect on hesitant people: We can see from the plots shown in Figure 3(b) that increasing the rate shifts the peak point slightly leftwards, which means the system is reaching the peak point a bit faster and we also can see that the maximum number of people reaching in a hesitant class decreases a little bit. • Effect on unwilling people: As we can see from the plots shown in Figure 3(c), increasing the parameter flattens the curve which means that the maximum number of people going to the unwilling class decreases that can be explained by the success rate of vaccination process. As in figure 3(d), we can see that the saturation time decreases with increasing vaccination rate that is, λ which states that this particular parameter will make the system to reach the saturation point earlier. Effect of Negative Rumors Negative rumors often sway people towards some risky and dangerous activity. In this vaccination process some negative rumors about vaccine exist that concerns with matters like, several side effects, cost effective, fear, misinformation regarding vaccine. Hesitant people often are getting influenced by them. The parameter that takes into account of this phenomena is β. The results related to this are shown in Figure 4. • Effect on vaccinated class: As we can see in Figure 4(a), increasing the parameter, β, shifts the saturation point of vaccinated people rightwards. This means as more people feel negative about vaccination, more time it will take for the system to reach the saturation point. • Effect on unwilling class: In Figure 4(b) we can see that many people are motivated from their neighbors, friends, relatives to believe on some negative rumors about vaccination and denied to get vaccine and along the way the maximum number of unwilling people increases. As in figure 4(d) we can see that the saturation time increases with increasing the negative peer influence rate that is, β. This means the system will take more time to reach the saturation point. Effect of Positive Peer Influence Peer influence occurs when people are motivated or influenced towards something by seeing their neighbour or friends. When someone's peers influence them to do something positive is considered as positive peer influence. In the vaccination context, positive peer influence occurs when someone who is vaccinated and he/she is motivating others who are not sure about vaccination. Here the vaccinated people influences the hesitant people to get vaccinated. Here we are considering two parameters α and γ that takes into account this phenomena. The results related to this are shown in Figure 5 and Figure 6. • Effect on Vaccinated people: As we can see in Figure 5(a) and 6(a), that increasing the parameter it shifts the saturation point of vaccinated people leftwards that means the more people feel positive about vaccination then sooner it will take the system to reach the saturation point. • Effect on Unwilling people: We can also see from Figure 5(b) and 6(b) that along the way the maximum number of unwilling people decreases as they are getting motivated by some positive rumors towards vaccination. So, we can draw this conclusion that if more people are well informed about the success rate of vaccination then that would fasten up the vaccination process and also decrease the amount of unwilling people. As shown in figure 5(d) and 6(d), for both parameters of positive peer influence that is, α and γ, the saturation time decreases with increasing parameters which means positive peer influence will make the system reach the saturation point earlier. Conclusions and perspectives Along with the infectious spread of SARS-COV2, the information and misinformation regarding available vaccines are also spreading person-to-person causing a large-scale effect on the vaccination coverage. Through computational analysis and network simulations, we have explored this contagion dynamics, and proposed a framework to analyze this decision process. From the model it has been observed that how the presence of the efficient vaccination system can run the entire dynamics with positive feedback. This is a significant result as we can see a lot of people not only in India even around the world are hesitant about getting vaccinated. In our model, we have shown that if the peer influence is overall positive then the saturation point shifts leftwards, so we can conclude that the spreading of awareness about the positive effects of vaccination should be carried out on the ground level. We have also shown in our model that if vaccination rate increases then the saturation point of V class shift leftwards, in real life we can explain it like this− if the vaccination rate in a particular area can be increased by awareness and smooth execution of the vaccination process by competent authority then the system would reach its saturation point sooner. A strong peer influence also might signify a strongly connected society. The shift the saturation point leftwards for high positive peer influence means that physically and virtually everyone is well connected. This may also indicate high vaccination rates in urban areas. In future scenario, we will focus on implementing further realistic terms in our model. For example, if someone from hesitant class goes to unwilling class it can be considered that the person was under the influence of someone who has been vaccinated, thus we can bring the non-linearity. Mathematical explorations to find out The fixed points and bifurcations if there are any. Considering coupled dynamics, or delayed dynamics of disease along with the infodemic might be another interesting future study. Figure 1 : 1Compartmental diagram of the proposed model (I-H-U -V ). All the transitions, considered in the proposed model are shown through blue arrows. Subpopulations are denoted by yellow boxes. Figure 2 : 2Population of different classes vs. time. Eventually most of the population gets vaccinated, but time of coverage depends on parameters. Figure 3 : 3(a) Vaccination coverage with time for different values of vaccination rate λ. (b) Time evolution of Unwilling people in the population with different values of vaccination rate parameter. (c) Time evolution of Hesitant people in the population with different values of vaccination rate parameter. (d) Vaccination coverage time(τ V ) for different values of vaccination rate. Figure 4 : 4(a) Vaccination coverage with time for different values of negative rumor influence rate β. (b) Time evolution of Unwilling people in the population with different values of negative rumor influence rate parameter. (c) Time evolution of Hesitant people in the population with different values of negative rumor influence rate parameter. (d) Vaccination coverage time(τ V ) for different values of negative rumor influence rate. Figure 5 : 5(a) Vaccination coverage with time for different values of positive peer influence rate γ. 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[ "SEMI-SUPERVISED LEARNING WITH CONTEXT-CONDITIONAL GENERATIVE ADVERSARIAL NETWORKS", "SEMI-SUPERVISED LEARNING WITH CONTEXT-CONDITIONAL GENERATIVE ADVERSARIAL NETWORKS" ]	[ "Emily Denton [email protected] \nDept. of Computer Science Courant Institute\nFacebook AI Research New York\nFacebook AI Research New York\nNew York University\n\n", "Sam Gross [email protected] \nDept. of Computer Science Courant Institute\nFacebook AI Research New York\nFacebook AI Research New York\nNew York University\n\n", "Rob Fergus [email protected] \nDept. of Computer Science Courant Institute\nFacebook AI Research New York\nFacebook AI Research New York\nNew York University\n\n" ]	[ "Dept. of Computer Science Courant Institute\nFacebook AI Research New York\nFacebook AI Research New York\nNew York University\n", "Dept. of Computer Science Courant Institute\nFacebook AI Research New York\nFacebook AI Research New York\nNew York University\n", "Dept. of Computer Science Courant Institute\nFacebook AI Research New York\nFacebook AI Research New York\nNew York University\n" ]	[]	We introduce a simple semi-supervised learning approach for images based on in-painting using an adversarial loss. Images with random patches removed are presented to a generator whose task is to fill in the hole, based on the surrounding pixels. The in-painted images are then presented to a discriminator network that judges if they are real (unaltered training images) or not. This task acts as a regularizer for standard supervised training of the discriminator. Using our approach we are able to directly train large VGG-style networks in a semi-supervised fashion. We evaluate on STL-10 and PASCAL datasets, where our approach obtains performance comparable or superior to existing methods.	null	[ "https://arxiv.org/pdf/1611.06430v1.pdf" ]	10,020,949	1611.06430	6139ed57b3fe91915991897723aed6a98cd32f82	SEMI-SUPERVISED LEARNING WITH CONTEXT-CONDITIONAL GENERATIVE ADVERSARIAL NETWORKS Emily Denton [email protected] Dept. of Computer Science Courant Institute Facebook AI Research New York Facebook AI Research New York New York University Sam Gross [email protected] Dept. of Computer Science Courant Institute Facebook AI Research New York Facebook AI Research New York New York University Rob Fergus [email protected] Dept. of Computer Science Courant Institute Facebook AI Research New York Facebook AI Research New York New York University SEMI-SUPERVISED LEARNING WITH CONTEXT-CONDITIONAL GENERATIVE ADVERSARIAL NETWORKS Under review as a conference paper at ICLR 2017 We introduce a simple semi-supervised learning approach for images based on in-painting using an adversarial loss. Images with random patches removed are presented to a generator whose task is to fill in the hole, based on the surrounding pixels. The in-painted images are then presented to a discriminator network that judges if they are real (unaltered training images) or not. This task acts as a regularizer for standard supervised training of the discriminator. Using our approach we are able to directly train large VGG-style networks in a semi-supervised fashion. We evaluate on STL-10 and PASCAL datasets, where our approach obtains performance comparable or superior to existing methods. INTRODUCTION Deep neural networks have yielded dramatic performance gains in recent years on tasks such as object classification (Krizhevsky et al., 2012), text classification (Zhang et al., 2015) and machine translation (Sutskever et al., 2014;Bahdanau et al., 2015). These successes are heavily dependent on large training sets of manually annotated data. In many settings however, such large collections of labels may not be readily available, motivating the need for methods that can learn from data where labels are rare. We propose a method for harnessing unlabeled image data based on image in-painting. A generative model is trained to generate pixels within a missing hole, based on the context provided by surrounding parts of the image. These in-painted images are then used in an adversarial setting (Goodfellow et al., 2014) to train a large discriminator model whose task is to determine if the image was real (from the unlabeled training set) or fake (an in-painted image). The realistic looking fake examples provided by the generative model cause the discriminator to learn features that generalize to the related task of classifying objects. Thus adversarial training for the in-painting task can be used to regularize large discriminative models during supervised training on a handful of labeled images. RELATED WORK Learning From Context: The closest work to ours is the independently developed context-encoder approach of Pathak et al. (2016). This introduces an encoder-decoder framework, shown in Fig. 1(a), that is used to in-paint images where a patch has been randomly removed. After using this as a pre-training task, a classifier is added to the encoder and the model is fine-tuned using the labeled examples. Although both approaches use the concept of in-painting, they differ in several important ways. First, the architectures are different (see Fig. 1): in Pathak et al. (2016), the features for the classifier are taken from the encoder, whereas ours come from the discriminator network. In practice this makes an important difference as we are able to directly train large models such as VGG (Simonyan & Zisserman, 2015) using adversarial loss alone. By contrast, Pathak et al. (2016) report difficulties in training an AlexNet encoder with this loss. This leads to the second difference, namely that on account of these issues, they instead employ an 2 loss when training models for classification and detection (however they do use a joint 2 and adversarial loss to achieve impressive in-painting results). Finally, the unsupervised learning task differs between the two models. The context-encoder learns a feature representation suitable for in-painting whereas our model learns a feature representation suitable for differentiating real/fake in-paintings. Notably, while we also use a neural network to generate the in-paintings, this model is only used as an adversary for the In (a-c) the blue network indicates the feature representation being learned (encoder network in the context-encoder model and discriminator network in the GAN and CC-GAN models). discriminator, rather than as a feature extractor. In section 4, we compare the performance of our model to the context-encoder on the PASCAL dataset. Other forms of spatial context within images have recently been utilized for representation learning. Doersch et al. (2015) propose training a CNN to predict the spatial location of one image patch relative to another. Noroozi & Favaro (2016) propose a model that learns by unscrambling image patches, essentially solving a jigsaw puzzle to learn visual representations. In the text domain, context has been successfully leveraged as an unsupervised criterion for training useful word and sentence level representations (Collobert et al., 2011;Mikolov et al., 2015;Kiros et al., 2015). Deep unsupervised and semi-supervised learning: A popular method of utilizing unlabeled data is to layer-wise train a deep autoencoder or restricted Botlzmann machine (Hinton et al., 2006) and then fine tune with labels on a discriminative task. More recently, several autoencoding variants have been proposed for unsupervised and semi-supervised learning, such as the ladder network (Rasmus et al., 2015), stacked what-where autoencoders (Zhao et al., 2016) and variational autoencoders . Dosovitskiy et al. (2014) achieved state-of-the-art results by training a CNN with a different class for each training example and introducing a set of transformations to provide multiple examples per class. The pseudo-label approach (Lee, 2013) is a simple semi-supervised method that trains using the maximumly predicted class as a label when labels are unavailable. Springenberg (2015) propose a categorical generative adversarial network (CatGAN) which can be used for unsupervised and semi-supervised learning. The discriminator in a CatGAN outputs a distribution over classes and is trained to minimize the predicted entropy for real data and maximize the predicted entropy for fake data. Similar to our model, CatGANs use the feature space learned by the discriminator for the final supervised learning task. Salimans et al. (2016) recently proposed a semi-supervised GAN model in which the discriminator outputs a softmax over classes rather than a probability of real vs. fake. An additional 'generated' class is used as the target for generated samples. This method differs from our work in that it does not utilize context information and has only been applied to datasets of small resolution. However, the discriminator loss is similar to the one we propose and could be combined with our context-conditional approach. More traditional semi-supervised methods include graph-based approaches (Zhou et al., 2004;Zhu, 2006) that show impressive performance when good image representations are available. However, the focus of our work is on learning such representations. Generative models of images: Restricted Boltzmann machines (Salakhutdinov, 2015), de-noising autoencoders (Vincent et al., 2008) and variational autoencoders optimize a maximum likelihood criterion and thus learn decoders that map from latent space to image space. More recently, generative adversarial networks (Goodfellow et al., 2014) and generative mo-ment matching networks (Li et al., 2015;Dziugaite et al., 2015) have been proposed. These methods ignore data likelihoods and instead directly train a generative model to produce realistic samples. Several extensions to the generative adversarial network framework have been proposed to scale the approach to larger images (Denton et al., 2015;Salimans et al., 2016). Our work draws on the insights of regarding adversarial training practices and architecture for the generator network, as well as the notion that the discriminator can produce useful features for classification tasks. Other models used recurrent approaches to generate images (Gregor et al., 2015;Theis & Bethge, 2015;Mansimov et al., 2016;van den Oord et al., 2016). Dosovitskiy et al. (2015) trained a CNN to generate objects with different shapes, viewpoints and color. Sohl-Dickstein et al. (2015) propose a generative model based on a reverse diffusion process. While our model does involve image generation, it differs from these approaches in that the main focus is on learning a good representation for classification tasks. Predictive generative models of videos aim to extrapolate from current frames to future ones and in doing so learn a feature representation that is useful for other tasks. In this vein, Ranzato et al. (2014) used an 2 -loss in pixel-space. Mathieu et al. (2015) combined an adversarial loss with 2 , giving models that generate crisper images. While our model is also predictive, it only considers interpolation within an image, rather than extrapolation in time. APPROACH We present a semi-supervised learning framework built on generative adversarial networks (GANs) of Goodfellow et al. (2014). We first review the generative adversarial network framework and then introduce context conditional generative adversarial networks (CC-GANs). Finally, we show how combining a classification objective and a CC-GAN objective provides a unified framework for semi-supervised learning. GENERATIVE ADVERSARIAL NETWORKS The generative adversarial network approach (Goodfellow et al., 2014) is a framework for training generative models, which we briefly review. It consists of two networks pitted against one another in a two player game: A generative model, G, is trained to synthesize images resembling the data distribution and a discriminative model, D, is trained to distinguish between samples drawn from G and images drawn from the training data. More formally, let X = {x 1 , ..., x n } be a dataset of images of dimensionality d. Let D denote a discriminative function that takes as input an image x ∈ R d and outputs a scalar representing the probability of input x being a real sample. Let G denote the generative function that takes as input a random vector z ∈ R z sampled from a prior noise distribution p Noise and outputs a synthesized imagex = G(z) ∈ R d . Ideally, D(x) = 1 when x ∈ X and D(x) = 0 when x was generated from G. The GAN objective is given by: min G max D E x∼X [log D(x)] + E z∼pNoise(z) [log(1 − D(G(z)))](1) The conditional generative adversarial network (Mirza & Osindero, 2014) is an extension of the GAN in which both D and G receive an additional vector of information y as input. The conditional GAN objective is given by: min G max D E x,y∼X [log D(x, y)] + E z∼pNoise(z) [log(1 − D(G(z, y), x))](2) CONTEXT-CONDITIONAL GENERATIVE ADVERSARIAL NETWORKS We propose context-conditional generative adversarial networks (CC-GANs) which are conditional GANs where the generator is trained to fill in a missing image patch and the generator and discriminator are conditioned on the surrounding pixels. In particular, the generator G receives as input an image with a randomly masked out patch. The generator outputs an entire image. We fill in the missing patch from the generated output and then pass the completed image into D. We pass the completed image into D rather than the context and the patch as two separate inputs so as to prevent D from simply learning to identify discontinuities along the edge of the missing patch. More formally, let m ∈ R d denote to a binary mask that will be used to drop out a specified portion of an image. The generator receives as input m x where denotes element-wise multiplication. The generator outputs x G = G(m x, z) ∈ R d and the in-painted image x I is given by: x I = (1 − m) x G + m x(3) The CC-GAN objective is given by: min G max D E x∼X [log D(x)] + E x∼X ,m∼M [log(1 − D(x I ))](4) COMBINED GAN AND CC-GAN While the generator of the CC-GAN outputs a full image, only a portion of it (corresponding to the missing hole) is seen by the discriminator. In the combined model, which we denote by CC-GAN 2 , the fake examples for the discriminator include both the in-painted image x I and the full image x G produced by the generator (i.e. not just the missing patch). By combining the GAN and CC-GAN approaches, we introduce a wider array of negative examples to the discriminator. The CC-GAN 2 objective given by: min G max D E x∼X [log D(x)](5)+ E x∼X ,m∼M [log(1 − D(x I ))] (6) + E x∼X ,m∼M [log(1 − D(x G ))](7) SEMI-SUPERVISED LEARNING WITH CC-GANS A common approach to semi-supervised learning is to combine a supervised and unsupervised objective during training. As a result unlabeled data can be leveraged to aid the supervised task. Intuitively, a GAN discriminator must learn something about the structure of natural images in order to effectively distinguish real from generated images. Recently, showed that a GAN discriminator learns a hierarchical image representation that is useful for object classification. Such results suggest that combining an unsupervised GAN objective with a supervised classification objective would produce a simple and effective semi-supervised learning method. This approach, denoted by SSL-GAN, is illustrated in Fig. 1(b). The discriminator network receives a gradient from the real/fake loss for every real and generated image. The discriminator also receives a gradient from the classification loss on the subset of (real) images for which labels are available. Generative adversarial networks have shown impressive performance on many diverse datasets. However, samples are most coherent when the set of images the network is trained on comes from a limited domain (eg. churches or faces). Additionally, it is difficult to train GANs on very large images. Both these issues suggest semi-supervised learning with vanilla GANs may not scale well to datasets of large diverse images. Rather than determining if a full image is real or fake, context conditional GANs address a different task: determining if a part of an image is real or fake given the surrounding context. Formally, let X L = {(x 1 , y 1 ), ..., (x n , y n )} denote a dataset of labeled images. Let D c (x) denote the output of the classifier head on the discriminator (see Fig. 1(c) for details). Then the semisupervised CC-GAN objective is: min G max D E x∼X [log D(x)] + E x∼X ,m∼M [log(1 − D(x I ))] + λ c E x,y∼X L [log(D c (y\|x))] (8) The hyperparameter λ c balances the classification and adversarial losses. We only consider the CC-GAN in the semi-supervised setting and thus drop the SSL notation when referring to this model. MODEL ARCHITECTURE AND TRAINING DETAILS The architecture of our generative model, G, is inspired by the generator architecture of the DCGAN . The model consists of a sequence of convolutional layers with subsampling (but no pooling) followed by a sequence of fractionally-strided convolutional layers. For the discriminator, D, we used the VGG-A network (Simonyan & Zisserman, 2015) without the fully connected layers (which we call the VGG-A' architecture). Details of the generator and discriminator are given conv(64, 4x4, 2x2) denotes a conv layer with 64 channels, 4x4 kernels and stride 2x2. Each convolution layer is followed by a spatial batch normalization and rectified linear layer. Dashed lines indicate optional pathways. in Fig. 2. The input to the generator is an image with a patch zeroed out. In preliminary experiments we also tried passing in a separate mask to the generator to make the missing area more explicit but found this did not effect performance. Even with the context conditioning it is difficult to generate large image patches that look realistic, making it problematic to scale our approach to high resolution images. To address this, we propose conditioning the generator on both the high resolution image with a missing patch and a low resolution version of the whole image (with no missing region). In this setting, the generators task becomes one of super-resolution on a portion of an image. However, the discriminator does not receive the low resolution image and thus is still faced with the same problem of determining if a given in-painting is viable or not. Where indicated, we used this approach in our PASCAL VOC 2007 experiments, with the original image being downsampled by a factor of 4. This provided enough information for the generator to fill in larger holes but not so much that it made the task trivial. This optional low resolution image is illustrated in Fig. 2(left) with the dotted line. We followed the training procedures of . We used the Adam optimizer (Kingma & Ba, 2015) in all our experiments with learning rate of 0.0002, momentum term β 1 of 0.5, and the remaining Adam hyperparameters set to their default values. We set λ c = 1 for all experiments. Method Accuracy Multi-task Bayesian Optimization (Swersky et al., 2013) 70.10 ± 0.6 Exemplar CNN (Dosovitskiy et al., 2014) 75.40 ± 0.3 Stacked What-Where Autoencoder (Zhao et al., 2016) 74.33 Supervised VGG-A' 61.19 ± 1.1 SSL-GAN 73.81 ± 0.5 CC-GAN 75.67 ± 0.5 CC-GAN 2 77.79 ± 0.8 EXPERIMENTS 3.1 STL-10 CLASSIFICATION STL-10 is a dataset of 96×96 color images with a 1:100 ratio of labeled to unlabeled examples, making it an ideal fit for our semi-supervised learning framework. The training set consists of 5000 labeled images, mapped to 10 pre-defined folds of 1000 images each, and 100,000 unlabeled images. The labeled images belong to 10 classes and were extracted from the ImageNet dataset and the unlabeled images come from a broader distribution of classes. We follow the standard testing protocol and train 10 different models on each of the 10 predefined folds of data. We then evaluate classification accuracy of each model on the test set and report the mean and standard deviation. We trained our CC-GAN and CC-GAN 2 models on 64×64 crops of the 96×96 image. The hole was 32×32 pixels and the location of the hole varied randomly (see Fig. 3(top)). We trained for 100 epochs and then fine-tuned the discriminator on the 96x96 labeled images, stopping when training accuracy reached 100%. As shown in Table 1, the CC-GAN model performs comparably to current state of the art (Dosovitskiy et al., 2014) and the CC-GAN 2 model improves upon it. We also trained two baseline models in an attempt to tease apart the contributions of adversarial training and context conditional adversarial training. The first is a purely supervised training of the VGG-A' model (the same architecture as the discriminator in the CC-GAN framework). This was trained using a dropout of 0.5 on the final layer and weight decay of 0.001. The performance of this model is significantly worse than the CC-GAN model. We also trained a semi-supervised GAN (SSL-GAN, see Fig. 1(b)) on STL-10. This consisted of the same discriminator as the CC-GAN (VGG-A' architecture) and generator from the DCGAN model . The training setup in this case is identical to the CC-GAN model. The SSL-GAN performs almost as well as the CC-GAN, confirming our hypothesis that the GAN objective is a useful unsupervised criterion. PASCAL VOC CLASSIFICATION In order to compare against other methods that utilize spatial context we ran the CC-GAN model on PASCAL VOC 2007 dataset. This dataset consists of natural images coming from 20 classes. The dataset contains a large amount of variability with objects varying in size, pose, and position. The training and validation sets combined contain 5,011 images, and the test set contains 4,952 images. The standard measure of performance is mean average precision (mAP). We trained each model on the combined training and validation set for ∼5000 epochs and evaluated on the test set once 1 . Following Pathak et al. (2016), we train using random cropping, and then evaluate using the average prediction from 10 random crops. Our best performing model was trained on images of resolution 128×128 with a hole size of 64×64 and a low resolution input of size 32×32. Table 2 compares our CC-GAN method to other feature learning approaches on the PASCAL test set. It outperforms them, beating the current state of the art (Wang & Gupta, 2015) by 3.8%. It is important to note that our feature extractor is the VGG-A' model which is larger than the AlexNet architecture (Krizhevsky et al., 2012) used by other approaches in Table 2. However, purely supervised training of the two models reveals that VGG-A' Method mAP Supervised AlexNet 53.3 % Visual tracking from video (Wang & Gupta, 2015) 58.4% Context prediction (Doersch et al., 2015) 55.3% Context encoders (Pathak et al., 2016) 56.5% Supervised VGG-A' 55.2% CC-GAN 62.2% CC-GAN 2 62.7% is less than 2% better than AlexNet. Furthermore, our model outperforms the supervised VGG-A' baseline by a 7% margin (62.2% vs. 55.2%). This suggests that our gains stem from the CC-GAN method rather than the use of a better architecture. Table 3 shows the effect of training on different resolutions. The CC-GAN improves over the baseline CNN consistently regardless of image size. We found that conditioning on the low resolution image began to help when the hole size was largest (64×64). We hypothesize that the low resolution conditioning would be more important for larger images, potentially allowing the method to scale to larger image sizes than we explored in this work. INPAINTING We now show some sample in-paintings produced by our CC-GAN generators. In our semisupervised learning experiments on STL-10 we remove a single fixed size hole from the image. The top row of Fig. 3 shows in-paintings produced by this model. We can also explored different masking schemes as illustrated in the remaining rows of Fig. 3 (however these did not improve classification results). In all cases we see that training the generator with the adversarial loss produces sharp semantically plausible in-painting results. Fig. 4 shows generated images and in-painted images from a model trained with the CC-GAN 2 criterion. The output of a CC-GAN generator tends to be corrupted outside the patch used to inpaint the image (since gradients only flow back to the missing patch). However, in the CC-GAN 2 model, we see that both the in-painted image and the generated image are coherent and semantically consistent with the masked input image. Fig. 5 shows in-painted images from a generator trained on 128×128 PASCAL images. Fig. 6 shows the effect of adding a low resolution (32×32) image as input to the generator. For comparison we also show the result of in-painting by filling in with a bi-linearly upsampled image. Here we see the generator produces high-frequency structure rather than simply learning to copy the low resolution patch. DISCUSSION We have presented a simple semi-supervised learning framework based on in-painting with an adversarial loss. The generator in our CC-GAN model is capable of producing semantically meaningful in-paintings and the discriminator performs comparable to or better than existing semi-supervised methods on two classification benchmarks. Since discrimination of real/fake in-paintings is more closely related to the target task of object classification than extracting a feature representation suitable for in-filling, it is not surprising that we are able to exceed the performance of Pathak et al. (2016) on PASCAL classification. Furthermore, since our model operates on images half the resolution as those used by other approaches (128×128 vs. 224×244), there is potential for further gains if improvements in the generator resolution can be made. Our models and code are available at https://github.com/edenton/cc-gan. Figure 1 : 1(a) Context-encoder of Pathak et al. (2016), configured for object classification task. (b) Semi-supervised learning with GANs (SSL-GAN). (c) Semi-supervised learning with CC-GANs. Figure 2 : 2Architecture of our context-conditional generative adversarial network (CC-GAN). Figure 3 : 3STL-10 in-painting with CC-GAN training and varying methods of dropping out the image. Figure 4 : 4STL-10 in-painting with combined CC-GAN 2 training. Figure 5 : 5PASCAL in-painting with CC-GAN. Figure 6 : 6PASCAL in-painting with CC-GAN conditioned on low resolution image. Top two rows show input to generator. Third row shows inpainting my bilinear upsampling. Bottom row shows inpainted image by generator. Table 1 : 1Comparison of CC-GAN and other published results on STL-10. Table 2 : 2Comparison of CC-GAN and other methods (as reported by Pathak et al. (2016)) on PAS-CAL VOC 2007.Method Image size Hole size Low res size mAP Supervised VGG-A' 64×64 - - 52.97% CC-GAN 64×64 32×32 - 56.79% Supervised VGG-A' 96×96 - - 55.22% CC-GAN 96×96 48×48 - 60.38% CC-GAN 96×96 48×48 24×24 60.98% Supervised VGG-A' 128×128 - - 55.2% CC-GAN 128×128 64×64 - 61.3% CC-GAN 128×128 64×64 32×32 62.2% Table 3 : 3Comparison of different CC-GAN variants on PASCAL VOC 2007. Hyperparameters were determined by initially training on the training set alone and measuring performance on the validation set. Acknowledgements: Emily Denton is supported by a Google Fellowship. Rob Fergus is grateful for the support of CIFAR. Neural machine translation by jointly learning to align and translate. D Bahdanau, K Cho, Y Bengio, The International Conference on Learning Representations. D. Bahdanau, K. Cho, and Y. Bengio. Neural machine translation by jointly learning to align and translate. In The International Conference on Learning Representations, 2015. 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[ "https://github.com/edenton/cc-gan." ]
[ "The dynamics of simple gene network motifs subject to extrinsic fluctuations", "The dynamics of simple gene network motifs subject to extrinsic fluctuations" ]	[ "Elijah Roberts \nDepartment of Biophysics\nJohns Hopkins University\n21218BaltimoreMDUSA\n", "Shay Be'er \nRacah Institute of Physics\nHebrew University of Jerusalem\n91904JerusalemIsrael\n", "Chris Bohrer \nDepartment of Biophysics\nJohns Hopkins University\n21218BaltimoreMDUSA\n", "Rati Sharma \nDepartment of Biophysics\nJohns Hopkins University\n21218BaltimoreMDUSA\n", "Michael Assaf \nRacah Institute of Physics\nHebrew University of Jerusalem\n91904JerusalemIsrael\n" ]	[ "Department of Biophysics\nJohns Hopkins University\n21218BaltimoreMDUSA", "Racah Institute of Physics\nHebrew University of Jerusalem\n91904JerusalemIsrael", "Department of Biophysics\nJohns Hopkins University\n21218BaltimoreMDUSA", "Department of Biophysics\nJohns Hopkins University\n21218BaltimoreMDUSA", "Racah Institute of Physics\nHebrew University of Jerusalem\n91904JerusalemIsrael" ]	[]	Cellular processes do not follow deterministic rules; even in identical environments genetically identical cells can make random choices leading to different phenotypes. This randomness originates from fluctuations present in the biomolecular interaction networks. Most previous work has been focused on the intrinsic noise (IN) of these networks. Yet, especially for high-copy-number biomolecules, extrinsic or environmental noise (EN) has been experimentally shown to dominate the variation. Here we develop an analytical formalism that allows for calculation of the effect of EN on gene expression motifs. We introduce a new method for modeling bounded EN as an auxiliary species in the master equation. The method is fully generic and is not limited to systems with small EN magnitudes. We focus our study on motifs that can be viewed as the building blocks of genetic switches: a non-regulated gene, a self-inhibiting gene, and a self-promoting gene. The role of the EN properties (magnitude, correlation time, and distribution) on the statistics of interest are systematically investigated, and the effect of fluctuations in different reaction rates is compared. Due to its analytical nature, our formalism can be used to quantify the effect of EN on the dynamics of biochemical networks and can also be used to improve the interpretation of data from single-cell gene expression experiments.	10.1103/physreve.92.062717	[ "https://arxiv.org/pdf/1509.01083v2.pdf" ]	1,524,040	1509.01083	9540ae8c0da0fb13e6d76447280651fa98697847	The dynamics of simple gene network motifs subject to extrinsic fluctuations Elijah Roberts Department of Biophysics Johns Hopkins University 21218BaltimoreMDUSA Shay Be'er Racah Institute of Physics Hebrew University of Jerusalem 91904JerusalemIsrael Chris Bohrer Department of Biophysics Johns Hopkins University 21218BaltimoreMDUSA Rati Sharma Department of Biophysics Johns Hopkins University 21218BaltimoreMDUSA Michael Assaf Racah Institute of Physics Hebrew University of Jerusalem 91904JerusalemIsrael The dynamics of simple gene network motifs subject to extrinsic fluctuations numbers: 8716Yc0250Ey0540-a8717Aa Cellular processes do not follow deterministic rules; even in identical environments genetically identical cells can make random choices leading to different phenotypes. This randomness originates from fluctuations present in the biomolecular interaction networks. Most previous work has been focused on the intrinsic noise (IN) of these networks. Yet, especially for high-copy-number biomolecules, extrinsic or environmental noise (EN) has been experimentally shown to dominate the variation. Here we develop an analytical formalism that allows for calculation of the effect of EN on gene expression motifs. We introduce a new method for modeling bounded EN as an auxiliary species in the master equation. The method is fully generic and is not limited to systems with small EN magnitudes. We focus our study on motifs that can be viewed as the building blocks of genetic switches: a non-regulated gene, a self-inhibiting gene, and a self-promoting gene. The role of the EN properties (magnitude, correlation time, and distribution) on the statistics of interest are systematically investigated, and the effect of fluctuations in different reaction rates is compared. Due to its analytical nature, our formalism can be used to quantify the effect of EN on the dynamics of biochemical networks and can also be used to improve the interpretation of data from single-cell gene expression experiments. I. INTRODUCTION Biochemical processes in cells are inherently noisy, because many molecular species such as genes, RNAs, and proteins that make up intracellular reaction networks are present in low copy numbers inside a cell, see e.g. Refs. [1,2]. One of the primary insights to emerge from studies on stochastic gene expression is the distinction between intrinsic noise (IN) and extrinsic noise (EN) [3][4][5][6][7][8]. Experimentally, EN is quantified using the correlation in fluctuations between two copies of an identical reporter gene expressed separately in the same cell. IN arises from fluctuations that are independent for each reporter [9]. Within a cell, then, IN is the variance due to the discreteness of biomolecules and the probabilistic nature of chemical reactions, while EN is the variance arising from the fact the genes share a common environment, the cell. Such noise in cellular reactions can have important consequences, e.g., on cellular decision making. In fact, noise can drive cells between distinct gene expression states corresponding to different decision phenotypes [10][11][12]. The ultimate stability of a decision state is then determined by fluctuations of mRNA and proteins, as well as other cellular components, during gene expression [3,[13][14][15][16][17]. These fluctuations can give rise to spontaneous switching between the states, with a switching time that depends on their strength and the switch's architecture. Genetic switches can regulate diverse decision making processes such as microbial environmental adaptation, developmental pathways, nutrient homeostasis, and bacteriophage lysogeny [18][19][20][21]. * email address: [email protected]; Corresponding author † email address: [email protected]; Corresponding author In recent years there have been numerous theoretical studies on genetic switches driven by IN, noise arising from within a closed system of interest [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Yet, these and most other studies of genetic switching, have neglected sources of EN due to interactions with other components in the cell and the environment. EN does not arise from a single well-defined process, but rather results from the complex chain of events that gave rise to a particular cellular state. Variation in a cell's number of ribosomes, transcription factors, and polymerases or fluctuations in the cell division time, as well as environmental fluctuations, can all affect the rates of a genetic process. These fluctuations in the reaction rates may dramatically affect the protein's statistics including its mean, variance and copy-number distribution. Importantly, in living cells a comparison of the relative contribution of EN versus IN to the protein distributions width has shown that EN dominates above copy numbers of O(10 − 100) [36][37][38]. Theoretically, EN has been shown to induce bistability [6,[39][40][41][42], vary the distribution tails [6], and modify switching times [43]. In signaling, EN limits the information transduction capacity of the pathways [44,45]. It has been shown that EN is present at multiple time scales during protein production in bacteria [46,47] and that negative feedback can filter EN [46]. Previously, in Ref. [48] the authors have studied the interplay between IN and EN noise in a genetic switch near bifurcation using a Fokker-Planck approximation and showed that EN can dramatically affect switching. However, their method could not be directly used to study questions regarding population heterogeneity in metastable systems nor with non-Gaussian EN statistics. Genetic switches and other more complex circuits using multiple positive and negative feedback links form the ba-sis of much of the transcriptional regulatory logic in bacteria. The overall regulatory network of these microorganisms is commonly composed of repeated patterns of relatively small circuits, called motifs. For a review, see [49]. As a prerequisite to understanding the effect of EN on complex regulatory networks, we would first like to understand the role of EN on simple genetic motifs. In this work we develop an analytical formalism that allows for the quantification of the effect of EN on intrinsic-noise-driven gene expression circuits. We introduce a new method for modeling bounded EN as an auxiliary species in a master equation that fluctuates according to non-Gaussian statistics. We then analyze three genetic motifs: a non-regulated gene, a self-inhibiting gene, and a self-promoting gene. These three motifs represent the simplest possible circuits, yet commonly occur in bacterial transcription networks. We study the properties of each motif as related to EN. All our analytical findings are tested and compared against numerical Monte Carlo simulations. II. THEORY A. Gene expression under intrinsic noise Our starting point is a simple gene-expression model without extrinsic noise (EN). Let us denote by n the protein's copy number and by N 1 the typical protein abundance in the steady state. Our gene-expression model will consist of two reactions: production of proteins at a rate of F (n) and degradation with rate νn. Here we assume that the mRNA lifetime is short compared to the cell cycle and momentarily ignore the mRNA fluctuations, which will be accounted for in the following. In the deterministic picture, the rate equation for the protein concentration x = n/N readṡ x = f (x) − x,(1) where f (x) = F (n)/N and we have rescaled time by the protein degradation rate ν. To account for intrinsic fluctuations due to the probabilistic reactions and the discreteness of the proteins, we write down the chemical master equation for P n (t) -the probability to find n proteins at time ṫ P n = F (n − 1)P n−1 + (n + 1)P n+1 − [F (n) + n]P n . (2) We look for the stationary PDF such thatṖ n = 0. This yields a set of recursive equations, whose solution can be found analytically. The solution reads [50] P n = P 0 n−1 m=0 F (m) m + 1 = P 0 exp n−1 m=0 ln F (m) m + 1 ,(3) where P 0 is a normalization factor such that ∞ n=0 P n = 1. The stationary solution of Eq. (2) can also be found by using a dissipative WKB approximation [51,52]. To this end, we assume n 1, treat n as a continuous variable, and search for P n as P n ≡ P (x) ∼ e −N S(x) . Here, N 1 is assumed to be a large parameter, and S(x) is called the action. Plugging this ansatz into the stationary master equation [Eq. (2) withṖ n = 0], we arrive in the leading O(N ) order at a stationary Hamilton-Jacobi equation H[x, S (x)] = 0 with Hamiltonian H(x, p x ) = f (x)(e px − 1) + x(e −px − 1), where we have denoted the associated momentum by p x = S (x). While the trivial zero-energy trajectory p x (x) = 0 of this Hamiltonian corresponds to the deterministic dynamics, in the leading order the statistics of interest are encoded in the nontrivial zero-energy trajectory of this Hamiltonian [52], which reads in this case p x (x) = ln[x/f (x)].(4) This allows us to calculate the action by integration S(x) = x p x (x )dx . Thus, the PDF and its variance [53], are found to be: due to IN, var IN = N S (x * ) −1P (x) S (x * ) 2πN e −N [S(x)−S(x * )] , var IN = N x * 1 − f (x * ) ,(5) where x * is the steady-state solution of Eq. (1), and the normalization was done over the Gaussian part of the PDF around x * . By transforming the sum into an integral, Eq. (3) coincides in the leading order in N 1 with the PDF in Eq. (5). B. Gene expression under intrinsic and extrinsic noise Next, we add to our model EN, which is commonly defined as intercellular variability due to fluctuations during gene expression that equally affect all genes within a cell. We thus introduce EN in the form of one or more fluctuating parameters. For concreteness we assume, e.g., that cell-to-cell variability in transcription and translation rates causes the protein degradation rate ν to fluctuate so that ν → ν(t) = ξ(t). (In Appendix C we consider other fluctuating parameters as well.) As a result, the degradation rate becomes nξ(t) where ξ(t) is a stochastic variable satisfying ξ(t) = 1. Many measured protein distributions appear to be well-fit by a negative binomial (or gamma) distribution. Without experimental knowledge of how rates fluctuate in vivo, we simply take ξ(t) to have a negative binomial statistics, as if being controlled by a single protein. In addition, ξ(t) has variance σ 2 ex and correlation time τ c , satisfying ξ(t)ξ(t ) = σ 2 ex e −\|t−t \|/τc . Other statistics are also possible [6,42]; in Appendix D we consider Ornstein-Uhlenbeck noise. Note that our choice of negative binomial statistics for the EN ensures that the rates are always positive. To model EN, we need a circuit that generates an auxiliary species whose copy number fluctuates with nega-tive binomial statistics and correlation time τ c . To create one, we use an auxiliary mRNA-protein circuit where mRNAs are transcribed at a rate α/τ c and degrade with rate ω/τ c , while proteins are translated at a rate ωβ/τ c and degrade at a rate 1/τ c , which ensures that the correlation time of the auxiliary proteins is τ c . As a result, the master equation describing the probability to find m auxiliary mRNAs and k auxiliary proteins satisfies: P m,k = α τ c (P m−1,k −P m,k ) + ω τ c [(m + 1)P m+1,k −mP m,k ] + ωβm τ c (P m,k−1 −P m,k ) + 1 τ c [(k + 1)P m,k+1 −kP m,k ]. (6) As shown in Appendix A, in the limit of short-lived mRNA such that ω 1, the stationary PDF of the auxiliary protein is [54]: P k = Γ(α + k) Γ(k + 1)Γ(α) β β + 1 k 1 β + 1 α ,(7) where P k is the probability to find k auxiliary proteins. Here k = Kξ, where K ≡ αβ is the PDF mean, while the variance is K(1 + β). Therefore, choosing α = 1/(σ 2 ex − 1/K) and β = Kσ 2 ex − 1 we find that ξ = αβ/K = 1 and the variance of ξ becomes K(1 + β)/K 2 = σ 2 ex as required by our EN stochastic variable. Note, that in the limit of large K such that β = Kσ 2 ex − 1 Kσ 2 ex and α 1/σ 2 ex , the negative binomial distribution can be well approximated by a gamma distribution P k β −α /Γ(α) k α−1 e −k/β [55]. To study the interplay between IN and EN, we combine the EN dynamics [Eq. (6)] with the underlying intrinsic noise dynamics [Eq. (2)]. This leads to a 3D master equation describing the evolution of the probability P n,m,k to find n proteins, m auxiliary mRNAs and k auxiliary proteins, where the death rate of the protein of interest depends on the auxiliary protein. To this end, by using the WKB theory and by adiabatically eliminating the shortlived auxiliary mRNA degree of freedom (see Appendix A), we arrive at a stationary Hamilton-Jacobi equation H = 0 with a reduced Hamiltonian for the protein of interest and auxiliary protein: H(x, p x ,ξ, pξ) = f (x)(e px − 1) + xξ ρ (e −px − 1) + ρ βτ c 1 1 + β(1 − e pξ ) − 1 +ξ τ c (e −pξ − 1).(8) Here we have defined a rescaled EN variableξ = ρξ, where ρ = K/N is the abundances ratio of the auxiliary protein and protein of interest. Also, p x and pξ are the momenta associated with the protein of interest and the auxiliary protein with corresponding concentrations x = n/N andξ = k/N , while α and β are defined above. Hamiltonian (8) encodes the stochastic dynamics of a protein when its degradation rate fluctuates due to negative binomial EN generated by another auxiliary protein. Note, that while the (arbitrary) copy number K of the auxiliary protein enters Hamiltonian (8), it does not enter the results below for the statistics of the protein of interest. Hamiltonian (8) can be theoretically analyzed by writing down the corresponding Hamilton equations, see Eqs. (B1) in Appendix B. These can be solved numerically for arbitrary τ c , see Methods section. Analytical progress can be made in two important limits: shortcorrelated "white" EN, and long-correlated "adiabatic" EN. In the white-noise limit, τ c 1, one arrives at a reduced white-noise Hamiltonian, which effectively takes into account the short-correlated EN. Solving the corresponding Hamilton-Jacobi equation we find (see Appendix B) p x = ln x 2f (x) 1−V τ c x+ (V τ c x−1) 2 + 4V f (x)τ c . (9) Here, V ≡ N σ 2 ex is the ratio between the relative EN and IN variances (where the IN variance is taken in the non-regulated case). From Eq. (9) we can calculate the action S(x) = x p x (x )dx , while the PDF is given by Eq. (5). The action function S(x) cannot be explicitly calculated without specifying f (x). Yet, a general result for the PDF variance can be derived. Differentiating p x (x) [Eq. (9)] once and plugging x = x * such that f (x * ) = x * , we find the observed PDF variance σ 2 obs = N S (x * ) −1 = N x * (1 + x * V τ c ) 1 − f (x * ) .(10) Comparing with Eq. (5), this indicates that shortcorrelated EN increases the variance by a factor of 1 + x * V τ c . In the adiabatic limit, τ c 1, we can assume that the EN is almost stationary [48]. As a result, the protein PDF can be written as P n = ∞ −∞ P (ξ)P (n\|ξ)dξ,(11) where P (n\|ξ) is the conditional probability to find n proteins given noise magnitude ξ, and P (ξ) is the probability to find EN magnitude ξ. In Eq. (11) we effectively optimize the "cost" of reaching a state with n proteins given EN magnitude ξ against the probability of choosing such ξ [48]. For simplicity, here we take gamma-distributed EN, P (ξ) =β −α /Γ(α) ξ α−1 e −ξ/β , where α 1/σ 2 ex and β = β/K σ 2 ex . This distribution has a mean of 1 and variance σ 2 ex as required, and is a good approximation to the negative binomial distribution for large K [55]. Performing the integration in (11) via the saddle-point approximation, see Appendix B, the PDF in the adiabatic limit reads P (x) C ∂ ξξ Φ[x, ξ = ξ * (x)] e −N Φ[x,ξ=ξ * (x)] ξ * (x) ,(12) where Φ(x, ξ) = x g(ξ) ln yξ f (y) dy + ξ − ln ξ − 1 V ,(13) is the cost function. Here, ξ * (x) is the solution of the saddle point equation ∂ ξ Φ(x, ξ) = 0, x = g(ξ) is the ξdependent stable fixed point found by solving the equation f (x) = xξ, and C is a normalization factor such that N ∞ 0 P (x)dx = 1. Similarly as in the white-noise case, here we can also calculate the variance of the PDF explicitly for any production rate f (x). After some algebra, see Appendix B, we find the observed variance of the PDF σ 2 obs = N x * 1 − f (x * ) 1 + V x * 1 − f (x * ) .(14) This indicates that adiabatic EN increases the variance compared to the IN-only case (5) by a factor of 1 + V x * /[1 − f (x * )]. Eqs. (10) and (14) for the variance are among our main results here. When EN is put in the production rate instead of the degradation rate the results for the variance in both the white-and adiabatic-EN cases remain the same, see Appendix C. III. RESULTS A. Unregulated gene expression We begin with a model for protein transcription given a constant birth rate, namely an unregulated gene. Here the rate equation is given by Eq. (1) with f (x) = 1, while the protein PDF is P n = e −N N n /n!. Now, we add EN to the protein's degradation as described above. In the white-noise limit, τ c 1, integrating over the momentum (9) with f (x) = 1, the action function be- comes S(x) = 1/(V τ c ) {Ω(x) + ln[V τ c x − 1 + Ω(x)] +V τ c x [ln ((x/2)(1 − V τ c x + Ω(x))) − 1]}, with Ω(x) = (V τ c x − 1) 2 + 4V τ c . Using S(x) , we find the PDF, given by Eq. (5), see Figure 1(a). Interestingly, in the presence of EN the far right tail of the PDF behaves as a power law. Indeed, taking the x 1 limit of the PDF, we find a power-law dependence in the leading order P (x) ∼ (2V τ c x) −N/(V τc) , in contrast to an exponential tail in the IN-only case. Nevertheless, because the power-law behavior appears only at x x * = 1, the corresponding probabilities are vanishingly small and thus, observing this behavior experimentally or even numerically is impractical. The variance of the PDF due to white EN [Eq. (10)] becomes σ 2 obs = N (1 + V τ c ) ( Figure 1(c,d)). That is, white EN increases the width of the PDF by a factor of √ 1 + V τ c . In the adiabatic limit, τ c 1, the PDF is given by Eq. (12) (Figure 1 becomes Φ(x, ξ) = (ξ − ln ξ − 1)/V + x 1/ξ ln(yξ)dy, where x = g(ξ) = 1/ξ is the solution to the equation f (x) = ξx with f (x) = 1. In addition, the saddle point is found at ξ * = (1 − V x)/2 1 + 1 + 4V /(1 − V x) , while ∂ ξξ Φ(x, ξ) = (2V + ξ − V xξ)/(V ξ 3 ). Plugging f (x) = 1 and x * = 1 into Eq. (14), the variance due to adiabatic EN is given by σ 2 obs = N (1 + V ) ( Figure 1(c,d)). That is, adiabatic EN increases the width of the PDF by a factor of √ 1 + V which can be significant when V O(1). To test our theory we performed Monte Carlo simulations of the full master equation describing all three species: the protein of interest n, the auxiliary mRNA m, and the auxiliary protein k, see Methods. Figure 1 shows example comparisons for N = 100 and σ ex ranging up to 0.5, which are typical values obtained from single-cell Escherichia coli protein distributions [36]. Good agreement is obtained between our theory and stochastic simulations for both the white and adiabatic cases, even for quite strong EN. We have also verified that the results hold for EN in the birth rate, see Appendix C and Figure S3 in [56]. Interestingly, when EN arises in the degradation term, the PDF mean shifts to the right due to the nonlinear dependence of the fixed point on the death rate. While it is negligible for weak and moderate EN, this shift in the mean becomes significant for very strong EN, when σ ex = O(1), see the end of Appendix B for details. In turn, this shift affects the IN of the system, see Appendix E. In such a case IN and EN cannot be independently separated, as is commonly assumed (see Figure S3 and Figure S4 in [56]). B. mRNA-protein model with no feedback Now we consider the more realistic case of an unregulated gene but with mRNA present in the model. Here mRNAs are transcribed at a rate a, decay with a rate γ, and translation of proteins occurs with rate γb while degradation of proteins occurs with rate 1. As in the auxiliary circuit, we take the ratio between the mRNA and protein degradation rates to be large γ 1. The mean protein number here is N ≡ ab. The rate equations describing the average mRNA and protein concentrations, r = l/N and x = n/N (with l and n being the respective copy-numbers of mRNA and proteins), areṙ = a/N −γr, andẋ = bγr − x. In the limit of short-lived mRNA, γ 1, the stochastic dynamics has been analyzed by various authors [31]. Using the WKB approximation one can find the protein PDF, see Appendix A, which coincides with the negative binomial distribution in the limit of n 1. In particular, the PDF variance becomes N (1 + b), indicating that mRNA noise increases the variance by a factor of 1 + b compared to the protein-only case [14]. We now proceed to calculate the observed variance of the proteins of interest under negative binomial adiabatic EN in the protein's degradation rate. We do so along the same lines done for the protein-only case. Here, accounting for mRNA noise, the momentum given noise magnitude ξ becomes p x (x, ξ) = ln[(1 + b)xξ/(1 + bxξ)], which reduces to the protein-only case for b → 0. Integrating over the momentum, we find the action to be S(x, ξ) = x ln [(1 + b)xξ/(1 + bxξ)] − 1/(bξ) ln(1 + bxξ). Now, similarly as done in Eq. (13), we can define the cost function Φ(x, ξ) = S(x, ξ) − S[g(ξ), ξ] + (ξ − ln ξ − 1)/V . Therefore, the variance of the PDF can be found using Eq. (B11), by repeating the calculations along the same lines as done in Appendix B for the protein-only case. As a result, we find the observed variance of the proteins of interest, while accounting for mRNA noise, to be σ 2 obs = N (1 + b + V ).(15) The gamma distribution is widely used to analyze single-cell protein abundance data [55]. The protein's PDF is fit to a gamma distribution and the a and b values resulting from the fit are interpreted as the gene's burst frequency and burst size, respectively. We wanted to study how EN would affect such interpretations. To this end, we performed a large number of stochastic simulations across a wide range of values for a, b, and σ 2 ex , again for biological ranges seen in single-cell experiments, and calculated stationary PDFs using 10 7 data points for each parameter set. We fit the PDFs to a gamma distribution to obtain estimates of the gene expression parameters a f it and b f it . To calculate the accuracy with which a f it and b f it recovered the actual parameters, we calculated the relative error as Err(a) = \|a − a f it \|/a and Err(b) = \|b−b f it \|/b using the known a and b values from the simulations. As can be seen in Figure 2(a+b), using a gamma distribution resulted in poor estimates even in the case of relatively weak EN. Given the sensitivity of the error to EN, gamma distribution estimates of gene expression parameters should be used with caution. We then instead used Eq. (15) along with the geneby-gene EN values of σ 2 ex to estimate a f it and b f it values from the simulated dataset. With the mean of the distribution N = a b and the observed variance σ obs = N (1 + b + V ) = a b(1 + b + a b σ 2 ex ), one can solve directly for a = N 2 /(σ 2 obs −N −N 2 σ 2 ex ) and b = σ 2 obs /N −1−N σ 2 ex . Using the calculated mean and variance of the PDF, along with the known σ 2 ex from the simulations, we recovered estimates for a f it and b f it . Figure 2(a+b) shows that if one knows the strength of the EN for a gene, Eq. (15) can reliably recover the true a and b values until the total variance becomes dominated by EN for V > 10. Next we attempted to recover the a f it and b f it values from a genome-scale protein abundance data set from E. coli [36]. Here, we made a simplifying assumption that a constant global EN of σ ex = 0.31 influenced all genes equally (see Figure S5 in [56]). We estimated the a f it and b f it values for each gene using the gamma distribution and also using Eq. (15) with this global EN. Figure 2(c-f) shows a comparison of the two methods. A few trends are apparent from the results. When accounting for EN, the global saturation in the burst frequency a disappears and instead we see a continuous linear increase in the burst frequency. Likewise, an observed global increase in b values at higher V disappears and a more uniform distribution of b values is seen with respect to V . Since V is correlated with overall expression levels (IN goes down as V goes up) this implies that burst frequency is a significant driver of protein expression levels in E. coli. Figure S6, see [56], shows the fits versus mean expression. Gene-by-gene estimates of EN, rather than a single global EN, would lead to even better estimates of gene expression parameters. C. Self-inhibiting gene Next we consider the case of a self-inhibiting gene. A self inhibiting gene is a simple yet common motif that is capable of filtering some types of IN [57], although it can lose effectiveness when multiple time scales are involved [58]. We were therefore interested to study the ability of a self-inhibiting gene to filter EN. The rate equation is given by Eq. (1) where we took the production rate to be f ficient h = 1 (below we will consider higher values of h as well), whose fixed point x * = 1 coincides with the non-regulated gene. To find the PDF in the IN-only case, we integrate over Eq. (4) using f (x) = (1+β)/(1+βx). This yields S(x) = −2x + (1/β) ln(1 + βx) + x ln [x(1 + βx)/(1 + β)], while the PDF is given by Eq. (5) with x * = 1. The PDF variance N S (x * ) −1 = N (1 + β)/(1 + 2β), indicates that such negative inhibition decreases the PDF variance by a factor of 2 at most. Adding negative binomial EN into the protein degradation rate, in the white noise limit, the momentum is given by Eq. (9) with f (x) = (1 + β)/(1 + βx), while the PDF is given by Eq. (5). Using Eq. (10) with f (x) = (1+β)/(1+βx) and x * = 1, the observed variance in this case becomes σ 2 obs = N (1 + β) 1 + 2β (1 + V τ c ).(16) This result indicates that negative inhibition can eliminate EN. Indeed, σ 2 obs returns to its non-regulated value without EN, N , when the inhibition strength satisfies β = V τ c /(1 − V τ c ), which holds as long as V τ c < 1. That is, our choice of negative inhibition with h = 1 can only attenuate moderate EN, and can reduce the observed variance at most by a factor of 2. In the adiabatic limit, we can find the PDF using Eqs. (12) and (13) (see Appendix B for details), with g(ξ) = 1/(2βξ)(−ξ + ξ 2 + 4β(β + 1)ξ). Using Eq. (14) with f (x) = (1 + β)/(1 + βx) and f (x * = 1) = −β/(β + 1), the observed variance is Figure 3(a-c) shows good agreement between theory and simulations over a wide range of parameters. Again, we see that negative inhibition can eliminate EN when β = ( σ 2 obs = N (1 + β) 1 + 2β 1 + V (1 + β) 1 + 2β .(17)√ 1 + 4V + 2V − 1)/[2(2 − V )]. Here, the maximum EN that can be attenuated for this particular choice of inhibition is V = N σ 2 ex = 2. D. Higher order inhibition We now consider a more generic inhibition function f (x) = (1 + β)/(1 + βx h ) with arbitrary Hill-coefficient h. Here, in the white-noise limit, we find σ 2 obs = N (1 + β) 1 + β(h + 1) (1 + V τ c ).(18) In this case, EN can be eliminated by taking β = V τ c /(h − V τ c ), which holds for V τ c < h. In the adiabatic limit we obtain σ 2 obs = N (1 + β) 1 + β(h + 1) 1 + V (1 + β) 1 + β(h + 1) .(19) Here, EN is eliminated when β = (h √ 1 + 4V + 2V − h)/[2(h(h + 1) − V )] , which can be achieved as long as V < V max = h(h + 1). One can see that as h is increased, this inhibition mechanism becomes more efficient in eliminating EN. We wanted to examine the relationship between the critical inhibition strength β cr that will exactly eliminate EN and the inhibition order. We calculated β cr across h values ranging from 0.1 to 100 for various values of V . Our analytical framework provides a significant advantage over simulations for studying such large parameters spaces. Figure 3(d) shows that cooperativity in inhibition is a necessary feature for systems that dampen strong EN. E. Self-promoting gene Lastly, we consider the case of a self-promoting gene. Self-promoting genes can serve as genetic switches, allowing the cell to change between two alternate expression states. The rate equation now satisfies Eq. (1) where we take the production rate to be f (x) = α 0 + (1 − α 0 )θ(x − x 0 ) -a step function imitating a Hill-like function with a high Hill coefficient of positive feedback [48]. Here α 0 < x 0 < 1, where α 0 is the protein concentration in the of f state, x 0 is the threshold concentration, and N is the protein abundance in the on state such that x = n/N . Unlike previously [48], our derivation here is generic and does not require x 0 to be near one of the metastable states α 0 or 1. In the absence of EN, the mean switching time (MST) from the of f to the on states and vice versa, τ of f →on and τ on→of f , can be calculated using the master equation (2) and employing the WKB approximation (see e.g., Ref. [48]). Indeed, assuming that we start from the vicinity of the of f or on metastable states we find the mo-menta p of f (x) = ln(x/α 0 ) and p on (x) = ln(x), where we have used the fact that f (x) = α 0 for x < x 0 and f (x) = 1 for x > x 0 . The corresponding action functions are S of f (x) = x ln(x/α 0 ) − x, and S on (x) = x ln x − x. Therefore, the (logarithm of the) MSTs are given in the leading order of N 1 by the accumulated action between the corresponding stable metastable state and unstable fixed point [52] ln τ of f →on N [S(x 0 )−S(α 0 )] = N x 0 ln x 0 α 0 −x 0 +α 0 ,(20) and τ on→of f coincides with τ of f →on upon replacing α 0 by 1. For brevity, below we only present the results for the of f → on switch. All the results related to the on → of f switch are identical upon replacing α 0 by 1. Note, that in the absence of EN, τ of f →on and τ on→of f are comparable when x 0 = (1 − α 0 )/ ln(1/α 0 ). Eq. (20) can be simplified in the bifurcation limit x 0 − α 0 α 0 . Here, we find ln τ of f →on (N/(2α 0 ))(x 0 − α 0 ) 2 [59,60]. Now, we add negative binomial EN to the protein's degradation rate. In the white-noise limit, integrating over the momentum (9) with f (x) = α 0 for x < x 0 , the action function becomes S of f (x) = 1/(V τ c ) {Ω α0 (x) + ln[V τ c x − 1 + Ω α0 (x)] +V τ c x [ln ((x/2)(1 − V τ c x + Ω α0 (x))) − 1]}, where Ω α0 (x) = (1 − τ c V x) 2 + 4α 0 τ c V . Therefore, the MST reads ln τ of f →on N [S of f (x 0 ) − S of f (α 0 )].(21) In the bifurcation limit x 0 − α 0 α 0 , we find ln τ of f →on [N/(2α 0 )](x 0 − α 0 ) 2 /(1 + α 0 V τ c ) [48]. In the adiabatic limit, we need to optimize the cost of switching from one metastable state to the other given noise magnitude ξ against the probability of choosing noise magnitude ξ [48]. To do so we use Eq. (13), where the upper integration limit is the unstable fixed point x 0 , and for the of f → on switch the lower limit is the stable fixed point given noise magnitude ξ, g(ξ) = α 0 /ξ. As a result, the cost function (13) is a function of ξ only, and reads Φ of f (ξ) = x 0 [ln(x 0 ξ/α 0 ) − 1] + α 0 /ξ + (ξ − ln ξ − 1)/V . Therefore, in the leading order, the MST reads ln τ of f →on N Φ of f (ξ of f * ),(22) where ξ of f * Figure 4(a-b) compares our analytical theory with stochastic simulations and the numerical solution of Hamilton equations (B1) (see Methods section). Good agreement is seen in the white noise limit (similar agreement is seen in the adiabatic case), where the numerical solution of the Hamilton equations allows us to explore parameter ranges that are inaccessible by stochastic simulation due to the long MSTs. Moreover, as can be seen in Figure 4 Finally, we were interested in studying the effect of EN on population dynamics. We used Eq. (22) to calculate the relative fraction of the population in the on vs of f state as a function of both the positive feedback threshold x 0 and of V , shown in Figure 4(d). With zero or low EN the behavior of the population with regard to x 0 is very homogeneous. Only for a very small range of x 0 values is a macroscopically bistable population observed (e.g., at least 1 part in 100). If x 0 is not tuned very precisely, no heterogeneity is observed. As the EN increases, though, the range of macroscopic bistability increases dramatically. For V ≥ 5, well within the range of EN observed in biological systems, the population exhibits macroscopic heterogeneity across the entire range of x 0 sampled. This effect is less pronounced, but still present for white EN ( Figure S7, see [56]). Here the EN is taken to be adiabatic. = 1 2 1 − V x 0 + (V x 0 − 1) 2 + 4V α 0 is the saddle point satisfying Φ of f (ξ of f * ) = 0. IV. CONCLUSIONS We have presented a new formalism for studying EN in gene expression circuits, allowing us to quantify how EN affects the PDFs, variances, and MSTs in various genetic circuits. EN is likely present in multiple forms and at multiple time scales [46] in a majority of processes in living cells. Understanding how EN alters the dynamics of gene networks is key for developing detailed models of genetic and regulatory processes. Our results from studying the effect of EN on simple genetic motifs have shown that EN has a dominant influence on the system's behavior. Analyzing experimental single-cell distributions without accounting for extrinsic noise is unlikely to provide meaningful interpretation. More work needs to be done to experimentally characterize the details of the extrinsic fluctuations that cellular reactions rates experience. Finally, EN seems to provide a distinct advantage for populations wishing to use bistability as a bet hedging strategy. Without EN, the parameters needed to have meaningful population heterogeneity in a given condition are exponentially sensitive. With EN, populations are able to explore a variety of states across a wide range of parameters. V. METHODS A. Monte Carlo simulations of the auxiliary circuit Monte Carlo simulations with negative binomial EN were performed using two auxiliary species, a 1 and a 2 , to model the EN. They can be thought of in terms of a generic mRNA and protein, respectively. The dynamics of these species are given by two birth and two death processes: ∅ v1 − → a 1 , a 1 v2 − → a 1 + a 2 , a 1 d1 − → ∅, a 2 d2 − → ∅, Here, ∅ is a symbol for the empty set, i.e., a species is created from nothing or destroyed into nothing. The rates are given by: v 1 = 1 τ c K Kσ 2 ex − 1 , v 2 = ω(Kσ 2 ex − 1) τ c d 1 = ω τ c , d 2 = 1 τ c . K is the mean copy number of species a 2 , σ 2 ex is the desired EN strength, and τ c is the desired EN correlation time. To remind the reader, the negative binomial parameters, α and β, defining the distribution (A7) are related to v 1 and v 2 in the following manner: v 1 = α/τ c , and v 2 = ωβ/τ c , where α = K/(Kσ 2 ex − 1) and β = Kσ 2 ex − 1. In all simulations ω was set to 100. To avoid negative rates, one must have Kσ 2 ex > 1. Therefore, the value used for K limits the lower bound of the EN that can be simulated using a particular set of parameters. In the limit as Kσ 2 ex → 1 the variance approaches the Poissonian variance of a birth death process centered on K. Larger K allows for smaller EN to be simulated. Outside of this limitation, K has no influence on the EN properties. It does, however, influence the computational efficiency of the simulation. Both larger and smaller values of K increase the runtime of the simulations. We used a value of K = 20, 000 throughout this work, which leads to a lower bound for the EN studied of σ ex > 0.007 and provides reasonable runtimes. Figure S1, see [56], shows that the mean and variance of the auxiliary species are as expected during the simulations. Figure S2, see [56], shows that the auxiliary species has the expected autocorrelation time. All simulations were performed using the standard Gillespie algorithm [61] using the Lattice Microbes software [62]. The auxiliary species a 2 was coupled to a reaction to be fluctuated by including it as an additional species participating in the reaction and adjusting the reaction rate constant such that the mean equals the original value. For example, to model a fluctuating birth rate for protein n with mean copy number N , the reaction ∅ In the above equations, a 2 appears on both sides to indicate that it is neither created nor destroyed by the reaction. B. Numerical solutions of the Hamilton equations In this section we use the shooting method [24,59,60] to find a numerical solution to the set of Hamilton equations (B1) in the case of arbitrary correlation time τ c . We focus on the case of the self-promoting gene, which gives rise to switching between metastable phenotypic states. Here there are three fixed points in the language of the deterministic rate equations, two stable points at x of f = α 0 and x on = 1 and one unstable point at x s = x 0 . We are interested to numerically compute the trajectories, z on (t) and z of f (t), corresponding to the optimal paths along which switching from the on → of f and of f → on occurs, respectively. Below we consider the trajectory z on (t), where the analysis of z of f (t) is similar. Let us denote by t i = 0 and t f the initial and final simulation times, respectively. The initial condition is given by z on (0) = x on + δv where x on = (x = x on , p x = 0, ξ = 1, p ξ = 0) is the corresponding fixed point in the 4D phase space and v is the initial direction of the trajectory, see below. Here δ is chosen to be small, but not too small to balance between simulation runtime and accuracy. The final condition is that the trajectory reaches the close vicinity of x s = (x s , 0, 1, 0), namely that \|z on (t f )−x s \| 1. [From there, the assumption is that the system flows almost deterministically to x of f = (α 0 , 0, 1, 0).] In order to find the initial direction of the trajectory we linearize the Hamilton equations (B1) in the vicinity of x on . This allows us to find the eigenvalues and eigenvectors in the vicinity of x on . Since the switching trajectory leaves the fixed point x on along its unstable manifold, we are only considering the eigenvectors v 1 and v 2 that correspond to the two positive eigenvalues λ 1 and λ 2 . (Note that the two other eigenvalues satisfy λ 3 = −λ 1 and λ 4 = −λ 2 such that i λ i = 0.) As a result, we take the initial direction to be v = v 1 cos(α) + v 2 sin(α), where it is assumed that each of the eigenvectors is normalized to unity, and 0 ≤ α ≤ 2π. Finally, we search over all possible values of α until we find the best choice that satisfies the above initial and final conditions. Note that along z on (t), the initial conditions ∂ t x(0) < 0 and ∂ t p x (0) < 0 are satisfied. This search over α is optimized by performing a binary search. Each time an initial condition is chosen, the set of equations is solved numerically by using a Matlab numerical solver, and we compare the final condition to x s . The search is terminated when we have sufficiently converged to the final condition. After successfully determining the trajectory we perform a numerical integration in order to find the accumulated action. We do so by using the formula ∆S = t f 0 [p x (t)ẋ + p ξ (t)ξ]dt N −1 i=0 [x(t i+1 ) − x(t i )] × [p x (t i+1 ) + p x (t i )]/2 + [ξ(t i+1 ) − ξ(t i )] × [p ξ (t i+1 ) + p ξ (t i )]/2, where t 0 = 0 and t N = t f . This result gives us the logarithm of the mean switching time divided by N . An example can be seen in Figure S8 in [56]. ACKNOWLEDGMENTS We thank Naftali R. Smith for useful discussions. This work was supported by Grant No. 300/14 of the Israel Science Foundation. Appendix A: mRNA-protein auxiliary circuit In this section, we derive the stationary PDF of the auxiliary protein. This PDF determines the extrinsic noise (EN) statistics of the degradation rate of the protein of interest. The choice of negative binomial statistics used in the main text for the reaction rate seems quite natural. Indeed, in genetic circuits of a non-regulated gene, if the mRNA is short lived, the proteins' stationary PDF is given by a negative binomial distribution [54,55]. As a result, if this auxiliary protein affects the degradation rate of our protein of interest, this rate will fluctuate with negative binomial statistics. In the auxiliary mRNA-protein circuit, mRNAs are transcribed at a rate α/τ c and degrade with rate ω/τ c , while proteins are translated at a rate ωβ/τ c and degrade at a rate 1/τ c , which insures that the correlation time of the auxiliary proteins is τ c . We assume a short-lived mRNA such that ω 1. The master equation describing the probability to find m mRNAs and k proteins satisfies: P m,k = α τ c (P m−1,k −P m,k ) + ω τ c [(m + 1)P m+1,k −mP m,k ] + ωβm τ c (P m,k−1 −P m,k ) + 1 τ c [(k + 1)P m,k+1 −kP m,k ].(A1) We denote the auxiliary mRNA and protein concentrations by z = m/K and ξ = k/K, respectively, where K = αβ is the auxiliary protein's abundance. We now use a dissipative version of the WKB approximation, see e.g., Refs. [51,52,63,64]. Employing the WKB ansatz P m,k = P (z, ξ) ∼ e −KS(z,ξ) , we arrive at a Hamilton-Jacobi equation H = 0 with Hamiltonian H(z, p z , ξ, p ξ ) = α K (e pz − 1) + ωz(e −pz − 1) +βωz(e p ξ − 1) + ξ(e −p ξ − 1),(A2) where p z = ∂ z S(z, ξ) and p ξ = ∂ ξ S(z, ξ) are the associated mRNA and protein momenta. This yields the following Hamilton equationṡ z = α K e pz − ωze −pz , p z = −ω(e −pz − 1) − βω(e p ξ − 1), ξ = βωze p ξ − ξe −p ξ , p ξ = 1 − e −p ξ .(A3) For ω 1, the mRNA lifetime is short compared to that of the protein. In this case, z and p z equilibrate much faster than ξ and p ξ , and we can adiabatically eliminate the mRNA species [65,66]. As a result, puttingż = p z = 0 we find z = z(ξ, p ξ ) and p z = p z (ξ, p ξ ). Plugging this into the Hamiltonian (A2) we arrive at the reduced Hamiltonian for the auxiliary protein only [21] H r (ξ, p ξ ) = 1 β 1 1 + β(1 − e p ξ ) − 1 + ξ(e −p ξ − 1), (A4) which effectively includes mRNA fluctuations. Solving the Hamilton-Jacobi equation H r (ξ, p ξ ) = 0 we find p ξ = ln[(1 + β)ξ/(1 + βξ)],(A5) and thus, the action becomes S(ξ) = ξ ln (1 + β)ξ 1 + βξ − 1 β ln(1 + ξβ).(A6) As a result, the stationary PDF to find k copies of the auxiliary protein is given by P (k) ∼ e −K[S(k/K)−S(1)] [see Eq. (5) in the main text], where K = αβ is the protein abundance. This distribution, when properly normalized, coincides at k 1 with the negative binomial distribution P k = Γ(α + k) Γ(k + 1)Γ(α) β β + 1 k 1 β + 1 α . (A7) Interestingly, these results can give us insight on the mRNA fluctuations that are implicitly incorporated in the protein-only model [Eq. (A4)], after eliminating the fast mRNA variable. Indeed, by comparing Eq. (A5) with the momentum in the protein-only model [Eq. (4)], we find that mRNA fluctuations emanating from this unregulated mRNA-protein circuit can be effectively accounted for by taking a protein-only model with a modified production rate f (x) = (1 + βx)/(1 + β). This production rate becomes 1 in the limit of small burst size β → 0, but becomes ξ + (1 − ξ)/β in the limit of large burst size β 1, which yields a much wider distribution with variance N β N . Importantly, this modified production rate gives rise to a protein PDF that coincides with the negative binomial distribution at k 1. Appendix B: Analysis of the Hamiltonian combining IN and EN In this section we will derive the stationary PDF of the proteins of interest for generic production rate f (x), where the degradation rate fluctuates due to EN with negative binomial statistics and correlation time τ c . In order to do so, we will analyze the Hamilton equations emanating from Hamiltonian (8) in the main text. In particular, we will find approximate solutions for the protein PDF in the limits of short-and long-correlated EN. Using Hamiltonian (8) the corresponding Hamilton equations reaḋ x = f (x)e px − xξ ρ e −px , p x = −f (x)(e px − 1) −ξ ρ (e −px − 1) ξ = ρe pξ τ c [1 + β(1 − e pξ )] 2 − e −pξξ τ c , pξ = − (e −pξ − 1) τ c − x ρ (e −px − 1). (B1) To remind the reader, ρ = K/N is the abundances ratio of the auxiliary protein and protein of interest andξ = ρξ = k/N is a rescaled noise variable, while α = 1/(σ 2 ex − 1/K) and β = Kσ 2 ex − 1. Hamilton equations (B1) can be solved numerically for any value of τ c , see Methods section. This numerical solution provides the statistics of interest in the leading order, and is far more efficient than performing numerical Monte-Carlo simulations, especially for short-correlated EN, see main text. Importantly, the cases of fast and slow dynamics of the EN can be studied analytically, see below. White-noise limit of EN In the white-noise limit, τ c 1, the dynamics of the auxiliary protein is fast. As a result,ξ(t) and pξ(t) equilibrate fast compared to x and p x , and we can look for slowly-varying x and p x dependent solutions of the third and fourth Hamilton equations (B1). This yields in the leading order of τ c 1 ξ ef f = 1 − 2(1 − e −px )V xτ c ,(B2) where we have defined V ≡ N σ 2 ex as the ratio between the relative EN and IN variances. Note, that the value of p ef f ξ = O(τ c ) 1 does not enter the equations foṙ x andṗ x . As expected, Eq. (B2) as well as the results below are independent of the arbitrary choice of the auxiliary protein abundance K. Plugging ξ ef f (x, p x ) from Eq. (B2) into the first and second Hamilton equations (B1) we arrive at an effective 1D white-noise Hamiltonian [48,67,68] H(x, p x ) = f (x)(e px −1)+x(e −px −1)+x 2 (e −px −1) 2 V τ c . (B3) Solving the Hamilton-Jacobi equation H = 0 we find the momentum p x = ln x 2f (x) 1−V τ c x+ (V τ c x−1) 2 + 4V f (x)τ c . (B4) The PDF can be formally found by integrating Eq. (B4) to find the corresponding action S(x) = x p x (x )dx , and by using Eq. (5). Adiabatic limit of EN In the adiabatic limit, τ c 1, we can assume the EN is almost stationary. As a result, the stationary PDF of the proteins satisfies [48,67,68] P n = ∞ −∞ P (ξ)P (n\|ξ)dξ,(B5) where P (n\|ξ) is the probability to find n proteins given noise magnitude ξ, and P (ξ) is the probability to find EN magnitude ξ. For simplicity we will take the EN to be gamma distributed, P (ξ) =β −α /Γ(α) ξ α−1 e −ξ/β . Here α = 1/σ 2 ex andβ = β/K σ 2 ex , which guarantees that the mean is 1 and the variance is σ 2 ex . As can be checked, the gamma distribution becomes a good approximation of the negative binomial distribution when K is sufficiently large. With these values of α andβ, we find P (ξ) 1 ξ 2πσ 2 ex e (1/σ 2 ex )(1+ln ξ−ξ) ,(B6) which holds as long as σ ex < 1. As a result, the PDF to find n proteins [Eq. (B5)] becomes P n = ∞ −∞ 1 ξ 2πσ 2 ex P (n\|ξ)e −(ξ−ln ξ−1)/σ 2 ex dξ, (B7) where P (n\|ξ) = Ae −N x g(ξ) ln yξ f (y) dy , and A = A(ξ) is a normalization constant. Here, we have used the fact that given ξ, the momentum along the optimal path (zero-energy Hamiltonian) satisfies p x (x, ξ) = ln[xξ/f (x)], and the fixed point given noise magnitude ξ satisfies the equation f (x) = xξ and is given by x(ξ) = g(ξ). To proceed, we rewrite the integral in Eq. (B7) as P n = ∞ −∞ B ξ e −N Φ(x,ξ) dξ,(B8) where B contains the preexponential factors including all normalization constants and Φ(x, ξ) = x g(ξ) ln yξ f (y) dy + ξ − ln ξ − 1 V ,(B9) is the cost function that we need to optimize, see main text. Now, we use the fact that N 1 and employ the saddle-point approximation. The saddle point is obtained at ∂ ξ Φ(x, ξ) = 0, which yields the following algebraic equation ∂Φ ∂ξ = x − g(ξ) ξ + 1 V − 1 V ξ = 0,(B10) where we have used the Leibniz integral rule when differentiating Eq. (B9), and g(ξ) is defined above. Solving the equation V [x − g(ξ)] + ξ − 1 = 0 for ξ yields the optimal noise magnitude ξ * (x). Plugging ξ * (x) into Φ(x, ξ) we find the PDF in the adiabatic limit, which is given by Eq. (12) in the main text, where ∂ ξξ Φ(x, ξ) = [1 − V g (ξ)]/(V ξ). The variance of this PDF can be explicitly calculated. It is given by the second derivative of Φ[x, ξ = ξ * (x)] [Eq. (B9) when plugging ξ = ξ * (x)] with respect to x, evaluated at x = x * N var −1 = d 2 Φ[x, ξ = ξ * (x)] dx 2 x=x * ,(B11) where x * is the unperturbed fixed point (with ξ = 1) satisfying x * = f (x * ). To carry out this calculation analytically we need to solve Eq. (B10) and find the optimal noise magnitude ξ * . We recall that the variance is calculated in the vicinity of the unperturbed fixed point x x * . Let us assume a-priori that \|ξ − 1\| 1 in the vicinity of x x * . Then, we can expand g(ξ) in the vicinity of ξ = 1, g(ξ) g(1) + g (1)(ξ − 1). However, since g(1) is the solution of the equation xξ = f (x) at ξ = 1, we have g(1) = x * . Therefore, we have g(ξ) x * + g (1)(ξ − 1). (B12) Plugging this into Eq. (B10) we find ξ * (x) 1 + V (x − x * ) V g (1) − 1 .(B13) This verifies our assumption that \|1 − ξ * (x)\| 1 as long as x is in the close vicinity of x * . Now, using Eqs. (B12) and (B13) in Eq. (B11), performing the differentiation, and evaluating the result at x = x * , we find the observed variance to be σ 2 obs = N x * [V g (1) − 1] 2 [f (x * ) − 1][2V g (1) − 1] − V x * ,(B14) where we have used the fact that x * = f (x * ). This expression can be further simplified if we recall that g(ξ) satisfies f [g(ξ)]/g(ξ) = ξ. Differentiating this with respect to ξ, evaluating the result at ξ = 1, and using the fact that g(1) = x * , we obtain g (1) = x * /[f (x * ) − 1]. Plugging this into Eq. (B14) we arrive at the final result σ 2 obs = N x * 1 − f (x * ) 1 + V x * 1 − f (x * ) . (B15) Note, that throughout these calculations we have assumed that the mean of the PDF remains at x = x * , and calculated the variance accordingly. This assumption is accurate as long as the EN magnitude is not too strong, σ 2 ex 1, see Figure S3(c) and Figure S4 in [56], which is within the range of EN observed in biological systems. Yet, for very strong EN, the mean of the PDF shifts to the right, due to the nonlinear dependence of the fixed point on the degradation rate, and due to the corresponding slowly-decreasing right tail of the protein PDF. We will now show this explicitly in the case of the unregulated gene, for which f (x) = 1. Let us assume σ ex = O(1) such that V = N σ 2 ex = O(N ) 1. In this strong-EN regime, the PDF is approximately given by P (n) C N n e N V (1− N n −ln n N ) ,(B16) where we have used Eqs. (12) and (13) in the main text, with g(ξ) = 1/ξ and ξ * (x) 1/x, and C = (2πN V ) −1/2 . In order to calculate the mean of this PDF we use the equality n = n nP n . Doing so, and using the saddle point approximation, we find n N (1 + 3σ 2 ex /2). (B17) This result for the PDF mean in the case of EN in the degradation rate agrees well with simulations, see Figure S3 and Figure S4 in [56]. the case of EN in the degradation rate [compare Eq. (C2) with Eq. (B4)], but for weak and moderate EN, the PDFs are indistinguishable, see Figure S3(a) in [56]. Differentiating the momentum with respect to x we find the observed variance to be σ 2 obs = N S (x * ) −1 = N x * (1 + x * V τ c ) 1 − f (x * ) ,(C3) which coincides with the variance when EN is in the degradation rate. In the adiabatic case, we again need to calculate the integral P n ∼ ∞ −∞ B ξ e −N Φ(x,ξ) dξ,(C4) where B contains the preexponential factors including all normalization constants. In this case, the cost function Φ(x, ξ) takes the form Φ(x, ξ) = x g(ξ) ln y ξf (y) dy + ξ − ln ξ − 1 V . (C5) Note, that the only difference between this equation and Eq. (B9) is that here ξ is in the denominator of the ln function, instead of the numerator. In addition, in this case x = g(ξ) solves the equation ξf (x) = x. Now, we use the fact that N 1 and solve the integral (C4) via the saddle-point approximation. The saddle point is obtained at ∂ ξ Φ(x, ξ) = 0, which yields the following algebraic equation ∂Φ ∂ξ = g(ξ) − x ξ + 1 V − 1 V ξ = 0.(C6) Solving the equation V [g(ξ) − x] + ξ − 1 = 0 for ξ we find ξ * (x), which allows finding the PDF according to Eq. (12), see Figure S3(b) in [56]. This figure emphasizes the lack of coincidence between the PDFs in the cases of adiabatic EN in the production and degradation rates. The variance of this PDF is given by Eq. (B11), where x = x * is the unperturbed fixed point satisfying x * = f (x * ). To carry out this calculation analytically we need to solve Eq. (C6) and find the optimal noise magnitude ξ * . We recall that the variance is calculated in the vicinity of the fixed point x x * . Assuming a-priori that \|ξ − 1\| 1 in the vicinity of x x * , we take Eq. (C6) and expand g(ξ) to first order in ξ around ξ = 1. By doing so, and using the fact that g(1) = x * , we have g(ξ) x * + g (1)(ξ − 1), which yields ξ * (x) 1 + V (x − x * ) V g (1) + 1 . (C7) f (x) = xξ. As a result, we arrive at Eq. (B8), where here, the cost function satisfies Φ(x, ξ) = x g(ξ) ln yξ f (y) dy + (ξ − 1) 2 2V . (D7) Now, we use the fact that N 1 and solve the integral in Eq. (B8) via the saddle-point approximation. The saddle point is obtained at ∂ ξ Φ(x, ξ) = 0. Using Eq. (D7), this yields the following algebraic equation ∂Φ ∂ξ = x − g(ξ) ξ + ξ − 1 V = 0.(D8) Solving the equation V [x−g(ξ)]+ξ(ξ −1) = 0 for ξ yields the optimal noise magnitude ξ * (x). Using this result and Eq. (D7), we find the PDF according to Eq. (12) in the main text. Note, that the resulting PDF here differs from the negative binomial case, since the cost function here [Eq. (D7)] differs from that in the case of negative binomial EN [Eq. (B9)]. The variance of this PDF can be explicitly calculated by using Eq. (B11), with x * being the unperturbed fixed point x * = f (x * ). Since the variance is calculated in the close vicinity of the fixed point x x * , similarly as for the negative binomial EN, we find the saddle point to be ξ * (x) 1 + V [x − g(1)] V g (1) − 1 ,(D9) which coincides with Eq. (B13). Repeating the calculations in the same manner as in the case of negative binomial EN, we find σ 2 obs = N x * 1 − f (x * ) 1 + V x * 1 − f (x * ) . (D10) This result again coincides with the variance in the negative binomial case [Eq. (B15)]. This indicates that to determine the variance of the protein PDF under EN (in both the white-and adiabatic-noise limits), the complete statistics of the EN is less relevant. The only relevant parameter here is the width of the EN distribution, or its magnitude, given by the parameter σ ex . Appendix E: Correction of analytical variance using the numerical mean In cases where the EN magnitude is large, the mean of the distribution can shift, as discussed above. In these cases we apply a correction to the observed variance to account for the change in the IN. For example, σ 2 obs = N (1 + V τ c ) is corrected to σ 2 obs = µ obs N N 1 + µ obs N V τ c ,(E1) where µ obs is the mean observed from numerical simulations, and V = N σ 2 ex . FIG. 1 . 1(b)). Here, the cost function [Eq. (13)] (Color online) Comparison of theory (lines) and numerics (symbols) for the non-regulated gene model with N = 100 and σex = 0.2. (a+b) Probability distributions for white (a; τc = 0.1) and adiabatic (b; τc = 1000) EN. Dotted lines show the Poisson distribution for the model with only intrinsic noise. (c) Observed variance vs EN strength for white (red) and adiabatic (blue) noise. (d) Observed variance vs EN correlation time for σex = 0.2. The left curve shows the white noise theory and the right shows the adiabatic theory. FIG. 2 . 2(x) = (1 + β)/(1 + βx), and β is the inhibition strength. Here we chose a simple form of nonlinear inhibitory Hill-like function with Hill coef-(Color online) (a+b) Relative error in a f it and b f it , respectively, from simulations spanning a wide range of parameters. Blue crosses show the values from fitting to a gamma distribution. Red circles show fits from Eq. (15).(c+d) The a f it and b f it values vs V from fitting the genome-scale protein abundance data from [36] using (blue crosses) the gamma distribution or (red circles) Eq. (15) with a global EN of σex = 0.31. (e+f) Relationship between a f it and b f it values obtained from a gamma distribution and Eq. (15). Points are colored by log10(V ) where V = σ 2 ex /σ 2 int . The solid y = x lines are a guide to the eye. FIG. 3 . 3(Color online) Comparison of theory (lines) and stochastic simulations (symbols) for the self-inhibited gene model with N = 100 and adiabatic EN τc = 100. (a) Probability distributions for (solid line) theory and (x's) numerics with β = βcr = 1.09 and h = 3. Dotted line shows the Poisson distribution for the model with only intrinsic noise and triangles show the distribution in the absence of negative inhibition. (b) Observed variance vs EN strength with β = 1.0 and various values of h. (c) Observed variance vs inhibition strength β for σex = 0.2 and for the same h values as in (b). (d) The critical β and h values that exactly cancel EN for a given relative EN strength V = σ 2 ex /σ 2 int . (c) the underlying Hamilton equations (B1) capture the correct dynamics well into intermediate correlation time ranges, τ c = O(1), where the white noise approximation breaks down. FIG. 4 . 4(Color online) Comparison of mean switching times from analytical theory (lines) given by Eq. (21), numerical solution of the Hamilton equations (o's), and stochastic simulations (x's) for the self-promoting gene model with N = 750, a0 = 0.63, and x0 = 0.80. (a) The MST from the of f to the on state vs EN strength for white noise τc = 0.1. (b) The MST from on to of f for τc = 0.1. (c) The MST vs EN correlation time for σex = 0.0365. (d) A heat map showing the relative probability for the system to be in the on vs of f metastable state [using Eq. (22)] according to the position of the barrier x0 and the relative strength V of the EN. Appendix C: The case of EN in the production rate In this section we consider EN in the production rate rather than in the degradation rate. We show that while the resulting protein PDF in this case differs from the case of EN in the degradation rate, the variance of the PDF coincides in the two cases, in both the white-and adiabatic-EN limits.We again consider EN with a negative binomial statistics and correlation time τ c . Our starting point is the 2D Hamiltonian which encodes the stochastic dynamics of the protein of interest under the influence of EN. Here, instead of EN in the degradation rate we have EN in the production rate in the form f (x) → ξf (x), where ξ satisfies ξ = 1, and fluctuates with negative binomial statistics. As a result, the Hamiltonian (8) in the case of EN in the degradation rate, gives way towhereξ and ρ are defined above. At this point, we can repeat the calculations done above for EN in the degradation rate. In the white noise limit we find the momentum to befrom which the PDF can be calculated via Eq. (5), with S(x) = x p(x )dx . This PDF does not coincide with Indeed, we find that \|1 − ξ * (x)\| 1 as long as x is in the close vicinity of x * . Now, we plug Φ(x, ξ) from Eq. (C5) and ξ = ξ * (x) from Eq. (C7) into Eq. (B11). Performing the differentiation twice with respect to x, plugging x = x * = f (x * ), using the fact that g(ξ) satisfies f [g(ξ)]/g(ξ) = 1/ξ, and evaluating the result at ξ = 1 which yields g (1) = x * /[1 − f (x * )], we find the observed variance to beThis result coincides with the variance in the case of EN in the degradation rate, seeFigure S3(d) in[56].Appendix D: The case of Ornstein-Uhlenbeck ENIn this section we consider EN with different statistics. We take Ornstein-Uhlenbeck (OU) extrinsic noise with mean ξ(t) = 1 and variance ξ(t)ξ(t ) = σ 2 ex e −\|t−t \|/τc with correlation time τ c > 0. Note that in our previous work[48]on the self-regulating-gene model, we have already used the OU noise when modeling EN. Here we develop a different and more generic formalism allowing to go beyond the bifurcation limit done previously, and to treat EN of arbitrary strength. Notably, EN with such statistics can give rise to zero or even negative reaction rates for sufficiently strong EN, which can cause, e.g., the divergence of the mean[69]. As a result, in our derivation below we implicitly assume that the noise statistics has a cutoff such that the reaction rates are always positive real numbers.The OU process satisfies the following Langevin equationξwhere η(t) is white noise η(t)η(t ) = δ(t − t ). Here η(t) can be defined as the dt → 0 limit of the temporally uncorrelated normal random variable with mean 0 and variance 1/dt. The stationary statistics of this noise is P (ξ) = 1/ 2πσ 2 ex e −(ξ−1) 2 /(2σ 2 ex ) . In order to go beyond the bifurcation limit[48], we are interested to describe the OU process via a discrete birth death process describable by a master equation. Defining k ≡ Kξ as the noise "copy number" in the OU process where K 1 is an arbitrary large number, the master equation describing the probability P k to find EN copy number k satisfies:(D2) Here λ k = 1/(2τ c )(2K 2 σ 2 ex − k + K) and ν k = 1/(2τ c )(2K 2 σ 2 ex + k − K) are the birth and death rates, respectively. One can check that using these birth and death rates one recovers the Langevin equation for the EN "copy number":k = −(k−K)/τ c + 2K 2 σ 2 ex /τ c η(t), which corresponds to Eq. (D1) with ξ = k/K.To study the interplay between IN and EN, we combine the EN dynamics [Eq. (D2)] with the underlying IN dynamics [Eq.(2)]. This yields a 2D master equation for the probability P (n, k, t) to find protein copy number n and noise copy number k, at time t. Similarly as in the case of negative binomial EN, using the WKB ansatz for the stationary PDF, P n,k ∼ e −N S(n/N,k/N ) , we arrive at a Hamilton-Jacobi equation H = 0 with a Hamiltonianwhere as before V = N σ 2 ex ,ξ = ρξ and ρ = K/N , while p x = ∂ x S, and pξ = ∂ξS are the associated momenta. This Hamiltonian encodes the stochastic dynamics of the protein of interest when its degradation rate fluctuates with OU noise.In order to proceed, we can write down the corresponding Hamilton equationṡSimilarly as in the negative binomial case, in the whitenoise limit, τ c 1,ξ(t) and pξ(t) equilibrate fast compared to x and p x . As a result, we can look for slowlyvarying x and p x dependent solutions of the third and fourth Hamilton equations (D4), which yields in the leading order of τ c(D5) Note, that the value of p ef f ξ = O(τ c ) 1 does not enter the equations forẋ andṗ x . Also, one can see that this result as well as the results below are independent of the arbitrary choice of K. 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[ "Interpreting Classifiers through Attribute Interactions in Datasets", "Interpreting Classifiers through Attribute Interactions in Datasets" ]	[ "Andreas Henelius ", "Kai Puolamäki ", "Antti Ukkonen " ]	[]	[]	In this work we present the novel ASTRID method for investigating which attribute interactions classifiers exploit when making predictions.Attribute interactions in classification tasks mean that two or more attributes together provide stronger evidence for a particular class label. Knowledge of such interactions makes models more interpretable by revealing associations between attributes. This has applications, e.g., in pharmacovigilance to identify interactions between drugs or in bioinformatics to investigate associations between single nucleotide polymorphisms. We also show how the found attribute partitioning is related to a factorisation of the data generating distribution and empirically demonstrate the utility of the proposed method.	null	[ "https://arxiv.org/pdf/1707.07576v1.pdf" ]	6,967,800	1707.07576	cbebafa6166aaab94a7e91a2c967b7c6add0125e	Interpreting Classifiers through Attribute Interactions in Datasets 24 Jul 2017 Andreas Henelius Kai Puolamäki Antti Ukkonen Interpreting Classifiers through Attribute Interactions in Datasets 24 Jul 2017 In this work we present the novel ASTRID method for investigating which attribute interactions classifiers exploit when making predictions.Attribute interactions in classification tasks mean that two or more attributes together provide stronger evidence for a particular class label. Knowledge of such interactions makes models more interpretable by revealing associations between attributes. This has applications, e.g., in pharmacovigilance to identify interactions between drugs or in bioinformatics to investigate associations between single nucleotide polymorphisms. We also show how the found attribute partitioning is related to a factorisation of the data generating distribution and empirically demonstrate the utility of the proposed method. Introduction A lot of attention has been on creating high-performing classifiers such as, e.g., support vector machines (SVMs) (Cortes & Vapnik, 1995) and random forest (Breiman, 2001), both of which are among the best-performing classifiers (Fernández-Delgado et al., 2014). However, the complexity of many state-of-the-art classifiers means that they are essentially opaque, black boxes, i.e., it is very difficult to gain insight into how the classifiers work. Gaining insight into machine learning models is a topic that will become more important in the future, e.g., due to possible legislative requirements (Goodman & Flaxman, 2016). Interpretability of machine learning models is a multifaceted problem, one aspect of which is post-hoc interpretability (Lipton, 2016), i.e., gaining insight into how the method reaches the given predictions. Interpreting black box machine learning models in terms 1 Finnish Institute of Occupational Health, Helsinki, Finland. Correspondence to: Andreas Henelius <[email protected]>. ICML Workshop on Human Interpretability in Machine Learning (WHI 2017), Sydney, NSW, Australia. Copyright by the author(s). of attribute interactions provides one form of post-hoc interpretability and is the focus of this paper. Given a supervised classification dataset D = (X, C), where X is a data matrix with m predictor attributes x 1 , . . . , x m (e.g., gender, age etc), and C is a vector with a target attributes (class), an interaction between a subset of these m attributes means that the attributes together provide stronger evidence concerning C than if the attributes are considered alone. We say that attributes interact whenever they are conditionally dependent given the class. We next motivate attribute interactions from the perspective of interpretability of real-world problems. Two difficult problems involving interactions concern drug-drug interactions in pharmacovigilance (e.g., Zhang et al., 2017;Cheng & Zhao, 2014) and investigating interactions between single nucleotide polymorphisms (SNPs) in bioinformatics (e.g., Lunetta et al., 2004;Moore et al., 2010). Recently, machine learning methods have been applied to investigate drug-drug (Henelius et al., 2015) and gene-gene interactions (Li et al., 2016). The benefit of using powerful classifiers, such as random forest, is that one does not need to specify the exact form of interactions between attributes (Lunetta et al., 2004), which is necessary in many traditional statistical methods (e.g., linear regression models that include interaction terms). To utilise classifiers in this manner for studying associations in the data requires that we have some method for revealing how the classifier perceives attribute interactions. A grouping of the attributes in a dataset is a partition where interacting attributes are in the same group, while non-interacting (i.e., independent) attributes are in different groups. In this paper we study two problems. Firstly we want to determine if a particular grouping of attributes represents the attribute interaction structure in a given dataset. Secondly, we want to automatically find a maximum cardinality grouping of the attributes in a given dataset. We approach these problems using the following intuition concerning classifiers, which are used as tools to investigate interactions. A classifier tries to model the class probabilities given the data, i.e., the probability P (C \| X) ∝ P (X \| C) P (C). Here P (X \| C) is the 8 class-conditional distribution of the attributes, which we focus on here. Formally, let S represent a factorisation of P (X \| C) into independent factors, i.e., P (X \| C; S) = S∈S P (X (·, S) \| C)(1) where X (·, S) only contains the attributes in the set S. In other words, interacting attributes are in the same group S ∈ S and, hence, in the same factor in P (X \| C; S). Assume that the dataset D is sampled from a factorised distribution of the form given in Eq. (1) for some S. Further assume that we can generate datasets D S that are exchangeable with D. Suppose now that we train a classifier f 1 using D and that we train a second classifier f 2 (of the same type as f 1 ) using D S . Now, if classifiers f 1 and f 2 cannot be distinguished from each other in terms of accuracy on the same test data, it means that the factorisation S captures the class-dependent structure in the data to the extent needed by the classifier. On the other hand, if f 2 performs worse than f 1 , some essential relationships in the data needed by the classifier are no longer present, i.e., D has not been sampled from a distribution of the form given by Eq. (1). To determine whether f 1 and f 2 are indistinguishable, we compute a confidence interval (CI) for the performance of f 2 by generating an ensemble of datasets D S . If the performance of f 1 is above the CI we conclude that the factorisation S is not valid. Related Work In this paper we combine the probabilistic approach of Ojala & Garriga (2010) studying whether a classifier utilises attribute interactions at all with the method of Henelius et al. (2014) allowing identification of groups of interacting attributes. For a review on attribute interactions in data mining see, e.g., Freitas (2001). Interactions have been considered in feature selection (Zhao & Liu, 2007;2009) . Mampaey & Vreeken (2013) partition attributes by a greedy hierarchical clustering algorithm based on Minimum Description Length (MDL). Their goal is similar to our, but we focus on supervised learning. Tatti (2011) ordered attributes according to their dependencies while Jakulin & Bratko (2003) quantified the degree of attribute interaction and Jakulin & Bratko (2004) factorised the joint data distribution and presented a method for significance testing of attribute interactions. Contributions We present and study the two problems of (i) assessing whether a particular grouping of attributes represents the class-conditional structure of a dataset (Sec. 2.2) and (ii) automatically discovering the attribute grouping of highest granularity (Sec. 2.3). We empirically demonstrate using synthetic and real data how the proposed ASTRID 1 (Automatic STRucture IDentification) method finds attribute interactions in data (Secs. 3-5). Methods In this section we consider (i) how to determine if a particular attribute grouping is a valid factorisation of the class-conditional joint distribution, and (ii) automatically finding the maximum cardinality attribute grouping. Preliminaries Let X be an n × m data matrix, where X(i, ·) denotes the ith row (item), X(·, j) the jth column (attribute) of X, and X(·, S) the columns of X given by S, where S ⊆ [m] = {1, . . . , m}, respectively. Let C be a finite set of class labels and let C be an n-vector of class labels, such that C (i) gives the class label for X(i, ·). We denote a dataset D by the tuple D = (X, C). We denote by P the set of disjoint partitions of [m] = {1, . . . , m}, where a partition S ∈ P satisfies ∪ S∈S S = [m] and for all S, S ′ ∈ S either S = S ′ or S ∩ S ′ = ∅, respectively. Here we assume that the dataset has been sampled i.i.d., i.e., the dataset D follows a joint probability distribution given by P (D) = i∈[n] P (X (i, ·) , C (i)) = P (X\|C) i∈[n] P (X (i, ·) \| C(i)) P (C (i)) ,(2) where P (X \| C) is the class-conditional distribution. We consider a factorisation of P (D) into class-conditional factors given by the grouping S ∈ P and write P (D) = S∈S P (X(·,S)\|C) i∈[n] S∈S P (X (i, S) \| C (i)) P (C (i)) . (3) Given an observed dataset D, we want to find the attribute associations in the data and ask: Has the observed dataset D been sampled from a distribution given by Eq. (3) with the grouping given by S ∈ P? Framework for Investigating Factorisations Our goal is to determine whether the data obeys the factorised distribution of Eq. (3). To do this we compare the accuracy of a classifier trained using the original data with the confidence interval (CI) formed from the accuracies of a collection of classifiers trained using permuted data. The permuted datasets are formed such that they are exchangeable with the original dataset if Eq. (3) holds. If the accuracy of the original data is above the CI we can conclude with high confidence that the data does not obey the factorised distribution. We denote a classifier trained using the dataset D by f D . Further assume that we have a separate independent test dataset from the same distribution as D, denoted by D test = (X test , C test ). Definition 1. Classification Accuracy Given the above definitions, the accuracy for a classifier trained using D is given by T (D) = 1 n test ntest i=1 I [f D (X test (i, ·)) = C test (i)], (4) where I [ ] is the indicator function and n test is the number of items in the test dataset. Note that T is not the accuracy of f on D, but the accuracy of f on X test when f is trained using D. Because direct sampling from Eq. (3) is not possible as the data generating model is unknown, we generate the permuted data matrices X S (defined below) so that they have same probability as X under the assumption that X is a sample from a factorised distribution as given in Eq. (3). This means that X and X S are exchangeable under the assumption of a joint distribution that is factorised in terms of S. We sample datasets using the permutation scheme described in Henelius et al. (2014). A new permuted dataset D S = X S , C is created by permuting the data matrix of the dataset D = (X, C) at random. The permutation is defined by m bijective permutation functions π j : [n] → [n] sampled uniformly at random from the set of allowed permutations functions. The new data matrix is then given by X S (i, j) = X (π j (i) , j). The allowed permutation functions satisfy the following constraints for all i ∈ [n], j, j ′ ∈ [m], and S ∈ S: 1. permutations are within-a class, i.e., C (i) = C (π j (i)), and 2. items within a group are permuted together, i.e., j ∈ S ∧ j ′ ∈ S =⇒ π j (i) = π j ′ (i). Let D S be the set of datasets that can be generated by the above permutation scheme using the grouping S. We note: Lemma 1. Each invocation of the permutation scheme produces each of the datasets in D S with uniform probability. We cast the above discussion as a problem: Problem 1. Given an observed dataset D, a grouping S and a classifier f , let a 0 be the accuracy of f (trained using the original data) on the test set. Determine if the upper end of the CI of Def. 2 for the accuracy of a classifier trained using factorised data is at least a 0 . If the above condition is met, we conclude that the factorisation correctly captures the structure of the data. Automatically Finding Groupings (ASTRID) In the previous section we examined whether a particular grouping S describes the structure of the data in terms of the factorisation in Eq. (3). A natural step is now to ask how to find the grouping best describing the associations in a dataset D? Here we choose best to be the grouping S of (i) maximum cardinality such that (ii) a classifier trained using data shuffled with S is indistinguishable in terms of accuracy from a classifier trained using the original, unfactorised data. Finding the maximum cardinality grouping is motivated by the fact that in this case there are no irrelevant interactions. Also, interpreting attribute interactions in small groups is easier than in large groups. The requirement on accuracy means that no essential information is lost and in practice this means that the upper end of the CI for the accuracy of the classifier f trained using D S is at least as large as the original accuracy a 0 of f trained using D. Exhaustive search of all groupings is in general impossible due to the size of the search space. Hence, to make our problem tractable we assume that accuracy decreases approximately monotonically with respect to breaking of groups in the correct solution, i.e., the more the interactions are broken, the more classification performance decreases. Using this property we use a top-down greedy algorithm termed ASTRID. For details see the extended description in Henelius et al. (2017). In practice, T in Eq. (4) is susceptible to stochastic variation and for stability we instead use expected accuracy V when optimising accuracy 10 in the greedy algorithm: V (S) = 1 N N i=1 T D S i ,(5) where N is the number of samples used to calculate the expectation, D S i (i ∈ [N ]) is a dataset generated by the permutation parametrised by S and T is defined as in Eq. (4). Experiments We use ASTRID to identify attribute interactions. We use a synthetic dataset and 11 datasets from the UCI machine learning repository (Bache & Lichman, 2013) 2 . All experiments were run in R (R Core Team, 2015) and our method is released as the ASTRID R-package, available for download 3 . We use a value of R = 250 in Def. 2 and N = 100 in Eq. (5). In all experiments the dataset was randomly split as follows: 50% for training (D) and the rest for testing (D test , see Eq. (4)): 25% for computing V (Eq. (5)), and 25% for computing CIs. As classifiers we use support vector machines (SVM) with RBF kernel, random forest (RF) and naïve Bayes (NB). The datasets are summarised in Table 1. The UCI datasets were chosen so that the SVM and random forest classifiers achieve reasonably good accuracy at default settings, since the goal here is to demonstrate the applicability of the method rather than optimise classifier performance. Rows with missing values and constant-value columns were removed from the UCI datasets. The synthetic dataset has two classes, each with 500 data points. Attributes 1 and 2 carry meaningful class information only when considered jointly, attribute 3 contains some class information and attribute 4 is random noise. The correct grouping is hence S = {{1, 2} , {3} , {4}}. Results The results are presented as tables where each row is a grouping and the columns represent attributes. Attributes belonging to the same group are marked with the same letter, i.e., attributes marked with the same letter on the same row are interacting. Table 2 shows the results for the synthetic dataset where the highest-cardinality grouping is highlighted and is also shown below the table. Using the SVM and RF classifiers ASTRID identifies the correct attribute interaction structure (k = 3). For k = 4 the accuracy is clearly lower. For naïve Bayes all groupings (all values of k) are equally valid Table 1: The datasets used in the experiments (2-10 from UCI). Columns as follows: Number of items (Ni) after removal of rows with missing values, number of classes (Nc) after removal of constant-value columns, number of attributes (Na). MCP is major class proportion. T SVM and T RF give the computation in minutes of the ASTRID method for the SVM and random forest, respectively. since the classifier assumes attribute independence. The results mean that the average accuracy of an SVM or RF classifier trained on the synthetic dataset permuted using S = {{1, 2} , {3} , {4}} is within CIs. ASTRID reveals the factorised form of the joint distribution of the data, which makes it possible to identify the attribute interaction structure exploited by the classifier in the datasets. This makes the models more interpretable and we, e.g., learn that NB does not exploit interactions (as expected!). The groupings for the UCI datasets are summarised in Table 3. SVM and RF are in general similar in terms of the cardinality (k), with the exception of kr-vs-kp and soybean. In many cases it appears that the classifiers utilise few interactions in the UCI datasets. To compare this finding with the results of Ojala & Garriga (2010), we calculated the value of their Test 2, denoted p OG in Table 3. This test investigates whether a classifier utilises attribute interactions. p OG ≥ 0.05 indicates that no attribute interactions are used by the classifier, which we find for diabetes and soybean for SVM and for diabetes and credit-a for random forest (highlighted in the table). This is in line with the findings from ASTRID, since for these datasets k equals N in Table 3 and no interactions are hence utilised as the dataset can be factorised into singleton groups. Finally, as an illustrative example of grouping attributes exploited by a classifier we consider the vote dataset. This dataset contains yes/no information on 16 issues with the target of classifying if a person is republican or democrat. Using SVM ASTRID finds that the maximum cardinality grouping is of size k = 8 (Tab. 3). The grouping consists of 7 singleton attributes (water-project-cost-sharing, synfuels-corporation-cutback, physicianfee-freeze, education-spending, duty-free- Table 2: The synthetic dataset. The cardinality of the grouping is k and CI is the confidence interval for accuracy. Original accuracy using unshuffled data (a 0 ) and the final grouping (S, highlighted row) shown above and below the table, respectively. An asterisk ( * ) denotes that the factorisation is valid. exports, export-administration-act-southafrica, immigration) and one group with 9 interacting attributes (crime, handicapped-infants, religious-groups-in-school, superfund-rightto-sue, adoption-of-the-budget-resolution, mx-missile, anti-satellite-test-ban, aid-tonicaraguan-contras, el-salvador-aid). It appears that the 9 attributes in the group roughly represent military and foreign policy issues, and economic and social issues. This means, that the SVM exploits relations between these 9 political issues when classifying persons into republicans or democrats. On the other hand, the singleton attributes seem to mostly represent domestic economic, economic and export issues. The classifier does not use any singleton attribute jointly with any other attribute when making predictions. Note that ASTRID is a randomised algorithm and the found groupings are hence not necessarily unique. The stability of the results depends on factors such as the used classifier, the size of the data and the strength of the interactions. Also, the results are affected by the number of random samples (R in Def. 2 and N in Eq. (5)) and for practical applications a trade-off between accuracy and speed must be made. Discussion and Conclusion Interpreting black box machine learning models is an important emerging topic in data mining and in this paper we present the ASTRID method for investigating classifiers. This method provides insight into generic, opaque classifier by revealing how the attributes are interacting. ASTRID automatically finds in polynomial time the maximum cardi- Table 3: Groupings for UCI datasets. Columns as follows: number of attributes in the dataset (N), size of the grouping (k), size of the largest (N 1 ) and second-largest (N 2 ) groups, baseline accuracy for the classifier trained with unshuffled data (a 0 ) and the CI. p OG is the p-value of Test 2 in Ojala & Garriga (2010) nality grouping such that the accuracy of a classifier trained using the factorised data cannot be distinguished (in terms of confidence intervals) from a classifier trained using the original data. The method makes no assumptions on the data distribution or the used classifier and hence has high generic applicability to different datasets and problems. This work extends previous research (Henelius et al., 2014;Ojala & Garriga, 2010) on studying attribute interactions in opaque classifiers. Knowledge of attribute interactions exploited by classifiers is important in, e.g., pharmacovigilance and bioinformatics (see Sec. 1) where powerful classifiers are used in data analysis, since they make it possible to simultaneously investigate multiple attributes instead of, e.g., just pairwise interactions. Here ASTRID allows the practitioner to automatically discover attribute groupings, providing insight into the data by making the classifiers more transparent. Lemma 2 . 2The datasets in D S have equal probability under the distribution of Eq. (3), parametrised by S. Proof. The proofs follow directly from the definition of the permutation and the probability distribution of Eq. (3). Definition 2. Confidence intervals Given a dataset D, a grouping S, a classifier f and an integer R, let A = T D S 1 , . . . , T D S R be a vector of accuracies where the datasets D S i are obtained by the permutation parametrised by S, and T is as in Eq. (4). The CI is the tuple C = (c lower , c upper ), where c lower and c upper are values corresponding to the 5% and 95% quantiles in A, respectively. S = {{1} , {2} , {3} , {4}} (p ≥ 0.05 highlighted).Dataset N k N1 N2 a0 CI pOG SVM balance-scale 4 3 2 1 0.891 [0.821, 0.897] 0.03 credit-a 15 12 4 1 0.871 [0.847, 0.871] 0.04 diabetes 8 8 1 1 0.714 [0.688, 0.740] 0.59 kr-vs-kp 36 33 4 1 0.917 [0.922, 0.924] 0.00 mushroom 21 15 7 1 0.995 [0.991, 0.995] 0.00 segment 18 3 16 1 0.948 [0.936, 0.948] 0.00 soybean 35 35 1 1 0.844 [0.820, 0.850] 0.26 vehicle 18 3 15 2 0.767 [0.719, 0.781] 0.00 vote 16 8 9 1 0.931 [0.897, 0.931] 0.00 vowel 13 3 11 1 0.806 [0.760, 0.806] 0.00 random forest balance-scale 4 3 2 1 0.821 [0.731, 0.833] 0.02 credit-a 15 15 1 1 0.877 [0.847, 0.883] 0.19 diabetes 8 8 1 1 0.703 [0.698, 0.740] 0.89 kr-vs-kp 36 16 21 1 0.982 [0.972, 0.982] 0.00 mushroom 21 14 8 1 1.000 [0.996, 1.000] 0.00 segment 18 4 15 1 0.986 [0.979, 0.986] 0.00 soybean 35 24 12 1 0.964 [0.946, 0.964] 0.00 vehicle 18 3 13 4 0.752 [0.710, 0.757] 0.00 vote 16 10 7 1 0.948 [0.897, 0.948] 0.00 vowel 13 3 11 1 0.917 [0.901, 0.917] 0.00 R-package available:https://github.com/bwrc/astrid-r Datasets obtained from http://www.cs.waikato.ac.nz/ml/weka/datasets.html 3 https://github.com/bwrc/astrid-r (R-package and source code for experiments) AcknowledgementsThis work was supported by Academy of Finland (decision 288814) and Tekes (Revolution of Knowledge Work project). 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[ "An Intelligent System for Multi-topic Social Spam Detection in Microblogging", "An Intelligent System for Multi-topic Social Spam Detection in Microblogging" ]	[ "Bilal Abu-Salih \nThe University of Jordan\n\n", "{b Busalih@ju Edu Jo } ", "Dana Al Qudah \nThe University of Jordan\n\n", "Malak Al-Hassan \nThe University of Jordan\n\n", "Seyed Mohssen Ghafari \nMacquarie University\n\n", "Tomayess Issa \nCurtin University\n\n", "Ibrahim Aljarah \nThe University of Jordan\n\n", "Amin Beheshti \nMacquarie University\n\n", "Sulaiman Alqahtani \nMacquarie University\n\n" ]	[ "The University of Jordan\n", "The University of Jordan\n", "The University of Jordan\n", "Macquarie University\n", "Curtin University\n", "The University of Jordan\n", "Macquarie University\n", "Macquarie University\n" ]	[]	The communication revolution has perpetually reshaped the means through which people send and receive information. Social media is an important pillar of this revolution and has brought profound changes to various aspects of our lives. However, the open environment and popularity of these platforms inaugurate windows of opportunities for various cyber threats, thus social networks have become a fertile venue for spammers and other illegitimate users to execute their malicious activities. These activities include phishing hot and trendy topics and posting a wide range of contents in many topics. Hence, it is crucial to continuously introduce new techniques and approaches to detect and stop this category of users. This paper proposes a novel and effective approach to detect social spammers. An investigation into several attributes to measure topic-dependent and topic-independent users' behaviours on Twitter is carried out. The experiments of this study are undertaken on various machine learning classifiers. The performance of these classifiers are compared and their effectiveness is measured via a number of robust evaluation measures. Further, the proposed approach is benchmarked against state-of-the-art social spam and anomalous detection techniques. These experiments report the effectiveness and utility of the proposed approach and embedded modules.	10.1177/01655515221124062	[ "https://arxiv.org/pdf/2201.05203v1.pdf" ]	245,986,615	2201.05203	4b6b7344223b3bf49d68a983fffae483592725fe	An Intelligent System for Multi-topic Social Spam Detection in Microblogging Bilal Abu-Salih The University of Jordan {b Busalih@ju Edu Jo } Dana Al Qudah The University of Jordan Malak Al-Hassan The University of Jordan Seyed Mohssen Ghafari Macquarie University Tomayess Issa Curtin University Ibrahim Aljarah The University of Jordan Amin Beheshti Macquarie University Sulaiman Alqahtani Macquarie University An Intelligent System for Multi-topic Social Spam Detection in Microblogging 1Social SpammersOnline Social NetworksMachine LearningSocial CredibilitySemantic AnalysisCyber Threats The communication revolution has perpetually reshaped the means through which people send and receive information. Social media is an important pillar of this revolution and has brought profound changes to various aspects of our lives. However, the open environment and popularity of these platforms inaugurate windows of opportunities for various cyber threats, thus social networks have become a fertile venue for spammers and other illegitimate users to execute their malicious activities. These activities include phishing hot and trendy topics and posting a wide range of contents in many topics. Hence, it is crucial to continuously introduce new techniques and approaches to detect and stop this category of users. This paper proposes a novel and effective approach to detect social spammers. An investigation into several attributes to measure topic-dependent and topic-independent users' behaviours on Twitter is carried out. The experiments of this study are undertaken on various machine learning classifiers. The performance of these classifiers are compared and their effectiveness is measured via a number of robust evaluation measures. Further, the proposed approach is benchmarked against state-of-the-art social spam and anomalous detection techniques. These experiments report the effectiveness and utility of the proposed approach and embedded modules. Introduction The rise of Online Social Networks (OSNs) has led to reinforce values of freedom of expression, communication, and breaking the monopoly of information. Therefore, these virtual platforms have been used to circulate information, support movements of rejections and protests, expose corrupt practices and other various activities. Despite the plethora of benefits brought by the continuous use of these platforms, the lack of 'gatekeepers' has opened wide the door to carrying out fraud, defamation and speeding of rumours, tarnishing the reputation of organisations and individuals, to all other types of false information that have permeated these platforms. This poses significant challenges as various malicious activities have degraded the quality of experience obtained by the members of these virtual communities [1]. Social spam is commonly referred to as nasty activities or unsolicited and low-quality content that spread over the OSNs. Examples of such activities include profile cloning, social phishing, fake reviews, bulk submissions, hashtags hijacking, clickjacking, sending malicious links, etc. Social spammers are those who inject spam and practice such activities over the OSNs [2]. Despite the ongoing efforts to cease such activities and keep the platforms clean, the social spam phenomenon continues to rise; almost 47 percent of respondents to a recent survey indicate perceiving more spam into their social media feeds [3]. To gain credibility, this category of users attempts to establish dialogues with people of dissimilar interests by attempting to allure non-spammers into befriending them or by carrying out hashtag hijacking or bulk messaging of a specific topic or a wide range of topics [1,4]. Topics of interest are particular areas of an individual's work, expertise, or specialisation within the scope of subject-matter knowledge such as science, politics, sports, education etc. [5][6][7]. This implies that a user's credibility in OSNs is topic-driven; users can be trustworthy and reliable in one or few topics, yet this does not apply to other topics of interest. In OSNs, we argue that there is an inverse relationship between the number of topics the user is interested in and the user's topic-based (domain-based) credibility. This argument is justified based on the following facts: (i) there is no well-informed legitimate user who has the intellectual capacity to publish contents in all topics [8]; (ii) users who publish contents on various topics of interest do not convey to other users the particular topic that they are interested in. This is evident as topics of interest can be intuitively deduced from content of users who commonly post wide-ranging content in one or few topics; (iii) it is likely that this user is a spammer or anomalous; this illegitimate category of users tends to post tweets about numerous topics [9,10]. This could end up with tweets being posted on all topics which do not convey a legitimate user's behaviour. Therefore, distinguishing users in a set of topics is a significant aspect. The existing social spam detection approaches are generic in terms of topic extraction, and they passively extract features based merely on users' data and metadata. In particular, the current approaches incorporate a probabilistic generative model, namely the Latent Dirichlet Allocation (LDA) [11] to distil users topics of interest from their textual data. LDA and similar other statistical models fail to capture high-level topics and to consider and integrate the semantic relationships between terms in textual data. Also, they are inadequate to extract correct topics from short textual messages such as tweets [12,13]. Another drawback of the existing works is that they neglect to extract sentiments of the tweets' conversations (tweets replies), thereby unable to listen to the subjective impressions of users' followers towards tweets' contents. On the other hand, our study proposes a novel and effective multi-topic social spammers detection model for microblogging. To distinguish users' topics of interest, we incorporate the idea of distinguishing/discriminating which was developed in the Information Retrieval (IR) domain through applying . formula [14]. "The intuition was that a query term which occurs in many documents is not a good discriminator" [15]. This implies that a term that occurs in many documents decreases its weight in general as this term does not show which document the user is interested in [16]. This heuristic aspect is incorporated into our model as an important driver to extract features from the users' contents based on their topics of interest. The users' topics of interest have been investigated and analysed using two approaches, namely: IBM Watson -Natural Language Understanding (NLU) API [17]) and a developed topic discovery model (based on 20 Newsgroups dataset [18]). IBM Watson NLU, formally known as Alchemy API, involves analysing textual content and extract semantic features incorporating machine learning, semantic web technologies, and linked open data. These semantic features include entities, emotion, keywords, relations, categories/taxonomies (currently 23 topics), sentiment analysis, etc. IBM Watson NLU provides both topic-independent and customized topic analysis that can be implemented using their Knowledge Studio. This designated API is used both to extract users' domains of knowledge as well as to infer sentiments of followers' replies as it will be discussed later. Further, we developed a topic classification module based on the 20 Newsgroups dataset and by implementing a number of machine learning classification models. The set of topics extracted by both models are mapped and aggregated, thereby inferring a unified model for topic discovery that will be used further in the proposed spammer detection model. In particular, we investigate several attributes to measure topic-dependent and topic-independent users' behaviours in Twitter. A total of 18 key features is attained and extracted from contents and user analysis. The experiment of this study was carried out on manually labelled Twitter datasets. The list of user_ids used for data collection is stemmed from two popular corpora: (i) topically anomalous dataset [19]. This graph is chosen since it includes the list of Twitter users who had less than 5,000 friends. This threshold was initially established to discover anomalous and other illegitimate users; (i) Social honeypot dataset [2]. This dataset comprises a long-term study of social honeypots through 60 honeypots on Twitter that resulted in collecting 36,000 candidate content polluters. The labelled dataset is trained and tested over six popular machine learning algorithms. The performance of these algorithms are compared and their effectiveness is measured via a number of robust evaluation metrics. Further, the proposed approach is benchmarked against state-of-the-art social spam and anomalous detection techniques. These experiments verify the effectiveness and utility of the proposed approach and embedded modules. The key contributions of this paper are summarised as follows: • An effective multi-topic social spammer detection model for microblogging is proposed. • The proposed model addresses the deficiencies in the existing approaches by conducting a fine-grained analysis to users data and metadata using semantic analysis and sentiment analysis., thereby extracting various topic-dependent and topic-independent features. • The proposed model introduces a topic mapping scheme between two well-known topic discovery approaches. • A comprehensive experimental analysis is conducted which verifies the utility of our model to detect social spammers in microblogging. The remaining of this paper is organised as follows: the following section lays the background on various approaches used to detect spam in social networks and provides an evaluation of the existing approaches. Section 3 presents and discusses the methodology and embedded modules. The experiments conducted in this study are explained in Section 4. Section 5 discusses the key insights of this paper before we conclude it with certain remarks on future research directions. Related Works This section reviews important research on social spam detection. First, we discuss the notion of social spam followed by a report on various types of social spam. Then, recent solutions provided to tackle the social spam issue are presented and discussed. Social spam -an overview The Internet has witnessed a paradigm shift since the emergence of the second generation of the Web (i.e., Web 2.0, a.k.a. Social Web). This has transformed the Web experience enabling people to interact and exchange ideas, thoughts, and beliefs leveraging the free and easy access to such virtual environments. However, the lack of a 'gatekeeper' in these technological means has established a new form of spam, commonly referred to as social spam. The notion of social spam refers to any form of undesirable information (including textual content, images, videos, URLs, follow/friend requests, etc.) which spreads through a network(s) of social media in a process referred to as spamming [20]. Social spam differs from a typical spam email or SMS spam in which the former manifests in multiple forms and modi operandi, and the latter can be mainly conveyed in one form (i.e. emails [21][22][23] or SMSs [24]). The following are examples of social spam: • Social anomalies: Despite the lack of consensus to provide a unified definition to the notion of an anomaly in OSNs, the term can be referred to as a deviation resulting from unexpected, illegal and irregular behaviour of users that is prevalent in these societies [25]. Producers of bad quality social data, such as anomalies, provide their content with anonymity and impunity. • Malicious links: Links that cause harm to individuals or computers. They are embedded in the spearphishing tweets and distributed through the networks [26]. • Bulk messaging(mass direct messaging or spam-bombs): A chunk of messages of the same content that are proliferated to a group of users in a relatively short period of time [27]. • Fake profiles (or Sybils/Socialbots): Social mock accounts that are created mainly to gain visibility or influence on social media. Social bots concoct various accounts of similar identities [28]. Twitter does not tolerate fake and suspicious accounts. They suspended more than 70 million accounts in 2018 [29]. Yet, it is expected that social bots will continue to propagate and social media manipulation remains undetected [30]. • Fake reviews (opinion spamming): Reviews are written to promote or to discredit products, services or businesses, thus do not reflect a genuine user experience [31]. Approaches for social spam detection The rapid increase in utilising OSNs along with the lack of the gatekeeper have simulated spammers to inject unsolicited contents into these platforms [32][33][34][35][36][37]. This has led the research community to pay more efforts to stop the spreading of spam and spammers. These endeavours are commonly divided into two main directions; (i) Machine Learning approaches: -The set of techniques that incorporate machine learning and artificial intelligence to detect and discover social spam; (ii) Graph-based approaches:-The techniques which rely on the social network structure and graph properties for spam detection. Machine learning approaches -un/semi-supervised Machine Learning algorithms have proven the ability to modernize technology enabling companies to design solutions to sophisticated problems as well as to make informed and better decisions [38][39][40][41][42][43][44]. The applications of machine learning span various domains such as healthcare [45], industry [46], recommender systems [47], and NLP systems [48]. The mechanisms of detecting spammers in OSNs incorporating machine learning are commonly determined based on the nature of training detection models [49,50]. Supervised detection methods extract a collection of features that can be used to train the machine learning classification model [51]. These features are mainly inferred from the analysis of users' metadata and user's contents. For example, Clark et al. [52] incorporated certain features such as time between tweets, #followers, etc. to detect fake and robotic accounts. With a robust textual classification scheme, that uses natural language structure, authors were able to obtain promising results on discovering accounts that send automatic messages. Various supervised learning approaches that are built on labelled datasets and using classification algorithms such as Support Vector Machine, Logistic Regression, Decision Tree, and Naïve Bayes have proven effectiveness in spam detection [53]. The availability of publicly labelled datasets is very infrequent; thus, unsupervised learning methodologies provide an alternative as they do not require labelled datasets. For example, Zhang et al. [54] applied the Twiceclustering approach on product reviews collected from 360buy.com, and the authors obtained 66% accuracy for detecting spam reviews. Another thread of efforts incorporated a k-means clustering algorithm for spam detection. These efforts reported good results on spam detection. For example, Liu [55] and Wu [56] reported 71% and 72% accuracy respectively by employing the k-means approach. Unsupervised learning is also used in real-time spam tweet detection using collective-based tweets analysis [57], and also used in spam reviews detection [58] and bot detection [59]. The semi-supervised learning provides a trade-off between the aforementioned machine learning techniques; thus it can work with a few labelled observations along with more unlabelled ones. Amongst various techniques reported in the literature, generic or wrapper and non-generic are mainly key types of semi-supervised learning. For example, using a semi-supervised generative active learning approach to automatically generate semantically similar texts for spam content detection [60]. Another attempt was undertaken by [61] in which authors used semi-supervised learning over a partially labelled dataset for Twitter spam drift problem. Semisupervised learning was further employed in [62][63][64] Graph-based approaches Social spam can also be detected using attributes and features captured from graph nodes of the social network. These features can be distilled from two categories; social features (i.e. account-based features) and structural features of the social graph [56]. Various studies were proposed in this direction. For example, Gupta et al. [65] developed a model to detect Twitter spammers who spread social spam by accessing phone numbers of Twitter users to deliver annoying advertisements of products and services. The authors proposed a Hierarchical Meta-Path Score (HMPS) metric to measure the similarity between two nodes in the network. Then, they framed a Twitter dataset as a heterogeneous network by leveraging diverse interconnections between different categories of nodes embedded in the dataset. Al-Thelaya et al. [66] proposed representation models for social interaction's graph-based datasets. The models were designed to detect social spam based on graph-based analysis and sequential processing of user interactions. Integrating both social content and network structure are present in the literature; Noekhah et al. [67] proposed MGSD (Multi-iterative Graph-based opinion Spam Detection) model to detect 'spamicity' effects of internal and external entities employing topic independent features Hybrid models were also demonstrated in [68][69][70]. Evaluation of current approaches Lack of incorporating semantic analysis: The difficulty of acquiring an accurate understanding of the contextual meaning of social textual content affects a further conducted analysis. The existing social spam detection approaches, into their textual analysis, adopt a probabilistic generative model, namely the Latent Dirichlet Allocation (LDA) [11] and its variations such as Labeled LDA, PhraseLDA, etc. In spite of the popularity of these models to infer a predefined set of topics, they suffer from the following drawbacks; (i) they fail to capture high-level topics/domains; (ii) they are unable to consider and integrate the semantic relationships between terms in the text; and (iii) they are inadequate to extract topics from short textual contents such as tweets as well as to extract semantically meaningful data from discrete textual content [12,13]. On the other hand, this study incorporates semantic analysis and semantic web techniques which enable the elicitation of meaningful information from social data, thereby enhancing its textual content to offer semantics and linking each message to a specific domain. IBM Watson NLU, for example, provides a list of taxonomies (categories) that are further subdivided into finer-grained subcategories based on ontologies and linked open data. Employing semantic analysis techniques have proven effective to address the brevity problem of textual messages such as tweets and to mitigate problems pertaining to features of linguistics such as polysemy, homonymy, and contronymy [37]. Lack of topic-specific spam detection models: Despite the ongoing endeavours to tackle the pressing problem of social spam and its widespread negative implications, the current approaches are inadequate to address multi-topic social spamming behaviours. The current social spam detection approaches are generic and passively extract features based merely on users' data and metadata. This study, on the other hand, proposes a novel and effective approach to detect social spammers incorporating a number of attributes to measure topicdependent and topic-independent users' behaviours in OSNs. In particular, our approach proposes a novel topic distinguishing mechanism based on . heuristic aspect of IR. This heuristic aspect is incorporated into our model to extract features from the users' contents based on their topics of interest. The extracted features from user's data are classified into two different groups (topic-dependent and topic-independent), thereby providing fine-grained topic-specific user behavioural analysis. The next section elaborates on the proposed methodology. Figure 1 illustrates the proposed framework and the embodied modules. As depicted in Figure 1, the system architecture comprises of three main phases which are detailed in the next sections, namely; (1) Tweets Acquisition and pre-processing; (2) Feature extraction and selection; and (3) Machine learning model building and tuning. Methodology Tweets acquisition and pre-processing This section presents the steps followed to collect and pre-process datasets. First, we provide an overview of the two Twitter sets used in this study, and then we discuss the undertaken steps for data preparation and integration. Dataset source This study focuses on social data generated from Twitter™. Twitter analytics is an evolving research field falling under the category of data mining, social big data analysis, and machine learning. Twitter has been chosen in this paper due to the following reasons: (i) Twitter platform has been studied broadly in the research communities [71], leveraging the vast volume of content (6,000 tweets/seconds) [72]; (ii) It facilitates retrieving public tweets through providing APIs; (iii) the Twitter messages' "max 140 characters" feature enables data analysis and prototype implementation for a proof of concept purpose. Twitter data access mechanisms have been harnessed in this study for data collection purposes. Users' information and their tweets and all related metadata were crawled using TwitterAPI [73]. A PHP script was implemented to crawl users' content and their metadata using the User_timeline API method. This API allows access and retrieve the collection of tweets posted by a certain user_id associated with each API request. This approach is used rather than a keyword search API due to the reasons as follows. Keyword-based search API has certain limitations listed in [71], i.e. Twitter index provides only tweets posted within 6-9 days thus it is hard to acquire a historical Twitter dataset before this period. Further, Search API retrieves results based on the relevance to the query caused by uncompleted results. This implies missing tweets and users in the search results. Using the user's timeline approach, on the other hand, retrieves up to 3,200 of the recent users' tweets. Last but not least, the purpose of this paper is to measure the users' credibility hence user-driven tweets collection is the suitable approach. Further, acTwitterConversation 1 API was used to retrieve all public conversations related to the tweets being fetched using Twitter API. Data acquisition is carried out using a PHP script triggered by running a cron job that selects a new user_id and starts collecting historical user information, tweets, replies and the related metadata. The list of Twitterers' user_ids used in the data acquisition phase is extracted from two key selected datasets; (1) Topically anomalous dataset: A Twitter graph dataset crawled by Akcora et al. [19]. This graph is chosen since it includes the list of users who had less than 5,000 friends. This threshold was established by Akcora et al. [19] to discover bots, spammers and robot accounts; (2) Social honeypot dataset: This dataset comprises a long-term study of social honeypots through 60 honeypots on Twitter that resulted in the collecting of 36,000 candidate content polluters [2]. It is worth indicating that we have obtained only the list of user_ids from the aforementioned datasets. But the tweets and all the related contents and metadata were collected by our script, thereby studying the recent social behavior of those users as will be discussed in the experiments section. The collection of tweets was carried out using supercomputing facilities provided and supported by Pawsey Supercomputing Centre 2 . Tweets pre-processing and storage Features of Big data should be tackled when handling large-scale dataset such as social data. To ensure the veracity of Big data in the context of this study, accuracy, correctness and credibility of data should be established. Although the origin of data and storage is critical to ensuring the veracity of Big data, the trustworthiness of the source does not guarantee data correctness and consistency. Data cleansing and integration should also be used to guarantee the veracity of data. Further improvements to the collected data quality will be discussed later in the analysis phase. The raw extracted tweets were subjected to a pre-processing phase to address the data veracity regarding data correctness. This phase includes the following steps: Data integration and temporary storage: Tweets are collected from the designated APIs in JSON format. Further, the tweets' replies collected from AcTwitterConversation API is obtained in arrays. To maximize the reuse of data and facilitate data analysis, data integration aims to address the format dissimilarity by blending data from different resources. In particular, the raw tweets (JSON) and tweets' replies (ARRAY) are extracted, reformatted, and unified to fit a predesigned relational database ( i.e. MySQL) that is used as a temporal storage for data for the following analysis phase. Data cleansing: Data at this stage may include various errors, meaningless and irrelevant data, redundant data, etc. Thus, data is cleansed to detect and remove corrupt, incomplete and noisy data, thus ensuring data consistency. For example, duplicate contents are removed. This comprises tweets, tweets' replies or any other metadata. Further, URLs to photos and videos are eliminated as those do not contain textual contents that can be extracted and examined in the proposed model. However, future research aims to provide a multimodal approach to tackle this issue. Topic classification & sentiment analysis In this module, we aim to segment users based on their topics of interest and to carry out opinion mining on the tweets' replies. For this purpose, we employ IBM Watson -NLU to be utilised for topic inference and sentiment analysis. Also, we develop a topic classification model using the 20 Newsgroups dataset. IBM Watson -Natural Language Understanding: IBM Watson NLU is a cloud-based service which is used to extract metadata from textual content such as entities, taxonomies/categories/high-level topics, sentiments, and other NLP components. IBM Watson analyses the given text or URL and categorises the content of the text or webpage according to various topics/categories/taxonomies with the corresponding scores values. Scores are calculated using IBM Watson, range from "0" to "1", and report the accuracy of an assigned taxonomy to the analysed text or webpage. IBM Watson is used further to determine the overall positive or negative sentiment of a given text (reply). Table 1 demonstrates an example of incorporating this API to extract taxonomies and the sentiment of a given tweet. As illustrated in Table 1, the content of the tweet is analysed by IBM Watson using two key components: Categories Inference and Sentiment Analysis. The scores are given for each component to represent the adequacy of the retrieved taxonomy and sentiment to the provided tweets. The taxonomy inference module is used in this research in the topic discovery, while opinion mining is used to detect the sentiments of tweets' replies [74,75]. A tweet is commonly comprised of two key elements, namely the text and the embedded URL. URLs are attached to tweets to convey further information on the topic discussed in the tweets, especially that a tweet is limited in length. URLs direct users to another website, or an image, video, etc. Twitter automatically scrutinises URLs against a set of harmful and malicious sites, then those URLs are shortened to http://t.co links to help users share lengthy URLs into their short tweets. This feature is unfortunately abused by spammers through hijacking trends or embedding unsolicited mentions and attaching clickjacked URLs. This tricks twitterers into carrying out undesired actions by clicking on such concealed links. Therefore, as an important credibility measure, it is crucial to examine embedded URLs, detect topics of the URLs' web pages, and validate that topics match with the textual contents of tweets handling those URLs. IBM Watson is utilised further in this study to stem the sentiment of a certain reply whether it is positive, natural, or negative with the corresponding sentiment score. Consequently, all of a tweet's set of replies are collected and the sentiments of these replies are combined in the analysis to enhance the credibility. Topic discovery using the 20 Newsgroup: The 20 Newsgroups data set is a collection of around 20,000 collected news documents which are grouped into 20 high-level categories (newsgroups) [18]. Each document in this corpus is labelled with one of the twenty categories. This dataset is a popular dataset and is commonly used to conduct textual clustering tasks. The documents embedded in this set are distributed almost evenly amongst the twenty domains, and each document is written in a form of an email. This dataset is further used to provide consolidate the topic discovery process. To use the dataset in the intended task, the textual contents of the dataset were vectorized into numerical vectors to be used for conducting the predictive task. In this regard, a Bag-of-Words technique, namely TF-IDF is used to extract feature vectors from each of the textual documents. The read-for-prediction dataset is then used to feed the incorporated classification modules. Further details will be discussed in the experimental results section. Topic mapping scheme: The aforementioned topic discovery and classification approaches provide different high-level domains, yet a unified model is required to facilitate features extraction to be used for the credibility module. Therefore, we propose a mapping scheme between all domains of the two approaches. In particular, we manually and carefully match domains in two approaches if they are semantically interrelated. Figure 2 shows the proposed mapping between all of the two high-level categories. This mapping establishes a unified model which will be used in the topic analysis of the users' textual content. Hence, textual contents of the tweets will be analysed using two approaches and only tweets that meet the following criteria will be selected: A tweet will be selected if the extracted topic in each module corresponds to the proposed mapping of Figure 2. Thus, tweets that infer different domains do not comply with the designated mapping scheme and will be neglected and not be incorporated in a further conducted analysis. The aim of incorporating two different mechanisms for topic discovery is twofold: (i) to consolidate the inferred topic by establishing an agreement between more than one topic discovery model; and (ii) to validate and verify the need for more than one topic discovery model in similar tasks. For example, poor correlation in the number of matched topics extracted by analysing the same tweets poses a question on the utility of incorporating a sole model for a topic discovery task. This aspect is further elaborated in the experimental results section. Features extraction and selection Understanding users' behaviour in the context of social networks is important to detect and identify various categories of users in these virtual environments. Spammers exhibit different behaviour than normal and legitimate users. This can be seen in different activities such as posting activities, following rate, quantity and quality of shared content and URLs, etc. In this study, we examine spammers behaviour both at the content level and at the user level. At the content level, we perform fine-grained analysis to users' content to examine and count all topics obtained from their textual content, hashtags and embedded URLs. Further, we listen to the voice of users' followers by conducting sentiment analysis to their followers' replies. At the user level, we examined the following to follower relationship incorporating the account's age. Also, other fine-grained metrics extracted from user metadata is examined and utilised in the endeavour of evaluating users' credibility. Features and mechanisms used to extract them are discussed in the following sections. Topic distinguishing mechanism The analysis of a user's content to discover the user's main topics of interest is an essential start to the process of measuring the user's credibility. In social networks, a user conveys to others his/her interest in a certain topic(s) by continuously posting tweets and attaching URLs with those tweets to discuss that particular topic(s). This category of users should be given a higher weight than the other category of users whose tweets discuss a various number of topics, thus their topics of interest can be difficult to be identified. To quantify this weight, the theoretical notion of TF-IDF has been used to distinguish users' topics of interest. TF-IDF is considered as a fundamental component embodied onto several Information Retrieval models, namely the vector space model (VSM) [76]. "The intuition was that a query term which occurs in many documents is not a good discriminator" [15]. This indicates that a word that occurs in several documents reduces its weight in the document and the corpus as this word does not help with the documents retrieval process as it does not reveal the document(s) of interest to the user [16]. TF-IDF is commonly used to quantify the significance of a word to a certain document in a collection of documents. It includes standard concepts which formulate its structure; Term Frequency (TF): is used to compute the number of times that a word appears in a certain document. TF, therefore expresses the importance of a word in the document; Document Frequency (DF): evaluates the significance of a word to a document in the overall corpus [77], and Inverse Document Frequency (IDF): is used as a distinguishing measure for a word in the text corpus to infer the term's importance in a certain document(s) [14]. Therefore, TF_IDF integrates the definitions of the significance of each index word in the document and the importance of the index term in the text collection to produce an aggregate weight for each word in each document. It assigns to a word a weight in document that is: (i) greatest when appears several times within a few numbers of documents; (ii) smaller when the word appears less times in a the document , or appears in several documents; and (iii) smallest when the word appears in all documents of the corpus. In the context of this research, this heuristic notion is assimilated into our framework to measure the credibility of users. Thus, it is argued that a user who posts on all topics has low credibility in general. This argument can be justified based on the following facts: (i) No one person is an expert in all topics [8]; (ii) A user who posts in all topics does not state or convey to other users which topic(s) s/he is interested in. The topic of interest can be conveyed by posting a wide range of content in that particular topic; (iii) There is a possibility that this user is a social spammer due to the behaviour of spammers posting tweets on several topics [9]. This could end up posting tweets on all topics which does not demonstrate a legitimate user's behaviour. Incorporated features A set of various features is used and extracted from both users' data (users' tweets) and their metadata (miscellaneous data collected from Twitter APIs). These features can be categorised into topic-dependent and topic-independent features. Table 2 shows the list of both topic-dependent and topic-independent features incorporated in this study along with their description. The count of words in each topic captured from the user's tweets. √ #Unique words The count of non-redundant words in each topic obtained from the user's tweets. √ #URLs The total count of URLs posted in user's tweets of each topic. √ #Unique URLs The total count of non-redundant URLs posted in user's tweets of each topic. √ #Unique URLs' hosts The total count of non-redundant URLs posted in user's tweets of each topic. √ Topic frequency The total number of topics the user has tweeted about. √ Inverse topic frequency Distinguishes users among the list of their topics of interest. It is computed as: = ( ) Where is the number of topics that a user discussed, is the topic frequency. √ #User's retweets The total number of retweets for user' content in each topic. √ #User's likes The total number of likes count for the users' content in each topic. √ #User's replies Embody the number of replies to the users' content in each topic. √ Sum of user positive sentiment replies Refers to the total sum of the positive scores of all replies to each user in each topic. Positive scores are achieved from IBM Watson NLU (sentiment analysis) with values greater than "0" and less than or equal to "1". The higher the positive score, the greater is the positive attitude the repliers have to the users' content. √ Sum of user negative sentiment replies Represents the sum of the negative scores of all replies to a user u in each topic. Negative scores are those values greater than or equal to "-1" and less than "0". The lower the negative score, the greater is the negative attitude the repliers have to the users' content. √ Users' followers The total count of users' followers. User's friends The total count of the user's friends (followees) Followers-friends ratio User followers-friends ratio. It is computed as: � − × 100 , − > 0 1 × 100 , − ≤ 0 , where is the age of user profile in years. #Hashtags The total number of hashtags in the user's tweets. #DistinctHashtags The total count of the unique number of hashtags in users' tweet. Topic-specific Hashtags Ratio Indicates the % between the number of hashtags mentioned in the tweets to the total number of tweets for each user in each topic. √ Experimental Results Dataset selection and exploration As discussed in the dataset source section, this study incorporates a set of users collected from two different sources, namely: the topically anomalous dataset [19] and the social honeypot dataset [2]. This experiment is undertaken on 4,000 users (2,000 from each source) amongst users with the highest number of tweets in each set. As depicted in the system architecture, Figure 1, all historical tweets of the selected users were collected and preprocessed as discussed in Section 3.1.2. Figure 3 shows the distribution of the collected tweets for the period between 2009 and 2020 which are related to the users obtained from the two datasets. In comparison with the continuous increase in the tweets count of the topically anomalous dataset, it is interesting to notice the decline in the posted tweets for users in the honeypot dataset. This can be interpreted as the latter dataset were intentionally containing polluters of social contents, thereby the number of social spammers of this dataset can be intuitively higher than the former dataset which contains users whose only suspicious feature is that the number of followers is less than 5,000. Therefore, we would expect this decline in posted tweets by users of the topically anomalous dataset as they can be detected by Twitter and thus banned from the Twitter community. This study supports this endeavour and provides another approach to detect this category of users that can hide from Twitter's radar. Figure 3: Tweets distribution between 2009 and 2020 for the two selected datasets All tweets and users' metadata of both datasets are examined through written python scripts which were used to shorten all embedded URLs, extract URLs' hosts and all hashtags, and conduct all the experiments of this study. Table 3 shows a range of statistics on the users and their metadata collected for this study. This contains the collected tweets with their replies/conversations, categories/topics captured for those tweets, #URLs and #unique host URLs embedded in the tweets. Also, both datasets contain the #words, #unique words, #hashtags, and #unique hashtags. All these figures are obtained from all historical tweets up to 2020. Topics extraction As discussed in Section 3, all historical tweets of all selected users were collected and examined through two topic discovery modules, namely IBM Watson NLU API as well as a developed classification model using 20-Newsgroup. IBM Watson NLU was used as an off-the-shelf API to infer the topic(s) of the examined preprocessed textual contents of the tweets and the websites of the embedded URLs. To consolidate the topics obtained by IBM Watson NLU API, we developed a topic classification model based on the 20-Newsgroup data set. In this task, the Python scikit-learn's implementation of textual classification using 20-Newsgroup 4 is used and enhanced. In particular, besides the Multinomial Naïve Based (Multinomial NB) classifier that is used by scikit-learn, two more classifiers are implemented, namely Logistic Regression (LR) and Stochastic Gradient Decent (SGD). Further, grid search for the optimal hyperparameters is carried out to find optimal and tuned parameters. The outcome of this experiment demonstrates that SGD outperforms other classifiers with an accuracy of 0.91 which is higher than the accuracy obtained by the scikit-learn's default implementation of Multinomial NB (≈ 0.77). Figure 5 illustrates the distribution of the tweets over topics captured using IBM Watson NLU for the same datasets. The next step is to capture the mapping between the two topic discovery approaches, thereby finding the tweets that match topics as illustrated in Figure 2 and discussed in section 3.2. Table 4 and Table 5 demonstrate the total number of tweets in each topic and corresponding topic using the incorporated domain discovery approaches on both datasets. Interestingly, these tables show a relatively poor correlation between domains captured using both topic discovery models. This divergence in capturing common topics in both models poses a question on the mechanism used by such well-known models to infer high-level topics. In future research, we will further examine this poor correlation by conduction a comprehensive study on similar high-level topic discovery models to verify the utility of such approaches in topic discovery designated tasks. The set of key features as depicted in Table 2 is extracted from users' data and metadata. The topic-dependent features are computed based on the 'Information Technology' topic. In future, we will carry out a cross-topic comparison between all features that are extracted based on all available topics. The Pearson correlation between incorporated features is computed as illustrated in Figure 6. The correlation matrix as depicted in Figure 6 is acceptable and most of the features are not highly positively correlated, thus no redundancy can be observed amongst features that might affect the employed models' performance. We incorporate lasso regularization [78] method to carry out automatic feature selection to enhance accuracy and interpretability. The collected users were then manually and carefully examined to label and identify those who demonstrate spamming behaviour, thereby establishing a ground truth data set for measuring the performance of the proposed system based on the extracted features. Figure 7 shows the percentage of spammers to nonspammers users in the collected and labelled datasets. As depicted in Figure 7, only 37.75 per cent of the examined users are labelled as spammer users even though the selected datasets embody users who should be highly exhibiting spamming behaviour. This indicates the significance of conducting further analysis on published spam datasets. The correlation depicted in Figure 8 demonstrates the importance of each feature and how each feature relates to the designated class label. As indicated in Figure 8 certain features play a significant role to infer those demonstrating spamming behaviour amongst the second category. For example, Topic frequency ( 6 ) shows that spammers tend to publish contents related to many topics of knowledge. Also, legitimate and spammers behaviours differ in terms of #retweets ( 8 ), #likes ( 9 ), and #replies ( 10 ). These conventional features have been continuously proving their utility in detecting users with spamming behaviours. The last three boxplots of Figure 8 namely, #followers ( 14 ) #friends ( 15 ), and followers-friend-ratio ( 16 ) demonstrate the importance of incorporating age of the user's profile. In particular, the median of both #followers ( 14 ) and #friends ( 15 ) are close in values -despite the observed outliers of non-spammer users which can be understood due to the fact that this is a legitimate category of users in which we can find users with a high number of followers. Yet the convergence in the median of #followers ( 14 ) and #friends ( 15 ) for both spammers and non-spammers categories poses a question on the utility of models that mainly rely on these features. To tackle this issue, we proposed the followers-friend-ratio ( 16 ) which involves the age of the profile in computing its value as depicted in Table 2. This ratio credits the older profiles as they likely belong to legitimate users, and the correlation depicted in the last boxplot of Figure 8 justifies this claim. To undertake further exploration, we also carry out bivariate distribution on 'topic frequency ( 6 )' feature to obtain a better understanding of the predictive power of topic frequency to the designated label. Figure 9 depicts dissimilar perspectives which conclude that the topic frequency feature is predictive. For example, the first chart of Figure 9 shows the densities of two categories in the dataset (spammers and non-spammers). These densities convey diverse distributions which imply pattern differences. It can be observed that most of the nonspammer users discuss an average number of topics which explains why the density of non-spammers is positioned in the middle of the chart. On the other hand, the spammers demonstrate an interest to post in few specific topics [79] or many topics [10], thus the density is positioned on the far-left and far-right sides of the chart. The subsamples in the middle chart (bins) establish the composition of various groups of topic frequencies. The dissimilarity in the proportion of spammers and non-spammers in each group validates again the significance of topic frequency as an important measure for prediction. Spammers tend to have a high topic frequency or low topic frequency which confirms the utility of incorporating this theoretically proven and intuitive aspect in this study. In particular, there is an inverse relationship between the number of topics the user is interested in and the user's credibility. This argument is justified based on the following facts: (i) there is no well-informed legitimate user who has the intellectual capacity to publish contents in all topics [8]; (ii) users who publish contents on various topics of interest do not convey to other users the particular topic that they are interested in. This is evident as topics of interest can be intuitively deduced from content of users who commonly post wide-ranging content in one or few topics; (iii) it is likely that this user is a spammer or anomalous; this illegitimate category of users tends to post tweets about numerous topics [9]. This could end up with tweets being posted on all topics which do not convey a legitimate user's behaviour. Therefore, distinguishing users in a set of topics is a significant aspect. The box plots appear in the third chart spot the behaviours of anomalies inferred from the dataset. These three charts prove the importance of topic frequency as an important variable in this study. density bins anomalies Figure 9: Bivariate distribution on 'topic frequency ( 6 )' Social spam system evaluation Evaluation metrics The proposed system framework can be incorporated to classify whether or not a user is anomalous. As depicted in Figure 10, the proposed system framework can be used to determine (classify) if a user is a spammer or nonspammer. Four scenarios are illustrated by the classification as (i) a true positive ( ): is an actual spammer user where the model correctly predicts it as a spammer user; (ii) a true negative ( ) is a non-spammer (legitimate) user where the model correctly predicts it as a non-spammer user; (iii) a false positive ( ) is a non-spammer user where the model incorrectly predicts it as a spammer user; (iv) a false negative ( ) is an spammer user where the model incorrectly predicts it as a non-spammer user. Although selecting a machine learning model for a certain classification task remains an ad-hoc process that requires further scrutiny, the selected models are well-proven learning algorithms and were selected due to their utility in various applications, especially in binary classification tasks [35,[80][81][82]. The performance of each of the implemented machine learning models in predicting multi-topic spammer users is measured based on theoretically proven key metrics, namely, accuracy, precision, recall and F-measure. • Accuracy: it is commonly used for determining the rate of correct classification obtained by the incorporated algorithm. It is computed as: = + + + + • Precision: indicates the proportion of correct anomalous predictions to those who are actually anomalous. Thus, it provides the rate of anomalous users who are classified correctly as anomalous. Precision can be calculated as: = + • Recall: indicates the proportion of actual anomalous who were classified correctly as anomalous. Recall is computed as: = + • F-score: conveys the trade-off between both Recall and Precision. Its formula is: − = . + = + ( + ) Accuracy is considered the most intuitive performance metric as it is used in this study to infer the ratio of correctly predicted observations to the total observations (samples). Accuracy is a good performance measure; thus a model of a high accuracy might indicate a well-performed model, yet this is true when the dataset is symmetric (class-balanced) and embodies an almost equal number of false positives and false negatives. The disparity between the number of spammers and non-spammers in our dataset requires incorporating Precision and Recall metrics. Precision indicates the ratio of the accurately predicted spammers to the total predicted spammers (accuracy of minority class predictions). This measure answers the question: Amongst all users labelled as spammers in the labelled dataset, how many users are actually spammers? A model of high precision conveys that it has a low false positive ( ) rate. Precision does not specify how many real spammers users were predicted as belonging to the non-spammers class -false negatives ( ). Recall or sensitivity deduces the ratio of accurately predicted spammers to all samples in the actual class. Recall answers the question: Amongst all users who are truly spammers, how many did we label? Neither recall nor precision can be solely used to evaluate the model performance in this binary classification task. Hence, F-Score is used that captures the properties of both recall and precision in one harmonic mean measure that is commonly used in training imbalanced datasets [83]. Experimental setting The incorporated machine learning models are implemented and their hyperparameters are tuned using a random search strategy as illustrated in Table 6. Cross-validation has been conducted using three different kfold values, namely 3,5 and 10. We found out that when k=10 the models perform best on an unseen dataset. We have not presented the results of each fold in each model due to the size of the paper and this is expected since a greater value of k commonly decreases the bias of the incorporated technique [84]. In each split, the total observations (i.e., 4000 users) are randomly divided into two datasets, namely the training dataset (80% of the total samples) and the validation dataset (20% of the total samples). Features to consider when observing for the best split. {"auto", "sqrt", "log2"} max_samples The number of samples to draw from original dataset none criterion On which attribute will be split. {"gini", "entropy"}, min_samples_leaf The minimum number of samples exist in the leaf node after splitting a certain node. The number of boosting stages to conduct. {100, 200} criterion Measure the quality of tree split. {"friedman_mse", "mse", "mae"} max threads Controls parallelism level of model building. minimal gain The gain of a node is calculated before splitting it 0.05 Naive Bayes (NB) var_smoothing Slice of the variance of all features that is appended to variances for control stability. 1e-9 laplace correction Prevents the occurrence of zero values. True Priors Prior probabilities of the classes. None Figure 11 shows the ROC curves for each fold (k=10 folds) in each incorporated classifier. It provides an aggregate measure and shows the true positive rates against the false-positive rates at various threshold settings. The Area Under the ROC curve -AUC demonstrates the probability that a certain classifier ranks a randomly chosen positive observation (spammer) higher than a randomly chosen negative one (non-spammer). As illustrated in Figure 11, MLFFNN , RFT, and GBT achieve the best ROC means among all other classification models. MLFFNN embodies a sophisticated underlying structure that can learn and make intelligent decisions on its own. RF is also known as it is less overfit and robust on prediction [85] and performs well on complex and nonlinear data [86]. GLM is usually flexible and capable of performing unconventional analysis and is often utilised to analyse categorical forecaster variables [87]. On the other hand, NB classifier shows relatively worse performance compared to other incorporated models. This is mainly due to some internal assumptions that might lead NB model to perform inadequately. NB commonly presumes the independence of features and is not able to cope with the interactions of these features. Performance comparison of Machine Learning models RF MLFFNN GBT MLFFNN GLM NB Figure 11: ROC curves of all folds for incorporated machine learning algorithms Error! Reference source not found. Figure 12 illustrates the performance comparison between all implemented classifiers respectively. It can be seen that despite some convergence on the performance of some of the models, MLFFNN performs better in the classification task of this study; around 14% of the dataset used for validating the experiment were classified incorrectly by MLFFNN. However, some other models such as NB algorithm for example wrongly classified more samples in the prediction validations. Figure 13 shows the highest estimated coefficient values computed for each feature using MLFFNN model. It shows that "topic frequency, x 6 " is the highest estimated coefficient. This indicates that x 6 has the highest impact when compared with the other features. This is due to the importance of this feature in distinguishing the topics of interest. In particular, users who demonstrate an interest in a broad range of topics or few numbers of topics do not commonly convey legitimate behaviour as discussed in section 3.3.1. values in topic frequency ( 6 ) and Inverse topic frequency ( 7 ) as well as users who obtain the lowest value in all other topic-dependent features as indicated in Table 2. • Topic-independent features (DIF): In this method, we implement a generic version of the proposed model. Therefore, we extract features of users with no consideration to users' topics of interest (i.e. extracted features are all topic-independent). The set of users retrieved based on this method are those who attain the lowest values in all extracted features ( 6 and 7 were excluded). • Low in-degree of topically anomalous (low in-degree TA) and social honeypot datasets (low indegree SH): The high number of followers, referred to high in-degree, was used in previous studies to indicate influential users as well as spammers [88,89]. We conduct this comparison on retrieved users who obtain the lowest number of followers (low in-degree) from the two selected datasets discussed in section 3.1.1. • Don't Follow Me (DFM) [88]: this is one of the well-known attempts to detect social spammers using both content-based features and graph-based features. The model incorporates a number of machine learning classifiers such as Decision Tree, Neural Networks, SVM, and Naïve Bayesian. The model proposed in [88] is implemented and the set of users who achieved the lowest values in all features are retrieved for comparison. Evaluation Metric: To report on the performance of each method, and due to the huge size of the dataset, we incorporate @ ( @ ). This measure is used as a weighted mean of precisions ( @ ) achieved at each arbitrary threshold . @ is computed based on how many spammers are present in the topretrieved users in each method. The average precision evaluation metric is computed as follows: @ = 1 ( ) �( @ × @ ) =1 Where refers to the true positives, ( ) is the total number of ground truth positives, and @ is an indicator function that equals '1' if the user at rank is a spammer, zero otherwise. Figure 14Error! Reference source not found. shows the retrieval precision of the topat 10, 20, 30, 40, 50 and the @ . As depicted in the figure, the evaluation results on the retrieved users of the proposed multi-topic social spammer detection verify the effectiveness of the developed approach to detect social spammer. This is evident as the topic-dependent features ( ) method overshadows other incorporated and proposed baseline models. For example, the first ten users retrieved by enquiring users who obtained low and high topic frequency ( ) were all exhibiting spamming behaviour. However, only three users of the first 10 retrieved users from the topically anomalous dataset, and whose in-degree features are the lowest, are spammers. Despite the good performance of the topic-independent features (DIF) method, along with don't follow me (DFM) approach, these approaches do not tackle the user's topic of interest, thus, the features are computed in general. However, the average precision accumulated using the proposed approach is promising for building spamming detection frameworks consolidated with the topic-based features proposed in this study. To add further clarification to the conducted benchmark comparison, we selected three true positive social spammers as identified by our model. Features of these users were extracted and the classification of these users based on other spammers classification approaches was inferred. As depicted in Table 7, low-in-degree models were unable to positively classify the users as spammers. This is evident as the idea of low-in-degree to identify spammers is built merely upon the followers-to-friends ratio ( ) which is not always an adequate mechanism. Legitimate users might also convey an interest in following other users while attaining a low number of friends. DFM model [88] is still capable to identify spammers although the followers-to-friends ratio is one of its core features. This is because the original algorithm incorporates other graph and content-based features which offer it further consolidation, yet it is inadequate to detect spammers with other sophisticated features. Although the generic version of our model (i.e., DIF) demonstrates the ability to capture two out of three spammers, its generic nature incapacitates it from detecting users who publish contents in various topics ( and ). On the other hand, our multi-topic social spammer detection model (i.e., DDF) is designed to be sensitive to capture such a spamming behaviour. Spammers attempt to gain credibility by establishing conversations addressing various topics, thereby persuading legitimate users to befriend them, or by hashtag hijacking or bulk messaging a certain topic or a variety of topics [1,4]. By incorporating the novel distinguishing measure, semantic analysis, sentiment analysis, and other fine-grained features, our model overshadows other existing approaches. For example, although Spammer3 has a relatively balanced followers-to-friends ratio, other features such as topic frequency ( ), Inverse topic frequency ( ), topic-based #retweet ( ), topic-based #likes ( ), sum of the positive replies ( 11 ), and sum of the negative replies ( ) conveys a suspicious behaviour. Model Comparison: Discussion In this paper, we have proposed a novel social spammer detection model for microblogging. We mainly focused on topic-dependent and topic-independent users' behaviours. We then conducted our experiments on the labelled twitter datasets using six popular machine learning algorithms (Random Forest Tree, Multi-Layer Feed-Forward Neural Network, Gradient Boosted Classifier, Generalised Linear Model, Decision Tree, and Naive Bayes). According to our experimental results, MLFFNN and RF approaches achieved the best ROC. The main reason behind the performance of MLFFNN in this experiment is the advantages that neural network-based approaches have compared to the traditional machine learning algorithms. They have their feature engineering capability and they focus on more relevant features for their predictions. Further, the implemented and trained RF model demonstrates an ability to capture spammers in the incorporated dataset. The good performance of the RF model is due to its adaptability and flexibility allowing it to search for the best features. RF operates on multiple and uncorrelated decision trees and the prediction of the model is based on the aggregated prediction values obtained by all trees which further consolidates the RF performance [90]. On the other hand, our analysis illustrates that NB has a poor prediction quality for this application. This is due to NB's too simplistic assumption that all features are independent. Although our experimental results demonstrate good performance for our proposed approach in this study, next we need to improve it as follows: • We will incorporate the shared textual contents by users in OSNs. There are a great set of features that we may extract from users' tweets/reviews/posts, e.g., their personality, emotional state, level of trustworthiness, and gender. • We will design a deep neural-based classifier to act as a spam detection approach. We plan to use Graph Convolutional Network (GCN), because it allows converting the data of users and their features to a graph-based environment, where users are the nodes, and their features are the nodes' features. • We will conduct further investigation on the users' credibility and trustworthiness level and incorporate them in our social spamming detection analysis. This will be attained by proposing context-aware social spammer detection model considering geographical location and temporal factors. Conclusion With the coronavirus COVID-19 pandemic continuing, the risk of spreading rumours, misinformation and spam through social media dominates. On the other side of the spectrum, efforts pursue to spot and stop spammers and their activities over the social networks. This paper attempts to consolidate these endeavours by implementing an effective and robust approach to detect social spammers. With a distinctive vector of features constructed by performing fine-grained topic-based analysis of users' contents, various machine learning algorithms are implemented and trained over a labelled dataset. The performance of these algorithms is evaluated using important evaluation indicators. Further, the model is benchmarked against existing techniques used to detect social spammers and anomalous users. The model overshadows the baseline models which verifies the utility of the proposed model in the designated task. In future work, we will incorporate the temporal dimension. Users' behaviours in OSNs may vary over time. It follows that their credibility changes as well; hence, the temporal factor should be considered. Further, a new topic-based graph model will be designed and implemented to proliferate the users' credibility throughout the entire social network. Therefore, we aim to study the structure of online social networks and consider integrating semantics of the textual content and the temporal factor. Figure 1 : 1System Architecture -prepared by the authors Figure 2 : 2Topic mapping between 20-Newsgroup and IBM Watson NLU-Categories Definition ( 1 ). 1Topic dependent features are the set of fine-grained features in which values are computed considering each topic of interest captured from the user's content. Definition (2). Topic independent features are the set of coarse-grained features in which values are computed with no consideration is taken regarding the user's topic of interest. Figure 4 4demonstrates the distribution of the tweets over each topic based on 20-Newgroup for two subsets: (A) Topically anomalous dataset, and (B) Social honeypot dataset, and Figure 4 : 4The distribution of the tweets in each topic based on 20-Newgroup for two subsets: (A) Topically anomalous dataset, and (B) Social honeypot dataset. Figure 5 : 5The distribution of the tweets in each topic based on IBM Watson NLU API for two subsets: (A) Topically anomalous dataset, and (B) Social honeypot dataset. Figure 6 : 6Correlation of incorporated features Figure 7 : 7The percentage of spammers to non-spammers in the labelled dataset Figure 8 : 8Correlation between selected incorporated features and class label. Figure 10 : 10Classification of spammers/non-spammers usersThe experiment of this study has been carried out incorporating various machine learning algorithms including Random Forest Classifier (RF), Multi-Layer Feed-Forward Neural Network (MLFFNN), Gradient Boosted Trees (GBT), Decision Tree (DT), Generalised Linear Model (GLM) and Naïve Bayes (NB). 5 n_estimators 5Specifies the number of trees in the forest of the model. {1, 2, 3}Multi-Layer Feed-Forward Neural Network (MLFFNN) activationThe function used in the hidden layers . {"Rectifier", "Tanh", "Maxout", "relu"} neurons/layerNumber and size of the hidden layer. {50, 100, 200} #epochs Iteration times over dataset. {10, 50} adaptive rate Unifies the benefits of momentum training and learning rate annealing {0.1, 0.2, 0.3} mean learning rate A non-negative scalar indicating step size 0.003772 L1 Regularization (absolute value of the weights) 1.0E-5 L2 Regularization (sum of the squared weights) 0.0 Loss function loss (error) function {"Quadratic", "CrossEntropy" Gradient Boosted Classifier (GB) #trees Number of generated trees. {10, 20, 30} loss loss function to be optimized. {"deviance", "exponential"} learning_rate Assist in reducing the contribution of each tree. 0.1 n_estimators Figure 12 : 12Performance comparison of six classifiers to predict multi-topic spammers Figure 13 :• 13Highest Positive Coefficients 4.3.4 Social spam detection models -a baseline comparison We undertake benchmark comparison with other spam detection approaches as well as with different versions of the proposed model over the curated labelled dataset. The list of models used in this comparison as well as the set of users retrieved from implementing each model are discussed as follows: Topic-dependent features (DDF): This method selects the set of users who attain the lowest and highest Figure 14 : 14Models comparison performance Table 1 : 1An example of incorporating IBM Watson for taxonomies inference and sentiment analysis"Even as we focus on fighting COVID-19, it's important to recognize that there's another pandemic raging right now-one that's decades in the making and unique to the United States. We need to treat gun violence with the same urgency and resolve."3 Tweet Categories Inference Sentiment Analysis Category Score Sentiment Score /health and fitness/disease/cold and flu 0.909 positive 0.53 /health and fitness/disease/epidemic 0.77 /society/unrest and war 0.65 Table 2 : 2Incorporated Features (domain-dependent and domain-independent) No. Feature Description Topic Dependent #Words Table 3 : 3Statistics of the collected datasetsData Source #users #tweets #tweets replies/ conversations #URLs #unique URLs #unique host URLS #words #unique words #hashtags #unique hashtags Topically anomalous dataset [19] 2,000 6,270,257 114,481 1,238,395 307,855 40,133 83,432,897 19,178,212 1,904,985 403,272 Honeypot dataset [2] 2,000 5,463,761 752,297 1,451,511 457,489 147,342 75,763,023 24,052,466 2,061,820 865,734 Table 4 : 4The tweets distribution over domains captured using two topic discovery approaches on the topically anomalous dataset.IBM Watson NLU topics #Tweets Corresponding topcis(s) using 20-Newsgroup #Tweets #Matched Tweets %Matching technology and computing 1,353,296 comp.graphics 305,921 473,627 35% comp.os.ms-windows.misc 183,961 comp.sys.ibm.pc.hardware 326,508 comp.sys.mac.hardware 197,839 comp.windows.x 310,996 sci.electronics 374,253 business and industrial 636,181 misc.forsale 593,046 60,126 10% automotive and vehicles 422,810 rec.autos 311,467 105,119 25% rec.motorcycles 524,771 sports 814,351 rec.sport.baseball 446,359 257,998 32% rec.sport.hockey 594,955 science 269,721 sci.crypt 216,677 64,203 24% sci.med 1,160,796 sci.space 313,337 news 226,418 talk.politics.guns 210,976 18,595 8% talk.politics.mideast 304,039 law, govt and politics 727,755 talk.politics.misc 283,544 72,693 26% religion and spirituality 200,769 talk.religion.misc 92,224 75,760 38% alt.atheism 74,758 soc.religion.christian 786,435 Table 5 : 5The tweets distribution over domains captured using two topic discovery approaches on the social honeypot dataset.IBM Watson NLU topics #Tweets Corresponding topcis(s) using 20-Newsgroup #Tweets #Matched Tweets %Matching technology and computing 1,510,056 comp.graphics 260,751 465,558 81% comp.os.ms-windows.misc 179,389 comp.sys.ibm.pc.hardware 132,104 comp.sys.mac.hardware 138,384 comp.windows.x 240,945 sci.electronics 277,344 business and industrial 895,182 misc.forsale 593,169 85,194 14% automotive and vehicles 498,293 rec.autos 263,877 137,915 28% rec.motorcycles 282,194 sports 757,087 rec.sport.baseball 262,202 199,620 31% rec.sport.hockey 373,535 science 196,987 sci.crypt 133,314 48,608 25% sci.med 791,032 sci.space 243,509 news 225,478 talk.politics.guns 125,208 17,108 8% talk.politics.mideast 166,428 law, govt and politics 507,471 talk.politics.misc 257,609 44,818 17% religion and spirituality 207,714 talk.religion.misc 56,162 66,791 32% alt.atheism 48,239 soc.religion.christian 449,312 Table 6 : 6Hyperparameter settings Hyperparameter Description Examined Values (Optimal values are in bold format) Random Forest Tree (RFT) max_depth The longest route between the root and the leaf of the tree. {10 , 20, 30, 40, 100} min_sample_split The minimum number of observations that are required in any given node in order to split it. {2 , 4, 6} max_terminal_nodes Restricts the growth of the tree. none max_features Table 7 : 7Features extracted for three identified social spammers.Spammer1 Spammer2 Spammer3 https://github.com/farmisen/acTwitterConversation 2 https://pawsey.org.au/ This is a tweet by Barak Obama on 24/02/2021 https://scikit-learn.org/0.19/datasets/twenty_newsgroups.html Uncovering social spammers. 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[ "A Survey on Semantic Communications for Intelligent Wireless Networks", "A Survey on Semantic Communications for Intelligent Wireless Networks" ]	[ "Sridhar Iyer [email protected] ", "Rajashri Khanai ", "Dattaprasad Torse ", "Rahul Jashvantbhai Pandya ", "Khaled Rabie ", "Krishna Pai ", "Wali Ullah Khan ", "Zubair Fadlullah ", "\nDepartment of ECE\nDepartment of ECE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology\nDepartment of CSE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology\nDepartment of EE, Indian Institute of Technology-Dharwad, WALMI\nDepartment of Engineering\nKLE Dr. M.S. Sheshgiri College of Engineering and Technology\nCampus, PB RoadBelagavi, Belagavi, BelagaviKarnataka, Karnataka, Karnataka, KarnatakaIndia, India, India, India\n", "\nDepartment of ECE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology\nInterdisciplinary Centre for Security, Reliability and Trust (SnT)\nManchester Metropolitan University\nManchester, BelagaviKarnatakaUK., India\n", "\nDepartment of Computer Science\nUniversity of Luxembourg\nATAC Building, Room AT50241855Luxembourg CityLuxembourg\n", "\nLakehead University\nP7B 5E1ONCanada\n" ]	[ "Department of ECE\nDepartment of ECE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology\nDepartment of CSE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology\nDepartment of EE, Indian Institute of Technology-Dharwad, WALMI\nDepartment of Engineering\nKLE Dr. M.S. Sheshgiri College of Engineering and Technology\nCampus, PB RoadBelagavi, Belagavi, BelagaviKarnataka, Karnataka, Karnataka, KarnatakaIndia, India, India, India", "Department of ECE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology\nInterdisciplinary Centre for Security, Reliability and Trust (SnT)\nManchester Metropolitan University\nManchester, BelagaviKarnatakaUK., India", "Department of Computer Science\nUniversity of Luxembourg\nATAC Building, Room AT50241855Luxembourg CityLuxembourg", "Lakehead University\nP7B 5E1ONCanada" ]	[]	With the upsurge in next-generation intelligent applications and exponentially increasing heterogeneous data traffic both, the industry and the research community have already begun conceptualizing the 6G technology. With deployment of the 6G technology, it is envisioned that the competitive edge of wireless technology will be sustained and the next decade's communication requirements will be stratified. Also, the 6G technology will aim to aid the development of a human society which is ubiquitous, intelligent, and mobile, simultaneously providing solutions to key challenges such as, coverage, capacity, providing intelligence, computing, etc. In addition, 6G technology will focus on providing the intelligent use-cases and applications using higher data-rates over the millmeter (mm) waves and Tera-Hertz (THz) frequency. However, at higher frequencies multiple non-desired phenomena such as atmospheric absorption, blocking, etc., occur which create a bottleneck owing to resource (spectrum and energy) scarcity. Hence, following the same trend of making efforts towards reproducing at the receiver, the exact information which was sent by the transmitter, will result in a never ending need for higher bandwidth. A possible solution to such a challenge lies in semantic communications which focuses on the meaning (relevance/context) of the received data as opposed to only reproducing the correct transmitted data. This in turn will require less bandwidth, and will reduce the bottleneck due to the various undesired phenomenon. In this respect, the current article presents a detailed survey on the recent technological trends in regard to semantic communications for intelligent wireless networks. We focus on the semantic communications architecture including the model, and source and channel coding. Next, we detail the cross-layer interaction, and the various goal-oriented communication applications. We also present the overall semantic communications trends in detail, and identify the challenges which need timely solutions before the practical implementation of semantic communications within the 6G wireless technology. Our survey article is an attempt to significantly contribute towards initiating future research directions in the area of semantic communications for the next-generation intelligent 6G wireless networks.	10.1007/s11277-022-10111-7	[ "https://export.arxiv.org/pdf/2202.03705v2.pdf" ]	246,652,329	2202.03705	ea3a70a99ee031314e4bc3fc193b5479d9ec41e4	A Survey on Semantic Communications for Intelligent Wireless Networks Sridhar Iyer [email protected] Rajashri Khanai Dattaprasad Torse Rahul Jashvantbhai Pandya Khaled Rabie Krishna Pai Wali Ullah Khan Zubair Fadlullah Department of ECE Department of ECE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology Department of CSE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology Department of EE, Indian Institute of Technology-Dharwad, WALMI Department of Engineering KLE Dr. M.S. Sheshgiri College of Engineering and Technology Campus, PB RoadBelagavi, Belagavi, BelagaviKarnataka, Karnataka, Karnataka, KarnatakaIndia, India, India, India Department of ECE, KLE Dr. M.S. Sheshgiri College of Engineering and Technology Interdisciplinary Centre for Security, Reliability and Trust (SnT) Manchester Metropolitan University Manchester, BelagaviKarnatakaUK., India Department of Computer Science University of Luxembourg ATAC Building, Room AT50241855Luxembourg CityLuxembourg Lakehead University P7B 5E1ONCanada A Survey on Semantic Communications for Intelligent Wireless Networks Manchester Institute, Manchester, UK (TO BE ADDED)Semantic communications6G wireless networkssemantic learningartificial intelligencemachine learningintelligent networks With the upsurge in next-generation intelligent applications and exponentially increasing heterogeneous data traffic both, the industry and the research community have already begun conceptualizing the 6G technology. With deployment of the 6G technology, it is envisioned that the competitive edge of wireless technology will be sustained and the next decade's communication requirements will be stratified. Also, the 6G technology will aim to aid the development of a human society which is ubiquitous, intelligent, and mobile, simultaneously providing solutions to key challenges such as, coverage, capacity, providing intelligence, computing, etc. In addition, 6G technology will focus on providing the intelligent use-cases and applications using higher data-rates over the millmeter (mm) waves and Tera-Hertz (THz) frequency. However, at higher frequencies multiple non-desired phenomena such as atmospheric absorption, blocking, etc., occur which create a bottleneck owing to resource (spectrum and energy) scarcity. Hence, following the same trend of making efforts towards reproducing at the receiver, the exact information which was sent by the transmitter, will result in a never ending need for higher bandwidth. A possible solution to such a challenge lies in semantic communications which focuses on the meaning (relevance/context) of the received data as opposed to only reproducing the correct transmitted data. This in turn will require less bandwidth, and will reduce the bottleneck due to the various undesired phenomenon. In this respect, the current article presents a detailed survey on the recent technological trends in regard to semantic communications for intelligent wireless networks. We focus on the semantic communications architecture including the model, and source and channel coding. Next, we detail the cross-layer interaction, and the various goal-oriented communication applications. We also present the overall semantic communications trends in detail, and identify the challenges which need timely solutions before the practical implementation of semantic communications within the 6G wireless technology. Our survey article is an attempt to significantly contribute towards initiating future research directions in the area of semantic communications for the next-generation intelligent 6G wireless networks.  Discussion and exploration of the state of art advancements towards obtaining a semantic communication architecture relevant for the wireless networks.  Devising a taxonomy, as shown in Fig. 1, of the semantic communication enabled wireless networks via crucial enablers, use-cases, AI/ML schemes which are emerging, and technologies pertaining to communication, networking and computing.  Presenting and discussing multiple current and future research challenges, and their possible solutions with an outlook towards enhancing research in regard to semantic communications for intelligent wireless networks. The rest of the article is organized as follows. Section II details the need for incorporating the semantic and effectiveness problems within a novel architecture for facilitating efficient designs for cross-layer interactions with simultaneous focus on goal-oriented communications. Here, we detail the semantic architecture, and source and channel coding models. In Section III, we detail the cross-layer interaction for semantic communications enabled systems. Section IV describes the various state-of-art applications of semantic communications. Section V presents the various research activities which have been conducted concerning the semantic communications in wireless networks. The open research challenges with potential directions are presented in Section VI. Finally, the conclusions are presented in Section VII. II. Semantic Communication Architecture With a move towards the 5G technology, wireless networks are evolving towards a system which involves both, communication and computation wherein, edge cloud supports the tasks requiring communication, computation, and control. In effect, these advanced tasks are equipped to sense, compute, control, and actuate, so as to lay a foundation for the intelligent machines. The 6G wireless networks will further ensure a drastic change by significantly advancing the infrastructure for communication and computation. This will mainly be a result of the increasing pervasive use of the AI/ML tools within all layers of the network which will need a joint coordination of the resources required for computation, communication, and control. As a result, there will be a tremendously fast accumulation of the data which will require almost ideal filtering, transmission, and processing via intelligence of both, natural of intelligent type. Hence, considering the limits on the resources which are available, the key challenge will be the designing of a new network which is able to deliver the next-generation intelligent services without requiring an exponential increase in the capacity, computation, storage, and energy. Also, observing the capacity limits already reached due to the existing and emerging service requirements, it is evident that operating at higher data-rates to achieve larger bandwidths is not the solution to cater to the needs of the advanced intelligent services. As an alternative, to ensure that the wireless network is qualitatively more efficient, instead of efforts towards increasing the resources, there is a need to ensure that the network is capable of more intelligently utilizing the limited resources. The solution to the aforementioned challenge exists in the work conducted by Shannon [8] which, in addition to the Level 1 or technical problem aiming to only accurately transmit the data, articulates the Level 2 or semantic problem focusing on the precise meaning conveyed by the transmitted data, and the Level 3 or effectiveness problem which overviews the impact of the received meaning in a desired manner. The incorporation of the aforementioned three levels together will ensure that the new services demanding an interconnection of the humans and the machines will possess various intelligence degrees. In addition to the technical level, which exists at the bottom of current communication protocol stack, new stack will also include the semantic and the effectiveness levels. With the aforementioned, the new architecture, shown in Fig. 2, consists of a protocol stack which includes all the three levels with Level 1 occupying the bottom layer. In 6G technology, in addition to NFV and SDN, the Level 1 layer will rely heavily on network virtualization which will be able to distribute the tasks involving communication and computation to the virtual machines [12], and this can be further coordinated efficiently through Level 2 and Level 3. On top of the Level 1 layer, the Level 2 layer is placed which will provision such services which require and have a set role for semantics. The Level 3 layer placed on top of Level 2 will ensure that the lower levels function in a coordinated manner so that the available resources can be optimized in their usage and the important performance parameters also achieve the desired values. For instances which do not require any Level 2 aspects, Level 3 will be able to have a direct interaction with Level 1. Such a new protocol stack will enable the layers to conduct an effective cross layer interaction which will be key to allocate the available resources. Specifically, Level 2, if implemented accurately, will ensure that the data which has be transmitted will be interpreted correctly by the receiver irrespective of whether the entire data is decoded without errors. This will in turn offer significant improvement in the performance levels as the communication will now depend on the sharing of a-priori information between the transmitter and the receiver. In regard to Level 3, the aim will be to ensure that communication occurs to achieve a specific goal via correct implementation of the optimized resources with least latency. For this, Level 3 will use the resources available at Level 1 and will coordinate with Level 2, if and when required. In view of the Level 1 problem, multiple studies have presented alternate methods from an information theory viewpoint which are consolidated in [13,14]. There also exist studies in regard to semantic communication system [11,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. In what follows, we present a summary of the most recent studies, and in Table 1, we present the consolidated review of all the studies which exist in literature in regard to the semantic communications. In view of extended connectivity, authors in [11] have extended the scope of semantic communications to design both, the objectives and the constraints. Authors have encompassed semantic information of transmitted data for any application/use case, and have provided a platform to obtain semantic-aware connectivity solutions via a semantic-effectiveness plane. This enhances the existing stack of protocol by provisioning interfaces which follow standards so as to enable the filtering of data and to directly control the function(s) at all the layers of protocol stack. The authors have demonstrated that the semantic-effectiveness plane replaces current architecture in wireless networks with a framework which demonstrates performance enhancements. Lastly, authors have presented open research problems in regard to cross-layer recovery mechanisms, and security. In [15], the authors have proposed and analysed an end to end framework for communication via semantics which incorporates issues of inference of semantics and communication via the physical layer. The authors have considered the semantic aspect by considering the similarities which exist between individual words. The authors in [16] have envisioned the data semantics as foundation for the process of communication. The authors have pointed out that such a foundation will ensure that data is generated and transmitted with an aim of goal-oriented communication. By taking advantages of the semantics empowered sampling and communication policies, the authors have shown that significant minimization of both, reconstruction error and actuation error cost can be achieved in addition to the generated number of uninformative samples. In [17], the authors have used a deep neural network (DNN) for learning and jointly encoding a semantic channel encoder under the consideration of similarities which exists between the complete sentences. As a key aspect, the study focuses on recovering the transmitted message's context/meaning instead of trying to prevent data (bits/symbols) errors. The authors have proposed a novel metric namely, sentence similarity for justifying performance of semantic communications. It is shown that, compared to exiting communication technique, the proposed method achieves higher performance. The authors in [18] have designed a deep learning (DL) based system for semantic communication in regard to speech signals. For improving speech signals' accuracy to recover, the proposed system operates on an attention mechanism by resorting to the use of a squeeze and excitation network. For enabling system to tackle dynamic channel environments, the authors have proposed a generic model for coping with multiple channel conditions without requiring any re-training. It is demonstrated that proposed system offers better performance compared to traditional communications, and also shows higher robustness to variations in the channel. In [19], the authors have proposed a model to investigate the issue of audio semantic communications over the wireless networks. The model uses semantic communication methods for transmitting, to server, audio data which is large sized through wireless edge devices. For extracting semantics from audio data, authors propose an auto-encoder based on wave to vector architecture which uses convolutional neural networks (CNNs). For further improving accuracy of extracted semantic information, the authors have implemented federated learning (FL) technique considering many devices and one server. It is demonstrated that new technique converges efficiently and is able to minimize mean squared error of the audio transmission by approximately 100 times in comparison to the traditional schemes of coding. The authors in [20] have proposed a route to boost network capability(s) method in view of enabling 6G wireless network. The authors have (i) conceived a novel semantic framework, named as semantic base, and (i) established an intelligent and efficient semantic architecture which integrates AI and network technologies for enabling intelligent interactions among various 6G communication objects. The authors have also presented a survey of recent advances in semantic communications simultaneously highlighting potential use cases. In [21], the authors have discussed the relationship between semantic communications and IoE, and have introduced the basic models and fundamental components of semantic communications. The authors have discussed the major limitations of the point-to-point semantic communications, and have hence put forth that a knowledge sharing and resource convergence enabled semantic communication networking is ideal to support the future massive scales of IoE systems. In addition, the authors also discuss the basic components of the semantic communication networking system, and a federated edge intelligence enabled semantic communication networking architecture is investigated as a case study. The results demonstrate the potential of semantic communication networking to further minimize the resource demand(s) and result in an improvement of the semantic communication efficiency. Lastly, the authors have discussed the open problems for future research. The authors in [22] have explored the opportunities which are offered by the semantic communications for the next generation wireless networks. Specifically, the authors have focused on the benefits of semantic compression, and have presented a detailed new architecture which aids in enabling the semantic symbols in view of effective semantic communications. The authors have also discussed the theoretical aspects and have designed the objective functions which aid in learning the effective semantic encoders and decoders. Lastly, the authors demonstrate promising results considering transmission of text scenario when transmitter and receiver converse in varied languages. In [23], the authors have reviewed the classic semantic communication frameworks following which they have summarized the key challenges which hinder the popularity of semantic communications. The authors observe that few semantic communication processes result in excessive resource consumption and are hence inefficient. As a solution, the authors have proposed a new architecture which is based on intelligence via federated edge for supporting a semantic enabled network which is resource efficient. The proposed architecture provides security to the model related information by coordinating through the intermediate results. The obtained results demonstrate that proposed architecture is able to minimize resource consumption simultaneously demonstrating significant improvement in the communication efficiency. The authors in [24] have introduced the concept of physical elements including genie, which also features with the 6G technology concept. The authors have focused on genie realization in view of intelligent transmission/access within the 6G networks in conjunction with semantic information theory and AI joint transceiver design. Further, AI is integrated with transmitter and receiver design including multiple granularities, and a complete end to end AI transceiver is designed to optimize parameter via DL within estimation of channel, detection of signal, etc. The authors demonstrate that a genie enabled wireless system operates with high intelligence and demonstrates better performance compared to manual control. Lastly, the authors present a related comprehensive survey and provide a scope for future research with relevant suggestions. In [29], the authors have incorporated the data semantics in a system of networks. Further, the authors have developed advanced semantic metrics, an optimal sampling theory, semantic compressed sensing techniques, and generation of data which is semantic aware, coding for channel, etc. It is demonstrated that proposed architecture enables to generate appropriate data amount and to transmit correct content to apt place at the correct duration. This is possible via a conjunction of process involving redesign of data generation, transmission and utilization. Overall, authors have concluded that semantic networks require a combination of concepts and tools which are developed separately. Lastly, as a scope of further research, authors have identified the following issues: scheduling, random access, packetization, and operation at higher levels. The authors in [30], have implemented semantics for solving the issues of spectrum and energy by proposing a framework for transmission under high semantic fidelity. The authors have introduced the framework for semantic fidelity for improving the efficiency following which they have introduced transforming of semantics to convert input to semantic bits/symbols. In comparison to the existing transforms, the proposed transform incurs data loss which in turn saves on bandwidth simultaneously provisioning high semantic fidelity. The authors have conducted performance evaluation under the consideration of semantic noise and have presented an audio transmission case study for the evaluation of effectiveness. Lastly, the authors have discussed various applications and multiple open research problems. In the following sub-sections, firstly, we present a detailed description on the semantic communications model and representation following which, the source-and channel-coding are described. The reader must note that in what follows, the term 'semantic data' refers to the meaning associated with the data, and the term 'syntactic data' refers to the data's probabilistic model which is used to encode the data [24]. Identify the specific resource whose usage the language wants to optimize. [28] General theory of goal oriented communication Proposed model accounts for all the aims. Identify concept of sensing which will allow goals to be achieved even where there is any misunderstanding Any communication aim is modelled mathematically via a referee. When user senses progress, communication aim can be achieved despite initial misunderstanding. Sensing may be required to overcome initial misunderstanding. Focusing on the aim helps in detection and possible correction of the misunderstanding. [29] Incorporate the data semantics to communicate and control in a networks. i. Model and Representation A semantic communications model is efficient if it ensures the correct communication between source and receiver provided that the receiver (i) recovers true meaning of the data sent by the transmitter from the data which has been received, and (ii) increases the related knowledge from the data which has been received. The aforementioned implies that there will occur a semantic equivalence when the receiver infers the same meaning from the received data as was intended by the transmitter. Such an inference requires a model which deviates from the level 1 aspect of the Shannon's theory in the following manners: (i) semantic content decides the data amount rather than the probability of symbol generation for encoding the data, (ii) exact content of the data is important rather than the mean data which has an association with the most likely information that can be transmitted, and (iii) in addition to the information which is conveyed by the data depending on the data itself, it also depends on the knowledge level which is available at the transmitter and the receiver. Further, as semantic communication is related to the meaning of the data, a system of knowledge will be required which will formally represent knowledge associated with the semantics of the data. This representation of the knowledge, which is the base of AI mechanisms [31], will aim to efficiently represent and interpret the data via appropriate definitions of the information carried by the data thereby, also possibly creating new knowledge. Therefore, a base for knowledge representation and interpretation will be required at the transmitter and the receiver which may differ from one another. It must be noted that for the base of knowledge, incomplete description will be a key feature which will distinguish it from a database. The incompleteness will arise mostly owing to the constraints posed by computations since to completely finish reasoning will consume large amounts of time. Also, data will be correctly received only when the associated meaning is same as was intended by the transmitter, or it results in a value addition of the base of knowledge. Further, the received data will be depended only on the semantics rather than the syntactics which implies that the data will have multiple encoding options which could give rise to similar semantic meaning which is an open problem for research. ii. Source and Channel Coding The multiple level communication flow diagram is shown in Fig. 3 which shows the three layers in regard to three communication levels viz., level 1, level 2, and level 3. At layer 3, there will be a transmitter and a receiver which will interact with one another via an environment. The nodes may correspond to humans or machines, also referred to as the 'agents', which will operate as detailed in [31]. These agents may also be rational by implementing an interaction which may include data exchange, control, sending, etc. Specifically, for interaction, the transmitter will generate a message T M m  (where T M represents a transmitter alphabet), following the rules set by the base of knowledge which exists at the transmitter, such that it will deliver the intended meaning to the receiver. Further, for physically conveying the data to the receiver through the channel, the data ( m ) will be converted to a symbols sequence, S s  (where S represents the alphabet of the symbols), which will then be converted to physical signal which is apt to propagate over channel. Also, it must be noted that the mapping from S M T  , which is denoted as ) (m f s  , will always not be a one-to-one mapping, and could also be a one-tomany mapping since any data is represented via many symbols which convey similar meaning. The aforementioned will result in ambiguity which in one of the key research challenges in the domain of Natural Language Processing (NLP) [32]. Following the Shannon's theorem, a source encoder is required to translate s to m such that redundancy within the data can be minimized. This will be followed by a channel encoder, which will implement only the most required redundancy, which will aim to increase the reliability of the communication. It must be noted that this combination of the aforementioned encoder for source and channel will form syntactic encoder since it will only focus on correctness of data and not the related meaning (semantics). Lastly, complete sequence s is converted to physical signal which is compatible with the channel being used for propagation. Consider a semantic encoder in which the transmitter is random and transmits the data with a probability T T M m m pM  , ) ( . Therefore, following the Shannon's theory, the data entropy will be given as     T M m T T T m pM m pM m H ) ( log ) ( ) ( 2 . (1) Further, following the study in [25], the probability that a transmitter is able to transmit a symbol s can be evaluated as     T M m m f s m T T m pm s p ) ( : ) ( ) ( .(2) where, the probability of a transmitter being able to transmit a message m is denoted by ) (m pm T . Thus, for any symbol i s , the corresponding semantic information is given as ) ( log ) ( 2 i T i T s p s H   .(3) and for the symbols transmitted by the transmitter, the corresponding semantic entropy is given as     S s i T i T i s p s p S H ) ( log ) ( ) ( 2 .(4)) / ( ) / ( ) ( ) ( S M H M S H M H S H T T T T    .(5) where, ) / ( M S H T is entropy of S conditioned by M whereas, ) / ( S M H T is entropy of M conditioned by S. However, in semantic communications, the aim of source coding is to preserve the meaning (semantics) of the data rather than the bits/symbols sequence which is generated by the transmitter. With this understanding, ) / ( M S H T denotes the semantic redundancy since it will differ from zero only when many bits/symbols which are associated to a similar message exist, and ) / ( S M H T denotes the ambiguity in semantics since it will differ from zero only when many contexts associated to similar bits/symbols exist. Further, the study in [26] has shown that for such a scenario, there will exist a semantic encoder which will require, on an average, ) ; ( S M I number of bits for encoding data which is transmitted by the transmitter. Hence, the mutual data which exists between the transmitted messages and the transmitted symbols is denoted as ) / ( ) ( ) / ( ) ( ) ; ( M S H S H S M H M H S M I T T T T     .(6) Also, few practical semantic transmitter encoders proposed in [26] exploit the shared knowledge between the transmitter and the receiver. Overall, as the main aim, semantic encoder detects and extracts meaning from the transmitted data and then compresses or removes information which is not relevant. To do so, initially, through the knowledge existing at transmitter and receiver, the encoder will have to identify the related entities from the transmitted image/text following which, it will have to infer the closest relationship in accordance with a common model. At the receiver end, the signal r which is received is decoded syntactically for generating ' s symbols sequence. Next, depending on the base of knowledge which exists at the receiver, ' s is interpreted to generate the message ' m . From the aforementioned, having received the signal r , aim of semantic decoder is the recovery of message ' m which is same as transmitted message m . Equivalence here will imply that ' m and m deliver exact same meaning and not necessarily same structure. Hence, decoder will have to interpret the information which is transmitted by the transmitter, and then recover the signal which is received in a form which is understood by user at receiver. Further, decoder will aim to evaluate satisfaction level of user at the receiver based on which it will decide the success of the semantic data which has been received. However, as in the syntactic communication case, there will be sources of errors in the semantic communications case also. The error will occur at the semantic level if ' m is unequal to m , and at the syntactic level if ' s differs from s . Further, at the syntactic level, the main source of error will random noise or interference during the transmission or propagation of the signal whereas, at the semantic level, the error will mainly occur if the base of knowledge at the transmitter and receiver differ or if there is misinterpretation of the data. This implies that semantic layer is reliant on the syntactic layer wherein, multiple errors in received signal decoding may affect data recovery. However, this does not necessarily imply that there will errors in the interpretation of the data i.e., errors at the syntactic level do not necessarily result in errors within the semantic level since, even with few errors, the interpreter at the semantic level can still recover the meaning from the received data by exploiting the base of knowledge which exists at the receiver. Also, there could occur no errors at the syntactic layer but at the semantic layer when the received message is decoded correctly; however, it is not interpreted appropriately due to the difference in the base of knowledge which exists at the transmitter and the receiver. The aforementioned is demonstrated as follows: let there be a channel which is modelled such that the conditional probability is ) / ( s r p to receive r when s has been transmitted. For semantic communication, the aim is to recover the meaning and not the symbol s which was transmitted, and hence, a semantic decoder can be used which will select ' m such that it can increase (or maximize) the a-posterior probability which is conditioned to the received symbol, and is given as      s m f s m m f s m r s m p r m p m ) , , ( max arg ) / ( max arg ) ( : ) ( : ' .(7) Further, using property of Markov i.e., ) / ( ) , / ( s r p s m r p  , In the aforementioned, the values of ) (m p and ) / ( s r p are known, and optimizing the complete performance of the system will involve a searching of that ) / ( m s p which will reduce the probability of the semantic error under the multiple physical layer constraints. In fact, the ) / ( m s p value has a key role within the functioning of the semantic encoder wherein, in the case of few errors at the syntactic level will imply that the performance can be significantly improved via semantic decoding as most of the received data can be corrected using the base of knowledge at the receiver using natural of artificial intelligence. III. Cross Layer Interaction With increased usage of real-time applications over the wireless networks, high performance and efficient quality of service (QoS) are essential to the end user. Hence, existing TCP/IP method will not suffice in view of meeting the increasing heterogeneous demands, and therefore, much research must be focused on various cross-layer design approaches for increasing the overall performance and QoS by allowing data sharing across the various layers. To meet the increasing demands, timely exchange of the data across layers, and periodic reconfiguration of modes due to increased mobility cross-layer designs (CLD) have been proposed [33]. It is a common misconception that CLD destroys the traditional 5-layer architecture; however, CLD also works in conjunction with the layered architecture to enable cross-layer communication between non-adjacent layers. Using CLD, the layers may share parameters and internal details to enable effective troubleshooting of the root cause with lesser processing power and reduced cost. Also, CLD can achieve reduced latency, increased throughput, and lesser error rate which are vital for next-generation wireless communication applications [33]. The coordination plane is a model used to showcase the issues which can be solved by CLDs and the corresponding advantages. Here, each coordination plane represents an issue which can be solved by CLD. Four different types of coordination plane are labelled, as shown in Fig. 4.  Security: Layered TCP/IP protocol leads to multiple levels of encryption across various network layers with increased processing and costs.  Quality of Service: The coordination plane emphasizes on increased QoS which is possible by sharing certain information across the non-adjacent layers also.  Mobility: CLD is built to provide seamless and uninterrupted communication in situations of augmented mobility.  Wireless Link Adaptation: This coordination plane emphasizes on reducing the bit-error rate (BER) and channel fading in wireless networks using the CLDs. Further, as detailed in [34], a probable solution to CLD is to conduct a cross-layer design to shield the upper layers from operational details of the handover; however, it is also important to inform the upper layers about the handover(s) to enable them to adjust to the handover. In what follows, we present few optimization techniques which could be adopted with CLD enabling the essential communication between different network layers which is required for the optimizations. Specifically, we detail the CLD approaches and the corresponding execution problems that require upcoming investigations. i. Satellite Protocol Reference Model The European Telecommunication Standards Institute Technical Committee-Satellites Standard Earth Stations and Systems/Broadband Satellite Multimedia (ETSI TC-SES/BSM) has demarcated IP-primarily based satellite system structure containing bottom layer air interfaces [34]. Fig. 5 demonstrations this sort of protocol architecture in which, the bottom layers rely upon the satellite structure for putting into practice and the top layers are standard of the IP suite (satellite-free layers). Such two protocols layers are interrelated due to satellite-freeservice access point (SF-SAP) interface, and standardization envisions small range of typical functions which pass SF-SAP. Precisely, those features are deal with resolution, aid management, and visitors training QoS. The mission exists in the implementation of a move-layer technique (i.e., linking satellite system for pc-based and satellite independent layers) which debts for the aforementioned protocol shape and the SF-SAP interface. More specifically, appropriate primitives should be included within the standardization for supporting such a prolonged signalling. Transport Layer Application Layer Network Layer Data Layer Security QOS Mobility Wireless Link ii. Fractal Cross-Layer Service with Semantics Pervasive assembly, developed platform for service, and precise scientific model of service depending on logic of service in order to yield quick and clever actions are the most significant characteristics of services in the next-generation IoT [35]. IoT can achieve ubiquitous connection, and hence, the manner in which this technology can be used to build cross-layer platform to provide service with active integration and interoperability is a major challenge. For example, in sensing enabled technology, IoT serves as a service oriented system having a core value of "smart services" [36,37]. In addition, to including common services in traditional internet, IoT services comprise pervasive services in different network setting such as, mobile networks and wireless sensor networks (WSNs), etc. [38]. Further, IoT services are considerable and diversified in nature, and typically face morphological changes, environmental change, outward expansion, business restructuring, altered sharing and interoperability levels problems, and other situational dynamic adaptabilities. These issues present challenges to the IoT service platform, with possible solutions such as, web services, semantics, etc., which aid in achieving services with efficient integration, sharing, discovery, and interoperability [39]. With this, an effective service discovery and personalized delivery can be achieved [40,41]. iii. Semantic-based and Cross-layer service Platform for IoT Service In this sub-section, following the comprehensive analysis of web service architecture and semantic based sensor network architecture [42] projected in EU FP7 [43], the new semantic-dependent IoT service architecture is presented in Fig. 6. Such an architecture comprises of two types viz., the service based concept involving both, semantic description and IoT ontology, and corresponding service operation. The architecture includes handler ontology which aids the description of IoT service resource. Finding practical requirements includes service and quality ontology to attain semantic depiction of user's practical needs. Context ontology will be used to confirm semantic annotation on characteristics of context that are attained by context aware computing. Semantic-related characteristics of user helps to meet the user's adaptive and individual needs of the improved service. Further, service ontology creates service, then publishes it, and it exits throughout the entire platform of process [44]. Further, this framework contains service discovery, selection and combination. Service composition process is attained by dividing requirements, and service discovery involves the examining and comparing process on the original advertising services set based on service requester's service requirement. Thus, it could return service set which can meet functional requirements. The service resource can also be organized and then managed by decentralization. Finally, the service selection corresponds to a process of selecting the personalized service depending on service requester. In specific, it needs to select the service discovery and choice strategy to adapt in any dissimilar case, with strategy, service ontology, quality ontology, context ontology, etc., which ensure a semantic assisting role. During course of the outmost part of the framework architecture, there exist privacy, security and trust that offers the basic assurance for the execution of the complete process. iv. Scalable Semantic Image Compression with CLD Approach In an intelligent society, image compression aims to serve human imaginative, prescient and, MV. Traditional picture compression schemes will not ignore visual fine for viewer. The process of complete decoding of images is necessary earlier to their application to the semantic analysis. These elements make traditional schemes semantically inefficient. Hence, it is advised to compress and then transmit the photo indicators and features concurrently in order to efficiently serve the needs of HV and device imaginative and prescient. A unique semantic scalable image compression approach has been proposed. The method gradually compresses the "coarsegrained" semantic capabilities, the first-class-grained semantic functions, and the photograph alerts as shown in Fig. 7. To make use of the pass-layer correlation among functions and alerts for image, a "pass-layer" context model has been recommended to lessen the information redundancy. This approach ensures enhanced layer capabilities as move-layer appears to predict the distribution parameters. This is possible due to use of the entropy form of "decrease-layer" functions. Furthermore, by considering the region of interest (ROI) compression system, in which the equipments using high semantic records are used and the background are compressed one after the other in order to enhance the compression performance. It has been experimental shown at the CUB-200-2011 and FGVC-Aircraft datasets are efficient in assessment of the methods which one by one compress the picture indicators and capabilities. Currently, intelligent multimedia system applications square measure taking part in a crucial role in daily life like, in good cities and in intelligent police investigation. It is not possible to process and analyse the increasing image/video knowledge simply via human vision. Therefore, there is surge in application of machine vision techniques that are responsible for the speedy development of decilitre, and DNNs. The major role of these devices is in performing visual investigation by machine. However, it is not possible to substitute human understanding and decision-making by the MV algorithms. Thus, in order to facilitate human-machine interaction, compression techniques have to work for both HV and MV needs. A technique of first compress and then analyse is used in conventional multimedia system. The image signals square measure compressed as bit stream for transmission and storage. In the next step, the receivers on works on the decoded signals to rewrite the bit stream to perform visual study tasks. Further, traditional compression ways primarily target the image quality for HV [45,46]. The linguistics fidelity isn't thought of in such a scenario. Therefore, compression of artifacts that appears below low bitrates extremely worsen the performance of the linguistics conditions [47]. Additionally, the visual analysis method is sometimes resource overwhelming, particularly using DNNs [48][49][50]. The result of this is observed in terms of many procedures overhead for "back-end devices or servers". Hence existing compression approaches fail to address matter of "human-machine co-judgment" expeditiously, and so, for enhancing potency of linguistics tasks and cut back, complexness of linguistics analysis, another technique is "analyse-then-compress". In this technique, option square measures computed and compressed are used for analysis in a very simple manner by the receivers. Within the theme of decilitre, options square measure sometimes extracted by DNNs, which might be considered as a heap of "feature extractors". Therefore, options here are even having a "multi-layer structure", and also the learning method of DNN is neglecting task-independent info increasingly [51]. An existence of additional abstract and task-specific info results in higher-layer options that are troublesome to generalize to different analysis tasks. Recently, intermediate function compression has gained immense interest [52][53][54][55] because of the reality that computational load may be transformed to "front-end" whereas, ability to generalize compressed capabilities is preserved. Nevertheless, because of records loss inside function extraction procedure, it is challenging to restructure the unique photographs from the restructured functions, which limits its possibility of application as human imaginative and prescient is also crucial. To help each, human imaginative and prescient, and system imaginative and prescient, and additionally discover a better alternate among the computational load and the generalization ability, the best scenario occurs when, simultaneous transmission and compression of photo indicators and functions occurs. Consequently, receivers may request appropriate features consistent with necessities of analysis obligations and may also request photo indicators for presentation. A powerful method is compression of picture alerts and features for my part, as shown in Fig. 7(a), wherein, generated bit streams are simultaneously transmitted. Nevertheless, these manner desertions the correlation among the functions and the photos, and as end effect, the efficiency of compression techniques may be actually poor. The preceding sections and this section has made it clear that, in evaluation to a syntactic level conventional receiver which requires re-transmission of the packets which are in errors, on the semantic level, the receiver re-transmits handiest whilst any mistakes occur on the semantic stage. Also, further to the traditional approach of regarding the transmitted bits/symbols sequence corresponding to the syntactic stage, a framework related to the semantic stage might be capable of presenting remarks from the receiver to the transmitter thru the involvement of semantics of the transmitted records. However, the principle mission to be addressed is that of devising such mechanisms in order to be capable of hit upon the errors which arise on the semantic ranges. A possible technique to this problem will be that a semantic orientated feedback stating the requirement of a retransmission may be sent by means of the interpreter on the receiver to the source generator at the transmitter in case when the meaning of the received statistics is not clear. Further, in view of pass-layer interactions, there can be a constant statistics trade among the syntactic and the semantic ranges. As an instance, if the received information is being always interpreted and decoded effortlessly within the favoured time body, the interpreter (semantic) on the receiver will ship a comments to the encoder (syntactic) on the transmitter declaring that no longer all the facts desires to be transmitted, hence being able to limit the statistics-rate (bandwidth), and power (strength). Such an interaction between the diverse ranges paves manner to new designs for conversation structures in which, in spite of errors at the syntactic layer, correction can be performed on the semantic layer without the requirement of re-transmission of information. As an instance, inside the 5G communication running at mm-waves wherein blocking off and absorption because of atmospheric consequences is excessive, the transmitted statistics won't reach the intended receiver; however, the usage of appropriately tuned models for prediction, the interpreter at the semantic stage will nonetheless be capable of reconstruct the semantic records. Overall, the common knowledge which is shared among the transmitter and the receiver enable multiple errors correction without having to retransmit the statistics; however, at increased receiver complexity. IV. Applications of Semantic Communications In this section, we detail the various applications which can be enabled via semantic communications to improve their performance. i. Holographic Communications Holographic communications, a challenging new use-case foreseen for 6G networks, is a technique in which many views of one scene are transmitted to create a hologram at the receiver end. In this regard, semantic communications can play a major role in the advancement of holographic technique by incorporating the semantic aspects which will be shared by the transmitter and the receiver to ensure a common base of knowledge. Also, for serving the next generation use cases such as, wireless brain-to-computer interactions (WBCI), mixed reality (XR), Internet of robots, etc., semantic communications enabled applications will be required. However, to attain the advantages provided by including the semantic aspects, the price to be paid is that of additional computational complexity at the receiver end which will result in being a major bottleneck for few applications. To minimize the delay, semantic communication enabled systems may look to work as the human brain functions as detailed in [56]. The brain is in a continuous process to create the external world image in accordance with the data which it already knows and what it has already observed. This is done through a model which generates the signals hierarchically with an aim of reducing the errors in prediction via a bi-directional cascading of the cortical processing. Specifically, the brain selects, in terms of what it expects, a minute subset of signal(s) multitude which is sent by the retina. In this manner, majority of the signals which are generated in retina are not required to propagate via optical nerve as only those signals (observations) which deviate from the prediction are transmitted from the retina to the brain. In this manner, much energy is saved and this process could be replicated to the next generation semantic communications enabled systems. ii. Speech Communications The incorporation of NLP within the speech communications will be beneficial in addition to including a step of speech recognition which will aid in the translation of speech to text. Further, to minimize the efforts of the forward error correction codes, automated word(s)/sentence(s) corrections can be included so that any error(s) at the bit/symbol level is compensated by the word(s)/sentence(s) technique. As an alternate, instead of retransmission, speech bits which are lost due to interference/fading/crosstalk can be retrieved through context only. Also, in regard to speech signal processing, an intelligent task will be the conversion of the speech signals to the corresponding text data which, in comparison to the usual automatic speech recognition technique, will account for speech signals characteristics. The process will include the mapping of every phoneme to a corresponding individual alphabet following which, there will be a requirement of concatenating all the alphabets to the corresponding sequence of words through a model of language which can be understood by all. Further, for this scenario, the semantic features which will be extracted will only contain the text characteristics whereas, other features are not transmitted by source. Hence, network traffic is considerably minimized ensuring no degradation in the network performance. iii. Video Communications This use case expects significant improvements presently and in the future as it consumes much of the network resources. The open research problem is the manner in which the semantic aspect can be integrated with the existing video communications setup. A possible solution is to incorporate the meaning/context of the content(s) within the video. Specifically, using the previously existing frames of the video, an interpreter will be able to predict the current and future frames via an appropriately trained model for prediction. Using the previous frames, interpreter is able to predict next frames via an appropriately trained model for prediction. The study in [57] has performed the video coding after the incorporation of a frame predicting method which is enabled by deep neural networks. In this study, when there occurs no significant change due to fading, the complete flow of the video is constricted with no changes at the semantic level. The interpreter at the receiver is able to reproduce a video, which may not be syntactically matching with the transmitted video; however, is a semantic equivalent. As a result, this technique generates major savings in regard to the transmitted power and/or the bandwidth. Another concept in regard to the video communications is the process of segmentation of the semantics i.e., to implement machine vision tasks to segment all the objects within a scene so as to classify them based on a concept. The aforementioned is in turn advantageous since it will serve as a pre-processing step to further tasks such as, object detection, scene understanding, and scene parsing. In effect, semantic segmentation will analyse and classify the objects' nature and concept in addition to being able to recognize the objects and the corresponding shape within the scene. Therefore, semantics segmentation will be considered as one of the three basic steps of object detection, shape recognition and classification. iv. Goal Oriented Communications In our view, the goal-oriented communications will be part of the opportunities which are offered by the Level 3 (effectiveness) layer. The success of this communication type will require the specification of a goal with immense clarity such that all the data is not transmitted; rather, only the most relevant data is sent by the transmitter resulting in optimized network performance and effective goal-oriented communication. Existing work on goaloriented communication has focused on the concept of misunderstanding between the communicating parties arising due to the absence of an agreement on the protocol/language type to be used for communication [27,28,58]. However, we present a different view of goal-oriented communication in which the aim to communicate must be to fulfil a goal in which case the system performance will be dictated by the completion of the specific goal. In specific, effectiveness will have to evaluated based on the goal completion with the constraints on the resources. As an example, for the 6G networks, after the allocation of a chunk of the THz frequency, the aim will be to ensure that the transmitter and receiver send/receive the bits/symbols to complete the communication within the allocated frequency chunk only. Such an issue is addressed in [59][60][61]. The authors in [59] have addressed this issue for the case of edge learning in which the tools for learning are much closer to the user equipment to provision the applications with acute constraints of delay. The authors state that in such cases there will occur multiple trade-offs such as, between delay and consumption of power, between delay and accuracy, etc. In [60], the authors have addressed the trade-off between delay and accuracy considering an edge ML system enabled via an edge processor implementing the stochastic gradient descent (SGD) algorithm. Given every data packet's overhead and ration of computation and communicating rates, the authors have optimized size of the packet payload. The authors in [61] have proposed an algorithm which aims at maximizing the accuracy of learning under the constraints of delay. In [62], authors have proposed a distributed ML algorithm for edge operation of wireless equipment which aim to reduce an empirical function indicating loss via a remote server. In view of addressing the aforementioned issue, we propose the following formulation. Considering our proposal, shown in Fig. 8 . With the aforementioned, the aim will be to transmit the data, D , to a decision centre for making a decision, and the issue to be resolved will the manner in which the data must be encoded i.e., the mapping of M to D simultaneously provisioning the applications with highest accuracy and lowest power consumption. Further, initially, the data will have to be encoded at the transmitter for the removal of any redundancy simultaneously permitting ideal reconstruction or to reach a permissible compromise between the rate of encoding and distortion. In comparison to the aforementioned, we propose a strategy in which if communication occurs to complete a specific goal, then the source encoder must be designed such that it is able to achieve the accuracy which is desired over the classification simultaneously reducing the resources which are used or minimizing the decision time. Further, in comparison to reducing the errors while reconstructing the data, the source encoder will have to be tuned such that it falls in line with the learners' performance at the decision centre. To state the problem formally, we assume that the parameter (random variable) to be estimated is denoted as  . The problem is analogous to searching a map of ) (M f D  such that there occurs a maximum compression of D . However, it must be noted that in this process of moving from M to D there should occur no loss of the data in regard to the recovery of  . In other words, the problem can be stated as: However, there could be cases when multiple sufficient statistics exist. In this regard, considering the proposal in this study, it will be key to identify the minimal sufficient statistics. In specific, relative to ) ; ( M . In turn, this will aid in minimization of the transmitted bits simultaneously ensuring that there is no loss in the meaning (semantics) of the data.   ) ( ) ( ) ; ( m h m f g m p    .(9) Further, for cases when it is difficult to find the minimal sufficient statistics, we extend our proposal for general cases which follows the bottleneck principle as detailed in [64]. The formulation searches for an encoding function ) (M f D  which aims to achieve the best trade-off which exists between data loss in regard to estimation of which is obtained via the compression of M and the bits amount required to encode ) (M f . Mathematically, the generalized formulation is as follows [63]: ) ; ( ) ; ( min ) (   D I D M I M f  .(10) where  denotes a positive real number which assigns multiple weights to the two terms in the formulation. The data can be compressed to maximum value if the value of  is small; however, this compromises the accuracy of learning. To improve the accuracy of learning, the value of  must be large; however, this compromises the compression at the transmitter. The solution to the generalized problem can be achieved through the iterative data bottleneck technique which has been shown to be effective within the learning problems via the use of autoencoders [64,65]. Lastly,  can be adapted dynamically to evaluate best compromise between power consumption, latency, and learning accuracy. This will depend on both, the channel state and accuracy level which is achieved by the learner. v. Online Learning Communications In 6G networks, compared to the previous generation networks, machines will be contributing significantly via the enabling of AI at the network edge which will exist much closer to the end users [66,67]. This in turn will ensure that (i) ML can be implemented for communication, computation, control and infrastructure in view of optimizing the network resource(s) usage, and (ii) communication can be implemented for ML with an aim to enable semantic aware latency critical services via the ML algorithms' distributed implementation. For supporting the aforementioned, 6G networks design will have to include learning tools so as to tune the network in response to changes in the requirements and the constraints [68]. In addition, we also envision that the inductive ML techniques will be integrated with the deductive semantic aspects to achieve the further development of (i) ML algorithms through semantic communication for exploiting the context aspects for improving the learning capabilities, using the semantic data efficiently, and enhancing the robust nature of the system in response to adverse attacks, and (ii) semantic communications through ML techniques to infer the context efficiently so as to improve the system efficiency. vi. Intelligent Communications 6G wireless networks will provision next-generation services which will require the system to take effective decisions under the constraints of stringent latency and jitter bounds. In view of the aforementioned, edge computing has been identified as an emerging technology which will ensure that the computations will occur at the network edge instead of a distant cloud server. This will in turn ensure that the end users will be able to access all the computations and resources incurring very low delays. In general, latency occurs due to communication delay, computation delay, and infrastructure (resource(s)) access delay. In addition, if the procedures which are latency critical also need to be controlled, then a control delay also occurs. Hence, for the 6G networks, a joint inclusion of communication, computation, infrastructure, and control must be considered for the design [69]. Such a proposal is presented by the authors in [70,71] considering case of cellular edge computing and further, in [72], authors have proposed an edge controller to implement a strategy which aims at optimized controlling of the systems which enable advanced driving and controlling. However, in all these studies the authors have not considered the latency and power consumption by the data which is to be transmitted/received and processed. In this regard, the authors in [73] have considered the joint optimization of the resources relevant for communication and computation considering that the offloading of these resources will result in latency. Hence, the authors have proposed a setup which is multi-user i.e., one edge computing host is deployed for serving many small cells. Also, in [74,75], the authors have proposed an algorithm which aims to optimizing communication and computation resources together in a dynamic environment. As an extension, the authors in [76] have proposed the use of a dynamic convex optimization technique in cases when the optimizer is not aware of the exact system state. Lastly, in addition to joint consideration of communication and computation, infrastructure and communication must also be considered together. This is mainly required in cases wherein; the applications demand the contents dynamically to ensure that the desired latency constraints are satisfied. Further, to deliver the contents in an intelligent manner at the edge, dynamic access of the infrastructure will be key in view of minimizing the request initialization and the corresponding content delivery. In the networks which allow for infrastructure access at the edge, majority of the content will be present at the edge for access thereby, resulting in the ubiquitous visibility of the data at the user equipment. In specific, the infrastructure access rules will be a key enabler if it allows for the dynamic data storage in response to the demands estimated through an efficient prediction algorithm using multiple variables. Further, the variables will include the infrastructure deployment, content storage, and content routing [77]. In regard to infrastructure, thus far, the main issue has been the formulation of policies for moving the relevant content throughout the network. However, in the semantic communications, we envision that, in addition to only moving the relevant contents, it will also be required to move the base of knowledge systems and machines for dynamic provisioning of the demands much closer to the user equipment. The aforementioned has received much research attention as it poses the challenging issue in regard to the computation and infrastructure access for applications which are latency sensitive. Specifically, when contents are distributed within the network, it results in multiple issues in terms of bandwidth, latency, privacy, security, etc. As a solution, the coded distributed computing may be employed which resorts to the use of coding theory to introduce structured computation with redundancy [78]. vii. Machine Learning for Wireless Networks Generally, when the supervised learning type of ML technique is implemented, there is trade-off between requirement of large labelled data amount and the achievement of the desired high performance. This implies that there is immense manual intervention with the supervised learning in view of providing the labelled data/examples. In comparison, the unsupervised learning type of ML technique does not require any labelled data/examples, and the learner aims to search for specific patterns within the data. In addition, few applications also implement the graph enabled approaches which aim at capturing the pair-wise relations among the variables; however, this is not efficient as it does not extract the complete information. For the effective operation of systems which are enabled by wireless network, mechanisms must be proactive which will need efficient ML prediction methods to operate at the network edge. Over the past decade, deep neural networks (DNNs), a class of supervised learning, have been demonstrated to provide higher performance in comparison to other techniques, especially for applications such as, recognition of the sound and classification of the image [79]. Recently, the aspect of explainability has gained much importance owing to the widespread use of DNN, which find key applications in multiple fields. However, the manner in which an input of DNN generates output is not often evident. In specific, even when clarity exists in relation between input and output, final output of network weights, which result from training phase involving highly non-linear optimization, does not match with the theoretical guarantees. A key issue to be addressed in the explainable ML technique is whether the algorithm provides information which permits the user to relate the characteristics of the input features with the corresponding output. Also, the convolutional neural networks (CNN) have also shown suitability in regard to the sound and images owing to the technique's ability to intrinsically sparsify the edges amount from one layer to the next layer. As an extension, studies such as [80] have addressed the combination of graph-enabled representation with the learning aspect in view of operating with the data residing on the graphs rather than on the regular grids. In specific, the application of supervised learning, in which the testing and learning phases have a clear distinction, to any communication network provides immense opportunity for training the learner through aggregated models which are stochastic in nature via a simulator which can generate the input data, the channels, and the output [81,82]. Thus, the learner learns through large number of labelled examples. However, for the wireless networks with varying channel over time, dynamic (online) learning techniques are better suited as they have a combined testing and learning phase which updates with time. In this regard, the reinforcement Learning (RL) methods and the random optimization methods may be implemented. In RL, learning of an agent occurs through the observation of its own action(s) without the assumption of any a-priori observation model [83]. In the random optimization technique, an adaptive online process is implemented which updates a per the actions and the knowledge which is available at that instant regarding the variables which are involved [84]. As an extension, an advance is demonstrated in [85] by incorporating the multi-way relations via topological signal processing (TSP) which is helpful when dealing with observations which are associated to graph edges. Further, the ML methods are applicable to physical layer of the network via auto-encoder (AE) [86,87], a recurrent neural network (RNN) [88], and a generative adversarial network (GAN) [89], and it can also be extended to higher layers through a complex AI framework as detailed in [90]. In the 6G wireless networks, there will exist tools for learning pervasively at the network edges which will pose numerous challenges. Further, within the network, multiple user equipment will generate large data which will require collaborative learning for the analysis in view of improving the learning technique's performance. These collaborative learning methods will demand data exchange which will in turn raise serious privacy/security issues which need to be addressed. As a solution, federated learning (FL) can be implemented in which the model parameter(s) learning occurs at a central unit/data centre/edge host, and the data is stored at the peripheral nodes [91,92]. In the FL of centralized type, rather than sending the data to a remote server, the user equipment share parameters' local estimates which are to be learned. Therefore, the privacy issue is resolved and the user equipment performance is also improved. Considering the aforementioned, a typical FL enabled network will have the following objective function:   N i i i i m m d f p 1 ) ; ( min .(11) where, the user equipment's ( i ) empirical loss function is denoted by It must be noted that FL follows an iterative technique wherein, for each iteration n , as opposed to transmitting the local data i m , every user equipment transmits the local estimate ] [ n m i to a hub which in turn transmits back an update factor which accounts for the data which is received via all the cooperating nodes. This objective function converges to a global optimum under the conditions of convexity [92]. Also, this setting is simple; however, it faces the major challenges owing to communication channel heterogeneity, and behaviour of the local user equipment' model. Therefore, an extension technique multi-task FL can be implemented [93,94]. From the aforementioned, it is clear that the ML technique is driven through data which aids in learning which is mostly inductive. However, human learning is deductive wherein, accumulation of experience and knowledge over a period of time helps in the continuous updation of the base of knowledge. Further, it is known that the inductive learning models such as, DL enabled systems result in ambiguity(s) as they are always searching for correlations within the data whereas, search for semantics (meaning) has much more value [95]. In this context, we believe that ML techniques will take significant leaps forward by incorporating the knowledge from external world and including the context within the decision making process. Further, the 6G networks will facilitate the merging of ML techniques and systems which are enabled knowledge representation [96,97]. As a novelty, semantic communication will ensure that a base of knowledge exists at all the network nodes so as to enable the interpretation of semantics. This will require new ML techniques which will include the schemes for representation and reasoning in effect offering multiple capabilities to effectively optimize the network resource(s) usage in view of semantic and goal-oriented communications. Overall, the new ML techniques are envisioned to be more efficient and reliable owing to the inclusion of transfer learning and knowledge sharing mechanisms. V. Issues and Challenges In this section, we present the various challenges and issues which exist in the research over semantic communications and networks. 1. Immersive XR is a combination of AR, VR, and mixed reality (MR) which includes the advantages of the three methods. XR technology will integrate with wireless networking, edge computing, and AI/ML for offering complete immersive experiences to the humans for multiple applications. The designs for XR are similar to the ones required for AR/VR; however, there will be stringent requirements in regard to the accuracy and the sensing human characteristics diversity, data rates, and latency [98]. In addition, large scale AI/ML models will be required to be implemented in view of efficient training and inference support. 2. Holographic Communication will involve the transmission and reception of 3D holograms of human or physical objects [99]. Depending on the high resolution, wearables, and AI/ML, user equipment will render the 3D holograms for displaying the remote users or machines local presence, to create increased realistic local presence of a remote human or object. This will require an enhancement in the users' visual perception for improving the virtual interaction efficiency. In this regard, haptic information's high-resolution encoding, colours, positions, and human/object tilts will be highly desired which will need extremely high data rates, and stringent delay constraints [100]. Hence, it will be crucial to boost semantic communication's processes viz., encoding, decoding, transmission, reception, etc., efficiency and speed to higher levels such that the holographic communications will be able to satisfy the main aim of real-time and seamless integration of human-to-machine and machine-to-machine techniques. 3. All Sense Communication will be included within the 6G communications via a sensors ensemble which will be a wearable or will be mounted over every device [101]. In combination with holographic communications, all sense data will be effectively integrated for realizing a close-to-real feelings of the remote environments thereby, facilitating the tactile communications and haptic control [102]. The challenge will be to provision the exponential complexity within the applications which will arise due to diversified types of sensing signals which will create multiple new data dimensions. 4. In comparison to the sampling and reconstruction method used for syntactic communication, in semantic communications compressed sensing will be implemented for performing specific signal processing methods which will occur directly in compressed domain without complete signal reconstruction [103]. Therefore, a major challenge will be to develop an optimal sampling technique (theory) which will combine signal sparsity and semantics in view of real-time prediction and/or reconstruction considering the various constraints and latency. For achieving the optimal trade-offs between the metrics used for semantic communications and the network energy consumption, major revision is required in the feedback structure, packetization, scheduling and resource assignment. This further implies that there is a need to abandon simplistic assumptions and simple metrics, and in favour of the advanced ML techniques. 5. In view of savings on miscellaneous operations over packet data, the optimization of packet structure i.e., a clear differentiation between data and miscellaneous data is required. Also, keeping the aims of semantic communications in mind, the following require further research (i) the role which feedback will play considering real time channel environment, (ii) the retransmissions over the link layer under real channel conditions, and (iii) the error rates which can be achieved considering lengths of the block which are nonasymptotic in nature. 6. Another key challenge is related to scheduling and resource allocation policies for semantic communications which will also consider the optimization of energy within the system. Further, advanced access techniques, flow control mechanisms, and much broader metric for semantic communications remains an open research problem which requires timely solutions. Specifically, in regard to the metrics, it must also be noted that considering the diversity of humans, new composite metrics need to be developed so as to cover the key experiences. 7. For a semantic enabled network, the complexity will be high as the network will have to provision a system in which the knowledge is shared very close to all the users [104]. This will require the development of a framework which is mathematically enabled, and which will aid in the evaluation of performance limits imposed by the semantic network. Further, tracking and modelling the upgrade in human knowledge to continuously update the base of knowledge will aid in the improvement of communication efficiency, and the minimization of error probability within the semantic network. This aspect requires further investigation. Finally, in Table 2, we suggest the various directions which can be adopted in response to the existing research challenges on semantic communications. Provide enhancement in encoding, decoding, transmission, reception, etc., to increase the efficiency and speed to higher levels. Develop a model which ensure that holographic communications satisfy real-time and seamless integration of human-to-machine and machine-to-machine techniques. XR Satisfy stringent requirements in regard to accuracy and sensing human characteristics diversity, data rates, and latency. Implement large scale AI/ML models for efficient training and inference support. All Sense Communications Develop models to provision the exponential complexity within applications arising due to varied sensing signals types which will create multiple new data dimensions. Sampling and Reconstruction Develop an optimal sampling technique to combine signal sparsity and semantics for real-time prediction and/or reconstruction considering various constraints and latency. Metrics Develop advances metrics in favour of the advanced ML techniques. Examples: freshness, relevance, value, etc. [29]. 6 Semantic Network Develop a mathematical framework which will aid in performance evaluation limits imposed by the semantic network. 7 Scheduling and resource allocation policies Formulate new random/scheduled access policies in regard to broader semantic metrics. Knowledge modelling and tracking Mechanism to continuously update the base of knowledge automatically for improving communication efficiency, and reduce error probability within the semantic network. 9 Multi-objective Stochastic Optimization To develop a model which will solve the multiple criteria optimization with goal-oriented, end-user perceived utilities to enable relative priority degree among various attributes of information. 10 Flow control techniques To develop optimal operating protocols for higher layer operation to improve the network resource(s) usage. VI. Conclusion In this article, we have surveyed the most recent advances which will aid the enabling of semantic communications for intelligent wireless networks. We reviewed the concept of semantic communications and the application to intelligent wireless networks, which presents a paradigm deviation from the conventional method of following the Shannon's information theory for communication. From the survey, we gathered that the semantic method of communication offers a potential to generate enormous benefits, increased effectiveness and reliability in addition to presenting a goal-oriented manner of information exchange. Further, the aforementioned is possible without the need to allocate increased resources (bandwidth, energy, etc.), and requires only the correct identification of the appropriate data. This implies that the semantic communication enabled wireless networks will only require to extract the correct meaning/context of the transmitted data, and use the same to decode the received data to ensure that goal-oriented communication occurs. This in turn focuses on the sustainability of the wireless networks and breaks away from the convention of providing more resources for provisioning advanced applications. Conversely, semantic communication aims to utilize untapped capability(s) such as, communication, computation, infrastructure and control of a communication system to represent the knowledge so as to communicate only the relevant data between the parties which interact with one another. In regard to the challenges, we found that semantic communications will require much research in regard to the (i) distributed computing techniques' implementation so as to learn, and then extract the precise context from the received data, (ii) exploitation of the appropriate system(s) to represent the knowledge, and (iii) identification and exploitation of the most relevant data in view of goal oriented communications. Once implemented, such a new framework will enable a form of learning which will be gather enhanced benefits by the introduced aspects of semantics thereby, migrating from a completely inductive technique to a strategy which will encompass an interplay of both, inductive and deductive techniques. Further, this will also enable a learning mechanism which, in addition to learning through examples, will construct abstract models to guide further learning. Lastly, we have also presented future directions to the multiple existing challenges with a hope that this will spur further research on semantic communications for next generation intelligent wireless networks. Declarations Funding Fig. 1 . 1Taxonomy of the Semantic Communications for Intelligent Wireless Networks survey. Fig. 2 . 2The new three levels architecture for intelligent wireless networks. Fig. 3 . 3Multiple level communication system involving the three levels of communications. Fig. 4 . 4CLD Coordination planes. Fig. 5 . 5Satellite protocol architecture by ETSI TC-SES/BSM. Fig. 6 .Fig. 7 . 67Framework of IoT service based on semantics. (a) Individual compression together. (b) New scalable semantic image compression. , the novelty in our formulation exists in the feedback which is provided to the source encoder for goal oriented communication by the decision maker block. To elaborate, consider the goal of communication being either to conduct a classification or use parameters set  comprised of an observation(s the mutual information from M to Y , and the corresponding solution is the reduced sufficient statistics of M . Further, given m , ) (m f is a sufficient statistics for  and the joint pdf ) , (  m p is evaluated as [63]: Fig. 8 . 8Block diagram of goal-oriented communication. parameter vector which is to be learnt is denoted as m , and i p denotes the weighting coefficient which highlights the data importance which has been collected by user i within s m' estimation. Also, the amount of examples which are observed by the th i device. Table 1 . 1Summary of various existing studies on semantic communications.Reference Aim Problem addressed Method Major Results Future Scope [11] Extend scope in regard to designing of the of the aim and constraints. Also, include the transmitted data's semantics for any application or use-case. Provision a framework to obtain solutions which are enabled by semantics via a semantic- effectiveness plane. Augment the existing stack of protocol through interfaces which follow standards. Semantic- effectiveness plane replaces the current version by a framework which regularly improves and extends both, the system and the standards. Cross-Layer Recovery Mechanisms, Security [15] Propose a framework for communication which accounts for context of the transmitted data under the consideration of a channel which is affected by noise.. Characterize optimal transmission policy(s) for reducing end to end mean semantic error Formulation of the communication issue in the form of a Bayesian game. Investigation of conditions for which there is existence of the Bayesian Nash equilibrium. Consideration of an online communication case wherein, actions are taken sequentially so as to form beliefs about other party. Finally, evaluate sequential game, structure for system of belief and profiles of strategy. Semantics of word can be used to assess performance of communication. Also, the beliefs of transmitter and receiver influence the optimal strategies which are adopted for communication. Investigate the trade-off between encoding and decoding optimality functions and words amount. Consider the cases of incorrect data and/or manipulated received data. [16] Capitalize on semantics driven sampling and communication policies. Apply a method which is new in structure and has synergy to ensure that receiver must perform real-time Semantic-enabled system with foundations of goal-oriented data aggregation, and Major minimization in errors owing to reconstruction and actuation error. Multi-objective Stochastic optimization, Semantics-aware Multiple Access, Goal-oriented decoding equation is re-written as[25]    s m f s m m p m s p s r p m ) ( ) / ( ) / ( max arg ) ( : ' . Table 2 . 2A summary of various existing challenges in research on semantic communications and possible directions to obtain solutions. Sl. No.Existing Challenges Proposed Directions 1 Holographic Communications The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.Conflict of InterestsThe authors have no conflict of interests to disclose.Data AvailabilityData sharing not applicable to this article as no datasets were generated or analysed during the current study.The pre-print version of this article is available at: http://arxiv.org/abs/2202.03705Code Availability Not ApplicableAuthor ContributionsAll authors contributed to the study conception and design. All authors read and approved the final manuscript." 5G: A tutorial overview of standards, trials, challenges, deployment, and practice. M Shafi, A F Molisch, P J Smith, T Haustein, P Zhu, P D Silva, F Tufvesson, A Benjebbour, G Wunder, IEEE J SeL Area Comm. 356M. Shafi, A. F. Molisch, P. J. Smith, T. Haustein, P. Zhu, P. D. Silva, F. Tufvesson, A. Benjebbour, and G. 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[ "Federated Asymptotics: a model to compare federated learning algorithms", "Federated Asymptotics: a model to compare federated learning algorithms" ]	[ "Gary Cheng [email protected] ", "Karan Chadha [email protected] ", "John Duchi [email protected] " ]	[]	[]	We propose an asymptotic framework to analyze the performance of (personalized) federated learning algorithms. In this new framework, we formulate federated learning as a multi-criterion objective, where the goal is to minimize each client's loss using information from all of the clients. We analyze a linear regression model where, for a given client, we may theoretically compare the performance of various algorithms in the high-dimensional asymptotic limit. This asymptotic multicriterion approach naturally models the high-dimensional, many-device nature of federated learning. These tools make fairly precise predictions about the benefits of personalization and information sharing in federated scenarios-at least in our (stylized) model-including that Federated Averaging with simple client fine-tuning achieves the same asymptotic risk as the more intricate metalearning and proximal-regularized approaches and outperforming Federated Averaging without personalization. We evaluate these predictions on federated versions of the EMNIST, CIFAR-100, Shakespeare, and Stack Overflow datasets, where the experiments corroborate the theoretical predictions, suggesting such frameworks may provide a useful guide to practical algorithmic development.	null	[ "https://arxiv.org/pdf/2108.07313v3.pdf" ]	246,996,493	2108.07313	a1ea84887c4ebcd418ff0104e65600bdb8971e30	Federated Asymptotics: a model to compare federated learning algorithms February 21, 2022 Gary Cheng [email protected] Karan Chadha [email protected] John Duchi [email protected] Federated Asymptotics: a model to compare federated learning algorithms February 21, 2022 We propose an asymptotic framework to analyze the performance of (personalized) federated learning algorithms. In this new framework, we formulate federated learning as a multi-criterion objective, where the goal is to minimize each client's loss using information from all of the clients. We analyze a linear regression model where, for a given client, we may theoretically compare the performance of various algorithms in the high-dimensional asymptotic limit. This asymptotic multicriterion approach naturally models the high-dimensional, many-device nature of federated learning. These tools make fairly precise predictions about the benefits of personalization and information sharing in federated scenarios-at least in our (stylized) model-including that Federated Averaging with simple client fine-tuning achieves the same asymptotic risk as the more intricate metalearning and proximal-regularized approaches and outperforming Federated Averaging without personalization. We evaluate these predictions on federated versions of the EMNIST, CIFAR-100, Shakespeare, and Stack Overflow datasets, where the experiments corroborate the theoretical predictions, suggesting such frameworks may provide a useful guide to practical algorithmic development. Introduction In Federated learning (FL), a collection of client machines, or devices, collect data and coordinate with a central server to fit machine-learned models, where communication and availability constraints add challenges [KMA + 19]. A natural formulation here, assuming a supervised learning setting, is to assume that among m distinct clients, each client i has distribution P i , draws observations Z ∼ P i , and wishes to fit a model-which we represent abstractly as a parameter vector θ ∈ Θ-to minimize a risk, or expected loss, L i (θ) := E Pi [ (θ; Z)], where the loss (θ; z) measures the performance of θ on example z. Thus, at the most abstract level, the federated learning problem is to solve the multi-criterion problem minimize θ1,...,θm (L 1 (θ 1 ), . . . , L m (θ m )) . (1) At this level, problem (1) is both trivial-one should simply minimize each risk L i individually-and impossible, as no individual machine has enough data locally to effectively minimize L i . Consequently, methods in federated learning typically take various departures from the multicriterion objective (1) to provide more tractable problems. Many approaches build off of the empirical risk minimization principal [Vap92, Vap95, HTF09], where we seek a single parameter θ that does well across all machines and data, minimizing a (weighted) average loss m i=1 p i L i (θ)(2) over θ ∈ Θ, where p ∈ R m + satisfies p T 1 = 1. This "zero personalization" approach has the advantage that data is (relatively) plentiful, and has led to a substantial literature. Much of this work focuses on developing efficient methods that limit possibly expensive and unreliable communication between centralized servers and distributed devices [HM19, RCZ + 21, MMR + 17, KKM + 20, MSS19, LSZ + 20]. Given (i) the challenges of engineering such large-scale systems, (ii) the success of large-scale machine learning models without personalization, and (iii) the plausibility that individual devices have similar distributions P i , the zero personalization approach is natural. However, as distributions across individual devices are typically non-identical, it is of interest to develop methods more closely targeting problem (1). One natural assumption to make is that the optimal client parameters are "close" to one another and thus must be "close" to the minimizer of the zero-personalization objective (2). Approaches leveraging this assumption [DTN20, SCST17, WMK + 19, FMO20] regularize client parameters towards the global parameter. While these methods are intuitive, most focus on showing convergence rates to local minima. While convergence is important, these analyses do little to characterize the performance of solutions attained-to what the methods actually converge. These issues motivate our paper. Contributions: 1. New model (Sec. 2): We propose and analyze a (stylized) high-dimensional linear regression model, where, for a given client, we can characterize the performance of collaborate-then-personalize algorithms in the high-dimensional asymptotic limit. Precise risk characterization: We use our stylized model to evaluate the asymptotic test loss of several procedures. These include simple fine-tuned variants of Federated Averaging [MMR + 17], where one learns an average global model (2) and updates once using local data; meta-learning variants of federated learning; and proximal-regularized personalization in federated learning. Precise predictions and experiments: Our theory makes several percise predictions, including that fairly naive methods-fine-tuning variants-should perform as well as more sophisticated methods, as well as conditions under which federated methods improve upon zero-personalization (2) or zero-collaboration methods. To test these predicted behaviors, we perform several experiments on federated versions of the EMNIST, CIFAR-100, Shakespeare, and Stack Overflow datasets. Perhaps surprisingly, the experiments are quite consistent with the behavior the theory predicts. Our choice to study linear models in the high-dimensional asymptotic setting (when dimension and samples scale proportionally) takes as motivation a growing phenomenological approach to research in machine learning, where one develops simple models that predict (perhaps unexpected) complex behavior in model-fitting. Such an approach has advantages: by developing simpler models, one can isolate causative aspects of behavior and make precise predictions of performance, leveraging these to provide insights in more complex models. Consider, for instance, [HMRT19], who show that the "double-descent" phenomenon, where (neural-network) models show decreasing test loss as model size grows, exists even in linear regression. In a robust (adversarial) learning setting, [CRS + 19] use a two-class Gaussian linear discriminant model to suggest ways that self-supervised training can circumvent hardness results, using the predictions (on the simplified model) to inform a full deep training pipeline substantially outperforming (then) state-of-the-art. [Fel19] develops clustering models where memorization of data is necessary for good learning performance, suggesting new models for understanding generalization. We view our contributions in this intellectual tradition: using a stylized high-dimensional asymptotics to develop statistical insights underpinning Federated Learning (FL). This allows direct comparison between different FL methods-not between upper bounds, but actual losses-and serving as a basic framework to motivate new methodologies in Federated Learning. Related Work. The tried-and-true method to adapt to new data distributions is fine-tuning [HR18]. In Federated Learning (FL), this broadly corresponds to fine-tuning a global model (e.g., from FedAvg) on a user's local data [WMK + 19, YBS20, LHBS21]. While fine-tuning's simplicity and practical efficacy recommend it, we know of little theoretical analysis. A major direction in FL is to design personalization-incentivizing objectives. [SCST17], for example, build out of the literature on multitask and transfer learning [Car97,PY09] to formulate a multi-task learning objective, treating each machine as an individual task; this and other papers [FMO20,MMRS20,DTN20] show rates of convergence for optimization methods on these surrogates. These methods use the heuristic that personalized, local models should lie "close" to one another, and the authors provide empirical evidence for their improved performance. Yet it is not always clear what conditions are necessary (or sufficient) for these specialized personalization methods to outperform naive zero collaboration-fully local training on available data on each individual device-and zero personalization (averaged) methods. In a related vein, meta-learning approaches [FAL17,FMO20,JKRK19] seek a global model that can quickly "adapt" to local distributions P i , typically by using a few gradient steps. This generally yields a sophisticated non-convex objective, making it hard to give even heuristic guarantees, leading authors instead to emphasize worst-case convergence rates of iterative algorithms to (near) stationary points. Other methods of personalization have also been proposed. In contrast to using a global model to help train the local model, [MMRS20] and [ZMM + 20] use a mixture of global and local models to incorporate personalized features. [CZLS21], like we do, propose evaluating federated algorithms via the formulation (1); they give minimax bounds to distinguish situations in which zero collaboration and zero personalization (averaged) methods (2) are, respectively, worst-case optimal. The Linear Model We consider a high-dimensional asymptotic model, where clients solve statistically related linear regression problems, and each client i ∈ [m] has a local dataset size n i smaller than (but comparable to) the dimension d of the problem. This choice models the empirical fact that the data on a single client is typically small relative to model dimension (e.g., even training the last layer of a deep neural network). More concretely, each client i ∈ [m] will use an overparameterized linear regression problem to recover an unknown parameter θ i ∈ R d . Client i has n i i.i.d. observations (x i,k , y i,k ) ∈ R d × R, y i,k = x T i,k θ i + ξ i,k , x i,k iid ∼ P i x and ξ i,k iid ∼ P i ξ . We make the routine assumption that the features are centered, with E[x i,k ] = 0 and Cov(x i,k ) = Σ i . We also assume that the noise is centered with finite variance, i.e., E[ξ i,k ] = 0 and Var(ξ i,k ) = σ 2 i . For convenience, we let X i ∈ R ni×d and y i ∈ R ni denote client i's data, and X := [X T 1 , . . . , X T m ] T . We also let N := m j=1 n j . A prior P i θ on the parameter θ i relates tasks on each client, where conditional on θ 0 , θ i is supported on r i S d−1 + θ 0 -the sphere of radius r i (bounded by a constant for all i ∈ [m]) centered at θ 0 -with E[θ i ] = θ 0 . The variation between clients is captured by differences in r i (label shift) and Σ i (covariate shift), while the similarity is captured by the shared center θ 0 . Intuitively, data from client j is useful to client i as it provides information on the possible location of θ 0 . Lastly, we assume that the distributions of x, θ , and ξ are independent of each other and across clients. Every client i seeks to minimize its local population loss-the squared prediction error of a new sample x i,0 independent of the training set-conditioned on X. The sample loss is (θ; (x, y)) = (x T θ − y) 2 − σ 2 i , giving per client test loss L i (θ i \| X) := E[(x T i,0θi − x T i,0 θ i ) 2 \| X] = E θ i − θ i 2 Σi \| X where the expectation is taken over (x i,0 , θ i , ξ i ) ∼ P i x × P i θ × P i ξ and x 2 Σ = x T Σx. It is essential here that we focus on per client performance: the ultimate goal is to improve performance on any given client, as per eq. (1). For analysis purposes, we often consider the equivalent bias-variance decomposition L i (θ i \|X) = E[θ i \|X] − θ 2 Σi Bi(θi\|X) + tr(Cov(θ i \|X)Σ i ) Vi(θi\|X) . (3) Our main asymptotic assumption, which captures the high-dimensional and many-device nature central to modern federated learning problems, follows: Assumption A1. As m → ∞, both d = d(m) → ∞ and n j = n j (m) → ∞ for j ∈ [m], and lim m d nj = γ j . Moreover, 1 < γ min ≤ lim m inf j∈[m] d nj ≤ lim m sup j∈[m] d nj ≤ γ max < ∞. Importantly, individual devices are overparameterized: we always have γ j > 1, as is common, when the dimension of models is large relative to local sample sizes, but may be smaller than the (full) sample. Intuitively, γ j captures the degree of overparameterization of the network for user j. We also require control of the eigenspectrum of our data [cf. HMRT19, Assumption 1]. Definition 2.1. The empirical distribution of the eigenvalues of Σ is the function µ(·; Σ) : R → R + with µ(s; Σ) := 1 d d j=1 1 {s ≥ s i } ,(4) where s 1 ≥ s 2 ≥ · · · ≥ s d are the eigenvalues of Σ. Assumption A2. For each user i, data x ∼ P i x have the form x = Σ 1 2 i z. For some q > 2, κ q < ∞, and M < ∞, (a) The vector z ∈ R d has independent entries with E[z i ] = 0, E[z 2 i ] = 1, and E[\|z i \| 2q ] ≤ κ q < ∞ (b) s 1 = \|\|\|Σ i \|\|\| op ≤ M , s d = λ min (Σ i ) ≥ 1/M , and s −1 dµ(s; Σ i ) < M . (c) µ(·; Σ i ) converges weakly to ν i These conditions are standard, guaranteeing sufficient moments for convergence of covariance estimates, that the eigenvalues of Σ i do not accumulate near 0, and a mode of convergence for the spectrum of Σ i . Locally fine-tuning a global solution In this section, we describe and analyze fine-tuning algorithms that use the FedAvg solution (a minimizer of the objective (2)) as a warm start to find personalized models. We compare the test loss of these algorithms with naive, zero personalization and zero collaboration approaches. Among other things, we show that a ridge-regularized locally fine-tuned method outperforms the other methods. Fine-tuned Federted Averaging (FTFA) Fine-tuned Federated Averaging (FTFA) approximates minimizing the multi-criterion loss (1) using a two-step procedure in Algorithm 1 (see Section 7 for detailed pseudocode). Let S i denote client i's sample. The idea is to replace the local expected risks L i in (2) with the local empirical risks L i (θ) := 1 n i z∈Si (θ; z), using the FedAvg solutionθ F A 0 as a warm-start for local training in the second step. Intuitively, FTFA interpolates between zero collaboration and zero personalization algorithms. Each client i can run this local training phase independently of (and in parallel with) all others, as the data is fully local; this separation makes FTFA essentially no more expensive than Federated Averaging. For the linear model in Section 2, FTFA first minimizes the average weighted loss m j=1 p j 1 2nj X j θ − y j 2 2 over all clients. As the local linear regression problem to minimize X i θ − y i 2 2 is overparameterized, first-order methods on it (including the stochastic gradient method) correspond to solving minimum Algorithm 1 FTFA & RTFA (details in appendix) n j → 0 as m, d, n j → ∞. In Assumption A3, p j is the weight associated with the loss of j th client when finding the global model using federated averaging. To ground the assumption, consider two particular cases of interest: (i) p j = 1/m, when every client is weighted equally, and (ii) p j = n j /N , when each data point is weighted equally. When p j = 1/m, we have (log d) cq m j=1 p q/2+1 j n j = N m (log d) cq m q/2 . When p j = n j /N , using Assumption A1 we have (log d) cq m j=1 p q/2+1 j n j = (log d) cq m j=1 n q/2+2 j N q/2+1 ≤ ( γmax γmin ) q/2+1 N m (log d) cq m q/2 .B i (θ F A i \|X) p = lim m→∞ Π i [θ 0 − θ i ] 2 Σi lim m→∞ V i (θ F A i \|X) p = lim m→∞ σ 2 i n i tr(Σ † i Σ i ), where Π i := I −Σ † iΣ i and z 2 Σi := z T Σ i z. The exact expressions of these limits for general choice Σ in the implicit form can be found in the appendix. For the special case when Σ i = I, the closed form limits are B i (θ F A i \|X) p → r 2 i 1 − 1 γ i V i (θ F A i \|X) p → σ 2 i γ i − 1 , where p → denotes convergence in probability. Ridge-tuned FedAvg (RTFA) Minimum-norm results provide insight into the behavior of popular algorithms including SGM and mirror descent. Having said that, we can also analyze ridge penalized versions of FTFA. In this algorithm, the server finds the same global model as FTFA, but each client uses a regularized objective to find a local personalized model as in 2b of Algorithm 1. More concretely, in the linear regression setup, for appropriately chosen step size and as the number of steps taken goes to infinity, this corresponds to the two step procedure with the first step (6) and second step θ R i (λ) = argmin θ 1 2n i X i θ − y i 2 2 + λ 2 θ F A 0 − θ 2 2 ,(8) where RTFA outputs the modelθ R i (λ) for client i. Under the same assumptions as Theorem 1, we can again calculate the asymptotic test loss. Theorem 2. Let the conditions of Theorem 1 hold. Then for client i, the asymptotic prediction bias and variance of RTFA are lim m→∞ B i (θ R i (λ)\|X) p = lim m→∞ λ 2 (Σ i + λI) −1 (θ 0 − θ i ) 2 Σi lim m→∞ V i (θ R i (λ)\|X) p = lim m→∞ σ 2 i n i tr(Σ iΣi (λI +Σ i ) −2 ), The exact expressions of these limits for general choice Σ in the implicit form can be found in the appendix. For the special case when Σ i = I, the closed form limits are B i (θ R i (λ)\|X) p → r 2 i λ 2 m i (−λ) V i (θ R i (λ)\|X) p → σ 2 i γ i (m i (−λ) − λm i (−λ)), where m i (z) is the Stieltjes transform of the limiting spectral measure ν i of the covariance matrix Σ i . When Cov(x j,k ) = I, m i (z) has the closed form expression m i (z) = (1−γ i −z− (1 − γ i − z) 2 − 4γ i z)/(2γ i z). For each client i ∈ [m], when λ is set to be the minimizing value λ i = σ 2 i γ i /r 2 i , the expression simplifies to L i (θ R i (λ i )\|X) → σ 2 i γ i m i (−λ i ). With the optimal choice of hyperparameter λ, RTFA has lower test loss than FTFA; indeed, in overparameterized linear regression, the ridge solution with regularization λ → 0 converges to the minimum 2 -norm interpolant (7). L i (θ R i (σ 2 i γ i /r 2 i ) \| X) − L i (θ N i (σ 2 i γ i /ρ 2 i ) \| X) = C · (ρ 2 i − r 2 i ) + o(ρ 2 i − r 2 i ), where C depends on all the problem parameters. With appropriate regularization λ, RTFA mitigates the weaknesses of FTFA. Thus, formally, we may show that RTFA with the optimal hyperparameter always outperforms the zero-personalization estimatorθ F A 0 (see the appendices). Furthermore, since ρ i ≥ r i , its straightforward to see that RTFA outperforms ridgeless zero-collaboration estimatorθ N i , and the ridge-regularized zero-collaboration estimatorθ N i (λ ) as well. Meta learning and Proximal Regularized Algorithms The fine-tuning procedures in the previous section provide a (perhaps) naive baseline, so we consider a few alternative federated learning procedures, both of which highlight the advantages of the highdimensional asymptotics in its ability to predict performance. While we cannot survey the numerous procedures in FL, we pick two we consider representative: the first adapting meta learning [FMO20] and the second using a proximal-regularized approach [DTN20]. In both cases, the researchers develop convergence rates for their methods (in the former case, to stationary points), but no results on the predictive performance or their statistical behavior exists. We develop these in this section, showing that these more sophisticated approaches perform no better, in our asymptotic framework, than the FTFA and RTFA algorithms we outline in Section 3. Model-Agnostic Meta-Learning Model-Agnostic Meta-Learning (MAML) [FAL17] was learns models that adapt to related tasks by minimizing an empirical loss augmented evaluated not at a given parameter θ but at a "one-step-updated" parameter θ − α∇L(θ), representing one-shot learning. [FMO20], contrasting this MAML approach to the standard averaging objectives (2), adapt MAML to the federated setting, developing a method we term MAML-FL. We describe their two step procedure in Algorithm 2 (see Section 7 for detailed pseudocode). Algorithm 2 has two variants [FMO20]; in one, the Hessian term is ignored, and in the other, the Hessian is approximated using finite differences. [FMO20] showed that these these algorithms converge to a stationary point of eq. (11) (with p j = 1/m) for general non-convex smooth functions. Algorithm 2 MAML-FL (details in Appendix 7) 1. Server and clients coordinate to (approximately) solvê θ M 0 (α) = argmin θ m j=1 p j L j (θ − α∇ L j (θ)),(11) where p j ∈ (0, 1) are weights such that m j=1 p j = 1 and α denotes stepsize. Server broadcasts global modelθ M 0 (α) to clients. 2. Client i learns a modelθ M i (α) by optimizing its empirical risk, L i (·), using SGM initialized atθ M 0 (α) In our linear model, for appropriately chosen hyperparameters and as the number of steps taken goes to infinity, this personalization method corresponds to the following two step procedure: θ M 0 (α) (12) = argmin θ m j=1 p j 2n j X j θ − α n j X T j (X j θ − y j ) − y j 2 2 θ M i (α) = argmin θ θ M 0 (α) − θ 2 s.t. X i θ = y i(13) As in Section 3.1, we assume that the client model in step 2 of Alg. 2 has fully converged; any convergent first-order method converges to the minimum norm interpolant (13). The representations (12) and (13) allow us to analyze the test loss of the MAML-FL personalization scheme in our asymptotic framework. Theorem 3. Consider the observation model in Section 2 and the estimatorθ M i (α) in (13). Let Assumption A1 hold, Assumption A2 hold with q = 3v where v > 2, and Assumption A3 hold with c = 5 and q = v. Additionally, assume that for each m and all j ∈ [m], λ min (E[Σ j (I − αΣ j ) 2 ]) ≥ λ 0 > 0 and \|\|\|E[Σ 6 j ]\|\|\| op ≤ τ 6 < ∞. Then for client i, the asymptotic prediction bias and variance of MAML-FL are lim m→∞ B i (θ M i (α)\|X) p = lim m→∞ Π i [θ 0 − θ i ] 2 Σi lim m→∞ V i (θ M i (α)\|X) p = lim m→∞ σ 2 i n i tr(Σ † i Σ i ), where Π i := I −Σ † iΣ i and z 2 Σi := z T Σ i z. The exact expressions of these limits for general choice Σ in the implicit form are in the appendices. For the special case that Σ i = I, the closed form limits are B i (θ M i (α)\|X) p → r 2 i 1 − 1 γ i V i (θ M i (α)\|X) p → σ 2 i γ i − 1 . In short, the asymptotic test risk of MAML-FL matches that of FTFA (Theorem 1). In general, the MAML-FL objective (11) is typically non-convex even when L j is convex, making convergence subtle. Even ignoring convexity, the inclusion of a derivative term in the objective can make the standard smoothness conditions [Nes04] upon which convergence analyses (and algorithms) repose fail to hold. Additionally, computing gradients of the MAML-FL objective (11) requires potentially expensive second-order derivative computations or careful approximations to these, making optimization more challenging and expensive irrespective of convexity. We provide more discussion in the appendices. Theorem 3 thus suggests that one might be circumspect about choosing MAML-FL or similar algorithms over simpler baselines that do not require such complexity in optimization. Remark The algorithm [FMO20] propose performs only a single stochastic gradient step for personalization, which is distinct from our analyzed procedure (13), as it is essentially equivalent to running SGM until convergence from the initializationθ M 0 (α)M α (see step 2 of Algorithm 2). We find two main justifications for this choice: first, experimental work of [JKRK19], in addition to our own experiments (see Figures 5 and 6) empirically suggest that the more (stochastic gradient) steps of personalization, the better performance we expect. Furthermore, as we mention above earlier, performing personalization SGM steps locally, in parallel, and asynchronously is no more expensive than running the first step of Algorithm 2. This guaranteed of convergence also presents a fair point of comparison between the algorithms we consider. Proximal-Regularized Approach Instead of a sequential, fine-tuning approach, an alternative approach to personalization involves jointly optimizing global and local parameters. In this vein, [DTN20] propose the pFedMe algorithm (whose details we provide in the appendices) to solve the following coupled optimization problem to find personalized models for each client: θ P 0 (λ),θ P 1 (λ), . . . ,θ P m (λ) = argmin θ0,θ1,...,θm m j=1 p j L i (θ j ) + λ 2 θ j − θ 0 2 2 . The proximal penalty encourages the local models θ i to be close to one another. In our linear model, for appropriately chosen hyperparameters and as the number of steps taken goes to infinity, the proposed optimization problem simplifies to θ P 0 (λ),θ P 1 (λ), . . . ,θ P m (λ) = (14) argmin θ0,θ1,...,θm m j=1 p j 1 2n j X j θ j − y j 2 2 + λ 2 θ j − θ 0 2 2 , whereθ P 0 (λ) denotes the global model andθ P i (λ) denote the local models. We can again use our asymptotic framework to analyze the test loss of this scheme. For this result, we use an additional condition on sup j∈[m] P(λ max (Σ j ) > R) that gives us uniform control over the eigenvalues of all the users. Theorem 4. Consider the observation model n section 2 and the estimatorθ P i (λ) in (14). Let Assumption A1 hold, and let Assumptions A3 and A2 hold with c = 2 and the same q > 2. Ad- ditionally, assume that E[\|\|\|Σ 2 j \|\|\| op ] ≤ τ 3 < ∞. Further suppose that there exists R ≥ M such that lim sup m→∞ sup j∈[m] P(λ max (Σ j ) > R) ≤ 1 16M 2 τ3 . Then for client i, the asymptotic prediction bias and variance of pFedMe are lim m→∞ B i (θ P i (λ)\|X) p = lim m→∞ λ 2 (Σ i + λI) −1 (θ 0 − θ i ) 2 Σi lim m→∞ V i (θ P i (λ)\|X) p = lim m→∞ σ 2 i n i tr(Σ iΣi (λI +Σ i ) −2 ), See the appendices for exact expressions of these limits for general Σ. For the special case that Σ i = I, the limits are B i (θ P i (λ)\|X) p → r 2 i λ 2 m i (−λ) V i (θ P i (λ)\|X) p → σ 2 i γ(m i (−λ) − λm i (−λ)),where m i (z) is as in Theorem 2. For each client i ∈ [m], the minimizing λ is λ i = σ 2 i γ i /r 2 i and L i (θ P i (λ i )\|X) → σ 2 i γ i m i (−λ i ). The asymptotic test loss of the proximal-regularized approach is thus identical to the locally ridgeregularized (RTFA) solution; see Theorem 2. [DTN20]'s algorithm to optimize eq. (14) is sensitive to hyperparameter choice, meaning significant hyperparameter tuning may be needed for good performance and even convergence of the method (of course, both methods do require tuning λ). Moreover, a local update step in pFedMe requires approximately solving a proximal-regularized optimization problem, as opposed to taking a single stochastic gradient step. This can make pFedMe much more computationally expensive depending on the properties of L. This is not to dismiss more complex proximal-type algorithms, but only to say that, at least in our analytical framework, simpler and embarassingly parallelizable procedures (RTFA in this case) may suffice to capture the advantages of a proximal-regularized scheme. Experiments While the statistical model we assume in our analytical sections is stylized and certainly will not fully hold, it suggests some guidance in practice, and make precise predictions about the error rates of different methods: that the simpler fine-tuning methods should exhibit performance comparable to more complex federated methods, such as MAML-FL and pFedMe. With this in mind, we turn to several datasets, performing experiments on federated versions of the Shakespeare [MMR + 17], CIFAR-100 [KH09], EMNIST [CATvS17], and Stack Overflow [MRR + 19] datasets; dataset statistics and details of how we divide the data to make effective "users" are in Section 8. For each dataset, we compare the performance of the following algorithms: Zero Communication (Local Training), Zero Personalization (FedAvg), FTFA, RTFA, MAML-FL, and pFedMe [DTN20]. For each classification task, we use each federated learning algorithm to train the last layer of a pre-trained neural network. We run each algorithm for 400 communication rounds, and we compute the test accuracy (the fraction of correctly classified test data points across machines) every 50 communication rounds. FTFA, RTFA, and MAML-FL each perform 10 epochs of local training for each client before the evaluation of test accuracy. For each client, pFedMe uses the local models to compute test accuracy. We first hyperparameter tune each method using training and validation splits; again, see Appendix 8 for details. We track the test accuracy of each tuned method over 11 trials using two different kinds of randomness: 1. Different seeds: We run each hyperparameter-tuned method on 11 different seeds. This captures how different initializations and batching affect accuracy. 2. Different training-validation splits: We generate 11 different training / validation splits (same test data) and run each hyperparameter-tuned method on each split. This captures how variations in user data affect test accuracy. Experimental setting Our experiments are "semi-synthetic" in that in each, we re-fit the top layer of a pre-trained neural network. While this differs from some practice with experimental work in federated learning, several considerations motivate our choices to take this tack, and we contend they may be valuable for other researchers: (i) our (distributed) models are convex, that is, can be fit via convex optimization. In the context of real-world engineering problems, it is important to know when a method has converged and, if it does not, why it has not; in this vein, non-convexity can be a bugaboo, as it hides the causes of divergent algorithms-is it non-convexity and poor optimization or engineering issues (e.g. communication bugs)? This choice thus can be valuable even in real, large-scale systems. (ii) In our experiments, we achieve state-of-the-art or near state-of-the-art results; using federated approaches to fit full deep models appears to lead to substantial degradation in performance over a single centralized, pre-trained model (see, e.g., [RCZ + 21, Table 1], where accuracies on CIFAR-10 using a ResNet 18 are at best 78%, substantially lower than current state-of-the-art). A question whose answer we do not know: if a federated learning method provides worse performance than a downloadable model, what does the FL method's performance tell us about good methodologies in federated learning? (iii) Finally, computing with large-scale distributed models is computationally expensive: the energy use for fitting large distributed models is substantial and may be a poor use of resources [SGM19]. In effort to better approximate the use of a pre-trained model in real federated learning applications, we use held-out data to pre-train a preliminary network in our Stack Overflow experiments, doing the experimental training and validation on an independent dataset. Results Figures 1 to 4 plot test accuracy against communication rounds. The performance of MAML-FL is similar to that of FTFA and RTFA, and on the Stack Overflow and EMNIST datasets, where the total dataset size is much larger than the other datasets, the accuracies of MAML-FL, FTFA and RTFA are nearly identical. This is consistent with our theoretical claims. The performances of the naive, zero communication and zero personalization algorithms are worse than that of FTFA, RTFA and MAML-FL in all figures. This is also consistent with our theoretical claims. The performance of pFedMe in Figures 1 to 3 is worse than that of FTFA, RTFA and MAML-FL. In Figures 5 and 6, we plot the test accuracy of FTFA and MAML-FL and vary the number of personalization steps each algorithm takes. In both plots, the global model performs the worst, and performance improves monotonically as we increase the number of personalization steps. As personalization steps are cheap relative to the centralized training procedure, this suggests benefits for clients to locally train to convergence. LG], 2020. Proofs Additional Notation To simplify notation, we define some aggregated parameters, X i := [x i,1 , . . . , x i,ni ] T ∈ R ni×d , y i = [y i,1 , . . . , y i,ni ] T ∈ R ni , X := [X T 1 , . . . , X T m ] T ∈ R N ×d , and y := [y T 1 , . . . , y T m ] T ∈ R N . Additionally, we defineΣ i := X T i X i /n i ∈ R d×d . We use the notation a b to denote a ≤ Kb for some absolute constant K. Useful Lemmas Lemma 6.1. Let x j be vectors in R d and let ζ j be Rademacher (±1) random variables. Then, we have E   m j=1 ζ j x j p 2   1/p ≤ p − 1   m j=1 x j 2 2   1/2 , where the expectation is over the Rademacher random variables. Proof Using Theorem 1.3.1 of [dlPnG99], we have E   m j=1 ζ j x j p 2   1/p ≤ p − 1E    m j=1 ζ j x j 2 2    1/2 = p − 1E   m i,j=1 ζ j ζ i x T j x i   = p − 1   m j=1 x j 2 2   1/2 Lemma 6.2. For all clients j ∈ [m], let the data x j,k ∈ R d for k ∈ [n] be such that x j,k = Σ 1/2 j z j,k for some Σ j , z j,k , and p > 2 that satisfy Assumption A2. Let (x j,k ) l ∈ R denote the l ∈ [d] entry of the vector x j,k ∈ R d . DefineΣ j = 1 nj k∈ [nj ] x j,k x T j,k . Then, we have E Σ j p op ≤ K(e log d) p n j , where the inequality holds up to constant factors for sufficiently large m. Proof We first show a helpful fact that E[(z j,k ) 2p l ] ≤ κ p < ∞ implies E[ x j,k 2p 2 ] 1/(2p) √ d. For any j ∈ [m], we have by Jensen's inequality E[ x j,k 2p 2 ] ≤ M 2p E[ z j,k 2p 2 ] = M 2p d p E 1 d d l=1 (z j,k ) 2 l p ≤ M 2p d p 1 d d l=1 E[(z j,k ) 2p l ] ≤ M 2p κ p d p We define some constant C 4 > M 2p κ p . With this fact and Theorem A.1 from [CGT12], we have E Σ j p op = E   n k=1 x j,k x T j,k n p op   ≤ 2 2p−1 \|\|\|Σ j \|\|\| p op + (e log d) p n p j E max k x j,k x T j,k p op ≤ 2 2p−1 C + (e log d) p n p−1 j E x j,k 2p 2 ≤ 2 2p−1 C + C 4 (e log d) p d p n p−1 j Now, 2 2p−1 C + C 4 (e log d) p d p n p−1 j ≤ K(e log d) p n j for some absolute constant K since d nj → γ i . Lemma 6.3. For all clients j ∈ [m], let the data x j,k ∈ R d for k ∈ [n] be such that x j,k = Σ 1/2 j z j,k for some Σ j , z j,k , and q > 2 that satisfy Assumption A2. Further let q = pq where p ≥ 1 and q ≥ 2. LetΣ j = 1 nj k∈[nj ] x j,k x T j,k and µ j = E[Σ p j ]. Additionally assume that E[Σ 2p j ] op ≤ C 3 for some constant C 3 . Let d, n j grow as in Assumption A1. Then we have for sufficiently large m, P   m j=1 p j Σ p j − µ j op > t   ≤ 2 q−1 C 2 t q   (log d) q/2 m j=1 p q/2+1 j + (log d) pq+q m j=1 p q j n j   . Further supposing that (log d) pq+q m j=1 p q j n j → 0 , we get that m j=1 p j Σ p j − µ j op p → 0 . Proof Using Markov's inequality, Jensen's inequality, and symmeterization, we have with ζ j iid Rademacher P   m j=1 p j Σ p j − µ j op > t   ≤ E m j=1 p j (Σ p j − µ j ) q op t q ≤ 2 q E m j=1 p jΣ p j ζ j q op t q We use the second part of Theorem A.1 with q ≥ 2 from [CGT12] to bound the RHS. E   m j=1 p jΣ p j ζ j q op   ≤   e log d E[ m j=1 p 2 jΣ 2p j ] 1/2 op + (2e log d)E max j p jΣ p j q op 1/q   q ≤ 2 q−1 (e log d) q/2 E[ m j=1 p 2 jΣ 2p j ] 1/2 q op + 2 q−1 (e log d) q E max j p q j Σ p j q op ≤ 2 q−1 (e log d) q/2 E   m j=1 p 2 jΣ 2p j   q/2 op + 2 q−1 (e log d) q E   m j=1 p q j Σ j pq op  E Σ 2p j q/2 op   + 2 q−1 (e log d) q m j=1 p q j K(e log d) pq n j ≤ C 2   (log d) q/2 m j=1 p q/2+1 j + (log d) pq+q m j=1 p q j n j   , where in the first term of the first inequality, we use Jensen's inequality to pull out m j=1 p j of the expectation. To prove the second part of the lemma, we observe that if (log d) (p+1)q m j=1 p q j n j → 0 as m → ∞ such that d/n i → γ i > 1 for all devices i ∈ [m], then (log d) q/2 m j=1 p q/2+1 j → 0. To see this, we first observe (log d) (p+1)q m j=1 p q j n j ≥ ( max j∈[m] p j (log d) p+1 ) q , so we know that max j∈[m] p j (log d) p+1 → 0. Further, by Holder's inequality, we know that (log d) q/2 m j=1 p q/2+1 j ≤ ( max j∈[m] p j log d) q/2 . By the continuity of the q/2 power, we get the result. Proof For any t > 0, we have by Theorem 2.5 (from Section III) of [SS90] P V −1 − U −1 op > t ≤ P V −1 − U −1 op > t ∩ \|\|\|V − U \|\|\| op < 1 \|\|\|U −1 \|\|\| op + P \|\|\|V − U \|\|\| op ≥ λ 0 ≤ P U −1 (V − U ) op > t t + \|\|\|U −1 \|\|\| op + o(1) ≤ P \|\|\|V − U \|\|\| op > tλ 0 t + λ −1 0 + o(1) We know this quantity goes to 0 by assumption. Some useful definitions from previous work In this section, we recall some definitions from [HMRT19] that will be useful in finding the exact expressions for risk. The expressions for asymptotic risk in high dimensional regression problems (both ridge and ridgeless) are given in an implicit form in [HMRT19]. It depends on the geometry of the covariance matrix Σ and the true solution to the regression problem θ . Let Σ = d i=1 s i v i v T i denote the eigenvalue decomposition of Σ with s 1 ≥ s 2 · · · ≥ s d , and let (c, . . . , v T d θ ) denote the inner products of θ with the eigenvectors. We define two probability distributions which will be useful in giving the expressions for risk: H n (s) := 1 d d i=1 1{s ≥ s i } , G n (s) := 1 θ 2 2 d i=1 (v T i θ ) 2 1{s ≥ s i } . Note that G n is a reweighted version of H n and both have the same support (eigenvalues of Σ). Definition 6.1. For γ ∈ R + , let c 0 = c 0 (γ, H n ) be the unique non-negative solution of 1 − 1 γ = 1 1 + c 0 γs d H n (s), the predicted bias and variance is then defined as B( H n , G n , γ) := θ 2 2    1 + γc 0 s 2 (1+c0γs) d H n (s) s (1+c0γs) d H n (s)    · s (1 + c 0 γs) d G n (s),(15)V ( H n , γ) := σ 2 γ s 2 (1+c0γs) d H n (s) s (1+c0γs) d H n (s) .(16) Definition 6.2. For γ ∈ R + and z ∈ C + , let m n (z) = m(z; H n , γ) be the unique solution of m n (z) := 1 s[1 − γ − γzm n (z)] − z d H n (s). Further, define m n,1 (z) = m n,1 (z; H n , γ) as m n,1 (z) := s 2 [1−γ−γzmn(z)] [s[1−γ−γzmn(z)]−z] 2 d H n (s) 1 − γ zs [s[1−γ−γzmn(z)]−z] 2 d H n (s) The definitions are extended analytically to Im(z) = 0 whenever possible, the predicted bias and variance are then defined by B(λ; H n , G n , γ) := λ 2 θ 2 (1 + γm n,1 (−λ)) s [λ + (1 − γ + γλm n (−λ))s] 2 d G n (s), V (λ; H n , γ) := σ 2 γ s 2 ((1 − γ + γλm n (−λ))) [λ + (1 − γ + γλm n (−λ))s] 2 d H n (s). (18) Proof of Theorem 1 On solving (6) and (7), the closed form of the estimatorsθ F A 0 andθ F A i is given bŷ θ F A 0 = argmin θ m j=1 p j 1 2n j X j θ − y j 2 2 =   m j=1 p jΣj   −1 m j=1 p j X T j y j n j =   m j=1 p jΣj   −1 m j=1 p jΣj θ j +   m j=1 p jΣj   −1 m j=1 p j X T j ξ j n j(19) and θ F A i = (I −Σ † iΣ i )θ F A 0 + X † i y i = (I −Σ † iΣ i )θ F A 0 +Σ † iΣ i θ i + 1 n iΣ † i X T i ξ i = Π i      m j=1 p jΣj   −1 m j=1 p jΣj θ j +   m j=1 p jΣj   −1 m j=1 p j X T j ξ j n j    +Σ † iΣ i θ i + 1 n iΣ † i X T i ξ i We now calculate the risk by splitting it into two parts as in (3), and then calculate the asymptotic bias and variance. Bias: B i (θ F A i \|X) := E[θ F A i \|X] − θ i 2 Σi = Π i      m j=1 p jΣj   −1 m j=1 p jΣj (θ j − θ i )    2 Σi = Σ 1/2 i Π i   θ 0 − θ i +   m j=1 p jΣj   −1 m j=1 p jΣj (θ j − θ 0 )    2 2 The idea is to show that the second term goes to 0 and use results from [HMRT19] to find the asymptotic bias. For simplicity, we let ∆ j := θ j − θ 0 , and we define the event: B t :=        m j=1 p jΣj   −1 −   m j=1 p j Σ j   −1 op > t      A t :=        m j=1 p jΣj   −1 m j=1 p jΣj ∆ j Σi > t      The proof proceeds in the following steps: Bias Proof Outline Step 1. We first show for any t > 0, the P(B t ) → 0 as d → ∞ Step 2. Then we show for any t > 0, the P(A t ) → 0 as d → ∞ Step 3. We show that for any t ∈ (0, 1]on event A c t , B i (θ F A i \|X) ≤ Π i [θ 0 − θ i ] 2 Σi + ct and B i (θ F A i \|X) ≥ Π i [θ 0 − θ i ] 2 Σi − ct Step 4. Show that lim d→∞ P( \|B i (θ F A i \|X) − Π i [θ 0 − θ i ] 2 Σi \| ≤ ε) = 1 Step 5. Finally, using the asymptotic limit of Π i [θ 0 − θ i ] 2 Σi from Theorem 1 of [HMRT19], we get the result. Step 1 Since we have λ min ( m j=1 p j Σ j ) > 1/M > 0, it suffices to show by Lemma 6.4 that the probability of C t :=    m j=1 p jΣj − m j=1 p j Σ j op > t    goes to 0 as d, m → ∞ (obeying Assumption A1). Using Lemma 6.3 with p = 1, we have that P(C t ) ≤ 2 q−1 C 2 t q   (log d) q/2 m j=1 p q/2+1 j + (log d) 2q m j=1 p q j n j   Since (log d) 2q m j=1 p q j n j → 0, this quantity goes to 0. Step 2 Fix any t > 0, P(A t ) ≤ P           m j=1 p jΣj   −1 m j=1 p jΣj ∆ j Σi > t      ∩ B c c1    + P (B c1 ) ≤ P   M (c 1 + M ) m j=1 p jΣj ∆ j 2 > t   + P (B c1 ) By Step 1, we know that P (B c1 ) → 0. The second inequality comes from Ax 2 ≤ \|\|\|A\|\|\| op x 2 and triangle inequality. Now to bound the first term, we use Markov and a Khintchine inequality (Lemma 6.1). We have that P   M (c 1 + M ) m j=1 p jΣj ∆ j 2 > t   ≤ (M (c 1 + M )) q E m j=1 p jΣj ∆ j q 2 t q ≤ 2M (c 1 + M ) √ q q E m j=1 p jΣj ∆ j 2 2 q/2 t q Using Jensen's inequality and the definition of operator norm, we have 2M (c 1 + M ) √ q q E m j=1 p 2 j Σ j ∆ j 2 2 q/2 t q = 2M (c 1 + M ) √ q q E m j=1 p j · p j Σ j ∆ j 2 2 q/2 t q ≤ 2M (c 1 + M ) √ q q m j=1 p q/2+1 j E Σ j ∆ j q 2 t q ≤ 2M (c 1 + M ) √ q q m j=1 p q/2+1 j E Σ j q op E ∆ j q 2 t q Lastly, we can bound this using Lemma 6.2 as follows. P   M (c 1 + M ) m j=1 p jΣj ∆ j 2 > t   ≤ K 2M (c 1 + M ) √ q q (e log d) q m j=1 n j p q/2+1 j E[ ∆ j q 2 ] t q → 0, using (log d) q m j=1 p q/2+1 j n j r q j → 0. Step 3 For any t ∈ (0, 1], on the event A c t , we have that B(θ F A i \|X) = Π i [θ 0 − θ i + E] 2 Σi for some vector E where we know E 2 ≤ t(which means E 2 ≤ t √ M ). Thus, we have Π i [θ 0 − θ i + E] 2 Σi ≤ Π i [θ 0 − θ i ] 2 Σi + Π i E 2 Σi + 2 Π i E Σi Π i [θ 0 − θ i ] Σi ≤ Π i [θ 0 − θ i ] 2 Σi + M 2 t 2 + 2tM 3/2 r 2 i Π i [θ 0 − θ i + E] 2 Σi ≥ Π i [θ 0 − θ i ] 2 Σi + Π i E 2 Σi − 2 Π i E 2 Π i [θ 0 − θ i ] Σi ≥ Π i [θ 0 − θ i ] 2 Σi − 2tM 2 r 2 i Since t ∈ (0, 1], we have that t 2 ≤ t and thus we can choose c = M 2 + 2M 3/2 r 2 i Step 4 Reparameterizing ε := ct, we have that for any ε > 0 lim n→∞ P(\|B i (θ F A i \|X) − Π i [θ 0 − θ i ] 2 Σi \| ≤ ε) ≥ lim n→∞ P(\|B i (θ F A i \|X) − Π i [θ 0 − θ i ] 2 Σi \| ≤ ε ∧ c) ≥ lim n→∞ P(A c ε c ∧1 ) = 1 Step 5 Using Theorem 3 of [HMRT19], as d → ∞, such that d ni → γ i > 1, we know that the limit of Π i [θ 0 − θ i ] 2 Σi is given by (15) with γ = γ i and H n , G n be the empirical spectral distribution and weighted empirical spectral distribution of Σ i respectively. In the case when Σ i = I, using Theorem 1 of [HMRT19] we have B i (θ F A i \|X) = Π i [θ 0 − θ i ] 2 2 → r 2 i 1 − 1 γi . Variance: We let ξ i = [ξ i,1 , . . . , ξ i,n ] denote the vector of noise. V i (θ F A i \|X) = tr(Cov(θ F A i \|X)Σ i ) = E θ F A i − E θ F A i \|X 2 Σi \|X = E    Π i      m j=1 p jΣj   −1 m j=1 p j X T j ξ j n j    + 1 n iΣ † i X T i ξ i 2 Σi \|X    = m j=1 p 2 j n j tr   ΠiΣiΠi   m j=1 p jΣj   −1Σ j   m j=1 p jΣj   −1    σ 2 j (i) + 2 tr   ΣiΣ † i X T i X i n 2 i   m j=1 p jΣj   −1 Π i    σ 2 i (ii) + 1 n 2 i tr Σ † i X T i X iΣ † i Σ i σ 2 i (iii) We now study the asymptotic behavior of each of the terms (i), (ii) and (iii) separately. (i) Using the Cauchy Schwartz inequality on Schatten p−norms and using the fact that the nuclear norm of a projection matrix is at most d, we get m j=1 p 2 j σ 2 j n j tr   ΠiΣiΠi   m j=1 p jΣj   −1Σ j   m j=1 p jΣj   −1    ≤ m j=1 p 2 j σ 2 j n j \|\|\|Π i \|\|\| 1 \|\|\|Σ i \|\|\| op \|\|\|Π i \|\|\| op   m j=1 p jΣj   −1 op Σ j op   m j=1 p jΣj   −1 op ≤ C 3 σ 2 max γ max   m j=1 p 2 j Σ j op   ,(20) where the last inequality holds with probability going to 1 for some constant C 3 because P(B t ) → 0. Lastly, we show that P m j=1 p 2 j Σ j op > t → 0. Using Markov's and Jensen's inequality, we have P   m j=1 p 2 j Σ j op > t   ≤ E m j=1 p 2 j Σ j op q t q ≤ m j=1 p q+1 j E Σ j q op t q Using Lemma 6.2, we have P   m j=1 p 2 j Σ j op > t   ≤ K m j=1 p q+1 j (e log d) q n j t q Finally, since we know that m j=1 p q+1 j (e log d) q n j → 0, we have m j=1 p 2 j Σ j op p → 0. Thus, m j=1 p 2 j σ 2 j n j tr   ΠiΣiΠi   m j=1 p jΣj   −1Σ j   m j=1 p jΣj   −1    p → 0 (ii) Using the Cauchy Schwartz inequality on Schatten p−norms and using the fact that the nuclear norm of a projection matrix is d − n, we get 2p i σ 2 n i tr   ΠiΣiΣ † iΣ i   m j=1 p jΣj   −1    ≤ 2p i σ 2 n i \|\|\|Π i \|\|\| 1 \|\|\|Σ i \|\|\| op Σ † iΣ i op   m j=1 p jΣj   −1 op ≤ C 4 p i , where the last inequality holds with probability going to 1 for some constant C 4 because P(B t ) → 0 and using Assumption A2. Since p i → 0, we have 2p i σ 2 n i tr   ΠiΣiΣ † iΣ i   m j=1 p jΣj   −1    → 0 (iii) 1 n 2 i tr(Σ † i X T i X iΣ † i Σ i )σ 2 i = 1 n i tr(Σ † i Σ i )σ 2 i Using Theorem 3 of [HMRT19], as d → ∞, such that d ni → γ i > 1, we know that the limit of σ 2 i ni tr(Σ † i Σ i ) is given by (16) with γ = γ i and H n , G n be the empirical spectral distribution and weighted empirical spectral distribution of Σ i respectively. In the case when Σ i = I, using Theorem 1 of [HMRT19] we have V i (θ F A i \|X) = σ 2 i ni tr(Σ † i ) → σ 2 i γi−1 . Proof of Theorem 2 We use the global model from (6) and the personalized model from (8). The closed form of the estimatorsθ F A 0 andθ R i (λ) is given bŷ θ F A 0 = argmin θ m j=1 p j 1 2n j X j θ − y j 2 =   m j=1 p jΣj   −1 m j=1 p j X T j y j n j =   m j=1 p jΣj   −1 m j=1 p jΣj θ j +   m j=1 p jΣj   −1 m j=1 p j X T j ξ j n j andθ R i (λ) = argmin θ 1 2n i X i θ − y i 2 2 + λ 2 θ F A 0 − θ 2 2 = (Σ i + λI) −1 λθ F A +Σ i θ i + 1 n i X T i ξ i We now calculate the risk by splitting it into two parts as in (3), and then calculate the asymptotic bias and variance. Bias: B(θ R i (λ)\|X) := E[θ R i (λ)\|X] − θ i 2 Σi = λ 2 (Σ i + λI) −1      m j=1 p jΣj   −1 m j=1 p jΣj (θ j − θ i )    2 Σi = λ 2 (Σ i + λI) −1   θ 0 − θ i +   m j=1 p jΣj   −1 m j=1 p jΣj (θ j − θ 0 )    2 Σi The idea is to show that the second term goes to 0 and use results from [HMRT19] to find the asymptotic bias. For simplicity, we let ∆ j := θ j − θ 0 , and we define the event: B t :=        m j=1 p jΣj   −1 −   m j=1 p j Σ j   −1 op > t      A t :=        m j=1 p jΣj   −1 m j=1 p jΣj ∆ j Σi > t      The proof proceeds in the following steps: Bias Proof Outline Step 1. We first show for any t > 0, the P(B t ) → 0 as d → ∞ Step 2. We show for any t > 0, the P(A t ) → 0 as d → ∞ Step 3. We show that for any t ∈ (0, 1] on event A c t , B(θ R i (λ)\|X) ≤ λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi + ct and B(θ R i (λ)\|X) ≥ λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi − ct Step 4. Show that lim d→∞ P( \|B(θ R i (λ)\|X) − λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi \| ≤ ε) = 1 Step 5. Finally, using the asymptotic limit of λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi from Corollary 5 of [HMRT19], we get the result. Step 1 and Step 2 follow from Step 1 and Step 2 of proof of Theorem 1. Step 3 For any t ∈ (0, 1], on the event A c t where T −1 i = (Σ i + λI) −1 , we have that B(θ R i (λ)\|X) = λ 2 T −1 i [θ 0 − θ i + E] 2 Σi for some vector E where we know E Σi ≤ t (which means E 2 ≤ t √ M ) . We can form the bounds T −1 i [θ 0 − θ i + E] 2 Σi ≤ T −1 i [θ 0 − θ i ] 2 Σi + T −1 i E 2 Σi + 2 T −1 i E Σi T −1 i [θ 0 − θ i ] Σi ≤ T −1 i [θ 0 − θ i ] 2 Σi + M 2 λ −2 t 2 + 2M 3/2 tλ −2 r 2 i T −1 i [θ 0 − θ i + E] 2 Σi ≥ T −1 i [θ 0 − θ i ] 2 Σi + T −1 i E 2 Σi − 2 T −1 i E Σi T −1 i [θ 0 − θ i ] Σi ≥ T −1 i [θ 0 − θ i ] 2 Σi − 2M 2 tλ −2 r 2 i . Since t ∈ (0, 1], we have that t 2 ≤ t and thus we can choose c = λ −2 (M 2 + 2M 3/2 r 2 i ). Step 4 Reparameterizing ε := ct, we have that for any ε > 0 lim n→∞ P(\|B(θ R i (λ)\|X) − λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi \| ≤ ε) ≥ lim n→∞ P(\|B(θ R i (λ)\|X) − λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi \| ≤ ε ∧ c) ≥ lim n→∞ P(A c ε c ∧1 ) = 1. Step 5 Using Theorem 6 of [HMRT19], as d → ∞, such that d ni → γ i > 1, we know that the limit of λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi is given by (17) with γ = γ i and H n , G n be the empirical spectral distribution and weighted empirical spectral distribution of Σ i respectively. In the case when Σ i = I, using Corollary 5 of [HMRT19] we have B i (θ R i (λ)\|X) = Π i [θ 0 − θ i ] 2 2 → r 2 i λ 2 m i (−λ). Variance: We let ξ i = [ξ i,1 , . . . , ξ i,n ] denote the vector of noise. Substituting in the variance formula and using E[ξ i ξ T j ] = 0 and E[ξ i ξ T i ] = σ 2 I, we get Var(θ R i (λ)\|X) = E    (Σ i + λI) −1    1 n i X T i ξ i + λ   m j=1 p jΣj   −1 m j=1 p j X T j ξ j n j    2 Σi X    = m j=1 λ 2 p 2 j n j tr   (Σi + λI) −1 Σ(Σ i + λI) −1   m j=1 p jΣj   −1Σ j   m j=1 p jΣj   −1    σ 2 j (i) + 2λp i tr    X T i X i n 2 i   m j=1 p jΣj   −1 (Σ i + λI) −1 Σ(Σ i + λI) −1    σ 2 i (ii) + tr (Σ i + λI) −1 Σ(Σ i + λI) −1Σ i σ 2 i n i (iii) We now study the asymptotic behavior of each of the terms (i), (ii) and (iii) separately. (i) Using the Cauchy Schwartz inequality on Schatten p−norms, we get m j=1 p 2 j λ 2 σ 2 j n j tr   (Σi + λI) −1 Σ(Σ i + λI) −1   m j=1 p jΣj   −1Σ j   m j=1 p jΣj   −1    ≤ m j=1 λ 2 p 2 j σ 2 j n j \|\|\|(Σ i + λI) −1 \|\|\| 1 \|\|\|Σ\|\|\| op (Σ i + λI) −1 op   m j=1 p jΣj   −1 op Σ j op   m j=1 p jΣj   −1 op ≤ C 5 σ 2 max γ max   m j=1 p 2 j Σ j op   , where the last inequality holds with probability going to 1 for some constant C 5 because P(B t ) → 0. Note that this expression is same as (20) and hence the rest of the analysis for this term is same as the one in the proof of FTFA (Section 6.4). (ii) Using the Cauchy Schwartz inequality on Schatten p−norms, we get 2p i λσ 2 i n i tr   (Σi + λI) −1Σ i   m j=1 p jΣj   −1 (Σ i + λI) −1 Σ    ≤ 2p i λσ 2 i n i \|\|\|(Σ i + λI) −1 \|\|\| 1 Σ i op   m j=1 p jΣj   −1 op \|\|\|Σ\|\|\| op (Σ i + λI) −1 op ≤ C 6 σ 2 i dp i λn i , where C 6 is an absolute constant which captures an upper bound on the operator norm of the sample covariance matrixΣ i using Bai Yin Theorem [BY93], and an upper bound on the operator norm of m j=1 p jΣj −1 , which follows from P(B t ) → 0. Since p i → 0, we have 2piλσ 2 i ni tr (Σ i + λI) −1Σ i m j=1 p jΣj −1 (Σ i + λI) −1 Σ p → 0 (iii) Using Theorem 3 of [HMRT19], as d → ∞, such that d ni → γ i > 1, we know that the limit of tr((Σ i + λI) −2Σ i Σ i ) σ 2 i ni is given by (18) with γ = γ i and H n , G n be the empirical spectral distribution and weighted empirical spectral distribution of Σ i respectively. In the case when Σ i = I, using Theorem 1 of [HMRT19] we have V i (θ R i (λ)\|X) = σ 2 i ni tr((Σ i + λI) −2Σ † i Σ i ) p → σ 2 i γi−1 . Proof of Theorem 3 On solving (12) and (13), the closed form of the estimatorsθ M 0 (α) andθ M i (α) is given bŷ θ M 0 (α) := argmin θ m j=1 p j 2n j X j θ − α n j X T j (X j θ − y j ) − y j 2 2 = argmin θ m j=1 p j 2n j I n − α n X j X T j (X j θ − y j ) 2 2 =   m j=1 p j n j X T j W 2 j X j   −1 m j=1 p j n j X T j W 2 j y j where W j := I − α nj X j X T j and θ M i (α) := argmin θ θ M 0 (α) − θ 2 s.t. X i θ = y i = (I −Σ † iΣ i )θ M 0 (α) +Σ † iΣ i θ i + 1 n iΣ † i X T i ξ i We now calculate the risk by splitting it into two parts as in (3), and then calculate the asymptotic bias and variance. Bias: B(θ M i (α)\|X) := E[θ M i (α)\|X] − θ i 2 Σi = Π i      m j=1 p j n j X T j W 2 j X j   −1 m j=1 p j n j X T j W 2 j X j (θ j − θ i )    2 2 = Π i   θ 0 − θ i +   m j=1 p j n j X T j W 2 j X j   −1 m j=1 p j n j X T j W 2 j X j (θ j − θ 0 )    2 Σi For simplicity, we let ∆ j := θ j − θ 0 , and we define the events: B t :=        m j=1 p j n j X T j W 2 j X j   −1 − E   m j=1 p j 1 n j X T j W 2 j X j   −1 op > t      (21) A t :=        m j=1 p j n j X T j W 2 j X j   −1 m j=1 p j n j X T j W 2 j X j ∆ j Σi > t     (22) The proof proceeds in the following steps: Bias Proof Outline Step 1. We first show for any t > 0, the P(B t ) → 0 as d → ∞ Step 2. Then, we show for any t > 0, the P(A t ) → 0 as d → ∞ Step 3. We show that for any t ∈ (0, 1] 1 on event A c t , B(θ M i (α)\|X) ≤ Π i [θ 0 − θ i ] 2 Σi + ct and B(θ M i (α)\|X) ≥ Π i [θ 0 − θ i ] 2 Σi − ct Step 4. Show that lim d→∞ P( \|B(θ M i (α)\|X) − Π i [θ 0 − θ i ] 2 Σi \| ≤ ε) = 1 Step 5. Finally, using the asymptotic limit of Π i [θ 0 − θ i ] 2 Σi from Theorem 1 of [HMRT19], we get the result. We now give the detailed proof: Step 1 Since λ min (E 1 nj X T j W 2 j X j ) ≥ λ 0 , it suffices to show by Lemma 6.4 that the probability of C t :=    m j=1 p j 1 n j X T j W 2 j X j − E 1 n j X T j W 2 j X j op > t    goes to 0 as d, m → ∞ under Assumption A1. P(C t ) = P   m j=1 p j Σ j − 2α 2Σ2 j + α 2Σ3 j − E 1 n j X T j W 2 j X j op > t   ≤ P   m j=1 p j Σ j − µ 1,j op > t/3   + P   2α m j=1 p j Σ 2 j − µ 2,j op > t/3   + P   α 2 m j=1 p j Σ 3 j − µ 3,j op > t/3   ,(23) where µ p,j := E[Σ p j ]. We repeatedly apply Lemma 6.3 for p = 1, 2, 3 to bound each of these three terms. It is clear that if E[ x j,k 6q 2 ] 1/(6q) √ d, E[Σ 6 j ] op ≤ C 3 for some constant C 3 , (log d) 4q m j=1 p q j n j → 0, and (log d) q/2 m j=1 p q/2+1 j → 0, then (23) goes to 0. Step 2 P(A t ) ≤ P(A t ∩ B c c1 ) + P(B c1 ) ≤ P   M c 1 + 1 λ 0 m j=1 p j n j X T j W 2 j X j ∆ j 2 > t   + P(B c1 ) From Step 1, we know that lim n→∞ P(B c1 ) = 0. The second inequality comes from the fact that Ax 2 ≤ \|\|\|A\|\|\| op x 2 . To handle the first term, we use Markov's inequality. P   c 2 m j=1 p j n j X T j W 2 j X j ∆ j 2 > t   ≤ c q 2 t q E   m j=1 p j n j X T j W 2 j X j ∆ j q 2   ≤ (2c 2 √ q) q t q E      m j=1 p j n j X T j W 2 j X j ∆ j 2 2   q/2    ≤ (2c 2 √ q) q t q m j=1 p j E   p j 1 n j X T j W 2 j X j ∆ j 2 2 q/2   = (2c 2 √ q) q t q m j=1 p q/2+1 j E Σ j (I − αΣ j ) 2 ∆ j q 2 ≤ (8c 2 √ q) q 2t q m j=1 p q/2+1 j E Σ j q op + α 2 Σ j 3q op E[ ∆ j q 2 ] ≤ (8c 2 √ q) q 2t q m j=1 p q/2+1 j α 2 K(e log d) 3q n j r q j , where the last step follows from Lemma 6.2 and the final expression goes to 0 since (log d) 3q m j=1 p q/2+1 j n j r q j → 0. Step 3, 4 and 5 are same as the bias calculation of proof of Theorem 1. Variance: We let ξ i = [ξ i,1 , . . . , ξ i,n ] denote the vector of noise. V i (θ M i (α); θ i \|X) = tr(Cov(θ M i (α)\|X)Σ) = E[ θ M i (α) − E[θ M i (α)\|X] 2 Σi \|X] = E[ Π i      m j=1 p j n j X T j W 2 j X j   −1 m j=1 p j n j X T j W 2 j ξ j    + 1 n iΣ † i X T i ξ i 2 Σi \|X] = m j=1 p 2 j n 2 j tr   ΠiΣiΠi   m j=1 p j n j X T j W 2 j X j   −1 X T j W 4 j X j   m j=1 p j n j X T j W 2 j X j   −1    σ 2 j (i) + 2 tr   Σ † i X T i W 2 j X i n 2 i   m j=1 p j n j X T j W 2 j X j   −1 Π i Σ i    σ 2 i (ii) + 1 n 2 i tr(Σ † i X T i X iΣ † i Σ i )σ 2 i (iii) We now study the asymptotic behavior of each of the terms (i), (ii) and (iii) separately. (i) Using the Cauchy Schwartz inequality on Schatten p−norms and using the fact that the nuclear norm of a projection matrix is at most d, we get m j=1 p 2 j σ 2 j n j tr   ΠiΣiΠi   m j=1 p j n j X T j W 2 j X j   −1Σ j (I − αΣ j ) 4   m j=1 p j n j X T j W 2 j X j   −1    ≤ m j=1 p 2 j σ 2 j n j \|\|\|Π i \|\|\| 1 \|\|\|Σ i \|\|\| op \|\|\|Π i \|\|\| op   m j=1 p j n j X T j W 2 j X j   −1 op Σ j (I − αΣ j ) 4 op   m j=1 p j n j X T j W 2 j X j   −1 op ≤ C 7 σ 2 max γ max   m j=1 p 2 j Σ j (I − αΣ j ) 4 op   , where the last inequality holds with probability going to 1 for some constant C 7 because P(C t ) → 0. Lastly, we show that P Σ j (I −Σ j ) 4 op > t   ≤ E m j=1 p 2 j Σ j (I − αΣ j ) 4 op q t q ≤ m j=1 p q+1 j E Σ j (I − αΣ j ) 4 q op t q ≤ m j=1 p q+1 j E Σ j op (I − αΣ j ) 4q op t q ≤ m j=1 p q+1 j E Σ j op \|\|\|I\|\|\| op + α Σ j op 4q t q ≤ 2 4q−1 m j=1 p q+1 j E Σ j op + α 4 Σ j 5q op t q Using Lemma 6.2 and Markov's inequality, we have P   m j=1 p 2 j Σ j op > t   ≤ 2 4q−1 K 1 m j=1 p q+1 j (e log d)n j t q + K 2 α 4 m j=1 p q+1 j (e log d) 5q n j t q . Finally, since we know that m j=1 p q+1 j (log d) 5q n j → 0, we have P m j=1 p 2 j Σ j (I − αΣ j ) 4 op > t → 0. (ii) Using the Cauchy Schwartz inequality on Schatten p−norms and using the fact that the nuclear norm of a projection matrix is d − n, we get   Σ † i X T i W 2 i X i n 2 i   m j=1 p j n j X T j W 2 j X j   −1 Π i Σ i    σ 2 i = 2 tr   ΠiΣiΣ † iΣ i (I −Σ i ) 2 n i   m j=1 p j n j X T j W 2 j X j   −1    σ 2 i ≤ 2p i σ 2 n i \|\|\|Π i \|\|\| 1 \|\|\|Σ i \|\|\| op Σ † iΣ i op (I −Σ i ) 2 op   m j=1 p j n j X T j W 2 j X j   −1 op ≤ C 4 p i , where the last inequality holds with probability going to 1 for some constant C 4 because P(B t ) → 0 and using Assumption A2. Since p i → 0, we have 2 tr   Σ † i X T i W 2 i X i n 2 i   m j=1 p j n j X T j W 2 j X j   −1 Π i Σ i    σ 2 i → 0 (iii) 1 n 2 i tr(Σ † i X T i X iΣ † i Σ i )σ 2 i = 1 n i tr(Σ † i Σ i )σ 2 i Using Theorem 3 of [HMRT19], as d → ∞, such that d ni → γ i > 1, we know that the limit of σ 2 i ni tr(Σ † i Σ i ) is given by (16) with γ = γ i and H n , G n be the empirical spectral distribution and weighted empirical spectral distribution of Σ i respectively. In the case when Σ i = I, using Theorem 1 of [HMRT19] we have V i (θ M i (α)\|X) = σ 2 i ni tr(Σ † i ) → σ 2 i γi−1 . Proof of Theorem 4 The solution to this minimization problem in (14) is given bŷ θ P 0 (λ) = θ 0 + Q −1   m j=1 p j T −1 jΣ j ∆ j + m j=1 p j T −1 j 1 n j X T j ξ j   , where ∆ j = θ j − θ 0 , T j =Σ j + λI and Q = I − λ m j=1 p j T −1 j . The personalized solutions are then given byθ P i (λ) = T −1 i λθ P 0 (λ) +Σ i θ i + 1 n i X T i ξ i We now calculate the risk by splitting it into two parts as in (3), and then calculate the asymptotic bias and variance. Bias: Let ∆ j := θ j − θ 0 , then we have B(θ P i (λ)\|X) := T −1 i   λθ 0 − λθ i + λQ −1   m j=1 p j T −1 jΣ j ∆ j     2 Σi The idea is to show that the second term goes to 0 and use results from [HMRT19] to find the asymptotic bias. To do this, we first define the events: C t :=    m j=1 p j (T −1 j − E[T −1 j ]) op > t    A t :=    Q −1   m j=1 p j T −1 jΣ j ∆ j   Σi > t    The proof proceeds in the following steps: Bias Proof Outline Step 1. We first show for any t > 0, the P(C t ) → 0 as d → ∞ Step 2. Then, we show for any t > 0, the P(A t ) → 0 as d → ∞. Step 3. We show that for any t ∈ (0, 1], B(θ P i (λ)\|X) ≤ T −1 i [λθ 0 − λθ i ] 2 2 + ct and B(θ, X) ≥ T −1 i [λθ 0 − λθ i ] 2 2 − ct Step 4. Show that lim d→∞ P(\|B(θ P i (λ)\|X) − T −1 i [λθ 0 − λθ i ] 2 2 \| ≤ ε) = 1 Step 5. Finally, using the asymptotic limit of T −1 i [λθ 0 − λθ i ] 2 2 from Corollary 5 of [HMRT19], we get the result. We now give the detailed proof: Step 1 P(C t ) = P   m j=1 p j (T −1 j − E[T −1 j ]) op > t   ≤ 2 q E m j=1 ξ j p j T −1 j q op t q , We use Theorem A.1 from [CGT12] to bound this object. where we use the fact that T −1 j op = (Σ j + λI) −1 op ≤ 1 λ sinceΣ j is always positive semidefinite. Since (log d) q/2 m j=1 p q/2+1 j and (log d) q m j=1 p q j , we get that P(C t ) → 0 for all t > 0. E   m j=1 ξ j T −1 j q op   ≤    e log d   m j=1 p 2 j E[T −1 j ]   1/2 op + (e log d)(E max j p j T −1 j q op ) 1/q    q ≤ 2 q−1    e log d q   m j=1 p 2 j E[(T −1 j ) 2 ]   1/2 q op + (e log d) q (E max j p j T −1 j q op )    ≤ 2 q−1    e log d q m j=1 p 2 j E[(T −1 j ) 2 ] q/2 op + (e log d) q max j p q j λ q    ≤ 2 q−1   e log d Step 2 To prove this step, we will first use a helpful lemma, Lemma 6.5. Suppose that Σ = E[Σ] ∈ R d,d has a spectrum supported on [a, b] where 0 < a < b < ∞. Further suppose that E Σ 2 op ≤ τ and there exists an R ≥ b such that P(λ max (Σ) > R) ≤ a 2 8τ , then E[(Σ + λI) −1 ] op ≤ 1 λ 1 − a 3 16τ (R + λ) ≤ 1 λ Proof Fix an arbitrary vector u ∈ R d with unit 2 norm. We fix δ = a/2 > 0, we define the event A := {u TΣ u ≥ δ} and B := {λ max (Σ) ≤ R} u T E[(Σ + λI) −1 ]u ≤ E[1 {A ∩ B} u T (Σ + λI) −1 u] + 1 λ (1 − P(A ∩ B)) Let σ 2 i and v i denote the ith eigenvalue and eigenvector ofΣ respectively sorted in descending order with respect to eigenvalue (σ 2 1 ≥ σ 2 2 ≥ . . . ≥ σ 2 d ). On the event A, we have that u T (Σ + λI) −1 u has value no larger than max α∈R d d i=1 1 σ 2 i + λ α i s.t. α ≥ 0 1 T α = 1 d i=1 σ 2 i α i ≥ δ The dual of this problem is min θ max j∈[d] θσ 2 j + 1 σ 2 j + λ − θδ s.t. θ ≥ 0 It suffices to demonstrate that there exists a θ which satisfies the constraints of the dual and has objective value less than 1 λ . We can verify that selecting θ = 1 λ(σ 2 1 +λ) has an objective value of 1 λ − δ λ(σ 2 1 + λ) which is less than the desired 1 λ . All that remains is to lower bound P(A ∩ B) ≥ P(A) − P(B c ). We know by Paley-Zygmund P(A) ≥ P u TΣ u ≥ δ a u T Σu ≥ P u TΣ u ≥ 1 2 u T Σu ≥ (u T Σu) 2 4E[(u TΣ u) 2 ] ≥ a 2 4τ Note that a 2 /4τ < 1 because the second moment of a random variable is no smaller than the first moment squared of the random variable. Moreover, by construction, R is large enough such that P(B c ) ≤ P(A)/2, thus, u T E[(Σ + λI) −1 ]u ≤ 1 λ 1 − a 2(R + λ) a 2 8τ + 1 λ 1 − a 2 8τ = 1 λ 1 − a 3 16τ (R + λ) Recall that we have the assumptions that for sufficiently large m, for all j ∈ [m] we have Σ j has a spectrum supported on [a, b] where a = 1/M and b = M and E Σ 2 j op ≤ τ 3 . Moreover, since we have the assumption that there exists an R ≥ b such that lim sup m→∞ sup j∈[m] P(λ max (Σ j ) > R) ≤ a 2 16τ3 , by Lemma 6.5 there exists and 1 > ε > 0 such that for sufficiently large m, for all j ∈ [m], E[(Σ j + λI) −1 ] op ≤ 1−ε λ . P(A t ) ≤ P(A t ∩ C c c1 ) + P(C c1 ) Since we know P(C c1 ) → 0, it suffices to bound the first term. P(A t ∩ C c c1 ) ≤ P   √ M Q −1 op   m j=1 p j T −1 jΣ j ∆ j   2 > t ∩ C c c1   = P    √ M   1 − λ m j=1 p j T −1 j op   −1   m j=1 p j T −1 jΣ j ∆ j   2 > t ∩ C c c1    ≤ P    √ M   1 − λ m j=1 p j E[T −1 j ] + E c1 op   −1   m j=1 p j T −1 jΣ j ∆ j   2 > t    ≤ P    √ M   1 − λ m j=1 p j E[T −1 j ] op − c 1   −1   m j=1 p j T −1 jΣ j ∆ j   2 > t    where we used Jensen's inequality in the last step. E c1 is a matrix error term which on the event C c c1 has operator norm bounded by c 1 . As discussed, we have that m j=1 p j E[T −1 j ] op is less than 1−ε λ , which shows there exists a constant c 2 , such that √ M (1 − λ m j=1 p j E[T −1 j ] op − c 1 ) −1 < c 2 . Now, we have, using Lemma 6.1, P   c 2 m j=1 p j T −1 jΣ j ∆ j 2 > t   ≤ c q 2 E m j=1 p j T −1 jΣ j ∆ j q 2 t q ≤ (2c 2 √ q) q E m j=1 p j T −1 jΣ j ∆ j 2 2 q/2 t q Using Jensen's inequality and the definition of operator norm, we have (2c 2 √ q) q E m j=1 p 2 j T −1 jΣ j ∆ j 2 2 q/2 t q = (2c 2 √ q) q E m j=1 p j · p j T −1 jΣ j ∆ j 2 2 q/2 t q ≤ (2c 2 √ q) q m j=1 p q/2+1 j E T −1 jΣ j ∆ j q 2 t q ≤ (2c 2 √ q) q m j=1 p q/2+1 j E T −1 j q op E Σ j q op E ∆ j q 2 t q Lastly, we can bound this using Lemma 6.2 as follows and using the fact that T −1 j op ≤ 1 λ . P   c 2 m j=1 p j T −1 jΣ j ∆ j 2 > t   ≤ (2c 2 √ q) q m j=1 p q/2+1 j Kn j (e log d) q r q j λ q t q → 0, using (log d) q m j=1 n j p q/2+1 j → 0 Step 3 For any t ∈ (0, 1], on the event A c t , we have that B(θ P i (λ)\|X) = T −1 i [λθ 0 − λθ i + E] 2 Σi for some vector E where we know E Σi ≤ t (which means E 2 ≤ t √ M ). We can form the bounds T −1 i [λθ 0 − λθ i + E] 2 Σi ≤ λ 2 T −1 i [θ 0 − θ i ] 2 Σi + T −1 i E 2 Σi + 2λ T −1 i E Σi T −1 i [θ 0 − θ i ] Σi ≤ λ 2 T −1 i [θ 0 − θ i ] 2 Σi + λ −2 t 2 M 2 + 2tλ −1 r 2 i M 3/2 T −1 i [λθ 0 − λθ i + E] 2 Σi ≥ λ 2 T −1 i [θ 0 − θ i ] 2 Σi + T −1 i E 2 Σi − 2λ T −1 i E Σi T −1 i [θ 0 − θ i ] Σi ≥ λ 2 T −1 i [θ 0 − θ i ] 2 Σi − 2tλ −1 r 2 i M 3/2 Since t ∈ (0, 1], we have that t 2 ≤ t and thus we can choose c = λ −2 M 2 + 2r 2 i λ −1 M 3/2 Step 4 Reparameterizing ε := ct, we have that for any ε > 0 lim n→∞ P(\|B(θ P i (λ), X) − T −1 i [θ 0 − θ i ] 2 Σi \| ≤ ε) ≥ lim n→∞ P(\|B(θ P i (λ)\|X) − T −1 i [θ 0 − θ i ] 2 Σi \| ≤ ε ∧ c) ≥ lim n→∞ P(A c ε c ∧1 ) = 1 Step 5 Using Theorem 6 of [HMRT19], as d → ∞, such that d ni → γ i > 1, we know that the limit of λ 2 (Σ i + λI) −1 [θ 0 − θ i ] 2 Σi is given by (17) with γ = γ i and H n , G n be the empirical spectral distribution and weighted empirical spectral distribution of Σ i respectively. In the case when Σ i = I, using Corollary 5 of [HMRT19] we have B i (θ P i (λ)\|X) = Π i [θ 0 − θ i ] 2 2 → r 2 i λ 2 m i (−λ). Variance Var(θ P i (λ)\|X) = E    T −1 i   λQ −1   m j=1 p j T −1 j 1 n X T j ξ j   + 1 n i X T i ξ i   2 Σi    = m j=1 λ 2 p 2 j n j tr T −1 i Σ i T −1 i Q −1 T jΣj T j Q −1 σ 2 j (i) + 2λσ 2 i p i n i 2 tr Σ i T −1 i Q −1 T −1 iΣ i T −1 i (ii) + tr T −1 i Σ i T −1 iΣ i σ 2 i n i (iii) We now study the asymptotic behavior of each of the terms (i), (ii) and (iii) separately. In these steps, we will have to bound Q −1 op . To do this, we observe that there exists a sufficiently large constant t such that the following statement is true. P( Q −1 op > t) = P( Q −1 op > t ∩ C c c1 ) + P(C c1 ) = P      1 − λ m j=1 p j T −1 j op   −1 > t ∩ C c c1    + o(1) ≤ P      1 − λ m j=1 p j E[T −1 j ] + E c1 op   −1 > t    + o(1) ≤ P      1 − λ m j=1 p j E[T −1 j ] op − c 1   −1 > t    + o(1) ≤ o(1). This is true because of Lemma 6.5. (i) Using the Cauchy Schwartz inequality on Schatten p−norms and using the high probability bounds from the bias proof, we get that for some constant C 8 , the following holds with probability going to 1. Server samples a subset of clients S r uniformly at random such that \|S r \| = D m j=1 λ 2 p 2 j n j tr T −1 i Σ i T −1 i Q −1 T jΣj T j Q −1 σ 2 3: Server sendsθ F A 0,r to all clients in S r 4: for i ∈ S r do 5: Setθ F A i,r+1,0 ←θ F A 0,r 6: for k ← 1 to K do 7: Sample a batch D i k of size B from user i's data D i 8: Server samples a subset of clients S r uniformly at random such that \|S r \| = D Compute Stochastic Gradient g(θ F A i,r+1,k−1 ; D i k ) = 1 B S∈D i k ∇F (θ F A i,r+1,k−1 ; S) Setθ F A i,r+1,k ←θ F A i,r+1,k−1 − ηg(θ F A i, 3: Server sendsθ M 0,r (α) to all clients in S r 4: for i ∈ S r do 5: Setθ M i,r+1,0 (α) ←θ M 0,r (α) 6: for k ← 1 to K do 7: Sample a batch D i k of size B from user i's data D i 8: Server samples a subset of clients S r uniformly at random such that \|S r \| = D Compute Stochastic Gradient g(θ M i,r+1,k−1 (α); D i k ) = 1 B S∈D i k ∇F (θ M i,r+1,k−1 (α); S) Setθ M i,r+1,k (α) ←θ M i,r+1,k−1 (α) − αg(θ M i,r+1,k−1 (α); D i k ) 9 3: Server sendsθ P 0,r (λ) to all clients in S r 4: for i ∈ S r do 5: Setθ P i,r+1,0 (λ) ←θ P 0,r (λ) 6: for k ← 1 to K do 7: Sample a batch D i k of size B from user i's data D i 8: Server updates the central model usingθ P 0,r+1 (λ) = (1 − β)θ P 0,r (λ) + β D j=1 nj D j=1 njθ P i,r+1,K (λ). 14: end for 15: returnθ P 0,R (λ) Compute θ i (θ P i,r+1,k−1 (λ)) = argmin θ 1 B S∈D i k ∇F (θ; S) + λ 2 θ −θ P i,r+1,k−1 (λ) 2 2 9: Setθ P i,r+1,k (λ) ←θ P i,r+1,k−1 (λ) − ηλ(θ P i,r+1,k−1 (λ) − θ i (θ P i,r+1,k−1 (λ))) Shakespeare Shakespeare is a language modeling dataset built using the works of William Shakespeare and the clients correspond to a speaking role with at least two lines. The task here is next character prediction. The way lines are split into sequences of length 80, and the description of the vocabulary size is same as [RCZ + 21] (Appendix C.3). Additionally, we filtered out clients with less than 3 sequences of data, so as to have a train-validation-test split for all the clients. This brought the number of clients down to 669. More information on sample sizes can be found in Table 1. The models trained on this dataset are trained on two Tesla P100-PCIE-12GB GPUs. CIFAR-100 CIFAR-100 is an image classification dataset with 100 classes and each image consisting of 3 channels of 32x32 pixels. We use the clients created in the Tensorflow Federated framework [MRR + 19] -client division is described in Appendix F of [RCZ + 21]. Instead of using 500 clients for training and 100 clients for testing as in [RCZ + 21], we divided each clients' dataset into train, validation and test sets and use all the clients' corresponding data for training, validation and testing respectively. The models trained on this dataset are trained on two Titan Xp GPUs. EMNIST EMNIST contains images of upper and lower characters of the English language along with images of digits, with total 62 classes. The federated version of EMNIST partitions images by their author providing the dataset natural heterogenity according to the writing style of each person. The task is to classify images into the 62 classes. As in other datasets, we divide each clients' data into train, validation and test sets randomly. The models trained on this dataset are trained on two Tesla P100-PCIE-12GB GPUs. Stack Overflow Stack Overflow is a language model consisting of questions and answers from the StackOverflow website. The task we focus on is next word prediction. MAML-FL-HF This is the hessian free version of the algorithm, i.e., the hessian term is approximated via finite differences (details can be found in [FMO20]). This algorithm is described in Algorithm 7. We hyperparameter tune over the step size η MAML-FL-FO This is the first order version of the algorithm, i.e., the hessian term is set to 0 (details can be found in [FMO20]). This algorithm is described in Algorithm 7. We hyperparameter tune over the step size η pFedMe This algorithm is described in Algorithm 8. We hyperparameter tune over the step size η [0.0005, 0.005, 0.05], and the weight β [1, 2]. The proximal optimization step size, hyperparameter K, and prox-regularizer λ associated with approximately solving the prox problem is set to 0.05, 5, and 15 respectively. We chose these hyperparameters based on the suggestions from [DTN20]. We were unable to hyperparameter tune pFedMe as much as, for example, RTFA because each run of pFedMe takes significantly longer to run. Additionally, for this same reason, we were unable to run pFedMe on the Stack Overflow dataset. While we do not have wall clock comparisons (due to multiple jobs running on the same gpu), we have observed that pFedMe, with the hyperparameters we specified, takes approximately 20x the compute time to complete relative to FTFA, RTFA, and MAML-FL-FO. The ideal hyperparameters for each dataset can be found in the tables below: Additional Results In this section, we add additional plots from the experiments we conducted, which were omitted from the main paper due to length constraints. In essence, these plots only strengthen the claims made in the experiments section in the main body of the paper. Figure 1 . 1CIFAR-100. Best-average-worst intervals created from different train-val splits. Figure 2 . 2EMNIST. Best-average-worst intervals created from different random seeds. Figure 3 . 3Shakespeare Figure 4 .Figure 6 . 46Stack Overflow. Best-average-worst intervals created from different train-val splits. CIFAR. Gains of personalization for Hessian free MAML-FL Lemma 6. 4 . 4Let U ∈ R d×d and V ∈ R d×d be positive semidefinite matrices such that λ min (U ) ≥ λ 0 for some constant λ 0 . Let d, n j , m → ∞ as in Assumption A1. Suppose \|\|\|V − U \|\|\| op [ 0 . 00001, 0.001, 0.01, 0.1, 1], the step size of the personalization SGM steps α [0.0001, 0.001, 0.01, 0.1, 1], and the hessian finite-difference-approximation parameter δ [0.001, 0.00001]. We used only two different values of δ because the results of preliminary experiments suggested little change in accuracy with changing δ. Figure 7 . 7CIFAR-100. Best-average-worst intervals created from different random seeds. Figure 8 . 8EMNIST. Best-average-worst intervals created from different train-val splits. Figure 9 :Figure 10 .Figure 11 .Figure 12 .Figure 13 . 910111213Shakespeare. Best-average-worst intervals created from different random train-val splits. EMNISTCIFAREMNIST. Gains of personalization for HF-MAML-FL CIFAR. Gains of personalization for FTFA Thus, in both the cases, ignoring the polylog factors, if we have N m (log d) cq m q/2 → 0 i.e., m q/2 grows faster than the average client sample size, N/m, then Assumption A3 holds. With these assumptions defined, we are able to compute the asymptotic test loss of FTFA.Theorem 1. Consider the observation model in Sec. 2 and the estimatorθ F A i in (7). Let Assumption A1 hold, and let Assumptions A2 and A3 hold with c = 2 and q > 2. Additionally, assume that for each m and j ∈ [m], \|\|\|E[Σ 2 j ]\|\|\| op ≤ τ 2 , where τ 2 < ∞. Then for client i, the asymptotic prediction bias and variance of FTFA arelim m→∞ [ KH09 ] KH09Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009. [KKM + 20] Sai Praneeth Karimireddy, Satyen Kale, Mehryar Mohri, Sashank Reddi, Sebastian Stich, and Ananda Theertha Suresh. SCAFFOLD: Stochastic controlled averaging for federated learning. In Proceedings of the 37th International Conference on Machine Learning, 2020. [KMA + 19] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Keith Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al. Advances and open problems in federated learning. arXiv:1912.04977 [cs.LG], 2019. [LHBS21] Tian Li, Shengyuan Hu, Ahmad Beirami, and Virginia Smith. Ditto: Fair and robust federated learning through personalization. In ICML, 2021. [LSZ + 20] Tian Li, Anit Kumar Sahu, Manzil Zaheer, Maziar Sanjabi, Ameet Talwalkar, and Virginia Smith. Federated optimization in heterogeneous networks. In Proceedings of Machine Learning and Systems, volume 2, pages 429-450, 2020. [MMR + 17] H. Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Agüera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, 2017. [MMRS20] Yishay Mansour, Mehryar Mohri, Jae Ro, and Ananda Theertha Suresh. Three approaches for personalization with applications to federated learning. arXiv:2002.10619 [cs.LG], 2020. [MRR + 19] Brendan McMahan, Keith Rush, Michael Reneer, Zachary Garrett, and TensorFlow Federated Team. Tensorflow federated stack overflow dataset. https://www.tensorflow.org/federated/ api_docs/python/tff/simulation/datasets/stackoverflow/load_data, 2019. [MSS19] Mehryar Mohri, Gary Sivek, and Ananda Theertha Suresh. Agnostic federated learning. In Proceedings of the 36th International Conference on Machine Learning, 2019. [Nes04] Y. Nesterov. Introductory Lectures on Convex Optimization. Kluwer Academic Publishers, 2004. 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For each j ∈ [m], we have by Lemma 6.2 for sufficiently large m,E Σ j pq op ≤ K(e log d) pq n j ,for some absolute constant K. Supposing that E Σ 2pj op ≤ C 3 exist for all j. Combining all the inequalities, we have for sufficiently large m,   m j=1 p jΣ p j ξ j q op   ≤ 2 q−1 (e log d) q/2   m j=1 p q/2+1 j Algorithm 3 Naive local training Require: m: number of users, K: epochs 1: for i ← 1 to m do Each client runs K epochs of SGM with personal stepsize α 3: end for Algorithm 4 Federated Averaging [MMR + 17] Require: R: Communication Rounds, D: Number of users sampled each round, K: Number of local update steps,θ F A 0,0 : Initial iterate for global model 1: for r ← 0 to R − 1 do2: 2: Client i sendsθ F A i,r+1,K back to the server. Server updates the central model usingθ F A 0,r+1 = In this section, we give all steps of the exact algorithms used to implement all algorithms in the experiments section. Algorithm 5 FTFA Require: P : Personalization iterations 1: Server sendsθ F A =θ F A 0,R (using Algorithm 4 with stepsize η) to all clients 2: for i ← 1 to m do Run P steps of SGM on L i (·) usingθ F A as initial point with learning rate α and outputθ F A Algorithm 6 RTFA Require: P : Personalization iterations 1: Server sendsθ F A =θ F A 0,R (using Algorithm 4 with stepsize η) to all clients 2: for i ← 1 to m do Run P steps of SGM on L i (θ) + λ 2 θ −θ F A Algorithm 7 MAML-FL-HF [FMO20] Require: R: Communication Rounds, D: Number of users sampled each round, K: Number of local update steps,θ M 0,0 (α): Initial iterate for global model 1: for r ← 0 to R − 1 dor+1,k−1 ; D i k ) 9: end for 10: 11: end for 12: D j=1 nj D j=1 njθ F A i,r+1,K . 13: end for 14: returnθ F A 0,R 7 Algorithm implementations 0 3: 0 i,P 4: end for 5: returnθ F A i,P 0 3: 0 2 2 with learning rate α and outputθ F A i,P 4: end for 5: returnθ F A i,P 2: : end for 10 : end10Client i sendsθ M i,r+1,K (α) back to the server.Server updates the central model usingθ M 0,r+1 (α) = r+1,K (α). 13: end for 14: Server sendsθ M 0,R (α) to all clients 15: for i ← 1 to m do Run P steps of SGM on L i (·) usingθ M 0 (α) as initial point with learning rate α and outputIn this section, we provide detailed descriptions on datasets and how they were divided into users.We perform experiments on federated versions of the Shakespeare [MMR + 17], CIFAR-100 [KH09], EMNIST [CATvS17], and Stack Overflow [MRR + 19] datasets. We download all datasets using FedML APIs [HLS + 20] which in turn get their datasets from [MRR + 19]. For each dataset, for each client, we divide their data into train, validation and test sets with roughly a 80%, 10%, 10% split. The information regarding the number of users in each dataset, dimension of the model used, and the division of all samples into train, validation and test sets is given in Table 1.Table 1: Dataset Information Algorithm 8 pFedMe [DTN20] Require: R: Communication Rounds, D: Number of users sampled each round, K: Number of local update steps,θ P 0,0 (λ): Initial iterate for global model 1: for r ← 0 to R − 1 do11: end for 12: D j=1 nj D j=1 njθ M i,16: θ M i,P (α) 17: end for 18: returnθ M 0,R (α) 8 Experimental Details 8.1 Dataset Details Dataset Users Dimension Train Validation Test Total Samples CIFAR 100 600 51200 48000 6000 6000 60000 Shakespeare 669 23040 33244 4494 5288 43026 EMNIST 3400 31744 595523 76062 77483 749068 Stackoverflow-nwp 300 960384 155702 19341 19736 194779 2: Client i sendsθ P i,r+1,K (λ) back to the server.10: end for 11: 12: end for 13: As described in Appendix C.4 of [RCZ + 21], we also restrict to the 10000 most frequently used words, and perform padding/truncation to ensure each sentence to have 20 words. Additionally, due to scalability issues, we use only a sample of 300 clients from the original dataset from [MRR + 19] and for each client, we divide their data into train, validation and test sets randomly. The models trained on this dataset are trained on two Titan Xp GPUs. RTFA This algorithm is described in Algorithm 6. We hyperparameter tune over the step size of FedAvg η [0.0001, 0.001, 0.01, 0.1, 1], the step size of the personalization SGM steps α [0.0001, 0.001, 0.01, 0.1, 1], and the ridge parameter λ [0.001, 0.01, 0.1, 1, 10]. Table 2 : 2Shakespeare Best Hyperparameters Table 3 : 3CIFAR-100 Best HyperparametersAlgorithm η α λ δ β Naive Local - 0.001 - - - FedAvg 0.01 - - - - FTFA 0.1 0.01 - - - RTFA 0.1 0.01 0.1 - - MAML-FL-HF 0.1 0.01 - 0.00001 - MAML-FL-FO 0.1 0.01 - - - pFedMe 0.05 - - - 2 Table 4 : 4EMNIST Best Hyperparameters Table 5 : 5Stack Overflow Best Hyperparameters Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. 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Surprises in highdimensional ridgeless linear least squares interpolation. Trevor Hastie, Andrea Montanari, Saharon Rosset, Ryan Tibshirani, arXiv:1903.08560math.STTrevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan Tibshirani. Surprises in high- dimensional ridgeless linear least squares interpolation. arXiv:1903.08560 [math.ST], 2019. Universal language model fine-tuning for text classification. Jeremy Howard, Sebastian Ruder, arXiv:1801.06146[cs.LG]Jeremy Howard and Sebastian Ruder. Universal language model fine-tuning for text classification. arXiv:1801.06146 [cs.LG], 2018. The Elements of Statistical Learning. Trevor Hastie, Robert Tibshirani, Jerome Friedman, Springersecond editionTrevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning. Springer, second edition, 2009. Deep residual learning for image recognition. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 770-778, 2016. Improving federated learning personalization via model agnostic meta learning. Yihan Jiang, Jakub Konečný, Keith Rush, Sreeram Kannan, arXiv:1909.12488cs.LGYihan Jiang, Jakub Konečný, Keith Rush, and Sreeram Kannan. Improving federated learning personalization via model agnostic meta learning. arXiv:1909.12488 [cs.LG], 2019. Hyperparameter Tuning Details. Hyperparameter Tuning Details Pretrained Model We now describe how we obtain our pretrained models. First, we train and hyperparameter tune a neural net classifier on the train and validation sets in a non-federated manner. The details of the hyperparameter sweep are as followsPretrained Model We now describe how we obtain our pretrained models. First, we train and hyperparameter tune a neural net classifier on the train and validation sets in a non-federated manner. The details of the hyperparameter sweep are as follows: It has an embedding layer, an LSTM layer and a fully connected layer. We use the StepLR learning rate scheduler of PyTorch , and we hyperparameter tune over the step size. Shakespeare For this dataset we use the same neural network architecture as used for Shakespeare in. 0.0001, 0.001, 0.01, 0.1, 1] and the learning rate decay gamma [0.1, 0.3, 0.5] for 25 epochs with a batch size of 128Shakespeare For this dataset we use the same neural network architecture as used for Shakespeare in [MMR + 17]. It has an embedding layer, an LSTM layer and a fully connected layer. We use the StepLR learning rate scheduler of PyTorch , and we hyperparameter tune over the step size [0.0001, 0.001, 0.01, 0.1, 1] and the learning rate decay gamma [0.1, 0.3, 0.5] for 25 epochs with a batch size of 128. For training images, we perform a random crop to shape (32, 32, 3) with padding size 4, followed by a horizontal random flip. For all training, validationn and testing images, we normalize each image according to their mean and standard deviation. We use the hyperparameters specified by. CIFAR-100 For this dataset we use the Res-Net18 architecture. We perform the standard preprocessing for CIFAR datasets for train, validation and test data. wei20] to train our nets for 200 epochsCIFAR-100 For this dataset we use the Res-Net18 architecture [HZRS16]. We perform the standard preprocessing for CIFAR datasets for train, validation and test data. For training images, we perform a random crop to shape (32, 32, 3) with padding size 4, followed by a horizontal random flip. For all training, validationn and testing images, we normalize each image according to their mean and standard deviation. We use the hyperparameters specified by [wei20] to train our nets for 200 epochs. EMNIST For this dataset, the architecture we use is similar to that found in [RCZ + 21]; the exact architecture can be found in our code. We use the StepLR learning rate scheduler of PyTorch, and we hyperparameter tune over the step size. 0.0001, 0.001, 0.01, 0.1, 1] and the learning rate decay gammaEMNIST For this dataset, the architecture we use is similar to that found in [RCZ + 21]; the exact architecture can be found in our code. We use the StepLR learning rate scheduler of PyTorch, and we hyperparameter tune over the step size [0.0001, 0.001, 0.01, 0.1, 1] and the learning rate decay gamma . Pytorch, and we hyperparameter tune over the step size [0.0001, 0.001, 0.01, 0.1, 1] and the learningPyTorch, and we hyperparameter tune over the step size [0.0001, 0.001, 0.01, 0.1, 1] and the learning Using these representations, we do multi-class logistic regression with each of the federated learning algorithms we test; we adapt and extend this code base [DTN20] to do our experiments. For all of our algorithms, the number of global iterations R is set to 400, and the number of local iterations K is set to 20. The number of users sampled at global iteration r, D, is set to 20. The batch size per local iteration, B, is 32. The random seed is set to 1. For algorithms FTFA, RTFA, MAML-FL-FO, and MAML-FL-HF, we set the number of personalization epochs P to be 10. We fix some hyperparameters due to computational resource restrictions and to avoid conflating variables. After selecting the best hyperparameters for each net, we pass our data through said net and store their representations (i.e. These representations are the data we operate on in our federated experiments. we choose to fix these ones out of precedence, see experimental details of [RCZ + 21]. 1, 10After selecting the best hyperparameters for each net, we pass our data through said net and store their representations (i.e., output from penultimate layer). These representations are the data we operate on in our federated experiments. Using these representations, we do multi-class logistic regression with each of the federated learning algorithms we test; we adapt and extend this code base [DTN20] to do our experiments. For all of our algorithms, the number of global iterations R is set to 400, and the number of local iterations K is set to 20. The number of users sampled at global iteration r, D, is set to 20. The batch size per local iteration, B, is 32. The random seed is set to 1. For algorithms FTFA, RTFA, MAML-FL-FO, and MAML-FL-HF, we set the number of personalization epochs P to be 10. We fix some hyperparameters due to computational resource restrictions and to avoid conflating variables; we choose to fix these ones out of precedence, see experimental details of [RCZ + 21]. 1, 10]. FTFA This algorithm is described in Algorithm 1. We hyperparameter tune over the step size of FedAvg η. 0.0001, 0.001, 0.01, 0.1, 1], and the step size of the personalization SGM steps α [0.0001, 0.001, 0.01, 0.1, 1FTFA This algorithm is described in Algorithm 1. We hyperparameter tune over the step size of FedAvg η [0.0001, 0.001, 0.01, 0.1, 1], and the step size of the personalization SGM steps α [0.0001, 0.001, 0.01, 0.1, 1].	[ "https://github.com/weiaicunzai/pytorch-cifar100," ]
[ "Improving SGD convergence by online linear regression of gradients in multiple statistically relevant directions", "Improving SGD convergence by online linear regression of gradients in multiple statistically relevant directions" ]	[ "Jarek Duda [email protected] \nJagiellonian University\nGolebia 2431-007KrakowPoland\n" ]	[ "Jagiellonian University\nGolebia 2431-007KrakowPoland" ]	[]	Deep neural networks are usually trained with stochastic gradient descent (SGD), which minimizes objective function using very rough approximations of gradient, only averaging to the real gradient. Standard approaches like momentum or ADAM only consider a single direction, and do not try to model distance from extremum -neglecting valuable information from calculated sequence of gradients, often stagnating in some suboptimal plateau. Second order methods could exploit these missed opportunities, however, beside suffering from very large cost and numerical instabilities, many of them attract to suboptimal points like saddles due to negligence of signs of curvatures (as eigenvalues of Hessian).Saddle-free Newton method (SFN) [1] is a rare example of addressing this issue -changes saddle attraction into repulsion, and was shown to provide essential improvement for final value this way. However, it neglects noise while modelling second order behavior, focuses on Krylov subspace for numerical reasons, and requires costly eigendecomposion.Maintaining SFN advantages, there are proposed inexpensive ways for exploiting these opportunities. Second order behavior is linear dependence of first derivative -we can optimally estimate it from sequence of noisy gradients with least square linear regression, in online setting here: with weakening weights of old gradients. Statistically relevant subspace is suggested by PCA of recent noisy gradients -in online setting it can be made by slowly rotating considered directions toward new gradients, gradually replacing old directions with recent statistically relevant. Eigendecomposition can be also performed online: with regularly performed step of QR method to maintain diagonal Hessian. Outside the second order modeled subspace we can simultaneously perform gradient descent.	null	[ "https://export.arxiv.org/pdf/1901.11457v11.pdf" ]	118,648,685	1901.11457	61034c0c448b54af635c8d55847b2380041208b4	Improving SGD convergence by online linear regression of gradients in multiple statistically relevant directions Jarek Duda [email protected] Jagiellonian University Golebia 2431-007KrakowPoland Improving SGD convergence by online linear regression of gradients in multiple statistically relevant directions 1non-convex optimizationstochastic gradient de- scentconvergencedeep learningHessianlinear regressionsaddle-free Newton Deep neural networks are usually trained with stochastic gradient descent (SGD), which minimizes objective function using very rough approximations of gradient, only averaging to the real gradient. Standard approaches like momentum or ADAM only consider a single direction, and do not try to model distance from extremum -neglecting valuable information from calculated sequence of gradients, often stagnating in some suboptimal plateau. Second order methods could exploit these missed opportunities, however, beside suffering from very large cost and numerical instabilities, many of them attract to suboptimal points like saddles due to negligence of signs of curvatures (as eigenvalues of Hessian).Saddle-free Newton method (SFN) [1] is a rare example of addressing this issue -changes saddle attraction into repulsion, and was shown to provide essential improvement for final value this way. However, it neglects noise while modelling second order behavior, focuses on Krylov subspace for numerical reasons, and requires costly eigendecomposion.Maintaining SFN advantages, there are proposed inexpensive ways for exploiting these opportunities. Second order behavior is linear dependence of first derivative -we can optimally estimate it from sequence of noisy gradients with least square linear regression, in online setting here: with weakening weights of old gradients. Statistically relevant subspace is suggested by PCA of recent noisy gradients -in online setting it can be made by slowly rotating considered directions toward new gradients, gradually replacing old directions with recent statistically relevant. Eigendecomposition can be also performed online: with regularly performed step of QR method to maintain diagonal Hessian. Outside the second order modeled subspace we can simultaneously perform gradient descent. I. INTRODUCTION In many machine learning situations we search for parameters θ ∈ R D (often in millions) minimizing some objective function F : R D → R averaged over given (size n) dataset: F (θ) = 1 n n i=1 F i (θ)(1) where F i (θ) is objective function for i-th object of dataset. We would like to minimize F through gradient descent, however, dataset can be very large, and calculation of ∇ θ F i is often relatively costly, for example through backpropagation of neural network. Therefore, we would like to work on gradients calculated from succeeding subsets of dataset (mini-batches), or even single objects (original SGD). Additionally, objective function can have some intrinsic randomness. In stochastic gradient descent (SGD) framework [2], we can ask for some approximation of gradient for a chosen point θ t in consecutive times t: g t = ∇ θ F t (θ t ) (≈ ∇ θ F (θ t ))(2) where F t corresponds to average over succeeding subset (minibatch) or even a single element (F t = F i ) -we can assume that g t averages to the real gradient over time. In this philosophy, instead of using the entire dataset in a single point of the space of parameters, we do it in online setting: improve the point of question on the way to use more accurate local information. To handle resulting noise we need to extract statistical trends from g t sequence while optimizing evolution of θ t , providing fast convergence to local minimum, avoiding saddles and plateaus. As calculation of g t is relatively costly and such convergence often stagnates in a plateau, improving it with more accurate (and costly) modelling might provide significant benefits especially for deep learning applications -very difficult to train efficiently due to long reason-result relation chains. A natural basic approach is working on momentum [3]: evolve θ toward some (e.g. exponential moving) average of g t . There are also many other approaches, for example popular AdaGrad [4] and ADAM [5] estimating both average gradient and coordinate-wise squared gradient to strengthen underrepresented coordinates. Such standard simple methods leave two basic opportunities for improved exploitation of statistical trends from g t sequence, discarding valuable information this sequence contains. We would like to practically use them here: 1) They do not try to estimate distance from local extremum (∇F = 0), which is suggested in linear behavior of gradients, and could allow to optimize the crucial choice of step size: which should be increased in plateaus, decreased to avoid jumping over valleys. It can be exploited in second order methods, however, directly calculating inverse Hessian from noisy gradients is numerically problematic. Instead, we will extract it from statistics of noisy gradients in an optimal way: estimate this linear behavior of gradients by least squares linear regression, reducing weights of old gradients. Controlling sign of curvatures we can handle saddles this way -correspondingly attracting (λ < 0) or repelling (λ < 0). 2) First order methods focus on information in only a single considered direction, discarding the remaining. Modelling based on information in multiple directions allows for optimization step in all of them simultaneously, arXiv:1901.11457v11 [cs. LG] 13 Mar 2023 for example attracting in some and repelling in others to efficiently pass saddles. As full Hessian is often too costly, a natural compromise is second order modelling in d dimensional linear subspace, usually d D. One difficulty is locally choosing subspace where the action is, for example as largest eigenvalues subspace of PCA of recent noisy gradients. It can be turned into online update of considered directions by slowly rotating them toward new noisy gradients, extracting their statistical trends. While not necessary, maintaining nearly diagonal Hessian in the considered subspace should improve performanceit can be made e.g. by making linear regression in entire subspace and periodically performing diagonalization, or regularly QR method [6] step. Both these opportunities could be exploited if trying to model full Hessian H, but it is usually much too costly: requiring at least O(D 2 ) memory and regular computations. However, we can focus on some chosen number d ≤ D of looking most promising directions (as modelled eigenvectors), reducing this memory and computational cost to ≈ O(dD). Many second order methods neglect signs of curvatures, e.g. natural gradient [7]: θ ← θ−H −1 g attracting to near point with zero gradient, which is usually a saddle. It also concerns quasi-Newton methods like L-BFGS [8]. Many other methods approximate Hessian with positive-defined, trying to pretend that non-convex function is locally convex, again attracting not only to local minima -for example using Gauss-Newton, Levenberg-Marquardt, or analogous Fisher information matrix (e.g. K-FAC [9]). Negative curvature is also neglected in covariance matrix based methods like TONGA [10]. Nonlinear conjugated gradient method assumes convectiveness. However, saddles are believed to be very problematic in such training. While there is belief that nearly all local minima are practically equally good, they are completely dominated by saddles -which number is exponentially (with dimension) larger than of minima. Additionally, plateaus are often formed near them -requiring much larger steps to be efficiently passed. Hence it is beneficial to not only use second order methods, but those which can handle saddles -which include signs of curvatures into considerations, instead of just ignoring it. Very rare example of such methods is saddle-free Newton (SFN) [1], changing step sign for negative curvature directions: from attraction to saddle into repulsion. It has shown to lead to significantly better parameters, literally a few times lower error rate on MNIST while compared with other methods: stagnating in suboptimal solutions with strong negative curvatures (saddles, plateaus). SFN was able to weaken these negative eignevalues a few orders of magnitude. The importance of not neglecting negative curvature is also presented e.g. in [11], showing that such rare negative curvature directions allow for significant improvements of value. However, SFN leaves some improvement opportunities, we would like to exploit here: 1) SFN tries to directly calculate second order behavior, neglecting stochastic nature of base of this calculation, what results in numerical instabilities for required Hessian inversion. To extract it from statistics instead, we can see second order behavior as linear trend in the first order behavior, which can be found in an optimal way by linear regression of the noisy gradients, for example with weakening weights of the older ones. 2) While we need to restrict modelling to a subspace, SFN uses Krylov subspace for convenience of numerical procedure. Instead, recent noisy gradients suggest locally interesting directions, which can be extracted through PCA or its online analogue discussed here. 3) Instead of calculating eigendecomposition in a given point, we can do it in online setting: split into regularly performed steps maintaining diagonal Hessian, like of QR method. 4) As we can practically use second order model only for low dimensional subspace, in the remaining directions we can simultaneously perform gradient descent. This paper is work in progress, requiring experimental investigation for choosing the details. The v4 version is completely rewritten -previous were based on gradient agreement, what is problematic for stability, only 1D linear regression was suggested. This version is completely based on gradient linear regression, including multidimensional. II. 1D CASE WITH LINEAR REGRESSION OF DERIVATIVES Estimating second order behavior from noisy gradients is a challenging task, especially if we would like to distinguish signs of curvatures -what is required to avoid saddle attraction. Close to zero Hessian eigenvalues can change sing due to this randomness, changing predicted zero gradient position from one infinity to another. Hence this estimation needs to be performed in a very careful way, preferably extracting statistics from a large number of such noisy gradients. What we are in fact interested in is position of zero derivative point in each considered direction, determined by liner trend of first derivative. An optimal way to extract linear trend is least squares linear regression -applied to sequence of gradients here, preferably with reduced weights of old noisy gradients to include two approximations: that function is only locally modeled as second degree polynomial, and that we would like to slowly rotate the considered directions. Let us focus on 1D case in this Section, then we will do it in multiple directions (e.g. a few from a million) -separately in each direction for diagonalized, or combined to maintain nearly diagonal Hessian. The choice of such subspace will explore recent statistically relevant directions, in the remaining we can perform casual gradient descent. A. 1D static case -parabola approximation Let us start with 1D case, with parabola model first: f (θ) = h + 1 2 λ(θ − p) 2 f (θ) = λ(θ − p) and MSE optimizing its parameters for (θ t , g t ) sequence: arg min λ,p t w t (g t − λ(θ t − p)) 2 for some weights (w t ) For parabola and t = 1, . . . , T times we can choose uniform weights w t = 1/T . Later we will use exponential moving average -reducing weights of old noisy gradients, seeing such parabola as only local approximation. The necessary ∂ p = ∂ λ = 0 condition (neglecting λ = 0 case) becomes: t w t (g t − λ(θ t − p)) = 0 = t w t (θ t − p)(g t − λθ t + λp) g − λθ + sλp = 0 = θg − λθ 2 + 2λpθ − pg − sλp 2 for averages: s = t w t θ = t w t θ t g = t w t g t θg = t w t θ t g t θ 2 = t w t (θ t ) 2(3) Their solution is (least squares linear regression): λ = s gθ − g θ s θ 2 − θ 2 clipped e.g.: λ = c(s gθ − g θ) s θ 2 − θ 2 p = λθ − g sλ = θ 2 g − θ gθ θ g − s θg(4) Where some "clipping" is required to avoid λ ≈ 0, e.g. for some minimal value > 0, c(x) = sign(x) min(\|x\|, ). Observe that λ estimator is (g, θ) covariance divided by variance of θ (positive if not all equal). B. 1D online update by exponential moving average Objective function is rather not exactly parabola -should be only locally approximated this way. It can be handled by increasing recent weights in the above averages, weakening influence of the old noisy gradients. Its simplest realization is through exponential moving averages w t ∝ β −t for β ∈ (0, 1) usually β ∈ (0.9, 0.999), allowing to inexpensively update such averages for a given moment, starting with 0 in t = 0: θ t = β θ t−1 + (1 − β) θ t = (1 − β) t t =1 β t−t θ t g t = β g t−1 + (1 − β) g t θg t = β θg t−1 + (1 − β) θ t g t θ 2 t = β θ 2 t−1 + (1 − β) (θ t ) 2 s t = (1 − β) t t =1 β t−t = 1 − β t(5) The s t is analogous e.g. to bias in ADAM, in later training it can be assumed as just s = 1. It might be worth modifying β, e.g. increasing it for larger step to reduce weights of far gradients. C. 1D linear regression-based SGD method Linear regression requires values in at least two points, hence there is needed at least one step (better a few) of a different method to start using linear regression, for example just SGDgoing toward stochastic gradients, updating averages (5) starting from initial θ 0 = g 0 = θg 0 = θ 2 0 = s 0 = 0. Then we can start using linear model for derivative: f (θ) ≈ λ(θ − p), using updated parameters from (4) regression. Getting λ > 0 curvature, the modeled optimal position would be θ = p. However, as we do not have complete confidence in this models, and would like to work in online setting, a safer step is θ ← θ + α(p − θ) for α ∈ (0, 1] parameter describing trust in the model, which generally can vary e.g. depending on estimated uncertainty of parameters. Natural gradient method corresponds to α = 1 complete trust. Getting λ < 0, minimizing modelled parabola would take us to infinity, hence we need some arbitrary choice for these negative curvature directions. Saddle-free Newton kind of chooses −α in these directions, experiments in [11] suggest to use ∼ 0.1 of gradient projection for such directions. The λ ≈ 0 case can correspond to plateau, or to inflection point: switching between convex and concave behavior. These are very different situations: in the former we should maintain larger step size for a longer time, in the latter we need to be more careful as λ = 0 would correspond to p in infinity. These two cases could be distinguished for example by looking at evolution of λ (higher order method), or just at its local variance: reduce step if it is large. While it leaves opportunities for improvements, for simplicity we can for example use SFN-like step: just switching sign for λ < 0 directions. Applied clipping prevents λ ≈ 0 cases, alternatively we could e.g. replace sign with tanh: θ t+1 = θ t + α sign(λ t ) (p t − θ t )(6) III. MULTIDIMENSIONAL CASE We can now take it higher dimensions, what will be done in 3 steps here: first directly for the entire space, then with periodic or online diagonalization updating basis rotation, and finally as a linear subspace of a higher dimensional spaceadditionally rotated toward new statistically relevant directions. The next Section contains algorithm for such final method for improving convergence of SGD. A. Direct multivariate approach Second degree polynomial parametrization in d dimensional space analogously becomes: f (θ) = h + 1 2 (θ − p) T H(θ − p) ∇f = H(θ − p) For Hessian H ∈ R d×d and p ∈ R d position of saddle or extremum. Least square linear regression would like to analogously minimize: arg min H,p i,t w t g t i − k H ik (θ t k − p k ) 2 Getting analogous necessary conditions. First for ∂ pj = 0 (neglecting generic case of getting to kernel of H): ∀ j t,i w t g t i − k H ik (θ t k − p k ) H ij = 0 ∀ i g i − k H ik θ k + s k H ik p k = 0 g = Hθ − s Hp = H(θ − s p)(7) for g i = g i , θ i = θ i vectors of averaged values as previously, s = t w t . For ∂ Hij = 0 we get: ∀ i,j t w t (θ t j − p j ) g t i − k H ik (θ t k − p k ) = 0 g i θ j − g i p j = k H ik θ k θ j − p k θ j − θ k p j + s p k p j gθ − gp T = Hθθ − Hpθ T − H(θ − sp)p T where the last is matrix equation with gθ ij = g i θ j , θθ ij = θ i θ j averages. Substituting (7) twice (Hp = s −1 (Hθ − g)) we get: gθ (7) = Hθθ − Hpθ T (7) = Hθθ − s −1 (Hθ − g)θ T sgθ = sHθθ − Hθθ T + gθ T H = sgθ − g θ T sθθ − θ θ T −1(8) analogous to λ formula (4) in 1D, replacing covariance with covariance matrices, denominator is positive defined. Using p = (θ − H −1 g)/s(9) we get the ∇f = 0 position: extremum or saddle of degree 2 polynomial modelling our function. To treat it only as local model, in online setting we can analogously calculate averages as exponential moving average, e.g. θ i θ j t+1 = β θ i θ j t + (1 − β) θ t i θ t j Now SFN-like approach would be calculating eigendecomposition H = O T ΛO, then performing θ t+1 = θ t + O T sign(Λ) O (θ t − p t ) Where sign(Λ) means applying sign to each coordinate of this diagonal matrix -it turns attraction into repulsion for negative curvature directions to handle saddles. There is also needed some clipping to handle λ ≈ 0. To avoid costly diagonalization in every step, for online method let us now discuss doing it periodically. B. Periodic diagonalization -currently main approach Diagonalization is relatively costly and using inaccurate one seems not a problem, e.g. completely omitting it is analogous to using gradient method instead of conjugated gradients. Hence we can perform it periodically -every some number of steps, modifying the considered basis: in which we can assume that Hessian is nearly diagonal. Let (v t i ) i=1..d be basis in moment t, which is approximately orthonormal (no need for perfect): v t i · v t j ≈ δ ij . Denote: θ t ·i = θ t · v t i g t ·i = g t · v t i(10) as coordinates in current basis. We can update multidimensional averages for linear regression in these coordinates, e.g. θ i θ j t+1 = β θ i θ j t + (1 − β) θ t ·i θ t ·j Then update 1D regression (4) independently for each coordinate i = 1, . . . , d, using diagonals: g i θ i , θ i θ i , and perform step (6) in each coordinate (with reduced trust α). The next Section contains complete algorithm. Additionally, while in most steps the basis is unchanged, every some number of steps we should improve digitalization: estimate Hessian from averages using (8), find its eigendecom- position H = O T ΛO, use it to rotate the basis [v 1 , . . . , v d ] T ← O[v 1 , . . . , v d ] T (matrix with v i as columns) and calculated averages: θ ← Oθ g ← Og θθ ← OθθO T gθ ← OgθO T C. Online diagonalization There is also a possibility to perform diagonalization in online setting -split it into regular less expensive steps maintaining nearly diagonal Hessian. Looking at Hessian formula (8), we can start from diagonal H and ask for let say first order correction during step of updating the averages -introducing tiny nondiagonal terms. Then we can perform step for example of QR algorithm [6]: decomposing matrix A = QR and multiplying in opposite order: B = RQ = Q T QRQ = Q T AQ, this way reducing nondiagonal terms. Its cost can be reduced if neglecting products of non-diagonal terms, also for rotating the basis with such Q close to identity matrix. To avoid directly calculating Hessian with (8), we could try to separately evolve sgθ − g θ T and sθθ − θ θ T −1 , using (AB −1 ) = A B −1 − AB −1 B B −1 formula and first order steps for updating the averages. However, it seems still relatively costly, would need to recalculate Hessian sometimes due to inaccuracies, brings additional complications -the details of possible improvements it can bring are left for further work. D. Modeling subspace for very high dimensions As the original space of parameters has often huge dimension (D) in machine learning applications, often in millions, for practical optimization we would like to model Hessian only for some of them (d D), e.g. a few. In the remaining we can simultaneously e.g. perform gradient descent. Modelling just d = 2 directions, in contrast to d = 1, has a chance to see both attracting and repelling direction near a saddle to efficiently handle them. As negative eigenvalues have often lower absolute values, what we can see e.g. in [11], there is rather required larger d to include some negative curvature directions in the model, like d = 10. To work in a linear subspace, analogously as for periodic online diagonalization, we can consider evolving (v t i ) i=1..d basis, this time with vectors from the large space: v t i ∈ R D . The question is how to choose these d directions in much larger D dimensional space? We would like to find where the action locally is, what is suggested by directions of the fastest change: (noisy) gradients. To extract multiple relevant directions from their statistics, a natural way is performing PCA and taking subspace spanned by eigenvectors corresponding to the largest d absolute eigenvalues -getting d dimensional subspace with the lowest average Euclidean distance from projections. However, PCA would require construction and diagonalization of huge D × D covariance matrix. Hence we would like to use only (v t i ) i=1..d basis as the current description, and modify it accordingly to part of gradient it cannot represent:g t := g t − d i=1 (g t · v t i ) v t i(11) which can be also directly used for simultaneous gradient descent. A simple way to update the basis is just adding v i + = γ ig t to each vector. This way recent statistically relevant directions would gradually become represented by the used basis, replacing locally insignificant directions. To maintain v i ·v j ≈ δ ij , sometimes improved with orthonormalization step, let us assume it in time t and find matrix Γ ≈ I to satisfy it in t + 1 in below form: v t+1 i = Γ ii v t i + j =i Γ ij v t j + γ ig (12) 1 ≈ v t+1 i · v t+1 i ≈ Γ 2 ii + j =i (Γ ij ) 2 + γ 2 i g 2 i = j : 0 ≈ v t+1 i · v t+1 j ≈ Γ ii Γ ji + Γ ij Γ jj + γ i γ j g 2 Neglecting higher order terms (e.g. Γ ii Γ ji ≈ Γ ji ), and taking symmetric Γ ij = Γ ji , we can choose (i = j): Γ ij = − 1 2 γ i γ j g 2 Γ ii = 1 − 1 2 γ 2 i g 2 Further pseudocode simplifies it for γ i = γ choice -rotating all basis vectors with the same strength. It might be also worth to consider e.g. being more conservative for large \|λ\| directions -try to mostly rotate those of low \|λ\|, what can be obtained e.g. by using γ i ∝ \|λ i \| −κ e.g. for κ = 1/2. We slowly loose orthonormality this way, hence orthonormalization should be performed from time to time, e.g. every some number of steps, or if not passing some test of orthonormality. While Gram-Schmidt depends on vector order, we can e.g. use approximate but symmetric orthonormalization step: ∀ i u i ← v i − j =i (v i · v j ) v j then ∀ i v i ← u i u i 2(13) As we neglect all but d direction, we can additionally make gradient descent θ ← θ + ηg for a standard choice of η. IV. ALGORITHM EXAMPLE This Section summarizes a basic choice of algorithm using periodic diagonalization. For simplicity it neglects time index. Initialization -choose: • d number of considered directions (could vary), • α ∈ (0, 1] describing confidence in model, step size -can be increased in the beginning, • β ∈ (0, 1) constant in exponential moving average, weight of old gradients drops ∝ β ∆t . Generally can depend e.g. on step size -be increased for larger steps. • tiny γ > 0 describing speed of exploration of new directions, can be e.g. increased in the beginning, • tiny > 0 for clipping -handling λ ≈ 0 situations, • optional η > 0 for neglected directions gradient descent, • θ -initial parameters, e.g. chosen randomly using probability distribution of parameters of given type of network, • s = 0 ∈ R, g = θ = 0 ∈ R d , gθ = θθ = 0 ∈ R d×d • (v 1 , . . . , v d ) initial basis -for example as random size d orthonormal set of vectors v i ∈ R D . Initial model training -perform some number of steps of a different method like SGD (θ ← θ − ηg), in each step updating averages (2 below) and basis (5 below, γ can be increased). Finally diagonalize Hessian (6 below) and orthonormalize the basis (7 below). Until some convergence condition do optimization step: 1) Calculate stochastic gradient g ← g t in current postion θ t = θ for current minibatch, 2) Update averages for linear regression in (v i ) basis: ∀ i=1..d θ ·i ← θ · v i g ·i ← g · v i ∀ i=1..d θ i ← β θ i + (1 − β) θ ·i ∀ i=1..d g i ← β g i + (1 − β) g ·i ∀ i,j=1..d θθ ij ← β θθ ij + (1 − β) θ ·i θ ·j ∀ i,j=1..d gθ ij ← β gθ ij + (1 − β) g ·i θ ·j s ← βs + (1 − β) 3) Calculate curvatures and positions assuming diagonal Hessian: ∀ i=1..d λ i ← c(s gθ ii − g i θ i ) s θθ ii − (θ i ) 2 p i ← λ i θ i − g i sλ i with example of clipping: c(x) = sign(x) min(\|x\|, ). 4) Perform proper step for position, for example: θ ← θ + α d i=1 sign(λ i ) (p i − θ ·i ) v i 5) Explore new directions -outside current subspace: g := g − d i=1 (g · v i )v iv = 1 2 γ 2 g 2 d i=1 v i ∀ i=1..d v i ← v i + γg −v optionally do gradient descent: θ ← θ − ηg 6 ) Every some number of steps diagonalize Hessian: H ← sgθ − g θ T sθθ − θ θ T −1 diagonalize: H = O T ΛO θ ← Oθ g ← Og θθ ← OθθO T gθ ← OgθO T ∀ i u i ← j O ij v j then ∀ i v i ← u i 7) Every some number of steps improve orthonormality: ∀ i=1..d u i ← v i − j =i (v i · v j )v j v i ← u i u i 2 V. CONCLUSIONS AND FURTHER WORK There were presented general ideas for second order optimization methods: • estimating second order behavior from linear trend of (noisy) gradients -using least square linear regression, • focused on online setting: evolution split into regular inexpensive steps exploiting local behavior, • using inexpensive adaptation of subspace to statistically relevant directions in sequence of gradient, • optimized also for non-convex situation, handling saddles by considering signs of curvatures, • hybrid with first order model -simultaneously using stochastic gradient outside second order model subspace. While they can be useful also for other situations, the main purpose here is optimizing SGD -algorithm for such application is finally suggested. Choosing the details like parameters or minibatch size will require further experimental work. There are also many questions and opportunities to explore, for example: • Beside the question of choosing parameters including minibatch size, it might be worth to evolve them -e.g. increase steps α in the beginning, lower β for larger steps for faster forgetting of far gradients, increase γ in the beginning for faster search of relevant directions. • While for λ > 0 we should just go toward minimum of modeled parabola, the λ < 0 case needs some arbitrary choice of step size. As in SFN we just switch sign here, however, it seems unlikely that it is the optimal way, there is flexibility to customize it. • Improving way of handling λ ≈ 0 situations, e.g. various ways for clipping, maybe using higher order behavior to distinguish inflection point from plateau. • Online diagonalization might offer improvements. • It might be worth weakening external basis rotation for higher absolute eigenvalues e.g. γ i = \|λ i \| −κ , the question is how much: e.g. what power to choose. • It might be worth adding estimation of uncertainty especially of positions p, and modify step size α accordingly. • Subspace dimension d might evolve depending on local situation, e.g. by removing low curvature directions, or adding new random ones -first only updating their model before including into proper step. • Having the gradients, we can by the way use them for some first order optimization -like mentioned gradient descent in directions not included in second order model (g). It might be worth to explore more sophisticated hybrids of different order methods. • Least squares linear regression could analogously provide 3rd order (or higher) local situation by additionally updating e.g. θθθ and gθθ averages -it might be worth considering e.g. using 2nd order model in a few directions, additionally 3rd order in let say one dominant direction (e.g. as coefficient of its orthogonal polynomial), and 1st order gradient descent in the remaining. Their choice of dimensions could very adaptively. • There are successful mechanisms for strengthening underrepresented coordinates, for example in AdaGrad or ADAM -they can be also applied in second order methods, what might be also worth exploring for example by increasing weights of such rare coordinates here. APPENDIX The above article was written in 2019, this Appendix contains additional remarks (mainly from late 2022) for the proposed Hessian estimation like diagonal variants, Hessian symmetrization, corr=1 approximation, also simple implementations and low dimensional tests of the proposed OGR (online gradient regression) family of optimizers. A. Mean subtraction, subspace dimension and OGR variants The central for discussed OGR (online gradient regression) is Hessian estimator, which can be seen as being built of (co)variance estimators, requiring to subtractθ,ḡ means (exponential moving here) -what is often missing in other approaches like ADAM (also σ(θ) in nominator), it offers improvements. Figure 1. Examples of 2D situations for 2nd order polynomial -having full Hessian (estimated by fOGR, in subspaces by sOGR) and gradient, in one step we could reach the minimum (red dot). In contrast, using approximation of Hessian as diagonal (dOGR), we would use two blue parabolas, for each going (e.g. 1/div) toward its minimum (dot). Using line (subspace) approximation e.g. along gradient (s1OGR ≡ lOGR), we focus on locally interesting direction. In example on the right, we can see that such directional approaches are imperfect -it would be beneficial to model Hessian in evolving subspace (sOGR) covering most of recent (local) gradient activity. Denoting means with overline, as for covariance we get: θ := θ −θĝ := g −ḡ gθ T = (g −ḡ)(θ −θ) T = gθ T − g θ T θθ T = (θ −θ)(θ −θ) T = θθ T − θ θ T H =ĝθ TθθT −1 = cov(g, θ) (cov(θ, θ)) −1(14) It seems beneficial to directly work onθ,ĝ by first subtractinḡ θ,ḡ, then updatingθθ T ,ĝθ T (later project to subspace), used in all further discussed simple implementations. 1) Full Hessian (fOGR, cfOGR): A basic approach is estimating full dimensional Hessian, with example simple implementation in Fig. 4, practical rather only for low dimensional (D) problems. In this case we can just use the above mean subtraction, and update D × D matrices:ĝθ T ,θθ T (or in later corr=1 approximation of cfOGR:ĝĝ T ,θθ T ). 2) Evolving d dimensional subspace (sOGR, lOGR, csOGR): For high dimension we can use second order model in evolving d < D dimensional local subspace, defined with evolving d×D projection matrix V = (v 1 , .., v d ) regularly shifted toward new gradients and (e.g. Gram-Schmidt) orthonormalized. Its simple implementation is shown in Fig. 5, using d number in name, e.g. s2OGR denotes d = 2 dimensional subspace. For s = 1 it becomes model along evolving line (as e.g. in ADAM), simplifying calculations (no eigendecompositions), so it might be worth treating separately: denoting lOGR ≡ s1OGR. As in this implementation, it is suggested to updateḡ,θ averages in full dimension D, subtract them from current g, θ then make projection to subspace, then update d × d matriceŝ θθ T ,ĝθ T using subtracted-then-projectedĝ,θ. This intuitively fixes to 0 center of subspace rotation -leading to smaller disturbance ofθθ T ,ĝθ T averages with subspace rotation. Initial basis could be chosen randomly, or better from the first gradients. Initial values of averages can be chosen as 0, with exception ofθθ T which can be chosen e.g. as identity matrix times some small value. A more sophisticated initialization can improve behavior of the first steps. Estimated Hessian allows to rotate inside basis to diagonalize it, allowing for less frequent eigendecomposition. 3) Choice of d dimension, its potential evolution: There is a difficult question of choosing considered dimension d, article [12] suggests most of evolution happens in d ∼ 10 subspace we could use, maybe trying to tune, optimize it. It would be beneficial to automatically choose d, maybe evolve it through optimization process. A natural evaluation of subspace activity is through (absolute values of) eigenvalues of modelled (sub)space Hessian. We can for example sort them \|λ 1 \| ≥ \|λ 2 \| ≥ . . ., and choose the largest ones as above a threshold for minimal percentage of the largest eigenvalue, e.g. \|λ\| > 0.01\|λ 1 \|. This kind of evaluation would also allow to automatically evolve d to adapt to local situation. We can e.g. remove direction from Hessian eigenbasis when its eigenvalue drops below the chosen threshold. Adding (increasing d) is more difficult: we first need to add a new vector to basis (e.g. momentum minus its projection on V ), for a few steps just update averages for it, and then treat it as the remaining vectors. A condition for such basis increase could be e.g. last eigenvalue being above some larger threshold e.g. \|λ d \| > 0.02\|λ 1 \|. 4) Diagonal Hessian approximation (dOGR,cdOGR): Calculations are much simpler if approximating Hessian as diagonal (dOGR variants) -zeroing nondiagonal dependencies, using separate parabola models. It allows to work on averages and Hessian as size D vectors -update for i = 1, . . . , D becomes: gθ T ii = (1−β)ĝθ T ii +βĝ iθiθθ T ii = (1−β) θθ T ii +βθ iθi It allows to use very fast coordinate-wise multiplications of size D vectors to model all D parabolas, as e.g. in Fig. 5, 6. This simple variant builds independent parabola models for each of D dimensions simultaneously in canonical basis, and use them to optimize step for each parameter separatelymaking it inexpensive and promising approach as least in low dimensions. Such independent parameter modelling might be also compatible with neural network. The suggested 1D learning rate is λ −1 =θθ/ĝθ. The problem is that its denominator can be very small. As discussed, mathematically it is covariance, which can be written using correlation:ĝθ = corr(g, θ) σ(g)σ(θ) for standard deviations: σ(g) = ĝĝ, σ(θ) = θθ . To prevent this covariance being low, as noticed in [13], we can use corr(g, θ) = 1 approximation, giving safe learning rate λ −1 = σ(θ)/σ(g). Such simple "corr = 1" cdOGR variant is tested in further Fig. 6 -getting better behavior thanks to safer choice (longer steps), also stuck less frequently -suggesting better generalization. Fig. 5, 6 suggest superiority of dOGR variants, however, sometimes low dimensional subspace models are misleading e.g. as in Fig. 1, also neural network training often turns out focusing on a low dimensional subspace. 5) Diagonal+subspace combinations (ldOGR/sdOGR): Low dimension evaluations in It suggests to combine (c)dOGR variant with sOGR: update both models simultaneously, and e.g. use a weighted average of their predicted step in the subspace (in perpendicular directions we can use dOGR step) -like in implementation in Fig. 5. The weigh w is additional hyperparameter, but usually we can choose w ≈ 1/2, also we could test which Hessian model has locally better prediction for difference of gradients (∆g ≈ H · ∆θ) -increasing weight of the more trusted one. While evaluation in Fig. 5 has used common hyperparameters for 's' and 'd', it might be also worth optimize them separately. 6) Online basis diagonalization: The subspace sOGR variants have freedom of internal rotation of the used basis as d × D orthonormal projection V . Having d × d Hessian model inside, we can diagonalize it H = O T DO and internally rotate V → OV , making Hessian nearly diagonal inside such optimized basis. To reduce computational cost, instead of eigendecomposition, we can regularity perform step of e.g. QR method reducing non-diagonal coefficients. The basic motivation is cost reduction, also removing eigendecomposition from Hessian eigenvalue handling (e.g. div&cut here). Also, we can use some additional methods to separately optimize behavior in such optimized (Hessian eigen-)directions, e.g. using evaluated trust levels, or higher order behavior. B. Symmetrized Hessian estimator The (14) Hessian estimator is usually close to symmetric, but not exactly. We could manually symmetrize it e.g. using: H = 1 4 θθT −1 ĝθ T +ĝθ T T + ĝθ T +ĝθ T T θθT −1(15)C = DH + H D C ij = H ij (σ 2 i + σ 2 j ) which is symmetric, finally allowing to calculate H as: H = O O T ĝθ T +ĝθ T T O ij /(σ 2 i + σ 2 j ) ij O T (16) with internal coordinate-wise division by (σ 2 i +σ 2 j ). Experimentally (16) usually gives better optimization performance than (15), at cost of required eigendecomposition -which can be avoided if online diagonalizing (e.g. step of QR method) C. Hessian uncertainty While we know the exact positions θ, unfortunately we rather do not know the exact gradients g, only estimate them from mini-batches, for randomized sample with uncertainty ("noise") of standard deviation σ n ∝ 1/ √ mini-batch size. While generally it can be direction-dependent, for simplicity let us assume here it is homogeneous noise -spherically symmetric with σ n standard deviation for each coordinate. Now in time t,ḡ t = i≥0 β(1 − β) i g t−i with weights summing to 1 hence it has the same σ n noise level. Forĝ t = g t −ḡ t = β(ḡ t−1 − g) it grows to ≈ √ 2βσ n . For Hessian estimator we need uncertainty ofĝθ T . Approximating trajectory with the meanθ, standard deviation ofĝθ T ij is approximately √ 2βσ nθj . Getting uncertainty of H ij as ≈ √ 2βσ n diag(θ)θθ T N (µ, σ)). Working on values, we would naively assume it is close to 1/c, what is true only for large c = µ/σ. Generally the expected value is infinite, there is shown density maximum (orange) and median (green) -we could use instead of 1/µ as regularized inversion, maybe including absolute value 1/\|N (µ, σ)\|. Replacing value with a density model might be beneficial. −1 ij . Of H −1 ij approximately √ 2βσ n H −1 diag(θ)ĝθ T −1 ij . 1) Division by uncertain value: Let us look closer at 1D case (e.g. dOGR or in eigendirections), where estimated learning rate isθθ/ĝθ along such single direction. The problem is that denominator is noisy here. Figure 2 shows probability density functions for division by value from N (µ, σ) Gaussian distribution. If µ >> σ we can rather neglect the noise, but for small µ it becomes problematic, e.g. suggesting to increase mini-batch size. As in this Figure, instead of just dividing by such value with uncertainty, as a regularization we can use some characteristic position of 1/N (µ, σ) density, e.g. its maximum or median. D. Value calculation, augmented line search Beside gradient calculation, it seems worth to also regularly calculate values -on training set, or on validation/test set in early stopping -maybe also to help with generalization. Another basic standard technique is line search -using e.g. 2nd order method to determine local direction, but not trusting it sufficiently to determine the distance -instead performing line search: calculating value in a few positions on this line and choosing position of the smallest value. In contrast, as in shown simple implementations (in Fig. 4, 5), the prosed approach is able to reasonably choose distance. However, being able to calculate values could help with speed of optimization process, for example using longer step (e.g. smaller div), but then measuring value and e.g. finally using a smaller step, especially if the value has turned out larger (worse). There could be used smaller step of fixed e.g. 1/2 length, maybe more steps in such simple binary, line search. It would be also worth to use the current estimated Hessian to augment such line search (one or a few steps), maybe also use the values to update the model e.g. through averages. Assume such first step was suggested to be θ = θ − ∆ θ , e.g. with SFN: ∆ θ = α\|H\| −1 m for H current Hessian estimation and m momentum. Line search would test θ = θ − a∆ θ for various a ∈ R. Taking absolute value as in SFN, we can assume m T ∆ θ > 0, the current model suggests behavior: f l (a) := f (θ−a∆ θ ) ≈ f (θ)−a m T ∆ θ +a 2 \|∆ T θ H∆ θ \|/2 (17) Its inaccuracy has various sources: higher derivatives (∆H ≈ f · ∆θ), inaccuracy of momentum, and of gradient estimation. Additionally, we usually calculate values on some finite size (sub)set of training or test dataset, also making them noisy (contain inaccuracy). Then if this value was worse or not satisfactory, we can test/perform a shorter step in this line f (θ − a∆ θ ) e.g. for a = 1/(2 − 2b) when b < 1/2, maybe also update the parabola model. Assuming we have calculated such (noisy) f l (0) and then f l (1), if the latter has turned out worse that the former, then we can e.g. try f l (1/2), and maybe continue such binary search. One open question is making more educated guess than a = 1/2, which intuitively seems perfect if f l (0) = f l (1). However, if f l (1) turns out much worse than f l (0), then we could start with a smaller a. The optimal choice of such a would need to control both uncertainties, but an heuristic should be close, e.g. by comparing prediction with calculation -define their division as b: b = (f l (0) − f l (1))/(m T ∆ θ − \|∆ T θ H∆ θ \|/2) Then b = 1 means perfect prediction -if ∆ θ was chosen as minimum of modeled parabola then we can stay there, also for b > 1 meaning even better value drop than predicted. However, for b < 1 we could try to improve. For b < 0 we got worsening -should not make such step, at least try to improve with line search. For b = 0 we get symmetric parabola -suggesting to use a = 1/2, reduced for smaller b. Finally a → 0 for b → −∞. Example of a simple formula agreeing with these constraints (to be optimized) is: a = 1/(2 − 2b) for e.g. b ≤ 1/2(18) This way for b > 1/2 we just accept the step, for lower we can e.g. use the above formula for the next step position, maybe further continuing e.g. using binary search or the same formula for modified ∆ θ → a∆ θ in such augmented line search. 1) Online value-based model update: While above we have discussed the use of calculated values for augmented line search, it is tempting to also use these values as additional local information to improve the model -e.g. the averages we use, especiallyḡ (and m if separate), andḡθ. While gradient-based model update is at heart of discussed Hessian estimator, trying to analogously derive for values leads to more averages. We can instead try approach: slightly modify the model to reduce its inaccuracy. Denote current inaccuracy as ∆ f : ∆ f = f (θ ) − f (θ) + m T ∆ θ − 1 2 ∆ T θ H∆ θ One reason of ∆ f = 0 are higher order derivatives, others are various inaccuracies -we can try modifying m and H to reduce ∆ f inaccuracy, let say by some small ξ percentage: • we can modify m → m − ξ∆ f ∆ θ / ∆ θ 2 , • H → H + ∆ H for ∆ H = 2ξ∆ f ∆ θ ∆ T θ / ∆ θ 4 Comparing with (16), for symmetrized estimator the latter can be obtained by modifying/updatingĝθ T s : gθ T s →ĝθ T s + O O T ∆ H O ij · (σ 2 i + σ 2 j ) ij O T For dOGR we can modifyĝθ T coordinate-wise. The details are yet to be polished, but it could offer improvement over line search -additionally updating the model. Also, in case of nonsatisfying value, model update could suggest another promising step no longer being restricted to the θ − a∆ θ line. E. Implicit OGR -separating gradient calculation position There are popular implicit e.g. Euler numerical scheme, in optimization e.g. as Nesterov Adaptive Optimizer [14], where we calculate gradient (maybe also value?) in a slightly shifted position (from θ). Let us call the latter as θ g , which usually was θ g = θ, but now we will separate them. The main suggestion here is that the estimated Hessian is delayed: while being in θ t position, the Hessian is estimated for current exponential moving average:θ t = i≥0 (1 − β)β i θ t−i g (assuming gradients calculated in θ t g ), and generallyθ t = θ t . Hence the basic motivation for separation of θ g from θ is trying to shiftθ closer to θ, hopefully reducing the effect of higher derivatives. Here are two basic approaches for that: • After calculating the step θ = θ − ∆ θ , we can calculate gradient and update the averages for slightly different position, e.g. for θ g = θ − χ∆ θ with χ hyperparameter slightly larger than 1. • Optimize θ g to shift new mean to the new positionθ = θ : θ =θ = (1 − β)θ + βθ g ⇒ θ g = θ − (1 − β)θ β(19) There were performed some low dimensional tests for such simple modification, but not bringing essential improvements, what might change in higher dimensions. F. Online monitoring for hyperparameter update In practice, as benchmarks e.g. [15] show, it is often beneficial both to tune hyperparameters for specific tasks, also additionally schedule: evolve them through the optimization process (usually using fixed time evolution) -both can and should be considered for the discussed approach, e.g. reducing adaptation rates (especially Γ) in later phase of optimization. However, it should be more beneficial to automatize it inside the optimization process, e.g. through some online monitoring trying to update -improve the current hyperparameters based on local situation, this way hopefully also properly choosing them in the first steps for a given specific task. Let us discuss here some possibilities for such difficult but promising improvement directions. 1) Trust level for gradient agreement: Having current Hessian estimation, we can estimate change of gradients and compare it with the calculated difference -the better the agreement, the more trust we can put in the model -like using longer steps (e.g. div in implementations closer to 1). The Hessian estimation is forθ position, suggesting to use change of gradient from here: g −ḡ ≈ H(θ −θ). Let us denote such trust evaluation as T , to handle noise it can e.g. undergo exponential moving average -below with adaptation rate ν, to normalize it we can e.g. divide difference by sum of norms (like l 2 or l 1 ) -getting T ∈ [0, 1] local trust level evaluation (the lower the better): T = (1 − ν)T + ν H(θ −θ) − (g −ḡ) H(θ −θ) + g −ḡ (20) The big question is how to translate such trust level into hyperparameters, mostly in the step size, e.g. something like div= 1 + T in suggested implementation, or some more sophisticated formula, up to applying a neural network here. 2) Coordinate-wise trust levels e.g. for dOGR: In dOGR variant (or sOGR with online Hessian diagonalization) we can treat all coordinates (or Hessian eigendirections) independently, also choose their trust levels separately. For this purpose we can just calculate (20)-like formula for 1D values with coordinatewise vector functions, and somehow use the trust levels e.g. to choose separate div step sizes. 3) Updating separate models for various parameters: A basic universal approach is maintaining multiple models, here especially averages for various adaptation rates β, and continuously monitoring their accuracies -e.g. choosing locally the best model, or weighting between suggestions of separate modelsthe better recent accuracy, the higher the weight. For example using above trust level (20) to evaluate multiple Hessian models for various β, and using 1 − T as weights divided by their sums, and choosing step as such weighted average of steps proposed by individual models. Such set of hyperparameters could be also modified, e.g. through parameter evolution, removing those having the worst evaluations, adding new ones, maybe also using some crossover between parameter sets as in evolutionary algorithms. Analogous trust level application could be choice of weight in dsOGR -updating two models, the locally more accurate one should be assigned larger weight. 4) Direct optimization of adaptation rates like β: We could try to analogously optimize adaptation rate β parameter, which controls speed of change of estimated Hessian (H → H ) -we can try to optimize it to make new Hessian in better agreement with the observed change of gradients (g −ḡ ≈ H (θ −θ)). Let us denote next values with prime, previous without: g = (1 − β)g + βgĝ = g −ḡ (21) θ = (1 − β)θ + βθθ = θ −θ gθ T = (1 − β)ĝθ T + βĝθ Tθθ T = (1 − β)θθ T + βθθ T Let us use the last to look at update of Hessian estimator, using H =ĝθ T θθT −1 = ĝθ T +βĝθ T θθT +βθθ T −1 ≈ ≈ ĝθ T +βĝθ T θθT −1 I −βθθ Tθθ T −1 H ≈ H +β ĝθ T − Hθθ T θθT −1 + O(β 2 )(22) From the other side, Hessian defines change of gradient with position: ∆g ≈ H∆θ. We can for example use differences from the means here: g −ḡ ≈ H +β ĝθ T − Hθθ T θθT −1 (θ −θ) g −ḡ − H(θ −θ) ≈β ĝθ T − Hθθ T θθT −1 (θ −θ) (23) Calculating vectors on the left and right, their good/bad agreement suggests that we can/cannot trust the model e.g. decreasing/increasing β. Natural evaluation of vector agreement is scalar product, cosine -the open question are details of using it here, to evolve β or maybe modify it for single steps. Also step size and change of Hessian might be included. G. Adding regularizer to Hessian estimator In Section III-A there was derived central here MSE Hessian estimator from 4 (e.g. exponential moving) averages. Let us add here regularizer preferring use of low Hessian coefficients, by adding the r(H) penalty to the previously optimized: arg min H,p i,t w t g t i − k H ik (θ t k − p k ) 2 + r(H) Repeating the previous derivation with this additional term r(H) adds its derivation, usingĝ = g −ḡ we get: gθ T = Hθθ T + ∂ H r(H)(24) for ∂ H ≡ [∂ Hij ] ij matrix of derivatives. 1) Hessian l 2 regularizer: a natural choice: r(H) = 1 2 η ij (H ij ) 2 gives ∂ H r(H) = ηH, getting simple natural formula for l 2 regularized Hessian estimator: H =ĝθ T θθT + ηI −1(25) While it might be valuable for Hessian estimation, Newton step uses inverted H −1 g, suggesting e.g. to add ηI to nominator instead -let us now try to formalize it. 2) Inverted Hessian l 2 regularizer: replacing H → H −1 : r(H) = 1 2 η ij ((H −1 ) ij ) 2 and using standard formula: (M + ∆) −1 = M −1 − M −1 ∆M −1 + O( ∆ 2 )(26)leads to ∂ H r(H) = −η(H −1 ) T H −1 (H −1 ) T . While Hessian is symmetric, this MSE estimation is not exactly (further we discuss symmetrization), but we can use approximation ∂ H r(H) ≈ −ηH −3 making (24): gθ T ≈ Hθθ T − ηH −3 H −1 ≈ θθT − ηH −4 ĝθ T −1 Which could be used with the right hand side H −1 as some approximation, e.g. H −1 from the previous step. However, it is difficulty to avoid infinities inĝθ T −1 , suggesting to shift η to denominator -what can be approximately done using (26): H −1 ≈θθ T ĝθ T + ηH −3 −1(27) Still to avoid singularities, we would need to replaceĝθ T with a nonnegative matrix, e.g. replacing eigenvalues with their absolute values as in SFN. H. Practical bound for H −1 Let us discuss here extension of cdOGR "corr = 1" bound to higher dimension. The Newton step uses H −1 g, which can go to infinity -it is crucial to prevent such behavior, somehow clipping the step, bounding H −1 which eigenvalues work as learning rates in corresponding eigendirections. It hasĝθ T covariance matrix in denominator: H −1 =θθ TĝθT −1 = cov(θ, θ) (cov(g, θ)) −1 We can perform PCA (principle component analysis): find cov(θ, θ) = O θ diag(σ 2 θ )O T θ cov(g, g) = O g diag(σ 2 g )O T g with σ as vectors of standard deviations, σ 2 is coordinatewise. Then transformθ = O T θθ O θ from approximately normal distribution with standard deviations given by coordinates of σ θ , and analogously forg = O T gĝ O g with σ g standard deviations. This way Hessian estimator becomes: H = O gg O T g O θθ T diag(σ −2 θ )O T θ(28) The difficult part is O T g O θ difference of PCA basis for gradients and positions. However, changes of positions are made accordingly to gradients, suggesting to approximate O θ ≈ O g as single O from gradients, which can be used in the discussed online basis diagonalization to reduce cost (matrix → diagonal as vector). H −1 safe ≈ O diag(σ θ /σ g ) O T(29) This is maximal correlation approximation, as suggested for 1D in [13], now extended to multiple directions. It essentially bounds H −1 allowing for σ θ /σ g safe choice of learning rates. This approximation neglects non-diagonal dependencies, but it is done in locally optimized basis from PCA of gradients. Some its improvements could be obtained e.g. using SVD, Canonical Correlation Analysis. 1) Direct corr=I multidimensional approximation: While there are various ways to take corr=1 approximation to higher dimensions, brief numerical exploration suggests to just directly use cov(g, θ) ≈ var(g) var(θ) for covariance matrices (30) with square root acting on matrix -its eigenvalues. Then for such cov(g, θ) approximation we can use symmetrized Hessian estimator (16). Thanks to avoiding the division by 0 problem, the improvement turned out excellent -available in GitHub, shown in Figure 7. Low dimension numerical experiments suggests this approximation allows to have nearly compete trust in parabola model: div = 1/η ≈ 1. However, sometimes steps are too large -suggesting to use "cut" mechanism, or adding small value to denominator, or maybe some line search. I. Higher order methods While full 2nd order method is rather impractical in high dimensions: requiring D 2 size Hessian, full higher order methods would require even higher powers of D for tensor of derivatives -making them even less practical. However, as e.g. dOGR here, using separate models for each coordinate brings excellent performance, and analogously could We can see usually in ≈ 10 steps (yellow) it starts approaching the minimum in (3, 0.5). It uses separate learning rates for momentum (γ) and symmetrized (16) Hessian estimator (β). Suggested 1/λ learning rates undergo absolute value as in SFN, then are divided by "div", then bounded by "1/cut". For neutral network training it should evolve in locally interesting e.g. 10 dimensional subspace [12], chosen e.g. by online PCA [16] be extended to include higher derivatives e.g. as d3OGR for separate 3rd order model in each direction. Alternatively, we could use the evolving d << D dimensional subspace approach (line for d = 1): d 2 size Hessian inside, d 3 size tensor for 3rd power, and so on. Intermediate approach could use Hessian to diagonalize subspace basis, and for its coordinates (Hessian eigendirections) use separate 3rd order models to improve choice of step sizes. To find analogous 1D higher order formulas, let us consider linear regression of gradients g using family of functions e.g. h i (x) = x i for polynomials, and overline as the average (exponential moving here) for varying (θ, g) ∈ R 2 pairs: optimization problem: arg min (ai) g − order i=0 a i h i (θ) 2 ∀ j 0 = −δ aj = h j (θ) g − i h i (θ) ∀ j h j (θ)g = i a i h i (θ)h j (θ) coef. vector: (a i ) i = h i (θ)h j (θ) ij −1 (h j (θ)g) j (31) For order 2 polynomials it leads to the previous formulas. For order 3 we would need updating additional 3 averages: θ 3 , θ 4 , gθ 2 , and the formulas are much more complicated due to matrix inversion in (31), even if using mean subtraction. In practice it seems worth to treat higher order information as additional helping with choice of step sizes -e.g. vector of div Figure 5. Simple implementations of further OGR variants (dsOGR.nb file in github.com/JarekDuda/SGD-OGR-Hessian-estimator/) intended for neural network training (d and s, but not f) -to be used in high dimension D. The diagonal version dOGR models separate parabolas for each direction, what requires updating 4 (m=mg) or here 5 size D vectors, using coordinate-wise vector multiplications/divisions. The subspace sOGR variants model Hessian in (written in name) d < D dimensional locally interesting subspace -regularly updated by adding Γ times gradient to basis vectors and applying Gram-Schmidt orthogonalization. There is also shown dsOGR variant updating both models, and inside the subspace taking weighted (w) average of two predicted steps. Shown evaluation has used written 3D Beale function starting from {−3, . . . 3} 3 size 343 set of points, there are shown sorted 343 values after 50 steps, with individual hyperparameter tuning for geometric mean evaluation -the lower, the better. Simple dOGR gave excellent results, which can be slightly improved augmenting with subspace models -could be beneficial for NN training, where evolution mostly happens in nearly constant low dimensional subspace [12]. for cdOGR predicted e.g. with some machine learning methods from trust levels, but also e.g. (g −ḡ) 3 , (g −ḡ) 4 . J. Adding a neural network optimizing use of local information While there were discussed various approaches for improvements, finding the details seems very difficult and problem/architecture dependant. It suggests to maybe train a small neural network (or a different machine learning technique) trying to optimize behavior based on all gathered local information. Especially for diagonal 'd' variants -updating averageŝ g p ,θ q for various powers p, q (not necessarily natural), maybe alsoĝ pθq , maybe also such averages for various adaptation rates, maybe also trust levels (evaluations of ∆g ≈ H∆θ agreement), numbers of step (scheduling), maybe some additional e.g. architecture information, and trying to predict a perfect learning rates e.g. for each canonical direction. To train such neural network (or other model), we can start with trajectories for a standard method like ADAM: gathering such local information along trajectory, and also finding perfect step lengths: from line search in each direction. Then train this small neural network based on such pairs: (local information, perfect step), and use it to choose steps. As such trained model would change the trajectory, we can later gather new training data for its trajectories, retrain and so on. It might be also worth to consider more sophisticated e.g. LSTM neural networks here -trying to continuously adapt to the current specific problem during optimization. K. Simple implementation and additional remarks There were prepared simple Mathematica implementations for OGR family 1 , together with results for popular 2/3D Beale function (https://en.wikipedia.org/wiki/Test functions for optimization) shown in Fig. 4, 5. They use mean subtraction (21) (before subspace projection). For Newton H −1 m step they use coordinate-wise dOGR or symmetric Hessian estimator (16), and separate learning rates for m momentum (γ, fastest adaptation), H Hessian estimator (β, slower), and subspace update (Γ, the slowest). For subspace update it just adds gradient times Γ to all basis vectors (might be worth normalizing before), and apply Gram-Schmidt orthonormalization. For saddle repulsion they use absolute values of Hessian eigenvalues as SFN, also divide suggested by Hessian learning rates by div, and bound them by 1/cut. Further optimization might e.g. use separate div, cut for positive and negative eigenvalues -reduced learning rates for negative as suggested e.g. by simulations in [11]. There could be optimized more sophisticated than such div&cut behaviors, e.g. with some nonlinearity like based on discussed above regularized estimators. It seems worth to modify hyperparameters during optimization (scheduling), e.g. reduce div toward 1 in late stage to speedup convergence in safe situations, reduce Γ for slower subspace rotations. Other mentioned improvement directions are to be considered in the future. Figure 6. Improved "correlation=1" diagonal dOGR variant (separate parabola models in all directions) using corr(g, θ) = 1 approximation (hence λ ≈ σ(g)/σ(θ)) to avoid division by small values, what has allowed for more confident steps (smaller div ≈ 1 and cut). Using γ = β we can remove one average (m=mg), however, separating them with larger γ (faster adaptation of momentum in Newton step) usually gives slightly better performance. Interestingly, it turned out to much less frequently stuck in false solutions, bringing hope for being better at generalization. While these f/s implementations use two eigendecompositions, its computational cost in e.g. 10 dimensions (suggested in [12]) is rather negligible in comparison to batch gradient calculation. In much higher dimension, as discussed such Hessian estimation can focus on locally interesting e.g. 10 dimensional subspace (and tiny learning rate in perpendicular directions): obtained e.g. by online PCA of gradients or discussed here cheaper exploration of new dimensions. It can be combined with more or less frequent online Hessian diagonalization e.g. regular QR method steps for savings in eigendecompostion. The Figures also contain simple tests to compare with other approaches, like momentum -which often escapes to infinity: requiring small learning rate, slow evolution. In contrast popular ADAM has longer jumps, but we can zig-zags as it locally thinks one-dimensionally, also the formulas are heuristic (e.g. missing mean subtraction) -in OGR replaced by real 2nd order method: formally derived from linear regression of gradients, also simultaneously optimizing in multiple dimensions -allowing for long confident jumps. Figure 6 contains later approach for corr = 1 approximation for separate coordinates, later extended to multidimensional approximation and combined with subspace variants in Figure 7 -providing essential improvements. Some additional remarks: • At least in low dimensions dOGR, cdOGR give excellent performance, could be improved combining with sOGR for additional optimization in locally active directions, maybe also e.g. with trust level evaluation, higher order information e.g. (g −ḡ) 3 , (g −ḡ) 4 and other mentioned techniques. • The symmetrized Hessian estimator (16) experimentally gives significant improvement for optimizer. • The corr=1 approximation of "c" variants essentially improves behavior by avoiding division by low values. However, in theory avoiding this approximation might allow for even better behavior -e.g. using line search or even some small neural network optimizing use of various local information, trying to predict perfect step sizes. • It is worth to continuously update the Hessian model, preferably with small batches, maybe with some uncertainty control, in let say d = 10 dimensional locally interesting subspace (e.g. [12]). In directions perpendicular to considered subspace, we can still use momentum method with tiny learning rate (α in Fig. 5), or dOGR in dsOGR. • Details of Newton step for estimated Hessian is an open question worth optimization, e.g. including regularizer, choose separate behaviors for positive and negative eigendirections ( [11]), trust evaluation, etc. • It seems worth to continuously monitor agreement of our model with calculated gradients and maybe values, e.g. trying to update step lengths (up to separately for each direction in dOGR), adaptation rates, maybe use values to modify averages (especiallyĝθ), or locally change behavior e.g. to some (augmented) line search. • It is be worth to use multiple adaptation rates (maybe e.g. given by function of a single parameter): the fastest for gradient used in Newton step (momentum), slower for 4 averages in Hessian estimator, and the slowest for the (v i ) vectors defining locally interesting subspace (Γ in Fig. 5). It might be also worth to evolve adaptation rates through some scheduling, or even use multiple simultaneously and combine their predictions by some weighting. • It might be worth separating positions for gradient/value calculation, as discussed for implicit OGR. • For generalization we would like to find wide minimum, avoiding narrow ones. Such focus could be enforced by smoothing the function, for example by adding Laplacian times some constant -it could be done here inside the considered low dimensional subspace, e.g. using trace of estimated Hessian as approximated Laplacian. Experimentally some variants are better in avoiding fake local minima (especially cdsOGR) -might have generalization. To find the proper formula, let us return to derivation from Section III-A assuming H = H T , also summing the necessity equations: ∂ Hij + ∂ Hji = 0 leads tô gθ T +θĝ T = Hθθ T +θθ T H To find H, let us decomposeθθ T = ODO T for D = diag(σ 2 ), also denote C = O T ĝθ T +ĝθ T T O and H = O T HO. It allows to transform the above equation to Figure 2 . 2Regularization of noise in denominator (for 1/ĝθ): examples of probability density functions of 1/N (c, 1) inverted Gaussian distributions (generally 1/ Figure 3 . 3Augmented line search: if the used model suggests ∆ θ step, we can calculate value in this position (f (θ −∆ θ )) and compare with model prediction -getting evaluation b. Figure 4 . 4Beale function optimization using shown fOGR variant, with various starting points: {−2, −1, 0, 1, 2} × {−1, 0, 1}. of gradients or proposed here exploration of new directions. 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Francis, "The QR transformation a unitary analogue to the lr transformation -part 1," The Computer Journal, vol. 4, no. 3, pp. 265- 271, 1961. for optimization) using written trivial hyperparameters, and for 3D Beale function as previously in Fig. 5. We can see large benefits for multi-dimensional Hessian model of cfOGR in 2D, often ∼1000x smaller values than ADAM in 20 steps. For 3D Beale there is good convergence for nearly all starting points (generalization?) especially when combining diagonal and (d = 1 or 2 dimensional) subspace Hessian models in cdsOGR variants. Figure 7. Various 'c' variants: using corr=1 approximation: cov(g, θ) ≈ var(g) var(θ). Now such approximation is also used for multidimensional Hessian e.g. in cfOGR, csOGR -with square roots as matrix functions (acting on eigenvalues), approximating cov(g, θ) as var(g), var(θ) eigenvectors being perfectly correlated. There are also shown evaluations (GitHub) as previously (sorted values starting in lattice after fixed numbers of step) -in 2D for popular test functions. then weighting their predicted steps with trivial w = 1/2 weight here. Surprisingly, η = 1/div= 1 worked perfectly here. what means complete trust in such parabola models with corr=1 approximationFigure 7. Various 'c' variants: using corr=1 approximation: cov(g, θ) ≈ var(g) var(θ), previously in standard 1D case in cdOGR (separate for each canonical direction). Now such approximation is also used for multidimensional Hessian e.g. in cfOGR, csOGR -with square roots as matrix functions (acting on eigenvalues), approximating cov(g, θ) as var(g), var(θ) eigenvectors being perfectly correlated. 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Smoothing technique and its applications in semidefinite optimization. Y Nesterov, Mathematical Programming. 1102Y. Nesterov, "Smoothing technique and its applications in semidefinite optimization," Mathematical Programming, vol. 110, no. 2, pp. 245-259, 2007. Descending through a crowded valley-benchmarking deep learning optimizers. R M Schmidt, F Schneider, P Hennig, International Conference on Machine Learning. PMLR, 2021. R. M. Schmidt, F. Schneider, and P. Hennig, "Descending through a crowded valley-benchmarking deep learning optimizers," in International Conference on Machine Learning. PMLR, 2021, pp. 9367-9376. Online principal components analysis. C Boutsidis, D Garber, Z Karnin, E Liberty, Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms. the twenty-sixth annual ACM-SIAM symposium on Discrete algorithmsC. Boutsidis, D. Garber, Z. Karnin, and E. 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[]	[ "John E Norris \nDepartment of Physics & Astronomy\nResearch School of Astronomy & Astrophysics\nThe Australian National University\nPrivate Bag, Weston Creek Post Office, ACT 2611Australia\n", "Timothy C Beers [email protected] \nPhysics Department\nMichigan State University\n48824East LansingMI\n", "Sean G Ryan [email protected] \nThe Open University\nWalton Hall, Milton KeynesMK7 6AAUnited Kingdom\n" ]	[ "Department of Physics & Astronomy\nResearch School of Astronomy & Astrophysics\nThe Australian National University\nPrivate Bag, Weston Creek Post Office, ACT 2611Australia", "Physics Department\nMichigan State University\n48824East LansingMI", "The Open University\nWalton Hall, Milton KeynesMK7 6AAUnited Kingdom" ]	[]	We report high-resolution, high-signal-to-noise, observations of the extremely metal-poor double-lined spectroscopic binary CS 22876-032. The system has a long period : P = 424.7 ± 0.6 days. It comprises two main sequence stars having effective temperatures 6300 K and 5600 K, with a ratio of secondary to primary mass of 0.89 ± 0.04. The metallicity of the system is [Fe/H] = -3.71 ± 0.11 ± 0.12 (random and systematic errors) -somewhat higher than previous estimates.We find [Mg/Fe] = 0.50, typical of values of less extreme halo material. [Si/Fe], [Ca/Fe], and [Ti/Fe], however, all have significantly lower values, ∼ 0.0-0.1, suggesting that the heavier elements might have been underproduced relative to Mg in the material from which this object formed. In the context of the hypothesis that the abundance patterns of extremely metal-poor stars are driven by individual enrichment events and the models ofWoosley and Weaver (1995), the data for CS 22876-032 are consistent with its having been enriched by a zero-metallicity supernova of mass 30 M ⊙ .As the most metal-poor near-main-sequence-turnoff star currently known, the primary of the system has the potential to strongly constrain the primordial -2lithium abundance. We find A(Li) (= log (N(Li)/N(H)) + 12.00) = 2.03 ± 0.07, which is consistent with the finding ofRyan et al. (1999)that for stars of extremely low metallicity A(Li) is a function of [Fe/H].	10.1086/309320	[ "https://arxiv.org/pdf/astro-ph/0004350v1.pdf" ]	17,983,200	astro-ph/0004350	e5a4bc214d3bcffe96c8c1b11ce381be5a0b917a	26 Apr 2000 John E Norris Department of Physics & Astronomy Research School of Astronomy & Astrophysics The Australian National University Private Bag, Weston Creek Post Office, ACT 2611Australia Timothy C Beers [email protected] Physics Department Michigan State University 48824East LansingMI Sean G Ryan [email protected] The Open University Walton Hall, Milton KeynesMK7 6AAUnited Kingdom 26 Apr 2000arXiv:astro-ph/0004350v1Subject headings: stars : nuclear reactionsnucleosynthesisabundances -stars : abundances -stars : Population II -subdwarfs We report high-resolution, high-signal-to-noise, observations of the extremely metal-poor double-lined spectroscopic binary CS 22876-032. The system has a long period : P = 424.7 ± 0.6 days. It comprises two main sequence stars having effective temperatures 6300 K and 5600 K, with a ratio of secondary to primary mass of 0.89 ± 0.04. The metallicity of the system is [Fe/H] = -3.71 ± 0.11 ± 0.12 (random and systematic errors) -somewhat higher than previous estimates.We find [Mg/Fe] = 0.50, typical of values of less extreme halo material. [Si/Fe], [Ca/Fe], and [Ti/Fe], however, all have significantly lower values, ∼ 0.0-0.1, suggesting that the heavier elements might have been underproduced relative to Mg in the material from which this object formed. In the context of the hypothesis that the abundance patterns of extremely metal-poor stars are driven by individual enrichment events and the models ofWoosley and Weaver (1995), the data for CS 22876-032 are consistent with its having been enriched by a zero-metallicity supernova of mass 30 M ⊙ .As the most metal-poor near-main-sequence-turnoff star currently known, the primary of the system has the potential to strongly constrain the primordial -2lithium abundance. We find A(Li) (= log (N(Li)/N(H)) + 12.00) = 2.03 ± 0.07, which is consistent with the finding ofRyan et al. (1999)that for stars of extremely low metallicity A(Li) is a function of [Fe/H]. INTRODUCTION In the past two decades a handful of stars with heavy element abundance less than 1/1000 that of the sun ([Fe/H] 1 < -3.0) have been analyzed at high spectral resolution, with a view to understanding the production of the elements at the earliest times. Such stars have the potential to constrain Big Bang Nucleosynthesis (Ryan, Norris & Beers 1999), the nature of the first supernovae (McWilliam et al. 1995;Nakamura et al. 1999), and the manner in which the ejecta from the first generations were incorporated into subsequent ones Ikuta & Arimoto 1999;Tsujimoto, Shigeyama & Yoshii 1999). In special cases, they can even be used to determine the age of the Galaxy (Cowan et al. 1999). As relatively simple objects, formed at redshifts ∼ > 4-5, they nicely complement and constrain abundance results from the more complicated and less well-understood Lyman-α clouds and Damped Lyman-α systems currently studied at redshifts z < 3.5 (see eg Ryan 2000). The most metal-poor objects are generally faint, and studies to date have been undertaken with somewhat limited signal-to-noise (S/N). For the six stars having [Fe/H] < -3.5 and so far analyzed at high resolution (see Table 6 of Ryan et al. 1996), perusal of the sources suggests that the data were obtained with a representative S/N per equivalent 0.04Å pixel of roughly 30, corresponding to a 3σ detection limit for weak lines of ∼ 10 mÅ. In some cases this has led to uncertainty as to the existence of important elements in these objects 2 . As one moves to the lowest abundances (or higher temperatures), the problem of determining reliable abundances can only deteriorate: for example, in the two near-main-sequence-turnoff dwarfs in the above-mentioned sample there are fewer than 20 non-Fe lines detected in the preferred wavelength range (3700-4700Å), of which fewer than half are stronger than 20 mÅ. To improve the accuracy of the existing abundance analyses we are attempting to obtain high resolution (R = 40,000) data with S/N = 100 of objects with [Fe/H] < -3.5. The purpose of the present work is to present results for CS 22876-032. In a subsequent paper we shall present results for CD-24 o 17504, CD-38 o 245, CS 22172-002, and CS 22885-096. CS 22876-032 was discovered by Beers, Preston, & Shectman (1985), and with a high-resolution abundance determination of [Fe/H] = -4.3 (Molaro & Castelli 1990;Norris, Peterson, & Beers 1993) it is currently the most metal-poor dwarf for which such analysis exists. The abundance determinations for this object are, however, compromised to some extent by the fact that it is a double-lined spectroscopic binary (Nissen 1989) of long but undetermined period. We set out to determine the nature of the two components and thus improve the reliability of the abundances in the system. CS 22876-032 has the added attraction that it is the main-sequence dwarf closest in abundance to the material which emerged from the Big Bang. It therefore has a potentially important role in constraining the primordial abundance of Li, especially if the Li abundance in higher metallicity stars has been elevated by Galactic chemical evolution, as we contend elsewhere (Ryan et al. 1999). However, only after its atmospheric parameters have been reliably determined will one be in a position to fully realize that potential. The outline of the paper is as follows. The observational material upon which our analysis is based is presented in § 2, while in § 3 we use the available material to present a first estimate of the period and orbital elements of the system. These results are coupled with the observed BVRI colors and available isochrones in § 4 to determine the atmospheric parameters T eff , log g, [Fe/H], and ξ of the components, together with element ratios [X/Fe]. Finally, in § 5 we discuss the results and revisit the question of the Li abundance of the system, and its role in constraining the primordial value. OBSERVATIONAL MATERIAL 2.1. High-resolution, High-signal-to-noise Spectroscopy CS 22876-032 was observed with the University College London coudééchelle spectrograph, Anglo-Australian Telescope combination during sessions in August 1996August , 1997August , and 1998. The instrumental setup and data reduction techniques were as described by , and will not be discussed in detail here, except in particular instances where different techniques were dictated by the binarity of the system. Suffice it to say that the material covers the wavelength range 3700-4700Å and was obtained with resolving power 40000. The numbers of detected photons in the spectra were 2700, 5800, and 3300 per 0.04Å pixel at 4300Å at the three epochs, respectively. A comparison of the spectra in the region of the Ca II K line, registered to the rest frame of the primary of the system, is presented in Figure 1. Note the binary motion of the secondary as seen in the Fe I lines and the asymmetry of the Ca II line. In an effort to best use the effective S/N ∼100 of the combined data set, equivalent widths were determined for each binary component in two ways. First, since the lines of the two stars are well separated in the 1996 and 1997 data, the spectra from these epochs were co-added in the rest frame of each component in turn, and equivalent widths measured using techniques similar to those described in Norris et al. (1996). The second method utilized the relationship between central line depth and equivalent width. All three spectra were co-added in the rest frame of the relevant component, and central line depths measured. A quadratic was then fit to the dependence of these central depths on the equivalent widths which had been determined from the first method. This relationship was then used to determine equivalent widths for all lines having measured central depths. The final adopted equivalent widths, W A and W B , are the average of the two methods, and are presented in the final two columns of Table 1 3 . Throughout this work we shall denote the observed properties of primary and secondary with subscripts A and B, respectively, while for the inferred intrinsic parameters we shall use subscripts P and S. Based on the number of photons detected, we expect the equivalent widths to be accurate to 1-2 mÅ. The first three columns of the table present wavelength, excitation potential, and log gf values from sources identified in Norris et al. (1996). For undetected species, limits of 5-10 mÅ are indicated in the table: these are a little larger than ∼3σ limits based on photon statistics, and are commensurate with the detectability of lines on the spectra. A comparison of the present results for CS 22876-032 with those of Molaro & Castelli (1990) and Norris et al. (1993) is presented in Figure 2, where one sees that the presently derived equivalent widths are somewhat smaller than reported by them. We conjecture that this may have resulted from the continuum having been placed too high on the lower S/N spectra in these earlier investigations. For lines stronger than 10 mÅ in both members of the system, Figure 3 compares the ratio of primary to secondary equivalent widths as a function of primary line strength, wavelength, and excitation potential. One sees little, if any, dependence on wavelength and excitation, but an apparent correlation with line strength. Note too the much smaller scatter in the W A /W B vs W A diagram, which is more commensurate with our error estimates, shown in the figure for σ W = 2 mÅ. This may be understood in terms of curve-of-growth effects, to which we shall return in § 4.1. In § 5.2 we shall introduce spectra taken at longer wavelengths to determine the strength of the Li I 6707Å line. These data permitted us to determine the strength of the Mg I b lines in both components, which are also presented in Table 1. The S/N per 0.04Å bin at λ5180Å was 55. Table 2 contains the radial velocities for CS 22786-032 which we have been able to glean from the archives of the Anglo-Australian Observatory (principally from our own observations), literature material kindly provided by Drs. P.E. Nissen and J.A. Thorburn, and a service observation from the William Herschel Telescope. For the AAO and WHT data we measured velocities using techniques similar to those described by Norris et al. (1996), which we shall not repeat here and to which we refer the reader. The information in the table should be self-explanatory, except perhaps for the errors, which provide an estimate of combined internal and external uncertainties. Radial Velocity Data BVRI and Strömgren Photometry, and Interstellar Reddening Broad-band BVRI photometry is available for CS 22876-032 from the work of Norris et al. (1993), following whom we adopt B-V = 0.395, V-R = 0.28, and R-I = 0.31. These values have uncertainties of 0.01 mag. Strömgren photometry is available from Schuster, Parrao, & Contreras Martinez (1993) and Schuster et al. (1996): b-y = 0.334, m 1 = 0.036, and c 1 = 0.245, with uncertainties ∼0.005. Following the latter authors we adopt E(B-V) = 0.00, based on the analysis of their Strömgren data. Beers, Preston, & Shectman (1992) report E(B-V) = 0.01 for CS 22876-032 in good agreement with this value, suggesting that the reddening is well-determined, with uncertainty 0.01 mag. ORBITAL ELEMENTS The 21 radial velocity measurements are sufficient to constrain considerably the orbital elements of this system. In Figure 4a we present the periodogram of the primary, determined using the procedure of Lafler & Kinman (1965). Our best estimate of the period from this method is ∼425 days. Values corresponding to other less prominent dips in the figure result in the phased velocity diagram having considerably larger scatter than is seen for this value. An initial estimate of the orbital elements for the two components was obtained by first setting the orbital period fixed at 425 days, and solving only for the elements of the primary using the method of Russell-Wilsing (Russell 1902; re-discussed by Binnendijk 1960). We then obtained estimates of the orbital elements for both the primary and secondary of CS 22876-032 following the differential corrections procedure of Lehmann-Filhés (1894; re-discussed by Underhill 1966), using code kindly made available to us by Dr. R.D. Mathieu (private communication). The code we employed, SBOP, is an adaptation of a FORTRAN II code originally listed by Wolfe, Horak, & Storer (1967). Solutions were attempted with an application of equal weights to all observations, followed by a de-weighting of the two observations made with the RGO spectrograph, in order to reflect their lower velocity accuracy. The elements which were obtained were quite similar in both cases. Results are presented in columns (2) and (3) of Table 3. The observational data, phased to these elements, together with the model radial velocity curves for primary and secondary are presented in Figure 4b. The agreement is quite satisfactory. At the suggestion of the referee, we have attempted to obtain orbital solutions with starting values for the period centered around the less prominent dips in the periodogram, at P ∼316 and P ∼500 days, respectively. In both instances, SBOP failed to converge to a satisfactory solution, consistent with our expectations. As a check on the sensitivity to the initial guess of the orbital period, we have also sought solutions with starting values for the period in the range 405 to 450 days. Outside this range of initial periods, SBOP again failed to converge to a solution. For starting points within the above range, SBOP converged to the same final period estimate, and the same set of orbital parameters as when we chose a starting period of 425 days. We conclude that our final solutions are stable, and accurate to the limit which the present data allow. Future spectroscopic study of CS 22876-032 will allow us to more fully populate the phase diagram, and hence enable refinement of the derived orbital elements. The errors on all of the estimated elements are also listed in Table 3. As can be seen from inspection of the table, the elements are well constrained with the exception of the time of periastron passage (T 0 ) and the angle of periastron passage (ω). This is perhaps to be expected given the low eccentricity of this system (e ∼ 0.12) and the relatively sparse phase coverage of our measurements. The mass ratio of the system M S /M P = 0.89 ± 0.04 provides a useful confirmation of our attempt at deriving appropriate atmospheric parameters for the two components by comparison with model isochrones, discussed below. Even though the error in T 0 is larger than we would like, it corresponds to only 0.03 in phase, so one still obtains useful predictions of optimal observing opportunities when the components are expected to show the greatest velocity separation. For heuristic purposes we present in column (4) of the table the changes to the elements which were obtained when the two most discrepant data points (which have large observational uncertainties) were removed from the data set. While this leads to elements with slightly improved errors, it is interesting to note that the differences are well within the cited errors in column (3). We now consider whether CS 22876-032 is an eclipsing system. Given the large separation of this pair of dwarfs the likelihood is not great. Assuming solar diameters, we estimate that the chance of eclipse is less than 0.02, and its duration less than one day. We also note that while the period of the system is established to ∼ 0.6 day, the zero point of the ephemeris is known to only 14 days, which will make the search for eclipse more difficult. That said, the additional insight to be obtained from eclipse information would seem to warrant such an effort. ATMOSPHERIC PARAMETERS T eff , log g, [Fe/H] One may use the observed BVRI and Strömgren colors together with stellar evolutionary isochrones to constrain the atmospheric parameters of the components of the system. With (B-V) 0 = 0.395, CS 22876-032 could in principle comprise a system having either two objects below the halo main-sequence turnoff, or a system with one or both objects above it. The Strömgren c 1 value of 0.245, however, which is typical of metal-poor main sequence dwarfs (see Schuster et al. 1996), clearly requires that both objects lie below the turnoff. If, for example, one forms composite colors from pairs of models on the VandenBerg & Bell (1985) isochrones with Y = 0.20, Z = 0.0001, and age = 14 Gyr, one finds that with B-V ∼ 0.40, c 1 ∼ 0.35 for those containing at least one star above the turnoff, and c 1 ∼ 0.25 for those with both below it. In what follows, therefore, we shall assume that both components of CS 22876-032 lie below the main sequence turnoff. For convenience we illustrate our method using the Revised Yale Isochrones (Green, Demarque, & King 1987 (RYI)), for Y = 0.20, Z = 0.00001 ([Fe/H] ∼ -2.3), and age = 13 Gyr, and later comment on its sensitivity to these choices. Figure 5a,b show the locus of isochrone pairs of B-V and of T eff values which together combine to produce the value (B-V) 0 = 0.395 observed for CS 22876-032. One sees that only a narrow range of permitted color and T eff exists for the primary, and one can thus constrain the parameters for this component quite well. The situation is not so favorable for the secondary, but once the primary is specified one can constrain the secondary by requiring that both stars have the same metallicity. An iterative procedure, which converged quickly, is described below. For any isochrone pair one may also determine the equivalent width correction factors one should apply to observed line strengths to obtain intrinsic values. (The corrected equivalent widths are obtained by multiplying the observed ones by these factors.) The correction factors f 1 and f 2 are the reciprocals of the fractional flux contributions of the two components, so 1/f 1 + 1/f 2 = 1.0, and the primary:secondary luminosity ratio is simply f 2 /f 1 . The values are inferred directly from the isochrones. Strictly, the (f 1 , f 2 ) values are wavelength dependent, but in practice a single pair suffices throughout the blue-violet spectral region. We note for completeness that the values adopted here are determined at the effective wavelength of the B band. Fluxes are sufficiently different in the red spectral region, however, that red-specific ratios must be utilized for the analysis of Li 6707Å(see below). In Figure 5c we show the locus of pairs of correction factors applicable in the blue-violet region. Similar diagrams may be constructed for V-R and R-I. Given the observational data of Table 1, one is now in a position to determine abundances for model pairs which reproduce the observed colors. We refer the reader to for the details of our model atmosphere abundance determination technique, which are quite standard and which we shall not repeat here, except to note that we employ Bell (1983) models in our analysis. Figure 6 shows the dependence of [Fe/H] on T eff for several pairs of permitted primary, secondary pairs, determined when using atmospheric parameters based on B-V. By adopting the requirement that both components have the same value of [Fe/H] (to within 0.01 dex) we then remove the degeneracy of the procedure. Similar diagrams were constructed using V-R and R-I rather than B-V to constrain the atmospheric parameters. Table 4 contains the results, where column (1) presents the observed composite color, columns (2), (3), (4), and (5) contain the mass, T eff , color, and equivalent width correction factor for the primary, respectively, and columns (6)-(9) give the same parameters for the secondary. Assigning equal weight to the values for each of the three cases we find that the system has [Fe/H] = -3.71 and that the masses are 0.83 and 0.73 M ⊙ . The average mass ratio is 0.88, in pleasing accord with the analysis of orbital parameters in § 3. Finally, in Table 5 we present our adopted atmospheric parameters for the components of the system. In Figure 3 we showed that the ratio of observed equivalent widths, W A /W B , increases with line strength. The equivalent width correction factors derived in the decomposition of the binary, f 4300,P and f 4300,S , are listed in Table 4 and aid in explaining the observed variation of the ratio of the observed equivalent widths. Specifically, the large value of f 4300,S indicates that the apparently weak lines in the secondary (W B in the range 8-24 mÅ) are in reality stronger by a factor ∼ 4.6 in the uncontaminated spectrum, and thus occupy the moderately strong range 37 ≤ W S ≤ 110 mÅ, partially on the flat part of the curve of growth. The primary, on the other hand, has only a factor 1.28 difference between its observed (W A ) and intrinsic (W P ) values, confirming that most of its lines do indeed lie on the linear portion of the curve of growth. The increase in the ratio W A /W B is therefore understood as the progressive saturation of the secondary's lines with increasing line strength, while those of the primary are not so affected. We illustrate this in Figure 7, which compares the observations with theoretical line ratios computed for two stars having the adopted parameter set. The theoretical curve reproduces the observed behavior quite well, confirming the origin of this effect. The reader might comment that the agreement is less than satisfactory for W A > 40 mÅ, and we would have to concur. We note, however, that it is in this regime that the lines of the secondary have W S ∼ 100-140 mÅ and experience strong saturation effects. It is not unlikely that uncertainties in the relatively poorly-defined adopted microturbulence of the secondary and the treatment of van der Waals broadening (see e.g. Ryan 1998) might lead to significant error in the theoretical strengths of these lines. Error Budget How robust are the abundances and atmospheric parameters in Tables 4 and 5? The answer is somewhat complicated because of the interplay of the two components of the binary. We discuss random and systematic errors, including the impact of one star upon the other, and summarize the situation in Table 6. First consider the uncertainties in the adopted atmospheric parameters. Errors in color-effective temperature transformations of ∼ 100 K are (unfortunately) common in the analyses of metal-poor stars. If the two components of the system were of comparable spectral type, both would be affected similarly by this error source. Since, however, their temperatures differ by 700 K, independent errors of 100 K are allocated to each. Additionally, errors in photometry and reddening are 0.01 mag, each yielding an additional uncertainty of 40-60 K. The adopted error in each star's T eff , summing in quadrature, is therefore 130 K. We note that this is consistent with the range of temperatures derived in Table 4 for the three photometric colors. The error in T eff for the secondary, however, has almost no impact on the derived [Fe/H] of the system, as can be seen in Figure 6, since an incorrect temperature will be compensated for by a change in the derived f S factor. We assign 0.10 dex and 0.03 dex as the error in [Fe/H] induced by the primary and secondary stars' temperature uncertainties, respectively. We adopt the sensitivity to errors in microturbulence and surface gravity from the calculations, but apply only those due to the primary, again because of the weak sensitivity to the secondary's parameters. Taking all atmospheric parameter uncertainties into account the quadratic sum of errors in [Fe/H] is 0.11 dex. Errors in the parameters of the primary and secondary also change the inferred mass ratio of the system. There is little latitude for the mass of the primary, as indicated by the small range of T eff and f P values in Figure 6. However, the temperature and mass inferred for the secondary will vary if the primary is revised. Changes in the primary corresponding to ∆[Fe/H] = 0.11 dex (from above) would induce a change in the secondary corresponding to 0.02 in the mass ratio. The small sensitivity of the mass ratio to the decomposition of the system is confirmed in Table 4, where the range of values is 0.015. Additional errors are associated with the adoption of a particular model atmosphere grid, which can lead to differences of 0.10 dex (Ryan et al. 1996, § 3.2), and in the isochrones used in the decomposition of the binary. The latter effect was assessed by changing the adopted RYI isochrone and viewing the impact on the derived parameters of the binary. Reasonable changes in helium, heavy element abundance, and age, of ∆Y = 0.05, ∆log Z = 0.5, and ∆Age = 2 Gyr, respectively, lead to changes in the abundance of the primary ∆[Fe/H] < 0.04. The corresponding change in the mass ratio is also small, ≃ 0.02. We also repeated the temperature and abundance determination exercise at Z = 0.0001 for the isochrones of the RYI, of Bergbusch & VandenBerg (1992) and those of the Padua group available in 1999 October at http://dns2.pd.astro.it/. Using B-V to constrain the temperature we obtained [Fe/H] P = -3.52. -3.51, and -3.38, respectively. The difference between the value obtained by adoption of the Padua and the others is driven essentially by a difference in their adopted B-V, T eff transformation, the role of which has already been considered above. We conclude, therefore, that our abundance determination for CS 22876-032 is insensitive to the adoption of the Revised Yale Isochrones. The summary of the error budget is shown in Table 6, where column (1) lists the error source, column (2) the representative error in the parameter, and column (3) the error in the abundance of the primary (and hence of the system) or M S /M P , as appropriate. Given these errors we conclude that the abundance of CS 22876-032 is -3.71 ± 0.11 ± 0.12, where the first and second uncertainties refer to random and systematic errors, respectively. It is of interest to compare this value with the significantly lower abundances reported by Molaro & Castelli (1990) and Norris et al. (1993), who found [Fe/H] = -4.29 and -4.31, respectively. For illustrative purposes we show in Table 7 abundance changes which result from various sources between the present value and that of Norris et al. (1993). In all but two cases the parameters and assumptions made in the present work lead to a higher abundance, accumulating to ∼ +0.4 when taken one at a time -not too much below the reported difference of +0.6. Although CS 22876-032 is still the most metal-poor dwarf known, our current best estimate of its metallicity is considerably less extreme than the first measurements. We complete the discussion by noting that in our high-resolution studies and those of McWilliam et al. (1995) no halo dwarf or giant is known with [Fe/H] < − 4.0. The low-resolution surveys of Beers and his co-workers also strongly suggest that the metallicity distribution of the Galactic halo cuts off near this value (e.g. Beers et al. 1998;Norris 1999). Relative Abundances, [X/Fe] Using the techniques described in some detail in Ryan et al. (1996) we next computed the abundances of the other elements. Results for the two components are presented in Table 8, where the column headers should be self-evident except perhaps for column (2), which gives details of the features used in the abundance determination. At the metal weakness of CS 22876-032, one has in most cases only one or two lines from which to determine the abundance. If only one feature is available we give its wavelength in column (2); otherwise the number of lines involved is tabulated. The standard errors given in columns (5) and (7) were determined using techniques described by Ryan et al. (1996, § 4.3). In brief, they represent the quadratic addition of abundance errors due to uncertainties in the atmospheric parameters and the measurement of line strengths. The reader should note that our knowledge of the abundances of elements, other than Fe, observed in CS 22876-032 is based on only eight atomic lines. Table 8 also contains abundance limits for a number of elements which are more stringent than previously available, given the higher S/N of the present investigation. We note that in most cases limits are given only for the primary of the system, since they are not very useful for the secondary. The present data permit us to revisit the Mn abundance of CS 22876-032. The contrast between the previous Mn I 4030Å measurement of W = 20 mÅ (Molaro & Castelli 1990; S/N ≃ 55-100 at R ≃ 20000), and ours at W A < 5 mÅ(S/N ≃ 100 at R ≃ 40000) highlights yet again (if such a comment is necessary) the need for high S/N and resolving power to study weak lines. In the absence of a definitive test, we note that the new measurement leads to a Mn abundance which more closely resembles that of other stars at this [Fe/H]. With one exception the abundances determined for the two components of CS 22876-032 are in excellent agreement, consistent with the atmospheric parameters of both objects having been well determined. The exception is Si for which the relative abundances disagree by 0.50, a difference formally significant at the 1.9σ level. In comparison, the mean absolute difference for the four other elements observed in both components is 0.05 dex. Although a 1.9σ difference is not highly significant, we have scrutinized all of our spectra of this line. We find no reason beyond the formal errors for mistrusting this measurement. However, we remain also at a loss to explain the different Si abundances in the two components. DISCUSSION Heavy Element Abundances Evidence has mounted over the past decade that below [Fe/H] = −3.0, Galactic chemical enrichment was a rather patchy business (e.g. Ryan, Norris & Bessell 1991;McWilliam et al. 1995;Ryan et al. 1996), leading to models in which such metal-poor objects are formed in the interaction of supernova remnants with material in their immediate vicinity Tsujimoto et al. 1999). In such models, one expects the abundance patterns of the most metal-poor stars to be representative of individual supernova, rather than of a time-averaged ensemble. Hence, high-resolution spectroscopic analyses of extremely metal-poor stars provide a powerful probe of the nature of the first supernovae in the Milky Way. Given the paucity of lines in CS 22876-032 we have only Al and the α-elements to discuss. In the context of star-to-star scatter we shall defer further consideration until we present results on the other four extremely metal-poor stars for which we have data, and which we mentioned in § 1 -except to note here that the value [Al/Fe] = -0.44 for CS 22876-032 lies within ∼ 0.20 of the mean line determined for the sample of metal-poor stars discussed in the investigation of Ryan et al. (1996, Figure 3). In the spirit of associating the yields of particular supernovae models with the observed abundances in extremely metal-poor stars, we conclude by discussing the α-elements. For Mg, which is produced mainly during hydrostatic carbon and neon burning we find [Mg/Fe] = 0.50. For Si and Ca we find a weighted mean value, which we designate [ Si,Ca /Fe], of -0.05 (the data were weighted by the inverse square of their errors). These elements are produced by a combination of hydrostatic oxygen shell burning and explosive oxygen burning, which varies from star to star (Weaver & Woosley 1993). As discussed by Woosley & Weaver (1995), the predicted yields of Si and Ca are expected to vary relative to Mg, depending sensitively on several physical effects. (These include among others the treatment of convection, the density structure near the iron core, the location of the mass cut, and the amount of material which falls back in the explosion.) For CS 22876-032 we have [ Si,Ca /Mg] = -0.55. It is then interesting to consider the production factors of the zero-heavy-element model supernovae of Woosley & Weaver (1995, Table 17). For those models which produce significant amounts of Mg, Si, and Ca we find the results presented in Figure 8. If one were to seek to interpret the data for CS 22876-032 in terms of enrichment from a single zero-heavy-element supernova one could identify it, quite reasonably, with the Woosley & Weaver (1995) 30 M ⊙ model Z30B. Titanium is produced under more extreme conditions than Si and Ca, as discussed by Woosley & Weaver (1995 Is our result of [Al/Mg] = -0.94 ± 0.17 consistent with the hypothesis? Before comparison with theory one must correct the present LTE abundance for Al for non-LTE effects. According to Baumüller & Gehren (1997), LTE abundances for Al based on the line we have used underestimate the abundance by ∼ 0.65 dex at [Fe/H] ∼ -2.5−-3.0. Adopting this correction for CS 22876-032, one therefore has [Al/Mg] ∼ -0.3 ± 0.2. For the above-mentioned models of Woosley & Weaver (1995), in comparison, one finds [Al/Mg] in the range ∼ -0.65−-0.35. That is, the models do not strongly constrain the situation, and within the observational errors one has reasonable agreement between theory and observation. Lithium Abundance Given the improved atmospheric parameters for the components of CS 22876-032, one may revisit the question of its lithium abundance. The best currently available observational material is that of Thorburn & Beers (1993). They obtained data with S/N = 150 and reported W A (Li I 6707Å) = 11 ± 1.3 mÅ at JD 2448850.8, when the lines of the two components were well separated. They did not detect the Li line in the secondary. Norris et al. (1994) have also observed this object and report W A (Li I 6707Å) = 15 ± 2.1 mÅ from a spectrum with S/N = 70, obtained at JD 2448490.5 when the lines of the primary and secondary would have been well separated, by some 20 km s −1 . As part of our continuing interest in the lithium problem we attempted to remeasure the equivalent width during an observing run on the Anglo-Australian Telescope in 1999 September, using techniques similar to those described by Norris et al. (1994) and Ryan, Norris, & Beers (1999). From a spectrum having S/N = 70 per 0.04Å increment we obtain W A (Li I 6707Å) = 14.4 ± 2.3 mÅ. At the time of observation the lines of the primary and secondary were separated by 21 km s −1 , and the line in the secondary was below our threshold of detectability, which we estimate to be 7 mÅ (3σ). We seek to compare these data with the recent accurate and homogeneous observational material of Ryan et al. (1999). We first ask if there are systematic equivalent width errors between the various authors. For 16 stars in common between Ryan et al. (1999) and Thorburn (1994) (who used the same equipment and techniques as those of Thorburn & Beers) one finds W RNB -W T = -1.9 ± 1.1 (s.e.) mÅ, while for 5 stars in common between Ryan et al. and Norris et al. (1994) one finds W RNB -W NRS = +2.3 ± 1.7 (s.e.) mÅ. We do not regard these differences as statistically significant, and make no correction to the reported equivalent widths. We then adopt W A (Li I 6707Å) = 12.5 ± 1.4 mÅ as the best currently available value for CS 22876-032, having weighted the data by the inverse square of their estimated errors. (The error is based on small sample statistics following Keeping (1962) without cognisance of the weights.) At λ6707Å our model shows that to obtain the line strength intrinsic to the primary this should be corrected by the factor f 6707,P = 1.35 4 , which leads to W P (Li I 6707Å) = 16.9 ± 1.9 mÅ. The recent study of Li by Ryan et al. (1999) used an effective temperature scale different from the RYI one that accompanies the isochrones used in the decomposition of the binary. It would be inappropriate to compare the Li abundance of CS 22876-032 computed using the RYI temperature scale, with a set of data based on an entirely different one. For this reason, we use the decomposed colors of the binary, as given in Table 4, to infer the temperature on the same scale as our 1999 Li work. We obtain T eff = 6223 ± 38 K. The inferred lithium abundance 5 A P (Li) (= log (N P (Li)/N P (H)) + 12.00) = 2.03 ± 0.07 6 . (The uncertainty in the quadratic addition of errors corresponding to ∆f 6707,P = 0.01, ∆T eff = 38 K, and ∆W P (Li I 6707Å) = 1.9 mÅ. We recall that here 'P' by convention refers to 'Primary' and not necessarily 'Primordial'.) The non-detection of the secondary by Thorburn & Beers (1993) and the present work is also important. If this component has T eff = 5600 K and the same Li abundance as the primary, one would expect an intrinsic line strength W S (Li I 6707Å) = 45 mÅ. Applying a model correction factor f 6707,S = 3.9 (applicable at λ6707Å for the secondary), the observed line strength would then be 12 mÅ. The most likely explanation for the non-detection of Li in the secondary is that with T eff = 5600 K it lies at the cool edge of the Spite Plateau and has experienced modest depletion to below its primordial value. The uncertainty in the temperature of the secondary is also greater than that of the primary, as is evident from Figure 6, and could explain part of the problem. From their investigation of an unbiased sample of 23 near-main-sequence-turnoff stars having 6050 K < T eff < 6350 K and -3.6 < [Fe/H] < -2.3 Ryan et al. (1999) reported a dependence of A(Li) on [Fe/H]. Their "best fit" was A(Li) = 2.447(±0.066) + 0.118(±0.023)×[Fe/H]. They argued that the most likely explanation of the positive trend with metallicity was that the primordial lithium abundance had been increased by early Galactic chemical enrichment involving the interaction of cosmic rays with the interstellar medium (see also Ryan et al. (2000)). Their material and its "best fit" trend, together with the present result for CS 22876-032, are presented in Figure 9. It is clear that the data for the latter object, which is currently the lowest metallicity dwarf known, are completely consistent with the trend reported by Ryan et al. and support their hypothesis. Equally clearly, as noted by those authors, more data are required at [Fe/H] < -3.5 to place their result, and its implications for the primordial Li abundance, on a firmer basis. We also note that to infer the primordial abundance from the observed one, a range of potentially systematic as well as random uncertainties needs to be taken into consideration. Furthermore, the computations of the primordial Li abundance as a function of baryon-to-photon ratio, η, or equivalently Ω B , are themselves subject to significant uncertainties. The impact of these issues is discussed more fully elsewhere . We are grateful to the Director and staff of the Anglo-Australian Observatory, and the Australian Time Allocation Committee for providing the observational facilities used in this study. We are likewise grateful to Dr D. Folha for obtaining the WHT spectrum as a Service Observation. It is a pleasure to thank Drs. R.D. Mathieu, P.E. Nissen, and J.A. Thorburn, for their generous assistance. J.E.N. gratefully acknowledges the support of the Department of Astronomy, University of Texas at Austin during the preparation of the manuscript, and T.C.B. acknowledges support of NSF grants AST 92-22326, AST 95-29454, and INT 94-17547. We also gratefully acknowledge the thorough report from Dr. P. Bonifacio, which led to a number of improvements to the text. Thorburn & Beers (1993) e WHT, Utrechtéchelle spectrograph combination; this closely matches the AAT, UCLES setup Norris et al.(1993) b Present measures are 0.08 dex smaller than those of NPB, but were multiplied by f 4300,P = 1.28. Arpigny & Magain (1983) this line suffers from potential blending with a feature of CH, and is accordingly excluded from many analyses (e.g. Ryan et al. 1996), including the present work. For CS 22876-032 this may explain the higher abundance derived from the λ 3944Å line in the secondary. b Following Magain (1988) and Ryan et al. (1991Ryan et al. ( , 1996 we have applied a correction of +0.18 dex to the abundances determined from Ca I 4226Å. This line consistently yields a lower abundance than other Ca I lines. Table 3, while the horizontal line represents the center-of-mass velocity of the system. Woosley & Weaver (1995). See text for discussion. λ χ log gf W A W B (Å) (eV) (mÅ) (mÅ) (1) (2) (3) (4)(5) Fig. 1 . 1-Spectra of CS 22876-032 in the region of the Ca II K line. The vertical lines indicate the positions of Fe I lines in the primary component. Fig. 2 . 2-Comparison of the equivalent widths presented here for CS 22876-032A with those of (a)Molaro and Castelli (1990) and (b)Norris et al. (1993). Fig. 3 . 3-The dependence of W A /W B on W A , wavelength, and excitation potential. Fig. 4 . 4-(a) Periodogram of radial velocities for CS 22876-032. ('Theta' is as defined by Lafler & Kinman (1965, Eqn. 1).) (b) Radial velocity curves for CS 22876-032A (filled symbols) and CS 22876-032B (open symbols) for period 424.71 days. 1σ error bars, comparable in size to the symbols, are also shown. The curves are determined by the elements in Fig. 5 . 5-Locus of primary, secondary isochrone pairs of (a) B-V, (b) T eff , and (c) equivalent width correction factors, f (applicable at λ4300Å), consistent with B-V = 0.395 observedFig. 6.-The dependence of [Fe/H] on T eff for pairs of primary and secondary isochrone components defined by the observed B-V and model atmosphere analysis of the Fe I line strengths. Filled and open symbols refer to primary and secondary, respectively, and members of pairs are joined by lines. The equivalent width correction factors, f 4300 , are appended to the data points. See text for discussion. Fig. 7.-A comparison of (a) the observed and (b) predicted dependence of W A /W B on W A for CS 22876-032. In both panels only Fe I lines stronger than 10 mÅ in both components -35 -Fig. 8.-The dependence of [ Si,Ca /Mg] on supernova mass, from the models of Fig. 9 . 9-A comparison of the lithium abundance of CS 22876-032 (filled symbol) with those of the unbiased near-main-sequence-turnoff sample of Ryan et al. (1999) (open symbols). The regression line A(Li) = 2.447 + 0.118×[Fe/H] of Ryan et al. is also shown. ). In CS 22876-032, as for Si and Ca, [Ti/Fe] is low relative to [Mg/Fe]: we find [Ti/Mg] = -0.39. This is also best reproduced by Woosley & Weaver model Z30B. Table 1 . 1EQUIVALENT WIDTHS FOR CS 22876-032A,Bλ χ log gf W A W B (Å) (eV) (mÅ) (mÅ) (1) (2) (3) (4) (5) Mg I 3829.35 2.71 -0.48 57 29 5172.70 2.71 -0.38 51 24 5183.62 2.72 -0.16 59 29 Al I 3944.01 0.00 -0.64 11 11 3961.52 0.00 -0.34 15 11 Si I 3905.52 1.91 -1.09 15 17 Ca I 4226.73 0.00 +0.24 34 18 Sc II 4246.82 0.32 +0.32 <5 <5 Ti II 3759.30 0.61 +0.20 28 14 3761.32 0.57 +0.10 21 9 Cr I 4254.33 0.00 -0.11 <5 <5 Mn I 4030.75 0.00 -0.62 <5 <5 Fe I 3727.63 0.96 -0.62 24 12 3743.37 0.99 -0.78 27 <10 3758.24 0.96 -0.02 44 17 3763.80 0.99 -0.23 33 19 3787.88 1.01 -0.85 19 <10 3812.96 0.96 -1.03 13 13: 3815.84 1.48 +0.24 33 20 3820.43 0.86 +0.14 57 26 3825.88 0.92 -0.03 47 23 3827.82 1.56 +0.08 28 19 3849.97 1.01 -0.87 17 7 3856.37 0.05 -1.28 38 17 Table 2 . 2RADIAL VELOCITIES FOR CS 22876-032A,B AAT, RGO spectrograph combination b AAT, University College London coudééchelle spectrograph combination c Private communication from P.E. Nissen. See Nissen (1989) d Private communication from J.A. Thorburn. SeeDate JD V A ∆V A V B ∆V B Source (1) (2) (3) (4) (5) (6) (7) 1985 Sep 6 2446315.1 -83.9 2.0 ... ... AAT, RGO a 1985 Dec 16 2446416.0 -96.8 2.0 ... ... AAT, RGO a 1989 Sep 13 2447783.2 -107.8 1.0 -74.8 1.0 AAT, UCLES b 1989 Oct 16 2447815.7 -103.1 1.0 -81.2 1.0 P.E. Nissen c 1989 Dec 6 2447867.0 -93.3 1.0 ... ... AAT, UCLES b 1990 Sep 27 2448162.1 -109.6 1.0 -76.2 1.0 AAT, UCLES b 1991 Aug 22 2448490.5 -87.0 2.0 ... ... AAT, UCLES b 1992 Aug 17 2448851.8 -78.6 1.0 -107.6 1.0 J.A. Thorburn d 1996 Aug 7 2450303.1 -110.2 1.0 -76.2 1.0 AAT, UCLES b 1997 Aug 23 2450683.8 -104.2 1.0 -81.3 1.0 AAT, UCLES b 1998 Aug 12 2451038.3 -87.8 1.0 -101.1 1.0 AAT, UCLES b 1999 Jul 29 2451388.7 -77.0 2.0 ... ... WHT, UES e 1999 Sep 23 2451444.6 -84.3 1.0 -105.5 1.0 AAT, UCLES b a Table 3 . 3ORBITAL ELEMENTS OF CS 22876-032 Center-of-mass radial velocity (km s −1 ) Element change when two most discrepant points are removed from the sample.Parameter Value s.e ∆ a (1) (2) (3) (4) Orbital period (days) 424.71 0.60 +0.18 T 0 (JD-2400000) 48576.37 13.51 +2.67 Eccentricity 0.12 0.03 -0.01 ω (degrees) 144.96 12.40 +2.95 −93.36 0.28 -0.02 Primary: Radial vel. semi-amplitude (K P ) (km s −1 ) 15.13 0.51 +0.03 Projected semi-major axis (a P sin(i)) (km) 8.77 10 7 3.00 10 6 0.03 10 7 M P sin 3 (i) (M ⊙ ) 0.76 0.04 0.01 Secondary: Radial vel. semi-amplitude (K S ) (km s −1 ) 17.06 0.56 0.08 Projected semi-major axis (a S sin(i)) (km) 9.89 10 7 3.25 10 6 0.06 10 7 M S sin 3 (i) M ⊙ 0.68 0.04 0.01 M S /M P 0.89 0.04 0.00 a Table 4 . 4PHYSICAL PARAMETERS FOR CS 22876-032 Color A+B M P T P Color P f 4300,P a M S T S Color S f 4300,S a [Fe/H] M S /M P Equivalent width correction factor at λ4300Å.(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) Individual: B-V = 0.395 0.841 6425 0.360 1.28 0.743 5700 0.515 4.6 -3.60 0.883 V-R = 0.280 0.825 6250 0.258 1.29 0.723 5575 0.345 4.5 -3.75 0.876 R-I = 0.310 0.819 6225 0.287 1.26 0.711 5500 0.375 4.9 -3.78 0.868 Mean: 0.828 6300 1.28 0.726 5592 4.7 -3.71 0.876 a Table 5 . 5ADOPTED ATMOSPHERIC PARAMETERS FOR CS 22876-032Component T eff log g [Fe/H] ξ a f 4300 (1) (2) (3) (4) (5) (6) Primary 6300 4.5 -3.71 1.4 1.28 Secondary 5600 4.5 -3.71 1.0 4.60 Table 8 . 8RELATIVE ABUNDANCES, [X/Fe], FOR CS 22876-032 a From Al 3944Å, [Al/Fe] P = -0.32 and [Al/Fe] S = -0.13, respectively. As emphasized first byElement Feature log (N/N H ) ⊙ [X/Fe] P s.e [X/Fe] S s.e (1) (2) (3) (4) (5) (6) (7) Mg 3 -4.42 0.50 0.12 0.51 0.15 Al a 3961Å -5.53 -0.45 0.12 -0.42 0.23 Si 3905Å -4.45 -0.21 0.13 0.29 0.24 Ca 4226Å -5.64 +0.01 b 0.13 -0.01 b 0.24 Sc 4246Å -8.90 < 0.27 0.07 ... ... Ti 2 -7.01 0.08 0.11 0.23 0.23 Cr 4254Å -6.33 <-0.22 0.04 ... ... Mn 4030Å -6.61 <-0.18 0.04 <-0.26 0.06 Co 3873Å -7.08 < 1.25 0.04 ... ... Ni 3858Å -5.75 < 0.00 0.04 ... ... Sr 4077Å -9.10 <-0.65 0.05 ... ... Ba 4554Å -9.87 < 0.38 0.07 ... ... [Fe/H] = log (N Fe /N H ) star -log (N Fe /N H ) ⊙ 2 For example, fromNorris et al. (1996) : "Ba 4554Å measurements of CD-38 o 245 are rather discordant, ranging from detection at 18 mÅ and 19 mÅ(Bessell & Norris 1984;Primas et al. 1994) to upper limits < 10 mÅ and < 7 mÅ byMolaro & Castelli (1990) andPeterson & Carney (1989)." It may be of interest to some readers to compare our final equivalent widths with those determined with the first, more standard, method. The mean difference between final and first estimates is less than 1% for both primary and secondary. The mean absolute differences are 4 and 6 %, respectively. Based on B-R results from the RYI and on model atmosphere synthetic spectra, which give consistent results. The Li abundance was calculated using exactly the same formalism as inRyan et al. (1999), i.e. computing synthetic spectra forBell (1983) models using four 7 Li components, and measuring the equivalent width of the synthesized line.6 The perceptive reader will note that in this work we have combined the higher S/N data ofThorburn & Beers (1993; S/N = 150) with our lower S/N data (two spectra each with S/N = 70), whereas in a recent work(Ryan et al. 1999) on the spread about the plateau we advocated the importance of homogeneity. 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[ "The apparent eta Carinae's long-term evolution and the critical role played by the strengthening of P Cygni absorption lines", "The apparent eta Carinae's long-term evolution and the critical role played by the strengthening of P Cygni absorption lines" ]	[ "A Damineli ", "D J Hillier ", "F Navarete ", "A F J Moffat ", "G Weigelt ", "T R Gull ", "M F Corcoran ", "N D Richardson ", "T P Ho ", "T I Madura ", "D Espinoza-Galeas ", "H Hartman ", "P Morris ", "C S Pickett ", "I R Stevens ", "C M P Russell ", "K Hamaguchi ", "F J Jablonski ", "M Teodoro ", "P Mcgee ", "P Cacella ", "B Heathcote ", "K Harrison ", "M Johnston ", "T Bohlsen ", "G Di Scala " ]	[]	[ "Mon. Not. R. Astron. Soc" ]	Over the entire 20th century, Eta Carinae (η Car) has displayed a unique spectrum, which recently has been evolving towards that of a typical LBV. The two competing scenarios to explain such evolution are: (1) a dissipating occulter in front of a stable star or (2) a decreasing mass loss rate of the star. The first mechanism simultaneously explains why the central star appears to be secularly increasing its apparent brightness while its luminosity does not change; why the Homunculus' apparent brightness remains almost constant; and why the spectrum seen in direct light is becoming more similar to that reflected from the Homunculus (and which resembles a typical LBV). The second scenario does not account for these facts and predicts an increase in the terminal speed of the wind, contrary to observations. In this work, we present new data showing that the P Cygni absorption lines are secularly strengthening, which is not the expected behaviour for a decreasing wind-density scenario. CMFGEN modelling of the primary's wind with a small occulter in front agrees with observations. One could argue that invoking a dissipating coronagraphic occulter makes this object even more peculiar than it already appears to be. However, on the contrary, it solves the apparent contradictions between many observations. Moreover, by assigning the long-term behaviour to circumstellar causes and the periodic variations due to binarity, a star more stable after the 1900s than previously thought is revealed, contrary to the earlier paradigm of an unpredictable object.	null	[ "https://export.arxiv.org/pdf/2211.01445v1.pdf" ]	253,265,059	2211.01445	b4c7183073e83043a30a3b33584ab2d4f56f61fc	The apparent eta Carinae's long-term evolution and the critical role played by the strengthening of P Cygni absorption lines 2022 A Damineli D J Hillier F Navarete A F J Moffat G Weigelt T R Gull M F Corcoran N D Richardson T P Ho T I Madura D Espinoza-Galeas H Hartman P Morris C S Pickett I R Stevens C M P Russell K Hamaguchi F J Jablonski M Teodoro P Mcgee P Cacella B Heathcote K Harrison M Johnston T Bohlsen G Di Scala The apparent eta Carinae's long-term evolution and the critical role played by the strengthening of P Cygni absorption lines Mon. Not. R. Astron. Soc 0002022Printed 4 November 2022 Accepted XXX. Received YYY; in original form ZZZ(MN L A T E X style file v2.2)Stars: winds, outflows -stars: individual: η Carinae -stars: massive -stars: mass-loss -stars: binaries Over the entire 20th century, Eta Carinae (η Car) has displayed a unique spectrum, which recently has been evolving towards that of a typical LBV. The two competing scenarios to explain such evolution are: (1) a dissipating occulter in front of a stable star or (2) a decreasing mass loss rate of the star. The first mechanism simultaneously explains why the central star appears to be secularly increasing its apparent brightness while its luminosity does not change; why the Homunculus' apparent brightness remains almost constant; and why the spectrum seen in direct light is becoming more similar to that reflected from the Homunculus (and which resembles a typical LBV). The second scenario does not account for these facts and predicts an increase in the terminal speed of the wind, contrary to observations. In this work, we present new data showing that the P Cygni absorption lines are secularly strengthening, which is not the expected behaviour for a decreasing wind-density scenario. CMFGEN modelling of the primary's wind with a small occulter in front agrees with observations. One could argue that invoking a dissipating coronagraphic occulter makes this object even more peculiar than it already appears to be. However, on the contrary, it solves the apparent contradictions between many observations. Moreover, by assigning the long-term behaviour to circumstellar causes and the periodic variations due to binarity, a star more stable after the 1900s than previously thought is revealed, contrary to the earlier paradigm of an unpredictable object. INTRODUCTION Eta Carinae (η Car) underwent the so-called Great Eruption (GE) in the 1840s (Innes & Kapteyn 1903;Smith & Frew 2011) when it ejected the well-known Homunculus bipolar flow. The mechanism of the GE is still not well understood. The recent detection of light echoes at speeds of up to 20,000 km s −1 is in line with a supernova-like explosion, but without the disruption of the core, as the star continued to exist (Smith et al. 2018a). The most popular current H i absorptions, resembling an early-F supergiant star (Walborn & Liller 1977). After 1895, the H i and Fe ii lines went into strong emission, without absorption. No significant changes were observed until 1944, at least for the strong lines. Those old spectra are of low resolution, and little can be said about faint lines, like those associated with He i, especially when blended with the strong Fe ii lines. Humphreys et al. (2008) carried out a detailed analysis of the spectra taken between 1892 and 1941. An excellent digitised spectrum from a plate taken in 1938 shows the presence of He i λ4712 (see their Fig. 10). An intriguing event, which has not been adequately discussed, occurred in the 1940s. In just two years, the brightness of the whole object (core plus nebula) brightened by ∼ 1.2 mag while the central star remained almost constant -with at most 0.1 mag of brightness increase - (Thackeray 1953b;O'Connell & S.J. 1956). The stellar core brightness dominated the light curve before 1940, and only recently (2010) it became again brighter than the nebula (Damineli et al. 2019). The decoupled behaviour of the star and its reflection nebula indicates a circumstellar cause. The dust extinction inside the Homunculus seems to have dropped in all directions in 1940, except to our line-of-sight (LOS). Thackeray (1953a) identified forbidden lines of [Fe iii], [Ne iii and [S iii] in spectra taken in 1951-52. Gaviola (1953) recorded the forbidden line of [Ne iii] at similar intensity (relatively to H ii λ3835) in spectra taken in 1944-52, what indicates that the other three high excitation forbidden lines were present also in 1944, but were not recognised/recorded by Gaviola. Contrary to the claim by Abraham et al. (2014), and in agreement with , this event was not a mass ejection but a rearrangement of breaches in the circumstellar medium that enabled the UV radiation from the secondary companion star to ionise the Weigelt clumps (Weigelt & Ebersberger 1986), where the narrow emission line components originate (Davidson et al. 1995). Those narrow features strongly impact the visibility of the He i lines, since they are more readily identified than their broad emission components, which are inconspicuous and difficult to recognise in low-resolution spectra taken before 1940 (if they were present at all; . Since the 1940 event, the object's apparent brightness has gradually increased. It might have been because of the Homunculus' expansion or slow changes in the general circumstellar extinction. The impact of the coronagraphic occulter was revealed again in 1992 (Hillier & Allen 1992). Around 2000, the object's brightness exhibited an accelerated increase (Martin et al. 2006;Damineli et al. 2019;Fernández-Lajús et al. 2010;Martin et al. 2010;Davidson et al. 1999;Davidson et al. 2005). The cause of this welldocumented episode was the brightness increase of the stellar core as opposed to Homunculus, which remained at almost constant brightness (Damineli et al. 2019(Damineli et al. , 2021. The η Car spectrum recorded in the 1980s was very unusual for an LBV star. It exhibited broad wind lines (FWHM∼ 800 km s −1 ) of H i, Fe ii, and He i which are now understood to be associated with the primary stellar wind, although the He i is also likely to be influenced by the windwind collision (WWC) zone and the radiation of the secondary. Very prominent narrow emission lines (FWHM 20-80 km s −1 ) of H i, He i, Fe ii and [Fe ii], superimposed on the spectrum, arise in the Weigelt clumps (Davidson et al. 1995). Both the Fe ii and [Fe ii] lines exhibit a broad base, with a similar FWHM to the primary wind lines. The broad [Fe ii] lines are unusual and are not seen in other stars with strong stellar winds at similar excitation regimes (e.g. HDE 316285, Hillier et al. 2001b). Many photometric and spectroscopic features have been reported to vary periodically (P = 5.534 yr). These "lowexcitation events" (Damineli et al. 1998) are known to be produced by the periastron approach of the companion star in a very eccentric binary system (Damineli et al. 1998(Damineli et al. , 2000. The periastron passages produce large variations in all features in the time frame between one month before to three months after the periastron. Some features suffer large variations on scales shorter than a week ( e.g., Hα, He ii, Si ii), and some are affected by instabilities in the wind-wind collision (e.g., He i, N ii). Such phases should be avoided when studying the long-term evolution of the system. Mid-cycle phases also are affected by orbital variations, but with much lower amplitude. In general, it is sufficient to compare data taken at the same orbital phase at distinct epochs to measure the long-term trend. Long-term spectroscopic evolution was reported in several works like Mehner et al. (2010); Mehner et al. (2011Mehner et al. ( , 2012Mehner et al. ( , 2015Mehner et al. ( , 2019; Martin et al. (2021); Davidson et al. (2018). Those authors suggest that the variations are due to the intrinsic evolution of the central star, and assign the secular brightness increase to a decrease in η Car's primary wind density or a decrease in circumstellar extinction. Secular evolution in photometric measurements was described in Damineli et al. (2019, hereafter paper I) and direct versus reflected line emission analysis is reported in Damineli et al. (2021, hereafter paper II). Long-term variability has accelerated since 1998-2000 and is well documented in spectra taken with HST/STIS, in which the central star is separated from the ejecta at scales of ∼0. 1. Since the longterm variability is affected by significant random fluctuations, and since the low cadence of observations precludes an adequate characterisation of the secular evolution, we present in this paper high cadence monitoring obtained over the last 3 decades. This work is focused on a few selected spectral lines that have ground-based spectroscopic observations with a temporal sampling of at least monthly over the last six orbital cycles. Ground-based spectroscopy is useful because it collects light from the inner circumstellar region (out to a few arcseconds) from which the bulk of the stellar and circumstellar emission arises. The spatial resolution is set by the seeing, and the measurements are similar in coeval observations from different observatories, regardless of the spatial and spectral resolutions. A localised occulter, covering the central part of the system in the direction of the observer, was proposed by Hillier & Allen (1992) to explain the differences between the spectrum seen directly from the central source and that reflected from the dust in the surrounding Homunculus nebula. This model could also explain the presence of strong and broad forbidden lines of Fe ii that is not normally seen in spectra of P Cygni stars such as HDE 316285 (Hillier et al. 2001a). Many additional observations lend support to the presence of a localised occulter. Observations of Mehner et al. (2012) and Damineli et al. pcygabs-midcycle 3 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 4 Damineli et al. (2021) show that the equivalent width (EW) of wind emission lines in the direct light of the star is decreasing with time, becoming increasingly similar to the EW of lines observed in the light reflected from dust in the south pole of the Homunculus (whose system axis tilted by ∼ 45 degrees from our LOS; Smith 2006). The unexpected intense brightness of the Weigelt clumps (at the time when they were discovered in 1983, 1985Weigelt & Ebersberger 1986) as compared to the central star was later attributed to an enhanced extinction centred on the star, and that was restricted to a few tenths of an arcsecond Davidson et al. 1995). Gull et al. (2009) andMehner et al. (2014) reported that while the central star brightened by a substantial amount, the Weigelt clumps, which reflect its light, remained at constant flux. We interpret this as a diminishing effect of the enhanced extinction, mentioned earlier. Polarisation maps from Falcke et al. (1996) exhibited a sub-arcsecond knife-edged structure partially covering the star. CMFGEN models by Hillier et al. (2001bHillier et al. ( , 2006 required extra extinction in front of the star to match the wind model, and the observed UV spectrum could only be fitted after excluding the central 0. 033 region. Damineli et al. (2019) showed that the stellar core brightened by 2.5 magnitudes in the V band in the period 1998-2018, while the nebular brightness did not increase more than a few tenths of a magnitude. This mimics the behaviour reported when comparing the fluxes of the central star and the Weigelt blobs, which are now more difficult to see than when they were discovered (Weigelt & Ebersberger 1986), despite having maintained the same absolute flux, because the central core brightened so much. Mehner et al. (2019) reported that the near-and midinfrared luminosity of the whole object (star plus its nebula) fluctuated by only a small amount in the period 1968-2018 while the stellar core brightened by more than a factor of 10. Since the emission at those wavelengths is set by dust absorption of UV and optical radiation from the central stars (and its subsequent re-emission at infrared wavelengths), the stability in brightness at long wavelengths constrains the luminosity evolution of the central sources. Based on high-resolution ALMA maps, Bordiu & Rizzo (2019) and Morris et al. (2020) reported molecular absorption in front of the central star with velocities that indicate a localised clump of relatively cold material close to the star in our LOS. HCN was detected at -60 km s −1 , which suggests a connection with the Little Homunculus. A fading absorption by an unidentified carrier at λ 10792Å was reported by Damineli et al. (2021), probably associated with the same molecular absorber detected at longer wavelengths, as revealed by ALMA. Most of the data on η Car obtained in recent times have been collected in monitoring campaigns focused on the lowexcitation periastron events. Those data can also be useful for monitoring the long-term evolution of the η Car system, provided that comparison is made at the exact same orbital phases. Close to periastron, the spectrum suffers large variation in a couple of days. The long-term appearance of absorption components due to the increased transparency of the occulter can be differentiated from the low-excitation (periodic) periastron events based on the line-profile velocities. In the long-term evolution, the centroid velocity is less than the terminal wind velocity of the primary (420 km s −1 ), while the components associated with the periastron events typically show higher velocities and shorter timescales (Groh et al. 2012). For further clarity, variations between phases 0.98 and 1.18 (during which the influences of periastron passage are the largest) are excluded from the long-term evolution analysis. However, measurements in these intervals are still presented for completeness. For a description of the 2003, 2009, and 2014 periastron-related events occurring in the light from our direct LOS to the central star, and in light reflected from the Homunculus nebula (which samples different lines of sight to the central star); see Mehner et al. (2011Mehner et al. ( , 2015. Gull et al. (2021) studied the FUV photometric and spectroscopic evolution at periastron at the same phases from 2003 to 2020. In 2020.2 (# 14 periastron), the UV continuum suffered a short-lived peak, coeval with the one seen at longer wavelengths, which corresponds to the alignment of the "borehole" (Madura & Owocki 2010) to our LOS. Those authors interpret the recent visibility of the "borehole" in the FUV as due to a dissipation of absorbing clumps in our LOS. Up to date, the long-term increase in the P Cygni absorption profiles in η Car has not been discussed yet and is the main subject of this work. These features are crucial for understanding the long-term evolution of the primary wind as they carry signatures of the wind density and terminal speed. In this work, we adopted the period P = 2022.7 days measured from the periodic variation in the EW of He ii λ4686 over the last four orbital cycles, and use the Groh & Damineli (2004) orbital cycle numbering scheme, in which cycle #14 begins at the last periastron passage on February 17, 2020. We also updated phase zero, the time at which the EW of He ii λ4686 is at minimum, by subtracting 0.6 days. Averaged over four cycles, phase zero occurs at T0 = 2458896.5 (Navarete et al. -in preparation). As in paper II we use the intensity (I ) normalised to the continuum (Ic) where the continuum flux was fitted by a low-order polynomial passing through points selected far from strong emission lines. This paper is organised as follows. In Sect. 1, we included a Sub-section 1.1 summarising the reports of previous works related to the idea of the dissipating coronagraphic occulter. In Sect. 2, we present the origin and general characteristics of our data; in Sect. 3, we present our analysis of the measurements; in Sect. 4, we report semi-quantitative modelling. Finally, in Sect. 5, we present discussions about the results of this work. A qualitative scenario to interpret the spectroscopic and photometric evolution of eta Carinae in the last century Here we summarise the properties of a dissipating occulter as discussed in previous works (Hillier & Allen 1992;Weigelt et al. 1995;Damineli et al. 2021;Pickett et al. 2022;Gull et al. 2021Gull et al. , 2022. The semi-quantitative model is discussed in Sect. 4. To understand the observed spectrum of η Car, we need to identify the formation region of the different components, their relative flux, and the impact of the occulter on them. Figure 1 shows a schematic view of the three main formation regions of the spectrum. With STIS/HST it is possible to isolate the spectrum of the Weigelt clumps with its nebular pcygabs-midcycle 5 emission lines and continuum reflection -which are relatively faint. The reflected spectrum on the Homunculus can also be resolved (for example, at the FOS 4 position) even from the ground. The central region itself is unresolved, and is composed of: a) the central star plus its inner wind -the major source of light in the system, the source of both the stellar continuum and the very broad P Cygni spectral features profiles; and b) the outer wind which contributes with relatively broad lines and some continuum flux. The occulter covers the inner parts of the wind region, leaving the Weigelt clumps outside. The FOS 4 region at the SE pole of the Homunculus is located on the bipolar axis, which is tilted by ∼ 45 degrees from our direct view of the star. This region receives the unobstructed light from the central object (see the black line in the upper plot of Fig. 2). This spectrum is typical for an LBV star, without a significant contribution from narrow-line components. It was taken on 2006.5 and has remained very stable since then (see Fig. 4 from Damineli et al. 2021) 1 A coeval spectrum was taken in direct viewred line overplotted in Fig. 2. The direct view of the star is blocked by the occulter, which depresses the flux originating from the central region, increasing the EW of narrow lines in rectified spectra. The bottom plot of Fig. 2 shows two ground-based spectra taken at approximately the same orbital phase, one in 1993.5 (magenta line) and another in 2021.2 (blue). The older one exhibits plenty of very strong narrow lines from the Weigelt clumps, whose apparent strength decreased in 2021.2, due to the long-term decrease of the occulter's extinction. At the same time, the narrow-line fluxes remained constant (Damineli et al. 2021). As the occulter dissipates with time, the spectrum becomes more similar to that of the reflected view (black line in the upper plot of Fig. 2). The broad base associated with many of the Fe ii and [Fe ii] lines is also much less apparent in the later spectrum. The impact of the occulter on the line intensities is strongly dependent on the radial profile of the line-formation zone plus the size, geometry and radial-opacity profile of the occulter. A rough idea of the stellar wind structure can be obtained from the model reported by Hillier et al. (2001b) -see their Fig. 9. Fe ii broad lines are formed in the outer regions of the primary's wind, and their relative intensities have been decreasing with time at a pace slower than the narrow lines formed in the Weigelt clumps. As the broad [Fe ii] lines are formed at larger radii than of Fe ii, they have also decreased by a larger factor, as indicated by the magenta line in the bottom panel of Fig. 2. The higher energy excitation lines (He i, N ii, high members of H i line series) are formed close to the central star, and some of them are entirely covered by the occulter. For instance, the H γ line shown in Fig. 2 has not significantly changed over the last 28 years. The high-excitation permitted lines formed behind the occulter would maintain their normalised intensity if there was no continuum source outside the occulter. However, the wind region emits free-free radiation, veiling the absorption lines. As the flux from the occulted region increases due to the decreasing extinction, the contribution from the central region dominates more, making the absorption lines more visible. The presence of the hot secondary star complicates this simple scenario, as it excites the inner wind of the primary, causing significant changes on the lines formed in the wind, especially at phases close to the periastron (Mehner et al. 2012(Mehner et al. , 2011, when instabilities appear in the wind-wind collision zone. Recently, the stellar wind of eta Car was directly resolved by infrared interferometry across several emission lines (Weigelt et al. 2021). These observations agree with the CMFGEN model. DATA The data used in this work are almost all from the same facilities as those described in Damineli et al. (2021). To avoid repetition, we list those facilities in Table 1, and for details, we direct the reader to Sect. 2 of paper II. There are a few observations from amateur astronomers, listed in Table 1 and partly described in Teodoro et al. (2016). The equivalent widths we have measured, which are plotted in Figs. 7 and 8 are displayed in Tables 2-5. STIS spectra give much cleaner measurements of the central star, since they are not contaminated by the Weigelt blobs seated a few hundred milliarcsec away. To exemplify how STIS and ground-based spectra are nevertheless comparable for most of the spectral lines, we present in Fig. 3, two spectra separated by three cycles (bottom plot). In the upper plot, two ground-based spectra separated by 28 years are shown. Although the narrow emission lines have a lower impact on the ground-based spectrum at present times, they still have a significant influence. Figure 11 shows how much STIS spectra were already cleaned from the emission components in the past orbital cycle. Unfortunately, STIS spectra well separated in time are rare and have not been taken during a high excitation state in the last three orbital cycles. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + RESULTS We measured the EW of the absorption component of a group of spectral lines in about one thousand ground-based spectra obtained between 1989 and 2022. Measurements are reported in Tables 2-5, and presented in Figs. 7 and 8. The absorption components exhibit four different types of variability: • A peak in EW absorption P Cygni components around periastron. • Variations along the orbital cycle due to the distorted shape of the primary star's wind, caused by the WWC cavity. In most of the lines, these variations have a relatively low amplitude when compared to the low-excitation event. • Episodic low-amplitude variations near mid-cycle, lasting from days to months. • A continuous and smooth progression of the absorption over the long term, with a low amplitude modulation along a single orbit, far from periastron. The events caused by periastron passage are easily identifiable by the strong peaks in the EW light curves -the most remarkable variations start ten days before phase zero (assumed to be periastron) and last for about two months. pcygabs-midcycle 7 These events are characterised by an increase in the radial velocities of the absorbing components to values much greater (in absolute value) than the terminal wind speed of the primary star (−420 km s −1 ; Groh et al. 2012). Photometric modulation along the orbital cycle, whose maximum occurs around phase 0.5, but is not exactly repeatable from cycle to cycle, has been described in paper I (see Fig. 16 therein). Emission lines also vary along the orbital cycle, although without a simple pattern. In general, they get stronger just before the end of the cycle. Episodic variations are difficult to identify as they require frequent monitoring, with accurate measurements of EW over long time scales (years to decades). They do not show changes in the velocity of the line profiles in absorption at mid-cycle. The safest way to identify those sporadic variations is by analysing the time series after subtracting off the long-term trend. Long-term variations in absorption line strength can be gleaned by comparing spectra taken at the same phase of widely-spaced orbital cycles, as such variations are generally larger than the sporadic low-amplitude cycle-to-cycle variations. The long-term increase in the strength of the P Cygni absorption component is easy to see in some lines, such as He i λ10830, but more difficult in other lines such as He i λ5876 and H i Pa14 λ8750. 3.1 Comparison of spectra taken at the same phase, separated by several orbits We first compare in the upper plot of Fig. 3 two highresolution spectra (R∼48,000) taken 28 years apart (five orbital cycles). They were taken one year after the periastron events, the first in June 1993 (phase 9.18) and the second in March 2021 (phase 14.18). At this phase, the spectrum has already recovered from the "low-excitation event". The 1993 spectrum is representative of what has been called the "η Car-like" spectrum seen since the beginning of the last century: It shows very strong lines of H i (H-alpha I ∼150 Ic), Fe ii and [Fe ii] (I ∼15 Ic) which exhibit both broad and narrow components. In the 9.18-phase spectrum, absorption profiles only appeared in lines from very high-excitation levels of the Balmer and Paschen series, in Si ii 6347 and in He i lines, such as He i 4026, He i 4712 and He i 10830. During cycles 9 and 10, strong absorption components appeared only around the periastron passage. A detailed analysis shows that the long-term weakening of emission lines is closely correlated with the secular increase in brightness of the central star, consistent with both being produced by the dissipation of the occulter in front of η Car (paper II). The same spectral range observed with STIS at phases 10.60 and 13.64 is shown at the bottom plot in Fig. 3. It essentially shows the broad emission lines, but also the deepening of the P Cygni absorption of He iλ5876. Figure 11 shows the increasing strengthening of N iiλ5668-5712 absorptions from 2001 to 2018. The fact that the N ii lines in the models are stronger than those observed could indicate that the dissipation of the coronagraph is still in progress. However, it is also possible that the lines from the model are stronger than in reality. A set of N ii lines exhibiting variable emission/absorption is N iiλ4601-4643. This multiplet was high-lighted by Davidson et al. (2015), who showed that these lines were weak/absent at 2003.5 and 2009.0 periastra, but clearly present in 2014.5. Figure 4 focuses on typical line-profile differences at phases 9.18 and 14.18. In Fig. 4 d (bottom left panel), it can be seen that the Hδ line did not exhibit P Cygni absorption at phase 9.18, but did at five orbital cycles later. In Some spectral lines, such as He i λ10830 and Si ii λ6347, were monitored sufficiently often to allow a comparison of their line profiles over five or six orbital cycles. Figure 5 shows that emission components are decreasing and absorptions are increasing. An expanded view of the P Cygni absorption evolution is shown in the inset. The same variation pattern is seen Si ii λ6347 (left panel, Fig. 6). The right panel of Fig. 6 shows that far from periastron, velocities in the absorption profiles are always smaller than the terminal velocity of the primary's wind, which is compatible with line formation at smaller radii. From 10 days before to a month after periastron the profile shows very large speeds, as expected for the passage of the trailing arm of the WWC wall through our LOS. This set of two line-profiles evolution shows that the amplitude of both emission and absorption variations seem to have decreased in the last three cycles, in line with the slowing down of the secular brightness increase observed in the V -band (to be modelled in a future work); the almost complete disappearance of many "circumstellar" absorption lines in the UV (Gull et al. 2021), and in the blue-shifted absorption of NaD (Pickett et al. 2022). Time series of components in P Cygni absorption Comparing spectra at the same phase in successive cycles is very useful because, in principle, they would be identical if no long-term evolution occurred. However, as we are measuring low amplitude effects, even small irregularities in the orbital variations can impact the comparisons. The alternative is to assemble time series in which the irregularities can be isolated. The number of spectral lines in which this is possible, over 5-6 cycles, is small, especially as there are only a few lines that show an absorption component during the middle of a cycle. The P Cygni absorption component that shows the simplest variation is that of He i λ10830, as shown in the top panel of Fig. 7. The absorption EW is measured by direct integration below the projected stellar continuum. As readily apparent, the absorption EW at mid-cycle, which was very weak in cycle #9, has increased relatively smoothly with 8 Damineli et al. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Pa 14 Hd Hg + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + He I + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Normalized Intensity (I/Ic) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + time. An anomalous peak occurred in 1994.5 (phase ∼ 9.38, which was recorded in many other spectral lines, will not be discussed further here. While it shows a behaviour similar to that of He i λ10830, it exhibits irregularities even at mid-cycle. This is due to the contribution of the WWC in this specific spectral line as shown by Richardson et al. (2016). This is even more pronounced at periastron passages, and such variability is not strictly periodic. Even so, it is possible to identify a long-term strengthening trend in the mid-cycle from cycle #10 onward. The absolute values of the centroid radial velocity of the P Cygni absorption component are ∼ 500 km s −1 at mid-cycle and up to 900 km s −1 at periastron. Absorption wings in this line are much larger, reaching 1000 km s −1 at periastron, indicating that they are mostly produced in the WWC walls and subject to shock instabilities. This is in line with the findings of Richardson et al. (2016). The evolution of the absorption component of the H i Pa14 line (λ8750) is shown in the bottom panel of Fig. 7. Although measurements of this weak line are subject to large relative errors, they also show a secular increase in the EW of the absorption component. Since it is formed in the innermost parts of the primary's wind, the emission contribution from radii outside the occulter is smaller than in another H i lines that involve lower energy levels, and hence its EW progression is also smaller. The importance of studying this spectral line is that it has a permanent absorption component and is little affected by blends with other spectral lines. Figure 8 (green line) shows a clear correlation between the V-band light curve and the Si ii λ6347 EW-phase curve. These two quantities share long-term brightness increases due to occulter dissipation, as well as periodic variations due to orbital modulation arising from the distortion caused by the WWC cavity, and periastron passage. The last of these three are caused by the WWC crossing our LOS in the case of the Si ii λ6347 absorption and by the borehole effect in the V-band (Madura & Owocki 2010). The intriguing fact is that the P Cygni absorption is formed only in our LOS to the central star, while the V -band flux is emitted by the whole (distorted) pseudo-photosphere of the primary star. Some spectral lines, although following this general behaviour, are affected by important anomalies probably arising from the wind-wind collision shock, which is subject to instabilities. The lines of He i λ5876 and He i λ4712, for example, show uncorrelated variations as a function of the orbital phase of sufficiently high amplitude that they can dominate variations arising from long-term evolution, and only extensive monitoring can separate the two effects. A SEMI-QUANTITATIVE CMFGEN MODELLING OF THE OCCULTER'S IMPACT ON THE OBSERVED SPECTRUM One of the main goals of this paper is to show that the decrease in extinction of the coronagraph is correlated with the decrease of the emission line intensity and with the increase in the P Cygni absorption. The CMFGEN code has been very successful to model the wind spectrum of many kinds of massive stars, but we do not have access to the intrinsic spectrum of η Car because it has been always covered by the occulter. The main limitations to calculate the full spectrum (unobstructed from the occulter's extinction) are presented in Appendix A: Model limitations. Regarding the occulter's model the main limitations are the lack of constraints: the shape, size, radial opacity profile and physical constitution of the occulter (dust, gas) are not constrained by observations. However, we still can derive some constraints. We developed a procedure we call semiquantitative because it does not aim to fit the fine details of the spectrum and its time variations, but the relative temporal changes in the intensity of a representative set of lines formed at different stellar radii. To calculate the model, we used our experience on earlier works (Hillier et al. 2001b(Hillier et al. , 2006 and set the following parameters: R * = 240 R ; M = 6.2 × 10 −4 M yr −1 ; L = 4 × 10 6 L ; and N(H)/N(He) = 10. Figure 9 shows the observed spectrum (as a black line) taken at the FOS 4 position using UVES/ESO in 2015. The model spectrum is represented by a green line in the same figure, and we call it the "full" spectrum (in the sense that it is unobstructed). The model is very good for the Hydrogen emission lines. Regarding the P Cygni absorptions, they are a little too large for Hα and Hβ. See zoomed views in Fig. 10 0.0 0.5 1.0 1.5 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + worse agreement. The largest discrepancy occurs for [Fe ii] lines which are formed at the largest radii. The N ii lines are formed in the very inner regions, close to the secondary hot star. Figure 11 shows STIS observations taken in 2001 (magenta line) and 2018 (blue line) on top of the full spectrum (green line). To study the impact of an occulter, we adopted a very simple geometry of a uniform "disc" occulter centered on the primary star with a radius r = 22.7 Rast. It is semi-transparent, reducing the flux of the occulter area by 1.6 mag, which is the extinction measured by the brightness increase in the V-band from 1993.5 to 2021.2 in the extended region around the central star. We admit that the extinction is gray, as reported by Hillier et al. (2001b). The model spectrum after the correction by this semi-transparent occulter is represented by a light blue line in Figs. 9 and 10. This occulter's model matches very well the observed broad Hydrogen lines' behaviour from 1993.5 (magenta) to 2021.2 (blue), and it agrees reasonably well with the Fe ii broad line components. However, there is a large disagreement with [Fe ii] broad lines. Regarding the lines formed in the inner regions of the primary's wind, they are perturbed by the ionising radiation and the CMFGEN models are not designed to deal with them. We tried to understand if a different occulter's model could improve the situation. We used an occulter with radius pcygabs-midcycle 11 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + r = 10.2 R * and increased the opacity (10 times reduction in flux, instead of 4.6). The result can be seen as an orange line in Fig. 11. In fact, for N ii and probably other lines formed in the inner region, a smaller occulter has a larger impact, reducing the excess absorption. Although wind and occulter models do not match the observations at the level we would like, they still do a good job for the sake of the differential behaviour of the line variations. For this, we measure the relative change of the broad line intensities for lines of different excitation energies in the 1993.5 and the 2021.2 spectra. The observations and model are reported in Table 6. Emission lines are identified by "e" and absorption line-depth by "a" before the name of the ion (upper row in the table). We report in the last column of that Table the intensity of the N iiλ5665 line P Cygni depth taken by STIS in 2001 and 2018. Figure 12 presents these values normalised to those in the year-2021.2 spectrum. Observations are represented by solid lines (and filled polygons), and models by dashed lines (and void polygons). The important result is that almost all the observed and modelled emission lines have decreased in this time interval, as seen in Fig. 12. Regarding the emission lines (upper plot) there are additional details about the agreement between the model and observation. The Weigelt clump narrow line Fe ii λ8610 (black), formed outside the occulter's radius, is the one that has mostly become fainter. The Fe ii λ4585 (dark green) formed in a range of external radii in the wind decreased by a smaller amount and presents a good correlation with the model. The correlation between the gradients is almost perfect for Hα (brown). The lines formed close to the centre, like Hδ, Pa14 and N ii He i λ5876 and Si ii, presented a small rate of decrease, as expected, because their formation regions are largely covered by the occulter. Regarding the P Cygni components (bottom panel), both the model and the observations show increasing depth as a function of the extinction decrease. The observed rate of decrease, however, is larger for the observations as compared to the model. When we decrease the radius of the occulter's model, the P Cygni features show better agreement with the observations, but then the emission components lose their good agreement. A realistic model should explore the asymmetry in the stellar wind (highly deformed by the wind-wind collision), and a much larger number of spectral lines, which are not so many with P Cygni absorption in the past observations. This requires paramount work to be faced in future studies. DISCUSSION AND CONCLUSIONS We have used spectral data taken over 30 years to monitor the long-term evolution of Eta Carina. In particular, we have studied the variations in EWs of P Cygni absorption components of selected lines. Such measurements have their difficulties. In modern astronomy, it is virtually impossible to have the same observational setup (detector, telescope, spectrograph) for three decades. Consequently, a scatter in EW measurements from instrumental origin is unavoidable even though the measurement procedure is stable. For isolated lines, the EW is roughly independent of the spectral resolution. Further, seeing will not cause large changes as long as the main emitting region of the object is as concentrated as in η Car, which has a scattering region of ∼1 around the unresolved central core (Hillier et al. 2006). Of course, such data cannot be mixed with data taken using a very narrow slit width under good seeing, or data taken using adaptive optics. Each spectral line in η Car displays a different behaviour + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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As the occulter dissipates, the EW of lines formed far out in the wind (such as this Hα and Fe ii) decrease while the EW of P Cygni absorption components increases, again in good agreement with observations. Better agreement would be achieved by allowing for more complex occulter shapes, and perhaps by allowing for a spatial variation in the dissipation of the occulter. An occulter with a shape of a semiinfinite uniform slab crossing over the central star did not produce results in agreement with the observations. The reason for this is that the lines formed close and far from the star are reduced in the same proportion maintaining their relative intensity. An offset slab can produce similar results to a disc, but with some differences. In the FUV range, absorption lines are seen to the SE of the star, suggesting a cometary shape of the occulter (Gull et al. 2016). The extinction must be larger closer to the central star than in the outside region, as observations clearly show that the continuum emitting region suffered a much larger extinction than did the outer wind (and the Weigelt blobs). Our main conclusions, including results from previous works, are: (i) The brightness of the central star has been increasing secularly at a fast rate, while the light from η Car reflected off and emitted by dust in the circumstellar nebula has stayed reasonably stable (Mehner et al. 2019 and paper I). + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 5660 5680 5700 5720 (ii) Stellar wind emission lines observed by direct pointing to the star have larger absolute EWs than those reflected from the Homunculus in coeval observations (Hillier & Allen 1992;Mehner et al. 2011, 2015. (iii) The absolute EWs of the emission lines formed in the wind along our LOS have decreased systematically from cycle to cycle. While emission lines from the Weigelt blobs (located at a projected distance of a fraction of an arcsec from the central star) show a similar decrease in EW, they have maintained a constant flux and ionization level (Gull et al. 2009;Damineli et al. 2021). (iv) After correction for the secular brightness increase in the stellar continuum, the broad emission lines formed in the stellar wind exhibit a constant line profile -which translates into a constant line flux (paper II -figure 2.c). (v) Emission lines that are formed at larger radii from the star (e.g. Hα, Fe ii) suffer larger EW secular variation than those formed at smaller radii (e.g. upper members of the Balmer and Paschen series). (vi) P Cygni absorption components are deeper in reflected than in direct light, further indicating that the stellar component suffers additional extinction along our LOS along with veiling effects. (vii) As the long-term evolution progresses, emission and absorption lines in direct view become more similar to those reflected in the Homunculus (FOS 4 position). (viii) The observed constancy of the central star during the 1940s brightness jump (Thackeray 1953b;O'Connell & S.J. 1956) indicates that the coronagraphic occulter was already in place before that time. (ix) The long-term light-curve variability after the 1900s looks contrary to the prediction of a stellar merger by Schneider et al. (2019). However, if these variations are driven by the circumstellar medium, as suggested by the dissipating coro-nagraph, the star is much more constant, and the binary merger in a triple system is not ruled out; neither is an 1847 supernova explosion. Intrinsic evolution of the primary star in η Car A is not required to explain the observations reported here, in papers I and II, and in Mehner et al. (2011Mehner et al. ( , 2015. A decrease in the mass-loss rate, as has been claimed elsewhere, is usually accompanied by changes in V∞, as seen for S Doradus stars (Leitherer et al. 1985;Groh et al. 2009;Hillier et al. 2001b); it would produce a decrease in the emission line intensity and also in the P Cygni absorption strength; it would change the X-ray light curve contrary to what has been observed (Espinoza et al. 2022 -submitted). None of these changes has been observed in the long term. STIS would be very valuable to take the pulse of the coronagraph dissipation, by comparing spectra taken in 2023-24 with the same gratings/setups as in 2001.3 as a critical test. Some observations indicate that the occulter may have already ceased to dissipate during the last cycle (2014.6-2020.2). pcygabs-midcycle DATA AVAILABILITY The data underlying this article are available in the article and in its online supplementary material. Figure 1 . 1Scenario of the coronagraphic occulter in front of η Car and how it modulates the total contribution from three main regions: a) the central region around η Car A, where most of the continuum and line flux and the broad P Cygni profiles of H i and He i are formed; b) the outer wind which produces the relatively broad permitted lines (with faint P Cygni absorption) and forbidden lines; c) the Weigelt clumps which form narrow nebular emission lines (permitted and forbidden). Without the coronagraph in our line-of-sight, the spectrum is of a typical LBV, as seen in the reflected spectrum in the Homunculus (FOS 4) -see the black line in the upper plot of Fig. 2. The occulter depresses the lines from the central region, enhancing the outer wind and the nebular emission lines (red line in Fig. 2). As the occulter dissipates, the contribution of the external regions is diminished -bottom plot of Fig. 2. 6 Damineli et al. Figure 3 . 3Spectral evolution: a) upper plot: ground-based spectra separated by 5 orbital cycles at the same orbital phase b) bottom plot: STIS spectra separated by 3 orbital cycles. A decrease in emission line intensity (narrow and broad) of [Fe ii], [N ii] and NaD. An increase in the absorption P Cygni component of He i is clearly seen in ground-and space-based observations. Fig. 4 c (top right), the Pa 14 line (H i λ8750) shows the permanent absorption component in both cycles, to be compared with Hγ (H i λ4340) in the upper left, which shows little variation over five cycles. The Fe ii λ6455 line in Fig. 4 (bottom right) shows a large decrease in the emission component because that feature is formed at larger radii, potentially outside the region covered by the occulter. The He i λ5876 line (Fig. 4 b, top middle) and He i λ4712 (bottom middle) present a mild increase in the strength of the absorption component while the emission component shows a small decrease. These two features are mostly formed in the inner regions of the WWC walls and so are not as variable as the narrow emission component, which is formed in the Weigelt blobs. Figure 4 .Figure 5 . 45Representative line profiles at phase 9.18 -June 1993 (red lines) and phase 14.18 -March 2021 (blue lines), showing the increase in P Cygni absorption over the 28-year time interval and decrease of (almost all) emission lines at mid-cycle. Also evident is the weakening of the narrow nebular lines, and the broad base associated with forbidden and permitted Fe ii lines. For flux-calibrated spectra, seeFig. 2of Paper II. ++++++++++++++ ++++++++++++++++ +++++ ++ +++++++++++++++++++++++ +++++ ++ ++ +++++ +++ +++++ + + + + + + + + ++ ++ + Strengthening of the P Cygni absorption component in the line of He i λ10830 at mid-cycle. Line profiles show the progress of P Cygni absorption in spectra taken at the same phase, over 30 years. The inset shows an expanded view of the P Cygni absorption component. The range of variations seems to have decreased in the last three cycles. Figure 7 ( 7middle panel) shows the time series for the absorption component of He i λ5876. Figure 6 . 6Strengthening of P Cygni absorption components in the line Si iiλ6347 at mid-cycle (left panel). Line profiles show the progress of P Cygni absorption in spectra taken at the same phase, over 33 years. The right panel shows representative line profiles along the orbit. Close to periastron the absolute absorption velocity is larger than the terminal speed of the primary's wind when the WWC trailing arm crosses our LOS. Figure 7 . 7Time series of EWs of the P Cygni component in absorption. a) Top: He i λ10830. The EW of this absorption line has been increasing smoothly at mid-cycle since periastron #10. Short-lived peaks at periastron are displayed for cycles #9 through #14 and can easily be separated from the mid-cycle component. The mid-cycle peak around phase ∼ 9.38 was a transient event that occurred in 1994.5 and affected in a remarkable way many other spectral lines but has not been reported previously. b) Middle: He i λ5876. Note the complex variations in the vicinity of periastron and even at mid-cycle. This line is strongly influenced by the wind-wind collision. c) Bottom: Paschen 14 showing its secular increase at mid-cycle phases. Transient variations occur in the vicinity of periastron. The dashed green lines are general trends, made to guide the eye through the time interval of more remarkable evolution: phase 10.0 (1998.0) to phase 14.0 (2021.1). Figure 8 . 8Time series of: a) the EW of the P Cygni absorption associated with Si ii λ6347 line (red). In the phases preceding periastron, absorption disappears when the wind collision apex crosses our LOS because the WWC cone is opened towards us and is basically void, with no or little material to absorb light in our direction. Mid-cycle maximum events are similar to the 1994.5 (phase 9.38) transient event, but they do not always occur at phase = 0.5. These peaks are rising with time. b) V-band magnitudes of the whole object (green), which are also secularly increasing in brightness. The orbital modulation in the optical flux(Damineli et al. 2019) seems to be well correlated with the Si ii 6347 absorption-component EW intensity-curve. Figure 11 . 11N ii lines at 5600Å. STIS observations: 2001-magenta line and 2018 -blue line. Models: full spectrum -green line and disc occulter with r = 10 R * and 10 mag extinction -orange line Figure 12 . 12Line intensity variation between 1993.5 and 2021.2. The peaks of the emission lines in 1993.5 are divided by those in 2021.2. The same for the depth of the P Cygni absorption. Observations and models are displayed as filled polygons (and solid lines) and empty polygons (and dashed lines), respectively. The model for 2021.2 is the full spectrum for 1993.5; a disc occulter with r = 22.7 R * and opacity 1.6 magnitudes covers the central region. All are broad line profiles (from the stellar wind), except for Fe iiλ 8610 (which is narrow and formed in the Weigelt clumps). Figure 2. Upper panel: The black line shows the spectrum reflected at the FOS 4 position on the Homunculus, representing the unobstructed view. The red line shows the direct-view spectrum of our LOS impacted by the occulter (spectra taken in 2006.5). Bottom panel: this pair of spectra shows the evolution from the year 1993.5 (magenta) to 2021.2 (blue). The direct spectrum has evolved towards the spectrum in reflected light (typical LBV) as the occulter's extinction decreases.+ + + + + + + + + + + 4200 4300 4400 4500 4600 4700 0 5 10 15 20 Wavelenght (A) Normalized Intensity (I/Ic) Hg HeI [FeII]+[FeII] FeII HeI [FeIII] [FeII] [FeII] FeII + FeII [FeII] FeII [FeII]+[FeII] direct ---1993.5 ---2021.2 2006.5 ---direct ---reflected Table 1 . 1ObservatoriesObservatory telescope Resolving diameter power ESO/FEROS 2.2 m 48,000 ESO/hexapod 1.5 m 48,000 ESO/UVES 8 m 90,000 CTIO/CHIRON 1.5 m 90,000 CTIO/SMARTS 1.5 m 40,000 LCOGT/NRES 1 m 48,000 MJUO/Hercules 1 m 48,000 OPD/Coudé 1.6 m 6,000-22,000 HST/STIS 2.5 m 10,000 Gemini S/GMOS 8 m 4,400 SASER/D. B. Heatcote 0.28 m 16,000 SASER/P. Cacella 0.30 m 5,500 SASER/P. McGee 0.35 m 11,000 SASER/T. Bohlsen 0.27 m 15,000 SASER/K. Harrison 0.28 m 10,000 SASER/M. Johnston 0.60 m 17,000 SASER/G. Di Scala 0.20 17,000 Table 2 . 2Equivalent widths of the 5875Å absorption feature.HJD EW 5875 σ EW Facility (Å) (Å) 48793.994 0.18 0.09 OPD/Coudé 48830.020 0.07 0.06 OPD/Coudé 48838.996 0.05 0.04 OPD/Coudé 48843.988 0.32 0.06 OPD/Coudé 49063.609 0.05 0.04 ESO/FEROS 49063.609 0.10 0.08 ESO/FEROS 49065.609 0.10 0.08 ESO/FEROS 49066.621 0.10 0.08 ESO/FEROS 49067.605 0.10 0.08 ESO/FEROS 49068.609 0.09 0.07 ESO/FEROS Note. The first 10 rows of the table are presented. The full version is available online at the CDS. Table 3 . 3Equivalent widths of the 6347Å absorption feature.HJD EW 6347 σ EW Facility (Å) (Å) 47918.500 0.16 0.03 OPD/Coudé 48780.500 0.01 0.00 OPD/Coudé 48793.990 0.94 0.19 OPD/Coudé 48798.500 0.77 0.15 OPD/Coudé 48825.010 0.04 0.01 OPD/Coudé 48830.010 0.03 0.01 OPD/Coudé 48838.300 0.01 0.00 OPD/Coudé 48843.990 0.02 0.00 OPD/Coudé 49063.100 0.12 0.02 OPD/Coudé 49063.109 0.11 0.02 ESO/FEROS Note. The first 10 rows of the table are presented. The full version is available online at the CDS. Table 4 . 4Equivalent widths of the 10830Å absorption feature.HJD EW 10830 σ EW Facility (Å) (Å) 47613.590 0.72 0.12 OPD/Coudé 47916.730 0.54 0.09 OPD/Coudé 48020.800 0.32 0.05 OPD/Coudé 48059.390 0.79 0.13 OPD/Coudé 48255.760 1.00 0.17 OPD/Coudé 48285.670 2.15 0.24 OPD/Coudé 48402.410 1.46 0.25 OPD/Coudé 48701.760 3.00 0.33 OPD/Coudé 48776.420 14.00 0.70 OPD/Coudé 48820.400 9.60 0.77 OPD/Coudé Note. The first 10 rows of the table are presented. The full version is available online at the CDS. Table 5 . 5Equivalent widths of the Pa-14 8750Å absorption feature.HJD EW Pa-14 σ EW Facility (Å) (Å) 50833.750 0.74 0.15 LCOGT/NRES 50854.583 1.33 0.20 OPD/Coudé 50946.375 0.74 0.15 OPD/Coudé 51131.815 0.33 0.16 ESO/FEROS 51135.819 0.45 0.23 ESO/FEROS 51135.819 0.54 0.27 ESO/FEROS 51135.822 0.41 0.21 ESO/FEROS 51135.825 0.50 0.25 ESO/FEROS 51142.820 0.46 0.23 ESO/FEROS 51142.820 0.65 0.32 ESO/FEROS Note. The first 10 rows of the table are presented. The full version is available online at the CDS. . The Fe ii wind lines also are in reasonable agreement with observations and He i emission and absorption lines are in + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +9 10 11 12 13 14 0 5 10 20 ++ + ++ + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + EW (A) 15 HeI 10830 A Pa14 8750 A Phase (14.0 = 2020.2) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 9 10 11 12 13 14 Table 6 . 6Broad line peak intensities in observations and modelsSpec eHα eFeII eHδ aHδ eHeI aHeI eFeII aPa14 eSiII aSiII aNII λ (Å) 6563 4585 4101 4101 5876 5876 8610 8750 6347 6347 5668 1993.5 62 1.9 5.0 1.00 2.8 1.00 5.6 0.93 1.30 0.95 0.99* 2021.2 32 1.3 4.5 0.86 2.5 0.63 2.0 0.76 1.20 0.79 0.95** occulter 67 3.1 4.2 0.2 1.9 0.65 - 0.61 1.20 0.85 0.68 full 39 1.7 3.7 0.05 1.9 0.65 - 0.61 1.20 0.66 0.68 © 2022 RAS, MNRAS 000, 1-17 Some intrinsic, low excitation, line emission (such as [Ni ii] λ7378) arises in the Homunculus but the exhibit distinct velocity shift from the scattered emission; Hillier & Allen 1992. © 2022 RAS, MNRAS 000, 1-17 10 Damineli et al. ACKNOWLEDGEMENTS AD thanks to CNPq (301490/2019-8) and to FAPESP (2011/51680-6) for support. The work of FN is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. AFJM is grateful for financial aid from NSERC (Canada). This work is partially based on observations collected with the facilities listed below. DJH gratefully acknowledges support from STScI grants HST-AR-14568.001-A and HST-AR-16131.001-A. Based on observations collected at the European Southern Observatory (Chile) Based on observations made at the Observatório do Pico dos Dias/LNA (Brazil). Based in part on data from Mt. John University Observatory: MJUO -University of Canterbury -New Zealand). Based in part on observations obtained at the Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation and the SMARTS Consortium Based on observations obtained at the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). ESO/UVES and Gemini S/GMOS spectra used in this paper were downloaded from the HST Treasury Project archive.APPENDIX B: MODEL DISCUSSIONThe modelling of Eta Carinae is complicated by several factors. These include the following:(i) While we have an estimate of the primary's luminosity (albeit somewhat uncertain due to the unknown efficiency of the conversion of optical/UV light to IR wavelengths and the luminosity of the companion) we do not have a measurement of the intrinsic optical flux emitted by the system. This occurs because of the influence of the coronagraph, and the unusual reddening law.(ii) There is a degeneracy between the mass-loss rate and the H/He ratio, such that similar H and He i spectra can be produced for different mass-loss rates and the H/He ratio(Hillier et al. 2001b). Metal lines, such as He ii can break the degeneracy, but there are difficulties modelling these lines. In models with a higher H/He ratio, the metal lines are weaker since N(H)/N(Z) is higher (the metals are assumed to have a solar mass fraction).(iii) The He I line strengths are most likely affected by the ionising field of the companion(Nielsen et al. 2007). This limits our ability to use the strength of these lines as constraints. Therefore, in the present modelling, we demanded that the He I lines should be weaker than in the observations.(iv) The companion star ionises the outer wind, affecting the strength of the P Cygni profiles on Balmer lines, with low series members (such as Hα) being most affected. The emission, and especially the P Cygni absorption associated with Fe ii lines are also affected.(v) The wind is optically thick, and hence the core radius is difficult to constrain. Core radii between 60 and 240 R are roughly compatible with the observations. In the present modelling, we adopted 240 R since this allowed us to get a reasonable match to the higher Balmer series members without having too much He i emission. Note: At small core radii, the wind dominates the formation of the optical spectrum, but as we increase the radius, the influence of the winds declines, and the shape of the optical continuum changes.(vi) If the primary associated with Eta Car is a fast rotator, there could be significant asymmetries as suggested by(Smith et al. 2003). However, direct spectra taken of the central star, and those using the reflected light seen at FOS 4 are now much more similar than in the past. Our interpretation is that this is caused by the weakening of the coronagraph, rather than an intrinsic change in the primary star.(vii) The coronagraph may still be influencing (albeit more weakly than in the past) the ground-based spectra. 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[]	[ "\nSATORU FUKASAWA\n\n" ]	[ "SATORU FUKASAWA\n" ]	[ "Mathematics Subject Classification. 14H37" ]	This paper presents a new characterisation of the Fermat curve, according to the arrangement of Galois points.function field extension k(C)/π * P k(P 1 ) induced by π P is Galois. Obviously, noncollinear points (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1) are outer Galois points for the Fermat curve, namely, the "if part" of Theorem is confirmed.	null	[ "https://export.arxiv.org/pdf/2210.02076v3.pdf" ]	252,716,047	2210.02076	63f68d2aacfd42ee6c9a7d54fc805be08946fdba	2020 SATORU FUKASAWA Mathematics Subject Classification. 14H37 2020A NEW CHARACTERISATION OF THE FERMAT CURVE The author was partially supported by JSPS KAKENHI Grant Numbers JP19K03438 and JP22K03223.and phrases Fermat curveGalois pointautomorphism groupHurwitz's theorem This paper presents a new characterisation of the Fermat curve, according to the arrangement of Galois points.function field extension k(C)/π * P k(P 1 ) induced by π P is Galois. Obviously, noncollinear points (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1) are outer Galois points for the Fermat curve, namely, the "if part" of Theorem is confirmed. Introduction Let k be an algebraically closed field of characteristic zero and let C ⊂ P 2 be an irreducible plane curve of degree d ≥ 3 over k. The genus of the smooth model of C is denoted by g. This paper proves the following. Theorem. Assume that g ≥ 1. Then there exist non-collinear outer Galois points P 1 , P 2 , P 3 ∈ P 2 \ C for C if and only if C is projectively equivalent to the Fermat curve X d + Y d + Z d = 0, where (X : Y : Z) is a system of homogeneous coordinates of P 2 . We recall the definition of a Galois point, which was introduced by Hisao Yoshihara in 1996 ( [2,8,11]). We consider the projection π P : C → P 1 ; Q → (F 0 (Q) : F 1 (Q)) from a point P ∈ P 2 \ C, where F 0 , F 1 are homogeneous polynomials of degree one defining P . The point P ∈ P 2 \ C is called an outer Galois point for C if the Preliminaries For different points P, Q ∈ P 2 , the line passing through P and Q is denoted by P Q. Let X be the smooth model of C and let g be the genus of X. In this paper, we assume that g ≥ 1. The following well known fact on Galois extensions is used frequently (see, for example, [10, III.7.1, III.8.2]). Fact 1. Let π : X → X ′ be a surjective morphism between smooth projective curves such that k(X)/π * k(X ′ ) is a Galois extension, and let G ⊂ Aut(X) be the induced Galois group. Then the following hold. (a) The Galois group G acts on any fibre transitively. (b) If Q ∈ X, then e Q = \|G(Q)\|, where e Q is a ramification index at Q, and G(Q) is the stabiliser subgroup of Q under G. A well known theorem of Hurwitz (and its proof) on the order of the automorphism group plays an important role in this paper (see, for example, [7,Theorem 11.56]). The following version is needed later. Fact 2 (Hurwitz's theorem). Let G ⊂ Aut(X). Assume that g ≥ 2 and X/G ∼ = P 1 . The number of short orbits under G is denoted by n, and the orders of their stabiliser subgroups are denoted by r 1 , . . . , r n . Let T := −2 + n i=1 1 − 1 r i . Then the following hold. (a) T > 0 and \|G\| = (2g − 2)/T . (b) n ≥ 3. (c) If n ≥ 5, then \|G\| ≤ 4(g − 1). For an outer Galois point P ∈ P 2 \ C, the induced Galois group is denoted by G P , which admits an injective homomorphism G P ֒→ Aut(X). For the case where two Galois points exist, the following holds. Fact 3 (Lemma 7 in [3]). Let P 1 , P 2 be different outer Galois points. Then G P 1 ∩ G P 2 = {1}. Several important results in the theory of Galois points are needed for the proof. Fact 4 (Yoshihara [11]). If C is smooth and there exist three outer Galois points, then C is projectively equivalent to the Fermat curve. Fact 5 (Duyaguit-Miura [1]). If g ≥ 1, d is a prime and there exist three outer Galois points, then C is projectively equivalent to the Fermat curve. Fact 6 (Fukasawa [4]). If g ≥ 1 and there exist three outer Galois points for C, then the morphism X → P 2 induced by the normalisation X → C is unramified. Fact 7 (Fukasawa [6]). Assume that points P 1 = (1 : 0 : 0), P 2 = (0 : 1 : 0) are outer Galois points, and that \| G P 1 , G P 2 \| < ∞. Then there exist two polynomials f 1 (x), f 2 (x) ∈ k[x] of degree \| G P 1 , G P 2 \|/d such that C is an irreducible component of the curveC defined by f 1 (x) − f 2 (y) = 0. Since points ofC ∩ {Z = 0} are given by X degC + cY degC = 0 for some c ∈ k \ {0}, it follows thatC ∩ {Z = 0} consists of degC points. This implies the following. Corollary 1. Let P 1 , P 2 be different outer Galois points. Assume that \| G P 1 , G P 2 \| < ∞. Then the set C ∩ P 1 P 2 consists of d points, namely, all points of C ∩ P 1 P 2 are smooth points of C. are outer Galois points. If \| G P 1 , G P 2 \| = d 2 , \| G P 1 , G P 3 \| = d 2 , and \| G P 2 , G P 3 \| < ∞, then C is projectively equivalent to the Fermat curve. Proof. Since \| G P 1 , G P 2 \| = d 2 , it follows from Fact 7 that there exist polynomials f 0,1 (x), f 0,2 (x) ∈ k[x] of degree d such that C is defined by f 0,1 (x) − f 0,2 (y) = 0. Let c 1 = f 0,1 (0) and let c 2 = f 0,2 (0). Since P 3 ∈ P 2 \ C, it follows that c 1 − c 2 = 0. We take f 1 (x) := (f 0,1 (x) − c 1 )/(c 1 − c 2 ) and f 2 (x) := (−f 0,2 (x) + c 2 )/(c 1 − c 2 ). Then C is defined by f (x, y) := f 1 (x) + f 2 (y) + 1 = 0. Since \| G P 1 , G P 3 \| = d 2 , there exist polynomials g 1 (u), g 2 (u) ∈ k[u] of degree d with g 1 (0) = g 2 (0) = 0 such that C is defined by g(u, v) := g 1 (u) + g 2 (v) + 1 = 0, where u = X/Y = x/y and v = Z/Y = 1/y. Then y = 1 v , x = u v and define h(u, v) := v d f (u/v, 1/v) = v d f 1 (u/v) + v d f 2 (1/v) + v d . The degree d homogeneous parts of g(u, v) and of h(u, v) are c 1 u d + c 2 v d for some c 1 , c 2 ∈ k \ {0} and v d f 1 (u/v) + v d respectively. Since g(u, v) and h(u, v) are the same up to a constant, it follows that polynomials Assume that points P 1 , P 2 , P 3 ∈ P 2 \ C are non-collinear outer Galois points. Fix a triple i, j, k such that {i, j, k} = {1, 2, 3}. We consider the group c 1 u d + c 2 v d and v d f 1 (u/v) + v dG P i , G P j ∩ G P k . The order is denoted by m k . The following proposition is proved by the same method as in [5, Proof of Theorem 1.5]; however, the proof is given for the convenience of the readers. Proposition 2. Let P 1 = (1 : 0 : 0), P 2 = (0 : 1 : 0), P 3 = (0 : 0 : 1). Assume that P 1 , P 2 , P 3 are Galois points, and that \| G P 1 , G P 2 \| < ∞. Then the group G P 1 , G P 2 ∩ G P 3 is a cyclic group, there exists an injective homomorphism G P 1 , G P 2 ∩ G P 3 ֒→ P GL(3, k), and a generator σ ∈ G P 1 , G P 2 ∩ G P 3 is represented by a matrix A σ =    1 0 0 0 1 0 0 0 ζ    , where ζ ∈ k is a primitive m 3 -th root of unity. In particular, for a point Q ∈ P 2 and an integer i with 1 ≤ i ≤ m 3 − 1, σ i (Q) = Q if and only if Q ∈ P 1 P 2 ∪ {P 3 }. Proof. By Corollary 1, the set C ∩ P 1 P 2 consists of different d points Q 1 , . . . , Q d . Let D 3 := Q 1 + · · · + Q d . Let σ ∈ G P 1 , G P 2 ∩ G P 3 . Since σ ∈ G P 1 , G P 2 , it follows that σ(Q i ) ∈ P 1 P 2 for i = 1, . . . , d, namely, σ * D 3 = D 3 . Let D 1 and D 2 be divisors coming from C ∩ P 3 P 2 and C ∩ P 3 P 1 respectively. We take a function f ∈ k(X) with k(f ) = k(X) G P 3 such that (f ) = D 1 − D 2 . Similarly, we can take a function g ∈ k(X) G P 1 such that (g) = D 3 − D 2 . Since P 1 P 2 does not pass through P 3 , it follows that g ∈ 1, f ⊂ L(D 2 ). It follows from the condition σ * D 3 = D 3 that σ * g = a(σ)g for some a(σ) ∈ k. Therefore, a linear space 1, f, g ⊂ L(D 2 ) is invariant under the action of σ. Since the embedding X → C ⊂ P 2 is represented by (1 : f : g), there exists an injective homomorphism G P 1 , G P 2 ∩ G P 3 ֒→ P GL(3, k); σ →    1 0 0 0 1 0 0 0 a(σ)    . The map G P 1 , G P 2 ∩ G P 3 → k \ {0}; σ → a(σ) is an injective homomorphism. This implies that G P 1 , G P 2 ∩ G P 3 is a cyclic group, and that C is invariant under the linear transformation (X : Y : Z) → (X : Y : ζZ). 3. Proof of Theorem: The case d = 4 Hereafter, we assume that points P 1 , P 2 , P 3 ∈ P 2 \C are non-collinear outer Galois points. According to Fact 4, we can assume that C is a singular curve of degree d ≥ 3, namely, g < (d − 1)(d − 2)/2. By Fact 5, we can assume that d is not a prime. In the proof below, we would like to prove that d < 6. The proof for the case where d = 4 is carried out in the next section. For any i, j with i = j, the group G P i , G P j acts on C ∩P i P j . It follows form Fact 3 that \| G P i , G P j \| ≥ d 2 . The orbit-stabiliser theorem implies that \| G P i , G P j \| < ∞ (even if g = 1), and that the order of the stabiliser subgroup G P i , G P j (Q) of a point Q ∈ C ∩ P 1 P 2 is at least d. By Proposition 1, if \| G P i , G P j (Q)\| = d for any pair i, j ∈ {1, 2, 3} with i = j, then C is projectively equivalent to the Fermat curve. Therefore, there exist i, j such that \| G P i , G P j (Q)\| ≥ d + 1. Assume that g = 1. A well known theorem on the automorphism group of an elliptic curve implies that d + 1 ≤ 6 (see, for example, [9, III. Theorem 10.1]). Therefore, we can assume that g ≥ 2. Let G := G P 1 , G P 2 , G P 3 . There are two main steps in the proof; m k = 1 for some k, and m k ≥ 2 for any k. For each step, we take into consideration the various possibilities for the number of short orbits and the orders of their stabiliser subgroups. Case 1: m k = 1 for some k. In this case, \|G\| ≥ d 3 . We can assume that k = 3, namely, G P 1 , G P 2 ∩ G P 3 = {1}. As we saw above, for a point Q ∈ C ∩ P 1 P 2 , \|G(Q)\| ≥ d. Assume that the number of short orbits under G is at least five. It follows from Fact 2 (c) that d 3 ≤ \|G\| ≤ 4(g − 1) ≤ 2d(d − 3). Then d ≤ 2 × d − 3 d < 2 × 1 = 2. Assume that the number of short orbits is four. On the estimate of T in Fact 2, the assumption that three stabilisers have order 2 and the fourth has order d, from \|G(Q)\| ≥ d, gives T ≥ −2 + 1 − 1 2 × 3 + 1 − 1 d = 1 2 − 1 d . It follows from Fact 2 that d 3 ≤ \|G\| = 2g − 2 T ≤ 2d 2 (d − 3) d − 2 . Then d ≤ 2 × d − 3 d − 2 < 2. Assume that the number of short orbits is three. We take a smooth point Q k ∈ C ∩ P i P j for any {i, j, k} = {1, 2, 3}. We consider the case where three orbits G · Q 1 , G · Q 2 , G · Q 3 are different. By Proposition 1, two of \|G(Q 1 )\|, \|G(Q 2 )\|, \|G(Q 3 )\| are at least d + 1. It follows from Fact 2 that T ≥ −2 + 1 − 1 d + 1 − 1 d + 1 × 2 = d 2 − 2d − 1 d(d + 1) , and d 3 ≤ \|G\| = 2g − 2 T ≤ d 2 (d + 1)(d − 3) d 2 − 2d − 1 . Then d ≤ d 2 − 2d − 3 d 2 − 2d − 1 < 1. We consider the case where there exist i, j, k with {i, j, k} = {1, 2, 3} such that G · Q i = G · Q j and G · Q i = G · Q k . By Proposition 1, \|G(Q i )\| = \|G(Q j )\| ≥ d + 1 and \|G(Q k )\| ≥ d. Then T ≥ −2 + 1 − 1 2 + 1 − 1 d + 1 − 1 d + 1 = d 2 − 3d − 2 2d(d + 1) , and d 3 ≤ \|G\| = 2g − 2 T ≤ 2d 2 (d + 1)(d − 3) d 2 − 3d − 2 . It follows that if d ≥ 5, then d ≤ 2 × d 2 − 2d − 3 d 2 − 3d − 2 < 4. Finally, we consider the case where G · Q 1 = G · Q 2 = G · Q 3 . By Proposition 1, \|G(Q 3 )\| ≥ d + 1. Assume that the projection π P 3 from P 3 is ramified at some point Q ∈ C ∩ P 1 P 2 . Since G P 1 , G P 2 ∩ G P 3 = {1}, it follows that \|G(Q)\| ≥ 2d. It follows that T ≥ −2 + 1 − 1 2 + 1 − 1 3 + 1 − 1 2d = d − 3 6d , and d 3 ≤ \|G\| = 2g − 2 T ≤ d(d − 3) × 6d d − 3 = 6d 2 . This implies that d ≤ 6. If d = 6, then g = 10, namely, C is smooth. This is a contradiction. It follows that d < 6. Therefore, we can assume that π P 3 is not ramified at any point in C ∩ P 1 P 2 . Then G P 3 · Q contains d points for any point Q ∈ C ∩ P 1 P 2 . Note that C ∩ P 1 P 2 consists of d points, according to Corollary 1. Since the orbit G · Q contains the set C ∩ (P 2 P 3 ∪ P 3 P 1 ), it follows that the orbit G · Q contains at least d 2 + 2d points. The orbit-stabiliser theorem implies that \|G\| ≥ (d + 1)(d 2 + 2d). Then T ≥ −2 + 1 − 1 2 + 1 − 1 3 + 1 − 1 d + 1 = 1 6 − 1 d + 1 , and d(d + 1)(d + 2) ≤ \|G\| = 2g − 2 T ≤ 6d(d + 1)(d − 3) d − 5 . It follows that d + 2 ≤ 6 × d − 3 d − 5 , and that d ≤ 8. If d = 8, then g = 21, namely, C is smooth. This is a contradiction. We consider the case where d = 6. For a triple i, j, k such that \| G P i , G P j (Q k )\| ≥ d + 1 = 7, we assume that \| G P i , G P j (Q k )\| = 7. Then the cyclic group G P i , G P j (Q k ) of order seven acts on six points of C ∩ P i P j . This implies that G P i , G P j (Q k ) fixes all points of C ∩ P i P j . With Hurwitz's theorem applied to the covering X → X/( G P i , G P j (Q k )), this is a contradiction. Therefore, \| G P i , G P j (Q k )\| ≥ 8, namely, \|G(Q)\| ≥ 8. It follows that the orbit G · Q contains at least d 2 + 2d = 48 points. The orbit-stabiliser theorem implies that \|G\| ≥ 8 × 48. Then T ≥ −2 + 1 − 1 2 + 1 − 1 3 + 1 − 1 8 = 1 24 , and 8 × 48 ≤ \|G\| = 2g − 2 T ≤ (g − 1) × 48. It follows that g = 9 or g = 10. If g = 10, then C is smooth. Therefore, g = 9. Then \|G\| = 8 × 48, the length of the orbit G · Q is equal to d 2 + 2d = 48, and \| G P i , G P j (Q k )\| = 8. Note that the group G P i , G P j is of order 48, and acts on the set G·Q\(C ∩P i P j ), which consists of 48−6 = 42 points. Then the group G P i , G P j has three short orbits other than C ∩ P i P j . Note that for such short orbits, there exist at most one such that the order of stabiliser subgroup is two. It follows from Fact 2 that T ≥ −2 + 1 − 1 2 + 1 − 1 3 × 2 + 1 − 1 8 = 17 24 , and 48 = \| G P i , G P j \| = 2g − 2 T ≤ 16 × 24 17 . This is a contradiction. We have d < 6. Case 2: m k ≥ 2 for any k. According to Proposition 2, for k = 3, there exists a generator σ ∈ G P 1 , G P 2 ∩ G 3 considered as a linear transformatioñ σ(X : Y : Z) = (X : Y : ζZ), where ζ is a primitive m 3 -th root of unity, and C is defined by a polynomial of the form Z d + G m 3 (X, Y )Z d−m 3 + · · · + G d−m 3 (X, Y )Z m 3 + G d (X, Y ), where G i (X, Y ) ∈ k[X, Y ] is a homogeneous polynomial of degree i. Since G d (X, Y ) has no multiple component by Corollary 1, it follows that there exists a point Q ∈ C ∩ P 1 P 2 with \|G P 3 (Q)\| = m 3 . If m 3 = d, then C is smooth. Therefore, m 3 < d. Since the set of all fixed points ofσ coincides with P 1 P 2 ∪ {P 3 }, it follows that σ acts on d m 3 − 1 points, namely, m 3 \| d m 3 − 1 . On the other hand, m 3 ≤ (d/m 3 ) − 1. Then m 2 3 + m 3 ≤ d. In particular, m 3 < √ d and d m 3 > √ d. The same discussion can be applied to m 1 , m 2 . It follows from Proposition 2 that G P k (Q) ⊃ G P i , G P j ∩ G P k for any point Q ∈ C ∩ P i P j . Let e k := max{\|G P k (Q)\|; Q ∈ C ∩ P i P j }, and let l k = e k /m k . Then, for a point Q ∈ C ∩ P i P j with \|G P k (Q)\| = e k , \|G(Q)\| ≥ \| G P i , G P j (Q)\| × l k ≥ dl k , and the length of an orbit G · Q is at least d × d m k l k . The orbit-stabiliser theorem implies that \|G\| ≥ dl k × d × d m k l k = d 2 × d m k ≥ d 2 √ d. Assume that the number of short orbits under G is at least five. It follows from Fact 2 (c) that d 2 √ d ≤ \|G\| ≤ 4(g − 1) ≤ 2d(d − 3) . Then √ d ≤ 2 × d − 3 d < 2 × 1 = 2. This implies d < 4. Assume that the number of short orbits is four. It follows from Fact 2 that T ≥ −2 + 1 − 1 2 × 3 + 1 − 1 d = 1 2 − 1 d , and d 2 √ d ≤ \|G\| = 2g − 2 T ≤ 2d 2 (d − 3) d − 2 . Then √ d ≤ 2 × d − 3 d − 2 < 2. This implies d < 4. Assume that the number of short orbits is three. We take a smooth point Q k ∈ C ∩ P i P j for any {i, j, k} = {1, 2, 3}. We consider the case where three orbits G · Q 1 , G · Q 2 , G · Q 3 are different.T ≥ −2 + 1 − 1 d + 1 − 1 d + 1 × 2 = d 2 − 2d − 1 d(d + 1) , and (d + 1)d √ d ≤ \|G\| = 2g − 2 T ≤ d 2 (d + 1)(d − 3) d 2 − 2d − 1 . Then √ d ≤ d 2 − 3d d 2 − 2d − 1 < 1. This implies d < 1. We consider the case where there exist i, j, k with {i, j, k} = {1, 2, 3} such that G · Q i = G · Q j and G · Q i = G · Q k . By Proposition 1, \|G(Q i )\| = \|G(Q j )\| ≥ d + 1 and \|G(Q k )\| ≥ d. The orbit-stabiliser theorem implies that \|G\| ≥ (d + 1)d √ d. Then T ≥ −2 + 1 − 1 2 + 1 − 1 d + 1 − 1 d + 1 = d 2 − 3d − 2 2d(d + 1) , and (d + 1)d √ d ≤ \|G\| = 2g − 2 T ≤ 2d 2 (d + 1)(d − 3) d 2 − 3d − 2 . It follows that √ d ≤ 2d 2 − 6d d 2 − 3d − 2 ≤ 9 4 . This implies that d < 6. Finally, we consider the case where G · Q 1 = G · Q 2 = G · Q 3 . By Proposition 1, \|G(Q k )\| ≥ d + 1 for k = 1, 2, 3. Let e := max{e 1 , e 2 , e 3 }. We take k ∈ {1, 2, 3} with e k = e and a point Q ∈ C ∩ P i P j with \|G P k (Q)\| = e k . Then \|G(Q)\| ≥ dl k ≥ 2d if l k ≥ 2, where l k = e k /m k . It follows from Fact 6 that if the tangent line of a point R ∈ X contains P i , then the tangent line does not contain P j for j = i. This implies that   R∈C∩P j P k G P i · R     R∈C∩P i P k G P j · R   = ∅ if i = j. Assume that l k ≥ 2. The length of an orbit G · Q is at least 3 × d × d m k l k . The orbit-stabiliser theorem implies that \|G\| ≥ dl k × 3d × d m k l k = 3d 2 × d m k ≥ 3d 2 √ d. It follows that T ≥ −2 + 1 − 1 2 + 1 − 1 3 + 1 − 1 dl k ≥ 1 6 − 1 2d , and 3d 2 √ d ≤ \|G\| = 2g − 2 T ≤ d(d − 3) × 6d d − 3 = 6d 2 . Then √ d ≤ 2. This implies d ≤ 4. We consider the case where l k = 1. The length of an orbit G · Q is at least 3 × d × d m k . The orbit-stabiliser theorem implies that \|G\| ≥ (d + 1) × 3d × d m k ≥ 3d(d + 1) × √ d. It follows that T ≥ −2 + 1 − 1 2 + 1 − 1 3 + 1 − 1 d + 1 = 1 6 − 1 d + 1 , and 3d(d + 1) √ d ≤ \|G\| = 2g − 2 T ≤ 6d(d + 1)(d − 3) d − 5 . Then √ d ≤ 2 × d − 3 d − 5 , and d ≤ 9. We recall that m k < d, and the integer m k divides the integers d and (d/m k ) − 1. For the case d = 9, m k = 3 divides 9 3 − 1 = 2. This is a contradiction. For the case d = 8, m k = 2 divides 8 2 − 1 = 3. This is a contradiction. We consider the case where d = 6. Then we have m k = 2. This implies that \|G(Q k )\| is even, namely, \|G(Q k )\| ≥ 8. The length of an orbit G · Q is at least 3 × d × d m k = 54. The orbit-stabiliser theorem implies that \|G\| ≥ 8 × 54. It follows that T ≥ −2 + 1 − 1 2 + 1 − 1 3 + 1 − 1 8 = 1 24 , and 8 × 54 ≤ \|G\| = 2g − 2 T ≤ (g − 1) × 48. Then g = 10, namely, C is smooth. This is a contradiction. We have d < 6. 4. Proof of Theorem: The case d = 4 It follows from Fact 4 that g = 1 or g = 2. Assume that g = 2. Then there exists a unique singular point Q ∈ C with multiplicity 2. It follows from Corollary 1 that any line containing Q does not contain two outer Galois points. Since three outer Galois points P 1 , P 2 , P 3 exist, there exists i such that C ∩ P i Q \ {Q} consists of exactly two points. Let C ∩ P i Q \ {Q} = {R 1 , R 2 }. The two points over Q for the normalisation X → C are denoted by Q 1 , Q 2 . Since the smooth model X is hyperelliptic, the projection from Q corresponds to the canonical linear system, namely, R 1 + R 2 is a canonical divisor. Note that R i + Q j is not a canonical divisor for i = 1, 2 and j = 1, 2. Therefore, for any element σ ∈ G P i , σ({R 1 , R 2 }) = {R 1 , R 2 } or {Q 1 , Q 2 }. Since \|G P i \| = 4, it follows that there exists σ ∈ G P i with σ({R 1 , R 2 }) = {Q 1 , Q 2 }, namely, Q 1 + Q 2 is a canonical divisor. This implies that a tangent line T Q of C at Q is uniquely determined, which corresponds to the effective divisor D := 2Q 1 + 2Q 2 . According to Riemann-Roch's theorem, dim \|D\| = 2, namely, the linear system corresponding to a birational embedding into P 2 is complete. Since any σ ∈ G P i fixes the divisor induced by any line passing through P i , any σ ∈ G P i is the restriction of some linear transformationσ of P 2 . Thenσ(T Q ) = T Q , namely, G P i acts on the set {Q 1 , Q 2 }. As we saw above, there exists σ ∈ G P i with σ({R 1 , R 2 }) = {Q 1 , Q 2 }. This is a contradiction. Assume that g = 1. As we saw in the previous section, there exist i, j such that \| G P i , G P j (Q)\| ≥ d + 1 for a point Q ∈ C ∩ P i P j . A well known theorem on the automorphism group of an elliptic curve implies that \| G P i , G P j (Q)\| = 6 (see, for example, [9, III. Theorem 10.1]). If G P i is a cyclic group of order 4, then there exists a point Q ′ ∈ X such that \|G P i (Q ′ )\| = 4. This is a contradiction, because there does not exist an elliptic curve admitting two cyclic coverings of degree 6 and of degree 4 with totally ramified points (see, for example, [9, III. Theorem 10.1]). Therefore, G P i ∼ = G P j ∼ = (Z/2Z) ⊕2 . Note that there exist two involutions σ ∈ G P i with X/ σ ∼ = P 1 , since there exist eight ramification points for the covering X → X/G P i . Since G P i , G P j act on four points of C ∩ P i P j transitively, there exist involutions σ ∈ G P i , τ ∈ G P j such that X/ σ ∼ = P 1 , X/ τ ∼ = P 1 , and σ\| C∩P i P j = τ \| C∩P i P j . We take four points Q 1 , Q 2 , Q 3 , Q 4 ∈ C ∩ P i P j so that σ(Q 1 ) = Q 2 , σ(Q 3 ) = Q 4 , τ (Q 1 ) = Q 2 and τ (Q 3 ) = Q 4 . Then the double coverings X → X/ σ ∼ = P 1 and X → X/ τ ∼ = P 1 are given by rational functions f ∈ k(X) and g ∈ k(X) such that (f ) = Q 1 + Q 2 − Q 3 − Q 4 and (g) = Q 1 + Q 2 − Q 3 − Q 4 respectively. Then k(f ) = k(g), namely, σ = τ , and G P i ∩ G P j ⊃ {σ}. This is a contradiction. Proposition 1 . 1Assume that points P 1 = (1 : 0 : 0), P 2 = (0 : 1 : 0), P 3 = (0 : 0 : 1) are the same up to a constant. Therefore, polynomials c 1 x d + c 2 and f 1 (x) + 1 are the same up to a constant. By taking into account the definition of f (x, y), C is defined by a polynomial of the formx d + f 3 (y),where f 3 (y) ∈ k[y] is of degree d. This implies that all singular points of C are contained in the union of two lines {X = 0} and {Z = 0}. According to Corollary 1, C is smooth. It follows from Fact 4 that C is projectively equivalent to the Fermat curve. AcknowledgementsThe author is grateful to Professor Takeshi Harui for helpful comments enabling the author to prove Theorem in the case d = 6. On the number of Galois points for plane curves of prime degree. C Duyaguit, K Miura, Nihonkai Math. J. 14C. Duyaguit and K. Miura, On the number of Galois points for plane curves of prime degree, Nihonkai Math. J. 14 (2003), 55-59. Galois points for a plane curve in arbitrary characteristic. S Fukasawa, Proceedings of the IV Iberoamerican Conference on Complex Geometry. the IV Iberoamerican Conference on Complex Geometry139S. Fukasawa, Galois points for a plane curve in arbitrary characteristic, Proceedings of the IV Iberoamerican Conference on Complex Geometry, Geom. Dedicata 139 (2009), 211-218. Classification of plane curves with infinitely many Galois points. S Fukasawa, J. Math. Soc. Japan. 63S. Fukasawa, Classification of plane curves with infinitely many Galois points, J. Math. Soc. Japan 63 (2011), 195-209. On the number of Galois points for a plane curve in characteristic zero. S Fukasawa, arXiv:1604.01907preprintS. Fukasawa, On the number of Galois points for a plane curve in characteristic zero, preprint, arXiv:1604.01907. Algebraic curves admitting non-collinear Galois points. S Fukasawa, Rend. Sem. Mat. Univ. Padova. to appearS. Fukasawa, Algebraic curves admitting non-collinear Galois points, Rend. Sem. Mat. Univ. Padova, to appear. Galois points and rational functions with small value sets. S Fukasawa, Hiroshima Math. J. to appearS. Fukasawa, Galois points and rational functions with small value sets, Hiroshima Math. J., to appear. Algebraic Curves over a Finite Field. J W P Hirschfeld, G Korchmáros, F Torres, Princeton Univ. PressPrincetonJ. W. P. Hirschfeld, G. Korchmáros and F. Torres, Algebraic Curves over a Finite Field, Princeton Univ. Press, Princeton, 2008. Field theory for function fields of plane quartic curves. K Miura, H Yoshihara, J. Algebra. 226K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), 283-294. The Arithmetic of Elliptic Curves. J H Silverman, Graduate Texts in Mathematics. 106Springer-VerlagJ. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986. H Stichtenoth, Algebraic Function Fields and Codes. BerlinSpringer-VerlagUniversitextH. Stichtenoth, Algebraic Function Fields and Codes, Universitext, Springer-Verlag, Berlin, 1993. Function field theory of plane curves by dual curves. H Yoshihara, J. Algebra. 239H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), 340-355. Kojirakawa-machi 1-4-12. Faculty of Science, Yamagata UniversityJapan Email address: [email protected] of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560, Japan Email address: [email protected]	[]
[ "Estimating Patterns of Classical and Quantum Skyrmion States", "Estimating Patterns of Classical and Quantum Skyrmion States" ]	[ "Vladimir V Mazurenko \nTheoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia\n", "Ilia A Iakovlev \nTheoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia\n", "Oleg M Sotnikov \nTheoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia\n", "Mikhail I Katsnelson \nInstitute for Molecules and Materials\nRadboud University\nNijmegenNetherlands\n" ]	[ "Theoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia", "Theoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia", "Theoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia", "Institute for Molecules and Materials\nRadboud University\nNijmegenNetherlands" ]	[]	In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements.	10.7566/jpsj.92.081004	[ "https://export.arxiv.org/pdf/2304.02201v1.pdf" ]	257,952,366	2304.02201	b48ac4c389abc8e14dba99da00b74bc94646df38	Estimating Patterns of Classical and Quantum Skyrmion States Vladimir V Mazurenko Theoretical Physics and Applied Mathematics Department Ural Federal University Mira Str. 19620002EkaterinburgRussia Ilia A Iakovlev Theoretical Physics and Applied Mathematics Department Ural Federal University Mira Str. 19620002EkaterinburgRussia Oleg M Sotnikov Theoretical Physics and Applied Mathematics Department Ural Federal University Mira Str. 19620002EkaterinburgRussia Mikhail I Katsnelson Institute for Molecules and Materials Radboud University NijmegenNetherlands Estimating Patterns of Classical and Quantum Skyrmion States (Dated: April 6, 2023) In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements. In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements. I. INTRODUCTION History of science unambiguously evidences that the development of new theoretical concepts and physical models is impossible without insights from experiments and observations, which is crucial for exploring physical systems of any scale, from planetary objects to atoms and less than atoms. One of the bright examples of this insight is magnetic measurements 1 of a seemingly standard magnet, hematite α-Fe 2 O 3 performed by Smith in 1916, which, at the very end, paved the way to the concept of anisotropic inter-atomic interactions formulated by Igor Dzyaloshinskii 2 and Toru Moriya 3 . In his work Smith has found a ferromagnetic response by applying magnetic field perpendicular to the trigonal axis of α-Fe 2 O 3 . Remarkably, this experimental observation playing the crucial role in the theory of the inter-spin interaction was done about 10 years before the concept of the electron spin itself was introduced. Further experiments 4 confirmed Smith's findings for Fe 2 O 3 and showed robust weak ferromagnetism in other antiferromagnets 5,6 (MnCO 3 , CoCO 3 , NiF 2 ) characterized absence of inversion symmetry center between nearest magnetic ions. This relation with crystallography assumes that we deal with an intrinsic property rather than something related to crystal lattice imperfections (impurities, violation of the stoichiometric composition and others) 7 . By 1957, a critical mass of experimental data that evidence existence of a symmetry-dependent spontaneous magnetization in a number of antiferromagnets had been accumulated. In this year Igor Dzyaloshinskii proposed an elegant way to explain the magnetic moment by using the symmetry arguments based on the twisting of the spin arrangement due to inversion symmetry breaking 2 . For that anisotropic exchange interaction in the form D ij [S i × S j ] was introduced into the spin Hamiltonian. Such an interaction is antisymmetric with respect to interchange of the spins and favors noncollinear magnetic order. Toru Moriya has proposed a micro-scopic mechanism for this newborn coupling by developing its superexchange theory 3, 8,9 with taking the spinorbit coupling into account. Subsequently, the original Moriya's microscopic theory was consistently improved and refined in papers [10][11][12] . Moreover, different numerical schemes based on the density functional theory calculations were developed to estimate Dzyaloshinskii-Moriya interaction (DMI) from first-principles calculations [13][14][15][16][17] . All these methodological results facilitate modeling magnetic properties of completely different materials and make estimation of DMI to be a routine procedure in the modern computational physics. Remarkably, even after half century our understanding of the Dzyaloshinskii-Moriya interaction is far from being complete. It can be justified by the example that standard magnetic measurements techniques allow to estimate the magnitude and symmetry of DMI in concrete correlated materials, however, they do not provide the information on the DMI sign, which defines the local twist of the magnetic structure with respect to the atomic rearrangement due to inversion symmetry breaking. Such a problem was solved in 2014 when a new experimental technique, suggested earlier in Ref. 18, was developed in Ref. 19. This technique is based on the interference between two X-ray scattering processes, where one acts as a reference wave allowing to determine the sign of another. Experimentally, the sign of the DMI in correlated materials can be controlled with the occupation of the 3d shell. For instance, as it was shown in Ref. 20 there is a DMI sign change in the series of isostructural weak ferromagnets, MnCO 3 (DMI < 0), FeBO 3 (DMI < 0), CoCO 3 (DMI > 0) and NiCO 3 (DMI > 0). These experimental results agree with magnetic structure features obtained from the DFT calculations and can be explained Moriya's microscopic theory taking into account the occupation change of the correlated states. Over time, it became clear that the scope of Dzyaloshinskii-Moriya interaction is not limited to antiferromagnetic insulators with weak ferromagnetism. It has been used for prediction of long-range spiral structures in certain magnets 21 without inversion symmetry. Later on such spiral structures were experimentally observed in metallic MnSi and FeGe magnets 22,23 and Fe 1−x Co x Si alloys 24,25 with the B20 crystal group. For these metallic systems, the period of the spiral structures varies in wide ranges from 175Å to 700Å. In turn, magnetic critical temperatures for stabilization of the spin-spiral ground state can be also very different and include technologically important regimes that are close to the room temperature. In addition, it was found that there is a complex interplay between magnetic properties and electronic structure of long-range spin-spiral metallic magnets. For instance, first-principles calculations 26 of the B20 crystal group systems have revealed a strong renormalization of the electronic spectra near the Fermi level due to the dynamical electron-electron correlations, which can also affect values of the magnetic moments as well as isotropic and anisotropic magnetic exchange interactions 27 and, therefore, should be taken into account when one describing these systems theoretically. The exploration of the materials hosting the DMI spin spirals played an important role in establishing new research field of topologically-protected magnetic structures. It was first shown theoretically 28 in 1989 and then confirmed experimentally 29,30 that the DMI is responsible for forming long-range topologically protected chiral structures, magnetic skyrmions in metallic ferromagnets. The possibility to stabilize and manipulate skyrmions with magnetic and electric fields at the room temperature makes them very promising in numerous technological applications including next-generation memory devices and quantum computing 31 . Undoubtedly, further strides in the field of the magnetic skyrmions as well as creating new technologies that will make of use topological properties of the materials require implementation of the most advanced techniques for generation, detection, exploration and control. In this respect machine learning and quantum computing are of special interest. While the former allows to automatically classify and characterize magnetic structures the latter facilitates imitation of new phases of matter including topological ones. Keeping in mind the recent progress in developing machine learning and quantum computing techniques for scientific research in this review paper we first focus on the latest activity concerning the implementation of computing methods for exploration of the magnetic skyrmion phases. A special accent will be given on the recently introduced renormalization procedure for calculating structural complexity of an object 32 , which allows straightforward estimation of the phase boundaries in non-collinear magnets in a purely unsupervised manner by using a few magnetic snapshots of the system in question. The second part of the paper is devoted to theoretical analysis of the quantum skyrmions that are ground states of quantum systems with Dzyaloshinskii-Moriya interaction. Our abilities in simulation of such quantum states with classical computers are limited due to the expo-nential growth of the Hilbert space with the number of particles. At the same time quantum computing can be considered as the most promising technology for further exploration of the quantum skyrmion states. In this respect the development of approaches for certification and identification of quantum states of large-scale quantum systems are in demand and attract considerable attention. We discuss the generalization of the procedure for calculating structural complexity of object onto the case of quantum states and report on the classification of the quantum phases in DMI magnet on the basis of the bitstrings obtained after a limited number of projective measurements. The problems concerning imitation of the quantum skyrmions with quantum computers are discussed. II. CLASSICAL SKYRMIONS A typical spin Hamiltonian used to simulate skyrmion structures can be written in the following form: H = ij J ij S i S j + ij D ij [S i × S j ] + B i S z i (1) where J ij and D ij are the isotropic interaction and Dzyaloshinskii-Moriya vector, respectively, S i is a unit vector along the direction of the ith spin and B denotes the out-of-plane magnetic field. To stabilize a skyrmion state for Hamiltonian (1) the Dzyaloshinskii-Moriya interaction for each bond should be in-plane. In our work we consider D ij that points in the direction perpendicular to the bond between neighboring i and j sites. At low temperatures, different phases realized with Hamiltonian Eq.1 can be identified using conventional techniques, that is, calculating skyrmion number and spin structure factors. The skyrmion number (topological charge) Q is defined as Q = 1 8π ijk S i · [S j × S k ],(2) where the summation runs over all nonequivalent elementary triangles that connect neighboring i, j, and k sites. In the case of skyrmions of a few atoms in size 33 one can use an approach proposed in Refs. 34 and 35. In turn, the spin structure factors are given by χ q = S z q S z -q ,(3)χ ⊥ q = S x q S x −q + S y q S y −q ,(4) where q is the reciprocal space vector. As it was previously shown 30,36 , phase diagram of such system (Eq.1) consists of three clear phases: spin spirals, skyrmion crystal and ferromagnetic state, and two significant intermediate regions, namely, skyrmion-bimeron state and skyrmion gas. Some examples of these spin textures are presented in Fig. 1. By bimeron here we mean a spin texture composed of two half-disk meron domains having Q = ±1/2 divided by neutral rectangular stripe domain 37 . Such quasi-particle can be associated with either elongated skyrmions or broken helix segments, since its length strongly depend on the Hamiltonian parameters. We should note that term bimeron is rather used to describe particles, composed of a pair of merons of different vorticity [38][39][40] . However, to observe them one has to change the symmetry of the DM vector. The detailed description of these quasi-particles is given in Ref. 41. Unfortunately, the skyrmion number and spin structural factors are very sensitive to temperature and give us inappropriate results even in case when the spin structures still remain visually recognisable 42 (see Fig. 2). This fact aroused significant interest in the development of machine methods for conducting phase classification in this system. Below we will discuss such techniques. A. Complexity One of the basic concept which is usually used to analyze various patterns, systems and processes is their structural complexity. Although being intuitively clear since it reflects human's perception of reality, this value is very difficult to describe quantitatively. However, a lot of domains from geology to social sciences require a robust mathematical notion that properly reflects complexity of hierarchical non-random structures. Despite numerous attempts to give a formal definition of this quantity 43-50 , our understanding of these matters is still far from being complete. Recently, some of us have proposed an easy to compute, robust and universal definition of structural (effective) complexity based on inter-scale dissimilarity of patterns 32 . Besides meeting an intuitive perception of what is "complex" and what is "simple", this measure has been shown to be a suitable tool for determining phase transitions in various types of systems, including skyrmion structures. In its simplest form, the algorithm to compute structural complexity of a given magnetic configuration consisting of L × L atoms can be formulated in the following way 32 : at each iteration, the whole system is divided into blocks of Λ × Λ size, and each block is substituted with a single spin which is calculated as s ij (k) = 1 Λ 2 l m s Λi+m,Λj+l (k − 1) , where the lm indices enumerate the spins belonging to the same block, and k is the number of iteration. Then, one can compute overlaps between patterns separated by one step of such an averaging procedure: O k,k−1 = 1 L 2 L i=1 L j=1 s ij (k) · s ij (k − 1),(5) with k = 0 corresponding to the original pattern, and O k,k is an overlap of the pattern at scale k with its own self. Defining structural complexity C as an integral characteristic accounting for features emerging at every new scale, we obtain C = N −1 k=0 C k = N −1 k=0 \|O k+1,k − 1 2 (O k,k + O k+1,k+1 ) \|,(6) where N is the total number of averaging steps. Fig.3 shows an example of the implementation of the structural complexity approach to the skyrmion problem. More specifically, it gives the resulting dependence of structural complexity on magnetic field for two-dimensional triangular lattice system described by Hamiltonian Eq.(1). Remarkably, for each value of B the complexity appears to be very robust, fluctuating within 0.01% error range for independent Monte Carlo runs. It means that one can safely use a single magnetization image for each magnetic field value to define the complexity. The extrema of complexity derivatives dC/dB reflect very well both the melting of spin spirals (magnetic labyrinths) into skyrmion crystals, with the transition point being exactly the bimeron phase, as well as the transition between skyrmion crystals and ferromagnets. Recently, the structural complexity was used to find the phase boundary between self-induced spin-glass and noncollinear magnetically ordered states in elemental Nd at low temperatures 51 . B. Machine learning methods for phase classification By the construction the method for calculating structural complexity can be classified as unsupervised one, since it does not use any apriori information on the system in question. At the same time, we would like to stress the advances of various supervised techniques that involve a learning with pre-prepared and labeled data. After the inspiring work of Carrasquilla and Melko 52 , who had demonstrated the ability of neural networks to define phase transitions in magnetic systems, significant efforts have been made in this field. Here we give a brief overview of such approaches aimed to study the properties of the magnetic skyrmions. In Ref. 42 we have demonstrated the possibility to use machine learning algorithms for exploration phases of non-collinear magnets that can host skyrmionic structures (see Fig. 4). In the work 42 , a square lattice system described by a spin Hamiltonian (1) was considered. Given the fact, that all presented spin textures have distinct magnetization profiles, we decided to use only z components of spins as an input for machine learning algorithms. We have found out, that a simple singlehidden-layer feed-forward neural network (FFN) with only 64 hidden neurons, trained on a moderate number of clear-phase Monte Carlo configurations, was able to successfully reproduce the entire phase diagram, including intermediate regions. Moreover, it demonstrated good results on unseen data, namely, high-temperature configurations, larger skyrmions and configurations obtained for triangular lattice system. Unfortunately, we found that such a network relies mostly on total magnetization and therefore cannot distinguish spin spirals and paramagnetic state. However, such an inconvenience can be easily overcome by simple sorting of the input vector even in case of 3D systems 53 (see Fig. 5). It was also shown, that standard machine learning techniques like k-nearest neighbours, nearest centroids and supportvector machine work well in case of all clear phases and paramagnetic state. Later, more complicated convolutional neural networks (CNN) were used to study the effect of uniaxial magnetocrystalline anisotropy pointing in the z direction on the phase diagram of a disk-shaped system 54 , and to construct detailed phase diagrams for skyrmion systems including intermediate regions and paramagnetic state 55 . Moreover, it was shown that such an architecture is able to not only determine phase boundaries but also restore various parameters. The authors of Ref. 56 have demon- strated, that a CNN trained on ground state configurations successfully recovers the chirality and magnetization of a given spin texture, as well as the temperature and external magnetic field at which it was stabilized. It is interesting to note that the accuracy of the algorithm remains remarkably high in presence of disorder caused by the randomly generated site-dependent uniaxial anisotropy. The authors of Ref. 57 have addressed an important problem of finding a topological charge of a given system based on its time-integrated spacedependent magnetization. They demonstrated that, being analytically inaccessible, this quantity can be extracted using a CNN with almost 100% accuracy. It was shown, that such an approach works well on systems of different confined geometries, including random islands, which looks very promising from the point of view of potential application to real experimental data. Recently, considerable attention has also been paid to the dynamic properties of skyrmion structures. Some of us have shown that the simplest recurrent neural network (RNN) is able to automatically detect different processes occurring with an isolated skyrmion under the influence of picosecond magnetic field pulses 58 (see Fig. 6). Such an approach is promising as a technique which performs an autonomous control of the system's dynamics in case of a prototype of skyrmionic data storage elements 35 . The authors of Ref. 59 have studied a dynamic phase diagram of a particle model for skyrmions in metallic chiral magnets with using CNN-RNN architecture. It was shown, that the network is able to not only draw the correct phase boundaries but also define the exact number of the order parameters of the system in question. III. QUANTUM SKYRMIONS The progress in the development of experimental techniques 29,30 for the observation of magnetic skyrmions, topologically protected spin structures, poses new challenges for the theory and numerical simulations of ordered magnetic phases 60 . Nowadays, skyrmions are mostly discussed in the context of spintronics, where these stable magnetic structures are proposed as bits in magnetic memory devices 61 . The need to store more and more information requires the development of ultradense memories. This fact motivates the investigation of skyrmions of a nanoscale size, with recent significant progress. Skyrmions with the characteristic size of a few nanometers have already been observed in real experiments 62 and were theoretically predicted in magnets with DMI 63 , in frustrated magnets 64 , as well as in narrow band Mott insulators under high-frequency light irradiation 65 , and others. On such small characteristic length scales compared to the lattice constant, quantum effects cannot be neglected. Given this, the numerical study of classical spin models can no longer be considered as an comprehensive solution of the problem. Quantum fluctuations play a crucial role, because, strictly speak- ing, the spin itself is a quantum characteristic of an electron. A common way to approach this problem is to force the quantum system to behave as a classical one. As the result, description of a quantum skyrmionic problem is either done semiclassically assuming that the magnetization dynamics is dominated by classical magnetic excitations that emerge on top of the symmetry-broken ground state of the system 66 , or by means of the Holstein-Primakoff transformation, which only allows to compute quantum corrections to the classical solution 67 . Besides, in paper 68 topological states of small clusters embedded in the ferromagnetic environment were investigated. Recently, some of us have developed an approach for a characterisation of the quantum skyrmion state 69 in an infinite magnetic systems for which, in contrast to the classical case, the magnetization density is uniform. For that, the following spin model defined on the 19-spin supercell with periodic boundary conditions was considered,Ĥ = ij J ijŜiŜj + ij D ij [Ŝ i ×Ŝ j ] + B iŜ z i . (7) HereŜ i is the spin-1/2 operator. For characterization of the quantum ground states obtained at different magnetic field values, the local three-spin correlation function, Q Ψ = N π Ŝ 1 · [Ŝ 2 ×Ŝ 3 ] defined on neighboring lattice sites (here N is the number of non-overlapping triangles in the supercell) was used. Such a correlator gives information about the topology of the entire quan-tum system, for instance, from Fig.7 one can see that the scalar chirality is characterized by non-zero constant value for the magnetic fields 0.3 < B < 0.66, which clear signature of the quantum skyrmion phase. Theoretically, exponential growth of the Hilbert space is the main factor preventing one from simulating larger systems than 19-site clusters discussed in Ref. 69 and from exploring quantum skyrmions of different kinds and sizes. The topological spin structures such as skyrmions emerge as the result of a competition between different magnetic interactions, leading to a magnetic frustration that restricts the applicability of quantum Monte Carlo methods due to the notorious sign problem 70 . In turn, exact diagonalization (ED) based methods have a severe restriction on the cluster size. For instance, for spin-1/2 Heisenberg-type models, the current limit is 50 lattice sites 71 . Account of the anisotropic terms such as Dzyaloshinskii-Moriya interaction leads to mixing of the sectors of the Hamiltonian with different total spins, which significantly limits our opportunities to use symmetry of the system in question to reduce the size of the Hamiltonian matrix. Thus, this supercell of 19 sites with isotropic and anisotropic exchange interactions between spins defines the current limit on the system size for simulating quantum skyrmions with exact diagonalization approach. In this tough situation quantum computing provides a promising alternative to the standard approaches aimed at the search for ground and low-lying excited states of quantum Hamiltonian. Over the last decades there has been a fantastic progress in quantum computing on the level of constructing complete operating devices of up to 65 qubits with on-line access as well as in developing numerous algorithms for different fields of research including material science and condensed matter physics 72 . The first attempt 73 of the Google team to demonstrate a quantum supremacy by the example of quantum chaos states have additionally heat up the interest of scientific community to the work on quantum states that are significantly delocalized in Hilbert space. Our preliminary results show that the quantum skyrmion and spin spiral states being solutions of the quantum spin model, Eq.7 are significantly spread over Hilbert space, which is an indication that their further theoretical exploration may be effective with quantum computers. However, it calls for development of the methods for characterization and identification of the quantum states. Below we will discuss such a procedure based on the estimating structural complexity of the bit-string patterns. Calculating inter-scale dissimilarity of bit-string arrays Generally, the exploration of the magnetic skyrmions with quantum devices or simulators should include the following steps. First of all, one needs to define a quantum circuit that transforms the initial trivial quantum state \|000..0 into desire Ψ Skyrm (Fig.8) being the ground state of the quantum Hamiltonian, Eq.7. There are two alternative solutions of this problem. (i) One can use a variational approach 72,74,75 in which the quantum state is represented with a fixed sequence of one-and two-qubit gates. The parameters of these gates are tuned to get the best approximation of the desire state. Unfortunately, there is no a universal sequence of gates that can approximate the ground state of an arbitrary Hamiltonian. Another problem is that one obtains not a true target state but only its approximation, which is of crucial importance for certain tasks. (ii) On the other hand, in the case of the small-size problems, the exact decomposition of the quantum ground state over basis functions can be found with exact diagonalization. It allows to employ Least Significant Bit (LSB) procedure 76 or similar procedure to find a sequence of gates realizing such a state. In the case of the 19-site quantum skyrmion ground state the LSB circuit contains thousands of gates, which is appropriate for creation and manipulation of such a state on a quantum simulator that imitates quantum logical operations on classical computer, but not on a real quantum device subjected to decoherence. For the latter one needs to develop new approaches aiming at generating quantum circuits that are as compact as possible. In this work we explore the quantum skyrmion state on the quantum simulator by using the LSB procedure. When quantum is initialized with target state, the measurements in a basis are performed (Fig.8). As it was shown in our previous work 77 to characterize a quantum state with dissimilarity procedure it is necessary to perform measurements in at least two bases because measurements in one fixed basis give the information only about amplitudes of wave function coefficients, but not the local phases. Following Ref. 77 the measurements in σ z and random bases were performed. For each basis the measurement outputs are concatenated together into one string which can be considered as binary array of length L = N × N shots . After sampling of bit-string arrays, we estimate its patterns using the procedure presented in Ref. 77. Below we reproduce the main steps of this procedure. At every step of coarse-graining k, a vector of the same length L is constructed as b k i = 1 Λ k Λ k l=1 b k−1 Λ k [(i−1)/Λ k ]+l ,(8) where b 0 is initial bit-string array containing 0 and 1 elements, square brackets denote taking integer part. According to this expression at each iteration the whole array is divided into blocks of Λ k size, and elements within a block are substituted with the same value resulting from averaging all elements of the block (Fig.9). Index l denotes elements belonging to the same block. In our recent work 77 we have shown that in some cases it is enough to measure only part of the system to extract information about phase transitions. Following this way, for simplicity, we measured only 16 qubits in 19 site system to make the bit-string length an integer power of filter size Λ: log Λ N ∈ N. Dissimilarity between scales k and k + 1 is defined as D k = \|O k+1,k − 1 2 (O k,k + O k+1,k+1 ) \|,(9) where O m,n is the overlap between vectors at scales m and n: O m,n = 1 L (b m · b n ) .(10) This expression can be considered as a modification of the structural complexity 32 discussed above in the context of classical magnetic patterns to our quantum problem. There are two quantities of our principal interest: D k that contains scale-resolved information on the pattern structure of the generated bit-string array and overall dissimilarity, D = k D k , where the sum goes over all the renormalization steps. D and {D k } can be used for a unambiguous identification of a quantum state. Fig.10 gives the dissimilarity calculated in σ z and random bases for the ground state of the spin Hamiltonian, Eq.7 taken with \|D ij \| = 1, J ij = 0.5 and B= 0.4. In these calculations of D we used 16 of 19 bits from each measurement. The calculated D z reveals a smooth transition between spin spiral and skyrmion phases. On the other hand, the transition between skyrmion and ferromagnetic states is abrupt. Figure 11 shows partial dissimilarities {D k } calculated for 19-qubit system in σ z and random bases. Although partial dissimilarities of different non-collinear magnetic phases measured in σ z basis have similar shape, they still have different absolute values. Note, that in σ z basis the value of partial and total dissimilarity is equal to zero for ferromagnetic state. This happens because in σ z basis all measurements result in 0000. . . 0 or 1111. . . 1 depending on the direction of magnetic field which gives exactly zero for Eq. 9. The D r demonstrates different behaviour for ferromagnetic phase, and, peak at k = 4 emerges when filter size Λ k becomes larger than the length of individual measurement bit-string effectively mixing different measurements. Thus, the dissimilarity metric facilitates the search for phase boundaries of quantum models. Formally, this problem can be solved by choosing an appropriate order parameter, a correlation function characterized by a specific non-zero value for the definite range of model parameters 78,79 . However, in this case one faces the problem that different phases cannot be described with a single order parameter. For instance, by using the scalar chirality we cannot distinguish between the cases of zero and high magnetic fields in the considered DMI magnet, Fig.7. We would like to stress that one can consider the dissimilarity as a worthy alternative to other approaches for detecting phase boundaries in strongly correlated systems. First, the calculation of the dissimilarity is considerably cheaper in computational resources than traditional methods based on the calculation of the correlation functions. Then, the dissimilarity allows the accurate description of the phase boundaries in the cases when the choice of the order parameter is not obvious. It was demonstrated by the examples of the Shastry-Sutherland model of the orthogonal spin dimers and one-dimensional bond-alternating XXZ model hosting topological order 77 . IV. CONCLUSION To conclude, implementation of the machine learning methods considerably facilitates and accelerates theoretical characterization of non-collinear magnetic structures originated from the competition between anisotropic Dzyaloshinskii-Moriya and isotropic Heisenberg exchange interactions. As we have shown in this review by the example of the magnetic skyrmions the range of tasks to be solved is very wide, from constructing phase diagrams to estimating parameters of the parent Hamiltonians. In contrast to the traditional approach for description of magnetic systems that assumes accumulation of significant amount of the statistical data and calculations correlation functions of different orders, machine learning approaches do not require a complete information on the system in question. For instance, the temperature-magnetic-field phase diagrams can be constructed on the basis of few snapshots containing only z projection of the spins. In this respect, we would like to emphasize a crucial role of the data preprocessing that can realize in different ways and forms. For instance, it could be simple sorting the magnetization vectors that considerably improves the quality of the supervised neural network classification. The procedure for estimating structural complexity of an object we described in the main text can be also considered as that performing a kind of preprocessing. It al-lows unsupervised identification of the phase boundaries of physical systems at a smooth varying external parameters such as temperature, magnetic field and others. As a result the complete phase space of the system in question can be divided onto separate phases, whose origin and properties could be further clarified with traditional and machine learning approaches. The same procedure for estimating pattern complexity can be applied in the case of quantum systems, for which one explores sequences of the bitstrings obtained from the projective measurements. As a prominent example we analyzed the pattern structure of the quantum ferromagnet with Dzyaloshinskii-Moriya interaction. Depending on the value of the external magnetic field one can observe different quantum phases including recently introduced quantum skyrmion one. Our results presented in this paper complete the picture of the quantum skyrmion properties and can be useful for characterization of such a state on the quantum computing devices. V. ACKNOWLEDGEMENTS We thank Vladimir Dmitrienko for fruitful discussions. FIG. 1 . 1(Color online) Monte Carlo solutions of the spin model, Eq. (1) on the rhombic plaquette, corresponding to the spin spiral (a), bimeron (b), skyrmion (c) and ferromagnetic states. This figure is reproduced from Ref. 36. (c) [2018] American Physical Society. FIG. 2 . 2(Color online) (A) Magnetic labyrinth on a triangular lattice at low temperature (T = 0.02J), (B) spin spirals, (C) mixed skyrmion-bimeron magnetic configuration and (D) pure skyrmions on a square lattice at high temperature (T = 0.4J), and the corresponding spin structure factors. This figure is reproduced from Ref. 32. (c) [2020] National Academy of Sciences. online) (Top) Structural complexity calculated for the classical two-dimensional triangular lattice magnetic configurations obtained with Hamiltonian Eq.(1) as a function of external magnetic field. The error bars are smaller than the symbol size. (Bottom) Complexity derivative we used for accurate detection of the phases boundaries. Squares and circles correspond to the low (T = 0.02J) and high (T = 0.4J) temperature configurations obtained with using all spin components, meanwhile triangles and diamonds represent results obtained with using only z component. All the results are obtained for \|D\| = J, only the interaction between the nearest neighbours is taken into account. FIG. 4 . 4(Color online) (Top) Schematic representation of the machine learning process. Neural network with single hidden layer of sigmoid neurons performs phase classification based on z components of spins of the magnetic configuration. (Bottom) Phase triptych obtained by using the neural network with 64 hidden neurons for \|D\| = 0.72J. White circles denote the phase boundaries defined with the spin structure factors. This figure is reproduced from Ref. 42. (c) [2018] American Physical Society. online) Magnetization profiles of configurations belonging to different phases obtained with the spin Hamiltonian Eq. (1) on the two-dimensional triangular lattice (top) and three-dimensional cubic lattice (bottom). This figure is reproduced from Ref. 53. (c) [2019] American Physical Society. FIG. 6 . 6(Color online) (Top) Illustration of idea of the ultrafast skyrmionic process recognition. Magnetization dynamics is used frame by frame as an input for recurrent neural network providing the process classification. (Bottom) Obtained process diagram with the θ = 40 • magnetic pulses. Phase boundaries determined by means of RNN are indicated by brown dashed lines. This figure is reproduced from Ref. 58. (c) [2019] American Physical Society. FIG . 7. (Color online) (a) Calculated scalar chirality and magnetization as functions of the magnetic field. (b) Magnetization density. (c) Calculated spin structural factors. This figure is reproduced from Ref. 69. (c) [2021] American Physical Society. FIG. 8 .FIG. 9 . 89(Color online) Schematic representation of initialization of the quantum skyrmion state on a quantum simulator(computer). The measurement basis is chosen with onequbit rotational gate U0 that is described in the text. (Color online) Schematic representation of the renormalization procedure for bitstring array. First three steps are shown. 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[ "Generalized Self-Concordant Analysis of Frank-Wolfe algorithms", "Generalized Self-Concordant Analysis of Frank-Wolfe algorithms" ]	[ "Pavel Dvurechensky [email protected] \nWeierstrass Institute for Applied Analysis and Stochastics\nMohrenstr. 3910117BerlinGermany\n", "Kamil Safin [email protected] \nMoscow Institute of Physics and Technology, Dolgoprudny\nRussia\n", "Shimrit Shtern [email protected] \nFaculty of Industrial Engineering and Management\nTechnion -Israel Institute of Technology\nHaifaIsrael\n", "Mathias Staudigl [email protected] \nDepartment of Data Science and Knowledge Engineering\nMaastricht University\nP.O. Box 6166200 MDMaastrichtNLThe Netherlands\n" ]	[ "Weierstrass Institute for Applied Analysis and Stochastics\nMohrenstr. 3910117BerlinGermany", "Moscow Institute of Physics and Technology, Dolgoprudny\nRussia", "Faculty of Industrial Engineering and Management\nTechnion -Israel Institute of Technology\nHaifaIsrael", "Department of Data Science and Knowledge Engineering\nMaastricht University\nP.O. Box 6166200 MDMaastrichtNLThe Netherlands" ]	[]	Projection-free optimization via different variants of the Frank-Wolfe method has become one of the cornerstones of large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with self-concordance like properties. Such generalized self-concordant functions do not necessarily feature a Lipschitz continuous gradient, nor are they strongly convex, making them a challenging class of functions for first-order methods. Indeed, in a number of applications, such as inverse covariance estimation or distance-weighted discrimination problems in binary classification, the loss is given by a generalized self-concordant function having potentially unbounded curvature. For such problems projection-free minimization methods have no theoretical convergence guarantee. This paper closes this apparent gap in the literature by developing provably convergent Frank-Wolfe algorithms with standard O(1/k) convergence rate guarantees. Based on these new insights, we show how these sublinearly convergent methods can be accelerated to yield linearly convergent projection-free methods, by either relying on the availability of a local liner minimization oracle, or a suitable modification of the away-step Frank-Wolfe method.	10.1007/s10107-022-01771-1	[ "https://arxiv.org/pdf/2010.01009v3.pdf" ]	222,125,310	2010.01009	513b52e73a54bc52e683f6b96f989c1a15228d6b	Generalized Self-Concordant Analysis of Frank-Wolfe algorithms August 3, 2021 Pavel Dvurechensky [email protected] Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr. 3910117BerlinGermany Kamil Safin [email protected] Moscow Institute of Physics and Technology, Dolgoprudny Russia Shimrit Shtern [email protected] Faculty of Industrial Engineering and Management Technion -Israel Institute of Technology HaifaIsrael Mathias Staudigl [email protected] Department of Data Science and Knowledge Engineering Maastricht University P.O. Box 6166200 MDMaastrichtNLThe Netherlands Generalized Self-Concordant Analysis of Frank-Wolfe algorithms August 3, 2021 Projection-free optimization via different variants of the Frank-Wolfe method has become one of the cornerstones of large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with self-concordance like properties. Such generalized self-concordant functions do not necessarily feature a Lipschitz continuous gradient, nor are they strongly convex, making them a challenging class of functions for first-order methods. Indeed, in a number of applications, such as inverse covariance estimation or distance-weighted discrimination problems in binary classification, the loss is given by a generalized self-concordant function having potentially unbounded curvature. For such problems projection-free minimization methods have no theoretical convergence guarantee. This paper closes this apparent gap in the literature by developing provably convergent Frank-Wolfe algorithms with standard O(1/k) convergence rate guarantees. Based on these new insights, we show how these sublinearly convergent methods can be accelerated to yield linearly convergent projection-free methods, by either relying on the availability of a local liner minimization oracle, or a suitable modification of the away-step Frank-Wolfe method. Introduction Statistical analysis using generalized self-concordant (GSC) functions as a loss function is gaining increasing attention in the machine learning community [1,39,40,44]. Beyond machine learning, GSC loss functions are also used in image analysis [38] and quantum state tomography [27]. This class of loss functions allows to obtain faster statistical rates similar to least-squares [31]. At the same time, the minimization of empirical risk in this setting is a challenging optimization problem in high dimensions. Thus, without knowledge of specific structure, interior point, or other polynomial time methods, are unappealing. Moreover, large-scale optimization models in machine learning often depend on noisy data and thus high-accuracy solutions are not really needed or obtainable. All these features make simple optimization algorithms with low implementation costs the preferred methods of choice. In this paper we focus on projection-free methods which rely on the availability of a Linear minimization oracle (LMO). Such algorithms are known as Conditional Gradient (CG) or Frank-Wolfe (FW) methods. These classes of gradient-based algorithms belong to the oldest convex optimization tools, and their origins can be traced back to [16,26]. For a given convex compact set X ⊂ R n , and a convex objective function f , FW methods solve the smooth convex optimization problem min x∈X f (x),(P) by sequential calls of a LMO, returning at point x the target vector s(x) ∈ arg min d∈X ∇ f (x), d . (1.1) The selection s(x) is determined via some pre-defined tie breaking rule, whose specific form is of no importance for the moment. Computing this target state is the only computational bottleneck of the method. Progress of the algorithm is monitored via a merit function. The standard merit function in this setting is the Frank-Wolfe (dual) gap Gap(x) max s∈X ∇ f (x), x − s . (1.2) It is easy to see that Gap(x) ≥ 0 for all x ∈ X, with equality if and only if x is a solution to (P). The vanilla implementation of FW (Algorithm 1) aims to reduce the gap function by sequentially solving linear minimization subproblems to obtain the target point s(x). As always, the general performance of an algorithm depends heavily on the availability of practical step-size policies {α k } k∈N . Two popular choices are either α k = 2 k+2 (FW-Standard), or an exact line-search (FW-Line Search). Under either choice, the algorithm exhibits an O(1/k) rate of convergence for solving (P) in case where f is convex and either possess a Lipschitz continuous gradient, or a bounded curvature constant. The latter concept is a slight weakening of the classical Lipschitz gradient assumption, and is the key quantity in the modern analysis of FW due to Jaggi [22]. The curvature constant is defined as κ f sup x,s∈X,t∈[0,1] The main difficulties one faces in minimizing functions with self-concordance like properties can be easily illustrated with a basic, in some sense minimal, example: Example 1.1. Consider the function f (x, y) = − ln(x) − ln(y) where x, y > 0 satisfy x + y = 1. This function is the standard self-concordant barrier for the positive orthant (the log-barrier) and thus (2, 3)-generalized self-concordant (see Definition 2.1). Its Bregman divergence is easily calculated as D f (u, v) = 2 i=1 − ln u i v i + u i v i − 1 u = (u 1 , u 2 ), v = (v 1 , v 2 ). Neither the function f , nor its gradient, is Lipschitz continuous over the set of interest. In particular the curvature constant is unbounded, i.e. κ f = ∞. Moreover, if we start from u 0 = (1/4, 3/4) and apply the standard 2/(k + 2)-step size policy, then α 0 = 1, which leads to u 1 = s(u 0 ) = (1, 0) dom f . Clearly, the standard method fails. The logarithm is one of the canonical members of (generalized) self-concordant functions, and thus the above example is quite representative for the class of optimization problems of interest in this paper. It is therefore clear that the standard analysis of [22], and all subsequent investigations relying on estimates of the Lipschitz constant of the gradient or the curvature, cannot be applied straightforwardly to the problem of minimizing a GSC function via projection-free methods. Related literature The development of FW methods for ill-conditioned problems has received quite some attention recently. [36] requires the gradient of the objective function to be Hölder continuous and similar results for this setting are obtained in [6,43]. Implicitly it is assumed that X ⊆ dom f . This would also not be satisfied in important GSC minimization problems, and hence we do not impose it (e.g. 0 ∈ X, but 0 dom f in the Covariance Estimation problem in Section 6.4). Specialized to solving a quadratic Poisson inverse problem in phase retrieval, [38] provided a globally convergent FW method using the convex and self-concordant (SC) reformulation, based on the PhaseLift approach [7]. They constructed a provably convergent FW variant using a new step size policy derived from estimate sequence techniques [2,34], in order to match the proof technique of [36]. Very recently, two other FW-methods for ill-conditioned problems appeared. [28] employed a FW-subroutine for computing the Newton step in a proximal Newton framework for minimizing self-concordant (SC)-functions over a convex compact set. After the first submission of this work, Professor Robert M. Freund sent us the preprint [51], in which the SC-FW method from our previous conference paper [13] is refined to minimize a logarithmically homogeneous barrier [33] over a convex compact set. They also propose new stepsizes for FW for minimizing functions with Hölder continuous gradient. None of these recent contributions develop FW methods for the much larger class of GSC-functions, nor do they consider linearly convergent variants. Linearly convergent Frank-Wolfe methods Given their slow convergence, it is clear that the application of projection-free methods can only be interesting if projections onto the feasible set are computationally expensive. Various previous papers worked out conditions under which the iteration complexity of projection-free methods can be potentially improved. [19] obtained linear convergence rates in well conditioned problems under the a-priori assumption that the solution lies in the relative interior of the feasible set, and the rate of convergence explicitly depends on the distance of the solution from the boundary (see also [5,15]). If no a-priori information on the location of the solution is available, there are essentially two known twists of the vanilla FW to boost the convergence rates. One twist is to modify the search directions via corrective, or away search directions [17,19,20,42,49]. The Away-Step Frank Wolfe (ASFW) method can remove weight from "bad" atoms in the active set. These drop steps have the potential to circumvent the well-known zigzagging phenomenon of FW when the solution lies on the boundary of the feasible set. When the feasible set X is a polytope, [23] derived linear convergence rates for ASFW using the "pyramidal width constant" in the well-conditioned optimization case. Unfortunately, the pyramidal width is the optimal value of a complicated combinatorial optimization problem, whose value is unknown even on simple sets such as the unit simplex. [4] improved their construction by replacing the pyramidal width with a much more tractable gradient bound condition, involving the "vertex-facet distance". In many instances, including the unit simplex, the 1 -ball and the ∞ -ball, the vertexfacet distance can be computed (see Section 3.4 in [4]). In this paper we develop a corresponding away-step FW variant for the minimization of a GSC function (Algorithm 8 (ASFWGSC)), extending [4] to ill-conditioned problems. While we were working on the revision of this paper, Professor Sebastian Pokutta shared with us the recent preprint [8], where a monotone modification of FW-Standard applied to GSCminimization problems is proposed. They derive a O(1/k) convergence rate guarantee for minimizing GSC. Moreover, they exhibit a linearly convergent variant using away-steps. These results have been achieved independently from our work, and they give a nice complementary view on our away-step variant ASFWGSC. The basic difference between our analysis and [8] is that we exploit the vertex-facet distance instead of the pyramidal width. As already said, this gives explicit and efficiently computable error bounds for some important geometries, and thus allows for a more in-depth complexity assessment. The alternative twist to obtain linear convergence is to change the design of the LMO [18,21,24] via a well-calibrated localization procedure. Extending the work by Garber and Hazan [18], we construct another linearly convergent FW-variant based on local linear minimization oracles (Algorithm 7, FWLLOO). Main contributions and outline of the paper In this paper, we demonstrate that projection-free methods extend to a large class of potentially illconditioned convex programming problems, featuring self-concordant like properties. Our main contributions can be succinctly summarized as follows: (i) Ill-Conditioned problems: We construct a set of globally convergent projection-free methods for minimizing generalized self-concordant functions over convex compact domains. (ii) Detailed Complexity analysis: Algorithms with sublinear and linear convergence rate guarantees are derived. (iii) Adaptivity: We develop new backtracking variants in order to come up with new step size policies which are adaptive with respect to local estimates of the gradient's Lipschitz constant, or basic parameters related to the self-concordance properties of the objective function. The construction of these backtracking schemes fully exploits the basic properties of GSCfunctions. Specifically, Algorithm 3 (LBTFWGSC) builds on a standard quadratic upper model over which a local search for the Lipschtiz modulus of the gradient, restricted to level sets, can be performed. This local search method is inspired by [41], but our convergence proof is much simpler and direct. Our second backtracking variant (Algorithm 5, MBTFWGSC) performs a local search for the generalized self-concordance constant. To the best of our knowledge this is the first algorithm which adaptively adjusts the self-concordance parameters on-the-fly. We thus present three new sublinearly converging FW-variants which are all adaptive, and share the standard sublinear O(1/ε) complexity bound which is proved in Section 4. On top of that, we derive two new linearly converging schemes, either building on the availability of Local Linear minimization oracle (LLOO) (Algorithm 7 (FWLLOO)), or suitably defined Away-Steps (Algorithm 8 (ASFWGSC)). (iv) Detailed Numerical experiments: We test the performance of our method on a set of challenging test problems, spanning all possible GSC parameters over which our algorithms are provably convergent. This paper builds on, and significantly extends, our conference paper [13]. This previous work exclusively focused on the minimization of standard self-concordant functions. The extension to generalized self-concordant functions requires some careful additional steps and a detailed caseby-case analysis that are not simple corollaries of [13]. On top of that, in this paper we derive two completely new projection-free algorithms, and new proofs of existing algorithms we already introduced in our first publication. In light of these contributions, this paper significantly extends the results of [13]. Outline Section 2 contains necessary definitions and properties for the class of GSC functions in a self-contained way. Our algorithmic analysis starts in Section 3 where a new FW variant with an analytic step-size rule is presented (Algorithm 2, FWGSC). This algorithm can be seen as the basic template from which the other methods are subsequently derived. Section 4 presents the convergence analysis for the three sublinearly convergent variants presented in Section 3. Section 5 presents the two linearly convergent variants and their convergence analysis. Section 6 reports results from extensive numerical experiments using the proposed algorithms and their comparison with the baselines. Section 7 concludes the paper. Notation Given a proper, closed, and convex function f : R n → (−∞, ∞], we denote by dom f {x ∈ R n \| f (x) < ∞} the (effective) domain of f . For a set X, we define the indicator function δ X (x) = ∞ if x X, and δ X (x) = 0 otherwise. We use C k (dom f ) to denote the class of functions f : R n → (−∞, ∞] which are k-times continuously differentiable on their effective domain. We denote by ∇ f the gradient map, and ∇ 2 f the Hessian map. Let R + and R ++ denote the set of nonnegative, and positive real numbers, respectively. We use S n {x ∈ R n×n \|x = x} the set of symmetric matrices, and S n + , S n ++ to denote the set of symmetric positive semi-definite and positive definite matrices, respectively. Given Q ∈ S n ++ we define the weighted inner product u, v Q Qu, v for u, v ∈ R n , and the corresponding norm u Q u, u Q . The associated dual norm is v * Q v, v Q −1 . For Q ∈ S n , we let λ min (Q) and λ max (Q) denote the smallest and largest eigenvalues of the matrix Q, respectively. Generalized self-concordant functions Following [44], we briefly introduce the basic properties of the class of GSC functions. Let ϕ : R → R be a three-times continuously differentiable function on dom ϕ. Recall that ϕ is convex if and only if ϕ (t) ≥ 0 for all t ∈ dom ϕ. ϕ (M ϕ , ν) generalized self-concordant (GSC) if \|ϕ (t)\| ≤ M φ ϕ (t) ν 2 ∀t ∈ dom ϕ. (2.1) If ϕ(t) = a 2 t 2 + bt + c for any constant a ≥ 0 we get a (0, ν)-generalized self-concordant function. Hence, any convex quadratic function is GSC for any ν > 0. Standard one-dimensional examples are summarized in Table 1 (based on [44]). Function name Form of [44]). ϕ(t) ν M ϕ dom ϕ Lipschitz smooth Burg entropy − ln(t) 3 2 (0, ∞) No Logistic ln(1 + e −t ) 2 1 (−∞, ∞) Yes Exponential e −t 2 1 (−∞, ∞) Yes Negative Power t −q , q > 0 2(q+3) q+2 q+2 q+2 √ q(q+1) (0, ∞) No Arcsine distribution 1 √ 1−t 2 14 5 < 3.25 (−1, 1) No This definition generalizes to multivariate functions by requiring GSC along every straight line. Specifically, let f : R n → (−∞, +∞] be a closed convex, lower semi-continuous function with effective domain dom f which is an open nonempty subset of R n . For x ∈ dom f and u, v ∈ R n , define the real-valued function ϕ(t) := ∇ 2 f (x + tv)u, u . For t ∈ dom ϕ, one sees that φ (t) = D 3 f (x + tv)[v]u, u , where D 3 f (x)[v] denotes the third-derivative tensor at (x, v), viewed as a bilinear mapping R n × R n → R. The Hessian of the function f defines a semi-norm u x u, u ∇ 2 f (x) for all x ∈ dom f, with dual norm a * x sup d∈R n {2 d, a − d 2 x }. If ∇ 2 f (x) ∈ S n ++ then · x is a true norm, and d * x = d, d [∇ 2 f (x)] −1 .≥ 0, if for all x ∈ dom f \| D 3 f (x)[v]u, u \| ≤ M f u 2 x v ν−2 x v 3−ν 2 ∀u, v ∈ R n . (2.2) We denote this class of functions as F M f ,ν . In the extreme case ν = 2 we recover the definition \| D 3 f (x)[v]u, u \| ≤ M f u 2 x v 2 , which is the generalized self-concordance definition proposed by Bach [1]. If ν = 3 and u = v the definition becomes \| D 3 f (x)[u]u, u \| ≤ M f u 3 x , which is the standard self-concordance definition due to [33]. Given ν ∈ [2, 3] and f ∈ F M f ,ν , we define the distance-like function d ν (x, y) M f y − x 2 if ν = 2, ν−2 2 M f y − x 3−ν 2 · y − x ν−2 x if ν ∈ (2, 3],(2.3) and the Dikin Ellipsoid W(x; r) {y ∈ R n : d ν (x, y) < r} ∀(x, r) ∈ dom f × R. (2.4) Since f ∈ F M f ,} n∈N ⊂ dom f with dist x n , bd(dom f ) → 0 we have f (x n ) → ∞. This fact allows us to use the Dikin Ellipsoid as a safeguard region within which we can perturb the current position x without falling off dom f . Lemma 2.3 ([44], Prop. 7). Let f ∈ F M f ,ν with ν ∈ (2, 3]. We have W(x; 1) ⊂ dom f for all x ∈ dom f . The inclusion W(x; 1) ⊂ dom f for ν ∈ (2, 3] is a generalization of a well-known classical property of self-concordant functions [33]. It gains relevance for the case ν > 2, since when ν = 2, we have dom f = R n , making the statement trivial. The next Lemma gives a-priori local bounds on the function values. Lemma 2.4 ([44], Prop. 10). Let x, y ∈ dom f for f ∈ F M f ,ν and ν ∈ [2,3]. Then f (y) ≥ f (x) + ∇ f (x), y − x + ω ν (−d ν (x, y)) y − x 2 x , and (2.5) f (y) ≤ f (x) + ∇ f (x), y − x + ω ν (d ν (x, y)) y − x 2 x , (2.6) where, if ν > 2, the right-hand side of (2.6) holds if and only if d ν (x, y) < 1. Here ω ν (·) is defined as ω ν (t)              1 t 2 (e t − t − 1) if ν = 2, −t−ln(1−t) t 2 if ν = 3, ν−2 4−ν 1 t ν−2 2(3−ν)t ((1 − t) 2(3−ν) 2−ν − 1) − 1 if ν ∈ (2, 3). (2.7) The function ω ν (·) is strictly convex and one can check that ω ν (t) ≥ 0 for all t ∈ dom(ω ν ). These bounds on the function values can be seen as local versions of the standard approximations valid for strongly convex functions, respectively for functions with a Lipschitz continuous gradient (see e.g. [35], Def. 2.1.3 and Lemma 1.2.3). In particular, the upper bound (2.6) corresponds to a local version of the celebrated descent lemma, a fundamental tool in the analysis of first-order methods [14]. To emphasize this analogy, we will also refer to (2.6) as the GSC-descent lemma. Algorithm 2: FWGSC Input: x 0 ∈ dom f ∩ X initial state, ε > 0 error tolerance, and f ∈ F M,ν . for k = 0, . . . do if Gap(x k ) > ε then Obtain s k = s(x k ) from (1.1) and define v k = v FW (x k ); Obtain α k = α ν (x k ) from (3.5); Set x k+1 = x k + α k v k end if end for Frank-Wolfe works for generalized self-concordant functions In this section we describe three provably convergent modifications of Algorithm 1, displaying sublinear convergence rates. Preliminaries Assumption 1. The following assumptions shall be in place throughout this paper: • The function f in (P) belongs to the class F M f ,ν with ν ∈ [2,3]. • The solution set X * of (P) is nonempty, with x * ∈ X * representing a solution and f * = f (x * ) the corresponding objective function value. • X is convex compact and the search direction (1.1) can be computed efficiently and accurately. • ∇ 2 f is continuous and positive definite on X ∩ dom f . Define the Frank-Wolfe search direction as v FW (x) s(x) − x. (3.1) We also declare the functions e(x) v FW (x) x and β(x) v FW (x) 2 for all x ∈ dom f. A Frank-Wolfe method with analytical step-size Our first Frank-Wolfe method (Algorithm 2, FWGSC) for minimizing generalized self-concordant functions builds on a new adaptive step-size rule, which we derive from a judicious application of the GSC-descent Lemma (2.6). An attractive feature of this new step size policy is that it is available in analytical form, which allows us to do away with any globalization strategy (e.g. line search). This has significant practical impact when function evaluations are expensive. Given x ∈ X, set x + t x + tv FW (x), and assume that e(x) 0. Moving from the current position x to the point x + t , we know that d ν (x, x + t ) = tM f δ ν (x), where δ ν (x) β(x) if ν = 2, ν−2 2 β(x) 3−ν e(x) ν−2 if ν > 2. (3.2) Choosing t ∈ (0, 1 M f δ ν (x) ), the GSC-descent lemma (2.6) gives us the upper bound f (x + t ) ≤ f (x) + ∇ f (x), x + t − x + ω ν (d ν (x, x + t )) x + t − x 2 x = f (x) + ∇ f (x), x + t − x + ω ν tM f δ ν (x) t 2 e(x) 2 = f (x) − t Gap(x) + ω ν tM f δ ν (x) t 2 e(x) 2 For x ∈ dom f ∩ X, define η x,M,ν : R + → (−∞, +∞] by η x,M,ν (t) Gap(x) t − ω ν (tMδ ν (x)) t 2 e(x) 2 Gap(x) . (3.3) Note that η x,M,ν (t) is strictly concave on dom(η x,M,ν ) ⊆ [0, 1 Mδ ν (x) ] . This leads to the per-iteration change in the objective function value as f (x + t ) − f (x) ≤ −η x,M f ,ν (t) ∀t ∈ (0, 1 M f δ ν (x) ). Since η x,M f ,ν (t) > 0 for t ∈ (0, 1 M f δ ν (x) ), we are ensured that we make progress in reducing the objective function value when choosing a step size within the indicated range. Given the triple (x, M, ν), we search for a value t such that the per-iteration decrease is as big as possible. Hence, we aim to find t ≥ 0 which solves the concave maximization problem sup t≥0 η x,M,ν (t). (3.4) Call t M,ν (x) a solution of this program. Since we have to stay within the feasible set, we cannot simply use the number t M,ν (x) as our step size as it might lead to an infeasible point. Consequently, we propose the truncated step-size α M,ν (x) min 1, t M,ν (x) ∀x ∈ dom f. (3.5) In Section 4 we show that this step-size policy guarantees feasibility and a sufficient decrease. Remark 3.1. We emphasize that the basic step-size rule is derived by identifying a suitable local majorizing model f (x) − η x,M f ,ν (t). Minimization with respect to t aligns the model as close as possible to the effective progress we are making in reducing the objective function value. This upper model holds for all GSC functions with the same characteristic parameter (M f , ν), and thus, our derived step size strategy is universally applicable to all functions within the class F M f ,ν . Therefore, akin to [44,46], the derived adaptive step size policy can be regarded as an optimal choice in the analytic worst-case sense. Backtracking Frank-Wolfe variants Algorithm FWGSC comes with several drawbacks. First, it relies on the minimization of a universal upper model derived from the GSC-descent Lemma. This over-estimation strategy leads to a worst-case performance estimate, relying on various state-dependent quantities, such as the local norm e(x k ), and the GSC parameters (M f , ν). Evaluating the local norm requires the computation of the matrix-vector product between the Hessian ∇ 2 f (x k ), and the FW search direction v FW (x k ). 1 Algorithm 3: FWGSC with backtracking over the Lipschitz parameter (LBTFWGSC) Input: x 0 ∈ dom f ∩ X initial state, f ∈ F M,ν , L −1 > 0 initial Lipschitz estimate, γ u > 1, γ d < 1 fixed scaling parameters for the backtracking routine. for k = 0, . . . do if Gap(x k ) > ε then Obtain s k = s(x k ) and set v k = v FW (x k ) Obtain (α k , L k ) = step L ( f, v k , x k , L k−1 ) Update x k+1 = x k + α k v k end if end for Algorithm 4: Function step L ( f, v, x, L) rrwefawefawetawgawegwefwefw ChooseL ∈ [γ d L, L] α = min{1, Gap(x) L v 2 2 } if x + αv dom f or f (x + αv) > Q L (x, α,L) theñ L ← γ uL α ← min{ Gap(x) L v 2 2 , 1} end if Return α,L The GSC parameter M f is a global quantity, relating the second and third derivative over the entire domain of the function f . Additionally, it restricts the interval of admissible step sizes (0, 1 M f δ ν (x) ). Consequently, a local search for this parameter could lead to larger step-sizes, which may improve the performance. Motivated by these facts, this section presents two backtracking variants of the basic Frank-Wolfe method. Both methods are based on the assumption that we can easily answer the question whether a given candidate search point x belongs to the domain of the function f , or not. Assumption 2 (Domain Oracle). Given a point x, it is easy to decide if x ∈ dom f , or not. Remark 3.2. For many problems such domain oracles are easy to construct. As a concrete example, consider the problem of minimizing the log-barrier function over a compact domain in R n + , which is a standard routine in interior-point methods (e.g. the computation of the analytic center). For this problem, a simple domain oracle is a single pass through all the coordinates of the vector x and checking if each entry is positive. The complexity of such an oracle is linear in the number of variables. Backtracking over the Lipschitz constant Our first backtracking variant of FWGSC preforms a local search over the Lipschitz modulus of the gradient over level sets. This produces a nested sequence of level sets visited by the algorithm successively. This kind of backtracking is inspired by the recent paper [41]. However, our proof is both simpler and much more direct. Consider the quadratic model Q L (x, t, L) f (x) − t Gap(x) + t 2 L 2 v FW (x) 2 2 = f (x) − t Gap(x) + t 2 L 2 β(x) 2 , (3.6) where x ∈ X is the current position of the algorithm, and t, L > 0 are parameters. From the complexity analysis of FWGSC, we know that there exists a range of step-size parameters t > 0 that guarantee decrease in the objective function value. Denote by S(x) {x ∈ X\| f (x ) ≤ f (x)}, and set γ k sup{t > 0\|x k + t(s k − x k ) ∈ S(x k )} as well as L k max x∈S(x k ) λ 2 max (∇ 2 f (x)). Then, for all t ∈ [0, γ k ], it holds true that f (x k + t(s k − x k )) ≤ f (x k ). Therefore, by the mean-value-theorem ∇ f (x k + t(s k − x k )) − ∇ f (x k ) ≤ L k t s k − x k 2 ∀t ∈ (0, γ k ). Hence, for all t ∈ (0, γ k ), f (x k + t(s k − x k )) − f (x k ) ≤ −t Gap(x k ) + L k t 2 2 s k − x k 2 2 = Q L (x k , t, L k ) − f (x k ), (3.7) The idea is to dispense with the computation of the local Lipschitz estimate L k over the level set S(x k ), and replace it by the backtracking procedure step L ( f, v k , x k , L k−1 ) (Algorithm 4) as an inner-loop within Algorithm 3 (LBTFWGSC). In particular, using Assumption 2, the implementation of LBTFWGSC does not require the evaluation of the Hessian matrix ∇ 2 f (x k ), and simultaneously determines a step size which minimizes the quadratic model under the prevailing local Lipschitz estimate. Backtracking over the GSC parameter M f Our second backtracking variant performs a local search for the GSC parameter M f . Our goal is to construct a backtracking procedure for the constant M f such that for a given candidate GSC parameter µ > 0 and search point x + t = x + tv FW (x), we have feasibility: x + t ∈ dom f , and sufficient decrease: f (x + t ) ≤ f (x) − t Gap(x) + t 2 e(x) 2 ω ν (tµδ ν (x)) Q M (x, t, µ). (3.8) Optimizing the new upper model Q M (x, t, µ) with respect to t ≥ 0 yields a step-size t µ,ν (x), whose definition is just like the maximizer in (3.4), but using the parameters (x, µ, ν) as input. This approach allows us to define a localized step-size, exploiting the analytic structure of the step-size policy associated with the base algorithm FWGSC. The main merit of this backtracking method can be seen by revisiting the analytical step-size criterion attached with FWGSC, defined in eq. (3.5). It is clear from the definition of the function α M,ν (x) that a larger M cannot lead to a larger step size. Hence, a precise local estimate of the GSC parameter M opens up possibilities to make larger steps and thus improve the practical performance of the method. We will see in our numerical experiments in Section 6 that this claim has some substance in important machine learning problems. Algorithm 5: FWGSC with backtracking over the GSC parameter M f (MBTFWGSC) Input: x 0 ∈ dom f ∩ X initial state, f ∈ F M f ,ν , µ −1 > 0 initial GSC parameter. γ u > 1, γ d < 1 fixed scaling parameters for the backtracking routine. for k = 0, . . . do if Gap(x k ) > ε then Obtain s k = s(x k ) and set v k = v FW (x k ) Obtain (α k , µ k ) = step M ( f, v k , x k , µ k−1 ) Update x k+1 = x k + α k v k end if end for Algorithm 6: Function step M ( f, v, x, µ) asdfasdgjlasdlkfjlasdk ChooseM ∈ [γ d µ, µ] α = αM ,ν (x) defined in (3.5) if x + αv dom f or f (x + αv) > Q M (x, α,M) theñ M ← γ uM α ← αM ,ν (x) end if Return α,M A technical analysis of the optimization problem (3.4), relegated to Appendix B, yields the following explicit expression for t M f ,ν (x). t M f ,ν (x) =                    1 M f δ 2 (x) ln 1 + Gap(x)M f δ 2 (x) e(x) 2 if ν = 2, 1 M f δ ν (x) 1 − 1 + M f δ ν (x) Gap(x) e(x) 2 4−ν ν−2 − ν−2 4−ν if ν ∈ (2, 3), Gap(x) M f δ 3 (x) Gap(x)+e(x) 2 if ν = 3. (4.1) where δ ν (x), ν ∈ [2,3], is defined in eq. (3.2). Next we show that FWGSC is well-defined using the step size policy (3.5). Proposition 4.2. Let {x k } k≥0 be generated by FWGSC with step size policy {α M f ,ν (x k )} k≥0 defined in (3.5). Then x k ∈ X ∩ dom f for all k ≥ 0. Proof. The proof proceeds by induction. By assumption, x 0 ∈ dom f ∩ X. To perform the induction step, assume that x k ∈ X ∩ dom f for some k ≥ 0. We consider two cases. • If ν = 2, then since α M f ,2 (x k ) ≤ 1, feasibility follows immediately from convexity of X (recall that dom f = R n in this case). • If ν ∈ (2, 3], then whenever x k ∈ X, we deduce from (4.1) that t M f ,ν (x k )M f δ ν (x k ) < 1. If t M f ,ν (x k ) > 1, then α M f ,ν (x k )M f δ ν (x k ) = M f δ ν (x k ) < t M f ,ν (x k )M f δ ν (x k ) < 1. The claim then follows thanks to Lemma 2.3. In order to simplify the notation, let us introduce the sequences α k ≡ α M f ,ν (x k ) and ∆ k ≡ η x k ,M f ,ν (α M f ,ν (x k )). Along the sequence {x k } k≥0 , we have d ν (x k , x k+1 ) = M f α k δ ν (x k ) < 1, and we know that we reduce the objective function value by at least the quantity ∆ k > 0. Whence, Proof. S(x 0 ) ⊆ X and therefore it is bounded. Moreover, since x 0 ∈ dom f ∩X, f is closed and convex and X is also closed. S(x 0 ) is closed as the intersection of two closed sets, and therefore compact. f (x k+1 ) ≤ f (x k ) − ∆ k < f (x k ), (4.2) so that f (x k ) ≤ f (x 0 ), or equivalently, {x k } k≥0 ⊂ S(x 0 ) {x ∈ dom f ∩ X\| f (x) ≤ f (x 0 )}. Accordingly, S(x 0 ) ⊂ dom( f ) and the numbers L ∇ f max x∈S(x 0 ) λ max (∇ 2 f (x)) and σ f min x∈S(x 0 ) λ min (∇ 2 f (x)) are well defined and finite. Furthermore, since the level set S(x 0 ) is compact, Assumption 1 guarantees ∇ 2 f (x) 0 for all x ∈ S(x 0 ), and hence σ f > 0. By [35,Thm.2.1.11], for any x ∈ S(x 0 ) it holds that f (x) − f * ≥ σ f 2 x − x * 2 2 . (4.3) Proposition 4.4 below shows asymptotic convergence to a solution along subsequences. We omit the proof, as it follows from [13]. (a) { f (x k )} k≥0 is non-increasing; (b) k≥0 ∆ k < ∞, and hence the sequence {∆ k } k≥0 converges to 0; (c) For all K ≥ 1 we have min 0≤k<K ∆ k ≤ 1 K ( f (x 0 ) − f * ). In order to assess the iteration complexity of FWGSC, we need a lower bound on the sequence {∆ k } k≥0 . We start with a bound at iterations satisfying t M f ,ν (x k ) > 1. Lemma 4.5. If t M f ,ν (x k ) > 1, we have ∆ k ≥ 1 2 Gap(x k ). Proof. See Appendix C.1. Next, we turn to iterates for which t M f ,ν (x k ) ≤ 1. In this case, the per-iteration progress reads as ∆ k = η x k ,M f ,ν (t M f ,ν (x k )), and enjoys the following lower bound: 4.1. It can be checked that lim ν→3γν = 1 − ln (2), so that the lower bound∆ k is continuous in the parameter range ν ∈ (2, 3]. Lemma 4.6. If t M f ,ν (x k ) ≤ 1, we have ∆ k ≥∆ k                      2 ln(2)−1 diam(X) min Gap(x k ) M f , Gap(x k ) 2 diam(X)L ∇ f if ν = 2, γ ν diam(X) min Gap(x k ) ( ν 2 −1)M f L (ν−2)/2 ∇ f , −1 b Gap(x k ) 2 L ∇ f diam(X) if ν ∈ (2, 3), 2(1−ln(2)) √ L ∇ f diam(X) min Gap(x k ) M f , Gap(x k ) 2 √ L ∇ f diam(X) if ν = 3. Combining Lemma 4.5 together with Lemma 4.6 and estimates summarized in Appendix C.2, we get the next fundamental relation. Proposition 4.7. Suppose Assumption 1 holds. Let {x k } k≥0 be generated by FWGSC. Then, for all k ≥ 0, we have Proof. We only illustrate the lower bound for the case ν = 2. All other claims can be verified in exactly the same way. From Lemma 4.5, we know that ∆ k ≥ 1 2 Gap(x k ) whenever t M f ,2 (x k ) > 1. Moreover, from Lemma 4.6 we have that t M f ,2 (x k ) ≤ 1, then ∆ k ≥ 2 ln 2−1 diam(X) min ∆ k ≥ min{c 1 (M f , ν) Gap(x k ), c 2 (M f , ν) Gap(x k ) 2 }, where, for (M, ν) ∈ (0, ∞) × [2, 3], we define c 1 (M, ν)                    min 1 2 , 2 ln(2)−1 M diam(X) if ν = 2, min 1 2 ,γ ν diam(X)(ν/2−1)ML (ν−2)/2 ∇ f if ν ∈ (2, 3),min 1 2 , 2(1−ln 2) M √ L ∇ f diam(X) if ν = 3.             2 ln(2)−1 L ∇ f diam(X) 2 if ν = 2, −1 bγ ν diam(X) 2 L ∇ f if ν ∈ (2, 3), 2(1−ln 2) L ∇ f diam(X) 2 if ν = 3.Gap(x k ) M f , Gap(x k ) 2 diam(X)L ∇ f . Consequently, ∆ k ≥ min min 1 2 , 2 ln(2) − 1 M f diam(X) Gap(x k ), 2 ln(2) − 1 diam(X) 2 L ∇ f Gap(x k ) 2 . With the help of the lower bound in Proposition 4.7, we are now able to establish the O(1/ε) convergence rate in terms of the approximation error h k f (x k ) − f * . Theorem 4.8. Suppose that Assumption 1 holds. Let {x k } k≥0 be generated by FWGSC. For x 0 ∈ X ∩ dom f and ε > 0, define N ε (x 0 ) inf{k ≥ 0\|h k ≤ ε}. Then, for all ε > 0, N ε (x 0 ) ≤ ln c 1 (M f ,ν) h 0 c 2 (M f ,ν) ln(1 − c 1 (M f , ν)) + 1 c 2 (M f , ν)ε . (4.7) Proof. To simplify the notation, let us set c 1 ≡ c 1 (M f , ν) and c 2 ≡ c 2 (M f , ν). By convexity, we have Gap(x k ) ≥ h k . Therefore, Proposition 4.7 shows that ∆ k ≥ min{c 1 h k , c 2 h 2 k }. This implies h k+1 ≤ h k − min{c 1 h k , c 2 h 2 k } ∀k ≥ 0. From this inequality we see that h k is decreasing and there are two potential phases of convergence: Phase I. c 1 h k < c 2 h 2 k , which is equivalent to h k > c 1 c 2 . Phase II. c 1 h k ≥ c 2 h 2 k , which is equivalent to h k ≤ c 1 c 2 . For fixed initial condition x 0 ∈ dom f ∩ X, we can thus subdivide the time domain into the set K 1 (x 0 ) {k ≥ 0\|h k > c 1 c 2 } (Phase I) and K 2 (x 0 ) {k ≥ 0\|h k ≤ c 1 c 2 } (Phase II). Since {h k } k∈K 1 (x 0 ) is decreasing and bounded from below by the positive constant c 1 /c 2 , the set K 1 (x 0 ) is bounded. Let us set T 1 (x 0 ) inf{k ≥ 0\|h k ≤ c 1 c 2 }, (4.8) the first time at which the process {h k } k enters Phase II. To get a worst-case estimate on this quantity, we assume without loss of generality that 0 ∈ K 1 (x 0 ), so that K 1 (x 0 ) = {0, 1, . . . , T 1 (x 0 ) − 1}. Then, for all k = 1, . . . , T 1 (x 0 ) − 1 we have c 1 c 2 < h k ≤ h k−1 − min{c 1 h k−1 , c 2 h 2 k−1 } = h k−1 − c 1 h k−1 . Note that c 1 ≤ 1/2, so we make progressions like a geometric series, i.e. we have linear convergence in this phase. Hence, h k ≤ (1 − c 1 ) k h 0 for all k = 0, . . . , T 1 (x 0 ) − 1. By definition h T 1 (x 0 )−1 > c 1 c 2 , so we get c 1 c 2 ≤ h 0 (1 − c 1 ) T 1 (x 0 )−1 iff (T 1 (x 0 ) − 1) ln(1 − c 1 ) ≥ ln c 1 h 0 c 2 . Hence, T 1 (x 0 ) ≤ ln c 1 h 0 c 2 ln(1 − c 1 ) + 1. (4.9) After these number of iterations, the process will enter Phase II, at which h k ≤ c 1 c 2 holds. Therefore, h k ≥ h k+1 + c 2 h 2 k , or equivalently, 1 h k+1 ≥ 1 h k + c 2 h k h k+1 ≥ 1 h k + c 2 . (4.10) Pick N > T 1 (x 0 ) an arbitrary integer. Summing (4.10) from k = T 1 (x 0 ) up to k = N − 1, we arrive at 1 h N ≥ 1 h T 1 (x 0 ) + c 2 (N − T 1 (x 0 ) + 1). By definition h T 1 (x 0 ) ≤ c 1 c 2 , so that for all N > T 1 (x 0 ), we see 1 h N ≥ c 2 c 1 + c 2 (N − T 1 (x 0 ) + 1). Consequently, h N ≤ 1 c 2 c 1 + c 2 (N − T 1 (x 0 ) + 1) ≤ 1 c 2 (N − T 1 (x 0 ) + 1) . (4.11) By definition of the stopping time N ε (x 0 ), it is true that h N ε (x 0 )−1 > ε. Consequently, evaluating (4.11) at N = N ε (x 0 ) − 1, we obtain ε ≤ 1 c 2 (N ε (x 0 ) − T 1 (x 0 )) ⇔ N ε (x 0 ) ≤ T 1 (x 0 ) + 1 c 2 ε . Combining this upper bound with (4.9) shows the claim. and c 2 (M, ν) in (4.6), we can see that, neglecting the logarithmic terms and using that − 1 ln(1−x) ≤ 1 x for x ∈ [0, 1], the iteration complexity of FWGSC can be bounded as max c 1 , c 2 M f L (ν−2)/2 ∇ f diam(X) + c 3 L ∇ f diam(X) 2 ε ,(4.12) where c 1 , c 2 , c 3 are numerical constants. The first term corresponds to Phase I where one observes the linear convergence, the second term corresponds to the Phase II with sublinear convergence. Interestingly, the second term has the same form as the standard complexity bound for FW methods. The only difference is that the global Lipschitz constant of the gradient is changed to the Lipschitz constant over the level set defined by the starting point. Complexity Analysis of Backtracking versions The complexity analysis of both backtracking-based algorithms (LBTFWGSC and MBTFWGSC) use similar ideas, which all essentially rest on the specific form of the employed upper model Q L and Q M , respectively. We will first derive a uniform bound on the per-iteration decrease of the objective function value, and then deduce the complexity analysis from Theorem 4.8. In both algorithms we use a generic bound on the backtracking parameter. Lemma 4.9. Let {L k } k∈N be the sequence of Lipschitz estimates produced by procedure step L ( f, v k , x k , L k−1 ) and {µ k } k∈N the sequence of GSC-parameter estimates produced by step M ( f, v k , x k , µ k−1 ), respectively. We have L k ≤ max{L −1 , γ u L ∇ f } and µ k ≤ max{µ −1 , γ u M f }. Proof. We proof the statement only for the sequence {L k } k . The claim for {µ k } k∈N can be shown in the same way. By construction of the backtracking procedure we know that if the sufficient decrease condition is evaluated successfully at the first run, then L k−1 ≥ L k ≥ γ d L k−1 . If not, then it is clear that L k ≤ γ d L ∇ f . Hence, for all k ≥ 0, L k ≤ max{γ d L ∇ f , L k−1 }. By backwards induction, it follows then L k ≤ max{L −1 , γ u L ∇ f }. Analysis of LBTFWGSC Calling Algorithm LBTFWGSC at position x k generates a step size α k and a local Lipschitz estimate L k via (α k , L k ) = step L ( f, v FW (x k ), x k , L k−1 ). The thus produced new search point satisfies x k+1 = x k + α k v k ∈ dom f ∩ X, and f (x k+1 ) ≤ f (x k ) − α k Gap(x k ) + L k α 2 k 2 β 2 k where β k ≡ β(x k ). The reported step size is α k = min 1, Gap(x k ) L k β 2 k . For each of these possible realizations of this step size, we will provide a lower bound of the achieved reduction in the objective function value. Case 1: If α k = 1, then L k β 2 k ≤ Gap(x k ) and x k+1 = x k + v k ∈ dom f ∩ X. Hence, f (x k+1 ) ≤ f (x k ) − Gap(x k ) + L k 2 β 2 k ≤ f (x k ) − Gap(x k ) 2 . Case 2: If α k = Gap(x k ) L k β 2 k , then f (x k+1 ) ≤ f (x k ) − Gap(x k ) 2 2L k β 2 k . Since L k ≤ max{γ u L ∇ f , L −1 } ≡L (Lemma 4.9), we obtain the performance guarantee f (x k ) − f (x k+1 ) ≥ min        Gap(x k ) 2 , Gap(x k ) 2 2L k β 2 k        ≥ min Gap(x k ) 2 , Gap(x k ) 2 2L diam(X) 2 . Set c 1 ≡ 1 2 and c 2 ≡ 1 2L diam(X) 2 , it therefore follows that f (x k ) − f (x k+1 ) ≥ min c 1 Gap(x k ), c 2 Gap(x k ) 2 . In terms of the approximation error, this implies h k − h k+1 ≥ min{c 1 h k , c 2 h 2 k }. Thus, we can use a similar analysis as in the one in the proof of Theorem 4.8, and obtain the following O(1/ε) iteration complexity guarantee for method LBTFWGSC. Theorem 4.10. Suppose that Assumptions 1 and 2 hold. Let {x k } k≥0 be generated by LBTFWGSC. For x 0 ∈ X ∩ dom f and ε > 0, define N ε (x 0 ) inf{k ≥ 0\|h k ≤ ε}. Then, for all ε > 0, N ε (x 0 ) ≤ ln(L diam(X) 2 /h 0 ) ln(1/2) + 2L diam(X) 2 ε , (4.13) whereL = max{γ u L ∇ f , L −1 }. Analysis of MBTFWGSC The complexity analysis of this algorithm is completely analogous to the one corresponding to Algorithm LBTFWGSC. The main difference between the two variants is the upper model employed in the local search. Calling MBTFWGSC at position x k , generates the pair (α k , µ k ) = step M ( f, v FW (x k ), x k , µ k−1 ) such that f (x k+1 ) ≤ f (x k ) − α k Gap(x k ) + α 2 k e 2 k ω ν (µ k α k δ ν (x k )), where e k ≡ e(x k ). The step size parameter α k satisfies α k = min{1, t µ k ,ν (x k )}. We can thus apply Proposition 4.7 in order to obtain the recursion h k+1 ≤ h k − min{c 1 (µ k , ν)h k , c 2 (µ k , ν)h 2 k }, involving the constants defined in (4.5) and (4.6). By construction of the backtracking step, we know that µ k ≤ max{γ u M f , µ −1 } ≡M (Lemma 4.9). Hence, after setting c 1 ≡ c 1 (M, ν), c 2 ≡ c 2 (M, ν), we arrive at h k+1 ≤ h k − min{c 1 h k , c 2 h 2 k } ∀k ≥ 0. From here the complexity analysis proceeds as in Theorem 4.8. The only change that has to be made is to replace the expressions c 1 (M f , ν) and c 2 (M f , ν) by the numbers c 1 (M, ν) and c 2 (M, ν), respectively. Theorem 4.11. Suppose that Assumption 1 and 2 hold. Let {x k } k≥0 be generated by MBTFWGSC. For x 0 ∈ X ∩ dom f and ε > 0, define N ε (x 0 ) inf{k ≥ 0\|h k ≤ ε}. Then, for all ε > 0, N ε (x 0 ) ≤ ln c 1 (M,ν) h 0 c 2 (M,ν) ln(1 − c 1 (M, ν)) + 1 c 2 (M, ν)ε , (4.14) whereM = max{γ u M f , µ −1 }. Note that a similar remark to Remark 4.2 can be made in this case. Algorithm 7: FWLLOO Input: A(x, r, c)-LLOO with parameter ρ ≥ 1 for polytope X, f ∈ F M f ,ν . σ f > 0 convexity parameter. x 0 ∈ dom f ∩ X, and let h 0 = f (x 0 ) − f * , and c 0 = 1. r 0 = 2 Gap(x 0 ) σ f for k = 0, 1, . . . do if Gap(x k ) > ε then Set r 2 k = r 2 0 c k ; Obtain u k = u(x k , r k , ∇ f (x k )) by querying procedure A(x k , r k , ∇ f (x k )); Set α k = α ν (x k ) by evaluating (5.5); Set x k+1 = x k + α k (u k − x k ); Set c k+1 = c k exp(− 1 2 α k ). end if end for Linearly convergent variants of Frank-Wolfe for GSC functions In the development of all our linearly convergent variants, we assume that the feasible set is a polytope described by a system of linear inequalities. Assumption 3. The feasible set X admits the explicit representation X {x ∈ R n \|Bx ≤ b}, (5.1) where B ∈ R m×n and b ∈ R m . Local Linear Minimization Oracles In this section we show how the local linear minimization oracle of [18] can be adapted to accelerate the convergence of FW-methods for minimizing GSC functions. In particular, we work out an analytic step-size criterion which guarantees linear convergence towards a solution of (P). The construction is a non-trivial modification of [18], as it exploits the local descent properties of GSC functions. In particular, we neither assume global Lipschitz continuity, nor strong convexity of the objective function. Instead, our working assumption in this section is the availability of a local linear minimization oracle, defined as follows: Definition 5.1 ([18], Def. 2.5). A procedure A(x, r, c), where x ∈ X, r > 0, c ∈ R n , is a Local Linear minimization oracle (LLOO) with parameter ρ ≥ 1 for the polytope X if A(x, r, c) returns a point u(x, r, c) = u ∈ X such that ∀y ∈ B(x, r) ∩ X : c, y ≥ c, u , and x − u 2 ≤ ρr. (5.2) We refer to [18] for illustrative examples for oracles A(x, r, c). In particular, [18] provide an explicit construction of the LLOO for a simplex and for general polytopes. We further redefine the local norm as e(x) u(x, r, ∇ f (x)) − x x ∀x ∈ dom f. With an obvious abuse of notation, we also redefine δ ν (x) u(x, r, ∇ f (x)) − x 2 if ν = 2, ν−2 2 u(x, r, ∇ f (x)) − x 3−ν 2 u(x, r, ∇ f (x)) − x ν−2 x if ν ∈ (2, 3]. (5.3) As in the previous sections, our goal is to come up with a step-size policy guaranteeing feasibility and a sufficient decrease. As will become clear in a moment, our construction relies on a careful analysis of the function ψ ν (t) t − ξω ν (tδ)t 2 t ∈ [0, 1/δ), where ξ, δ ≥ 0 are free parameters. This function is also used in the complexity analysis of FWGSC, and thoroughly discussed in Appendix B. In particular, the analysis in Appendix B shows that t → ψ ν (t) is concave, unimodal with ψ ν (0) = 0, increasing on the interval [0, t * ν ) and decreasing on [t * ν , ∞), where the cut-off value t * ν is defined in eq. (B.2). Moreover, ψ ν (t) ≥ 0 for t ∈ [0, t * ν ]. To facilitate the discussion, let us redefine this cut-off value in a way which emphasizes its dependence on structural parameters. We call t * ν = t * ν (δ, ξ)                1 δ ln 1 + δ ξ if ν = 2, 1 δ 1 − 1 + δ ξ 4−ν ν−2 − ν−2 4−ν if ν ∈ (2, 3), 1 δ+ξ if ν = 3. (5.4) We construct our step size policy iteratively. Suppose we are given the current iterate x k ∈ dom f ∩ X, produced by k sequential calls of FWLLOO, using a finite sequence {α i } k−1 i=0 of stepsizes and search radii {r i } k−1 i=0 . Set c k = exp − k−1 i=0 α i . Call the LLOO to obtain the target state u k = u(x k , r k , ∇ f (x k )), using the updated search radius r k = r 0 c k . We define the next step size α k = α ν (x k ) by setting α ν (x k ) min 1, t * ν M f δ ν (x k ), 2e(x k ) 2 Gap(x 0 )c k . (5.5) Update the sequence of search points to x k+1 = x k +α k (u k −x k ). By construction of t k ν ≡ t * ν M f δ ν (x k ), 2e(x k ) 2 Gap(x 0 )c k , this point lies in dom f ∩ X. To see this, consider first the case in which α k = 1 < t k ν . Then, d ν (x k+1 , x k ) = α k M f δ ν (x k ) = M f δ ν (x k ) < t k ν M f δ ν (x k ) < 1. On the other hand, if α k = t k ν , then it follows from the definition of the involved quantities that d ν (x k+1 , x k ) = α k M f δ ν (x k ) < 1. Repeating this procedure iteratively yields a sequence {x k } k∈N , whose performance guarantees in terms of the approximation error h k = f (x k ) − f * are described in the Theorem below. Theorem 5.2. Suppose Assumption 1 holds. Let {x k } k≥0 be generated by FWLLOO. Then, for all k ≥ 0, we have x * ∈ B(x k , r k ) and h k ≤ Gap(x 0 ) exp        − 1 2 k−1 i=0 α i        (5.6) where the sequence {α k } k is constructed as in (5.5). Proof. Let us define P(x 0 ) x ∈ X\| f (x) ≤ f * + Gap(x 0 ) . We proceed by induction. For k = 0, we have x 0 ∈ dom f ∩ X by assumption and x 0 ∈ P(x 0 ) by definition. (4.3) gives f (x 0 ) − f * = h 0 ≥ σ f 2 x 0 − x * 2 2 . (5.7) Let u 0 ≡ u(x 0 , r 0 , ∇ f (x 0 )), δ 0 ≡ δ ν (x 0 ), ξ 0 = 2e(x 0 ) 2 Gap(x 0 ) and α 0 = α ν (x 0 ) obtained by evaluating (5.5) with the cut-off value t * ν (M f δ 0 , ξ 0 ). Since r 0 = 2 Gap(x 0 ) σ f ≥ 2h 0 σ f , (5.7) implies that x * ∈ B(x 0 , r 0 ). The definition of the LLOO gives us ∇ f (x 0 ), u 0 − x 0 ≤ ∇ f (x 0 ), x * − x 0 . (5.8) Set x 1 = x 0 + α 0 (u 0 − x 0 ) ∈ dom f ∩ X. The GSC-descent lemma (2.6) gives then f (x 1 ) ≤ f (x 0 ) + α 0 ∇ f (x 0 ), u 0 − x 0 + α 2 0 e(x 0 ) 2 ω ν (α 0 M f δ 0 ) (5.8) ≤ f (x 0 ) + α 0 ∇ f (x 0 ), x * − x 0 + α 2 0 e(x 0 ) 2 ω ν (α 0 M f δ 0 ) ≤ f (x 0 ) + α 0 ( f * − f (x 0 )) + α 2 0 e(x 0 ) 2 ω ν (α 0 M f δ 0 ) Hence, writing the above in terms of the approximation error h k = f (x k ) − f * , we obtain h 1 ≤ h 0 (1 − α 0 ) + α 2 0 e(x 0 ) 2 ω ν (α 0 M f δ 0 ) ≤ (1 − α 0 ) Gap(x 0 ) + α 2 0 e(x 0 ) 2 ω ν (α 0 M f δ 0 ) = 1 − α 0 2 Gap(x 0 ) − Gap(x 0 ) 2 α 0 − α 2 0 2e(x 0 ) 2 Gap(x 0 ) ω ν (α 0 M f δ 0 ) . We see that the second summand in the right-hand side above is just the value of the function ψ ν (α 0 ), with the parameters δ = M f δ 0 and ξ = ξ 0 = 2e(x 0 ) 2 Gap(x 0 ) . Hence, by construction, the second summand is nonnegative, which gives us the bound h 1 ≤ (1 − α 0 2 ) Gap(x 0 ) ≤ exp(−α 0 /2) Gap(x 0 ). To perform the induction step, assume that for some k ≥ 1 it holds h k ≤ Gap(x 0 )c k , c k exp        − 1 2 k−1 i=0 α i        . (5.9) Since c k ∈ (0, 1), we readily see that x k ∈ P(x 0 ). Call δ k = δ ν (x k ) and ξ k = 2e(x k ) 2 Gap(x 0 )c k . (4.3) leads to x k − x * 2 2 ≤ 2h k σ f ≤ 2 Gap(x 0 ) σ f c k = r 2 0 c k ≡ r 2 k ⇒ x * ∈ B(x k , r k ). (5.10) Call the LLOO to obtain the target point u k = A(x k , r k , ∇ f (x k )). Using the definition of the LLOO, (5.10) implies ∇ f (x k ), u k − x k ≤ ∇ f (x k ), x * − x k . (5.11) Define the step size α k = α ν (x k ), and declare the next search point x k+1 = x k + α k (u k − x k ) ∈ dom f ∩ X. By the discussion preceeding the Theorem, it is clear that x k+1 ∈ X ∩ dom f . Via the GSC-descent lemma and the induction hypothesis we arrive in exactly the same way as for the case k = 0 to the inequality h k+1 ≤ 1 − α k 2 Gap(x 0 )c k − Gap(x 0 )c k 2 α k − α 2 k 2e(x k ) 2 Gap(x 0 )c k ω ν (α k M f δ k ) . The construction of the step size α k ensures that the expression in the brackets on the right-handside is non-negative. Consequently, we obtain h k+1 ≤ (1 − α k /2) Gap(x 0 )c k ≤ Gap(x 0 )c k exp(−α k /2) = Gap(x 0 )c k+1 , which finishes the induction proof. To obtain the final linear convergence rate, it remains to lower bound the step size sequence α k = α ν (x k ). Note that for all values ν ∈ [2, 3], t * ν (δ, ξ) is an increasing function of 1 δ and δ ξ . Thus, our next steps are to lower bound the values of the non-negative sequences { 1 M f δ k } k and { M f δ k ξ k } k , where δ k = δ ν (x k ) and ξ k = 2e(x k ) 2 Gap(x 0 )c k for all k ≥ 0. We have 1 M f δ k =          1 M f u k −x k 2 if ν = 2, 1 ν−2 2 M f u k −x k 3−ν 2 u k −x k ν−2 x k if ν ∈ (2, 3]. By definition of the LLOO, we have u k − x k 2 ≤ min{ρr k , diam(X)}. Thus, if ν = 2, we have 1 M f δ k ≥ 1 M f min{ρr k , diam(X)} ≥ 1 M f ρr k , while if ν > 2, we observe 1 M f δ k ≥ 1 ν−2 2 M f u k − x k 3−ν 2 L ν−2 2 ∇ f u k − x k ν−2 2 = 1 ν−2 2 M f L ν−2 2 ∇ f u k − x k 2 ≥ 1 ν−2 2 M f L ν−2 2 ∇ f min{ρr k , diam(X)} ≥ 1 ν−2 2 M f L ν−2 2 ∇ f ρr k . Furthermore, from the identity 2 Gap(x 0 )c k σ f = r 2 k , we conclude Gap(x 0 )c k = σ f r 2 k 2 . Hence, M f δ k ξ k = M f δ ν (x k ) Gap(x 0 )c k 2e(x k ) 2 =              M f u k −x k 2 σ f r 2 k 2 2 u k −x k 2 x k if ν = 2, ν−2 2 M f u k −x k 3−ν 2 e(x k ) ν−2 σ f r 2 k 2 2e(x k ) 2 if ν ∈ (2, 3]. If ν = 2, we see that M f δ k ξ k ≥ M f u k − x k 2 σ f r 2 k 4L ∇ f u k − x k 2 2 = M f σ f r 2 k 4L ∇ f u k − x k 2 ≥ M f σ f r 2 k 4L ∇ f min{ρr k , diam(X)} ≥ M f σ f r k 4ρL ∇ f , while if ν > 2, we have in turn M f δ k ξ k = (ν − 2)M f u k − x k 3−ν 2 σ f r 2 k 8e(x k ) 4−ν ≥ (ν − 2)M f u k − x k 3−ν 2 σ f r 2 k 8L 4−ν 2 ∇ f u k − x k 4−ν 2 = (ν − 2)M f σ f r 2 k 8L 4−ν 2 ∇ f u k − x k 2 ≥ (ν − 2)M f σ f r 2 k 8L 4−ν 2 ∇ f min{ρr k , diam(X)} ≥ (ν − 2)M f σ f r k 8ρL 4−ν 2 ∇ f = (ν − 2)M f L ν−2 2 ∇ f σ f r k 8ρL ∇ f . Denoting γ ν = ν−2 2 M f L ν−2 2 ∇ f for ν > 2 and γ ν = M f for ν = 2, and substituting these lower bounds to the expression for t * ν , we obtain t k ν ≡ t * ν M f δ ν (x k ), 2e(x k ) 2 Gap(x k )c k ≥ t k                      1 γ ν ρr k ln 1 + γ ν σ f r k 4ρL ∇ f if ν = 2, 1 γ ν ρr k 1 − 1 + γ ν σ f r k 4ρL ∇ f 4−ν ν−2 − ν−2 4−ν if ν ∈ (2, 3), 1 γ ν ρr k 1 1+ 4ρL ∇ f γνσ f r k if ν = 3. For all ν ∈ [2, 3], the minorizing sequence {t k } k has a limit σ f 4ρ 2 L ∇ f as r k → 0. Moreover,t k ν ≥ t                      1 γ ν ρr 0 ln 1 + γ ν σ f r 0 4ρL ∇ f if ν = 2, 1 γ ν ρr 0 1 − 1 + γ ν σ f r 0 4ρL ∇ f 4−ν ν−2 − ν−2 4−ν if ν ∈ (2, 3) 1 γ ν ρr 0 1 1+ 4ρL ∇ f γνσ f r 0 if ν = 3.h k ≤ Gap(x 0 ) exp(−kᾱ/2) ∀k ≥ 0, whereᾱ = min{t, 1} with t defined in (5.12). Proof. It is clear that α k ≥ᾱ = min{t, 1} for all k ≥ 0. Hence exp − 1 2 k−1 i=0 α i ≤ exp(−kᾱ/2) , and the claim follows. The obtained bound can be quite conservative since we used a uniform bound for the sequence t k . At the same time, since r k geometrically converges to 0 and for all ν ∈ [2, 3], the minorizing sequence {t k } k has a limit σ f 4ρ 2 L ∇ f as r k → 0, we may expect that after some burn-in phase, the sequence α k can be bounded from below by σ f 8ρ 2 L ∇ f . This lower bound leads to the linear convergence as h k ≤ Gap(x 0 ) exp(−k 0ᾱ /2)) exp(−(k − k 0 ) σ f 16ρ 2 L ∇ f ) for k ≥ k 0 , where the length of the burn-in phase k 0 is up to logarithmic factors equal to 1 α . This corresponds to the iteration complexity k 0 + 16ρ 2 L ∇ f σ f ln Gap(x 0 ) exp(−k 0ᾱ /2)) ε . Interestingly, the second term has the same form as the complexity bound for FW method under the LLOO proved in [18] with ρ 2 L ∇ f σ f playing the role of condition number. The only difference is that the global Lipschitz constant of the gradient is changed to the Lipschitz constant over the level set defined by the starting point. Away-Step Frank-Wolfe (ASFW) We start with some preparatory remarks. Recall that in this section Assumption 3 is in place. Hence, X is a polytope of the form (5.1). By compactness and the Krein-Milman theorem, we know that X is the convex hull of finitely many vertices (extreme points) U {u 1 , . . . , u q }. Let ∆(U) denote the set of discrete measures µ (µ u : u ∈ U) with µ u ≥ 0 for all u ∈ U and u∈U µ u = 1, µ u ≥ 0. A measure µ x ∈ ∆(U) is a vertex representation of x if x = u∈U µ x u u. Given µ ∈ ∆(U), we define supp(µ) {u ∈ U\|µ u > 0} and the set of active vertices U(x) {u ∈ U\|u ∈ supp(µ x )} of point x ∈ X under the vertex representation µ x ∈ ∆(U). We use I(x) {i ∈ {1, . . . , m}\|B i x = b i } to denote the set of binding constraints at x. For a given set V ⊂ U, we let I(V) = u∈V I(u). For the linear minimization oracle generating the target point s(x), we invoke an explicit tie-breaking rule in the definition of the linear minimization oracle. Assumption 4. The linear minimization procedure s(x) ∈ argmin d∈X ∇ f (x), d returns a vertex solution, i.e. s(x) ∈ U for all x ∈ X. Remark 5.1. [4] refer to this as a vertex linear oracle. ASFW needs also a target vertex which is as much aligned as possible with the same direction of the gradient vector at the current position x. Such a target vertex is defined as u(x) ∈ argmax u∈U(x) ∇ f (x), u (5.13) At each iteration, we assume that the iterate x k is represented as a convex combination of active vertices x k = u∈U µ k u u, where µ k ∈ ∆(U). In this case, the sets U k = U(x k ) and the carrying measure µ k = µ x k provide a compact representation of x k . The ASFW scheme updates the thus described representation (U k , µ k ) via the vertex representation updating (VRU) scheme, as defined in [4]. A single iteration of ASFW can perform two different updating steps: Forward Step: This update is constructed in the same way as FWGSC. Away Step: This is a correction step in which the weight of a single vertex is reduced, or even nullified. Specifically, the away step regime builds on the following ideas: Let x ∈ X be the current position of the algorithm with vertex representation x = u∈U µ x u u. Pick u(x) as in (5.13). Define the away direction v A (x) x − u(x),(5.14) and apply the step size t > 0 to produce the new point x + t = x + tv A (x) = u∈U(x)\{u(x)} (1 + t)µ x u u + µ x u(x) (1 + t) − t u(x). Algorithm 8: ASFWGSC x 0 ∈ dom f ∩ U where µ 1 u = 0 for all u ∈ U \ {x 1 } and U 1 = {x 1 }. for k = 0, 1, . . . do Set s k = s(x k ), u k = u(x k ), and v A (x k ) = x k − u k , v FW (x k ) = s k − x k if ∇ f (x k ), s k − x k ≤ ∇ f (x k ), x k − u k then Set v k = v FW (x k ) else Set v k = v A (x k ) end if Set β k = v k 2 , e k = v k x k ,t k ≡t(x k ) defined in (5.17) Find α k = argmin t∈[0,t k ] t ∇ f (x k ), v k + t 2 e 2 k ω ν (tM f δ ν (x k )) Update x k+1 = x k + α k v k if v k = v FW (x k ) then Update U k+1 = U k ∪ {s k } else if v k = v A (x k ) and α k =t k then Update U k+1 = U k \ {u k } and µ k+1 via the VRU of [4]. else Update U k+1 = U k end if end if end for Choosing t ≡t(x) µ x u(x) 1−µ x u(x) eliminates the vertex u = u(x) from the support of the current point x and leaves us with the new position x + = x + t(x) = u∈U(x)\{u(x)} µ x u 1−µ x u(x) u. This vertex removal is called a drop step. For the complexity analysis of ASFWGSC, we introduce some convenient notation. Define the vector field v : X → R n by v(x) v FW (x) if a Forward Step is performed, v A (x) if an Away Step is performed. (5.15) The modified gap function is G(x) − ∇ f (x), v(x) = max{ ∇ f (x), x − s(x) , ∇ f (x), u(x) − x }. (5.16) One observes that G(x) ≥ 0 for all x ∈ dom f ∩ X. To construct a feasible method, we need to impose bounds on the step-size. To that end, definē t(x) 1 if a Forward Step is performed, µ u(x) 1−µ u(x) if an Away Step is performed, (5.17) where {µ u } u∈U ∈ ∆(U) is a given vertex representation of the current point x, and u(x) is the target state identified under the away-step regime (5.13). The construction of our step size policy is based on an optimization argument, similar to the one used in the construction of FWGSC. In order to avoid unnecessary repetitions, we thus only spell out the main steps. Recall that if d ν (x, x + tv(x)) < 1, then we can apply the generalized self-concordant descent lemma (2.6): f (x + tv(x)) ≤ f (x) + t ∇ f (x), v(x) + t 2 v(x) 2 x ω ν (tM f δ ν (x)), where δ ν (x) is defined as in (3.2), modulo the change β(x) = v(x) 2 and e(x) = v(x) x . Using the modified gap function (5.16), this gives the upper model for the objective function f (x + tv(x)) ≤ f (x) − G(x) t − t 2 e(x) 2 G(x) ω ν (tM f δ ν (x)) , provided that G(x) > 0. This upper model is structurally equivalent to the one employed in the step-size analysis of FWGSC. Hence, to obtain an adaptive step-size rule in Algorithm 8, we solve the concave program max t≥0η x,ν (t) t − t 2 e(x) 2 G(x) ω ν (tM f δ ν (x)). (5.18) As in Section 3.2, and with some deliberate abuse of notation, let us denote the unique solution to this maximization problem by t ν (x) (dependence on M f is suppressed here, since we consider this parameter as given and fixed in this regime). Building on the insights we gained from proving Proposition 4.1, we thus obtain the familiarly looking characterization of the unique maximizer of the concave program (5.18): Theorem 5.4. The unique solution to program (5.18) is given by t ν (x) =                    1 M f δ 2 (x) ln 1 + G(x)M f δ 2 (x) e(x) 2 if ν = 2, 1 M f δ ν (x) 1 − 1 + M f δ ν (x)G(x) e(x) 2 4−ν ν−2 − ν−2 4−ν if ν ∈ (2, 3), G(x) M f δ 3 (x)G(x)+e(x) 2 if ν = 3, (5.19) where δ ν (x) is defined in eq. (3.2), with β(x) = v(x) 2 and e(x) = v(x) x considering the vector field (5.15). Analogously to Proposition 4.2, we see that when applying the step-size policy α ν (x) min{t(x), t ν (x)},(5.20) we can guarantee that x k ∈ X for all k ≥ 0. Indeed, inspecting the expression (5.19) for each value ν ∈ [2, 3], it is easy to see that M f δ ν (x)t ν (x) < 1. Hence, ift(x) ≤ t ν (x), it is immediate thatt(x)M f δ ν (x) < 1. Consequently, x + α ν (x)v(x) ∈ X ∩ dom f for all x ∈ X ∩ dom f . Therefore, the sequence generated by Algorithm 8 is always well defined. In terms of the thus constructed process {x k } k≥0 , we can quantify the per-iteration progress ∆ k ≡η x k ,ν (α k ), setting α k ≡ α ν (x k ), via the following modified version of Lemma 4.6: Lemma 5.5. If t ν (x) ≤t(x), we have ∆ k ≥∆ k                      2 ln(2)−1 diam(X) min G(x k ) M f , G(x k ) 2 diam(X)L ∇ f if ν = 2, γ ν diam(X) min G(x k ) ( ν 2 −1)M f L (ν−2)/2 ∇ f , −1 b G(x k ) 2 L ∇ f diam(X) if ν ∈ (2, 3), 2(1−ln(2)) √ L ∇ f diam(X) min G(x k ) M f , G(x k ) 2 √ L ∇ f diam(X) if ν = 3, (5.21) whereγ ν 1 + 4−ν 2(3−ν) 1 − 2 2(3−ν)/(4−ν) and b 2−ν 4−ν . This means that at each iteration of Algorithm 8 in which α k = t ν (x k ), we succeed in reducing the objective function value by at least f (x k+1 ) ≤ f (x k ) −∆ k . To proceed further with the complexity analysis of ASFWGSC, we need the following technical angle condition, valid for polytope domains: Lemma 5.6 (Corollary 3.1, [4]). For any x ∈ X \ X * with support U(x), we have max u∈U(x),w∈U ∇ f (x), u − w ≥ Ω X \|U(x)\| max x * ∈X * ∇ f (x), x − x * x − x * , (5.22) where ζ min u∈U,i∈{1,...,m}:b i >(Bu) i (b i − B i u), ϕ max i∈{1,...,m}\I(x) B i , and Ω X ζ ϕ . To assess the overall iteration complexity of Algorithm 8 we consider separately the following cases: (a) If the step size regime α k = t ν (x k ) applies, then from Proposition 4.7 we deduce that f ( x k+1 ) − f (x k ) ≤ −∆ k , were ∆ k ≥ min{c 1 (M f , ν)G(x k ), c 2 (M f , ν)G(x k ) 2 }. The multiplicative constants c 1 (M f , ν), c 2 (M f , ν) are the ones defined in (4.5) and (4.6). Hence, f (x k+1 ) − f (x k ) ≤ − min{c 1 (M f , ν)G(x k ), c 2 (M f , ν)G(x k ) 2 }. (b) Else, we apply the step size α k =t k . Then, there are two cases to consider: (b.i) If a Forward Step is applied, then we know thatt k = 1. Since 1 < t ν (x k ), we can apply Lemma 4.5, but now evaluating the functionη x,ν (t) at t = 1, to obtain the bound η x k ,ν (t k ) G(x k ) ≥ 1 2 . This gives the per-iteration progress f (x k+1 ) − f (x k ) ≤ − 1 2 G(x k ). (b.ii) If an Away Step is applied, then we do not have a lower bound ont k . However, we know that f (x k+1 ) − f (x * ) ≤ f (x k ) − f (x * ). As in [4], we know that such drop steps can happen at most half of the iterations. Collecting these cases, we are ready to state and prove the main result of this section. h k ≤ (1 − θ) k/2 h 0 ≤ exp −θ k 2 h 0 . (5.23) where θ min 1 2 , c 1 (M f ,ν)Ω 2 diam(X) , c 2 (M f ,ν)Ω 2 σ f 8 , Ω ≡ Ω X \|U\| . Proof. We say that iteration k is productive if it is either a Forward step or an Away step, which is not a drop step. Based on the estimates developed by inspecting thes cases (a) and (b.i) above, we see that at all productive steps we reduce the objective function value according to f (x k+1 ) − f (x k ) ≤ − min min{ 1 2 , c 1 (M f , ν)}G(x k ), c 2 (M f , ν)G(x k ) 2 . We now develop a uniform bound for this decrease. First, we recall that on the level set S(x 0 ), we have the strong convexity estimate f (x k ) − f * ≥ σ f 2 x k − x * 2 2 . Using Lemma 5.6 and the definition of an Away-Step, we obtain the bound ∇ f (x k ), u k − s k ≥ Ω x k − x * ∇ f (x k ), x k − x * , where Ω ≡ Ω X \|U\| ≤ Ω X \|U(x k )\| . At the same time, ∇ f (x k ), u k − s k = ∇ f (x k ), u k − x k + ∇ f (x k ), x k − s k ≤ 2 max ∇ f (x k ), u k − x k , ∇ f (x k ), x k − s k = 2G(x k ). Consequently, G(x k ) ≥ 1 2 ∇ f (x k ), u k − s k ,(5.24) and G(x k ) ≥ 1 2 ∇ f (x k ), u k − s k ≥ Ω 2 x k − x * ∇ f (x k ), x k − x * ≥ Ω 2 x k − x * ( f (x k ) − f * ) ≥ Ω 2 diam(X) ( f (x k ) − f * ). Furthermore, G(x k ) 2 ≥ Ω 2 4 x k − x * 2 ( f (x k ) − f * ) 2 ≥ Ω 2 4 ( f (x k ) − f * ) 2 2 σ f ( f (x k ) − f * ) = Ω 2 σ f 8 ( f (x k ) − f * ). Hence, in the cases (a) and (b.i), we can lower bound the per-iteration progress in terms of the approximation error h k = f (x k ) − f * as h k+1 − h k ≤ − min        1 2 , c 1 (M f , ν)Ω 2 diam(X) , c 2 (M f , ν)Ω 2 σ f 8        h k ≡ −θh k . Since we are making a full drop step in at most k/2 iterations (recall that we initialize the algorithm from a vertex), we conclude from this that h k ≤ (1 − θ) k/2 h 0 ≤ exp −θ k 2 h 0 . Remark 5.2. We would like to point out that Algorithm ASFWGSC does not need to know the constants σ f , L ∇ f which may be hard to estimate. Moreover, the constants in Lemma 5.6 are also used only in the analysis and are not required to run the algorithm. Compared to [8], our ASFW does not rely on the backtracking line search, but requires to evaluate the Hessian, yet without its inversion. Furthermore, our method does not involve the pyramidal width of the feasible set, which is in general extremely difficult to evaluate. Numerical Results We provide four examples to compare our methods with existing methods in the literature. As competitors we take Algorithm 1, with its specific versions FW-Standard and FW-Line Search. Recall that no general convergence proof for generalized self-concordance functions exists for either method. As further benchmarks, we implement the self-concordant Proximal-Newton (PN) and the Proximal-Gradient (PG) of [44,46], as available in the SCOPT package 2 . All codes are written in Python 3, with packages for scientific computing NumPy 1.18.1 and SciPy 1.4.1. The experiments were conducted on a Intel(R) Xeon(R) Gold 6254 CPU @ 3.10 GHz server with a total of 300 GB RAM and 72 threads, where each method was allowed to run on a maximum of two threads. We ran all first order methods for a maximum 50,000 iterations and PN, which is more computationally expensive, for a maximum of 1,000 iterations. FW-Line Search is run with a tolerance of 10 −10 . In order to ensure that FW-standard generates feasible iterates for ν > 2, we check if the next iterate is inside the domain; If not we replace the step-size by 0, as suggested in [8]. PG was only used in instances where ν = 3 as this method has been developed for standard self-concordant functions only [46]. Within PN we use monotone FISTA [3], with at most 100 iterations and a tolerance of 10 −5 to find the Newton direction. The step size used in PG is determined by the Barzilai-Borwein method [37] with a limit of 100 iterations, similar to [46]. Our comparison is made by the construction of versions of performance profiles [12]. In order to present the result, we first estimate f * by the best function value achieved by any of the algorithms, and compute the relative error attained by each of the methods at iteration k. More precisely, given the set of methods S, test problems P and initial points I, denote by F i jl the function value attained by method j ∈ S on problem i ∈ P starting from starting point l ∈ I . We define the estimate of the optimal value of problem j by f * j = min{F i jl \|j ∈ S, l ∈ I}. Denoting {x k i jl } k the sequence produced by method j on problem i starting from point l, we define the relative error as r k i jl = f (x k ijl )− f * j f * j . Now, for all methods j ∈ S and any relative error ε, we compute the proportion of data sets that achieve a relative error of at most ε (successful instances). We construct this statistic as follow: Let N j denote the maximum allowed number of iterations for method j ∈ S (i.e for first-order methods 50,000 and for PN 1,000). Define I i j (ε) {l ∈ I : ∃k ≤N j , r k i jl ≤ ε}. Then, the proportion of successful instances is ρ j (ε) 1 \|P\|\|I\| i∈P,l∈I \|I i j (ε)\| (average success ratio). We are also interested in comparing the iteration complexity (IC) and CPU time. For that purpose, we define N i jl (ε) min{0 ≤ k ≤N j \|r k i jl ≤ ε} as the first iteration in which method j ∈ S achieves a relative error ε on problem i ∈ P starting from point l ∈ I. Analogously, T i jl (ε) measures the minimal CPU time in which method j ∈ S achieves a relative error ε on problem i ∈ P starting from point l ∈ I. For comparing IC and the CPU time across methods we construct the statistics ρ j (ε) 1 \|P\| i∈P 1 \|I i j \| l∈I ij ( ) N i jl (ε) min{N isl (ε)\|s ∈ S} (average iteration ratio), ρ j (ε) 1 \|P\| i∈P 1 \|I i j \| l∈I ij T i jl (ε) min{T isl (ε)\|s ∈ S} (average time ratio). Besides average performance, we also report the mean and standard deviation of N i jl (ε) and T i jl (ε) across starting points, for specific values of relative error ε for all tested methods and data sets. Logistic regression Starting with [1], the logistic regression problem has been the main motivation from the perspective of statistical theory to analyze self-concordant functions in detail. The objective function involved in this standard classification problem is given by f (x) = 1 p p i=1 ln 1 + exp −y i ( a i , x + µ) + γ 2 x 2 2 . (6.1) Here µ is a given intercept, y i ∈ {−1, 1} is the label attached to the i-th observation, and a i ∈ R n are predictors given as input data for i = 1, 2, . . . , p. The regularization parameter γ > 0 is usually calibrated via cross-validation. The task is to learn a linear hypothesis x ∈ R n . According to [44], we can treat (6.1) as a (M f . This gains relevance, since usually the regularization parameter is negatively correlated with the sample size p. Hence, for p 1, the GSC constant M f could differ by orders of magnitude, which suggests considerable differences in the performance of numerical algorithms. We consider the elastic net formulation of the logistic regression problems, by enforcing sparsity of the estimators via an added 1 penalty. The resulting optimization problem reads as min x∈R n f (x) s.t. x 1 ≤ R This introduces another free parameter R > 0, which can be treated as another hyperparameter just like γ. We test our algorithms using R = 10, µ = 0 and γ = 1/p, where a i and y i are based on data sets a1a-a9a from the LIBSVM library [10], where the predictors are normalized so that a i = 1. Hence, M (2) f /M (3) f = p −1/2 . For each data set, the methods were ran for 10 randomly generated starting points, where each starting point was chosen as a random vertex of the 1 ball with radius 10. We first compare the methods that are affected by the value of ν ∈ {2, 3} and M f ∈ {M (2) f , M(3) f }, i.e. FWGSC, MBTFWGSC, ASFWGSC, and PN. We display the comparison of the average relative error over the starting points versus iteration and time for data set a9a in Figure 1. Note that for this data set we have p = 32, 561. It is apparent that the linearly convergent methods ASFWGSC and PN gain the most benefit from the lower M f associated with the shift from ν = 3 to ν = 2, reducing both iteration complexity and time. Moreover, for FWGSC and MBTFWGSC the change of ν only seems to benefit the method in earlier iteration, but does not create any asymptotic speedup. Specifically, the benefit for MBTFWGSC is very small, probably since the backtracking procedure already takes advantage of (a) Average iteration ratioρ(ε). (b) Average time ratioρ(ε). (c) Average success ratio ρ(ε). the possible increase in the step-size that is partially responsible for the improved performance in the other methods. We observed the same behavior for all other data sets considered. Thus, we next compare these methods with ν = 2 to the MBTFWGSC, FW-standard, FW-Line Search, and PG and display the performance of all tested methods using the aggregate statistics ρ(ε),ρ(ε),ρ(ε), in Figure 2. Table 2 reports statistics for N(ε) and T(ε) for each individual data set. The PG has the best performance in terms of time to reach a certain value of relative error, followed by FW-standard and ASFWGSC, where FW-standard is slightly better for relative error higher than 10 −5 but becomes inferior to ASFWGSC for lower error values. Portfolio optimization with logarithmic utility We study high-dimensional portfolio optimization problems with logarithmic utility [11]. In this problem there are n assets with returns r t ∈ R n + in period t of the investment horizon. More precisely, r t measures the return as the ratio between the closing price of the current day R t,i and the previous day R t−1,i , i.e. r t,i = R t,i /R t−1,i , 1 ≤ i ≤ n. The utility function of the investor is given as f (x) = − p t=1 log(r t x). Our task is to design a portfolio x solving the problem min x∈R n f (x) s.t.: x i ≥ 0, n i=1 x i = 1. (6.2) Since f is the sum of n standard self-concordant functions, we know that f ∈ F 2,3 with effective domain dom f = {x ∈ R n \|r t x > 0 for all 1 ≤ t ≤ p}. We remark that this self-concordant minimization problem gains also relevance in the universal prediction problem in information theory [32] and online optimization [9]. For this example, computing a LLOO with ρ = √ n is simple and a complete description can be found in [18]. Therefore, we also ran algorithm FWLLOO, where σ f is evaluated by the lowest eigenvalue of the Hessian observed at the initial point. If due to numerical errors, this number is nonpositive, we take σ f = 10 −10 . For conducting numerical experiments, we generated synthetic data, as in Section 6.4 in [44]. We generate a matrix [r t,i ] 1≤t≤p,1≤i≤n ∈ R p×n with given price ratios as: r t,i = 1 + N(0, 0.1) for any i ∈ {1, . . . , n} and t ∈ {1, . . . , p}, which allows the closing price to vary by about 10% between two (a) Average iteration ratioρ(ε). (b) Average time ratioρ(ε). (c) Average success ratio ρ(ε). Table 2: Results for logistic regression problem (6.1). Mean (standard deviation) across starting point realizations of number of iterations and CPU time in seconds to achieve a certain relative error or best relative error achieved by methods, as well as the relative error achieved at that iteration. We highlight in bold the best performance among all competitors. * Maximum iteration number was reached without obtaining the desired relative error for at least one of the starting points. consecutive periods. We used (p, n) = (1000, 800), (1000, 1200), and (1000, 1500) with 4 samples for each size. Hence, there are 12 data sets in total. For each data set, all methods were initialized from 10 randomly chosen vertices from the unit simplex. Figure 3 collects results on the average performance of our methods and Table 3 reports numerical values obtained for each individual data set. MBTFWGSC and ASFWGSC outperforms all other methods considered in terms of time to reach a certain relative error, including PN and PG. Moreover, the advantage of ASFWGSC becomes more significant as the relative error decreases. Interestingly the iteration complexity of MBTFWGSC is almost identical to FW-Line Search while having superior time complexity. Additionally, despite its theoretical linear convergence, FWLLOO has inferior performance to both MBTFWGSC and ASFWGSC, indicating the strong convexity parameter σ f here is very small resulting in a large convergence coefficient. Distance weighted discrimination In the context of binary classification, an interesting modification of the classical support-vector machine is the distance weighted discrimination (DWD) problem, introduced in [29]. In that problem, the classification loss attains the form f (x) = 1 n p i=1 (a i w + µy i + ξ i ) −q + c ξ, over the convex compact set X = {x = (w, µ, ξ)\| w 2 ≤ 1, µ ∈ [−u, u], ξ 2 ≤ R, ξ ∈ R p + }, where R > 0 is a hyperparameter that has to be learned via cross-validation. The parameter q ≥ 1 calibrates the statistical loss function, and (a i , y i ) ∈ R d × {−1, 1}, (i = 1, 2, . . . , p) is the observed sample. The decision variable is decoded as x = (w, µ, ξ) ∈ R n , where n = d + 1 + p, corresponding to a normal vector w ∈ R d , an intercept µ ∈ R and a slack variable ξ ∈ R p . Since ϕ(t) = t −q , q ≥ 1 is generalized self-concordant with parameters M ϕ = q+2 q+2 √ q(q+1) and ν = 2(q+3) q+2 ∈ (2, 3) (cf. Table 1) we get a GSC minimization problem over the compact set X, with parameters ν = 2(q+3) q+2 and M f = q+2 q+2 √ q(q+1) n 1/(q+2) max (a i , y i , e i ) q/(q+2) 2 : 1 ≤ i ≤ n . The special case q = 1 corresponds to the loss function of [29], who solved this problem via a second-order cone reformulation. We test our algorithms using q = 2, and the observations a i and y i are based on data sets a1a-a9a from the LIBSVM library [10], where a i are normalized. For each data set, the methods were ran 10 Table 3: Results for portfolio selection problem (6.2). Mean (standard deviation) across starting point realizations of number of iterations and CPU time in seconds to achieve a certain relative error or best relative error achieved by methods, as well as the relative error achieved at that iteration. We highlight in bold the best performance among all competitors. * Maximum iteration number was reached without obtaining the desired relative error for at least one of the starting points. times, one for each randomly generated starting point of the structure (0, 0, ξ) where ξ is sampled uniformly from its domain. The results presented are averages across these realizations. We set c i = 1 for all i = 1, . . . , p, u = 5, and R = 10. PG cannot be applied to this problem, since 2 < ν < 3. We also do not apply ASFWGSC, since X is not a polyhedral set. Figure 4 collects results on the average performance of our methods and Table 4 shows the results obtained for each individual data set. Here we see that for all data sets and all starting points all FW based methods reach a minimal relative error 10 −3 , with the exception of standard-FW which reaches a relative error of 10 −4 for the smaller instances a1a-a4a but obtains a relative error higher than 10 2 for the larger instances a5a-a9a. The poor performance of FW-Standard on the largest instances is due to the monotonically decreasing step sizes and the fact that it requires very small step size in order to keep the iterates in the domain in the first iteration. From the other methods, MBTFWGSC and FWGSC perform the best, with MBTFWGSC having a slight advantage for lower accuracy due to the use of a smaller M f values. Inverse covariance estimation Undirected graphical models offer a way to describe and explain the relationships among a set of variables, a central element of multivariate data analysis. The principle of parsimony dictates that we should select the simplest graphical model that adequately explains the data. The typical approach to tackle this problem is the following: Given a data set, we solve a maximum likelihood problem with an added low-rank penalty to make the resulting graph as sparse as possible. We consider learning a Gaussian graphical random field of p nodes/variables from a data set {φ 1 , . . . , φ N }. Each random vector φ j is an iid realization from a p-dimensional Gaussian distribution with mean µ and covariance matrix Σ. Let Θ = Σ −1 be the precision matrix. To satisfy conditional dependencies between the random variables, Θ must have zero in Θ i j if i and j are not connected in the underlying graphical model. To learn the graphical model via an 1 -regularization framework in its constrained formulation, we minimize the loss function f (x) = − log det(mat(x)) + tr(Σ mat(x)) (6.3) over set of symmetric matrices with 1 -ball restriction, that is X = {x ∈ R n \| x 1 ≤ R, mat(x) ∈ S n } where R = √ p . The decision variables are vectors x ∈ R n for n = p 2 , so that mat(x) represents the p × p matrix constructed from the p 2 -dimensional vector x. It can be seen that f is standard self-concordant with domain S n ++ . Hence, M f = 2 and ν = 3. One can see that the gradient ∇ f (x) =Σ − mat(x) −1 and Hessian ∇ 2 f (x) = mat(x) −1 ⊗ mat(x) −1 . Since mat(x) is positive definite, we can compute the inverse via a Cholesky decomposition, which in the worst case needs O(p 3 ) arithmetic steps. To compute the search direction, we have to solve the LP Table 4: Results for distance weighted discrimination (DWD) problem. Mean (standard deviation) across starting point realizations of number of iterations and CPU time in seconds to achieve a certain relative error or best relative error achieved by method after 50,000 iterations, as well as the relative error achieved at that iteration. * Maximum iteration number was reached without obtaining the desired relative error for at least one of the starting points. s(x) ∈ argmin s∈X Σ − mat(x) −1 , mat(s) , where A, B = tr(AB) for A, B ∈ S n . This Linear minimization oracle requires to identify the minimal elements of the matrixΣ − mat(x) −1 . Moreover, for the backtracking procedures as well as line search, we also need to construct a domain oracle. This requires to find the maximal step size t > 0 for which x + t(s(x) − x) 0, which is equivalent to finding the maximal t ∈ (0, 1] such that 1 t mat(x) mat(x) − mat(s(x)) or 1 t > λ max (I − mat(x) −1/2 mat(s(x)) mat(x) −1/2 ). Note that this step oracle is not needed when using the theoretical step size in FWGSC and ASFWGSC. We test our method on synthetically generated data sets. We generated the data by first creating the matrixΣ randomly, by generating a random orthonormal basis or R p , B = {v 1 , . . . , v p }, and then set Σ = p i=1 σ i v i v i , where σ i are independently and uniformly distributed between 0.5 and 1. We generated 10 such data sets, for p ranging between 50 and 300. For each data set, the methods were ran for 10 randomly generated starting points. Each starting point has been chosen as a diagonal matrix where the diagonal was randomly chosen from the R-simplex. Figure 5 collects results on the average performance of our methods and Table 5 shows the results obtained for each individual data set. We observe that ASFWGSC has the lowest time of obtaining any relative error below 10 −3 . Moreover, though PG has a lower iteration complexity in some instances, the higher computational cost of projection vs. linear oracle computations, makes it significantly inferior to ASFWGSC. Conclusion Motivated by the recent interest in computational statistics and machine learning in functions displaying generalized self-concordant properties, this paper develops a set of projection-free algorithms for minimizing generalized self-concordant functions as defined in [44]. This function class covers several well-known examples, including logistic, power, reciprocal and, of course, standard self-concordant functions. In particular, members of this function class are potentially ill-conditioned: they may neither have a Lipschitz continuous gradient nor be strongly convex on their domain. Hence, no provably convergent Frank-Wolfe method has been available so far for minimizing generalized self-concordant functions. This paper fills this important gap by developing a set of new provably convergent FW algorithms with sublinear convergence rates. The key innovation of this paper is the design of new adaptive step-size policies and backtracking formulations, exploiting the specific problem structure of GSC-minimization problems. This paper also derives new linearly convergent projection-free methods for the minimization of GSC functions. Specifically, we show how to adapt the local linear minimization ideas of [18] to the current, potentially ill-conditioned, setup. Together with the concurrent paper [8], which appeared on arXive after this work has been submitted for publication, we also derive a new linearly convergent variant of the FW method featuring linear global convergence rates for GSC functions. With the help of extensive numerical experiments, we demonstrate the practical efficiency of our approach. We conclude by mentioning some interesting potential extensions. First, our theory could be used to derive distributed versions of the algorithms presented in this paper in order to develop a generalized and projection-free variant of the DISCO algorithm [50]. DISCO Table 5: Results for covariance estimation example. Number of iterations and CPU time in seconds to achieve a certain relative error or best relative error achieved by methods, as well as the relative error achieved at that iteration. designed to minimize a self-concordant function using distributed computations. Projection-free methods which are able to handle the same problem, but now including generalized self-concordant functions, have the potential to be serious competitors in practice. Second, it will be interesting to incorporate gradient sliding techniques [25], and stochastic versions of our algorithms. Recently, a Newton Frank-Wolfe method has been introduced in [28]. It seems natural to us that their algorithm can be extended to GSC functions. All these are important extensions, which we are planning to pursue in the near future. A Additional Facts about GSC functions In order to make this paper self-contained we are collecting in this appendix finer estimates provided by generalized self-concordance. For a complete treatise the reader should consult the seminal paper [44]. An important feature of GSC functions is their invariance under affine transformations. This is made precise in the following Lemma. When we apply FW to the minimization of a function f ∈ F M , the search direction at position x is determined by the target state s(x) = s defined in (1.1). If A :X → X is a surjective linear re-parametrization of the domain X, then the new optimization problem minXf (x) = f (Ax) is still within the frame of problem (P). Furthermore, the updates produced by FW are not affected by this re-parametrization since ∇f (x),ŝ = ∇ f (Ax), Aŝ = ∇ f (x), s for x = Ax ∈ X, s = Aŝ ∈ X. Beside affine invariance, we will use some stability properties of GSC functions. As corollary of this Proposition and invariance under linear transformations, we obtain the next characterization theorem, which is of particular importance in machine learning applications. Given N functions ϕ i ∈ F M ϕ i ,ν . For (a i , b i ) ∈ R n × R, q ∈ R n and Q ∈ R n×n a positive definite and symmetric matrix, consider the finite-sum model where ξ, δ ≥ 0 are parameters and ν ∈ [2,3]. For all ν ∈ [2, 3], the function t → ψ ν (t) is concave and differentiable. The unique maximum of this function is achieved at 2−ν − 1) − 1 , some simple algebra shows that f (x) N i=1 ϕ i ( a i , x + b i ) + q, x + 1 2 Qx,t * ν                1 δ ln 1 + δ ξ if ν = 2, 1 δ 1 − 1 + δ ξ 4−ν ν−2 − ν−2 4−ν if ν ∈ (2, 3),ψ ν (t) = t 1 + ξ δ ν − 2 4 − ν − ξ δ 2 (ν − 2) 2 2(3 − ν)(4 − ν) (1 − tδ) 2(3−ν) 2−ν − 1 . Setting ψ ν (t) = 0, yields the value t * ν = 1 δ       1 − 1 + δ ξ 4 − ν ν − 2 − ν−2 4−ν       . It is easy to check that ψ ν (t) = −ξ(1 − tδ) 2 2−ν < 0, so that t * is the global maximum of ψ ν (t). If ν > 2 we know that M f δ ν (x) ≤ t ν (x)M f δ ν (x) < 1, and the expression above is well-defined. If ν = 2, the domain of the function ω 2 is full, and again the expression above is well-defined. Set ζ ν (t) ω ν (tM f δ ν (x))t 2 and ξ(x) e(x) 2 Gap(x) , so that η x,M f ,ν (t) Gap(x) = t − ζ ν (t)ξ(x), where t ∈ (0, ∞) if ν = 2 and t ∈ (0, 1 M f δ ν (x) ) for ν ∈ (2, 3]. By definition, t M f ,ν (x) is the unconstrained maximizer of the right-hand-side above. Therefore, 1 − ξ(x)ζ ν (t M f ,ν (x)) = 0. Since t → ζ ν (t) is convex, its derivative is a non-decreasing function. Thus, since we assume that 1 < t M f ,ν (x), it follows ξ(x) = 1 ζ ν (t M f ,ν (x)) ≤ 1 ζ ν (1) . Moreover, ζ ν (1) ≥ 0, so that η x,M f ,ν (1) Gap(x) = 1 − ξ(x)ζ ν (1) = 1 − ζ ν (1) ζ ν (t M f ,ν (x)) ≥ 1 − ζ ν (1) ζ ν (1) = 1 − ω ν (M f δ ν (x)) 2ω ν (M f δ ν (x)) + M f δ ν (x)ω ν (M f δ ν (x)) ≥ 1 2 . where we used that ω ν (t) ≥ 0 for t > 0. C.2 Proof of Lemma 4.6 We first prove a general lower estimate on the per-iteration progress. Lemma C.1. Suppose that t ν (x k ) ≤ 1. Then, the per-iteration progress in the objective function value is lower bounded by whereγ ν 1 + 4−ν 2(3−ν) 1 − 2 2(3−ν)/(4−ν) and b 2−ν 4−ν . We demonstrate this result as a corollary of the technical lemma below. ∆ k ≥                  Lemma C.2. Consider function t → ψ ν (t) defined in eq. (B.1) with unique maximum t * ν as described in eq. (B.2). It holds that ψ ν (t * ν ) =              1 δ (1 + ξ δ ) ln 1 + δ ξ − 1 if ν = 2, 1 δ 1 − abξ δ + abξ δ 1 − 1 b δ ξ b+1 if ν ∈ (2, 3), 1 δ 1 − ξ δ ln 1 + δ ξ if ν = 3. (C.2) where a 4−ν 2(3−ν) and b 2−ν 4−ν < 0. Moreover, the following lower bound holds Whence, h(u) ≤ h(2) for all u ∈ [2, +∞). It remains to show that h(2) ≤ 0. Let us consider h(2) = ϕ(b) := 1 − (b + 1)2 b + b2 b+1 = 1 + b2 b − 2 b as a function of b ∈ (−1, 0). Clearly, ϕ(−1) = ϕ(0) = 0, and it is easy to check via the intermediate value theorem that ϕ(b) < 0 for all b ∈ (−1, 0). We conclude that for u ≥ 2 we get ψ ν (t * 2 ) ≥ 1 δ γ(2). 2. t ∈ (1, 2]. We will show that d du γ(u)/(u − 1) ≤ 0, whence γ(u) ≥ (u − 1)γ (2). Thus, we need to show that 0 ≥ d dt ψ ν (t * ν ) ≥            2 ln 2−1 δ min{1, δ ξ } if ν = 2,1 u − 1 + a (u − 1) 2 − au b+1 (u − 1) 2 = 1 (u − 1) 3 −u + 1 − 2a + a(b + 1)u b − a(b − 1)u b+1 ≡ 1 (u − 1) 3 h(u). Therefore, our next step is to show that h(u) ≤ 0. We have h (u) = −1 + a(b + 1)bu b−1 − a(b − 1)(b + 1)u b , h (u) = ab(b + 1)(b − 1)u b−2 − a(b − 1)b(b + 1)u b−1 = ab(b + 1)(b − 1)u b−2 (1 − u). By definition, a(b + 1) = 1. Hence, since u > 1 and b ∈ (−1, 0), we observe that h (u) ≤ 0. Thus, h (u) ≤ h (1) = 0, and consequently, h(u) ≤ h(1) = 0, for all u ∈ (1,2]. This proves the claim γ(u)/(u − 1) ≥ γ(2) for u ∈ (1, 2]. Combining both cases, we obtain that γ(u) ≥ min{γ (2), (u − 1)γ(2)}, where γ(2) = 1 − a + a2 1/a , using the fact that b + 1 = 1/a. Unraveling this expression by using the definition of the constant a, we see that γ(2) depends only on the self-concordance parameter ν ∈ (2, 3). In light of this, let us introduce the constant γ ν 1 + 4 − ν 2(3 − ν) 1 − 2 2(3−ν)/(4−ν) . (C.6) Observe thatγ 2 = 0 and, by a simple application of l'Hôpital's rule, lim ν↑3γν = 1 − log(2) ∈ (0, 1). Hence γ(2) ≡γ ν ∈ (0, 1) for all ν ∈ (2, 3). We conclude, ψ ν (t * ν ) ≥γ ν δ min 1, −1 b δ ξ (C.7) The case ν = 3: A direct substitution for ψ 3 (t) gives us ψ 3 (t * 3 ) = 1 δ + ξ δ 2 ln ξ δ + ξ . (C.8) Denote u = ξ/δ. Then t * 3 = 1 δ+ξ , so that ψ 3 (t * 3 ) = 1 δ 1 + u ln u u + 1 . Consider the function φ : (0, ∞) → (0, ∞), given by φ(t) := 1 + t ln t 1+t . Then, ψ 3 (t * 3 ) = 1 δ φ(ξ/δ). For t ∈ (0, 1), one sees φ (t) = ln t 1 + t + t 1 + t t 1 1 + t − t (1 + t) 2 = ln 1 − 1 1 + t + 1 1 + t < 0. Consequently, φ(t) is decreasing for t ∈ (0, 1). Hence, φ(t) ≥ φ(1) = 1 − ln 2, for all t ∈ (0, 1). On the other hand, if t ≥ 1, d dt φ(t) 1/t = d dt (tφ(t)) = 1 + 2t ln t 1 + t + t 1 + t ≥ 0. Hence, t → φ(t) 1/t is an increasing function for t ≥ 1, and thus φ(t) ≥ 1−ln 2 t , for all t ≥ 1. Summarizing these two cases we see Proposition 4. 1 . 1The unique solution to program (3.4) is given by Lemma 4. 3 . 3The set S(x 0 ) is compact. Proposition 4. 4 . 4Suppose Assumption 1 holds. Then, the following assertions hold for FWGSC: ν 1 + 4−ν 2(3−ν) 1 − 2 2(3−ν)/(4−ν) and b 2−ν 4−ν . Proof. See Appendix C.2. Remark Remark 4. 2 . 2Combining the result of Theorem 4.8 and the definitions of the constants c 1 (M, ν) in (4.5) . 3 . 3Suppose Assumption 1 holds. Algorithm FWLLOO guarantees linear convergence in terms of the approximation error: max{ a i 2 \|1 ≤ i ≤ p}.On the other hand, we can also consider it as a i 2 \|1 ≤ i ≤ p}. It is important to observe that the regularization parameter γ > 0 affects the self-concordant parameter M Figure 1 : 1Comparison between ν = 3 and ν = 2 for data set a9a. Figure 2 : 2Performance profile for the logistic regression problem (6.1) obtained after averaging over 9 binary classification problems. Figure 3 : 3Performance Profile for the portfolio selection problem (6.2) obtained after averaging over 12 synthetically generated data sets. (a) Average iteration ratioρ(ε).(b) Average time ratioρ(ε). (c) Average success ratio ρ(ε). Figure 4 : 4Performance Profile for the DWD problem averaged over binary classification problems. (a) Average iteration ratioρ(ε).(b) Average time ratioρ(ε). (c) Average success ratio ρ(ε). Figure 5 : 5Performance Profile for Covariance estimation problem (6.3) averaged on 10 data sets. Lemma A. 1 1([44], Prop. 2). Let f ∈ F M f ,ν and A(x) = Ax + b : R n → R p a linear operator. Then(a) If ν ∈ [2, 3], thenf (x) f (A(x)) is (Mf , ν)-GSC with Mf = M f A 3−ν . (b) If ν > 3 and λ min (A A) > 0, thenf (x) = f (A(x)) is (Mf , ν)-GSC with Mf = M f λ min (A A) where λ min (A A) is the smallest eigenvalue of A A. Proposition A.2 ([44], Prop. 1). Let f i ∈ F M f i ,ν where M f i ≥ 0 and ν ≥ 2 for i = 1, . . . , N. Then, given scalars w i > 0, 1 ≤ i ≤ N, the function f N i=1 w i f i is well defined on dom f N i=1 dom f i and belongs to F M f ,ν , where M f max 1≤i≤N w Proposition A.3 ([44], Prop. 5). If ϕ i ∈ F M ϕ i ,ν for ν ∈ (0, 3], then f : R n → (−∞, ∞] defined in (A.1) belongs to F M f ,3 , where M f λ min (Q) (ν−3)/2 max 1≤i≤N M ϕ ofProposition 4.1 is an application of the technical Lemma below. Lemma B.1. Consider the function ψ ν (t) t − ξω ν (tδ)t 2 , (B.1) We will organize the proof of Lemma B.1 according to the generalized self-concordance parameter ν ∈ [2, 3].The case ν = 2: For this parameter we have ω 2 (t) = 1 t 2 [e t − t − 1], and thusψ 2 (t) = t − ξ δ 2 [e tδ − tδ − 1].This is a strictly concave function with unique maximum at 4 − ν 2(3 − ν) 1 − 2 2(3−ν)/(4−ν) . (C.4)Proof. We organize the proof according to the value of ν ∈[2,3]. min{1, δ/ξ}(1 − ln(2)) = (1 − ln(2)) min{1/δ, 1/ξ}. (C.9) Table 1 : 1Examples of univariate GSC functions (based on Definition 2.2 ([44]). A closed convex functionf ∈ C 3 (dom f ), with dom f open, is called (M f , ν)generalized self-concordant of the order ν ∈[2,3] and with constant M f ν are closed convex functions with open domain, it follows that they are barrier functions for dom f : Along any sequence {x n as the search radii sequence {r k } k∈N is decreasing, basic calculus shows that the sequence {t k } k is monotonically increasing. Whence, we get a uniform lower bound of the cut-off values {t k ν } k as Theorem 5.7. Let {x k } k≥0 be the trajectory generated by Algorithm 8 (ASFWGSC). Suppose that Assumption 1, Assumption 3 and Assumption 4 are in place. Then, for all k ≥ 0 we have is a Newton methodProblem FW-Standard FWGSC LBTFWGSC MBTFWGSC FW-Line Search ASFWGSC PN PG n p iter time[s] error iter time[s] error iter time[s] error iter time[s] error iter time[s] error iter time[s] error iter time[s] error iter time[s] error Relative error = 1e-04 cov_50 2500 50 1370.7 (20.6) 0.66 (0.12) 9.93e-05 (6.86e-07) 429.7 (84.3) 0.29 (0.07) 9.96e-05 (2.99e-07) 457.4 (109.5) 0.85 (0.22) 9.98e-05 (1.97e-07) 370.4 (79.2) 0.79 (0.18) 9.96e-05 (2.72e-07) 276.4 (72.3) 2.05 (0.71) 9.96e-05 (2.40e-07) 137.5 (11.2) 0.20 (0.03) 9.25e-05 (2.90e-06) 15.2 (2.0) 80.19 (13.33) 3.37e-05 (2.95e-05) 120.8 (87.4) 0.96 (0.59) 2.21e-05 (1.87e-05) cov_80 6400 80 2100.6 (11.1) 2.46 (0.39) 9.96e-05 (2.03e-07) 665.6 (129.0) 0.97 (0.22) 9.98e-05 (1.25e-07) 719.1 (162.6) 3.27 (0.82) 9.98e-05 (9.62e-08) 575.2 (118.7) 2.75 (0.60) 9.96e-05 (1.68e-07) 429.9 (103.8) 6.60 (1.63) 9.98e-05 (2.06e-07) 210.0 (11.5) 0.74 (0.14) 9.40e-05 (1.98e-06) 18.1 (3.0) 309.32 (74.70) 3.98e-05 (3.02e-05) 325.0 (234.9) 7.19 (5.05) 1.54e-05 (1.24e-05) cov_120 14400 120 3056.3 (21.7) 5.75 (0.30) 9.99e-05 (8.43e-08) 1035.5 (120.5) 2.42 (0.30) 9.98e-05 (1.03e-07) 1712.1 (265.0) 12.04 (1.91) 9.99e-05 (3.51e-08) 897.6 (114.1) 5.55 (1.43) 9.99e-05 (8.34e-08) 677.6 (105.9) 17.14 (3.03) 9.98e-05 (1.08e-07) 310.0 (19.0) 1.26 (0.20) 9.71e-05 (1.13e-06) 19.4 (2.0) 569.46 (33.40) 4.11e-05 (3.13e-05) 314.9 (172.8) 10.71 (5.35) 2.83e-05 (2.41e-05) cov_150 22500 150 3787.5 (13.6) 10.89 (2.64) 9.99e-05 (7.50e-08) 1188.0 (103.6) 4.21 (0.41) 9.98e-05 (1.02e-07) 1720.2 (199.1) 10.60 (1.27) 9.99e-05 (4.92e-08) 1025.0 (97.0) 6.82 (0.71) 9.99e-05 (7.49e-08) 763.1 (88.7) 41.25 (5.69) 9.99e-05 (8.65e-08) 382.7 (20.3) 2.17 (0.24) 9.82e-05 (1.20e-06) 20.0 (2.4) 665.23 (96.11) 5.13e-05 (2.74e-05) 362.5 (201.7) 18.49 (10.58) 5.20e-05 (3.28e-05) cov_170 28900 170 4268.2 (7.2) 18.48 (1.99) 9.99e-05 (5.84e-08) 1414.9 (197.1) 8.04 (1.62) 9.99e-05 (4.47e-08) 1834.6 (333.0) 29.92 (5.11) 9.99e-05 (3.57e-08) 1223.4 (184.3) 21.83 (4.67) 9.99e-05 (8.93e-08) 916.1 (167.1) 69.58 (11.15) 9.99e-05 (7.73e-08) 447.3 (33.0) 6.12 (0.97) 9.78e-05 (8.26e-07) 22.6 (2.9) 1930.93 (316.94) 4.05e-05 (2.75e-05) 611.8 (395.7) 64.61 (38.46) 5.63e-05 (2.64e-05) cov_200 40000 200 4963.6 (12.8) 45.46 (3.27) 9.99e-05 (3.43e-08) 1728.0 (180.7) 17.93 (2.45) 9.99e-05 (4.24e-08) 1824.3 (215.1) 53.20 (8.39) 9.99e-05 (4.35e-08) 1493.0 (163.0) 51.21 (8.58) 9.99e-05 (5.37e-08) 1116.0 (137.7) 130.65 (18.04) 9.99e-05 (6.64e-08) 530.4 (28.6) 9.21 (0.61) 9.87e-05 (7.32e-07) 23.7 (2.5) 1794.82 (207.54) 3.51e-05 (2.32e-05) 582.4 (266.1) 57.60 (24.23) 5.45e-05 (2.88e-05) cov_220 48400 220 5348.4 (13.6) 58.24 (5.08) 1.00e-04 (2.18e-08) 1782.2 (196.5) 23.17 (2.83) 9.99e-05 (4.50e-08) 2449.2 (258.4) 68.85 (7.65) 9.99e-05 (4.17e-08) 1541.7 (180.9) 53.43 (9.05) 9.99e-05 (6.03e-08) 1155.5 (159.0) 178.02 (26.72) 9.99e-05 (5.25e-08) 557.4 (27.2) 10.40 (1.23) 9.85e-05 (1.02e-06) 23.6 (1.6) 1855.80 (135.15) 5.43e-05 (2.21e-05) 655.6 (216.1) 73.64 (23.54) 6.98e-05 (2.29e-05) cov_250 62500 250 6022.7 (6.3) 92.52 (14.39) 1.00e-04 (2.38e-08) 2009.6 (175.2) 36.01 (5.14) 9.99e-05 (4.97e-08) 3363.5 (406.1) 204.46 (22.85) 1.00e-04 (2.34e-08) 1725.2 (165.4) 114.56 (21.08) 9.99e-05 (5.92e-08) 1267.0 (154.8) 324.70 (63.34) 9.99e-05 (5.10e-08) 662.7 (39.0) 27.99 (3.76) 9.88e-05 (7.57e-07) 27.6 (3.7) 5039.51 (753.80) 5.28e-05 (2.30e-05) 2454.3 (4178.9) 438.30 (637.84) 7.57e-05 (1.93e-05) cov_270 72900 270 6495.2 (4.6) 112.35 (16.95) 1.00e-04 (2.04e-08) 2211.2 (193.3) 48.04 (9.08) 9.99e-05 (4.54e-08) 3636.2 (408.0) 241.96 (36.67) 1.00e-04 (1.13e-08) 1907.5 (180.4) 121.16 (31.03) 9.99e-05 (3.46e-08) 1419.6 (163.4) 426.59 (65.65) 9.99e-05 (4.84e-08) 690.2 (28.2) 23.27 (3.75) 9.89e-05 (7.85e-07) 28.2 (4.6) 4963.79 (736.25) 5.06e-05 (1.47e-05) 3337.9 (5712.1) 589.54 (1004.08) 6.08e-05 (1.53e-05) cov_300 90000 300 7183.0 (11.1) 145.69 (1.04) 1.00e-04 (2.18e-08) 2338.4 (237.4) 57.24 (6.23) 9.99e-05 (4.38e-08) 3330.5 (416.7) 259.57 (44.82) 1.00e-04 (1.87e-08) 2006.8 (216.5) 145.92 (19.94) 9.99e-05 (2.49e-08) 1472.0 (186.6) 560.21 (66.08) 9.99e-05 (4.10e-08) 775.4 (36.5) 25.96 (2.28) 9.89e-05 (5.45e-07) 27.1 (2.0) 4651.52 (468.64) 4.82e-05 (2.66e-05) 971.1 (291.6) 214.63 (62.11) 6.46e-05 (1.72e-05) Relative error = 1e-06 cov_50 2500 50 13178.1 (416.7) 6.56 (0.75) 9.95e-07 (2.45e-09) 1650.5 (412.4) 1.08 (0.35) 9.99e-07 (8.02e-10) 2288.4 (614.9) 4.28 (1.38) 9.99e-07 (4.89e-10) 1584.7 (410.5) 3.29 (1.07) 9.99e-07 (5.24e-10) 1481.5 (407.5) 11.12 (3.76) 9.99e-07 (6.58e-10) 172.8 (10.9) 0.25 (0.03) 9.11e-07 (4.26e-08) 16.6 (2.3) 84.52 (13.55) 4.45e-08 (4.76e-08) 122.8 (86.9) 0.98 (0.58) 7.53e-08 (1.59e-07) cov_80 6400 80 19819.2 (494.7) 23.01 (3.69) 9.97e-07 (1.79e-09) 3380.9 (678.2) 4.86 (1.10) 9.99e-07 (2.97e-10) 4930.8 (1058.8) 21.94 (4.75) 1.00e-06 (2.23e-10) 3277.8 (677.6) 16.10 (4.96) 9.99e-07 (3.25e-10) 3116.0 (677.3) 47.82 (10.03) 9.99e-07 (2.50e-10) 267.9 (11.7) 0.91 (0.17) 9.51e-07 (2.96e-08) 19.5 (3.1) 325.16 (73.64) 1.96e-07 (1.66e-07) 326.0 (235.0) 7.22 (5.06) 9.70e-08 (2.07e-07) cov_120 14400 120 29325.2 (567.2) 54.18 (2.16) 9.99e-07 (1.09e-09) 4514.2 (851.3) 10.77 (2.21) 1.00e-06 (2.38e-10) 10587.9 (2156.0) 74.15 (14.24) 1.00e-06 (8.35e-11) 4358.0 (846.9) 25.02 (5.85) 1.00e-06 (2.61e-10) 4112.2 (840.5) 99.25 (23.82) 1.00e-06 (2.03e-10) 399.6 (19.6) 1.56 (0.25) 9.73e-07 (1.96e-08) 21.0 (2.2) 603.12 (32.05) 1.02e-07 (8.71e-08) 316.4 (172.8) 10.76 (5.35) 4.52e-08 (8.35e-08) cov_150 22500 150 36789.8 (544.6) 116.11 (18.38) 9.99e-07 (6.05e-10) 5061.4 (1030.4) 17.72 (3.83) 1.00e-06 (2.51e-10) 10517.3 (2303.2) 64.70 (13.95) 1.00e-06 (5.64e-11) 4876.0 (1022.6) 38.89 (10.48) 1.00e-06 (2.19e-10) 4583.5 (1011.6) 246.57 (56.16) 1.00e-06 (2.12e-10) 495.0 (20.6) 2.73 (0.31) 9.69e-07 (1.57e-08) 21.8 (2.6) 717.20 (105.85) 1.60e-07 (2.48e-07) 364.3 (201.4) 18.58 (10.56) 6.98e-09 (4.58e-09) cov_170 28900 170 41651.4 (237.0) 179.79 (21.21) 9.99e-07 (9.11e-10) 6865.5 (1430.3) 39.43 (12.59) 1.00e-06 (1.54e-10) 12660.4 (2797.2) 204.40 (47.65) 1.00e-06 (8.71e-11) 6646.7 (1419.3) 119.90 (38.07) 1.00e-06 (1.79e-10) 6301.6 (1403.9) 477.78 (106.69) 1.00e-06 (1.70e-10) 575.3 (33.0) 7.51 (1.11) 9.79e-07 (1.52e-08) 24.2 (3.2) 2038.72 (344.85) 3.16e-07 (2.88e-07) 614.3 (395.6) 64.89 (38.41) 1.16e-08 (2.66e-08) cov_200 40000 200 48118.1 (683.4) 439.85 (26.28) 9.99e-07 (4.86e-10) 7773.0 (1347.1) 82.98 (16.96) 1.00e-06 (1.47e-10) 10872.1 (1983.0) 300.09 (49.33) 1.00e-06 (6.17e-11) 7505.3 (1333.1) 289.89 (58.84) 1.00e-06 (1.25e-10) 7084.0 (1312.4) 838.13 (176.41) 1.00e-06 (8.71e-11) 681.4 (29.7) 11.31 (0.69) 9.78e-07 (1.64e-08) 25.2 (2.9) 1881.89 (217.82) 3.69e-07 (3.18e-07) 584.9 (265.9) 57.86 (24.19) 6.69e-08 (1.94e-07) cov_220 48400 220 50001.0 (0.0) 552.84 (44.79) 1.16e-06 (3.46e-08) 7832.2 (1012.3) 100.95 (11.50) 1.00e-06 (1.56e-10) 8826.2 (1085.6) 246.51 (29.06) 1.00e-06 (7.30e-11) 7556.7 (1005.9) 335.30 (51.38) 1.00e-06 (1.58e-10) 7121.9 (996.9) 1091.47 (142.32) 1.00e-06 (8.41e-11) 724.1 (29.2) 12.66 (1.38) 9.82e-07 (1.32e-08) 25.5 (1.6) 1977.83 (140.60) 2.54e-07 (2.34e-07) 658.3 (215.5) 73.91 (23.49) 7.18e-08 (1.94e-07) cov_250 62500 250 50001.0 (0.0) 767.94 (119.51) 1.46e-06 (3.57e-08) 8705.8 (1498.3) 159.21 (44.27) 1.00e-06 (1.08e-10) 21310.9 (3991.2) 1289.64 (221.73) 1.00e-06 (2.94e-11) 8381.2 (1492.7) 566.67 (171.05) 1.00e-06 (1.44e-10) 7868.1 (1486.4) 2067.88 (583.44) 1.00e-06 (1.04e-10) 851.6 (35.8) 34.29 (4.60) 9.87e-07 (5.59e-09) 29.6 (3.6) 5324.53 (762.52) 2.35e-07 (2.41e-07) 2457.6 (4179.1) 438.91 (637.89) 7.91e-08 (2.35e-07) cov_270 72900 270 50001.0 (0.0) 869.79 (131.12) 1.71e-06 (3.12e-08) 9900.4 (1396.7) 217.19 (57.43) 1.00e-06 (9.54e-11) 23551.3 (3621.0) 1511.20 (248.37) 1.00e-06 (3.08e-11) 9553.1 (1394.7) 702.23 (198.56) 1.00e-06 (1.02e-10) 9004.9 (1393.2) 2750.95 (605.23) 1.00e-06 (9.68e-11) 895.6 (27.2) 28.86 (4.55) 9.80e-07 (5.52e-09) 30.2 (4.6) 5256.39 (756.05) 2.50e-07 (2.22e-07) 3340.9 (5711.7) 589.97 (1004.03) 1.02e-07 (2.36e-07) cov_300 90000 300 50001.0 (0.0) 1017.68 (9.11) 2.06e-06 (3.52e-08) 10531.1 (1653.3) 258.92 (40.44) 1.00e-06 (6.88e-11) 22015.4 (3712.3) 1578.08 (312.05) 1.00e-06 (5.01e-11) 10151.0 (1640.5) 847.26 (146.52) 1.00e-06 (1.22e-10) 9549.6 (1622.9) 3655.96 (604.43) 1.00e-06 (1.16e-10) 1002.3 (38.5) 33.18 (2.92) 9.92e-07 (5.68e-09) 29.2 (2.0) 4986.63 (453.33) 2.12e-07 (3.05e-07) 973.7 (291.6) 215.23 (62.14) 2.08e-07 (2.69e-07) In fact, evaluating the local norm requires the Hessian matrix ∇ 2 f (x), and thus FWGSC is actually second-order method. At the same time, no inversion of the Hessian is needed. For instance, the matrix-vector product can be efficiently computed when the objective belongs to the class of generalized linear models, where the Hessian is given as a sum of rank 1 matrices. Complexity analysis4.1 Complexity Analysis of FWGSCBased on the preliminary discussion of Section 3.2, our strategy to determine the step-size policy is to first compute t M f ,ν (x) defined as the solution to program (3.4) and then clip the value accordingly. https://www.epfl.ch/labs/lions/technology/scopt/ AcknowledgementsThe authors sincerely thank Professor Shoham Sabach for his contribution in the early stages of this project, including his part in developing the basic ideas developed in this paper. We would also like to thank Professor Quoc Tranh-Dinh for fruitful discussions on this topic and in sharing MATLAB codes of SCOPT with us. Feedback from Professors Robert M. Freund and Sebastian Pokutta are also gratefully acknowledged. Finally, we would like to thank the Associate Editor and the Reviewers for their valuable remarks and suggestions. M. Staudigl acknowledges support by the COST Action CA16228 "European Network for Game Theory". The research by P. Dvurechensky is supported by the Ministry of Science and Higher Education of the Russian Federation (Goszadaniye) No. 075-00337-20-03, project No. 0714-2020-0005 and by RFBR grant 18-29-03071_mk.The case ν = 3: For this case, we have ω3It is easy to see that ψ 3 (t) = t + ξ δ 2 [tδ + ln(1 − tδ)] t ∈ (0, 1/δ).Therefore, for t ∈ (0, 1/δ), we see thatThe unique maximum is attained atB.2 Proof of Theorem 4.1Identifying the parameters involved in (B.1) as δ = M f δ ν (x), and ξ = e(x) 2Gap(x) gives usHence, the following explicit expressions for the step-size parameters are immediate consequences of Lemma B.1.Gap(x) , we getThis completes the proof of Theorem 4.1.C Auxiliary Results needed in the proof of Theorem 4.8C.1 Proof of Lemma 4.5Set x ≡ x k . Since t M f ,ν (x) > 1, the decrease of the objective function is, once we plug in t * 2 from eq. (B.3) we arrive, after some computations, atLet us now consider the function t →Hence,Combining these two cases, we see thatThe case ν ∈ (2, 3): A computation shows thatSet aTo verify the lower bound, we rewrite ψ ν (t * ν ) as follows:where γ(u) 1 + a u−1 − au b+1 u−1 . Our next goal is to show that, for u ∈ [2, +∞), γ(u) is below bounded by some positive constant and, for u ∈ (1, 2], γ(u) is below bounded by some positive constant multiplied by u − 1.1. u ∈ [2, +∞). We will show that γ (u) ≥ 0, whence γ(u) ≥ γ(2). Thus, we need to show thatSince a > 1, to show that γ (u) ≥ 0 it is enough to show that h(u) ≤ 0. Since b ∈ (−1, 0) and t ≥ 2,Proof of Lemma C.1. Recall that η x,M f ,ν (t) = Gap(x)ψ ν (t). By identifying the parameters appropriately, we can give the proof of Lemma C.1 as a straightforward exercise derived from Lemma C.2. 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[ "Entropy gain in p-adic quantum channels", "Entropy gain in p-adic quantum channels" ]	[ "Evgeny Zelenov " ]	[]	[]	The construction of the p-adic quantum Gaussian channel is suggested. Entropy gain is calculated. The adelic formula for the entropy gain is obtained.	10.1134/s1063779620040814	[ "https://export.arxiv.org/pdf/2010.02750v1.pdf" ]	222,142,199	2010.02750	ff28fb58366b512e36ea66ce8aa348154c47bfdb	Entropy gain in p-adic quantum channels 4 Oct 2020 October 3, 2020 Evgeny Zelenov Entropy gain in p-adic quantum channels 4 Oct 2020 October 3, 2020 The construction of the p-adic quantum Gaussian channel is suggested. Entropy gain is calculated. The adelic formula for the entropy gain is obtained. Introduction Starting from [1,2], non-Archimedean analysis has been actively used to build physical models. Thereby, the branch of mathematical physics (p-adic mathematical physics) has arisen. One can find a bibliography on this subject in the book [3] and reviews [4,5]. The article is organized as follows. Firstly, we give the necessary facts of the p-adic analysis. Secondly, we define the p-adic Gaussian state and the p-adic Gaussian linear bosonic channel and also give their properties. This information follows [6]; the results are presented without proof. New results are shown below. Thirdly, we prove a formula for the magnitude of the entropy gain in the p-adic Gaussian channel. Fourthly, we give the adelic formula for the entropy gain and its possible applications. p-Adic numbers and symplectic geometry In this section, some known facts concerning the geometry of lattices in a two-dimensional symplectic space over the field Q p of p-adic numbers are presented without proof. The necessary information from the p-adic analysis is contained, for example, in [7]. Most statements regarding the geometry of lattices can be found in [8]. Let F be a two-dimensional vector space over the field Q p . A nondegenerate symplectic form ∆ : F × F → Q p is given in the space F . A free module of rank two over the ring Z p of p-adic integers considered as a subset of F will be called a lattice in F . A lattice is a compact set in the natural topology in the space F . We introduce the duality relation on the set of lattices. Let L be a lattice; then the dual lattice L * means the following subset of the space F : u ∈ L * if and only if the condition ∆(u, v) ∈ Z p is satisfied for all v ∈ L. If L coincides with L * , then the lattice L will be called a self-dual lattice. For any self-dual lattice L, there exists a symplectic basis (e 1 , e 2 ) in the space F such that the lattice L has the form L = Z p e 1 Z p e 2 . By Sp(F, ∆) denote the symplectic group (the group of nondegenerate linear transformations of the space F that preserve the form ∆). The group Sp(F, ∆) is isomorphic to the group SL 2 (Q p ). In the space F , there exists a unique translation-invariant measure (Haar measure) up to normalization. We will normalize the measure in such a way that the measure of a self-dual lattice is equal to unity. The action of the symplectic group preserves the measure; therefore, the measure of any self-dual lattice is equal to unity. The measure of the lattice L will be denoted by \|L\|. If L is a self-dual lattice, then, as noted above, \|L\| = 1; the converse is also true, if \|L\| = 1, then the lattice L is self-dual. It is easy to verify the validity of the relation \|L\|\|L * \| = 1. Let S ∈ Sp(F, ∆), L ⊂ F be an arbitrary lattice. As already noted, the action of the symplectic group preserves measure, that is, \|SL\| = \|L\| . The converse is also true, if L 1 , L 2 are arbitrary lattices in F with the same measure, \|L 1 \| = \|L 2 \| , then there exists a symplectic transformation S ∈ Sp(F, ∆), such that SL 1 = L 2 . p-Adic linear bosonic channels Let H be a separable Hilbert space over the field C of complex numbers. The scalar product in H will be denoted by ·, · , and we will consider it antilinear in the first argument. The state of the system is described by the density matrix ρ in the space H. We do not go beyond the framework of the standard statistical model of quantum mechanics ( [9]), since the set of real numbers and the set of p-adic numbers are Borel isomorphic. Let W be a mapping from the space F to the set of unitary operators in the space H, satisfying the relation W (z)W (z ′ ) = χ 1 2 ∆(z, z ′ ) W (z + z ′ ) for all z, z ′ ∈ F . Here we use the notation χ(x) = exp (2πi{x} p ), where {x} p denotes the p-adic fractional part of the number of x ∈ Q p . Besides, we assume that mapping W is continuous in the strong operator topology. The mapping W is called the representation of canonical commutation relations (CCRs) in the Weyl form. As noted above, the state of a quantum system is described by a density matrix ρ in a Hilbert space H. Let an irreducible representation of the CCR {W (z), z ∈ F } be given in this space. Each density operator can be associated with a function π ρ in the space F by the following formula. π ρ (z) = Tr (ρW (z)) . The function π ρ is called the characteristic function of the quantum state ρ and defines this state uniquely. The characteristic function is used to reconstruct the state ρ according to the relation ρ = F π ρ (z)W (−z)dz. The characteristic function of a quantum state has the property of ∆-positive definiteness: for any finite sets z 1 , z 2 , . . . , z n of points in the space F and complex numbers c 1 , c 2 , . . . , c n , the inequality n i,j=1 c icj π ρ (z i − z j )χ − 1 2 ∆(z i , z j ) ≥ 0 is satisfied. As in the case of the representation of CCR over a real symplectic space, a noncommutative analogue of the Bochner-Khinchin theorem holds for the p-adic case; this theorem establishes one-to-one correspondence between states and ∆-positive definite functions ( [10]) . We give the following definition. Definition 1 A state ρ will be called a p-adic Gaussian state if its characteristic function is the indicator function of some lattice, that is, π ρ (z) = Tr (ρW (z)) = h L (z) = 1, z ∈ L; 0, z / ∈ L. This definition is natural in the following context. Let F be the Fourier transform in L 2 (F ) defined by the formula F [f ] (z) = F χ (∆(z, s)) f (s)ds, L is a lattice in F . Then, the formula \|L\| −1/2 F [h L ] = \|L * \| −1/2 h L * is satisfied. In other words, the Fourier transform turns the characteristic function of the lattice into the characteristic function of the dual lattice up to a factor. In particular, the characteristic function of the self-dual lattice is invariant under the action of the Fourier transform. In this context, the characteristic function of the lattice is an analogue of the Gaussian function in real analysis. p-Adic Gaussian states are elementary in structure. Namely, the following statements are true. Proposition 1 The characteristic function h L of the lattice L determines the quantum state, if and only if the condition \|L\| ≤ 1 is satisfied. A Gaussian state ρ having a characteristic function π ρ = h L is \|L\|P L , where P L is an orthogonal projector of rank 1/\|L\|. Some obvious properties of Gaussian states are given below. Proposition 2 The following statements are true. • A Gaussian state is pure if and only if the corresponding lattice is selfdual. • The entropy of the Gaussian state defined by the lattice L is − log \|L\|. • Gaussian states ρ 1 and ρ 2 are unitarily equivalent if and only if the corresponding lattices L 1 and L 2 have the same measure. • The entropy of the Gaussian state determines this state uniquely up to unitary equivalence. • The Gaussian state has the maximum entropy among all states with a fixed rank p m , m ∈ N. We will use the notation γ(L) for the density operator of the Gaussian state defined by the lattice L. We considered only centred Gaussian states. We can similarly view general Gaussian states γ(L, α), which are defined by a characteristic function of the form π γ(L,α) = χ (∆(α, z)) h L (z). It is easy to see that γ(L, α) = W (α)γ(L)W (−α). Let W be the irreducible representation of the CCR in the Hilbert space H; S(H) is the set of states. By analogy with the real case ( [11]), a linear bosonic channel (in the Schrodinger representation) is a linear completely positive trace-preserving mapping Φ : S(H) → S(H) such that the characteristic function π ρ of any state ρ ∈ S(H) is transformed by the formula π Φ[ρ] (z) = π ρ (Kz)k(z)(1) for some linear transformation K of the space F and some complex-valued function k in F . Generally speaking, expression (1) does not always determine the channel; to do this, additional conditions on the transformation K and the function k are necessary. A p-adic Gaussian channel is a linear bosonic channel for which the function k is the characteristic function of some lattice L ⊂ F , that is, k(z) = h L (z), z ∈ F . The following proposition holds. Proposition 3 Let K be a nondegenerate linear transformation of the space F , L be a lattice in the space F , k(z) = h L (z). In this case, expression (1) defines a channel if and only if the inequality \|1 − detK\| p \|L\| ≤ 1(2) is satisfied. Note that in the case of detK = 1, the transformation is symplectic and, therefore, is unitarily representable. Next, we consider the case of detK = 1. Entropy gain The entropy H(ρ) of the state ρ is defined by the following expression H(ρ) = − Tr ρ log ρ. According to the definition, the entropy gain is G(Φ) = inf ρ {H (Φ[ρ]) − H(ρ)} . Theorem 1 The following formula is valid for the entropy gain of the p-adic Gaussian linear bosonic channel G(Φ) = log \| det K\| p . Let K be a nondegenerate linear transformation and L be a lattice. Moreover, we assume that K and L define the channel, that is, they satisfy condition (2). By L n , n ∈ N we denote the lattice L n = p n L. Note that L n ⊂ L, and for sufficiently large n, the condition \|L n \| < 1 is satisfied, since \|L n \| = p −n \|L\|. According to Proposition 1, the Gaussian state γ(L n ) is defined for all such n. We also note that there exists such N ∈ N, that for all n ≥ N, the conditions L n ⊂ L and K −1 L n ⊂ L are simultaneously satisfied. It is not difficult to calculate the entropy gain for state γ(L n ), n ≥ N. Indeed, the condition π Φ[γ(Ln)] (z) = π γ(Ln) (Kz)h L (z) = h Ln (Kz)h L (z) = h K −1 p n L (z) follows directly from the definition of the Gaussian channel (1). Consequently, Φ [γ(L n )] = γ K −1 L n(3) and, in accordance with Proposition 1, H (Φ [γ(L n )]) − H (γ(L n )) = log \| det K\| p .(4) The latter formula immediately implies an evaluation for the minimum entropy gain G(Φ) ≤ log \| det K\| p . Further, we recall (Proposition 1) that operator \|L n \| −1 γ(L n ) is an orthogonal projection, and the sequence of these operators strongly converges to the unit operator I when n → ∞. Therefore, we can correctly determine Φ[I] (the channel is regular); moreover, formula (3) and Proposition 1 imply Φ[I] = \| det K\| −1 p . Next, we use the evaluation of the minimum entropy gain for a regular channel − log Φ[I] ≤ G(Φ) obtained in [12]. The theorem is proved. Note that the expression for the entropy gain in the p-adic linear bosonic Gaussian channel is given by a a formula similar to the corresponding expression for the real linear bosonic Gaussian channel obtained in [12]. Adelic channels and entropy gain Now let F be a two-dimensional vector space over the field Q of rational numbers, ∆ be a nondegenerate symplectic form taking values in the field Q, and K be a nondegenerate linear transformation of the space F . For each prime p, we construct the corresponding linear bosonic Gaussian channel Φ p and calculate the entropy gain G(Φ p ) of such a channel. We can also construct a real linear bosonic Gaussian channel Φ ∞ and calculate the entropy gain G(Φ ∞ ). Let P denotes the set of all prime numbers. The following statement is true. Theorem 2 G(Φ ∞ ) + p∈P G(Φ p ) = 0. Note that det K ∈ Q. The further follows from a simple adelic formula \| det K\| p∈P \| det K\| p = 1, which is valid for an arbitrary nonzero rational number (for example, see [7]). Theorem 2 can be interpreted as follows. Consider the adelic linear bosonic Gaussian channel generated by the linear transformation K of a two-dimensional vector space over the field of rational numbers, that is, the tensor product of the channels Φ p over all primes p and the real channel Φ ∞ . A nontrivial entropy gain G(Φ p ) is possible in each component of this adelic channel. However, the total entropy gain in the adelic channel is zero. There is a nontrivial exchange of information between the components of the adelic channel when the total entropy is conserved. If we accept the hypothesis that ¡¡at the fundamental level, our world . . . is adelic . . . ¿¿, then Theorem 2 can be used, for example, to interpret the black hole information paradox. Denote by S the set of all states. By B(F ) we denote the σ-algebra of Borel subsets of F ; by B(H) we denote the algebra of bounded operators on space H. A quantum observable M : B(F ) → B(H) is a projection-valued measure on B(F ). Denote by M the set of quantum observables. The Born-von Neumann formula gives the probability distribution of the observable M in a state ρ µ M ρ = Tr (ρM(B)) , B ∈ B(F ). Number theory as the ultimate physical theory. I V Volovich, CERN-TH-4781-87 --CernGenevaVolovich I.V. Number theory as the ultimate physical theory // Preprint CERN-TH-4781-87 --Cern, Geneva, 1987. I Volovich, 10.1088/0264-9381/4/4/003adic string // Classical and Quantum Gravity--1987. 4Volovich I.V. p-adic string // Classical and Quantum Gravity--1987.-- V. 4, no. 4--P. L83-L87. Zelenov E.I. p-Adic analysis and mathematical physics. V S Vladimirov, I V Volovich, World ScientificSingapureVladimirov V.S., Volovich I.V., Zelenov E.I. p-Adic analysis and mathe- matical physics.--World Scientific, Singapure, 1994. . 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W H Schikhof, Cambridge University PressSchikhof W.H. Ultrametric calculus. An introduction to p-adic analysis. -- Cambridge University Press, 1984. . J.-P Serre, Trees, Springer-VerlagSerre J.-P. Trees. --Springer-Verlag, 1980. Statistical structure of quantum theory. A S Holevo, Springer-VerlagHolevo A.S. Statistical structure of quantum theory. --Springer-Verlag, 2001. Representations of commutations relations for p-adic systems of infinitely many degrees of freedom // Journal of Mathematical Physics --1992. E I Zelenov, 10.1063/1.52994233Zelenov E.I. Representations of commutations relations for p-adic systems of infinitely many degrees of freedom // Journal of Mathematical Physics --1992. --V. 33, no. 1. --P. 178- 188. -- Quantum systems, channels, information. A mathematical introduction. A S Holevo, De GruyterBerlin-BostonHolevo A.S. Quantum systems, channels, information. A mathematical introduction. --De Gruyter, Berlin-Boston, 2019. 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[ "AN O-ACYCLIC VARIETY OF EVEN INDEX", "AN O-ACYCLIC VARIETY OF EVEN INDEX" ]	[ "John Christian ", "Fumiaki Suzuki ", "Olivier Wittenberg " ]	[]	[]	We give the first examples of O-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over P 1 such that any multi-section has even degree over the base P 1 and show moreover that we can find such a family defined over Q. This answers affirmatively a question of Colliot-Thélène and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel-Jacobi maps.One can then ask: Question 1.1. Does an O-acyclic smooth projective geometrically connected variety Y over the function field of a complex curve always have I(Y ) = 1?In other words, we ask whether Serre's question has a positive answer if we replace a rational point on Y with a 0-cycle of degree 1.It is important to note that there is no local obstruction here: the Riemann-Roch theorem implies that Y as in Question 1.1 always has indices one everywhere	10.1007/s00208-023-02581-2	[ "https://export.arxiv.org/pdf/2010.06079v3.pdf" ]	222,310,215	2010.06079	c7be10b7f260e800cfb68721d5046aaa74b28c04	AN O-ACYCLIC VARIETY OF EVEN INDEX 14 Apr 2023 John Christian Fumiaki Suzuki Olivier Wittenberg AN O-ACYCLIC VARIETY OF EVEN INDEX 14 Apr 2023arXiv:2010.06079v3 [math.AG] We give the first examples of O-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over P 1 such that any multi-section has even degree over the base P 1 and show moreover that we can find such a family defined over Q. This answers affirmatively a question of Colliot-Thélène and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel-Jacobi maps.One can then ask: Question 1.1. Does an O-acyclic smooth projective geometrically connected variety Y over the function field of a complex curve always have I(Y ) = 1?In other words, we ask whether Serre's question has a positive answer if we replace a rational point on Y with a 0-cycle of degree 1.It is important to note that there is no local obstruction here: the Riemann-Roch theorem implies that Y as in Question 1.1 always has indices one everywhere Introduction In a letter to Grothendieck [10, p. 152], Serre asked whether a smooth projective geometrically connected variety Y over the function field of a complex curve should always have a rational point if it is O-acyclic, that is, H i (Y, O Y ) = 0 for all i > 0. This indeed holds for rationally connected varieties, as proved by Graber-Harris-Starr [15], generalizing a classical theorem of Tsen. However, Graber-Harris-Mazur-Starr [14] gave a counterexample for the general case; in fact, they showed that there exist Enriques surfaces with no rational points over the function field of a complex curve. Later, more explicit constructions of such Enriques surfaces were given by Lafon [19] and Starr [26]. It is remarkable that the example of Lafon is defined over Q(t) and has no rational point over the local field C((t)). In light of these examples, one might still hope that a weaker statement could be true. We recall that the index of a proper variety Y over a field F is defined to be I(Y ) = gcd {deg F (α) \| α ∈ CH 0 (Y )} . locally, or equivalently, that Y gives a one-parameter family X → C of O-acyclic varieties with no multiple fiber (see also [9,Proposition 7.3] and [12,Theorem 1]). Nevertheless, it was expected by several mathematicians that Question 1.1 would have a negative answer (see [26] for expectations of Esnault on the indices of the examples of Graber-Harris-Mazur-Starr and Lafon). In particular, Colliot-Thélène and Voisin asked [9,Question 7.9] whether one can construct an O-acyclic surface of index not equal to one. The aim of this paper is to give the first counterexamples to Question 1.1 and thereby to answer affirmatively the question raised by Colliot-Thélène and Voisin. Our main result is the following: Theorem 1.2 (=Theorem 3.1, 4.1). Let X ⊂ P 1 × P 2 × P 2 be the rank one degeneracy locus of a map of vector bundles O ⊕3 → O(2, 2, 0) ⊕ O(2, 0, 2). If X is very general, then the first projection gives a family X → P 1 of Enriques surfaces such that any multi-section has even degree over the base P 1 . That is, the index I(X η ) is even, where X η is the generic fiber. Moreover, we can find threefolds with these properties defined over Q. Remark 1.3. Our construction can be generalized to give a counterexample to Question 1.1 when dim Y = 2n for any positive integer n (besides ones obtained from Theorem 1.2 by taking the product with a projective space); see Theorem 3.3. Remark 1.4. It would be natural to ask an analogue of Question 1.1 over the function field of a curve over the algebraic closure of a finite field. We will prove some conditional positive results in Proposition 4.4 and Corollary 4.6. Our construction has consequences for certain questions in number theory. We say that the Hasse principle holds for 0-cycles of degree 1 on a smooth projective geometrically connected variety Y over the function field F = C(C) of a complex curve C if there is a 0-cycle of degree 1 on Y whenever there is such a cycle on Y Fp for any point p ∈ C, where F p ∼ = C((t)) is the completion of F at p. The reciprocity obstruction to the Hasse principle for 0-cycles of degree 1 on a variety over the function field of a complex curve, which is an analogue of the Brauer-Manin obstruction for rational points on a variety over a number field, was defined and pointed out to the authors by Colliot-Thélène (see also [7,Section 5]). As a consequence of our construction, we prove that the failure of the Hasse principle for 0-cycles of degree 1 on an Enriques surface over C(P 1 ) cannot always be accounted for by the reciprocity obstruction. Theorem 1.5 (=Theorem 5.2). Let X η be the generic fiber of a very general family X → P 1 of Enriques surfaces as in Theorem 1.2. Then the Hasse principle fails for 0-cycles of degree 1 on X η , while there is no reciprocity obstruction for X η . Question 1.1 is also related to the integral Hodge conjecture. We recall that the integral Hodge conjecture in degree 2i on a smooth complex projective variety X is the statement that degree 2i integral Hodge classes on X are algebraic, i.e., the image H 2i alg (X, Z) ⊆ H 2i (X, Z) of the cycle class map cl i : CH i (X) → H 2i (X, Z) generates the entire group Hdg 2i (X, Z) = H i,i (X) ∩ H 2i (X, Z) of integral Hodge classes. While the statement holds for i = 0, 1, dim X, it is known to fail in general for 2 ≤ i ≤ dim X − 1. The first counterexample was constructed by Atiyah-Hirzebruch [1] and many others have been found since then [2,3,9,20,23,25,29]. As pointed out by Colliot-Thélène and Voisin [9, Theorem 7.6], a counterexample to Question 1.1 gives a one-parameter family X → C of O-acyclic varieties for which the integral Hodge conjecture fails in degree 2d − 2, where d = dim X. This means that the defect of the integral Hodge conjecture in degree 2d − 2, defined as Z 2d−2 (X) = Hdg 2d−2 (X, Z)/H 2d−2 alg (X, Z), is non-zero. It follows that the integral Hodge conjecture fails in degree 4 for the threefold X in Theorem 1.2, and that the defect Z 4 (X) is non-zero. In the last part of the paper, we determine completely the 2-torsion subgroup Z 4 (X) [2]. In addition, this allows us to compute explicitly the degree 3 unramified cohomology group H 3 nr (X, Z/2), a stable birational invariant of smooth complex projective varieties defined in the framework of the Bloch-Ogus theory [5]. A key input is a theorem of Colliot-Thélène and Voisin [9,Theorem 3.9] together with the fact that we have CH 0 (X) = Z (this can be deduced from a result of Bloch-Kas-Lieberman [4]). Theorem 1.6 (=Theorem 6.1, Corollary 6.2). Let X be the total space of a very general family of Enriques surfaces as in Theorem 1.2. Then we have H 3 nr (X, Z/2) = Z 4 (X)[2] = (Z/2) 46 . Remark 1.7. Note that there is a 2-torsion element in the Néron-Severi group of the geometric generic fiber of the family X → P 1 . In contrast, Colliot-Thélène and Voisin proved that if X → C is a family of O-acyclic surfaces such that the geometric generic fiber has torsion free Néron-Severi group, then the degree 3 unramified cohomology group with torsion coefficients is conjecturally of rank at most one [9,Theorem 7.7,8.21,Remark 8.22]. We note that Theorem 1.2 also has an application to universality of the Abel-Jacobi maps. A classical question of Murre asks whether the Abel-Jacobi map is universal among all regular homomorphisms (see [20,Section 4] and [27,Section 1] for more precise statements). Recently, a negative answer to the question was given by a fourfold constructed by the authors [20]. In fact, the threefold X of Theorem 1.2 can be used to construct another such fourfold. We refer the reader to the papers [20] and [27] for the details of the argument. This paper is organized as follows. In Section 2, we introduce certain families of Enriques surfaces parametrized by P 1 and study their basic properties. In Section 3, we prove the main theorem over C using an explicit geometric construction. The proof involves a combination of monodromy and specialization arguments, and a key congruence obtained previously by the authors in [20]. In Section 4, we refine this construction to get counterexamples defined over Q. In Section 5, we discuss the failure of the Hasse principle and the reciprocity obstruction on our examples. In Section 6, we compute the defect of the integral Hodge conjecture in degree 4 on the total space of the family of Enriques surfaces of the main theorem, and in addition, its degree 3 unramified cohomology group with Z/2 coefficients. Finally, in the Appendix, Olivier Wittenberg proves that the vanishing of the reciprocity obstruction obtained in Theorem 5.2 is in fact a completely general phenomenon. Notation. We work over the complex numbers in Section 2, 3, 5, and 6. In Section 4, we work over Q. We use Grothendieck's notation for projective bundles: for a vector bundle E, P(E) parameterizes one-dimensional quotients of E. We write O P(E) (1) for the relative hyperplane bundle. We will let O P r ×P s (a, b) and O P r ×P s ×P t (a, b, c) denote line bundles on products of projective spaces (i.e., these are pr * 1 O P r (a)⊗pr * 2 O P s (b) and pr * 1 O P r (a)⊗pr * 2 O P s (b)⊗ pr * 3 O P t (c) respectively). Similarly, we will write O P 1 ×P(E) (a, b) for the line bundle pr * 1 O P 1 (a) ⊗ pr * 2 O P(E) (b) on P 1 × P(E). To simplify notation we will usually drop the subscripts when the context is clear. Families of Enriques surfaces parametrized by P 1 We will fix the following notation: • P A = P P 2 ×P 2 (O(2, 0) ⊕ O(0, 2)), E 1 = P P 2 ×P 2 (O(2, 0)), E 2 = P P 2 ×P 2 (O(0, 2)) • P B = P P 2 ×P 2 (O(1, 0) ⊕ O(0, 1)), F 1 = P P 2 ×P 2 (O(1, 0)), F 2 = P P 2 ×P 2 (O(0, 1)) • P C = P(H 0 (P B , O(1))), P 1 = P(H 0 (P 2 ×P 2 , O(1, 0))), P 2 = P(H 0 (P 2 ×P 2 , O(0, 1))). As is explained in [20], these spaces are related by the following geometric construction: P C is a 5-dimensional projective space, and P 1 and P 2 define disjoint planes in it via the isomorphism H 0 (P B , O(1)) = H 0 (P 2 × P 2 , O(1, 0)) ⊕ H 0 (P 2 × P 2 , O(0, 1)). The projective bundle P B is then identified with the blow-up of P C along the union of P 1 and P 2 , and F 1 and F 2 are the corresponding exceptional divisors. Furthermore, there is an involution ι on P C induced by the involution on H 0 (P B , O(1)) with the (±1)-eigenspaces H 0 (P 2 × P 2 , O(1, 0)) and H 0 (P 2 × P 2 , O(0, 1)), respectively. The involution ι lifts to an involution on the blow-up P B , and we have P A = P B /ι. Thus there is a double cover P B → P A over P 2 × P 2 , which is ramified along F 1 ∪ F 2 , and the divisors F i are mapped isomorphically onto E i for i = 1, 2. The varieties P A , P B , P C were used in [20] to give projective models of Enriques surfaces. In this paper, we will use them to study the threefolds X in Theorem 1.2; these are Enriques surface fibrations over P 1 . We now explain the main construction. Let X ⊂ P 1 × P 2 × P 2 be the rank one degeneracy locus of a general map of vector bundles (1) O ⊕3 → O(2, 2, 0) ⊕ O(2, 0, 2). Then X is a smooth threefold and the first projection X → P 1 defines a family of Enriques surfaces (see [20,Lemma 2.1]). There is a natural diagram P 1 × P B / / P 1 × P A P 1 × P C P 1 × P 2 × P 2 in which P 1 × P A → P 1 × P 2 × P 2 is the natural projection; P 1 × P B → P 1 × P A is the quotient map by the involution ι (which acts trivially on P 1 ); and P 1 ×P B → P 1 ×P C is the blow-up along the union of P 1 × P 1 and P 1 × P 2 . The above diagram restricts to a diagram Y / / X ′ ≃ Y min X where X ′ ⊂ P 1 × P A , Y ⊂ P 1 × P B , and Y min ⊂ P 1 × P C are respectively defined by a section of O(2, 1) ⊕3 on P 1 × P A and ι-invariant sections of O(2, 2) ⊕3 on P 1 × P B and P 1 × P C induced by the map of vector bundles defining X. Note that each of the intersections Y min ∩ (P 1 × P i ) is a complete intersection of three divisors of type (2, 2) on P 1 × P 2 ; thus they consist of 24 points y i,1 , . . . , y i, 24 . Moreover, the map Y → Y min is the blow-up of Y min along the 48 points y i,j , with the corresponding exceptional divisors F i,j being the components of Y ∩ (P 1 × F i ). The double cover Y → X ′ is ramified exactly along the union of the F i,j , and each F i,j is mapped isomorphically onto E i,j (the components of X ′ ∩ (P 1 × E i )). If X is general, the map P 1 × P A → P 1 × P 2 × P 2 restricts to an isomorphism X ′ → X. Remark 2.1. The minimal model X min of X can be obtained by contracting the projective planes E i,j to points; X min is singular exactly at the images of E i,j , and at each of the singular points the tangent cone is the affine cone over a Veronese surface. Lemma 2.2. The threefold X has the following properties: (1) The degree homomorphism deg : CH 0 (X) → Z is an isomorphism. (2) The canonical divisor of X is of the form K X = 4F + 1 2 2 i=1 24 j=1 E i,j , where F is the class of a fiber of the projection X → P 1 . Thus X has Kodaira dimension κ(X) = 1. O ⊕3 → O(1, 2, 0) ⊕ O(1, 0, 2) is considered. The properties (1) to (4) We will also need the following: Lemma 2.3. Let X be the threefold defined by (1). Then we have H 2 (X, Z) = Z[F ] ⊕ Z[H 1 ] ⊕ Z[H 2 ] ⊕ 2 i=1 24 j=1 Z[E i,j ] −2[H 1 ] + 24 j=1 [E 1,j ] = −2[H 2 ] + 24 j=1 [E 2,j ] ) , where F is the class of a fiber of the first projection X → P 1 and H 1 (resp. H 2 ) is the pullback of the class of a line in P 2 via the composition X → P 2 × P 2 pr1 − − → P 2 (resp. X → P 2 × P 2 pr2 − − → P 2 ). Proof. Let X • = X \ 2 i=1 24 j=1 E i,j . The long exact sequence for cohomology with supports yields (2) 0 → 2 i=1 24 j=1 Z[E i,j ] → H 2 (X, Z) → H 2 (X • , Z) → 0. Let Y • = Y \ 2 i=1 24 j=1 F i,j . Since X • is the quotient of Y • by the group ι = Z/2, which acts freely, we can apply the Cartan-Leray spectral sequence E p,q 2 = H p (Z/2, H q (Y • , Z)) ⇒ H p+q (X • , Z). We have H 1 (Y • , Z) = H 1 (Y min , Z) = 0 by the Lefschetz hyperplane section theorem, so we have a short exact sequence 0 → Z/2 → H 2 (X • , Z) → H 2 (Y • , Z) ι → 0. The long exact sequence for cohomology with supports yields 0 → 2 i=1 24 j=1 Z[F i,j ] → H 2 (Y, Z) → H 2 (Y • , Z) → 0. Applying the Lefschetz hyperplane section theorem to Y min , it is straightforward to compute H 2 (Y, Z) = Z[F ] ⊕ Z[H 1 ] ⊕ Z[H 2 ] ⊕ 2 i=1 24 j=1 Z[F i,j ] −[H 1 ] + 24 j=1 [F 1,j ] = −[H 2 ] + 24 j=1 [F 2,j ] . Thus we obtain H 2 (Y • , Z) = Z[F ] ⊕ Z[H 1 ] ⊕ Z[H 2 ] [H 1 ] = [H 2 ] and H 2 (Y • , Z) is ι-invariant. This, combined with the equality −2[H 1 ] + 24 j=1 [E 1,j ] = −2[H 2 ] + 24 j=1 [E 2,j ] in H 2 (X, Z), implies that H 2 (X • , Z) = Z[F ] ⊕ Z[H 1 ] ⊕ Z[H 2 ] 2[H 1 ] = 2[H 2 ] , and the claim follows immediately from (2). Proof of the main theorem We will now prove Theorem 1.2 over the complex numbers. Theorem 3.1. Let X ⊂ P 1 × P 2 × P 2 be the rank one degeneracy locus of a very general map of vector bundles O ⊕3 → O(2, 2, 0) ⊕ O(2, 0, 2). Then the first projection gives a family X → P 1 of Enriques surfaces such that any multi-section has even degree over the base P 1 . That is, the index I(X η ) is even, where X η is the generic fiber. Proof. The first goal will be to prove that for any 1-cycle α on X and for any 12-tuple of integers 1 ≤ j 1 < · · · < j 12 ≤ 24, there is a congruence deg(α/P 1 ) ≡ 12 k=1 α · E 1,j k mod 2.(3) These congruences will imply the theorem. Indeed, from (3) we obtain α · E 1,1 ≡ · · · ≡ α · E 1,24 mod 2,(4) which in turn implies that deg(α/P 1 ) is even. To prove the congruence (3), we combine monodromy and specialization arguments. First, we prove that a certain monodromy group acts on the set of 24 planes E 1,1 , . . . , E 1,24 by permutations, and every permutation of the E 1,j is realized by this action. This will allow us to reduce to proving (3) for a fixed 12-tuple 1 ≤ j 1 < · · · < j 12 ≤ 24. Consider the universal family X → G = Gr(3, H 0 (P 1 × P A , O(2, 1))) of complete intersections in P 1 × P A of three divisors of type (2,1). Let E 1 denote the pullback of the Cartier divisor E 1 via the projection map X → P A . The corresponding family E 1 → G is the union of the planes E 1,1 , . . . , E 1,24 in the fibers of X → G. Let G → G be the Stein factorization of E 1 → G, which is a finite morphism of degree 24, and let U ⊂ G be the largest open set over which G → G isétale. We will now prove the following: Lemma 3.2. The monodromy representation ρ : π 1 (U ) → S 24 , uniquely determined up to the choice of a base point, is surjective. Proof. Recall from Section 2 that the planes E 1,1 , . . . , E 1,24 are parameterized by the 24 intersection points of three divisors of type (2, 2) in P 1 × P 2 . To prove the lemma, we restrict over a certain line l on G defined as follows. Let l ⊂ P 1 × P 2 be a general complete intersection of two divisors of type (2,2). Taking a general pencil in \|O l (2, 2)\|, we obtain a Lefschetz pencil l → P 1 by [11, Theorem XVII. 2.5]. This defines a line l ⊂ Gr(3, H 0 (P 1 × P 2 × P 2 , O(2, 2, 0))) ⊂ G such that l = l × G G, where the inclusion between the Grassmannians is via the identification H 0 (P 1 × P A , O(2, 1)) = H 0 (P 1 × P 2 × P 2 , O(2, 2, 0)) ⊕ H 0 (P 1 × P 2 × P 2 , O(2, 0, 2)). We let l • = l ∩ U ; this is the maximal open set where l → l isétale. We claim that the induced monodromy representation ρ l • : π 1 (l • ) → S 24 is surjective. Indeed, π 1 (l • ) is generated by loops around branch points b ∈ B of l → l, and the image of each such loop is a transposition in S 24 . The image of ρ l • is moreover transitive since l is irreducible. Any transitive subgroup of S 24 which is generated by transpositions must be S 24 itself, so it follows that ρ : π 1 (U ) → S 24 is surjective. By the above lemma, we reduce to proving the congruence (3) for a single 12tuple 1 ≤ j 1 < · · · < j 12 ≤ 24. Indeed, if g ∈ π 1 (U ) is a lift of a permutation σ ∈ S 24 , then it will imply for any 1-cycle α on X, there is a congruence deg(α/P 1 ) = deg(g * (α)/P 1 ) ≡ 12 k=1 g * (α) · E j k ≡ 12 k=1 α · E σ −1 (j k ) mod 2. Here we have used g * (α)·E σ(j) = α·E j for each j, and the fact that g * (α) is again an algebraic cycle because X is very general. We also have deg(g * (α)/P 1 ) = deg(α/P 1 ), because the degree is obtained by intersecting with the class of a fiber over P 1 , which is invariant under monodromy. Letting σ run over all permutations, we see that the congruence will hold for all 12-tuples. To finish the proof of the congruence (3), we use a specialization argument. We consider X as the complete intersection of three divisors D 1 , D 2 , D 3 in \|O(2, 1)\| on P 1 × P A . If we degenerate each D i to a union D ′ i + D ′′ i , where D ′ i ∈ \|O(1, 1)\| and D ′′ i ∈ \|O(1, 0)\| are very general divisors, we obtain a family of threefolds X T → T , with special fiber equal to X 0 ∪ R 1 ∪ R 2 ∪ R 3 , where X 0 is a very general intersection of three divisors in \|O(1, 1)\|, and R 1 , R 2 , R 3 are intersections of two divisors of type O(1, 1) and one of type O(1, 0). In particular, the R i are pairwise disjoint and can be regarded as complete intersections of two relative hyperplane sections in P A . By the geometric construction in Section 2, we may regard X 0 as the rank one degeneracy locus in P 1 × P 2 × P 2 of a very general map of vector bundles O ⊕3 → O(1, 2, 0) ⊕ O(1, 0, 2). By construction, X 0 is also the only dominant component with respect to the projection X 0 ∪R 1 ∪R 2 ∪R 3 → P 1 . Furthermore, again by genericity, we may assume that X 0 ∪ R 1 ∪ R 2 ∪ R 3 is a simple normal crossing variety and the intersection (X 0 ∪ R 1 ∪ R 2 ∪ R 3 ) ∩ (P 1 × E 1 ) is transversal. This degeneration allows us to specialize cycles on X to cycles on X 0 ∪ R 1 ∪ R 2 ∪ R 2 . On the level of divisors, the union of 24 components E 1,1 , . . . , E 1,24 on X specializes to the union of 12 components E 1,4 on R l for l = 1, 2, 3 given by the intersections with P 1 × E 1 . Thus the chosen specialization gives a 12-tuple 1 ≤ j 1 < · · · < j 12 ≤ 24 such that E 1,j1 , . . . , E 1,j12 specialize to E CH 1 (X) → CH 1 (X 0 ∪ R 1 ∪ R 2 ∪ R 3 ) which is compatible with intersections with Cartier divisors. If α 0 is the specialization of a 1-cycle α on X, we may write α 0 = α (0) 0 + α (R) 0 , where α (0) 0 is a 1-cycle on X 0 and α (R) 0 is supported in R 1 ∪ R 2 ∪ R 3 . Now Note that deg(α/P 1 ) = deg(α (0) 0 /P 1 ) and α · E 1,j k = α 0 · E (0) 1,k = α (0) 0 · E (0) 1,k since E (0) 1,k is disjoint from R 1 , R 2 and R 3 . Thus from the congruence (5), we deduce the congruence (3) for 1 ≤ j 1 < · · · < j 12 ≤ 24. This completes the proof. Theorem 3.1 can be generalized to higher dimensions: Theorem 3.3. For a positive integer n, we let X ⊂ P 1 × P 2n × P 2n be the rank one degeneracy locus of a very general map of vector bundles O ⊕(2n+1) → O(2, 2, 0) ⊕ O(2, 0, 2). Then the first projection gives a family X → P 1 of O-acyclic 2n-folds such that any multi-section has even degree over the base P 1 . That is, the index I(X η ) is even, where X η is the generic fiber. Proof. The geometry of the family of O-acyclic 2n-folds is similar to that of Lemma 2.2. An alternative projective model of X is given by a complete intersection in P 1 × P P 2n ×P 2n (O(2, 0) ⊕ O(0, 2)) of (2n + 1) divisors of type (2, 1), and the intersection X ∩ (P 1 × P P 2n ×P 2n (O(2, 0))) consists of (2n + 1)2 2n+1 components E 1,1 , . . . , E 1,(2n+1)2 2n+1 . The theorem follows from a congruence deg(α/P 1 ) ≡ (2n+1)2 2n k=1 α · E 1,j k mod 2 for any 1-cycle α on X and for any (2n + 1)2 2n -tuple 1 ≤ j 1 < · · · < j (2n+1)2 2n ≤ (2n + 1)2 2n+1 . We leave the details of the proof to the reader. Degenerations and examples over Q We now explain how to give examples as in Theorem 1.2 defined over the rational numbers. The construction is similar to the one used in the previous section, but the degeneration argument now uses Enriques fibrations defined in terms 2 × 3-minors, rather than complete intersections of three divisors. We will work over Q and set P 1 × P 2 × P 2 = Proj Q[s, t] × Proj Q[x 0 , x 1 , x 2 ] × Proj Q[y 0 , y 1 , y 2 ]. The goal is to prove the following: given by the matrix M = sp 0 + pr 0 (s − t)p 1 + pr 1 (s + t)p 2 + pr 2 stq 0 + ps 0 t(s − t)q 1 + ps 1 t(s + t)q 2 + ps 2 , then the first projection gives a family X → P 1 of Enriques surfaces such that any multi-section has even degree over the base P 1 . That is, the index I(X η ) is even, where X η is the generic fiber. Note that for general p i , q i , r i , s i defined over Q and large p, the threefold X is smooth and irreducible. In order to prove Theorem 4.1, it will be convenient to introduce the following 1-dimensional family of degeneracy loci of vector bundles O ⊕3 → O(2, 2, 0) ⊕ O(2, 0, 2) on P 1 × P 2 × P 2 . We set B = Proj Q[λ, µ] and define the total space X as the subvariety of B × P 1 × P 2 × P 2 defined by the maximal minors of the matrix M (λ,µ) = λsp 0 + µr 0 λ(s − t)p 1 + µr 1 λ(s + t)p 2 + µr 2 λstq 0 + µs 0 λt(s − t)q 1 + µs 1 λt(s + t)q 2 + µs 2 , where the p i , q i , r i , s i have tridegrees (1, 2, 0), (0, 0, 2), (2,2,0) and (2, 0, 2) respectively. Let X → B denote the natural projection map onto the first factor. By construction, the generic fiber X ηB is a smooth threefold with an Enriques surface fibration X ηB → P 1 ηB . The morphism X → B is flat outside of the fiber (λ, µ) = (1, 0); we will compute the flat closure of X ηB in B × P 1 × P 2 × P 2 below. In any case, in order to prove Theorem 4.1, we will mainly be interested in the fiber over (λ, µ) = (1, p). For now, let E 1 ⊂ X denote the codimension 1 subscheme defined by the top row of M (λ,µ) , i.e., λsp 0 + µr 0 = λ(s − t)p 1 + µr 1 = λ(s + t)p 2 + µr 2 = 0. Proof. Note that B is defined by (8) inside B × P 1 × P 2 . It is straightforward to check that the cover B → B is Lefschetz for general p i , r i . Now the assertion follows from an argument similar to that in the proof of Theorem 3.1. We note that πé t 1 (B • ) is generated by loops around the branch points of B → B [16, XIII, Corollaire 2.12] and the image of each loop is a transposition in S 24 . The parameter space for the families of threefolds given by (7) is a certain rational variety, hence has a Zariski dense set of Q-rational points. As a consequence, we can choose p i , q i , r i , s i defined over Q such that ρ is surjective. We will therefore in the following choose p i , q i , r i , s i satisfying the above conditions: thus for the family X → B over Q, the generic fiber is smooth and irreducible; E 1 is smooth and irreducible; and the monodromy map ρ : πé t 1 (B • ) → S 24 is surjective.ρ x : πé t 1 (x, x) = Gal(Q/Q) → S 24 is surjective. Proof. The setting resembles that of [28, Section 1] (but is more classical). By Hilbert's irreducibility theorem, the set {x ∈ B • (Q) \| ρ x is surjective} is the complement of a thin set in B(Q) = P 1 (Q). Moreover, the complement of a thin set in P 1 (Q) contains infinitely many points with (λ, µ) = (1, p) for some prime number p (see [24, Section 9.6, Theorem]), which gives us the desired conclusion. To conclude the proof of Theorem 4.1, we again use a specialization argument as in Theorem 3.1. We begin by computing the flat limit of the family X → B over (λ, µ) = (1, 0). Note that X contains {µ = t = 0} as a component. Removing this component reveals that the flat closure of X ηB in B × P 1 × P 2 × P 2 is defined by the 3 × 3-minors of the matrix   λsp 0 + µr 0 λ(s − t)p 1 + µr 1 λ(s + t)p 2 + µr 2 0 λsq 0 λ(s − t)q 1 λ(s + t)q 2 µ s 0 s 1 s 2 −t   The corresponding family X → B is flat and has special fiber X 0 over (λ, µ) = (1, 0) given by a union X 0 ∪ R 0 ∪ R 1 ∪ R 2 ∪ R 3 , where X 0 is given by the minors of the matrix N = p 0 p 1 p 2 q 0 q 1 q 2 , R 0 is given by t = det   p 0 p 1 p 2 q 0 q 1 q 2 s 0 s 1 s 2   = 0, and R 1 , R 2 , R 3 are respectively given by s = p 2 q 1 − p 1 q 2 = 0, s − t = p 2 q 0 − p 0 q 2 = 0, s + t = p 1 q 0 − p 0 q 1 = 0. Note that the R i are pairwise disjoint, and X 0 is regular if the p i , q i are general. Similarly, the subfamily E 1 → B, given by (7), has a special fiber over (λ, µ) = (1, 0) which consists of the union of 12 components E 1,4 supported on R l for l = 1, 2, 3 respectively given by s = p 1 = p 2 = 0, s − t = p 0 = p 2 = 0, s + t = p 0 = p 1 = 0. It is important to note that E X 0 \ (R 0 ∪ R 1 ∪ R 2 ∪ R 3 ) which is regular. Let p and x ∈ B be as in Lemma 4.3. For any valuation ring R ⊂ Q whose maximal ideal contains p, we have the following diagram of restrictions: X x = X x / / (X R ) (λ,µ)=(1,p) / / X R X o o x / / Spec R (λ,µ)=(1,p) / / B R B o o Spec F p O O / / Spec R (λ,µ)=(1,0) O O . Proof of Theorem 4.1. Let p i , q i , r i , s i be general and defined over Q. Let p be a sufficiently large prime number which satisfies Lemma 4.3 and let X = X x . We prove that any multi-section of X → P 1 has even degree over the base P 1 . As in the proof of Theorem 3.1, it is enough to prove, for any 1-cycle α on X and for any 12-tuple 1 ≤ j 1 < · · · < j 12 ≤ 24, a congruence deg(α/P 1 ) ≡ 12 k=1 α · E 1,j k mod 2. By Lemma 4.3, it suffices to verify this congruence for some 12-tuple 1 ≤ j 1 < · · · < j 12 ≤ 24. To establish this, we use the above family over Spec R, which allows us to specialize cycles from X to cycles on ((X R ) (λ,µ)=(1,p) ) Fp . For a sufficiently large valuation ring R ⊂ Q whose maximal ideal contains p, the specialization ((E 1 ) R ) (λ,µ)=(1,p) ⊂ (X R ) (λ,µ)=(1,p) is a disjoint union of 24 components E 1,1 , . . . , E 1,24 , each of which is isomorphic to P 2 R . Let E 1,j1 , . . . , E 1,j12 be the components which restrict to E (0) 1,1 , · · · , E (0) 1,12 on the special fiber ((X R ) (λ,µ)=(1,p) ) Fp . Then E 1,j1 , . . . , E 1,j12 ⊂ (X R ) (λ,µ)=(1,p) are Cartier divisors since they are supported on (X R ) (λ,µ)=(1,p) \ (R 0 ∪ R 1 ∪ R 2 ∪ R 3 ) which is regular. Now by the specialization homomorphism for Chow groups [13,Ex. 20.3.5], the desired congruence follows from a congruence in the proof of [20, Theorem 3.1]: we have, for any 1-cycle α 0 on X 0 , a congruence deg( α 0 /P 1 ) ≡ 12 j=1 α 0 · E (0) 1,j mod 2. The proof is complete. The above proof uses a specialization argument which does not extend in general to other fields. One natural question is whether one can find such examples defined over the algebraic closure of a finite field. In contrast to the examples above, we prove some positive results in this situation, conditional on the Tate conjecture. We recall that the Tate conjecture in degree 2i on a smooth projective variety V over a finite field k of characteristic p asserts that the image of the cycle class map Proposition 4.4. Let X be a smooth projective variety over F p with fibration X → C over a smooth projective curve C. Assume that (1) the generic fiber X η is smooth with χ(O Xη ) = 1; cl i ⊗Q l : CH i (V k ) ⊗ Q l → H 2í et (V k , Q l (i)) (2) b 2 = ρ on X, where b 2 is the second Betti number and ρ is the Neron-Severi rank; (3) the Tate conjecture holds in degree 2 on surfaces over finite fields of characteristic p. Then the fibration X → C admits multi-sections whose degrees over the base C add up to a power of p. That is, the index I(X η ) is a power of p, where X η is the generic fiber. Remark 4.5. A similar assertion was proved by Colliot-Thélène and Szamuely [8, Theorem 6.1], where, among other things, the torsion-freeness of the Picard group Pic(X η ) of the geometric generic fiber X η is assumed. Proof of Proposition 4.4. Let X → C be a fibration as in the statement and d = dim X. Under the assumption (1), the Riemann-Roch formula together with the Poincaré duality shows that the push-forward homomorphism H 2d−2 (X, Z l (d − 1)) → H 0 (C, Z l ) = Z l is surjective for any prime l = p (the arguments are analogous to the proofs of Proposition A.6 and Corollary A.7 due to Wittenberg). On the other hand, if b 2 = ρ on X, the cokernel of the cycle class map cl 2d−2 ⊗Z l : CH 2d−2 (X) ⊗ Z l → H 2d−2 (X, Z l (d − 1)) is finite by the hard Lefschetz theorem due to Deligne. If we further assume that the Tate conjecture holds in degree 2 on surfaces over finite fields of characteristic p, then the integral Tate conjecture holds in degree 2d − 2 on X (viewed as the base extension of a smooth projective variety over a finite field of characteristic p), according to a theorem of Schoen [22,Theorem 0.5]. This implies that the cokernel of cl 2d−2 ⊗Z l is torsion-free, hence cl 2d−2 ⊗Z l is surjective. Combined with the argument in the previous paragraph, the statement now follows. Corollary 4.6. Let X → C be a one-parameter family of O-acyclic varieties over Q. Assume that the Tate conjecture holds in degree 2 on surfaces over finite fields. Then the reduction X p → C p over F p admits multi-sections with coprime degrees over the base C p for any large prime number p. That is, I((X p ) η ) = 1, where (X p ) η is the generic fiber. Proof. We note that the O-acyclicity of fibers of the family X → C implies b 2 = ρ on X by [9, Proposition 7.3], thus we also have b 2 = ρ on the good reductions of X by specialization. Now the statement is immediate from Proposition 4.4 by observing that there exist 1-cycles on X p obtained by spreading out 1-cycles on X over valuations rings inside Q, whose degrees over the base C p do not depend on p. Failure of the Hasse principle and the reciprocity obstruction The reciprocity obstruction to the Hasse principle for 0-cycles of degree 1 on a smooth projective geometrically connected variety Z over the function field F = C(C) of a complex curve C was defined and pointed out to the authors by Colliot-Thélène (see also [7,Section 5]). We explain the construction in the following. We will assume that H 1 et (Z F , Z/2) = Z/2 for simplicity. The Leray spectral sequence for theétale sheaf Z/2 and the morphism Z → Spec F yields a short exact sequence for any p ∈ C, where F p ∼ = C((t)) is the completion of F at p. The local evaluation maps are identically zero for all but finitely many p ∈ C by an argument of good reduction. The diagonal embedding 0 → H 1 et (F, Z/2) → H 1 et (Z, Z/2) → H 1 et (Z F , Z/2) → 0.F ֒→ p∈C F p yields a complex H 1 et (F, Z/2) → p∈C H 1 et (F p , Z/2) → Z/2 that is F * /F * 2 → p∈C Z/2 → Z/2, where the first map is induced by the divisor map and the second map is the summation map. Then it follows that the image of the diagonal map CH 0 (Z) → p∈C CH 0 (Z Fp ) is contained in the kernel of the sum of the local evaluations θ : As a consequence of our construction in Section 3, we prove that the failure of the Hasse principle for 0-cycles of degree 1 on an Enriques surface over C(P 1 ) cannot always be accounted for by the reciprocity obstruction. p∈C CH 0 (Z Fp ) → Z/2. Theorem 5.2. Let X η be the generic fiber of the family X → P 1 of Enriques surfaces of Theorem 3.1. Then the Hasse principle fails for 0-cycles of degree 1 on X η , while the assumption of Proposition 5.1 is not satisfied. Remark 5.3. In fact, a direct computation shows that X η has rational points everywhere locally. Hence the Hasse principle already fails for rational points on X η . The proof in the following also shows that there is no reciprocity obstruction to the Hasse principle for rational points on X η . Therefore it follows that the reciprocity obstruction to the Hasse principle for rational points on an Enriques surface over C(P 1 ) is not the only obstruction. Proof of Theorem 5.2. Let F = C(P 1 ). Theorem 3.1 shows that there is no 0-cycle of degree 1 on X F . On the other hand, it is automatic from the O-acyclicity of Enriques surfaces and the Riemann-Roch theorem that the family X → P 1 has no multiple fiber (in fact this is easy to see directly from the defining equations). It then follows from Hensel's lemma that there is a 0-cycle of degree 1 on X Fp for any p ∈ P 1 . Therefore the Hasse principle fails for 0-cycles of degree 1 on X F . By choosing a lift ξ ∈ H 1 et (X F , Z/2) of the non-zero class in H 1 et (X F , Z/2) = Z/2, we obtain the map θ in Proposition 5.1. To see that X F does not satisfy the assumption of Proposition 5.1, it is enough to verify the following: for each i and j, if p i,j ∈ P 1 is the image of E i,j , then the local evaluation map CH 0 (X Fp i,j ) → H 1 et (F pi,j , Z/2) = Z/2 restricts to a surjection on 0-cycles of degree 1; this will then provide a family {α p } p∈C of 0-cycles of degree 1 such that θ({α p }) = 0 ∈ Z/2. Recall that by construction in Section 2, X admits a natural double cover Y → X over P 1 and the cover is ramified along F i,j and branched over E i,j . Then one can in fact assume that ξ is given by theétale double cover Y • → X • , where X • = X \ E i,j and Y • = Y \ F i,j , since evaluation maps only differ by classes in H 1 et (F, Z/2). For each i and j, working locally around p i,j , we consider the base change of the Enriques fibration X → P 1 X × P 1 Spec O P 1 ,pi,j → Spec O P 1 ,pi,j , where O P 1 ,pi,j is the completion of the local ring O P 1 ,pi,j . One can compute that the special fiber is reduced and consists of E i,j and a residual component R i,j . Then, by Hensel's lemma, there is a section S 1 (resp. S 2 ) which intersects transversally with E i,j (resp. R i,j ) at one point. Now we consider the double cover Y × P 1 Spec O P 1 ,pi,j → X × P 1 Spec O P 1 ,pi,j , whose branched locus is E i,j . Then it is straightforward to see that the inverse image of S 1 gives degree 2 integral multi-section, while that of S 2 splits into two disjoint sections. Therefore the F pi,j -rational points of X Fp i,j corresponding to the sections S 1 and S 2 take values 1 and 0 in Z/2 respectively under the local evaluation map. This concludes that X F does not satisfy the assumption of Proposition 5.1, hence the proof of the theorem. Proof. By Lemma 2.2, the Hodge structure of H 4 (X, Z) is trivial and H 4 (X, Z) is free of rank 50. By the Tor long exact sequence, we have Z 4 (X) [2] = Ker(H 4 alg (X, Z)/2 → H 4 (X, Z/2)). We define H 4 alg (X, Z/2) = Im cl 2 ⊗Z/2 : CH 2 (X)/2 → H 4 (X, Z/2) . Since H 4 alg (X, Z)/2 = (Z/2) 50 , we are reduced to proving that H 4 alg (X, Z/2) = (Z/2) 4 . We first prove that Im CH 2 (X)/2 cl ⊗Z/2 − −−−− → H 4 (X, Z/2) (iX ) * −−−→ H 10 (P 1 × P A , Z/2) = (Z/2) 2 , where i X : X → P 1 × P A is the inclusion map. The rank of the image is ≤ 2 as a result of Theorem 3.1 and the congruence (4) in the proof. Thus it suffices to find two linearly independent classes in the image. It is easy to see that lines l 1 ⊂ E 1,1 and l 2 ⊂ E 2,1 give such classes. We define H 4 van (X, Z/2) = Ker (i X ) * : H 4 (X, Z/2) → H 10 (P 1 × P A , Z/2) . We have rank H 4 van (X, Z/2) = 46. Indeed, it is enough to observe that the pushforward homomorphism (i X ) * : H 4 (X, Z) → H 10 (P 1 × P A , Z) is surjective, which follows from the fact that the pullback homomorphism (i X ) * : H 2 (P 1 × P A , Z) → H 2 (X, Z) is injective with torsion-free cokernel by Lemma 2.3. We prove that H 4 van (X, Z/2) is generated by classes c i,j1,j2 ∈ H 4 (X, Z) (i = 1, 2, 1 ≤ j 1 < j 2 ≤ 24) with intersection properties c i,j1,j2 · E i ′ ,j ′ = δ i,i ′ · (δ j1,j ′ − δ j2,j ′ ), c i,j1,j2 · F = c i,j1,j2 · H 1 = c i,j1,j2 · H 2 = 0. It is enough to show that H 4 van (X, Z) = Ker (i X ) * : H 4 (X, Z) → H 10 (P 1 × P A , Z) , which is of rank 46, is generated by the above classes. Let i Y : Y → P 1 × P B be the inclusion map and let H 4 van (Y, Z) = Ker (i Y ) * : H 4 (Y, Z) → H 10 (P 1 × P B , Zd i,j1,j2 · F i ′ ,j ′ = δ i,i ′ · (δ j1,j ′ − δ j2,j ′ ), d i,j1,j2 · F = d i,j1,j2 · H 1 = d i,j1,j2 · H 2 = 0, which is immediate. We prove that H 4 alg (X, Z/2) ∩ H 4 van (X, Z/2) = (Z/2) 2 . We note that we have the congruence (4) in the proof of Theorem 3.1, and moreover, we may also assume a congruence α · E 2,1 ≡ · · · ≡ α · E 2,24 mod 2 (9) for any 1-cycle α on X. Then the congruences (4) and (9) imply rank H 4 alg (X, Z/2) ∩ H 4 van (X, Z/2) ≤ 2. Now it is enough to find two linearly independent classes in H 4 alg (X, Z/2)∩H 4 van (X, Z/2). It is a simple matter to check that C 1 = (H 1 ) 2 and C 2 = (H 2 ) 2 indeed give such classes. It follows that Let n be a positive integer. We recall that the degree 3 unramified cohomology group H 3 nr (X, Z/n) for a smooth projective variety X is defined to be H 3 nr (X, Z/n) = H 0 (X Zar , H 3 (Z/n)), where H 3 (Z/n) is the Zariski sheaf associated to the presheaf U → H 3 (U, Z/n) [5]. The group H 3 nr (X, Z/n) is a stable birational invariant of smooth projective varieties [5,Theorem 4.2]. As an application of the Bloch-Kato conjecture settled by Voevodsky, it was proved by Colliot-Thélène and Voisin [9, Theorem 3.9] that we have H 3 nr (X, Z/n) = Z 4 (X)[n] if CH 0 (X) is supported on a surface. This theorem, together with Lemma 2.2 (1) and Theorem 6.1, implies: In this appendix, we prove that the vanishing of the reciprocity obstruction to the existence of a 0-cycle of degree 1 is a general fact that holds for all O-acyclic varieties over the function field F of a complex curve, and, in fact, for all smooth proper varieties Y over F such that χ(Y, O Y ) = 1. We actually prove the following slightly more general statement, in the spirit of [12]. In the situation of Theorem A.1, local 0-cycles of degree χ(Y, E) had previously been shown to exist in [12,Theorem 1]. It may seem surprising that the existence of a collection of local 0-cycles that globally survives the reciprocity obstruction comes "for free", without having to make any additional assumption on Y , especially in view of the negative answer to Question 1.1 now provided by Ottem and Suzuki. A.1. Recollections on the reciprocity obstruction. Let us first recall how the pairing (10), introduced by Colliot-Thélène and Gille in [7, §5], is defined. For p ∈ C(C), the Galois cohomology group H 1 (F p , Q/Z(1)), where Q/Z(1) denotes the torsion subgroup of C * , is canonically isomorphic to Q/Z. We denote this canonical isomorphism by inv p : H 1 (F p , Q/Z(1)) ∼ −→ Q/Z. Mapping a closed point q ∈ Y Fp and a class β ∈ H 1 et (Y Fp , Q/Z(1)) to inv p Cores Fp(q)/Fp β(q) ∈ Q/Z, Cartier divisor X p,i ⊂ X p . The map Z h 1 → Z n , z → ((z · X p,1 ), . . . , (z · X p,n )) is surjective as a consequence of [6, 9.1/9], and fits into a commutative diagram Z n Z h 1 o o o o / / / / CH 0 (Y Fp ) cl (2 ′′ ) Same as (2 ′ ), except that we impose, in addition, that the image of α in The Hirzebruch-Riemann-Roch theorem applied to the locally free sheaves V and V ⊗ OX O X (H) on X therefore gives us the equality χ(X, V ⊗ OX O X (H)) − χ(X, V ) = deg(ch(V ) · (H + H 2 /2 + · · · ) · Td(T X )) = deg(H · Z) = 1 n deg(F · Z), where Z ∈ CH 1 (X) ⊗ Z Q denotes the 1-dimensional component of ch(V ) · Td(T X ). By the Hirzebruch-Riemann-Roch theorem applied to the locally free sheaf E on Y , we also have the equality χ(Y, E) = deg(ch(E) · Td(T Y )), which can be rewritten as χ(Y, E) = deg(F · Z) since Td(T X )\| Y = Td(T Y ) and ch(V )\| Y = ch(E). Hence χ(X, V ⊗ OX O X (H)) − χ(X, V ) = 1 n χ(Y, E)(22) and we conclude that n divides χ(Y, E) since the left-hand side is an integer. Corollary A.7. For any coherent sheaf E on Y , there exists α ∈ H 2d (X(C), Z(d)) such that f * α = χ(Y, E) in H 0 (C(C), Z) = Z. Proof. According to Proposition A.6, the integer χ(Y, E) annihilates the kernel of f * : NS(C) ⊗ Z Q/Z → NS(X) ⊗ Z Q/Z. As NS(C) ⊗ Z Q/Z = H 2 (C(C), Q/Z(1)) and NS(X) ⊗ Z Q/Z ⊂ H 2 (X(C), Q/Z(1)), the latter kernel coincides with the kernel of f * : H 2 (C(C), Q/Z(1)) → H 2 (X(C), Q/Z(1)). Thus, Poincaré duality implies that χ(Y, E) also annihilates the cokernel of f * : H 2d (X(C), Z(d)) → H 0 (C(C), Z). Combining Proposition A.4 with Corollary A.7 now yields Theorem A.1. ( 3 )( 4 ) 34The topological Euler characteristic equals χ top (X) = −96 and Hodge dia-X is simply connected and the cohomology groups H i (X, Z) are torsion-free for all i.Proof. The arguments are entirely analogous to those in[20, Section 2], where the case of the rank one degeneracy locus of a general map of vector bundles 13, Section 20.3] there is moreover a specialization map of Chow groups we recall a key congruence obtained in the course of the proof of [20, Theorem 3.1]: we have, for any 1-cycle α Theorem 4. 1 . 1Let p i , q i , r i , s i (i = 0, 1, 2) be general homogeneous polynomials of tridegree (1, 2, 0), (0, 0, 2), (2, 2, 0), (2, 0, 2) in variables s, t, x i , y i defined over Q. Then there exists a prime number p such that, if X ⊂ P 1 × P 2 × P 2 is the rank one degeneracy locus of a map of vector bundles O ⊕3 → O(2, 2, 0) ⊕ O(2, 0, 2) , E 1 is smooth and irreducible for general p i , r i . Let E 1 → B → B denote the Stein factorization of E 1 → B. The morphism B → B is finite of degree 24; over a general point b ∈ B the fiber corresponds to the 24 distinct planes E 1,1 , . . . , E 1,24 in X b . We let B • ⊂ B denote the maximal open set over which B → B isétale. There is an associated monodromy representation ρ : πé t 1 (B • ) → S 24 . Lemma 4. 2 . 2For general p i , r i as above, the map ρ is surjective. Lemma 4 . 3 . 43There are infinitely many prime numbers p such that if x ∈ B is given by (λ, µ) = (1, p), then the induced map 12 are Cartier divisors on X 0 since they are supported on generates the subspace of classes in H 2í et (V k , Q l (i)) fixed by some open subgroup of Gal(k/k) for any prime number l = p. The integral Tate conjecture is an integral analogue of the Tate conjecture (with Z l instead of Q l ). Note that the Galois group Gal(F /F ) acts trivially on H 1 et (Z F , Z/2) = Z/2. We then choose a lift ξ ∈ H 1 et (Z, Z/2) of the non-trivial class in H 1 et (Z F , Z/2) = Z/2. The evaluation pairing Z(F ) × H 1 et (Z, Z/2) → H 1 et (F, Z/2) extends to an evaluation pairing on the Chow group of 0-cycles CH 0 (Z) × H 1 et (Z, Z/2) → H 1 et (F, Z/2). Thus we get the evaluation map of ξ CH 0 (Z) → H 1 et (F, Z/2). Similarly, we get the local evaluation map of ξ CH 0 (Z Fp ) → H 1 et (F p , Z/2) = Z/2 Proposition 5. 1 ( 1Reciprocity obstruction). If for each family {α p } p∈C of 0-cycles of degree 1, we have θ({α p }) = 1 ∈ Z/2, then there is no 0-cycle of degree 1 on Z. 6 . 6Defect of the integral Hodge conjecture in degree 4 and degree 3 unramified cohomology with Z/2 coefficients Theorem 6.1. Let X be the total space of the family of Enriques surfaces of Theorem 3.1. Then we have Z 4 (X)[2] = (Z/2) 46 . H 4 alg (X, Z/2) = Z/2[l 1 ] ⊕ Z/2[l 2 ] ⊕ Z/2[C 1 ] ⊕ Z/2[C 2 ] = (Z/2) 4 .The proof is complete. Corollary 6 . 2 . 62Let X be the total space of the family of Enriques surfaces of Theorem 3.1. Then we have H 3 nr (X, Z/2) = (Z/2) 46 .Appendix A. Vanishing of the reciprocity obstruction by Olivier Wittenberg Theorem A. 1 . 1Let F = C(C) be the function field of a smooth proper irreducible complex curve C. Let Y be a smooth proper variety over F and E be a coherent sheaf on Y . Then there exists a collection (α p ) p∈C(C) ∈ p∈C(C) CH 0 (Y Fp ) of local 0-cycle classes of degree χ(Y, E) that belongs to the left kernel of the natural pairing p∈C(C) CH 0 (Y Fp ) × H 1 et (Y, Q/Z(1)) → Q/Z. (10) In other words, there is no reciprocity obstruction to the existence of a 0-cycle of degree χ(Y, E) on Y . Theorem A.1 builds on a purely cohomological reinterpretation of the reciprocity obstruction (presented in § §A.1-A.4) and on a variant of an argument of Colliot-Thélène and Voisin itself based on the Riemann-Roch theorem (see §A.5). ) . The group H 4 van (Y, Z) has rank 46. Using Lemma 2.3, it is straightforward to see that Coker f * :H 4 van (Y, Z) → H 4 van (X, Z) = (Z/2) 46 , where f : Y → X is the natural map, thus the push-forward homomorphism f * : H 4 van (Y, Z) → H 4van (X, Z) can be identified with the multiplication homomorphism Z 46 ×2 − − → Z 46 . Now it is enough to observe that H 4 van (Y, Z) is generated by classes d i,j1,j2 ∈ H 4 (Y, Z) (i = 1, 2, 1 ≤ j 1 < j 2 ≤ 24) with intersection properties Acknowledgements. We would like to thank Lawrence Ein, János Kollár, Jørgen Vold Rennemo, and Burt Totaro for interesting discussions and useful suggestions. We wish to thank Jean-Louis Colliot-Thélène for many helpful correspondences and for encouraging us to check whether there is any reciprocity obstruction on our examples, which led to Theorem 5.2. We are grateful to Olivier Wittenberg for his remarks and for kindly agreeing to write an appendix for our paper. Finally, we thank the referee for careful reading and valuable comments.where Cores denotes the corestriction map in Galois cohomology, uniquely extends to a bilinear pairingDenoting the latter by angle brackets, the pairing (10) is then defined as the sumwhich can be checked to have only finitely many non-zero terms.The "reciprocity law", in this context, is the equalityvalid for any global class γ ∈ H 1 (F, Q/Z(1)), and which amounts to the assertion that any principal divisor on C has degree 0. Applied to γ = Cores F (q)/F β(q) for a closed point q ∈ Y , it implies that the diagonal map CH 0 (Y ) → p∈C(C) CH 0 (Y Fp ) takes values in the left kernel of (10). Equivalently, an element of p∈C(C) CH 0 (Y Fp ) that does not belong to the left kernel of (10) cannot come from CH 0 (Y ); in this situation one says that there is a "reciprocity obstruction".A.2. From Chow groups to cohomology. Let us fix a smooth and proper variety X over C and a morphism f : X → C with generic fibre Y . Let X p denote the fibre of X above p ∈ C. For any scheme Z of finite type over C, over F p or over O C,p , and all integers q, j, we set H q et (Z,Ẑ(j)) = limgiven by the proper base change theorem with the canonical identification between singular andétale cohomology H 2d (X p (C), Z(d)) ⊗ ZẐ = H 2d et (X p ,Ẑ(d)), we obtain a canonical injectionWe shall consider the pull-back mapand its compositionwith this injection.Proposition A.2. For any p ∈ C(C), the image of the cycle class map toétale cohomology cl :is equal to the image of (16).Proof. Let X p,1 , . . . , X p,n denote the irreducible components of X p , endowed with the reduced subscheme structure. Let Z h 1 be the group of horizontal 1-cycles on the scheme X × C Spec( O C,p ), that is, the group of those 1-cycles whose support is flatwhose middle vertical arrow is the cycle class map (see[21, (1.12)] for its definition), whose lower horizontal arrows are the injection(14)and the pull-back map(15), and whose leftmost vertical map comes from the canonical isomorphisms. The desired statement now follows from the diagram. A.3. Extension to a pairing between cohomology classes. Let us denote byconsisting of those families (α p ) p∈C(C) such that for all but finitely many p ∈ C(C), the class α p belongs to the image of the pull-back map(15). Letting f also stand for the morphisms Y → Spec(F ) and Y Fp → Spec(F p ) obtained from f : X → C by base change, we consider the pairingis induced by the trace morphism associated with f (see[where C ′ = RHom(C, Q/Z(1)), where D denotes the Pontrjagin dual, and where we still denote by C (resp. C ′ ) the pull-back of C (resp. C ′ ) to any of U , Spec(F ), Spec( O C,p ), Spec(F p ). The lemma now follows by considering the exact sequences associated in this way with C = Rf * Z/nZ(d)[2d] for n ≥ 1 and applying lim − →U lim ← −n to these sequences, in view of the canonical isomorphism C ′ = Rf * Z/nZ(1) given by Poincaré duality; as the groups H 0 et (U, C) = H 2d et (X U , Z/nZ(d)) are all finite, the resulting sequence is still exact (Mittag-Leffler criterion).A.5. Applying the Riemann-Roch theorem. The next statement and its proof are a variation on a result of Colliot-Thélène and Voisin [9, Proposition 7.3 (ii)], in the style of[12]. When E = O Y , its formulation is parallel to[12,Proposition 2.4].Proposition A.6. Let E be a coherent sheaf on Y and n ≥ 1 be an integer. If the class of the fibres of f : X → C in NS(X)/(torsion) is divisible by n, then χ(Y, E) is divisible by n.Proof. As E is the restriction of a coherent sheaf on X and as any coherent sheaf on X admits a finite resolution by locally free sheaves, we may assume that E is the restriction of a locally free sheaf V on X, which we henceforth fix.Let F be a fibre of f . By assumption, there exist divisors H and M on X such that the equality(21)holds in NS(X) and such that [M ] belongs to the torsion subgroup of NS(X). By the last condition, the divisor M is numerically trivial. Hence so is the cycle class H 2 ∈ CH 2 (X), in view of(21)and of the equality F 2 = 0 ∈ CH 2 (X). Analytic cycles on complex manifolds. M F Atiyah, F Hirzebruch, Topology. 1Atiyah, M. F., Hirzebruch, F.: Analytic cycles on complex manifolds, Topology 1 (1962), 25-45. E Ballico, F Catanese, C Ciliberto, Classification of irregular varieties. BerlinSpringer-Verlag1515Ballico, E., Catanese, F., Ciliberto, C. (eds): Classification of irregular varieties, Lecture Notes in Mathematics, 1515, Springer-Verlag, Berlin, 1992. 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[]	[ "Artur D'avila Garcez \nCity\nUniversity of London\nUK\n", "Luís C Lamb [email protected] \nFederal University of Rio Grande do Sul\nBrazil\n", "Artur D'avila Garcez \nCity\nUniversity of London\nUK\n", "Luís C Lamb [email protected] \nFederal University of Rio Grande do Sul\nBrazil\n" ]	[ "City\nUniversity of London\nUK", "Federal University of Rio Grande do Sul\nBrazil", "City\nUniversity of London\nUK", "Federal University of Rio Grande do Sul\nBrazil" ]	[]	Current advances in Artificial Intelligence (AI) and Machine Learning (ML) have achieved unprecedented impact across research communities and industry. Nevertheless, concerns about trust, safety, interpretability and accountability of AI were raised by influential thinkers. Many have identified the need for well-founded knowledge representation and reasoning to be integrated with deep learning and for sound explainability. Neural-symbolic computing has been an active area of research for many years seeking to bring together robust learning in neural networks with reasoning and explainability via symbolic representations for network models. In this paper, we relate recent and early research results in neurosymbolic AI with the objective of identifying the key ingredients of the next wave of AI systems. We focus on research that integrates in a principled way neural network-based learning with symbolic knowledge representation and logical reasoning. The insights provided by 20 years of neural-symbolic computing are shown to shed new light onto the increasingly prominent role of trust, safety, interpretability and accountability of AI. We also identify promising directions and challenges for the next decade of AI research from the perspective of neural-symbolic systems.	10.1007/s10462-023-10448-w	[ "https://arxiv.org/pdf/2012.05876v2.pdf" ]	228,083,996	2012.05876	d5901f15a0214b50e6a0085337e49a9b966775a7	December, 2020 Artur D'avila Garcez City University of London UK Luís C Lamb [email protected] Federal University of Rio Grande do Sul Brazil December, 2020arXiv:2012.05876v2 [cs.AI] 16 Dec 2020 Neurosymbolic AI: The 3 rd WaveNeurosymbolic ComputingMachine Learning and ReasoningExplainable AIAI Fast and SlowDeep Learning Current advances in Artificial Intelligence (AI) and Machine Learning (ML) have achieved unprecedented impact across research communities and industry. Nevertheless, concerns about trust, safety, interpretability and accountability of AI were raised by influential thinkers. Many have identified the need for well-founded knowledge representation and reasoning to be integrated with deep learning and for sound explainability. Neural-symbolic computing has been an active area of research for many years seeking to bring together robust learning in neural networks with reasoning and explainability via symbolic representations for network models. In this paper, we relate recent and early research results in neurosymbolic AI with the objective of identifying the key ingredients of the next wave of AI systems. We focus on research that integrates in a principled way neural network-based learning with symbolic knowledge representation and logical reasoning. The insights provided by 20 years of neural-symbolic computing are shown to shed new light onto the increasingly prominent role of trust, safety, interpretability and accountability of AI. We also identify promising directions and challenges for the next decade of AI research from the perspective of neural-symbolic systems. Introduction Over the past decade, Artificial Intelligence and in particular deep learning have attracted media attention, have become the focus of increasingly large research endeavours, and have changed businesses. This led to influential debates on the impact of AI both on academia and industry [52], [66]. It has been claimed that Deep Learning (DL) caused a paradigm shift not only in AI, but in several Computer Science fields, including speech recognition, computer vision and image understanding, natural language processing (NLP) and machine translation [49]. The 2019 Montréal AI Debate between Yoshua Bengio and Gary Marcus, mediated by Vincent Boucher [52], and the AAAI-2020 fireside conversation with Economics Nobel Laureate Daniel Kahneman, mediated by Francesca Rossi and including the 2018 Turing Award winners and DL pioneers Geoffrey Hinton, Yoshua Bengio and Yann LeCun, have pointed to new perspectives and concerns on the future of AI. It has now been argued eloquently that if the aim is to build a rich AI system, that is, a semantically sound, explainable and ultimately trustworthy AI system, one needs to include with it a sound reasoning layer in combination with deep learning. Kahneman corroborated this point at AAAI-2020 by stating that ...as far as I'm concerned, System 1 certainly knows language... System 2 does involve certain manipulation of symbols [41]. Kahneman's comments at AAAI-2020 go to the heart of the matter, with parallels having been drawn many times by AI researchers between Kahneman's research on human reasoning and decision making -reflected in his book "Thinking, Fast and Slow" [40] -and the so-called "AI systems 1 and 2", which would in principle be modelled by deep learning and symbolic reasoning, respectively. 1 In this paper, we place 20 years of research from the area of neurosymbolic AI, known as neural-symbolic integration, in the context of the recent explosion of interest and excitement about the combination of deep learning and symbolic reasoning. We revisit early theoretical results of fundamental relevance to shaping the latest research, and identify bottlenecks and the most promising technical directions for the sound representation of learning and reasoning in neural and symbolic systems. As well as pointing to the various related and promising techniques within AI, ML and Deep Learning, this article seeks to help organise some of the terminology commonly used around AI. This seems important at this exciting time when AI becomes popularized and more people from other areas of Computer Science and from other fields altogether turn to AI: psychology, cognitive science, economics, medicine, engineering and neuroscience to name a few. In Section 2, we position the current debate in the context of the necessary and sufficient building blocks of AI and long-standing challenges of variable grounding and commonsense reasoning. In Section 3, we seek to organise the debate, which can become vague if defined around the concepts of neurons versus symbols, around the concepts of distributed and localist representations. We argue for the importance of this focus on representation since representation precedes learning as well as reasoning. We also analyse a taxonomy for neurosymbolic AI proposed by Henry Kautz at AAAI-2020 from the angle of localist and distributed representations. In Section 4, we delve deeper into a more technical discussion of current neurosymbolic systems and methods with their pros and cons. In Section 5, we identify promising approaches and directions for neurosymbolic AI from the perspective of learning, reasoning and explainable AI. In Section 6, we return to the debate that was so present at AAAI-2020 to conclude the paper and identify exciting challenges for the third wave of AI. Neurons and Symbols: Context and Current Debate Deep learning researchers and AI companies have achieved groundbreaking results in areas such as computer vision, game playing and natural language processing [49,83] Despite the impressive results, deep learning has been criticised for brittleness (being susceptible to adversarial attacks), lack of explainability (not having a formally defined computational semantics or even intuitive explanation, leading to questions around the trustworthiness of AI systems), and lack of parsimony (requiring far too much data, computational power at training time or unacceptable levels of energy consumption) [52]. Against this backdrop, leading entrepreneurs and scientists such as Bill Gates and the late Stephen Hawking have voiced concerns about AI's accountability, impact on humanity and the future of the planet [74]. The need for a better understanding of the underlying principles of AI has become generally accepted. A key question however is that of identifying the necessary and sufficient building blocks of AI, and how systems that evolve automatically based on machine learning can be developed and analysed in effective ways that make AI trustworthy. Turing award winner and machine learning theory pioneer Leslie Valiant pointed out that a key challenge for Computer Science is the principled combination of reasoning and learning, building a rich semantics and robust representation language for intelligent cognitive behaviour [92]. In Valiant's words: "The aim is to identify a way of looking at and manipulating commonsense knowledge that is consistent with and can support what we consider to be the two most fundamental aspects of intelligent cognitive behaviour: the ability to learn from experience and the ability to reason from what has been learned. We are therefore seeking a semantics of knowledge that can computationally support the basic phenomena of intelligent behaviour." Neuralsymbolic computing seeks to offer such a principled way of studying AI by establishing provable correspondences between neural models and logical representations [4,23,16,20,12]. In neural-symbolic computation, logic can be seen as a language with which to compile a neural network, as discussed in more detail later in this paper. 2 The success of deep learning along with a number of drawbacks identified more recently such as a surprising lack of robustness [88] has prompted a heated debate around the value of symbolic AI by contrast with neural computation and deep learning. A key weakness, as Bengio et al. state in a recent article, is that current machine learning methods seem weak when they are required to generalize beyond the training distribution, which is what is often needed in practice [6]. In the recent AI debate between Yoshua Bengio and Gary Marcus, Marcus argues the case for hybrid systems [52] and seeks to define what makes an AI system effectively hybrid: "Many more drastic approaches might be pursued. Yoshua Bengio, for example, has made a number of sophisticated suggestions for significantly broadening the toolkit of deep learning, including developing techniques for statistically extracting causal relationships through a sensitivity to distributional changes and techniques for automatically extracting modular structure, both of which I am quite sympathetic to. But for reasons that will become apparent, I worry that even these sorts of tools will not suffice on their own for getting us to robust intelligence. Instead, I will propose that in order to get to robust artificial intelligence, we need to develop a framework for building systems that can routinely acquire, represent, and manipulate abstract knowledge, with a focus on building systems that use that knowledge in the service of building, updating, and reasoning over complex, internal models of the external world." Key to the appreciation of the above statement by Marcus, also advocated in [46] and [28], is an understanding of the representational value of the symbolic manipulation of variables in logic and the compositionality of language. It is probably fair to assume that the next decade will be devoted to researching specific methods and techniques which seek to address the above issues of representation, robustness and extrapolation. Such techniques will be drawn from a broader perspective of neurosymbolic machine learning and AI which embraces hybrid systems, including: (a) Variable Grounding and Symbol Manipulation: Embracing hybrid systems requires the study of how symbols may emerge and become useful in the context of what deep learning researchers have termed disentanglement. Once symbols emerge (which may happen at different levels of abstraction, ideally within a modular network architecture), it may be more productive from a computational perspective to refer to such symbols and manipulate (i.e. compute) them symbolically rather than numerically. Once it becomes known that a complex neural network serves to calculate, for example, the sum of two handwritten digits provided as input images, or equally that a complex neural network has learned the function f (x) = x, then it is probably the case that one would prefer such a calculation to be precise and to extrapolate well to any value of x. This is easily achieved symbolically. Reasoning, in many cases too, is preferred to be precise and not approximate, although there are cases where approximate or human-like reasoning become more efficient than logical deduction [34]. (b) Commonsense and Combinatorial Reasoning: Another key distinction that is worth making explicit refers to the difference between commonsense knowledge and expert knowledge. While the former is approximate and difficult to specify, the latter strives to be as precise as possible and to prove its properties. We believe that, once equipped with a solid understanding of the value of hybrid systems, variable manipulation and reasoning, the debate will be allowed to progress from the question of symbols versus neurons to the research question: How to compute and learn with symbols, inside or outside of a neural network, and how efficiently computationally, in a precise or approximate reasoning setting? Foundational work about neurosymbolic models and systems such as [16,17,20] will be relevant as we embark in this journey. In [20], correspondences are shown between various logical-symbolic systems and neural network models. The current limits of neural networks as essentially a propositional 3 system are also evaluated. In a nutshell, current neural networks are capable of representing propositional logic, nonmonotonic logic programming, propositional modal logic and fragments of first-order logic, but not full first-order or higher-order logic. This limitation has prompted the recent work in the area of Logic Tensor Networks (LTN) [79,53,95] which, in order to use the language of full first-order logic with deep learning, translates logical statements into the loss function rather than into the network architecture. First-order logic statements are therefore mapped onto differentiable real-valued constraints using a many-valued logic interpretation in the interval [0,1]. The trained network and the logic become communicating modules of a hybrid system, instead of the logic computation being implemented by the network. This distinction between having neural and symbolic modules that communicate in various ways and having translations from one representation to the other in a more integrative approach to reasoning and learning should be at the centre of the debate in the next decade. 4 Among the recent neurosymbolic systems, one can identify quite a variety in range from integrative to hybrid systems: [51] can be seen as a loosely-coupled hybrid approach where image classification is combined with reasoning from text data; [50] offers further integration by allowing a node in the probabilistic inference tree of a symbolic ML system (ProbLog) to be replaced by a neural network; [79] takes another step towards integration by using a differentiable many-valued logic in the loss function of a neural network (in LTN, theorem proving is left for the symbolic counterpart of the system); [54] proposes to perform differentiable unification and theorem proving inside the neural network. Out of the systems and techniques now available, some more integrative others more loosely-coupled, a common question clearly emerges: what are the fundamental building blocks, the necessary and sufficient components of neurosymbolic AI? For example, is the use of an attention layer necessary [96] or can it be replaced by richer structure such as graph networks [47]? Is the explicit use of probability theory necessary, and in this case inside the network or at the symbolic level or both? Is there a real computational gain in combinatorial problem solving by theorem proving using neural networks or is this task better left to the devices of a symbolic system? One thing is now very clearer: there is great practical value in the use of gradient-based learning on distributed representations [45]. In this paper, we also seek to bring attention to another perhaps less attractive but equally if not more relevant question of adopting a distributed versus a localist representation. In a localist representation the relevant concepts have an associated identifier. This is typically a discrete representation. By contrast, in a distributed representation, concepts are denoted by vectors with continuous values. This is therefore an issue of which representation is adequate or most appropriate. Symbolic machine learning takes a localist approach while neural networks are distributed, although neural networks can also be localist [57]. The next section will be devoted to the pros and cons of distributed and localist representations. Forms of Neurosymbolic Integration: Within neurosymbolic AI one may identify systems that translate and encode symbolic knowledge in the set of weights of a network [27], or systems that translate and encode symbolic knowledge into the loss function of the network [79]. The neural-symbolic cycle translating symbolic knowledge into neural networks and vice-versa perform logical unification exactly or approximately using a neural network, although at present the most practical way may be to adopt a hybrid approach whereby unification is computed symbolically. offers a kind of compiler for neural networks 5 , whereby prior knowledge is translated into the network, and a decompiler whenever symbolic descriptions are extracted from a trained network. The compiler can either set-up the network's initial weights akin to a one-shot learning algorithm which is guided by knowledge, or define a knowledge-based penalty or constraint which is added to the network's loss function. A third form of integration has been proposed in [6] which is based on changing the representation of neural networks into factor graphs. The value of this particular representation deserves to be studied. Change of representation is a worthwhile endeavour on its own right in that it may help us understand the strengths and limitations of different neural models and network architecture choices. This third form of integration, however, proposes to create an intermediate representation with factor graphs in between neural networks and logical representations. A note about terminology: In [58], Turing award winner Judea Pearl offers a critique of machine learning which, unfortunately, conflates the terms machine learning and deep learning. Similarly, when Geoffrey Hinton refers to symbolic AI, the connotation of the term tends to be that of expert systems dispossessed of any ability to learn. The use of the terminology is in need of clarification. Machine learning is not confined to association rule mining, c.f. the body of work on symbolic ML and relational learning [55] (the differences to deep learning being the choice of representation, localist logical rather than distributed, and the non-use of gradient-based learning algorithms). Equally, symbolic AI is not just about production rules written by hand. A proper definition of AI concerns knowledge representation and reasoning, autonomous multi-agent systems, planning and argumentation, as well as learning. In what follows, we elaborate on the above misunderstandings one at a turn. Symbolic Machine Learning and Deep Learning: In [58], Pearl proposes a hierarchy consisting of three levels: association, intervention and counterfactual reasoning, and claims that ML is only capable of achieving association. A neurosymbolic or purely symbolic ML system should be capable of satisfying the requirements of all three of Pearl's levels, e.g. by mapping the neural networks onto symbolic descriptions. It is fair to say in relation to Pearl's top level in the hierarchy -counterfactual reasoning -that progress has only been made recently and that much research is still needed, although good progress is being made towards the extraction of local, measurable counterfactual explanations from black box ML systems [99]. Once a neural network has been endowed with a symbolic interpretation, one has no reason to doubt the ability of a neural system to ask what if questions. In fact, the very algorithm for extracting symbolic logic descriptions of the form A → B from trained neural networks [15] uses a form of interrogation of the network akin to the intervention of Bayesian models advocated by Pearl. We argue therefore that a more important question is representational: which representation is most effective, deep networks or Bayesian networks? While attempting to answer this question, as well as considering the demands of the practical applications, it is important to recognise that neural networks offer a concrete model of computation, one which can be implemented efficiently by message passing or propagation of activation, differently from Bayesian networks, and trained by differentiable learning algorithms. A limitation of having such a concrete computational model, however, may be a difficulty of pure neural networks at modelling rich forms of abstraction which are not dependent on the data (images, audio, etc.) but which exist instead at a higher conceptual level. We shall return to this challenge later in the paper. Knowledge Representation and Reasoning in AI: Complex problem solving using AI requires a much richer language than that of expert systems as suggested by Hinton [36]. AI requires a language that can go well beyond Horn clauses to include relational knowledge, time and other modalities, negation by failure, variable substitution and quantification, etc. In statistical relational learning, the use of first-order logic does not require instantiating (or grounding) all possible combinations of the values of the variables (e.g. X and Y in a relation R(X, Y )). In relational reasoning with neural networks, borrowing from the field of relational databases, it is typically the grounded (and therefore propositional rather than first-order) representation that is learned and reasoned about. For the avoidance of confusion, we would term this latter task relationship learning. Two other equally important attributes of a rich language for complex problem solving are compositionality, in the sense of the compositionality of the semantics of a logical language, and modularity. It is worth noting that in the original paper about deep learning [37], before much of the attention turned to convolutional networks, modularity was a main objective of the proposed semisupervised greedy learning of stacks of restricted Boltzmann machines. The recently-proposed stacked capsule autoencoders [44] and neural-symbolic approaches such as Logic Tensor Networks [79] as well as other weakly-supervised approaches revive the important stance of modularity in neural computation. Earlier efforts towards modularity in neurosymbolic AI can be traced back to the system for Connectionist Modal and Intuitionistic Logics [18,19]. Modal logics with a possible-world semantics have been shown to offer a natural approach to modularity in neural computation [20]. With AI understood as a superset of ML which in turn is a superset of DL, we shall argue for the combination of statistical machine learning, knowledge representation (KR) and logical reasoning. By logical reasoning, we shall mean not only classical logic reasoning with the traditional true-false interpretation, but non-classical reasoning including nonmonotonic, modal and many-valued logics. In the study of the interplay between learning and reasoning and how best to implement it (e.g. in a continuous or discrete system), it shall become clear that universal quantification is easy to reason about and hard to learn using neural networks; existential quantification is easy to learn and harder to reason about in a symbolic system. Such limitations on either side of the spectrum will dictate a few practical design decisions to be discussed in this paper. In a nutshell, we claim that neurosymbolic AI is well placed to address concerns of computational efficiency, modularity, KR + ML and even causal inference. More researchers than ever on both sides of the connectionist-symbolic AI divide are now open to studying and learning about each others' tools and techniques. This was not the case until very recently. The use of different terminology alongside some preconceived opinion or perhaps idleness, fuelled by the way that science normally rewards research that is carried out in silos, have prevented earlier progress. The fact that this is now changing will lead to faster progress in the overall field of AI. It is reassuring to see it happening in this way: the neural information processing community have shown the value of neural computation in practice, which has attracted the curiosity of great minds from symbolic AI. We hope that further collaboration in neurosymbolic AI will help solve many of the issues which are still outstanding. Distributed and Localist Representation The integration of learning and reasoning through neurosymbolic systems requires a bridge between localist and distributed representations. The success of deep learning indicates that distributed representations with gradientbased methods are more adequate than localist ones for learning and optimization. At the same time, the difficulty of neural networks at extrapolation, explainability and goal-directed reasoning point to the need of a bridge between distributed and localist representations for reasoning. Neural-symbolic computing has been an active area of research seeking to establish such a bridge for several years [4,26,16,20,35,43,80,93]. In neural-symbolic computation, knowledge learned by a neural network can be represented symbolically. Reasoning takes place either symbolically or within the network in distributed form. Despite their differences, both the symbolic and connectionist paradigms share common characteristics, offering benefits when put together in a principled way (see e.g. [17,20,84,93]). Change of representation also offers a way of making sense of the value of different neural models and architectures with respect to what is a more formal and better understood area of research: symbolic logic. Neural network-based learning and inference under uncertainty have been expected to address the brittleness and computational complexity of symbolic systems. Symbolism has been expected to provide additional knowledge in the form of constraints for learning [23,32], which ameliorate neural network's well-known catastrophic forgetting or difficulty with extrapolation in unbounded domains or with out-of-distribution data. The integration of neural models with logic-based symbolism is expected therefore to provide an AI system capable of explainability, transfer learning and a bridge between lower-level information processing (for efficient perception and pattern recognition) and higher-level abstract knowledge (for reasoning, extrapolation and planning). Suppose that a complex neural network learns a function f (x). Once this function is known, or more precisely a simplified description of f (x) is known, computationally it makes sense to use such a representation, not least for the sake of extrapolation, as exemplified earlier with the f (x) = x function. One could argue that at this point the neural network has become superfluous. Symbol manipulation (once symbols have been discovered) is key to further learning at new levels of abstraction. This is exemplified well in [52] with the use of the concept of a container which may be learned from images. Among the most promising recent approaches to neural-symbolic integration, so-called embedding techniques seek to transform symbolic representations into vector spaces where reasoning can take place through matrix computations over distance functions [7,85,87,80,75,11,26,100,24,70]. In such systems, learning of an embedding is carried out using backpropagation [98,73]. Most of the research in this area is focused on the art of representing relational knowledge such as P (X, Y ) in a distributed neural network. The logical predicate P relating variables X and Y could be used to denote, for example, the container relation between two objects in an image such as a violin and its case, which are in turn described by their embedding. This process is known as relational embedding [7,75,85,87]. For representing more complex logical structures such as first order-logic formulas, e.g. ∀X, Y, Z : (P (X, Y ) → Q(Y, Z)), a system named Logic Tensor Networks (LTN) [80] was proposed by extending Neural Tensor Networks (NTN) [85], a state-of-the-art relational embedding method. Related ideas are discussed formally in the context of constraint-based learning and reasoning in [32]. Two powerful concepts of LTN are (1) the grounding of logical concepts onto tensors with the use of logical statements which act as constraints on the vector space to help learning of an adequate embedding, and (2) the modular and differentiable organisation of knowledge within the neural network which allows querying and interaction with the system. Any userdefined statement in first-order logic can be queried in LTN which checks if that knowledge is satisfied by the trained neural network. With such a tool, a user can decide when to keep using a distributed connectionist representation or switch to a localist symbolic representation. This last aspect brings the question of the emergence of symbols and their meaning in neural networks to the fore: recent work using the weakly supervision of auto-encoders and ideas borrowed from disentanglement have been showing promise in the direction of learning relevant concepts which can in turn be re-used symbolically [10]. Related work seeking to explore the advantages of distributed representations of logic include [11], which is based on stochastic logic programs, [26,100,24], with a focus on inductive programming, and [70], based on differentiable theorem proving. A taxonomy for neurosymbolic AI: with an understanding of the role of localist and distributed approaches, we now provide an analysis of Henry Kautz's taxonomy for neurosymbolic AI [42], which was introduced at AAAI 2020: In Kautz's taxonomy, a Type 1 neural-symbolic integration is standard deep learning, which some may argue is a stretch, but which is included by Kautz to note that the input and output of a neural network can be made of symbols, e.g. text in the case of language translation or question answering applications. Type 2 are hybrid systems such as DeepMind's Al-phaGo and other systems where the core neural network is loosely-coupled with a symbolic problem solver such as Monte Carlo tree search. Type 3 is also a hybrid system whereby a neural network focusing on one task (e.g. object detection) interacts via its input and output with a symbolic system specialising in a complementary task (e.g. query answering). Examples include the neuro-symbolic concept learner [51] and deepProbLog [50]. In a Type 4 neural-symbolic system, symbolic knowledge is compiled into the training set of a neural network. Kautz offers [48] as an example (to be read alongside the critique in [21]). An approach to learn and reason over mathematical constructions is proposed in [2], and in [1] a learning architecture that extrapolates to harder symbolic maths reasoning problems is introduced. We would also include in Type 4 other tightly-coupled but localist neural-symbolic systems where various forms of symbolic knowledge, not restricted to if-then rules, is translated into the initial architecture and set of weights of a neural network, in some cases with guarantees of correctness [20], as well as Logical Neural Networks, where the key concept is to create a 1-to-1 correspondence between neurons and the elements of logical formulas [69]. Type 5 are those tightly-coupled but distributed neuralsymbolic systems where a symbolic logic rule is mapped onto an embedding which acts as a soft-constraint (a regularizer) on the network's loss function. Examples of these include Logic Tensor Networks [79] and Tensor Product Representations [39], referred to in [13] as tensorization methods. Finally, a Type 6 system should be capable, according to Kautz, of true symbolic reasoning inside a neural engine. This is what one could refer to as a fullyintegrated system. Early work in neural-symbolic computing has achieved this (see [20] for a historical overview). Some Type 4 systems are also capable of it, but using a localist rather than a distributed representation and using much simpler forms of embedding than Type 5 systems. Kautz adds that a Type 6 system should be capable of combinatorial reasoning, possibly by using an attention schema to achieve it effectively. Recent efforts in this direction include [8,47,63], although a fully-fledged Type 6 system for combinatorial reasoning does not exist yet. Further research into Type 5 systems will likely focus on the provision of rich embeddings and the study of the extent to which such embeddings may correspond either to pre-defined prior knowledge or to learned attention mechanisms. Further research onto Type 6 systems is highly relevant to the theory of neural-symbolic computing, as discussed in more detail in the next section. In practical terms, a tension exists between effective learning and sound reasoning, which may prescribe the use of a more hybrid approach of Types 3 to 5, or variations thereof such as the use of attention with tensorization. Orthogonal to the above taxonomy, but mostly associated thus far with Type 4, is the study of the limits of reasoning within neural networks, which was already of interest since the first efforts by Valiant at providing a foundation for computational learning [91]. Recently, this has been the focus of experimental analyses of deep learning in symbolic domains [89], and it should include the study of first-order logic, higher-order, manyvalued and non-classical logic. Neurosymbolic Computing Systems: Technical Aspects In symbolic ML, symbols are manipulated as part of a discrete search for the best representation to solve a given classification or regression task. The most well-known form of symbolic ML are decision trees, but richer forms of representation exist, in particular relational representations using first-order logic to denote concepts ranging over variables X, Y, Z... within a (possibly infinite) domain, e.g. ∀X, Y, Z : grandf ather(X, Y ) ← (f ather(X, Z) ∧ mother(Z, Y )) (the father of someone's mother is that person's grandfather). Probabilistic extensions of this approach seek to learn probability distributions for such logical rules (or functional programs) as a way of accounting for uncertainty in the training data. Work in these areas is probably best characterised by the conference series on Inductive Logic Programming [65,56], Statistical Relational Learning [68,25,3,81] and Probabilistic or Inductive Programming [77]. All of the excitement and industrial interest in the past 10 years surrounding AI and Machine Learning, though, have come from an entirely separate type of ML: deep learning. Deep learning uses neural networks and stochastic gradient descent to search through a continuous space, also to solve a given classification or regression task, but creating vector-based, distributed representations, rather than logical or symbolic ones. For this reason, such systems are called sub-symbolic. Whilst it is clear now that AI will not be achieved by building expert systems by hand from scratch (GOFAI), but by learning from large collections of data, one would be misguided to conflate all of machine learning or dismiss the role of symbolic logic, which remains the most powerful and adequate representation for the analysis of computational systems. As put simply by Moshe Vardi "Logic is the Calculus of Computer Science" and, differently from statistics, machine learning can only exist within the context of a computational system. Specifically, deep neural networks will require a language for description, as also advocated by Leslie Valiant. Neural network-based AI is distributed and continuous, deals well with large-scale multimodal noisy perceptual data such as text and audio, handles symbol grounding better than symbolic systems since concepts are grounded on feature vectors, and is by definition a computational model, frequently implemented efficiently using propagation of activation and tensor processing units. 6 Symbolic AI is generally localist and discrete, capable of sophisticated reasoning, including temporal, epistemic and nonmonotonic reasoning, planning, extrapolation and reasoning by analogy. Neurosymbolic AI has shown that non-classical logics, in particular many-valued logics, offer an adequate language for describing neural networks [79,69]. As the field of AI moves towards agreement on the need for combining the strengths of neural and symbolic AI, it should turn next to the question: what is the best representation for neurosymbolic AI? To answer this question, one should seek to be informed by developments in neural-symbolic computing of the past 20 years, and to evaluate in a precise manner the methods, algorithms and applications of neurosymbolic AI. For instance, it is known that current recurrent neural networks are capable of computing the logical consequences of propositional modal logic programs and other forms of non-classical reasoning and fragment of first order logic programs [5,20]. Obtaining results for full first-order logic has not been possible thus far, which reinforces John McCarthy's claim that neural networks are essentially propositional. In terms of applications of AI, these have been largely focused on perceptual or pattern matching tasks such as image and audio classification. Recent efforts at question answering and language translation as well as protein folding classification have highlighted the importance of the neurosymbolic approach. The ideal type of application for a neurosymbolic system, however, should be that where abstract information is required to be reasoned about at different levels beyond that what can be perceived from data alone, such as complex concept learning whereby simpler concepts are required to be organised systematically as part of the definition of a higher concept. Such a conceptual structure, which is still to be discovered using data, also requires knowledge which is governed by general rules and exceptions to the rules, allowing for sound generalization in the face of uncertainty but also capable of handling specific cases (the many exceptions, which may be important for the sake of robustness although not necessarily statistically relevant). Similarly, it seems hard to achieve true relational learning using only neural networks. A useful but simple example can be borrowed from the area of Inductive Logic Programming: learning the concept of ancestor from a few examples of the mother, father and grandparent relations. Grounding the entire knowledge-base in this case would not be productive since the chain of reasoning to derive the concept of ancestor may be arbitrarily large depending on the data available. In this case, one is better off learning certain relations by jumping to conclufrequently requiring simplification of graphs into tree-based representations. sions, such as e.g. ∀X, Y : f ather(X, Y ) → ancestor(X, Y ), from relatively few examples and using similarity measures to infer new relations, at the same time deriving symbolic descriptions which can be used for reasoning beyond the distribution of the data, allowing in turn for extrapolation. In this example, once a description for ancestor is obtained, one should be able to reason about arbitrarily long chains of family relationships. Notice that key to this process is the ability to revise the conclusion taken once new evidence to the contrary of what has been inferred is made available from the data. In other words, the reasoning here is nonmonotonic [14]. In summary, at least two options exist for neurosymbolic AI. In Option 1, symbols are translated into a neural network and one seeks to perform reasoning within the network. In Option 2, a more hybrid approach is taken whereby the network interacts with a symbolic system for reasoning. A third option, which would not require a neurosymbolic approach, exists when expert knowledge is made available, rather than learned from data, and one is interested in achieving precise sound reasoning as opposed to approximate reasoning. We discuss each option briefly next. In Option 1, it is desirable still to produce a symbolic description of the network for the sake of improving explainability (discussed later) or trust, or for the purpose of communication and interaction with the system. In Option 2, by definition, a neurosymbolic interface is needed. This may be the best option in practice given the need for combining reasoning and learning in AI, and the apparent different nature of both tasks (discrete and exact versus continuous and approximate). However, the value of distributed approximate reasoning using neural networks is only starting to be investigated as in the case of differentiable neural computers [33] and neural theorem proving [97], although early efforts did not prove to be promising in terms of practical efficiency [62,38,82]. In Option 3, a reasonable requirement nowadays would be to compare results with deep learning and the other options. This is warranted by the latest practical results of deep learning showing that neural networks can offer, at least from a computational perspective, better results than purely symbolic systems. In practice, the choice between Options 1 and 2 above may depend on the application at hand and the availability of quality data and knowledge. A comparatively small number of scientists will continue to seek to make sense of the strengths and limitations of both neural and symbolic approaches. On this front, the research advances faster on the symbolic side due to the clear hierarchy of semantics and language expressiveness and rigour that exists at the foundation of the area. By contrast, little is known about the expressiveness of the latest deep learning models in relation to established neural models beyond data-driven comparative empirical evaluations. As advocated by Paul Smolensky, neurosymbolic computing can help map the latest neural models into existing symbolic hierarchies, thus helping organise the extensively ad-hoc body of work in neural computation. Challenges for the Principled Combination of Reasoning and Learning For a combined perspective on reasoning and learning, it is useful to note that reasoning systems may have difficulties computationally when reasoning with existential quantifiers and function symbols, such as ∃xP (f (x)). Efficient logic-based programming languages such as Prolog, for example, assume that every logical statement is universally quantified. By contrast, learning systems may have difficulty when adopting universal quantification over variables. To be able to learn a universally quantified statement such as ∀xP (x), a learning systems needs in theory to be exposed to all possible instances of x. This simple duality points to a possible complementary nature of the strengths of learning and reasoning systems. To learn efficiently ∀xP (x), a learning system needs to jump to conclusions, extrapolating ∀xP (x) given an adequate amount of evidence (the number of examples or instances of x). Such conclusions may obviously need to be revised over time in the presence of new evidence, as in the case of nonmonotonic logic. In this case, a statement of the form ∀xP (x) becomes a data-dependent generalization, which is not to be assumed equivalent to a statement ∀yP (y), as done in classical logic. Such statements may have been learned from different samples of the overall potentially infinite population. On the other hand, a statement of the form ∃xP (x) is trivial to learn from data by identifying at least one case P (a), although reasoning from ∃xP (x) is more involved, requiring the adoption of an arbitrary constant b such that P (b) holds. It is now accepted that learning takes place on a continuous search space of (sub)differentiable functions; reasoning takes place in general on a discrete space as in the case of goal-directed theorem proving. The most immediate way of benefiting from the combination of reasoning and learning, therefore, is to adopt a hybrid approach whereby a neural network takes care of the continuous search space and learning of probabilities, while a symbolic system consisting of logical descriptions of the network uses discrete search to achieve extrapolation and goal-directed reasoning. 7 As hinted already in this paper, a property of early deep learning may hold the key to the above hybrid perspective: modularity. In the original paper on deep learning [37], a modular system is proposed consisting of a stack of restricted Boltzmann machines (RBMs). The extraction of symbolic descriptions from each RBM is thus made considerably easier [90]. Each RBM learns a joint probability distribution while their symbolic description reflects the result of learning without manipulating probabilities explicitly, thus avoiding the complexity of probabilistic inference found in symbolic AI. In [90], an efficient algorithm is presented that extracts propositional rules enriched with confidence values from RBMs, similar to what was proposed with Penalty Logic for Hopfield networks in [62]. When RBMs are stacked onto a deep belief network, however, the modular extraction of compositional rules may be accompanied by a compounding loss of accuracy, indicating that knowledge learned by the neural network might not have been as modular as one would have wished. Systems that impose a more explicit separation of modules may hold the answer to this problem, in particular systems where unsupervised learning is combined with weakly-supervised classification at distinct levels of abstraction, such as with the use of variational auto-encoders [86] or generative adversarial networks [31]. In the area of Reinforcement Learning too, deep learning systems combined with symbolic rules were first proposed in [29], and there is promise of considerable progress with the use of neurosymbolic approaches especially in the case of model-based Reinforcement Learning. We therefore do not advocate the adoption of monoblock networks with millions of parameters. Even though this may be how the human brain works, loss of modularity seems to be, at least at present from a computational perspective, a price that is too high to pay. Modularity remains a fundamentally relevant property of any computing system. Applications of Neurosymbolic AI: One major way of driving advances in AI continues to be through challenging applications, be it language translation, computer games or protein folding competitions. Language understanding in the broadest sense of the term, including question-answering that requires commonsense reasoning, offers probably the most complete application area for neurosymbolic AI. As an example, consider this question and its commonsense answer from the COPA data set [72]: It got dark outside. What happened as a result? (a) Snowflakes began to fall from results in both of these directions [71,50,100,64]. Computational complexity issues remain a challenge though for the combination of first-order logic and probabilities in symbolic AI. the sky; (b) The moon became visible in the sky. Another very relevant application domain is planning, which requires learning and reasoning over time, as in this example adapted from [76]: Daniel picks up the milk; Daniel goes to the bedroom; Daniel places the milk on the table; Daniel goes to the bathroom. Where is the milk? Finally, an area where machine learning and knowledge representation and reasoning have complementary strengths is knowledge engineering, including knowledge-base completion and datadriven ontology learning. In this area of application, rich and large-scale symbolic representations exist alongside data, including knowledge graphs to be combined with neural networks such as graph neural networks. A common thread across the above examples and applications is the need for modelling cause and effect with the use of implicit information. This requires learning of general rules and exceptions to the rules that evolve over time. In such cases, deep learning alone fails when presented with examples from outside the distribution of the training data. This motivated Judea Pearl's critique of Machine Learning [58] which we shall address in some detail next. In Pearl's 3-level causal hierarchy (association, intervention and counterfactuals), association involves purely statistical relationships. In Pearl's words, observing a customer who buys toothpaste makes it more likely that this customer will also buy floss. Such associations can be inferred directly from the observed data using standard conditional probabilities and conditional expectation or other standard non-probabilistic ML model. Questions asked at this level require no causal information and, for this reason, this layer is placed at the bottom of the hierarchy. Answering such questions is the hallmark of current machine learning methods. Pearl's hierarchy may unintentionally give the impression that machine learning is confined to this bottom layer, since no reference is made in [58] to the body of work on symbolic machine learning which is unequivocally not confined to association rules [55,70]. Pearl continues: the second level, intervention, ranks higher than association because it involves not just seeing what is but changing what we see. A typical question at this level would be: what will happen if we double the price? Such a question cannot be answered from sales data alone, as it involves a change in customers' choices in reaction to the new pricing. These choices may differ substantially from those taken in previous price-raising situations, unless we replicate precisely the market conditions that existed when the price reached double its current value. At this level there is a need for inference that can reach beyond the data distribution. Finally, the top level invokes counterfactuals, a mode of reasoning that reverts to the philosophers David Hume and John Stuart Mill and that has been given computational semantics in the past two decades. A typical question in the counterfactual category is: what if I had acted differently?, thus necessitating retrospective reasoning. As noted earlier in the paper, neural-symbolic computing can implement all three of Pearl's levels. Once a symbolic description of the form if A then B has been associated with a neural network, surely the idea of intervention [9] and counterfactual reasoning become possible, c.f. for example, [99] on the measurement and extraction of counterfactual knowledge from trained neural networks. Our conclusion from the above discussion is that in neurosymbolic AI: • Knowledge should be grounded onto vector representations for efficient learning from data based on message passing in neural networks as an efficient computational model. • Symbols should become available as a result of querying and knowledge extraction from trained networks, and offer a rich description language at an adequate level of abstraction, enabling infinite uses of finite means, but also compositional discrete reasoning at the symbolic level allowing for extrapolation beyond the data distribution. • The combination of learning and reasoning should offer an important alternative to the problem of combinatorial reasoning by learning to reduce the number of effective combinations, thus producing simpler symbolic descriptions as part of the neurosymbolic cycle. As an example, consider an Autoencoder which learns in unsupervised fashion to maximise mutual information between pixel inputs and a latent code or some other embedding consisting of fewer relevant features than pixels. Suppose that this neural network has learned to find regularities such as e.g. when it sees features of type A, it also sees features of type B but not features of type C. At this point, such regularities can be converted into symbols: ∀xA(x) → (∃yB(y) ∧ ¬∃zC(z)). As a result of the use of variables x, y, z at the symbolic level, one can extrapolate the above regularity to any features of type A, B or C. 8 A symbolic description is also a constraint on the neurosymbolic cycle. It is generalised from data during learning and it certainly includes an ability to ask what-if questions. Reasoning about what has been learned allows for extrapolation beyond the data distribution, and finally the symbolic description can serve as prior knowledge (as a constraint) for further learning in the presence of more data, which includes the case of knowledge-based transfer learning [90]. Further training can now take place at the perceptual or sub-symbolic level, or at the conceptual or symbolic level. This is when having a distributed (sub-symbolic) and a localist (symbolic or sub-symbolic) representation becomes relevant. Assuming that probabilities are dealt with at the sub-symbolic distributed level (as in RBMs) and that the symbolic level is used for a more qualitative representation of uncertainty in the form of general rules with exceptions, we avoid the complications of having to deal simultaneously with discrete and continuous learning of rules and probabilities. A Note about Explainable AI (XAI): Knowledge extraction is an integral part of neural-symbolic integration and a major ingredient towards explainability of black-box AI systems. The main difficulty in XAI is the efficient extraction of compact and yet correct and complete knowledge. It can be argued that a large knowledge-base is not more explainable than a large neural network. 9 Although this may be true at the level of the entire model explainability, in the case of local explanations, i.e. explanations of individual cases, a knowledge-base is certainly more explainable than a neural network because it offers a trace (a proof history) showing why an outcome was obtained, as opposed to simply showing how propagation of activation through the network has led to that outcome. It is a main goal of knowledge extraction algorithms to seek to derive compact relevant representations from large complex networks. This is not always possible to achieve efficiently, in which case one may need to resort to having local explanations only. Many aspects of XAI are being investigated at present. Some of the research questions include: Is the explanation intended for an expert or lay person? Is an explanation required because one does not trust the system, expected a different outcome/would like to induce a different outcome, or would like to question the normative system that has led to the outcome? Is an explanation intended to try and improve system performance, reduce bias/increase fairness, or is it the case that one would simply like to be able to understand the decision process? The answers to these questions are likely to be application and user specific. While some stakeholders may be happy without an explanation, as in the case of a patient faced with a medical diagnosis, in most cases some form of explanation is needed to improve system performance or trust. In some cases, producing an explanation is in fact the main goal of the system as in the application of ML to responsible gambling in [59,60]. Early efforts at knowledge extraction in neurosymbolic AI as a form of explanation were always evaluated w.r.t. fidelity: a measure of the accuracy of the extracted knowledge in relation to the neural network rather than the data, or in today's terminology, a measure of the accuracy of the student model w.r.t. the teacher model. High fidelity is therefore fundamental whenever a student model is to be claimed to offer a good explanation for a (more complex) teacher model. Unfortunately, many recent XAI methods have abandoned fidelity as a measure of the quality of an explanation, making it easier for an apparently excellent explanation to be simply wrong, in that it may not be at all an explanation of the ML model in question. This is particularly problematic for very popular local XAI methods such as LIME [67]. Local explanation systems currently in use by consultancy firms and available in many AI toolboxes can be shown to achieve very low levels of fidelity. Without high fidelity results, an apparently perfectly good explanation produced by an XAI system is likely not to be an explanation of the underlying ML system which it is expected to explain. In [99] a way of measuring fidelity of local methods was introduced which we argue should be adopted by all XAI methods. The same paper exemplifies how LIME's explanations may achieve very low fidelity. Even better than fidelity, if an XAI method can be shown to be sound [15] then it will provably converge to a high-fidelity. Soundness however is normally associated with exponential complexity and so in practice a measure of fidelity may be all that is available. Knowledge extraction should also allow communication between users and the ML system. Current interactive ML is insufficient to the extent that it proposes to replace sound statistical evaluation by subjective user evaluation. Communication with the system implies an ability to ask questions (query the system) and check one's understanding (obtain a rationale for the outcome). The user can then either agree with the outcome and the rationale, agree with the outcome but not the rationale, or disagree with the outcome, thus providing useful feedback or direct intervention to change the system and its outcomes, probably through the system's symbolic description language in the case of neurosymbolic AI. Knowledge extraction also offers a way of identifying and correcting for bias in the ML system, which is a serious and present problem [25]. As a result of the General Data Protection Regulation (GDPR), many companies have decided as a precaution to remove protected variables such as gender and race from their ML system. It is well known, however, that proxies exist in the data which will continue to bias the outcome so that the removal of such variables may serve only to hide a bias that otherwise could have been revealed via knowledge extraction [61]. Current AI-based decision support systems process very large amounts of data which humans cannot possibly evaluate in a timely fashion. Thus, even with a so-called human-in-the-loop approach where technology domain experts or end-users may become accountable to the decisions, domain experts or end-users are likely to quickly feel progressively less capable of over-riding recommendations which were deemed accurate and based on so much more data than they can handle. Even when such AI systems are portrayed as decision support systems, the current reality is that in order to function well with Big Data, the system must execute a form of triage of the data to be presented to the expert, thus only offering partial information to the decision maker. Without knowledge extraction and a capacity for system communication, the decision maker will, by the very nature of the automated data triage, not be in control. Finally, the simple extraction of rules from trained networks may be insufficient. One may need to extract also confidence values so as to be able to rank extracted rules. This offers a system that knows when it does not know. As a simple example, consider a typical neural network trained to classify the well-known MNIST hand-written digits from 0 to 9. Faced with an image of an obvious non-digit such as an image of a cat, this system must surely provide a very low confidence value to any of the outcomes 0 to 9. The use of adversarial approaches alongside knowledge extraction for robustness has a contribution to make here. In summary, for the many reasons discussed above, neurosymbolic AI with a measurable form of knowledge extraction is a fundamental part of XAI. AAAI 2020, a Turning Point We now return to the debate around System 1 and System 2 that motivated the introduction of this paper and provide a short summary of the AAAI 2020 conference. Not only did the AAAI 2020 conference contained a larger than usual number of papers proposing to combine neural networks and symbolic AI, there were a number of keynote addresses and debates directly relevant to neural-symbolic computing, notably: (1) Fireside conversation between 2019 Turing award winners, Francesca Rossi, and Nobel Laureate Daniel Kahneman on thinking fast and slow and its relation to neural networks and symbol manipulation; (2) The Third AI Summer, Henry Kautz's AAAI 2020 Robert S. Engelmore Memorial Award Lecture, which introduced a taxonomy for neurosymbolic computing; (3) The Director of MIT/IBM Watson AI laboratory David Cox IAAI keynote address, which focused on neurosymbolic AI and applications on vision and language understanding, machine commonsense, question answering, argumentation and XAI. At the Turing award session and fireside conversation with Daniel Kahneman, there was a clear convergence towards integrating symbolic reasoning and deep learning. Kahneman made his point clear by stating that on top of deep learning, a System 2 symbolic layer is needed. This is reassuring for neurosymbolic AI going forward as a single more cohesive research community that can agree about definitions and terminology, rather than a community divided as AI has been up to now. AI is still in its infancy, so perhaps some of the early disputes can be understood. The debate around symbols versus neurons is unlikely to produce concrete results unless it prompts researchers on either side of the divide to learn about each others' methods and techniques. As the saying goes, "all vectors are symbols, but not all symbols are vectors". Kahneman made the point that System 1 (S1) and System 2 (S2) are terms not coined by him which have a long history in psychology research [40], and that he prefers to use implicit versus explicit thinking and reasoning [41]. He argued that S1 (as the intuitive parallel system) is capable of understanding language, in contradiction with Yoshua Bengio's account of deep learning's S1 and S2. Kahneman also stated that S2 (as the sequential deliberative system) is most probably performing symbol manipulation as argued by Gary Marcus in [52]. In his "next decade of AI" paper, Marcus argues strongly in favour of hybrid systems, and seeks to define what makes a system hybrid. Marcus's definition is important inasmuch as a main problem of deep learning is a lack of definition. By contrast, Yann LeCun's recent attempt at defining deep learning, also presented at the AAAI 2020 debate [41], falls short of what is a useful formal definition. For example, LeCun's definition fails to distinguish deep and shallow networks. All attempts to create such a bridge between S1 and S2 are at this point useful and should be commended given our state of lack of understanding of how the brain works. For example, attempts to create differentiable reasoning are useful, although we would require an important distinction be made, whether the purpose is to achieve brain-like systems or to create robust AI. It is possible that these two goals may soon lead into two quite separate research directions: those who seek to understand and model the brain and those who seek to achieve or improve AI. Maybe from that perspective the field is too broad and will require further specialization. A common challenge that will persist, however, is embodied in the question: how symbolic meaning emerges from large networks of neurons? Perhaps an important choice for neurosymbolic AI is the choice between combinatorial (exact) reasoning and commonsense, approximate reasoning. While learning is always approximate, reasoning can be approximate or precise. In a neurosymbolic system, it is possible to envisage the combination of efficient approximate reasoning (jumping to conclusions) with more deliberative and precise or normative symbolic reasoning [94]. Conclusions may be revised through learning from new observations and via communication with the system through knowledge extraction and precise reasoning. One might expect commonsense to emerge as a result of this process of reasoning and learning, although the modelling and computing of commonsense knowledge continues to be another challenge. From a practical perspective, a recipe for neurosymbolic AI might be: learning is carried out from data by neural networks which use gradientdescent optimization; efficient forms of propositional reasoning can also be carried out by the network, c.f. neural-symbolic cognitive reasoning [20]; rich first-order logic reasoning and extrapolation needs to be done symbolically from descriptions extracted from the trained network; once symbolic meaning has emerged from the trained network, symbols can be manipulated easily by the current computer and can serve as constraints for further learning from data as done in [79]; this establishes a practical form of neurosymbolic cycle for learning and reasoning which is feasible with the current technology. Ingredients of neurosymbolic AI: Narrow AI based on neural networks is already successful and useful in practice with big data. There is obvious value in this as shown by the flourishing of the Machine Learning community and the growing NeurIPS conference community. Data scientists will do this work. In this paper, however, we have been discussing the science of what constitutes the fundamental ingredients of an intelligent system [92]. One such ingredient, current results show, is gradient-based optimization used by deep learning to handle large amounts of data, but other ingredients are surely needed. At the AAAI-2020 fireside conversation, a question was asked about the beauty and value of abstract compact symbolic representations such as F = ma or e = mc 2 . Yoshua Bengio's answer was to point out that these must have come out of someone's brain, Isaac Newton and Albert Einstein to be precise. As Stephen Muggleton noted at another debate on the future of AI, his goal is to shorten the wait for the next Newton or Einstein, or Alan Turing. Muggleton's bet is on the use of higher-order logic representations and meta-interpretive learning. 10 With this example we seek to illustrate that among highly respected researchers the choice of ingredients may vary widely, from the need for much more realistic models of the brain to the need for ever more sophisticated forms of higher-order computation. With the neural-symbolic methodology, the goal is to develop neural network models with a symbolic interpretation. The key is how to learn representations neurally and make them available for use symbolically (as for example when an AI system is asked to explain itself). In this paper, we have argued for modularity as an important ingredient, allowing one to refer to large parts of the network by the composition of symbols and relations among them. Having an adequate language for describing knowledge encoded in such networks is another important ingredient. We have argued for the use of first-order logic as this language, as a canonical form of representation, but also other forms of non-classical representation such as nonmonotonic and modal logic and logic programming. Once a complex network can be described symbolically, ideally in an abstract compact form as in e = mc 2 , any style of deductive reasoning becomes possible. Reasoning is obviously another fundamental ingredient, either within or outside the network, exact or approximate. Finally, symbolic meaning can serve to improve performance of S1. In other words, symbols which have been learned, derived or even invented can act as constraints on the large network and help improve learning performance as part of a positive cycle of learning and reasoning. Constraint satisfaction as part of the interplay between learning and reasoning is therefore another ingredient. Summary and Future Directions: With the above five ingredients of neurosymbolic AI -gradient-based optimization, modularity, symbolic language, reasoning and constraint satisfaction -the reader will not be surprised to know that there are many outstanding challenges for neurosymbolic AI. First, no agreement exists on the best way of achieving the above combination of language and structure, of knowledge acquired by agents acting in an environment and the corresponding reasoning that an agent must implement to achieve its goals. It is highly desirable, though, that the study of how to achieve the combination of (symbolic) language and (neural) structure be principled, in that both language and structure should be formally specified with theorems proven about their correspondence or lack thereof. As done in the case of Noam Chomsky's language hierarchy, proofs are needed of the capability of different neural architectures at representing various logical languages. Proofs of correspondence have been shown between neural networks and propositional, nonmonotonic, modal, epistemic and temporal logic in [20]. Similar proofs are required for first-order and higher-order logic. Henry Kautz spoke of 6 types of neurosymbolic AI and said that what is important next is to work out which specific technique is best [42]. Kautz made a distinction between expert knowledge and commonsense knowledge and noted that one should not necessarily want to backpropagate through expert knowledge. In this case, exact reasoning is neededà la neural-symbolic computing with knowledge extraction. An equally valid argument exists for differentiable reasoning in the case of commonsense knowledge. The use of probabilistic languages or higher-order (functional or logical) languages may also have a central role in the technical debate, including on the best place for probability theory: in S1 or S2 or both? In the meantime, we say: translate back and forth between representations, take a principled approach, adopt a language as a constraint on the structure, seek to provide explanations, combine reasoning and learning, and repeat. We also set out three immediate challenges for neurosymbolic AI, each capable of spinning out multiple research strands which may become area defining in the next decade: • Challenge 1: First-order logic and higher-order knowledge extraction from very large networks that is provably sound and yet efficient, explains the entire model and local network interactions and accounts for different levels of abstraction. • Challenge 2: Goal-directed commonsense and efficient combinatorial reasoning about what has been learned by a complex deep network trained on large amounts of multimodal data. • Challenge 3: Human-network communication as part of a multi-agent system that promotes communication and argumentation protocols between the user and an agent that can ask questions and check her understanding. Whether or not an AI system truly understands what it does is another recurring theme in the current debate. A point made recently by Geoff Hinton on this issue was that: "the goal posts keep changing. Before, if an AI system could translate a text, it would have been deemed as having understood the text, or if the system could sustain a conversation or describe the scene on an image. Now, none of these count as proper understanding". One may argue that it is the very definition of AI and the recent success of deep learning itself that have been responsible for this situation. Perhaps, instead of proper "understanding", a more forgiving approach might be to specify comprehensibility tests of the kind used in schools to evaluate the performance of students on various subjects, including e.g. foreign language comprehensibility tests, as proposed by Stephen Muggleton. Neurosymbolic AI is in need of standard benchmarks and associated comprehensibility tests which could in a principled way offer a fair comparative evaluation with other approaches with a focus on learning from fewer data, reasoning about extrapolation, computational complexity and energy consumption. Just as the field of AI progressed when challenging applications were set such as chess playing, robotic football, self-driving vehicles and protein folding, neurosymbolic AI should benefit from a similar challenge and benchmark being set by the AI community specifically for the next decade. Over the years, the terminology "neural-symbolic" (integration, computing, system, etc.) was used predominantly by the research community to indicate a combination of two paradigms: neural and symbolic AI, see e.g.[20] More recently, the more colloquial terminology "neuro-symbolic" (AI, approach, system, etc) has become more commonly used in publications and the printed press. In this paper, we use the term "neuralsymbolic" when referring to the combination of paradigms, and we introduce the term "neurosymbolic" as a single word to symbolise the coming of age of a new area of research. The current limitation of neural networks, which John McCarthy referred to as propositional fixation, is of course based on the current simple models of neuron. Although this may be about to change through important work on understanding the mind and brain which may produce richer models of neural networks[30], one should note that the recent state-of-the-art results obtained by deep networks using large amounts of data are predicated on the notion of a simple neuron[49,78].4 Having worked for two decades on integrative neurosymbolic AI and more recently on hybrid neural-symbolic systems, we are acutely aware of the tension between principled integration and practical value and application. Scientifically, there is obvious value in the study of the limits of integration to improve our understanding of the power of neural networks using the well-studied structures and algebras of computer science logic. When seeking to solve a specific problem, however, one may prefer to take, for example, an existing knowledge-base and find the most effective way of using it alongside the tools available from deep learning and software agents. As a case in point, take the unification algorithm, which is an efficient way of computing symbolic substitutions. It is notoriously difficult to implement in neural networks. One may, of course, wish to study how to In the study of programming languages it is accepted that different levels of abstraction and different representations are needed -e.g. java bytecode and a java programfor the purpose of efficiency, system maintenance, user interaction and verification. We argue that in AI, neural-symbolic systems will provide equally important forms of abstract representation. Contrast with Bayesian networks which may be inefficient as a computational model The investigation of continuous reasoning or discrete learning approaches is of course worth pursuing too. There have been a number of recent developments with relevant As another example, consider a neural network trained to classify graphs into those which contain a Hamiltonian cycle and those which do not, given a fixed range of available graph sizes as training examples. Contrast this network with a symbolic description of the definition of Hamiltonian cycle which therefore applies to graphs of any size. The now apparent lack of explainability of Random Forests, which amount to a collection of Decision Trees and therefore propositional logic formulas with probabilities, serves as a good reminder that XAI is not confined to neural networks. 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[ "Coreference-aware Double-channel Attention Network for Multi-party Dialogue Reading Comprehension", "Coreference-aware Double-channel Attention Network for Multi-party Dialogue Reading Comprehension", "Coreference-aware Double-channel Attention Network for Multi-party Dialogue Reading Comprehension", "Coreference-aware Double-channel Attention Network for Multi-party Dialogue Reading Comprehension" ]	[ "Yanling Li \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n", "Bowei Zou [email protected] \nInstitute for Infocomm Research\nASTARSingapore\n", "Yifan Fan [email protected] \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n", "Mengxing Dong \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n", "Yu Hong \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n", "Yanling Li \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n", "Bowei Zou [email protected] \nInstitute for Infocomm Research\nASTARSingapore\n", "Yifan Fan [email protected] \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n", "Mengxing Dong \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n", "Yu Hong \nSchool of Computer Science and Technology\nSoochow University\nSoochowChina\n" ]	[ "School of Computer Science and Technology\nSoochow University\nSoochowChina", "Institute for Infocomm Research\nASTARSingapore", "School of Computer Science and Technology\nSoochow University\nSoochowChina", "School of Computer Science and Technology\nSoochow University\nSoochowChina", "School of Computer Science and Technology\nSoochow University\nSoochowChina", "School of Computer Science and Technology\nSoochow University\nSoochowChina", "Institute for Infocomm Research\nASTARSingapore", "School of Computer Science and Technology\nSoochow University\nSoochowChina", "School of Computer Science and Technology\nSoochow University\nSoochowChina", "School of Computer Science and Technology\nSoochow University\nSoochowChina" ]	[]	We tackle Multi-party Dialogue Reading Comprehension (abbr., MDRC). MDRC stands for an extractive reading comprehension task grounded on a batch of dialogues among multiple interlocutors. It is challenging due to the requirement of understanding cross-utterance contexts and relationships in a multi-turn multi-party conversation. Previous studies have made great efforts on the utterance profiling of a single interlocutor and graph-based interaction modeling. The corresponding solutions contribute to the answer-oriented reasoning on a series of wellorganized and thread-aware conversational contexts. However, the current MDRC models still suffer from two bottlenecks. On the one hand, a pronoun like "it" most probably produces multiskip reasoning throughout the utterances of different interlocutors. On the other hand, an MDRC encoder is potentially puzzled by fuzzy features, i.e., the mixture of inner linguistic features in utterances and external interactive features among utterances. To overcome the bottlenecks, we propose a coreference-aware attention modeling method to strengthen the reasoning ability. In addition, we construct a two-channel encoding network. It separately encodes utterance profiles and interactive relationships, so as to relieve the confusion among heterogeneous features. We experiment on the benchmark corpora Molweni and FriendsQA. Experimental results demonstrate that our approach yields substantial improvements on both corpora, compared to the fine-tuned BERT and ELECTRA baselines. The maximum performance gain is about 2.5% F 1-score. Besides, our MDRC models outperform the state-of-the-art in most cases.	10.48550/arxiv.2305.08348	[ "https://export.arxiv.org/pdf/2305.08348v2.pdf" ]	258,685,747	2305.08348	38e91b38fbccbd8211c9d83462fc6ef4b458df5d	Coreference-aware Double-channel Attention Network for Multi-party Dialogue Reading Comprehension Yanling Li School of Computer Science and Technology Soochow University SoochowChina Bowei Zou [email protected] Institute for Infocomm Research A*STARSingapore Yifan Fan [email protected] School of Computer Science and Technology Soochow University SoochowChina Mengxing Dong School of Computer Science and Technology Soochow University SoochowChina Yu Hong School of Computer Science and Technology Soochow University SoochowChina Coreference-aware Double-channel Attention Network for Multi-party Dialogue Reading Comprehension Index Terms-Multi-party dialogue reading comprehensionCoreference-aware attentionUtterance profilingInteraction modeling We tackle Multi-party Dialogue Reading Comprehension (abbr., MDRC). MDRC stands for an extractive reading comprehension task grounded on a batch of dialogues among multiple interlocutors. It is challenging due to the requirement of understanding cross-utterance contexts and relationships in a multi-turn multi-party conversation. Previous studies have made great efforts on the utterance profiling of a single interlocutor and graph-based interaction modeling. The corresponding solutions contribute to the answer-oriented reasoning on a series of wellorganized and thread-aware conversational contexts. However, the current MDRC models still suffer from two bottlenecks. On the one hand, a pronoun like "it" most probably produces multiskip reasoning throughout the utterances of different interlocutors. On the other hand, an MDRC encoder is potentially puzzled by fuzzy features, i.e., the mixture of inner linguistic features in utterances and external interactive features among utterances. To overcome the bottlenecks, we propose a coreference-aware attention modeling method to strengthen the reasoning ability. In addition, we construct a two-channel encoding network. It separately encodes utterance profiles and interactive relationships, so as to relieve the confusion among heterogeneous features. We experiment on the benchmark corpora Molweni and FriendsQA. Experimental results demonstrate that our approach yields substantial improvements on both corpora, compared to the fine-tuned BERT and ELECTRA baselines. The maximum performance gain is about 2.5% F 1-score. Besides, our MDRC models outperform the state-of-the-art in most cases. I. INTRODUCTION MDRC [1], [2] is an increasingly attractive task in the field of multi-party dialogue research [1], [3]- [5]. Essentially, it serves as an extractive Machine Reading Comprehension task (MRC), automatically extracting proper answers from the given context for a series of questions. Though, it is more challenging than the conventional MRC because the context consists of multi-party dialogues, instead of a narrative or two-party conversations. The information flow that an MDRC model needs to treat is generally characterized by irregular and unstructured utterances, i.e., all sorts of gossip among different interlocutors (see the examples in Fig. 1-(a) and (b)). Similar to nearly the majority of current MRC solutions, the studies of MDRC leverage encoder-decoder neural networks. The encoder is generally implemented using Pre-trained Language Models (PLMs) [6]- [10], which learns to represent the sequentially-occurred utterances with attention mechanisms [11], [12]. The decoder is developed to conduct token-level boundary discrimination, i.e., positioning the boundaries of answers grounded on the attentive representations of utterances. Recently, researchers tend to give deeper insights into the unique characteristics of MDRC and, accordingly, concentrate on two task-specific schemes, including utterance profiling and interaction modeling: • Utterance profiling is designed to collect the utterances of a single interlocutor, and encode them separately. The goal is to represent distinguishable internal features (e.g., coherence) in the dialogue history of the same interlocutor. It helps to bridge discrete clues for answer prediction, with less distraction from the utterances of other interlocutors. • Interaction modeling is conditioned on the analysis of discourse dependency relationships among utterances of all interlocutors, instead of that of a single interlocutor. Interactively modeling this kind of global structural features (e.g., posting and response) enables the connection of clues hidden in different interlocutors' utterances. Both of the aforementioned approaches contribute to the enhancement of the existing MDRC models (as overviewed in Section II). However, they still suffer from two bottlenecks. First, some of the crucial clues for reasoning are actually coreferred across utterances, or maintaining close relationships with pronouns. For example, the highlighted clues like "running PowerPC" in Fig. 1-(c) is co-referred by the pronoun "that" in the utterance U 2 , while the answer "No" in this case syntactically depends on the pronoun heavily (see Q 1 , U 1 and U 2 in Fig. 1-(a) and (b)). Though, the co-reference resolution has neither been incorporated into the utterance profiling process nor interaction modeling. Second, most of : Is neighborlee running PowerPC? : What should Peter do first to make the pc work? : No. : Download off an archive. ( the current studies fail to simultaneously take advantage of utterance profiling and interaction modeling. Instead, they are conducted separately and independently. This unavoidably causes the omission of dependency features of either internal clues or external ones. To overcome the bottlenecks, we propose a Coreference-Aware Double-channel Attention network (abbr., CADA). CADA is first characterized by the awareness of coreference at the initial encoding stage. Coreference resolution is performed to formulate the one-hot coreference graph. Graph-based attention modeling is further used to enhance token-level semantic encoding over all the utterances, where PLMs are considered as the fundamental encoders. CADA is additionally equipped with two parallel encoding channels. One performs utterance profiling to model attentive dependency features of internal clues within the utterances of each interlocutor, where a role-based utterance coherence graph is utilized. The other channel conducts interaction modeling to strengthen the interaction representation among multi-party utterances. Essentially, it is applied to perceive the attentive dependency features of external clues. Discourselevel dialogue graph is used for interaction modeling. We finally combine the three encoding results for answer prediction. In our experiments, we construct two CADA models using BERT [6] and ELECTRA [9] respectively. We evaluate models on the benchmark corpora Molweni [5] and FriendsQA [4], in terms of the canonical data partition schemes. Experimental results show that our approach yields significant improvements compared to the BERT and ELECTRA baselines. The maximum performance gain is about 2.5% F 1-score. In particular, CADA outperforms the state-of-the-art MDRC models. The rest of the paper is organized as follows. Section II overviews the related work. We present CADA in Section III. Section IV gives the experimental results and corresponding discussions on them. We conclude the paper in Section V . II. RELATED WORK We briefly overview two categories of related work, including that of dialogue machine reading comprehension and dialogue discourse dependency. The study of dialogue discourse analysis provides fundamental theories and supportive techniques for modeling cross-utterance interaction. A. Dialogue Machine Reading Comprehension Recently, the research of dialogue-oriented linguistic computing is undergoing a new trend, i.e., performing MRC over complicated multi-party dialogues [1], [4], [5]. This raises a new research topic-MDRC. The most crucial challenge of MDRC is that the interactions among different interlocutors produce an intricate discourse information flow, and thus the current semantics encoding approaches suffer from the confusion of internal features (i.e., the ones occurring in the utterances of a single interlocutor) and external features (i.e., the ones caused by interaction with other interlocutors). Liu et al. [13] propose a decoupled fusion network to decouple interlocutor information from utterance receiver and sender, where masked attention mechanisms are utilized. Ma et al. [14] strengthen the aforementioned mechanism using a speaker-aware graph. The graph explicitly represents the consanguineous relations for the utterances of the same speaker. Li et al. [15] design a self-supervised task to model interlocutors' information flows implicitly. All the above studies prove the awareness of interlocutors' characteristics and internal features of their own utterances contributes to dialogue understanding. More importantly, Ma et al. [14] and Li et al. [15] successfully use utterance profiling to improve the current MDRC models. It is noteworthy that our approach is different from the previous work. We enhance utterance profiling conditioned on the interlocutor role information. More importantly, we combine internal features which are pronouns and their corresponding entities, where coreference resolution is used. B. Dialogue Discourse Dependency In general, an interlocutor may respond to various utterances of other interlocutors in a multi-party dialogue flow. This allows the utterance-hopping discourse dependency structure to be formed implicitly. The previous studies illustrate that discourse dependency parsing helps to perceive the interaction mode among interlocutors and, accordingly, the potentially connected clues for reasoning [16]- [20]. Recently, a variety of utterance-level discourse parsers have been proposed. Shi and Huang [21] propose a deep sequential model for dependency parsing over utterances. Liu et al. [22] design a transformer-based discourse parser. It improves cross-domain performance using prior language knowledge and cross-domain pre-training. These parsers provide explicit discourse structures of utterances for downstream applications. In addition, considerable efforts have been made to integrate discourse structure into dialogue understanding models. Li et al. [23] develop a graph-based model (namely DADgraph) to fuse discourse dependency features, where a graph neural network is used. Jia et al. [18] design a thread encoder that incorporates dialogue dependency information by threads. On this basis, PLM is leveraged to generate corresponding representations. Gu et al. [24] develop an MPC-BERT model trained in several self-supervised tasks, where discourse structural features are used as the objectives for self-supervised learning. In this paper, we incorporate discourse structure information into an attention-based encoding channel of CADA. The goal is to enhance the capacity of CADA in perceiving the close interaction and, accordingly, assign higher attention to the interactively-dependent constituents among utterances. III. APPROACH We develop a coreference-aware two-channel neural network (namely CADA) for MDRC. CADA encodes the input utterances in terms of the raised question as usual. Though, it performs encoding to represent two categories of features, including internal and external features. The internal features are taken by utterance profiling in the coreference-aware embedding layer and a separate encoding channel, where coreference resolution and interlocutor role information are used to enhance the attention modeling of token-level semantic relationships, respectively. The external features are extracted by interaction modeling in another separate encoding channel, where explicit discourse dependency structures are used for modeling attentive information that is bridged by strong interactive relationships (among different interlocutors). Structurally, the two-channel neural encoders are built over a PLMbased embedding layer. PLMs like BERT and ELECTRA are used in our experiments. Besides, the encoders are connected with a fusion layer and the Fully-Connected linear layer (FC layer). The FC layer plays the role of a decoder which predicts the answers. We show the architecture of CADA in Fig. 2. A. Input and Output of MDRC We follow the previous studies [4], [5] to set up the input and output formations. Given a multi-party dialogue context D = {U 1 , U 2 , ..., U n } with n utterances and a question Q, MDRC aims to extract a text span A from D, and specify it as the answer for the question Q. Each utterance U i = {S i , W i } includes a interlocutor name S i and utterance contents W i issued by S i . We denote W i as a word sequence W i = {w i1 , w i2 , ..., w il } with l tokens. It is noteworthy that, in some benchmark MDRC corpus, the answer A can be answerable or unanswerable. The answerable A is constituted with a concrete text span (one token at least) and the unanswerable A has been officially designated with a tag "Truly Impossible". B. Coreference-aware Embedding Layer We conduct the first-stage encoding at the embedding layer, where PLM is utilized for computing the initial distributed representations of tokens in both Q and D. Different from the previous work, we additionally use coreference resolution at this stage, where pronouns correspond to the referred entities (nouns in some cases). This enables the sharing of attentive information among the co-referred tokens. Specifically, we first concatenate Q and utterances in D to form the input sequence [10], [25] to cluster pronouns and entities they referred (namely coreference cluster for short). For each dialogue D, in general, there are a variety of coreference clusters {C 1 , C 2 , ..., C u } that can be discovered and taken out of the utterances. For each cluster C i , the mentions (pronouns) referring to the same entity are organized into the sequence e i = {e i1 , e i2 , ..., e im }, where e ij is a mention or the referred entity. We use the graph to represent the coreference relationships of all the sequences e i s, i.e., one subgraph per e i and one global graph obtained by concatenating all subgraphs. Mathematically, the one-hot adjacent matrix M 1 ∈ R N ×N is used to represent the global graph of the input sequence X, where N denotes the number of tokens in X. We show an example of M 1 in Fig. 3-(a), where two subgraphs exist. Each corresponds to a coreference cluster. The coreference relations are highlighted by green colors which are signaled by the value "1" mathematically in the adjacent matrix (Note the coreference relations in the subgraphs are highlighted with different depths of green color). X = {[CLS] Q [SEP] U 1 [SEP] U 2 ... [SEP] U n [SEP]}, To enhance token-level coreference-aware encoding, we use M 1 to incorporate additional graph-based attention into the encoding process. Inspired by the previous study [26], we set up the trainable graph-based coreference-aware attention mechanism, and deploy it together with the canonical multihead self-attention mechanism. The goal is to enable the direct learning of structural coreference information conditioned on the query and key representations. Specifically, we utilize biaffine transformation [27] to construct coreference attentive score λ. It is computed as (1) and b serves as a prior bias which is utilized for differentiating coreference dependency from other features in the dialogue context. Accordingly, the coreference-aware attention score between the i-th and j-th token in each attention head is calculated as (2). λ ij = q i W ij k T j + b ij ,(1)e ij = q i k T j + λ ij M 1,ij √ d ,(2) By the aforementioned computation, we finally obtain the coreference-aware distributed representation H 1 ∈ R N ×F , where F denotes hidden states dimension. C. Double-channel Attention Modeling Grounded on the coreference-aware embedding layer, we develop a double-channel attention-based encoding model for utterance profiling and interaction modeling. Specifically, it is implemented using an interlocutor property module and discourse dependency module, respectively. 1) Interlocutor Property Module: Due to the uncertainty of the interaction process, there are intricate interlocutor information flows in a multi-party conversation. For example, five interlocutors participate in the dialogue in a relatively random order in Fig. 1. Meanwhile, the specific interlocutor information of "neighborlee" and "Peter" in the case is required during answering Q 1 and Q 2 , respectively. However, the information actually exchanges among interlocutors. As a result, the essential information flows back and forth across interlocutors and utterances. This phenomenon makes it difficult to either reason about key clues or capture the specific property of each interlocutor. Therefore, we set up an independent encoding channel to model the interlocutor property information. It conducts utterance profiling to represent the internal property of each interlocutor in the dialogue. All the utterances issued by the same interlocutor imply individual (personal) features, and they naturally form a rolebased utterance coherence subgraph. Accordingly, we build a role-based utterance matrix M 2 ∈ R N ×N , which is illustrated in Fig. 3-(b). More specifically, we denote s k as the interlocutor of the k-th utterance, and k i as the utterance where the i-th token is located. We set M 2 [i, j] = 1 if s ki = s kj . Hence, M 2 is formulated as follows: M 2 [i, j] = 1, s ki = s kj 0, otherwise(3) The graph-based attention is used to regulate the interaction with the interlocutor property information. In detail, we employ biaffine transformation [27] and the aforementioned M 2 to construct interlocutor property attentionė ij . The calculation method ofė ij is similar to (1) and (2). The only change is that M 1,ij is necessarily substituted by M 2,ij . Further, we use L 1 -layer transformer f tran to compute the representations H 2 ∈ R N ×F of interlocutor properties: H 2 = f tran (H 1 ,ė). 2) Discourse Dependency Module: As discussed in Section II(B), discourse dependency structure exhibits the interactive dependencies among utterances of different interlocutors. Fig. 1-(d) shows an example of the discourse structure graph, where each utterance serves as a node, and each node is connected by at least one specific-type relation arc. Such structure is crucial for perceiving valuable clues. For example, the fifth and seventh utterances (i.e., U 5 and U 7 ) hold a QAP relation (i.e., Question-Answering-Pair relation), which potentially contributes to the perception of clues for answering Q 2 . Accordingly, we intend to use the discourse dependency structure of dialogue to attentively weigh the utterances that possess key clues. Correspondingly, we design a decoupling module and deploy it at the second encoding channel, so as to impose the effects of discourse structure information upon the token-level representations. To be specific, we follow Ying et al. [28] to compute the distance between utterances in the dependency graph, and use it to establish a discourse dependency matrix G. For example, in Fig. 1-(b), U 7 → U 5 → U 3 is a path being launched from U 7 to U 3 , which holds a length l of 2 (l 37 =2) in terms of the relation arcs "QAP" and "Clari q (i.e., Clarification question)". Besides, we set l ij to 1 for the path between an utterance and itself. It is noteworthy that we set G ij to 0 when building G if the length l ij is larger than 0, while we set G ij to −∞ when there is no path between utterances (i.e., G ij =−∞ if l ij =0). For the utterances that are far apart from each other, we assume that their dependency intensity is relatively weak and negligible. Accordingly, we also set G ij to −∞ if the length between utterances is larger than the threshold γ. Fig. 3-(c) shows an example for matrix G. We formulate the dependency matrix G as follows: G[i, j] = 0, 0 < l i,j ≤ γ −∞, otherwise(4) Note that the connectivity itself (i.e., whether dependent) is more crucial than the dependency type (i.e., marks on dependency arcs) for our method. Hence, we omit the representation of dependency types when building dependency graphs. We utilize L 2 -layer transformer blocks to incorporate discourse dependency information into the interaction encoding channel. Following the scheme of masking attention [13], we adopt a mask-based multi-head self-attention mechanism to emphasize correlations between utterances: Atten (Q, K, V ) = sof tmax QK T √ d k + G V,(5) where Q, K, V denote the query, key, and value representation, respectively, and d k is the dimension of the key. Eventually, we obtain the distributed representation H 3 ∈ R N ×F that involves the attention-weighted effects of discourse dependency information. D. Answer Prediction and Training We combine the aforementioned three representations (i.e., coreference-aware representation H 1 , as well as the updated representations H 2 and H 3 ), and feed the fused representation to the decoder for predicting answers. Specifically, we concatenate H 1 , H 2 , H 3 to form the final representation H o ∈ R N ×3F : H o = [H 1 , H 2 , H 3 ],(6) On this basis, we use two fully-connected linear layers to compute the probability p s of the start position of a possible answer span over all tokens in the dialogue, as well as the probability p e of the end position. Answer extraction is conducted by taking out tokens between start and end positions. Besides, we employ a linear classifier to estimate the binary probability p t of whether the question is answerable. Given the start and end position of the answer span [A s , A e ] and answer type A t , the computation is formulated as follows: p s = sof tmax (H o W s + b s ) , p e = sof tmax (H o W e + b e ) , p t = sigmoid (H o W t + b t ) ,(7) where W s , W e , W t , b s , b e , b t are trainable matrices and biases. We use cross-entropy loss of both answer type and answer itself to train CADA. The loss is computed as (8), where K denotes the number of examples in a batch. Loss = − 1 K K [log(p s As ) + log(p e Ae ) + log(p t At )],(8) IV. EXPERIMENTATION We report the test results in this section, along with the experimental settings including the benchmark corpora, evaluation metrics, and implementation details (i.e., hyperparameter settings). We also conduct a series of analyses and discussions over the experimental results in this section. A. Datasets We experiment on two benchmark MDRC corpora, including Molweni [5] and FriendsQA [4]. Molweni is the first MDRC dataset that annotates the discourse dependency structure of dialogues, which is derived from Ubuntu Chat Corpus [29]. The interlocutor number per dialogue in Molweni is 3.51 on average. Besides, Molweni includes both answerable extractive questions and unanswerable cases. Due to the exact annotation of discourse dependency structure and the real scenario of multiple-interlocutor dialogue, Molweni appears as an appropriate corpus for the research of MDRC that is conditioned on the ground-truth structural information. FriendsQA is a question-answering dataset excerpted from the TV show Friends. Most of the dialogues in it are colloquial everyday conversations. FriendsQA contains 1,222 dialogues and 10,610 answerable extractive questions. Different from Molweni, FriendsQA does not provide the ground-truth annotations of discourse dependency structure among utterances. Therefore, We use Liu et al. [22]'s parser to pretreat the dialogues in FriendsQA for acquiring the structural information. We follow the canonical partition schemes to divide both corpora into the training, validation (Dev), and test sets. The statistics of the datasets are shown in Table I. B. Comparison and Evaluation We use two PLMs as the baselines, including BERT large [6] and ELECTRA large [9]. Both are connected with multiple perceptions for decoding the answers. In addition, we compare our CADA to two state-of-the-art neural MDRC models, including SKIDB [15] and ESA [14]. We follow the common practice [5], [15] to use F 1-score and exact matching score (EM for short) to evaluate all the models in our experiments. C. Implementation Details We implement our CADA based on Transformers Library [30]. We employ AdamW [31] as our optimizer. For Molweni, we set the batch size as 16 and the learning rate as 5e-5 when BERT is used as the embedding layer. When we employ the ELECTRA backbone, the batch size and learning rate are set to 1e-5 and 12, respectively. For FriendsQA, we set the batch size as 8. The learning rate is set to 6e-6 and 4e-6 for the BERTbased CADA and the ELECTRA-based version respectively. In addition, the number of interlocutor property layers L 1 and discourse dependency layers L 2 ranges from 2 to 4 according to different PLMs. We train our models for 2 or 3 epochs. We Table II shows the test results obtained on both Molweni and FriendsQA. It can be observed that our CADA produces substantial improvements, compared to both BERT large and ELECTRA large baselines. More importantly, CADA achieves state-of-the-art performance for F 1-score and EM in most cases, compared to the existing MDRC models. D. Main Results Specifically, we carry out the comparison with the neural models ESA [14] and SKIDB [15]. ESA is a strong graphbased neural model which leverages both the self-connected interlocutor graph and the mutual discourse dependency graph. Graph Convolutional Network (GCN) [32] is employed to strengthen the encoder for weighting available clues attentively. Particularly, ESA applies the masked attention modeling method conditioned on the structural information of the interlocutor graph. By contrast, SKIDB [15] employs a multi-task model which not only conducts the self-supervised interlocutor prediction but a key-utterance prediction. The shareable encoder for both tasks contributes to the joint incorporation of interlocutor property and key utterance information. Our CADA is different from ESA and SKIDB. On the one hand, CADA straightforwardly refines the attention computation in the transformers conditioned on 1) utterance subgraphs of different individual interlocutors and 2) discourse dependency graphs, where GCN is not used. On the other hand, CADA additionally updates the PLM embedding layer using coreference resolution. According to the results in Table II, ESA and SKIDB perform better than the baselines in most cases, except that SKIDB performs worse than the baselines on FriendsQA. On the contrary, they fail to outperform our CADA except that EM is considered for evaluation on FriendQA. Therefore, we suggest that, within the task-specific modeling method that is deployed upon the PLM layer, CADA has a relatively obvious superiority. Briefly, CADA outperforms ESA due to the fine-grained structural information integration, while it performs better than SKIDB because of the additional utilization of cross-utterance dependency features. More importantly, the awareness of coreference relationships in CADA enhances token-level encoding fundamentally. used as the baseline. It can be found that CADA stably outperforms the baseline for most question types except "Why"type questions. It proves that the perception of heterogeneous features (coreference features, internal features of an individual interlocutor, or interaction features among interlocutors) is beneficial for extractive MDRC. However, it is ineffective for answering "Why"-type questions, which are more likely treated well using generative models (instead of extractive ones). E. Ablation Study We conduct an ablation analysis on Molweni and FriendsQA to verify the effects of each module of CADA. It is conducted by verifying performance variation when a module is removed, where the hyperparameters remain unchanged. Table IV shows the results of ablation experiments. It can be observed that the three modules have different domain-specific positive effects. The coreference-aware embedding layer is more effective than interlocutor property and discourse dependency modules when FriendsQA is considered. The condition is thoroughly reversed when Molweni is considered. It is most probably because that most of the dialogues in FriendsQA are colloquial conversations and are full of a larger number of pronouns. F. Effectiveness of CADA on Long Dialogues A long dialogue usually comprises a mass of utterances issued by a larger number of interlocutors. It possesses more complicated self-coherent and mutually-interactive relationships. We provide a study on the effects of MDRC models on different lengths of dialogues by using ELECTRA-based CADA. To be specific, we divide a test set into different subsets according to the number of utterances or interlocutors in dialogues and verify the performance of CADA on each subset. Molweni is considered in the experiments instead of FriendQA. It is because there are 40% of dialogues in FriendQA contain overly-long contexts. External sliding windows are necessarily utilized for the separate encoding over them, which fundamentally obstructs the verification of the diversity of utterances and interlocutors. Table V shows the performance variation over the subsets. It can be found that the performance of CADA degrades significantly when a dialogue involves more than 10 utterances or there are more than 3 interlocutors joining the multi-party dialogue. More seriously, the performance is reduced more severely when the Interlocutor Property Module (i.e., IPM for utterance profiling) is removed from CADA. Therefore, we suggest dealing with the long dialogues for MDRC is still challenging and should be paid more attention to in a real application scenario. G. Case Study We analyze a series of dialogues in Molweni and FriendsQA manually, so as to intuitively investigate the impacts of CADA compared to baselines. Fig. 4 shows two representative cases, for which CADA extracts precise answers while the ELEC-TRA baseline yields incorrect ones. Note that the dialogue context in the first case has been illustrated in Fig. 1-(b), and thus it is omitted in Fig. 4 due to the page limitation. For the first case, the baseline predicts the question to be unanswerable, though CADA extracts the accurate answer. This is because CADA is able to better capture the discourse dependency from U 5 to U 7 , as well as the co-referred clues (e.g., "Peter" in U 5 and "you" in U 7 ). This results from the coreference-aware encoder and discourse dependency module. In the second case, for the question Q 1 , the baseline predicts the answer to be "Joey". It mistakenly implies that "Ross" communicates with "Joey". On the contrary, CADA precisely extracts the true answer "Rachel Green" since it correctly captures interlocutor properties and interaction information. For the question (Q 2 ), the baseline gives the answer "you're so great". It is the closely-related context to "Ross", though it is uttered by Rachel Green and has nothing to do with "How Ross feels about himself ". By contrast, CADA better captures the coreferential relation between Ross and himself. This clue supports the extraction of the correct answer. V. CONCLUSION We propose a novel MDRC model (namely CADA). CADA enables not only coreference-aware encoding but doublechannel attention-weighted encoding. More importantly, it supports the separate implementation of utterance profiling (internal feature extraction) and interaction modeling (external feature extraction), as well as the combination of internal and external features. Experimental results on benchmark corpora demonstrate that CADA yields substantial improvements compared to the PLM-based baselines, and achieves state-of-theart performance. The codes are available at https://github.com/ YanLingLi-AI/CADA. However, CADA fails to maintain a stable performance on long multi-party dialogues. The sufferings comprise two main aspects, including the unawareness of complicated coherent relations within the long-term iteratively-occurred utterances of a single interlocutor, as well as that among a larger number of interlocutors. In the future, accordingly, we will study the modeling of memories, including the ones of self-coherent and interactively-coherent contents in historical utterances. Fig. 1 : 1(a) Two question-answering pairs of the dialogue in (b). (b) An multi-party dialogue from Molweni [5]. Different highlight colors indicate different coreference clusters required for (a). (c) Different coreference clusters of the example dialogue. (d) The discourse graph of the example dialogue, where utterances with the same interlocutor are filled with the same color. Fig. 2 : 2The main architecture of Coreference-Aware Double-channel Attention network (CADA). Fig. 3 : 3(a) An example of coreference adjacent matrix M1, where different depths of green color indicate different coreference clusters. For instance, {t1, t4} belong to one cluster and {t3, t5, tN } relate to another, while t2 has no coreference link. (b) and (c) show the role-based utterance matrix M2 and discourse dependency matrix G for dialogue inFig. 1, respectively. For the sake of simplicity, we display in form of utterance units in (b). Different colors in (c) denote different path lengths between utterances in the discourse graph. We set the threshold γ as 2 in (c) for convenience. CADA on 32G NVIDIA V100 GPUs. It takes about 1 hour to train CADA for one epoch on Linux. : Who did Ross talk with at Monica and Rachel's ? ELECTRA (Our baseline): Joey CADA: Rachel Green : How did Ross feel about himself ? ELECTRA (Our baseline): You're so great CADA: Pleased with himself Dialogue context see Fig.1 (from Molweni) : What should Peter do first to make the pc work? ELECTRA (Our baseline): Unanswerable CADA: Download off an archive Fig. 4 : 4Two selected cases from the Molweni and FriendsQA, where the correct answers are marked in purple and incorrect ones in blue. #NOTE# refers to the utterance that provides additional information (location, time, character activities, etc.). You are not running PowerPC, are you? Neighborlee: Heh that would be a no. Tritium: I don't think we're all experiencing those problems. Ollie: Then something about an extended bios functions failing. Peter: If I got the name of the kernel and used apt-get, would it work? Tritium: But I've never tried anything like that. Are all the kernel listed amd64 specific? Tritium: I think you'll have to download off an archive then, to get it.a) (b) QAP Clari_q QAP Narration Comment Clari_q You_ you_ Neighborlee_ I_ you_ Peter_ it_ it_ PowerPC_ (c) (d) running PowerPC_ that_ Shock: where [CLS] and [SEP] are special tokens. The token [SEP] is used to separate adjacent utterances. Further, we employ the coreference resolution toolkit , where W is a trainable matrix,Internal Feature Encoding Channel External Feature Encoding Channel Coreference-aware Embedding Layer X X Fusing Layer Transformer Layer PLM Span Prediction Layer Answer Prediction Double-channel Attention Modeling Utterance Profiling Interaction Modeling TABLE I : IStatistics in Molweni and FriendsQA.Datasets Molweni FriendsQA Train Dev Test Train Dev Test Dialogues 8,771 883 100 977 122 123 Utterances 77,374 7,823 2,513 21,607 2,847 2,336 Questions 24,682 2,513 2,871 8,535 1,010 1,065 TABLE II : IIThe test results on Molweni and FriendsQA. The mark " †" denotes a statistically significant improvement in the F 1-score (p < 0.05) compared with baselines.Model Molweni FriendsQA EM F1 EM F1 BERT (Devlin et al., 2019) [6] BERT (Baseline) 50.5 65.1 46.0 63.1 SKIDB (Li and Zhao, 2021) [15] 51.1 66.0 46.9 63.9 ESA (Ma et al., 2021) [14] 52.9 66.9 49.0 64.0 CADA † (Ours) 52.9 67.6 47.4 65.6 ELECT RA (Clark et al., 2020) [9] ELECTRA (Baseline) 57.4 71.8 56.9 74.9 SKIDB (Li and Zhao, 2021) [15] 58.0 72.9 55.8 72.3 ESA (Ma et al., 2021) [14] 58.6 72.2 58.7 75.4 CADA † (Ours) 59.8 73.6 59.2 76.7 TABLE III : IIIPerformance of ELECTRA-based CADA on different question types. The marks ↑ and ↓ denote performance gain and induction compared to the ELECTRA baseline. Dist signals the proportion of each question type in the dataset (%).Types Molweni FriendsQA Dist. EM F1 Dist. EM F1 Who 4.7 80.1 (↑ 4.2) 82.1 (↑ 4.8) 18.8 66.4 (↑ 2.2) 78.4 (↑ 1.9) When 1. Table III provides a direct insight into the performance of CADA for different question types, where ELECTRA large is TABLE IV : IVAblation study on both corpora. IPM, DDM, and CAE are abbreviations of Interlocutor Property, Discourse Dependency modules, and Coreference-Aware Embedding layer, respectively.Model Molweni FriendsQA EM F1 EM F1 CADA (ELECTRA) 59.8 73.6 59.2 76.7 w/o IPM 58.5 72.6 58.9 75.8 w/o DDM 58.8 72.5 58.2 75.5 w/o CAE 59.2 73.0 58.2 75.4 TABLE V : VPerformance of CADA (ELECTRA) on Molweni under different numbers of utterances or interlocutors. #NOTE#: [Scene: Monica and Rachel's, Ross and Rachel are talking about passion.] Ross Geller: See, I see... Big passion in your future. Rachel Green: Really? Ross Geller: I do. Rachel Green: Oh Ross. 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[ "Privacy-aware Process Performance Indicators: Framework and Release Mechanisms", "Privacy-aware Process Performance Indicators: Framework and Release Mechanisms" ]	[ "Martin Kabierski \nHumboldt-Universität zu Berlin\nGermany\n", "Stephan A Fahrenkrog-Petersen \nHumboldt-Universität zu Berlin\nGermany\n", "Matthias Weidlich \nHumboldt-Universität zu Berlin\nGermany\n" ]	[ "Humboldt-Universität zu Berlin\nGermany", "Humboldt-Universität zu Berlin\nGermany", "Humboldt-Universität zu Berlin\nGermany" ]	[]	Process performance indicators (PPIs) are metrics to quantify the degree with which organizational goals defined based on business processes are fulfilled. They exploit the event logs recorded by information systems during the execution of business processes, thereby providing a basis for process monitoring and subsequent optimization. However, PPIs are often evaluated on processes that involve individuals, which implies an inevitable risk of privacy intrusion. In this paper, we address the demand for privacy protection in the computation of PPIs. We first present a framework that enforces control over the data exploited for process monitoring. We then show how PPIs defined based on the established PPINOT meta-model are instantiated in this framework through a set of data release mechanisms. These mechanisms are designed to provide provable guarantees in terms of differential privacy. We evaluate our framework and the release mechanisms in a series of controlled experiments. We further use a public event log to compare our framework with approaches based on privatization of event logs. The results demonstrate feasibility and shed light on the trade-offs between data utility and privacy guarantees in the computation of PPIs.	10.1007/978-3-030-79382-1_2	[ "https://arxiv.org/pdf/2103.11740v1.pdf" ]	232,307,539	2103.11740	7c508ae4add59799b676798ba3f385186db2106a	Privacy-aware Process Performance Indicators: Framework and Release Mechanisms Martin Kabierski Humboldt-Universität zu Berlin Germany Stephan A Fahrenkrog-Petersen Humboldt-Universität zu Berlin Germany Matthias Weidlich Humboldt-Universität zu Berlin Germany Privacy-aware Process Performance Indicators: Framework and Release Mechanisms Performance Indicators · Process Monitoring · Differential Privacy Process performance indicators (PPIs) are metrics to quantify the degree with which organizational goals defined based on business processes are fulfilled. They exploit the event logs recorded by information systems during the execution of business processes, thereby providing a basis for process monitoring and subsequent optimization. However, PPIs are often evaluated on processes that involve individuals, which implies an inevitable risk of privacy intrusion. In this paper, we address the demand for privacy protection in the computation of PPIs. We first present a framework that enforces control over the data exploited for process monitoring. We then show how PPIs defined based on the established PPINOT meta-model are instantiated in this framework through a set of data release mechanisms. These mechanisms are designed to provide provable guarantees in terms of differential privacy. We evaluate our framework and the release mechanisms in a series of controlled experiments. We further use a public event log to compare our framework with approaches based on privatization of event logs. The results demonstrate feasibility and shed light on the trade-offs between data utility and privacy guarantees in the computation of PPIs. Introduction Many companies improve their operation by applying process-oriented methodologies. In this context, Business Process Management (BPM) provides methods and techniques to aid in the monitoring, analysis, and optimization of business processes [4]. Important means to enable the continuous optimization of processes are process performance indicators (PPIs), i.e. numerical measures computed based on data recorded during process execution [3]. PPIs assess whether predefined goals set by the process owner are fulfilled, e.g., related to the mean sojourn time of a business process. Fig. 1 illustrates a simple insurance claim handling process and respective PPIs. Each indicator comprises a definition of a measure, a target value, and an observation period, called scope. The data used to calculate PPIs often includes personal data. In Fig. 1, such data relates to the knowledge workers handling the claims or the customers who submitted them. Processing of personal data is strictly regulated. The GDPR [7], as an example, prohibits the use of personal data without explicit consent and especially restricts their secondary use, i.e., the processing of data beyond the purpose for which they were originally recorded. Process optimization typically represents such a secondary use of process execution data [14]. To motivate, why unregulated access to process execution data may be problematic, we turn back to the example model and PPI in Fig. 1b. Assume that data is recorded about three claims handled by Alice with sojourn times of 4, 4, and 5 days; three claims handled by Bob within 2, 6, and 6 days; and three claims handled by Sue lasting for 7, 8, and 8 days. Here, the mean sojourn time of these nine process instances is~5.5 days and thus fulfils PPI 1 set by management. Yet, considering this data directly would reveal Sue's generally slower processing times, which may be prohibited by privacy regulations. Here, privacy-protected PPI schemes, i.e., techniques that incorporate data anonymization in the computation of PPIs, would allow for the evaluation of PPIs, while protecting the privacy of the recorded individuals in the log file, thus lifting these privacy regulations. Yet, data anonymization commonly leads to a trade-off between the strength of a privacy guarantee and a loss in data utility, thus a privacy-protected PPI scheme needs to minimize the accuracy loss introduced. Models for privacy-aware computation of traditional aggregates [16,20] have limited applicability for PPIs, though. Since these models do not take into account the highly structured nature of data generated by processes and PPIs defined on them, these methods are not suitable for privatizing PPIs. Approaches for privacy-aware publishing and querying of process execution data [8,13], in turn, are too coarse-grained. Handling comprehensive execution data, these techniques cannot be tailored to minimize the loss in data utility for a given set of PPIs. Against this background, we identify the research question of how to design a framework for the evaluation of privacy-protected PPIs. In this paper, we address the above question, by proposing PaPPI, a first framework for privacy-aware evaluation of PPIs. It separates trusted and untrusted environments to handle process execution data. They are connected by a dedicated interface that serves as a privacy checkpoint, ensuring -differential privacy [5]. We then instantiate this framework with data release mechanisms for PPIs that are defined based on the established PPINOT meta-model [3]. This way, we enable organizations to compute expressive PPIs without risking privacy violations. Finally, we explore the impact of privacy-aware evaluation of PPIs on their quality. We report on controlled experiments using synthetic data and a case study with a publicly available event log. Our results demonstrate the feasibility of the framework and its instantiation through specific release mechanisms, given that a reasonable amount of process execution data has been recorded. In the remainder, Section 2 provides background on PPIs and privacy guarantees. Section 3 introduces our framework for privacy-aware evaluation of PPIs, which is instantiated with specific release mechanisms in Section 4 and evaluated in Section 5. Finally, we review related work in Section 6, before we conclude in Section 7. Background We introduce a basic model for event logs (Section 2.1) and process performance indicators (Section 2.2). Finally, we review the concept of differential privacy (Section 2.3). Notions and Notations for Event Logs We consider ordered, finite datasets, each being a set of elements X = {x 1 , . . . , x n } that carry a numeric value and are partially ordered by ≤. 1 The cardinality of the dataset is denoted as \|X\| = n. For one of the (potentially many) elements of X that are minimal and maximal according to ≤, we write X and X, respectively. An interval of the dataset is defined by I = (x lower , x upper ) with x lower , x upper ∈ X and x lower ≤ x upper . Lifting the notation for minima and maxima to I, we define I = x lower and I = x upper . Our notion of an event log is based on a relational event model [1]. That is, an event schema is defined by a tuple of attributes A = (A 1 , . . . , A n ), so that an event is an instance of the schema, i.e., a tuple of attribute values e = (a 1 , . . . , a n ). An event schema consists of at least three attributes, the case that identifies the process instance to which an event belongs, the timestamp for the point in time an event has been recorded, and the activity, for which the execution is signalled by an event. The timestamp-ordered list of events corresponding to a single case is called a trace. Such a trace represents the execution of a single process instance. An event log is a set of traces. Process Performance Indicators A key performance indicator (KPI) is a metric that quantifies, to which extent the goals set for an organisation are fulfilled. A process performance indicator (PPI) is a KPI, which is related to a single business process and which is evaluated solely based on the traces recorded during process execution. The Process Performance Indicator Notation (PPINOT) [3] is a meta-model for the definition and evaluation of PPIs. At its core, the PPINOT model relies on the composition of measures, i.e., simple, well-defined functions that enable the definition and automated evaluation of more complex PPIs: Base measures concern a single instance of a process and include event counts (e.g., to count activity executions), timestamp differences between events, the satisfaction of conditions, or aggregations over the events' attribute values. Aggregation measures are multi-instance measures that combine values from multiple process instances into a single value. PPINOT includes aggregation measures to calculate the minimum, maximum, mean, and sum of a set of input values. Derived measures are user-defined functions of arbitrary form, applied to a single process instance, or a set thereof. r(x, y) = x y · 100 sum 1 count('RejC') sum 2 count('RecC') x y Fig. 2: The function composition tree of PPI 2 in Fig. 1b. A PPI defined using the PPI meta-model is represented as a function composition tree. Fig. 2 exemplifies such a tree for the PPI 2 of our example from Fig. 1b. It calculates the fraction of rejected insurance claims and received claims. Here, count is a base measure; sum is an aggregation measure; and the fraction r(x, y) is a derived measure. Differential Privacy Differential privacy [5] is a privacy guarantee that limits the impact a single element may have on the output of a function f that is computed over a set of elements. Therefore, it limits an adversary to conclude on the set of used input elements (or the presence of a certain input element) from the result of the function. This obfuscation is usually achieved by adding noise, for which the magnitude depends on the sensitivity ∆f of function f , i.e., the maximal impact any element x ∈ X may have on f (X). A randomized mechanism K is a randomized function that can be applied to a dataset, with range(K) as the set of possible results. Let D 1 , D 2 be two neighbouring datasets, i.e., they differ in exactly one element. The randomized mechanism K provides ( , δ)-differential privacy, if the following inequality holds for the probabilities of the function result falling into a sub-range of all possible results: ∀ S ⊆ range(K) : P (K(D 1 ) ∈ S) ≤ e P (K(D 2 ) ∈ S) + δ Differential privacy enforces an upper bound on the difference in result probabilities of neighbouring datasets. If δ = 0, K is -differentially private (or is omitted altogether). Larger values imply weaker privacy, while the contrary holds true for smaller values. A specific mechanism to achieve differential privacy is the Laplace mechanism [5]. It adds noise sampled from a Laplace distribution with parameters µ = 0 and b = ∆f / onto f (X). Due to the symmetric nature of the Laplacian and the exponential falloff, results are expected to lie close to f (X). The symmetrical monotonous falloff of the Laplace mechanism may yield undesirable results, e.g., if values close to the true result have a disproportionally negative effect on the utility. The exponential mechanism [15] avoids this problem, by constructing a probability space based on a function q(D, r), which assigns a score to all results r ∈ range(K) based on the input dataset D. Here, a higher score is assigned to more desirable results. The mechanism then chooses a result r ∈ range(K) with a probability proportional to e ( q(D,r))/(2∆q) , where ∆q is the sensitivity of the scoring function, i.e., the maximum change in assigned scores possible for two neighbouring datasets. Both above mechanisms assume that ∆f and ∆q are known beforehand. The sampleand-aggregate framework [17] drops this assumption by sampling subsets of the input set and evaluating the given function per sample. The obtained results are then combined using a known differentially private aggregator. If function f can be approximated well on small sub-samples, then the results per sample are close to f (X). By aggregating these approximated results using a differentially private mean, i.e., by computing the mean and adding noise drawn from a Laplacian calibrated with ∆(mean), one achieves a differentially private result for f (X). A Framework for Privacy-Aware PPIs In this section, we introduce a generic framework for the evaluation of privacy-protected PPIs, thereby addressing the research question raised above. Specifically, we discuss design decisions, as well as the underlying assumptions and limitations of the framework. The evaluation of PPIs is usually conducted on event data recorded and administered by the process owner. As such, we consider a centralized model and assume that the entity collecting and persistently storing the event data is trustworthy. However, as illustrated in Fig. 3, the actual information demand regarding the PPIs is external to this environment. Following existing models for the flow of data in the analysis of information systems, the evaluation of PPIs is conducted in an untrusted environment [2,14]. Considering the handling of event data in the trusted environment in more detail, the following phases are distinguished. First, the Data Capture phase concerns the collection of process-related event data, i.e., whenever an activity has been executed, a respective event is created. Subsequently, the phase of Primary Use represents that the captured event data is exploited for the purposes for which it was recorded, which is commonly the proper execution of an individual instance of the process. For example, the recorded events may be used to invoke services, trigger notifications, or schedule tasks for knowledge workers. At the same time, the event data is made persistent, which is modelled as a Data Storage phase. The persisted event data may then be used for Secondary Use, such as process improvement initiatives conducted by process analysts. Eventually, the event data may be deleted from the persistent storage, in a Data Deletion phase. All phases, except the Secondary Use, are conducted within the realms of the trusted environment. The Secondary Use is part of the untrusted environment, since the data was recorded without having any consent on their use for these applications. Unlike common primary use of process-related event data, process improvement in general, and the computation of PPIs in particular, aim at generalizing the observations made for individual process instances in some aggregated measures. Thus, in these contexts, the privacy of an involved individual would be compromised, if their contribution to the published aggregate would be revealed to the process analyst. To enable such secondary use without compromising an individual's privacy, we need to prevent a process analyst to assess the impact of a single process instance on the aggregated result. Hence, we consider each trace and the information inferred from it as sensitive information. For example, when a PPI is based on the mean sojourn time of all process instances, we aim to protect the specific sojourn time of each instance. To achieve this protection, any access to the event data from the untrusted environment must be restricted. Therefore, we propose to design an interface for the evaluation of PPIs, thereby realizing an explicit privacy checkpoint. The interface receives PPI queries stated in PPINOT syntax and answers them while ensuring differential privacy. To this end, the interface fetches relevant event data from the persistent storage based on the scope of the PPI query, calculates the result, and adds noise to the result, before releasing the result to the process analyst. Any such release reduces a privacy budget, which is chosen based on the desired strength of the privacy guarantee to implement. Since the noise added to the result is calibrated based on the specifics of a PPINOT query and the event data retrieved for evaluating it, we ensure -differential privacy. By the above, we achieve plausible deniability: An analyst cannot distinguish between query results that contain a particular process instance and those that do not. While access to the event data is restricted, our framework assumes that an analyst has access to models of the respective processes in order to specify the PPIs. Another assumption of our framework is that, for a given time scope, an upper bound for the appearances of an individual in the recorded process instances is known. An individual appearing in n process instances dilutes -differential privacy by at most e n . Knowing an upper bound for n, however, enables mitigation of this effect by changing the privacy parameter accordingly. For our example in Fig. 1, we would need to know the maximal number of claims that can be handled by a knowledge worker within a single month. Lastly, we acknowledge that, while we focus on the evaluation of PPIs, further privacy threats in the trusted environment require additional protection mechanisms [2]. Release Mechanisms for Privacy-Aware PPIs In this section, we instantiate the above framework and introduce a specific realization of the interface for the evaluation of PPIs. We first show how the interface leverages the compositional structure of PPIs defined based on the PPINOT meta-model in Section 4.1. We then provide a set of -differentially private release mechanisms in Section 4.2. Using Function Composition Trees for Privacy Protection Our idea is to exploit the compositional nature of PPIs defined in the PPINOT meta-model for privacy protection. Instead of adding noise to the final query result, we introduce noise, with smaller magnitude, at the inner functions of a PPI. Such a compositional approach still guarantees -differential privacy of the result. At the same time, it enables us to minimize the overall introduced error. Hence, data utility is preserved to a higher degree, which leads to more useful process analysis, under the same privacy guarantees. We aim to protect the privacy of individuals, of whom personal data is materialized in a trace. Hence, the results of single-instance measures (base or derived measures) shall be protected. However, common PPIs assess the general performance of process execution by aggregating these results in multi-instance measures (aggregation or derived measures), so that guarantees in terms of differential privacy may be given for these measures. This raises the question of selecting a subset of the multi-instance measures for privatization. On the one hand, this selection shall ensure that the results of all aforementioned single-instance measures are protected. On the other hand, the selection shall be minimal to keep the introduced noise to the absolutely necessary magnitude. We capture the above intuition with the notion of an admissible set of measures of a PPI. Let (F, ρ) be the function composition tree of a PPI, with F as the set of measures and ρ : F → 2 F as the function assigning child measures to measures. With ρ * as the transitive closure of ρ, a set of measures F ⊆ F is admissible, if it: contains only multi-instance measures: f ∈ F implies that f ∈ dom(ρ); covers all trace-based measures: ∀ f ∈ (F \ dom(ρ)) : ∃ f ∈ F : f ∈ ρ * (f ); -is minimal: ∀ F ⊂ F : ∃ f ∈ (F \ dom(ρ)) : ∀ f ∈ F : f / ∈ ρ * (f ). The first condition of an admissable set applies, as differential privacy may only be used for the aggregation of multiple inputs, thus single-instance measures cannot be privatized with the given privacy framework. The second condition ensures, that the selected set of functions privatizes all single-instance derived or base measures, that directly access trace information. Finally, the third condition ensures, that only the minimum amount of noise to achieve -differential privacy is added onto the intermediate results. The function composition tree in Fig. 2 has two sets of admissible measures, {r} and {sum 1 , sum 2 }, which both cover the single-instance measures count('RejC') and count('RecC'). In contrast, the set {sum 1 } is not admissible, as count('RecC') is not covered (second constraint). Likewise, selecting both base measures or {r, sum 1 , sum 2 } is not admissible, as this would violate the first and third constraint, respectively. Once a set of admissible measures is selected, the evaluation of the PPI is adapted by incorporating a release mechanism, as defined next, for the chosen measures. Release Mechanisms for Multi-Instance Measures The design of a release mechanism for a specific multi-instance measure is influenced by (i) the ability to assess the domain of input values over which the measure is evaluated, and (ii) the ability to assess the sensitivity of the measure. As for the first aspect, the PPI interface of our framework, see Section 3, can rely on an estimation of the respective domain. Here, a simple estimation is based on the minimal and maximal values, X and X, of the dataset X used as input for the measure (i.e., the result of the child measures). The bounds may be extended by constant offsets to account for the fact that the dataset X is merely a sample of an unknown domain. The sensitivity of the measure, in turn, depends on the semantics of the measure. While for the aggregation functions of PPINOT, this sensitivity may be estimated, it is unknown in the general case of derived measures. Against this background, this section first introduces three release mechanisms for aggregation measures: an instantiation of the Laplace mechanism; an interval-based mechanism based on the traditional exponential mechanism; and a threshold-sensitive mechanism that extends the interval-based one to preserve the significance of a measure related to a threshold. Finally, we discuss how derived measures, in the absence of an estimate of their sensitivity, can be privatized using a sample-and-aggregate strategy. Laplace Mechanism for Aggregation Measures. Privatization of an aggregation measure can be based on the addition of Laplace noise to the actual result. As mentioned, this requires to estimate the sensitivity ∆f of the given aggregation function, i.e., the maximal impact any element x ∈ X may have on f (X). Since the sensitivity ∆f directly influences the magnitude of added noise, for mean measures, this mechanism potentially leaks information about the number \|X\| of process instances (and hence, individuals) within the given scope. An adversary may conclude on the difference \|X − X\| based on the magnitude of noise from another PPI incorporating a min or max measure and, based thereon, derive \|X\| from the magnitude of noise in a PPI with a mean measure. However, in practice, \|X\| may be revealed explicitly to enable a process analyst to assess the statistical reliability of the PPI result. Interval-based Mechanism for Aggregation Measures. The drawback of the Laplace mechanism to privatize aggregation measures is the inherently high sensitivity, which scales linearly with the domain of input values. Our idea, therefore, is to group similar result values into intervals and score them using the exponential mechanism. This way, we obtain a release mechanism with a score function sensitivity ∆q = 1, which ultimately leads to a smaller magnitude of noise for large domains of input values. To realize this idea, our interval-based release mechanisms consists of three phases: (1) Interval creation: We partition the domain and the range of the aggregation function into a set of intervals. (2) Interval probability construction: Scores are assigned to these intervals, which are then converted to result probabilities. (3) Result sampling: Using these probabilities, an interval is chosen as the output interval, from which the result value is sampled. The interval creation is based on the range of the aggregation function, given as range(f (X)) = (X, X) for f ∈ {min, max, mean} and range(f (X)) = (X · \|X\|, X · \|X\|) for f = sum. This range is split into non-overlapping intervals I = {I 0 , . . . , I n }, with I 0 ∩ . . . ∩ I n = ∅ and I 0 ∪ . . . ∪ I n = range(f (X)). Let τ (x i , x j ) = (x i + x j )/2 be the mean of x i , x j and let I f be the interval containing the result value, i.e. f (X) ∈ I f . For mean and sum, the range of f (X) is divided into evenly spaced intervals of size ∆f , so that f (X) = τ (I f , I f ) is the mean of its containing interval. For min and max, the range of f (X) is divided into n intervals of different size, for which the boundaries are the means of neighbouring values τ ( The interval probability construction relies on a scoring function that assigns higher scores to intervals that are closer to the interval containing f (X). Let I 1 , . . . , I n be the intervals in the order induced by ≤ over their boundaries, and let 1 ≤ k ≤ n be the index of interval I f containing the result value. Then, the score for each interval I i is defined as q(i) = −\|k − i\|, as illustrated in Fig. 4 for the example. Here, intervals, that lie closer to f (X), denoted by the blue dashed lines, are scored higher, than those further away. Since each interval I i corresponds to a set of potential result values, we incorporate the size of this set in the probability computation. Hence, the probability for I i is defined as: x i , x i+1 ) with x i , x i+1 ∈ X.P (I i ) = \|I i \| · e ( ·q(i) /2·∆q) Σ 1≤j≤n \|I j \| · e ( ·q(j) /2·∆q) Result sampling chooses one interval based on their probabilities. From this interval one specific value is drawn based on a uniform distribution over all interval values. Threshold-Sensitive Mechanism for Aggregation Measures. The interval-based mechanism is problematic, if a PPI is tested against a threshold, as often done in practice. Consider the dataset X and assume that the sum function is the root of a PPI's function composition tree, i.e., f (X) = 30 as shown in Fig. 4. Assume that it is important whether the PPI is less or equal than 30. Then, adding noise may change the actual interpretation of the PPI, since the release mechanism will sometimes publish values larger than 30. To mitigate this effect, we present a threshold-sensitive release mechanism that extends the previous mechanism in terms of interval creation and interval probability construction. Let χ be a Boolean function formalizing a threshold, e.g., χ(x) = x ≤ 30. Then, the Boolean predicate φ(x, f (X), χ) ⇔ χ(x) ≡ χ(f (X)) describes, whether the possible result value x ∈ range(f (X)) leads to the same outcome of χ as the true result f (X). For our example, φ(20, 30, χ) holds true (20 ≤ 30 and 30 ≤ 30), whereas φ(40, 30, χ) is false (40 30, but 30 ≤ 30). Using this predicate, we adapt the intervals I = I 1 , . . . , I n obtained during interval creation, so that interval boundaries coincide with changes in φ. Let B(φ) be the boundary values of φ, i.e., the values x ∈ range(f (X)) with lim y<x,y→x φ(y, f (X), χ) = lim y>x,y→x φ(y, f (X), χ). For our example, we arrive at B(φ) = {30}. Based thereon, we split each interval I i containing a boundary value b ∈ B(φ) into two new intervals (I i , b), (b, I i ). Hence, each interval contains only values that share the outcome of the Boolean function χ. In our example, the interval (25, 35) is split into (25, 30) and (30, 35), as shown in Fig. 5. Finally, the scoring function used for interval probability construction is adapted. Let d(i) be the minimal inter-interval-distance of interval I i to any other interval I j with φ(x, f (X), χ) = φ(y, f (X), χ) for all I i ≤ x ≤ I i and I j ≤ y ≤ I j . As before, let k be the index of interval I f containing the result value. Then, scores assigned to intervals that preserve the outcome of the Boolean function χ remain unchanged. For all other intervals I i , the score is reduced by ξ · d(i), i.e., by the distance to the closest interval preserving the outcome multiplied by a falloff factor ξ ∈ N. The adapted scoring function is defined as: We obtain d(4) = 1, d(5) = 2, and d(6) = 3 for those intervals, given that the third interval (25, 30) is the closest one retaining φ to any of those three. Thus, we arrive at q(4) = −4, q(5) = −8, and q(6) = −12. As the largest possible change in scores assigned to a possible result value in neighbouring input sets is never larger than ξ and as the interval sizes are determined based on ∆f , we conclude that ∆q = ξ. q(i) = −\|k − i\| if φ(x, f (X), χ) for all I i ≤ x ≤ I i , −\|k − i\| − ξ · d(i) otherwise. Sample-and-aggregate Mechanism for Derived Measures. Since the sensitivity of a derived multi-instance measures is unknown in the general case, the above mechanisms are not applicable. However, many derived measures may be approximated using small samples, since their range is often independent of the domain of their input values. Functions that compute a normalized result are an example of this class of measures. For instance, the derived measure that denotes the root of the function composition tree of the example in Fig. 2 yields a percentage, i.e., it is normalized to 0% to 100%. For such measures, the sample-and-aggregate-framework mentioned in Section 2.3 may be instantiated. That is, the actual result f (X) is computed on n partitions of X. The obtained results per sample are then aggregated using a differentially private mean function to achieve privatization of the derived measure. Experimental Evaluation To assess the feasability and utility of the proposed approach, we realized the PPI interface on top of an existing PPINOT implementation. 2 We conducted controlled experiments using synthetic data (Section 5.1) and a case study with the Sepsis Cases log (Section 5.2). The latter compares the proposed tree-based privatization with the direct evaluation of PPIs on logs that have been anonymized with the PRIPEL framework [10] beforehand. Our implementation and evaluation scripts are publicly available. 3 Controlled Experiments In a first series of experiments, we assessed the impact of different properties of the dataset X used as input. Specifically, we consider the impact of the estimation of the domain of input values, its size and underlying value distribution, and the privacy parameter . We sampled sets of 10, 50, 100, and 200 random values from a Gaussian distribution, a Pareto distribution, and a Poisson distribution. We chose these distributions, as they are often observed in event data recorded by business processes. We performed 200 runs per experiment. Unless noted otherwise, the input domain is estimated using the minimal and maximal element of X, the dataset comprises 200 values drawn from a Gaussian distribution, and the privacy parameter is set as = 0.1. Input Boundary Estimation. First, we compare the boundary estimation using the minimal and maximal elements in X with extensions of these boundaries by 15% and 30% at either boundary. The results for the interval-based mechanism, see Fig. 6, show that an extension of the domain increases the introduced magnitude of noise for all functions due to an increase in sensitivity. These observations are confirmed for the Laplace mechanism. Yet, for min and max, there is a shift of the expected result towards the true result f (X) (denoted by the blue line). The reason is that, without the extension, f (X) coincides with boundary values of X. The extension increases the size of the interval containing f (X), which increases the probability of this interval to be chosen. Input Size and Distribution. For the Laplace and interval-based mechanisms, we identify a dependency of ∆f on the input size for mean functions. This dependency coincides with smaller noise magnitudes for larger input sizes, as illustrated in Fig. 7. These trends were confirmed for the interval-based mechanism for min and max. Here, the increased number of intervals and a more fine-grained differentiation between result values leads to higher utility, i.e., the expected result is close to the actual one. Yet, the trends are only visible for distributions with small inter-value distances, such as the Gaussian. For the Pareto-and Poisson-distributions, there was a significant reduction in utility for larger inputs using max. These distributions preserve most of their probability mass on the smaller values, This inadvertently results in the creation of disproportionally large intervals and the same output probability for large portions of the output space. Epsilon. The results obtained when changing the privacy parameter are shown in Fig. 8a for mean. Both the Laplace and interval-based mechanism show a similar increase in the introduced noise. The Laplace mechanism yields better results for larger . For min and max, however, the interval-based mechanism clearly outperforms the Laplace mechanism for all values of , see Fig. 8b for the maximum function. Here, the large sensitivity for the Laplace mechanism completely obfuscates the actual result f (X), rendering the mechanism inappropriate for these functions. Threshold-sensitive Mechanism. For the extension of the interval-based mechanism that aims to preserve the significance for thresholds, the general trends remain unaffected. However, the threshold-sensitive mechanism shifts large portions of the probability mass of the output space, as shown in Fig. 9. Here, the threshold to preserve is φ(x) : x < f (X) ± y, with y being 100 for sum and 10 for the other aggregation functions. For comparison, the results for the interval-based mechanism without threshold preservation are also given. There is a clear shift in output probabilities, depending on which values preserve the same properties as f (X). Note that the results should not be interpreted in absolute terms, but serve as a binary indicator regarding the threshold. Derived Measures. The sample-and-aggregate mechanism for derived measures mirrored the trends of the Laplace mechanism for mean. This is expected since the mechanism is based on the privatized mean. Yet, due to the use of m buckets of size n, the magnitude of noise is larger. The mechanism requires m times as many values in X to achieve the same sensitivity as the mechanism for the mean. Since the mean is computed using n values per bucket, the result estimation is accurate only for large datasets. Case Study: Process for Sepsis Cases To explore how the presented mechanisms perform in a real-world application, we conducted a study using the Sepsis Cases log. As part of that, we compare our approach to a state-of-the-art privatization approach for event logs. That is, we evaluated the same PPIs non-anonymously using logs that have been anonymized with PRIPEL [10]. The PPIs used in our case study were created based on criteria and guidelines presented in [12,21] and are listed in Table 1, together with the employed mechanism used for privatization. Some concern the lengths of stays and treatments for patients (PPI 1-4), wile others target the adherence to treatment guidelines (PPI 5-6). To illustrate the behaviour of our release mechanism, we calculated each PPI 10 times using = 0.1 and report aggregate values. While results for all PPIs are available online, due to space constraints, we here focus on PPI 1 and PPI 6, see Fig. 10 and Fig. 11, respectively. For PPIs 1 to 3, we were able to reconstruct the general trends of the non-privatized analysis (exemplified for PPI 1 in Fig. 10). Yet, we also observed specific months with high result variances. For PPI 1 and 2 (mean functions), the variance stems from the The results obtained with our framework are in sharp contrast to those achieved when privatizing the event log with PRIPEL before computing the PPIs in a regular manner. As shown in Fig. 10 (right), the latter approach accumulates an error over the recorded time period. The steadily increasing deviation from the true value is caused by traces that represent outlier behaviour, which was artificially created by PRIPEL. PPIs 4 to 6 were calculated using privatized sum functions. Due to the relatively low number of traces recorded per month, the application of the sample-andaggregate-framework for the calculation of the final percentage value led to worse results. Here, the buckets contained not enough values to approximate the true result well. However, using privatized sum functions, the results for PPIs 4 to 6 follow the general trends of the true values, see Fig. 11 for PPI 6. Similarly, also the computation based on logs privatized with PRIPEL yields comparable results. In months, in which few traces are selected for a PPI, e.g., at the beginning and end of the covered time period, the variance is notably larger for our proposed framework, an effect that is avoided by the approach based on event log privatization. Turning to research question RQ3, our results provide evidence that the proposed framework enables the computation of privacy-aware PPIs that mirror the general trends of their true values. Only for time periods, in which the PPI computation is based solely on a few traces, our framework does not yield sensible results. Thus, given a sufficiently large number of traces as the basis for the evaluation of PPIs, we can expect our framework to retain the trends. Related Work For a general overview of privacy-preserving data mining, as mentioned in Section 1, we refer to [16] and [20]. However, data anonymization commonly leads to a tradeoff between the strength of a privacy guarantee and a loss in data utility. This calls for anonymization schemes that minimize the accuracy loss of PPI queries, so that management may still assess the fulfilment of operational goals, while the privacy of involved individuals is protected. To define PPIs, it was suggested to rely on ontology-based systems [24] or resort to predicate logic to enable formal verification [18]. In this work, we followed the PPINOT meta-model, which is very expressive due to its compositional approach. The compositionality is also the reason why we opted for the adoption of differential privacy in our approach. Other privacy models include k-anonymity [22] and its derivatives [9,11], which statically mask recorded data points. Yet, since the evaluation of PPIs is driven by queries and processes continuously record data, these techniques are not suitable. In the context of data-driven business process analysis, the re-identification risk related to event data was highlighted empirically in [23]. To mitigate this risk, various directions have been followed, including the addition of noise to occurrence frequencies of activities in event logs [13], transformations of logs to ensure k-anonymity or t-closeness before publishing them [8,19], and the adoption of secure multi-party computation [6]. However, since these approaches focus on the control-flow perspective of processes, they cannot be employed for the privacy-aware evaluation of PPIs in the general case. Conclusion In this paper, we proposed the first approach to privacy-aware evaluation of process performance indicators based on event logs recorded during the execution of business processes. We presented a generic framework that includes an explicit interface to serve as the single point of access for PPI evaluation. In addition, for PPIs that are defined following the PPINOT meta-model, we showed how to design release mechanisms that ensure -differential privacy. We evaluated our mechanisms on both synthetic data and in a case study using the Sepsis Cases log. The results highlight the feasibility of our approach, given that a sufficiently large number of process executions is available. In future work, we aim to extend the evaluation of the mechanisms, in order to recommend which functions to privatize, for a given function tree. This would aid process analysts in receiving PPI results, with minimal quality loss. Fig. 1 : 1(a) Model of a claim handling process; (b) PPIs defined on the model. Fig. 3 : 3Overview of the framework for privacy-aware PPIs. For the aggregation functions of the PPINOT meta-model, the sensitivity is derived as ∆(min) = ∆(max) = \|X − X\|, ∆(sum) = X and ∆(mean) = \|X − X\|/\|X\|. Based thereon, noise from a Laplacian (with parameters µ = 0 and b = ∆f / , see Section 2.3) is added to f (X). Fig. 4 : 4Intervals and scores for the aggregation functions for dataset X = {2, 3, 7, 8, 10}. Fig. 4 4exemplifies the intervals for a dataset X = {2, 3, 7, 8, 10}. For min and max, the interval boundaries are 2.5, 5, 7.5, and 9. For mean and sum, the intervals have size ∆f = 1.6 and ∆f = 10, and are centred around mean(X) = 6 and sum(X) = 30. Fig. 5 5illustrates the adapted scores for our running example, using ξ = 3. The scores of the right-most three intervals are reduced, as all of their values lead to a different outcome compared to the true result, f (X) = 30, when testing against χ(x) = x ≤ 30. 6XP Fig. 5 : 6XP5Adapted intervals and scores. Fig. 6 : 6Impact of the input boundary estimation on the results. Fig. 7 : 7Impact of the cardinality of the dataset X on the results for mean. Fig. 8: Sensitivity of mean and maximum function towards . Fig. 9 : 9Results for the threshold-sensitive mechanism using differing result thresholds. Fig. 11: Evaluation Results of PPI6 Table 1 : 1PPIs defined for the Sepsis Cases log.Fig. 10: Evaluation Results for PPI1, only PaPPI (left), and PaPPI and PRIPEL (right).large domain of input values, resulting in higher sensitivities. For PPI 3 (max function), variances were relatively small, except for one month, which represents a notable outlier.ID Measure Target Values Scope Mechanism 1: Avg waiting time until admission <24 hours Monthly Mean -Interval 2: Avg length of stay <30 days Monthly Mean -Interval 3: Max length of stay <35 days Monthly Max -Interval 4: Returning patient within 28 days <5% Monthly Sum -Laplace 5: Antibiotics within one hour >95% Monthly Sum -Laplace 6: Lactic acid test within three hours >95% Monthly Sum -Laplace 0RQWK 7LPHPLQ 33,$YJZDLWLQJWLPHXQWLODGPLVVLRQ 0RQWK 7LPHPLQ 33,$YJZDLWLQJWLPHXQWLODGPLVVLRQ 3D33, 35,3(/ For ease of presentation, we exemplify datasets as sets of integers or real numbers, even though in practice, a dataset may contain multiple elements referring to the same numeric value. 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[ "Reduced-order Deep Learning for Flow Dynamics", "Reduced-order Deep Learning for Flow Dynamics" ]	[ "Min Wang ", "Siu Wun ", "Cheung † Wing ", "Tat Leung ", "Eric T Chung ", "Yalchin Efendiev ", "Mary Wheeler " ]	[]	[]	In this paper, we investigate neural networks applied to multiscale simulations and discuss a design of a novel deep neural network model reduction approach for multiscale problems. Due to the multiscale nature of the medium, the fine-grid resolution gives rise to a huge number of degrees of freedom. In practice, low-order models are derived to reduce the computational cost. In our paper, we use a non-local multicontinuum (NLMC) approach, which represents the solution on a coarse grid[18]. Using multi-layer learning techniques, we formulate and learn input-output maps constructed with NLMC on a coarse grid. We study the features of the coarse-grid solutions that neural networks capture via relating the input-output optimization to 1 minimization of PDE solutions. In proposed multi-layer networks, we can learn the forward operators in a reduced way without computing them as in POD like approaches. We present soft thresholding operators as activation function, which our studies show to have some advantages. With these activation functions, the neural network identifies and selects important multiscale features which are crucial in modeling the underlying flow. Using trained neural network approximation of the input-output map, we construct a reduced-order model for the solution approximation. We use multi-layer networks for the time stepping and reduced-order modeling, where at each time step the appropriate important modes are selected. For a class of nonlinear problems, we suggest an efficient strategy. Numerical examples are presented to examine the performance of our method.	null	[ "https://arxiv.org/pdf/1901.10343v2.pdf" ]	119,720,833	1901.10343	9cc74372b2aab9b4b1088648f9d7d32a0120d5b3	Reduced-order Deep Learning for Flow Dynamics January 31, 2019 Min Wang Siu Wun Cheung † Wing Tat Leung Eric T Chung Yalchin Efendiev Mary Wheeler Reduced-order Deep Learning for Flow Dynamics January 31, 2019 In this paper, we investigate neural networks applied to multiscale simulations and discuss a design of a novel deep neural network model reduction approach for multiscale problems. Due to the multiscale nature of the medium, the fine-grid resolution gives rise to a huge number of degrees of freedom. In practice, low-order models are derived to reduce the computational cost. In our paper, we use a non-local multicontinuum (NLMC) approach, which represents the solution on a coarse grid[18]. Using multi-layer learning techniques, we formulate and learn input-output maps constructed with NLMC on a coarse grid. We study the features of the coarse-grid solutions that neural networks capture via relating the input-output optimization to 1 minimization of PDE solutions. In proposed multi-layer networks, we can learn the forward operators in a reduced way without computing them as in POD like approaches. We present soft thresholding operators as activation function, which our studies show to have some advantages. With these activation functions, the neural network identifies and selects important multiscale features which are crucial in modeling the underlying flow. Using trained neural network approximation of the input-output map, we construct a reduced-order model for the solution approximation. We use multi-layer networks for the time stepping and reduced-order modeling, where at each time step the appropriate important modes are selected. For a class of nonlinear problems, we suggest an efficient strategy. Numerical examples are presented to examine the performance of our method. Introduction Modeling of multiscale process has been of great interest in diverse applied fields. These include flow in porous media, mechanics, biological applications, and so on. However, resolving the fine-scale features of such processes could be computationally expensive due to scale disparity. Researchers have been working on developing model reduction methods to bridge the information between the scales, in particular, bring the fine-grid information to coarse grid. This generally reduces to constructing appropriate approximation spaces when solving a governing PDEs numerically. Local model reduction is an important tool in designing reduced-order models for practical applications. Many approaches have been proposed to perform model reduction, which include [3,24,5,23,15,10,11,21,2,20,26,27,34,32,29,14,17,12,13], where the coarsegrid equations are formed and the parameters are computed or found via inverse problems [7,31,8,33,39]. Adopting upscaling ideas, methods such as Generalized Multiscale Finite Element Method (GMsFEM) [25,16], Non-local Multicontinuum Methods (NLMC) [19] have been developed. They share similar ideas in constructing the approximation space, where local problems are solved to obtain the basis functions. The numerical solution is then searched within the space that constituted by such basis. With extra local computation, these multiscale spaces can capture the solution using small degrees of freedom. These methods have been proved to be successful in a number of multiscale problems. Global model reduction methods seeks a reduced dimensional solution approximation using global basis functions. When seeking numerical solutions within a high-dimensional approximation space V h , the solutions could be sparse. Proper Orthogonal Decomposition (POD) [9] is often used to find a low-dimensional approximate representation of the solutions. To exploit the advantages of local and global model reduction techniques, global-local model reduction approaches have been explored in the past (see, e.g., [4,28,22,38]). In global-local model reduction methods, the input-output map is approximated using POD or other approaches in combination with multiscale methods. Multiscale methods allow adaptive computing and re-computing global modes [38]. In this paper, we investigate learning strategies for multiscale methods by approximating the input-output map using multi-layer neural networks. Nowadays, neural networks have been commonly used in many applications [36]. In our paper, we use neural networks in conjunction with local reduced-order models. Our studies show that the neural networks provide some type of reduced-order global modes when considering linear problems. In the paper, we investigate these global modes and their relation to spectral modes of inputoutput map. With a local-global model reduction technique, we limit our search of solution to a smaller zone (see Figure 1 for an illustration) that fits the solution populations better. Our studies show that the multi-layer network provides a good approximation of coarse-grid input-output map and can be used in conjunction with observation data. Moreover, in multilayer networks, we can learn the forward operators in a reduced way without computing them as in POD like approaches. In our paper, we use multi-layer network for combined time stepping and reduced-order modeling, where at each time step the appropriate important modes are selected. A soft thresholding neural network is suggested based on our analysis. Our previous work on neural networks and multiscale methods [37] have mostly focused on incorporating the observed data. In this paper, the key tasks are: (1) reproducing the coarse-scale multiscale model; (2) simultaneously determining the sparse representation of solution; and, (3) identifying the dominant multiscale modes and provide a new basis system which can be used to reduce the multiscale model dimension. There are a few advantages of using neural networks to accomplish these tasks. For (1) & (2), the computation is inexpensive and robust. For (3), the dominant multiscale modes found by the neural network can be generalized to conduct model reduction solutions. Our studies suggest the use of soft thresholding and we observe a relation between the soft thresholding and 1 minimization. To handle the nonlinearities in PDEs, we propose the use of clustering. Clustering is an effective pretreatment for learning and prediction. Specifically, the solution of nonlinear PDEs clusters around some nonlinear values of the solution (see Figure 1 for an illustration). Sometimes, solution clusters can be predicted without computing solutions. For example, if we are interested in the solutions to the flow equation ∂u ∂t − div(κ(t, x, u)∇u) = f,(1) where κ(t, x, u) =κ(x)u 2 , andκ(x) has a limited number of configurations as shown in Figure 2, then we could anticipate the solutions to accumulate for each configuration of κ(x). This is an important example of channelized permeability fields that arise in many subsurface applications. Thus, if we can use this prejudged information and divide the training samples into clusters, we are likely to get an accurate model with significantly less training resources. The main contributions of the paper are the following. (1) We study how neural network captures the important multiscale modes related to the features of the solution. (2) We relate 1 minimization to model reduction and derive a more robust network for our problems using a soft thresholding. (3) We suggest an efficient strategy for some class of nonlinear problems that arise in porous media applications (4) We use multi-layer networks for combined time stepping and reduced-order modeling, where at each time step the appropriate important modes are selected. We remark that we can use observed data to learn multiscale model as in [37]. The paper is organized as follows. Section 2 will be a preliminary introduction of the multiscale problem with a short review on multiscale methods. A description for general neural network is also provided. Section 3 mainly focuses on discussions on the reducedorder neural network. The structure of the neural network is presented. Section4 later discuss the proposed neural network from different aspects and propose a way to conduct model reduction with the neural network coefficient. We also present the relation between the soft thresholding neural network and a 1 minimization problem in this section. Section 5 provides various numerical examples to verify the predictive power of our proposed neural network and provide support to the claims in Section 4. Lastly, the paper is concluded with Section 6. Preliminaries Governing equations and local model reduction We consider a nonlinear flow equation ∂u ∂t − div(κ(t, x, u)∇u) = f,( 1) in the domain (0, T ) × Ω. Here, Ω is the spatial domain, κ is the permeability coefficient which can have a multiscale dependence with respect to space and time, and f is a source function. The weak formulation of (1) involves finding u ∈ H 1 (Ω) such that ∂u ∂t , v + (κ(t, x, u)∇u, ∇v) = (f, v), ∀v ∈ H 1 (Ω).(2) If we numerically solve this problem in a m-dimensional approximation space V h ⊂ H 1 (Ω), and use an Euler temporal discretization, the numerical solution can be written as u h (tn, x) = u n h = m j=1 α n j φj(x),(3) where {φj} m j=1 is a set of basis for V h . Moreover, the problem (2) can then be reformulated as: find u h ∈ V h such that u n+1 h − u n h ∆t , v h + (κ(tn, x, u ν h )∇u h , ∇v h ) = (f, v h ), ∀v h ∈ V h ,(4) where ν = n or n + 1 corresponds to explicit and implicit Euler discretization respectively. Here, (·, ·) denotes the L2 inner product. For problems with multiple scales, a multiscale basis function for each coarse node is computed following the idea of upscaling, i.e., the problem can be solved with a local model reduction. Instead of using the classic piece-wise polynomials as the basis functions, we construct the local multiscale basis following NLMC [37] and use the span of all such basis function as the approximation space V h . More specifically, for a fractured media (Figure 3), the basis functions of (1) can be constructed by the following steps: Assume T H is a coarse-grid partition of the domain Ω with a mesh size H which is further refined into a fine mesh T h . Denote by {Ki\| i = 1, · · · , N } the set of coarse elements in T H , where N is the number of coarse blocks. We also define the oversampled region K + i for each Ki, with a few layers of neighboring coarse blocks, see Figure 3 for the illustration of Ki and K + i . We further define the set of all fracture segments within a coarse block Kj as F j = {f (j) n \|1 ≤ n ≤ Lj} = {∪nΩ f,n } ∩ Kj and Lj = dim{F j }. • Step 2: Computation of local basis function in K + i . The basis for each over-sampled region ψ (i) m solves the following local constraint minimizing problem on the fine grid where a(u, v) = Dm κm∇u · ∇v a(ψ (i) m , v) + K j ⊂K + i   µ (j) 0 K j v + 1≤m≤L j µ (j) m f (j) m v   = 0, ∀v ∈ V0(K + i ), K j ψ (i) m = δijδ0m, ∀Kj ⊂ K + i , f (j) n ψ (i) m = δijδnm, ∀f (j) n ∈ F (j) , ∀Kj ⊂ K + i ,(6)+ i D f,i κi∇ f u · ∇ f v, µ (j) 0 , µ (j) m are Lagrange multi- pliers while V0(K + i ) = {v ∈ V (K + i )\|v = 0 on ∂K + i } and V (K + i ) is the fine grid space over an over-sampled region K + i . By this way of construction, the average of the basis ψ Thus, the coefficient U n = (α n 1 , α n 2 , . . . , α n m ) T satisfies the recurrence relation U n+1 = (M + ∆tA ν ) −1 (M U n + ∆tF n ),(7) where M and A ν are the mass matrix and the stiffness matrix with respect to the basis {φj} m j=1 , ν can be n or n + 1 depending on the temporal scheme that we use. We have [M ]ij = Ω φi(x)φj(x) dx, [A ν ]ij = Ω κ(tn, x, u ν h )∇φi(x) · ∇φj(x) dx. We claim that a global model reductions can be conducted to the problem described above, as solution u n h (x) in many cases can be sparse in V h even if V h is a reduced-order space. For instance, u h (tn, x) is strongly bonded to initial condition u h (t0, x). It can be foreseen that if initial conditions are chosen from a small subspace of V h (Ω), u n h (x) = u h (tn, x) is also likely to accumulate somewhere in V h . In other words, the distribution of coefficients U n can hardly expand over the entire R m space but only lies in a far smaller subspace. Other physical restrictions to the problem could also narrow down the space of solution. This indicates that u h (tn, x) can be closely approximated with less degrees of freedom. Section 3 and Section 4 will be discussing how to identify dominant modes in the space of U n using a neural network. Neural Network A neural network is usually used to learn a map from existing data. For example, if we are given samples {(xi, yi)} L i=1 from the map F : X → Y , i.e., F(xi) = yi for 1 ≤ i ≤ L, and we would like to predict the value of F(xi) for i = L + 1, · · · , L + M . With the help of neural network, this problem can be reformulated as the optimization problem. The optimization takes {(xi, yi)} L i=1 as training samples and produces a proper network coefficient θ * starting from some initialization chosen at random. More specifically, if the neural network N N (·) has a feed-forward fully-connected structure, then N N (x; θ) = Wnσ(· · · σ(W2σ(W1x + b1) + b2) · · · ) + bn.(8) Here, θ represents all tuneable coefficients in neural network N N (·) and σ(·) is some nonlinear activation function. There are many choices of such nonlinear functions [35], while the most common ones used are ReLU and tanh. For the network defined above, θ is indeed defined as θ = {W1, b1, W2, b2, · · · , Wn, bn}. We will use the output N N (xi) to approximate the desired output yi. The difference between them will be measured using a cost function C(·). For example, we can take the mean square error and the loss function is defined as C(θ) = 1 N N i=1 (yi − N N (xi)) 2 ,(9) which measures the average squared difference between the estimated values and the observed values. The neural network is then optimized by seeking θ * to minimize the loss function. θ * = argmin θ C(θ).(10) Numerically, this optimization problem can be solved with a stochastic gradient descent(SGD) type method [30]. By calculating the gradient of the loss function, the coefficient θ is iteratively updated in an attempt to minimize the loss function. This process is also referred as "training". Once the loss is minimized to a satisfactory small level, the neural network parameters θ * is decided, and further, the overall neural network architecture (15) is constructed. The predictions can then be given as N N (xi; θ * ) for i = L + 1, · · · , L + M . Reduced-order neural network In this section, we present the construction of reduced-order neural network. We propose a reduced-order neural network that can model a time series. Moreover, if there exists a basis that approximates the solution for each time step with sparse coefficients, then the proposed neural network can identify such basis from the training samples. Specifically, Subsection 3.1 will discuss the macroscopic structure of the proposed neural network while Subsection 3.2 will discuss two designs of the sub-network of the network with more details. Subsection 3.3 later assembles the full multi-layer neural network. Reduced-order neural network structure We propose a reduced-order neural network as shown in Figure 4a. Here, the full network is constituted by several sub-networks. Each sub-network N n (·) is expected to model a one-step temporal evolution from x n to x n+1 in a time series x = [x 0 , x 1 , · · · , x n ]. Sub-networks should have a general structure as shown in Figure 4b. The specific design will vary depending on the problem we are modeling (see discussions in Subsection 3.2). The sub-network is built in a way that the input x n will be first feed into a multi-layer fullyconnected network named as "operation layer". This layer is intended for the neural network to capture the map between two consecutive time steps. The output of the operation layer is then fed into a soft thresholding function to impose sparsity to the solution coefficient. Lastly, the data will be processed with a "basis transform layer" in which a new basis set will be learned. With the basis set, one can represent the solution with sparse coefficients assuming such representation exists. (a) Multi-layer reduced-order neural network N N (·) (b) Sub-network N n (·) Sub-network In this subsection, we present two designs of the sub-network N n (·). One is for modeling linear dynamics while the other is designed for nonlinear dynamics. One can choose from these two options when assembling the full network depending on the dynamics of interest. Both sub-network designs are intended to learn a new set of basis and then impose the sparsity to the solution coefficient in the new system while learning dynamics. Sub-network for linear process We first present the sub-network for modeling linear dynamics. It can be used to model the one-step flow described in (1), where we define the sub-network for the n-th time step as N n (x n ; θn) := W 2 n Sγ(W 1 n · x n + bn). Here, for the sub-network parameter θn = (W 1 n , W 2 n , bn), W 1 n and bn are in the operation layer and W 2 n works as the basis transformation layer. Sγ is the soft thresholding function defined point-wise as Sγ : R → R Sγ(x) = sign(x)(\|x\| − γ)+      x − γ if x ≥ γ, 0 if − γ < x < γ, x + γ if x ≤ −γ.(12) We further require W 2 n to be an orthogonal matrix, i.e. (W 2 n ) T · (W 2 n ) = I. To this end, we train the network (11) with respect to the cost function C n (θn) = x n+1 − N n (x n ; θn) 2 2 + ηn (W 2 n ) T · (W 2 n ) − I 1(13) with a penalty on the orthogonality constraint of W 2 n . η is a hyper coefficient for adjusting the weight of the 1 regularization term. Here, x n is the input of N n (·; θn), and C n (·) measures the mismatch between true solution x n+1 and the prediction N n (x n ) while forcing W 2 n to be orthogonal. We remark that this cost functions is only a part of the full cost function that we will be discussed in Subsection 3.3. With such a design, the trained neural sub-network N n (·; θ * n ) will be able to model the input-output map specified by the training data while producing a matrix W 2 n whose columns forms an orthogonal basis in R m . Sub-network for nonlinear process Similar to the linear case, a sub-network for nonlinear process can also be designed. When the output x n+1 is non-linearly dependent on input x n , we make use of a dn-layer feedforward neural networkÑ (·) to approximate the input-output map.Ñ (·) will work as the operation layer of the sub-network. The output ofÑ (·) is then processed with softthresholding and a basis transformation layer W 2 n . We define the sub-network to be N n (x n ; θn) := W 2 n Sγ(Ñ (x n ; θn)), N (x n ; θn) := W 1,dn n (· · · (σ(W 1,3 n σ(W 1,2 n σ(W 1,1 n x n + b 1 n ) + b 2 n ) · · · + b dn n ),(14) where θn = W 1,1 n , W 1,2 n , . . . , W 1,dn n , W 2 n , b 1 n , b 2 n , . . . , b dn n is the sub-network parameter, and σ(·) is a nonlinear activation function. The cost function for training the sub-network parameter is again defined in (13). We also remark that if dn = 1 and σ = 1 is the point-wise identify function, then the network structure (14) is reduced to (11). Another approach to reduce the difficulty of reproducing a nonlinear process is clustering. Instead of using a single network to approximate complicated nonlinear relations, we use different networks for different data cluster. As discussed in the Section 1, the clusters of the solutions can be predicted. Thus, separate the training samples by cluster can be an easy and effective way to accurately recover complicated process. Discussions on clustering and numerical examples are presented in Section 5.5. Multi-layer reduced order network Now we construct the full neural network N N (·; θ) by stacking up the sub-network N n (·; θn). More precisely, N N (·; θ) : R m → R m is defined as: N N (x 0 ; θ) := N n (· · · N 1 (N 0 (x 0 ; θ0); θ1) · · · ; θn),(15) where N n (·; θn) is defined as in (11) or (14) depending the linearity of the process, and θ = (θ0, θ1, . . . , θn) is the full network parameter. We use such N N (·; θ) to approximate time series x = [x 0 , x 1 , · · · , x n+1 ]. Denote the output of (t + 1)-fold composition of subnetwork N t as o t+1 := N t (· · · N 1 (N 0 (x 0 ; θ0); θ1) · · · ; θt). Then o t+1 works as a prediction of the solution at time t + 1. The full cost function is then defined as C(θ) = n t=0 o t+1 − x t+1 2 2 + ηt (W 2 t ) T · (W 2 t ) − I 1,(17) where x is the true time sequence, and ηt is hyper-parameter stands for the weight of the regularizer while θ represents all tuneable parameters of N N (·). Each layer (sub-network) corresponds to a one-step time evolution of the dynamics. Suppose we have L training samples, xi = [(x 0 i ), (x 1 i , x 2 i , · · · , x n+1 i )], 1 ≤ i ≤ L. The optimal parameter θ * of N N (·) is then determined by optimizing the cost function C(θ) subject to this training set { xi} L i=1 as discussed Section 2.2. Once θ * is decided, predictions can be made for testing samples [ x 1 i , · · · , x n+1 i ] by N N (x 0 i ), i > L. Discussions and applications In this section, we discuss some theoretical aspects of proposed neural networks. Specifically, we use the proposed network to model fluid dynamics in heterogeneous media, as described in (1). First, we relate the soft thresholding network with 1 optimization problem. Secondly, we explore how learned coefficients of neural network are related to the map that is being approximated. Thirdly, based on the understanding of proposed neural network, we present a way to utilize the trained network coefficient to construct a reduced-order model. Specifically, we consider a one-layer neural network for single-step linear dynamics N N (x; θ) = N (x; θ) := W 2 Sγ(W 1 x + b),(18) omitting the indices for time step n. We train the neural network N N (·; θ) with data pairs {(U n i , U n+1 i )} L i=1 , which are NLMC solution coefficients (see Section 2.1 for more details details of data generation) to (1) at t n and t n+1 , respectively. More precisely, we consider the linear case of (1), when κ(t, x) is independent of u n h . In this case, the one step fluid dynamics (7) is indeed a linear revolution such that A n is only dependent on time. Further, we denoteŴn andbn byŴ = (M + ∆tA n ) −1 M, b = (M + ∆tA n ) −1 ∆tF n .(19) Then U n+1 i =Ŵ U n i +b, 1 ≤ i ≤ L,(20) for all training samples that we use in this section. We further define the linear map between U n i and U n+1 i asL(·) : R m → R mL (x) :=Ŵ x +b.(21) From now on, when there is no risk of confusion, we drop the optimal network parameter θ * in the trained network N N (·). fz We expect N N (·) to learn the mapL(·) from the data while extracting a system W 2 in which the data U n+1 i can be represented sparsely. Sparsity and 1 minimization To understand the trained neural network N N (·), we first assume the following. Assumption 1. For a subspace S ⊂ R m , there exist some orthogonal matrixŴ 2 ∈ R m×m such that (Ŵ 2 ) TL (x) can be approximated by a sparse vector in R m for any x ∈ S. More precisely, there exist an approximation error function (·) : N → R + , such that for any x ∈ S, there exist a corresponding s-sparse vector ys ∈ R m satisfies (Ŵ 2 ) TL (x) − ys 2 ≤ (s).(22) We then take {U n i } L i=1 from S and define U n+1,Ŵ 2 i := (Ŵ 2 ) T U n+1 i for all training pairs(U n i , U n+1 i ) of N N (·). By Assumption 1, there exist s-sparse vector U s i ∈ R m such that \|\|U n+1,Ŵ 2 i − U s i \|\|2 ≤ (s)(23)u n+1 h,i = Φ(W 2 )U n+1,W 2 i = Φ W 2 U n+1,W 2 i ,(24)with Φ W 2 defined as Φ W 2 := ΦW 2 .(25) We further denote the columns of Φ W 2 as φ W 2 j 's. Then {φ W 2 j } m j=1 is actually a new set of multiscale basis functions such that u n+1 h,i can be written as u n+1 h,i = m j=1 α n+1,W 2 i,j φ W 2 j .(26) If W 2 is taken to beŴ 2 as in Assumption 1, we will obtain u n+1 h,i = m j=1 α n+1,Ŵ 2 i,j φŴ 2 j , where the coefficient U n+1,Ŵ 2 i = [α n+1,Ŵ 2 i,1 , α n+1,Ŵ 2 i,2 , · · · , α n+1,Ŵ 2 i,m ] T can be closely approximated by a sparse vector U s i . We then claim that our proposed neural network N N (·) is able to approximate suchŴ 2 and a sparse approximation of the output from data. This is guaranteed by the following lemma from [6]: Lemma 4.1. We defineN (·), R m → R m aŝ N (x) := Sγ(W x + b). The output ofN (x) is the solution of the 1-regularized problem y * = argmin y∈R m 1 2 y − (W x + b) 2 2 + γ y 1(27) by proximal gradient update. Proof. It is straightforward to see that the directional derivative of the residual with respect to y is given by y − (W x + b). On the other hand, the soft thresholding operator is the proximal operator of the 1 norm, i.e. Sγ(x) = prox γ · 1 (x) = argmin y∈R m 1 2 y − x 2 2 + γ y 1 . With a zero initial guess, the 1-step proximal gradient update of (27) with a step size γ is therefore y * =N (x) = Sγ(W x + b). Thus, for the one-step neural network N N (·) defined in (18), Lemma 4.1 implies that N N (x) = argmin y∈R m 1 2 y − W 2 W 1 x + W 2 b 2 2 + γ (W 2 ) T y 1.(28) That is to say, the output of the trained neural network N N (·) is actually the solution to a 1 optimization problem. We further define the linear operatorL(·) : R m → R m as L(x) := W 2 W 1 x + W 2 b.(29) Equation actually (28) implies that L(·) ≈ N N (·) (30) as N N (x) minimizes y − W 2 W 1 x + W 2 b 2 2 . Moreover, the output of N N (·) is sparse in the coordinate system W 2 as it also minimized the (W 2 ) T y 1 term. Sγ is widely used in 1-type optimization for promoting sparsity and extracting important features as discussed above. It is therefore also brought into network to extract sparsity from the training data. For other network defined with activation functions such as ReLU, we also remark there's an correlation between Sγ and ReLU. Recall its definition in (12) : Sγ(x) = sign(x)(\|x\| − γ)+      x − γ if x ≥ γ, 0 if − γ < x < γ, x + γ if x ≤ −γ. and that of ReLU : R → R ReLU(x) = max{x, 0} = x if x ≥ 0, 0 if x < 0. We have the following: One can explicitly represent the soft thresholding operator Sγ by the ReLU function as Sγ(x) = ReLU(x − γ) − ReLU(−x − γ) ∀x ∈ R,(31) or in an entry-wise sense, one can write Sγ(x) = JmReLU(J T m x − γ12m) ∀x ∈ R m ,(32) where Jm = [Im, −Im]. Activation functions Sγ(·) can thus be easily implemented with the help of ReLU. Further, it also means that our proposed neural network is only a special class of neural networks that are defined with ReLU. linear operatorL ≈ N N For neural network N N (·) as defined in (18) N N (x) = W 2 Sγ(W 1 x + b), we claim the following Lemma 4.2. We assume Assumption 1 holds. There exist a set of parameters (W 1 , W 2 , b) ∈ R m×m × R m×m × R m such that N N (x) −L(x) 2 ≤ 2 (s) + s 1 2 γ, ∀x ∈ S.(33) Proof. By Assumption 1, there exist some orthogonal matrixŴ 2 ∈ R m×m such that for all x ∈ S, we have (Ŵ 2 ) TL (x) − ys 2 ≤ (s),(34) where ys is an s-sparse vector. Next, we consider W 1 = (Ŵ 2 ) TŴ , W 2 =Ŵ 2 and b = (Ŵ 2 ) Tb . We recall the definition ofL(·) L(x) :=Ŵ x +b. The difference between N N (x) andL(x) can then be estimated by N N (x) −L(x) 2 = Ŵ 2 Sγ (Ŵ 2 ) TŴ x + (Ŵ 2 ) Tb −L(x) 2 = Ŵ 2 Sγ (Ŵ 2 ) TŴ x + (Ŵ 2 ) Tb − (Ŵ 2 ) TL (x) 2 = Ŵ 2 Sγ (Ŵ 2 ) TL (x) − (Ŵ 2 ) TL (x) 2 = Sγ (Ŵ 2 ) TL (x) − (Ŵ 2 ) TL (x) 2. Since \|Sγ(z2) − Sγ(z1)\| ≤ \|z2 − z1\|, ∀z1, z2 ∈ R, we have Sγ (Ŵ 2 ) TL (x) − Sγ(ys) 2 ≤ (Ŵ 2 ) TL (x) − ys 2 ≤ (s). Thus, we obtain Sγ (Ŵ 2 ) TL (x) − (Ŵ 2 ) TL (x) 2 ≤ Sγ (Ŵ 2 ) TL (x) − Sγ(ys) 2 + Sγ(ys) − ys 2 + ys − (Ŵ 2 ) TL (x) 2 ≤2 (s)+ ys − Sγ(ys) . Since \|(Sγ(ys) − ys)i\| ≤ γ, we have Sγ(ys) − ys 2 2 = (ys) i =0 \|(Sγ(ys) − ys)i\| 2 ≤ sγ 2 , and therefore we have N N (x) −L(x) 2 ≤ 2σ(s) + s 1 2 γ, letting W 1 = (Ŵ 2 ) TŴ , W 2 =Ŵ 2 and b = (Ŵ 2 ) Tb . Since N N (·) is trained with (U n i , U n+1 i ), where U n i ∈ S andL(U n i ) = U n+1 i , we have N N (x) ≈L(x), x ∈ S.(35) More specifically, this approximation error N N (x) −L(x) 2 is small for all x ∈ S providing sufficient training. Therefore, by Lemma 4.2, we claim the trained parameters closely approximate the optimal choice to guarantee the small error indicated in (33) , i.e. W 2 ≈Ŵ 2 , W 2 W 1 ≈Ŵ , and W 2 b =b. However, due to the high dimension ofŴ , full recovery ofŴ requires enormous number of training and is thus impractical. However, by enforcing N N (x) =L(x) for x ∈ S, the neural network learns a set of parameters W 2 W 1 =Ŵ , and W 2 b =b such that they functions similarly asL(·) on the subset S in the sense of linear operator. A validation of this is later provided in Subsection 5.1. Recall definition of L(·) in (29) and the fact that it can approximate N N (·) as in (30), we claim the linear operator have the following property: L(x) ≈L(x) ∀x ∈ S.(36) In the following subsection, we further construct a reduced-order model with the help of L(·). Model reduction with W 2 In this subsection, we further assume the s-sparse vector ys in Assumption 1 has non-zero entries only at fixed coordinates for all x in S. That is to say, we have a fix reordering {j k } s k=1 for {1, 2 · · · m}, such that (ys)j k = 0 for s + 1 ≤ k ≤ m. Then, we will be able to utilize the coordinate system W 2 ≈Ŵ 2 learned through training network to construct a reduced-order operator Ls(·), such that it can approximate the linear mapL(·) and maps x in S to a s-sparse vector in R m . To do so, we first define L(·) form learned coefficients of N N (·) as in last subsection, and Ls(·) will be exactly a truncation of it. Moreover, let the new basis set {φ W 2 j } m j=1 be defined with trained coefficient W 2 as in (25). When truncating W 2 in L(·), we also determine the dominant basis among {φ W 2 j } m j=1 simultaneously. Thus, we can view the model reduction from another aspect that we actually drop the basis with less significance and represent the solution with only the dominant multiscale modes. To construct such Ls(·), we follow the steps: 1. Find the dominant coordinates of outputs N N (·) in the system W 2 . (a) Compute W 2 system coefficients of N N (U n i ) for all training samples by O n+1,W 2 i := (W 2 ) T N N (U n i ), 1 ≤ i ≤ L, where i refers to the sample index. Notice (W 2 ) T N N (x) is sparse for x ∈ S by (28), therefore O n+1,W 2 i is also sparse. (b) Calculate the quadratic mean of {O n+1,W 2 i } L i=1 over all samples, coordinate by coordinate: Sj := 1 L ( L i=1 \|O n+1,W 2 i,j \| 2 ) 1 2 , 1 ≤ j ≤ m. (c) Sort the quadratic mean value Sj in descending order and denoted the reordered sequence as {Sj k } m k=1 . 2. Keep the dominant j k -th columns of W 2 for k = 1, · · · , s. Then let the rest columns be zero. Thus, we construct a reduced-order coordinate system W 2,s ∈ R m×m . Consequently, y =L(x) for any x ∈ S can be approximated with the reduced-order system W 2 as an s-sparse vector y W 2,s := (W 2,s ) T y, and y W 2,s ≈ (W 2 ) T y.(37)u n+1 h,i ≈ Φ W 2 U n+1,W 2,s i = s k=1 α n+1,W 2 i,j k φ W 2 j k .(38) 3. We finally define the reduced linear operator Ls(·) : R m → R m as Ls(x) = W 2,s W 1 · x + W 2,s b, x ∈ R m .(39) Here, the output of Ls(·) is an s-sparse vector in R m . This algorithm is designed based on the fact that O n+1, W 2 i ≈ (W 2 ) T U n+1 i = U n+1,W 2 i as N N (·) is fully trained with (U n i , U n+1 i ). Thus, the order of Sj can reflect not only the significance of the coordinates of the output of N N (·) but also that of (W 2 ) T L(x) for all x in S. Moreover, the existence of the sparse approximation ys to (Ŵ 2 ) TL (x) as described in Assumption 1 guarantees the effectiveness of the ordering. We then claim that this reduced-order linear operator Ls(·) can approximate the true input-output mapL(·) on S: Since Ls(·) is simply a truncation of L(·), we have: Ls −→ L, as s → m.(40)Moreover, recall (36)L (x) ≈ L(x), ∀x ∈ S, it implies the followingL (x) ≈ Ls(x), ∀x ∈ S.(41) This property of Ls(·) provides us a way to represent the projected vectorsL(x) for x ∈ S using a vector with only s nonzero coefficients, which corresponds to a reduced multiscale model to represent the class of solution u n+1 h that we are interested in. Numerical examples are presented in Section 5.3 to verify this claim, from which we actually observe that s can be taken as a fraction of the original number of multiscale basis m to give the approximation in (41). Numerical experiment In this section, we present numerical examples in support of the previous discussion on the reduced-order neural network. Specifically, Section 5.1 demonstrates the L(·) andL(·) functions similarly on a subspace by comparing the eigenvalues of the two operators; Section 5.2 demonstrates that a one-layer soft thresholding neural network can accurately recover a linear dynamics with a sparse coefficient vector; Section 5.3 then uses the learned coefficient W 2 from the one-layer neural network to conduct model reduction as describe in Section 4.3; Section 5.4 later presents the predicting results for multi-layer reduced-order neural network which corresponds to Section 3.3; and Section 5.5 applies the clustering scheme to nonlinear process modeling. All the network training are performed using the Python deep learning API TensorFlow [1]. 5.1L ≈ L in a subspace of R m We recall the one-layer neural network for a single-step linear dynamics in (18) N N (x) := W 2 Sγ(W 1 x + b), and the definition ofL(·) and L(·) in (21) and (29) respectively: L(x) :=Ŵ x +b, L(x) = W 2 W 1 x + W 2 b, whereŴn andbn are defined as in (19), while W 2 n , W 1 n are trained parameters of N N (·). We also recall (36): L\|S ≈L\|S, for S ⊂ R m . To support this claim, we design a special subspace S ⊂ R m . For r < m, we then let S = Vr := span{vi, 1 ≤ i ≤ r} ⊂ span{vi, 1 ≤ i ≤ m} = R m ,(42) where {vi} m i=1 are eigenvectors ofŴ corresponding to eigenvalues λi in descending order. We also define matrix V as V := [v1, v2, · · · , vm]. We then randomly pick training input U ∈ Vs such that U = r i=1 civi. The N N (·) is then trained with (U,L(U ))-like training pairs, and we obtain a corresponding operator L(·) with trained coefficients. The linear operators of L(·) andL(·) are compared by their eigenvalues, i.e. V TŴ V and V T W 2 W 1 V . By the definition of V , the former will exactly be a diagonal matrix with λi be its diagonal value. We expect W 2 W 1 functions similarly toŴ on Vr, and further the r-by-r sub-matrix of V T W 2 W 1 V should be similar to that of V TŴ V . Figure 5 compares V T W 2 W 1 V and V TŴ V for the case when Vr is constructed letting r = 30. We can tell that the first 30 × 30 submatrix are very much alike. That is to say, despite the fact that the operator L(·) andL(·) are different on R m , their behavior on the subspace Vr are the same. Moreover, Figure 6, shows that such similarity only exist in Vr as for the i th diagonal values of V T W 2 W 1 V and that of V TŴ V distinct when i > s. This also makes sense as the operator L(·) is defined from the trained parameters of N N (·) where only subspace Vr is visible to the network. (a) V TŴ V (b) 30 × 30 sub-matrix of V TŴ V (c) V T W 2 W 1 V (d) 30 × 30 sub-matrix of V T W 2 W 1 V One-layer reduced order neural network In this example, we consider the one-layer reduced-order neural network as defined in (18). We use this neural network to predict a one-step fluid dynamics, where the data are taken to be NLMC solution coefficients to (1) in the form (U 0 i , U 1 i ). We fix κ(t, x) and f (t, x) among samples, thus all data describes linear dynamics for different initial conditions. We take 2% out of all data pairs as testing samples and the remaining 98% as training samples and use only the training sample to train N N (·). We then evaluate the neural network by examining the accuracy of the following approximation for the unseen testing samples: U 1 ≈ N N (U 0 ). The 2 relative error of the prediction is computed by From Table 1, we can see that the prediction of our proposed network N N (U 0 ) is rather effective with an average 2 error of 5.34%. \|\|U 1 − N N (U 0 )\|\|2 \|\|U 1 \|\|2 .(44) We also verify that U 1 is sparse in the learned W 2 -system for all data(training and testing). We first reorder the columns of W 2 by their dominance as discussed in Section 4.3, then compute the corresponding W 2 -system coefficients U 1,W 2 , which should be a roughly decreasing vector. From Figure 7, we can tell that the W 2 -system coefficients U 1,W 2 are sparse. This can be an reflection of successful learning ofŴ 2 in Assumption 1. Moreover, only a few dominant modes are needed to recover the solution as the quadratic averages of coordinates \|U 1,W 2 j \| decays fast when j > 100. In a word, the proposed neural network can indeed learn the dominant multiscale modes needed to represent U 1 from training data while properly reproduce the map between U 0 and U 1 . Model reduction with W 2 As discussed in Section 4.3, we would like to used the reduced-order system W 2,s to further conduct model reduction. The reduced-order solution coefficient is defined with the reducedorder linear operator Ls(·): U n+1 sN = Ls(U n ). is sparse and has maximum s nonzero elements. The numerical experiments is conduced based on a one-layer neural network as defined in (18) for a one-step linear dynamics. We would like to compare the following coefficient vectors: U 1 sN := Ls(U 0 ) = W 2,s W 1 · U 0 + W 2,s b, U 1 true =Ŵ U 0 +b,\|\|U 1 sN −U 1 true \|\|2 \|\|U 1 true \|\|2 as s grows. Figure 8 shows the error decay of U 1 sN compared to U 1 true . As s grows, the error gets smaller. This figure actually verified (41), i.e. the reduced operator Ls(·) can approximatê L(·).Moreover, this approximation gets more accurate as s gets larger. The error in Figure 8 at s = m = 445 is also expected that it is a consequence of the training error. Besides, we observe that the error decays fast when s > 40 for our training samples. with respect to number of selected dominant modes in W 2,s . sacrifice in the solution accuracy. We notice that the order of the reduce operator Ls(·) is only around 18% that of the original multiscale model. We lastly present the comparison between U 1 true , U 1 N , and U 1 sN .From Table 3, we can tell that for a single testing sample, we have \|\|U 1 sN − U 1 true \|\|2 \|\|U 1 true \|\|2 > \|\|U 1 N − U 1 true \|\|2 \|\|U 1 true \|\|2 , \|\|U 1 sN − U 1 true \|\|2 \|\|U 1 true \|\|2 > \|\|U 1 sN − U 1 N \|\|2 \|\|U 1 N \|\|2 , which is as expected since \|\|U 1 N −U 1 true \|\| 2 \|\|U 1 true \|\| 2 and \|\|U 1 sN −U 1 N \|\| 2 \|\|U 1 N \|\| 2 are exactly the two components of error \|\|U 1 sN −U 1 true \|\| 2 \|\|U 1 true \|\| 2 . They stands for neural network prediction error and Ls truncation error, respectively. The latter can be reduce by increasing s, while the former one can only be improved with more effective training. Table 3: Relative error percentage of solutions obtained in full W 2 system and reduced-order system W 2,s for s = 100. Sample Index \|\|U 1 N −U 1 true \|\|2 \|\|U 1 true \|\|2 \|\|U 1 sN −U 1 true \|\|2 \|\|U 1 true \|\|2 \|\|U 1 sN −U 1 N \|\|2 \|\|U 1 N \|\|2 #1 Multi-layer reduced order Neural Network In this example, we use a multi-layer reduced-order neural network N N (·) to predict multistep fluid dynamics. Recall (15), it is defined as N N (x 0 ) := N n (· · · N 1 (N 0 (x 0 ))). The input of N N (·) is taken to be U 0 , the initial condition, while the outputs are the collection of outputs at n th -layer sub-network N n (·) which correspond to the true values [U 1 , U 2 , · · · , U 9 ]. U n+1 are all taken to be NLMC solutions of (1) at time step n for n = 0, · · · , 8. Prediction accuracy is measured with 2 relative error that defined similar to (44). Sample Index U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 #1 1.62 Table 4: 2 relative error of prediction of U n+1 using N N (·). In Table 4, the columns shows the prediction error for U n+1 , n = 0, 1, · · · 8 where the average error is computed for all time steps among testing samples which are less than 10% in average. Therefore, we claim that the proposed multi-layer reduced order neural network N N (·) is effective in the aspect of prediction. We also claim that the coefficient U n+1,W 2 n for n = 0, 1, · · · , 8 are sparse in the independent systems W 2 n . These systems are again learned simultaneously by training N N (·). Clustering In this experiment, we aim to model the fluid dynamics correspond to two different fractured media as shown in Figure 9. More specifically, the permeability coefficient of matrix region have κm = 1 and the permeability of the fractures are κ f = 10 3 . The one-step NLMC solutions pairs (U 0 , U 1 ) are generated following these two different configurations of fracture are then referred as "Cluster 1" and "Cluster 2" (see Figure 10 for an illustration). We will then compare the one-step prediction of networks N N 1, N N 2 with that of N N mixed . The input are taken taken to be U 0 , which are chosen to be the terminal solution of mobility driven 10-step nonlinear dynamics, while the output is an approximation of U 1 . N N 1, N N 2 and N N mixed share same one-layer soft thresholding neural network structure as in (18) while the first two network are trained with data for each cluster separately and the latter one is trained with mixed data for two clusters. Figure 9 shows the fracture network we use to generate data for two clusters, while Figure 10 shows an example of solution U 0 and U 1 for each cluster. As observed from the profiles of U 0 and U 1 , we can see that these the solutions are very different due to the translation of the fractures. Moreover, since the data resulted from both clusters have non-uniform map between U 0 and U 1 , the mixed data set can be considered as obtained from a nonlinear map. Table 5 demonstrates the comparison of the prediction accuracy when the network is fed with a single cluster data and when the network is fed with a mixed data. This simple Conclusion In this paper, we discuss a novel deep neural network approach to model reduction approach for multiscale problems. To numerically solve multiscale problems, a fine grid needs to be used and results in a huge number of degrees of freedom. To this end, non-local multicontinuum (NLMC) upscaling [18] is used as a dimensionality reduction technique. In flow dynamics problems, multiscale solutions at consecutive time instants are regarded as an input-output mechanism and learnt from deep neural networks techniques. By exploiting a relation between a soft-thresholding neural network and 1 minimization, multiscale features of the coarse-grid solutions are extracted using neural networks. This results in a new neural-network-based construction of reduced-order model, which involves extracting appropriate important modes at each time step. We also suggest an efficient strategy for a class of nonlinear flow problems. Finally, we present numerical examples to demonstrate the performance of our method applied to some selected problems. Figure 1 : 1Illustration of sparsity of our solution. Figure 2 : 2Illustration of different configurations ofκ(x). • Step Figure 3 : 3Illustration of coarse and fine meshes. 1 in the matrix part of coarse element Ki, and equals 0 in other parts of the coarse blocks Kj ⊂ K + i as well as any fracture inside K + i . As for ψ (i) m , m ≥ 1, it has average 1 on the m-th fracture continua inside the coarse element Ki, and average 0 in other fracture continua as well as the matrix continua of any coarse block Kj ⊂ K + i . It indicates that the basis functions separate the matrix and fractures, and each basis represents a continuum.The resulting multiscale basis space is finally written as V h = span{ψ m \|0 ≤ m ≤ Li, 1 ≤ i ≤ N }. By renumbering the basis, we denote all basis function as {φj} m j=1 , where m = N i=1 (Lj + 1). Figure 4 : 4Reduced-order neural network structure. Figure 5 : 5Ŵ and W 2 W 1 function similarly on S, where r = 30. Figure 6 : 6Comparison of eigen-values ofŴ and W 2 W 1 when r = 30. Figure 7 : 7Sparsity of the output of N N (U 0 ) in W 2 . Noticing that U n+1 sN is the coefficient of u n+1 h in the original basis system {φj} m j=1 while it is sparse in W 2 -system, i.e, U n+1,W 2 sN and U 1 N 1:= N N (U 0 ). Here U 1 true is the true solution to (20), while U 1 N is the prediction of N N (·). Figure 8 : 8Decay of relative error Figure 9 : 9Fracture networks for two clusters. Figure 10 : 10) Coarse-scale solution of pressure u 1 -Cluster 1 (d) Coarse-scale solution of pressure u 1 -Cluster 2 Example solution pairs for two different clusters. for 1 ≤ i ≤ L.Additionally, for any orthogonal matrix W 2 ∈ R m×m , we denote define U n+1,W 2 i := (W 2 ) T U n+1 i and denote it as [α n+1,W 2 i,1 , α n+1,W 2 i,2 , · · · , α n+1,W 2 i,m ] T . Recall (3), and the corre- sponding numerical solution u n+1 h,i at time step n + 1 can be written as u n+1 h,i = ΦU n+1 i , letting Φ = [φ1, φ2, · · · , φm] be the multiscale basis functions, and U n+1 i be the correspond- ing coefficient vector. By the orthogonality of W 2 , u n+1 h,i can be further written as Consequently, for training/testing samples, we have U n+1,W 2,s can be approximated with only basis {φ W 2 j k \|1 ≤ k ≤ s} correspond to dominant multiscale modes.i ≈ U n+1,W 2 i . Thus, corresponding function u n+1 h,i Table 1 1is the error table for the case when we use 500 data pairs with 490 to be training samples and 10 to be testing samples. These data are generated with different choice of initial condition U 0 i . To match the realistic physical situation, we took all initial conditions to be the NLMC terminal pressure of a mobility driven nonlinear flow process.Sample Index Error(%) #1 0.25 #2 0.43 #3 10.02 #4 9.91 #5 3.90 #6 8.18 #7 17.27 #8 1.57 #9 1.13 #10 0.76 Mean 5.34 Table 1: 2 relative error of N N (·) prediction. Table 2 2further facilitates such conclusion. The error of U 1 sN is less than 12% when s ≥ 80. We can thus represent the multiscale solution u 1 h using s = 80 basis with littleSample Index s 5 10 20 40 80 160 320 #1 66.71 46.80 29.02 17.01 8.44 3.53 1.11 #2 66.55 46.58 29.03 17.41 9.48 4.62 1.59 #3 66.35 46.77 29.70 18.59 12.26 10.92 10.16 #4 67.95 48.58 31.23 19.87 12.59 10.03 9.90 #5 66.22 46.31 28.99 17.73 10.49 6.49 4.34 #6 66.47 46.76 29.40 17.89 10.86 9.12 8.33 #7 69.59 51.33 36.01 27.42 22.33 18.62 17.29 #8 66.60 46.61 28.81 16.39 7.47 3.86 1.96 #9 66.90 47.13 29.21 16.94 8.02 3.58 1.67 #10 67.14 47.39 29.44 17.23 8.23 3.19 1.32 Mean 67.05 47.43 30.08 18.65 11.02 7.40 5.77 Table 2 : 2Decay of error\|\|U 1 sN −U 1 true \|\|2 \|\|U 1 true \|\|2 (b) Relative 2 prediction error(%) of N mixed .Sample Index Cluster 1 Error Cluster 2 Error #1 0.87 0.26 #2 3.19 0.45 #3 13.62 0.79 #4 12.36 0.49 #5 7.64 0.43 #6 10.69 1.00 #7 19.08 0.53 #8 2.57 0.92 #9 0.77 0.46 #10 3.70 0.38 Mean 7.45 0.57 (a) Relative 2 prediction error(%) of N1 and N2. Sample Index Cluster 1 Error Cluster 2 Error # 1 36.25 84.83 #2 35.08 84.20 #3 28.20 111.05 #4 44.60 69.35 #5 32.32 89.30 #6 30.03 106.41 #7 48.61 57.76 #8 35.63 88.46 #9 36.89 88.67 #10 38.78 83.85 Mean 36.64 86.39 Table 5 : 5Prediction error of N 1 , N 2 and N mixed . can significantly improve the accuracy. For Cluster 1, the average prediction error of N N 1 is around 7.45% while that of the N N mixed is around 36.64%. Similar contrast can also be observed for Cluster 2.treatment TensorFlow: Large-scale machine learning on heterogeneous systems. Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mané. 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[ "Fisher matrix forecasts for astrophysical tests of the stability of the fine-structure constant", "Fisher matrix forecasts for astrophysical tests of the stability of the fine-structure constant" ]	[ "C S Alves \nCentro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nFaculdade de Ciências\nUniversidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal\n", "T A Silva \nCentro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nFaculdade de Ciências\nUniversidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal\n", "C ", "J A P Martins \nCentro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nInstituto de Astrofísica e Ciências do Espaço\nCAUP\nRua das Estrelas4150-762PortoPortugal\n", "A C O Leite \nCentro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nFaculdade de Ciências\nUniversidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal\n\nInstituto de Astrofísica e Ciências do Espaço\nCAUP\nRua das Estrelas4150-762PortoPortugal\n" ]	[ "Centro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal", "Faculdade de Ciências\nUniversidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal", "Centro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal", "Faculdade de Ciências\nUniversidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal", "Centro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal", "Instituto de Astrofísica e Ciências do Espaço\nCAUP\nRua das Estrelas4150-762PortoPortugal", "Centro de Astrofísica\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal", "Faculdade de Ciências\nUniversidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal", "Instituto de Astrofísica e Ciências do Espaço\nCAUP\nRua das Estrelas4150-762PortoPortugal" ]	[]	We use Fisher Matrix analysis techniques to forecast the cosmological impact of astrophysical tests of the stability of the fine-structure constant to be carried out by the forthcoming ESPRESSO spectrograph at the VLT (due for commissioning in late 2017), as well by the planned high-resolution spectrograph (currently in Phase A) for the European Extremely Large Telescope. Assuming a fiducial model without α variations, we show that ESPRESSO can improve current bounds on the Eötvös parameter-which quantifies Weak Equivalence Principle violationsby up to two orders of magnitude, leading to stronger bounds than those expected from the ongoing tests with the MICROSCOPE satellite, while constraints from the E-ELT should be competitive with those of the proposed STEP satellite. Should an α variation be detected, these measurements will further constrain cosmological parameters, being particularly sensitive to the dynamics of dark energy.	10.1016/j.physletb.2017.03.053	[ "https://arxiv.org/pdf/1704.08728v1.pdf" ]	119,224,866	1704.08728	db58f6ff1c57a49cf4dcc363d9b831213e15693d	Fisher matrix forecasts for astrophysical tests of the stability of the fine-structure constant 27 Apr 2017 C S Alves Centro de Astrofísica Universidade do Porto Rua das Estrelas4150-762PortoPortugal Faculdade de Ciências Universidade do Porto Rua do Campo Alegre 6874169-007PortoPortugal T A Silva Centro de Astrofísica Universidade do Porto Rua das Estrelas4150-762PortoPortugal Faculdade de Ciências Universidade do Porto Rua do Campo Alegre 6874169-007PortoPortugal C J A P Martins Centro de Astrofísica Universidade do Porto Rua das Estrelas4150-762PortoPortugal Instituto de Astrofísica e Ciências do Espaço CAUP Rua das Estrelas4150-762PortoPortugal A C O Leite Centro de Astrofísica Universidade do Porto Rua das Estrelas4150-762PortoPortugal Faculdade de Ciências Universidade do Porto Rua do Campo Alegre 6874169-007PortoPortugal Instituto de Astrofísica e Ciências do Espaço CAUP Rua das Estrelas4150-762PortoPortugal Fisher matrix forecasts for astrophysical tests of the stability of the fine-structure constant 27 Apr 2017arXiv:1704.08728v1 [astro-ph.CO]CosmologyFundamental couplingsFine-structure constantFisher Matrix analysis We use Fisher Matrix analysis techniques to forecast the cosmological impact of astrophysical tests of the stability of the fine-structure constant to be carried out by the forthcoming ESPRESSO spectrograph at the VLT (due for commissioning in late 2017), as well by the planned high-resolution spectrograph (currently in Phase A) for the European Extremely Large Telescope. Assuming a fiducial model without α variations, we show that ESPRESSO can improve current bounds on the Eötvös parameter-which quantifies Weak Equivalence Principle violationsby up to two orders of magnitude, leading to stronger bounds than those expected from the ongoing tests with the MICROSCOPE satellite, while constraints from the E-ELT should be competitive with those of the proposed STEP satellite. Should an α variation be detected, these measurements will further constrain cosmological parameters, being particularly sensitive to the dynamics of dark energy. Introduction Astrophysical tests of the stability of fundamental couplings are an extremely active area of observational research [1,2]. The deep conceptual importance of carrying out these tests has been complemented by recent (even if somewhat controversial [3]) evidence for such a variation [4], coming from high-resolution optical/UV spectroscopic measurements of the fine-structure constant α in absorption systems along the line of sight of bright quasars. The forthcoming ESPRESSO spectrograph [5], due for commissioning at the combined Coudé focus of ESO's VLT in late 2017, should significantly improve the sensitivity of these tests, as well as the degree of control over possible systematics. Moreover, the results of these tests-whether they are detections of variations or null results-have a range of additional cosmological implications. They provide competitive constraints on Weak Equivalence Principle (WEP) violations [1,6,7] and, in the more natural scenarios where the same dynamical degree of freedom is responsible both for the dark energy and the α variation, can also be used in combination with standard cosmological observables to constrain the dark energy equation of state [8,9] and indeed to reconstruct its redshift dependence [10,11]. While current data already provides useful constraints, the imminent availability of more precise measurements from the ESPRESSO spectrograph will have a significant impact in the field. In this work we apply standard Fisher Matrix analysis techniques to forecast the improvements that may be expected from ESPRESSO, but we also take the opportunity to look further ahead and discuss additional gains in sensitivity from the European Extremely Large Telescope (E-ELT), whose first light will be in 2024. Varying α, dark energy and the Weak Equivalence principle Dynamical scalar fields in an effective fourdimensional field theory are naturally expected to couple to the rest of the theory, unless a (still unknown) symmetry is postulated to suppress these couplings [12,13,14]. We will assume that this coupling does exist for the dynamical degree of freedom responsible for the dark energy, assumed to be a dynamical scalar field denoted φ. Specifically the coupling to the electromagnetic sector is due to a gauge kinetic function B F (φ) L φF = − 1 4 B F (φ)F µν F µν .(1) This function can be assumed to be linear, B F (φ) = 1 − ζκ(φ − φ 0 ) ,(2) (where κ 2 = 8πG) since, as has been pointed out in [13], the absence of such a term would require the presence of a φ → −φ symmetry, but such a symmetry must be broken throughout most of the cosmological evolution. The dimensionless parameter ζ quantifies the strength of the coupling. With these assumptions one can explicitly relate the evolution of α to that of dark energy [6,15]. The evolution of α can be written ∆α α ≡ α − α 0 α 0 = B −1 F (φ) − 1 = ζκ(φ − φ 0 ) ,(3) and defining the fraction of the dark energy density (the ratio of the energy density of the scalar field to the total energy density, which also includes a matter component) as a function of redshift z as follows Ω φ (z) ≡ ρ φ (z) ρ tot (z) ≃ ρ φ (z) ρ φ (z) + ρ m (z) ,(4) where in the last step we have neglected the contribution from the radiation density (we will be interested in low redshifts, z < 5, where it is indeed negligible), the evolution of the scalar field can be expressed in terms of Ω φ and of the dark energy equation of state w φ as 1 + w φ = (κφ ′ ) 2 3Ω φ ,(5) with the prime denoting the derivative with respect to the logarithm of the scale factor. Putting the two together we finally obtain ∆α α (z) = ζ z 0 3Ω φ (z ′ ) 1 + w φ (z ′ ) dz ′ 1 + z ′ . (6) The above relation assumes a canonical scalar field, but the argument can be repeated for phantom fields, leading to ∆α α (z) = −ζ z 0 3Ω φ (z ′ ) 1 + w φ (z ′ ) dz ′ 1 + z ′ ; (7) the change of sign stems from the fact that one expects phantom fields to roll up the potential rather than down. Note that in these models the evolution of α can be expressed as a function of cosmological parameters plus the coupling ζ, without explicit reference to the putative underlying scalar field. In these models the proton and neutron masses are also expected to vary-by different amounts-due to the electromagnetic corrections of their masses. Therefore, local tests of the Equivalence Principle also constrain the dimensionless coupling parameter ζ [1], and (more to the point for our present purposes) they provide us with a prior on it. We note that there is in principle an additional source term driving the evolution of the scalar field, due to the derivative of the gauge kinetic function, i.e. a term proportional to F 2 B ′ F . By comparison to the standard (kinetic and potential energy) terms, the contribution of this term is expected to be subdominant, both because its average is zero for a radiation fluid and because the corresponding term for the baryonic density is constrained by the aforementioned Equivalence Principle tests. For these reasons, in what follows we neglect this term (which would lead to environmental dependencies). We nevertheless note that this term can play a role in scenarios where the dominant standard term is suppressed. A light scalar field such as we are considering inevitably couples to nucleons due to the α dependence of their masses, and therefore it mediates an isotopedependent long-range force. This can be quantified through the dimensionless Eötvös parameter η, which describes the level of violation of the WEP [1]. One can show that for the class of models we are considering the Eötvös parameter and the dimensionless coupling ζ are simply related by [1,13,14] η ≈ 10 −3 ζ 2 ; (8) we note that while this relation is correct for the simplest canonical scalar field models we will consider in what follows, it is somewhat model-dependent (for example, it is linear rather than quadratic in ζ for Bekenstein-type models [7]). Forecasting tools and fiducial models We will be considering three fiducial dynamical dark energy models where the scalar field also leads to α variations according to Eq. 6, as follows • A constant dark energy equation of state, w 0 = const. • A dilaton-type model where the scalar field φ behaves as φ(z) ∝ (1 + z); this is well motivated in string theory inspired models [16], but for our purposes it also has the advantage that despite the fact that it leads to a relatively complicated dark energy equation of state w(z) = [1 − Ω φ (1 + w 0 )]w 0 Ω m (1 + w 0 )(1 + z) 3[1−Ω φ (1+w 0 )] − w 0 ,(9) (where we are assuming flat universes, so the present-day values of the matter and dark energy fractions satisfy Ω m + Ω φ = 1); in this case Eq. 6 simplifies to [6] ∆α α (z) = ζ 3Ω φ (1 + w 0 ) ln (1 + z) .(10) Thus this case allows us to carry out analytic calculations, which we have used to validate our numerical pipeline. • The well-known Chevallier-Polarski-Linder (CPL) parametrization [17,18], where the redshift dependence of the dark energy equation of state is described by two separate parameters, w 0 (which is still its present-day value) and w a describing its evolution, as follows w(z) = w 0 + w a z 1 + z .(11) All of these have been used in previous works to obtain constraints from current data [6,8,9] or to forecast dark energy equation of state reconstructions [10,11], and therefore these previous works can easily be compared with ours. Our forecasts were done with a Fisher Matrix analysis [19,20]. If we have a set of M model parameters (p 1 , p 2 , ..., p M ) and N observables-that is, measured quantities-( f 1 , f 2 , ..., f N ) with uncertainties (σ 1 , σ 2 , ..., σ N ), then the Fisher matrix is F i j = N a=1 ∂ f a ∂p i 1 σ 2 a ∂ f a ∂p j .(12) For an unbiased estimator, if we don't marginalize over any other parameters (in other words, if all are assumed to be known) then the minimal expected error is θ = 1/ √ F ii . The inverse of the Fisher matrix provides an estimate of the parameter covariance matrix. Its diagonal elements are the squares of the uncertainties in each parameter marginalizing over the others, while the off-diagonal terms yield the correlation coefficients between parameters. Note that the marginalized uncertainty is always greater than (or at most equal to) the non-marginalized one: marginalization can't decrease the error, and only has no effect if all other parameter are uncorrelated with it. It is also useful to define a Figure of Merit (denoted FoM for brevity in the results section) for each pair of parameters [20] which is the inverse of the area of their onesigma confidence ellipse: a small area (meaning small uncertainties in the parameters) corresponds to a large figure of merit. Previously known uncertainties on the parameters, known as priors, can be trivially added to the calculated Fisher matrix. This is manifestly the case for us: a plethora of standard cosmological datasets provide priors on our previously defined cosmological parameters (Ω m , w 0 , w a ), while local constraints on the Eötvös parameter η from torsion balance and lunar laser ranging experiments [21,22] provide priors on the dimensionless coupling ζ. Specifically, we will assume the following fiducial values and prior uncertainties for our cosmological parameters Ω m, f id = 0.3 , σ Ω m = 0.03 (13) w 0, f id = −0.9 , σ w 0 = 0.1 (14) w a, f id = 0.3 , σ w a = 0.3 ,(15) while for the coupling ζ we will consider three different scenarios ζ f id = 0 , ζ f id = 5 × 10 −7 , ζ f id = 5 × 10 −6 ,(16) always with the same prior uncertainty σ ζ = 10 −4 .(17) Thus we will consider both the case where there are no α variations (ζ = 0), and the case where they exist: the case ζ = 5 × 10 −6 corresponds to a coupling which saturates constraints from current data [6,9], while ζ = 5 × 10 −7 illustrates an intermediate scenario. The first ESPRESSO measurements of α should be obtained in the context of the consortium's Guaranteed Time Observations (GTO). The target list for these measurements has recently been selected: full details can be found in [23]. Bearing this in mind we have studied the following three scenarios: • ESPRESSO Baseline: we assumed that each of the targets on the list can be measured by ESPRESSO with an uncertainty of σ ∆α/α = 0.6 × 10 −6 ; this represents what we can currently expect to achieve on a time scale of 3-5 years (though this expectation needs to be confirmed at the time of commissioning of the instrument); • ESPRESSO Ideal: in this case we assumed a factor of three improvement in the uncertainty, σ ∆α/α = 0.2 × 10 −6 ; this represents somewhat optimistic uncertainties. This provides a useful comparison point, but in any case such an improved uncertainty should be achievable with additional integration time; • ELT-HIRES: We will also provide forecasts for a longer-term dataset, on the assumption that the same targets can be observed with the ELT-HIRES spectrograph [24]; in this case we assume an improvement in sensitivity by a factor of six relative to the ESPRESSO baseline scenario, coming from the larger collecting area of the telescope and additional improvements at the level of the spectrograph. Although at present not all details of the instrument and the telescope have been fixed, this is representative of the expected sensitivity of measurements on a 10-15 year time scale. We note that our choices of possible theoretical and observational parameters span a broad range of possible scenarios. As a simple illustration of this point, let us consider a single measurement of α at redshift z = 2. In the case of the dilaton model we have the simple relation ∆α/α(z = 2) ∼ 0.5ζ. Thus if ζ = 5 × 10 −7 a single precise and accurate measurement of α with ESPRESSO baseline sensitivity would not detect its variation, while ELT-HIRES would detect it at 2.5 standard deviations. On the other hand, for ζ = 5 × 10 −6 (which as previously mentioned saturates current bounds) a single z = 2 ESPRESSO baseline measurement would detect a variation at 4σ and ELT-HIRES would detect it at 25σ. Before proceeding with our general analysis it is instructive to provide a simple analytic illustration for the dilaton model, in which case the α variation is given by Eq. 10. Let's further assume that Ω φ (or equivalently Ω m ) is perfectly known, so we are left with a two-dimensional parameter space (ζ, w 0 ). Including priors on both ζ and w 0 (respectively denoted σ ζ and σ w ), the Fisher matrix is [F(ζ, w 0 )] =         Q 2 (1 + w 0 ) + 1 σ 2 ζ 1 2 Q 2 ζ 1 2 Q 2 ζ Q 2 ζ 2 4(1+w 0 ) + 1 σ 2 w         ,(18) where we have defined Q 2 = 3Ω φ i log(1 + z i ) σ αi 2 .(19) The un-marginalized uncertainties are θ ζ = σ ζ 1 + (1 + w 0 )Q 2 σ 2 ζ(20)θ w = σ w 1 + ζ 2 4(1+w 0 ) Q 2 σ 2 w(21) while the determinant of F is detF = Q 2        1 + w 0 σ 2 w + ζ 2 4(1 + w 0 )σ 2 ζ        + 1 σ 2 w σ 2 ζ ;(22) this would be zero in the absence of priors-a point already discussed in [15]-but as mentioned above cosmological data and local tests of the WEP do provide us with these priors. As expected, if ζ = 0 the two parameters decorrelate, and there is no new information on the equation of state (θ w = σ w ): if ζ = 0 we will always measure ∆α/α = 0 regardless of the experimental sensitivity. Now we can calculate the covariance matrix [C(ζ, w 0 )] = 1 detF         Q 2 ζ 2 4(1+w 0 ) + 1 σ 2 w − 1 2 Q 2 ζ − 1 2 Q 2 ζ Q 2 (1 + w 0 ) + 1 σ 2 ζ         ,(23) and the correlation coefficient ρ can be written ρ =        1 + 4(1 + w 0 ) Q 2 ζ 2 σ 2 w + 1 (1 + w 0 )Q 2 σ 2 ζ + 4 ζ 2 Q 4 σ 2 ζ σ 2 w        −1/2 .(24) We thus confirm the physical intuition that in the limit ζ −→ 0, the two parameters become independent (ρ → 0). The general marginalized uncertainties are 1 σ 2 ζ,new = 1 σ 2 ζ + 1 σ 2 w (1 + w 0 )Q 2 ζ 2 Q 2 4(1+w 0 ) + 1 σ 2 w (25) 1 σ 2 w,new = 1 σ 2 w + 1 σ 2 ζ ζ 2 Q 2 4(1 + w 0 ) 1 (1 + w 0 )Q 2 + 1 σ 2 ζ ;(26) In the particular case where the fiducial model is ζ = 0 the former becomes 1 σ 2 ζ,new = 1 σ 2 ζ + (1 + w 0 )Q 2(27) while the latter trivially gives σ w,new = σ w . As previously mentioned, we have used these analytic results to validate our more generic numerical code (where furthermore Ω m will also be allowed to vary). Results We start with the case where there is no coupling between the scalar field and the electromagnetic sector of the theory, such that ζ f id = 0 . We emphasize that if the cosmological model Lagrangian does contain a dynamical scalar field, the suppression of such a coupling will require a (still unknown) symmetry [12,13,14]. In this case precise α measurements will find null results which can be translated into bounds on ζ, whose one-sigma uncertainties, marginalized over Ω m , w 0 and (for the case of the CPL model), w a , are displayed in Table 1 and in Fig. 1. For comparison, the current two-sigma bound on ζ is \|ζ\| < 5 × 10 −6 , with a mild dependence on the choice of fiducial dark energy model [6,9]. Thus in this case we expect ESPRESSO to improve current bounds on ζ by about one order of magnitude. Naturally these improvements also lead to stronger bounds on the Eötvös parameter: we note that constraints from ESPRESSO should be stronger than those expected from the ongoing tests with the MI-CROSCOPE satellite [25], while those from ELT-HIRES should be competitive with those of the proposed STEP satellite [26] (though at present the wavelength coverage and sensitivity of the latter are relatively uncertain). Table 1 also shows that there is a mild dependence on the choice of underlying dark energy model. This has been previously studied, and is well understood-refer to [6,9,15] for further discussion of this point. The dilaton model is a 'freezing' dark energy model. Thus, according to Eq. 6, a dilaton model with a given value of w 0 will have a value of ∆α/α(z) that is larger than the corresponding value for a model with a constant equation of state with the same value of w 0 . Thus, for similar cosmological priors, null measurements of α will provide slightly stronger constraints for the dilaton case. The same argument applies for the CPL case, where the additional free parameter w a further enlarges the range of possible values of α. Now we consider the case where an α variation does exist, corresponding to a non-zero fiducial value of the dimensionless coupling ζ. In this case the marginalized sensitivity on the parameter ζ will be weakened due to its correlations with other parameters. On the other hand, the α measurements can themselves help in constraining the cosmological parameters. The results of our analysis are summarized in Tables 2, 3 and 4, respectively for the constant equation of state, dilaton and CPL fiducial models, and also shown in Figs. 2, 3 and 4. Starting with the constant equation of state and dilaton models, we confirm the strong anticorrelation between ζ and w 0 (which naturally is weaker for smaller values of the coupling): since the α variation depends both on the strength of the coupling and on how fast the scalar field is moving-which depends on (1+w(z)), cf. Eq. 5-to a first approximation one can increase one and decrease the other and still get similar α variations. On the other hand, the present-day value of the matter density is not significantly correlated with the other parameters, as is clear from the right-hand side panels of Figs. 2 and 3. Comparing the two models the correlations are somewhat weaker in the dilaton case, thus leading to a better sensitivity on the coupling ζ. Overall, with the range of assumed couplings the ESPRESSO GTO measurements would detect a non-zero ζ at between one and two stan-dard deviations, while the same observations with the foreseen ELT-HIRES would ensure a two-sigma detection. We also note that for the largest permissible values of the coupling, ELT-HIRES measurements can improve constraints on the dark energy equation of state w 0 by up to ten percent. The case of the CPL parametrization is particularly illuminating. Here the behavior of the dark energy equation of state depends on two parameters, w 0 and w a (while in the case of the other models we assumed that it just depended on the former). Each of these parameters is still anticorrelated with ζ, for the reasons already explained, but this anticorrelation is now weaker than in the dilaton or the constant equation of state cases, enabling stronger constraints on ζ. This is manifest in the left-hand side panels of Figs. 2 and 3. Thus in the case of the largest currently allowed value ζ = 5 × 10 −6 ELT-HIRES observations of the ESPRESSO GTO sample would detect a non.zero ζ at the 99.7% (3σ) confidence level. It is particularly worthy of note that the two dark energy equation of state parameters are not significantly correlated. This occurs because measurements of α typically span a sufficiently large redshift range (in our case roughly 1 < z < 3) to make the roles of both in the redshift dependence of α sufficiently distinct. The practical result of this is that in the case of large values of ζ these measurements can significantly improve constraints on w a -by more than a factor of two for the case of ELT-HIRES, and by about 30% for ESPRESSO ideal data, in the case of a large coupling-see the last line in Table 4. Thus α measurements can ideally complement cosmological probes in mapping the behavior of dynamical dark energy. Table 3: Same as Table 2 for the case of the dilaton model. Discussion and conclusions We have used standard Fisher Matrix analysis techniques to study the cosmological impact of short and medium-term astrophysical tests of the stability of the fine-structure constant. The ESPRESSO spectrograph will be commissioned at the VLT in late 2017, and since it will be located at the combined Coudé focus it will be able to incoherently combine light from the four VLT unit telescopes. On the other hand, the European Extremely Large Telescope, with first light expected in 2024, will have a 39.3m primary mirror. The larger telescope collecting areas are one of the reasons behind the expected improvements in the sensitivity of these measurements (which are photon-starved). The other such reason pertains to technological improvements in the spectrographs themselves, enabling, among others, higher resolution and stability [5,24]. Our analysis demonstrates that whether these measurements lead to detections of variations or to improved null results, they will have important implications for cosmology as well as for fundamental physics. In the scenario where there are no α variations, ESPRESSO can improve current bounds on WEP violations by up to two orders of magnitude: such bounds would be stronger bounds than those expected from the MICROSCOPE satellite. Similarly, constraints from the high-resolution spectrograph at the E-ELT should be competitive with those of the proposed STEP satellite (although in this case one should be mindful of the caveat that both facilities are currently still in early stages of development). In the opposite case where an α variation is detected, and quite apart from the direct implications (direct evidence of Einstein Equivalence Principle violation, falsifying the notion of gravity as a purely geometric phenomenon, and of a fifth interaction in nature [1]) there are additional implications for cosmology. While the anticorrelations between the scalar field electromagnetic coupling ζ and the dark energy equation of state parameters mean that constraints on ζ will in this case be weaker than in the null case, the new facilities will extend the range of couplings that can be meaningfully probed by at least one order of magnitude. Moreover, these measurements are particularly sensitive to the dynamics of dark energy, and could conceivably improve constraints on w a by more than a factor of two. We emphasize that the analysis we have presented is conservative in at least one sense: our sample of α measurements consisted only of the 14 measurements in the range 1 < z < 3 foreseen for the fundamental physics part of the ESPRESSO GTO [11,23]. This is to be compared to the 293 archival measurements of Webb et al. [4], in the approximate redshift range 0.5 < z < 4.2. The latter contains data gathered over a period of about ten years from two of the world's largest telescopes, while the 14 GTO targets were chosen on the grounds that they are the best currently known targets for these measurements (and are visible from the location of the VLT, at Cerro Paranal in Chile) and improving the measurements on these targets will have a significant impact in the field. Nevertheless it is clear that in a time scale of 5-10 years a significantly larger dataset could be obtained, leading to even stronger constraints. Although our analysis focused on time (redshift) variations of α, the 14 ESPRESSO GTO measurements will also test possible spatial variations. In particular, if one assumes that α varies as a pure spatial dipole with the best-fit parameters given in the analysis of [4], then along the 14 ESPRESSO GTO lines of sight one expects variations ranging between ∆α/α = −3.8 × 10 −6 and ∆α/α = +4.1 × 10 −6 . Moreover, in 11 of the 14 lines of sight those best-fit parameters predict variations whose absolute value is larger than the nominal statistical uncertainty of the ESPRESSO baseline measurements, σ ∆α/α = 0.6 × 10 −6 . Thus the measurements will provide a test of such spatial variations at a high level of statistical significance, at least at the claimed level of parts-permillion in amplitude. Finally, although our constraints are mildly dependent on the choice of fiducial cosmological model-a result that confirms analyses of current data [6,8,9]-this is actually a desirable feature. Should ESPRESSO or ELT-HIRES detect α variations, these measurements will ideally complement other canonical observables in selecting between otherwise indistinguishable cosmological models. A more detailed study of this procedure is left for subsequent work. Figure 1 : 1One sigma forecasted uncertainties on ζ, marginalizing over the remaining model parameters, for the various choices of fiducial cosmological model (shown in different colors) and dataset of α measurements (with solid, dotted and dashed lines respectively depicting ESPRESSO baseline, ESPRESSO ideal and ELT-HIRES, cf. the main text). The fiducial value of the coupling is ζ f id = 0 in all cases. 1 : 1The first three lines show the one sigma forecasted uncertainties on the dimensionless coupling parameter ζ, marginalizing over the remaining model parameters, for the various choices of fiducial cosmological model and dataset of α measurements. The fiducial value of the coupling is ζ f id = 0 in all cases. The last line shows the corresponding one-sigma uncertainty on the Eötvös parameter η, in the least constraining case of the w 0 = const. modelModelESPRESSO baseline ESPRESSO ideal ELT-HIRES w 0 = const. 2 : 2Results of the Fisher Matrix analysis for the case of the constant dark energy equation of state model. The first three lines show the correlation coefficients ρ for each pair of parameters, the following three the Figure of Merit for each pair of parameters, and the last two the one-sigma marginalized uncertainties for the coupling ζ and the present-day value of the dark energy equation of state w 0 . ESPRESSO baseline ESPRESSO ideal ELT-HIRES Parameter ζ = 5 × 10 −7 ζ = 5 × 10 −6 ζ = 5 × 10 −7 ζ = 5 × 10 −6 ζ = 5 × 10 −7 ζ = 5 × 10 Table 4 : 4ζ = 5 × 10 −7 ζ = 5 × 10 −6 ζ = 5 × 10 −7 ζ = 5 × 10 −6 ζ = 5 × 10 −7 ζ = 5 × 10 Results of the Fisher Matrix analysis for the case of the CPL parametrization. The first six lines show the correlation coefficients ρ for each pair of parameters, the following six the Figure of Merit for each pair of parameters, and the last three the one-sigma marginalized uncertainties for the coupling ζ and the dark energy equation of state parameters w 0 and w a .ESPRESSObaseline ESPRESSO ideal ELT-HIRES Parameter ζ = 5 × 10 −7 ζ = 5 × 10 −6 ζ = 5 × 10 −7 ζ = 5 × 10 −6 ζ = 5 × 10 −7 ζ = 5 × 10 Figure 2 :Figure 3 :Figure 4 : 234Forecasted uncertainties in the ζ − w 0 and ζ − Ω m planes (left and right panels), marginalizing over the remaining model parameters, for the various choices of fiducial cosmological model (shown in different colors) and dataset of α measurements (solid lines for the ESPRESSO baseline case and dashed lines for ELT-HIRES, cf. the main text), for a fiducial value of the coupling ζ = 5 × 10 −7 . Same as Fig. 2, for a fiducial values of the coupling ζ = 5 × 10 −6 . 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[ "Controllable nonlinear propagation of partially incoherent Airy beams", "Controllable nonlinear propagation of partially incoherent Airy beams" ]	[ "Kaijian Chen \nCenter on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina\n", "Peiyu Zhang \nCenter on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina\n", "Nana Liu \nCenter on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina\n", "Liu Tan \nCenter on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina\n", "Peilong Hong [email protected]@gxu.edu.cn \nSchool of Optoelectronic Science and Engineering\nUniversity of Electronic Science and Technology of China (UESTC)\n610054ChengduChina\n\nThe MOE Key Laboratory of Weak-Light Nonlinear Photonics\nTEDA Applied Physics Institute and School of Physics\nNankai University\n300457TianjinChina\n", "Bingsuo Zou \nSchool of Physical Science and Technology and School of Resources\nEnvironment and Materials\nKey Laboratory of new Processing Technology for Nonferrous Metals and Materials\nGuangxi University\n530004NanningChina\n", "Jingjun Xu \nThe MOE Key Laboratory of Weak-Light Nonlinear Photonics\nTEDA Applied Physics Institute and School of Physics\nNankai University\n300457TianjinChina\n", "Y I Liang \nCenter on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina\n\nThe MOE Key Laboratory of Weak-Light Nonlinear Photonics\nTEDA Applied Physics Institute and School of Physics\nNankai University\n300457TianjinChina\n\nState Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures\n530004NanningChina\n" ]	[ "Center on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina", "Center on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina", "Center on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina", "Center on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina", "School of Optoelectronic Science and Engineering\nUniversity of Electronic Science and Technology of China (UESTC)\n610054ChengduChina", "The MOE Key Laboratory of Weak-Light Nonlinear Photonics\nTEDA Applied Physics Institute and School of Physics\nNankai University\n300457TianjinChina", "School of Physical Science and Technology and School of Resources\nEnvironment and Materials\nKey Laboratory of new Processing Technology for Nonferrous Metals and Materials\nGuangxi University\n530004NanningChina", "The MOE Key Laboratory of Weak-Light Nonlinear Photonics\nTEDA Applied Physics Institute and School of Physics\nNankai University\n300457TianjinChina", "Center on Nanoenergy Research\nSchool of Physical Science and Technology\nGuangxi Key Lab for Relativistic Astrophysics\nGuangxi University\n530004NanningGuangxiChina", "The MOE Key Laboratory of Weak-Light Nonlinear Photonics\nTEDA Applied Physics Institute and School of Physics\nNankai University\n300457TianjinChina", "State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures\n530004NanningChina" ]	[]	The self-accelerating beams such as the Airy beam show great potentials in many applications including optical manipulation, imaging and communication. However, their superior features during linear propagation could be easily corrupted by optical nonlinearity or spatial incoherence individually. Here we investigate how the interaction of spatial incoherence and nonlinear propagation affect the beam quality of Airy beam, and find that the two destroying factors can in fact balance each other. Our results show that the influence of coherence and nonlinearity on the propagation of PIABs can be formulated as two exponential functions that have factors of opposite signs. With appropriate spatial coherence length, the PIABs not only resist the corruption of beam profile caused by self-focusing nonlinearity, but also exhibits less anomalous diffraction caused by the self-defocusing nonlinearity. Our work provides deep insight into how to maintain the beam quality of self-accelerating Airy beams by exploiting the interaction between partially incoherence and optical nonlinearity. Our results may bring about new possibilities for optimizing partially incoherent structured field and developing related applications such as optical communication, incoherent imaging and optical manipulations.	null	[ "https://export.arxiv.org/pdf/2304.02326v1.pdf" ]	257,952,418	2304.02326	1953c96431f2b570d4d527da3f4c46f4d42a9c20	Controllable nonlinear propagation of partially incoherent Airy beams Kaijian Chen Center on Nanoenergy Research School of Physical Science and Technology Guangxi Key Lab for Relativistic Astrophysics Guangxi University 530004NanningGuangxiChina Peiyu Zhang Center on Nanoenergy Research School of Physical Science and Technology Guangxi Key Lab for Relativistic Astrophysics Guangxi University 530004NanningGuangxiChina Nana Liu Center on Nanoenergy Research School of Physical Science and Technology Guangxi Key Lab for Relativistic Astrophysics Guangxi University 530004NanningGuangxiChina Liu Tan Center on Nanoenergy Research School of Physical Science and Technology Guangxi Key Lab for Relativistic Astrophysics Guangxi University 530004NanningGuangxiChina Peilong Hong [email protected]@gxu.edu.cn School of Optoelectronic Science and Engineering University of Electronic Science and Technology of China (UESTC) 610054ChengduChina The MOE Key Laboratory of Weak-Light Nonlinear Photonics TEDA Applied Physics Institute and School of Physics Nankai University 300457TianjinChina Bingsuo Zou School of Physical Science and Technology and School of Resources Environment and Materials Key Laboratory of new Processing Technology for Nonferrous Metals and Materials Guangxi University 530004NanningChina Jingjun Xu The MOE Key Laboratory of Weak-Light Nonlinear Photonics TEDA Applied Physics Institute and School of Physics Nankai University 300457TianjinChina Y I Liang Center on Nanoenergy Research School of Physical Science and Technology Guangxi Key Lab for Relativistic Astrophysics Guangxi University 530004NanningGuangxiChina The MOE Key Laboratory of Weak-Light Nonlinear Photonics TEDA Applied Physics Institute and School of Physics Nankai University 300457TianjinChina State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures 530004NanningChina Controllable nonlinear propagation of partially incoherent Airy beams The self-accelerating beams such as the Airy beam show great potentials in many applications including optical manipulation, imaging and communication. However, their superior features during linear propagation could be easily corrupted by optical nonlinearity or spatial incoherence individually. Here we investigate how the interaction of spatial incoherence and nonlinear propagation affect the beam quality of Airy beam, and find that the two destroying factors can in fact balance each other. Our results show that the influence of coherence and nonlinearity on the propagation of PIABs can be formulated as two exponential functions that have factors of opposite signs. With appropriate spatial coherence length, the PIABs not only resist the corruption of beam profile caused by self-focusing nonlinearity, but also exhibits less anomalous diffraction caused by the self-defocusing nonlinearity. Our work provides deep insight into how to maintain the beam quality of self-accelerating Airy beams by exploiting the interaction between partially incoherence and optical nonlinearity. Our results may bring about new possibilities for optimizing partially incoherent structured field and developing related applications such as optical communication, incoherent imaging and optical manipulations. Introduction Partially incoherent structured light has attracted a lot of interests recently [1][2][3][4], since the control of coherence length provides new features into the structured light that typically originates from optical interference. Interestingly, partially incoherent structured light can present better robustness in turbulent atmosphere and random media [5,6], leading to superior performance in many intriguing applications such as optical communication [5], incoherent imaging [7], optical manipulations [8], optical nonlinear solitons or lattices [9] and so on. Airy beams have been widely employed in optical manipulations [10], plasmons [11] and optical image transmissions [12,13] due to their intriguing and counterintuitive properties [14,15]: self-accelerating, non-diffracting and self-healing. As one of the typical self-accelerating beams, Airy beams in the linear systems have been studied fully in previous works [16,17]. Beyond the linear systems, Yi Hu et al reported the diffraction behavior of Airy beams in photorefractive crystals [18], which had led to the discussion of the nonlinear propagation properties of Airy beams, such as self-trapped waves and spatial solitons [9,19]. Then Noémi Wiersma et al [20] explored the spatial solitons generated by one-dimension Airy beams passing through photorefractive crystals and discussed the influence of an off-shooting soliton on self-focusing nonlinearity. In fact, These results indicate that a strong self-focusing nonlinearity corrupts the beam structure and self-accelerating propagation.Thus, it is challenging yet meaningful to optimize the self-accelerating beams for maintaining their excellent features during nonlinear propagation. However, to the best of our konwledge, in compare with the thorough investigation of nonlinear dynamics of fully coherent self-accelerating beams, self-accelerating beams with a low spatial coherence, i.e. partially incoherent self-accelerating beams, have not been investigated in nonlinear media. By appropriately controlling the coherence length, R. Martínez-Herrero et al [21] theoretically demonstrated that partially incoherent Airy beams can still maintain their own unique propagation properties by using cross-spectral-density. Nonetheless, similar to nonlinear propagation, partially coherence was usually thought corrupting the beam quality of self-focusing beam [21,22]. Here, by thoroughly studying the nonlinear propagation of partially incoherent airy beam, we found that, surprisingly, under proper condition the partially coherence could resist the corruption of beam quality caused by nonlinear propagation. To demonstrate such an interesting results, we first study the propagation of partially incoherent Airy beams driven by nonlinearity experimentally. The results show that, compared with the nonlinear propagation of fully-coherent airy beams, PIABs can reduce the nonlinear effect and keep their shape better. In other words, we demonstrate that a partially incoherent self-accelerating beam can maintain its shape easier in a nonlinear media and exhibit a good self-accelerating performance in subsequent propagation. For self-focusing case, within the appropriate range, with more spatial incoherence and the less nonlinearity, the beams present better self-accelerating and shape-preserving propagation. Counter to that, for self-defocusing case, the beams show a better performance until reaching a saturable state when self-defocusing nonlinearity is too large. Further theoretical analysis shows that the nonlinearity and spatial coherence works as two exponential functions that influences the beam profiles. Our work is beneficial for the understanding the role of the interaction between spatial coherence and optical nonlinearity in the propagation of self-accelerating beams, which provides a new way to design reliable self-accelerating beams in nonlinear media. Our results thus can promote for further applications in optical manipulation and optical communication of complex environment. Theory In theory, the electric field of a typical coherent Airy beam at the source plane can be described as [14] ( , , = 0) = 0 Ai 0 exp 0 × Ai 0 exp 0 (1) where ( ) is the Airy function, is a truncation factor, 0 is a scale factor called characteristic length, and 0 is a constant controlling the amplitude of Airy beam. In fact, PIABs can be superposed by coherent modes (ì) [23][24][25]. Therefore, the coherent function ( ì 1 , ì 2 ) of PIABs can be deifined as ( ì 1 , ì 2 ) = ∑︁ * ( ì 1 ) ( ì 2 )(2) where are real values representing the power associated with coherent mode (ì). ì 1 and ì 2 represent the position vectors of two points in space, respectively. Clearly, a fully coherent beam is comprised of a single spatial mode, while a partially incoherent beam is comprised of an arbitrary number of modes. According to Eq.(2) and (ì) = \| (ì)\| 2 , the spatial correlation function is ( ì 1 , ì 2 ) = ( ì 1 , ì 2 ) / √︁ ( ì 1 ) ( ì 2 ). Then is the coherence length related to the spatial coherence function, which can be defined as (¯) = ∫ +∞ −∞ \| (¯− 1 2 Δ ,¯+ 1 2 Δ )\| 2 dΔ [26]. For arbitrary coherent modes (ì), they all obey the paraxial wave equation by using slowly varying envelope approximation (ì) = 2 0 2 (ì) 2 + 2 (ì) 2 + 0 Δ ( ) (ì)(3) where 0 is the vacuum wave vector, and is the refractive index. Δ ( ) is the nonlinear refractive index varying with , where is the time average intensity of the beam. Based on Eq. (2) and Eq.(3), one can simulate the propagation of the PIAB in linear or nonlinear media. Experiment Our experimental setup for generation and propagation of PIAB is shown as Fig. 1(a): a linearly polarized Gaussian beam ( =532nm) transmits through a rotating ground glass disk, such that a spatially incoherent light is generated [23,24]. The partially incoherent light is then launched onto a spatial light modulator (SLM, Holoeye Pluto-VIS), where a cubic-phase pattern is loaded on the wave front. To produce a PIAB, a Fourier transform lens 3 ( = 150 ) is placed in front of the SLM, and the PIAB is obtained at the front focal plane of 3 . Here, the coherence length of the PIAB can be controlled by adjusting the diameter of the laser beam incident on the rotating diffuser [27].The PIAB is then launched into a biased 1-cm-long photorefractive media strontium-barium niobate (SBN:60) crystal for investigating its nonlinear propagation. By switching the polarity of the bias field 0 , self-focusing and self-defocusing nonlinearity can be achieved [18]. The PIAB and its Fourier spectrum are monitored by two CCD cameras. In all experiments, the characteristic length 0 and the truncation factor of PIABs are adopted as 14 and 0.08, respectively. Results and Discussion Linear propagation of PIABs As for comparison, we first studied the linear propagation properties of the PIABs with different spatial coherence lengths. Here, there is no bias field on the SBN such that Δ = 0, and the results of PIAB are shown in Figs. 1(b-d). Apparently, as the coherence length decreases, the main lobe of the resulting Airy beam becomes larger with less side lobes, suggesting the limited coherence length working as an effective truncation factor of the Airy beam as shown in eq.(1). In other words, even though we set the truncation factor as = 0.08 in all cases, the spatial incoherence enhances the truncating influence, which can be formulated as an additional exponential function exp( ) with is a negetive value. For the PIABs, the fine structures of Airy beam are blurred as the coherence length decreases, and finally the beam profile becomes close to that of a Gaussian-like beam if the beam is fully incoherent. Nonetheless, the generated PIABs keep the self-accelerating property, and the transverse positions of the main lobes are shifted along the propagation direction as shown in Figs. 1(b-d) . Besides, the PIABs exhibit nearly the same transverse self-acceleration in the three cases, implying that the accelerating property of the PIAB weakly depends on the spatial coherence. However, the peak intensity changes during propagation look different. After comparing the results of the three columns , we find that the peak intensity decreases more with a shorter spatial coherence length. Therefore, the PIAB is more difficult to keep its non-diffracting features and the intensity decay is larger. To further prove above results, we also employed Eq.(3) to do the numerical simulation via multiple-phase-screen methods [23], and the results are as presented in Fig. 2. Clearly, the numerical results agree well with the experimental result, demonstrating that PIABs still preserve the self-accelerating properties and the intensity profiles in Fourier space keep unchanged. To quantitatively analyze the change of the diffracting property of the beams with or without the bias field, we define two factors for simple discussion: one factor is called peak intensity ratio = / , where and refer to the final and initial peak intensity of PIABs with nonlinear propagation, respectively; Another factor is named intensity profile similarity, = ( , )/ √︃ ( ) ( ), where ( , ) is the covariance between random distribution and , ( ) is the variance of random distribution , and refer to the final and initial intensity distributions of PIABs with nonlinear propagation, respectively. In the linear case, the calculated results of and with different coherence lengths are plotted in Figs.2(d, e). When coherence length is larger than 30 , the decrease of incoherence does not affect beams structure of Airy beams much. The peak intensity ratio is almost unchanged (first goes down little until around 50 and then goes up), as well as the intensity profile similarity . PIABs possess a stable transverse accelerating, as presented in Fig. 2(f). However, when coherence length is smaller than 30 , quickly increases with a decreasing coherence length while decreases. This behavior is same as that by increasing the truncating factor a for a fully coherent Airy beam. Especially, when the coherence length is very small (16 ), the intensity distributes randomly (the 1 inset of Fig.2(d)) and is nearly zero though is close to one, as a result of the strong distortions of the incoherence, i.e., a large effective truncation factor . Beams have evolved into Gaussian-like beams and the diffracting is too strong. PIABs cannot keep self-accelerating and close-nondiffracting. By employing an additional exponential function to the wavefunction of a fully coherent Airy beam, the simulation results can predict the results for PIABs, confirming our theoretical assumption that spatial incoherence can be formulated as an effective exponential function. Nonlinear propagation of PIABs Next, we experimentally study the nonlinear propagation of the PIABs in a biased SBN:60 crystal. The saturable nonlinearity of SBN for an e-polarized beam can be determined by Δ = −0.5 3 33 0 /(1+ ), in which = 2.3 is the unperturbed refractive index, 33 = 280 / is the nonlinear coefficient, 0 is the amplitude of the bias field and is the time average intensity of the beam. When a positive bias field of 0 = 3 × 10 4 / is applied, the PIABs experience a self-focusing nonlinearity. As a result, the lobes of the beams with different spatial coherence all self-trap into smaller lobes [ Figs. 3(a1-c1)]. To further observe how the incoherence affects propagation property of PIABs, we let these self-focused beams propagate in a subsequent free-space (5-mm long air). The final results are shown in Figs.3(a2-c2). For the fully coherent case, the shape of Airy beam is strongly deformed and peak intensity cannot keep at in the main lobe (Airy "head") though the acceleration of main lobe still exists [ Fig. 3(a2)]. Its Fourier spectrum in k-space is partially focused at the center shown in the inset of Fig. 3(a1), suggesting that the Airy beam exhibits normal diffraction. In contrast, a PIAB can keep somewhat Airy-like pattern [ Figs. 3(b2-c2)]. The peak intensity not only stays at the main lobe, but also maintains the obvious self-acceleration. Modulated by more incoherence, PIAB driven by a self-focusing nonlinearity keep the shape and self-accelerating propagation properties better. Moreover, PIAB with more incoherence has a larger output peak intensity and intensity ratio after the subsequent 5-mm propagation in free space. Less intensity in k-space is found to be focused toward the center [the insets of Figs. 3(b1-c1)], verifying that spatial incoherence reduces the degradation of the beam quality caused by self-focusing nonlinearity. It is kown that when self-focusing nonlinearity is employed, it would firstly balance the diffraction of the beams and then break down the original interference of the plane waves that contribute to the nondiffraction and self-acceleration of Airy beams [28,29]. Thus, the shape of an Airy beam is deformed and the beam cannot keep self-acceleration or a good intensity output. However, if the diffraction of Airy beam becomes stronger, self-focusing nonlinearity would be difficult to break down the beam structure and propagation properties of Airy beams [18]. Diffraction of a PIAB is determined by its coherence length while diffracting of coherent case depends on its beam diameter. So, the more incoherent, the more difficult it is for self-focusing nonlinearity to break down the beam structure and propagation properties of Airy beams. Breaking down the shape of a PIAB requires stronger self-focusing nonlinearity to balance the stronger diffraction. Clearly, appropriate spatial incoherence can resist the corruption of beam quality caused by self-focusing nonlinearity. By changing the polarity of the bias field( 0 = −3 × 10 4 / ), PIABs experience a self- defocusing nonlinearity. In this case, the shapes of all Airy beams spread much more but the peak intensity after subsequent linear propagation in the air decreases less as compared to the linear case [ Figs. 3(d-f)] while the beams keep a good transversal acceleration. Especially, the Fourier spectrum in all the cases reshapes into a diamond-like pattern in k-space, as shown in the insets of Fig. 3(d1-f1), resembling the first Brillouin zone of an asymmetric square lattice as in Ref. [18]. It implies that PIABs will experience an anomalous diffraction when driven by a self-defocusing nonlinearity and their output peak intensity is stronger than input peak intensity during subsequent linear propagation. However, such effect gets weaker for the beams of more incoherent, and the beams can maintain the Airy-like pattern better after the subsequent linear propagation. Clearly, increasing the spatial incoherence indeed resist the degradation of beam quality caused by self-defocusing nonlinearity. To further confirm our experimental observations, we did simulations with the same parameters as those in experiment. As shown in Fig. 4, when 0 = 3 × 10 4 / is applied, numerical results are highly consistent with the experimental results. PIABs driven by a self-focusing nonlinearity keep their shapes better and self-accelerating propagation with a stronger spatial incoherence. In addition, less intensity in k-space is as expected to focused onto the center, further verifying that spatial incoherence reduces the self-focusing nonlinear effect. Moreover, from Figs.4(a3-c3), one can find that incoherent cases always keep the transverse peak intensity stay at main lobe longer and the longitudinal peak intensity along propagation directions appears latter, indicating that spatial coherence length can control the transversal and longitudinal positions of peak intensity. With a reversed bias field of 0 = −3 × 10 4 / , self-defocusing exhibits a weaker corruption of the beam structures of Airy beams and the beams maintain their patterns and self-accelerating well, as shown in Fig.5. Furthermore, as the spatial coherence length decreases, the PIABs keep the pattern better after the subsequent free-space propagation, exhibiting a better propagation effect compared with the linear case. These results can be also seen clearly from the sideviews of beam propagation [Figs. 4(a3-c3) and 5(a3-b3)]. Similar to the linear case, we also observe the quantitative variation of the intensity ratio , intensity profile similarity and transverse shifting of main lobe Δ of main lobe under different coherence and nonlinearity [ Fig. 6]. When the coherence length increases, intensity ratio in a fixed self-focusing case ( 0 = 3 × 10 4 / ) shows a sharper and larger exponential fall due to the further attenuation of breaking down of beam structure by the self-focusing effect [ Fig. 6(a)]. Meanwhile, as a result of the same self-focusing effect, the change of the intensity profile similarity is much different from that of the linear case [ Fig. 6(b)]. At the beginning, still keeps going up with coherence length. However, from 26 m, does not preserve as linear case. It will quickly declines from 0.9 to 0.3 until 200 m and then keep almost unchanged. For a fixed self-defocusing case ( 0 = −3 × 10 4 / ), a similar but much less obvious phenomenon happens in intensity profile similarity . However, due to the enhancement of self-defocusing on the output intensity (anomalous diffraction) shown in Fig. 6(a), the intensity ratio experience a quick rise and turns into a stable state after 70 . For both of the two nonlinear cases, the transverse accelerating shift maintains stable while coherence varies [ Fig. 6(c)]. Generally, self-focusing with 0 = 3 × 10 4 / enhances the accelerating while self-defocusing does not affect it unless the incoherence is strong. In addition, for the self-focusing nonlinearity, when the coherence length is fixed to 48 (> 30 ), which means the truncating effect of incoherence is still weak, the intensity ratio and sharply go down with the increasing biased field and will have an obvious jumping at a certain value (here, ∼ 5 × 10 4 / ) [Figs. 6(d,e)]. This jump is caused by the soliton shedding of Airy beams [18] generated by strong self-focusing nonlinearity, i.e., there is a new self-focusing Airy soliton beginning to form at this jumping biased field, as shown in the insets in Fig. 6(d). Interestingly, while the biased field is larger than 7.5 × 10 4 / , goes up, implying that the beam pattern changes back to an Airy profile. As mentioned in Ref. [18], this is possible to be a result of soliton shedding and the self-healing property of Airy beams. Note that, the transverse accelerating shift during subsequent linear propagation in free space also changes with the periodic Airy solitons and shows snake oscillation [Figs. 6(f)]. Contrary to that, for the self-defocusing case, no jumping or snake oscillation happens. As the biased field increases, keeps increasing and can be larger than one [Figs. 6(d)], indicating the anomalous diffraction is getting stronger with an increasing self-defocusing nonlinearity. However, the increase of is getting smaller and smaller and may reach a saturated state finally. Meanwhile, decreases slowly and the transverse accelerating shift has no change, demonstrating the Airy-like structure and profiles of PIABs keep good, and self-defocusing nonlinearity produces a very weak influence. Definitely, this influence gets stronger with a bigger biased field. To further investigate the combined interaction of incoherence and nonlinearity, we plotted the detailed results of PIABs with different coherence and nonlinearity in Figs. 6(g-i). From Figs. 6(g,h), one can see that the larger coherence or biased field can enhance the anomalous diffraction caused by self-defocusing, leading to an increasing intensity ratio and intensity similarity . However, for the focusing case, more coherence or nonlinearity causes more serious corruption of propagation properties and beam structure, resulting in smaller and . Especially, the biased field inducing jumping decreases with a larger coherence length (See the black lines in Fig. 6). In this case, , , and Δ present quicker and stronger periodic fluctuations though and still keep a decreasing trend while Δ keeps an increasing trend. Interestingly, for self-defocusing nonlinearity, the decline of and enhancement of also get stronger and stronger while incoherence becomes weaker, and Δ exhibit a jumping at coherence lengths larger than around 60 . Definitely, when coherence length is smaller than 30 , the PIABs are seriously broken down by the random distortion of huge incoherence. In general, PIABs with proper spatial incoherence always have the better robustness for nonlinear effect. Conclusion In summary, we have quantitatively investigated the nonlinear propagation of PIABs both in theory and experiment. It is found that, the influence of coherence and nonlinearity on the propagation of PIABs can be formulated as two effective exponential functions. Moreover, spatial incoherence and nonlinearity have opposite effects on the propagation properties of PIABs, and thus their interaction can be used to improve beam quality of Airy beam. By controlling the spatial incoherence, PIABs can resist the corruption on the beam quality caused by nonlinear propagation. Under a appropriate balance between incoherence and nonlinearity, PIABs present much better beam quality after a long propagation distance. Our results provide a comprehensive understanding on the nonlinear propagation dynamics of PIABs, beneficial to the development and control of Airy beam and the corresponding applications. Fig. 1 . 1(a) Schematic of experimental setup. SLM, spatial light modulator; SBN, strontium-barium-niobate crystal; BS, beam splitter; PBS, polarized beam splitter; L, lens; D, rotating ground-glass disk; Mask, the cubic-phase pattern. (b-d) The experimental results of the PIABs with different coherence lengths after 10 mm through crystal plus another 5mm through air. The 1 , 2 and the 3 columns are the input intensity pattern, output intensity pattern after 10mm through crystal and output intensity pattern after 15 mm through crystal and air, respectively. The coherence length of (b)-(d) is infinity, = 48 and = 32 , respectively. The illustrations in (b2)-(d2) show corresponding Fourier spectra. Fig. 2 . 2(a-c) The simulation results correspond to Figs. 1. (b)-(d) and (d-f) the changes of the intensity ratio , intensity profile similarity and transverse shifting of main lobe Δ with different coherence lengths. The insets in (d) show the initial intensity of PIAB with coherence length 16 , 26 , 50 , 165 , 975 , respectively. Fig. 3 . 3The output intensity patterns of PIABs with different spatial coherence after 10 mm through crystal (1 column and 3 column) plus another 5mm through air (2 column and 4 ℎ column). The illustrations in the upper right corner in (a1)-(f1) show corresponding Fourier spectra. (a)-(c) self-focusing nonlinearity ( 0 = 3 × 10 4 / ), (d)-(f) self-defocusing nonlinearity ( 0 = −3 × 10 4 / ). 1 , 2 and 3 rows correspond to = ∞ ,48 and = 32 , respectively. Fig. 4 . 4Numerical simulation of propagation of PIABs under an initial self-focusing nonlinearity ( 0 = 3 × 10 4 / ). (a1)-(c1) and (a2-c2) correspond to the experimantal reuslts in Figs. 3(a1-c1) and (a2-c2), respectively. (a3)-(c3) are the corresponding sideviews. Fig. 5 . 5Numerical simulation of PIABs propagation under an initial self-defocusing nonlinearity( 0 = −3 × 10 4 / ). (a1)-(c2) correspond to Figs. 3(d-f). (a3)-(c3) are the corresponding sideviews. Fig. 6 . 6Changes of peak intensity ratio , intensity profile similarity and transverse shifting of main lobe of PIABs with different conditions: (a, b) different coherence length with the same initial nonlinearity (\| 0 \| = 300 × 10 4 / ); (c, d) different nonlinearity with = 48 ; (g-i) different spatial incoherence and driven by different initial nonlinearity. Black curves indicate the changes of , , and Δ with 0 at = 200 . Funding. the National Natural Science Foundation of China (11604058), the Guangxi Natural Science Foundation (2020GXNSFAA297041, 2020GXNSFDA238004, 2016GXNSFBA380244), Innovation Project of Guangxi Graduate Education(YCSW2022041), Sichuan Science and Technology Program (2023NS-FSC0460), and Open Project Funding of the Ministry of Education Key Laboratory of Weak-Light Nonlinear Photonics (OS22-1). Acknowledgments.Disclosures. The authors declare that there are no conflicts of interest related to this article. Data Availability Statement. 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[ "Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes", "Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes" ]	[ "Guanghui Lan " ]	[]	[]	We present new policy mirror descent (PMD) methods for solving reinforcement learning (RL) problems with either strongly convex or general convex regularizers. By exploring the structural properties of these overall highly nonconvex problems we show that the PMD methods exhibit fast linear rate of convergence to the global optimality. We develop stochastic counterparts of these methods, and establish an O(1/ǫ) (resp., O(1/ǫ 2 )) sampling complexity for solving these RL problems with strongly (resp., general) convex regularizers using different sampling schemes, where ǫ denote the target accuracy. We further show that the complexity for computing the gradients of these regularizers, if necessary, can be bounded by O{(log γ ǫ)[(1−γ)L/µ] 1/2 log(1/ǫ)} (resp., O{(log γ ǫ)(L/ǫ) 1/2 }) for problems with strongly (resp., general) convex regularizers. Here γ denotes the discounting factor. To the best of our knowledge, these complexity bounds, along with our algorithmic developments, appear to be new in both optimization and RL literature. The introduction of these convex regularizers also greatly enhances the flexibility and thus expands the applicability of RL models.Address(es) of author(s) should be given	10.1007/s10107-022-01816-5	[ "https://arxiv.org/pdf/2102.00135v6.pdf" ]	231,740,467	2102.00135	f64548680d22cb03a94bab5ebeb06b24f372206e	Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes Guanghui Lan Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes Submitted: Feb 5, 2021; Revised: Oct 26, 2021; Accepted: April 5, 2022.arXiv:2102.00135v6 [cs.LG] 6 Apr 2022 Noname manuscript No. (will be inserted by the editor) We present new policy mirror descent (PMD) methods for solving reinforcement learning (RL) problems with either strongly convex or general convex regularizers. By exploring the structural properties of these overall highly nonconvex problems we show that the PMD methods exhibit fast linear rate of convergence to the global optimality. We develop stochastic counterparts of these methods, and establish an O(1/ǫ) (resp., O(1/ǫ 2 )) sampling complexity for solving these RL problems with strongly (resp., general) convex regularizers using different sampling schemes, where ǫ denote the target accuracy. We further show that the complexity for computing the gradients of these regularizers, if necessary, can be bounded by O{(log γ ǫ)[(1−γ)L/µ] 1/2 log(1/ǫ)} (resp., O{(log γ ǫ)(L/ǫ) 1/2 }) for problems with strongly (resp., general) convex regularizers. Here γ denotes the discounting factor. To the best of our knowledge, these complexity bounds, along with our algorithmic developments, appear to be new in both optimization and RL literature. The introduction of these convex regularizers also greatly enhances the flexibility and thus expands the applicability of RL models.Address(es) of author(s) should be given Introduction In this paper, we study a general class of reinforcement learning (RL) problems involving either covex or strongly convex regularizers in their cost functions. Consider the finite Markov decision process M = (S, A, P, c, γ), where S is a finite state space, A is a finite action space, P : S × S × A → R is transition model, c : S × A → R is the cost function, and γ ∈ (0, 1) is the discount factor. A policy π : A × S → R determines the probability of selecting a particular action at a given state. For a given policy π, we measure its performance by the action-value function (Q-function) Q π : S × A → R defined as Q π (s, a) := E ∞ t=0 γ t [c(s t , a t ) + h π (s t )] \| s 0 = s, a 0 = a, a t ∼ π(·\|s t ), s t+1 ∼ P(·\|s t , a t )] . (1.1) Here h π is a closed convex function w.r.t. the policy π, i.e., there exist some µ ≥ 0 s.t. h π (s) − [h π ′ (s) + (h ′ ) π ′ (s, ·), π(·\|s) − π ′ (·\|s) ] ≥ µD π π ′ (s), (1.2) where ·, · denotes the inner product over the action space A, (h ′ ) π ′ (s, ·) denotes a subgradient of h(s) at π ′ , and D π π ′ (s) is the Bregman's distance or Kullback-Leibler (KL) divergence between π and π ′ (see Subsection 1.1 for more discussion). Clearly, if h π = 0, then Q π becomes the classic action-value function. If h π (s) = µD π π0 (s) for some µ > 0, then Q π reduces to the so-called entropy regularized action-value function. The incorporation of a more general convex regularizer h π allows us to not only unify these two cases, but also to greatly enhance the expression power and thus the applicability of RL. For example, by using either the indicator function, quadratic penalty or barrier functions, h π can model the set of constraints that an optimal policy should satisfy. It can describe the correlation among different actions for different states. h π can also model some risk or utility function associated with the policy π. Throughout this paper, we say that h π is a strongly convex regularizer if µ > 0. Otherwise, we call h π a general convex regularizer. Clearly the latter class of problems covers the regular case with h π = 0. We define the state-value function V π : S → R associated with π as V π (s) := E ∞ t=0 γ t [c(s t , a t ) + h π (s t )] \| s 0 = s, a t ∼ π(·\|s t ), s t+1 ∼ P(·\|s t , a t )] . (1. 3) It can be easily seen from the definitions of Q π and V π that V π (s) = a∈A π(a\|s)Q π (s, a) = Q π (s, ·), π(·\|s) , (1.4) Q π (s, a) = c(s, a) + h π (s) + γ s ′ ∈S P(s ′ \|s, a)V π (s ′ ). (1.5) The main objective in RL is to find an optimal policy π * : S × A → R s.t. V π * (s) ≤ V π (s), ∀π(·\|s) ∈ ∆ \|A\| , ∀s ∈ S. (1. 6) for any s ∈ S. Here ∆ \|A\| denotes the simplex constraint given by ∆ \|A\| := {p ∈ R \|A\| \| \|A\| i=1 p i = 1, p i ≥ 0}, ∀s ∈ S. (1.7) By examining Bellman's optimality condition for dynamic programming ( [3] and Chapter 6 of [19]), we can show the existence of a policy π * which satisfies (1.6) simultaneously for all s ∈ S. Hence, we can formulate (1.6) as an optimization problem with a single objective by taking the weighted sum of V π over s (with weights ρs > 0 and s∈S ρs = 1): minπ Es∼ρ[V π (s)] s.t. π(·\|s) ∈ ∆ \|A\| , ∀s ∈ S. (1.8) While the weights ρ can be arbitrarily chosen, a reasonable selection of ρ would be the stationary state distribution induced by the optimal policy π * , denoted by ν * ≡ ν(π * ). As such, problem (1.8) reduces to minπ {f (π) := E s∼ν * [V π (s)]} s.t. π(·\|s) ∈ ∆ \|A\| , ∀s ∈ S. (1. 9) It has been observed recently (eg., [14]) that one can simplify the analysis of various algorithms by setting ρ to ν * . As we will also see later, even though the definition of the objective f in (1.9) depends on ν * and hence the unknown optimal policy π * , the algorithms for solving (1.6) and (1.9) do not really require the input of π * . Recently, there has been considerable interest in the development of first-order methods for solving RL problems in (1.8) - (1.9). While these methods have been derived under various names (e.g., policy gradient, natural policy gradient, trust region policy optimization), they all utilize the gradient information of f (i.e., Q function) in some form to guide the search of optimal policy (e.g., [21,9,7,1,20,5,23,15]). As pointed out by a few authors recently, many of these algorithms are intrinsically connected to the classic mirror descent method originally presented by Nemirovski and Yudin [17,2,16], and some analysis techniques in mirror descent method have thus been adapted to reinforcement learning [20,23,22]. In spite of the popularity of these methods in practice, a few significant issues remain on their theoretical studies. Firstly, most policy gradient methods converge only sublinearly, while many other classic algorithms (e.g., policy iteration) can converge at a linear rate due to the contraction properties of the Bellman operator. Recently, there are some interesting works relating first-order methods with the Bellman operator to establish their linear convergence [4,5]. However, in a nutshell these developments rely on the contraction of the Bellman operator, and as a consequence, they either require unrealistic algorithmic assumptions (e.g., exact line search [4]) or apply only for some restricted problem classes (e.g., entropy regularized problems [5]). Secondly, the convergence of stochastic policy gradient methods has not been well-understood in spite of intensive research effort. Due to unavoidable bias, stochastic policy gradient methods exhibit much slower rate of convergence than related methods, e.g., stochastic Q-learning. Our contributions in this paper mainly exist in the following several aspects. Firstly, we present a policy mirror descent (PMD) method and show that it can achieve a linear rate of convergence for solving RL problems with strongly convex regularizers. We then develop a more general form of PMD, namely approximate policy mirror descent (APMD) method, obtained by applying an adaptive perturbation term into PMD, and show that it can achieve a linear rate of convergence for solving RL problems with general convex regularizers. Even though the overall problem is highly nonconvex, we exploit the generalized monotonicity [6,13,11] associated with the variational inequality (VI) reformulation of (1.8)-(1.9) (see [8] for a comprehensive introduction to VI). As a consequence, our convergence analysis does not rely on the contraction properties of the Bellman operator. This fact not only enables us to define h π as a general (strongly) convex function of π and thus expand the problem classes considered in RL, but also facilitates the study of PMD methods under the stochastic settings. Secondly, we develop the stochastic policy mirror descent (SPMD) and stochastic approximate policy mirror descent (SAPMD) method to handle stochastic first-order information. One key idea of SPMD and SAPMD is to handle separately the bias and expected error of the stochastic estimation of the action-value functions in our convergence analysis, since we can usually reduce the bias term much faster than the total expected error. We establish general convergence results for both SPMD and SAPMD applied to solve RL problems with strongly convex and general convex regularizers, under different conditions about the bias and expected error associated with the estimation of value functions. Thirdly, we establish the overall sampling complexity of these algorithms by employing different schemes to estimate the action-value function. More specifically, we present an O(\|S\|\|A\|/µǫ) and O(\|S\|\|A\|/ǫ 2 ) sampling complexity for solving RL problems with strongly convex and general convex regularizers, when one has access to multiple independent sampling trajectories. To the best of our knowledge, the former sampling complexity is new in the RL literature, while the latter one has not been reported before for policy gradient type methods. We further enhance a recently developed conditional temporal difference (CTD) method [12] so that it can reduce the bias term faster. We show that with CTD, the aforementioned O(1/µǫ) and O(1/ǫ 2 ) sampling complexity bounds can be achieved in the single trajectory setting with Markovian noise under certain regularity assumptions. Fourthly, observe that unless h π is relatively simple (e.g., h π does not exist or it is given as the KL divergence), the subproblems in the SPMD and SAPMD methods do not have an explicit solution in general and require an efficient solution procedure to find some approximate solutions. We establish the general conditions on the accuracy for solving these subproblems, so that the aforementioned linear rate of convergence and new sampling complexity bounds can still be maintained. We further show that if h π is a smooth convex function, by employing an accelerated gradient descent method for solving these subproblems, the overall gradient computations for h π can be bounded by O{(log γ ǫ) (1 − γ)L/µ log(1/ǫ)} and O{(log γ ǫ) L/ǫ}, respectively, for the case when h π is a strongly convex and general convex function. To the best of our knowledge, such gradient complexity has not been considered before in the RL and optimization literature. This paper is organized as follows. In Section 2, we discuss the optimality conditions and generalized monotonicity about RL with convex regularizers. Sections 3 and 4 are dedicated to the deterministic and stochastic policy mirror descent methods, respectively. In Section 5 we establish the sampling complexity bounds under different sampling schemes, while the gradient complexity of computing ∇h π is shown in Section 6. Some concluding remarks are made in Section 7. Notation and terminology For any two points π(·\|s), π ′ (·\|s) ∈ ∆ \|A\| , we measure their Kullback-Leibler (KL) divergence by KL(π(·\|s) π ′ (·\|s)) = a∈A π(a\|s) log π(a\|s) π ′ (a\|s) . Observe that the KL divergence can be viewed as is a special instance of the Bregman's distance (or proxfunction) widely used in the optimization literature. Let the distance generating function ω(π(·\|s)) := a∈A π(a\|s) log π(a\|s) 1 . The Bregman's distance associated with ω is given by D π π ′ (s) := ω(π(·\|s)) − [ω(π ′ (·\|s)) + ∇ω(π ′ (·\|s)), π(·\|s) − π ′ (·\|s) ] = a∈A π(a\|s) log π(a\|s) − π ′ (a\|s) log π ′ (a\|s) −(1 + log π ′ (a\|s))(π(a\|s) − π ′ (a\|s)) = a∈A π(a\|s) log π(a\|s) π ′ (a\|s) , (1.10) 1 It is worth noting that we do not enforce π(a\|s) > 0 when defining ω(π(·\|s)) as all the search points generated by our algorithms will satisfy this assumption. where the last equation follows from the fact that a∈A (π(a\|s) − π ′ (a\|s)) = 0. Therefore, we will use the KL divergence KL(π(·\|s) π ′ (·\|s)) and Bregman's distance D π π ′ (s) interchangeably throughout this paper. It should be noted that our algorithmic framework allows us to use other distance generating functions, such as · 2 p for some p > 1, which, different from the KL divergence, has a bounded prox-function over ∆ \|A\| . Optimality Conditions and Generalized Monotonicity It is well-known that the value function V π (s) in (1.3) is highly nonconvex w.r.t. π, because the components of π(·\|s) are multiplied by each other in their definitions (see also Lemma 3 of [1] for an instructive counterexample). However, we will show in this subsection that problem (1.9) can be formulated as a variational inequality (VI) which satisfies certain generalized monotonicity properties (see [6], Section 3.8.2 of [13] and [11]). Let us first compute the gradient of the value function V π (s) in (1.3). For simplicity, we assume for now that h π is differentiable and will relax this assumption later. For a given policy π, we define the discounted state visitation distribution by d π s0 (s) := (1 − γ) ∞ t=0 γ t Pr π (s t = s\|s 0 ), (2.1) where Pr π (s t = s\|s 0 ) denotes the state visitation probability of s t = s after we follow the policy π starting at state s 0 . Let P π denote the transition probability matrix associated with policy π, i.e., P π (i, j) = a∈A π(a\|i)P(j\|i, a), and e i be the i-th unit vector. Then Pr π (s t = s\|s 0 ) = e T s0 (P π ) t es and d π s0 (s) = (1 − γ) ∞ t=0 γ t e T s0 (P π ) t es. (2.2) Lemma 1 For any (s 0 , s, a) ∈ S × S × A, we have ∂V π (s0) ∂π(a\|s) = 1 1−γ d π s0 (s) [Q π (s, a) + ∇h π (s, a)] , where ∇h π (s, ·) denotes the gradient of h π (s) w.r.t. π. Proof. It follows from (1.4) that ∂V π (s0) ∂π(a\|s) = ∂ ∂π(a ′ \|s) a ′ ∈A π(a ′ \|s 0 )Q π (s 0 , a ′ ) = a ′ ∈A ∂π(a ′ \|s0) ∂π(a\|s) Q π (s 0 , a ′ ) + π(a ′ \|s 0 ) ∂Q π (s0,a ′ ) ∂π(a\|s) . Also the relation in (1.5) implies that ∂Q π (s0,a ′ ) ∂π(a\|s) = ∂h π (s0) ∂π(a\|s) + γ s ′ ∈S P(s ′ \|s 0 , a ′ ) ∂V π (s ′ ) ∂π(a\|s) . Combining the above two relations, we obtain ∂V π (s0) ∂π(a\|s) = a ′ ∈A ∂π(a ′ \|s0) ∂π(a\|s) Q π (s 0 , a ′ ) + π(a ′ \|s 0 ) ∂h π (s0) ∂π(a\|s) + γ a ′ ∈A π(a ′ \|s 0 ) s ′ ∈S P(s ′ \|s 0 , a ′ ) ∂V π (s ′ ) ∂π(a\|s) = x∈S ∞ t=0 γ t Pr π (s t = x\|s 0 ) a ′ ∈A ∂π(a ′ \|x) ∂π(a\|s) Q π (x, a ′ ) + π(a ′ \|x) ∂h π (x) ∂π(a\|s) = 1 1−γ x∈S d π s0 (x) a ′ ∈A ∂π(a ′ \|x) ∂π(a\|s) Q π (x, a ′ ) + ∂h π (x) ∂π(a\|s) = 1 1−γ d π s0 (s) Q π (s, a) + ∂h π (s) ∂π(a\|s) , where the second equality follows by expanding ∂V π (s ′ ) ∂π(a\|s) recursively, and the third equality follows from the definition of d π s0 (s) in (2.1), and the last identity follows from ∂π(a ′ \|x) ∂π(a\|s) = 0 for x = s or a ′ = a, and ∂h π (x) ∂π(a\|s) = 0 for x = s. In view of Lemma 1, the gradient of the objective f (π) in (1.9) at the optimal policy π * is given by ∂f (π * ) ∂π(a\|s) = E s0∼ν * ∂V π * (s0) ∂π(a\|s) = 1 1−γ E s0∼ν * d π * s0 (s)[Q π * (s, a) + ∇h π * (s, a)] = ∞ t=0 γ t (ν * ) T (P π * ) t es [Q π * (s, a) + ∇h π * (s, a)] = 1 1−γ (ν * ) T es [Q π * (s, a) + ∇h π * (s, a)] = 1 1−γ ν * (s) [Q π * (s, a) + ∇h π * (s, a)],(2.3) where the third identity follows from (2.2) and the last one follows from the fact that (ν * ) T (P π * ) t = (ν * ) T for any t ≥ 0 since ν * is the steady state distribution of π * . Therefore, the optimality condition of (1.9) suggests us to solve the following variational inequality E s∼ν * Q π * (s, ·) + ∇h π * (s, ·), π(·\|s) − π * (·\|s) ≥ 0. (2.4) However, the above VI requires h π to be differentiable. In order to handle the possible non-smoothness of h π , we instead solve the following problem E s∼ν * Q π * (s, ·), π(·\|s) − π * (·\|s) + h π (s) − h π * (s) ≥ 0. (2.5) It turns out this variational inequality satisfies certain generalized monotonicity properties thanks to the following performance difference lemma obtained by generalizing some previous results (e.g., Lemma 6.1 of [9]). Lemma 2 For any two feasible policies π and π ′ , we have V π ′ (s) − V π (s) = 1 1−γ E s ′ ∼d π ′ s A π (s ′ , ·), π ′ (·\|s ′ ) + h π ′ (s ′ ) − h π (s ′ ) , where A π (s ′ , a) := Q π (s ′ , a) − V π (s ′ ). (2.6) Proof. For simplicity, let us denote ξ π ′ (s 0 ) the random process (s t , a t , s t+1 ), t ≥ 0, generated by following the policy π ′ starting with the initial state s 0 . It then follows from the definition of V π ′ that V π ′ (s) − V π (s) = E ξ π ′ (s) ∞ t=0 γ t [c(s t , a t ) + h π ′ (s t )] − V π (s) = E ξ π ′ (s) ∞ t=0 γ t [c(s t , a t ) + h π ′ (s t ) + V π (s t ) − V π (s t )] − V π (s) (a) = E ξ π ′ (s) ∞ t=0 γ t [c(s t , a t ) + h π ′ (s t ) + γV π (s t+1 ) − V π (s t )] + E ξ π ′ (s) [V π (s 0 )] − V π (s) (b) = E ξ π ′ (s) ∞ t=0 γ t [c(s t , a t ) + h π ′ (s t ) + γV π (s t+1 ) − V π (s t )] = E ξ π ′ (s) ∞ t=0 γ t [c(s t , a t ) + h π (s t ) + γV π (s t+1 ) − V π (s t ) +h π ′ (s t ) − h π (s t )] (c) = E ξ π ′ (s) ∞ t=0 γ t Q π (s t , a t ) − V π (s t ) + h π ′ (s t ) − h π (s t ) , where (a) follows by taking the term V π (s 0 ) outside the summation, (b) follows from the fact that E ξ π ′ (s) [V π (s 0 )] = V π (s) since the random process starts with s 0 = s, and (c) follows from (1.5). The previous conclusion, together with (2.6) and the definition d π ′ s in (2.1), then imply that V π ′ (s) − V π (s) = 1 1−γ s ′ ∈S a ′ ∈A d π ′ s (s ′ )π ′ (a ′ \|s ′ ) A π (s ′ , a ′ ) + h π ′ (s ′ ) − h π (s ′ ) = 1 1−γ s ′ ∈S d π ′ s (s ′ ) A π (s ′ , ·), π ′ (·\|s ′ ) + h π ′ (s ′ ) − h π (s ′ ) , which immediately implies the result. We are now ready to prove the generalized monotonicity for the variational inequality in (2.5). Lemma 3 The VI problem in (2.5) satisfies E s∼ν * Q π (s, ·), π(·\|s) − π * (·\|s) + h π (s) − h π * (s) = E s∼ν * [(1 − γ)(V π (s) − V π * (s))]. (2.7) Proof. It follows from Lemma 2 (with π ′ = π * ) that (1 − γ)[V π * (s) − V π (s)] = E s ′ ∼d π * s A π (s ′ , ·), π * (·\|s ′ ) + h π * (s ′ ) − h π (s ′ ) . Let e denote the vector of all 1's. Then, we have A π (s ′ , ·), π * (·\|s ′ ) = Q π (s ′ , ·) − V π (s ′ )e, π * (·\|s ′ ) = Q π (s ′ , ·), π * (·\|s ′ ) − V π (s ′ ) = Q π (s ′ , ·), π * (·\|s ′ ) − Q π (s ′ , ·), π(·\|s ′ ) = Q π (s ′ , ·), π * (·\|s ′ ) − π(·\|s ′ ) ,(2.8) where the first identity follows from the definition of A π (s ′ , ·) in (2.6), the second equality follows from the fact that e, π * (·\|s ′ ) = 1, and the third equality follows from the definition of V π in (1.3). Combining the above two relations and taking expectation w.r.t. ν * , we obtain (1 − γ)E s∼ν * [V π * (s) − V π (s)] = E s∼ν * ,s ′ ∼d π * s Q π (s ′ , ·), π * (·\|s ′ ) − π(·\|s ′ ) + h π * (s ′ ) − h π (s ′ ) = E s∼ν * Q π (s, ·), π * (·\|s) − π(·\|s) + h π * (s) − h π (s) , where the second identity follows similarly to (2.3) since ν * is the steady state distribution induced by π * . The result then follows by rearranging the terms. Since V π (s) − V π * (s) ≥ 0 for any feasible policity π, we conclude from Lemma 3 that E s∼ν * Q π (s, ·), π(·\|s) − π * (·\|s) + h π (s) − h π * (s) ≥ 0. Therefore, the VI in (2.5) satisfies the generalized monotonicity. In the next few sections, we will exploit the generalized monotonicity and some other structural properties to design efficient algorithms for solving the RL problem. Deterministic Policy Mirror Descent In this section, we present the basic schemes of policy mirror descent (PMD) and establish their convergence properties. Prox-mapping In the proposed PMD methods, we will update a given policy π to π + through the following proximal mapping: π + (·\|s) = arg min p(·\|s)∈∆ \|A\| η[ G π (s, ·), p(·\|s) + h p (s)] + D p π (s). (3.1) Here η > 0 denotes a certain stepsize (or learning rate), and G π can be the operator for the VI formulation, e.g., G π (s, ·) = Q π (s, ·) or its approximation. It is well-known that one can solve (3.1) explicitly for some interesting special cases, e.g., when h p (s) = 0 or h p (s) = τ D p π0 (s) for some τ > 0 and given π 0 . For both these cases, the solution of (3.1) boils down to solving a problem of the form p * := arg min p(·\|s)∈∆ \|A\| \|A\| i=1 (g i p i + p i log p i ) for some g ∈ R \|A\| . It can be easily checked from the Karush-Kuhn-Tucker conditions that its optimal solution is given by p * i = exp(−g i )/[ \|A\| i=1 exp(−g i )]. (3.2) For more general convex functions h p , problem (3.1) usually does not have an explicit solution, and one can only solve it approximately. In fact, we will show in Section 6 that by applying the accelerated gradient descent method, we only need to compute a small number of updates in the form of (3.2) in order to approximately solve (3.1) without slowing down the efficiency of the overall PMD algorithms. Basic PMD method As shown in Algorithm 1, each iteration of the PMD method applies the prox-mapping step discussed in Subsection 3.1 to update the policy π k . It involves the stepsize parameter η k and requires the selection of an initial point π 0 . For the sake of simplicity, we will assume throughout the paper that π 0 (a\|s) = 1/\|A\|, ∀a ∈ A, ∀s ∈ S. In this case, we have D π π0 (s) = a∈A π(a\|s) log π(a\|s) + log \|A\| ≤ log \|A\|, ∀π(·\|s) ∈ ∆ \|A\| . (3.4) Observe also that we can replace Q π k (s, ·) in (3.5) with A π k (s, a) defined in (2.6) without impacting the updating of π k+1 (s, ·), since this only introduces an extra constant into the objective function of (3.5). Algorithm 1 The policy mirror descent (PMD) method Input: initial points π 0 and stepsizes η k ≥ 0. for k = 0, 1, . . . , do π k+1 (·\|s) = arg min p(·\|s)∈∆ \|A\| η k [ Q π k (s, ·), p(·\|s) + h p (s)] + D p π k (s) , ∀s ∈ S. Below we establish some general convergence properties about the PMD method. Different from the classic policy iteration or value iteration method used in Markov Decision Processes, our analysis does not rely on the contraction properties of the Bellman's operator, but on the so-called three-point lemma associated with the optimality condition of problem (3.5) (see Lemma 4). Our analysis also significantly differs from the one for the classic mirror descent method in convex optimization (see, e.g., Chapter 3 of [13]). First, the classic mirror descent method requires the convexity of the objective function, while the analysis of PMD utilizes the generalized monotonicity in Lemma 3. Second, the classic mirror descent utilizes the Lipschitz or smoothness properties of the objective function, while in the PMD method, we show the progress made in each iteration of this algorithm (see Lemma 5) by using the performance difference lemma (c.f., Lemma 2) and the three-point lemma (c.f., Lemma 4). As a result, we make no assumptions about the smoothness properties of the objective function at all. The following result characterizes the optimality condition of problem (3.5) (see Lemma 3.5 of [13]). We add a proof for the sake of completeness. Lemma 4 For any p(·\|s) ∈ ∆ \|A\| , we have η k [ Q π k (s, ·), π k+1 (·\|s) − p(·\|s) + h π k+1 (s) − h p (s)] + D π k+1 π k (s) ≤ D p π k (s) − (1 + η k µ)D p π k+1 (s). Proof. By the optimality condition of (3.5), η k [Q π k (s, ·) + (h ′ ) π k+1 (s, ·)] + ∇D π k+1 π k (s, ·), p(·\|s) − π k+1 (·\|s) ≥ 0, ∀p(·\|s) ∈ ∆ \|A\| , where (h ′ ) π k+1 denotes the subgradient of h at π k+1 and ∇D π k+1 π k (s, ·) denotes the gradient of D π k+1 π k (s) at π k+1 . Using the definition of Bregman's distance, it is easy to verify that D p π k (s) = D π k+1 π k (s) + ∇D π k+1 π k (s, ·), p(·\|s) − π k+1 (·\|s) + D p π k+1 (s). (3.6) The result then immediately follows by combining the above two relations together with (1.2). Lemma 5 For any s ∈ S, we have V π k+1 (s) ≤ V π k (s), (3.7) Q π k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) ≥ V π k+1 (s) − V π k (s). (3.8) Proof. It follows from Lemma 2 (with π ′ = π k+1 , π = π k and τ = τ k ) that V π k+1 (s) − V π k (s) = 1 1−γ E s ′ ∼d π k+1 s A π k (s ′ , ·), π k+1 (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) . (3.9) Similarly to (2.8), we can show that A π k (s ′ , ·), π k+1 (·\|s ′ ) = Q π k (s ′ , ·) − V π k (s ′ )e, π k+1 (·\|s ′ ) = Q π k (s ′ , ·), π k+1 (·\|s ′ ) − V π k τ k (s ′ ) = Q π k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) . Combining the above two identities, we then obtain V π k+1 (s) − V π k (s) = 1 1−γ E s ′ ∼d π k+1 s Q π k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) +h π k+1 (s ′ ) − h π k (s ′ ) . (3.10) Now we conclude from Lemma 4 applied to (3.5) with p(·\|s ′ ) = π k (·\|s ′ ) that Q π k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) ≤ − 1 η k [(1 + η k µ)D π k π k+1 (s ′ ) + D π k+1 π k (s ′ )]. (3.11) The previous two conclusions then clearly imply the result in (3.7). It also follows from (3.11) that E s ′ ∼d π k+1 s Q π k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) ≤ d π k+1 s (s) [ Q π k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s)] ≤ (1 − γ) [ Q π k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s)] ,(3.12) where the last inequality follows from the fact that d π k+1 s (s) ≥ (1 − γ) due to the definition of d π k+1 s in (2.1). The result in (3.7) then follows immediately from (3.10) and the above inequality. Now we show that with a constant stepsize rule, the PMD method can achieve a linear rate of convergence for solving RL problems with strongly convex regularizers (i.e., µ > 0). Theorem 1 Suppose that η k = η for any k ≥ 0 in the PMD method with 1 + ηµ ≥ 1 γ . (3.13) Then we have f (π k ) − f (π * ) + µ 1−γ D(π k , π * ) ≤ γ k [f(π 0 ) − f (π * τ ) + µ 1−γ log \|A\|] for any k ≥ 0, where D(π k , π * ) := E s∼ν * [D π * π k (s)]. (3.14) Proof. By Lemma 4 applied to (3.5) (with η k = η and p = π * ), we have η[ Q π k (s, ·), π k+1 (·\|s) − π * (·\|s) + h π k+1 (s) − h π * (s)] + D π k+1 π k (s) ≤ D π * π k (s) − (1 + ηµ)D π * π k+1 (s), which, in view of (3.8), then implies that η[ Q π k (s, ·), π k (·\|s) − π * (·\|s) + h π k (s) − h π * (s)] + η[V π k+1 (s) − V π k (s)] + D π k+1 π k (s) ≤ D π * π k (s) − (1 + ηµ)D π * π k+1 (s). Taking expectation w.r.t. ν * on both sides of the above inequality and using Lemma 3, we arrive at E s∼ν * [η(1 − γ)(V π k (s) − V π * τ (s))] + ηE s∼ν * [V π k+1 (s) − V π k (s)] + E s∼ν * [D π k+1 π k (s)] ≤ E s∼ν * [D π * π k (s) − (1 + ηµ)D π * π k+1 (s)]. Noting V π k+1 (s) − V π k (s) = V π k+1 (s) − V π * (s) − [V π k (s) − V π * (s)] and rearranging the terms in the above inequality, we have E s∼ν * [η(V π k+1 (s) − V π * (s)) + (1 + ηµ)D π * π k+1 (s)] + E s∼ν * [D π k+1 π k (s)] ≤ γE s∼ν * [η(V π k (s) − V π * (s)) + 1 γ D π * π k (s)], (3.15) which, in view of the assumption (3.13) and the definition of f in (1.9) f (π k+1 ) − f (π * ) + µ 1−γ E s∼ν * [D π * π k+1 (s)] ≤ γ (f(π k ) − f (π * )) + µ 1−γ E s∼ν * [D π * π k (s)] . Applying this relation recursively and using the bound in (3.4) we then conclude the result. According to Theorem 1, the PMD method converges linearly in terms of both function value and the distance to the optimal solution for solving RL problems with strongly convex regularizers. Now we show that a direct application of the PMD method only achieves a sublinear rate of convergence for the case when µ = 0. Theorem 2 Suppose that η k = η in the PMD method. Then we have f (π k+1 ) − f (π * ) ≤ ηγ[f (π0)−f (π * )]+log \|A\| η(1−γ)(k+1) for any k ≥ 0. Proof. It follows from (3.15) with µ = 0 that E s∼ν * [η(V π k+1 (s) − V π * (s)) + D π * π k+1 (s)] + E s∼ν * [D π k+1 π k (s)] ≤ ηγE s∼ν * [V π k (s) − V π * (s)] + E s∼ν * [D π * π k (s)]. Taking the telescopic sum of the above inequalities and using the fact that V π k+1 (s) ≤ V π k (s) due to (3.7) , we obtain (k + 1)η(1 − γ)E s∼ν * [V π k+1 (s) − V π * (s)] ≤ E s∼ν * [ηγ(V π0 (s) − V π * (s)) + D π * π0 (s)], which clearly implies the result in view of the definition of f in (1.9) and the bound on D π * π0 in (3.4). The result in Theorem 2 shows that the PMD method requires O(1/(1 − γ)ǫ) iterations to find an ǫ-solution for general RL problems. This bound already matches, in terms of its dependence on (1−γ) and ǫ, the previously best-known complexity for natural policy gradient methods [1]. We will further enhance the PMD method so that it can achieve a linear rate of convergence for the case when µ = 0 in next subsection. Approximate policy mirror descent method In this subsection, we propose to enhance the basic PMD method by adding adaptively a perturbation term into the definition of the value functions or the proximal-mapping. For some τ ≥ 0 and a given initial policy π 0 (a\|s) > 0, ∀s ∈ S, a ∈ A, we define the perturbed action-value and state-value functions, respectively, by Q π τ (s, a) := E ∞ t=0 γ t [c(s t , a t ) + h π (s t ) + τ D π π0 (s t )] \| s 0 = s, a 0 = a, a t ∼ π(·\|s t ), s t+1 ∼ P(·\|s t , a t )] , (3.16) V π τ (s) := Q π τ (s, ·), π(·\|s) . (3.17) Clearly, if τ = 0, then the perturbed value functions reduce to the usual value functions, i.e., Q π 0 (s, a) = Q π (s, a) and V π 0 (s) = V π (s). The following result relates the value functions with different τ . Lemma 6 For any given τ, τ ′ ≥ 0, we have V π τ (s) − V π τ ′ (s) = τ −τ ′ 1−γ E s ′ ∼d π s [D π π0 (s ′ )]. (3.18) As a consequence, if τ ≥ τ ′ ≥ 0 then V π τ ′ (s) ≤ V π τ (s) ≤ V π τ ′ (s) + τ −τ ′ 1−γ log \|A\|. (3.19) Proof. By the definitions of V π τ and d π s , we have V π τ (s) = E ∞ t=0 γ t [c(s t , a t ) + h π (s t ) + τ D π π0 (s)] \| s 0 = s, a t ∼ π(·\|s t ), s t+1 ∼ P(·\|s t , a t ) = E ∞ t=0 γ t [c(s t , a t ) + h π (s t ) + τ ′ D π π0 (s)] \| s 0 = s, a t ∼ π(·\|s t ), s t+1 ∼ P(·\|s t , a t ) + E ∞ t=0 γ t (τ − τ ′ )D π π0 (s)] \| s 0 = s, a t ∼ π(·\|s t ), s t+1 ∼ P(·\|s t , a t ) = V π τ ′ (s) + τ −τ ′ 1−γ E s ′ ∼d π s [D π π0 (s ′ )], which together with the bound on D π π0 in (3.4) then imply (3.19). As shown in Algorithm 2, the approximate policy mirror descent (APMD) method is obtained by replacing Q π k (s, ·) with its approximation Q π k τ k (s, ·) and adding the perturbation τ k D π π0 (s t ) for the updating of π k+1 in the basic PMD method. As discussed in Subsection 3.1, the incorporation of the perturbation term does not impact the difficulty of solving the subproblem in (3.20). In fact, the APMD method can be viewed as a general form of the PMD method since it reduces to the PMD method when τ k = 0. In fact, the perturbation parameter τ k used to define the action-value function Q π k τ k (s, ·) is not necessarily the same as the one used in the regularization term τ k D p π0 (s t ), yielding more flexibility to the design and analysis for this class of algorithms. Algorithm 2 The approximate policy mirror descent (APMD) method Input: initial points π 0 , stepsizes η k ≥ 0 and perturbation τ k ≥ 0. for k = 0, 1, . . . , do π k+1 (·\|s) = arg min p(·\|s)∈∆ \|A\| η k [ Q π k τ k (s, ·), p(·\|s) + h p (s) + τ k D p π 0 (st)] + D p π k (s) , ∀s ∈ S. (3.20) end for Our goal in the remaining part of this subsection is to show that the APMD method, when employed with proper selection of τ k , can achieve a linear rate of convergence for solving general RL problems. Note that in the classic mirror descent method, adding a perturbation term into the objective function usually would not improve its rate of convergence from sublinear to linear. However, the linear rate of convergence in PMD depends on the discount factor rather than the strongly convex modulus of the regularization term, which makes it possible for us to show a linear rate of convergence for the APMD method. First we observe that Lemma 3 can still be applied to the perturbed value functions. The difference between the following result and Lemma 3 exists in that the RHS of (3.21) is no longer nonnegative, i.e., V π τ (s)−V π * τ (s) 0. However, this relation will be approximately satisfied if τ is small enough. Lemma 7 The VI problem in (2.5) satisfies E s∼ν * Q π τ (s, ·), π(·\|s) − π * (·\|s) + h π (s) − h π * (s) + τ [D π π0 (s) − D π * π0 (s)] = E s∼ν * [(1 − γ)(V π τ (s) − V π * τ (s))]. (3.21) Proof. The proof is the same as that for Lemma 3 except that we will apply the performance difference lemma (i.e., Lemma 2) to the perturbed value function V π τ . Next we establish some general convergence properties about the APMD method. Lemma 8 below characterizes the optimal solution of (3.20) (see, e.g., Lemma 3.5 of [13]). Lemma 8 Let π k+1 (·\|s) be defined in (3.20). For any p(·\|s) ∈ ∆ \|A\| , we have η k [ Q π k τ k (s, ·), π k+1 (·\|s) − p(·\|s) + h π + (s) − h p (s)] + η k τ k [D π k+1 π0 (s t ) − D p π0 (s t )] + D π k+1 π k (s) ≤ D p π k (s) − (1 + ητ k )D p π k+1 (s). Lemma 9 below is similar to Lemma 5 for the PMD method. Lemma 9 For any s ∈ S, we have Q π k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + τ k [D π k+1 π0 (s) − D π k π0 (s)] ≥ V π k+1 τ k (s) − V π k τ k (s). (3.22) Proof. By applying Lemma 2 to the perturbed value function V π τ and using an argument similar to (3.10), we can show that V π k+1 τ k (s) − V π k τ k (s) = 1 1−γ E s ′ ∼d π k+1 s Q π k τ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) +h π k+1 (s ′ ) − h π k (s ′ ) + τ k [D π k+1 π0 (s) − D π k π0 (s)] . (3.23) Now we conclude from Lemma 8 with p(·\|s ′ ) = π k (·\|s ′ ) that (3.24) which implies that Q π k τ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + τ k [D π k+1 π0 (s ′ ) − D π k π0 (s ′ )] ≤ − 1 η k [(1 + η k τ k )D π k π k+1 (s ′ ) + D π k+1 π k (s ′ )],E s ′ ∼d π k+1 s Q π k τ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) +τ k [D π k+1 π0 (s ′ ) − D π k π0 (s ′ )] ≤ d π k+1 s (s) [ Q π k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) +τ k [D π k+1 π0 (s) − D π k π0 (s)] ≤ (1 − γ) [ Q π k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) +τ k [D π k+1 π0 (s) − D π k π0 (s)] ,(3.25) where the last inequality follows from the fact that d π k+1 s (s) ≥ (1 − γ) due to the definition of d π k+1 s in (2.1). The result in (3.22) then follows immediately from (3.23) and the above inequality. The following general result holds for different stepsize rules for APMD. Lemma 10 Suppose 1 + η k τ k = 1/γ and τ k ≥ τ k+1 in the APMD method. Then for any k ≥ 0, we have E s∼ν * [V π k+1 τ k+1 (s) − V π * τ k+1 (s) + τ k+1 1−γ D π * π k+1 (s)] ≤ E s∼ν * [γ[V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π * π k (s)] + τ k −τ k+1 1−γ log \|A\|. (3.26) Proof. By Lemma 8 with p = π * , we have η k Q π k τ k (s, ·), π k+1 (·\|s) − π * (·\|s) + h π k+1 (s) − h π * (s) + η k τ k [D π k+1 π0 (s t ) − D π * π0 (s t )] + D π k+1 π k (s) ≤ D π * π k (s) − (1 + η k τ k )D π * π k+1 (s). Moreover, by Lemma 9, Q π k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + τ k [D π k+1 π0 (s t ) − D π k π0 (s t )] ≥ V π k+1 τ k (s) − V π k τ k (s). Combining the above two relations, we obtain η k Q π k τ k (s, ·), π k (·\|s) − π * (·\|s) + h π k (s) − h π * (s) + η k τ k [D π k π0 (s t ) − D π * π0 (s t )] + η k [V π k+1 τ k (s) − V π k τ k (s)] + D π k+1 π k (s) ≤ D π * π k (s) − (1 + η k τ k )D π * π k+1 (s). Taking expectation w.r.t. ν * on both sides of the above inequality and using Lemma 7, we arrive at E s∼ν * [η k (1 − γ)(V π k τ k (s) − V π * τ k (s))] + η k E s∼ν * [V π k+1 τ k (s) − V π k τ k (s)] + E s∼ν * [D π k+1 π k (s)] ≤ E s∼ν * [D π * π k (s) − (1 + η k τ k )D π * π k+1 (s)]. Noting V π k+1 τ k (s) − V π k τ k (s) = V π k+1 τ k (s) − V π * τ k (s) − [V π k τ k (s) − V π * τ k (s)] and rearranging the terms in the above inequality, we have E s∼ν * [η k (V π k+1 τ k (s) − V π * τ k (s)) + (1 + η k τ k )D π * π k+1 (s) + D π k+1 π k (s)] ≤ η k γE s∼ν * [V π k τ k (s) − V π * τ k (s)] + E s∼ν * [D π * π k (s)]. (3.27) Using the above inequality, the assumption τ k ≥ τ k+1 and (3.19), we have E s∼ν * [η k (V π k+1 τ k+1 (s) − V π * τ k+1 (s)) + (1 + η k τ k )D π * π k+1 (s) + D π k+1 π k (s)] ≤ E s∼ν * [η k γ(V π k τ k (s) − V π * τ k (s)) + D π * π k (s)] + η k (τ k −τ k+1 ) 1−γ log \|A\|, (3.28) which implies the result by the assumption 1 + η k τ k = 1/γ. We are now ready to establish the rate of convergence of the APMD method with dynamic stepsize rules to select η k and τ k for solving general RL problems. Theorem 3 Suppose that τ k = τ 0 γ k for some τ 0 ≥ 0 and that 1 + η k τ k = 1/γ for any k ≥ 0 in the APMD method. Then for any k ≥ 0, we have f (π k ) − f (π * ) ≤ γ k f (π 0 ) − f (π * ) + τ 0 2 1−γ + k γ log \|A\| . (3.29) Proof. Applying the result in Lemma 10 recursively, we have (3.18), and that V π0 τ0 (s) = V π0 (s) due to D π0 π0 (s) = 0, we conclude from the previous inequality that E s∼ν * [V π k τ k (s) − V π * τ k (s)] ≤ γ k E s∼ν * [V π0 τ0 (s) − V π * τ0 (s) + τ0 1−γ D π * π0 (s)] + k i=1 (τi−1−τi)γ k−i 1−γ log \|A\|. Noting that V π k τ k (s) ≥ V π k (s), V π * τ k (s) ≤ V π * (s) + τ k 1−γ log \|A\|, and V π * τ0 (s) ≥ V π * (s) due toE s∼ν * [V π k (s) − V π * (s)] ≤ γ k E s∼ν * [V π0 (s) − V π * (s) + τ0 1−γ D π * π0 (s)] + τ k 1−γ + k i=1 (τi−1−τi)γ k−i 1−γ log \|A\|. (3.30) The result in (3.29) immediately follows from the above relation, the definition of f in (1.9), and the selection of τ k . According to (3.29), if τ 0 is a constant, then the rate of convergence of the APMD method is O(kγ k ). If the total number of iterations k is given a priori, we can improve the rate of convergence to O(γ k ) by setting τ 0 = 1/k. Below we propose a different way to specify τ k for the APMD method so that it can achieve this O(γ k ) rate of convergence without fixing k a priori. We first establish a technical result that will also be used later for the analysis of stochastic PMD methods. Lemma 11 Assume that the nonnegative sequences {X k } k≥0 , {Y k } k≥0 and {Z k } k≥0 satisfy X k+1 ≤ γX k + (Y k − Y k+1 ) + Z k . (3.31) Let us denote l = log γ 1 4 . If Y k = Y · 2 −(⌊k/l⌋+1) and Z k = Z · 2 −(⌊k/l⌋+2) for some Y ≥ 0 and Z ≥ 0, then X k ≤ 2 −⌊k/l⌋ (X 0 + Y + 5Z 4(1−γ) ). (3.32) Proof. Let us group the indices {0, . . . , k} intop ≡ ⌊k/l⌋+1 epochs with each of the firstp−1 epochs consisting of l iterations. Let p = 0, . . . ,p be the epoch indices. We first show that for any p = 0, . . . ,p − 1, X pl ≤ 2 −p (X 0 + Y + Z 1−γ ). (3.33) This relation holds obviously for p = 0. Let us assume that (3.33) holds at the beginning of epoch p ad examine the progress made in epoch p. Note that for any indices k = pl, . . . , (p+1)l−1 in epoch p, we have Y k = Y ·2 −(p+1) and Z = Z · 2 −(p+2) . By applying (3.31) recursively, we have X (p+1)l ≤ γ l X pl + Y pl − Y (p+1)l + Z pl l−1 i=0 γ i = γ l X pl + Y (p+1)l + Z pl 1−γ l 1−γ ≤ γ l X pl + Y · 2 −(p+2) + Z·2 −(p+2) 1−γ ≤ 1 4 X pl + Y · 2 −(p+2) + Z·2 −(p+2) 1−γ ≤ 1 4 2 −p (X 0 + Y + Z 1−γ ) + Y · 2 −(p+2) + Z·2 −(p+2) 1−γ ≤ 2 −(p+1) (X 0 + Y + Z 1−γ ), where the second inequality follows from the definition of Z pl and γ l ≥ 0, the third one follows from γ l ≤ 1/4, the fourth one follows by induction hypothesis, and the last one follows by regrouping the terms. Since k = (p−1)l+k (mod l), we have X k ≤ γ k (mod l) X (p−1)l + Z (p−1)l k (mod l)−1 i=0 γ i ≤ 2 −(p−1) (X 0 + Y + Z 1−γ ) + Z·2 −(p+1) 1−γ = 2 −(p−1) (X 0 + Y + 5Z 4(1−γ) ), which implies the result. We are now ready to present a more convenient selection of τ k and η k for the APMD method. Theorem 4 Let us denote l := log γ 1 4 . If τ k = 2 −(⌊k/l⌋+1) and 1 + η k τ k = 1/γ, then f (π k ) − f (π * ) ≤ 2 −⌊k/l⌋ [f(π 0 ) − f (π * ) + 2 log \|A\| 1−γ ]. Proof. By using Lemma 10 and Lemma 11 ( (3.18), and that V π0 τ0 (s) = V π0 (s) due to D π0 π0 (s) = 0, we conclude from the previous inequality and the definition of τ k that with X k = E s∼ν * [V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π * π k (s)] and Y k = τ k 1−γ log \|A\|), we have E s∼ν * [V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π * π k (s)] ≤ 2 −⌊k/l⌋ E s∼ν * [V π k τ0 (s) − V π * τ0 (s) + τ0 1−γ D π * π0 (s)] + log \|A\| 1−γ . Noting that V π k τ k (s) ≥ V π k (s), V π * τ k (s) ≤ V π * (s) + τ k 1−γ log \|A\|, V π * τ0 (s) ≥ V π * (s) due toE s∼ν * [V π k (s) − V π * (s) + τ k 1−γ D π * π k (s)] ≤ 2 −⌊k/l⌋ E s∼ν * [V π k (s) − V π * τ0 (s) + τ0 1−γ D π * π0 (s)] + log \|A\| 1−γ + τ k log \|A\| 1−γ ≤ 2 −⌊k/l⌋ E s∼ν * [V π0 (s) − V π * τ0 (s) + 2 log \|A\| 1−γ . In view of Theorem 4, a policyπ s.t. f (π) − f (π * ) ≤ ǫ will be found in at most O(log(1/ǫ)) epochs and hence at most O(l log(1/ǫ)) = O(log γ (ǫ)) iterations, which matches the one for solving RL problems with strongly convex regularizers. However, for general RL problems, we cannot guarantee the linear convergence of D π * π k+1 (s) since its coefficient τ k will become very small eventually. By using the continuity of the objective function and the compactness of the feasible set, we can possibly show that the solution sequence converges to the true optimal policy asymptotically as the number of iterations increases. On the other hand, the rate of convergence associated with the solution sequence of the PMD method for general RL problems cannot be established unless more structural properties of the RL problems can be further explored. Stochastic Policy Mirror Descent The policy mirror descent methods described in the previous section require the input of the exact action-value functions Q π k . This requirement can hardly be satisfied in practice even for the case when P is given explicitly, since Q π k is defined as an infinite sum. In addition, in RL one does not know the transition dynamics P and thus only stochastic estimators of action-value functions are available. In this section, we propose stochastic versions for the PMD and APMD methods to address these issues. Basic stochastic policy mirror descent In this subsection, we assume that for a given policy π k , there exists a stochastic estimator Q π k ,ξ k s.t. E ξ k [Q π k ,ξ k ] =Q π k , (4.1) E ξ k [ Q π k ,ξ k − Q π k 2 ∞ ] ≤ σ 2 k , (4.2) Q π k − Q π k ∞ ≤ ς k ,(4.3) for some σ k ≥ and ς k ≥ 0, where ξ k denotes the random vector used to generate the stochastic estimator Q π k ,ξ k . Clearly, if σ k = 0, then we have exact information about Q π k . One key insight we have for the stochastic PMD methods is to handle separately the bias term ς k from the overall expected error term σ k , because one can reduce the bias term much faster than the total error. This makes the analysis of the stochastic PMD method considerably different from that of the classic stochastic mirror descent method. While in this section we focus on the convergence analysis of the algorithms, we will show in next section that such separate treatment of bias and total error enables us to substantially improve the sampling complexity for solving RL problems by using policy gradient type methods. The stochastic policy mirror descent (SPMD) is obtained by replacing Q π k in (3.5) with its stochastic estimator Q π k ,ξ k , i.e., π k+1 (·\|s) = arg min p(·\|s)∈∆ \|A\| Φ k (p) := η k [ Q π k ,ξ k (s, ·), p(·\|s) + h p (s)] + D p π k (s) . (4.4) In the sequel, we denote ξ ⌈k⌉ the sequence of random vectors ξ 0 , . . . , ξ k and define δ k := Q π k ,ξ k − Q π k .(4.5) By using the assumptions in (4.1) and (4.3) and the decomposition Q π k ,ξ k (s, ·), π k (·\|s) − π * (·\|s) = Q π k (s, ·), π k (·\|s) − π * (·\|s) + Q π k (s, ·) − Q π k (s, ·), π k (·\|s) − π * (·\|s) + Q π k ,ξ k (s, ·) −Q π k (s, ·), π k (·\|s) − π * (·\|s) , we can see that E ξ k [ Q π k ,ξ k (s, ·), π k (·\|s) − π * (·\|s) \| ξ ⌈ξ k−1 ⌉ ] ≥ Q π k (s, ·), π k (·\|s) − π * (·\|s) − 2ς k . (4.6) Similar to Lemma 5, below we show some general convergence properties about the SPMD method. Unlike PMD, SPMD does not guarantee the non-increasing property of V π k (s) anymore. Lemma 12 For any s ∈ S, we have V π k+1 (s) − V π k (s) ≤ Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + 1 η k D π k+1 π k (s) + η k δ k 2 ∞ 2(1−γ) . (4.7) Proof. Observe that (3.10) still holds, and hence that V π k+1 (s) − V π k (s) = 1 1−γ E s ′ ∼d π k+1 s Q π k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) = 1 1−γ E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) − δ k , π k+1 (·\|s ′ ) − π k (·\|s ′ ) ≤ 1 1−γ E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + 1 2η k π k+1 (·\|s ′ ) − π k (·\|s ′ ) 2 1 + η k δ k 2 ∞ 2 ≤ 1 1−γ E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + 1 η k D π k+1 π k (s ′ ) + η k δ k 2 ∞ 2 ,(4.8) where the first inequality follows from Young's inequality and the second one follows from the strong convexity of D π k π k+1 w.r.t. to · 1 . Moreover, we conclude from Lemma 4 applied to (4.4) with Q π k replaced by Q π k ,ξ k and p(·\|s ′ ) = π k (·\|s ′ ) that Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + 1 η k D π k+1 π k (s ′ ) ≤ − 1 η k [(1 + η k µ)D π k π k+1 (s ′ )] ≤ 0, which implies that E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + 1 η k D π k π k+1 (s ′ ) ≤ d π k+1 s (s) Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + 1 η k D π k+1 π k (s) ≤ (1 − γ) Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + 1 η k D π k+1 π k (s) , where the last inequality follows from the fact that d π k+1 s (s) ≥ 1 − γ due to the definition of d π k+1 s in (2.1). The result in (4.7) then follows immediately from (4.8) and the above inequality. We now establish an important recursion about the SPMD method. Lemma 13 For any k ≥ 0, we have E ξ ⌈k⌉ [f(π k+1 ) − f (π * ) + ( 1 η k + µ)D(π k+1 , π * )] ≤ E ξ ⌈k−1⌉ [γ(f(π k ) − f (π * )) + 1 η k D(π k , π * )] + 2ς k + η k σ 2 k 2(1−γ) . Proof. By applying Lemma 4 to (3.5) (with Q π k replaced by Q π k ,ξ k and p = π * ), we have η k [ Q π k ,ξ k (s, ·), π k+1 (·\|s) − π * (·\|s) + h π k+1 (s) − h π * (s)] + D π k+1 π k (s) ≤ D π * π k (s) − (1 + η k µ)D π * π k+1 (s), which, in view of (4.7), then implies that Q π k ,ξ k (s, ·), π k (·\|s) − π * (·\|s) + h π k (s) − h π * (s) + V π k+1 (s) − V π k (s) ≤ 1 η k D π * π k (s) − ( 1 η k + µ)D π * π k+1 (s) + η k δ k 2 ∞ 2(1−γ) . Taking expectation w.r.t. ξ ⌈k⌉ and ν * on both sides of the above inequality, and using Lemma 3 and the relation in (4.6), we arrive at E s∼ν * ,ξ ⌈k⌉ (1 − γ)(V π k (s) − V π * τ (s)) + V π k+1 (s) − V π k (s) ≤ E s∼ν * ,ξ ⌈k⌉ [ 1 η k D π * π k (s) − ( 1 η k + µ)D π * π k+1 (s)] + 2ς k + η k σ 2 k 2(1−γ) . Noting V π k+1 (s)−V π k (s) = V π k+1 (s)−V π * (s)−[V π k (s)−V π * (s)] , rearranging the terms in the above inequality, and using the definition of f in (1.9), we arrive at the result. We are now ready to establish the convergence rate of the SPMD method. We start with the case when µ > 0 and state a constant stepsize rule which requires both ς k and σ k , k ≥ 0, to be small enough to guarantee the convergence of the SPMD method. Theorem 5 Suppose that η k = η = 1−γ γµ in the SPMD method. If ς k = 2 −(⌊k/l⌋+2) and σ 2 k = 2 −(⌊k/l⌋+2) for any k ≥ 0 with l := log γ (1/4) , then E ξ ⌈k−1⌉ [f(π k ) − f (π * ) + µ 1−γ D(π k , π * )] ≤ 2 −⌊k/l⌋ f (π 0 ) − f (π * ) + 1 1−γ (µ log \|A\| + 5 2 + 5 8γµ ) . (4.9) Proof. By Lemma 13 and the selection of η, we have E ξ ⌈k⌉ [f(π k+1 ) − f (π * ) + µ 1−γ D(π k+1 , π * )] ≤ γ[E ξ ⌈k−1⌉ [f(π k ) − f (π * ) + µ 1−γ D(π k , π * )] + 2ς k + σ 2 k 2γµ , which, in view of Lemma 11 with X k = E ξ ⌈k−1⌉ [f(π k ) − f (π * ) + µ 1−γ D(π k , π * ) and Z k = 2ς k + σ 2 k 2γµ , then implies that E ξ ⌈k−1⌉ [f(π k ) − f (π * ) + µ 1−γ D(π k , π * ) ≤ γ ⌊k/l⌋ f (π 0 ) − f (π * ) + µD(π0,π * ) 1−γ + 5 4 ( 2 1−γ + 1 2γ(1−γ)µ ) ≤ γ ⌊k/l⌋ f (π 0 ) − f (π * ) + 1 1−γ (µ log \|A\| + 5 2 + 5 8γµ ) . We now turn our attention to the convergence properties of the SPMD method for the case when µ = 0. Theorem 6 Suppose that η k = η for any k ≥ 0 in the SPMD method. If ς k ≤ ς and σ k ≤ σ for any k ≥ 0, then we have E ξ ⌈k⌉ ,R [f(π R ) − f (π * )] ≤ γ[f (π0)−f (π * )] (1−γ)k + log \|A\| η(1−γ)k + 2ς 1−γ + ησ 2 2(1−γ) 2 ,(4. 10) where R denotes a random number uniformly distributed between 1 and k. In particular, if the number of iterations k is given a priori and η = ( 2(1−γ) log \|A\| kσ 2 ) 1/2 , then E ξ ⌈k⌉ ,R [f(π R ) − f (π * )] ≤ γ[f (π0)−f (π * )] (1−γ)k + 2ς 1−γ + σ √ 2 log \|A\| (1−γ) 3/2 √ k . (4.11) Proof. By Lemma 13 and the fact that µ = 0, we have E ξ ⌈k⌉ [f(π k+1 ) − f (π * ) + 1 η D(π k+1 , π * )] ≤ E ξ ⌈k−1⌉ [γ(f(π k ) − f (π * )) + 1 η D(π k , π * )] + 2ς k + ησ 2 k 2(1−γ) . Taking the telescopic sum of the above relations, we have (1 − γ) k i=1 E ξ ⌈k⌉ [f(π i ) − f (π * )] ≤ [γ(f(π 0 ) − f (π * )) + 1 η D(π 0 , π * )] + 2kς + kησ 2 2(1−γ) . Dividing both sides by (1 − γ)k and using the definition of R, we obtain the result in (4.10). We add some remarks about the results in Theorem 6. In comparison with the convergence results of SPMD for the case µ > 0, there exist some possible shortcomings for the case when µ = 0. Firstly, one needs to output a randomly selected π R from the trajectory. Secondly, since the first term in (4.11) converges sublinearly, one has to update π k+1 at least O(1/ǫ) times, which may also impact the gradient complexity of computing ∇h π if π k+1 cannot be computed explicitly. We will address these issues by developing the stochastic APMD method in next subsection. Stochastic approximate policy mirror descent The stochastic approximate policy mirror descent (SAPMD) method is obtained by replacing Q π k τ k in (3.20) with its stochastic estimator Q π k ,ξ k τ k . As such, its updating formula is given by π k+1 (·\|s) = arg min p(·\|s)∈∆ \|A\| η k [ Q π k ,ξ k τ k (s, ·), p(·\|s) + h p (s) + τ k D π π0 (s t )] + D p π k (s) . (4.12) With a little abuse of notation, we still denote δ k := Q π k ,ξ k τ k − Q π k τ k and assume that E ξ k [Q π k ,ξ k τ k ] =Q π k τ k , (4.13) E ξ k [ Q π k ,ξ k τ k − Q π k τ k 2 ∞ ] ≤ σ 2 k , (4.14) Q π k τ k − Q π k τ k ∞ ≤ ς k ,(4.15) for some σ k ≥ and ς k ≥ 0. Similarly to (4.6) we have E ξ k [ Q π k ,ξ k τ k (s, ·), π k (·\|s) − π * (·\|s) \| ξ ⌈ξ k−1 ⌉ ] ≥ Q π k τ k (s, ·), π k (·\|s) − π * (·\|s) − 2ς k .V π k+1 τ k (s) − V π k τ k (s) ≤ Q π k ,ξ k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + τ k [D π k+1 π0 (s) − D π * π0 (s)] + 1 η k D π k+1 π k (s) + η k δ k 2 ∞ 2(1−γ) . (4.17) Proof. The proof is similar to the one for Lemma 12 except that we will apply Lemma 2 to the perturbed value functions V π τ k instead of V π . Lemma 15 If 1 + η k τ k = 1/γ and τ k ≥ τ k+1 in the SAPMD method, then for any k ≥ 0, E s∼ν * ,ξ ⌈k⌉ [V π k+1 τ k+1 (s) − V π * τ k+1 (s) + τ k 1−γ D π * π k+1 (s)] ≤ E s∼ν * ,ξ ⌈k−1⌉ [γ[V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π * π k (s)] + τ k −τ k+1 1−γ log \|A\| + 2ς k + σ 2 k 2γτ k . (4.18) Proof. By Lemma 8 with p = π * and Q π k τ k replaced by Q π k ,ξ k τ k , we have Q π k ,ξ k τ k (s, ·), π k+1 (·\|s) − π * (·\|s) + h π k+1 (s) − h π * (s) + τ k [D π k+1 π0 (s) − D π * π0 (s)] + 1 η k D π k+1 π k (s) ≤ 1 η k D π * π k (s) − ( 1 η k + τ k )D π * π k+1 (s), which, in view of (4.17), implies that Q π k ,ξ k τ k (s, ·), π k (·\|s) − π * (·\|s) + h π k (s) − h π * (s) + τ k [D π k π0 (s) − D π * π0 (s)] + V π k+1 τ k (s) − V π k τ k (s) ≤ 1 η k D π * π k (s) − ( 1 η k + τ k )D π * π k+1 (s) + η k δ k 2 ∞ 2(1−γ) . Taking expectation w.r.t. ξ ⌈k⌉ and ν * on both sides of the above inequality, and using Lemma 7 and the relation in (4.16), we arrive at E s∼ν * ,ξ ⌈k⌉ [(1 − γ)(V π k τ k (s) − V π * τ k (s))] + E s∼ν * ,ξ ⌈k⌉ [V π k+1 τ k (s) − V π k τ k (s)] ≤ E s∼ν * ,ξ ⌈k⌉ [ 1 η k D π * π k (s) − ( 1 η k + τ k )D π * π k+1 (s)] + 2ς k + η k σ 2 k 2(1−γ) . Noting V π k+1 τ k (s) − V π k τ k (s) = V π k+1 τ k (s) − V π * τ k (s) − [V π k τ k (s) − V π * τ k (s)] and rearranging the terms in the above inequality, we have E s∼ν * ,ξ ⌈k⌉ [V π k+1 τ k (s) − V π * τ k (s) + ( 1 η k + τ k )D π * π k+1 (s)] ≤ γE s∼ν * ,ξ ⌈k−1⌉ [V π k τ k (s) − V π * τ k (s)] + E s∼ν * ,ξ ⌈k−1⌉ [ 1 η k D π * π k (s)] + 2ζ k + η k σ 2 k 2(1−γ) , which, in view of the assumption τ k ≥ τ k+1 and (3.19), then implies that V π k+1 τ k (s) − V π k τ k (s) ≤ Q π k ,ξ k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + τ k [D π k+1 π0 (s) − D π * π0 (s)] + 1 η k D π k+1 π k (s) + η k δ k 2 ∞ 2(1−γ) . (4.19) The result then immediately follows from the assumption that 1 + η k τ k = 1/γ. We are now ready to establish the convergence of the SAPMD method. E ξ ⌈k−1⌉ [f(π k ) − f (π * )] ≤ 2 −⌊k/l⌋ [f(π 0 ) − f (π * ) + 3 √ log \|A\| (1−γ) √ γ + 5 2(1−γ) ]. (4.20) Proof. By Lemma 15 and the selection of τ k , ς k and σ k , we have E s∼ν * ,ξ ⌈k⌉ [V π k+1 τ k+1 (s) − V π * τ k+1 (s) + τ k 1−γ D π * π k+1 (s)] ≤ E s∼ν * ,ξ ⌈k−1⌉ [γ[V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π * π k (s)]] + τ k −τ k+1 1−γ log \|A\| + (2 + √ log \|A\| 2 √ γ )2 −(⌊k/l⌋+2) . (4.21) Using the above inequality and Lemma 11 ( (3.18), and that V π0 τ0 (s) = V π0 (s) due to D π0 π0 (s) = 0, we conclude from the previous inequality and the definition of τ k that with X k = E s∼ν * ,ξ ⌈k−1⌉ [γ[V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π * π k (s)]], Y k = τ k 1−γ log \|A\| and Z k = (2 + √ log \|A\| 2 √ γ )2 −(⌊k/l⌋+2) ), we conclude E s∼ν * ,ξ ⌈k−1⌉ [V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π * π k (s)] ≤ 2 −⌊k/l⌋ {E s∼ν * [V π k τ0 (s) − V π * τ0 (s) + √ log \|A\| 2(1−γ) √ γ ] + √ log \|A\| (1−γ) √ γ + 5 2(1−γ) + 5 √ log \|A\| 8(1−γ) √ γ } = 2 −⌊k/l⌋ {E s∼ν * [V π k τ0 (s) − V π * τ0 (s)] + 17 √ log \|A\| 8(1−γ) √ γ + 5 2(1−γ) }. Noting that V π k τ k (s) ≥ V π k (s), V π * τ k (s) ≤ V π * (s) + τ k 1−γ log \|A\|, V π * τ0 (s) ≥ V π * (s) due toE s∼ν * ,ξ ⌈k−1⌉ [V π k (s) − V π * (s)] ≤ 2 −⌊k/l⌋ {E s∼ν * [V π0 (s) − V π * (s) + 3 √ log \|A\| (1−γ) √ γ + 5 2(1−γ) }, from which the result immediately follows. A few remarks about the convergence of the SAPMD method are in place. First, in view of Theorem 7, the SAPMD method does not need to randomly output a solution as most existing nonconvex stochastic gradient descent methods did. Instead, the linear rate of convergence in (4.20) has been established for the last iterate π k generated by this algorithm. The convergence for the last iterate indicates that the SAMPD method will continuously improve the policy deployed by the system for implementation and evaluation. This is not the case for the convergence of the average or random iterate, since the average iterate will not be implemented and evaluated, and the convergence of the random iterate does not warrant continuous improvement of the generated policies. Second, both Theorems 5 and 7 allow us to establish some strong large-deviation properties associated with the convergence of SPMD and SAPMD. Let us focus on the SAPMD method. For a given confidence level λ ∈ (0, 1) and accuracy level ǫ > 0, if the number of iterations k satisfies ⌊k/l⌋ ≥ log 2 1 λǫ f (π 0 ) − f (π * ) + 3 √ log \|A\| (1−γ) √ γ + 5 2(1−γ) , then by (4.20) and Markov's inequality, we have Prob{f(π k ) − f (π * ) > ǫ} ≤ 1 ǫ 2 −⌊k/l⌋ [f(π 0 ) − f (π * ) + 3 √ log \|A\| (1−γ) √ γ + 5 2(1−γ) ] ≤ λ. In other words, with probability greater than 1 − λ, we have f (π k ) − f (π * ) ≤ ǫ. On the other hand, it is more difficult to derive a similar large deviation result for SPMD directly applied to unregularized problems (c.f. Theorem 6). Due to the sublinear rate of convergence and random selection of output, we need to run the algorithm for a few times to general several candidate solutions and apply a post-optimization procedure to choose from these candidate solutions in order to improve the the reliability of the algorithm (see Chapter 6 of [13] for more discussions). Stochastic Estimation for Action-value Functions In this section, we discuss the estimation of the action-value functions Q π or Q π τ through two different approaches. In Subsection 5.1, we assume the existence of a generative model for the Markov Chains so that we can estimate value functions by generating multiple independent trajectories starting from an arbitrary pair of state and action. In Subsection 5.2, we consider a more challenging setting where we only have access to a single trajectory observed when the dynamic system runs online. In this case, we employ and enhance the conditional temporal difference (CTD) method recently developed in [12] to estimate value functions. Throughout the section we assume that c(s, a) ≤c, ∀(s, a) ∈ S × A, (5.1) h π (s) ≤h, ∀s ∈ S, π ∈ ∆ \|A\| . (5.2) Multiple independent trajectories In the multiple trajectory setting, starting from state-action pair (s, a) and following policy π k , we can generate M k independent trajectories of length T k , denoted by ζ i k ≡ ζ i k (s, a) := {(s i 0 = s, a i 0 = a); (s i 1 , a i 1 ), . . . , (s i T k −1 , a i T k −1 )}, i = 1, . . . , M k . Let ξ k := {ζ i k (s, a), i = 1, . . . , M k , s ∈ S, a ∈ A} denote all these random variables. We can estimate Q π k in the SPMD method by Q π k ,ξ k (s, a) = 1 M k M k i=1 T k −1 t=0 γ t [c(s i t , a i We can show that Q π k ,ξ k satisfy (4.1)-(4.3) with ς k = (c+h)γ T k 1−γ and σ 2 k = (c+h) 2 (1−γ) 2 γ 2T k + κ(log(\|S\|\|A\|)+1) M k , (5.3) for some absolute constant κ > 0 (see Proposition 7 in the Appendix). By choosing T k and M k properly, we can show the convergence of the SPMD method employed with different stepsize rules as stated in Theorems 5 and 6. Proposition 1 Suppose that η k = 1−γ γµ in the SPMD method. If T k and M k are chosen such that T k ≥ l 2 (⌊k/l⌋ + log 2c +h 1−γ + 2) and M k ≥ (c+h) 2 κ(log(\|S\|\|A\|)+1) (1−γ) 2 2 ⌊k/l⌋+4 with l := log γ (1/4) , then the relation in (4.9) holds. As a consequence, an ǫ-solution of (1.9), i.e., a solutionπ s.t. E[f(π) − f (π * ) + µ 1−γ D(π, π * )] ≤ ǫ, can be found in at most O(log γ ǫ) SPMD iterations. In addition, the total number of samples for (s t , a t ) pairs can be bounded by O( \|S\|\|A\| log \|A\| log(\|S\|\|A\|) log γ (1/2) log γ ǫ µ(1−γ) 3 ǫ ). (5.4) Proof. Using the fact that γ l ≤ 1/4, we can easily check from (5.3) and the selection of T k and M k that (4.1)-(4.3) hold with ς k = 2 −(⌊k/l⌋+2) and σ 2 k = 2 −(⌊k/l⌋+2) . Suppose that an ǫ-solutionπ will be found at thē k iteration. By (4.9), we have ⌊k/l⌋ ≤ log 2 {[f (π 0 ) − f (π * ) + 1 1−γ (µ log \|A\| + 5 2 + 5 8γµ )]ǫ −1 }, which implies that the number of iterations is bounded by O(l⌊k/l⌋) = O(log γ ǫ). Moreover by the definition of T k and M k , the total number of samples is bounded by \|S\|\|A\|l ⌊k/l⌋ p=0 [ l 2 (p + log 2c +h 1−γ + 2) (c+h) 2 (1−γ) 2 2 p+4 ] = O{\|S\|\|A\|l 2 (⌊k/l⌋ + log 2c +h 1−γ ) (c+h) 2 κ(log(\|S\|\|A\|)+1) (1−γ) 2 2 ⌊k/l⌋ } = O( \|S\|\|A\| log \|A\| log(\|S\|\|A\|) log γ (1/2) log γ ǫ µ(1−γ) 3 ǫ ). To the best of our knowledge, this is the first time in the literature that an O(log(1/ǫ)/ǫ) sampling complexity, after disregarding all constant factors, has been obtained for solving RL problems with strongly convex regularizers, even though problem (1.9) is still nonconvex. The previously best-known sampling complexity for RL problems with entropy regularizer wasÕ(\|S\|\|A\| 2 /ǫ 3 ) [20], and the author was not aware of anÕ(1/ǫ) sampling complexity results for any RL problems. Below we discuss the sampling complexities of SPMD and SAPMD for solving RL problems with general convex regularizers. Proposition 2 Consider the general RL problems with µ = 0. Suppose that the number of iterations k is given a priori and η k = ( 2(1−γ) log \|A\| kσ 2 ) 1/2 . If T k ≥ T ≡ log γO( \|S\|\|A\| log \|A\| log γ ǫ (1−γ) 5 ǫ 2 ). (5.5) Proof. We can easily check from (5.3) and the selection of T k and M k that (4.1)-(4.3) holds with ς k = ǫ/3 and σ 2 k = 2( ǫ 2 3 2 + 2(c+h) 2 (1−γ) 2 ). Using these bounds in (4.10), we conclude that an ǫ-solution will be found in at most k = 4[(ǫ/3) 2 +(c+h) 2 /(1−γ) 2 )] log \|A\| (1−γ) 3 (ǫ/3) 2 + γ[f (π0)−f (π * )] (1−γ)(ǫ/3) (5.6) iterations. Moreover, the total number of samples is bounded by \|S\|\|A\|Tk and hence by (5.5). We can also establish the iteration and sampling complexities of the SAPMD method, in which we estimate Q π k τ k by Q π k ,ξ k τ k (s, a) = 1 M k M k i=1 T k −1 t=0 γ t [c(s i t , a i Since τ 0 ≥ τ k , similar to (5.3), we can show that Q π k ,ξ k τ k satisfy (4.13)-(4.15) with ς k = (c+h+τ0 log \|A\|)γ T k 1−γ and σ 2 k = 2(c+h+τ0 log \|A\|) 2 (1−γ) 2 (γ 2T k + κ(log(\|S\|\|A\|)+1) M k ) (5.7) for some absolute constant κ > 0. with l := log γ (1/4) , then the relation in (4.20) holds. As a consequence, an ǫ-solution of (1.9), i.e., a solutionπ s.t. E[f(π) − f (π * )] ≤ ǫ, can be found in at most O(log γ ǫ) SAPMD iterations. In addition, the total number of samples for (s t , a t ) pairs can be bounded by O( \|S\|\|A\| log 2 \|A\| log(\|S\|\|A\|) log γ (1/2) log γ ǫ (1−γ) 4 ǫ 2 ). (5.8) Proof. Using the fact that γ l ≤ 1/4, we can easily check from (5.7) and the selection of T k and M k that (4.13)-(4.15) hold with ς k = 2 −(⌊k/l⌋+2) and σ 2 k = 4 −(⌊k/l⌋+2) . Suppose that an ǫ-solutionπ will be found at thē k iteration. By (4.20), we have ⌊k/l⌋ ≤ log 2 {[f (π 0 ) − f (π * ) + 3 √ log \|A\| (1−γ) √ γ + 5 2(1−γ) ]ǫ −1 }, which implies that the number of iterations is bounded by O(l⌊k/l⌋) = O(log γ ǫ). Moreover by the definition of T k and M k , the number of samples is bounded by \|S\|\|A\|l ⌊k/l⌋+1 p=1 [ l 2 (p + log 2c +h+τ0 log \|A\| 1−γ + 4) (c+h+τ0 log \|A\|) 2 κ(log(\|S\|\|A\|)+1) (1−γ) 2 4 p+3 ] = O{\|S\|\|A\|l 2 (⌊k/l⌋ + log 2c +h+τ0 log \|A\| 1−γ ) (c+h+τ0 log \|A\|) 2 log(\|S\|\|A\|) (1−γ) 2 4 ⌊k/l⌋ } = O( \|S\|\|A\| log 2 \|A\| log(\|S\|\|A\|) log γ (1/2) log γ ǫ (1−γ) 4 ǫ ). To the best of our knowledge, the results in Propositions 2 and 3 appear to be new for policy gradient type methods. The previously best-known sampling complexity for policy gradient methods for RL problems wasÕ(\|S\|\|A\| 2 /ǫ 4 ) (e.g., [20]) although some improvements have been made under certain specific settings (e.g., [25]). Observe that the sampling complexity in (5.16) is slightly better than the one in (5.5) in the logarithmic terms. In fact, one can possibly further improve the dependence of the sampling complexity on γ in (5.5) by a factor of 1/(1 − γ) by allowing a slightly worse iteration complexity than the one in (5.6). This indicates that one needs to carefully consider the tradeoff between iteration and sampling complexities when implementing PMD type algorithms. Conditional temporal difference In this subsection, we enhance a recently developed temporal different (TD) type method, i.e., conditional temporal difference (CTD) method, and use it to estimate the action-value functions in an online manner. We focus on estimating Q π in SPMD since the estimation of Q π τ in SAPMD is similar. For a given policy π, we denote the Bellman operator T π Q(s, a) := c(s, a) + h π (s) + γ s ′ ∈S P(s ′ \|s, a) a ′ ∈A π(a ′ \|s ′ )Q(s ′ , a ′ ). (5.9) The action value function Q π corresponding to policy π satisfies the Bellman equation Q π (s, a) = T π Q π (s, a). (5.10) We also need to define a positive-definite weighting matrix M π ∈ R n×n to define the sampling scheme to evaluate policies using TD-type methods. A natural weighting matrix is the diagonal matrix M π = Diag(ν(π))⊗ Diag(π), where ν(π) is the steady state distribution induced by π and ⊗ denotes the Kronecker product. Assumption 1 We make the following assumptions about policy π: (a) ν(π)(s) ≥ ν for some ν > 0, which holds when the Markov chain employed with policy π has a single ergodic class with unique stationary distribution, i.e., ν(π) = ν(π)P π ; and (b) π is sufficiently random, i.e., π(s, a) ≥ π for some π > 0, which can be enforced, for example, by adding some corresponding constraints through h π . Note that Assumption 1.a) is widely accepted for evaluating policies using TD type methods in the RL literature, and that Assumption 1.b) requires that π assigns a non-zero probability to each action. We will discuss how to possibly relax these assumptions, especially Assumption 1.b) later in Remark 1. In view of Assumption 1 we have M π ≻ 0. With this weighting matrix M π , we define the operator F π as F π (θ) := M π θ − T π θ , where T π is the Bellman operator defined in (5.9). Our goal is to find the root θ * ≡ Q π of F (θ), i.e., F (θ * ) = 0. We can show that F is strongly monotone with strong monotonicity modulus bounded from below by Λ min := (1 − γ)λ min (M π ). Here λ min (A) denotes the smallest eigenvalue of A. It can also be easily seen that F π is Lipschitz continuous with Lipschitz constant bounded by Λmax := (1 − γ)λmax(M π ), where λmax(A) denotes the largest eigenvalue of A. At time instant t ∈ Z + , we define the stochastic operator of F π as F π (θ t , ζ t ) = ( e(s t , a t ), θ t − c(s t , a t ) − h π (s t ) − γ e(s t+1 , a t+1 ), θ t ) e(s t , a t ), where ζ t = (s t , a t , s t+1 , a t+1 ) denotes the state transition steps following policy π and e(s t , a t ) denotes the unit vector. The CTD method uses the stochastic operatorF π (θ t , ζ t ) to update the parameters θ t iteratively as shown in Algorithm 3. It involves two algorithmic parameters: α ≥ 0 determines how often θ t is updated and β t ≥ 0 defines the learning rate. Observe that if α = 0, then CTD reduces to the classic TD learning method. When applying the general convergence results of CTD to our setting, we need to handle the following possible pitfalls. Firstly, current analysis of TD-type methods only provides bounds on E[ θ t − θ * 2 2 ], which gives an upper bound on E[ θ t − Q π 2 ∞ ] and thus the bound on the total expected error (c.f., (4.2)). One needs to develop a tight enough bound on the bias E[θ t ] − θ * ∞ (c.f., (4.3)) to derive the overall best rate of convergence for the SPMD method. Secondly, the selection of α and {β t } that gives the best rate of convergence in terms of E[ θ t − θ * 2 2 ] does not necessarily result in the best rate of convergence for SPMD, since we need to deal with the bias term explicitly. The following result can be shown similarly to Lemma 4.1 of [12]. Lemma 16 Given the single ergodic class Markov chain ζ 1 1 , . . . , ζ α 1 , ζ 2 2 , . . . , ζ α 2 , . . ., there exists a constant C > 0 and ρ ∈ [0, 1) such that for every t, α ∈ Z + with probability 1, F π (θ t ) − E[F π (θ t , ζ α t )\|ζ ⌈t−1⌉ ] 2 ≤ Cρ α θ t − θ * 2 . We can also show that the variance ofF π is bounded as follows. E[ F π (θ t , ζ α t ) − E[F π (θ t , ζ α t )\|ζ ⌈t−1⌉ 2 2 ] ≤ 2(1 + γ) 2 E[ θ t 2 2 ] + 2(c +h) 2 ≤ 4(1 + γ) 2 E[ θ t − θ * 2 2 ] + θ * 2 2 + 2(c +h) 2 . (5.12) The following result has been shown in Proposition 6.2 of [12]. E[ θ t+1 − θ * 2 2 ] ≤ 2(t0+1)(t0+2) θ1−θ * 2 (t+t0)(t+t0+1) + 12tσ 2 F Λ 2 min (t+t0)(t+t0+1) , where σ 2 F := 4(1 + γ) 2 R 2 + θ * 2 2 + 2(c +h) 2 and R 2 : = 8 θ 1 − θ * 2 2 + 3[ θ * 2 2 +2(c+h) 2 ] 4(1+γ) 2 . Moreover, we have E[ θ t − θ * 2 2 ] ≤ R 2 for any t ≥ 1. We now enhance the above result with a bound on the bias term given by E[θ t+1 ] − θ * 2 . The proof of this result is put in the appendix since it is more technical. Lemma 18 Suppose that the algorithmic parameters in CTD are set according to Lemma 17. Then we have E[θ t+1 ] − θ * 2 2 ≤ (t0−1)(t0−2)(t0−3) θ1−θ * 2 2 (t+t0−1)(t+t0−2)(t+t0−3) + 8CR 2 ρ α 3Λmin + C 2 R 2 ρ 2α Λ 2 min . We are now ready to establish the convergence of the SMPD method by using the CTD method to estimate the action-value functions. We focus on the case when µ > 0, and the case for µ = 0 can be shown similarly. Proposition 4 Suppose that η k = 1−γ γµ in the SPMD method. If the initial point of CTD is set to θ 1 = 0 and the number of iterations T and the parameter α in CTD are set to T k = t 0 (3θ2 ⌊k/l⌋+2 ) 2/3 + (4t 2 0θ 2 2 ⌊k/l⌋+2 ) 1/2 + 24σ 2 F Λ −2 min 2 ⌊k/l⌋+2 , (5.14) α k = max{2(⌊ k l ⌋ + 2) log ρ 1 2 + log ρ Λmin 24CR 2 , (⌊ k l ⌋ + 2) log ρ 1 2 + log ρ Λmin 3CR 2 },(5. 15) where l := log γ (1/4) andθ := √ nc +h 1−γ , then the relation in (4.9) holds. As a consequence, an ǫ-solution of (1.9), i.e., a solutionπ s.t. E[f(π) − f (π * ) + µ 1−γ D(π, π * )] ≤ ǫ, can be found in at most O(log γ ǫ) SPMD iterations. In addition, the total number of samples for (s t , a t ) pairs can be bounded by O{(log γ 1 2 )(log 2 1 ǫ )(log ρ Λmin CR 2 )( t0θ 2/3 (µ(1−γ)ǫ) 2/3 + t0θ √ µ(1−γ)ǫ + σ 2 F µ(1−γ)Λ 2 min ǫ )}. (5.16) Proof. Using the fact that γ l ≤ 1/4, we can easily check from Lemma 17, Lemma 18, and the selection of T and α that (4.1)-(4.3) hold with ς k = 2 −(⌊k/l⌋+2) and σ 2 k = 2 −(⌊k/l⌋+2) . Suppose that an ǫ-solutionπ will be found at thek iteration. By (4.9), we have ⌊k/l⌋ ≤ log 2 {[f (π 0 ) − f (π * ) + 1 1−γ (µ log \|A\| + 5 2 + 5 8γµ )]ǫ −1 }, which implies that the number of iterations is bounded by O(l⌊k/l⌋) = O(log γ ǫ). Moreover by the definition of T k and α k , the number of samples is bounded by ⌊k/l⌋ p=0 lα k T k = O{log γ 1 2 ⌊k/l⌋ p=0 (p log ρ 1 2 + log ρ Λmin CR 2 )(t 0θ 2/3 2 2p/3 + t 0θ 2 p/2 + σ 2 F Λ −2 min 2 p )} = O{log γ 1 2 (⌊k/l⌋ log ρ 1 2 + log ρ Λmin CR 2 )(t 0θ 2/3 2 2⌊k/l⌋/3 + t 0θ 2 ⌊k/l⌋/2 + σ 2 F Λ −2 min 2 ⌊k/l⌋ )} = O{log γ 1 2 (log 2 1 ǫ log ρ 1 2 + log ρ Λmin CR 2 )( t0θ 2/3 (µ(1−γ)ǫ) 2/3 + t0θ √ µ(1−γ)ǫ + σ 2 F µ(1−γ)Λ 2 min ǫ )}. The following result shows the convergence properties of the SAMPD method when the action-value function is estimated by using the CTD method. Proof. The proof is similar to that of Proposition 4 except that we will show that (4.1)-(4.3) hold with ς k = 2 −(⌊k/l⌋+2) and σ 2 k = 4 −(⌊k/l⌋+2) . Moreover, we will use (4.20) instead of (4.9) to bound the number of iterations. To the best of our knowledge, the complexity result in (5.16) is new in the RL literature, while the one in (5.18) is new for policy gradient type methods. It seems that this bound significantly improves the previously best-known O(1/ǫ 3 ) sampling complexity result for stochastic policy gradient methods (see [25] and Appendix C of [10] for more explanation). Remark 1 In this subsection we focus on the more restrictive assumption M π ≻ 0 in order to compare our results with the existing ones in the literature. Here we discuss how one can possibly relax this assumption. If ν(π)(s) · π(s, a) = 0 for some (s, a) ∈ S × A, one may define the weighting matrix M π = (1 − λ)Diag(ν(π)) ⊗ Diag(π) + λ n I for some sufficiently small λ ∈ (0, 1) which depends on the target accuracy for solving the RL problem, where n = \|S\|×\|A\|. As a result, the algorithmic frameworks of CTD and SPMD, and their convergence analysis are still applicable to this more general setting. Obviously, the selection of λ will impact the efficiency estimate for policy evaluation. An alternative approach that can relax Assumption 1.b), would be to first run the enhanced CTD method to the following equation V π (s) = a π(a\|s)[c(s, a) + h π (s) + γ s ′ ∈S P(s ′ \|s, a)V π (s ′ )] to evaluate the state-value function V π . Then we estimate the action-value function Q π by using (1.5), i.e., Q π (s, a) = c(s, a) + h π (s) + γ s ′ ∈S P(s ′ \|s, a)V π (s ′ ). In order to use the above identity, we need to define an estimator of P(s ′ \|s, a) by using a uniform policy π 0 (·\|s) := {1/\|A\|, . . . , 1/\|A\|}. The sample size required to estimate the transition kernel from a single trajectory is an active research topic (see [24] and references therein). Current research has been focused only on bounding on the total error for estimating P(s ′ \|s, a) for a given sample size, and there do not exist separate and tighter bounds on the bias for these estimators. Therefore, it is still not evident whether the same sampling complexity bounds in Propositions 4 and 5 can be maintained using this alternative approach to relax the assumption of non-zero probability to each action. Remark 2 For problems of high dimension (i.e., n ≡ \|S\| × \|A\| is large), one often resorts to a parametric approximation of the value function. In this case it is possible to define a more general operator F π (θ) := Φ T M π Φθ − T π Φθ for some feature matrix Φ to evaluate the value functions (see Section 4 of [12] for a discussion about CTD with function approximation). Unless the column space of Φ spans the true value functions, an additional bias term will be introduced into the computation of gradients, resulting into an extra error term in the overall rate of convergence of PMD methods. In other words, these methods can only be guaranteed to converge to a neighborhood of the optimal solution. Nevertheless, the application of function approximation will significantly reduce the dependence of gradient computation on the problem dimension, i.e., from \|S\| × \|A\| to the number of columns of Φ. Efficient Solution for General Subproblems In this section, we study the convergence properties of the PMD methods for the situation where we do not have exact solutions for prox-mapping subprobems. Throughout this section, we assume that h π is differentiable and its gradients are Lipschitz continuous with Lipschitz constant L. We will first review Nesterov's accelerated gradient descent (AGD) method [18], and then discuss the overall gradient complexity of using this method for solving prox-mapping in the PMD methods. We will focus on the stochastic PMD methods since they cover deterministic methods as certain special cases. Review of accelerated gradient descent Let us denote X ≡ ∆ \|A\| and consider the problem of min x∈X {Φ(x) := φ(x) + χ(x)}, (6.1) where φ : X → R is a smooth convex function such that µ φ D x x ′ ≤ φ(x) − [φ(x ′ ) + ∇φ(x ′ ), x − x ′ ] ≤ L φ 2 x − x ′ 1 . Moreover, we assume that χ : X → R satisfies χ(x) − [χ(x ′ ) + χ ′ (x ′ ), x − x ′ ] ≥ µχD x x ′ for some µχ ≥ 0. Given (x t−1 , y t−1 ) ∈ X × X, the accelerated gradient method performs the following updates: x t = (1 − q t )y t−1 + q t x t−1 ,(6. 2) x t = arg min x∈X {r t [ ∇φ(x t ), x + µ φ D x x t + χ(x)] + D x xt−1 }, (6.3) y t = (1 − ρ t )y t−1 + ρ t x t ,(6.4) for some q t ∈ [0, 1], r t ≥ 0, and ρ t ∈ [0, 1] . Below we slightly generalize the convergence results for the AGD method so that they depend on the distance D x x0 rather than Φ(y 0 )−Φ(x) for any x ∈ X. This result better fits our need to analyze the convergence of inexact SPMD and SAPMD methods in the next two subsections. Lemma 19 Let us denote µ Φ := µ φ + µχ and t 0 : = ⌊2 L φ /µ Φ − 1⌋. If ρ t = 2 t+1 t ≤ t 0 µ Φ /L φ o.w. , q t = 2 t+1 t ≤ t 0 √ µΦ/L φ −µΦ/L φ 1−µΦ/L φ o.w. , r t =    t 2L φ t ≤ t 0 1 √ L φ µΦ−µΦ o.w. , then for any x ∈ X, Φ(y t ) − Φ(x) + µ Φ D x xt ≤ ε(t)D x x0 , (6.5) where ε(t) := 2L φ min 1 − µ Φ /L φ t−1 , 2 t(t+1) . (6.6) Proof. Using the discussions in Corollary 3.5 of [13] (and the possible strong convexity of χ), we can check that the conclusions in Theorems 3.6 and 3.7 of [13] hold for the AGD method applied to problem (6.1). It then follows from Theorem 3.6 of [13] that Φ(y t ) − Φ(x) + ρt rt D x * x k ≤ 4L φ t(t+1) D x x0 , ∀t = 1, . . . , t 0 . (6.7) Moreover, it follows from Theorem 3.7 of [13] that for any t ≥ t 0 , Φ(y t ) − Φ(x) + µ Φ D x x k ≤ 1 − µ Φ /L φ t−t0 [Φ(y t0 ) − Φ(x) + µ Φ D x xt 0 ] ≤ 2 1 − µ Φ /L φ t−1 L φ D x x0 , where the last inequality follows from (6.7) (with t = t 0 ) and the facts that ρt rt ≥ ρt 0 rt 0 ≥ µ Φ and 2 t(t+1) = t i=2 (1 − 2 i+1 ) ≤ (1 − µ Φ /L φ ) t−1 for any 2 ≤ t ≤ t 0 . The result then follows by combining these observations. Convergence of inexact SPMD In this subsection, we study the convergence properties of the SPMD method when its subproblems are solved inexactly by using the AGD method (see Algorithm 4). Observe that we use the same initial point π 0 whenever calling the AGD method. To use a dynamic initial point (e.g., v k ) will make the analysis more complicated since we do not have a uniform bound on the KL divergence D π v k for an arbitrary v k . To do so probably will require us to use other distance generating functions than the entropy function. Algorithm 4 The SPMD method with inexact subproblem solutions Input: initial points π 0 = v 0 and stepsizes η k ≥ 0. for k = 0, 1, . . . , do Apply T k AGD iterations (with initial points x 0 = y 0 = π 0 ) to π k+1 (·\|s) = arg min p(·\|s)∈∆ \|A\| Φ k (p) := η k [ Q π k ,ξ k (s, ·), p(·\|s) + h p (s)] + D p v k (s) . (6.8) Set (π k+1 , v k+1 ) = (y T k +1 , x T k +1 ). end for In the sequel, we will denote ε k ≡ ε(T k ) to simplify notations. The following result will take place of Lemma 4 in our convergence analysis. Lemma 20 For any π(·\|s) ∈ X, we have η k [ Q π k ,ξ k (s, ·), π k+1 (·\|s) − π(·\|s) + h π k+1 (s) − h π (s)] + D π k+1 v k (s) + (1 + µη k )D π v k+1 (s) ≤ D π v k (s) + ε k log \|A\|. (6.9) Moreover, we have η k [ Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s)] + D π k+1 v k (s) + (1 + µη k )D π k+1 v k+1 (s) ≤ (ε k + ε k−1 1+µη k−1 ) log \|A\|. (6.10) Proof. It follows from Lemma 19 (with µ Φ = 1 + µη k and L φ = L) that Φ k (π k+1 ) − Φ k (π) + (1 + µη k )D π v k+1 (s) ≤ ǫ k D π π0 (s) ≤ ε k log \|A\|. Using the definition of Φ k , we have η k [ Q π k ,ξ k (s, ·), π k+1 (·\|s) − π(·\|s) + h π k+1 (s) − h π (s)] + D π k+1 v k (s) − D π v k (s) + (1 + µη k )D π v k+1 (s) ≤ ε k log \|A\|, which proves (6.9). Setting π = π k and π = π k+1 respectively, in the above conclusion, we obtain η k [ Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s)] + D π k+1 v k (s) + (1 + µη k )D π k v k+1 (s) ≤ D π k v k (s) + ε k log \|A\|, (1 + µη k )D π k+1 v k+1 (s) ≤ ε k log \|A\|. Then (6.10) follows by combining these two inequalities. Proposition 6 For any s ∈ S, we have V π k+1 (s) − V π k (s) ≤ Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + 1 η k D π k+1 π k (s) + η k δ k 2 ∞ 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+µη k−1 ) log \|A\| − 1 1−γ E s ′ ∼d π k+1 s [ δ k , v k (·\|s ′ ) − π k (·\|s ′ ) ]. (6.11) Proof. Similar to (4.8), we have V π k+1 (s) − V π k (s) = 1 1−γ E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) − δ k , π k+1 (·\|s ′ ) − v k (·\|s ′ ) − δ k , v k (·\|s ′ ) − π k (·\|s ′ ) ≤ 1 1−γ E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + 1 2η k π k+1 (·\|s ′ ) − v k (·\|s ′ ) 2 1 + η k δ k 2 ∞ 2 − δ k , v k (·\|s ′ ) − π k (·\|s ′ ) ≤ 1 1−γ E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + 1 η k D π k+1 v k (s ′ ) + η k δ k 2 ∞ 2 − δ k , v k (·\|s ′ ) − π k (·\|s ′ ) . (6.12) It follows from (6.10) that Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + 1 η k D π k+1 v k (s) + (1 + µη k )D π k+1 v k+1 (s) − (ε k + ε k−1 1+µη k−1 ) log \|A\| ≤ 0, (6.13) which implies that E s ′ ∼d π k+1 s Q π k ,ξ k (s ′ , ·), π k+1 (·\|s ′ ) − π k (·\|s ′ ) + h π k+1 (s ′ ) − h π k (s ′ ) + 1 η k D π k π k+1 (s ′ ) − (ε k + ε k−1 1+µη k−1 ) log \|A\| ≤ d π k+1 s (s) Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + 1 η k D π k+1 π k (s) − (ε k + ε k−1 1+µη k−1 ) log \|A\| ≤ (1 − γ) Q π k ,ξ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + 1 η k D π k+1 π k (s) − (ε k + ε k−1 1+µη k−1 ) log \|A\| , where the last inequality follows from the fact that d π k+1 s (s) ≥ (1 − γ) due to the definition of d π k+1 s in (2.1). The result in (6.11) then follows immediately from (6.12) and the above inequality. We now establish an important recursion about the inexact SPMD method in Algorithm 4. Lemma 21 Suppose that η k = η = 1−γ γµ and ε k ≤ ε k−1 for any k ≥ 0 in the inexact SPMD method, we have E ξ ⌈k⌉ [f(π k+1 ) − f (π * ) + µ 1−γ D(π k+1 , π * )] ≤ γ[E ξ ⌈k−1⌉ [f(π k ) − f (π * ) + µ 1−γ D(π k , π * )] + 2(2−γ)ς k 1−γ + σ 2 k 2γµ + µγ 2 (1+γ) log \|A\|ε k−1 (1−γ) 2 . Proof. By (6.9) (with p = π * ), we have Q π k ,ξ k (s, ·), π k+1 (·\|s) − π * (·\|s) + h π k+1 (s) − h π * (s) + 1 η k D π k+1 v k (s) ≤ 1 η k D π * v k (s) − ( 1 η k + µ)D π * v k+1 (s) + ε k η k log \|A\|, which, in view of (4.7), then implies that Q π k ,ξ k (s, ·), π k (·\|s) − π * (·\|s) + h π k (s) − h π * (s) + V π k+1 (s) − V π k (s) ≤ 1 η k D π * v k (s) − ( 1 η k + µ)D π * v k+1 (s) + η k δ k 2 ∞ 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+µη k−1 ) log \|A\| − 1 1−γ E s ′ ∼d π k+1 s Taking expectation w.r.t. ξ ⌈k⌉ and ν * on both sides of the above inequality, and using Lemma 3 and the relation in (4.6), we arrive at E s∼ν * ,ξ ⌈k⌉ (1 − γ)(V π k (s) − V π * τ (s)) + V π k+1 (s) − V π k (s) ≤ E s∼ν * ,ξ ⌈k⌉ [ 1 η k D π * v k (s) − ( 1 η k + µ)D π * v k+1 (s)] + 2ς k + η k σ 2 k 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+µη k−1 ) log \|A\| + 2 1−γ ς k . Noting V π k+1 (s)−V π k (s) = V π k+1 (s)−V π * (s)−[V π k (s)−V π * (s)] , rearranging the terms in the above inequality, and using the definition of f in (1.9), we arrive at E ξ ⌈k⌉ [f(π k+1 ) − f (π * ) + ( 1 η k + µ)D(v k+1 , π * )] ≤ E ξ ⌈k−1⌉ [γ(f(π k ) − f (π * )) + 1 η k D(v k , π * )] + 2(2−γ)ς k 1−γ + η k σ 2 k 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+µη k−1 ) log \|A\|. The result then follows immediately by the selection of η and the assumption ε k ≤ ε k−1 . We now are now ready to state the convergence rate of the SPMD method with inexact prox-mapping. We focus on the case when µ > 0. Theorem 8 Suppose that η k = η = 1−γ γµ in the inexact SPMD method. If ς k = (1 − γ)2 −(⌊k/l⌋+2) , σ 2 k = 2 −(⌊k/l⌋+2) and ε k = (1 − γ) 2 2 −(⌊(k+1)/l⌋+2) for any k ≥ 0 with l := log γ (1/4) , then E ξ ⌈k−1⌉ [f(π k ) − f (π * ) + µ 1−γ D(π k , π * )] ≤ 2 −⌊k/l⌋ f (π 0 ) − f (π * ) + 1 1−γ (µ log \|A\| + 5(2−γ) 2 + 5 8γµ + 5µγ 2 (1+γ) log \|A\| 4 ) . (6.14) Proof. The result follows as an immediate consequence of Proposition 21 and Lemma 11. In view of Theorem 8, the inexact solutions of the subproblems barely affect the iteration and sampling complexities of the SPMD method as long as ε k ≤ (1−γ) 2 2 −(⌊(k+1)/l⌋+2) . Notice that an ǫ-solution of problem (1.9), i.e., a solutionπ s.t. E[f(π) − f (π * )] ≤ ǫ, can be found at thek-th iteration with ⌊k/l⌋ ≤ log 2 {ǫ −1 [f(π 0 ) − f (π * ) + 1 1−γ (4µ log \|A\| + 5 + 5 8γµ )]}. Also observe that the condition number of the subproblem is given by Lη k µη k +1 = L(1−γ) µ . Combining these observations with Lemma 19, we conclude that the total number of gradient computations of h can be bounded by l ⌊k/l⌋ p=0 Lη µη+1 log(Lη/ε k ) = l ⌊k/l⌋ p=0 L(1−γ) µ log 4L2 p+1 γ(1−γ)µ = O{l(⌊k/l⌋) 2 L(1−γ) µ log L γ(1−γ)µ } = O (log γ 1 4 )(log 2 1 ǫ ) L(1−γ) µ (log L γ(1−γ)µ ) . Convergence of inexact SAPMD In this subsection, we study the convergence properties of the SAPMD method when its subproblems are solved inexactly by using the AGD method (see Algorithm 5). Algorithm 5 The Inexact SAPMD method Input: initial points π 0 = v 0 , stepsizes η k ≥ 0, and regularization parameters τ k ≥ 0. for k = 0, 1, . . . , do Apply T k AGD iterations (with initial points x 0 = y 0 = π 0 ) to π k+1 (·\|s) = arg min p(·\|s)∈∆ \|A\| Φ k (p) := η k [ Q π k ,ξ k τ k (s, ·), p(·\|s) + h p (s) + τ k D p π 0 (s)] + D p v k (s) . (6.15) Set (π k+1 , v k+1 ) = (y T k +1 , x T k +1 ). end for In the sequel, we will still denote ε k ≡ ε(T k ) to simplify notations. The following result has the same role as Lemma 20 in our convergence analysis. Lemma 22 For any π(·\|s) ∈ X, we have η k [ Q π k ,ξ k τ k (s, ·), π k+1 (·\|s) − π(·\|s) + h π k+1 (s) − h π (s) + τ k (D π k+1 π0 (s) − D π π0 (s))] + D π k+1 v k (s) + (1 + τ k η k )D π v k+1 (s) ≤ D π v k (s) + ε k log \|A\|. (6.16) Moreover, we have η k [ Q π k ,ξ k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + τ k (D π k+1 π0 (s) − D π k π0 (s))] + D π k+1 v k (s) + (1 + τ k η k )D π k+1 v k+1 (s) ≤ (ε k + ε k−1 1+τ k−1 η k−1 ) log \|A\|. (6.17) Proof. The proof is the same as that for Lemma 20 except that Lemma 19 will be applied to problem (6.15) (with µ Φ = 1 + τ k η k and L φ = L). Lemma 23 For any s ∈ S, we have V π k+1 τ k (s) − V π k τ k (s) ≤ Q π k ,ξ k τ k (s, ·), π k+1 (·\|s) − π k (·\|s) + h π k+1 (s) − h π k (s) + τ k (D π k+1 π0 (s) − D π π k (s)) + 1 η k D π k+1 π k (s) + η k δ k 2 ∞ 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+τ k−1 η k−1 ) log \|A\| − 1 1−γ E s ′ ∼d π k+1 s [ δ k , v k (·\|s ′ ) − π k (·\|s ′ ) ]. (6.18) Proof. The proof is similar to that for Lemma 6 with the following two exceptions: (a) we will apply Lemma 2 (i.e., the performance difference lemma) to the perturbed value functions V π τ k instead of V π to obtain a result similar to (6.12); and (b) we will use use (6.17) in place of (6.10) to derive a bound similar to (6.13). Lemma 24 Suppose that 1 + η k τ k = 1/γ and ε k ≤ ε k−1 in the SAPMD method. Then for any k ≥ 0, we have Proof. By (6.16) (with p = π * ), we have Q π k ,ξ k τ k (s, ·), π k+1 (·\|s) − π * (·\|s) + h π k+1 (s) − h π * (s) E s∼ν * ,ξ ⌈k⌉ [V π k+1 τ k+1 (s) − V π * τ k+1 (s) + τ k 1−γ D π * π k+1 (s)] ≤ E s∼ν * ,ξ ⌈k−1⌉ [γ[V π k τ k (s) − V π * τ k (s) + τ k 1−γ D π + τ k [D π k+1 π0 (s t ) − D π π0 (s t )] + 1 η k D π k+1 π k (s) ≤ 1 η k D π * v k (s) − ( 1 η k + τ k )D π * v k+1 (s) + ε k η k log \|A\|, which, in view of (6.18), implies that Q π k ,ξ k τ k (s, ·), π k (·\|s) − π * (·\|s) + h π k (s) − h π * (s) + τ k [D π k π0 (s t ) − D π * π0 (s t )] + V π k+1 τ k (s) − V π k τ k (s) ≤ 1 η k D π * v k (s) − ( 1 η k + τ k )D π * v k+1 (s) + η k δ k 2 ∞ 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+τ k−1 η k−1 ) log \|A\| − 1 1−γ E s ′ ∼d π k+1 s [ δ k , v k (·\|s ′ ) − π k (·\|s ′ ) ]. Taking expectation w.r.t. ξ ⌈k⌉ and ν * on both sides of the above inequality, and using Lemma 3 (with h π replaced by h π + τ k D π π0 (s t ) and Q π replaced by Q π τ ) and the relation in (4.6), we arrive at E s∼ν * ,ξ ⌈k⌉ [(1 − γ)(V π k τ k (s) − V π * τ k (s))] + E s∼ν * ,ξ ⌈k⌉ [V π k+1 τ k (s) − V π k τ k (s)] ≤ E s∼ν * ,ξ ⌈k⌉ [ 1 η k D π * π k (s) − ( 1 η k + τ k )D π * π k+1 (s)] + 2ς k + η k σ 2 k 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+τ k−1 η k−1 ) log \|A\| + 2ς k 1−γ . Noting V π k+1 τ k (s) − V π k τ k (s) = V π k+1 τ k (s) − V π * τ k (s) − [V π k τ k (s) − V π * τ k (s)] and rearranging the terms in the above inequality, we have E s∼ν * ,ξ ⌈k⌉ [V π k+1 τ k (s) − V π * τ k (s) + ( 1 η k + τ k )D π * v k+1 (s)] ≤ γE s∼ν * ,ξ ⌈k−1⌉ [V π k τ k (s) − V π * τ k (s)] + E s∼ν * ,ξ ⌈k−1⌉ [ 1 η k D π * v k (s)] + 2(2−γ)ζ k 1−γ + η k σ 2 k 2(1−γ) + γ (1−γ)η k (ε k + ε k−1 1+τ k−1 η k−1 ) log \|A\|. (6.20) The result then follows from 1 + η k τ k = 1/γ, the assumptions τ k ≥ τ k+1 , ε k ≤ ε k−1 and (3.19). Theorem 9 Suppose that η k = 1−γ γτ k in the SAPMD method. If τ k = 1 √ γ log \|A\| 2 −(⌊k/l⌋+1) , ς k = 2 −(⌊k/l⌋+2) , σ 2 k = 4 −(⌊k/l⌋+2) , and ε k = (1−γ) 2 2γ 2 (1+γ) with l := log γ (1/4) , then E ξ ⌈k−1⌉ [f(π k ) − f (π * )] ≤ 2 −⌊k/l⌋ [f(π 0 ) − f (π * ) + 1 1−γ ( 3 √ log \|A\| √ γ + 5(2−γ) 2 + 5 √ log \|A\| 4 √ γ ) ]. (6.21) Proof. The result follows as an immediate consequence of Lemma 24, Lemma 11, and an argument similar to the one to prove Theorem 7. In view of Theorem 9, the inexact solution of the subproblem barely affect the iteration and sampling complexities of the SAPMD method as long as ε k ≤ (1−γ) 2 2γ 2 (1+γ) . Notice that an ǫ-solution of problem (1.9), i.e., a solutionπ s.t. E[f(π) − f (π * )] ≤ ǫ, can be found at thek-th iteration with ⌊k/l⌋ ≤ log 2 {ǫ −1 [f(π 0 ) − f (π * ) + 5 1−γ ( √ log \|A\| √ γ + 1)]}. Also observe that the condition number of the subproblem is given by Lη k τ k η k +1 = L(1−γ) τ k = (1 − γ)L γ log \|A\|2 ⌊k/l⌋+1 . Combining these observations with Lemma 19, we conclude that the total number of gradient computations of h can be bounded by Concluding Remarks In this paper, we present the policy mirror descent (PMD) method and show that it can achieve the linear and sublinear rate of convergence for RL problems with strongly convex or general convex regularizers, respectively. We then present a more general form of the PMD method, referred to as the approximate policy mirror descent (APMD) method, obtained by adding adaptive perturbations to the action-value functions and show that it can achieve the linear convergence rate for RL problems with general convex regularizers. We develop the stochastic PMD and APMD methods and derive general conditions on the bias and overall expected error to guarantee the convergence of these methods. Using these conditions, we establish new sampling complexity bounds of RL problems by using two different sampling schemes, i.e., either using a straightforward generative model or a more involved conditional temporal different method. The latter setting requires us to establish a bound on the bias for estimating action-value functions, which might be of independent interest. Finally, we establish the conditions on the accuracy required for the prox-mapping subproblems in these PMD type methods, as well as the overall complexity of computing the gradients of the regularizers. In the future, it will be interesting to study how to incorporate exploration into policy mirror descent to handle rarely visited states and actions. Moreover, since this paper focuses on the theoretical studies, it will be also rewarding to derive simplified PMD algorithms and conduct numerical experiments to demonstrate possible advantages of the proposed algorithms. Proof. By the property of sub-exponential random variables (Section 2.7 of [?]), we know that Y i = X i −E [X i ] is also sub-exponential with Y i ψ1 ≤ C 1 X i ψ1 ≤ C 1 σ for some absolute constant C 1 > 0. Hence by Proposition 2.7.1 of [?], there exists an absolute constant C > 0 such that E[exp(λY i )] ≤ exp(C 2 σ 2 λ 2 ), ∀\|λ\| ≤ 1/(Cσ). Using the previous observation, we have exp(E[λ max i Y i ]) ≤ E[exp(λ max i Y i )] ≤ E[ n i=1 exp(λY i )] ≤ n exp(C 2 σ 2 λ 2 ), ∀\|λ\| ≤ 1 Cσ , which implies E[max i Y i ] ≤ log n/λ + C 2 σ 2 λ, ∀\|λ\| ≤ 1/(Cσ). Choosing λ = 1/(Cσ), we obtain E [max i Y i ] ≤ Cσ(log n+1). By combining this relation with the definition of Y i , we conclude that E[max i X i ] ≤ E[max i Y i ]+v ≤ Cσ(log n + 1) + v. and Lemma 15 below show the improvement for each SAPMD iteration. Lemma 14 For any k ≥ 0, we have Theorem 7 7Suppose that η k = 1−γ γτ k in the SAPMD method. If τ k = 1 √ γ log \|A\| 2 −(⌊k/l⌋+1) , ς k = 2 −(⌊k/l⌋+2) , and σ 2 k = 4 −(⌊k/l⌋+2) with l := log γ (1/4) , then c+h) and M k = 1, then an ǫ-solution of problem of (1.9), i.e., a solutionπ s.t. E[f(π) − f (π * )] ≤ ǫ, can be found in at most O(log \|A\|/[(1 − γ) 5 ǫ 2 ]) SPMD iterations.In addition, the total number of state-action samples can be bounded by Proposition 3 3Suppose that η k = 1−γ γτ k and τ k = 1 √ γ log \|A\| 2 −(⌊k/l⌋+1) in the SAPMD method.If T k and M and M k ≥ (c+h+τ0 log \|A\|) 2 κ(log(\|S\|\|A\|)+1) Algorithm 3 3Conditional Temporal Difference (CTD) for evaluating policy π Let θ 1 , the nonnegative parameters α and {βt} be given. for t = 1, . . . , T do Collect α state transition steps without updating {θt}, denoted as {ζ 1 t , ζ 2 t , . . . , ζ α t }. Set θ t+1 = θt − βtF π (θt, Proposition 5 2 F 52Suppose that η k = 1−γ γτ k and τ k = 1 √ γ log \|A\| 2 −(⌊k/l⌋+1) in the SAPMD method. If the initial point of CTD is set to θ 1 = 0, the number of iterations T is set toT k = t 0 (3θ2 ⌊k/l⌋+2 ) parameter α inCTD is set to (5.15), where l := log γ (1/4) andθ :=c +h+τ0 log \|A\| 1−γ , then the relation in(4.20) holds. As a consequence, an ǫ-solution of (1.9), i.e., a solutionπ s.t. E[f(π) − f (π * )] ≤ ǫ, can be found in at most O(log γ ǫ) SPMD iterations. In addition, the total number of samples for (s t , a t ) pairs can be bounded by O{(log γ 1 2 )(log 2 1 ǫ )(log ρ Λmin CR 2 )( t0θ (1−γ)ǫ + σ (1−γ) 2 Λ 2 min ǫ 2 )}. (5.18) η k +1 log(Lη k /ε k ) = l ⌊k/l⌋ p=0 Lη k τ k η k +1 log L(1−γ) 3 2γ 2 (1+γ)τ k = O{l⌊k/l⌋[(1 − γ)L] 1/2 2 ⌊k/l⌋/2 log L( t ) + h π k (s i t )]. t ) + h π k (s i t ) + τ k D π k π0 (s i t )]. [ δ k , v k (·\|s ′ ) − π k (·\|s ′ ) ]. Acknowledgement: The author appreciates very much Caleb Ju, Sajad Khodaddadian, Tianjiao Li, Yan Li and two anonymous reviewers for their careful reading and a few suggested corrections for earlier versions of this paper.Appendix A: Concentration Bounds for l∞-bounded NoiseWe first show how to bound the expectation of the maximum for a finite number of sub-exponential variables.Lemma25 Let X ψ1 := inf{t > 0 : exp(\|X\|/t) ≤ exp(2)} denote the sub-exponential norm of X. For a given sequence of sub-exponential variables {X i } n i=1 with E[X i ] ≤ v and X i ψ1 ≤ σ, we havewhere C denotes an absolute constant.where κ > 0 denotes an absolute constant.Proof. To proceed, we denote δ k s,a := Q π k ,ξ k (s, a) − Q π k (s, a), and henceNote that by definition, for each (s, a) pair, we have M k independent trajectories of length T k starting from (s, a). Let us denote Z i :Since each Z i − Q π k (s, a) is independent of each other, it is immediate to see that Ys,a := (δ k s,a ) 2 is a subexponential with Ys,a ψ1 ≤ (c+h)2(1−γ) 2 M k . Also note thatThus in view of Lemma 25, withAppendix B: Bias for Conditional Temporal Difference MethodsProof of Lemma 18.Proof. For simplicity, let us denoteθ t ≡ E[θ t ], ζ t ≡ (ζ 1 t , . . . , ζ α t ) and ζ ⌈t⌉ = (ζ 1 , . . . , ζ t ). Also let us denoteIt follows from Jensen's ienquality and Lemma 17 thatAlso by Jensen's inequality, Lemma 16 and Lemma 17, we haveNow conditional on ζ ⌈t−1⌉ , taking expectation w.r.t. ζ t on (5.11), we haveTaking further expectation w.r.t. ζ ⌈t−1⌉ and using the linearity of F , we haveθ t+1 =θ t − β t F π (θ t ) + β tδ F t , which impliesThe above inequality, together with (7.1), (7.2) and the facts thatthen imply thatwhere the last inequality follows fromdue to the selection of β t in (5.13). 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[ "Object Detection in Optical Remote Sensing Images: A Survey and A New Benchmark", "Object Detection in Optical Remote Sensing Images: A Survey and A New Benchmark" ]	[ "Ke Li \nZhengzhou Institute of Surveying and Mapping\n450052ZhengzhouChina\n", "Gang Wan \nZhengzhou Institute of Surveying and Mapping\n450052ZhengzhouChina\n", "Liqiu Meng \nSchool of Automation\nNorthwestern Polytechnical University\n710072Xi'anChina\n\nDepartment of Cartography\nTechnical University of Munich\nArcisstr.2180333MunichGermany\n", "Junwei Han \nSchool of Automation\nNorthwestern Polytechnical University\n710072Xi'anChina\n" ]	[ "Zhengzhou Institute of Surveying and Mapping\n450052ZhengzhouChina", "Zhengzhou Institute of Surveying and Mapping\n450052ZhengzhouChina", "School of Automation\nNorthwestern Polytechnical University\n710072Xi'anChina", "Department of Cartography\nTechnical University of Munich\nArcisstr.2180333MunichGermany", "School of Automation\nNorthwestern Polytechnical University\n710072Xi'anChina" ]	[]	Substantial efforts have been devoted more recently to presenting various methods for object detection in optical remote sensing images. However, the current survey of datasets and deep learning based methods for object detection in optical remote sensing images is not adequate. Moreover, most of the existing datasets have some shortcomings, for example, the numbers of images and object categories are small scale, and the image diversity and variations are insufficient. These limitations greatly affect the development of deep learning based object detection methods. In the paper, we provide a comprehensive review of the recent deep learning based object detection progress in both the computer vision and earth observation communities. Then, we propose a large-scale, publicly available benchmark for object DetectIon in Optical Remote sensing images, which we name as DIOR. The dataset contains 23463 images and 192472 instances, covering 20 object classes. The proposed DIOR dataset 1) is large-scale on the object categories, on the object instance number, and on the total image number; 2) has a large range of object size variations, not only in terms of spatial resolutions, but also in the aspect of inter-and intra-class size variability across objects; 3) holds big variations as the images are obtained with different imaging conditions, weathers, seasons, and image quality; and 4) has high inter-class similarity and intra-class diversity. The proposed benchmark can help the researchers to develop and validate their data-driven methods. Finally, we evaluate several state-of-theart approaches on our DIOR dataset to establish a baseline for future research.	10.1016/j.isprsjprs.2019.11.023	[ "https://export.arxiv.org/pdf/1909.00133v2.pdf" ]	202,541,171	1909.00133	2dd479af74395f5731db202934c9cb02fae6e97b	Object Detection in Optical Remote Sensing Images: A Survey and A New Benchmark Ke Li Zhengzhou Institute of Surveying and Mapping 450052ZhengzhouChina Gang Wan Zhengzhou Institute of Surveying and Mapping 450052ZhengzhouChina Liqiu Meng School of Automation Northwestern Polytechnical University 710072Xi'anChina Department of Cartography Technical University of Munich Arcisstr.2180333MunichGermany Junwei Han School of Automation Northwestern Polytechnical University 710072Xi'anChina Object Detection in Optical Remote Sensing Images: A Survey and A New Benchmark 1Object detectionDeep learningConvolutional Neural Network (CNN)Benchmark DatasetOptical remote sensing images 1 Substantial efforts have been devoted more recently to presenting various methods for object detection in optical remote sensing images. However, the current survey of datasets and deep learning based methods for object detection in optical remote sensing images is not adequate. Moreover, most of the existing datasets have some shortcomings, for example, the numbers of images and object categories are small scale, and the image diversity and variations are insufficient. These limitations greatly affect the development of deep learning based object detection methods. In the paper, we provide a comprehensive review of the recent deep learning based object detection progress in both the computer vision and earth observation communities. Then, we propose a large-scale, publicly available benchmark for object DetectIon in Optical Remote sensing images, which we name as DIOR. The dataset contains 23463 images and 192472 instances, covering 20 object classes. The proposed DIOR dataset 1) is large-scale on the object categories, on the object instance number, and on the total image number; 2) has a large range of object size variations, not only in terms of spatial resolutions, but also in the aspect of inter-and intra-class size variability across objects; 3) holds big variations as the images are obtained with different imaging conditions, weathers, seasons, and image quality; and 4) has high inter-class similarity and intra-class diversity. The proposed benchmark can help the researchers to develop and validate their data-driven methods. Finally, we evaluate several state-of-theart approaches on our DIOR dataset to establish a baseline for future research. Introduction The rapid development of remote sensing techniques has significantly increased the quantity and quality of remote sensing images available to characterize various objects on the earth surface, such as airports, airplanes, buildings, etc. This naturally brings a strong requirement for intelligent earth observation through automatic analysis and understanding of satellite or aerial images. Object detection plays a crucial role in image interpretation and also is very important for a wide scope of applications, such as intelligent monitoring, urban planning, precision agriculture, and geographic information system (GIS) updating. Driven by this requirement, significant efforts have been made in the past few years to develop a variety of methods for object detection in optical remote sensing images (Aksoy, 2014;Bai et al., 2014;Cheng et al., 2013a;Cheng and Han, 2016;Cheng et al., 2013b;Cheng et al., 2014;Cheng et al., 2019;Cheng et al., 2016a;Das et al., 2011;Han et al., 2015;Han et al., 2014;Li et al., 2018;Long et al., 2017;Tang et al., 2017b;Yang et al., 2017;Zhang et al., 2016;Zhang et al., 2017;Zhou et al., 2016). More recently, deep learning based algorithms have been dominating the top accuracy benchmarks for various visual recognition tasks Cheng et al., 2018a;Clément et al., 2013;Ding et al., 2017;Hinton et al., 2012;Hou et al., 2017;Krizhevsky et al., 2012;Mikolov et al., 2012;Tian et al., 2017;Tompson et al., 2014;Wei et al., 2018) because of their powerful feature representation capabilities. Benefiting from this and some publicly available natural image datasets such as Microsoft Common Objects in Context (MSCOCO) (Lin et al., 2014) and PASCAL Visual Object Classes (VOC) (Everingham et al., 2010), a number of deep learning based object detection approaches have achieved great success in natural scene images (Agarwal et al., 2018;Dai et al., 2016;Girshick, 2015;Girshick et al., 2014;Han et al., 2018;Liu et al., 2018a;Liu et al., 2016a;Redmon et al., 2016;Redmon and Farhadi, 2017;Ren et al., 2017). However, despite the significant success achieved in natural images, it is difficult to straight-forward transfer deep learning based object detection methods to optical remote sensing images. As we know, high-quality and large-scale datasets are greatly important for training deep learning based object detection methods. However, the difference between remote sensing images and natural scene images is significant. As shown in Fig. 1, the remote sensing images generally capture the roof information of the geospatial objects, whereas the natural scene images usually capture the profile information of the objects. Therefore, it is not surprising that the object detectors learned from natural scene images are not easily transferable to remote sensing images. Although several popular object detection datasets, such as NWPU VHR-10 (Cheng et al., 2016a), UCAS-AOD (Zhu et al., 2015a), COWC (Mundhenk et al., 2016), and DOTA (Xia et al., 2018), are proposed in the earth observation community, they are still far from satisfying the requirements of deep learning algorithms. To date, significant efforts (Cheng and Han, 2016;Cheng et al., 2016a;Das et al., 2011;Han et al., 2015;Li et al., 2018;Razakarivony and Jurie, 2015;Tang et al., 2017b;Xia et al., 2018;Yokoya and Iwasaki, 2015;Zhang et al., 2016;Zhu et al., 2017) have been made for object detection in remote sensing images. However, the current survey of the literatures concerning the datasets and deep learning based object detection methods is still not adequate. Moreover, most of the existing publicly available datasets have some shortcomings, for example, the numbers of images and object categories are small scale, and the image diversity and variations are also insufficient. These limitations greatly block the development of deep learning based object detection methods. In order to address the aforementioned problems, we attempt to comprehensively review the recent progress of deep learning based object detection methods. Then, we propose a large-scale, publicly available benchmark for object DetectIon in Optical Remote sensing images, which we name as DIOR. Our proposed dataset consists of 23463 images covered by 20 object categories and each category contains about 1200 images. We highlight four key characteristics of the proposed DIOR dataset when comparing it with other existing object detection datasets. First, the numbers of total images, object categories, and object instances are large-scale. Second, the objects have a large range of size variations, not only in terms of spatial resolutions, but also in the aspect of inter-and intra-class size variability across objects. Third, our dataset holds large variations because the images are obtained with different imaging conditions, weathers, seasons, and image quality. Fourth, it possesses high inter-class similarity and intraclass diversity. Fig. 2 shows some example images and their annotations from our proposed DIOR dataset. Our main contributions are summarized as follows: 1) Comprehensive survey of deep learning based object detection progress. We review the recent progress of existing datasets and representative deep learning based methods for object detection in both the computer vision and earth observation communities, which covers more than 110 papers. 2) Creation of large-scale benchmark dataset. This paper proposes a large-scale, publicly available dataset for object detection in optical remote sensing images. The proposed DIOR dataset, to our best knowledge, is the largest scale on both the number of object categories and the total number of images. The dataset enables the community to validate and develop data-driven object detection methods. 3) Performance benchmarking on the proposed DIOR dataset. We benchmark several representative deep learning based object detection methods on our DIOR dataset to provide an overview of the state-of-the-art performance for future research work. The remainder of this paper is organized as follows. Sections 2-3 review the recent object detection progresses of benchmark datasets and deep learning methods in computer vision and earth observation community, respectively. Section 4 describes the proposed DIOR dataset in detail. Section 5 benchmarks several representative deep learning based object detection methods on the proposed dataset. Finally, Section 6 concludes this paper. Review on Object Detection in Computer Vision Community With the emergence of a variety of deep learning models, especially Convolutional Neural Networks (CNN), and their great success on image classification Krizhevsky et al., 2012;Luan et al., 2018;Simonyan and Zisserman, 2015;Szegedy et al., 2015), numerous deep learning based object detection frameworks have been proposed in the computer vision community. Therefore, we will first provide a systematic survey of the references about the datasets as well as deep learning based methods for the task of object detection in natural scene images. Object Detection Datasets of Natural Scene Images Large-scale and high-quality datasets are very important for boosting object detection performance, especially for deep learning based methods. The PASCAL VOC (Everingham et al., 2010), MSCOCO (Lin et al., 2014), and ImageNet object detection dataset (Deng et al., 2009) are three widely used datasets for object detection in natural scene images. These datasets are briefly reviewed as follows. 1) PASCAL VOC Dataset. The PASCAL VOC 2007 (Everingham et al., 2010) and VOC 2012 (Everingham et al., 2015) are two most-used datasets for natural scene image object detection. Both of them contain 20 object classes, but with different image numbers. Specifically, the PASCAL VOC 2007 dataset contains a total of 9963 images with 5011 images for training and 4952 images for testing. The PASCAL VOC 2012 dataset extends the PASCAL VOC 2007 dataset, resulting in a larger scale dataset that consists of 11540 images for training and 10991 images for testing. 2) MSCOCO Dataset. The MSCOCO dataset was proposed by Microsoft in 2014 (Lin et al., 2014). The scale of MSCOCO dataset is much larger than the PASCAL VOC dataset on both the number of object categories and object instances. Specifically, the dataset consists of more than 200000 images covered by 80 object categories. The dataset is further divided into three subsets: training set, validation set and testing set, which contain about 80k, 40k, and 80k images, respectively. 3) ImageNet Object Detection Dataset. This dataset was released in 2013 (Deng et al., 2009), which has the most object categories and the largest number of images among all object detection datasets. Specifically, this dataset includes 200 object classes and more than 500000 images, with 456567 images for training, 20121 images for validation, and 40152 images for testing, respectively. Deep Learning Based Object Detection Methods in Computer Vision Community Recently, a number of deep learning based object detection methods have been proposed, which significantly improve the performance of object detection. Generally, the existing deep learning methods designed for object detection can be divided into two streams on the basis of whether or not generating region proposals. They are region proposal-based methods and regression-based methods. Region Proposal-based Methods In the past few years, region proposal-based object detection methods have achieved great success in natural scene images (Dai et al., 2016;Girshick, 2015;Girshick et al., 2014;He et al., 2017;He et al., 2014;Lin et al., 2017b;Ren et al., 2017). This kind of approaches divides the framework of object detection into two stages. The first stage focuses on generating a series of candidate region proposals that may contain objects. The second stage aims to classify the candidate region proposals obtained from the first stage into object classes or background and further fine-tune the coordinates of the bounding boxes. The Region-based CNN (R-CNN) proposed by Girshick et al. (Girshick et al., 2014) is one of the most famous approaches in various region proposal-based methods. It is the representative work to adopt the CNN models to generate rich features for object detection, achieving breakthrough performance improvement in comparison with all previously works, which are mainly based on deformable part model (DPM) (Felzenszwalb et al., 2010). Briefly, R-CNN consists of three simple steps. First, it scans the input image for possible objects by using selective search method (Uijlings et al., 2013), generating about 2000 region proposals. Second, these region proposals are resized into a fixed size (e.g., 224×224) and the deep features of each region proposal are extracted by using a CNN model finetuned on the PASCAL VOC dataset. Finally, the features of each region proposal are fed into a set of class-specific support vector machines (SVMs) to label each region proposal as object or background and a linear regressor is used to refine the object localizations (if object exist). While R-CNN surpasses previous object detection methods, the low efficiency is its main shortcoming due to the repeated computation of abundant region proposals. In order to obtain better detection efficiency and accuracy, some recent works, such as SPPnet (He et al., 2014) and Fast R-CNN (Girshick, 2015), were proposed for sharing the computation load of CNN feature extraction of all region proposals. Compared with R-CNN, Fast R-CNN and SPPnet perform feature extraction over the whole image with a region of interest (RoI) layer and a spatial pyramid pooling (SPP) layer, respectively, in which the CNN model runs over the entire image only once rather than thousands of times, thus they need less computation time than R-CNN. Although SPPnet and Fast R-CNN work at faster speeds than R-CNN, they need obtaining region proposals in advance, which are usually generated with hand-engineering proposal detectors such as EdgeBox and selective search method (Uijlings et al., 2013). However, the handcrafted region proposal mechanism is a severe bottleneck of the entire object detection process. Thus, Faster R-CNN (Ren et al., 2017) was proposed in order to fix this problem. The main insight of Faster R-CNN is to adopt a fast module to generate region proposals instead of the slow selective search algorithm. Specifically, the Faster R-CNN framework consists of two modules. The first model is the region proposal network (RPN), which is a fully convolutional network used to generate region proposals. The second module is the Fast R-CNN object detector used for classifying the proposals which are generate with the first module. The core idea of the Faster R-CNN is to share the same convolutional layers for the RPN and Fast R-CNN detector up to their own fully connected layers. In this way, the image only needs to pass through the CNN once to generate region proposals and their corresponding features. More importantly, thanks to the sharing of convolutional layers, it is possible to use a very deep CNN model to generate higher-quality region proposals than traditional region proposal generation methods. In addition, some researchers further extend the work of Faster R-CNN for better performance. For example, Mask R-CNN builds on Faster R-CNN and adds an additional branch to predict an object mask in parallel with the existing branch for bounding box detection. Thus, Mask R-CNN can accurately recognize objects and simultaneously generate high-quality segmentation masks for each object instance. In order to further speed up object detection of Faster R-CNN, region-based fully convolutional network (R-FCN) (Dai et al., 2016) was proposed. It uses a position-sensitive region of interest (RoI) pooling layer to aggregate the outputs of the last convolutional layer and produce scores for each RoI. In contrast to Faster R-CNN that applies a costly per-region sub-network hundreds of times, R-FCN shares almost all computation load on the entire image, resulting in 2.5-20× faster than Faster R-CNN. Besides, Li et al. proposed a Light Head R-CNN to further speed up the detection speed of R-FCN (Dai et al., 2016) by making the head of the detection network as light as possible. Also, Singh et al. proposed a novel detector, called R-FCN-3000 (Singh et al., 2018a), towards large-scale real-time object detection for 3000 object classes. This approach is a modification of R-FCN (Dai et al., 2016) used to learn shared filters to perform localization across different object classes. In 2017, a feature pyramid network (FPN) (Lin et al., 2017b) was proposed by building feature pyramids inside CNNs, which shows significant improvement as a generic feature extractor for object detection with the frameworks of Faster R-CNN (Ren et al., 2017) and Mask R-CNN . Also, a path aggregation network (PANet) was proposed to boost the entire feature hierarchy with accurate localization information in lower layers by bottom-up path augmentation, which can significantly shorten the information path between lower layers and topmost feature. More recently, Singh et al. proposed two advanced and effective data argumentation methods for object detection, including Scale Normalization for Image Pyramids (SNIP) (Singh and Davis, 2018) and SNIP with Efficient Resampling (SNIPER) (Singh et al., 2018b). These two methods present detailed analysis of different techniques for detecting and recognizing objects under extreme scale variation. To be specific, SNIP (Singh and Davis, 2018) is a novel training paradigm that builds image pyramids at both of training and detection stages and only selectively back-propagates the gradients of objects of different sizes as a function of the image scale. Thus, it significantly benefits from reducing scale-variation during training but without reducing training samples. SNIPER (Singh et al., 2018b) is an efficient multi-scale training approach proposed to adaptively generate training samples from multiple scales of an image pyramid, conditioned on the image content. Under the same conditions, SNIPER performs as well as SNIP while reducing the number of pixels processed by a factor of 3 during training. Here, it should be pointed out that SNIP (Singh and Davis, 2018) and SNIPER (Singh et al., 2018b) are generic and thus can be broadly applied to many detectors, such as Faster R-CNN (Ren et al., 2017), Mask R-CNN , R-FCN (Dai et al., 2016), deformable R-FCN (Dai et al., 2017), and so on. Regression-based Methods This kind of methods uses one-stage object detectors for object instance prediction, thus simplifying detection as a regression problem. Compared with region proposal-based methods, regression-based methods are much simpler and more efficient, because there is no need to produce candidate region proposals and the subsequent feature resampling stages. OverFeat (Sermanet et al., 2014) is the first regression-based object detector based on deep networks using sliding-window paradigm. More recently, You Only Look Once (YOLO) (Redmon et al., 2016;Farhadi, 2017, 2018), Single Shot multibox Detector (SSD) (Fu et al., 2017;Liu et al., 2016a), and RetinaNet (Lin et al., 2017c) have renewed the performance in regression-based methods. YOLO (Redmon et al., 2016) is a representative regression-based object detection method. It adopts a single CNN backbone to directly predict bounding boxes and class probabilities from the entire images in one evaluation. It works as follows. Given an input image, it is firstly divided into S×S grids. If the center of an object falls into a grid cell, that grid is responsible for the detection of that object. Then, each grid cell predicts B bounding boxes together with their confidence scores and C class probabilities. YOLO achieves object detection in real-time by reframing it as a single regression problem. However, it still struggles to precisely localize some objects, especially small-sized objects. In order to improve both the speed and accuracy, SSD (Liu et al., 2016a) was proposed. Specifically, the output space of bounding boxes is discretized into a set of default boxes over different scales and aspect ratios per feature map location. At prediction process, the confidence scores for the presence of each object class in each default box are generated based on the SSD model and the adjustments to the box are also produced to better match the object shape. Furthermore, in order to address the problem of object size variations, SSD combines the predictions obtained from multiple feature maps with different resolutions. Compared with YOLO (Redmon et al., 2016), SSD achieved better performance for detecting and locating small-sized objects via introducing default boxes mechanism and multi-scale feature maps. Another interesting work is RetinaNet detector (Lin et al., 2017c), which is essentially a feature pyramid network with the traditional cross-entropy loss being replaced by a new Focal loss (Lin et al., 2017c), and thereby increasing the accuracy significantly. The insight of YOLOv2 model (Redmon and Farhadi, 2017) is to improve object detection accuracy while still being an efficient object detector. To this end, it proposes various improvements to the original YOLO method. For example, in order to prevent over-fitting without using dropout, YOLOv2 adds the batch normalization on all of the convolutional layers. It accepts higher-resolution images as input by adjusting the input image size from 224×224 (YOLO) to 448×448 (YOLOv2), thus the objects with smaller sizes can be detected effectively. Additionally, YOLOv2 removes the fully-connected layers from the original YOLO detector and predicts bounding boxes based on anchor boxes, which shares the similar idea with SSD (Liu et al., 2016a). More recently, YOLOv3 model (Redmon and Farhadi, 2018), which has similar performance but is faster than YOLOv2, SSD, and RetinaNet, was proposed. YOLOv3 adheres to YOLOv2's mechanism. To be specific, the bounding boxes are predicted using dimension clusters as anchor boxes. Then, independent logistic classifiers instead of softmax classifier are adopted to output an object score for each bounding box. Sharing a similar concept with FPN (Lin et al., 2017b), the bounding boxes are predicted at three different scales through extracting features from these scales. YOLOv3 uses a new backbone network, named Darketnet-53, for performing feature extraction. It has 53 convolutional layers, and is a newfangled residual network. Due to the introduction of Darketnet-53 and multi-scale feature maps, YOLOv3 achieves great speed improvement and also improves detection accuracy of small-sized objects when compared with the original YOLO or YOLOv2. In addition, Law and Deng proposed CornerNet (Law and Deng, 2018), a new and effective object detection paradigm that detects object bounding boxes as pairs of corners (i.e., the top-left corner and the bottom-right corner) by using a single CNN. By detecting objects as paired corners, CornerNet eliminates the need for designing a set of anchor boxes widely used in regression-based object detectors. This work also introduces corner pooling, a new type of pooling layer that helps the network better localize corners. In general, region proposal-based object detection methods have better accuracies than regression-based algorithms, while regression-based algorithms have advantages in speed (Lin et al., 2017c). It is generally accepted that CNN framework plays a crucial role in object detection task. CNN architectures serve as network backbones used in various object detection frameworks. Some representative CNN model architectures include AlexNet (Krizhevsky et al., 2012), ZFNet (Zeiler and Fergus, 2014), VGGNet (Simonyan and Zisserman, 2015), GoogLeNet , Inception series (Ioffe and Szegedy, 2015;Szegedy et al., 2017;Szegedy et al., 2016), ResNet , DenseNet (Huang et al., 2017) and SENet (Hu et al., 2018). Also, some researches have been widely explored to further improve the performance of deep learning based methods for object detection, such as feature enhancement (Cai et al., 2016;Cheng et al., 2019;Cheng et al., 2016b;Kong et al., 2016;Liu et al., 2017b), hard negative mining (Lin et al., 2017c;, contextual information fusion (Bell et al., 2016;Gidaris and Komodakis, 2015;Zhu et al., 2015b), modeling object deformations (Mordan et al., 2018;Ouyang et al., 2017;Xu et al., 2017), and so on. Review on Object Detection in Earth Observation Community In the past years, numerous object detection approaches have been explored to detect various geospatial objects in the earth observation community. Cheng et al (Cheng and Han, 2016) provide a comprehensive review in 2016 on object detection algorithms in optical remote sensing images. However, the work of (Cheng and Han, 2016) does not review various deep learning based object detection methods. Different from several previously published surveys, we focus on reviewing the literatures about datasets and deep learning based approaches for object detection in the earth observation community. Object Detection Datasets of Optical Remote Sensing Images During the last decades, several different research groups have released their publicly available earth observation image datasets for object detection (see Table 1). These datasets will be briefly reviewed as follows. 1) TAS: The TAS dataset (Heitz and Koller, 2008) is designed for car detection in aerial images. It contains a total of 30 images and 1319 manually annotated cars with arbitrary orientations. These images have relatively low spatial resolution and a lot of shadows caused by buildings and trees. 2) SZTAKI-INRIA: The SZTAKI-INRIA dataset (Benedek et al., 2011) is created for benchmarking various building detection methods. It consists of 665 buildings, manually annotated with oriented bounding boxes, distributed throughout nine remote sensing images derived from Manchester (U.K.), Szada and Budapest (Hungary), Cot d′Azur and Normandy (France), and Bodensee (Germany). All of the images contain only red (R), green (G), and blue (B) three channels. Among them, two images (Szada and Budapest) are aerial images and the rest seven images are satellite images from QuickBird, IKONOS, and Google Earth. 3) NWPU VHR-10: The NWPU VHR-10 dataset (Cheng and Han, 2016;Cheng et al., 2016a) has 10 geospatial object classes including airplane, baseball diamond, basketball court, bridge, harbor, ground track field, ship, storage tank, tennis court, and vehicle. It consists of 715 RGB images and 85 pan-sharpened color infrared images. To be specific, the 715 RGB images are collected from Google Earth and their spatial resolutions vary from 0.5m to 2m. The 85 pansharpened infrared images, with a spatial resolution of 0.08m, are obtained from Vaihingen data (Cramer, 2010). This dataset contains a total of 3775 object instances which are manually annotated with horizontal bounding boxes, including 757 airplanes, 390 baseball diamonds, 159 basketball courts, 124 bridges, 224 harbors, 163 ground track fields, 302 ships, 655 storage tanks, 524 tennis courts, and 477 vehicles. This dataset has been widely used in the earth observation community Cheng et al., 2018b;Farooq et al., 2017;Guo et al., 2018;Han et al., 2017a;Li et al., 2018;Yang et al., 2018b;Yang et al., 2017;Zhong et al., 2018). 4) VEDAI: The VEDAI (Razakarivony and Jurie, 2015) dataset is released for the task of multi-class vehicle detection in aerial images. It consists of 3640 vehicle instances covered by nine classes including boat, car, camping car, plane, pick-up, tractor, truck, van, and the other category. This dataset contains totally 1210 1024×1024 aerial images acquired from Utah AGRC (http://gis.utah.gov/), with a spatial resolution of 12.5 cm. The images in the dataset were captured during spring 2012 and each image has four uncompressed color channels including three RGB color channels and one near infrared channel. 5) UCAS-AOD: The UCAS-AOD dataset (Zhu et al., 2015a) is designed for airplane and vehicle detection. Specifically, the airplane dataset consists of 600 images with 3210 airplanes and the vehicle dataset consists of 310 images with 2819 vehicles. All the images are carefully selected so that the object orientations in the dataset distribute evenly. 6) DLR 3K Vehicle: The DLR 3K Vehicle dataset (Liu and Mattyus, 2015) is another dataset designed for vehicle detection. It contains 20 5616×3744 aerial images, with a spatial resolution of 13 cm. They are captured at a height of 1000 meters above the ground using the DLR 3K camera system (a near real time airborne digital monitoring system) over the area of Munich, Germany. There are 14235 vehicles that are manually labeled by using oriented bounding boxes in the images. 7) HRSC2016: The HRSC2016 dataset (Liu et al., 2016b) contains 1070 images and a total of 2976 ships collected from Google Earth used for ship detection. The image sizes change from 300×300 to 1500×900, and most of them are about 1000×600. These images are collected with large variations of rotation, scale, position, shape, and appearance. 8) RSOD: The RSOD dataset (Xiao et al., 2015) contains 976 images downloaded from Google Earth and Tianditu, and the spatial resolutions of these images range from 0.3m to 3m. It consists of totally 6950 object instances, covered by four object classes, including 1586 oil tanks, 4993 airplanes, 180 overpasses, and 191 playgrounds. 9) DOTA: The DOTA (Xia et al., 2018) is a new large-scale geospatial object detection dataset, which consists of 15 different object categories: baseball diamond, basketball court, bridge, harbor, helicopter, ground track field, large vehicle, plane, ship, small vehicle, soccer ball field, storage tank, swimming pool, tennis court, and roundabout. This dataset contains a total of 2806 aerial images obtained from different sensors and platforms with multiple resolutions. There are 188282 object instances labeled by an oriented bounding box. The sizes of images range from about 800×800 to 4000×4000 pixels. Each image contains multiple objects of different scales, orientations and shapes. To date, this dataset is the most challenging. Deep Learning Based Object Detection Methods in Earth Observation Community Inspired by the great success of deep learning-based object detection methods in computer vision community, extensive studies have been devoted recently to object detection in optical remote sensing images. Different from object detection in natural scene mages, most of the studies use region proposal-based methods to detect multi-class objects in the earth observation community. We therefore no longer distinguish them between region proposal-based methods or regression-based methods in the earth observation community. Here, we mainly review some representative methods. Driven by the excellent performance of R-CNN for natural scene image object detection, a number of earth observation researchers adopt R-CNN pipeline to detect various geospatial objects in remote sensing images (Cheng et al., 2016a;Long et al., 2017;Salberg, 2015;Ševo and Avramović, 2017). For instance, Cheng et al. (Cheng et al., 2016a) proposed to learn a rotation-invariant CNN (RICNN) model in R-CNN framework used for multi-class geospatial object detection. The RICNN is achieved by adding a new rotation-invariant layer to the off-the-shelf CNN model such as AlexNet (Krizhevsky et al., 2012). In order to further boost the state-of-the-arts of object detection, (Cheng et al., 2019) proposed a new method to train rotation-invariant and Fisher discriminative CNN (RIFD-CNN) model by imposing a rotation-invariant regularizer and a Fisher discrimination regularizer on the CNN features. To achieve accurate localization of geospatial objects in high-resolution earth observation images, Long et al. (Long et al., 2017) presented an unsupervised score-based bounding box regression (USB-BBR) method based on R-CNN framework. Although the aforementioned methods have achieved good performance in the earth observation community, they are still time-consuming because these approaches depend on human-designed object proposal generation methods which domain most of the running time of an object detection system. In addition, the quality of region proposals generated based on hand-engineered low-level features is not good, thus thereby degenerating object detection performance. In order to further enhance the detection accuracy and speed, a few research works extend the framework of Faster R-CNN to the earth observation community Guo et al., 2018;Han et al., 2017b;Li et al., 2018;Tang et al., 2017b;Xu et al., 2017;Yang et al., 2018a;Yang et al., 2017;Yao et al., 2017;Zhong et al., 2018). For instance, Li et al. (Li et al., 2018) presented a rotation-insensitive RPN by introducing multi-angle anchors into the existing RPN based on Faster R-CNN pipeline, which can effectively handle the problem of geospatial object rotation variations. Furthermore, in order to tackle the problem of appearance ambiguity, a double-channel feature combination network is designed to learn local and contextual properties. Zhong et al. (Zhong et al., 2018) utilized a position-sensitive balancing (PSB) method to enhance the quality of generated region proposal. In the proposed PSB framework, a fully convolutional network (FCN) (Long et al., 2015) was introduced, based on the residual network , to address the dilemma between translation-variance in object detection and translation-invariance in image classification. Xu et al. presented a deformable CNN to model the geometric variations of objects. In , non-maximum suppression constrained by aspect ratio was developed to reduce the increase of false region proposals. Aiming at vehicle detection, Tang et al. (Tang et al., 2017b) proposed a hyper region proposal network (HRPN) to find vehicle-like regions and use hard negative mining to further improve the detection accuracy. Although adapting region proposal-based methods, such as R-CNN, Faster R-CNN, and their variants, to detect geospatial objects in earth observation images shows greatly promising performance, remarkable efforts have been made to explore different deep learning based methods (Lin et al., 2017a;Liu et al., 2017a;Liu et al., 2018c;Tang et al., 2017a;Yu et al., 2015;Zou and Shi, 2016), which do not follow the pipeline of region proposal-based approaches to detect objects in remote sensing images. For example, Yu et al. (Yu et al., 2015) presented a rotation-invariant method to detect geospatial objects, in which super-pixel segmentation strategy is firstly used to produce local patches, then, deep Boltzmann machines are adopted to construct high-level feature representations of local patches, and finally a set of multi-scale Hough forests is built to cast rotation-invariant votes to locate object centroids. Zou et al. (Zou and Shi, 2016) used a singular value decompensation network to obtain ship-like regions and adopt feature pooling operation and a linear SVM classifier to verify each ship candidate for the task of ship detection. Although this detection framework is interesting, the training process is still clumsy and slow. More recently, in order to achieve real-time object detection, a few studies attempt to transfer regression-based detection methods developed for natural scene images to remote sensing images. For instance, sharing the similar idea with SSD, Tang et al. (Tang et al., 2017a) used a regression-based object detector to detect vehicle targets. Specifically, the detection bounding boxes are generated by adopting a set of default boxes with different scales per feature map location. Moreover, for each default box the offsets are predicted to better fit the object shape. Liu et al. (Liu et al., 2017a) replaced the traditional bounding box with rotatable bounding box (RBox) embedded into SSD framework (Liu et al., 2016a), thus being rotation invariant due to its ability of estimating the orientation angles of objects. Liu et al. (Liu et al., 2018c) designed a framework for detecting arbitrary-oriented ships. This model can directly predict rotated/oriented bounding boxes by using YOLOv2 architecture as the fundamental network. In addition, hard example mining (Tang et al., 2017a;Tang et al., 2017b), multi-feature fusion (Zhong et al., 2017), transfer learning (Han et al., 2017b), non-maximum suppression , etc., are often used in geospatial object detection to further boost the performance of deep learning based approaches. Although most of the existing deep learning based methods have demonstrated significant achievement on the task of object detection in the earth observation community, they are transferred from the methods (e.g., R-CNN, Faster R-CNN, SSD, etc.) designed for natural scene images. In fact, as we have pointed out above, earth observation images significantly differ from natural scene images is significant, especially in the aspects of rotation, scale variation and the complex and cluttered background. Although the existing methods partially addressed these issues through introducing prior knowledge or designing proprietary models, the task of object detection in earth observation images is still an open problem deserved to further research. Proposed DIOR Dataset In the last few years, remarkable efforts have been made to release various object detection datasets (reviewed in Section 3.2) in the earth observation community. However, most of existing object detection datasets in earth observation domain shares some common shortcomings, for example, the number of images and the number of object categories are small scale, and the image diversity and object variations are insufficient. These limitations significantly affect the development of deep learning based object detection methods. In such a situation, creating a large-scale object detection dataset by using remote sensing images is highly desirable for the earth observation community. This motivates us to create a large-scale dataset named DIOR. It is publicly available 2 and can be used freely for object detection in optical remote sensing images. Object Class Selection Selecting appropriate geospatial object classes is the first step of constructing the dataset and is crucial for the dataset. In our work, we first investigated the object classes of all existing datasets (Benedek et al., 2011;Cheng and Han, 2016;Cheng et al., 2016a;Heitz and Koller, 2008;Liu and Mattyus, 2015;Liu et al., 2016b;Razakarivony and Jurie, 2015;Xia et al., 2018;Xiao et al., 2015;Zhu et al., 2015a) to obtain 10 object categories which are commonly used in both NWPU VHR-10 dataset and DOTA dataset. We then further extend the object categories of our dataset by searching the keywords of "object detection", "object recognition", "earth observation images", and "remote sensing images" on Google Scholar and Web of Science to carefully select other 10 object classes, according to whether a kind of objects is common or its value for real-world applications. For example, some traffic infrastructures that are common and play an important role in transportation analysis, such as train stations, expressway service areas, and airports, are selected mainly because of their values in real applications. In addition, most of the object categories in existing datasets are selected from the urban areas. Therefore, dam and wind mill, which are common in the suburb as well as important infrastructures, are also chosen to improve the variations and diversity of geospatial objects. In such a situation, a total of 20 object classes are selected to create the proposed DIOR dataset. These 20 object classes are airplane, airport, baseball field, basketball court, bridge, chimney, dam, expressway service area, expressway toll station, harbor, golf course, ground track field, overpass, ship, stadium, storage tank, tennis court, train station, vehicle, and wind mill. Characteristics of Our Proposed DIOR Dataset The DIOR dataset is one of the largest, most diverse, and publicly available object detection dataset in earth observation community. We use LabelMe (Russell et al., 2008), an open-source image annotation tool, to annotate object instances. Each object instance is manually labeled by a horizontal bounding box which is typically used for object annotation in remote sensing images and natural scene images. Fig. 3 reports the number of object instances per class. In the proposed DIOR dataset, the object classes of ship and vehicle have higher instance counts, while the classes of train station, expressway toll station and expressway service area have lower instance counts. The diversity of object size is more helpful for real-world tasks. As shown in Fig. 4, we achieve a good balance between small-sized instances and big-sized instances. In addition, the significant object size differences across different categories makes the detection task more challenging, because which requires that the detectors have to be flexible enough to simultaneously handle small-sized and large-sized objects. Compared with existing object detection datasets including (Benedek et al., 2011;Cheng and Han, 2016;Cheng et al., 2016a;Heitz and Koller, 2008;Liu and Mattyus, 2015;Liu et al., 2016b;Razakarivony and Jurie, 2015;Tanner et al., 2009;Xia et al., 2018;Xiao et al., 2015;Zhu et al., 2015a), the proposed DIOR dataset has the following four remarkable characteristics. 1) Large scale. DIOR consists of 23463 optimal remote sensing images and 192472 object instances that are manually labeled with axis-aligned bounding boxes, covered by 20 common object categories. The size of images in the dataset is 800×800 pixels and the spatial resolutions range from 0.5m to 30m. Similar with most of the existing datasets, this dataset is also collected from Google Earth (Google Inc.), by the experts in the domain of earth observation interpretation. Compared with all existing remote sensing image datasets designed for object detection, the proposed DIOR dataset, to our best knowledge, is the largest scale on both the number of images and the number of object categories. The release of this dataset will help the earth observation community to explore and evaluate a variety of deep learning based methods, thus thereby further improving the state of the arts. 2) A large range of object size variations. Spatial size variation is an important feature of geospatial objects. This is not only because of the spatial resolutions of sensors, but also due to between-class size variation (e.g., aircraft carriers vs. cars) and within-class size variation (e.g., aircraft carriers vs. hookers). There are a large range of size variations of object instances in the proposed DIOR dataset. To increase the size variations of objects, images with different spatial resolutions of objects are collected and the images which contain rich size variations coming from the same object category and different object categories are also collected in our dataset. As shown in Fig. 5 (a), "vehicle" and "ship" instances present different sizes. Besides, due to different spatial resolutions, the object sizes of "stadium" instances are also obviously different. 3) Rich image variations. A highly desired characteristic for any object detection system is its robustness to image variations. However, most of the existing datasets are lack of image variations totally or partially. For example, the widely used NWPU VHR-10 dataset only consists of 800 images, which is too small to possess much richer variations in various weathers, seasons, imaging conditions, scales, etc. On the contrary, the proposed DIOR dataset contains 23463 remote sensing images covered more than 80 countries. Moreover, these images are carefully collected under different weathers, seasons, imaging conditions, and image quality (see Fig. 5 (b)). Thus, our proposed DIOR dataset holds richer variations in viewpoint, translation, illumination, background, object pose and appearance, occlusion, etc., for each object class. 4) High inter-class similarity and intra-class diversity. Another important characteristic of our proposed dataset is that it has high inter-class similarity and intra-class diversity, thus making it much challenging. To obtain big interclass similarity, we add some fine-grained object classes with high semantic overlapping, such as "bridge" vs. "overpass", "bridge" vs. "dam", "ground track field" vs. "stadium", "tennis court" vs. "basketball court", and so on. To increase intra-class diversity, all kinds of factors, such as different object colors, shapes and scales, are taken into account when collecting images. As shown in Fig. 5 (c), "chimney" instances present different shapes, and "dam" and "bridge" instances have very similar appearances. Benchmarking Representative Methods This section focuses on benchmarking some representative deep learning based object detection methods on our proposed DIOR dataset in order to provide an overview of the state-of-the-art performance for future research work. Experimental Setup In order to guarantee the distributions of training-validation (trainval) data and test data are similar, we randomly selected 11725 remote sensing images (i.e., 50% of the dataset) as trainval set, and the remaining 11738 images are used as test set. The trainval data consists of two parts, the training (train) set and validation (val) set. For each object category and subset, the number of images that contains at least one object instance of that object class is reported in Table 2. Note that one image may contain multiple object classes, so the column totals do not simply equal the sums of each corresponding column. A detection is regarded as correct if its bounding box has more than 50% overlap with the ground truth; otherwise, the detection is seen as a false positive. We conducted all experiments on a computer with a single Intel core i7 CPU, 64 GB of memory, and an NVIDIA Titan X GPU for acceleration. A total of 12 representative deep learning based object detection methods, which are widely used for object detection in natural scene images and earth observation images, were selected as our benchmark testing algorithms. To be specific, our selections include eight region proposal-based approaches: R-CNN , RICNN (with R-CNN framework) (Cheng et al., 2016a), RICAOD (Li et al., 2018), Faster R-CNN (Ren et al., 2017), RIFD-CNN (with Faster R-CNN framework) (Cheng et al., 2019), Faster R-CNN (Ren et al., 2017) with FPN (Lin et al., 2017b), Mask R-CNN with FPN (Lin et al., 2017b) and PANet , and four regression-based methods: YOLOv3 (Redmon and Farhadi, 2018), SSD (Liu et al., 2016a), RetinaNet (Lin et al., 2017c), and CornerNet (Law and Deng, 2018). To make fair comparisons, we kept all the experiment settings the same as that depicted in corresponding papers. R-CNN , RICNN (Cheng et al., 2016a), RICAOD (Li et al., 2018), and RIFD-CNN (Cheng et al., 2019) are built on the Caffe framework (Jia et al., 2014). Faster R-CNN (Ren et al., 2017), Faster R-CNN (Ren et al., 2017) with FPN (Lin et al., 2017b), Mask R-CNN with FPN (Lin et al., 2017b), PANet , RetinaNet (Lin et al., 2017c), and CornerNet (Law and Deng, 2018) are based on the PyTorch re-implementation (Paszke et al., 2017). YOLOv3 uses Darknet-53 framework (Redmon and Farhadi, 2018) and SSD (Liu et al., 2016a) is implemented with TensorFlow (Abadi et al., 2016). Note that, the backbone network is VGG16 model (Simonyan and Zisserman, 2015) for R-CNN , RICNN (Cheng et al., 2016a), RICAOD (Li et al., 2018), Faster R-CNN (Ren et al., 2017), RIFD-CNN (Cheng et al., 2019), and SSD (Liu et al., 2016a). YOLOv3 (Redmon and Farhadi, 2018) uses Darknet-53 as its backbone network. For Faster R-CNN (Ren et al., 2017) with FPN (Lin et al., 2017b), Mask R-CNN with FPN (Lin et al., 2017b), PANet , and RetinaNet (Lin et al., 2017c), we use ResNet-50 and ResNet-101 as their backbone networks. As regards CornerNet (Law and Deng, 2018), its backbone network is Hourglass-104 (Newell et al., 2016). We used average precision (AP) and mean AP as measures for evaluating the object detection performance. One can refer to (Cheng and Han, 2016) for more details about these two metrics. Experimental Results The results of 12 representative methods are shown in Table 3. We have the following observations from Table 3. (1) The deeper the backbone network is, the stronger the representation capability of the network is and the higher detection accuracy we could obtain. It generally follows the order: ResNet-101 and Hourglass-104 > ResNet50 and Darknet-53 > VGG16. The detection results of RetinaNet (Lin et al., 2017c) with ResNet-101 and PANet with ResNet-101 both achieve the highest mAP of 66.1%. (2) Since CNNs naturally form a feature pyramid through its forward propagation, exploiting inherent pyramidal hierarchy of CNNs to construct feature pyramid networks, such as FPN (Lin et al., 2017b) and PANet , could significantly boost the detection accuracy. Using FPN in basic Faster R-CNN and Mask RCNN systems shows great advances for detecting objects with a wide variety of scales. And for this reason, FPN has now become a basic building block of many latest detectors such as RetinaNet (Lin et al., 2017c) and PANet . (3) YOLOv3 (Redmon and Farhadi, 2018) could always achieve higher accuracy than other methods for detecting small-sized object instances (e.g., vehicles, storage tanks and ships). Especially for ship class, the detection accuracy of YOLOv3 (Redmon and Farhadi, 2018) achieves 87.40%, which is much better than all other 11 methods. This is probably because that the backbone network of Darknet-53 is specifically designed for object detection task and also the new multi-scale prediction is introduced into YOLOv3, which allows it to extract richer features from three different scales (Lin et al., 2017b). (4) For ship, airplane, basketball court, vehicle, bridge, RIFD-CNN (Cheng et al., 2019), RICAOD (Li et al., 2018) and RICNN (Cheng et al., 2016a) improve the detection accuracies to some extent in comparison with the baseline approaches of Faster R-CNN (Ren et al., 2017) and R-CNN . This is mainly because these methods proposed different strategies to enrich feature representations for remote sensing images to address the issue of geospatial object rotation variations. Specifically, RICAOD (Li et al., 2018) designs a rotation-insensitive region proposal network. RICNN (Cheng et al., 2016a) presents a rotation-invariant CNN by adding a new fully-connected layer. RIFD-CNN (Cheng et al., 2019) learns a rotation-invariant and Fisher discriminative CNN by proposing new objective functions yet without changing the CNN model architecture. (5) CornerNet (Law and Deng, 2018) obtains the best results for 9 of 20 object classes, which demonstrates that detecting an object as a pair of bounding box corners is a very promising research direction. While the results on some object categories are promising, there exists substantial improvement space for almost all object categories. For some object classes, e.g., bridge, harbor, overpass, and vehicle, the detection accuracies are still very low, and the currently existing methods are difficult to obtain satisfactory results. This probably attributes to the relatively low image quality and the complex and cluttered background in aerial images, when compared with natural scene images. This also indicates that the proposed DIOR dataset is a challenging benchmark for geospatial object detection. In the future work, some novel training scheme including SNIP (Singh and Davis, 2018) and SNIPER (Singh et al., 2018b) can be applied to many existing detectors, such as Faster R-CNN (Ren et al., 2017), Mask R-CNN , R-FCN (Dai et al., 2016), deformable R-FCN (Dai et al., 2017), and so on, to obtain better results. Conclusions This paper first highlighted the recent progress of object detection, including benchmark datasets and the state-ofthe-art deep learning-based approaches, in both the computer vision and earth observation communities. Then, a large-scale and publicly available object detection benchmark dataset is proposed. This new dataset can help the earth observation community to further explore and validate deep learning based methods. Finally, the performances of some representative object detection methods are evaluated by using the proposed dataset and the experimental results can be regarded as a useful performance baseline for future research. Fig. 1 . 1Some examples, taken from (a) the PASCAL VOC dataset and (b) the proposed DIOR dataset, demonstrate the difference between natural scene images and remote sensing images. Fig. 2 . 2Example images taken from the proposed DIOR dataset, which were obtained with different imaging conditions, weathers, seasons, and image quality. Fig. 3 . 3Number of object instances per class. Fig. 4 . 4Object size distribution per class. Fig. 5 . 5Characteristics of our proposed DIOR dataset. Table 1 1Comparison between the proposed DIOR dataset and nine publicly available object detection datasets in earth observation community.Datasets # Categories # Images # Instances Image width Annotation way Year TAS 1 30 1319 792 horizontal bounding box 2008 SZTAKI-INRIA 1 9 665 ~800 oriented bounding box 2012 NWPU VHR-10 10 800 3775 ~1000 horizontal bounding box 2014 VEDAI 9 1210 3640 1024 oriented bounding box 2015 UCAS-AOD 2 910 6029 1280 horizontal bounding box 2015 DLR 3K Vehicle 2 20 14235 5616 oriented bounding box 2015 HRSC2016 1 1070 2976 ~1000 oriented bounding box 2016 RSOD 4 976 6950 ~1000 horizontal bounding box 2017 DOTA 15 2806 188282 800-4000 oriented bounding box 2017 DIOR (ours) 20 23463 192472 800 horizontal bounding box 2018 Table 2 2Number of images per object class and per subset.Train val Trainval Test Table 3 3Detection average precision (%) of 12 representative methods on the proposed DIOR test set. The entries with the best APs for each object category are bold-faced.c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 Airplane Airport Baseball field Basketball court Bridge Chimney Dam Expressway service area Expressway toll station Golf course c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 Ground track field Harbor Overpass Ship Stadium Storage tank Tennis court Train station Vehicle Wind mill Backbone c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 mAP R-CNN VGG16 35.6 43.0 53.8 62.3 15.6 53.7 33.7 50.2 33.5 50.1 49.3 39.5 30.9 9.1 60.8 18.0 54.0 36.1 9.1 16.4 37.7 RICNN VGG16 39.1 61.0 60.1 66.3 25.3 63.3 41.1 51.7 36.6 55.9 58.9 43.5 39.0 9.1 61.1 19.1 63.5 46.1 11.4 31.5 44.2 RICAOD VGG16 42.2 69.7 62.0 79.0 27.7 68.9 50.1 60.5 49.3 64.4 65.3 42.3 46.8 11.7 53.5 24.5 70.3 53.3 20.4 56.2 50.9 RIFD-CNN VGG16 56.6 53.2 79.9 69.0 29.0 71.5 63.1 69.0 56.0 68.9 62.4 51.2 51.1 31.7 73.6 41.5 79.5 40.1 28.5 46.9 56.1 Faster R-CNN VGG16 53.6 49.3 78.8 66.2 28.0 70.9 62.3 69.0 55.2 68.0 56.9 50.2 50.1 27.7 73.0 39.8 75.2 38.6 23.6 45.4 54.1 SSD VGG16 59.5 72.7 72.4 75.7 29.7 65.8 56.6 63.5 53.1 65.3 68.6 49.4 48.1 59.2 61.0 46.6 76.3 55.1 27.4 65.7 58.6 YOLOv3 Darknet-53 72.2 29.2 74.0 78.6 31.2 69.7 26.9 48.6 54.4 31.1 61.1 44.9 49.7 87.4 70.6 68.7 87.3 29.4 48.3 78.7 57.1 ResNet-50 54.1 71.4 63.3 81.0 42.6 72.5 57.5 68.7 62.1 73.1 76.5 42.8 56.0 71.8 57.0 53.5 81.2 53.0 43.1 80.9 63.1 Faster RCNN with FPN ResNet-101 54.0 74.5 63.3 80.7 44.8 72.5 60.0 75.6 62.3 76.0 76.8 46.4 57.2 71.8 68.3 53.8 81.1 59.5 43.1 81.2 65.1 ResNet-50 53.8 72.3 63.2 81.0 38.7 72.6 55.9 71.6 67.0 73.0 75.8 44.2 56.5 71.9 58.6 53.6 81.1 54.0 43.1 81.1 63.5 Mask-RCNN with FPN ResNet-101 53.9 76.6 63.2 80.9 40.2 72.5 60.4 76.3 62.5 76.0 75.9 46.5 57.4 71.8 68.3 53.7 81.0 62.3 43.0 81.0 65.2 ResNet-50 53.7 77.3 69.0 81.3 44.1 72.3 62.5 76.2 66.0 77.7 74.2 50.7 59.6 71.2 69.3 44.8 81.3 54.2 45.1 83.4 65.7 RetinaNet ResNet-101 53.3 77.0 69.3 85.0 44.1 73.2 62.4 78.6 62.8 78.6 76.6 49.9 59.6 71.1 68.4 45.8 81.3 55.2 44.4 85.5 66.1 ResNet-50 61.9 70.4 71.0 80.4 38.9 72.5 56.6 68.4 60.0 69.0 74.6 41.6 55.8 71.7 72.9 62.3 81.2 54.6 48.2 86.7 63.8 PANet ResNet-101 60.2 72.0 70.6 80.5 43.6 72.3 61.4 72.1 66.7 72.0 73.4 45.3 56.9 71.7 70.4 62.0 80.9 57.0 47.2 84.5 66.1 CornerNet Hourglass- 104 58.8 84.2 72.0 80.8 46.4 75.3 64.3 81.6 76.3 79.5 79.5 26.1 60.6 37.6 70.7 45.2 84.0 57.1 43.0 75.9 64.9 http://www.escience.cn/people/gongcheng/DIOR.html Acknowledgements TensorFlow: a system for large-scale machine learning. 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[ "Fine-Grained Entity Typing for Domain Independent Entity Linking", "Fine-Grained Entity Typing for Domain Independent Entity Linking" ]	[ "Yasumasa Onoe [email protected] \nDepartment of Computer Science\nThe University of Texas at Austin\n\n", "Greg Durrett [email protected] \nDepartment of Computer Science\nThe University of Texas at Austin\n\n" ]	[ "Department of Computer Science\nThe University of Texas at Austin\n", "Department of Computer Science\nThe University of Texas at Austin\n" ]	[]	Neural entity linking models are very powerful, but run the risk of overfitting to the domain they are trained in. For this problem, a "domain" is characterized not just by genre of text but even by factors as specific as the particular distribution of entities, as neural models tend to overfit by memorizing properties of frequent entities in a dataset. We tackle the problem of building robust entity linking models that generalize effectively and do not rely on labeled entity linking data with a specific entity distribution. Rather than predicting entities directly, our approach models fine-grained entity properties, which can help disambiguate between even closely related entities. We derive a large inventory of types (tens of thousands) from Wikipedia categories, and use hyperlinked mentions in Wikipedia to distantly label data and train an entity typing model. At test time, we classify a mention with this typing model and use soft type predictions to link the mention to the most similar candidate entity. We evaluate our entity linking system on the CoNLL-YAGO dataset(Hoffart et al. 2011) and show that our approach outperforms prior domain-independent entity linking systems. We also test our approach in a harder setting derived from the WikilinksNED dataset(Eshel et al. 2017)where all the mention-entity pairs are unseen during test time. Results indicate that our approach generalizes better than a state-of-the-art neural model on the dataset.	10.1609/aaai.v34i05.6380	[ "https://ojs.aaai.org/index.php/AAAI/article/download/6380/6236" ]	202,565,531	1909.05780	433346f09c2c6ffcfba73719dab441ff2dff7da3	Fine-Grained Entity Typing for Domain Independent Entity Linking Yasumasa Onoe [email protected] Department of Computer Science The University of Texas at Austin Greg Durrett [email protected] Department of Computer Science The University of Texas at Austin Fine-Grained Entity Typing for Domain Independent Entity Linking Neural entity linking models are very powerful, but run the risk of overfitting to the domain they are trained in. For this problem, a "domain" is characterized not just by genre of text but even by factors as specific as the particular distribution of entities, as neural models tend to overfit by memorizing properties of frequent entities in a dataset. We tackle the problem of building robust entity linking models that generalize effectively and do not rely on labeled entity linking data with a specific entity distribution. Rather than predicting entities directly, our approach models fine-grained entity properties, which can help disambiguate between even closely related entities. We derive a large inventory of types (tens of thousands) from Wikipedia categories, and use hyperlinked mentions in Wikipedia to distantly label data and train an entity typing model. At test time, we classify a mention with this typing model and use soft type predictions to link the mention to the most similar candidate entity. We evaluate our entity linking system on the CoNLL-YAGO dataset(Hoffart et al. 2011) and show that our approach outperforms prior domain-independent entity linking systems. We also test our approach in a harder setting derived from the WikilinksNED dataset(Eshel et al. 2017)where all the mention-entity pairs are unseen during test time. Results indicate that our approach generalizes better than a state-of-the-art neural model on the dataset. Introduction Historically, systems for entity linking to Wikipedia relied on heuristics such as anchor text distributions (Cucerzan 2007, Milne andWitten 2008), tf-idf (Ratinov et al. 2011), and Wikipedia relatedness of nearby entities (Hoffart et al. 2011). These systems have few parameters, making them relatively flexible across domains. More recent systems have typically been parameter-rich neural network models (Sun et al. 2015, Yamada et al. 2016, Francis-Landau, Durrett, and Klein 2016, Eshel et al. 2017). Many of these models are trained and evaluated on data from the same domain such as the CoNLL-YAGO dataset (Hoffart et al. 2011) or Wik-ilinksNED (Eshel et al. 2017, Mueller andDurrett 2018), for which the training and test sets share similar distributions of Copyright c 2020, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. entities. These models partially learn to attain high performance by memorizing the entity distribution of the training set rather than learning how to link entities more generally. As a result, apparently strong systems in one domain may not generalize to other domains without fine-tuning. In this work, we aim to use feature-rich neural models for entity linking 1 in a way that can effectively generalize across domains. We do this by framing the entity linking problem as a problem of prediction of very fine-grained entity types. Ambiguous entity references (e.g., different locations with the same name, the same movie released in different years) often differ in critical properties that can be inferred from context, but which neural bag-of-words and similar methods may not effectively tease out. We use an inventory of tens of thousands of types to learn such highly specific properties. This represents a much larger-scale tagset than past work using entity typing for entity linking, which has usually used hundreds of types (Gupta, Singh, and Roth 2017, Murty et al. 2018, Raiman and Raiman 2018. Critically, type prediction is the only learned component of our model: our final entity prediction uses a very simple heuristic based on summing posterior type probabilities. To train our typing model, we collect data from Wikipedia targeting a range of types in a domain of interest. This type set is lightly specialized to the target domain, but importantly, the set is determined on the basis of purely unlabeled data in the domain (lists of candidates for the identified mentions). Moreover, because we use such a large type inventory, our model captures a wide range of types and can handle entity linking in both narrow settings such as CoNLL-YAGO and broader domain settings such as WikilinksNED. Our typing model itself is adapted from past work on ultrafine grained entity typing in a different setting (Choi et al. 2018, Onoe andDurrett 2019). As a high-capacity neural network model, this model can train on millions of examples and effectively predict even rare types. Our contributions are as follows: (1) Formulating entity linking as purely an entity typing problem. (2) Constructing a distantly-supervised typing dataset based on Wikipedia Figure 1: Examples selected from the WikilinksNED development set (Eshel et al. 2017). The mention (in bold) resolves to the topmost entity in each case. These correct entities can be distinguished by their fine-grained Wikipedia categories. categories and training an ultra-fine entity typing model on it. (3) Showing through evaluation on two domains that our model is more effective than a range of other approaches trained from out-of-domain data, including Wikipedia data specialized to that particular domain. Figure 1 shows two examples from the WikilinksNED development set (Eshel et al. 2017) which motivate our problem setup. In the example (a), the most frequent Wikipedia entity given the mention "Ant" is the insect ant. In the Wikipedia dump, source anchors "Ant" point to the Wikipedia article about the insect Ant 2 96% of the time and points to Apache Ant 0.8% of the time. Since Apache Ant is very rare in the training data, models trained on broad domain data will often prefer Ant in many contexts. Motivation and Setup Predicting categories here is much less problematic. Our category processing (described in the Training Data for Typing section) assigns this mention several categories including Software. Predicting Software in this context is relatively easy for a typing model given the indicative words in the context. Knowing this type is enough to disambiguate between these two entities independent of other clues, and notably, it is not skewed by the relative rarity of the Apache Ant title. The category information is shared across many entities, so we can expect that predicting the category information would be much more efficient than learning rare entities directly. The example (b) adds another challenge since "Washington" can correspond to many different people or locations, some of which can occur in relatively similar contexts. Even a coarse-grained type inventory will distinguish between these mentions. However, more specific category information is needed to distinguish between Washington (state) and Washington, D.C. In this case, States of the West Coast of the United States would disambiguate between these, and context clues like "northwestern" can help identify this. This category is extremely fine-grained and we cannot assume an ability to predict it reliably; we discuss in the Training Data for Typing section how to get around this limitation by splitting categories into parts. Finally, we note that a global linking system (Hoffart et al. 2011) can sometimes exploit relevant context information from related entities like Java (programming language). In this model, we focus on a purely local approach for simplicity and see how far this can go; this approach is also the most general for datasets like WikilinksNED where other reliable mentions may not be present close by. Setup We focus on the entity linking (EL) task of selecting the appropriate Wikipedia entities for the mentions in context. We use m to denote a mention of an entity, s to denote a context sentence, e to denote a Wikipedia entity associated with the mention m, and C to denote a set of candidate entities. We also assume that we have access to a set of Wikipedia categories T corresponding to the Wikipedia entity e. Suppose we have an entity linking dataset D EL = {(m, s, e, C) (1) , . . . , (m, s, e, C) (k) }. In the standard entity linking setting, we train a model using the training set of D EL and evaluate on the development/test sets of D EL . In our approach, we also have an entity typing dataset collected from hyperlinks in English Wikipedia D Wiki = {(m, s, T ) (1) , . . . , (m, s, T ) (l) }. Since D Wiki is derived from Wikipedia itself, this data contains a large number of common Wikipedia entities. This enables us to train a general entity typing model that maps the mention m and its context s to a set T of Wikipedia categories: Φ : (m, s) → T . Then, we use a scoring function Ω to make entity linking predictions based on the candidate set: e = Ω(Φ(m, s), C). We evaluate our approach on the development/test sets of the existing entity linking data D EL . During training, we never assume access to labeled entity data D EL . Furthermore, by optimizing to predict Wikipedia categories T instead of an entity e, we can achieve a higher level of generalization across entities rather than simply memorizing our Wikipedia training data. Model Our model consists of two parts: a learned entity typing model and a heuristic (untrained) entity link predictor that depends only on the types. Entity Typing Model Figure 2 summarizes the model architecture of the entity typing model Φ. We use an attention-based model (Onoe and Durrett 2019) designed for the fine-grained entity typing tasks (Gillick et al. 2014, Choi et al. 2018. This model takes a mention span in a sentence context, uses span attention over vector representations of the mention span and its context, then aggregates that information to predict types. We follow Onoe and Durrett (2019) We also use a 1-D convolution over the characters of the mention to generate a character-level mention representation m char . The final representation of the mention and the sentence is a concatenation of the three vectors: v = s; m word ; m char ∈ R d . Unlike Onoe and Durrett (2019), we do not include the contextualized word vectors of the mention headword. 3 Decoder We use \|V t \| to denote the size of the category vocabulary. Following previous work (Choi et al. 2018, Onoe andDurrett 2019), we assume independence between the categories; thus, this boils down to a binary classification problem for each of the categories. The decoder is a single linear layer parameterized with W ∈ R \|V t \|×d . The probabilities for all categories in the vocabulary are obtained by t = σ(Wv), where σ(·) is an element-wise sigmoid operation. The probability vector t is the final output from the entity typing model Φ. Learning The entity typing model Φ is learned on the training examples consisting of (m, s, T ) triples. The loss is a sum of binary cross-entropy losses over all categories over all examples. That is, the typing problem is viewed as independent classification for each category, with the mentioncontext encoder shared across categories. Formally, we optimize a multi-label binary cross entropy objective: L = − i y i · log t i + (1 − y i ) · log(1 − t i ),(1) where i are indices over categories, t i is a score of the ith category, and y i takes the value 1 if the ith category applies to the current mention. Entity Linking Prediction Once the entity typing model Φ is trained, we use the model output t to make entity linking predictions. Assume we have an example from the test set of an entity linking dataset: x = (m, s, C), where C is a set of the candidate entities. We perform the forward computation of Φ and obtain the probability vector t = Φ(m, s). Then, we score all the candidates in C using a scoring function Ω. Our choice of Ω is defined as the sum of probabilities for each type exhibited by the selected entity: e c = i t i · Tc V t i e = arg max e e 1 , . . . , e \|C\| ,(2) where e c is a score for a candidate entity c, Tc (·) is an indicator function that is activated when ith category in the vocabulary V t i is in the set of categories of the candidate entity c, and e is a predicted entity. We observed that simply summing up the scores performed better than other options such as averaging or computing a log odds ratio. Intuitively, summing benefits candidates with many categories, which biases the model towards more frequent entities in a beneficial way. It also rewards models with many correlated types, which is problematic, but approaches we tried that handled type correlation in a more principled way did not perform better. There are certain types of entities in Wikipedia whose categories do not mesh well with our prediction task. For example, the page Employment about the general concept only has the category Employment, making resolution of this concept challenging. In these cases, we back off to a mentionentity prior (see in the Preprocessing Evaluation Data section). We call our combined system ET4EL (entity typing for entity linking). 4 Training Data for Typing To train the ET4EL model to cover a wide range of entities, we need access to a large set of entities labeled with types. We derive this data directly from Wikipedia: each hyperlinked occurrence of an entity on Wikipedia can be treated as a distantly supervised (Craven andKumlien 1999, Mintz et al. 2009) example from the standpoint of entity typing. The distant types for that mention are derived from the Wikipedia categories associated with the linked entity. Annotation First, we collect all sentences that contain hyperlinks, internal links pointing to other English Wikipedia articles, from all articles on English Wikipedia. Our data is taken from the March 2019 English Wiki dump. Given a sentence with a hyperlink, we use the hyperlink as a mention m, the whole sentence as a context sentence s, the destination of the hyperlink as an entity e, and the Wiki categories that are associated with e as a set of fine-grained types T . One sentence could have multiple hyperlinks. In this case, we create a tuple (m, s, e, T ) for each of the hyperlinks. This process results 88M examples. Importantly, our training examples for typing are tuples of (m, s, T ) since the ET4EL model is optimized towards the gold Wiki categories T and do not rely on the gold entity e. Category Set The original Wikipedia categories are mostly fine-grained and lack general categories. For example, the Wiki entity New York City has fine-grained categories such as Cities in New York (state) and Populated places established in 1624, but there are no general categories (e.g. Cities) that potentially useful to distinguish between two obviously different entities (e.g. location vs person). We expand the original categories if they contain prepositions. 5 We split the original category at the location of the first-occurring preposition. We chunk the left side into words and add them to the category set. We add the right side, a prepositional phrase, to the category set without modification; retaining the preposition helps keep the relation information. We also retain the original category. For the two original categories above, the new categories Cities, in New York (state), Populated, places, established, in 1624 would be added to the category set. 6 Past work has looked at deriving similar category sets over Wikipedia (Nastase and Strube 2008). Further improve-ments to our category set are possible, but we found the simple of rules we defined to be sufficient for our purposes. Training the Typing Model Since the total number of Wikipedia categories is very large (over 1 million), we train on a subset of the categories for efficiency. For a given test set, we only need access to categories that might possibly occur. We therefore restrict the categories to the most common n categories occurring with candidates in that dataset; note that this does not assume the existence of labeled targetdomain data, only unlabeled. To create the training set, we randomly select 6M examples that have at least one Wikipedia category from the restricted category vocabulary. We select 10k examples for the development set using the same procedure. The encoder may specialize somewhat to these types, but as we show later, it can handle large type sets and recognize diverse entity types (see the Results and Discussion section). Experiments We evaluate our approach on the development/test sets of the CoNLL-YAGO (Hoffart et al. 2011) dataset, which is a widely used entity linking benchmark. The CoNLL data consists of news documents and covers relatively narrow domains. Additionally, we test our model in a much harder setting where the mention-entity pairs are unseen during test time. We create the training, development, and test sets from the WikilinksNED dataset (Eshel et al. 2017), which contains a diverse set of ambiguous entities spanning more domains than the CoNLL data. We call this dataset Unseen-Mentions. The domain-specific training set is only used for the baseline models. The ET4EL model is still trained on the Wikipedia data. Unlike the CoNLL data, the examples in the Unseen-Mentions dataset are essentially single-sentence, meaning that resolution has to happen with limited context. Preprocessing Evaluation Data Candidate Selection For the CoNLL data, we use the publicly available candidate list, PPRforNED (Pershina, He, and Grishman 2015) that gives 99% gold recall on the testa (development) and the testb (test) sets. 7 For the Unseen-Mentions data, we use a mention-entity priorp(e\|m) to select candidate entities (Ganea and Hofmann 2017). We computep(e\|m) using the count statistics from the Wiki dump. We rank the candidate entities based onp(e\|m) and clip low frequency entities with a threshold 0.05. On average, this produces around 30 candidates per example and gives 88% gold recall on the development and test sets. 7 Other domain independent entity linking systems employ different resources to generate candidates, e.g., Gupta, Singh, and Roth (2017) use CrossWikis (Spitkovsky and Chang 2012) and restrict to 30 candidates per mention (for 95% gold recall). Lazic et al. (2015) use the Wikilinks corpus, Wikipedia articles, and Freebase (for 92% gold recall). Because all these systems make slightly different precision-recall tradeoffs in their candidate selection, direct comparison is difficult, but we believe the results are still reflective of the overall quality of the systems. Model Input Training Category Vocabulary To reduce more than 1 million total Wikipedia categories to a more tractable number for a given dataset, we use count statistics from the candidates of the training examples in that data set. Note that this process does not use the training labels at all; the data may as well be unlabeled. For each category, we count the number of associated unique mentions. We rank all the categories by the counts and select the top 60k categories as our vocabulary. Data Supervision ET4EL Baselines MOST FREQUENT ENTITY Given a mention m, we choose an entity with the highest mention-entity priorp(e\|m). We computep(e\|m) using the count statistics from the March 2019 Wiki dump. COSINE SIMILARITY This baseline selects an entity with the highest cosine similarity between the context and entity vectors using the pretrained word2vecf (Levy and Goldberg 2014). The context vector is obtained by mean pooling over the input word vectors. Note that this similarity is computed using distributed representations while traditional cosine similarity is based on word counts (Hoffart et al. 2011). GRU-ATTN Our implementation of the attention-based model introduced in Mueller and Durrett (2018). This model achieves state-of-the-art performance on the WikilinksNED dataset in the standard supervised setting. See Mueller and Durrett (2018) for more details. CBOW+WORD2VEC This simpler baseline model 8 uses the pretrained embeddings and a simple bag-of-words mentioncontext encoder. That is, the encoder is unordered bag-ofwords representations of the mention, the left context, and the right context. For each of three, the words are embedded and combined using mean pooling to give context representations. Similar to Eshel et al. (2017), we use word2vecf to initialize entity embeddings. We compare the context representations and the entity representation by following Mueller and Durrett (2018). The final representation is fed into a two-layer multilayer perceptron (MLP) with ReLU activations, batch normalization, and dropout. Training Data for CoNLL Baselines To train our baselines in a comparable fashion, we create training examples (m, s, e) from the Wikipedia data D Wiki to use for our learning-based baselines (GRU-ATTN and CBOW+WORD2VEC). We use the same mention-entity priorp(e\|m), explained in the previous section, to select candidates for each training example. We consider two variants of this training data. First, we train these baselines on a training set sampled uniformly from all of Wikipedia. Second, we give these baselines a more favorable transductive setting where the training entities from Wikipedia are restricted to only include entities that are candidates in the domain-specific training data. The CoNLL training set contains 2k unique entities. We collect 1.4M training examples from D Wiki that cover these 2k CoNLL entities; this training set should specialize these models to CoNLL fairly substantially, though they maintain our setting of not considering the training labels. Training Data for Unseen-Mentions Baselines To ensure that all mentions in the development and test sets do not appear in the training set, we split the WikilinksNED training set into train, development, and test sets by unique mentions (15.5k for train, 1k for dev, and 1k for test). This results 2.2M, 10k, and 10k examples 9 respectively. Our learningbased baselines (GRU-ATTN and CBOW+WORD2VEC) are trained on the 2.2M training examples, which do not share any entities with the dev or test sets. We also train the learning-based baselines on the Wikipedia data described in the Training Data for Typing section. Similar to the Unseen-Mentions data, we use a mention-entity priorp(e\|m) to select candidate entities. We obtain 2.5M training examples that have at least 2 candidate entities. Table 1 compares assumptions and resources for different systems. The ET4EL model is a so-called local entity linking model, as it only uses a single mention and context, rather than a global model which does collective inference over the whole document (Ratinov et al. 2011, Hoffart et al. 2011. However, this allows us to easily support entity linking with little context, as is the case for WikilinksNED. Comparison with other systems The chief difference from other models is that the ET4EL model is trained on data derived from Wikipedia, where the Results and Discussion CoNLL-YAGO drop by 7 points. The CoNLL testb set is slightly "out-ofdomain" for the training set with respect to the time period it was drawn from, indicating that our method may have better generalization than more conventional neural models in the transductive setting. We also list the state-of-the-art domain independent entity linking systems. Our model outperforms the full CDTE model of Gupta, Singh, and Roth (2017), as well as Plato in the supervised setting (Lazic et al. 2015), which is the same setting as ours. Our model is competitive with Plato in the semi-supervised setting, which additionally uses 50 million documents as unlabeled data. Le and Titov (2019)'s setting is quite different from ours that their model is a global model (requires document input) and trained on Wikipedia and 30k newswire documents from the Reuters RCV1 corpus (Lewis et al. 2004). Their model is potentially trained on domain-specific data since the CoNLL-YAGO dataset is derived from the RCV1 corpus. How much context information should we add? On the CoNLL dataset, sentences containing entity mentions are often quite short, but are embedded in a larger document. We investigate the most effective amount of context information to add to our typing model. Table 3 compares accuracy for the different amount of context. We test the context sentence only, the left and right 50 tokens of the mention span, and the first sentence of the document. Adding the left and right 50 tokens of the mention span improves the accuracy over the context sentence only. Adding the first sentence of the document improves the accuracy over the context sentence only (no additional context) by 4 points. 10 Since the documents are news articles, the first sentence usually has meaningful information about the topics. This is especially useful when the document is a list of sports results, and a sentence does not have rich context. For example, one sentence is "Michael Johnson ( U.S. ) 20.02", which is highly uninformative, but the first sentence of the document is "ATHLET-ICS -BERLIN GRAND PRIX RESULTS." Our model correctly predicts Michael Johnson (sprinter) after giving more context information about the sport. Does the category vocabulary size matter? We show the performance on the development set with different numbers Table 6: Macro-averaged P/R/F1. Entity typing performance on the categories grouped by frequency. (1-100) is the most frequent group, and (10001+) is the least frequent group. of categories. As we can see in Table 4, the development accuracy monotonically increases as the category size goes up. Even the 1k most frequent category set can achieve reasonable performance, 85% accuracy. However, the model is able to make use of even very fine-grained categories to make correct predictions. Table 5 compares accuracy of our model and baselines on this dataset. Our model achieves the best performance in this setting, better than all baselines. Notably, the GRU+ATTN model, which achieves state-of-the-art performance on WikilinksNED, performs poorly, underperforming the MOST FREQUENT ENTITY baseline. The simpler CBOW+WORD2VEC model with the frozen entity embeddings shows slightly better performance than the GRU+ATTN model, implying that the model suffers from overfitting to the training data. The poor performance of these two models trained on the domain-specific data suggests that dealing with unseen mention-entity pairs is challenging even for these vector-based approaches trained with similar domain data, indicating that entity generalization is a major factor in entity linking performance. The GRU+ATTN model trained on the Wikipedia data also performs poorly. The baseline models trained on the domain-specific data even make mistakes in easy cases such as disambiguating between PERSON and LOCATION entities. For example, a mention spans is [Kobe], and an associated entity could be Kobe Bryant, a former basketball player, or Kobe, a city in Japan. Those baseline models guess Kobe Bryant correctly but get confused with Kobe. Our model predict both entities correctly; the context is usually indicative enough. Unseen-Mentions Typing Analysis In the Training Data for Typing section, we described how we added more general types to the category set. We compare the original Wikipedia category set and the expanded category set on the CoNLL development set. Using 30k categories in both settings, the original set and expanded set achieve accuracies of 84.4 and 87.1 respectively, showing that our refined type set helps substantially. Table 6 shows the typing performance on the 60k categories grouped by frequency. The first group (1-100) consists of the 100 most frequent categories. The fourth group (10001+) is formed with the least frequent categories. Precision is relatively high for all groups. The first group (1-100) achieves the highest precision, recall, and F1, possibly leveraging the rich training examples. Recall drastically decreases between the first group and the subsequent groups, which suggests the model has difficulty accounting for the imbalanced nature of the classification of rare tags. We further look at the performance of selected individual categories. We observe that having rich training examples, in general, leads the high performance. For example, births occurs with more than 2k unique mentions in the training set and achieves P:99/R:89/F1:93.7. However, history has more than 900 unique mentions in the training set but only achieves P:76.9/R:6.1/F1:11.4. This might be related to the purity of mentions (and entities). Most of the mentions for births are person entities, and this category is consistently applied. On the other hand, history may not denote a well-defined semantic type. Related Work Wikipedia categories have been used to construct ontologies (Suchanek, Kasneci, and Weikum 2007) and predict general concepts (Syed, Finin, and Joshi 2008). The internal link information in Wikipedia as supervision has also been studied extensively in the field of entity linking and named entity disambiguation in the past decade (Bunescu and Paşca 2006, Mihalcea and Csomai 2007, Nothman, Curran, and Murphy 2008, McNamee et al. 2009). Another approach utilizes manually annotated domain-specific data, using either neural techniques (He et al. 2013, Sun et al. 2015, Francis-Landau, Durrett, and Klein 2016 or various joint models Klein 2014, Nguyen et al. 2016). Learning pretrained entity representations from knowledge bases has also been studied for entity linking (Hu et al. 2015, Yamada et al. 2016, Eshel et al. 2017). Many of these approaches are orthogonal to ours and could be combined in a real system. Conclusion In this paper, we presented an entity typing approach that addresses the issue of overfitting to the entity distribution of a specific domain. Our approach does not rely on labeled entity linking data in the target domain and models fine-grained entity properties. With the domain independent setting, our approach achieves strong results on the CoNLL dataset. In a harder setting of unknown entities derived from the WikilinksNED dataset, our approach generalizes better than a state-of-the-art model on the dataset. Figure 2 : 2Entity typing for entity linking (ET4EL) model. Given a mention m and a sentence s, the entity typing model Φ computes a binary probability for membership in each type. Then the entity linking predictor Ω makes the final prediction based on summed type posteriors: the model chooses Big Bang over Big Bang Theory based on these scores (1.75 vs 0.6). for our entity typing model design and hyperparameter choices. Encoder The mention m and the sentence s are converted into sequences of contextualized word vectors s and m using ELMo (Peters et al. 2018). The sentence vectors s are concatenated with the location embedding and fed into a bi-LSTM encoder followed by a span attention layer (Lee et al. 2017, Choi et al. 2018): s = Attention(bi-LSTM([s ; ])), where s is the final representation of the sentence s. The mention vectors m are fed into another bi-LSTM and summed by a span attention layer to obtain the word-level mention representation: m word = Attention(bi-LSTM(m )). Table 1 : 1Assumptions and resources for different entity linking systems. Our model only requires supervision from Wikipedia and trains using typing supervision (from categories) only. Table 3 : 3Accuracy on the CoNLL development set (testa) with different amounts of context fed to our model. Adding the first sentence of the document gives the best performance because this is often indicative of topic in this dataset (e.g., what sport is being discussed). Category Size 1k 5k 30k 60k Dev 85.1 85.6 87.1 88.1 Table 4 : 4Accuracy on the CoNLL development set (testa) with different numbers of categories. supervision comes from categories attached to entity men- tions. Moreover, we only use Wikipedia as a source of train- ing data; some other work like Le and Titov (2019) uses un- labeled data from the same domain as the CoNLL-YAGO test set. Table 2 2shows accuracy of our model and baselines. Our model outperforms all baselines by a substantial margin. The MOST FREQUENT ENTITY baseline performs poorly on both development and test set. Interestingly, the simpler CBOW+WORD2VEC model is the strongest baseline here, outperforming the GRU+ATTN model in both general and transductive settings. Our model achieves the strongest per- formance on both the dev and test data. Interestingly, our model also has a much smaller drop from dev to test, only losing 2 points, compared to the transductive models, which Table 5 : 5Accuracy on the Unseen-Mentions test set. Our model substantially outperforms neural entity linking mod- els in this setting. Throughout this work, when we say entity linking, we refer to the task of disambiguating a given entity mention, not the full detection and disambiguation task which this sometimes refers to. We use italics to denote Wikipedia titles and true type to represent Wikipedia categories. Compared to other models we considered, such as BERT(Devlin et al. 2019), this approach was more stable and more scalable to use large amounts of Wikipedia data. The code for experiments is available at https://github.com/ yasumasaonoe/ET4EL 5 We use 'in', 'from', 'for', 'of', 'by', 'for', 'involving.' 6 Other splits are possible, e.g. extracting 20th century from 20th century philosophers. However, these are more difficult to reliably identify. This model shows comparable performance to GRU-ATTN, achieving 76.0 accuracy on the original test set of the Wik-ilinksNED data, comparable to the performance of 75.8 reported inMueller and Durrett (2018). Development and test are subsampled from their "raw" sizes of 130k token-level examples. Our baselines use this setting as well since we found it to work the best. AcknowledgmentsThis work was partially supported by NSF Grant IIS-1814522, NSF Grant SHF-1762299, a Bloomberg Data Science Grant, and an equipment grant from NVIDIA. The authors acknowledge the Texas Advanced Computing 8582 Center (TACC) at The University of Texas at Austin for providing HPC resources used to conduct this research. Results presented in this paper were obtained using the Chameleon testbed supported by the National Science Foundation. 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[ "Self-Assembling Modular Networks for Interpretable Multi-Hop Reasoning", "Self-Assembling Modular Networks for Interpretable Multi-Hop Reasoning" ]	[ "Yichen Jiang [email protected] \nUNC Chapel Hill\n\n", "Mohit Bansal [email protected] \nUNC Chapel Hill\n\n" ]	[ "UNC Chapel Hill\n", "UNC Chapel Hill\n" ]	[ "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing" ]	Multi-hop QA requires a model to connect multiple pieces of evidence scattered in a long context to answer the question. The recently proposed HotpotQA dataset is comprised of questions embodying four different multi-hop reasoning paradigms (two bridge entity setups, checking multiple properties, and comparing two entities), making it challenging for a single neural network to handle all four. In this work, we present an interpretable, controllerbased Self-Assembling Neural Modular Network(Hu et al., 2017(Hu et al., , 2018for multi-hop reasoning, where we design four novel modules (Find, Relocate, Compare, NoOp) to perform unique types of language reasoning. Based on a question, our layout controller RNN dynamically infers a series of reasoning modules to construct the entire network. Empirically, we show that our dynamic, multi-hop modular network achieves significant improvements over the static, single-hop baseline (on both regular and adversarial evaluation). We further demonstrate the interpretability of our model via three analyses. First, the controller can softly decompose the multi-hop question into multiple single-hop sub-questions to promote compositional reasoning behavior of the main network. Second, the controller can predict layouts that conform to the layouts designed by human experts. Finally, the intermediate module can infer the entity that connects two distantly-located supporting facts by addressing the sub-question from the controller. 1	10.18653/v1/d19-1455	[ "https://www.aclweb.org/anthology/D19-1455.pdf" ]	202,565,945	1909.05803	2d41ee8cff39d105fb553eff782fdc55647d47c0	Self-Assembling Modular Networks for Interpretable Multi-Hop Reasoning November 3-7, 2019 Yichen Jiang [email protected] UNC Chapel Hill Mohit Bansal [email protected] UNC Chapel Hill Self-Assembling Modular Networks for Interpretable Multi-Hop Reasoning Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language ProcessingHong Kong, ChinaNovember 3-7, 20194474 Multi-hop QA requires a model to connect multiple pieces of evidence scattered in a long context to answer the question. The recently proposed HotpotQA dataset is comprised of questions embodying four different multi-hop reasoning paradigms (two bridge entity setups, checking multiple properties, and comparing two entities), making it challenging for a single neural network to handle all four. In this work, we present an interpretable, controllerbased Self-Assembling Neural Modular Network(Hu et al., 2017(Hu et al., , 2018for multi-hop reasoning, where we design four novel modules (Find, Relocate, Compare, NoOp) to perform unique types of language reasoning. Based on a question, our layout controller RNN dynamically infers a series of reasoning modules to construct the entire network. Empirically, we show that our dynamic, multi-hop modular network achieves significant improvements over the static, single-hop baseline (on both regular and adversarial evaluation). We further demonstrate the interpretability of our model via three analyses. First, the controller can softly decompose the multi-hop question into multiple single-hop sub-questions to promote compositional reasoning behavior of the main network. Second, the controller can predict layouts that conform to the layouts designed by human experts. Finally, the intermediate module can infer the entity that connects two distantly-located supporting facts by addressing the sub-question from the controller. 1 Introduction The task of multi-hop question answering (QA) requires the model to answer a natural language question by finding multiple relevant information pieces scattered in a given natural language context. It has attracted more attention recently and multiple datasets have been proposed, including the recent HotpotQA that is comprised of questions embodying four different multi-hop reasoning paradigms: inferring the bridge entity to complete the 2nd-hop question (first question in Fig. 1), inferring the answer through a bridge entity, checking multiple properties to select the answer, and comparing two entities (second question in Fig. 1). For the first question, it is necessary to first find the person "who portrayed Corliss Archer in the film Kiss and Tell", and then find out the "government position" she held. For the second question, one may arrive at the answer by finding the country where Scott and Ed are from, and then comparing the two nationalities to conclude whether they are the same. Multi-hop QA is more challenging than singlehop QA for two main reasons. First, the techniques used for single-hop QA are not sufficient to answer a multi-hop question. In single-hop QA tasks like SQuAD (Rajpurkar et al., 2016), the evidence necessary to answer the question is concentrated in a short context (Q: "What is the color of the grass", Context: "The grass is green.", An-swer: "Green"). Such datasets emphasize the role of matching the information between the question and the short context surrounding the answer ("sky→sky, color→blue"), and can be solved by building a question-aware context representation (Seo et al., 2017;Xiong et al., 2017). In contrast, for multi-hop QA, directly matching the semantics of the question and context leads to the entity that bridges the two supporting facts (e.g., "Shirley Temple"), or the entities that need to be compared against each other (e.g., nationalities of Scott and Ed). In both cases, further action is required on top of direct semantic matching in order to get to the final answer. Second, Hot-potQA is comprised of questions of four different types of multi-hop reasoning paradigms. In Fig. 1, the first question embodies a sequential reasoning path where the model has to solve a subquestion to get an entity "Shirley Temple", which then leads to the answer to the main question about her "government position". The second question, on the other hand, requires a tree-structured reasoning path where the model first locates two entities on the leaves and then compares them to get the answer. The difference in the required reasoning skills makes it hard for a single static model to handle all types of questions. To automatically discover the multiple elaborated reasoning steps as required in HotpotQA, a model needs to dynamically adopt a sequence of different reasoning behaviors based on specific questions, which is still unexplored in the field of large-scale text-based QA. In our work, we present a highly-interpretable self-assembling Neural Modular Network (Andreas et al., 2016a;Hu et al., 2017) with three novel modules designed for multi-hop NLP tasks: Find, Relocate, Compare, where each module embodies a unique type of reasoning behavior. The Find module is similar to the previously-introduced 1-hop biattention model (Seo et al., 2017;Xiong et al., 2017), which produces an attention map over the context words given the context and the question representation. For the first example in Fig. 1, a Find module is used to find the answer ("Shirley Temple") to the sub-question ("who portrayed Corliss Archer ..."). The Relocate module takes the intermediate answer to the previous sub-question ("Shirley Temple"), combines it with the current sub-question ("What government position was held by ..."), and then outputs a new attention map conditioned on the previous one. The Compare module intuitively "compares" the outputs from two previous modules based on the question. For the second example in Fig. 1, after previous modules find the nationality of "Scott" and "Ed" (both American), the Compare module outputs the answer "Yes" based on the word "same" in the question. We also use the NoOp module that keeps the current state of the network unchanged when the model decides no more action is needed. After all the reasoning steps are performed, the final attention map is used to predict the answer span, and the vector output from the Compare module is used to predict whether the answer should be a context span or "Yes/No". Next, to dynamically assemble these modules into a network based on the specific question, we use a controller RNN that, at each step, infers the required reasoning behavior from the question, outputs the sub-question, and predicts a soft combination of all available modules. As shown in Fig. 1, a modular network Relocate(Find()) is constructed for the bridge-type question and another network with the layout Compare(Find(), Find()) is built for the comparison-type question. In order to make the entire model end-to-end differentiable in gradient descent, we follow Hu et al. (2018) to use continuous and soft layout prediction and maintain a differentiable stack-based data structure to store the predicted modules' output. This approach to optimize the modular network is shown to be superior to using a Reinforcement Learning approach which makes hard module decisions. Fig. 2 visualizes this controller that predicts the modular layout and the network assembled with the selected modules. We further apply intermediate supervision to the first produced attention map to encourage finding the bridge entity ("Shirley Temple" in Fig. 1). The entire model is trained end-to-end using cross entropy loss for all supervision. Overall, our self-assembling controller-based Neural Modular Network achieves statistically significant improvements over both the single-hop bi-attention baseline and the original modular network (Hu et al., 2018) designed for visual-domain QA. We also present adversarial evaluation (Jiang and Bansal, 2019), where single-hop reasoning shortcuts are eliminated and compositional reasoning is enforced; and here our NMN again outperforms the BiDAF baseline significantly in the EM score (as well as after adversarial training). We further demonstrate the interpretability of our modular network with three analyses. First, the controller understands the multi-hop semantics of the question and can provide accurate step-bystep sub-questions to the module to lead the main network to follow the reasoning path. Second, the controller can successfully predict layouts that conform to the layouts designed by human experts. Finally, before arriving at the final answer, the intermediate module can infer the bridge entity that connects two distantly-located supporting facts by leveraging the step-specific sub-question from the controller. All of these analyses show that our Modular Network is not operating as a black box, but demonstrates highly interpretable compositional reasoning behavior, which is beneficial in transparent model development and safe, trustworthy real-world applications. In summary, the contribution of this work is three-fold: 1) This is the first work to apply self-assembling modular networks to text-based QA; 2) We design three novel modules to handle multi-hop questions in HotpotQA; 3) The resulting network is interpretable in terms of the model's intermediate outputs and the assembled layout. Related Works Multi-hop Reading Comprehension The last few years have witnessed significant progress on large-scale QA datasets including clozestyle tasks (Hermann et al., 2015), open-domain QA (Yang et al., 2015), and more (Rajpurkar et al., 2016(Rajpurkar et al., , 2018. However, all of the above datasets are confined to a single-document context per question domain. Joshi et al. (2017) introduced a multi-document QA dataset with some questions requiring cross-sentence inferences to answer. The bAbI dataset (Weston et al., 2016) requires the model to combine multiple pieces of evidence in the synthetic text. Welbl et al. (2017) uses Wikipedia articles as the context and a subject-relation pair as the query, and constructs the multi-hop QAngaroo dataset by traversing a directed bipartite graph so that the evidence required to answer a query could be spread across multiple documents. HotpotQA is a more recent multi-hop QA dataset that has crowd-sourced questions with more diverse syntactic and semantic features compared to QAngaroo. It includes four types of questions, each re-quiring a different reasoning paradigm. Some examples require inferring the bridge entity from the question (Type I in ), while others demand fact-checking or comparing subjects' properties from two different documents (Type II and comparison question). Concurrently to our work, Min et al. (2019b) also tackle HotpotQA by decomposing its multi-hop questions into singlehop sub-questions to achieve better performance and interpretability. However, their system approaches the question decomposition by having a decomposer model trained via human labels, while our controller accomplishes this task automatically with the soft attention over the questionwords' representation and is only distantly supervised by the answer and bridge-entity supervision, with no extra human labels. Moreover, they propose a pipeline system with the decomposers, an answer-prediction model, and a decomposition scorer trained separately on the previous stage's output. Our modular network, on the other hand, is an end-to-end system that is optimized jointly. Neural Modular Network Neural Modular Network (NMN) is a class of models that are composed of a number of sub-modules, where each sub-module is capable of performing a specific subtask. In NMN (Andreas et al., 2016b), N2NMN (Hu et al., 2017), PG+EE (Johnson et al., 2017) and TbD (Mascharka et al., 2018), the entire reasoning procedure starts by analyzing the question and decomposing the reasoning procedure into a sequence of sub-tasks, each with a corresponding module. This is done by either a parser (Andreas et al., 2016b) or a layout policy (Hu et al., 2017;Johnson et al., 2017;Mascharka et al., 2018) that turns the question into a module layout. Then the module layout is executed with a neural module network. Overall, given an input question, the layout policy learns what sub-tasks to perform, and the neural modules learn how to perform each individual sub-task. However, since the the modular layout is sampled from the controller, the controller itself is not end-to-end differentiable and has to be optimized using Reinforcement Learning Algorithms like Reinforce (Williams, 1992). Hu et al. (2018) used soft program execution where the output of each step is the average of outputs from all modules weighted by the module distribution, and showed its superiority over hard-layout NMNs. All previous works in NMN, including Hu et al. (2018) targeted visual question answering or structured knowledge-based GeoQA, and hence the modules are designed to process image or KB inputs. We are the first to apply modular networks to unstructured, text-based QA, where we redesigned the modules for language-based reasoning by using bi-attention (Seo et al., 2017;Xiong et al., 2017) to replace convolution and multiplication of the question vector with the image feature. Our model also has access to the full-sized bi-attention vector before it is projected down to the 1-d distribution. Architecture Learning Our work also shares the spirit of recent research on Neural Architecture Search (NAS) (Zoph and Le, 2016;Pham et al., 2018;Liu et al., 2018), since the architecture of the model is learned dynamically based on controllers instead of being manually-designed. However, Neural Architecture Search aims to learn the structure of the individual CNN/RNN cell with fixed inter-connections between the cells, while Modular Networks have preset individual modules but learns the way to assemble them into a larger network. Moreover, Modular Networks' architectures are predicted dynamically on each data point, while previous NAS methods learn a single cell structure independent of the example. Model In this section, we describe how we apply the Neural Modular Network to the multi-hop HotpotQA task . In Sec. 3.2 and Sec. 3.3, we describe the controller, which sketches the model layout for a specific example, and introduce how it assembles the network with the predicted modules. In Sec. 3.4, we go into details of every novel module that we design for this task. Encoding We use a Highway Network (Srivastava et al., 2015) to merge the character embeddings and GloVe word embeddings (Pennington et al., 2014), building word representations for the context and the question as x ∈ R S×v and q ∈ R J×v respectively, where S and J are the lengths of the context and the question. We then apply a bidirectional LSTM-RNN (Hochreiter and Schmidhuber, 1997) of d hidden units to get the contextualized word representations for the context and question: h = BiLSTM(x); u = BiLSTM(q) so that h ∈ R S×2d and u ∈ R J×2d . We also use a self attention layer (Zhong et al., 2019) to get qv, a fixed-sized vector representation of the question. Model Layout Controller In a modular network, the controller reads the question and predicts a series of our novel modules that could be executed in order to answer the given question. For multi-hop QA, each module represents a specific 1-hop reasoning behavior and the controller needs to deduce a chain of reasoning behavior that uncovers the evidence necessary to infer the answer. Similar to the model in Hu et al. (2018), at the step t, the probability of selecting module i is calculated as: q t = W 1,t · qv + b 1,t cq t = W 2 · [c t−1 ; q t ] + b 2 s t = W 3 · cq t p t,i = Softmax(s t )(1) where c t−1 is the controller's hidden state and q t is the vector representation of the question coming out of the encoding LSTM. W 3 projects the output to the dimension of 1×I, where I is the number of available modules. When predicting the next module, the controller also supplies the current module, which we design for the language domain, with a fixed-sized vector as the sub-question at the current reasoning step (the gray-shaded area of the controller in Fig. 2). Consider the first example in Fig. 1. At the first reasoning step, the Find module's sub-question is "the woman who portrayed Corliss Archer in the film Kiss and Tell". To generate this sub-question vector, the controller first calculates a distribution over all question words and then computes the weighted average of all the word representations: e t,j = W 4 · (cq t · u j ) + b 4 cv t = Softmax(e t ) c t = j cv t,j · u j(2) The sub-question vector c t also serves as the controller's hidden state and is used to calculate the module probability in the next reasoning step. Similarly, in the second reasoning step, the subquestion of the Relocate module is "What government position was held by" and the answer from the Find module at the first step. Stack NMN with Soft Program Execution In our NMN, some modules may interact with the attention output from previous modules to collect information that could benefit the current reasoning step. For tree-structured layouts, such as Compare(Find(), Find()) as shown in Fig. 1, the latter Compare module requires access to previous outputs. We follow Hu et al. (2018) to use a stack to store attention outputs generated by our modules. In the tree-structured layout, the two Find modules push two attention outputs onto the stack, and the Compare module pops these two attention outputs to compare the content they each focus on. To make the entire model differentiable, we also adopt soft program execution from Hu et al. (2018), where the output of each step is the sum of outputs from all modules weighted by the module distribution (blue-shaded area in Fig. 2) computed by the controller. NMN Modules We next describe all the modules we designed for our Neural Modular Network. All modules take the question representation u, context representation h, and sub-question vector c t as input. The core modules that produce the attention map over the context are based on bi-attention (Seo et al., 2017) between the question and context, instead of relying on convolution as in previous NMNs (Hu et al., 2017(Hu et al., , 2018. Every module outputs a fixedsize vector/zero vector as the memory state of the NMN, and is also able to push/pop attention maps onto/from the stack. These modules, each solving a single-hop sub-question, are chained together according to the layout predicted by the controller to infer the final answer to the multi-hop question. Find This module performs the bidirectional attention (Seo et al., 2017) between question u and context h = h · c t . By multiplying the original context representation by c t , we inject the sub-question into the following computation. The model first computes a similarity matrix M J×S between every question and context word and use it to derive context-to-query attention: M j,s = W 1 u j + W 2 h s + W 3 (u j h s ) p j,s = exp(M j,s ) J j=1 exp(M j,s ) c q s = J j=1 p j,s u j(3) where W 1 , W 2 and W 3 are trainable parameters, and is element-wise multiplication. Then the query-to-context attention vector is derived as: m s = max 1≤s≤S M j,s p s = exp(m s ) S s=1 exp(m s ) q c = S s=1 p s h S(4) We then obtain the question-aware context representation as: h = [h ; c q ; h c q ; c q q c ], and push this bi-attention result onto the stack. This process is visualized in the first two steps in Fig. 2. Relocate For the first example in Fig. 1, the second reasoning step requires finding the "government position" held by the woman, who is identified in the first step. We propose the Relocate module to model this reasoning behavior of finding the answer based on the information from the question as well as the previous reasoning step. The Relocate module first pops an attention map att 1 from the stack and computes the bridge entity's representation b as the weighted average over h using the popped attention. It then computes a bridge-entity-awared representation of context h b , and then applies a Find 4479 module between h b and question u: b = s att 1,s · h s h b = h b h = Find(u, h b , c t ) (5) The output h is then pushed onto the stack. Compare As shown in the final step in Fig. 2, the Compare module pops two attention maps att 1 and att 2 from the stack and computes two weighted averages over h using the attention maps. It then merges these two vectors with c t to update the NMN's memory state m. hs 1 = s att 1,s · h s ; hs 2 = s att 2,s · h s o in = [c t ; hs 1 ; hs 2 ; c t · (hs 1 − hs 2 )] m = W 1 · (ReLU(W 2 · o in ))(6) The intuition is that the Compare module aggregates the information from two attention maps and compares them according to the sub-question c t . NoOp This module updates neither the stack nor the memory state. It can be seen as a skip command when the controller decides no more computation is required for the current example. Bridge Entity Supervision Even with the multi-hop architecture to capture the hop-specific distribution over the question, there is no supervision on the controller's output distribution c about which part of the question is related to the current reasoning step, thus preventing the controller from learning the composite reasoning skill. To address this problem, we supervise the first Find module to predict the bridge entity ("Shirley Temple" in Fig. 1), which indirectly encourages the controller to look for question information related to this entity ("the woman who portrayed Corliss Archer...") at the first step. Since the original dataset does not label the bridge entity, we apply a simple heuristic to detect the bridge entity that is the title of one supporting document while also appearing in the other supporting document. 2 Prediction Layer After all the reasoning steps have completed, to predict a span from the context as the answer, we pop the top attention map from the stack, apply self-attention , and project it down to get the span's start index and end index. To predict yes/no, we take the final memory (m in Eqn. 6) and project it down to a 2-way classifier. We concatenate the question vector qv and memory m and then project down to a 2-way classifier to decide whether to output a span or yes/no. Optimization Previous works on modular network (Hu et al., 2017(Hu et al., , 2018 optimize the controller parameters θ jointly with the modules parameters ω on the training set. However, we found that our controller converges 3 in less than 20 iterations under this training routine. It is also likely that this setup causes the model to overfit to the single-hop reasoning shortcuts in the training set. Hence, to prevent such premature convergence and reasoningshortcut overfitting, we adopt the 2-phase training strategy that is widely used in Neural Architecture Search (Zoph and Le, 2016;Pham et al., 2018). The first phase updates ω for an entire epoch, followed by the second phase that updates θ over the entire dev set. We alternate between the two phases until the entire system converges. Experimental Setup Dataset and Metric We conduct our training and evaluation on the HotpotQA part-of-speech tags and NER tag of the question using Corenlp . For each question whose answer is either "yes" or "no", we generate a new question by randomly sampling two titles of the same type (based on POS and NER) from the training set to substitute the original entities in the question and corresponding supporting documents (1st row in Table 1). We then employ three strategies to generate 5,342 extra questions by mutating original questions. First, if the question contains the word "same" and the answer is yes or no, we substitute "same" with "different" and vice versa (2nd row in Table 1). Second, we detect the comparative and superlative adjectives/adverbs in the original question, transform them into their antonyms using wordnet, and then transform the antonyms back to their comparative form (3rd row in Table 1). Finally, if the question has a comparative adj./adv., we flip the order of the two entities compared (4th row in Table 1). In all three cases, the answer to the mutated question is also flipped. Training Details We use 300-d GloVe pretrained embeddings (Pennington et al., 2014). The model is supervised to predict either the start and end index of a span or "Yes/No" for specific questions. The entire model (controller + modular net-work) is trained end-to-end using the Adam optimizer (Kingma and Ba, 2014) with an initial learning rate of 0.001. Results Baseline Our baseline is the bi-attention (Seo et al., 2017) + self-attention model as in (Clark and Gardner, 2018;, which was shown to be able to achieve strong performance on single-hop QA tasks like SQuAD (Rajpurkar et al., 2016) and TriviaQA (Joshi et al., 2017). Our baseline shares the preprocessing and encoding layer with the modular network. Primary NMN Results We first present our model's performance on the HotpotQA dev and test set of our split. As shown in the first three rows of Table 2, our modular network achieves significant improvements over both the baseline and the convolution-based NMN (Hu et al., 2018) on our test set. In Table 3, we further break down the devset performance on different question types, 4 and our modular network obtains higher scores in both question types compared to the BiDAF baseline. Ablation Studies Data Augmentation: We also conduct ablation studies on our modeling choices. Augmenting the comparison-type questions in the training set boosts the performance on the comparison-type questions in the dev set (comparing row 3 and 4 in Table 3) without harming the scores on the bridgetype questions too much. Bridge-entity Supervision: Supervising the bridge entity (described in 3.5) proves to be Table 4: EM scores after training on the regular data or on the adversarial data from Jiang and Bansal (2019), and evaluation on the regular dev set or the adv-dev set. beneficial for the modular network to achieve good performance (row 5 in Table 2 and Table 3). Modules: As shown in the 6th to 8th row, removing either the Compare, Relocate or NoOp module also causes drops in the metric scores. Specifically, removing the Relocate module causes significant degrade in bridge-type questions, which solidifies our claim that relocating the attention based on the inferred bridge entity is important for compositional reasoning. Similarly, removing the Compare module harms the model's performance on comparisontype questions, suggesting the effectiveness of the module in addressing questions that require comparing two entities' properties. These results demonstrate the contribution of each module toward achieving a self-assembling modular network with the strong overall performance. Comparison with Original NMN Modules One primary contribution of this work is that we adapt neural modular networks (NMN) (Andreas et al., 2016a;Hu et al., 2017Hu et al., , 2018, which were designed for visual-domain QA, to text-domain QA by rebuilding every reasoning module. We substitute convolution and multiplication between question vectors and context features with biattention as the basic reasoning component in the Find and Relocate. Moreover, our model maintains a stack of attention outputs before it is projected down to 1-d, thus enabling skip connections when predicting the answer span. As shown in Table 2 and Table 3, our adapted modular network outperforms the original NMN significantly. Adversarial Evaluation Multiple previous works (Chen and Durrett, 2019;Min et al., 2019a) have shown that models performing strongly on HotpotQA are not necessarily capable of compositional reasoning. Jiang and Bansal (2019) proposed to construct adversarial distractor documents to eliminate the reasoning shortcut and necessitate compositional reasoning on HotpotQA dataset. To test whether our modular network can perform robust multi-hop reasoning against such adversaries, we evaluate our models on the adversarial dev set. The second column of Table 4 shows that our NMN outperforms the baseline significantly (+10 points in EM score) on the adversarial evaluation, suggesting that our NMN is indeed learning stronger compositional reasoning skills compared to the BiDAF baseline. We further train both models on the adversarial training set, and the results are shown in the last two columns of Table 4. We observe that after adversarial training, both the baseline and our NMN obtain significant improvement on the adversarial evaluation, while our NMN maintains a significant advantage over the BiDAF baseline. Analysis In this section, we present three analysis methods to show that our multi-hop NMN is highly interpretable and makes decisions similar to human actions. Combining the analyses below, we can understand how our model reasons to break the multi-step task into multiple single-step subtasks that are solvable by specific modules, and harnesses intermediate outputs to infer the answer. Controller's Attention on Questions Different from single-hop questions, a multihop question usually encodes a sequence of subquestions, among which the final sub-question requires an explicit answer. Therefore, the first step of solving a multi-hop question is to identify the hidden sub-questions and to sort them according to the correct reasoning order. Our controller handles this task by computing an attention distribution over all question words (see details in Sec. 3.2) at every reasoning step, which signifies which part of the question forms a sub-question that should be answered at the current step. Fig. 3 visualizes four W h a t g o v e rn m e n t p o s it io n w a s h e ld b y th e w o m a n w h o p o rt ra it C o rl is s A rc h e r in th e fi lm K is s a n d T e ll Step 1: Step 2: T h e a re n a w h e re th e L e w is to n M a in e ia c s p la y e d th e ir h o m e g a m e s c a n s e a t h o w m a n y p e o p le Step 1: Step 2 Step 2: W h o w a s k n o w n b y h is n a m e A la d in a n d h e lp e d o rg a n iz a ti o n s im p ro v e th e ir p e rf o rm a n c e a s s ta g e a c o n s u lt a n t Step 1: Step 2: Kiss and Tell is a 1945 American comedy film starring then 17-year-old Shirley Temple as Corliss Archer. ... Step 1: Shirley Temple Black was an American actress, ..., and also served as Chief of Protocol of the United States. Step 2: The Lewiston Maineiacs were a junior ice hockey team, played its home games at the Androscoggin Bank Colisée. Step 1: The Androscoggin Bank Colisée ... is a 4,000 capacity (3,677 seated) multi-purpose arena Step 2: Step 1: Step 2: Scott Derrickson is an American director. ... Edward Wood Jr. was an American filmmaker. ... Eenasul Fateh, also known by his stage name Aladin, ..., and former international management consultant. Step 1: Management consulting is the practice of helping organizations to improve their performance, Step 2: Figure 3: The controller's multi-hop attention map on the question (randomly chosen from the first 10 dev examples). This attention is used to compute the subquestion representation that is passed to the modules. attention maps from the controller when dealing with questions of different reasoning paradigms (randomly chosen from the first 10 dev examples), suggesting that our controller is able to accurately identify the sub-questions in the correct order. In the first example, the controller is able to attend to the first sub-question "who portrayed Corliss Archer in ...", and then relocate the attention to the next sub-question "What government position was held by ..." to complete the 2-step reasoning. Predicting Network Layouts After the controller decides the sub-question at the current reasoning step, it then predicts a soft layout of the the current step by outputting a distribution cv over all available modules (Find, Relocate, Compare, NoOp). As elaborated in Sec. 3.3, the output vector and attention map of all modules are weighted by cv. To quantitatively evaluate the layout predicted by the controller, we label the expert layout for each ques-tion in the dev set (Find-Relocate for bridgetype questions and Find-Find/Relocate-Compare for comparison-type questions). Next, we convert the soft module predictions into hard modular layouts by picking the module with the largest prediction probability at each step. Intuitively, we want the controller to assign higher weights to the module that can perform the desired reasoning. We then compute the percentage of dev set examples where the predicted layout matches the expert layout exactly. The empirical study shows that the layouts predicted by our NMN controller match the expert layout in 99.9% of bridge-type questions and 68.6% of comparison-type questions with Yes/No answer (examples shown in Fig. 1 and Fig. 2). Finding Intermediate Bridge-Entities The final action of an intermediate reasoning step is the predicted module reasoning through the question and context to answer the sub-question. For bridge-type questions (e.g., the first example in Fig. 3), the required action is to find the name ("Shirley Temple") of "the woman who portrait Corliss Archer ..." to complete the next subquestion "What government position was held by Shirley Temple". For comparison questions (e.g., the third example in Fig. 3), the model needs to infer the nationalities of "Scott" and "Ed" that will be compared in the following step. Conclusion In this work, we proposed a self-assembling neural modular network for multi-hop QA. We designed three modules that reason between the question and text-based context. The resulting model outperforms both the single-hop baseline and the original NMN on HotpotQA . Because of the interpretable nature of our model, we presented analyses to show that our model does in fact learn to perform compositional reasoning and can dynamically assemble the modular network based on the question. Figure 1 : 1Two HotpotQA examples and the modular network layout predicted by the controller. Figure 2 : 2Modular network with a controller (top) and the dynamically-assembled modular network (bottom). At every step, the controller produces a sub-question vector and predicts a distribution to weigh the averages of the modules' outputs. Table 1 : 1Examples of mutated comparison-type questions and answers from HotpotQA training set.Dev Test EM F1 EM F1 BiDAF Baseline 44.68 57.19 42.7 55.81 NMN 31.04 40.28 30.87 39.90 Our NMN 50.67 63.35 49.58 62.71 + Data aug. 50.63 63.29 49.46 62.59 -Bridge sup. 46.56 58.60 45.91 57.22 -Relocate 47.81 60.22 46.75 59.23 -Compare 50.29 63.30 48.87 62.52 -NoOp 49.11 61.79 48.56 62.10 Table 2 : 2EM and F1 scores on HotpotQA dev set and test set. All models are tuned on dev set. Table 3 : 3EM and F1 scores on bridge-type and comparison-type questions from HotpotQA dev set. Train Reg Reg Adv Adv Eval Reg Adv Reg Adv BiDAF Baseline 43.12 34.00 45.12 44.65 Our NMN 50.13 44.70 49.33 49.25 :W a s S c o tt D e rr ic k s o n a n d E d W o o d o f th e s a m e n a ti o n a li ty Step 1: Our code is publicly available at: https://github. com/jiangycTarheel/NMN-MultiHopQA There are two supporting documents per example. Generating modular layout with probability near 1. 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[ "How Deep Learning Sees the World: A Survey on Adversarial Attacks & Defenses", "How Deep Learning Sees the World: A Survey on Adversarial Attacks & Defenses" ]	[ "Joana C Costa ", "Tiago Roxo ", "Senior Member, IEEEHugo Proença ", "Senior Member, IEEEPedro R M Inácio " ]	[]	[ "ARXIV" ]	Deep Learning is currently used to perform multiple tasks, such as object recognition, face recognition, and natural language processing. However, Deep Neural Networks (DNNs) are vulnerable to perturbations that alter the network prediction (adversarial examples), raising concerns regarding its usage in critical areas, such as self-driving vehicles, malware detection, and healthcare. This paper compiles the most recent adversarial attacks, grouped by the attacker capacity, and modern defenses clustered by protection strategies. We also present the new advances regarding Vision Transformers, summarize the datasets and metrics used in the context of adversarial settings, and compare the state-of-the-art results under different attacks, finishing with the identification of open issues.	10.48550/arxiv.2305.10862	[ "https://export.arxiv.org/pdf/2305.10862v1.pdf" ]	258,762,374	2305.10862	d135e14810052e40d59857fb393f33e367b44149	How Deep Learning Sees the World: A Survey on Adversarial Attacks & Defenses MAY 2023 1 Joana C Costa Tiago Roxo Senior Member, IEEEHugo Proença Senior Member, IEEEPedro R M Inácio How Deep Learning Sees the World: A Survey on Adversarial Attacks & Defenses ARXIV MAY 2023 1Index Terms-Adversarial attacksadversarial defensesdatasetsevaluation metricssurveyvision transformers Deep Learning is currently used to perform multiple tasks, such as object recognition, face recognition, and natural language processing. However, Deep Neural Networks (DNNs) are vulnerable to perturbations that alter the network prediction (adversarial examples), raising concerns regarding its usage in critical areas, such as self-driving vehicles, malware detection, and healthcare. This paper compiles the most recent adversarial attacks, grouped by the attacker capacity, and modern defenses clustered by protection strategies. We also present the new advances regarding Vision Transformers, summarize the datasets and metrics used in the context of adversarial settings, and compare the state-of-the-art results under different attacks, finishing with the identification of open issues. I. INTRODUCTION M ACHINE Learning (ML) algorithms have been able to solve various types of problems, namely highly complex ones, through the usage of Deep Neural Networks (DNNs) [1], achieving results similar to, or better than, humans in multiple tasks, such as object recognition [2], [3], face recognition [4], [5], and natural language processing [6], [7]. These networks have also been employed in critical areas, such as self-driving vehicles [8], [9], malware detection [10], [11], and healthcare [12], [13], whose application and impaired functioning can severely impact their users. Promising results shown by DNNs lead to the sense that these networks could correctly generalize in the local neighborhood of an input (image). These results motivate the adoption and integration of these networks in real-time image analysis, such as traffic sign recognition and vehicle segmentation, making malicious entities target these techniques. However, it was discovered that DNNs are susceptible to small perturbations in their input [14], which entirely alter their prediction, making it harder for them to be applied in critical areas. These perturbations have two main characteristics: 1) invisible to the Human eye or slight noise that does not alter Human prediction; and 2) significantly increase the confidence of erroneous output, the DNNs predict the wrong class with higher confidence than all other classes. As a result of these assertions, the effect of the perturbations has been analyzed with more focus on object recognition, which will also be the main target of this survey. Papernot et al. [15] distinguishes four types of adversaries depending on the information they have access to: (i) training The authors are with Instituto de Telecomunicações, Universidade da Beira Interior, Portugal. Manuscript received XX, 2023; revised XX, 2023. data and network architecture, (ii) only training data or only network, (iii) oracle, and (iv) only pairs of input and output. In almost all real scenarios, the attacker does not have access to the training data or the network architecture, which diminishes the strength of the attack performed on a network, leaving the adversary with access to the responses given by the network, either by asking questions directly to it or by having pairs of input and prediction. Furthermore, the queries to a model are usually limited or very expensive [16], making it harder for an attacker to produce adversarial examples. Multiple mechanisms [17]- [20] were proposed to defend against legacy attacks, already displaying their weakened effect when adequately protected, which are clustered based on six different domains in this survey. Regardless of the attacks and defenses already proposed, there is no assurance about the effective robustness of these networks and if they can be trusted in critical areas, clearly raising the need to make the DNNs inherently robust or easy to be updated every time a new vulnerability is encountered. This motivates the presented work, whose main contributions are summarized as follows: • We present the most recent adversarial attacks grouped by the adversary capacity, accompanied by an illustration of the differences between black-box and white-box attacks; • We propose six different domains for adversarial defense grouping, assisted by exemplificative figures of each of these domains, and describe the effects of adversarial examples in ViTs; • We detail the most widely used metrics and datasets, present state-of-the-art results on CIFAR-10, CIFAR-100, and ImageNet, and propose directions for future works. The remaining of the paper is organized as follows: Section II provides background information; Section III compares this review with others; Section IV presents the set of adversarial attacks; Section V shows a collection of defenses to overcome these attacks; Section VII displays the commonly used datasets; Section VIII lists and elaborates on metrics and presents state-of-the-art results; and Section IX presents future directions, with the concluding remarks included in Section X. II. BACKGROUND FOR ADVERSARIAL ATTACKS A. Neural Network Architectures When an input image is fed into a CNN, it is converted into a matrix containing the numeric values representing the image or, if the image is colored, a set of matrices containing the numeric values for each color channel. Then, the Convolutions apply filters to these matrixes and calculate a set of reducedsize features. Finally, these features have an array format fed into the Fully Connected that classifies the provided image. Figure 1 shows an elementary example of CNNs used to classify images. Contrary to the CNN, ViT does not receive the image as a whole as input; instead, it is pre-processed to be divided into Patches, which are smaller parts of the original image, as displayed in Figure 2. These Patches are not fed randomly to the Transformer Encoder, they are ordered by their position, and both the Patches and their position are fed into the Transformer Encoder. Finally, the output resulting from the Transformer Encoder is fed into the Multi-Layer Perceptron (MLP) Head that classifies the image. B. Adversarial Example Misclassification might be justified if the object contained in the image is not visible even to Humans. However, adversarial examples do not fit this scope. These examples add a perturbation to an image that causes the DNNs to misclassify the object in the image, yet Humans can correctly classify the same object. The adversarial attacks described throughout this survey focus on identifying the adversarial examples that make DNNs misclassify. These attacks identify specific perturbations that modify the DNN classification while being correctly classified by Humans. The calculation of these perturbations is an optimization problem formally defined as: arg min δX δ X s.t. f(X + δ X ) = Y * ,(1) where f is the is the classifier, δ X is the perturbation, X is the original/benign image, and Y * is the adversarial output. Furthermore, the adversarial example is defined as: X * = X + δ X ,(2) Fig. 3. Adversarial Examples created using different state-of-the-art adversarial attacks. The first column represents the original image; the second represents the perturbation used to generate the adversarial images in the third column. The images were resized for better visualization. Images withdrawn from [14], [21], [22]. The first perturbation follows the edges of the building, the second is concentrated in the area of the whale, and the third is more smooth and greater in area. where X * is the adversarial image. Figure 3 displays adversarial examples generated using different attacks. Mainly, the first row is the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) [14] attack, the second row is the DeepFool [21] attack, and the third row is the SmoothFool [22] attack. When observing the L-BFGS, the perturbation applies noise to almost the entirety of the adversarial image. The DeepFool attack only perturbs the area of the whale but not all the pixels in that area. Finally, the SmoothFool attack slightly disturbs the pixels in the area of the image. These three attacks display the evolution of adversarial attacks in decreasing order of detectability and, consequently, increasing order of strength. To limit the noise that each perturbation can add to an image, the adversarial attacks are divided into L 0 , L 2 , and L p norms, known as Vector Norms. Furthermore, commonly used terminologies in the context of adversarial examples are defined in Table I. C. Vector Norms and Constraint Vector Norms are functions that take a vector as input and output a positive value (scalar). These functions are essential to ML and allow the backpropagation algorithms to compute the loss value as a scalar. The family of these functions is known as the p-norm, and, in the context of adversarial attacks, the considered values for p are 0, 2, and ∞. L 0 norm consists of counting the number of non-zero elements in the vector and is formally given as: \|\|x\|\| 0 = (\|x 1 \| 0 +\|x 2 \| 0 +... + \|x n \| 0 ),(3) where x 1 to x n are the elements of the vector x. L 2 norm, also known as the Euclidean distance, measures the vector distance to the origin and is formally defined as: \|\|x\|\| 2 = (\|x 1 \| 2 +\|x 2 \| 2 +... + \|x n \| 2 ) 1 2 ,(4) where x 1 to x n are the elements of the vector x. L ∞ norm represents the maximum hypothetical value that p can have and returns the absolute value of the element with the largest magnitude, formally as: \|\|x\|\| ∞ = max i \|x i \|,(5) where x i is each element of the vector x. A geometric representation of the area of exploitation for the three considered p-norm is displayed in Figure 4. One relevant property of the p-norm is: the higher p is, the more important the contribution of large errors; the lower p is, the higher the contribution of small errors. This translates into a large p benefiting small maximal errors (minimal perturbations along multiple pixels) and a small p encouraging larger spikes in fewer places (abrupt perturbations along minimal pixels). Therefore, l 2 and l 0 attacks have greater detectability than l ∞ attacks, with the latter being more threatening. Another constraint normally seen in the context of adversarial attacks is , which is a constant that controls the amount of noise, via generated perturbation, that can be added to an image. Usually, it is a tiny number and varies depending on the used dataset, decreasing when the task increases in difficulty. According to the literature, for MNIST, = 0.1, for CIFAR-10 and CIFAR-100, = 8/255, and for ImageNet, = 4/255. D. Adversary Goals and Capacity Besides the restriction imposed by the different Vector Norms, the adversarial attacks are also divided by their impact on the networks. Depending on the goals of the attacker, the designation is as follows: • Confidence Reduction, the classifier outputs the original label with less confidence; • Untargeted, the classifier outputs any class besides the original label; • Targeted, the classifier outputs a particular class besides the original label. Another important aspect of adversarial attacks is the amount of knowledge the attacker has access to. As defined by Papernot et al. [15], who proposed the first threat model for deep learning, the attackers can have access to: 1) data training and network architecture; 2) only network architecture; 3) 4) an oracle that replies to all the inputs given; and 5) only have pairs of input and corresponding output (samples). However, to simplify this classification, these capacities were divided into: • White-box, which considers that the attacker has access to either the architecture or data; • Black-box, when the attacker can only access samples from an oracle or in pairs of input and output. The attackers goals and capacity are essential to classify the strength of an attack. For example, the easiest is a Confidence Reduction White-box attack, and the strongest is a Targeted Black-box attack. III. RELATED SURVEYS The first attempt to summarize and display the recent developments in this area was made by Akhtar and Mian [23]. These authors studied adversarial attacks in computer vision, extensively referring to attacks for classification and providing a brief overview of attacks beyond the classification problem. Furthermore, the survey presents a set of attacks performed in the real world and provides insight into the existence of adversarial examples. Finally, the authors present the defenses distributed through three categories: modified training or input, modifying networks, and add-on networks. From a broader perspective, Liu et al. [24] studied the security threats and possible defenses in ML scope, considering the different phases of an ML algorithm. For example, the training phase is only susceptible to poisoning attacks; however, the testing phase is vulnerable to evasion, impersonation, and inversion attacks, making it harder to defend. The authors provide their insight on the currently used techniques. Additionally, focusing more on the object recognition task, Serban et al. [25] Qui et al. [26] extensively explains background concepts in Adversarial Attacks, mentioning adversary goals, capabilities, and characteristics. It also displays applications for adversarial attacks and presents some of the most relevant adversarial defenses. Furthermore, it explains a set of attacks divided by the stage in which they occur, referring to the most relevant attacks. Xu et al. [27] also describes background concepts, describing the adversary goals and knowledge. This review summarizes the most relevant adversarial attacks at the time of that work and presents physical world examples. Furthermore, the authors present a batch of defenses grouped by the underlying methodology. Finally, there is an outline of adversarial attacks in graphs, text, and audio networks, culminating in the possible applications of these attacks. Chakraborty et al. [28] provides insight into commonly used ML algorithms and presents the adversary capabilities and goals. The presented adversarial attacks are divided based on the stage of the attack (train or test). Additionally, the authors present relevant defenses used in adversarial settings. Long et al. [29] discusses a set of preliminary concepts of Computer Vision and adversarial context, providing a set of adversarial attacks grouped by adversary goals and capabilities. Finally, the authors provide a set of research directions that readers can use to continue the development of robust networks. Liang et al. [30] discuss the most significant attacks and defenses in the literature, with the latter being grouped by the underlying technique. This review finishes with a presentation of the challenges currently existing in the adversarial context. More recently, Zhou et al. [31] provides insight into Deep Learning and Threat Models, focusing on the Cybersecurity perspective. Therefore, the authors identify multiple stages based on Advanced Persistent Threats and explain which adversarial attacks are adequate for each stage. Similarly, the same structure is followed to present the appropriate defenses for each stage. Furthermore, this survey presents the commonly used datasets in adversarial settings and provides a set of future directions from a Cybersecurity perspective. From the analysis of the previous surveys, some concepts have already been standardized, such as adversary goals and capabilities and the existence of adversarial attacks and defenses. However, due to the recent inception of this area, there still needs to be more standardization in datasets and metrics. Therefore, with this survey, we also analyze datasets and met-rics to provide insight to novice researchers. Furthermore, this survey consolidates the state-of-the-art results and identifies which datasets can be further explored. Finally, similarly to other reviews, this paper provides a set of future directions that researchers and practitioners can follow to start their work. A comparison between the several surveys discussed in this section is summarized in Table II. IV. ADVERSARIAL ATTACKS Adversarial attacks are commonly divided by the amount of knowledge the adversaries have access to, white-box and black-box, as can be seen in Figure 5. A. White-box Settings Adversarial examples were first proposed by Szegedy et al. [14], which discovered that DNNs do not generalize well in the vicinity of an input. The same authors proposed L-BFGS, the first adversarial attack, to create adversarial examples and raised awareness in the scientific community for this generalization problem. Fast Gradient Sign Method (FGSM) [32] is a one-step method to find adversarial examples, which is based on the linear explanation for the existence of adversarial examples, and is calculated using the model cost function, the gradient, and the radius epsilon. This attack is formally defined as: x − · sign(∇loss F,t (x)),(6) where x is the original image, is the amount of changes to the image, and t is the target label. The value for should be very small to make the attack undetectable. Jacobian-based Saliency Maps (JSM) [15] explore the forward derivates to calculate the model gradients, replacing the gradient descent approaches, and discover which input regions are likely to yield adversarial examples. Then it uses saliency maps to construct the adversarial saliency maps, which display the features the adversary must perturb. Finally, to prove the effectiveness of JSM, only the adversarial examples correctly classified by humans were used to fool neural networks. DeepFool [21] is an iterative attack that stops when the minimal perturbation that alters the model output is found, exploiting its decision boundaries. It finds the minimum perturbation for an input x 0 , corresponding to the vector orthogonal to the hyperplane representing the decision boundary. Kurakin et al. [33] was the first to demonstrate that adversarial examples can also exist in the physical world, by using three different methods to generate the adversarial examples. Basic Iterative Method (BIM) applies the FGSM multiple times with a small step size between iterations and clips the intermediate values after each step. Iterative Least-likely Class Method (ILCM) uses the least-likely class, according to the prediction of the model, as the target class and uses BIM to calculate the adversarial example that outputs the target class. Carlini and Wagner (C&W) [34] attack is one of the most powerful attacks, which uses three different vector norms: 1) the L 2 attack uses a smoothing of clipped gradient descent approach, displaying low distortion; 2) the L 0 attack uses an iterative algorithm that, at each iteration, fixes the pixels that do not have much effect on the classifier and finds the minimum amount of pixels that need to be altered; and 3) the L ∞ attack also uses an iterative algorithm with an associated penalty, penalizing every perturbation that exceeds a predefined value, formally defined as: min c · f (x + δ) + i [(δ i − τ ) + ],(7) where δ is the perturbation, τ is the penalty threshold (initially 1, decreasing in each iteration), and c is a constant. The value for c starts as a very low value (e.g., 10 −4 ), and each time the attack fails, the value for c is doubled. If c exceeds a threshold (e.g., 10 10 ), it aborts the search. Gradient Aligned Adversarial Subspace (GAAS) [35] is an attack that directly estimates the dimensionality of the adversarial subspace using the first-order approximation of the loss function. Through the experiments, GAAS proved the most successful at finding many orthogonal attack directions, indicating that neural networks generalize linearly. Projected Gradient Descent (PGD) [36] is an iterative attack that uses saddle point formulation, viewed as an inner maximization problem and an outer minimization problem, to find a strong perturbation. It uses the inner maximization problem to find an adversarial version of a given input that achieves a high loss and the outer minimization problem to find model parameters that minimize the loss in the inner maximization problem. The saddle point problem used by PGD is defined as: min θ ρ(θ), where ρ(θ) = E (x,y)∼D max δ∈S L(θ, x + δ, y) , (8) where x is the original image, y is the corresponding label, and S is the set of allowed perturbations. AdvGAN [37] uses Generative Adversarial Networks (GAN) [38] to create adversarial examples that are realistic and have high attack success rate. The generator receives the original instance and creates a perturbation, the discriminator distinguishes the original instance from the perturbed instance, and the target neural network is used to measure the distance between the prediction and the target class. Motivated by the inability to achieve a high success rate in black-box settings, the Momentum Iterative FGSM (MI-FGSM) [39] was proposed. It introduces momentum, a technique for accelerating gradient descent algorithms, into the already proposed Iterative FGSM (I-FGSM), showing that the attack success rate in black-box settings increases almost double that of previous attacks. Croce and Hein [40] noted that the perturbations generated by l 0 attacks are sparse and by l ∞ attacks are smooth on all pixels, proposing Sparse and Imperceivable Adversarial Attacks (SIAA). This attack creates sporadic and imperceptible perturbations by applying the standard deviation of each color channel in both axis directions, calculated using the two immediate neighboring pixels and the original pixel. SmoothFool (SF) [22] is a geometry-inspired framework for computing smooth adversarial perturbations, exploiting the decision boundaries of a model. It is an iterative algorithm that uses DeepFool to calculate the initial perturbation and smoothly rectifies the resulting perturbation until the adversarial example fools the classifier. This attack provides smoother perturbations which improve the transferability of the adversarial examples, and their impact varies with the different categories in a dataset. In the context of exploring the adversarial examples in the physical world, the Adversarial Camouflage (AdvCam) [41], which crafts physical-world adversarial examples that are legitimate to human observers, was proposed. It uses the target image, region, and style to perform a physical adaptation (creating a realistic adversarial example), which is provided into a target neural network to evaluate the success rate of the adversarial example. Feature Importance-aware Attack (FIA) [42] considers the object-aware features that dominate the model decisions, using the aggregate gradient (gradients average concerning the feature maps). This approach avoids local optimum, represents transferable feature importance, and uses the aggregate gradient to assign weights identifying the essential features. Furthermore, FIA generates highly transferable adversarial examples when extracting the feature importance from multiple classification models. Meta Gradient Adversarial Attack (MGAA) [43] is a novel architecture that can be integrated into any existing gradient-based attack method to improve cross-model transferability. This approach consists of multiple iterations, and, in each iteration, various models are samples from a model zoo to generate adversarial perturbations using the selected model, which are added to the previously generated perturbations. In addition, using multiple models simulates both white-and black-box settings, making the attacks more successful. B. Universal Adversarial Perturbations Moosavi-Dezfooli et al. [44] discovered that some perturbations are image-agnostic (universal) and cause misclassification with high probability, labeled as Universal Adversarial Perturbations (UAPs). The authors found that these perturbations also generalize well across multiple neural networks, by searching for a vector of perturbations that cause misclassification in almost all the data drawn from a distribution of images. The optimization problem that Moosavi-Dezfooli et al. are trying to solve is the following: ∆v i ← − arg min r r 2 s.t.k(x i + v + r) =k(x i ),(9) where ∆v i is the minimal perturbation to fool the classifier, v is the universal perturbation, and x i is the original image. This optimization problem is calculated for each image in a dataset, and the vector containing the universal perturbation is updated. The Universal Adversarial Networks (UAN) [45] are Generative Networks that are capable of fooling a classifier when their output is added to an image. These networks were inspired by the discovery of UAPs, which were used as the training set and can create perturbations for any given input, demonstrating more outstanding results than the original UAPs. C. Black-box Settings Specifically considering black-box setup, Ilyas et al. [46] define three realistic threat models that are more faithful to real-world settings: query-limited, partial information, and label-only settings. The first one suggests the development of query-efficient algorithms, using Natural Evolutionary Strategies to estimate the gradients used to perform the PGD attack. When only having the probabilities for the top-k labels, the algorithm alternates between blending in the original image and maximizing the likelihood of the target class and, when the attacker only obtains the top-k predicted labels, the attack uses noise robustness to mount a targeted attack. Feature-Guided Black-Box (FGBB) [47] uses the features extracted from images to guide the creation of adversarial perturbations, by using Scale Invariant Feature Transform. High probability is assigned to pixels that impact the composition of an image in the Human visual system and the creation of adversarial examples is viewed as a two-player game, where the first player minimizes the distance to an adversarial example, and the second one can have different roles, leading to minimal adversarial examples. Square Attack [48] is an adversarial attack that does not need local gradient information, meaning that gradient masking does not affect it. Furthermore, this attack uses a randomized search scheme that selects localized square-shaped updates in random positions, causing the perturbation to be situated at the decision boundaries. D. Auto-Attack Auto-Attack [49] was proposed to test adversarial robustness in a parameter-free, computationally affordable, and user-independent way. As such, Croce et al. proposed two variations of PGD to overcome suboptimal step sizes of the objective function, namely APGD-CE and APGD-DLR, for a step size-free version of PGD using cross-entropy (CE) and Difference of Logits Ratio (DLR) loss, respectively. DLR is a loss proposed by Croce et al. which is both shift and rescaling invariant and thus has the same degrees of freedom as the decision of the classifier, not suffering from the issues of the cross-entropy loss [49]. Then, they combine these new PGD variations with two other existing attacks to create Auto-Attack, which is composed by: • APGD-CE, step size-free version of PGD on the crossentropy; • APGD-DLR, step size-free version of PGD on the DLR loss; • Fast Adaptive Boundary (FAB) [50], which minimizes the norm of the adversarial perturbations; • Square [48] Attack, a query-efficient black-box attack. Given the main motivation of the Auto-Attack proposal, the FAB attack is the targeted version of FAB [50] since the untargeted version computes each iteration of the Jacobian matrix of the classifier, which scales linearly with the number of classes of the dataset. Although this is feasible for datasets with a low number of classes (e.g., MNIST and CIFAR-10), it becomes both computationally and memory-wise challenging with an increased number of classes (e.g., CIFAR-100 and ImageNet). As such, Auto-Attack is an ensemble of attacks with important fundamental properties: APGD is a white-box attack aiming at any adversarial example within an L p -ball (Section II-C), FAB minimizes the norm of the perturbation necessary to achieve a misclassification, and Square Attack is a score-based black-box attack for norm bounded perturbations which use random search and do not exploit any gradient approximation, competitive with white-box attacks [48]. V. ADVERSARIAL DEFENSES A. Adversarial Training Szegedy et al. [14] proposed that training on a mixture of adversarial and clean examples could regularize a neural network, as shown in Figure 6. Goodfellow et al. [32] evaluated the impact of Adversarial Training as a regularizer by including it in the objective function, showing that this approach is a reliable defense that can be applied to every neural network. Kurakin et al. [51] demonstrates that it is possible to perform adversarial training in more massive datasets (ImageNet), displaying that the robustness significantly increases for onestep methods. When training the model with one-step attacks using the ground-truth labels, the model has significantly higher accuracy on the adversarial images than on the clean images, an effect denominated as Label Leaking, suggesting that the adversarial training should not make use of the groundtruth labels. Adversarial Training in large datasets implies using fast single-step methods, which converge to a degenerate global minimum, meaning that models trained with this technique remain vulnerable to black-box attacks. Therefore, Ensemble Adversarial Training [52] uses adversarial examples crafted on other static pre-trained models to augment the training data, preventing the trained model from influencing the strength of the adversarial examples. Shared Adversarial Training [53] is an extension of adversarial training aiming to maximize robustness against universal perturbations. It splits the mini-batch of images used in training into a set of stacks and obtains the loss gradients concerning these stacks. Afterward, the gradients for each stack are processed to create a shared perturbation that is applied to the whole stack. After every iteration, these perturbations are added and clipped to constrain them into a predefined magnitude. Finally, these perturbations are added to the images and used for adversarial training. TRadeoff-inspired Adversarial DEfense via Surrogateloss minimization (TRADES) [54] is inspired by the presumption that robustness can be at odds with accuracy [55], [56]. The authors show that the robust error can be tightly bounded by using natural error measured by the surrogate loss function and the likelihood of input features being close to the decision boundary (boundary error). These assumptions make the model weights biased toward natural or boundary errors. Based on the idea that gradient magnitude is directly linked to model robustness, Bilateral Adversarial Training (BAT) [57] proposes to perturb not only the images but also the manipulation of labels (adversarial labels) during the training phase. The adversarial labels are derived from a closed-form heuristic solution, and the adversarial images are generated from a one-step targeted attack. Adversarial Attack Classifier Adversarial Images Despite the popularity of adversarial training to defend models, it has a high cost of generating strong adversarial examples, namely for large datasets such as ImageNet. Therefore, Free Adversarial Training (Free-AT) [58] uses the gradient information when updating model parameters to generate the adversarial examples, eliminating the previously mentioned overhead. Considering the same issue presented in Free-AT, the authors analyze Pontryagin's Maximum Principle [59] of this problem and observe that the adversary update is only related to the first layer of the network. Thus, You Only Propagate Once (YOPO) [60] only considers the first layer of the network for forward and backpropagation, effectively reducing the amount of propagation to one in each update. Misclassification Aware adveRsarial Training (MART) [61] is an algorithm that explicitly differentiates the misclassified and correctly classified examples during training. This proposal is motivated by the finding that different maximization techniques are negligible, but minimization ones are crucial when looking at the misclassified examples. Defense against Occlusion Attacks (DOA) [62] is a defense mechanism that uses abstract adversarial attacks, Rectangular Occlusion Attack (ROA) [62], and applies the standard adversarial training. This attack considers including physically realizable attacks that are "normal" in the real world, such as eyeglasses and stickers on stop signs. The proposal of Smooth Adversarial Training (SAT) [63] considers the evolution normally seen in curriculum learning, where the difficulty increases with time (age), using two difficulty metrics. These metrics are based on the maximal Hessian eigenvalue (H-SAT) and the softmax Probability (P-SAT), which are used to stabilize the networks for large perturbations while having high clean accuracy. In the same context, Friendly Adversarial Training (Friend-AT) [64] minimizes the loss considering the least adversarial data (friendly) among the adversarial data that is confidently misclassified. This method can be employed by early stopping PGD attacks when performing adversarial training. Contrary to the idea of Free-AT [58], Cheap Adversarial Training (Cheap-AT) [65] proposes the use of weaker and cheaper adversaries (FGSM) combined with random initialization to train robust networks effectively. This method can be further accelerated by applying techniques that efficiently train networks. In a real-world context, the attacks are not limited by the imperceptibility constraint ( value); there are, in fact, multiple perturbations (for models) that have visible sizes. The main idea of Oracle-Aligned Adversarial Training (OA-AT) [66] is to create a model that is robust to high perturbation bounds by aligning the network predictions with ones of an Oracle during adversarial training. The key aspect of OA-AT is the use of Learned Perceptual Image Patch Similarity [67] to generate Oracle-Invariant attacks and convex combination of clean and adversarial predictions as targets for Oracle-Sensitive samples. Geometry-aware Instance-reweighted Adversarial Training (GI-AT) [68] has two foundations: 1) overparameterized models still lack capacity; and 2) a natural data point closer to the class boundary is less robust, translating into assigning the corresponding adversarial data a larger weight. Therefore, this defense proposes using standard adversarial training, considering that weights are based on how difficult it is to attack a natural data point. Adversarial training leads to unfounded increases in the margin along decision boundaries, reducing clean accuracy. To tackle this issue, Helper-based Adversarial Training (HAT) [69] incorporates additional wrongly labeled examples during training, achieving a good trade-off between accuracy and robustness. As a result of the good results achieved by applying random initialization, Fast Adversarial Training (FAT) [70] performs randomized smoothing to optimize the inner maximization problem efficiently, and proposes a new initialization strategy, named backward smoothing. This strategy helps to improve the stability and robustness of a model using single-step robust training methods, solving the overfitting issue. B. Modify the Training Process Gu and Rigazio [71] proposed using three preprocessing techniques to recover from the adversarial noise, namely, noise injection, autoencoder, and denoising autoencoder, discovering that the adversarial noise is mainly distributed in the high-frequency domain. Solving the adversarial problem corresponds to encountering adequate training techniques and objective functions to increase the distortion of the smallest adversarial examples. Another defense against adversarial examples is Defensive Distillation [72], which uses the predictions from a previously trained neural network, as displayed in Figure 7. This approach trains the initial neural network with the original training data and labels, producing the probability of the predictions, which replace the original training labels to train a smaller and resilient distilled network. Additionally, to improve the results obtained by Defensive Distillation, Papernot and Mc-Daniel [73] propose to change the vector used to train the distilled network by combining the original label with the first model uncertainty. To solve the vulnerabilities of the neural network to adversarial examples, the Visual Causal Feature Learning [74] method uses causal reasoning to perform data augmentation. This approach uses manipulator functions that return an image similar to the original one with the desired causal effect. Learning with a Strong Adversary [75] is a training procedure that formulates as a min-max problem, making the [72]. An Initial Network is trained on the dataset images and labels (discrete values). Then, the predictions given by the Initial Network are fed into another network, replacing the dataset labels. These predictions are continuous values, making the Distilled Network more resilient to adversarial attacks. classifier inherently robust. This approach considers that the adversary applies perturbations to each data point to maximize the classification error, and the learning procedure attempts to minimize the misclassification error against the adversary. The greatest advantage of this procedure is the significant increase in robustness while maintaining clean high accuracy. Zheng et al. [76] proposes the use of compression, rescaling, and cropping in benign images to increase the stability of DNNs, denominated as Image Processing, without changing the objective functions. A Gaussian perturbation sampler perturbs the benign image, which is fed to the DNN, and its feature representation of benign images is used to 1) minimize the standard CE loss; and 2) minimize the stability loss. Zantedeschi et al. [77] explored the standard architectures, which usually employ Rectified Linear Units (ReLU) [78], [79] to ease the training process, and discovered that this function makes a small perturbation in the input accumulate with multiple layers (unbounded). Therefore, the authors propose the use of bounded ReLU (BReLU) [80] to prevent this accumulation and Gaussian Data Augmentation to perform data augmentation. Zhang and Wang [19] suggest that adversarial examples are generated through Feature Scattering (FS) in the latent space to avoid the label leaking effect, which considers the interexample relationships. The adversarial examples are generated by maximizing the feature-matching distance between the clean and perturbed examples, FS produces a perturbed empirical distribution, and the DNN performs standard adversarial training. PGD attack causes the internal representation to shift closer to the "false" class, Triplet Loss Adversarial (TLA) [81] includes an additional term in the loss function that pulls natural and adversarial images of a specific class closer and the remaining classes further apart. This method was tested with different samples: Random Negative (TLA-RN), which refers to a randomly sampled negative example, and Switch Anchor (TLA-SA), which sets the anchor as a natural example and the positive to be adversarial examples. Kumari et al. [82] analyzes the previously adversarialtrained models to test their vulnerability against adversarial attacks at the level of latent layers, concluding that the latent layer of these models is significantly vulnerable to adversarial perturbations of small magnitude. Latent Adversarial Training (LAT) [82] consists of finetuning adversarial-trained models to ensure robustness at the latent level. Curvature Regularization (CR) [83] minimizes the curvature of the loss surface, which induces a more "natural" behavior of the network. The theoretical foundation behind this defense uses a locally quadratic approximation that demonstrates a strong relation between large robustness and small curvature. Furthermore, the proposed regularizer confirms the assumption that exhibiting quasi-linear behavior in the proximity of data points is essential to achieve robustness. Unsupervised Adversarial Training (UAT) [84] enables the training with unlabeled data considering two different approaches, UAT with Online Target (UAT-OT) that minimizes a differentiable surrogate of the smoothness loss, and UAT with Fixed Targets (UAT-FT) that trains an external classifier to predict the labels on the unsupervised data and uses its predictions as labels. Robust Self-Training (RST) [85], an extension of Self-Training [86], [87], uses a standard supervised training to obtain pseudo-labels and then feeds them into a supervised training algorithm that targets adversarial robustness. This approach bridges the gap between standard and robust accuracy, using the unlabeled data, achieving high robustness using the same number of labels as required for high standard accuracy. SENSEI [88] and SENSEI-SA [88] use the methodologies employed in software testing to perform data augmentation, enhancing the robustness of DNNs. SENSEI implements the strategy of replacing each data point with a suitable variant or leaving it unchanged. SENSEI-SA improves the previous one by identifying which opportunities are suitable for skipping the augmentation process. Bit Plane Feature Consistency (BPFC) [89] regularizer forces the DNNs to give more importance to the higher bit planes, inspired by the Human visual system perception. This regularizer uses the original image and a preprocessed version to calculate the l 2 norm between them and regularize the loss function, as the scheme shown in Figure 8. Adversarial Weight Perturbation (AWP) [90] explicitly regularizes the flatness of weight loss landscape and robustness gap, using a double-perturbation mechanism that disturbs both inputs and weights. This defense boosts the robustness of multiple existing adversarial training methods, confirming that it can be applied to other methods. Self-Adaptive Training (SAT) [91] dynamically calibrates the training process with the model predictions without extra computational cost, improving the generalization of corrupted data. In contrast with the double-descent phenomenon, SAT exhibits a single-descent error-capacity curve, mitigating the overfitting effect. HYDRA [92] is another technique that explores the effects of pruning on the robustness of models, which proposes using pruning techniques that are aware of the robust training objective, allowing this objective to guide the search for connections to prune. This approach reaches compressed models that are state-of-the-art in standard and robust accuracy. Based on the promising results demonstrated by previous distillation methods, the Robust Soft Label Adversarial Distillation (RSLAD) [ [89]. This method applies multiple operations to input images, simulating adversarial images. Then, the loss is changed to include a regularizer (new term), which compares the original images with these manipulated images. robust small student DNNs. This method uses the Robust Soft Labels (RSLs) produced by the teacher DNN to supervise the student training on natural and adversarial examples. An essential aspect of this method is that the student DNN does not access the original complex labels through the training process. The most sensitive neurons in each layer make significant non-trivial contributions to the model predictions under adversarial settings, which means that increasing adversarial robustness stabilizes the most sensitive neurons. Sensitive Neuron Stabilizing (SNS) [94] includes an objective function dedicated explicitly to maximizing the similarities of sensitive neuron behaviors when providing clean and adversarial examples. Dynamic Network Rewiring (DNR) [95] generates pruned DNNs that have high robust and standard accuracy, which employs a unified constrained optimization formulation using a hybrid loss function that merges ultra-high model compression with robust adversarial training. Furthermore, the authors propose a one-shot training method that achieves high compression, standard accuracy, and robustness, which has a practical inference 10 times faster than traditional methods. Manifold Regularization for Locally Stable (MRLS) [96] DNNs exploit the continuous piece-wise linear nature of ReLU to learn a function that is smooth over both predictions and decision boundaries. This method is based on approximating the graph Laplacian when the data is sparse. Inspired by the motivation behind distillation, Learnable Boundary Guided Adversarial Training (LBGAT) [97], assuming that models trained on clean data embed their most discriminative features, constrains the logits from the robust model to make them similar to the model trained on natural data. This approach makes the robust model inherit the decision boundaries of the clean model, preserving high standard and robust accuracy. Low Temperature Distillation (LTD) [98], which uses previous distillation frameworks to generate labels, uses relatively low temperatures in the teacher model and employs different fixed temperatures for the teacher and student models. The main benefit of this mechanism is that the generated soft labels can be integrated into existing works without additional costs. Recently, literature [99]- [101] demonstrated that neural Ordinary Differential Equations (ODE) are naturally more robust to adversarial attacks than vanilla DNNs. Therefore, Stable neural ODE for deFending against adversarial attacks (SODEF) [102] uses optimization formulation to force the Detector Network No Adversarial Example? Classifier Fig. 9. Schematic overview of the Use of Supplementary Networks. The Detector Network was previously trained to detect adversarial images and is included between the input images and the classifier. This network receives the input images and determines if these images are Adversarial or Not. If they are not, they are redirected to the Classifier; If they are, they are susceptible to Human evaluation. extracted feature points to be within the vicinity of Lyapunovstable equilibrium points, which suppresses the input perturbations. Self-COnsistent Robust Error (SCORE) [103] employs local equivariance to describe the ideal behavior of a robust model, facilitating the reconciliation between robustness and accuracy while still dealing with worst-case uncertainty. This method was inspired by the discovery that the trade-off between adversarial and clean accuracy imposes a bias toward smoothness. Analyzing the impact of activation shape on robustness, Dai et al. [104] observes that activation has positive outputs on negative inputs, and a high finite curvature can improve robustness. Therefore, Parametric Shifted Sigmoidal Linear Unit (PSSiLU) [104] combines these properties and parameterized activation functions with adversarial training. C. Use of Supplementary Networks MagNet [105] considers two reasons for the misclassification of an adversarial example: 1) incapacity of the classifier to reject an adversarial example distant from the boundary; and 2) classifier generalizes poorly when the adversarial example is close to the boundary. MagNet considers multiple detectors trained based on the reconstruction error, detecting significantly perturbed examples and detecting slightly perturbed examples based on probability divergence. Adversary Detection Network (ADN) [106] is a subnetwork that detects if the input example is adversarial or not, trained using adversarial images generated for a classification network which are classified as clean (0) or adversarial (1). Figure 9 displays a schematic overview of this network. However, this defense mechanism deeply correlates to the datasets and classification networks. Xu et al. found that the inclusion of Feature Squeezing (FS) [107] is highly reliable in detecting adversarial examples by reducing the search space available for the adversary to modify. This method compares the predictions of a standard network with a squeezed one, detecting adversarial examples with high accuracy and having few false positives. High-level representation Guided Denoiser (HGD) [17] uses the distance between original and adversarial images to guide an image denoiser and suppress the impact of adversarial examples. HGD uses a Denoising AutoEncoder [108] with additional lateral connections and considers the difference between the representations as the loss function at a specific layer that is activated by the normal and adversarial examples. Defense-GAN [18] explores the use of GANs to effectively represent the set of original training examples, making this defense independent from the attack used. Defense-GAN considers the usage of Wasserstein GANs (WGANs) [109] to learn the representation of the original data and denoise the adversarial examples, which start by minimizing the l 2 difference between the generator representation and the input image. Reverse Attacks [110] can be applied to each attack during the testing phase, by finding the suitable additive perturbation to repair the adversarial example similar to the adversarial attacks, which is highly difficult due to the unknown original label. Embedding Regularized Classifier (ER-Classifier) [111] is composed of a classifier, an encoder, and a discriminator, which uses the encoder to generate code vectors by reducing the dimensional space of the inputs and the discriminator to separate these vectors from the ideal code vectors (sampled from a prior distribution). This technique allows pushing adversarial examples into the benign image data distribution, removing the adversarial perturbations. Class Activation Feature-based Denoiser (CAFD) [112] is a self-supervised approach trained to remove the noise from adversarial examples, using a set of examples generated by the Class Activation Feature-based Attack (CAFA) [112]. This defense mechanism is trained to minimize the distance of the class activation features between the adversarial and natural examples, being robust to unseen attacks. Detector Graph (DG) [113] considers graphs to detect the adversarial examples by constructing a Latent Neighborhood Graph (LNG) for each original example and using Graph Neural Networks (GNNs) [114] to exploit the relationship and distinguish between original and adversarial examples. This method maintains an additional reference dataset to retrieve the manifold information and uses embedding representation of image pixel values, making the defense robust to unseen attacks. Images in the real world are represented in a continuous manner, yet machines can only store these images in discrete 2D arrays. Local Implicit Image Function (LIIF) [115] takes an image coordinate and the deep features around this coordinate as inputs, predicting the corresponding RGB value. This method of pre-processing input images can filter adversarial images by reducing their perturbations, which are subsequently fed to a classifier. ADversarIal defenSe with local impliCit functiOns (DISCO) [116] is an additional network to the classifier that removes adversarial perturbations using localized manifold projections, which receives an adversarial image and a query pixel location. This defense mechanism comprises an encoder that creates per-pixel deep features and a local implicit module that uses these features to predict the clean RGB value. D. Change Network Architecture To identify the type of layers and their order, Guo et al. [118] proposes the use of Neural Architecture Search (NAS) to identify the networks that are more robust to adversarial attacks, finding that densely connected patterns improve the robustness and adding convolution operations to direct connection edge is effective, combined to create the RobNets [118]. Feature Denoising [117] intends to address this problem by applying feature-denoising operations, consisting of nonlocal means, bilateral, mean, median filters, followed by 1x1 Convolution and an identity skip connection, as illustrated in Figure 10. These blocks are added to the intermediate layers of CNNs. Input Random [119] propose the addition of layers at the beginning of the classifier, consisting of 1) a random resizing layer, which resizes the width and height of the original image to a random width and height, and 2) a random padding layer, which pads zeros around the resized image in a random manner. Controlling Neural Level Sets (CNLS) [120] uses samples obtained from the neural level sets and relates their positions to the network parameters, which allows modifying the decision boundaries of the network. The relation between position and parameters is achieved by constructing a sample network with an additional single fixed linear layer, which can incorporate the level set samples into a loss function. Sparse Transformation Layer (STL) [121], included between the input image and the network first layer, transforms the received images into a low-dimensional quasinatural image space, which approximates the natural image space and removes adversarial perturbations. This creates an attack-agnostic adversarial defense that gets the original and adversarial images closer. Benz et al. [122] found that BN [123] and other normalization techniques make DNN more vulnerable to adversarial examples, suggesting the use of a framework that makes DNN more robust by learning Robust Features first and, then, Non-Robust Features (which are the ones learned when using BN). E. Perform Network Validation Most of the datasets store their images using the Joint Photographic Experts Group (JPEG) [124] compression, yet no one had evaluated the impact of this process on the network performance. Dziugaite et al. [125] (named as JPG) varies the magnitude of FGSM perturbations, discovering that smaller ones often reverse the drop in classification by a large extent and, when the perturbations increase in magnitude, this effect is nullified. Regarding formal verification, a tool [126] for automatic Safety Verification of the decisions made during the classification process was created using Satisfiability Modulo Theory (SMT). This approach assumes that a decision is safe when, after applying transformations in the input, the model decision does not change. It is applied to every layer individually in the network, using a finite space of transformations. DeepXplore [127] is the first white-box framework to perform a wide test coverage, introducing the concepts of neuron coverage, which are parts of the DNN that are exercised by test inputs. DeepXplore uses multiple DNNs as cross-referencing oracles to avoid manual checking for each test input and inputs that trigger different behaviors and achieve high neuron coverage is a joint optimization problem solved by gradientbased search techniques. DeepGauge [128] intends to identify a testbed containing multi-faceted representations using a set of multi-granularity testing criteria. DeepGauge evaluates the resilience of DNNs using two different strategies, namely, primary function and corner-case behaviors, considering neuron-and layer-level coverage criteria. Surprise Adequacy for Deep Learning Systems (SADL) [129] is based on the behavior of DNN on the training data, by introducing the surprise of an input, which is the difference between the DNN behavior when given the input and the learned training data. The surprise of input is used as an adequacy criterion (Surprise Adequacy), which is used as a metric for the Surprise Coverage to ensure the input surprise range coverage. The most recent data augmentation techniques, such as cutout [130] and mixup [131], fail to prevent overfitting and, sometimes, make the model over-regularized, concluding that, to achieve substantial improvements, the combination of early stopping and semi-supervised data augmentation, Overfit Reduction (OR) [132], is the best method. When creating a model, multiple implementation details influence its performance; Pang et al. [133] is the first one to provide insights on how these details influence the model robustness, herein named as Bag of Tricks (BT). Some conclusions drawn from this study are: 1) The robustness of the models is significantly affected by weight decay; 2) Early stopping of the adversarial attacks may deteriorate worst-case robustness; and 3) Smooth activation benefits lower capacity models. Overfitting is a known problem that affects model robustness; Rebuffi et al. [134] focuses on reducing this robust overfitting by using different data augmentation techniques. Fixing Data Augmentation (FDA) [134] demonstrates that model weight averaging combined with data augmentation schemes can significantly increase robustness, which is enhanced when using spatial composition techniques. Gowal et al. [135] systematically studies the effect of multiple training losses, model sizes, activation functions, the addition of unlabeled data, and other aspects. The main Adversarial Purified Diffusion Denoising Fig. 11. Overview of Adversarial Purification using Denoising Diffusion Probabilistic Models, adapted from [137]. The diffusion process is applied to an adversarial image, consisting of adding noise for a certain number of steps. During the denoising procedure, this noise is iteratively removed by the same amount of steps, resulting in a purified image (without perturbations). conclusion drawn by this analysis is that larger models with Swish/SiLU [136] activation functions and model weight averaging can reliably achieve state-of-the-art results in robust accuracy. F. Adversarial Purification Adversarial Purification consists of defense mechanisms that remove adversarial perturbations using a generative model. Improving Robustness Using Generated Data (IRUGD) [138] explores how generative models trained on the original images can be leveraged to increase the size of the original datasets. Through extensive experiments, they concluded that Denoising Diffusion Probabilistic Models (DDPM) [137], a progression of diffusion probabilistic models [139], is the model that more closely resembles real data. Figure 11 presents the main idea behind the DDPM process. Due to the great results in image synthesis displayed by the DDPM, Sehwag et al. [140] (Proxy) uses proxy distributions to significantly improve the performance of adversarial training by generating additional examples, demonstrating that the best generative models for proxy distribution are DDPM. Inspired by previous works on adversarial purification [141], [142], DiffPure [143] uses DDPM for adversarial purification, receiving as input an adversarial example and recovering the clean image through a reverse generative process. Since this discovery, multiple improvements regarding the use of DDPM for Adversarial Purification have been studied. Guided Diffusion Model for Adversarial Purification (GDMAP) [144] receives as initial input pure Gaussian noise and gradually denoises it with guidance to an adversarial image. DensePure [145] employs iterative denoising to an input image, with different random seeds, to get multiple reversed samples, which are given to the classifier and the final prediction is based on majority voting. Furthermore, Wang et al. [146] uses the most recent diffusion models [147] to demonstrate that diffusion models with higher efficiency and image quality directly translate into better robust accuracy. VI. ADVERSARIAL EFFECTS ON VISION TRANSFORMERS Like CNNs [148], the ViTs are also susceptible to adversarial perturbations that alter a patch in an image [149], and ViTs demonstrate higher robustness, almost double, compared with ResNet-50 [150]. To further evaluate the robustness of ViT to adversarial examples, Mahmood et al. [151] used multiple adversarial attacks in CNNs, namely FGSM, PGD, MIM, C&W, and MI-FGSM. The ViT has increased robustness (compared with ResNet) for the first four attacks and has no resilience to the C&W and MI-FGSM attacks. Additionally, to complement the results obtained from the performance of ViTs, an extensive study [152] using feature maps, attention maps, and Gradientweighted Class Activation Mapping (Grad-CAM) [153] intends to explain this performance visually. The transferability of adversarial examples from CNNs to ViTs was also evaluated, suggesting that the examples from CNNs do not instantly transfer to ViTs [151]. Furthermore, Self-Attention blended Gradient Attack (SAGA) [151] was proposed to misclassify both ViTs and CNNs. The Pay No Attention (PNA) [154] attack, which ignores the gradients of attention, and the PatchOut [154] attack, which randomly samples subsets of patches, demonstrate high transferability. To detect adversarial examples that might affect the ViTs, PatchVeto [20] uses different transformers with different attention masks that output the encoding of the class. An image is considered valid if all transformers reach a consensus in the voted class, overall the masked predictions (provided by masked transformers). Smoothed ViTs [155] perform preprocessing techniques to the images before feeding them into the ViT, by generating image ablations (images composed of only one column of the original image, and the remaining columns are black), which are converted into tokens, and droping the fully masked tokens. The remaining tokens are fed into a ViT, which predicts a class for each ablation, and the class with the most predictions of overall ablations is considered the correct one. Bai et al. [156] demonstrates that ViTs and CNNs are being unfairly evaluated because they do not have the same training details. Therefore, this work provides a fair and indepth comparison between ViTs and CNNs, indicating that ViTs are as vulnerable to adversarial perturbations as CNNs. Architecture-oriented Transferable Attacking (ATA) [157] is a framework that generates transferable adversarial examples by considering the common characteristics among different ViT architectures, such as self-attention and image-embedding. Specifically, it discovers the most attentional patch-wise regions significantly influencing the model decision and searches pixel-wise attacking positions using sensitive embedding perturbation. Patch-fool [158] explores the perturbations that turn ViTs more vulnerable learners than CNNs, proposing a dedicated attack framework that fools the self-attention mechanism by attacking a single patch with multiple attention-aware optimization techniques. This attack mechanism demonstrates, for the first time, that ViTs can be more vulnerable than CNNs if attacked with proper techniques. Gu et al. [159] evaluates the robustness of ViT to patch-wise perturbations, concluding that these models are more robust to naturally corrupted patches than CNNs while being more vulnerable to adversarially generated ones. Inspired by the observed results, the authors propose a simple Temperature Scaling based method that improves the robustness of ViTs. Fig. 12. Images withdrew from the MNIST dataset [162] in the first five columns and from the Fashion-MNIST dataset [163] in the last five columns. The images were resized for better visualization. Fig. 13. Images withdrew from the CIFAR-10 dataset [164] in the first five columns and from the CIFAR-100 dataset [164] in the last five columns. The images were resized for better visualization. As previously observed for CNNs, improving the robust accuracy sacrifices the standard accuracy of ViTs, which may limit their applicability in the real context. Derandomized Smoothing [160] uses a progressive smoothed image modeling task to train the ViTs, making them capture the more discriminating local context while preserving global semantic information, improving both robust and standard accuracy. VeinGuard [161] is a defense framework that helps ViTs be more robust against adversarial palm-vein image attacks, with practical applicability in the real world. Namely, VeinGuard is composed of a local transformer-based GAN that learns the distribution of unperturbed vein images and a purifier that automatically removes a variety of adversarial perturbations. VII. DATASETS A. MNIST and F-MNIST One of the most used datasets is the MNIST [162] dataset, which contains images of handwritten digits collected from approximately 250 writers in shades of black and white, withdrawn from two different databases. This dataset is divided into training and test sets, with the first one containing 60,000 examples and a second one containing 10,000 examples. Xiao et al. propose the creation of the Fashion-MNIST [163] dataset by using figures from a fashion website, which has a total size of 70,000 images, contains ten classes, uses greyscale images, and each image has a size of 28x28. The Fashion-MNIST dataset is divided into train and test sets, containing 60,000 and 10,000 examples, respectively. Fig. 12 displays the 10 digits (from 0 to 9) from the MNIST dataset in the first five columns and the 10 fashion objects from Fashion-MNIST dataset in the last five columns. MNIST is one of the most widely studied datasets in the earlier works of adversarial examples, with defense mechanisms already displaying high robustness on this dataset. The same does not apply to Fashion-MNIST, which has not been as widely studied, despite having similar characteristics to MNIST. B. CIFAR-10 and CIFAR-100 Another widely studied dataset is the CIFAR-10, which, in conjunction with the CIFAR-100 dataset, are subsets from a [166] in the first three columns and from the German Traffic Sign Recognition Benchmark dataset [167] in the last three columns. The images were resized for better visualization. vast database containing 80 million tiny images [165], 32x32, and three color channels 75,062 different classes. CIFAR-10 [164] contains only ten classes from this large database, with 6,000 images for each class, distributed into 50,000 training images and 10,000 test images. This dataset considers different objects, namely, animals and vehicles, usually found in different environments. CIFAR-100 [164] contains 100 classes with only 600 images for each one with the same size and amount of color channels as the CIFAR-10 dataset. CIFAR-100 groups its 100 classes into 20 superclasses, located in different contexts/environments, making this dataset much harder to achieve high results. Examples from the CIFAR-10 dataset are shown in Fig. 13 in the first five columns, and the remaining columns display examples of the superclasses from CIFAR-100. Due to the unsatisfactory results demonstrated by models trained on CIFAR-10, the CIFAR-100 dataset has not been included in most studies under the context of adversarial examples, suggesting that solving the issue of adversarial-perturbed images is still at its inception. C. Street View Datasets The Street View House Numbers (SVHN) [166] dataset provides the same challenge as MNIST: identifying which digits are present in a colored image, containing ten classes, 0 to 9 digits, and an image size of 32x32 centered around a single character, with multiple digits in a single image. Regarding the dataset size, it has 630,420 digit images, but only 73,257 images are used for training, 26,032 images are used for testing, and the remaining 531,131 images can be used as additional training data. German Traffic Sign Recognition Benchmark (GT-SRB) [167] is a dataset containing 43 classes of different traffic signs, has 50,000 images, and demonstrates realistic scenarios. The dataset has 51,840 images, whose size varies from 15x15 to 222x193, divided into training, validation, and test sets with 50%, 25%, and 25%, respectively, of the total images. The difficulties associated with the SVHN dataset are displayed in the first three rows of Fig. 14, showing unique digits that occupy the whole image and multiple digits on different backgrounds. Furthermore, the same figure presents the different types of traffic signs in the GTSRB dataset, such as prohibition, warning, mandatory, and end of prohibition. Fig. 15. Images withdrew from the ImageNet dataset [168] in the top left, from the ImageNet-A dataset [169] in the top right, from the ImageNet-C and ImageNet-P datasets [170] in the bottom left, and ImageNet-COLORDISTORT [171] in the bottom right. The images were resized for better visualization. D. ImageNet and Variants ImageNet [168] is one of the largest datasets for object recognition, containing 1,461,406 colored images and 1,000 classes, with images being resized to 224x224. This dataset collected photographs from Flickr, and other search engines, divided into 1.2 million training images, 50,000 validation images, and 100,000 test images. A possible alternative to ImageNet, when the dataset size is an important factor, is called Tiny ImageNet [172], a subset of ImageNet that contains fewer classes and images. This dataset contains only 200 classes (from the 1,000 classes in ImageNet), 100,000 training images, 10,000 validation images, and 10,000 test images. These classes include animals, vehicles, household items, insects, and clothing, considering the variety of contexts/environments that these objects can be found. Their images have a size of 64x64 and are colored. ImageNet-A [169] is a subset of ImageNet, containing only 200 classes from the 1,000 classes, covering the broadest categories in ImageNet. ImageNet-A is a dataset composed of real-world adversarially filtered images, which were obtained by deleting the correctly predicted images by ResNet-50 classifiers. Despite ImageNet-A being based on the deficiency of ResNet-50, it also demonstrates transferability to unseen models, making this dataset suitable for evaluating the robustness of multiple classifiers. Two additional benchmarks, ImageNet-C [170] and ImageNet-P [170], were designed to evaluate the robustness of DNNs. The ImageNet-C standardizes and expands the corruption robustness topic, consisting of 75 corruptions applied to each image in the ImageNet validation set. ImageNet-P applies distortions to the images, though it differs from ImageNet-C because it contains perturbation sequences using only ten common perturbations. Another benchmark to evaluate the model generalization capability is the ImageNet-COLORDISTORT (ImageNet-CD) [171], which considers multiple distortions in the color of an image using different color space representations. This dataset contains the 1,000 classes from ImageNet, removing images without color channels, and the same image considers multiple color distortions under the Red Green Blue (RGB), Hue-Saturation-Value (HSV), CIELAB, and YCbCr color spaces considered common transformations used in image processing. It is possible to observe a set of images withdrawn from ImageNet in the top left of Fig. 15. Additionally, some images misclassified by multiple classifiers (ImageNet-A) are shown in the top right of the same figure. The bottom represents the ImageNet with common corruptions and perturbations and is manipulated by multiple image techniques on the left and right, respectively. Table III summarizes the main characteristics of the datasets presented throughout this section. VIII. METRICS AND STATE-OF-THE-ART RESULTS A. Evaluation Metrics Due to the nature of adversarial examples, they need specific metrics to be correctly evaluated and constructed. Following this direction, multiple works have been proposing different metrics that calculate the percentage of adversarial examples that make a model misclassify (fooling rate), measure the amount of perturbation made in an image (destruction rate), and calculate the model robustness to adversarial examples (average robustness). 1) Accuracy: This metric measures the number of samples that are correctly predicted by the model, which is defined as: accuracy = T P + T N T P + T N + F P + F N ,(10) where T P refers to True Positive, T N to True Negative, F P to False Positive, and F N to False Negative. The True Positive and True Negative are the samples whose network prediction is the same as the label (correct), and the False Positive and False Negative are the samples whose network prediction differs from the label (incorrect). When considering original images, this metric is denominated as Clean Accuracy and, when using adversarial images, is named as Robust Accuracy. 2) Fooling Rate: After being perturbed to change the classifier label, the fooling rate F R [173] was proposed to calculate the percentage of images. 3) Average Robustness: To objectively evaluate the robustness to adversarial perturbations of a classifier f , the average robustnessp adv (f ) is defined as [21]: p adv (f ) = 1 D x∈D r(x) 2 x 2 ,(11) wherer(x) is the estimated minimal perturbation obtained using the attack, and D denotes the test set. 4) Destruction Rate: To evaluate the impact of arbitrary transformations on adversarial images, the notion of destruction rate d is introduced and formally defined as [33]: d = n k=1 C(X k , y k true )¬C(X k adv , y k true )C(T (X k adv ), y k true ) n k=1 (X k , y k true )C(X k adv , y k true ) ,(12) where n is the number of images, X k is the original image from the dataset, y k true is the true class of this image, X k adv is the adversarial image corresponding to that image, and T is an arbitrary image transformation. ¬C(X k adv , y k true ) is defined as the binary negation of C(X k adv , y k true ). Finally, the function C(X, y) is defined as [33]: C(X, y) = 1, if image X is classified as y; 0, otherwise.(13) B. Defense Mechanisms Robustness The metric used to evaluate models is accuracy, which evaluates the results on both original (Clean Accuracy) and adversarially perturbed (Robust Accuracy) datasets. One of the earliest and strongest adversarial attacks proposed was PGD, which was used by multiple defenses to evaluate their robustness. Table IV displays defenses evaluated on CIFAR-10 under multiple steps PGD attack, ordered by increasing robustness. For the PGD attack, the best performing defenses are from approaches that use supplementary networks (CAFD) or modify the training process (FS and AWP). Overall, Wide ResNets [175] have better robust accuracy, due to high-capacity networks exhibiting greater adversarial robustness [36], [51], suggesting the usage of these networks in future developments of adversarial attacks and defenses. To assess the robustness of defenses for white and blackbox settings, Auto-Attack has gained increased interest over PGD in recent works. Tables V, VI, and VII present a set of defenses that are evaluated under Auto-Attack, on CIFAR-10, CIFAR-100, and ImageNet, respectively, ordered by increasing Robust Accuracy. The most used networks are Wide ResNets with different sizes, with the biggest Wide ResNet displaying better results overall, and the most resilient defense derives from the use of supplementary networks (DISCO), followed by modifying the train process (SODEF) and changing network architecture (STL). The results suggest that the inclusion of additional components to sanitize inputs of the targeted model (use of supplementary networks) is the most resilient approach for model robustness in white and black-box settings. The updated results for defenses under Auto-Attack can be found on the RobustBench [176] website. IX. FUTURE DIRECTIONS Following the de facto standards adopted by the literature, we suggest that future proposals of defense mechanisms should be evaluated on Auto-Attack, using the robust accuracy as a metric for comparison purposes. The adversarial defense that demonstrates better results is Adversarial Training, which should be a requirement when evaluating attacks and defenses. The state-of-the-art results show that MNIST and CIFAR-10 datasets are already saturated. Other datasets should be further evaluated, namely: 1) CIFAR-100 and ImageNet since adversarial defenses do not achieve state-of-the-art clean accuracy (91% and 95%, respectively); 2) GTSRB and SVHN, depicting harder scenarios with greater variations of background, inclination, and luminosity; and 3) Fashion-MNIST that would allow better comprehension of which image properties influence DNNs performance (e.g., type of task, image shades, number of classes). Most works present their results using accuracy as the evaluation metric and, more recently, evaluate their defenses on the Auto-Attack. Furthermore, the values given for in each dataset were standardized by recurrent use. However, there should be an effort to develop a metric/process that quantifies the amount of perturbation added to the original image. This would ease the expansion of adversarial attacks to other datasets that do not have a standardized value. There has been a greater focus on the development of whitebox attacks, which consider that the adversary has access to the network and training data, yet this is not feasible in real contexts, translating into the need of focusing more on the development of black-box attacks. A unique black-box set, physical attacks, also require additional evaluation, considering the properties of the real world and perturbations commonly found in it. Considering the increasing liberation of ML in the real world, end-users can partially control the training phase of DNNs, suggesting that gray-box attacks will intensify (access only to network or data). The different network architectures are designed to increase the clean accuracy of DNNs in particular object recognition datasets, yet there should be further evaluation on the impact of the different layers and their structure. ViTs introduce a new paradigm in image analysis and are more robust against natural corruptions, suggesting that building ViT inherently robust to adversarial examples might be a possible solution. DDPM are generative models that perform adversarial purification of images, but they can not be applied in realtime since they take up to dozens of seconds to create a single purified image. Therefore, an effort on developing close to real-time adversarial purification strategies is a viable strategy for future works. X. CONCLUSIONS DNNs are vulnerable to a set of inputs, denominated as adversarial examples, that drastically modify the output of the considered network and are constructed by adding a perturbation to the original image. This survey presents background concepts, such as adversary capacity and vector norms, essential to comprehend adversarial settings, providing a comparison with existing surveys in the literature. Adversarial attacks are organized based on the adversary knowledge, highlighting the emphasis of current works toward white box settings, and adversarial defenses are clustered into six domains, with most works exploring the adversarial training strategy. We also present the latest developments of adversarial settings in ViTs and describe the commonly used datasets, providing the state-of-the-art results in CIFAR-10, CIFAR-100, and ImageNet. Finally, we propose a set of open issues that can be explored for subsequent future works. Fig. 1 .Fig. 2 . 12Schematic example of the Convolutional Neural Networks mechanism to classify images. Schematic example of a simplified vision transformer used to classify images. Fig. 4 . 4Geometric representation of the l 0 , l 2 , and l∞ norms, from left to right, respectively. Fig. 5 . 5Schematic overview of an Adversarial Attack under White-box Settings (left) and Black-box Settings (right). The first one uses the classifier predictions and network gradients to create perturbations (similar to noise), which can fool this classifier. These perturbations are added to the original images, creating adversarial images, which are fed to the network and cause misclassification. In the Black-box Settings, the same process is applied to a known classifier, and the obtained images are used to attack another classifier (represented as Target Architecture). Fig. 6 . 6Schematic overview of Adversarial Training. A subset of the original images of a dataset is fed into an adversarial attack (e.g., PGD, FGSM, or C&W), which creates adversarial images. Each batch contains original and adversarial images, with the Classifier being normally trained. Fig. 7 . 7Method proposed by Defensive Distillation Fig. 10 . 10Overview of a Feature Denoising Block[117], which can be included in the intermediate layers to make networks more robust. This method is an example of Change Network Architecture. Fig. 14 . 14Images withdrew from the Street View House Numbers dataset TABLE I COMMON ITERMINOLOGIES USED IN THE CONTEXT OF ADVERSARIAL ATTACKS AND THEIR DEFINITION.Terminology Definition Original/ Clean Example Original image presented in a dataset Adversarial/ Perturbed Example Image that an adversary has manipulated to fool the classifier Perturbation Set of changes (for each pixel and color channel) that are performed on the image Adversarial At- tack Technique used to calculate the perturbation that generates an adversarial example Transferability Capability of an adversarial example being transferred from a known network to an un- known network White-box Attacks that have access to DNN weights and datasets Black-box Attacks that do not have access to the DNN weights and datasets Adversarial Training Inclusion of adversarial examples in the training phase of the model only data training; extensively analyzed the adversarial attacks and defenses proposed under this context, providing conjectures for the existence of adversarial examples and evaluating the capacity of adversarial examples transferring between different DNNs. TABLE II CHARACTERISTICS IISHOWN IN STATE-OF-THE-ART SURVEYS ON ADVERSARIAL ATTACKS.Survey Year White & Survey Grouping of Future Datasets Metrics and State-of-the-art Vision Black-Box Comparison Defenses Directions Overview Architectures Comparison Transformers Akhtar and Mian [23] 2018 × × × × × Qiu et al. [26] 2019 × × × × × × × Serban et al. [25] 2020 × × × × × Xu et al. [27] 2020 × × × × × × Chakraborty et al. [28] 2021 × × × × × × × Long et al. [29] 2022 × × × × × Liang et al. [30] 2022 × × × × × Zhou et al. [31] 2022 × × × × This survey 2023 93] method uses soft labels to trainChange Loss Regularizer Quantization Shift Noise Clip Fig. 8. Schematic overview of the Bit Plane Feature Consistency TABLE III RELEVANT IIICHARACTERISTICS TO THE CONTEXT OF ADVERSARIAL EXAMPLES OF THE STATE-OF-THE-ART DATASETS. #CLASSES MEANS THE NUMBER OF CLASSES IN THE DATASET. EMPTY COLOR COLUMN MEANS THAT THE IMAGES IN THAT DATASET USE GREYSCALE OR BLACK AND WHITE SHADES. DATASETS WITH * ARE ONLY USED FOR TESTING PURPOSES.Dataset Size #Classes Classes Color MNIST 70,000 10 Digits Fashion-MNIST 70,000 10 Clothing CIFAR-10 60,000 10 Animals Vehicles SVHN 630,420 10 Digits GTSRB 51,840 43 Traffic Signs CIFAR-100 60,000 100 Household Items Outdoor Scenes Tiny ImageNet 120,000 200 Animals Household Items ImageNet-A * 7,500 200 Vehicles Food ImageNet-C * 3,750,000 200 Vehicles Food ImageNet-P * 15,000,000 200 Vehicles Food ImageNet 1,431,167 1,000 Vehicles Electronic devices ImageNet-CD * 736,515 1,000 Vehicles Electronic devices TABLE IV ACCURACY IVCOMPARISON OF DIFFERENT DEFENSE MECHANISMS ON CIFAR-10 UNDER PGD ATTACK, l∞ AND = 8/255. CLEAN AND ROBUST REFERS TO ACCURACY WITHOUT AND WITH ADVERSARIAL ATTACKS, RESPECTIVELY. DEFENSES WITH "-" ON CLEAN ACCURACY DO NOT HAVE A CLEAN ACCURACY REPORTED.Defense Method Year Architecture Accuracy Clean Robust BPFC [89] 2020 ResNet-18 82.4 34.4 SNS [94] 2021 VGG-16 86.0 39.6 AT-MIFGSM [51] 2017 Inception v3 85.3 45.9 AT-PGD [36] 2018 ResNet-18 87.3 47.0 RobNets [118] 2020 RobNet-free 82.8 52.6 HGD [17] 2018 DUNET 92.4 53.1 RSLAD [93] 2021 ResNet-18 83.4 54.2 MART [61] 2020 WRN-28-10 83.1 55.6 TRADES [54] 2019 WRN-34-10 84.9 56.4 BagT [133] 2020 WRN-34-10 - 56.4 RO [132] 2020 ResNet-18 - 56.8 DOA [62] 2019 VGGFace 93.6 61.0 AWP [90] 2020 WRN-28-10 - 63.6 FS [19] 2019 WRN-28-10 90.0 68.6 CAFD [112] 2021 DUNET 91.1 87.2 TABLE V ACCURACY VCOMPARISON OF DIFFERENT DEFENSE MECHANISMS ON CIFAR-10 UNDER AUTO-ATTACK ATTACK, l∞ AND = 8/255 . CLEAN AND ROBUST REFERS TO ACCURACY WITHOUT AND WITH ADVERSARIAL ATTACKS, RESPECTIVELY.Architecture Defense Method Year Accuracy Clean Robust WRN28-10 Input Random [119] 2017 94.3 8.6 BAT [57] 2019 92.8 29.4 FS [19] 2019 90.0 36.6 Jpeg [125] 2016 83.9 50.7 Pretrain [174] 2019 87.1 54.9 UAT [84] 2019 86.5 56.0 MART [61] 2020 87.5 56.3 HYDRA [92] 2020 89.0 57.1 RST [85] 2019 89.7 59.5 GI-AT [68] 2020 89.4 59.6 Proxy [140] 2021 89.5 59.7 AWP [90] 2020 88.3 60.0 FDA [134] 2021 87.3 60.8 HAT [69] 2021 88.2 61.0 SCORE [103] 2022 88.6 61.0 PSSiLU [104] 2022 87.0 61.6 Gowal et al. [135] 2020 89.5 62.8 IRUGD [138] 2021 87.5 63.4 Wang et al. [146] 2023 92.4 67.3 STL [121] 2019 82.2 67.9 DISCO [116] 2022 89.3 85.6 WRN34-10 Free-AT [58] 2019 86.1 41.5 AT-PGD [36] 2018 87.1 44.0 YOPO [60] 2019 87.2 44.8 TLA [81] 2019 86.2 47.4 LAT [82] 2019 87.8 49.1 SAT [63] 2020 86.8 50.7 FAT [70] 2022 85.3 51.1 LBGAT [97] 2021 88.2 52.3 TRADES [54] 2019 84.9 53.1 SAT [91] 2020 83.5 53.3 Friend-AT [64] 2020 84.5 55.5 AWP [90] 2020 85.4 56.2 LTD [98] 2021 85.2 56.9 OA-AT [66] 2021 85.3 58.0 Proxy [140] 2022 86.7 60.3 HAT [69] 2021 91.5 62.8 WRN-70-16 SCORE [103] 2022 89.0 63.4 IRUGD [138] 2021 91.1 65.9 Gowal et al. [135] 2020 88.7 66.1 FDA [134] 2021 92.2 66.6 Wang et al. [146] 2023 93.3 70.7 SODEF [102] 2021 93.7 71.3 TABLE VI ACCURACY VICOMPARISON OF DIFFERENT DEFENSE MECHANISMS ON CIFAR-100 UNDER AUTO-ATTACK ATTACK, l∞ AND = 8/255 . CLEAN AND ROBUST REFERS TO ACCURACY WITHOUT AND WITH ADVERSARIAL ATTACKS, RESPECTIVELY.Architecture Defense Method Year Accuracy Clean Robust WRN28-10 Input Random [119] 2017 73.6 3.3 LIIF [115] 2021 80.3 3.4 Bit Reduction [107] 2017 76.9 3.8 Pretrain [174] 2019 59.2 28.4 SCORE [103] 2022 63.7 31.1 FDA [134] 2021 62.4 32.1 Wang et al. [146] 2023 78.6 38.8 Jpeg [125] 2016 61.9 39.6 STL [121] 2019 67.4 46.1 DISCO [116] 2022 72.1 67.9 WRN34-10 SAT [63] 2020 62.8 24.6 AWP [90] 2020 60.4 28.9 LBGAT [97] 2021 60.6 29.3 OA-AT [66] 2021 65.7 30.4 LTD [98] 2021 64.1 30.6 Proxy [140] 2022 65.9 31.2 DISCO [116] 2022 71.6 69.0 WRN-70-16 SCORE [103] 2022 65.6 33.1 FDA [134] 2021 63.6 34.6 Gowal et al. [135] 2020 69.2 36.9 Wang et al. [146] 2023 75.2 42.7 TABLE VII ACCURACY VIICOMPARISON OF DIFFERENT DEFENSE MECHANISMS ON IMAGENET UNDER AUTO-ATTACK ATTACK, l∞ AND = 4/255. CLEAN AND ROBUST REFERS TO ACCURACY WITHOUT AND WITH ADVERSARIAL ATTACKS, RESPECTIVELY.Architecture Defense Method Year Accuracy Clean Robust ResNet-18 Bit Reduction [107] 2017 67.6 4.0 Jpeg [125] 2016 67.2 13.1 Input Random [119] 2017 64.0 17.8 Salman et al. [177] 2020 52.9 25.3 STL [121] 2019 65.6 32.9 DISCO [116] 2022 68.0 60.9 ResNet-50 Bit Reduction [107] 2017 73.8 1.9 Input Random [119] 2017 74.0 18.8 Cheap-AT [65] 2020 55.6 26.2 Jpeg [125] 2016 73.6 33.4 Salman et al. [177] 2020 64.0 35.0 STL [121] 2019 68.3 50.2 DISCO [116] 2022 72.6 68.2 WRN-50-2 Bit Reduction [107] 2017 75.1 5.0 Input Random [119] 2017 71.7 23.6 Jpeg [125] 2016 75.4 24.9 Salman et al. [177] 2020 68.5 38.1 DISCO [116] 2022 75.1 69.5 ACKNOWLEDGMENTSThis work was supported in part by the Portuguese FCT/MCTES through National Funds and co-funded by EU funds under Project UIDB/50008/2020; in part by the FCT Doctoral Grant 2020.09847.BD and Grant 2021.04905.BD; Deep learning. 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[ "Optimal sequential decision making with probabilistic digital twins", "Optimal sequential decision making with probabilistic digital twins" ]	[ "Christian Agrell \nGroup Research and Development\nDNV\nNorway\n\nDepartment of Mathematics\nUniversity of Oslo\nNorway\n", "Kristina Rognlien Dahl \nDepartment of Mathematics\nUniversity of Oslo\nNorway\n", "Andreas Hafver \nGroup Research and Development\nDNV\nNorway\n" ]	[ "Group Research and Development\nDNV\nNorway", "Department of Mathematics\nUniversity of Oslo\nNorway", "Department of Mathematics\nUniversity of Oslo\nNorway", "Group Research and Development\nDNV\nNorway" ]	[]	Digital twins are emerging in many industries, typically consisting of simulation models and data associated with a specific physical system. One of the main reasons for developing a digital twin, is to enable the simulation of possible consequences of a given action, without the need to interfere with the physical system itself. Physical systems of interest, and the environments they operate in, do not always behave deterministically. Moreover, information about the system and its environment is typically incomplete or imperfect. Probabilistic representations of systems and environments may therefore be called for, especially to support decisions in application areas where actions may have severe consequences. A probabilistic digital twin is a digital twin, with the added capability of proper treatment of uncertainties associated with the consequences, enabling better decision support and management of risks.In this paper we introduce the probabilistic digital twin (PDT). We will start by discussing how epistemic uncertainty can be treated using measure theory, by modelling epistemic information via σ-algebras. Based on this, we give a formal definition of how epistemic uncertainty can be updated in a PDT. We then study the problem of optimal sequential decision making. That is, we consider the case where the outcome of each decision may inform the next. Within the PDT framework, we formulate this optimization problem. We discuss how this problem may be solved (at least in theory) via the maximum principle method or the dynamic programming principle. However, due to the curse of dimensionality, these methods are often not tractable in practice. To mend this, we propose a generic approximate solution using deep reinforcement learning together with neural networks defined on sets. We illustrate the method on a practical problem, considering optimal information gathering for the estimation of a failure probability.	10.1007/s42452-023-05316-9	[ "https://arxiv.org/pdf/2103.07405v1.pdf" ]	232,222,459	2103.07405	cf139b982240f5522c0c537a77ca2ba7957e6891	Optimal sequential decision making with probabilistic digital twins Christian Agrell Group Research and Development DNV Norway Department of Mathematics University of Oslo Norway Kristina Rognlien Dahl Department of Mathematics University of Oslo Norway Andreas Hafver Group Research and Development DNV Norway Optimal sequential decision making with probabilistic digital twins Probabilistic digital twin Epistemic uncertainty Sequential decision making Partially observable Markov decision process Deep reinforcement learning Digital twins are emerging in many industries, typically consisting of simulation models and data associated with a specific physical system. One of the main reasons for developing a digital twin, is to enable the simulation of possible consequences of a given action, without the need to interfere with the physical system itself. Physical systems of interest, and the environments they operate in, do not always behave deterministically. Moreover, information about the system and its environment is typically incomplete or imperfect. Probabilistic representations of systems and environments may therefore be called for, especially to support decisions in application areas where actions may have severe consequences. A probabilistic digital twin is a digital twin, with the added capability of proper treatment of uncertainties associated with the consequences, enabling better decision support and management of risks.In this paper we introduce the probabilistic digital twin (PDT). We will start by discussing how epistemic uncertainty can be treated using measure theory, by modelling epistemic information via σ-algebras. Based on this, we give a formal definition of how epistemic uncertainty can be updated in a PDT. We then study the problem of optimal sequential decision making. That is, we consider the case where the outcome of each decision may inform the next. Within the PDT framework, we formulate this optimization problem. We discuss how this problem may be solved (at least in theory) via the maximum principle method or the dynamic programming principle. However, due to the curse of dimensionality, these methods are often not tractable in practice. To mend this, we propose a generic approximate solution using deep reinforcement learning together with neural networks defined on sets. We illustrate the method on a practical problem, considering optimal information gathering for the estimation of a failure probability. Introduction The use of digital twins has emerged as one of the major technology trends the last couple of years. In essence, a digital twin (DT) is a digital representation of some physical system, including data from observations of the physical system, which can be used to perform forecasts, evaluate the consequences of potential actions, simulate possible future scenarios, and in general inform decision making without requiring interference with the physical system. From a theoretical perspective, a digital pre-print: This is a pre-print version of this article twin may be regraded to consist of the following two components: • A set of assumptions regarding the physical system (e.g. about the behaviour or relationships among system components and between the system and its environment), often given in the form of a physics-based numerical simulation model. • A set of information, usually in the form of a set of observations, or records of the relevant actions taken within the system. In some cases, a digital twin may be desired for a system which attributes and behaviours are not deterministic, but stochastic. For example, the degradation and failure of physical structures or machin-ery is typically described as stochastic processes. A systems performance may be impacted by weather or financial conditions, which also may be most appropriately modelled as stochastic. Sometimes the functioning of the system itself is stochastic, such as supply chain or production chains involving stochastic variation in demand and performance of various system components. Even for systems or phenomena that are deterministic in principle, a model will never give a perfect rendering of reality. There will typically be uncertainty about the model's structure and parameters (i.e. epistemic uncertainty), and if consequences of actions can be critical, such uncertainties need to be captured and handled appropriately by the digital twin. In general, the system of interest will have both stochastic elements (aleatory uncertainty) and epistemic uncertainty. If we want to apply digital twins to inform decisions in systems where the analysis of uncertainty and risk is important, certain properties are required: 1. The digital twin must capture uncertainties: This could be done by using a probabilistic representation for uncertain system attributes. 2. It should be possible to update the digital twin as new information becomes available: This could be from new evidence in the form of data, or underlying assumptions about the system that have changed. 3. For the digital twin to be informative in decision making, it should be possible to query the model sufficiently fast: This could mean making use of surrogate models or emulators, which introduces additional uncertainties. These properties are paraphrased from Hafver et al. [1], which provides a detailed discussion on the use of digital twins for on-line risk assessment. In this paper we propose a mathematical framework for defining digital twins that comply with these properties. As Hafver et al. [1], we will refer to these as probabilistic digital twins (PDTs), and we will build on the Bayesian probabilistic framework which is a natural choice to satisfy (1)- (2). A numerical model of a complex physical system can often be computationally expensive, for instance if it involves numerical solution of nontrivial partial differential equations. In a probabilistic setting this is prohibitive, as a large number of evaluations (e.g. PDE solves) is needed for tasks involving uncertainty propagation, such as prediction and inference. Applications towards real-time decision making also sets natural restrictions with respect to the runtime of such queries. This is why property (3) is important, and why probabilistic models of complex physical phenomena often involve the use of approximate alternatives, usually obtained by "fitting" a computationally cheap model to the output of a few expensive model runs. These computationally cheap approximations are often referred to as response surface models, surrogate models or emulators in the literature. Introducing this kind of approximation for computational efficiency also means that we introduce additional epistemic uncertainty into our modelling framework. By epistemic uncertainty we mean, in short, any form of uncertainty that can be reduced by gathering more information (to be discussed further later on). In our context, uncertainty may in principle be reduced by running the expensive numerical modes instead of the cheaper approximations. Many interesting sequential decision making problems arise from the property that our knowledge about the system we operate changes as we learn about the outcomes. That is, each decision may affect the epistemic uncertainty which the next decision will be based upon. We are motivated by this type of scenario, in combination with the challenge of finding robust decisions in safety-critical systems, where a decision should be robust with respect to what we do not know, i.e. with respect to epistemic uncertainty. Although we will not restrict the framework presented in this paper to any specific type of sequential decision making objectives, we will mainly focus on problems related to optimal information gathering. That is, where the decisions we consider are related to acquiring information (e.g., by running an experiment) in order to reduce the epistemic uncertainty with respect to some specified objective (e.g., estimating some quantity of interest). A very relevant example of such a task, is the problem of optimal experimental design for structural reliability analysis. This involves deciding which experiments to run in order to build a surrogate model that can be used to estimate a failure probability with sufficient level of confidence. This is a problem that has received considerable attention (see e.g. [2,3,4,5,6,7,8]). These methods all make use of a myopic (one-step lookahead) criterion to determine the "optimal" experiment, as a multistep or full dynamic programming formulation of the optimization problem becomes numerically infeasible. In Agrell and Dahl [2], they consider the case where there are different types of experiments to choose from. Here, the myopic (one-step lookahead) assumption can still be justified, but if the different types of experiments are associated with different costs, then it can be difficult to apply in practice (e.g., if a feasible solution requires expensive experiments with delayed reward). We will review the mathematical framework of sequential decision making, and connect this to the definition of a PDT. Traditionally, there are two main solution strategies for solving discrete time sequential decision making problems: Maximum principles, and dynamic programming. We review these two solution methods, and conclude that the PDT framework is well suited for a dynamic programming approach. However, dynamic programming suffers from the curse of dimensionality, i.e. possible sequences of decisions and state realizations grow exponentially with the size of the state space. Hence, we are typically not able to solve a PDT sequential decision making problem in practice directly via dynamic programming. As a generic solution to the problem of optimal sequential decision making we instead propose an alternative based on reinforcement learning. This means that when we consider the problem of finding an optimal decision policy, instead of truncating the theoretical optimal solution (from the Bellman equation) by e.g., looking only one step ahead, we try to approximate the optimal policy. This approximation can be done by using e.g. a neural network. Here we will frame the sequential decision making setup as a Markov decision process (MDP), in general as a partially observed MDP (POMDP), where a state is represented by the information available at any given time. This kind of state specification is often referred to as the information state-space. As a generic approach to deep reinforcement learning using PDTs, we propose an approach using neural networks that operate on the information statespace directly. Main contributions. In this paper we will: (i) Propose a mathematical framework for modelling epistemic uncertainty based on measure theory, and define epistemic conditioning. (ii) Present a mathematical definition of the probabilistic digital twin (PDT). This is a mathematical framework for modelling physical systems with aleatory and epistemic uncertainty. (iii) Introduce the problem of sequential decision making in the PDT, and illustrate how this problem can be solved (at least in theory) via maximum principle methods or the dynamic programming principle. (iv) Discuss the curse of dimensionality for these solution methods, and illustrate how the sequential decision making problem in the PDT can be viewed as a partially observable Markov decision process. (v) Explain how reinforcement learning (RL) can be applied to find approximate optimal strategies for sequential decision making in the PDT, and propose a generic approach using a deep sets architecture that enables RL directly on the information state-space. We end with a numerical example to illustrate this approach. The paper is structured as follows: In Section 2 we introduce epistemic uncertainty and suggest modeling this via σ-algebras. We also define epistemic conditioning. In Section 3, we present the mathematical framework, as well as a formal definition, of a probabilistic digital twin (PDT), and discuss how such PDTs are used in practice. Then, in Section 4, we introduce the problem of stochastic sequential decision making. We discuss the traditional solution approaches, in particular dynamic programming which is theoretically a suitable approach for decision problems that can be modelled using a PDT. However, due to the curse of dimensionality, using the dynamic programming directly is typically not tractable. We therefore turn to reinforcement learning using function approximation as a practical alternative. In Section 5, we show how an approximate optimal strategy can be achieved using deep reinforcement learning, and we illustrate the approach with a numerical example. Finally, in Section 6 we conclude and sketch some future works in this direction. A measure-theoretic treatment of epistemic uncertainty In this section, we review the concepts of epistemic and aleatory uncertainty, and introduce a measure-theoretic framework for modelling epistemic uncertainty. We will also define epistemic conditioning. Motivation In uncertainty quantification (UQ), it is common to consider two different kinds of uncertainty: Aleatory (stochastic) and epistemic (knowledgebased) uncertainty. We say that uncertainty is epistemic if we foresee the possibility of reducing it through gathering more or better information. For instance, uncertainty related to a parameter that has a fixed but unknown value is considered epistemic. Aleatory uncertainty, on the other hand, is the uncertainty which cannot (in the modellers perspective) be affected by gathering information alone. Note that the characterization of aleatory and epistemic uncertainty has to depend on the modelling context. For instance, the result of a coin flip may be viewed as epistemic, if we imagine a physics-based model that could predict the outcome exactly (given all initial conditions etc.). However, under most circumstances it is most natural to view a coin flip as aleatory, or that it contains both aleatory and epistemic uncertainty (e.g. if the bias of the coin us unknown). Der Kiureghian et al. [9] provides a detailed discussion of the differences between aleatory and epistemic uncertainty. In this paper, we have two main reasons for distinguishing between epistemic and aleatory uncertainty. First, we would like to make decisions that are robust with respect to epistemic uncertainty. Secondly, we are interested in studying the effect of gathering information. Modelling epistemic uncertainty is a natural way of doing this. In the UQ literature, aleatory uncertainty is typically modelled via probability theory. However, epistemic uncertainty is represented in many different ways. For instance, Helton [10] considers four different ways of modelling epistemic uncertainty: Interval analysis, possibility theory, evidence theory (Dempster-Shafer theory) and probability theory. In this paper we take a measure-theoretic approach. This provides a framework that is relatively flexible with respect to the types of assumptions that underlay the epistemic uncertainty. As a motivating example, consider the following typical setup used in statistics: Example 2.1. (A parametric model) Let X = (Y, θ) where Y is a random variable representing some stochastic phenomenon, and assume Y is modelled using a given probability distribution, P (Y \|θ), that depends on a parameter θ (e.g. Y ∼ N (µ, σ) with θ = (µ, σ)). Assume that we do not know the value of θ, and we therefore consider θ as a (purely) epistemic parameter. For some fixed value of θ, the random variable Y is (purely) aleatory, but in general, as the true value of θ is not known, Y is associated with both epistemic and aleatory uncertainty. The model X in Example 2.1 can be decoupled into an aleatory component Y \|θ and an epistemic component θ. Any property of the aleatory uncertainty in X is determined by P (Y \|θ), and is therefore a function of θ. For instance, the probability P (Y ∈ A\|θ) and the expectation E[f (Y )\|θ], are both functions of θ. There are different ways in which we can choose to address the epistemic uncertainty in θ. We could consider intervals, for instance the minimum and maximum of P (Y ∈ A\|θ) over any plausible value of θ, or assign probabilities, or some other measure of belief, to the possible values θ may take. However, in order for this to be well-defined mathematically, we need to put some requirements on A, f and θ. By using probability theory to represent the aleatory uncertainty, we implicitly assume that the set A and function f are measurable, and we will assume that the same holds for θ. We will describe in detail what is meant by measurable in Section 2.2 below. Essentially, this is just a necessity for defining properties such as distance, volume or probability in the space where θ resides. In this paper we will rely on probability theory for handling both aleatory and epistemic uncertainty. This means that, along with the measurability requirement on θ, we have the familiar setup for Bayesian inference: Example 2.2. (A parametric model -Inference and prediction) If θ from Example 2.1 is a random variable with distribution P (θ), then X = (Y, θ) denotes a complete probabilistic model (capturing both aleatory and epistemic uncertainty). X is a random variable with distribution P (X) = P (Y \|θ)P (θ). Let I be some piece of information from which Bayesian inference is possible, i.e. P (X\|I) is well defined. We may then define the updated joint distribution P new (X) = P (Y \|θ)P (θ\|I), and the updated marginal (predictive) distribution for Y becomes P new (Y ) = P (Y \|θ)dP (θ\|I). Note that the distribution P new (X) in Example 2.2 is obtained by only updating the belief with respect to epistemic uncertainty, and that P new (X) = P (X\|I) = P (Y \|I, θ)P (θ\|I). For instance, if I corresponds to an observation of Y , e.g. I = {Y = y}, then P (Y \|I) = δ(y), the Dirac delta at y, whereas P (θ\|I) is the updated distribution for θ having observed one realization of Y . In the following, we will refer to the kind of Bayesian updating in Example 2.2 as epistemic updating. This epistemic updating of the model considered in Example 2.1 and Example 2.2 should be fairly intuitive, if 1. All epistemic uncertainty is represented by a single parameter θ, and 2. θ is a familiar object like a number or a vector in R n . But what can we say in a more general setting? It is common that epistemic uncertainty comes from lack of knowledge related to functions. This is the case with probabilistic emulators and surrogate models. The input to these functions may contain epistemic and/or aleatory uncertainty as well. Can we talk about isolating and modifying the epistemic uncertainty in such a model, without making reference to the specific details of how the model has been created? In the following we will show that with the measure-theoretic framework, we can still make use of a simple formulation like the one in Example 2.2. The probability space Let X be a random variable containing both aleatory and epistemic uncertainty. In order to describe how X can be treated like in Example 2.1 and Example 2.2, but for the general setting, we will first recall some of the basic definitions from measure theory and measure-theoretic probability. To say that X is a random variable, means that X is defined on some measurable space (Ω, F). Here, Ω is a set, and if X takes values in R n (or some other measurable space), then X is a so-called measurable function, X(ω) : Ω → R n (to be defined precisely later). Any randomness or uncertainty about X is just a consequence of uncertainty regarding ω ∈ Ω. As an example, X could relate to a some 1-year extreme value, whose uncertainty comes from day to day fluctuations, or some fundamental stochastic phenomenon represented by ω ∈ Ω. Examples of natural sources of uncertainty are weather or human actions in large scale. Therefore, whether modeling weather, option prices, structural safety at sea or traffic networks, stochastic models should be used. The probability of the event {X ∈ E}, for some subset E ⊂ R n , is really the probability of {ω ∈ X −1 (E)}. Technically, we need to ensure that {ω ∈ X −1 (E)} is something that we can compute the probability of, and for this we need F. F is a collection of subsets of Ω, and represents all possible events (in the "Ω-world"). When F is a σ-algebra 1 the pair (Ω, F) becomes a measurable space. So, when we define X as a random variable taking values in R n , this means that there exists some measurable space (Ω, F), such that any event {X ∈ E} in the "R n -world" (which has its own σ-algebra) has a corresponding event {ω ∈ X −1 (E)} ∈ F in the "Ω-world". It also means that we can define a probability measure on (Ω, F) that gives us the probability of each event, but before we introduce any specific probability measure, X will just be a measurable function 2 . -We start with assuming that there exists some measurable space (Ω, F) where X is a measurable function. The natural way to make X into a random variable is then to introduce some probability measure 3 P on F, giving us the probability space (Ω, F, P ). -Given a probability measure P on (Ω, F) we obtain the probability space (Ω, F, P ) on which X is defined as a random variable. We have considered here, for familiarity, that X takes values in R n . When no measure and σ-algebra is stated explicitly, one can assume that R n is endowed with the Lebesgue measure (which under-1 This means that 1) Ω ∈ F , 2) if S ∈ F then also the complement Ω \ S ∈ F, and 3) if S 1 , S 2 , · · · is a countable set of events then also the union S 1 ∪ S 2 ∪ . . . is in F . Note that if these properties hold, many other types of events (e.g. countable intersections) will have to be included as a consequence. 2 By definition, given two measure spaces (Ω, F ) and (X, X ), the function X : Ω → X is measurable if and only if X −1 (A) ∈ F ∀A ∈ X . 3 A function P : F → [0, 1] such that 1) P (Ω) = 1 and 2) P (∪E i ) = P (E i ) for any countable collection of pairwise disjoint events E i . lies the standard notion of length, area and volume etc.) and the Borel σ-algebra (the smallest σalgebra containing all open sets). Generally, X can take values in any measurable space. For example, X can map from Ω to a space of functions. This is important in the study of stochastic processes. The epistemic sub σ-algebra E In the probability space (Ω, F, P ), recall that the σ-algebra F contains all possible events. For any random variable X defined on (Ω, F, P ), the knowledge that some event has occurred provides information about X. This information may relate to X in a way that it only affects epistemic uncertainty, only aleatory uncertainty, or both. We are interested in specifying the events e ∈ F that are associated with epistemic information alone. It is the probability of these events we want to update as new information is obtained. The collection E of such sets is itself a σ-algebra, and we say that E ⊆ F(1) is the sub σ-algebra of F representing epistemic information. We illustrate this in the following examples. In Example 2.3, we consider the simplest possible scenario represented by the flip of a biased coin, and in Example 2.4 a familiar scenario from uncertainty quantification involving uncertainty with respect to functions. Example 2.3. (Coin flip) Define X = (Y, θ) as in Example 2.1, and let Y ∈ {0, 1} denote the outcome of a coin flip where "heads" is represented by Y = 0 and "tails" by Y = 1. Assume that P (Y = 0) = θ for some fixed but unknown θ ∈ [0, 1]. For simplicity we assume that θ can only take two values, θ ∈ {θ 1 , θ 2 } (e.g. there are two coins but we do not know which one is being used). Then Ω = {0, 1} × {θ 1 , θ 2 }, F = 2 Ω and E = {∅, Ω, {(0, θ 1 ), (1, θ 1 )}, {(0, θ 2 ), (1, θ 2 )}}. Example 2.4. (UQ) Let X = (x, y) where x is an aleatory random variable, and y is the result of a fixed but unknown function applied to x. We let y =f (x) wheref is a function-valued epistemic random variable. If x is defined on a probability space (Ω x , F x , P x ) andf is a stochastic process defined on (Ω f , F f , P f ), then (Ω, F, P ) can be defined as the product of the two spaces and E as the projection E = {Ω x × A \| A ∈ F f }. In the following, we assume that the epistemic sub σ-algebra E has been identified. Given a random variable X, we say that X is Emeasurable if X is measurable as a function defined on (Ω, E). We say that X is independent of E, if the conditional probability P (X\|e) is equal to P (X) for any event e ∈ E. With our definition of E, we then have for any random variable X on (Ω, F, P ) that -X is purely epistemic if and only if X is E- measurable, -X is purely aleatory if and only if X is independent of E. Epistemic conditioning Let X be a random variable on (Ω, F, P ) that may contain both epistemic and aleatory uncertainty, and assume that the epistemic sub σ-algebra E is given. By epistemic conditioning, we want to update the epistemic part of the uncertainty in X using some set of information I. In Example 2.3 this means updating the probabilities P (θ = θ 1 ) and P (θ = θ 2 ), and in Example 2.4 this means updating P f . In order to achieve this in the general setting, we first need a way to decouple epistemic and aleatory uncertainty. This can actually be made fairly intuitive, if we rely on the following assumption: Assumption 2.5. There exists a random variable θ : Ω → Θ that generates 4 E. If this generator θ exists, then for any fixed value θ ∈ Θ, we have that X\|θ is independent of E. Hence X\|θ is purely aleatory and θ is purely epistemic. We will call θ the epistemic generator, and we can interpret θ as a signal that reveals all epistemic information when known. That is, if θ could be observed, then knowing the value of θ would remove all epistemic uncertainty from our model. As it turns out, under fairly mild conditions one can always assume existence of this generator. One sufficient condition is that (Ω, F, P ) is a standard probability space, and then the statement holds up to sets of measure zero. This is a technical requirement to avoid pathological cases, and does not provide any new intuition that we see immediately useful, so we postpone further explanation to Appendix A. Example 2.6. (Coin flip -epistemic generator) In the coin flip example, the variable θ ∈ {θ 1 , θ 2 } which generates E is already specified. Example 2.7. (UQ -epistemic generator) In this example, when (Ω, F, P ) is the product of an aleatory space (Ω x , F x , P x ) and an epistemic space (Ω f , F f , P f ), we could let θ : Ω = Ω x × Ω f → Ω f be the projection θ(ω x , ω f ) = ω f . Alternatively, given only the space (Ω, F, P ) where both x andf are defined, assume thatf is a Gaussian process (or some other stochastic process for which the Karhunen-Loéve theorem holds). Then there exists a sequence of deterministic functions φ 1 , φ 2 , . . . and an infinite-dimensional vari- able θ = (θ 1 , θ 2 , . . . ) such thatf (x) = ∞ i=1 θ i φ i (x), and we can let E be generated by θ. The decoupling of epistemic and aleatory uncertainty is then obtained by considering the joint variable (X, θ) instead of X alone, because P (X, θ) = P (X \| θ)P (θ). (2) From (2) we see how the probability measure P becomes the product of the epistemic probability P (θ) and the aleatory probability P (X\|θ) when applied to (X, θ). Given new information, I, we will update our beliefs about θ, P (θ) → P (θ\|I), and we define the epistemic conditioning as follows: P new (X, θ) = P (X \| θ)P (θ \| I). (3) Two types of assumptions Consider the probability space (Ω, F, P ), with epistemic sub σ-algebra E. Here E represents epistemic information, which is the information associated with assumptions. In other words, an epistemic event e ∈ E represents an assumption. In fact, given a class of assumptions, the following Remark 2.8, shows why σ-algebras are appropriate structures. Remark 2.8. Let E be a collection of assumptions. If e ∈ E, this means that it is possible to assume that e is true. If it is also possible to assume that that e is false, then.ē ∈ E as well. It may then also be natural to require that e 1 , e 2 ∈ E ⇒ e 1 ∩ e 2 ∈ E, and so on. These are the defining properties of a σ-algebra. For any random variable X defined on (Ω, F, P ), when E is a sub σ-algebra of F, X\|e for e ∈ E is well defined, and represents the random variable under the assumption e. In particular, given any fixed epistemic event e ∈ E we have a corresponding aleatory distribution P (X\|e) over X, and the conditional P (X\|E) is the random measure corresponding to P (X\|e) when e is a random epistemic event in E. Here, the global probability measure P when applied to e, P (e), is the belief that e is true. In Section 2.4 we discussed updating the part of P associated with epistemic uncertainty. We also introduced the epistemic generator θ in order to associate the event e with an outcome θ(e), and make use of P (X\|θ) in place of P (X\|E). This provides a more intuitive interpretation of the assumptions that are measurable, i.e. those whose belief we may specify through P . Of course, the measure P is also based on assumptions. For instance, if we in Example 2.1 assume that Y follows a normal distribution. One could in principle specify a (measurable) space of probability distributions, from which the normal distribution is one example. Otherwise, we view the normality assumption as a structural assumption related to the probabilistic model for X, i.e. the measure P . These kinds of assumptions cannot be treated the same way as assumptions related to measurable events. For instance, the consequence of the complement assumption "Y does not follow a normal distribution" is not well defined. In order to avoid any confusion, we split the assumptions into two types: 1. The measurable assumptions represented by the σ-algebra E, and 2. the set M of structural assumptions underlying the probability measure P . This motivates the following definition. Definition 2.9 (Structural assumptions). We let M denote the set of structural assumptions that defines a probability measure on (Ω, F), which we may write P M (·) or P (· \| M ). We may also refer to M as the non-measurable assumptions, to emphasize that M contains all the assumptions not covered by E. When there is no risk of confusion we will also suppress the dependency on M and just write P (·). Stating the set M explicitly is typically only relevant for scenarios where we consider changes being made to the actual system that is being modelled, or for evaluating different candidate models, e.g. through the marginal likelihood P (I\|M ). In practice one would also state M so that decision makers can determine their level of trust in the probabilistic model, and the appropriate level of caution when applying the model. As we will see in the upcoming section, making changes to M and making changes to how P M acts on events in E are the two main ways in which we update a probabilistic digital twin. The Probabilistic Digital Twin The object that we will call probabilistic digital twin, PDT for short, is a probabilistic model of a physical system. It is essentially a (possibly degenerate) probability distribution of a vector X, representing the relevant attributes of the system, but where we in addition require the specification of epistemic uncertainty (assumptions) and how this uncertainty may be updated given new information. Before presenting the formal definition of a probabilistic digital twin, we start with an example showing why the identification of epistemic uncertainty is important. Why distinguish between aleatory and epistemic uncertainty? The decoupling of epistemic and aleatory uncertainty (as described in Section 2.4) is central in the PDT framework. There are two good reasons for doing this: 1. We want to make decisions that are robust with respect to epistemic uncertainty. 2. We want to study the effect of gathering information. Item 1. relates to the observation that decision theoretic approaches based on expectation may not be robust. That is, if we marginalize out the epistemic uncertainty (and considering only E θ [P (X\|θ)] = P (X\|θ)dP θ ). We give two examples of this below, see Example 3.1 and Example 3.2. Item 2. means that by considering the effect of information on epistemic uncertainty, we can evaluate the value of gathering information. This is discussed in further detail in Section 4.7. 3), we let θ 1 = 0.5, θ 2 = 0.99. Assume that you are given the option to guess the outcome of X. If you guess correct, you collect a reward of R = 10 6 $, otherwise you have to pay L = 10 6 $. A priori your belief about the bias of the coin is that P (θ = 0.5) = P (θ = 0.99) = 0.5. If you consider betting on X = 0, then the expected return, obtained by marginalizing over θ, becomes P (θ = 0.5)(0.5R− 0.5L) + P (θ = 0.99)(0.99R − 0.01L) = 490.000$. This is a scenario where decisions supported by taking the expectation with respect to epistemic uncertainty is not robust, as we believe that θ = 0.5 and θ = 0.99 are equally likely, and if θ = 0.5 we will lose 10 6 $ 50% of the time by betting on X = 0. In structural reliability analysis, we are dealing with an unknown function g with the property that the event {y = g(x) < 0} corresponds to failure. When g is represented by a random functionĝ with epistemic uncertainty, the failure probability is also uncertain. Or in other words, ifĝ is epistemic then g is a function of the generator θ. Hence, the failure probability is a function of θ. We want to make use of a conservative estimate of the failure probability, i.e., use a conservative value of θ. P (θ) tells us how conservative a given value of θ is. The attributes X To define a PDT, we start by considering a vector X consisting of the attributes of some system. This means that X is a representation of the physical object or asset that we are interested in. In general, X describes the physical system. In addition, X must contain attributes related to any type of information that we want to make use of. For instance, if the information consists of observations, the relevant observable quantities, as well as attributes related to measurement errors or noise, may be included in X. In general, we will think of a model of a system as a set of assumptions that describes how the components of X are related or behave. The canonical example here is where some physical quantity is inferred from observations including errors and noise, in which case a model of the physical quantity (physical system) is connected with a model of the data generating process (observational system). We are interested in modelling dependencies with associated uncertainty related to the components of X, and treat X as a random variable. The attributes X characterise the state of the system and the processes that the PDT represents. X may for instance include: • System parameters representing quantities that have a fixed, but possibly uncertain, value. For instance, these parameters may be related to the system configuration. • System variables that may vary in time, and which value may be uncertain. • System events i.e., the occurrence of defined state transitions. In risk analysis, one is often concerned with risk measures given as quantified properties of X, usually in terms of expectations. For instance, if X contains some extreme value (e.g. the 100-year wave) or some specified event of failure (using a binary variable), the expectations of these may be compared against risk acceptance criteria to determine compliance. The PDT definition Based on the concepts introduced so far, we define the PDT as follows: Definition 3.3 (Probabilistic Digital Twin). A Probabilistic Digital Twin (PDT) is a triplet (X, A, I) where X is a vector of attributes of a system, A contains the assumptions needed to specify a probabilistic model, and I contains information regarding actions and observations: A = ((Ω, F), E, M ), where (Ω, F) is a measure space where X is measurable, and E is the sub σ-algebra representing epistemic information. M contains the structural assumptions that defines a probability measure P M on (Ω, F). I is a set consisting of events of the form (d, o), where d encodes a description of the conditions under which the observation o was made, and where the likelihood P (o\|X, d) is well defined. For brevity, we will write this likelihood as P (I\|X) when I contains multiple events of this sort. When M is understood, and there is no risk on confusion, we will drop stating the dependency on M explicitly and just refer to the probability space (Ω, F, P ). It is important to note that consistency between I and P (X) is required. That is, when using the probabilistic model for X, it should be possible to simulate the type of observations given by I. In this case the likelihood P (I\|X) is well defined, and the epistemic updating of X can be obtained from Bayes' theorem. Finally, we note that with this setup the information Let (x 1 , x 2 ) denote two physical quantities where x 2 depends on x 1 , and let (y, ε) represent an observable quantity where y corresponds to observing x 2 together with additive noise ε. Set X = (x 1 , x 2 , y, ε). x 1 x 2 y ε We define a model M corresponding to x 1 ∼ p x1 (x 1 \|θ 1 ), x 2 = f (x 1 , θ 2 ), y = x 2 + ε and ε ∼ p ε , where p x1 is a probability density depending on the parameter θ 1 and f (·, θ 2 ) is a deterministic function depending on the parameter θ 2 . θ 1 and θ 2 are epistemic parameters for which we define a joint density p θ . Assume that I = {(d (1) , o (1) ), . . . , (d (n) , o (n) )} is a set of controlled experiments, where d (i) = (set x 1 = x (i) 1 ) and o (i) is a corresponding obser- vation of y\|(x 1 = x (i) 1 , ε = ε (i) ) for a selected set of inputs x (i) 1 , . . . , x (n) 1 and unknown i.i.d. ε (i) ∼ p ε . In this scenario, regression is performed by updating the distribution p θ to agree with the observations: p θ (θ\|I) = p θ (θ 1 \|θ 2 )p θ (θ 2 \|I) = 1 Z n i=1 p ε o (i) − f (x (i) 1 , θ 2 ) p θ (θ),(4) where Z is a constant ensuring that the updated density integrates to one. If instead I corresponds to direct observations, d (i) = (observe y (i) ), o (i) = y (i) , then p θ (θ\|I) cor- responds to using x 1 instead of x (i) 1 and multiplying with p x1 (x 1 \|θ 1 ) in (4). Note that the scenario with controlled experiments in Example 3.4 corresponds to a different model than the one in Figure 1. This is a familiar scenario in the study of causal inference, where actively setting the value of x 1 is the do-operator (see Pearl [11]) which breaks link between x 1 and x 2 . Corroded pipeline example To give a concrete example of a system where the PDT framework is relevant, we consider the following model from Agrell and Dahl [2]. This is based on a probabilistic structural reliability model which is recommended for engineering assessment of offshore pipelines with corrosion (DNV GL RP-F101 [12]). It is a model of a physical failure mechanism called pipeline burst, which may occur when the pipeline's ability to withstand the high internal pressure has been reduced as a consequence of corrosion. We will describe just a general overview of this model, and refer to [2, Example 4] for specific details regarding probability distributions etc. Later, in Section 5.6, we will revisit this example and make use of reinforcement learning to search for an optimal way of updating the PDT. Figure 2 shows a graphical representation of the structural reliability model. Here, a steel pipeline is characterised by the outer diameter D, the wall thickness t and the ultimate tensile strength s. The pipeline contains a rectangular shaped defect with a given depth d and length l. Given a pipeline (D, t, s) with a defect (d, l), we can determine the pipeline's pressure resistance capacity (the maximum differential pressure the pipeline can with-stand before bursting). We let p FE denote the capacity coming from a Finite Element simulation of the physical phenomenon. From the theoretical capacity p FE , we model the true pipeline capacity as a function of p FE and X m , where X m is the model discrepancy. For simplicity we have assumed that X m does not depend on the type of pipeline and defect, and we will also assume that σ m is fixed, and only the mean µ m can be inferred from observations. Finally, given the pressure load p d , the limit state representing the transition to failure is then given as g = p c − p d , and the probability of failure is defined as P (g ≤ 0). D t s d l σ m µ m X m p FE p c p d g Pipeline Defect Model discrepancy Load Capacity If we let X be the random vector containing all of the nodes in Figure 2, then X represents a probabilistic model of the physical system. In this example, we want to model some of the uncertainty related to the defect size, the model uncertainty, and the capacity as epistemic. We assume that the defect depth d has a fixed but unknown value, that can be inferred through observations that include noise. Similarly, the model uncertainty X m can be determined from experiments. Uncertainty with respect to p FE comes from the fact that evaluating the true value of p FE \|(D, t, s, d, l) involves a time-consuming numerical computation. Hence, p FE can only be known for a finite, and relatively small set of input combinations. We can letp FE denote a stochastic process that models our uncertainty about p FE . To construct a PDT from X we will letp FE take the place of p FE , and specify that d, µ m andp FE are epistemic, i.e. E = σ(d, µ m ,p FE ). If we want a way to update the epistemic uncer-tainty based on observations, we also need to specify the relevant data generating process. In this example, we assume that there are three different ways of collecting data: 1. Defect measurement: We assume that noise perturbed observations of the relative depth, d/t + ε, can be made. 2. Computer experiment: Evaluate p FE at some selected input (D, t, s, d, l). 3. Lab experiment: Obtain one observation of X m . As the defect measurements requires specification of an additional random variable, we have to include ε or (d/t) obs = d/t + ε in X as part of the complete probabilistic model. This would then define a PDT where epistemic updating is possible. The physical system that the PDT represents in this example is rarely viewed in isolation. For instance, the random variables representing the pipeline geometry and material are the result of uncertainty or variations in how the pipeline has been manufactured, installed and operated. And the size of the defect is the result of a chemical process, where scientific models are available. It could therefore be natural to view the PDT from this example as a component of a bigger PDT, where probabilistic models of the manufacturing, operating conditions and corrosion process etc. are connected. This form of modularity is often emphasized in the discussion of digital twins, and likewise for the kind of Bayesian network type of models as considered in this example. Sequential decision making We now consider how the PDT framework may be adopted in real-world applications. As with any statistical model of this form, the relevant type of applications are related to prediction and inference. Since the PDT is supposed to provide a one-to-one correspondence (including uncertainty) with a real physical system, we are interested in using the PDT to understand the consequences of actions that we have the option to make. In particular, we will consider the discrete sequential decision making scenario, where get the opportunity to make a decision, receive information related to the consequences of this decision, and use this to inform the next decision, and so on. In this kind of scenario, we want to use the PDT to determine an action or policy for how to act optimally (with respect to some case-specific criterion). By a policy here we mean the instructions for how to select among multiple actions given the information available at each discrete time step. We describe this in more detail in Section 4.4 where we discuss how the PDT is used for planning. When we make use of the PDT in this way, we consider the PDT as a "mental model" of the real physical system, which an agent uses to evaluate the potential consequences of actions. The agent then decides on some action to make, observes the outcome, and updates her beliefs about the true system, as illustrated in Figure 3. Whether the agent applies a policy or just an action (the first in the policy) before collecting information and updating the probabilistic model depends on the type of application at hand. In general it is better to update the model as often as possible, preferably between each action, but the actual computational time needed to perform this updating might make it impossible to achieve in practice. Mathematical framework of sequential decision making In this section, we briefly recap the mathematical framework of stochastic, sequential decision making in discrete time. We first recall the general framework, and in the following Section 4.2, we show how this relates to our definition of a PDT. Let t = 0, 1, 2, . . . , N − 1 and consider a discrete time system where the state of the system, {x t } t≥1 , is given by x t+1 = f t (x t , u t , w t ), t = 0, 1, 2, . . . , N − 1. (5) Here, x t is the state of the system at time t, u t is a control and w t is a noise, or random parameter at time t. Note that the control, u t , is a decision which can be made by an agent (the controller ) at time t. This control is to be chosen from a set of admissible controls A t (possibly, but not necessarily depending on time). Also, f t , t = 0, 1, 2, . . . , N − 1 are functions mapping from the space of state variables (state space), controls and noise into the set of possible states of {x t } t≥0 . The precise structure of the state space, set of admissible controls and the random parameter space depends on the particular problem under consideration. Note that due to the randomness in w t , t = 0, 1, 2, . . . , N − 1, the system state x t and control u t , t = 1, 2, . . . , N − 1 also become random variables. We remark that because of this, the state equation is sometimes written in the following form, x t+1 (ω) = f t (x t (ω), u t (ω), ω)(6) where ω ∈ Ω is a scenario in a scenario space Ω (representing the randomness). Sometimes, the randomness is suppressed for notational convenience, so the state equation becomes x t+1 = f t (x t , u t ), t = 0, 1, 2, . . . , N − 1. Note that in the state equation (5) (alternatively, equation (6)), x t+1 only depends on the previous time step, i.e., x t , u t , w t . This is the Markov property (as long as we assume that the distribution of w t does not depend on past values of w s , s = 0, 1, . . . t − 1, only x t , u t ). That is, the next system state only depends on the previous one. Since this Markovian framework is what will be used throughout this paper as we move on to reinforcement learning for a probabilistic digital twin, we focus on this. However, we remark that there is a large theory of sequential decision making which is not based on this Markovianity. This theory is based around maximum principles instead of dynamic programming, see the following Section 4.3 for more on this. The aim of the agent is to minimize a cost function under the state constraint (5) (or alternatively, (6)). We assume that this cost function is of the following, additive form, E g(x N ) + N −1 t=0 h t (x t , u t , w t )(7) where the expectation is taken with respect to an a priori given probability measure. That is, we sum over all instantaneous rewards h t (x t , u t , w t ), t = 0, 1, . . . , N − 1 which depend on the state of the system, the control and the randomness and add a terminal reward g(x N ) which only depends on the system state at the terminal time t = N . This function is called the objective function. Hence, the stochastic sequential decision making problem of the agent is to choose admissible controls u t , t = 0, 1, 2, . . . , N − 1 in order to, min ut∈At,t≥0 E g(x N ) + N −1 t=0 h t (x t , u t , w t )(8) such that x t+1 = f t (x t , u t , w t ), t = 0, 1, 2, . . . , N − 1. Typically, we assume that the agent has full information in the sense that they can choose the control at time t based on (fully) observing the state process up until this time, but that they are not able to use any more information than this (future information, such as inside information). This problem formulation is very similar to that of continuous time stochastic optimal control problem. Remark 4.1. (A note on continuous time) This framework is parallel to that of stochastic optimal control in continuous time. The main differences in the framework in the continuous time case is that the state equation is typically a stochastic differential equation, and the sum is replaced by an integral in the objective function. For a detailed introduction to continuous time control, see e.g., Øksendal [13]. Other versions of sequential decision making problems include inside information optimal control, partial information optimal control, infinite time horizon optimal control and control with various delay and memory effects. One can also consider problems where further constraints, either on the control or the state, is added to problem (8). In Bertsekas [14], the sequential decision making problem (8) is studied via the dynamic programming algorithm. This algorithm is based on the Bellman optimality principle, which says that an optimal policy chosen at some initial time, must be optimal when the problem is re-solved at a later stage given the state resulting from the initial choice. Sequential decision making in the PDT Now, we show how the sequential decision making framework from the previous section can be used to solve sequential decision making problems in the PDT. We may apply this sequential decision making framework to our PDT by letting x t := X t . That is, the state process for the PDT sequential decision making problem is the random vector of attributes X t . Note that in Definition 3.3, there is no time-dependency in the attributes X. However, since we are interested in considering sequential decision making in the PDT, we need to assume that there is some sort of development over time (or some indexed set, e.g. information) of the PDT. Hence, the stochastic sequential decision making problem of the PDT-agent is to choose admissible controls u t , t = 0, 1, 2, . . . , N − 1 in order to, min ut∈At,t≥0 E g(X N ) + N −1 t=0 h t (X t , u t , w t )(9) such that X t+1 = f t (X t , u t , w t ), t = 0, 1, 2, . . . , N − 1. Here, the set of admissible controls, {u t } t≥0 ∈ A, are problem specific. So are the functions h t , g t and f t for t ≥ 0. Given a particular problem, these functions are defined based on the goal of the PDTagent as well as the updating of the PDT given new input. Two solution methods for sequential decision making In the literature on (discrete time) stochastic sequential decision making, there are two main approaches: • The dynamic programming principle (DPP). • The Pontyagrin maximum principle (MP). The continuous time analogues are the Hamilton-Jacobi-Bellman equations (a particular kind of partial differential equation) and the stochastic maximum principle, respectively. The DPP is based on the Bellman optimality principle, and the resulting Bellman equation which can be derived from this. Dynamic programming has a few important advantages. It is tractable from an algorithmic perspective because of the backpropagation algorithm naturally resulting from the Bellman equation. Furthermore, the method is always well defined since it is based on working with the value function (the function that maps states to the optimal value given by (9), assuming that we start from the given state). However, there are some downsides to the DPP method as well. Firstly, DPP requires a Markovian framework (or that we can transform the problem to a Markovian framework). Also, the DPP requires that the Bellman equation holds. This may not be the case if we have problems with for example non-exponential discounting (with respect to time). In this case, we say that there are problems with time-inconsistency, see e.g., Rudloff et al. [15]. For instance, traditional risk measures such as value-at-risk (VaR) and conditional-value at risk (CVaR) are time-inconsistent in this sense, see Cheridito and Stadje [16], and Artzner et al. [17] respectively. Hence, we run into time-inconsistency issues when e.g., minimizing the conditional-valueat-risk of some financial position, if we are using the DPP method. Finally, we cannot have state constraints when using the DPP method, since this causes discontinuity of the value function, see Hao and Li [18]. The alternative approach to solving stochastic sequential decision making problems is via the Pontryagin maximum principle. This method does not require Markovianity or depend on Bellman equation. Hence, there are no problems with timeinconsistency. However, the MP approach is less tractable from an algorithmic point of view. Furthermore, the MP approach requires existence of a minimizing (or maximizing) control. This may not be the case, since it is possible that only limiting control processes converging to the minimum (maximum) exist. The pros and cons of dynamic programming and the maximum principle approach carry over in continuous time. From a computational point of view, the dynamic programming method suffers from the curse of dimensionality. When doing numerical backward induction in the DPP, the objective function must be computed for each combination of values. This makes the method too computationally demanding to be applicable in practice for problems where the state space is large, see Agrell and Dahl [2] for a discussion of this. Until recently, numerical algorithms based on the maximum principle were not frequently studied in the literature, an exception is Bonnans [19]. However, the MP approach leads to systems of backward differential equations, in the continuous case, which are often computationally demanding and also less tractable from an algorithmic point of view than the DPP method. However, with the advances of machine learning over the past decade, some new approaches based on the MP approach using deep learning have been introduced, see Li et al. [20]. Actually, reinforcement learning (RL) is essentially the DPP method. Hence, RL algorithms also suffer from the curse of dimensionality, see Sutton and Barto [21]. This means that most RL algorithms become less efficient when the dimension of the state space increases. However, by using function approximation the curse of dimensionality can often be efficiently handled, see Arulkumaran et al. [22]. The purpose of this paper is to build on this literature by connecting deep reinforcement learning (so essentially, the dynamic programming method) to probabilistic digital twins in order to do planning with respect to the PDT. This is the topic of the following section. Planning in the PDT In this section, we discuss how the PDT can be used for planning. That is, how we use the PDT to identify an optimal policy, without acting in the real world, but by instead simulating what will happen in the real world given that the agent chooses specific actions (or controls, as they are called in the sequential decision making literature, see Section 4.1). We use the PDT as a tool to find a plan (policy), or a single action (first action of policy), to perform in the real world. In order to solve our sequential decision making problem in the PDT, we have chosen to use a reinforcement learning formulation. As remarked in Section 4.3, this essentially corresponds to choosing the dynamic programming method for solving the optimal control problem (as opposed to a maximum principle approach). Because we will use a DPP approach, we need all the assumptions that come with this, see the discussion in Section 4.3: A Markovian framework, or the possibility of transforming the problem to something Markovian. We need the Bellman equation to hold in order to avoid issues with time-inconsistency. In order to ensure this, we for example need to use exponential discounting and not have e.g., conditional expectation of state process in a non-linear way in the objective function. Finally, our planning problem cannot have state constraints. Remark 4.2. Instead of using the DPP to solve the planning problem, we could use a maximum principle approach. One possible way of doing this in practice, is by using one of the MP based algorithms found in Li et al. [20], instead of using reinforcement learning. By this alternative approach, we avoid the Markovianity requirement, possible timeinconsistency issues and can allow for state constraints (via a Lagrange multiplier method -see e.g., Dahl and Stokkereit [23]). This topic is beyond the scope of this paper, but is a current work in progress. Starting with an initial PDT as a digital representation of a physical system given our current knowledge, we assume that there are two ways to update the PDT: 1. Changing or updating the structural assumptions M , and hence the probability measure P M . 2. Updating the information I. The structural assumptions M are related to the probabilistic model for X. Recall from Section 2.4, that these assumptions define the probability measure P M . Often, this probability measure is taken as given in stochastic modeling. However, in practice, probability measures are not given to us, but decided by analysts based on previous knowledge. Hence, the structural assumptions M may be updated because of new knowledge, external to the model, or for other reasons the analysts view as important. Updating the information is our main concern in this paper, since this is related to the agent making costly decisions in order to gather more information. An update of the information also means (potentially) reducing the epistemic uncertainty in the PDT. Optimal information gathering in the PDT will be discussed in detail in the following Section 4.7. MDP, POMDP and its relation to DPP In this section, we briefly recall the definitions of Markov decision processes, partially observable Markov decision processes and explain how these relate to the seuqntial decision making framework of Section 4.1. Markov decision processes (MDP) are discretetime stochastic control processes of a specific form. An MDP is a tuple (S, A, P a , R a ), where S is a set of states (the state space) and A is a set of actions (action space). Also, P a (s, s ) = P a (s t+1 = s \| a t = a, s t = s) is the probability of going from state s at time t to state s at time t + 1 if we do action a at time t. Finally, R a (s, s ) is the instantaneous reward of transitioning from state s at time t to state s at time t + 1 by doing action a (at time t). An MDP satisfies the Markov property, so given that the process is in state s and will be doing a at time t, the next state s t+1 is conditionally independent of all other previous states and actions. Remark 4.3. (MDP and DPP) Note that this definition of an MDP is essentially the same as our DPP framework of Section 4.1. In the MDP notation, we say actions, while in the control notation, it is common to use the word control. In Section 4.1, we talked about instantaneous cost functions, but here we talk about instantaneous rewards. Since minimization and maximization problems are equivalent (since inf{·} = − sup{−·}), so are these two concepts. Furthermore, the definition of the transition probabilities P a in the MDP framework corresponding to the Markov assumption of the DPP method. In both frameworks, we talk about the system states, though in the DPP framework we model this directly via equation (5). A generalization of MDP are partially observable Markov decision processes (POMDPs). While an MDP is a 4-tuple, a POMDP is a 6-tuple, (S, A, P a , R a ,Ω, O). Here (like before), S is the state space, A is the action space, P a give the conditional transition probabilities between the different states in S and R a give the instantaneous rewards of the transitions for a particular action a. In addition, we haveΩ, which is a set of observations. In contrast to the MDP framework, with POMDP, the agent no longer observes the state s directly, but only an observation o ∈Ω. Furthermore, the agent knows O which is a set of conditional observation probabilities. That is, O(o \| s , a) is the probability of observing o ∈Ω given that we do action a from state s . The objective of the agent in the POMDP sequential decision problem is to choose a policy, that is actions at each time, in order to max {at}∈A E T t=0 λ t r t(10) where r t is the reward earned at time t (depending on s t , a t and s t+1 ), and λ ∈ [0, 1] is a number called the discount factor. The discount factor can be used to introduce a preference for immediate rewards as opposed to more distant rewards, which may be relevant for the problem at hand, or used just for numerical efficiency. Hence, the agent aims to maximize their expected discounted reward over all future times. Note that is it also possible to consider problem (10) over an infinite time horizon or with a separate terminal reward function as well. This is similar to the DPP sequential decision making framework of Section 4.1. In order to solve a POMDP, it is necessary to include memory of past actions and observations. Actually, the inclusion of partial observations means that the problem is no longer Markovian. However, there is a way to Markovianize the POMDP by transforming the POMDP into a belief-state MDP. In this case, the agent summarizes all information about the past in a belief vector b(t), which is updated as time passes. See [24], Chapter 12.2.3 for details. MDP (and POMDP) in the PDT framework In this section, we show how the probabilistic digital twin can be incorporated in a reinforcement learning framework, in order to solve sequential decision problems in the PDT. In Section 4.2, we showed how we can use the mathematical framework of sequential decision making to solve optimal control problems for a PDT-agent. Also, in Section 4.5, we saw (in Remark 4.3) that the MDP (or POMDP in general) framework essentially corresponds to that of the DPP. In theory, we could use the sequential decision making framework and the DPP to solve optimal control problems in the PDT. However, due to the curse of dimensionality, this will typically not be practically tractable (see Section 4.3). In order to resolve this, we cast the PDT sequential decision making problem into a reinforcement learning, in particular a MDP, framework. This will enable us to solve the PDT optimal control problem via deep reinforcement learning, in which there are suitable tools to overcome the curse of dimensionality. To define a decision making process in the PDT as a MDP, we need to determine our state space, action space, (Markovian) transition probabilities and a reward function. • The action space A: These are the possible actions within the PDT. These may depend on the problem at hand. In the next Section 4.7, we will discuss optimal information gathering, where the agent can choose between different types of experiments, at different costs, in order to gain more information. In this case, the action space is the set of possible decisions that the agent can choose between in order to attain more information. • The state space S: We define a state as a PDT (or equivalently a version of a PDT that evolves in discrete time t = 0, 1, . . . ). A PDT represents our belief about the current physical state of a system, and it is defined by some initial assumptions together with the information acquired through time. In practice, if the structural assumptions are not changed, we may let the information available at the current time represent a state. This means that our MDP will consist of beliefstates, represented by information, from which inference about the true physical state can be made. This is a standard way of creating a MDP from a POMDP, so we can view the PDT state-space as a space of beliefs about some underlying partially observable physical state. Starting from a PDT, we define the state space as all updated PDTs we can reach by taking actions in the action space A. • The transition probabilities P a : Based on our chosen definition of the state space, the transition probabilities are the probabilities of going from one level of information to another, given the action chosen by the agent. For example, if the agent chooses to make decision (action) d, what is the probability of going from the current level of information to another (equal or better) level. This is given by epistemic conditioning of the PDT with respect to the given information set I = {(d, o)} based on the decisions d the new observation o. When it comes to updates of the structural assumptions M , we consider this as deterministic transitions. • The reward R a : The reward function, or equivalently, cost function, will depend on the specific problem at hand. To each action a ∈ A, we assume that we have an associated reward R a . In the numerical examples in Section 5, we give specific examples of how these rewards can be defined. As mentioned in Section 4.4, there are two ways to update the PDT: Updating the structural assumptions M and updating the information I. If we update the PDT by (only) adding to the information set I, we always have the Markov property. If we also update M , then the preservation of the Markov property is not given. In this case, using a maximum principle deep learning algorithm instead of the DPP based deep RL is a possibility, see [20]. Remark 4.4. Note that in the case where we have a very simple PDT with only discrete variables and only a few actions, then the RL approach is not necessary. In this case, the DPP method as done in traditional optimal control works well, and we can apply a planning algorithm to the PDT in order to derive an optimal policy. However, in general, the state-action space of the PDT will be too large for this. Hence, traditional planning algorithms, and even regular RL may not be feasible due to the curse of dimensionality. In this paper, we will consider deep reinforcement learning as an approach to deal with this. We discuss this further in Section 5. Note that what determines an optimal action or policy will of course depend on what objective the outcomes are measured against. That is, what do we want to achieve in the real world? There are many different objectives we could consider. In the following we present one generic objective related to optimal information gathering, where the PDT framework is suitable. Optimal information gathering A generic, but relevant, objective in optimal sequential decision making is simply to "improve itself". That is, to reduce epistemic uncertainty with respect to some quantity of interest. Another option, is to consider maximizing the Kullback-Leibler divergence with respect to epistemic uncertainty as a general objective. This would mean that we aim to collect the information that "will surprise us the most". By definition, a PDT contains an observational model related to the data generating process (the epistemic conditioning relies on this). This means that we can simulate the effect of gathering information, and we can study how to do this optimally. In order to define what we mean by an optimal strategy for gathering information, we then have to specify the following, • Objective: What we need the information for. For example, what kind of decision do we intend to support using the PDT? Is it something we want to estimate? What is the required accuracy needed? For instance, we might want to reduce epistemic uncertainty with respect to some quantity, e.g., a risk metric such as a failure probability, expected extreme values etc. • Cost: The cost related to the relevant information-gathering activities. Then, from the PDT together with a specified objective and cost, one alternative is to define the optimal strategy as the strategy that minimizes the (discounted) expected cost needed to achieve the objective (or equivalently achieves the objective while maximizing reward). Example 4.5. (Coin flip -information gathering) Continuing from Example 3.1, imagine that before making your final bet, you can flip the coin as many times as you like in order to learn about θ. Each of these test flips will cost 10.000 $. You also get the opportunity to replace the coin with a new one, at the cost of 100.000 $. An interesting problem is now how to select an optimal strategy for when to test, bet or replace in this game. And will such a strategy be robust? What if there is a limit on the total number of actions than can be performed? In Section 5.5 we illustrate how reinforcement learning can be applied to study this problem, where the coin represents a component with reliability θ, that we may test, use or replace. Deep Reinforcement Learning with PDTs In this section we give an example of how reinforcement learning can be used for planning, i.e. finding an optimal action or policy, with a PDT. The reinforcement learning paradigm is especially relevant for problems where the state and/or action space is large, or dynamical models where specific transition probabilities are not easily attainable but where efficient sampling is still feasible. In probabilistic modelling of complex physical phenomena, we often find ourselves in this kind of setting. Reinforcement Learning (RL) Reinforcement learning, in short, aims to optimize sequential decision problems through sampling from a MDP (Sutton and Barto [21]). We think of this as an agent taking actions within an environment, following some policy π(a\|s), which gives the probability of taking action a if the agent is currently at state s. Generally, π(a\|s) represents a (possibly degenerate) probability distribution over actions a ∈ A for each s ∈ S. The agent's objective is to maximize the amount of reward it receives over time, and a policy π that achieves this is called an optimal policy. Given a policy π we can define the value of a state s ∈ S as v π (s) = E T t=0 λ t r t \| s 0 = s (11) where r t is the reward earned at time t (depending on s t , a t and s t+1 ), given that the agent follows policy π starting from s 0 = s. That is, for P a and R a given by the MDP, a t ∼ π(a t \|s t ), s t+1 ∼ P at (s t , s t+1 ) and r t ∼ R at (s t , s t+1 ). Here we make use of a discount factor λ ∈ [0, 1] in the definition of cumulative reward. If we want to consider T = ∞ (continuing tasks) instead of T < ∞ (episodic task), then λ < 1 is generally necessary. The optimal value function is defined as the one that maximises (11) over all policies π. The optimal action at each state s ∈ S then corresponds to acting greedily with respect to this value function, i.e. selecting the action a t that in expectation maximises the value of s t+1 . Likewise, it is common to define the action-value function q π (s, a), which corresponds to the expected cumulative return of first taking action a in state s and following π thereafter. RL generally involves some form of Monte Carlo simulation, where a large number of episodes are sampled from the MDP, with the goal of estimating or approximating the optimal value of sates, state-action pairs, or an optimal policy directly. Theoretically this is essentially equivalent to the DPP framework, but with RL we are mostly concerned with problems where optimal solutions cannot be found and some form of approximation is needed. By the use of flexible approximation methods combined with adaptive sampling strategies, RL makes it possible to deal with large and complex state-and action spaces. Function approximation One way of using function approximation in RL is to define a parametric functionv(s, w) ≈ v π (s), given by a set of weights w ∈ R d , and try to learn the value function of an optimal policy by finding an appropriate value for w. Alternatively, we could approximate the value of a state-action pair, q(s, a, w) ≈ q π (s, a), or a policyπ(a\|s, w) ≈ π(a\|s). The general goal is then to optimize w, using data generated by sampling from the MDP, and the RL literature contains many different algorithms designed for this purpose. In the case where a neural network is used for function approximation, it is often referred to as deep reinforcement learning. One alternative, which we will make use of in an example later on, is the deep Q-learning (DQN) approach as introduced by van Hasselt et al. [25], which represents the value of a set of m actions at a state s using a multi-layered neural network q(s, w) : S → R m .(12) Note here thatq(s, w) is a function defined on the state space S. In general, any approximation of the value functions v or q, or the policy π are defined on S or S × A. A question that then arises, is how can we define parametric functions on the state space S when we are dealing with PDTs? We can assume that we have control over the set of admissible actions A, in the sense that this is something we define, and creating parametric functions defined on A should not be a problem. But as discussed in Section 4.6, S will consist of belief -states. Defining the state space We are interested in an MDP where the transition probabilities P a (s, s ) corresponds to updating a PDT as a consequence of action a. In that sense, s and s are PDTs. Given a well-defined set of admissible actions, the state space S is then the set of all PDTs that can be obtained starting from some initial state s 0 , within some defined horizon. Recall that going from s to s then means keeping track of any changes made to the structural assumptions M and the information I, as illustrated in Figure 3. From now on, we will for simplicity assume that updating the PDT only involves epistemic conditioning with respect to the information I. This is a rather generic situation. Also, finding a way to represent changes in M will have to be handled for the specific use case under consideration. Assuming some initial PDT s 0 is given, any state s t at a later time t is then uniquely defined by the set of information I t available at time t. Representing states by information in this way is something that is often done to transform a POMDP to a MDP. That is, although the true state s t at time t is unknown in a POMDP, the information I t , and consequently our belief about s t , is always know at time t. Inspired by the POMDP terminology, we may therefore view a PDT as a belief-state, which seems natural as the PDT is essentially a way to encode our beliefs about some real physical system. Hence, we will proceed with defining the state space S as the information state-space, which is the set of all sets of information I. Although this is a very generic approach, we will show that there is a way of defining a flexible parametric class of functions on S. But we must emphasize that that if there are other ways of compressing the information I, for instance due to conjugacy in the epistemic updating, then this is probably much more efficient. Example 5.1 below shows exactly what we mean by this. 3), all of our belief with respect to epistemic uncertainty is represented by the number ψ = P (θ = θ 1 ). Given some observation Y = y ∈ {0, 1}, the epistemic conditioning corresponds to ψ → β 1 (y)ψ β 1 (y)ψ + β 2 (y)(1 − ψ) , where, for j = 1, 2, β j (y) = θ j if y = 0 and β j (y) = 1 − θ j if y = 1. In this example, the information state-space consists of all sets of the form I t = {y 1 , . . . , y t } where each y i is binary. However, if the goal is to let I t be the representation of a PDT, we could just as well use ψ t , i.e. define S = [0, 1] as the state space. Alternatively, the number of heads and tails (0s and 1s) provides the same information, so we could also make use of S = {0, . . . , N } × {0, . . . , N } where N is an upper limit on the total number of flips we consider. Deep learning on the information state-space Let S be a set of sets I ⊂ R d . We will assume that each set I ∈ S consists of a finite number of elements y ∈ R d , but we do not require that all sets I have the same size. We are interested in functions defined on S. An important property of any function f that takes a set I as input, is permutation invariance. I.e. f ({y 1 , . . . , y N }) = f ({y κ(1) , . . . , y κ(N ) }) for any permutation κ. It can been shown that under fairly mild assumptions, that such functions have the following decomposition f (I) = ρ   y∈I φ(y)   .(13) These sum decompositions were studied by Zaheer et al. [26] and later by Wagstaff et al. [27], which showed that if \|I\| ≤ p for all I ∈ S, then any continuous function f : S → R can be written as (13) for some suitable functions φ : R d → R p and ρ : R p → R. The motivation in [26,27] was to enable supervised learning of permutation invariant and set-valued functions, by replacing ρ and φ with flexible function approximators, such as Gaussian processes or neural networks. Other forms of decomposition, by replacing the summation in (13) with something else that can be learned, has also been considered by Soelch et al. [28]. For reinforcement learning, we will make use of the form (13) to represent functions defined on the information states space S, such asv(s, w),q(s, a, w), or π(a\|s, w), using a neural network with parameter w. In the remaining part of this paper we present two examples showing how this works in practice. The "coin flip" example Throughout this paper we have presented a series of small examples involving a biased coin, represented by X = (Y, θ). In Example 4.5 we ended by introducing a game where the player has to select whether to bet on, test or replace the coin. As a simple illustration we will show how reinforcement learning can be applied in this setting. But now, we will imagine that the coin Y represents a component in some physical system, where Y = 0 corresponds to the component functioning and Y = 1 represents failure. The probability P (Y = 1) = 1 − θ is then the components failure probability, and we say that θ is the reliability. For simplicity we assume that θ ∈ {0.5, 0.99}, and that our initial belief is P (θ = 0.5) = 0.5. That is, when we buy a new component, there is a 50 % chance of getting a "bad" component (that fails 50 % of the time), and consequently a 50 % probability of getting a "good" component (that fails 1 % of the time). We consider a project going over N = 10 days. Each day we will decide between one of the following 4 actions: We will find a deterministic policy π : S → A that maps from the information state-space to one of the four actions. The information state-space S is here represented by the number of days left of the project, n = N − t, and the set I t of observations of the component that is currently in use at time t. If we let S Y contain all sets of the form I = {Y 1 , . . . , Y t }, for Y t ∈ {0, 1} and t < N , then S = S Y × {1, . . . , N }(14) represents the information state-space. In this example we made use of the deep Q-learning (DQN) approach described by van Hasselt et al. [25], where we define a neural network q(s, w) : S → R 4 , that represents the action-value of each of the four actions. The optimal policy is then obtained by at each state s selecting the action corresponding to the maximal component ofq. We start by finding a policy that optimizes the cumulative reward over the 10 days (without discounting). As it turns out, this policy prefers to "gamble" that the component works rather than performing tests. In the case where the starting component is reliable (which happens 50 % of the time), a high reward can be obtained by selection action 3 at every opportunity. The general "idea" with this policy, is that if action 3 results in failure, the following action is to replace the component (action 2), unless there are few days left of the project in which case action 0 is selected. We call this the "unconstrained" policy. Although the unconstrained policy givens the largest expected reward, there is an approximately 50 % chance that it will produce a failure, i.e. that action 3 is selected with Y = 1 as the resulting outcome. One way to reduce this failure probability, is to introduce the constraint that action 3 (using the component) is not allowed unless we have a certain level of confidence in that the component is reliable. We introduced this type of constraint by requiring that P (θ = 0.99) > 0.9 (a constraint on epistemic uncertainty). The optimal policy under this constraint will start with running experiments (action 1), before deciding whether to replace (action 2), use the component (action 3), or terminate the project (action 0). Figure 4 shows a histogram of the cumulative reward over 1000 simulated episodes, for the constrained and unconstrained policies obtained by RL, together with a completely random policy for comparison. : Total reward after 1000 episodes for a random policy, the unconstrained policy, and the agent which is subjected to the constraint that action 3 is not allowed unless P (θ = 0.99) > 0.9. In this example, the information state-space could also be defined in a simpler way, as explained in Example 5.1. As a result the reinforcement learning task will be simplified. Using the different statespace representations, we obtained the same results shown in Figure 4. Finally, we should note that in the case where defining the state space as in (14) is necessary, the constraint P (θ = 0.99) > 0.9 is not practical. That is, if we could estimate this probability efficiently, then we also have access to the compressed information state-space. One alternative could then be to instead consider the uncertain failure probability p f (θ) = P (Y = 1 \| θ), and set a limit on e.g. E[p f ] + 2 · Std(p f ). This is the approach taken in the following example concerning failure probability estimation. Corroded pipeline example Here we revisit the corroded pipeline example from Agrell and Dahl [2] which we introduced in Section 3.4. In this example, we have specified epistemic uncertainty with respect to model discrepancy, the size of a defect, and the capacity p FE coming from a Finite Element simulation. If we let θ be the epistemic generator, we can write the failure probability conditioned on epistemic information as p f (θ) = P (g ≤ 0 \| θ). In [2] the following objective was considered: Determine with confidence whether p f (θ) < 10 −3 . That is, when we consider p f as a purely epistemic random variable, we want to either confirm that the failure probability is less than the target 10 −3 (in which case we can continue operations as normal), or to detect with confidence that the target is exceeded (and we have to intervene). Will say the the objective is achieved if we obtain either E[p f ] + 2 · Std(p f ) < 10 −3 or E[p f ]−2·Std(p f ) > 10 −3 (where E[p f ] and Std(p f ) can be efficiently approximated using the method developed in [2]). There are three ways in which we can reduce epistemic uncertainty: 1. Defect measurement: Noise perturbed measurement that reduces uncertainty in the defect size d 2. Computer experiment: Evaluate p FE at some selected input (D, t, s, d, l), to reduce uncertainty in the surrogatep FE used to approximate p FE . 3. Lab experiment: Obtain one observation of X m , which reduces uncertainty in µ m . The set of information corresponding to defect measurements is I Measure ⊂ R as each measurement is a real valued number. Similarly, I Lab ⊂ R as well, and I FE ⊂ R 6 when we consider a vector y ∈ R 6 as an experiment [D, t, s, d, l, p FE ]. Actually, in this example we may exploit some conjugacy in in the representation of I Measure and I Lab as discussed in Example 5.1 (see [2] for details), so we can define the information state-space as S = S FE ×R 2 , where S FE consists of finite subsets of R 6 . We will use RL to determine which of the three types of experiment to perform, and define the action space A = {Measurement, FE, Lab}. Note that when we decide to run a computer experiment, we also have to specify the input (D, t, s, d, l). This is a separate decision making problem regarding design of experiments. For this we make use of the myopic (one-step lookahead) method developed in [2], although one could in principle use RL for this as well. This kind of decision making, where one first decides between different types of task to perform, and then proceed to find a way to perform the selected task optimally, is often referred to as hierarchical RL in the reinforcement learning literature. Actually, [2] considers a myopic alternative for also selecting between the different types of experiments, and it was observed that this might be challenging in practice if there are large differences in cost between the experiments. This was the motivation for studying the current example, where we now define the reward (cost) r as a direct consequence of a ∈ A as follows: r = −10 for a = Measurement, r = −1 for a = Lab and r = −0.1 for a = FE. In this example we also made use of the DQN approach of van Hasselt et al. [25], where we define a neural network q(s, w) : S = S FE × R 2 → R 3 , that gives, for each state s, the (near optimal) value of each of the three actions. We refrain from describing all details regarding the neural network and the specific RL algorithm, as the main purpose with this example is for illustration. But we note that two important innovations in the DQN algorithm, the use of a target network and experience replay as proposed in [29], was necessary for this to work. The objective in this RL example is to estimate a failure probability using as little resources as possible. If an agent achieves the criterion on epistemic uncertainty reduction, that the expected failure probability plus/minus two standard deviations is either above or below the target value, we say that the agent has succeeded and we report the sum of the cost of all performed experiments. We also set a maximum limit of 40 experiments. I.e. after 40 tries the agent has failed. To compare the policy obtained by RL, we consider the random policy that selects between the three actions uniformly at random. We also consider a more "human like" benchmark policy, that corresponds to first running 10 computer experiments, followed by one lab experiment then one defect measurement, then 10 new computer experiments, and so on. The final results from simulating 100 episodes with each of the three policies is shown in Figure 5. For the random and benchmark policy, the success rate was around 60% (to achieve the objective within 40 experiments in total), whereas 94 % was successful for the RL agent. Concluding remarks To conclude our discussion, we recall that in this paper, we have: • Given a measure-theoretic discussion of epistemic uncertainty and formally defined epistemic conditioning. • Provided a mathematical definition of a probabilistic digital twin (PDT). • Connected PDTs with sequential decision making problems, and discussed several solution approaches (maximum principle, dynamic programming, MDP and POMDP). • Argued that using (deep) RL to solve sequential decision making problems in the PDT is a good choice for practical applications today. • For the specific use-case of optimal information gathering, we proposed a generic solution using deep RL on the information state-space. Further research in this direction includes looking at alternative solution methods and RL algorithms in order to handle different PDT frameworks. A possible idea is to use a maximum principle approach instead of a DPP approach (as is done in RL). By using one of the MP based algorithms in [20], we may avoid the Markovianity requirement, possible time-inconsistency issues and can also allow for state constraints. For instance, this is of interest when the objective of the sequential decision making problem in the PDT is to minimize a risk measure such as CVaR or VaR. Both of these risk measures are known to cause time-inconsistency in the Bellman equation, and hence, the DPP (and also RL) cannot be applied in a straightforward manner. This is work in progress. Example 3.1. (Coin flip -robust decisions) Continuing from the coin flip example (see Example 2. I may contain observations made under different conditions than what is currently specified through M . The information I is generally defined as a set of events, given as pairs (d, o), where the d encodes the relevant action leading to observing o, as well as a description of the conditions under which o was observed. Here d may relate to modifications of the structural assumptions M , for instance if the the causal relationships that describes the model of X under observation of o is not the same as what is currently represented by M . This is the scenario when we perform controlled experiments. Alternatively, (d, o) may represent a passive observation, e.g. d = "measurement taken from sensor 1 at time 01:02:03", o = 1.7 mm. We illustrate this in the following example. Figure 1 : 1A standard regression model as a PDT. Figure 2 : 2Graphical representation of the corroded pipeline structural reliability model. The shaded nodes d, p FE and µm have associated epistemic uncertainty. Figure 3 : 3A PDT as a mental model of an agent taking actions in the real world. As new experience is gained, the PDT may be updated by changing the structural assumptions M that defined the probability measure P , or updating belief with respect to epistemic events through conditioning on the new set of information I. The changes in structural assumptions and epistemic information are represented by ∆M and ∆I respectively. As part of the planning process, the PDT may simulate possible scenarios as indicated by the inner circle. Example 5.1. (Coin flip -information statespace) In the coin flip example (Example 2. 1 . 1Test the component (flip the coin once). Cost r = −10.000$. 2. Replace the component (buy a new coin). Cost r = −100.000$. 3. Use the component (bet on the outcome). Obtain a reward of r = 10 6 $ if the component works (Y = 0) and a cost of r = −10 6 $ if the component fails (Y = 1). 4. Terminate the project (set t = N ), r = 0. Figure 4 4Figure 4: Total reward after 1000 episodes for a random policy, the unconstrained policy, and the agent which is subjected to the constraint that action 3 is not allowed unless P (θ = 0.99) > 0.9. Figure 5 : 5Total cost (negative reward) after 100 successful episodes. There exists some measurable space (Θ, T ) and a Fmeasurable function θ : Ω → Θ such that E = σ(θ), the smallest σ-algebra containing all of the sets θ −1 (T ) for T ∈ T . AcknowledgementsThis work has been supported by grant 276282 from the Research Council of Norway (RCN) and DNV Group Research and Development (Christian Agrell and Andreas Hafver). The work has also been supported by grant 29989 from the Research Council of Norway as part of the SCROLLER project (Kristina Rognlien Dahl). The main ideas presented in this paper is the culmination of input from recent research activities related to risk, uncertainty, machine learning and probabilistic modelling in DNV and at the University of Oslo. In particular we want to thank Simen Eldevik, Frank Børre Pedersen and Carla Ferreira for valuable input to this paper.AppendicesA. Existence of the epistemic generatorThe purpose of this section, is to explain why Assumption 2.5 (of the existence of a generating random variable for the epistemic σ-algebra) hold under some very mild assumptions.In order to do this, we consider a standard probability space. Roughly, this is a probability space consisting of an interval and/or a countable (or finite) number of atoms. Formally, a probability space is standard if it is isomorphic (up to P-null sets) with an interval equipped with the Lebesgue measure, a countable (or finite) set of atoms, or a disjoint union of both of these types of sets.The following proposition says that in a standard probability space, any sub σ-algebra is generated by a random variable up to P -null sets. For a proof, see e.g. Greinecker[30].Proposition A.1. Let (Ω, F, P ) be a standard probability space and E ⊆ F a sub σ-algebra.Then there exists an F-measurable random variable θ such that E = σ(θ) mod 0.Hence, as long as our probability space is standard (which is a mild assumption), we can assume that our sub σ-algebra of epistemic information, E, is generated (up to P -null sets) by a random variable θ without loss of generality. Note that for the purpose of this paper, the mod 0 (i.e., up to P -null sets) is not a problem. Since we are only considering conditional expectations (or in particular, expectations), the P -null sets disappear.Actually, this generating random variable, θ, can always be modified to another random variable,θ, which is E-measurable (purely epistemic) by augmenting the P -null sets. This means that θ andθ are the same with respect to conditional expectations.Furthermore, if X is a random variable on this standard probability space, X\|θ, is purely aleatory, i.e., independent of E. 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[ "A machine learning approach to drawing phase diagrams of topological lasing modes", "A machine learning approach to drawing phase diagrams of topological lasing modes" ]	[ "Stephan Wong \nSchool of Physics and Astronomy\nCardiff University\nCF24 3AACardiffUK\n", "Jan Olthaus \nInstitut für Festkörpertheorie\nUniversität Münster\n48149MünsterGermany\n", "Thomas K Bracht \nInstitut für Festkörpertheorie\nUniversität Münster\n48149MünsterGermany\n", "Doris E Reiter \nInstitut für Festkörpertheorie\nUniversität Münster\n48149MünsterGermany\n\nCondensed Matter Theory\nDepartment of Physics\nTU Dortmund\n44221DortmundGermany\n", "Sang Soon Oh \nSchool of Physics and Astronomy\nCardiff University\nCF24 3AACardiffUK\n" ]	[ "School of Physics and Astronomy\nCardiff University\nCF24 3AACardiffUK", "Institut für Festkörpertheorie\nUniversität Münster\n48149MünsterGermany", "Institut für Festkörpertheorie\nUniversität Münster\n48149MünsterGermany", "Institut für Festkörpertheorie\nUniversität Münster\n48149MünsterGermany", "Condensed Matter Theory\nDepartment of Physics\nTU Dortmund\n44221DortmundGermany", "School of Physics and Astronomy\nCardiff University\nCF24 3AACardiffUK" ]	[]	Identifying phases and analyzing the stability of dynamic states are ubiquitous and important problems which appear in various physical systems. Nonetheless, drawing a phase diagram in highdimensional and large parameter spaces has remained challenging. Here, we propose a data-driven method to derive the phase diagram of lasing modes in topological insulator lasers. The classification is based on the temporal behaviour of the topological modes obtained via numerical integration of the rate equation. A semi-supervised learning method is used and an adaptive library is constructed in order to distinguish the different topological modes present in the generated parameter space. The proposed method successfully distinguishes the different topological phases in the Su-Schrieffer-Heeger (SSH) lattice with saturable gain. This demonstrates the possibility of classifying the topological phases without needing for expert knowledge of the system and may give valuable insight into the fundamental physics of topological insulator lasers via reverse engineering.	10.1038/s42005-023-01230-z	[ "https://export.arxiv.org/pdf/2211.04373v1.pdf" ]	253,397,783	2211.04373	33dc640480de44fe59fbd91c5a553d819085e819	A machine learning approach to drawing phase diagrams of topological lasing modes Stephan Wong School of Physics and Astronomy Cardiff University CF24 3AACardiffUK Jan Olthaus Institut für Festkörpertheorie Universität Münster 48149MünsterGermany Thomas K Bracht Institut für Festkörpertheorie Universität Münster 48149MünsterGermany Doris E Reiter Institut für Festkörpertheorie Universität Münster 48149MünsterGermany Condensed Matter Theory Department of Physics TU Dortmund 44221DortmundGermany Sang Soon Oh School of Physics and Astronomy Cardiff University CF24 3AACardiffUK A machine learning approach to drawing phase diagrams of topological lasing modes (Dated: November 9, 2022) Identifying phases and analyzing the stability of dynamic states are ubiquitous and important problems which appear in various physical systems. Nonetheless, drawing a phase diagram in highdimensional and large parameter spaces has remained challenging. Here, we propose a data-driven method to derive the phase diagram of lasing modes in topological insulator lasers. The classification is based on the temporal behaviour of the topological modes obtained via numerical integration of the rate equation. A semi-supervised learning method is used and an adaptive library is constructed in order to distinguish the different topological modes present in the generated parameter space. The proposed method successfully distinguishes the different topological phases in the Su-Schrieffer-Heeger (SSH) lattice with saturable gain. This demonstrates the possibility of classifying the topological phases without needing for expert knowledge of the system and may give valuable insight into the fundamental physics of topological insulator lasers via reverse engineering. I. INTRODUCTION Over the last few years, significant research efforts have been made on photonic topological insulator (PTI) lasers. While the efforts have been concentrated on the spatial stability of the topologically protected edge modes, namely on the existence of such topological edge modes in non-Hermitian PTIs [1][2][3][4][5][6][7][8][9], the temporal stability has not been the focus of interest so far [10][11][12][13][14][15]. Due to the non-linear nature of PTI lasers, the temporal stability is an important characteristic to take into account for experimental demonstrations and real-life applications. Although the spatial stability of the topological modes, i.e., its robustness, may be guaranteed in active non-Hermitian PTIs, unstable behaviour may be present in the time domain. In this regard, the temporal dynamics of the topologically protected modes have been studied [10,12,13], mainly using linear stability analysis. It is, however, a challenging task to apply the same approach to more complex structures because of the highdimensional phase space and parameter space as well as the lack of analytical solutions [11]. Machine learning (ML) can be advantageous for the theoretical study of the stability of PTIs, which requires repetitive numerical simulations for several varying parameters. ML is a data-based method which can be implemented with different strategies, and the most appropriate ML strategy depends on the dataset under study. For instance, a supervised learning strategy relies on labelled data, a dataset of input-output pairs. This has been utilized in topological photonics to draw topological phase diagrams [16], calculate topological invariants [5], or explore topological band structures [17]. On the other hand, an unsupervised learning strategy consists of extracting information from the dataset for which we do not * Email: [email protected] have labels. This is used for dimensional reductions by keeping only the main features of the high-dimensional structure of the dataset or for clustering problems from which the data is divided into different types [18]. For instance, this has been successful in obtaining the phase transition in the Ising model [19], and clustering Hamiltonians that belong to the same symmetry classes [20]. In the unsupervised learning strategy, modal decomposition is a common and successful method which reduces the analysis of very high-dimensional data to a set of relatively few modes. Among the modal decomposition methods, principal component analysis (PCA) is a method which derives the eigenmodes or the main features based on their variance in the data [21]. These eigenmodes can then be utilized as a basis to represent the dataset [22]. This reduced-order model method has been extended to identify distinct non-linear regimes [23][24][25][26][27][28][29] by constructing a library composed of representatives of these regimes: this is known as representation classification. Nevertheless, the preliminary identification of the regimes composing the library and the construction of the library is a manual process and requires expert knowledge of the complex system. In this paper, we propose a representation classification method to study the spatio-temporal dynamics of non-linear topological systems. The results will be based on the phase diagram of the Su-Schrieffer-Heeger (SSH) lattice [30] with a domain wall and with saturable gain [12,13]. To remove the necessity of the required expertise on the complex system, we present an algorithm which constructs an appropriate library of the different phases automatically. For this goal, we propose two approaches: a top-down approach in which the library has numerous phases that are merged into the equivalent phases, and a bottom-up approach in which the library is completed on the fly to get the most accurate classification. Via reverse engineering, our proposed method can be used as a tool to find novel topological lasing modes in more complicated settings. For given rate equations of arXiv:2211.04373v1 [physics.optics] 8 Nov 2022 a lasing system, one would only need to integrate the differential equations in the desired parameter space region and then apply the adaptive representation classification to obtain the phase diagram. II. RESULTS A. Phase diagram of the SSH Model As a toy model, we will consider the domain-wall-type SSH lattice with saturable gain [ Fig. 1(a)]. The system has a domain wall, at the A site n = 0, which separates two SSH lattices, namely lattices composed of two sites per unit cell, A and B, and characterized by intraand inter-unit cell couplings, t intra and t inter , respectively. t intra = t 1 and t inter = t 2 (t intra = t 2 and t inter = t 1 ) for the lattice on the left (right) side of the domain wall, i.e., the sites with n < 0 (n ≥ 0). The dynamics of the system ψ(t) = (ψ −N (t), . . . , ψ −1 (t), ψ 0 (t), ψ 1 (t), . . . , ψ N (t)) ≡ x(t) , with N s = 2N + 1 sites, reads, for n = −N, . . . , N : i dψ n dt = i g n 1 + \|ψ n \| 2 − γ n ψ n + t intra,n+1 ψ n+1 + t inter,n-1 ψ n−1 (1) where ψ n is the amplitude of the n-th site. g n and γ n are the linear gain and linear loss at the n-th site, respectively. Using explicitly the amplitudes a p and b p of the A and B sites on the p-th unit cell, respectively, in ψ(t) = (. . . , a p (t), b p (t), . . .), Eq. (1) can be re-written as: i da p dt = i g A 1 + \|a p \| 2 − γ A a p + t intra b p + t inter b p−1 ,(2)i db p dt = i g B 1 + \|b p \| 2 − γ B b p + t intra a p + t inter a p+1 ,(3) where g σ and γ σ are the linear gain and linear loss at the site σ = A, B. In the passive setting (g A = g B = 0, γ A = γ B = 0), this configuration, with t 1 > t 2 , is known to give a single topologically protected zero-energy (nonoscillating) mode localized at the domain wall and with non-vanishing amplitudes only on the A sites [31]. This is due to the bulk-boundary correspondence at the domain wall between trivial and non-trivial topological SSH lattices. Indeed, an SSH lattice is topologically trivial (nontrivial) if the intra-unit cell coupling is lower (greater) than the inter-unit cell coupling. In the active setting, it has been shown that the topological phase of the system depends on the gain and coupling parameters [12,13]. As parameters, we use the values from Ref. [13] with t 1 = 1, t 2 = 0.7, g B = 0 and γ A = γ B ≡ γ AB . Figure 1(b) shows the phase diagram, by varying g A and γ AB in the parameter space, for a lattice composed of N s = 21 sites (N = 10). In this configuration, the system has two distinct topological phases: a non-oscillating phase [white area in Fig. 1(b)] and an oscillating phase [grey area in Fig. 1(b)]. The dynamics of the two topological phases can be visualized by plotting the total intensity I A = p \|a p \| 2 (I B = p \|b p \| 2 ) of the A (B) sites in Fig. 1(c) as in Ref. [12,13]. Alternatively, more details can be understood by plotting the space-time dynamics of the topological modes as shown in Figs. 1(d) and 1(e). The non-oscillating phase is similar to the zero-energy mode in the passive SSH lattice. We can see in Fig. 1(d) that the mode is localized at the interface and has the majority of its amplitudes on the A sublattice. On the other hand, the system with saturable gain exhibits a new topological phase with no counterpart in the passive setting. The new topological mode is characterized by an edge mode at the domain wall with an oscillating behaviour of the amplitudes on the A and B sites, as shown in Fig. 1(e). The classification of the new topological phases in nonlinear systems requires, so far, an expert knowledge of the given non-linear systems, for example, the known results derived in Ref. [12]. In fact, the phase diagram in Fig. 1(b) has been obtained solely by the fast Fourier transform of the time series in the parameter space. Thus, the main aim of this paper is to develop a tool to explore the topological phases of PTI lasers in more complicated settings for which we have little knowledge. SSH 1 SSH 2 (b) (c) (d) (e) ( The phase diagram shown in Fig. 1(b) will serve as a reference for our proposed method. The dataset we will utilize throughout this paper is composed of about 1000 samples which are randomly generated from the same coupling and gain parameters' range as in Fig. 1(b). The coupled-mode equations [Eqs. (2) and (3)] are integrated using the fourth-order Runge-Kutta method and with a p (t = 0) = b p (t = 0) = 0.01, ∀p, as an initial condition. Although the integration has been performed using a fixed time step dt = 0.01 until a final time at t = 1400, only 2000 time snapshots are uniformly retrieved in order to keep the time series at a reasonable size. For the parameters given above, this sample rate leaves about 10 time steps per period for the oscillating regime case [ Fig. 1 (c)]. The phase diagram is then obtained solely from the time series of the states within the given parameter space. B. Representation classification method To classify topological lasing modes based on their distinct non-linear regimes, we use a representation classification method [23,27,28]. The general idea of representation classification relies on the assumption, and common situations, that the dynamics of a high-dimensional system evolves on a low-dimensional attractor such as fixed points or periodic orbits [32]. The low-dimensional structure of the attractor allows for a reduced-order model that accurately approximates the underlying behaviour of the system: the dynamics of the complex system can thus be written using a basis that spans the lowdimensional space. Representation classification consists of constructing a library of appropriate basis, representative of the dynamical regimes of interest, and only then employ a filtering strategy to identify the regime corresponding to a given unknown time series. In the following, we will use the term "regime" to denote the different dynamical behaviours or the different topological phases in the non-linear SSH lattice with saturable gain. Besides, for convenience, we will plot only the total intensity on the A (I A = p \|a p \| 2 ) and B (I B = p \|b p \| 2 ) sublattices to represent the given regimes. Nevertheless, the time series of the complex amplitudes at each site will be considered for the construction of the library. As is common in complex dynamical systems, the dynamics of a system close to an attractor lie in a lowdimensional space. This means that a given spatiotemporal dynamics, denoted by the vector x(t), can be approximately written in terms of a basis Φ = {φ i } i=1,...,D spanning the low D-dimensional space, namely: x(t) ≈ D i=1 φ i β i (t) = Φβ(t)(4) where β i are the weighted coefficients in the above linear combination of basis states φ i . Using the terminology used in the literature [23,27,28], x(t) will, in the following, be referred to the state measured at time t. However, one of the main characteristics of non-linear systems is the drastically different dynamical behaviours with respect to the system's parameters. Therefore, the reduced-order modelling strategy using a single representative basis, i.e., corresponding to a single regime, is bound to fail. Instead of finding a global basis, we here construct a set of local bases, i.e., construct a library composed of the bases of each non-linear regime of interest: L = {Φ 1 , · · · , Φ J } = {φ j,i } j=1,...,J , i=1,...,D ,(5) where J is the number of regimes, Φ j 's are the bases of each of the dynamical regime j, and φ j,i 's are the corresponding basis states. This is the supervised learning part of the method, from which the data-driven method attempts to capture the dynamics of the system in the reduced-order model. Therefore, the library L contains the representative basis of each regime of interest, and corresponds to an overcomplete basis that approximates the dynamics of the system across the given parameter space. A better approximation of x(t), instead of using Eq. (4) for a single basis regime, then reads: x(t) ≈ J j=1 D i=1 φ j,i β j,i (t) = J j=1 Φ j β j (t)(6) where β j,i are the weighted coefficients in the above linear combination in the overcomplete basis library L. It is worth noting that the library modes φ j,i are not orthogonal to each other, but instead orthogonal in groups of modes for each different regime j. Throughout this paper, the bases used for constructing the library L will be generated by using a timeaugmented dynamical mode decomposition (aDMD) method [28] to consider both the spatial and temporal behaviours (see Supplementary section SI for additional information). Here, we use a classification scheme based on a simple hierarchical strategy [28]. The regime classification approach is fundamentally a subspace identification problem, where each regime is represented by a different subspace. Given the state x(t i : t i+Nw ) measured within the time window [t i , t i+Nw ], with N w the time step window size, the correct regime j * is identified as the corresponding subspace in the library L closest to the measurement in the L2-norm sense [28]. In other words, the classification strategy is to find the subspace that maximises the projection of the measurement onto the regime subspace: j * = arg max j=1,...,J P j x(t i : t i+Nw ) 2 ,(7) where P j is a projection operator given by: P j = Φ j Φ + j (8) with Φ + j being the pseudo-inverse of Φ j , · 2 the L2- norm of a vector, v 2 := i \|v i \| 2 , and arg max the function that returns the index of the maximum value. For the 1D system exemplified in this paper, the data collection is not too expensive because N s is reasonable. However, if N s is very large, sparse measurement might be desirable and a slight change in the methodology is then needed as explained in Supplementary section SII. C. Phase Diagrams In the following we will draw phase diagrams using different approaches to the representation classification. In the phase diagrams we will mark the different identified regimes by the color of the dots, where the (dark or light) blue dots always mark the oscillating regime, the green dots the non-oscillating regime, the red dots the transient regime and the orange dots the transition regime. The white and grey areas are overlays of the referenced phase diagram obtained in Fig. 1. The aDMD bases have been generated with N w = 25 from the time series starting at the 1800-th time step. Figure 2(a) displays the phase diagram derived from a library basis made of the two topological regimes known from Fig. 1. These two topological modes, used for the construction of the library, are represented by the two magenta dots. They are randomly chosen from the known regimes' region in Fig. 1(b) and the details of the resulted phase diagrams is dependent from that random choice. The remaining coloured dots in the plot represent the identified regime j * [Eq. (7)] of each sample depicted by the dots in the parameter space. However, we can see that the phase diagram fails to correctly predict all the dynamical behaviours. Indeed, we observe that many time series are not correctly identified. Using a different choice of topological regimes in the parameter space to construct the library could be a solution to find a better phase diagram, but our attempts only showed marginal improvement of the agreement. Testing every possible choices in the dataset with no guarantee of finding the accurate phase diagram is therefore not a practical solution. Instead, using four bases in the library instead of two bases, or equivalently considering four regimes from the given parameter space, the phase diagram in Fig. 2(b) shows better results: The identified oscillating and nonoscillating regimes are more separated and have a better fitting with the referenced phase diagram, even though they belong to a distinct regime index j * . Merging the three oscillating regimes present in the library would then give a more accurate derived phase diagram. Therefore, Figure 2 suggests that increasing the number of bases in the library L and merging some of them might help to get closer to the desired phase diagram, as we will see in the later sections. Fixed library Top-down adaptive library In the previous section, the construction of the library L was a manual process from which we already knew the different dynamical regimes. This, however, requires prior knowledge of the complex system considered. The strategy, here, is to adaptively construct the library based on the given data samples. Here, we employ a top-down approach in which we start with too many samples for the construction of the library, and then reduce the library size by merging some of them. Based on some measures in the decision process, this automated construction of the library thus removes the manual construction of the regimes. The underlying assumption of the classification scheme is based on the dissimilarity between the subspace of different regimes. We thus propose to consider regimes that are similar as equivalent regimes. This would, for instance, help us to merge the three phases in the nonoscillating region in Fig. 2(b), and consider them as a single regime. In other words, the regimes i and j are said to be equivalent, denoted by i ∼ j, if the fraction of information retained after the projection onto each other, γ ij ∈ [0, 1], is high enough: γ ij > γ th ,(9) where γ th ∈ [0, 1] is the hyper-parameter which decides the threshold value for merging different regimes and γ ij is the subspace alignment given by: γ ij := P i P j 2 F P i F P j F ,(10) with · F the Frobenius norm of a matrix, M F := ij \|M ij \| 2 . Importantly, the relation Eq. (9) is numerically computed in such a way that the transitivity property of the equivalence relation is satisfied, namely that if i ∼ j and j ∼ k then i ∼ k. The relation Eq. (9) is then indeed an equivalence relation because the reflexive (i ∼ i) and symmetric (i ∼ j ⇒ j ∼ i) property of the relation is automatically satisfied from the definition of γ ij [Eq. (10)]. Supplementary section SIII gives more details on the top-down library generation principle. The top-down representation classification strategy is to classify the time series according to a large library of bases, and only then merge the equivalent identified regimes via the calculated alignment subspace γ ij and the equivalence relation Eq. (9). Figure 3 shows the phase diagram obtained from the top-down algorithm that started with a library composed of J = 60 regimes randomly chosen, along with the representative of each phase. We observe in Fig. 3(a) that the derived phase diagram is able to distinguish the non-oscillating [top panel of Fig. 3(b)] and oscillating regimes [middle panel of Fig. 3(b)]. In addition, a third regime corresponding to a transient regime [bottom panel of Fig. 3(b)] is found close to the γ AB = 0 or g A − γ AB = 0 axis. This transient regime indicates that a longer simulation time might be needed to be considered either in the oscillating or non-oscillating regimes. However, we can see that the derived phase diagram is still failing in the low γ AB and low g A − γ AB region (bottom-left region of the present phase diagram), where some time series are interpreted as non-oscillating instead of oscillating regime. This shows the limitation of this method where the initially constructed library may lack some of the paths that may connect similar bases. For example, regimes i and k might not be similar enough to be considered as equivalent directly [Eq. (9)] but are both equivalent to the regime k, i.e., i ∼ j and j ∼ k, which is missing in the library. The natural workaround would be to increase the initial library size and ensure that the regimes in the library have no missing paths, as we will see in the next section. The hyper-parameter γ th is an important quantity in the algorithm since it dictates which regimes are equivalent or not. A low threshold γ th will easily merge regimes while a high γ th will barely reduce the size of the library as depicted in Fig. 4(a). The threshold is here arbitrarily chosen based on Fig. 4, and based on the refinement of the desired library. For example, we can see in Fig. 4(b) that the derived phase diagram with γ th = 0.55 has two different phases. For this coarse threshold, the transient regime is identified but the distinct non-oscillating and oscillating phases are merged together into a single phase. On the other hand, with the same library as in Fig. 4(b), Figure 4(c) displays the obtained phase diagram for a finer threshold value γ th = 0.95. The plot shows that the algorithm separates the parameter space into several regimes which can be grouped into four main regimes. In addition to the non-oscillating, the oscillating and the transient regimes, there is a regime corresponding to the transition between the two topological phases. Besides, this finer description allows us to see distinct sets of modes in the oscillating parameter space region [dark blue and light blue dots in Fig. 4(c)]. Nevertheless, we observe again that the initial library misclassifies some of the non-oscillating time series most likely because of some missing paths, as said previously. Bottom-up adaptive library We propose an alternative and dual approach which considers fewer samples in the library. The core idea of this bottom-up approach is then to add samples on the fly during the classification of the given sample if the library is not good enough. Here, the library is considered to be good enough if the maximal projection of the measurement onto the regimes' subspace is high enough. In other words, the library is said to be good enough if the worst relative reconstruction error, , is low enough: < th ,(11) where th is the hyper-parameter which decides the threshold quality of the library and := max j=1,...,J P j y(t) − y(t) 2 y(t) 2 ,(12) with · 2 the L 2 -norm of a vector. Supplementary section SIV gives more details on the bottom-up library generation principle. The advantage of this bottom-up approach is the full exploration of the parameter space region and the automatic construction of a library based on its quality. This method does not suffer from the randomly chosen samples used to construct the library, and the library composition is not restricted to a narrow parameter space region. Using a good enough library quality, the algorithm should therefore be able to sort the missing paths issue in the top-down method. The bottom-up representation classification scheme consists of classifying the time series according to a given library or adding this sample into the library if the library is not good enough, and only then merging the different phases obtained into groups of equivalent regimes using the top-down method. Figure 5 depicts the phase diagram derived from the bottom-up classification algorithm with a starting library composed of a single regime. Similarly to the top-down approach in Fig. 3(a), we observe three distinct regimes corresponding to the nonoscillating, oscillating and transient regimes [ Fig. 3(b)]. Nevertheless, the obtained phase diagram now better predicts the regimes. The misclassifications of the nonoscillating and oscillating regimes, which were due to missing paths in the library, are now reduced, and only very few dots are not correctly identified due to being close to the topological transition boundary. Likewise, the transient points are indications of longer simulations needed because of the long transient time. Along with the γ th hyper-parameter, the threshold hyper-parameter th is an important parameter since it tells us whether we want to add or not a given sample into the library. We observe in Fig. 6 that a low threshold th will add many samples to the library, whereas a high th will not add samples to the library at all. The threshold value th is, again, arbitrarily chosen but with a preference for a high-quality library, i.e., low th , in order to avoid missing paths. For example, we can see in Fig. 6(c) the phase diagram derived using th = 0.05 (and γ th = 0.95), namely with a library that gives less than 5% of the reconstruction error of the measurement. This set of hyper-parameter gives four main regimes that seem to correspond to the non-oscillating, oscillating, transition and transient regimes. Yet, there is some misidentification of the two topological phases most likely because of missing paths of the obtained library. On the other hand, with a better library quality, here th = 0.005 (and γ th = 0.95), the missing paths are retrieved and the derived phase diagram correctly predicts the topological phases [ Fig. 6(b)]. Figure 6(b) shows that the different regimes, obtained previously with a lower library quality, are now better defined. The non-oscillating and oscillating regimes are well located in their respective parameter space region, and the transition points follow the transition boundary between the two topological phases. In addition, the bottom-up representation classification is predicting distinct oscillating modes [dark blue and light blue dots in Fig. 6(b)]. The presence of distinct oscillating modes is an example of new insights given by the data-driven classification method. Indeed, the complex values of the amplitudes of x(t) are, here, taken into account instead of solely the total intensity of each sublattice A and B as in Ref. [12,13]. This allows for a finer description of the dynamic pattern based on the whole lattice with the relative phase difference of the sites or the absolute value of amplitudes. III. CONCLUSION We have proposed a data-driven approach to identify and classify topological phases of dynamical systems. By utilizing the representation classification strategy based on the aDMD, we have successfully drawn the phase diagrams of the domain-wall-type SSH lattice with saturable gain. To avoid manual labelling in the classification, we have proposed two automatic library construction schemes: top-down and bottom-up approaches that merge similar phases in a library or adaptively construct a library according to its quality, respectively. While the bottom-up method is preferred due to the missing paths issue, both methods allow exploration of parameter space without any expert knowledge of the complex non-linear system. Our approach is advantageous in doing the re-verse engineering to find new phases because only a small desired parameter space region is required for solving the laser rate equation, applying the bottom-up representation classification strategy and building a starting library. Our method opens the door for finding novel topological lasing modes by providing new insights into the dynamics of coupled lasers in more complicated settings. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. CODE AVAILABILITY The code that support the findings of this study are available from the corresponding author upon reasonable request. ACKNOWLEDGMENTS We acknowledge the support of the Sêr Cymru II Rising Star Fellowship (80762-CU145 (East)) which is partfunded by the European Regional Development Fund (ERDF) via the Welsh Government. COMPETING INTERESTS The authors declare that they have no competing interests. ADDITIONAL INFORMATION Correspondence and requests for materials should be addressed to Sang Soon Oh. SI. BASIS GENERATION METHODS This supplementary section review few methods for generating a basis from a time series. There are many ways to generate the low-order model of a given dynamical behaviour. An important quantity is the so-called data matrix X built from the data at hand. The data matrix is a (N s × N t )-matrix that collects the N t data snapshots x(t i ) into columns: X = [x(t 0 ), x(t 1 ), . . . , x(t Nt )] . (S1) Here, the vector x(t i ) is chosen to be the complex-valued amplitudes of the modes at the A and B sites. Other "observables", such as the absolute values or the total intensity per sublattices, can be used. This may give different (better or worse) results and is left for a future study. The bases can then be constructed using dimensional reduction techniques on the data matrix. Here, we will cover different methods in order to highlight their importance and limitations for the classification scheme used. A. Proper orthogonal decomposition Proper orthogonal decomposition (POD) [24] is a commonly used tool for dimensional reduction of physical systems. This decomposition relies on the singular value decomposition (SVD) of the data matrix, given by: X = U ΣV † (S2) where U and V † are (N s × N s ) and (N s × N t ) unitary matrices, respectively. Σ is a diagonal (N 0 × N 0 )-matrix diag = (σ 1 , . . . , σ N0 ), with N 0 = min(N s , N t ). The diagonal entries of Σ are the so-called singular values and are ordered in ascendant order σ 1 > σ 2 > . . . > σ N0 ≥ 0. The SVD gives us two orthonormal bases U and V † since the matrices U and V † are unitary matrices. The columns of U are ordered according to the variance σ i they capture in the data matrix and are called the singular vectors: these are the POD modes that are used in the basis Φ. Moreover, the POD modes are complex because of the complex data X. For a low-dimensional attractor, the POD basis can be safely truncated at a cut-off value r while retaining the main information of the data matrix. Explicitly, the SVD reads: X im = N0 n=0 U in σ n V † nm ,(S3) and keeping only the r highest terms in the decomposition [Eq. S3], we have the approximation: This is re-written, in a matrix form, as: X im r n=0 U in σ n V † nm . (S4) (d) (a) (b) (c) (e) (f)X U r Σ r V † r (S5) where U r , Σ r and V † r are the truncated matrix of U , Σ and V † , respectively. Although the cut-off value r can be chosen based on different criteria [33], r is typically chosen so that the POD modes retain a certain amount of the variance (or energy) σ X in the data, namely: r i=0 σ i > σ X . (S6) Figure S1 displays the POD method of the nonoscillating and oscillating regimes in the domain-wall SSH lattice with saturable gain [Fig. 1]. The truncation has been chosen such that σ X = 99% of the total variance is retained. We observe in Figs. S1(a) and S1(b) the normalised singular values, and that a single POD mode is retained for the zero-mode-like, whereas three POD modes are needed for the oscillating regime, as marked by the red dots. The real and imaginary parts of the field profile of the corresponding first few POD modes are plotted in Figs. S1(c) and S1(d) and Figs. S1(e) and S1(f) for the zero-mode-like non-oscillating and oscillating regimes, respectively. One can see that the main spatial feature of the zero-mode is captured in the single POD mode obtained after truncation, where the majority of its amplitudes are on the A sublattice. On the other hand, the POD modes of the oscillating dynamical regime also capture part of the information with some finite amplitudes on the A and B sublattices. Importantly, in this decomposition [Eq. S3], SVD is implicitly doing a space-time separation of the data matrix, where the POD modes U contain the spatial information while V have the temporal information at each spatial grid point. Therefore, the POD modes give a static basis and do not explicitly model the temporal dynamics of the time series. This method will therefore most likely fail to identify the correct dynamical regime in the classification step. B. Dynamical mode decomposition Dynamical mode decomposition (DMD) [24,34,35] is an alternative to the POD method for learning the dynamics of non-linear systems. DMD is an explicitly temporal decomposition that takes the sequences of snapshots into account, and is able to derive the spatiotemporal patterns of the data matrix X. The dynamics of the system are taken into account by considering a linear matrix A that maps a data matrix X 1 , starting at some time steps (t 1 ), to the data matrix X 2 , starting at the next time step (t 2 ). The matrix A is thus defined as: X 2 = AX 1 ,(S7) and the corresponding data matrices are given by: X 1 = [x(t 1 ), x(t 2 ), . . . , x(t Nt−1 )](S8) and X 2 = [x(t 2 ), x(t 3 ), . . . , x(t Nt )] .(S9) Interestingly, Equation S7 is similar to a linear stability analysis formulation for discrete maps if we think of the stability matrix as the linear matrix A. The DMD method thus consists of solving the following eigenvalues problem: AΦ = ΦΛ (S10) where the columns of Φ are the DMD modes φ i and the corresponding DMD eigenvalues λ i are the diagonal entry of Λ. The DMD modes φ i give us the spatial eigenmodes while their corresponding eigenvalues λ i have their temporal information. Using a change of units from the data snapshots, observed at every ∆t, to units in time, the eigenvalues are complex-valued scalars: ln (λ i ) ∆t = µ i + iω i (S11) where µ i gives the growth (decay) rate if µ i > 0 (µ i < 0) and ω i the oscillation frequency of the DMD modes φ i . However, the size of the data matrix usually makes the eigendecomposition not feasible. The goal, here, is therefore to approximate the eigenvalues and eigenvectors of A, using only the data matrices X 1 and X 2 . The idea is to start by the truncated SVD of X 1 = U r Σ r V † r in which Eq. S7 becomes: X 2 = AU r Σ r V † r .(S12) Then the linear matrix A is reduced by considering its projection onto the truncated POD subspace: A r := U † r AU r = U † r X 2 V r Σ −1 .(S13) The eigenvalue problem for A r is solved with: A r W = W Λ,(S14) from which we have the relation: Φ = X 2 V Σ −1 W. (S15) The key feature of the DMD method is that it decomposes the data into a set of coupled spatio-temporal modes. The DMD resembles a mixture of the POD in the spatial domain and the discrete Fourier transform (DFT) in the time domain. Figure S2 shows the DMD results for the oscillating regime. We can, indeed, see in Fig. S2(a) that the singular values are similar to that of the POD. Besides, we observe in Figs. S2 (c) and S2(d) that the field profile of the DMD modes closely resembles the POD modes in Fig. S1. The largest DMD modes not only look similar to the POD modes, but they also contain the oscillation frequencies from ω i , as in DFT. The DMD even goes beyond DFT by giving an estimate of the growth (decay) rate in time via µ i > 0 (µ i < 0). This can be seen by plotting the DMD modes, scaled by their contribution in the decomposition σ i , in the frequency plane of λ i . We can see in Fig. S2(b) that the dynamical regime has a single DMD mode with ω i = 0 akin to the offset of the oscillation amplitudes, and two DMD modes with opposite ω i = 0 corresponding to the oscillating behaviours. All the above three DMD modes have vanishing growth or decay rate µ i = 0. C. Time-augmented dynamical mode decomposition Although the DMD gives the temporal behaviours of the non-linear system, the temporal information is not fully incorporated into the DMD basis Φ since only the DMD modes are used. Exploiting the time evolution in the dynamical regime requires the use of DMD modes along with their eigenvalues. The idea is therefore to incorporate the dynamic information by augmenting the basis Φ [28]. This time-augmented DMD will be denoted by aDMD in the remaining of this chapter. Using the defining relation of the eigenvalues λ i as similar to a time evolution operator, i.e. multiplying by λ is the same as shifting by one time step, we have for a given DMD mode φ i , its evolution given by λ Nw i φ i at N w time step ahead in time. Therefore, the time-augmented basis vector reads:      φ i λ i φ i . . . λ Nw i φ i      . (S16) By considering a time window N w , the timeaugmentation of the DMD basis provides us with the dynamical information of the non-linear regime. (c) and S3(d), one of the first aDMD modes with ω i = 0 featuring some oscillating behaviour in time. The size of the basis mode is larger than the plain DMD, and can exhibit its time evolution. Nevertheless, the graphs do not exactly plot the temporal evolution of the DMD modes since the first N s entry of the basis state is for the N s sites; the next N s for again the N s sites but at the next time step, etc. D. Classification results from different decomposition methods The classification results [Eq. (7)] from the decomposition methods reviewed is shown in Fig. S4. We observe in Figs. S4(a) and S4(b) that the phase diagrams fail to correctly predict the distinct dynamical regimes. This is expected since the POD or DMD modes do not contain enough information about the temporal behaviours. Besides, the classification for these diagrams is based on a single snapshot (N w = 0). Thus, it is expected the classification fails to capture the correct dynamics since a single snapshot only relies on the spatial pattern of the regime. On the other hand, we can see in Figs. S4 (c) and S4(d) that the derived phase diagrams have better accuracy when the bases are time-augmented, or equivalently when the classification scheme uses several snapshots. By increasing the time window in the classification, the derived phase diagram is even better as illustrated in Figs. S4(c) and S4(d) for N w = 5 and N w = 25, respectively. The phase diagram will get improved until the time window is large enough to capture the dynamic behaviour. SII. SPARSE MEASUREMENT This section detailed the slight change in the methodology in case of sparse sensing. Sparse sensing is often desirable since the measurement and the data collection can be expensive for a complex system if the space grid is too fine, i.e. if N s is very large. The compressed mea-surement y(t) is derived from the full-state measurement x(t) and the measurement matrix C: y(t) = Cx(t),(S17) where C is a matrix of size (N p × N s ) with N p the number of measurements. Although the measurement matrix C can be represented by some advance and complex mapping [36], here we focus on point-wise measurements, namely the C ij entry in the matrix measurement corresponds to the i-th measurement at the j-th spatial grid point. Therefore the compressed basis is given by: Θ = CΦ (S18) where Φ is the basis obtained from the full-state data collection. The library of bases for the J distinct dynamical regimes is similarly re-written as: L = {Θ 1 , . . . , Θ J }.(S19) Nevertheless, the size of the current SSH lattice is, here, reasonable and allows us to choose the matrix measurement as the identity matrix C = 1 Ns . We will thus use the full-state instead of sparse measurements, but retain the Θ and y(t) notations to keep the general formalism. SIII. TOP-DOWN LIBRARY GENERATION PRINCIPLE This section supplement the top-down library generation principle. The general idea of the top-down construction of the library is to start with a library made of a high number of bases, generated from the time series randomly chosen in the given parameter space region, and then merge them into groups of equivalent regimes. SIV. BOTTOM-UP LIBRARY GENERATION PRINCIPLE This section supplement the bottom-up library generation principle. Figure S6 illustrates the bottom-up methodology proposed. We start with a single sample in the library, randomly chosen in the parameter space region [ Fig. S6(a)]. The library is then adaptively constructed according to the relative reconstruction error [ Fig. S6(b)]. Finally, with the large library at hand, the top-down approach is used to reduce the library size by merging equivalent regimes [ Fig. S6(c)]. The representative of the regimes is plotted in Fig. S6(d) and corresponds to the non-oscillating, oscillating and transient regimes, respectively. FIG. 2 . 2Representation classification based on a fixed library. Phase diagram obtained from the library composed of (a) two regimes (one non-oscillating and one oscillating) and (b) four regimes (three non-oscillating and one oscillating). The library is constructed by the magenta dots located at (γAB, gA − γAB) = (0.48, 0.06) and (0.16, 0.44), and for (b) additionally at (0.31, 0.11) and (0.17, 0.24). FIG. 3 . 3Representation classification based on a topdown adaptive library. (a) Phase diagram obtained using the top-down classification strategy with an initial library composed of J = 60 regimes randomly chosen and γ th = 0.75. (b) Representative total intensity IA (and IB) of the A (and B) sublattice in blue (and orange) for the different regimes. The black vertical dotted line indicates the starting time from which the bases are generated. FIG. 4 . 4γ th -hyper-parameter dependency. (a) Library size against γ th . Phase diagrams derived using the top-down classification strategy with (a) γ th = 0.55 and (b) γ th = 0.95. The initial library is composed of J = 60 regimes randomly chosen. FIG. 5 . 5Representation classification based on a bottom-up adaptive library. (a) Phase diagram obtained using the bottom-up classification strategy with a starting library composed of a single regime randomly chosen, th = 0.005 and γ th = 0.75. (b) Representative total intensity IA (and IB) of the A (and B) sublattice in blue (and orange) for the different regimes. th -hyper-parameter dependency. (a) Library size against th . The inset is a zoom-in of the plot. Phase diagram derived using the bottom-up classification strategy with γ th = 0.95 and (b) th = 0.005 and (c) th = 0.05. The initial library is composed of a single regime randomly chosen. W., D.R. and S.S.O. conceived the project. S.W developed the theoretical model and performed all the numerical simulations. All authors contributed to the discussions and the preparation of the manuscript. FIG. S1 . S1POD method. Singular values of the (a) nonoscillating and (b) oscillating regimes. The red dots correspond to the singular values accumulating 99% of the total variance of the data. (c) Real and (d) imaginary parts of the field profile of the first POD mode for the non-oscillating regime. (e) Real and (f) imaginary parts of the field profile of the (top) first and (bottom) second POD mode for the oscillating regime. The POD bases have been generated from the time series starting at the 1800-th time step. FIG. S2 . S2DMD method. (a) Singular values of the oscillating regime in the DMD. The red dots correspond to the singular values accumulating 99% of the total variance of the data. (b) Plot of the logarithm of the DMD values ln(λ) in the complex plane. The size of the open circle is proportional to their corresponding singular values. (c) Real and (d) imaginary parts of the field profile of the (top) first and (bottom) second DMD mode for the oscillating regime. The DMD basis has been generated from the time series starting at the 1800-th time step. Figure S3 shows the aDMD results for the same oscillating regime as in Fig. S2. We can see that the singular values and aDMD eigenvalues plots [Figs. S3(a) and S3(b)] are the same as in the DMD algorithm [Figs. S2(a) and . aDMD method. (a) Singular values of the oscillating regime in the aDMD. The red dots correspond to the singular values accumulating 99% of the total variance of the data. (b) Plot of the logarithm of the aDMD values ln(λ) in the complex plane. The size of the open circle is proportional to their corresponding singular values. (c) Real and (d) imaginary parts of the "field profile" of the (top) first and (bottom) second aDMD mode for the oscillating regime. Here, by "Site n", we mean the n-th entry of the eigenvector. The aDMD basis has been generated with Nw = 25 from the time series starting at the 1800-th time step. S2(b)] whereas the "field profile" of the aDMD modes [Figs. S3(c) and S3(d)] carry some temporal evolution information. In particular, we observe in the top panel of Figs. S3(c) and S3(d) the static behaviours of the aDMD mode with ω i = 0. On the other hand, we can see, in the bottom panel of Figs. S3 FIG. S4 . S4Phase diagram derived using different decomposition methods. Phase diagrams obtained from a library composed of two regimes (one non-oscillating and one oscillating) from which the bases have been generated using the (a) POD, (b) DMD, (c) aDMD with Nw = 5 and (d) aDMD with Nw = 25. The green and blue dots correspond respectively to the identified non-oscillating and oscillating regimes. The magenta dots represent the regimes used for the construction of the library. These are located at (γAB, gA − γAB) = (0.48, 0.06) and (0.16, 0.44). The white and grey areas are overlays of the referenced phase diagram obtained in Fig. 1. The bases have been generated from the time series starting at the 1800-th time step. Figure S5 illustrates the top-down algorithm. 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[ "FROM MINKOWSKI TO DE SITTER VACUA WITH VARIOUS GEOMETRIES", "FROM MINKOWSKI TO DE SITTER VACUA WITH VARIOUS GEOMETRIES" ]	[ "Constantinos Pallis [email protected] \nFaculty of Engineering\nLaboratory of Physics\nAristotle University of Thessaloniki\nGR-541 24ThessalonikiGREECE\n" ]	[ "Faculty of Engineering\nLaboratory of Physics\nAristotle University of Thessaloniki\nGR-541 24ThessalonikiGREECE" ]	[]	New no-scale supergravity models with F-term SUSY breaking are introduced, adopting Kähler potentials parameterizing flat or curved (compact or non-compact) Kähler manifolds. We systematically derive the form of the superpotentials leading to Minkowski vacua. Combining two types of these superpotentials we can also determine de Sitter or anti-de Sitter vacua. The construction can be easily extended to multi-modular settings of mixed geometry. The corresponding soft SUSY-breaking parameters are also derived.PACs numbers: 12.60.Jv, 04.65.+eCONTENTS	10.1140/epjc/s10052-023-11485-z	[ "https://export.arxiv.org/pdf/2211.05067v3.pdf" ]	253,420,718	2211.05067	e097ec748e60b60cbf337b93e2fd7a0ee1859911	FROM MINKOWSKI TO DE SITTER VACUA WITH VARIOUS GEOMETRIES 8 Apr 2023 Constantinos Pallis [email protected] Faculty of Engineering Laboratory of Physics Aristotle University of Thessaloniki GR-541 24ThessalonikiGREECE FROM MINKOWSKI TO DE SITTER VACUA WITH VARIOUS GEOMETRIES 8 Apr 2023PACs numbers: 1260Jv, 0465+e New no-scale supergravity models with F-term SUSY breaking are introduced, adopting Kähler potentials parameterizing flat or curved (compact or non-compact) Kähler manifolds. We systematically derive the form of the superpotentials leading to Minkowski vacua. Combining two types of these superpotentials we can also determine de Sitter or anti-de Sitter vacua. The construction can be easily extended to multi-modular settings of mixed geometry. The corresponding soft SUSY-breaking parameters are also derived.PACs numbers: 12.60.Jv, 04.65.+eCONTENTS Within Supergravity (SUGRA) [1,2], breaking Supersymmetry (SUSY) on a sufficiently flat background requires a huge amount of fine tuning, already at the classical level -see e.g. Ref. [3,4]. Besides remarkable exceptions presented recently [5], the so-called no-scale models [6][7][8][9][10][11][12][13][14] provide an elegant framework which alleviates the problem above since SUSY is broken with naturally vanishing vacuum energy along a flat direction. On the other hand, the discovery of the accelerate expansion of the present universe [15] motivates us to develop models with de Sitter (dS) -or even anti-dS (AdS) -vacua which may explain this expansion -independently of the controversy [16][17][18][19] surrounding this kind of (meta) stable vacua within string theory. In two recent papers [11,12], a systematic derivation of dS/AdS vacua is presented in the context of the noscale SUGRA without invoking any external mechanism of vacuum uplifting such as through the addition of anti-D3 brane contributions [20] or extra Fayet-Iliopoulos terms [21]. Namely, these vacua are achieved by combining two distinct Minkowski vacua taking as initial point the Kähler potential parameterizing the non-compact SU (1, 1)/U (1) Kähler manifold in half-plane coordinates, T and T * . Possible instabilities along the imaginary direction of the T field can be cured by introducing mild deformations of the adopted geometry. The analysis has been extended to incorporate more than one superfields in conjunction with the implementation of observationally successful inflation [13,22]. In this paper we show that the method above has a much wider applicability since it remains operational for flat spaces or curved ones. This is possible since the no-scale "character" of the models, as defined above, stems from the existence of a flat direction with SUSY broken along it, and not from the adopted moduli geometry. We parameterize the curved spaces of our models with the Poincaré disk coordinates Z α and Z * α which, although are widely adopted within the inflationary model building [23][24][25], they are not frequently employed for establishing SUSY-breaking models -cf. Ref. [5,6]. This parametrization gives us the opportunity to go beyond the non-compact geometry [11,12] and establish SUSY-breaking scenaria with compact [26] or "mixed" geometry. In total, we here establish three novel uni-modular no-scale models and discuss their extensions to the multi-modular level. In all cases, we show that a subdominant quartic term [5,11] in the Kähler potential stabilizes the sgoldstino field to a specific vacuum and provides mass for its scalar component without disturbing, though, the constant vacuum energy density. This can be identified with the present cosmological constant by finely tuning one parameter of the model whereas the others can be adjusted to perfectly natural values. If we connect, finally, our hidden sectors with some sample observable ones, non-vanishing soft SUSY-breaking (SSB) parameters [27], of the order of the gravitino mass can be readily determined at the tree level. We start our presentation with a simplified generic argument which outlines the transition from Minkowski to dS/AdS vacua in Sec. II. We then detail our models adopting first -in Sec. III -flat moduli geometry and then -see Sec. IV -two versions of curved geometry. Generalization of our findings displaying multi-modular models with mixed geometry is presented in Sec. V. We also study in Sec. VI the communication of the SUSY breaking to the observable sector by computing the SSB terms. We summarize our results in Sec. VII. Some useful formulae related to derivation of mass spectra in SUGRA with dS/AdS vacua is arranged in Appendix A. In Appendix B we show the consistency of our results with those in Ref. [11,12] translating the first ones in the language of the T − T * coordinates. Unless otherwise stated, we use units where the reduced Planck scale m P = 2.4 · 10 18 GeV is taken to be unity and the star ( * ) denotes throughout complex conjugation. Also, no summation convention is applied over the repeated Latin indices ℓ, i and j. II. START-UP CONSIDERATIONS The generation mechanism of dS/AdS vacua from a pair of Minkowski ones can be roughly established, if we consider a uni-modular model without specific geometry. In particular, we adopt a Kähler potential K = K(Z, Z * ) and attempt to determine an expression for the superpotential W = W (Z) so as to construct a no-scale scenario. The SUGRA potential V based on K and W from Eq. (A2) is written as V = e K g −1 K \|∂ Z W + W K Z \| 2 − 3\|W \| 2 ,(1) where g −1 K = K −1 ZZ * = K ZZ * . Suppose that there is an expression W = W 0 (Z) which assures that the direction Z = Z * is classically flat with V = 0, i.e., it provides a continuum of Minkowski vacua. The determination of W 0 , based on Eq. (1), entails g −1 K (W ′ 0 + W 0 K Z ) 2 = 3W 2 0 ⇒ dW 0 dZW 0 = ± 3g K − K Z (2) with Eq. (A3) being satisfied -the relevant conditions may constrain the model parameters once K is specified. Here prime stands for derivation with respect to (w.r.t) Z. Eq. (2) admits obligatorily two solutions W ± 0 = m exp ± dZ 3g K − dZK Z(3) with m a mass parameter. For the K's considered below, it is easy to verify that dZK Z = K/2(4) up to a constant of integration. E.g., if K = \|Z\| 2 , then K Z = Z for Z * = Z and dZK Z = Z 2 /2 = K/2. According to Ref. [11,12], the appearance of dS/AdS vacua is attained, if we consider the following linear combination of W ± 0 in Eq. (3) W Λ = C + W + 0 − C − W − 0 ,(5) where C − and C + are non-zero constants. As can be easily checked, W Λ does not consist solution of Eq. (2). It offers, however, the achievement of a technically natural dS/AdS vacuum since its substitution into Eq. (A2) yields V Λ = e K g −1 K (W ′ Λ + W Λ K Z ) 2 − 3W 2 Λ = 12e K C − C + W − 0 W + 0 = 12m 2 C − C + ,(6) where we take into account Eqs. (3) and (4). Rigorous validation and extension (to more superfields) of this method can be accomplished via its application to specific working models. This is done in the following sections. Let us, finally, note that V Λ can be identified with the present cosmological constant by demanding V Λ = Ω Λ ρ c0 = 7.2 · 10 −121 m 4 P ,(7) where Ω Λ = 0.6889 and ρ c0 = 2.4 · 10 −120 h 2 m 4 P with h = 0.6732 [15] is the density parameter of dark energy and the current critical energy density of the universe. III. FLAT MODULI GEOMETRY We focus first on the models with flat internal geometry and describe below their version for one -see Sec. III A -or more -see Sec. III B -moduli. A. UNI-MODULAR MODEL Our initial point is the Kähler potential K f = \|Z\| 2 − k 2 Z 4 v (8) where we include the stabilization term Z v = Z + − √ 2v with Z ± = Z ± Z * .(9) Here k and v are two real free parameters. Small k values are completely natural, according to the 't Hooft argument [28], since K f enjoys an enhanced U (1) symmetry which is exact for k = 0. It is evident that the Z space defined by K f is flat with metric K ZZ * = 1 along the stable configurations Z = Z * for k = 0 (10a) and Z = v/ √ 2 for k = 0.(10b) Hereafter, the value of a quantity Q for both alternatives above, -i.e. either along the flat direction of Eq. (10a) or at the (stable) minimum of Eq. (10b) -is denoted by the same symbol Q . Applying Eq. (A2) for K = K f , and an unknown W = W f for Z = Z * , we obtain V f = e Z 2 (ZW f + W ′ f ) 2 − 3W 2 f .(11) Following the strategy in Sec. II, we first find the required form of W f , W 0f , which assures the establishment of a Z-flat direction with Minkowski vacua. I.e. we require V f = 0 for any Z. Solving the resulting ordinary differential equation Z + dW 0f dZW 0f = ± √ 3(12) w.r.t W 0f , we obtain two possible forms of W 0f , W ± 0f (Z) = mwf ±1 with w = e −Z 2 /2 and f = e √ 3Z .(13) Note that a factor w appears already in the models of Ref. [12] associated, though, with a matter field and not with the goldstino superfield as in our case. The solutions in Eq. (13) above can be combined as follows -cf. Eq. (5) - W Λf = C + W + 0f − C − W − 0f = mwf C − f ,(14) where we introduce the symbols C ± f := C + ± C − f −2 .(15) Employing K = K f and W = W Λf from Eqs. (8) and (14), we find the corresponding V via Eq. (A2) V f = m 2 /(1 − 12k 2 Z 2 v ) \|f \| 2 exp −Z 2 − /2 − k 2 Z 4 v · √ 3C + f − Z − C − f − 4k 2 Z 3 v C − f 2 − 3 C − f 2 ,(16) which exhibits the dS/AdS vacua in Eqs. (10a) and (10b). Indeed, we verify that V f = V Λ , given in Eq. (6) and Eq. (A3) for V = V f and α = 1 is readily fulfilled. Indeed, decomposing Z α (with α = 1 suppressed when we have just one Z) in real and imaginary parts, -z 1 := z andz 1 :=z -i.e., Z α = (z α + iz α )/ √ 2,(17) we find that the eigenvalues of M 2 0 in Eq. (A4) are m 2 zf = 144k 2 m 2 3/2f C + f /C − f 2 and mz f = 4m 2 3/2f ,(18) where m 3/2f is the G mass along the configurations in Eqs. (10a) and (10b). This is found by replacing K and W from Eqs. (8) and (14) in Eq. (A7), with result m 3/2f = m f C − f = m    e √ 3Z C + − C − e −2 √ 3Z for k = 0 e √ 3/2v C + − C − e − √ 6v for k = 0. Note that, for k = 0 and unfixed Z, m 3/2f remains undetermined validating thereby the no-scale character of our models -cf. Ref. [8,13]. As a shown in Eq. (18), the real component z of Z remains massless due to the flatness of V f along the direction in Eq. (10a). However, the k-dependent term in Eq. (8) not only stabilizes Z but also provides mass to z. On the other hand, this term generates poles and so discontinuities in V f -see Eq. (6). We are obliged, therefore, to focus on Table I. The location of the dS vacuum in Eq. (10b) is also depicted by a thick black point. a local dS/AdS minimum as in Eq. (10b). Inserting Eqs. (18) and (19) into Eq. (A5a) we find STrM 2 f = m 2 zf + m 2 zf − 4m 2 3/2f = m 2 zf ,(20) which is consistent with Eq. (A12) given that R in Eq. (A13) is found to be R f = 24k 2 . Our analytic findings above can be further confirmed by Fig. 1, where the dimensionless quantity 10 2 V f /m 2 m 2 P is plotted as a function of z andz in Eq. (17). We employ the values of the parameters listed in column A of Table I -obviously k 1 there is identified with k in Eq. (8). We see that the dS vacuum in Eq. (10b) -indicated by the black thick point -is placed at (z,z) = (1, 0) and is stabilized w.r.t both directions. In the same column of Table I displayed are also the various masses of G and the scalar ( z) and pseudoscalar ( z) components of the sgoldstino Z given in GeV for convenience. For N = 1, the spectrum does not comprise any goldstino ( z) as explained in Appendix A. It is worth mentioning that the aforementioned masses may acquire quite natural values (of the order of 10 −15 ) for logical values of the relevant parameters despite the fact that the fulfilment of Eq. (7) via Eq. (6) requires a tiny C − . E.g., for the parameters given in Table I we need C − = 1.4 · 10 −90 . Let us, finally, note that performing a Kähler transformation K → K + Λ K + Λ * K and W → W e −ΛK ,(21) with K = K f and W = W Λf in Eqs. (8) and (14) respectively and Λ K = −Z 2 /2, the present model is equivalent with that described by the following K and W K f = − 1 2 Z 2 − − k 2 Z 4 v and W Λf = mf C − f .(22) From the form above we can easily infer that, for k → 0, K f enjoys the enhanced symmetries where c is a real number. These are more structured than the simple U (1) mentioned below Eq. (8) and underline, once more, the naturality of the possible small k values. In this limit, a similar model arises in the context of the α-scale SUGRA introduced in Ref. [10]. Z → Z + c and Z → −Z,(23)A B C D E F INPUT SETTINGS K K f K 2f K + K − K +− K f− W W Λf W Λ2f W Λ+ W Λ− W Λ+− W Λf− INPUT PARAMETERS vα = mP, C + = 0.01 and m = 5 TeV k1 0.3 0.3 0.3 0.3 0.3 0.3 k2 − 0.2 − − 0.2 0.2 a1 − 1 − − 1 1 n1 − − 1 4 1 4 n2 − − − − 4 − PARTICLE MASSES IN GeV G B. MULTI-MODULAR MODEL The model above can be extended to incorporate more than one modulus. In this case, the corresponding K is written as K N f = N f ℓ=1 \|Z ℓ \| 2 − k 2 ℓ Z 4 vℓ(24) where for any modulus Z α we include a stabilization term Z vα = Z α+ − √ 2v α with Z α± = Z α ± Z * α ,(25) with α = ℓ in the domain of the values shown in Eq. (24). As we verify below, for the same α values, we can obtain the stable configurations Z α− = 0 for k α = 0 (26a) and Z α = v α / √ 2 for k α = 0. (26b) Along them the Kähler metric is represented by a N f × N f diagonal matrix K αβ = diag(1, ..., 1).(27)Setting Z ℓ = Z * ℓ , K = K N f from Eq. (24) and W = W N f (Z ℓ ) in Eq. (A2), V takes the form V N f = e ℓ Z 2 ℓ ℓ (Z ℓ W N f + ∂ ℓ W N f ) 2 − 3W 2 N f .(28) Setting V N f = 0 and assuming the following form for the corresponding W N f W 0N f (Z 1 , ..., Z N f ) = ℓ W fℓ (Z ℓ ) ⇒ ∂ ℓ W 0N f = dW fℓ dZ ℓ W 0N f W fℓ ,(29) we obtain the separated differential equations ℓ Z ℓ + dW fℓ dZ ℓ W fℓ 2 = 3.(30) These can be solved w.r.t W fℓ , if we set Z ℓ + dW fℓ dZ ℓ W fℓ = \|a ℓ \| with ℓ a 2 ℓ = 3,(31) i.e., the a ℓ 's satisfy the equation of the hypersphere S N f −1 with radius √ 3. The resulting solutions take the form W ± fℓ = w ℓ f ±a ℓ / √ 3 ℓ with w ℓ = e −Z 2 ℓ /2 and f ℓ = e √ 3Z ℓ .(32) The total expression for W 0N f is found substituting the findings above into Eq. (29). Namely, W ± 0N f = m ℓ W ± fℓ = mWF ±1 ,(33) where we define the functions W = e − ℓ Z 2 ℓ /2 and F = e ℓ a ℓ Z ℓ .(34) As in the case with N f = 1, we combine both solutions above as follows W ΛN f = C + W + 0N f − C − W − 0N f = mWF C − F ,(35) where we introduce the "generalized" C symbols -cf. Eq. (15) C ± F := C + ± C − F −2 .(36) Substituting Eqs. (24) and (35) into Eq. (A2) we find that V takes the form V N f = m 2 \|F \| 2 exp − ℓ Z 2 ℓ− /2 + k 2 ℓ Z 4 vℓ · ℓ a ℓ C + F − Z ℓ− C − F − 4k 2 ℓ Z 3 vℓ C − F 2 1 − 12k 2 ℓ Z 2 vℓ − 3 C − F 2 . (37) We can confirm that V N f admits the dS/AdS vacua in Eqs. (26a) and (26b) for α = ℓ, since V N f = V Λ , given in Eq. (6). In addition, Eq. (A3) for V = V N f and α = ℓ is satisfied, since the 2N f masses squared of the relevant matrix in Eq. (A4) are positive. Indeed, analyzing Z ℓ in real and imaginary parts as in Eq. (17), we find m 2 zfℓ = 48k 2 ℓ a 2 ℓ m 2 3/2F C + F /C − F 2 ; (38a) m 2 zfℓ = 4m 2 3/2F 1 + (3 − a 2 ℓ )V Λ /6m 2 3/2F ,(38b) where m 3/2 for the present case is computed inserting Eqs. (24) and (35) into Eq. (A7) with result m 3/2F = m F C − F = m · e ℓ a ℓ Z ℓ C + − C − e −2 ℓ a ℓ Z ℓ for k ℓ = 0 e ℓ a ℓ v ℓ √ 2 C + − C − e − √ 2 ℓ a ℓ v ℓ for k ℓ = 0. (39) The expressions above conserve the basic features of the noscale models as explained below Eq. (19). We consider the stabilized version of these models (with k α = 0) as more complete since it offers the determination of m 3/2 and avoids the presence of a massless mode which may be problematic. We should note that the relevant 2N f × 2N f matrix M 2 0 of Eq. (A4) turns out to be diagonal up to some tiny mixings appearing in thez ℓ −zl positions. These contributions though can be safely neglected since these are proportional to V Λ . We also obtain N f − 1 Weyl fermions with masses mzl = m 3/2 wherel = 1, ..., N f − 1. Note that Eqs. (39), (38a) and (38b) reduce to the ones obtained for N f = 1, i.e. Eqs. (19) and (18), if we replace a ℓ = √ 3. Inserting the mass spectrum above into Eq. (A5a), we find STrM 2 N f = 2(N f − 1) m 2 3/2F + V Λ + ℓ m 2 zfℓ . (40) This result is consistent with Eq. (A5b) given that its last term turns out to be equal to the last term of Eq. (40). To highlight further the conclusions above we depict in Fig. 2 the dimensionless V N f for N = 2, i.e. V 2f , as a function of z 1 and z 2 forz 1 =z 2 = 0 and the other parameters displayed in column B of Table I. We observe that the dS vacuum in Eq. (26b) -indicated by a black thick point in the plot -is well stabilized against both directions. In the same column of Table I we also arrange some suggestive values of the particle masses for N = 2. Note that, due to the smallness of V Λ , the mz fℓ values are practically equal between each other. IV. CURVED MODULI GEOMETRY We proceed now to the models with curved internal geometry and describe below their version for one -see Sec. IV A -or more -see Sec. IV B -moduli. A. UNI-MODULAR MODEL The curved moduli geometry is described mainly by the Kähler potentials K ± = ±n ln Ω ± with Ω ± = 1 ± \|Z\| 2 − k 2 Z 4 v n(41) and Z v given in Eq. (9). Also n > 0 and k are real, free parameters. The positivity of the argument of logarithm in Eq. (41) implies Table I. The location of the dS vacuum in Eq. (26b) is also depicted by a thick black point. 1 ± \|Z\| 2 /n 0 ⇒ \|Z\| 2 −n for K = K + , \|Z\| √ n for K = K − .(42) The restriction for K = K + is trivially satisfied, whereas this for K = K − defines the allowed domain of Z values which lie in a disc with radius √ n and thus, the name disk coordinates. If we set k = 0 in Eq. (41), K − parameterizes [7,13,26] the coset space SU (1, 1)/U (1) whereas K + is associated [26] with SU (2)/U (1). Thanks to these symmetries, low k values are totally natural as we explained below Eq. (23). The Kähler metric and (the constant) R in Eq. (A13) are respectively K ZZ * = Ω −2 ± and R ± = ±2/n for K = K ± .(43) The last quantity reveals that the Kähler manifold is compact (spherical) or non-compact (hyperbolic) if K = K + or K = K − respectively. For this reason, the bold subscripts + or − associated with various quantities below are referred to K = K + or K = K − respectively. Repeating the procedure described in Sec. II, we find the form of V in Eq. (A2), V ± , as a function of K = K ± in Eq. (41) and W = W ± for Z = Z * . This is V ± = v ±n ± W 2 ± Z + v ± W ′ ± W ± 2 − 3 with v ± = 1 ± Z 2 n . (44) Setting V ± = 0 we see that the corresponding W ± = W 0± obeys the differential equation dW 0± W 0± = ± √ 3 − Z v ± dZ.(45) This can be resolved yielding two possible forms of W 0± , W ± 0± = mv ∓n/2 ± u ±1 ± for K = K ± ,(46) which assure the establishment of Minkowski minima -cf. Eq. (3). The corresponding functions u ± can be specified as follows u + = e √ 3natn(Z/n) and u − = e √ 3natnh(Z/n) ,(47) where atn and atnh stand for the functions arctan and arctanh respectively. The superscript ± in Eq. (46) correspond to the exponents of u ± and should not be confused with the bold subscripts ± with reference to K ± . Combining both Minkowski solutions, W ± 0± in Eq. (46) and introducing the shorthand notation -cf. Eq. (15) - C ± u± := C + ± C − u −2 ± for K = K ± ,(48) we can obtain the superpotential W Λ± = C + W + 0± − C − W − 0± = mv ∓n/2 ± u ± C − u± for K = K ±(49) which allows for dS/AdS vacua. To verify it, we insert K = K ± and W = W Λ± from Eqs. (41) and (49) in Eq. (A2) with result V ± = m 2 Ω ±n ± \|v ± \| ∓n \|u ± \| 2 − 3\|C − u± \| 2 + nΩ 2 ± · ( √ 3C + u± − ZC − u± )v −1 ± ± (Z * − 4k 2 Z 3 v )C − u± Ω −1 ± 2 · ∓\|4k 2 Z 3 v − Z\| 2 + nΩ ± (1 − 12k 2 Z 2 v ) −1 .(50) Given that for Z − = 0 we get Ω ± = v ± , we may infer that V ± = V Λ shown in Eq. (6) for the directions in Eqs. (10a) and (10b). Eq. (A3) for V = V ± and Z 1 := Z is also valid without restrictions for K = K + but only for n > 3 for K = K − . In fact, employing the decomposition of Z in Eq. (17) for α = 1, we can obtain the scalar spectrum of our models which includes the sgoldstino components with masses squared m 2 z± = 144k 2 m 2 3/2± v 3/2 ± C + u± /C − u± 2 ; (51a) m 2 z± = 4m 2 3/2± 1 ± (3/n) C + u± /C −m 3/2± = m u ± C − u± for K = K ± ,(52) which may be explicitly written if we use Eqs. (47) and (48)cf. Eq. (19). The stability of configurations in Eqs. (10a) and (10b) is protected for m 2 z− > 0 and m 2 z− > 0 provided that Z < √ n and n > 3 for K = K − .(53) Since we expect that Z ≤ 1, the latter restriction is capable to circumvent both requirements -see column D in Table I. Inserting the mass spectrum above into the definition of Eq. (A5a), we can find STrM 2 ± = 12m 2 3/2± C + u± /C − u± 2 12nk 2 v ± 3 ± 1 . (54) It can be easily verified that the result above is consistent with the expression of Eq. (A12) given that R in Eq. (A13) is Table I. Note that the selected n = n 1 = 4 for K = K − protects the stability of the vacuum in Eq. (10b) as dictated above. In columns C and D of Table I we display some explicit values of the particle masses encountered for K = K − and K + respectively. As a consequence of the employed n value in column D we accidentally obtain m 3/2− = mz 1 = mz − ; k 1 and n 1 obviously coincide with k and n in Eq. (41). R ± = 2 12k 2 n v ± 3 ± 1 .(55) B. MULTI-MODULAR MODEL The generalization of the model above to incorporate more than one modulus can be performed following the steps of Sec. III B. This generalization, however, is accompanied by a possible mixing of the two types of the curved geometry analyzed in Sec. IV A. More specifically, the considered here K, includes two sectors with N + compact components and N − non-compact ones. It may be written as K N + N − = N + i=1 n i ln Ω i + N ± j=N + +1 n j ln Ω j ,(56) where N ± = N + + N − and the arguments of the logarithms are identified as Ω α = Ω α+ for α = i, Ω α− for α = j.(57a) The symbols Ω α± can be collectively defined as Ω α± = 1 ± \|Z α \| 2 /n α ∓ k 2 α Z 4 vα (57b) with Z vα is given from Eq. (25) and α = i, j. When explicitly indicated, summation and multiplication over i and j is applied for the range of their values specified in Eq. (56). Given that i corresponds to compact geometry (+) and j to non-compact (−) we remove the relevant indices ± from the various quantities to simplify the notation. Under these assumptions, the positivity of the arguments of ln implies restrictions only to Ω j -cf. Eq. (42): Ω j > 0 ⇒ \|Z j \| < √ n j .(58) Along the configurations in Eqs. (26a) and (26b) for α = i, j, the Kähler metric is represented by a N ± × N ± diagonal matrix K αβ = diag(v −2 1 , ..., v −2 N+ , v −2 N++1 , ..., v −2 N ± ),(59) where we introduce the generalizations of the symbols v ± , defined in Eq. (44), as follows v i = (1 + Z 2 i /n i ) and v j = (1 − Z 2 j /n j ).(60) Also R in Eq. (A13) includes contributions for both geometric sectors, i.e, R N + N − = 2 i n −1 i − 2 j n −1 j .(61)Inserting K = K N + N − from Eq. (56) and W = W N ± (Z i , Z j ) with Z i = Z * i and Z j = Z * j in Eq. (A2), we obtain V N ± = V −2 i (Z i W N ± + ∂ i W N ± v i ) 2 + j (Z j W N ± + ∂ j W N ± v j ) 2 − 3W 2 N ± ,(62) where the prefactor V is defined as follows V = i,j v −ni/2 i v nj /2 j .(63) Setting V N ± = 0 and assuming the ansatz for the correspond- ing W N ± W 0N ± (Z 1 , ..., Z N ± ) = i,j W i (Z i )W j (Z j ) ⇒ ∂ i W 0N ± W 0N ± = dW i dZ i W i and ∂ j W 0N ± W 0N ± = dW j dZ j W j ,(64) we obtain the separated differential equations i Z i + dW i dZ i W i v i 2 + j Z j + dW j dZ j W j v j 2 = 3. (65) We can solve the equations above if we set Z i + dW i dZ i W i v i = \|a i \| and Z j + dW j dZ j W j v j = \|a j \| (66) imposing the constraint i a 2 i + j a 2 j = 3,(67) i.e., the a i and a j can be regarded as coordinates of the hypersphere S N ± −1 with radius √ 3. Solution of the differential equations above w.r.t W i and W j yields W ± i = v −ni/2 i u ±ai/ √ 3 i and W ± j = v ni/2 j u ±aj / √ 3 jW ± 0N ± = m i,j W ± i W ± j = mVU ±1 ,(70) where we define the function U = i,j u ai/ √ 3 i u aj/ √ 3 j .(71) Introducing the generalized C symbols -cf. Eq. (15) - C ± U := C + ± C − U −2 ,(72) we combine both solutions in Eq. (70) as follows W ΛN ± = C + W + 0N ± − C − W − 0N ± = mVUC − U .(73) Plugging K = K N + N − and W = W ΛN ± from Eqs. (56) and (73) in Eq. (A2) we find V N ± = m 2 ij Ω ni i Ω −nj j \|VU\| 2 − 3\|C − U \| 2 + i n i Ω 2 i · (a i C + U − Z i C − U )v −1 i + (Z * i − 4k 2 i Z 3 vi )C − U Ω −1 i 2 · − 4k 2 i Z 3 vi − Z i 2 + n i Ω i 1 − 12k 2 i Z 2 vi −1 + j n j Ω 2 j · (a j C + U − Z j C − U )v −1 j − (Z * j − 4k 2 j Z 3 vj )C − U Ω −1 j 2 · 4k 2 j Z 3 vj − Z j 2 + n j Ω j 1 − 12k 2 j Z 2 vj −1 . (74) Note that there are slight differences between the terms with subscripts i and j due to our convention in Eq. (57a) -cf. Eq. (50). The settings in Eqs. (26a) and (26b) consist honest dS/AdS vacua since V N ± = V Λ given in Eq. (6). However, the conditions in Eq. (A3) for V = V N ± and α = i, j are met only after imposing upper bound on v j and a j . To determine this, we extract the masses squared of the 2N ± scalar components of Z i and Z j in Eq. (17) which are where we restore the ± symbols for clarity and we neglect for simplicity terms of order (C − ) 2 in the two last expressions. We also compute m 3/2 upon substitution of Eqs. (56) and (73) into Eq. (A7) with result m 2 zi+ = 48k 2 i a 2 i m 2 3/2U v 3/2 i C + U /C − U 2 ; (75a) m 2 zj− = 48k 2 j a 2 j m 2 3/2U v 3/2 j C + U /C − U 2 ; (75b) m 2 zi+ ≃ 4m 2 3/2U 1 + a 2 i /n i ; (75c) m 2 zj− ≃ 4m 2 3/2U 1 − a 2 j /n j ,(75d)m 3/2U = m UC − U .(76) As in the case of Sec. III B, the relevant matrix M 2 0 in Eq. (A4) turns out to be essentially diagonal since the non-zero elements appearing in thez α −z β positions with α, β = i, j are proportional to V Λ and can be safely ignored compared to the diagonal terms. From Eqs. (75b) and (75d), we notice that positivity of m 2 zj− and m 2 zj− dictates Z j < √ n j and \|a j \| < √ n j . These restrictions together with Eq. (67) delineate the allowed ranges of parameters in the hyperbolic sector. We also obtain N ± − 1 Weyl fermions with masses mzα = m 3/2 with α = 1, ..., N ± − 1. Inserting the mass spectrum above into Eq. (A5a) we find STrM 2 N ± ≃ 6m 2 3/2U (N ± − 1) + 2 3 i a 2 i n i + j a 2 j n j + 8 C + U C − U 2 i a 2 i k 2 i v i 3 + j a 2 j k 2 j v j 3 . (78) It can be checked that this result is consistent with Eq. (A5b). For N = 2, N − = N + = 1 and the parameters shown in column E of Table I, we present in Fig. 4 the relevant V N ± , V 2± = V +− -conveniently normalized -versus z 1 and z 2 in Eq. (17) fixingz 1 =z 2 = 0. It is clearly shown that the vacuum of Eq. (26b), depicted by a bold point, is indeed stable. In column E of Table I we arrange also some representative masses (in GeV) of the particle spectrum for N ± = 2. From the parameters listed there we infer that a 2 = √ 2 < √ n 2 = 2 and so Eqs. (67) and (77) are met. V. GENERALIZATION It is certainly impressive that the models described in Sec. III B and IV B can be combined in a simple and (therefore) elegant way. We here just specify the utilized K and W of a such model and restrict ourselves to the verification of the results. In particular, we consider the following K K N f± = K N f + K N + N − ,(79) which incorporates the individual contributions from Eqs. (24) and (56). It is intuitively expected that the required W for achieving dS/AdS vacua has the form -cf. Eqs. (35) and (73) W ΛN f± = C + W + 0N f± + C − W − 0N f± ,(80) where the definitions of W ± 0N f± follow those in Eqs. (33) and (70) respectively. Namely, we set W ± 0N f± = mWV(F U) ±1 ,(81) where the parameters a ℓ , a i and a j , which enter the expressions of the functions F and U in Eqs. (34) and (71), satisfy the constraint -cf. Eqs. (31) and (67) ℓ a 2 ℓ + i a 2 i + j a 2 j = 3.(82) I.e., they lie at the hypersphere S Nt−1 with radius √ 3 and N t = N f + N ± . If we introduce, in addition, the C symbols -cf. Eqs. (36) and (72) - C ± F U := C + ± C − (F U) −2 ,(83) W ΛN f± in Eq. (80) is simplified as W ΛN f± = mWVF UC − F U .(84) Plugging K = K N f± and W = W ΛN f± from Eqs. (79) and (84) into Eq. (A2) we obtain 77) and (58). V N f± = m 2 e K N f ij Ω ni i Ω −nj j \|F VU\| 2 − 3\|C − F U \| 2 + i n i Ω 2 i · (a i C + F U − Z i C − F U )v −1 i + (Z * i − 4k 2 i Z 3 vi )C − F U Ω −1 i 2 · − 4k 2 i Z 3 vi − Z i 2 + n i Ω i 1 − 12k 2 i Z 2 vi −1 + j n j Ω 2 j · (a j C + F U − Z j C − F U )v −1 j − (Z * j − 4k 2 j Z 3 vj )C − F U Ω −1 j 2 · 4k 2 j Z 3 vj − Z j 2 + n j Ω j 1 − 12k 2 j Z 2 vj −1 + ℓ a ℓ C + F U − Z ℓ− C − F U − 4k 2 ℓ Z 3 vℓ C − F U 2 · 1 − 12k 2 ℓ Z 2 vℓ −1 .(85) The G mass is derived from Eq. (A7), after substituting K and W from Eqs. (79) and (84) respectively. The result is To provide a pictorial verification of our present setting, we demonstrate in Fig. 5 the three-dimensional plot of V N f± with N f = N − = 1 and N + = 0, i.e. V 1f− , versus z 1 and z 2 forz 1 =z 2 = 0 -see Eq. (17) -and the other parameters arranged in column F of Table I. Note that the subscripts 1 and 2 of z correspond to ℓ = 1 and j = 1 and the validity of Eqs. (77) and (82) is protected. It is evident that the ground state, depicted by a tick black point is totally stable. Some characteristic values of the masses of the relevant particles are also arranged in column F of Table I. m 3/2F U = m F UC − F U .(86) VI. LINK TO THE OBSERVABLE SECTOR Our next task is to study the transmission of the SUSY breaking to the visible world. Here we restrict for simplicity ourselves to the cases with just one Goldstino superfield, Z. To implement our analysis, we introduce the chiral superfields of the observable sector Φ α with α = 1, ..., 5 and assume the following structure -cf. Ref. [1,5,27] -for the total superpotential, W HO , of the theory W HO = W H (Z) + W O (Φ α ) ,(87) where W H is given by Eq. (14) or Eq. (49) for flat or curved Z geometry respectively whereas W O has the following generic form W O = hΦ 1 Φ 2 Φ 3 + µΦ 4 Φ 5 .(88) with h and µ free parameters. On the other hand, we consider three variants of the total K of the theory, K HO , ensuring universal SSB parameters for Φ α : K 1HO = K H (Z) + α \|Φ α \| 2 ; (89a) K 2HO = K H (Z) + N O ln 1 + α \|Φ α \| 2 /N O , (89b) where K H (Z) may be identified with K f in Eq. (8) or K ± in Eq. (41) for flat or curved Z geometry respectively whereas N O may remain unspecified. For curved Z geometry we may introduce one more variant K 3HO = ±n ln Ω ± ± α \|Φ α \| 2 /n .(89c) If we expand the K HO 's above for low Φ α values, these may assume the form K HO = K H (Z) + K H (Z)\|Φ α \| 2 ,(90a) with K H being identified as K H = 1 for K HO = K 1HO , K 2HO ; Ω −1 ± for K HO = K 3HO .(90b) Adapting the general formulae of Ref. [5,27] to the case with one hidden-sector field and tiny V , we obtain the SSB terms in the effective low energy potential which can be written as and the canonically normalized fields are Φ α = K H 1/2 Φ α . In deriving the values of the SSB parameters above, we distinguish the cases: V SSB = m 2 α \| Φ α \| 2 + A h Φ 1 Φ 2 Φ 3 + B µ Φ 4 Φ 5 + h.c. ,(91) (a) For flat Z geometry, i.e. K H = K f , we see from Eq. (90b) that K H is constant for both adopted K HO 's and so, the results are common. Substituting F Z = √ 3m 3/2f and ∂ Z K H = v,(93) into the relevant expressions [5] we arrive at m α = 1 + e √ 6v V Λ /m 2 C − f 2 m 3/2f ≃ m 3/2f , A = − 3 2 v C + f C − f m 3/2f ≃ 3 2 vm 3/2f , B = A − m 3/2f ,(94) where ( h, µ) = e v 2 /4 (h, µ) and the last simplified expressions are obtained in the realistic limit C − → 0 which implies C + f /C − f → −1. (b) For curved Z geometry, i.e. K H = K ± , we can distinguish two subcases depending on which K HO from those shown in Eq. (89a) -(89c) is selected. Namely, • If K HO = K 1HO or K 2HO , then K H in Eq. (90b) is Z independent. For K H = K ± respectively we find (m α , A, B) = 1, 3/2v, A/m 3/2± − 1 m 3/2± ,(95) where ( h, µ) = v ± ±n/2 (h, µ) and we take into account the following F Z = √ 3 v ± m 3/2± and ∂ Z K ± = Zv −1 ± . • If K HO = K 3HO , then K H in Eq. (90b) is Z dependent. Inserting the expressions ∂ Z ln K H = 1 n v ± 2 and ∂ Z ln K 2 H = 2v n v ± into the general formulae [5,27] we end up with the following results for K H = K ± correspondingly: m α = 1 ± 3/n m 3/2± ; A = 3/2v(1 ± 3/n)m 3/2± ; B = 3/2v(1 ± 2/n) − 1 m 3/2± ,(96) where h = v ± (3±n)/2 h and µ = v ± (2±n)/2 µ. In both cases above we take C − ≃ 0 for simplicity. Note that the condition n > 3 for K = K − which is imperative for the stability of the configurations in Eqs. (10a) and (10b) -see Eq. (51b) -implies non-vanishing SSB parameters too. Taking advantage from the numerical inputs listed in columns A, C and D of Table I (for the three unimodular models) we can obtain some explicit values for the SSB parameters derived above -restoring units for convenience. Our outputs are arranged in the three rightmost columns of Table II for the specific forms of W H , K H and K HO in Eqs. (87) and (90a) shown in the three leftmost columns. We remark that there is a variation of the achieved values of SSB parameters which remain of the order of the G mass in all cases. VII. CONCLUSIONS We have extended the approach of Ref. [11,12], proposing new no-scale SUGRA models which lead to Minkowski, dS and AdS vacua without need for any external uplifting mechanism. We first provided a simple but general enough argument which assists to appreciate the effectiveness of our paradigm. We then adopted specific single-field models and showed that the achievement of dS/AdS solutions using pairs of Minkowski ones works perfectly well for flat -see Eqs. (8) and (14) -and hyperbolic or spherical geometrysee Eqs. (41) and (49). We also broadened these constructions to multi-field models -see Sec. III B, IV B and V. Within each case we derived the SUGRA potential and the relevant mass spectrum paying special attention to the stability of the proposed solutions. Typical representatives of our results were illustrated in Fig. 1 -5 employing numerical inputs from Table I. We provided, finally, the set of the soft SUSY-breaking parameters induced by our unimodular models linking them to a generic observable sector -see Eqs. (94) -(96). We verified -see Table II -that their magnitude is of the order of the gravitino mass. As stressed in Ref. [12,22], this kind of constructions, based exclusively in SUGRA, can be considered as part of an effective theory valid below m P . However, the correspondence between Kähler and super-potentials which yields naturally Minkowski, dS and AdS (locally stable) vacua with broken SUSY may be a very helpful guide for string theory so as to establish new possible models with viable low energy phenomenology. As regards the ultraviolet completion, it would be interesting to investigate if our models belong to the string landscape or swampland [16]. Note that the swampland string conjectures are generically not satisfied in SUGRA-based models but there are suggestions [17,18] which may work in our framework too. One more open issue is the interface of our settings with inflation. We aspire to return on this topic soon taking advantage from other similar studies [22,[29][30][31]] -see Ref. [32]. At last but not least, let us mention that the achievement of the present value of the darkenergy density parameter in Eq. (7) requires an inelegant fine tuning, which may be somehow alleviated if we take into account contributions from the electroweak symmetry breaking and/or the confinement in quantum chromodynamics [12,22]. Despite the shortcomings above, we believe that the establishment of novel models for SUSY breaking with a natural emergence of Minkowski and dS/AdS vacua can be considered as an important development which offers the opportunity for further explorations towards several cosmophenomenological directions. DEDICATION I would like to dedicate the present paper to the memory of T. Tomaras, an excellent University teacher who let his imprint on my first post-graduate steps. APPENDIX A: MASS FORMULAE IN SUGRA We here generalize our formulae in Ref. [5] for N chiral multiplets and dS/AdS vacua. Let us initially remind that central role in the SUGRA formalism plays the Kähler-invariant function expressed in terms of the Kähler potential K and the superpotential W as follows G := K + ln \|W \| 2 . (A1) Using it we can derive the F-term scalar potential [1] V = e G G αβ G α Gβ − 3 = e K \|W \| 2 K αβ G α Gβ − 3 , (A2) where the subscripts of quantities G and K denote differentiation w.r.t the superfields Z α and G αβ = K αβ is the inverse of the Kähler metric K αβ . The spontaneous SUSY breaking takes place typically at a (locally stable) vacuum or flat direction of V which satisfies the extremum and minimum conditions ∂ α V = ∂ᾱV = 0 and m 2 A > 0.(A3) Here ∂ α := ∂/∂Z α and ∂ᾱ := ∂/∂Z * α with the scalar components of the superfields denoted by the same superfield symbol. Also m 2 A are the eigenvalues of the 2N × 2N masssquared matrix M 2 0 of the (canonically normalized) scalar fields which is computed applying the formula M 2 0 =        ∂ α ∂ β V ∂ α ∂ β V ∂ β ∂ α V ∂ β ∂ ᾱ V        ,(A4) where ∂ A := ∂/∂ Z A with A = α orᾱ and Z α = √ K αᾱ Z α given that the K's considered in our work are diagonal. The aforementioned M 2 0 is one of the mass-squared matrices M 2 J of the particles with spin J, composing the spectrum of the model. They obey the super-trace formula [1,2] STrM 2 := 3/2 J=0 (−1) 2J (2J + 1)TrM 2 J (A5a) = 2m 2 3/2 (N − 1)(1 + V /m 2 3/2 ) + G α G αβ R βγ G γδ Gδ ,(A5b) where R αβ is the (moduli-space) Ricci curvature which reads R αβ = −∂ α ∂β ln det G γδ .(A6) Note that Eq. (A5b) provides a geometric computation of STrM 2 which can be employed as an consistency check for the correctness of a direct computation via the extraction of the particle spectrum by applying Eq. (A5a). The factor N − 1 in the first term of Eq. (A5b) reflects the fact that we obtain one fermion with spin 1/2 less than the number N of the chiral multiplets. This is because one such fermion, known as goldstino, is absorbed by the gravitino ( G) with spin 3/2 according to the super-Higgs mechanism [1]. The G mass squared is evaluated as follows m 2 3/2 = e G = 1 3 G αβ F α F * β − V ,(A7) where the F terms are defined as [27] F α := e G/2 K αβ Gβ and F * ᾱ := e G/2 Kᾱ β G β . (A8) In our work we compute also the elements of M 1/2 , i.e., the masses of the (canonically normalized) chiral fermions, Z α , which can be found applying the formula m αβ = m 3/2 G αβ + (1 − 2/U )G α G β G αᾱ G ββ −1/2 ,(A9) where G αβ is defined in terms of the Kähler-covariant derivative D α as G αβ := D α G β = ∂ α G β − Γ γ αβ G γ ,(A10) with Γ γ αβ = K γγ ∂ α K βγ and U takes into account a possible non-vanishing V , i.e., U = 3 + V /m 2 3/2 .(A11) In Eq. (A9) care is taken so as to canonically normalize the various fields and remove the mass mixing between G and fields with spin 1/2 in the SUGRA lagrangian. Let us, finally, note that Eq. (A5b) can be significantly simplified for N = 1 since it can be brought into the form STrM 2 = 2m 2 3/2 G −2 ZZ * G Z G Z * R ZZ * = 2m 2 3/2 (3 + V /m 2 3/2 )R ,(A12) where we make use of Eq. (A2) and the definition of the scalar curvature R which is R = G αβ R αβ .(A13) Note that the first term of Eq. (A5b) vanishes for N = 1 due to the super-Higgs effect. APPENDIX B: HALF-PLANE PARAMETRIZATION In this Appendix we employ the half-plane parametrization of the hyperbolic geometry which allows us to compare our results in Sec. IV A with those established in Ref. [11,12]. The transformation from the disc coordinates Z and Z * , utilized in Sec. IV A, to the new ones T and T * is performed [8,24,26] via the replacement Z = − √ n T − 1/2 T + 1/2 with Re(T ) > 0.(B1) The last restriction -from which the name of the T − T * coordinates -is compatible with Eq. (42) for K = K − . Inserting Eq. (B1) into Eqs. (41) and (46), K − and W 0− may be expressed in terms of T and T * as follows K − = −n ln T + T * (T + 1/2)(T * + 1/2) (B2a) and W ± 0− = (2T ) n∓ (T + 1/2) −n ,(B2b) where we fix k = 0 in Eq. (41), define the exponents n ± = 1 2 n ± √ 3n (B3) and take into account the identity atnh Z √ n = 1 2 ln √ n + Z √ n − Z . (B4) Performing a Kähler transformation as in Eq. (21) with Λ K = −n ln (T + 1/2) (B5) we can show that the model described by Eqs. (B2a) and (B2b) is equivalent to a model relied on the following ingredients K − = −n ln T +T and W ± − = m(2T ) n∓ .(B6) We reveal the celebrated K and W analyzed in Ref. [11,12]. Contrary to the solutions proposed in Eqs. (13) and (47), the presence of the exponents in Eq. (B6) may require some special attention from the point of view of holomorphicity [11,12]. Considering, though, W ± − as an effective W , valid close to the non-zero vacuum of the theory, any value of n ± is, in principle, acceptable. Trying to achieve locally stable dS/AdS vacua with stabilized T we concentrate on the following K K − = −n ln Ω, where the argument of ln is introduced as Ω = T + T * + k 2 T 4 v /n with T v = T + T * − √ 2v. (B7b) As regards W , this can be generated by interconnecting the two parts in Eq. (B6). Namely, we define W Λ− = C + W + 0− − C − W − 0− = m(2T ) n+ C − T ,(B8) where the last short expression is achieved thanks to the new C symbols defined as C ± T := C + ± C − (2T ) − √ 3n .(B9) The resulting SUGRA potential V − , obtained after replacing Eqs. (B7a) and (B8) into Eq. (A2), is found to be V − = (m/2) 2 Ω −n \|2T \| 2n+ − 12\|C − T \| 2 + n Ω 2 · √ 3C + T + √ nC − T /T − 2 √ n(1 + 4k 2 T 3 v /n)C − T Ω −1 2 · n 1 + 4k 2 T 3 v /n 2 − 12k 2 T 2 v Ω −1 .(B10) For the directions in Eqs. (10a) and (10b) -with Z replaced by T -we obtain dS/AdS vacua since V − = V Λ given in Eq. (6). In addition, the conditions in Eq. (A3) for V = V − are satisfied after imposing n > 3. This is, because the sgoldstino components (t andt) -appearing by the decomposition of T as in Eq. (17) -acquire masses squared m 2 t = 288 √ 2k 2 n −1 v 3 C + T /C − T 2 m 2 3/2 ; (B11a) m 2 t = 4 1 − (3/n) C + T /C − T 2 m 2 3/2 . (B11b) Note that the expression for mt coincides with that for mz in Eq. (51b) for K = K − if we replace C ± T with C ± u− . As in that case, to ensure m 2 t > 0 we have to impose the aforementioned lower bound on n. Otherwise, an extra term of the form k(T − T * ) 4 [11,12] R T − = 24 √ 2k 2 v 3 /n − 1 /v 2 .(B14) Adopting the superpotential in Eq. (88) for the visiblesector fields Φ a and employing for simplicity C − ≃ 0 we below find the resulting SSB parameters. To this end, we identify K H in Eqs. (89a) and (89b) with K − in Eq. (B7a) and so we obtain the corresponding K 1HO and K 2HO . On the other hand, K 3HO in Eq. (89c) may be replaced with the following K 3HO = −n ln Ω − α \|Φ α \| 2 /n . For low Φ α values, the K HO 's above reduce to that shown in Eq. (90a), with K H being identified as K H = 1 for K HO = K 1HO , K 2HO ; Ω −1 for K HO = K 3HO . (B16) Using the standard formalism [5], we extract the following SSB masses squared h and µ = √ 2v m 2 α = m 2 3/2 for K HO = K 1HO , K 2HO ; (1 − 3/n) m 2 3/2 for K HO = K 3HO ,(B17a) (2−n)/2 µ . For K HO = K 3HO and n = 3 we recover the standard no-scale SSB terms as regards m α and A [8,13] but not for B -cf. Ref. [22]. The reason is that here W in Eq. (B8) is not constant as in the original no-scale models and this fact modifies the resulting F T which includes derivation of W w.r.t T . Comparing the above results with those in Eqs. (95) and (96) we remark that the expressions for m α are exactly the same. Extensions of the present model including more than one goldstini and also matter fields are extensively investigated in Ref. [11,12]. FIG. 1 : 1The (dimensionless) SUGRA potential 10 2 V f /m 2 m 2 P in Eq. (16) as a function of z andz in Eq. (17) for the inputs shown in column A of FIG. 2 : 2The (dimensionless) SUGRA potential V 2f /m 2 m 2 P in Eq. (37) as a function of z1 and z2 in Eq. (17) forz1 =z2 = 0 and the inputs shown in column B of = K ± respectively. The corresponding m 3/2 according to Eq. (A7) -with K and W given in Eqs. (41) and (49) -is FIG. 3 : 3The (dimensionless) SUGRA potential V /m 2 m 2 P as a function of z forz = 0 and the settings in columns A (solid line), C (dotdashed line) and D (dashed line) of Table I. The value z = mP is also indicated. Our analytic results are exemplified in Fig. 3, where we depict V + (dot-dashed line) V − (dashed line) together with V f (solid line) versus z forz = 0, k = k 1 = 0.3 and the other parameters shown in columns A, C and D of √ ni) and u j = e √ 3nj atnh(Zj / √ nj ) . (69) Upon substitution of Eq. (68) into Eq. (64) we obtain FIG. 4 : 4The (dimensionless) SUGRA potential V +− /m 2 m 2 P in Eq. (74) as a function of z1 and z2 in Eq. (17) forz1 =z2 = 0 and the inputs shown in column E ofTable I. The location of the dS vacuum in Eq. (26b) is also depicted by a thick black point. FIG. 5 : 5The (dimensionless) SUGRA potential V f− /m 2 m 2 P in Eq. (85) as a function of z1 and z2 forz1 =z2 = 0 -see Eq. (17)and the inputs shown in column F of Table I. The location of the dS vacuum in Eq. (26b) is also depicted by a thick black point. Once again, we infer that Eqs. (26a) and (26b) consist dS/AdS vacua since V N f± = V Λ -see Eq. (6) -and Eq. (A3) with V = V N f± and α = ℓ, i, j is fulfilled if we take into account the restrictions in Eqs. ( From Eqs. (A4) and (A9) with α = ℓ, i, j, we can obtain the mass spectrum of the present model which includes 2N t real scalars and N t − 1 Weyl fermions with masses mz α = m 3/2 where α = 1, ..., N t − 1. The masses squared of the 2N f scalars are given in the C − → 0 limit by Eqs. (38a) and (38b) for C + F /C − F = 1 and m 3/2F replaced by m 3/2F U . In the same limit the masses squared of the 2N ± scalars are given by Eq. (75a) -(75d) for C + U /C − U = 1 and m 3/2U replaced by m 3/2F U . where the rescaled parameters are h = e KH /2 K H −3/2 h and µ = e KH /2 K H −1 µ (92) added in Eq. (B7b) may facilitate the stabilization for lower n values. The expressions above contain the G mass be determined after inserting Eqs. (B7a) and (B8) into Eq. (A7). Upon substitution of the the mass spectrum above into Eq. the expression of Eq. (A12) given that R from Eq. (A13) is TABLE I : IA Case Study Overview CASE: TABLE II : IISSB Parameters: a Case Study for the Inputs ofTable I.INPUT SETTINGS SSB PARAMETERS IN GeV WH KH KH mα \|A\| \|B\| W Λf K f 1 170 208 378 W Λ+ K + 1 145 177 32 W Λ+ K + 1/Ω + 290 711 388 W Λ− K − 1 179 220 40 W Λ− K − 1/Ω − 90 55 69 HO = K 1HO and K 2HO ; 3/n(2 − n) − 1 for K HO = K 3HO .(B17c) To reach the results above we take into account the auxiliary expressions for K HO = K 1HO and K 2HO . 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[ "Long Exciton Dephasing Time and Coherent Phonon Coupling in CsPbBr 2 Cl Perovskite Nanocrystals", "Long Exciton Dephasing Time and Coherent Phonon Coupling in CsPbBr 2 Cl Perovskite Nanocrystals" ]	[ "Michael A Becker \nIBM Research−Zurich\nSäumerstrasse 48803RüschlikonSwitzerland\n\nOptical Materials Engineering Laboratory\nETH Zürich\n8092ZürichSwitzerland\n", "Lorenzo Scarpelli \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom\n", "Georgian Nedelcu \nInstitute of Inorganic Chemistry\nDepartment of Chemistry and Applied Bioscience\nETH Zürich\n8093ZürichSwitzerland\n\nLaboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology\n8600DübendorfSwitzerland\n", "Gabriele Rainò \nInstitute of Inorganic Chemistry\nDepartment of Chemistry and Applied Bioscience\nETH Zürich\n8093ZürichSwitzerland\n\nLaboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology\n8600DübendorfSwitzerland\n", "Francesco Masia \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom\n", "Paola Borri \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom\n\nCardiff University School of Biosciences\nMuseum AvenueCF10 3AXCardiffUnited Kingdom\n", "Thilo Stöferle \nIBM Research−Zurich\nSäumerstrasse 48803RüschlikonSwitzerland\n", "Maksym V Kovalenko \nInstitute of Inorganic Chemistry\nDepartment of Chemistry and Applied Bioscience\nETH Zürich\n8093ZürichSwitzerland\n\nLaboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology\n8600DübendorfSwitzerland\n", "Wolfgang Langbein \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom\n", "Rainer F Mahrt \nIBM Research−Zurich\nSäumerstrasse 48803RüschlikonSwitzerland\n", "\nAUTHOR ADDRESS\n\n" ]	[ "IBM Research−Zurich\nSäumerstrasse 48803RüschlikonSwitzerland", "Optical Materials Engineering Laboratory\nETH Zürich\n8092ZürichSwitzerland", "School of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom", "Institute of Inorganic Chemistry\nDepartment of Chemistry and Applied Bioscience\nETH Zürich\n8093ZürichSwitzerland", "Laboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology\n8600DübendorfSwitzerland", "Institute of Inorganic Chemistry\nDepartment of Chemistry and Applied Bioscience\nETH Zürich\n8093ZürichSwitzerland", "Laboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology\n8600DübendorfSwitzerland", "School of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom", "School of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom", "Cardiff University School of Biosciences\nMuseum AvenueCF10 3AXCardiffUnited Kingdom", "IBM Research−Zurich\nSäumerstrasse 48803RüschlikonSwitzerland", "Institute of Inorganic Chemistry\nDepartment of Chemistry and Applied Bioscience\nETH Zürich\n8093ZürichSwitzerland", "Laboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology\n8600DübendorfSwitzerland", "School of Physics and Astronomy\nCardiff University\nThe ParadeCF243AACardiffUnited Kingdom", "IBM Research−Zurich\nSäumerstrasse 48803RüschlikonSwitzerland", "AUTHOR ADDRESS\n" ]	[]	KEYWORDSPerovskite, nanocrystals, quantum dots, four wave mixing, photon echo, coherence, T2, phonons, lead halide, heterodyne detection 2 ABSTRACT Fully-inorganic cesium lead halide perovskite nanocrystals (NCs) have shown to exhibit outstanding optical properties such as wide spectral tunability, high quantum yield, high oscillator strength as well as blinking-free single photon emission and low spectral diffusion. Here, we report measurements of the coherent and incoherent exciton dynamics on the 100 fs to 10 ns timescale, determining dephasing and density decay rates in these NCs. The experiments are performed on CsPbBr2Cl NCs using transient resonant three-pulse four-wave mixing (FWM) in heterodyne detection at temperatures ranging from 5 K to 50 K. We found a low-temperature exciton dephasing time of 24.5±1.0 ps, inferred from the decay of the photon-echo amplitude at 5 K, corresponding to a homogeneous linewidth (FWHM) of 54±5 eV. Furthermore, oscillations in the photon-echo signal on a picosecond timescale are observed and attributed to coherent coupling of the exciton to a quantized phonon mode with 3.45 meV energy. TEXT Recently, a new type of colloidal nanocrystals (NCs) has emerged for opto-electronic applications which combines simplicity in synthesis with great spectral flexibility and exceptional optical properties. Fully inorganic cesium lead halide perovskite NCs (CsPbX3, where X = Cl, Br, I or mixture thereof) 1,2 have shown outstanding optical properties such as wide spectral tunability and high oscillator strength. 3 These NCs can be synthesized with precise compositional and size control, and show room-temperature photoluminescence (PL) quantum yields (QY) of 60 -90% (ref. 1). Moreover, perovskite NCs have attracted a lot of interest due to their large absorption coefficient and gain for optically pumped lasing devices. 4 At cryogenic temperatures, the PL decay is mostly radiative with lifetimes in the few hundred picosecond range, depending on size and	10.1021/acs.nanolett.8b03027	[ "https://arxiv.org/pdf/1808.06366v1.pdf" ]	53,212,192	1808.06366	f29c119a0e5a12361fa2edea1a8b04776404fa95	Long Exciton Dephasing Time and Coherent Phonon Coupling in CsPbBr 2 Cl Perovskite Nanocrystals Michael A Becker IBM Research−Zurich Säumerstrasse 48803RüschlikonSwitzerland Optical Materials Engineering Laboratory ETH Zürich 8092ZürichSwitzerland Lorenzo Scarpelli School of Physics and Astronomy Cardiff University The ParadeCF243AACardiffUnited Kingdom Georgian Nedelcu Institute of Inorganic Chemistry Department of Chemistry and Applied Bioscience ETH Zürich 8093ZürichSwitzerland Laboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology 8600DübendorfSwitzerland Gabriele Rainò Institute of Inorganic Chemistry Department of Chemistry and Applied Bioscience ETH Zürich 8093ZürichSwitzerland Laboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology 8600DübendorfSwitzerland Francesco Masia School of Physics and Astronomy Cardiff University The ParadeCF243AACardiffUnited Kingdom Paola Borri School of Physics and Astronomy Cardiff University The ParadeCF243AACardiffUnited Kingdom Cardiff University School of Biosciences Museum AvenueCF10 3AXCardiffUnited Kingdom Thilo Stöferle IBM Research−Zurich Säumerstrasse 48803RüschlikonSwitzerland Maksym V Kovalenko Institute of Inorganic Chemistry Department of Chemistry and Applied Bioscience ETH Zürich 8093ZürichSwitzerland Laboratory of Thin Films and Photovoltaics, Empa -Swiss Federal Laboratories for Materials Science and Technology 8600DübendorfSwitzerland Wolfgang Langbein School of Physics and Astronomy Cardiff University The ParadeCF243AACardiffUnited Kingdom Rainer F Mahrt IBM Research−Zurich Säumerstrasse 48803RüschlikonSwitzerland AUTHOR ADDRESS Long Exciton Dephasing Time and Coherent Phonon Coupling in CsPbBr 2 Cl Perovskite Nanocrystals 1Perovskitenanocrystalsquantum dotsfour wave mixingphoton echocoherenceT2phononslead halideheterodyne detection KEYWORDSPerovskite, nanocrystals, quantum dots, four wave mixing, photon echo, coherence, T2, phonons, lead halide, heterodyne detection 2 ABSTRACT Fully-inorganic cesium lead halide perovskite nanocrystals (NCs) have shown to exhibit outstanding optical properties such as wide spectral tunability, high quantum yield, high oscillator strength as well as blinking-free single photon emission and low spectral diffusion. Here, we report measurements of the coherent and incoherent exciton dynamics on the 100 fs to 10 ns timescale, determining dephasing and density decay rates in these NCs. The experiments are performed on CsPbBr2Cl NCs using transient resonant three-pulse four-wave mixing (FWM) in heterodyne detection at temperatures ranging from 5 K to 50 K. We found a low-temperature exciton dephasing time of 24.5±1.0 ps, inferred from the decay of the photon-echo amplitude at 5 K, corresponding to a homogeneous linewidth (FWHM) of 54±5 eV. Furthermore, oscillations in the photon-echo signal on a picosecond timescale are observed and attributed to coherent coupling of the exciton to a quantized phonon mode with 3.45 meV energy. TEXT Recently, a new type of colloidal nanocrystals (NCs) has emerged for opto-electronic applications which combines simplicity in synthesis with great spectral flexibility and exceptional optical properties. Fully inorganic cesium lead halide perovskite NCs (CsPbX3, where X = Cl, Br, I or mixture thereof) 1,2 have shown outstanding optical properties such as wide spectral tunability and high oscillator strength. 3 These NCs can be synthesized with precise compositional and size control, and show room-temperature photoluminescence (PL) quantum yields (QY) of 60 -90% (ref. 1). Moreover, perovskite NCs have attracted a lot of interest due to their large absorption coefficient and gain for optically pumped lasing devices. 4 At cryogenic temperatures, the PL decay is mostly radiative with lifetimes in the few hundred picosecond range, depending on size and composition of the NCs. This results from a "giant oscillator strength" in the intermediate confinement regime with an exciton Bohr radius of 5 -7 nm for cesium lead bromide-chloride (CsPbBr2Cl) NCs and a bright lowest triplet state manifold 5 . Furthermore, almost blinking-free single photon emission and marginal spectral diffusion have been reported for CsPbX3 quantum dots at low temperature. 3,6 These remarkable features make perovskite-type lead halide NCs a prime candidate for the observation of strong light-matter interaction, e.g., showing coherent cooperative emission 7 or creating exciton-polaritons by embedding them in high-finesse optical cavities, as shown for CVD-grown fully-inorganic lead halide perovskite nanowires 8,9 and nanoplatelets. 10 However, the timescale of such coherent coupling is limited by the exciton dephasing, which is still unknown for this material. Transient four-wave-mixing (TFWM) spectroscopy is a powerful method allowing to directly measure the loss of quantum coherence characterized by a dephasing time T2. It has been applied on various materials such as ruby crystals 11 , atoms 12 , molecules 13,14 and semiconductor nanostructures. [15][16][17] In general, the dephasing of excitons in semiconductor materials is caused by elastic and inelastic scattering processes with phonons and charge carriers, and by radiative population decay. 18 We studied the exciton dephasing and population dynamics using three-pulse degenerate TFWM spectroscopy on an ensemble of CsPbBr2Cl NCs. Previous measurements on colloidal CdSe-based NCs 19,20 and nanoplatelets 21 revealed a strong dependence of the 2 time on the material and its shape and size. The investigated cubic CsPbBr2Cl NCs with edge lengths of 10 ± 1 nm were synthesized as discussed in the Supplementary Information (SI), and possess the 3D-perovskite orthorhombic crystal structure (Pnma space group), shown in Figure 1a. Single quantum dot (QD) spectroscopy at cryogenic temperatures revealed that the emission of individual CsPbBr2Cl NCs exhibits a PL full-width at half-maximum (FWHM) below 1 meV, and a fine structure with an average energy splitting around 1 meV. 3,22 The exciton decay, measured at 5 K using non-resonant excitation, is mostly radiative with decay times of 180 -250 ps. This is 1000 times faster than in CdSe/ZnS QDs 23 at cryogenic temperatures, and attributed to high oscillator strength due to larger exciton coherence volume, and the absence of a low-energy dark state. 5 We performed TFWM experiments on films prepared by drop-casting a solution of NCs and polystyrene in toluene on c-cut quartz substrates (see SI). At room-temperature, the PL emission (see Figure 1c) is centered at a photon energy of 2.63 eV, and exhibits a Stokes shift of about 70 meV with respect to the ground-state exciton absorption resonance. At 5 K, the PL emission redshifts to 2.54 eV, which is a known feature of lead-based semiconductor NCs like PbS and PbSe 24 , and the PL FWHM decreases from 85 meV to 20 meV. The TFWM experiments have been performed by resonant excitation of the NCs at 2.6 eV with femtosecond pulses (120 fs intensity FWHM) from the second harmonic of a Ti:Sapphire oscillator with 76 MHz repetition rate (for details of the experimental setup see Naeem et al.,ref. 21). The first excitation pulse (P1) with wavevector 1 induces a coherent polarization of the emitters in the inhomogeneous sample, which is then subject to dephasing. After a time delay 12 , a second pulse (P2) converts the polarization into a population density grating. The third pulse (P3), that arrives on the sample after a time delay 23 , is diffracted by the density grating, creating a FWM signal with a wavevector of = 3 + 2 − 1 (refs. 20,21). To investigate the exciton population dynamics, we set the time delay between the first and the second pulse to zero ( 12 = 0 ps), and measure the FWM signal as a function of the time delay 23 . We use a spatial selection geometry to suppress the transmitted excitation pulses, and then further discriminate the FWM signal from the exciting pulses using a heterodyne technique, in which the pulse train Pi is radio-frequency shifted by i (i=1, 2, 3), resulting in a frequency-shifted FWM field which is detected by its interference with a reference pulse. 21 In Figure 2a, the measured FWM field amplitude (black) and phase (blue) at 5 K with their respective fits are shown. We fit amplitude and phase with a bi-exponential response function to quantify the population dynamics with decay rates Γ 1 > Γ 2 , as explained in the SI. Superimposed onto the population decay, we observe in the initial dynamics damped oscillations with a period of about 1.2 ± 0.1 ps, which we interpret as coherent phonon interactions, as we will discuss in more detail further below. At 5 K, the FWM field amplitude decays with two distinct time constants. The fast decay time 1 = 28.2 ± 0.8 ps with a relative amplitude of 1 1 + 2 = 0.89 ± 0.07 (see Figure S2 in the SI) corresponds to a decay rate Γ 1 = 1 1 = 35.5 ± 1.0 ns -1 , i.e. a linewidth of ℏΓ 1 = 23.4±2 µeV ( Figure 2b). This rate is independent of excitation power (see Figure S3 in the SI), and we note that the decay rate is higher compared to non-resonantly excited PL. 5 The second decay component is four orders of magnitude slower, Γ 2 = 0.037 ± 0.009 ns -1 at 5 K, which is below the repetition rate in the experiment, and has a low weight of about 10% of the first one. In contrast to the first decay component, Γ 2 is increasing with temperature, as shown in Figure 2c. We assign this component to trap or defect states present in a small fraction of the NCs, increasing their decay rate with temperature by thermal activation. The relative amplitude of the components is temperature independent within error (see SI). The highest excitation density is estimated to excite up to 0.08 excitons per excitation pulse per NC, ruling out significant multiexciton effects. The dephasing time can be extracted from the decay of the photon echo, which we measured using three-pulse FWM spectroscopy in a heterodyne detection scheme (see SI). We scan the time delay 12 between the first and the second pulse, while choosing a positive time delay 23 = 1 ps to avoid instantaneous non-resonant non-linearities. 21 The photon echo is then emitted at time 12 after the third pulse P 3 , as depicted in the inset of Figure 3a. The time-integrated FWM field amplitude as a function of delay time 12 is shown in Figure 3a for various temperatures. The FWM amplitude shows a bi-exponential behavior up to 41 K. For higher temperatures, it can be described with a fast mono-exponential decay. The initial fast decay of the amplitude proportional to exp (−2 1 12 ) in the time domain corresponds to a linewidth 2ℏ 1 = 4.37 ± 0.16 meV at 5 K. It is attributed to phonon-assisted transitions and a quantum beat of the fine-structure split states, which show a distribution of splittings in the meV energy range as they vary from NC to NC. Since the exciton-exciton interaction is significant between any of the states, the density gratings of the states are adding up, constructively interfering at 12 = 0, and the beat with a wide distribution of frequencies results in a decay over the timescale of the inverse splitting. Assuming we excite the three bright states uniformly, a decay of the signal by a factor of three over the timescale of about a picosecond, given by the inverse energy splitting, is expected. Additionally, phonon-assisted transitions will contribute, as observed in other 3D confined systems. 19,20,25 However, the large extension of the excitons inside the individual NCs, which is the origin of the giant oscillator strength, is reducing their weight in the signal. This is confirmed in low-temperature single NC PL spectra, which do not show significant phonon-assisted emission, with an estimated zero phonon line (ZPL) weight of 0.93 (see SI). At 5 K, we find a linewidth of 2ℏ 1 = 4.37 ± 0.16 meV, which increases slowly as a function of temperature ( Figure 3b). In general, the homogeneous linewidth of each fine-structure transition in the spectral domain is composed of a broad acoustic phonon band that corresponds to the fast initial dephasing, which is superimposed on a sharp Lorentzianshaped ZPL, corresponding to the long exponential dephasing in the time domain. 25 From the second decay component of the photon echo, we can therefore deduce the ZPL width 2ℏ 2 = 54 ± 5 μeV, corresponding to a dephasing time 2 = 24.5 ± 1.0 ps at 5 K. In PL measurements of single NCs at 5 K, linewidths of typically a few hundred μeV are found for CsPbBr2Cl NCs 3 (see SI). The value obtained by FWM is consistent with this, considering that spectral diffusion is typically affecting single NC PL. 26 The temperature dependence of the homogeneous linewidth is plotted in Figure 3c. The solid red line is a temperature-activated fit of the ZPL width using constant up to 10 K and then starts to decrease with increasing temperature. As discussed above we attribute the initial decay of the photon echo mostly to an overdamped beat between the finestructure states, so that the reported nominal ZPL weight is lower than the real ZPL weight of the individual bright transitions. For equal weights of three transitions in the fine-structure beat, the amplitude will decay by a factor of three. Taking into account this decay results in a ZPL weight of 0.63 at 5 K. If we furthermore assume that the upper two fine-structure states are dephasing fast due to phonon-assisted transitions to the lowest state, there is an additional decay by a factor of three. Taking all these corrections into account results in a ZPL weight of 0.91 at 5 K, as is shown in Figure 3d on the right y-axis. This weight is consistent with the ZPL weight 0.93 extracted from single NC PL (see SI). The overall temperature dependence of the corrected ZPL weight is similar to the one observed for epitaxial InGaAs QDs. 27 In CdSe colloidal quantum dots 19 Figure 4b as a function of time delay 12 , exhibiting an oscillating behavior. The red curve displays a fit with a damped squared-sine function. From this, an oscillation period of 1.22 ± 0.02 ps is obtained, which concurs with the period of the initial oscillations in the density decay. We therefore attribute these oscillations to coherent exciton-phonon coupling, resulting from the modulated polarization as a function of the harmonic nuclei displacement, as previously reported for CdSe 28,29 and PbS 30 NCs. The resulting vibrational energy of 3.45 ± 0.14 meV is in good agreement with the measured phonon energies of phonon-assisted transitions in single NC PL measurements (see SI), which have been attributed to TO-phonon modes in bulk CsPbCl3 31 and for individual CsPbBr3 NCs. 22 The oscillation damping could be due to decay of the phonon mode decay into acoustic phonons 29 and inhomogeneous broadening of the phonon mode in the QD ensemble, as observed in FAPbBr3 NCs. 32 The three-pulse photon-echo signal ( 12 , 13 ) in the direction = − 1 + 2 + 3 of an inhomogeneously broadened two-level system coupled to a single harmonic mode of angular frequency can be calculated as: 29,33 ( 12 , 13 ) ∼ exp[−4Δ 2 ( ( ) + 1) ⋅ (1 − cos ( 12 ))(1 − cos ( 13 ) For better comparison with experiment, the calculated photon-echo signals for the experimental time delays are plotted in Figure 4d and Figure S5 in the SI, from which good agreement with measurements can be inferred. In conclusion, we have investigated the coherence and density dynamics in fully inorganic CsPbBr2Cl NCs at cryogenic temperatures. Using three-beam FWM, we obtain a dephasing time and a density decay time of several tens of picoseconds at 5 K. Furthermore, we find excitation of Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes The authors declare no competing financial interest. 1 ) was swiftly injected. The reaction was stopped after 10 seconds by immersing the flask into a water-bath. The solution was centrifuged (4 min, 13,750 g) and the supernatant discarded. Hexane (0.3 ml) was added to the precipitate to disperse the NCs, and the mixture was then centrifuged again. The supernatant was collected separately, and 0.9 mL toluene was added. The NCs were precipitated by adding 0.24 mL acetonitrile and centrifuged (3 min, 6,740 g). The obtained precipitate was re-dispersed in 2 mL toluene and filtrated. For the sample preparation, we added 5 m% polystyrene in toluene in a 1:2 ratio to the solution of nanocrystals and immediately drop-casted on c-cut quartz substrates. The film had a thickness of 17 μm, and showed a certain degree of agglomeration, as can be seen in Figure S1. However, scattering was not an issue in the alignment of the box geometry in threepulse FWM (see SI: Transient Four-Wave Mixing and Heterodyne Detection). Figure S1. Fluorescence (left, excitation 410 -440nm) and transmission (right, condenser with numerical aperture NA=0.8) microscope image using a 100× NA=1.3 oil immersion objective. Transient Four-Wave Mixing and Heterodyne Detection The homogeneous linewidth of the spectral absorption is inversely proportional to the dephasing time 2 . In an ensemble measurement, the absorption linewidth is inhomogeneously broadened, thus making it impossible to deduce the microscopic 2 time from the absorption lineshape. Here, we make use of the third-order non-linearity of a material and perform four-wave mixing in the transient coherent domain after pulsed excitation to measure the 2 time in presence of inhomogeneous broadening. For the heterodyne detection scheme, the pulse train of the excitation laser is divided into the excitation pulses and a reference pulse. In the transient degenerate threebeam FWM configuration, three excitation pulses resonant to the absorption of CsPbBr2Cl ( = 2.6 eV) with a repetition rate Ω 2 = 76.11 MHz are used. To distinguish between the different non-linear orders, a so-called box geometry of the three excitation beams is used. According to the phase-matching conditions (i.e., momentum conservation), the third-order nonlinear signal is emitted in the direction = 3 + 2 − 1 , and is spatially selected by means of an iris. The signal is detected in a balanced heterodyne detection scheme, where the signal interferes with a reference beam at a beam-splitter, and the intensities of the two resulting beams are measured with two photodiodes. The photo-current is proportional to the square of the interfering incoming complex fields of the reference and signal pulse train. Furthermore, a frequency selection scheme allows to discriminate the order of the non-linear polarization. Hereby, the optical frequencies are slightly shifted by radio-frequency amounts Ω using acousto-optic modulators. Using pulse trains that exhibit controlled phase variations given by − Ω , the FWM signal can be detected at the frequency Ω = Ω 3 + Ω 2 − Ω 1 − Ω using lock-in amplifiers. This frequency selection scheme together with the interferometric detection of the FWM field amplitude constitutes the heterodyne detection scheme. For further information about the heterodyne detection scheme and the optical setup, we refer to refs. 21,35 . Single CsPbBr2Cl NC spectroscopy: Fine-structure splitting and phonon replica. Knowing the emission properties of single NCs helps to interpret the results of the FWM experiments on ensembles of CsPbBr2Cl NCs. Single NC spectroscopy of lead halide perovskite nanocrystals reveals a bright triplet emission with three orthogonal fine-structure states. On average, the fine-structure splitting for three emission peaks is Δ = 1.15 ± 0.26 meV with a large distribution ranging from several hundred eV up to meV. 5 In Figure S1, a spectrum of a single CsPbBr2Cl NC at 5 K is shown (see ref. 5 for details on the experiment) that exhibits two emission peaks with a fine-structure splitting of 2 meV. Additionally, we observe two phonon replica, red-shifted by 3.1 and 6.6 meV with Huang-Rhys factors of 0.038 and 0.033, respectively. In literature, these phonon replica were attributed to TO-phonon replica. 22 Figure S2. Single CsPbBr2Cl NC PL measurement. The nanocrystal exhibits a fine structure with two emission peaks split by Δ = 2.0 meV. Furthermore, the phonon replica at Δ 1 = 3.1 meV and Δ 2 = 6.6 meV are observed with Huang-Rhys factors of 1 = 0.038 and 2 = 0.033, respectively, resulting in a ZPL weight of 0.93. Heterodyne-Detected Four-Wave Mixing: Complex Fit to the Exciton Population Dynamics For a quantitative analysis, the population decay data is fitted with the complex multi-exponential response function, as explained in detail in ref. 35 : ( ) ∼ ( ) + ∑ Θ( ) ⋅ ( − ) . Here, , and are amplitude, phase and decay time of the n-th decay process. is a nonresonant instantaneous component to account for effects like two-photon absorption and Kerr effect. The above equation describes the FWM as a coherent superposition of exponential decay components with their respective phases, given by their relative effect on absorption and refractive index. Slow drifts of the setup, for example due to room temperature changes, can affect the relative phase of the reference and probe pulses over a timescale of minutes. The fitting procedure accounts for this phase drift with a prefactor − ( 0 + 0 ′ ) , corresponding to a linear timedependence of the phase, where is the real time during the measurement. The excitation pulse is taken into account by convoluting the response with a periodic Gaussian of full-width halfmaximum 2√ln(2) 0 in amplitude, given by the laser pulse auto-correlation. Furthermore, we include a pile-up of the signal ∼ ( − 1) −1 due to the finite repetition rate of the excitation pulses of 13 ns, relevant for decay components with lifetimes similar or longer than . Note that for lifetimes similar or longer than the modulation period of 1  1 − 2 = 160 ns, the pilep-up would be saturated; a regime which is not considered explicitly in the fit formula as it is not relevant to the presented fits. In the measurements, initial oscillations in the density decay can be seen. We model this by multiplying the decay components, excluding pile-up, by (1 + − ⋅ cos( )) with amplitude , damping rate and angular frequency of the oscillations . The fit function of the density decay uses two exponential decay processes, yielding ] . Furthermore, we measure the excitation-power dependent density grating decay at 5 K, shown in Figure S3. Again, we fit the data with the above described complex fit. The resulting decay times are independent of the excitation density within error. From the complex fits, we obtain an average initial decay time 1 = 35.6 ± 1.5 ps. Figure S3. Excitation-power dependent FWM Field amplitude. In Figure 2 in the main text, the temperature dependence of the first and the second decay component is shown. In Figure S4, we additionally plot the relative amplitude of the first decay component as a function of temperature, which stays almost constant within the temperature range measured. Figure S4. Relative amplitude of the first decay component of the bi-exponential complex fit as a function of temperature. One can also use three exponential decay processes, resulting in a better fit at intermediate delay times around 100 ps. However, this leads to two time-constants in the 11 ps and 62 ps range, and is evidence for an inhomogeneous distribution of decay constants in the sample. We have therefore opted to use only two time-constants to extract the average decay, consistent with the analysis of the exciton dephasing. For a better comparison, we plot the data of Figure 4a and d again in semi-log scale next to each other in Figure S5. As discussed in the main text, we use the model proposed by Mittelmann and Schoenlein et al. 28,29 to calculate the electric fields ( 12 , 23 )~√ ( 12 , 23 ) of the three-pulse photon-echo measurement in the presence of a single phonon mode. We added a damping term exp (− 12 ) exp (− 13 ) to the oscillations, and a rise-term the scattering energy = 0.23 ± 0.09 meV, and the zero-temperature extrapolated linewidth ℏ 0 = 39.6 ± 7.1 μeV. The activation energy Δ = 1.23 ± 0.35 meV is consistent with the reported fine-structure splitting from single NC spectroscopy measurements 5 (see SI), suggesting that phonon-assisted scattering into different fine-structure states is responsible for the dephasing. InFigure 3dthe temperature dependence of the nominal ZPL weight , where 1 and 2 are the decay amplitudes of the fast and long decay component of the photon echo, respectively. The nominal ZPL weight is = 0.44 and remains Figure 4a . 4ainstead, which have a much smaller exciton coherence volume, significantly stronger phonon assisted transition are present, with a ZPL weight of about 0.7 at 5 K, decaying to 0.4 at 20 K. However, the ZPL weight for temperatures below 12 K is found to be constant, in contrast to the two above mentioned systems, which both show a continuous decrease of the ZPL weight with temperature. Notably, the phonon-assisted transitions in these other QD systems are dominated by coupling to a continuum of acoustic phonons with linear dispersion around zero momentum (Γ-point), resulting in a broad phonon-assisted band. The single QD PL of the perovskite NCs (see SI) instead shows no such broad band, but rather two well-defined phonon energies of about 3 and 6 meV. These discrete phonon energies would be expected to lead to an activated behavior of the ZPL weight reduction, consistent with the observed constant ZPL weight below 12 K.To further investigate the initial oscillations in the density dynamics inFigure 2a, we took data for different delays 12 ≥ 0. The resulting FWM field amplitude as a function of 13 is shown in The amplitude of the oscillations changes while the oscillation period remains stable at 0 = 1.20 ± 0.05 ps. The relative amplitudes B of the damped oscillations are shown in Here, is the mode frequency and ( ) = 1/[exp ( ℏ ) − 1] is the Bose occupation factor. The function Φ( 12 , 13 ) = 1 + cos (Δ 2 [sin( 12 ) − sin( 13 ) + sin( 13 − 12 )]) arises from the superposition of different Liouville space pathways.34 Δ represents the coupling factor between excitonic state and phonon mode, and can be expressed by the Huang-Rhys parameter. For further details, we refer to the work of Mittelmann and Schoenlein et al.28,29 InFigure 4c, we show the modelled electric fields of the three-pulse photon echo with ( 12 , 23 )~√ ( 12 , 23 ) according to equation(1). The model uses a Huang-Rhys parameter of 1 2 Δ 2 = 0.038, a phonon mode at 3.45 meV (as measured in single NC PL measurements, see SI) and the above measured decay rates Γ 1 and 1,2 with their corresponding amplitudes. We have included an oscillation damping ( = 2.5 ) and a finite pulse duration by multiplying ( 12 , , where 0 = 72 fs describes the amplitude FWHM using the intensity FWHM 2√ln2 0 = 120 fs of the excitation pulses. The complete formula is provided in the SI. coherent phonons of 3.45 meV energy with a Huang-Rhys parameter of 0.038. The observed long dephasing time close to the lifetime limit is promising for applications in microcavity devices based on strong light-matter interaction. AUTHOR INFORMATION Corresponding Authors () [email protected] (M.V.K.) * [email protected] (W.W.L.) * [email protected] (R.F.M.) FIGURESFigure 1 . 1Crystal structure and optical properties of CsPbBr2Cl nanocrystals. (a) Fully-inorganic perovskite cesium lead halide unit cell with typical quasi-cubic crystal structure withorthorhombic distortion. (b) Transmission electron microscopy (TEM) image of CsPbBr2Cl nanocrystals (inset: single NC with high-resolution TEM). (c) Normalized ensemble PL and absorption spectra at 295 K and 5 K. Figure 2 .Figure 3 .Figure 4 . 234Temperature-dependent population density decay. (a) FWM field amplitude (black circles) and phase (blue circles) with their respective errors (shades) as a function of time delay23 between the first two pulses and the third pulse at 5 K. The solid lines represent fits to the data using the model discussed in the text. Inset: Sketch of the three-beam pulse sequence with 12 = 0 ps.(b) Decay rates 1 and (c) 2 of the two exponential decays extracted from the complex fit as function of temperature. Temperature-dependent photon echo. (a) Time-integrated FWM field amplitude as a function of the time delay 12 between the first and the second pulse in the temperatures range 5-51 K for fixed 23 = 1 ps. Inset: Sketch of the three-beam pulse sequence and resulting photon echo. (b) Linewidth 2ℏ 1 as a function of temperature. (c) Homogeneous linewidth 2ℏ 2 as a function of temperature. The line is a weighted fit to the data. (d) Temperature dependence of the nominal ZPL weight (left y-axis) and the corrected ZPL weight (right y-axis). Coherent coupling to phonon modes. (a) Three-beam photon-echo signal as a function of 13 for various time delays 12 at 5 K. Fits to the data (circles) are represented by solid lines. (b) Relative amplitude of the damped oscillation from the fits in (a) as a function of the time delay Synthesis and sample preparation. The CsPbBr2Cl NCs were prepared by following the procedure reported by Protesescu et al., 1 with some adjustments. PbBr2 (0.122 mmol) and PbCl2 (0.0658 mmol) were loaded, inside of a glovebox, into a 3-neck flask along with pre-dried OA (1 mL), OAm (1 mL), TOP (1 mL) and ODE (5 mL). The flask was transferred to a Schlenk line and evacuated for 10 minutes at 120°C. The reaction mixture was heated up to 200°C under N2 and 0.4 mL of hot Cs-oleate (prepared as described by Protesescu et al. Coherent exciton-phonon coupling: Measurement results and theoretical model. Figure S5 . S5Left panel: Three-beam photon-echo signal as a function of τ 13 for various time delays τ 12 at 5 K (left). Fits to the data (circles) are represented by solid lines. 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[ "Enriching the quantum toolbox of ultracold molecules with Rydberg atoms", "Enriching the quantum toolbox of ultracold molecules with Rydberg atoms", "Enriching the quantum toolbox of ultracold molecules with Rydberg atoms", "Enriching the quantum toolbox of ultracold molecules with Rydberg atoms" ]	[ "Kenneth Wang \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n", "Conner P Williams \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n", "Lewis R B Picard \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n", "Norman Y Yao \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Physics\nUC Berkeley\n94720BerkeleyCaliforniaUSA\n", "Kang-Kuen Ni \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n", "Kenneth Wang \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n", "Conner P Williams \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n", "Lewis R B Picard \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n", "Norman Y Yao \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Physics\nUC Berkeley\n94720BerkeleyCaliforniaUSA\n", "Kang-Kuen Ni \nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDepartment of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA\n\nHarvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA\n" ]	[ "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Physics\nUC Berkeley\n94720BerkeleyCaliforniaUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Physics\nUC Berkeley\n94720BerkeleyCaliforniaUSA", "Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Chemistry and Chemical Biology\nHarvard University\n02138CambridgeMassachusettsUSA", "Harvard-MIT Center for Ultracold Atoms\n02138CambridgeMassachusettsUSA" ]	[]	We describe a quantum information architecture consisting of a hybrid array of optically-trapped molecules and atoms. This design leverages the large transition dipole moments of Rydberg atoms to mediate fast, high-fidelity gates between qubits encoded in coherent molecular degrees of freedom. Error channels of detuning, decay, pulse area noise, and leakage to other molecular states are discussed. The molecule-Rydberg interaction can also be used to enable nondestructive molecule detection and rotational state readout. We consider a specific near-term implementation of this scheme using NaCs molecules and Cs Rydberg atoms, showing that it is possible to implement 300 ns gates with a potential fidelity of > 99.9%. arXiv:2204.05293v2 [physics.atom-ph]	10.1103/prxquantum.3.030339	[ "https://export.arxiv.org/pdf/2204.05293v2.pdf" ]	248,085,617	2204.05293	b8e48d916c6744de8c9fd2e0a0f53763c38c180c	Enriching the quantum toolbox of ultracold molecules with Rydberg atoms Kenneth Wang Department of Physics Harvard University 02138CambridgeMassachusettsUSA Department of Chemistry and Chemical Biology Harvard University 02138CambridgeMassachusettsUSA Harvard-MIT Center for Ultracold Atoms 02138CambridgeMassachusettsUSA Conner P Williams Department of Physics Harvard University 02138CambridgeMassachusettsUSA Department of Chemistry and Chemical Biology Harvard University 02138CambridgeMassachusettsUSA Harvard-MIT Center for Ultracold Atoms 02138CambridgeMassachusettsUSA Lewis R B Picard Department of Physics Harvard University 02138CambridgeMassachusettsUSA Department of Chemistry and Chemical Biology Harvard University 02138CambridgeMassachusettsUSA Harvard-MIT Center for Ultracold Atoms 02138CambridgeMassachusettsUSA Norman Y Yao Department of Physics Harvard University 02138CambridgeMassachusettsUSA Department of Physics UC Berkeley 94720BerkeleyCaliforniaUSA Kang-Kuen Ni Department of Physics Harvard University 02138CambridgeMassachusettsUSA Department of Chemistry and Chemical Biology Harvard University 02138CambridgeMassachusettsUSA Harvard-MIT Center for Ultracold Atoms 02138CambridgeMassachusettsUSA Enriching the quantum toolbox of ultracold molecules with Rydberg atoms (Dated: June 27, 2022) We describe a quantum information architecture consisting of a hybrid array of optically-trapped molecules and atoms. This design leverages the large transition dipole moments of Rydberg atoms to mediate fast, high-fidelity gates between qubits encoded in coherent molecular degrees of freedom. Error channels of detuning, decay, pulse area noise, and leakage to other molecular states are discussed. The molecule-Rydberg interaction can also be used to enable nondestructive molecule detection and rotational state readout. We consider a specific near-term implementation of this scheme using NaCs molecules and Cs Rydberg atoms, showing that it is possible to implement 300 ns gates with a potential fidelity of > 99.9%. arXiv:2204.05293v2 [physics.atom-ph] I. INTRODUCTION Individually trapped ultracold polar molecules [1][2][3][4][5][6][7] have emerged as a promising candidate system for scalable quantum computing due to their long-lived internal states and intrinsic tunable interactions. Long coherence times have been demonstrated for many molecular degrees of freedom, including nuclear spin [8,9], rotation [10][11][12], and vibration [13]. Molecular-frame dipole moments allow molecules to interact via the dipolar interaction, which has been observed for molecular gases prepared in opposite parity rotational states [14,15]. Early proposals of two-qubit gate schemes required external fields where field stability imposes a practical constraint to their viability [16][17][18][19]. Recently, robust schemes with the potential for greater than 99.99% fidelity have been proposed. These schemes directly take advantage of the intrinsic dipolar interaction between two field-free molecular rotors, using a dipolar exchange [20,21] or energy shifts created by the interaction [22]. However, the millisecond gate times in these schemes are long compared to what has been realized in superconducting [23], trapped ion [24], or trapped atom systems [25,26]. A path to achieving molecule dipolar interaction strengths larger than a kHz by reducing molecular separation to below the trap light wavelength in an optical tweezer system has been outlined, but is technically demanding [27]. Another challenge of using molecular systems for scalable quantum computing is state detection and measurement. Most molecules do not have a closed optical cycling transition, with the exception of a special set of molecules [3,28,29], making conventional fluorescence or absorption imaging techniques difficult for single-molecule detection [30]. A general route to overcoming such a challenge is to perform indirect detection * To whom correspondence should be addressed: [email protected] † [email protected] through state-sensitive coupling of a molecule to another quantum system such as an atom [31][32][33][34][35] or optical cavity [36] which can then be optically detected. Building upon these ideas, we present an approach which uses an atom to speed up molecular two-qubit gate times by several orders of magnitude, while also enabling nondestructive state-sensitive detection of single molecules. By transferring atoms to highly excited Rydberg states, they can be made to interact with polar molecules (1-5 Debye) through the dipolar interaction. When an atomic transition is brought into resonance with a molecule rotational transition, the Rydberg atom can mediate the interaction between molecules via its large transition dipole moment (∼ 10 kDebye), which amplifies this interaction by several orders of magnitude. This amplified interaction strength can be used to implement Rydberg-mediated entangling gates between molecules. Because one of the most widely-used schemes to create ultracold molecules is association of the constituent atoms [37][38][39], atoms are a readily available resource in many molecule experiments, making this scheme feasible to implement. As a concrete example, for a system of NaCs molecules and Cs atoms in optical tweezers [40], we show that sub-microsecond two-qubit gate times can be realized with high fidelity and at an interparticle spacing of 1 micron. We also outline a molecular detection scheme and avenues to extend the system to larger arrays, leveraging the mobility of optical tweezers and the many internal states of molecules. Because of the abundance of Rydberg atom transitions in the GHz range which can be brought into resonance with molecular rotational spacings, this scheme is general for a wide variety of polar molecules. II. DRIVEN EXCHANGE GATE We first describe the fast entangling gate resulting from a Rydberg-mediated interaction between two molecules. We consider a three-particle system, consisting of two molecules with an atom placed in between them, as FIG. 1. Relevant level structure for a driven Rydberg-mediated exchange between two molecules. (a) The relevant energy levels for the molecules (left and right) and the Rydberg atom (center). The energy spacing between the computational basis states for the molecule is ωm, and the spacing between Rydberg states \|r and \|R differs from the molecule spacing by ∆. The excitation laser, which excites an atom from \|g to \|r , has Rabi frequency Ω, and its detuning is denoted by δ. (b) The separation of the three-body Hilbert space (12 states) into distinct subspaces. The blue lines indicate the Rydberg excitation laser coupling, and the green lines depict the strong Rydberg-molecule interaction. shown in Fig. 1a. To capture the essence of the gate, the molecules are treated as two-level systems (\|0 and \|1 ) which have a transition dipole moment d m between them. In the atomic system, three states (\|g , \|r and \|R ) are used, where \|g is a ground electronic state of the atom, and \|r and \|R are opposite parity Rydberg states which have a large transition dipole moment d r between them. The interaction Hamiltonian arises from the dipolar interaction [14,41] between the particles separated by an interparticle spacing of a: H int = 1 2 [V mr (σ + 1 S − + σ − 1 S + + S + σ − 2 + S − σ + 2 ) +V mm (σ + 1 σ − 2 + σ − 1 σ + 2 )](1) where σ ± i are the Pauli ladder operators for molecules in the basis {\|0 , \|1 }, and S ± are the Pauli ladder operators for the Rydberg atom in the basis {\|r , \|R }. The molecule-Rydberg interaction and the molecule-molecule interaction are given by V mr = d m d r /(4π 0 a 3 ) and V mm = d 2 m /(32π 0 a 3 ), respectively. The ground state atom, \|g , is far off-resonance, and does not participate in the exchange interaction. Furthermore, we denote the molecular energy spacing as ω m , and the difference between the Rydberg energy spacing and the molecular energy spacing as ∆. The energy spacing between the atomic ground state and Rydberg state \|r is denoted ω gr . In addition to the intrinsic Hamiltonian arising from the dipolar interaction, we add a drive of the atom from \|g to \|r with Rabi frequency Ω, and detuning δ = ω L − ω gr , where ω L is the angular frequency of the laser. There are a total of twelve states, but the Hamiltonian is block diagonal in the sectors {\|0g0 , \|0r0 }, {\|0g1 , \|0r1 , \|1g0 , \|1r0 , \|0R0 }, {\|1g1 , \|1r1 , \|0R1 , \|1R0 }, and {\|1R1 }. These sectors can be characterized by the number of dipolar excitations n exc = 1 4 (σ + 1 σ − 1 + S + S − + σ + 2 σ − 2 ).(2) Neither dipolar interaction nor driving couple between these manifolds, and the dynamics within each sector are thus independent, with Hamiltonians given by: for Ω = 2/3Vmr/ . While molecule-molecule interaction is included in the evolution, we note that dr dm, such that it only introduces a perturbation to the molecule-Rydberg interaction of magnitude < 10 −9 . H nexc=0 = 0 Ω/2 Ω/2 δ ,(3)H nexc=1 =      − δ Ω/2 V mm /2 0 0 Ω/2 0 0 V mm /2 V mr /2 V mm /2 0 − δ Ω/2 0 0 V mm /2 Ω/2 0 V mr /2 0 V mr /2 0 V mr /2 ∆     (4)H nexc=2 =    − δ Ω/2 0 0 Ω/2 0 V mr /2 V mr /2 0 V mr /2 ∆ V mm /2 0 V mr /2 V mm /2 ∆   (5) We first explore the case where the laser detuning is zero (δ = 0) and the Rydberg transition is resonant with the molecule transition (∆ = 0). The gate is performed by driving the atom from the ground state for a time T = 2π/Ω. At the end of the drive, the \|0g0 state returns to itself with a phase of −1. For this particular T , it can be shown analytically, ignoring the much smaller moleculemolecule interaction, that the entangling gate in the basis {\|0g0 , \|0g1 , \|1g0 , \|1g1 } can be realized for a specific Rabi frequency of the drive, related to the molecule-Rydberg interaction by Ω = 2 4k 2 − 1 V mr /(7) where k is an integer larger than 0. For k = 1, this corresponds to a Rabi frequency of 2 3 V mr , allowing this gate to take advantage of the fast molecule-Rydberg interaction. The dynamics of this gate in the various manifolds are shown in Fig. 2. In the limit of large k, corresponding to Ω V mr , the locations of these resonant exchange drives get closer together, indicating a scheme robust to the exact drive Rabi frequency. In this limit, for the n exc = 1 manifold, the intermediate system consisting of {\|0r1 , \|0R0 , \|1r0 } can be diagonalized, where a zero energy mode will emerge [42]. The two edge states \|1g0 and \|0g1 will then be coupled through this mode and their states can swap after a particular time of unitary evolution. In the 2 excitation manifold, a zero energy mode in the {\|0R1 , \|1R0 , \|1r1 } manifold is also created, but consists only of a combination of the \|0R1 and \|1R0 states which has no matrix element with the \|1g1 ground state. Thus, no excitation is allowed and the system remains in the ground state with no phase accumulation. In the case where Ω is of the same order as V mr , the dynamics of the driven exchange can be further elucidated by an examination of the eigenvectors and eigenvalues of the system, provided in Appendix A. The full landscape of the fidelity as a function of Ω/V mr is shown in Fig. 3a. III. IMPLEMENTATION IN NACS + CS We now consider the implementation of this gate in a system of ground state NaCs molecules and Cs Rydberg atoms. The key requirement is to find a pair of Rydberg states that match the energy gap of a dipole-allowed transition in the molecule, typically a rotational transition. In NaCs, we measure the N = 0 to N = 1 rotational energy splitting to be h × 3471.8(1)MHz. Since Rydberg states are extremely sensitive to electric fields, and to a lesser extent, magnetic fields, external fields may be used to tune these states into resonance. Formation of ground state molecules from their constituent atoms is a well-established technique that has been successful both in bulk gasses and in optical tweezers, and often relies on magnetic field control to access Feshbach resonances. B(G) Transition(\|r → \|R ) ∆(MHz) \|dr\|(D) 0 64P 1/2 → 63D 3/2 −12.6 6488 859.3 72P 3/2,3/2 → 71D 5/2,5/2 ≈ 0 11220 769.9 57P 3/2,3/2 → 56D 5/2,3/2 ≈ 0 4329 908.4 49P 3/2,3/2 → 48D 3/2,1/2 ≈ 0 1280 A Feshbach resonance at 865 G is used to form NaCs molecules [43], and at magnetic fields of this order, Rydberg states can be tuned to resonance with the rotational transition in the molecule, as shown in Table I. For the molecule, the states \|0 = \|m INa , m ICs , N, m N = \|3/2, 5/2, 0, 0 and \|1 = \|3/2, 5/2, 1, 1 are chosen to be the qubit states. To maximize dipolar interaction, we choose the resonant pair \|72P 3/2,3/2 and \|71D 5/2,5/2 as our Rydberg states in a 859.3 G magnetic field. At 1 µm separation, this state choice results in interaction strengths V mr = 2π × 4. 64 MHz, compared to the molecule-molecule interaction strength of V mm = 2π × 142 Hz. A gate time of 263ns is achieved, with a fidelity of 0.9997, when accounting for the finite lifetimes of 221 µs and 118 µs for the Rydberg states [45]. This is four orders of magnitude faster than the molecule-molecule gate time of 3.5 ms without the enhancement of the coupling via the Rydberg atom. We now analyze the sensitivity of this gate to various parameters. Since Rydberg atoms are extremely sensitive to external fields, especially electric fields, the resonance condition may not be exactly met. In addition, differential light shifts for the rotational states of the molecule due to the trap can also result in an energy shift, but this effect can be lessened using a specific choice of elliptical trap polarization [46]. Fig. 3b shows the fidelity of the gate at the time of T due to fluctuations in the Rydberg state detuning ∆, the laser detuning δ, and laser Rabi frequency Ω. The Rabi frequency is the most sensitive parameter in this gate, where for k = 1, it must be stable to 1.94% [47]. Leakage to the many hyperfine states in the N = 0 and N = 1 rotational manifolds needs to be considered. The Possible states that may be accessed from the \|0 = \|mI Na , mI Cs , N, mN = \|3/2, 5/2, 0, 0 and \|1 = \|3/2, 5/2, 1, 1 computational basis states with a σ + polarization from N = 0 to N = 1 at 859.3 G. The energies of these transitions are also given, where a positive detuning is a transition which has larger energy separation. The relative strength of these transitions are also listed. closest rotational excited states are in the same hyperfine state, but with different m N , and are separated by only a few kHz. Exchange into these states is suppressed because the dipolar interaction preserves the total magnetic quantum number [48]. Excitation into a different m N of the final molecular state would require an exchange to a different Rydberg state, which is highly off-resonant at high magnetic field. Another leakage channel is other hyperfine states in the ground and excited rotational state manifolds, which at high field are separated by 100s of kHz. Transitions to these states are allowed, since the internal molecular Hamiltonian contains coupling between the nuclear spin and rotation through the electric quadrupole moment [49,50]. However, at high magnetic fields, these transitions are suppressed, since the Zeeman term in the internal molecular Hamiltonian begins to dominate the aforementioned mixing terms. The strongest polarization-allowed couplings to the states \|0 and \|1 are enumerated in Table II along with their detuning from the primary \|0 ↔ \|1 transition. The effect of leakage into the unwanted \|3/2, 7/2, 1, 0 state caps the fidelity to 0.9996 at the current gate time [51]. Although coupling to unwanted states limits our fidelity, this coupling can also be used for a hyperfine encoding of quantum information for enlarged coherence times [8,9], similar to the previously proposed iSWAP gate scheme in molecules [20]. In particular, the hyperfine qubit states \|0 h = \|0 = \|3/2, 5/2, 0, 0 and \|1 h = \|1/2, 7/2, 0, 0 can be used, and a π pulse from \|1 h to \|1 starts the driven exchange gate. The gate then proceeds in the {\|0 h , \|1 } basis as described in section II, and finally the population from \|1 is returned to the \|1 h state with another π pulse. IV. NONDESTRUCTIVE MOLECULE DETECTION In order to use molecules as part of a scalable quantum computing platform, it is also necessary to implement reliable state preparation and measurement schemes for the molecules themselves. Furthermore, detecting the state of the molecule nondestructively and projecting it into that state is important for use in quantum error correction [52] and measurement-based quantum computing [53]. Nondestructive state-sensitive detection of molecules, however, remains a major challenge, since most molecules, including bialkalis, do not have closed cycling transitions, so direct imaging of them is difficult. To detect bialkali molecules, they are dissociated into atoms, which can then be directly imaged. This technique is sensitive to the fidelity of the dissociation process and is also destructive, so it cannot be used for rearrangement in optical tweezer systems [54][55][56][57], which is important for realizing defect-free arrays of molecules. A hybrid system of molecules and atoms that can interact suggests the potential to perform nondestructive quantum state detection of the molecules via the atoms. Bringing a Rydberg atom transition into resonance with a molecular rotational transition as discussed above will create an energy shift due to the molecule-atom interaction. Consider a pair of Rydberg states \|r and \|R that are resonant with the N = 0 to N = 1 rotational transition of a molecule, as shown in Fig. 4. In the two-body basis, the states \|0R and \|1r are coupled via the dipolar interaction and acquire an energy splitting of V mr . Thus, in the presence of a molecule in state \|0 (\|1 ), the \|g → \|R (\|r ) will be blockaded [58], where \|g is a ground state of the atom, as long as the drive power is much weaker than the interaction strength, Ω drive V mr . Using this energy shift, the state of a molecular qubit in α \|0 + β \|1 can be mapped onto the hyperfine states of the atom. Consider two hyperfine ground states of the atom \|g 0 , and \|g 1 that can be selectively read out, where only state \|g 1 is coupled to the Rydberg states. The molecule is detected by preparing the atom in state 1 √ 2 (\|g 0 + \|g 1 ), and then driving a 2π pulse from \|g 1 to \|R , resulting in the state 1 √ 2 (α \|0 ⊗ (\|g 0 − \|g 1 ) + β \|1 ⊗ (\|g 0 + \|g 1 )). A Hadamard gate can then be performed on the atom in the {\|g 0 , \|g 1 } basis to obtain the state α \|0g 1 +β \|1g 0 . A state selective atom measurement will then project the molecule into state \|0 or \|1 . Repeated measurements of this form allow for state tomography to determine the populations \|α\| 2 and \|β\| 2 . In the case where there is no molecule, this procedure results in an atom in state \|g 0 , making it indistinguishable from having a molecule in state \|1 . To distinguish these cases, the same detection procedure can be performed again, but instead using a 2π pulse from \|g 1 to \|r . If the molecule were present, this would result in the atom state \|g 1 , in contrast to \|g 0 if there were no molecule. This second measurement can be used for postselection on the data for a background-free measurement of \|β\| 2 . The logic of this procedure is summarized in Table III. V. EXTENDING TO LARGER ARRAYS Extending the scheme to larger arrays is non-trivial due to the Rydberg atom's interaction with other molecules or atoms in the array. The resonant Rydberg-Rydberg interaction, which is enhanced by a factor of d r /d m , is much stronger than the molecule-Rydberg interaction. This interaction can only be suppressed by distance, which limits the number of Rydberg atoms that may be used at the same time to entangle separate pairs of molecules. One possible method of extending to larger arrays is to utilize the mobility of an optical tweezer platform, as has been recently demonstrated for neutral atom systems [59], selectively moving molecules to interact with distant Rydberg atoms, as shown in Fig. 5a. Another limitation is the interaction between a Rydberg atom and the next-nearest neighboring molecule, which is suppressed by a distance factor. Fig. 6 shows the effect of this molecule on the gate fidelity as function of distance [60]. A more robust method to ensure that only targeted molecules can interact with the Rydberg is to FIG. 6. In extending to larger arrays, neighboring molecules will affect the gate due to a suppressed dipolar interaction. In a uniform array, fidelities of just above 0.999 can be achieved . take advantage of the many internal states of molecules, particularly other rotational states, shown schematically in Fig. 5b. For example, if the Rydberg atom is made resonant with the N = 1 to N = 2 rotational transition, molecules can be stored in the N = 0 and N = 3 states to avoid interaction with the Rydberg. It is critical for the non-interacting states to be off-resonant with the Rydberg atom, which is possible due to the unequal and large spacing between rotational levels in molecules. The exchange rate of the gate is at the MHz level, so different hyperfine states in the same rotational manifold of a 1 Σ molecule may not be used to prevent interaction, as they are only off-resonant by tens of kHz. However, polarization offers a constraint and allows hy-perfine shelving starting from the N = 1 manifold. If the Rydberg atom transition from \|r → \|R is σ + and is resonant with the N = 0 to N = 1 transition then the \|N = 1, m N = 0 and \|N = 1, m N = −1 molecule states will not interact with the Rydberg atom and can store the quantum information of non-interacting molecules. VI. CONCLUSION AND OUTLOOK By introducing Rydberg atoms into a molecular system, it is possible to realize both high fidelity sub-microsecond entangling gates and nondestructive molecule detection. The large transition dipole moments in Rydberg atoms are used to facilitate dipolar exchange between two polar molecules. The gate only requires driving the Rydberg atom with a precise strength for a specific time and is general for all polar molecules with dipole-allowed GHz scale transitions, including diatomic and polyatomic species [29,34]. We detailed an implementation of the scheme with the bialkali molecule NaCs and Cs atoms, and analyzed its sensitivities to various experimental parameters, while also taking the hyperfine structure into account. Nondestructive projective measurement of the molecules can be performed via a blockade scheme, where detection ultimately occurs on the atoms. Using the atom as an ancillary qubit to detect gate errors shows promise and warrants further investigation. When compared to purely molecular systems, the spontaneous decay and Doppler effects of the untrapped Rydberg atom limit the fidelity of gates in this scheme. These limitations are common to Rydberg-Rydberg systems, but this scheme benefits from having only one qubit subject to these loss mechanisms during each entangling gate rather than two. Other benefits of using molecular qubits include relatively long coherence times of greater than 5 seconds in hyperfine states [8], and of around 100 ms in the interacting rotational states [10]. The molecule-Rydberg gates presented here can also be natively combined with higher fidelity, but slower, molecule-molecule gates [20], depending on the application. Molecules also offer a large number of internal states to selectively interact molecules in a larger array. These internal states can also be used as qudits [61] or as lattice sites in a synthetic dimension [62]. For the case of a synthetic dimension using rotational states, the Rydberg atom can be tuned to a particular rotational resonance enhancing excitation hopping between particular sites in the synthetic dimension. Using physical displacement, the enhancement of hopping is spatially-tunable and also allows for site-dependent interactions. Introducing neutral atoms into a molecular platform adds to the toolbox of polar molecule systems, enriching their potential for quantum science applications. Note added -While completing this work, we became aware of a related work [63]. ACKNOWLEDGMENTS We thank Mikhail Lukin, Jessie Zhang, Fang Fang, and Yu Wang for stimulating discussions. This work is supported by the AFOSR-MURI grant (FA9550-20-1-0323), NSF through the Harvard-MIT CUA, and the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. K. W. is supported by an NSF GRFP fellowship. Appendix A: Driven Exchange Gate Details Here, we work out the details for the driven exchange gate and derive the formula in equation 7, analyzing the 5 state n exc = 1 manifold and 4 state n exc = 2 manifold separately. nexc = 1 Manifold Diagonalizing the Hamiltonian for this manifold in equation 4, when δ = ∆ = V mm = 0,the eigenvalues 0, Ω/2, − Ω/2, − 2 Ω 2 + 2V 2 mr /2, 2 Ω 2 + 2V 2 mr /2 are obtained, and the states of interest \|0g1 and \|1g0 can be written in the eigenbasis \|0g1 =           Vmr √ 2 Ω 2 +2V 2 mr −1/2 −1/2 Ω 2 √ 2 Ω 2 +2V 2 mr Ω 2 √ 2 Ω 2 +2V 2 mr           , \|1g0 =           Vmr √ 2 Ω 2 +2V 2 mr 1/2 1/2 Ω 2 √ 2 Ω 2 +2V 2 mr Ω 2 √ 2 Ω 2 +2V 2 mr           . (A1) These two states only differ in the sign of their second and third component. In order to get a swap between these two in the time evolution, we need the second and third components to flip sign, while keeping the other components the same. From the eigenvalues, this can be accomplished provided that the system evolve for a time T = 2π/Ω and that 2 Ω 2 + 2V 2 mr 2 = k Ω (A2) where k is an integer. Solving this equation for Ω yields the result in equation 7. nexc = 2 Manifold We also need to verify that the Ω found above also works in the n exc = 2 manifold. Diagonalizing the Hamiltonian of this manifold in equation 5 when δ = ∆ = V mm = 0, the eigenvalues 0, 0, − 2 Ω 2 + 2V 2 mr /2, 2 Ω 2 + 2V 2 mr /2 are obtained, and the state of interest \|1g1 can be written in the eigenbasis \|1g1 =         − Vmr √ 2 Ω 2 +V 2 mr 2 Ω 2 +2V 2 mr − Vmr √ 2 Ω 2 +V 2 mr 2 Ω 2 +2V 2 mr Ω √ 2 2 Ω 2 +4V 2 mr Ω √ 2 2 Ω 2 +4V 2 mr         With the constraint found in equation A2, all 4 components will remain the same, and the state is unchanged as needed. Appendix B: Details of Gate Fidelity Calculations Fidelity Definition We use the following definition for the gate fidelity [64,65] F = Tr(U i U † p /n) (B1) where U p is the gate unitary as calculated with no error sources, and n is the size of the relevant Hilbert space. U i is the unitary of the operation with the error of interest, and the matrix elements (U i ) A,B are generated by applying a Hamiltonian with the error to a state A and using the coefficient of the resulting state B. For U i with no error, U i = U p and F = 1. Rydberg Decay An error source, which fundamentally limits the performance of this gate, is the decay of the Rydberg atom which facilitates our exchange between the molecules. Lifetimes at typical Rydberg levels used for quantum information are approximately 100 µs, and even the use of cooling to negate blackbody radiation will not increase this to more than a few hundred microseconds. Typical Rydberg blockade gates skirt around this error by minimizing the population in the Rydberg state, advantages that our scheme does not have. This gate, by operating outside the blockade regime, allows the excitation laser to pump to Rydberg levels and back faster than blockade based gates. The drivenexchange gate is therefore faster and less error from decay can occur. This advantage is twofold, as the gate only uses one Rydberg atom, rather than two. In the NaCs with Cs system, decay limits the gate to a fidelity of 0.9997. This number is calculated by adding non-Hermitian decay terms to our excitation-manifold Hamiltonians. H nexc=0 = H nexc=0 + 0 0 0 −i Γ r /2 (B2) H nexc=1 = H nexc=1 +        0 0 0 0 0 0 −i Γ r /2 0 0 0 0 0 0 0 0 0 0 0 −i Γ r /2 0 0 0 0 0 −i Γ R /2        (B3) H nexc=2 = H nexc=2 +      0 0 0 0 0 −i Γ r /2 0 0 0 0 −i Γ R /2 0 0 0 0 −i Γ R /2      (B4) Γ r and Γ R are the decay rates of the two chosen Rydberg levels. Experimental Errors We consider error sources resulting from experimental imperfections. These include magnetic and electric field amplitude noise, as well as excitation laser intensity and frequency error. Each source results in different errors in the Hamiltonian parameters: field amplitude changes ∆ and δ, laser intensity changes Ω, and laser frequency changes δ. To understand the worst-case effects of these errors, we simulate our gate evolution with a constant error over the entire gate time. These results are presented in Fig. 3b. The noise arising from detuning is dominated by phase error, while intensity will change the population still in the Rydberg state. At 3 mG magnetic field fluctuation, ∆ changes by 2π × 5 kHz and δ changes by 2π × 4 kHz, which gives a gate fidelity of 0.99998. To stay above 0.999, a fluctuation of up to 20 mG can be tolerated. For electric field fluctuations around zero field, this tolerance is 2mV/cm. One note to consider is that, as mentioned in the main text, this system is most sensitive to excitation drive Rabi frequency, and greater than 0.9999 fidelity requires greater than 1.94% stability. However, due to the dualspecies nature of the array, it is possible to measure the Rydberg atoms without measuring the corresponding qubits. This free measurement allows us to detect error-created Rydberg population and project into the correct ground state otherwise. At 10% drive Rabi frequency error, this error detection allows us to project from a 94.3% fidelity Hilbert space to a 99.9% fidelity subspace. Molecular Hyperfine State Leakage We provide details on how we account for hyperfine state leakage. As discussed in the main text, the \|0 = \|m INa , m ICs , N, m N = \|3/2, 5/2, 0, 0 state may exchange with not only the desired \|1 = \|3/2, 5/2, 1, 1 , but may also exchange weakly with the \|2 = \|3/2, 7/2, 1, 0 state, which is detuned away. To allow for this possible leakage, we perform a unitary simulation with an enlarged Hilbert space including the extra states. For instance, the Hilbert space of the n exc=1 manifold is given by {\|0R0 , \|1r0 , \|0r1 , \|2r0 , \|0r2 , \|1g0 , \|0g1 , \|2g0 , \|0g2 }. Then, the dynamics follow a Hamiltonian that includes coupling to the extra state and also includes the detuning of the extra state. To calculate the fidelity, the unitary U i in equation B1 is calculated for the computational basis states {\|0g0 , \|0g1 , \|1g0 , \|1g1 }. For this system, hyperfine state leakage limits fidelity to 0.9996. Interaction with a Next-Nearest Neighbor Molecule In a larger array, there will be nearby molecules that interact with the Rydberg atom and gate molecules, expanding the system to more than three particles. To characterize the additional error on our gate qubits, m 1 and m 2 , we consider a molecule, m 3 , on the same row, but distance 2a + d away. The atom is between the molecules m 1 and m 2 and will be denoted as r. The molecule m 3 will double the size of our Hilbert space and add states to our previous excitation manifolds, thus creating blocks H n exc =1 , H n exc =2 , and H n exc =3 where n exc is the number of excitations in the three molecules and Rydberg atom system. H n exc =0 and H n exc =4 have no coupling to the original states, so they are not considered. These four particle Hamiltonians can be described as 2 x 2 block matrices composed of our previous three particle Hamiltonians H nexc , and couplings Ω i,j caused by the molecule m 3 exchanging excitations with other particles in the system: H n exc=1 = H nexc=0,m3=\|1 Ω n=0,n=1 Ω n=1,n=0 H nexc=1,m3=\|0(B5)H n exc=2 = H nexc=1,m3=\|1 Ω n=1,n=2 Ω n=2,n=1 H nexc=2,m3=\|0(B6)H n exc=3 = H nexc=2,m3=\|1 Ω n=2,n=3 Ω n=3,n=2 H nexc=3,m3=\|0 .(B7) The Ω couplings are as such: Ω n=0,n=1 = 1 2 V m2m3 0 V m1m3 0 0 0 V m2m3 0 V m1m3 V m3r (B8) Ω n=1,n=2 = 1 2        V m1m3 0 0 0 0 V m1m3 V m3r 0 V m2m3 0 0 0 0 V m2m3 0 V m3r 0 0 V m2m3 V m1m3        (B9) Ω n=2,n=3 = 1 2      0 V m3r V m1m3 V m2m3      . (B10) In this case, Ω n=1,n=0 = Ω † n=0,n=1 , Ω n=2,n=1 = Ω † n=1,n=2 , and Ω n=3,n=2 = Ω † n=2,n=3 . We then evolve each excitation manifold under these enlarged Hamiltonians. To calculate error, we use equation B1 with an expanded U p and U i to encompass the dynamics of the additional states. When a = 1 µm, and d = 0, the fidelity is just above 0.999. This effect will limit the gate fidelity in larger arrays, however, as discussed in Section V, the next nearest neighbor molecules can be moved to an isolated region or shelved into non-interacting states. Atom Motion Since Rydberg atoms are typically anti-trapped by their tweezers, the trapping light is turned off during Rydberg excitation. During this period, the atomic motional wavefunction Ψ a (r, t) can evolve and thus lead to a time-varying interaction strength given by V mr (t) = d m d r 4π 0 d 3 r\|Ψ a (r, t)\| 2 1 \|(r − r m )\| 3 ,(B11) where r m is a fixed position of a motional ground-state cooled and trapped molecule. The atomic wavefunction is assumed to start in the ground state of a harmonic oscillator with trapping frequency ω = 2π × 80 kHz, and then evolved under a free-particle Hamiltonian. We recalculate B11 at each different time step and the timedependent problem is solved with a Lindblad master equation solver from the Quantum Toolbox in Python. The resulting gate fidelity is 0.99997. Van der Waals Interactions In the manuscript, the degeneracy of the \|0R and \|1r states leads to a strong resonant-exchange. In principle, off-resonant exchange processes exist for these states as well as for \|0r and \|1R . This leads to a 1/r 6 van der Waals interaction, which can affect the gate dynamics, especially when atom motion is considered. The contribution from nearby pair states is given by V 2 mr /4(∆E), where V mr is the interaction strength and ∆E is the energy gained or lost in the exchange process. We consider interacting pair states with ∆E ≤ h×100 GHz [66,67]. In addition, we also include off-resonant coupling to the second rotational state of the molecule. Adding all these contributions at a = 1 µm, the van der Waals interaction strength for each pair state is V vdW,0r = h × 2.3 kHz, V vdW,1r = h × 1.25 kHz, V vdW,0R = h × 863 Hz, V vdW,1R = h × 2.84 kHz. These interactions can be calibrated away at a particular interparticle spacing by carefully tuning the Rydberg resonance. If there are fluctuations in this interaction due to atom motion, errors will be introduced into the gate. Since these interaction strengths are much smaller than the MHz scale on-resonant interactions discussed in section B 6, they can be neglected. Appendix C: Quadratic Zeeman Shift At high magnetic fields and large principal quantum numbers, the quadratic Zeeman shift for atoms can be significant. The Hamiltonian for this effect is given by H QZeeman = e 2 B 2 8m (x 2 + y 2 ) = e 2 B 2 8m r 2 sin 2 θ,(C1) where θ is the polar angle [68,69]. In the basis \|n, l, j, m j , the matrix element n , l , j , m j \|H QZeeman \|n, l, j, m j (C2) is nonzero when l = l or l = l ± 2, and m j = m j . The diagonal contributions are relevant and can shift energy levels significantly. These terms consist of a radial integral and an angular integral. The radial integral is performed for a state \|n, l, j, m j with the wavefunctions in the Alkali Rydberg Calculator, which account for spinorbit coupling [44]. The angular integral is performed by decomposing the state into the uncoupled \|n, l, m l , s, m s basis, where the matrix elements of sin 2 θ in this basis are given in Ref. [69]. These diagonal matrix elements are taken into account in the selection of states in Table I. The off-diagonal contributions can mix different states, however they are only relevant when the coupling strength is on the same order as the energy separation between the states. Fortunately in Cs, the energy of the nearest states with l = l ± 2 to the states of interest (\|72P 3/2,3/2 and \|71D 5/2,5/2 ) are about an order of magnitude larger than the coupling strength between them. We note that for higher Rydberg states, these contributions are likely to grow larger and a full diagonalization may be required. FIG. 2 . 2Population evolution through the driven exchange gate, starting in (a) \|0g0 , (b) \|1g0 , and (c) \|1g1 FIG . 3. (a) Demonstration of driven exchange resonances at various integer k, as defined in equation 7. As k increases, the resonances become closer and shallower, leading to insensitivity to the exact drive Rabi frequency. (b) Fidelity loss due to fractional errors in detuning and drive Rabi frequency for k = 1. In this figure, dΩ refers to the error in Rabi frequency of the driving laser. FIG. 4 . 4The relevant energy levels for a blockade based detection of the molecule. 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[ "An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS", "An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS" ]	[ "Karsten Paul ", "Christopher Zimmermann ", "Kranthi K Mandadapu \nDepartment of Chemical and Biomolecular Engineering\nUniversity of California at Berkeley\n110A Gilman Hall\n94720-1460BerkeleyCAUSA\n\nChemical Sciences Division\nLawrence Berkeley National Laboratory\n94720CAUSA\n", "Thomas J R Hughes \nThe Oden Institute for Computational Engineering and Sciences\nThe University of Texas at Austin\n201 E. 24th Street, POB 4.102, 1 University Station (C0200)78712-1229AustinTXUSA\n", "Chad M Landis \nThe Oden Institute for Computational Engineering and Sciences\nThe University of Texas at Austin\n201 E. 24th Street, POB 4.102, 1 University Station (C0200)78712-1229AustinTXUSA\n", "Roger A Sauer ", "\nAachen Institute for Advanced Study in Computational Engineering Science (AICES)\nRWTH Aachen University\nTemplergraben 5552062AachenGermany\n" ]	[ "Department of Chemical and Biomolecular Engineering\nUniversity of California at Berkeley\n110A Gilman Hall\n94720-1460BerkeleyCAUSA", "Chemical Sciences Division\nLawrence Berkeley National Laboratory\n94720CAUSA", "The Oden Institute for Computational Engineering and Sciences\nThe University of Texas at Austin\n201 E. 24th Street, POB 4.102, 1 University Station (C0200)78712-1229AustinTXUSA", "The Oden Institute for Computational Engineering and Sciences\nThe University of Texas at Austin\n201 E. 24th Street, POB 4.102, 1 University Station (C0200)78712-1229AustinTXUSA", "Aachen Institute for Advanced Study in Computational Engineering Science (AICES)\nRWTH Aachen University\nTemplergraben 5552062AachenGermany" ]	[]	We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff-Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell's mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith's theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C 1 -continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton-Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.	10.1007/s00466-019-01807-y	[ "https://arxiv.org/pdf/1906.10679v1.pdf" ]	195,583,998	1906.10679	7e448752197d787ae87eb0a3dbe98def584a7f40	An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS Karsten Paul Christopher Zimmermann Kranthi K Mandadapu Department of Chemical and Biomolecular Engineering University of California at Berkeley 110A Gilman Hall 94720-1460BerkeleyCAUSA Chemical Sciences Division Lawrence Berkeley National Laboratory 94720CAUSA Thomas J R Hughes The Oden Institute for Computational Engineering and Sciences The University of Texas at Austin 201 E. 24th Street, POB 4.102, 1 University Station (C0200)78712-1229AustinTXUSA Chad M Landis The Oden Institute for Computational Engineering and Sciences The University of Texas at Austin 201 E. 24th Street, POB 4.102, 1 University Station (C0200)78712-1229AustinTXUSA Roger A Sauer Aachen Institute for Advanced Study in Computational Engineering Science (AICES) RWTH Aachen University Templergraben 5552062AachenGermany An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS Phase fieldsbrittle fractureisogeometric analysisadaptive local refinementLR NURBSnonlinear finite elementsKirchhoff-Love shells We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff-Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell's mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith's theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C 1 -continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton-Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching. Introduction The need for shortening development cycles of engineering components requires efficient computational methods. The robustness requirements for these components are increasing so that the prediction of structural defects and failure plays a major role in current development processes. It is therefore important to have efficient and reliable computational methods for predicting fracture. Several computational methods have been introduced to model crack growth. The most important ones in the framework of finite elements are described subsequently. Sharp interface models introduce discontinuities within the body in order to model cracks. In the extended finite element method by Moës et al. (1999), the basis functions are enriched by discontinuities to model the displacement jump across cracks. In contrast to this, a crack can be introduced by a modification of the finite element mesh as in the virtual crack closure technique (Krueger, 2002). Similar to the extended finite element method, Remmers et al. (2003) also enrich the basis in the cohesive segments method. Since the location of the crack has to be known, it has to be numerically tracked, which tends to be a tedious task, especially in three dimensions. Thus, diffuse interface models have gained popularity for modeling brittle fracture. In the phase field method no discontinuities are introduced within the body. Instead, the crack is smoothed out and described by a small transition zone that ranges between undamaged and fully fractured material. Phase field methods describe the evolving cracks by an additional partial differential equation (PDE) such that there is no need for tracking the interface. For complex crack patterns including nucleation, branching, and merging, phase field formulations have been shown to be very effective. Based on the thermodynamic considerations of brittle fracture by Griffith (1921), a variational formulation of brittle fracture has been introduced by Francfort and Marigo (1998). Their formulation includes the minimization of a global energy functional to model the quasi-static fracture process. A corresponding phase field implementation within the finite element method has been presented by Bourdin et al. (2000). The robustness and accuracy of the variational formulation in two and three dimensions using phase field methods have been demonstrated by e.g. Miehe et al. (2010a) and Miehe et al. (2010b). Successful extensions to dynamic problems have been presented by Larsen (2010), Larsen et al. (2010), Bourdin et al. (2011), Borden et al. (2012), Hofacker and Miehe (2012) and Schlüter et al. (2014). In contrast to the variational formulation of brittle fracture, Karma et al. (2001) and Kuhn and Müller (2010) use a phase transition framework based on the Ginzburg-Landau equation. The latter is more often used in the physics community, where a very general phase field formulation is applied to fracture mechanics. A stabilization for quasi-static simulations using a monolithic solution approach for the coupled system is proposed by Gerasimov and De Lorenzis (2016). apply a non-intrusive global/local approach in a phase field framework for brittle fracture, in which at first the structural analysis of the whole domain is performed and, afterwards, local regions where fracture is predicted are re-analyzed. These steps are then repeated until convergence is obtained. Ambati et al. (2015) summarize several phase field formulations for brittle fracture. In the work of Kuhn et al. (2015) the influence of different degradation functions on the solution has been investigated. Similar investigations have been made by Sargado et al. (2018) who have also studied parametric degradation functions. Possibilities to enforce irreversibility of the fracture process are presented in detail in the work of , especially focusing on the penalty method. The authors also derive a lower bound for the penalty parameter for a quasi-static second-order phase field model for brittle fracture. The majority of the published phase field methods for fracture use a second-order phase field formulation. The high order differential operators of the phase field PDE stemming from the crack density functional of Borden et al. (2014), which is used in this work, and the equation of motion of the shell framework require a spatial finite element discretization that is at least C 1 -continuous. Isogeometric Analysis (IGA), proposed by Hughes et al. (2005), allows for user-defined smoothness of the solution within the finite element framework. Within IGA, the smoothness is most commonly achieved through the use of B-Spline-and NURBS-based shape functions. Since phase field methods require a highly resolved finite element mesh in the transition zone, local refinement methods are commonly used in the context of phase field methods for fracture. The introduction of hierarchical B-splines by Forsey and Bartels (1988) has offered the possibility of local refinement within an IGA framework. The extension to the local refinement of NURBS is for instance given by Sederberg et al. (2003) by introducing T-Splines. Another approach that allows local refinement is Locally Refinable (LR) splines. LR B-splines were first introduced by Dokken et al. (2013) and further advanced by Johannessen et al. (2014). Their extension to LR NURBS is provided by Zimmermann and Sauer (2017). Isogeometric collocation methods (Gomez et al., 2014;Reali and Hughes, 2015) for phase field models of fracture are also introduced, for instance by Schillinger et al. (2015). Hesch et al. (2016b) employ a hierarchical refinement scheme within a higher order phase field model. Similarly, Hesch et al. (2016a) couple a model for frictional contact to a higher order phase field model using hierarchical NURBS. Kästner et al. (2016) investigate phase field models by comparing adaptive refinement based on locally refined hierarchical B-splines with uniformly refined discretizations. Borden et al. (2012) propose an adaptive refinement strategy using Tsplines and use the phase field value itself to identify the need for local refinement. Mesh adaptivity schemes, in which a predictor-corrector scheme is used, are employed by Zhou and Zhuang (2018) for modeling fracture in rocks and by Badnava et al. (2018) to model mechanically and thermo-mechanically induced cracks. In these approaches, the system is solved and then checked for the need of mesh refinement. In the work by Nagaraja et al. (2018) a multi-level hprefinement technique has been established using the finite cell method (Parvizian et al., 2007) to model brittle fracture in two dimensions. Many papers concerning the computational modeling of shells within an isogeometric framework have been published, for instance by Benson et al. (2013), Echter et al. (2013), Kiendl et al. (2015) and Duong et al. (2017). Since for shells the bending stress varies across the thickness, a suitable split of the energy within the fracture model has to be established. In the work by Ulmer et al. (2012) brittle fracture in thin plates and shells is modeled. They combine a plate and a standard membrane to model the shell but have only split the membrane and not the bending part of the elastic energy. Thus, the whole bending energy contributes to crack evolution and is degraded in regions of damage. Amiri et al. (2014) do not employ an energy split, which limits their model to shells under pure tension. In the work by the shell and the phase field are also discretized over the thickness. Areias et al. (2016) utilize two phase fields, one for the top and the other one for the bottom face of the shell. This framework is also used by Reinoso et al. (2017) for a 6-parameter shell model. Their formulation results in a non-constant phase field throughout the thickness. In contrast to this, Kiendl et al. (2016) use a constant phase field over the thickness but use thickness integration to split the whole energy into a tensile part, which contributes to crack growth, and a compressive part, which does not. Zimmermann et al. (2019) model Cahn-Hillard phase field equations on deforming surfaces based on the shell formulation of Duong et al. (2017). Even though a different physical process is modeled, the resulting coupled finite element formulation is similar to the one proposed here. In this paper we establish a dynamic brittle fracture framework within the nonlinear IGA thin shell formulation of Duong et al. (2017), in which shells with arbitrarily large curvature or doubly curved shells can be modeled. Its hyperelastic material model allows for large deformations and is given as a sum of membrane and bending contributions. The proposed higher order phase field model of Borden et al. (2014) is adopted because of its higher rate of convergence and it is formulated on the shell's mid-plane. Motivated by the work of Kiendl et al. (2016), bending effects on the fracture process are modeled based on thickness integration. Adaptive spatial refinement is based on LR NURBS (Zimmermann and Sauer, 2017) and temporal discretization is based on the generalized-α scheme (Chung and Hulbert, 1993). The time steps are adjusted based on the number of Newton-Raphson iterations required during the last time step. In summary, the proposed formulation contains the following features: • It couples a higher-order phase field model for fracture with a nonlinear shell formulation. • It is formulated in curvilinear coordinates, and applicable to general shell configurations. • The coupled system is solved within a monolithic, fully implicit solution approach. • It uses adaptive local refinement in space and time. • The spatial discretization is based on LR NURBS. • An energy split is used in which the membrane and bending energies are split separately. The subsequent sections are structured as follows: Sec. 2 summarizes the surface description and kinematics. The balance laws and the equation of motion are derived in Sec. 3. Sec. 4 introduces the energy minimization problem and the material model employed. Extensions to degradation, irreversibility and an energy split are also presented. Based on the Euler-Lagrange equation, the Helmholtz free energy is minimized, which leads to the governing equation for the phase field's evolution. The discretization of the coupled problem is described in Sec. 5. Numerical examples are presented in Sec. 6 to illustrate crack propagation on curved surfaces. Conclusions are drawn in Sec. 7. Deforming surfaces This section summarizes the thin shell formulation in the framework of curvilinear coordinates and Kichhoff-Love kinematics. A more detailed presentation can be found in Sauer (2018). Surface description A curved surface S in 3D space can be characterized by the parametric description at any time t by the function x = x(ξ α , t) , α = 1, 2 ,(1) where ξ α denote the curvilinear coordinates associated with a material point x ∈ S. ξ α are convected along with the material deformation of the surface and hence, they are also called convected coordinates. The co-variant tangent vectors at x are given by a α := ∂x ∂ξ α .(2) From these follow the surface metric a αβ := a α · a β ,(3) the surface normal n := a 1 × a 2 a 1 × a 2 , and the contra-variant tangent vectors a α = a αβ a β ,(5) where [a αβ ] = [a αβ ] −1 . All Greek indices range from 1 to 2 and are summed when repeated. Based on the second parametric derivative a α,β := ∂a α /∂ξ β , the curvature tensor components b αβ = a α,β · n , follow. The set of initial surface points X ∈ S 0 follows from X := x(ξ α , 0). In analogy to Eqs. (2)-(6), we define the surface quantities A α := ∂X/∂ξ α , A αβ := A α · A β , N := A 1 × A 2 / A 1 × A 2 , A α := A αβ A β , [A αβ ] := [A αβ ] −1 and B αβ := A α,β · N at t = 0 as a reference configuration, denoted S 0 . The surface gradient grad S φ = ∇ S φ := φ ;α A α ,(7) and surface Laplacian ∆ S φ := ∇ S · ∇ S φ = φ ;αβ A αβ ,(8) can be defined based on the parametrization in Eq. (1). Here, φ denotes a general scalar function and the subscript ';' indicates the co-variant derivative. It is equal to the parametric derivative for general scalars, i.e. φ ;α = φ ,α := ∂φ/∂ξ α . But, φ ;αβ = φ ,αβ and instead φ ;αβ = φ ,αβ −Γ γ αβ φ ,γ ,(9) whereΓ γ αβ = A α,β · A γ are the Christoffel symbols of the second kind on surface S 0 . On S, these read Γ γ αβ = a α,β · a γ . Surface kinematics The relation between reference surface S 0 and current surface S is described by the surface deformation gradient F = a α ⊗ A α .(10) The left surface Cauchy-Green tensor then follows as B = A αβ a α ⊗ a β ,(11) with its two invariants I 1 := A αβ a αβ and J := det[A αβ ] det[a αβ ] .(12) The latter characterizes the surface stretch between S 0 and S. The surface Green-Lagrange strain tensor and the symmetric relative curvature tensor are E = 1 2 (a αβ − A αβ ) A α ⊗ A β , K = (b αβ − B αβ ) A α ⊗ A β .(13) The material time derivative is denoted bẏ (...) := ∂... ∂t ξ α = fixed .(14) This leads to the material velocity at x v :=ẋ , and the ratesȧ α = v ,α = ∂v ∂ξ α , andȧ αβ = a α ·ȧ β +ȧ α · a β .(16) Surface variations The variation of various surface measures is required for the formulation of the weak form of the thin shell equation. Particularly important are the variations δa αβ = a α · δa β + δa α · a β , δb αβ = δa α,β − Γ γ αβ δa γ · n , δn = −(a α ⊗ n) δa α ,(17) where δa α = δx ,α and δa α,β = δx ,αβ . Here, δx denotes a kinematically admissible variation of the deformation. Additional variations of surface quantities are provided in . Thin shell theory The governing equations for the shell are summarized in the following. Equilibrium is given in strong and weak form. Considering Kirchhoff-Love kinematics, the constitutive behavior of thin shells can be fully characterized by the quantities a αβ and b αβ . Balance of linear and angular momentum The equation of motion ρv = T α ;α + f , ∀ x ∈ S ,(18) follows from the balance of linear momentum for surface S. f = f α a α + p n denotes prescribed body forces and T α = N αβ a β + S α n ,(19) are the stress vectors that include the in-plane membrane components N αβ and the out-ofplane shear components S α (Naghdi, 1973;Steigmann, 1999;. These are related to the stress tensor σ = N αβ a α ⊗ a β + S α a α ⊗ n ,(20) through Cauchy's formula T α = σ T a α . Given the outward pointing normal ν = ν α a α at a cut through S, the traction T = σ T ν = T α ν α acting on this cut follows. Likewise, the moment vector on the cut reads M = µ T ν with the moment tensor µ = −M αβ a α ⊗ a β ,(21) where M αβ denotes its in-plane components Sahu et al., 2017). The balance of angular momentum yields S α = −M βα ;β , σ αβ = σ βα ,(22) where σ αβ := N αβ − b β γ M γα . The stress components σ αβ and M αβ follow from constitution, which is discussed in Sec. 4.2. The component form of the equation of motion ρ a α = f α + N λα ;λ − S λ b α λ , ρ a n = p + N αβ b αβ + S α ;α ,(23) is obtained by combining Eqs. (18), (20) and (22.1). Here, a α :=v · a α , a n :=v · n, f α := f · a α and p := f · n. Weak form for deforming thin shells The weak form for Kirchhoff-Love shells is given by ) G kin + G int − G ext = 0 , ∀ δx ∈ U ,(24) with G kin := S δx · ρv da , G int := S 1 2 δa αβ σ αβ da + S δb αβ M αβ da , G ext := S δx · f da + ∂tS δx · T ds + ∂mS δn · M ds .(25) Here, U = δx ∈H 2 S(x, t) 3 \| δx = 0 on ∂ x S , δn = 0 on ∂ n S ,(26) is the space of suitable surface variations, whereH 2 is the Sobolev space of Lebesgue square integrable functions and ∂ x S and ∂ n S are the Dirichlet boundaries for displacements and rotations. The prescribed edge tractions T = σ T ν and edge moments M = µ T ν act on the boundaries ∂ t S and ∂ m S with the outward normal ν = ν α a α . We note that the torsional components of the moment M are perceived as an effective shear traction in Kirchhoff-Love shells, e.g. see . If desired, da = J dA and ρ da = ρ 0 dA can be used to map integrals to the reference surface S 0 . The components σ αβ and M αβ follow from the constitutive laws as outlined in Sec. 4.2. Fracture of deforming surfaces The formulation for the modeling of brittle fracture is based on Griffith's theory (Griffith, 1921), in which the energy release rate E G of a body, which describes the dissipated energy during crack evolution, is related to the fracture toughness G c [J m −2 ]. The latter is also referred to as the critical fracture energy density or critical energy release rate. The corresponding Kuhn-Tucker conditions read E G − G c ≤ 0,ċ ≥ 0, (E G − G c )ċ = 0 ,(27) withċ denoting the crack propagation velocity. Since crack nucleation and branching are not captured by this formulation, Griffith's theory has been reformulated as a global energy minimization problem (Francfort and Marigo, 1998). The corresponding energy functional is derived subsequently. Helmholtz free energy The total energy in the system is given by Π := Π int + Π kin − Π ext ,(28) where the three contributions denote the Helmholtz free energy Π int , the kinetic energy Π kin and the external energy Π ext , respectively. Based on the formulation of energy minimization by Francfort and Marigo (1998), the Helmholtz free energy contains elastic and fracture energy contributions in the form Π int = S 0 Ψ dA = S 0 g(φ)Ψ + el + Ψ − el + Ψ frac dA ,(29) where Ψ denotes the Helmholtz free energy per reference area. Cracks resemble discontinuities in the deformation that are smeared out in the phase field formulation. Therefore, an indicator φ ∈ [0, 1] is established that distinguishes between fully fractured, φ = 0, and undamaged, φ = 1, material. This field is referred to as the phase field or fracture field. Since it models the damage region, it is used to define the fracture energy appearing in Eq. (29). The higher order phase field model by Borden et al. (2014) is adopted here, which, expressed in variables of the present thin shell formulation, reads Ψ frac = G c 4 0 (φ − 1) 2 + 2 2 0 ∇ S φ · ∇ S φ + 4 0 (∆ S φ) 2 .(30) The length scale parameter 0 [m] controls the support width of the transition zone: supp(φ) ∼ 0 . Borden et al. (2014) have shown that the one-dimensional phase field approximation of the crack surface Γ = {0} has the form φ(x) = 1 − exp − \|x\| 0 1 + \|x\| 0 ,(31) which is illustrated in Fig. 1. An additive energy split is required in which the elastic energy density is split into a part that contributes to crack evolution ('+') and a part that has no effect on crack growth ('−'): −2 0 0 2 0 0 1 x φ(x) crack surface Γ φ(x)Ψ el = Ψ + el + Ψ − el . The two contributions are also referred to as the positive and negative part of the elastic energy density. The split is further motivated and derived in Sec. 4.2.1. According to Eq. (29), the positive part of the elastic energy density Ψ + el is degraded through g(φ) along the damage regions. Here, it is assumed to take the form (Borden et al., 2016) g(φ) = (3 − s)φ 2 − (2 − s)φ 3 ,(32) where s > 0 describes the slope of g(φ) at φ = 1. If s = 0, a surface without initial damage would fulfill the governing equation for crack evolution (60) for any deformation implying that crack nucleation would not occur. Thus, s is set to 10 −4 (Borden et al., 2016) in all subsequent computations to allow crack nucleation in the absence of initial damage. Degradation functions with g (1) = 0 could be used but they require a perturbation in the first Newton-Raphson iteration to allow for crack nucleation in sound materials (Kuhn et al., 2015). Hyperelastic material model The elastic energy density Ψ el is taken as an additive composition of dilatational, deviatoric and bending energy densities in the form Ψ el = Ψ dil (a αβ ) + Ψ dev (a αβ ) Ψmem(a αβ ) +Ψ bend (b αβ ) ,(33) where the first two terms describe the membrane part of Ψ el . A Neo-Hookean surface material model with Ψ dil = K 4 J 2 − 1 − 2 ln J ,(34) and Ψ dev = G 2 I 1 /J − 2 ,(35) is used to model the isotropic in-plane constitutive response. K refers to the 2D bulk modulus and G to the 2D shear modulus. The bending response follows from the Koiter model (Ciarlet, 1993) Ψ bend = c 2 b αβ − B αβ b αβ 0 − B αβ ,(36) with bending modulus c and b αβ 0 := A αγ b γδ A βδ . Differentiating the Helmholtz free energy with respect to metric and curvature components, yields the stress and moment components τ αβ = 2 ∂Ψ ∂a αβ , M αβ 0 = ∂Ψ ∂b αβ .(37) Here, these components are given with respect to the reference configuration but they can be mapped to the current configuration by dividing the expressions in Eq. (37) by the surface stretch J. The individual derivatives for the material model in Eqs. (34), (35) and (36) read Zimmermann et al., 2019) τ αβ = τ αβ dil + τ αβ dev , τ αβ dil = K 2 (J 2 − 1) a αβ , τ αβ dev = G 2J (2A αβ − I 1 a αβ ) , M αβ 0 = c (b αβ 0 − B αβ ) .(38) Split of the elastic energy density Crack evolution shows anisotropic behavior since cracks will not propagate for every state of stress. To avoid cracking in compression an energy split is required as follows Ψ el = Ψ + el + Ψ − el ,(39) where Ψ − el refers to the part of the elastic energy density that does not contribute to the fracture process. Amor et al. (2009) make use of a split into deviatoric and dilational parts in which crack evolution is not permitted in volumetric compression but allowed in states of volumetric expansion and shear. In the work of Miehe et al. (2010a), a spectral decomposition of the strain tensor is introduced in which only positive strains contribute to the fracture process. Likewise, Kiendl et al. (2016) establish a spectral decomposition within a small deformation framework in plates and shells. They have outlined that it is not possible to consider a split into tension and compression as well as a split into membrane and bending contributions at the same time if such a spectral decomposition of the total strain is used. In our formulation, the elastic energy density is already split into membrane and bending parts according to Eq. (33) such that these terms can be decomposed separately Ψ ± el = Ψ ± mem + Ψ ± bend .(40) In the following, we show an example taken from Kiendl et al. (2016) that they use to motivate the need for a thickness integration for the energy split. We use their example to motivate the proposed split of the bending energy density. The strain distribution over the shell's thickness is illustrated in Fig. 2. The total strainẼ = E − ξK with componentsẼ αβ and thickness coordinate ξ ∈ − T 2 , T 2 can have both, positive and negative parts over the thickness T . It follows that there is a region of compression, which must not contribute to the fracture process. The membrane strains (due to the surface Green-Lagrange strain tensor E) are purely positive in this example, whereas the strains associated with the curvature part are asymmetrically distributed around the mid-plane of the shell. Since Kiendl et al. (2016) are only interested in the tensile contributions, thickness effects for the elastic energy need to be considered to correctly distinguish between tensile and compressive contributions to the total strain. In contrast to this, the kinematical objects on the mid-plane include enough information for a suitable split of the membrane part. Subsequently, the individual splits of in-plane and out-of-plane parts are derived. Figure 2: Strains over the shell's thickness (Kiendl et al., 2016). In this example, the membrane part shows purely positive strains, whereas the strains of the bending part are skew-symmetric around the mid-plane. If the negative strains are not supposed to contribute to crack growth, the strain distribution over the thickness has to be taken into consideration. Kiendl et al. (2016) introduce a thickness integration and split the total strain with a spectral decomposition. Their work motivates the necessity of our thickness integration (cf. Eq. (44)). + −Ẽ αβ = + E αβ + + − K αβ ξ T 2 -T 2 As already mentioned, a spectral decomposition of the strain tensor is not suitable in the present formulation since our elastic energy density is given as a sum of membrane and bending contributions. Instead, we follow the decomposition introduced by Amor et al. (2009), which has also been used by e.g. and Borden et al. (2016). Corresponding to whether the surface stretch J is greater than/equal to 1 or smaller than 1, the dilatational part will contribute to crack growth or not. The split of the membrane energy density required in Eq. (40) then yields Ψ + mem = Ψ dev + Ψ dil , J ≥ 1 Ψ dev , J < 1 , Ψ − mem = 0 , J ≥ 1 Ψ dil , J < 1 .(41) Thus, crack evolution is not permitted in states of volumetric compression (J < 1) but allowed in states of pure shear (J = 1) or volumetric expansion (J > 1). For instance Ambati et al. (2015) have shown that this split works well for fracture prediction, but we note that a suitable split of the deviatoric energy density might be missing in Eq. (41). The thickness has to be taken into account in order to obtain a suitable split of the bending energy density in Eq. (36). This is obtained from following relation ) Ψ bend = T 2 − T 2Ψ bend (ξ) dξ ,(42) where the corresponding three-dimensional constitutive model 2 is given bỹ Ψ bend (K, ξ, T ) = ξ 2 12 T 3 c 2 tr K 2 .(43) The split of Ψ bend is then modeled as Ψ ± bend = T 2 − T 2Ψ ± bend (ξ) dξ .(44) Still, Eq. (43) has to be additively decomposed according toΨ bend =Ψ + bend +Ψ − bend . Already in Eq. (41) the surface stretch at the mid-plane has been employed as an indicator for a possible contribution to the fracture process. The surface stretch of other shell layers is obtained in analogy to Eq. (12.2) asJ = det[Ã αβ ] det[ã αβ ] .(45) The metricsÃ αβ andã αβ follow from the tangent vectorsÃ α andã α of the shell layer at points x + ξ n and X + ξ N , respectively . The split ofΨ bend then follows as Ψ + bend (ξ) =    ξ 2 12 T 3 c 2 tr K 2 ,J(ξ) ≥ 1 0 ,J(ξ) < 1 ,Ψ − bend (ξ) =    0 ,J(ξ) ≥ 1 ξ 2 12 T 3 c 2 tr K 2 ,J(ξ) < 1 .(46) Based on Eq. (44), the decomposition of the bending energy density follows from thickness integration of Eq. (46). Thickness integration is performed numerically using Gaussian quadrature. We note that an analytical integration of Eq. (46) over the thickness is in general not possible due to the strong nonlinear dependence of the surface stretchJ on ξ. But there are two special cases for which Eq. (44) can be solved analytically, i.e. J(ξ) ≥ 1 , ∀ ξ ∈ − T 2 , T 2 : Ψ + bend = c 2 b αβ − B αβ b αβ 0 − B αβ , Ψ − bend = 0 ,(47)andJ (ξ) < 1 , ∀ ξ ∈ − T 2 , T 2 : Ψ + bend = 0 , Ψ − bend = c 2 b αβ − B αβ b αβ 0 − B αβ .(48) These relations can then be used for an efficient FE implementation. Stresses and moments Based on the energy split from the previous section, the stress and moment components follow. In the reference configuration, the stress components read τ αβ = g(φ) τ αβ + + τ αβ − ,(49) with the individual contributions τ αβ + = τ αβ dev + τ αβ dil , J ≥ 1 τ αβ dev , J < 1 , τ αβ − = 0 , J ≥ 1 τ αβ dil , J < 1 .(50) The individual contributions in Eq. (50) are given in Eq. (38.2)-(38.3). The moment components read M αβ 0 = g(φ) M αβ 0,+ + M αβ 0,− ,(51) where the contributions are computed based on thickness integration via M αβ 0,± = T 2 − T 2M αβ 0,± (ξ) dξ ,(52)withM αβ 0,+ (ξ) =      ∂Ψ bend (ξ) ∂b αβ ,J(ξ) ≥ 1 0 ,J(ξ) < 1M αβ 0,− (ξ) =      0 ,J(ξ) ≥ 1 ∂Ψ bend (ξ) ∂b αβ ,J(ξ) < 1 .(53) The required derivative in Eq. (53) is given by ∂Ψ bend (ξ) ∂b αβ = ξ 2 12 T 3 c (b αβ 0 − B αβ ) ,(54) with b αβ 0 = A αγ b γδ A βδ . We note that we have assumed that the order of integration T /2 −T /2 (·) dξ and differentiation ∂(·)/∂b αβ can be exchanged. Irreversible fracture Crack evolution is an irreversible process since cracks cannot heal. Thus, the irreversibility condition Γ(t + ∆t) ⊆ Γ(t) , ∀∆t > 0 where Γ is the crack surface needs to be enforced algorithmically. As described in , several methods exist to enforce this constraint within a phase field model for fracture. The constraint is rewritten in terms of the phase field as φ(x, t + ∆t) ≤ φ(x, t) , ∀∆t > 0. In our work we make use of a history field H(x, t) := max τ ∈[0,t] Ψ + el (x, τ ) ,(55) which keeps track of the fracture contributing part of the elastic energy density (Miehe et al., 2010a). Ψ + el in Eq. (29) is then replaced by the history field H. Complex initial crack patterns can also be realized by means of the history field (Borden et al., 2012). Euler-Lagrange equation and strong form Combining Eqs. (28)-(29) and (55), the total energy in the system follows as Π := S 0 g(φ)H + Ψ − el + Ψ frac (φ) dA − Π ext + Π kin .(56) The kinetic energy Π kin and the potential energy Π ext do not depend on φ. The elastic energy density occurring from volumetric compression Ψ − el does not contribute to crack propagation and is thus, not degraded in the domain of fracture. In contrast to this, H is degraded by the degradation function g(φ), but is not a function of φ itself. Only the energy density Ψ frac depends on φ, as seen in Eq. (30). The minimization of the energy functional can be expressed by setting its variation to zero: δΠ = 0. The latter is solved by making use of the Euler-Lagrange equation, which then leads to the strong form for the phase field's evolution. Given the Helmholtz free energy per reference area Ψ = Ψ(φ, φ ,α , φ ;αβ ), its variation reads δΨ = ∂Ψ ∂φ δφ + ∂Ψ ∂φ ,α δ(φ ,α ) + ∂Ψ ∂φ ;αβ δ(φ ;αβ ) .(57) Integration over the reference surface and applying integration by parts twice, yields S 0 δΨdA = S 0 ∂Ψ ∂φ − ∂Ψ ∂φ ,α ,α + ∂Ψ ∂φ ;αβ ;αβ δφ dA + boundary terms .(58) Inserting the Helmholtz free energy per reference area described in Sec. 4 yields the strong form of the phase field fracture equation 2 0 G c g (φ) H + φ − 1 − 2 2 0 A αβ φ ,αβ + 4 0 A γδ A αβ φ ;αβ ;γδ = 0 , ∀ φ ∈ S ,(60) with g (φ) = ∂g(φ)/∂φ. Weak form for the phase field fracture equation Integrating Eq. (57) over the domain S 0 , the weak form for the phase field fracture equation becomes S 0 δφ f (φ) dA + S 0 ∇ S (δφ) · 2 2 0 ∇ S φ dA + S 0 ∆ S (δφ) 4 0 ∆ S φ dA = 0 , ∀ δφ ∈ V ,(61)with f (φ) := 2 0 G c g (φ)H + φ − 1 ,(62) and the space of suitable test functions V = δφ ∈ H 2 S(φ, t) . The boundary terms arising during the derivation of Eq. (61) vanish due to the choice of the following boundary conditions ∆ S φ = 0 , ∇ S 4 0 ∆ S φ − 2 2 0 φ · n = 0 ,(63) for all φ(x, t) with x ∈ ∂S. Discretization of the coupled problem This section presents the monolithic discretization of the coupled system consisting of the thin shell equation, the phase field evolution equation and their interaction. For the numerical examples presented in Sec. 6 the shell surface is discretized by isogeometric finite elements (Hughes et al., 2005) since the high order operators of the coupled weak form require at least global C 1 -continuity. For the spatial discretization, LR NURBS (Zimmermann and Sauer, 2017) are employed to construct locally refined meshes in the domain of fracture. For the temporal discretization, the generalized-α scheme of Chung and Hulbert (1993) is used. Adaptive local surface refinement LR NURBS The fundamental work of Dokken et al. (2013) and their introduction of LR B-splines has been extended to LR NURBS by Zimmermann and Sauer (2017). A knot vector Ξ of size n + p + 1 defines n linearly independent basis functions of order p. In the framework of LR NURBS, the global knot vector Ξ = [ξ 1 , ..., ξ n+p+1 ] is split into local knot vectors Ξ i = [ξ i , ..., ξ i+p+1 ] (i = 1, . . . , n) to represent local parameter domains. Each of these local knot vectors defines a single basis function. By construction the basis function has minimal support on the local knot vector. Local refinement is performed by mesh line extensions in the parameter space. This includes insertion of new mesh lines, joining or elongation of existing ones or an increase of their multiplicity. The latter results in a decrease of continuity. Local refinement is based on knot insertion (Dokken et al., 2013), which is described for LR NURBS in the work of Zimmermann and Sauer (2017). LR NURBS inherit several mathematical properties from standard NURBS: The basis forms a partition of unity, it is non-negative and the geometry lies within the convex hull of the control points. Criteria for surface refinement An accurate phase field approximation of the discontinuity across the crack is achieved by using a small length scale parameter 0 . This requires a highly resolved finite element mesh in the vicinity of the crack. The phase field φ is used as an indicator for refinement: As soon as a control point's phase field value is smaller or equal to φ bound , all elements that lie in the support domain of the corresponding basis functions will be flagged for refinement. If these elements are not yet refined up to a prescribed refinement depth, mesh line extensions are performed until the desired refinement depth is achieved. The latter can be computed based on the element areas. This refinement strategy is called Structured mesh (Johannessen et al., 2014) and is illustrated in Fig. 3. The blue shaded area in the parameter domain resembles the support domain of a basis function, that is flagged for refinement. The dashed red lines are then inserted into the parameter domain. This is done recursively for all newly created basis functions up to the prescribed refinement depth. The refinement based on mesh line insertion and modification is described in Sec. 5.1.1. We have found φ bound = 0.975 to be a suitable choice for the threshold. We note that in the case of crack nucleation, the last time step needs to be recomputed to ensure crack initiation in a region of highly resolved mesh. X h = N X e , and x h = N x e ,(64) for the reference and current surface, respectively. The corresponding shape function array reads N := [N i 1 1, N i 2 1, ..., N in e 1] . Here, the element-level vectors are denoted X e and x e and 1 refers to the (3×3) identity matrix. Likewise, the phase field is approximated via φ h =N φ e ,(66) with element-level nodal values φ e and shape function arraȳ N := [N i 1 , N i 2 , ..., N in e ] .(67) The local vectors contain the nodal values with indices i 1 , . . . , i ne . These can be extracted from the global ones X, x and φ which contain all nodal values. In analogy to Eqs. (64) and (66), the corresponding variations read δX h = N δX e , and δx h = N δx e ,(68) and δφ h =N δφ e .(69) Based on Eq. (64), the discretized tangent vectors follow as A h α = N ,α X e , and a h α = N ,α x e ,(70) with N ,α := ∂N/∂ξ α . From this, the discretized normals n h and N h follow according to Eq. (4). 3 The metric and curvature tensor components in the reference configuration are then given by A h αβ = X T e N T ,α N ,β X e , and B h αβ = N h · N ,αβ X e ,(71) and similarly for the current surface a h αβ = x T e N T ,α N ,β x e , and b h αβ = n h · N ,αβ x e .(72)δa h αβ = δx T e N T ,α N ,β + N T ,β N ,α x e , and δb h αβ = δx T e N T ;αβ n h ,(73) with N ;αβ := N ,αβ − Γ γ αβ N ,γ ,(74) and discretized Christoffel symbols (cf. Sec. 2.1) Γ γ αβ = x T e N T ,αβ a γδ h N ,δ x e .(75) Using Eqs. (7), (8) and (66), the derivatives of the phase field follow as φ h ;α =N ,α φ e , ∇ S φ h = A α hN ,α φ e , ∇ S δφ h = A α hN ,α δφ e , ∆ S φ h = ∆ SN φ e , ∆ S δφ h = ∆ SN δφ e ,(76)with A α h = A αβ h A h β andN ,α := ∂N/∂ξ α and ∆ SN := A αβ hN ;αβ ,(77) whereN ;αβ =N ,αβ −Γ γ αβN ,γ . Note that here the discretized Christoffel symbols need to be taken from the reference surface (cf. Sec. 2.1), i.e.Γ γ αβ = X T e N T ,αβ A γδ h N ,δ X e .(79) Spatial discretization of the mechanical weak form Inserting the above approximations into Eq. (24) yields the discretized mechanical weak form δx T f kin + f int − f ext = 0 , ∀ δx ∈ U h ,(80) with global force vectors f kin , f int and f ext . These are assembled from their respective elemental contributions f e kin := m eẍe , m e := Ω e ρ N T N da , f e int := Ω e g(φ h ) σ αβ + + σ αβ − N T ,α a h β da + Ω e g(φ h ) M αβ + + M αβ − N T ;αβ n h da , f e ext := Ω e N T p(φ) n h da + Ω e N T f α a h α da .(81) The terms σ αβ ± and M αβ ± are given by the energy split outlined in Sec. 4.2.1. In f e ext we have taken the boundary loads T and M acting on ∂S as zero. The extension to boundary loads can be found in Duong et al. (2017). Apart from the dependence on x e , the force f e int depends on φ e through the degradation of σ αβ + and M αβ + by g(φ h ). From a physical point of view the load-bearing capability vanishes in fully damaged regions where φ = 0. Thus, no pressure can act on the corresponding regions. We account for this by scaling the pressure linearly based on the phase field, i,.e. p(φ) = φp ,(82) withp denoting the pressure imposed on undamaged elements. Huge deformations and distorted elements at regions of full damage are prevented by means of the pressure function in Eq. (82). Putting everything together, the resulting equation system for the free nodes 4 reads f (x, φ) = Mẍ + f int (x, φ) − f ext (x, φ) = 0 .(83) The global mass matrix M is assembled from the elemental contributions m e . Spatial discretization of the phase field Inserting the approximations from Sec. 5.2 into the discretized weak form of Eq. (61) yields δφ T f kin +f int −f ext = 0 , ∀ δφ ∈ V h ,(84) where the global vectorsf kin ,f int ,f ext andf ir follow from the assembly of their corresponding elemental contributions Apart from the dependence on φ e , these expressions depend on x e through H. The resulting equations at the free nodes simplify tō f (x, φ) =f int (x, φ) = 0 .(86) Temporal discretization Generalized-α method The fully implicit generalized-α method of Chung and Hulbert (1993) is used as a monolithic time integration scheme. Given the quantities (x n ,ẋ n ,ẍ n , φ n ) at time t n , the new values (x n+1 ,ẋ n+1 ,ẍ n+1 , φ n+1 ) at time t n+1 need to be found. Additionally, equilibrium has to be fulfilled at intermediate states ( x n+α f ,ẋ n+α f ,ẍ n+αm , φ n+1 ), i.e. f x n+α f ,ẍ n+αm , φ n+1 f x n+α f , φ n+1 = 0 .(87) The complete scheme has been described in the work of Zimmermann et al. (2019). Since there are no temporal derivatives of the phase field in our framework, the corresponding equations simplify as outlined in Appendix A. Adaptive time-stepping The time step size should be chosen sufficiently small so that the crack does not propagate across too many elements in one time step. In contrast to this, large time steps can be used in cases of no crack propagation. This motivates the adaptive adjustment of the time step size. Since the phase field is not time-dependent, we cannot apply the adaptive time stepping scheme from Zimmermann et al. (2019). We therefore follow the subsequent approach: The need for smaller or the possibility of larger time steps can be indicated by the required number of Newton-Raphson iterations n NR during the last iteration, as for instance done by Schlüter et al. (2014). We adjust the new time step size at time step n + 1 as ∆t n+1 =            1.5 ∆t n , n NR < 4 1.1 ∆t n , n NR = 4 0.5 ∆t n , n NR > 4 0.2 ∆t n , local spatial refinement . The coefficients in Eq. (88) have been chosen based on the numerical examples presented in Sec. 6. Note that the time step size is also reduced after each spatial refinement step to ensure good convergence behavior. If not specified otherwise, a maximum time step size ∆t max = 0.1 T 0 and the initial time step size ∆t 0 = 10 −6 T 0 have been used for the numerical results 5 . Stabilization of jump conditions In Eqs. (46) and (52) and in the corresponding linearizations (cf. Appendix B), integrals of the form T 2 − T 2 ξ 2 χ J (ξ) dξ , with χ J (ξ) = 1,J(ξ) ≥ 1 0,J(ξ) < 1 ,(89) have to be computed. In the numerical examples presented in Sec. 6, we have observed that the jump function χ J (ξ) leads to convergence problems in which the Newton-Raphson iteration may alternate between different states. This occurs when the surface stretchJ(ξ) has values close to one so that χ J (ξ) may change its value after a Newton-Raphson update. We have tested two strategies to avoid these convergence problems: At first, an active set strategy can be employed. During a Newton-Raphson iteration the expressions in Eq. (89) are kept constant and the coupled system is solved for these values. Afterwards, the expressions are recomputed and another Newton-Raphson iteration is performed. This active set iteration is performed until either there is no change in the active set (the integral expressions), a maximum number of active set iterations is reached or the solution alternates again between different states. Since this strategy introduces another iteration it can increase the computational effort significantly. We thus propose another approach in which we smooth the discontinuity in χ J (ξ) bŷ χ J (ξ) := 1 1 + e −pχ J (ξ)−1 .(90) This regularization is illustrated in Fig. 4 for different values of the regularization parameter pχ ∈ (0, ∞). The black dashed line shows the discontinuous function. As the parameter pχ increases, the smoothed functionχ J (ξ) approximates the discontinuous function χ J (ξ) more precisely. By means of this smoothed function, the Newton-Raphson iteration does not alternate between different states and, in contrast to the active set strategy depicted above, no additional iteration is necessary. We note that an increase in the regularization parameter pχ leads to a decrease in the average time steps computed by the adaptive time-stepping scheme in Sec. 5.5.2. Dimensionless form The preceding formulation is normalized by introduction of the reference length L 0 , surface density ρ 0 6 and time T 0 . The corresponding dimensionless quantities are x = x L 0 , ρ = ρ ρ 0 , t = t T 0 .(91) The normalization quantities for the in-plane material parameters K and G, the bending modulus c and the critical energy density G c then follow as K = K E 0 , G = G E 0 , c = c E 0 L 0 , G c = G c E 0 L 0 ,(92) where E 0 := ρ 0 L 2 0 T −2 0 has units [N/m]. The surface stress σ αβ , the surface moment M αβ , the surface tension γ, the elastic energy density Ψ and potential Π are then given by σ αβ = σ αβ E 0 , M αβ = M αβ E 0 L 0 , γ = γ E 0 , Ψ = Ψ E 0 , Π = Π E 0 L 2 0 .(93) The temporal and spatial derivatives are (Zimmermann et al., 2019) ∂ . . . ∂t = T 0 . . . ∂t , ∇ S = L 0 ∇ S , ∆ S = L 2 0 ∆ S .(94) In the following, the superscript will be omitted for notational simplicity. Numerical examples This section shows several numerical examples of the proposed phase field formulation of brittle shells. The material parameters of the elastic energy density (cf. Sec. 4.2) are given via K = E ν (1 + ν) (1 − 2ν) , G = E 2 (1 + ν) , c = 0.1 E 0 L 0 ,(95) with stiffness E and Poisson's ratio ν. For all subsequently presented results, bi-quadratic LR NURBS are used and numerical integration on the bi-unit parent element is performed using Gaussian quadrature with 3×3 quadrature points. Numerical thickness integration is performed using four Gaussian quadrature points for all locations where the analytical integration formulae in Eqs. (47)-(48) do not apply. For the visualization, the surface tension γ = 1 2 N α α ,(96) is plotted, where N α α are the mixed components from the stress occurring in the equation of motion (18). All crack patterns are illustrated as follows: Red color resembles the fractured state (φ = 0) and blue color indicates undamaged material (φ = 1). In between these states, a transition based on the colors yellow-green-cyan is used. Remark: The examples in this section exhibit stress waves. The present formulation does not consider any damping such that stress waves do not dissipate but continue to propagate and reflect. An artificial damping, e.g. based on energy absorbing boundary elements, could be employed. Alternatively, physical viscosity can be introduced in the system, similar as is done by Zimmermann et al. (2019). The challenge for the latter is to correctly split the viscous terms in analogy to the elastic split outlined in Sec. 4.2.1. Especially, the propagation of stress waves over elements of different size needs to be investigated further. The stress waves may be emitted from the crack, thus they might start propagating in regions of the highest resolved meshes. As they cross mesh interfaces (where elements of different sizes meet) it can happen that very fine waves are not represented on the coarse mesh. A damping strategy across these mesh interfaces could be employed to capture the stress waves more accurately. 2D shear test The first example investigates crack evolution in a square two-dimensional membrane that is exposed to a shear load. The geometry including boundary and loading conditions is illustrated in Fig. 5. The mesh is initially constructed from 16 × 16 LR NURBS elements and the region next to the initial crack is refined by LR NURBS elements up to a refinement depth of d = 5, see Fig. 7. The material parameters are given in Tab. 1. The initial phase field distribution, which is induced by an initial history field, and the crack evolution are shown in Fig. 6. The crack evolves towards the bottom right corner on a curved path. The qualitative behavior resembles the results shown in the literature. For instance, in Borden et al. (2012) a quasi-static twodimensional shear test has been investigated where the crack path has been locally refined a E [E 0 ] ν [−] ∆ū [L 0 ] G c [E 0 L 0 ] 0 [L 0 ] T [L 0 ] 100 0.2 2 · 10 −6 0.001 0.0025 0.0125 priori based on analysis-suitable T-splines. Our results show that the split of the membrane energy from Sec. 4.2.1 works correctly since no branch is forming towards the specimen's top face. Based on the adaptive spatial refinement u = 0L0ū = 0.0094L0ū = 0.0128L0 Figure 6: 2D shear test: Crack propagation at various time steps. The energy split for the membrane part of the elastic energy density leads to the qualitatively correct crack path. strategy from Sec. 5.1.2, the LR mesh is refined as the crack evolves. The parametric domains of the LR meshes are illustrated in Fig. 7. Only the regions of damage are refined up to the prescribed refinement depth d = 5, while the periphery is kept coarse. Fig. 8 shows the time step sizes employed and the contributions to the total energy in the system. The latter have been computed from Π el Π frac Figure 8: 2D shear test: Computed time step sizes on the left and elastic and fracture energy over the prescribed deformationū on the right. As the crack evolves, the fracture energy increases whereas the elastic energy decreases due to the degradation of the contribution Ψ + el (cf. Eq. (29)). The fracture energy is non-zero atū = 0 L 0 since the initial crack is modeled by means of an initial phase field. the fracture energy is non-vanishing atū = 0 L 0 . The elastic energy increases steadily due to the applied deformation. As the crack evolves atū > 0.006 L 0 , the fracture energy increases, whereas the reduction of material stiffness leads to a decrease in elastic energy. Crack evolution takes place forū ∈ [0.006, 0.0128] L 0 . Π el = S g(φ)Ψ + el + Ψ − el da , and Π frac = S Ψ frac da .(97) Dynamic crack branching We next consider a rectangular 2D membrane with an initial crack at the top. The problem setup is shown in Fig. 9a. A displacement of constant velocity is applied on the top face upwards and on the bottom face downwards. At each time step we impose the deformation increment ∆ū =v ∆t where the maximum time step size is set to ∆t max = 10 −3 T 0 . The loading velocity is denotedv. The material parameters are depicted in Tab. 2. The initial mesh is constructed from 64 × 32 LR NURBS elements and refined around the prescribed initial damage up to E [E 0 ] ν [−] G c [E 0 L 0 ] 0 [L 0 ] T [L 0 ] 100 0.3 0.001 0.0025 0.0125 a refinement depth d = 3, see Fig. 9b. The initial crack is not located on the mid-line so Figure 10: Dynamic crack branching: Crack evolution at the final state for different loading velocitiesv. As the loading intensity is increased, crack branching occurs at an earlier time and closer to the left face of the membrane. (a)v = 1.25 · 10 −3 L0 T −1 0 (b)v = 5 · 10 −3 L0 T −1 0 (c)v = 1 · 10 −2 L0 T −1 0 (d)v = 2 · 10 −2 L0 T −1 0 that the resulting asymmetric stress distribution leads to a deflection of the crack towards the bottom face, see Fig. 10. As the figure also shows, a higher loading velocityv leads to more complex fracture patterns with branching occurring sooner and more often. This makes their prediction a priori to the simulation very difficult. Fig. 11 shows the final LR meshes in the undeformed configuration for the different crack patterns. There are large elements in regions of no fracture, whereas a highly resolved mesh is only obtained in the domain of fracture. Fig. 12 shows three snapshots of the crack evolution and the corresponding LR meshes for the loading intensityv = 2 · 10 −2 L 0 T −1 0 . The final states for these are shown in Figs. 10d and 11d. Only the periphery around the crack tip is refined, whereas no refinement is performed ahead of the crack tip. This adaptivity in space leads to an efficient prediction of fracture patterns. (a)v = 1.25 · 10 −3 L0 T −1 0 (b)v = 5 · 10 −3 L0 T −1 0 (c)v = 1 · 10 −2 L0 T −1 0 (d)v = 2 · 10 − Pressurized cylinder In this example we study crack propagation on a curved surface. In the previous sections plane membranes without bending energy have been studied. The new problem setup is illustrated in Fig. 13. The corresponding parameters, including the imposed pressurep (cf. Eq. (82)), are listed in Tab. 3. We note that the pressure is not ramped up over time but imposed as an initial pressure shock in the interior of the cylinder. The maximum pressure is then kept constant over time. Fig. 14 illustrates the phase field evolution over time. Elements with φ < 0.001 have been removed for visualization. The crack propagates in axial direction until it branches into two cracks at each end. These branches propagate towards the cylinder ends. The radius at these ends is fixed, which serves as a stiffener of the structure in these regions. Thus, the cracks are deflected and continue propagating in radial direction. This shows the ability of our model to capture crack evolution, branching and deflection on curved surfaces. Additionally, it proves that it is able to handle large deformations: The last state shown in Fig. 14 includes maximum stretches of approximately 130.49%. In Fig. 15 the LR meshes for three different time steps are shown. In between the branches it is not refined as much as in the areas of fracture. The regions of no damage are kept coarse completely. As the crack is deflected in E [E 0 ] ν [−]p [E 0 L −1 0 ] G c [E 0 L 0 ] 0 [L 0 ] T [L 0 ] 10 0.3 −0.1 0.00075 0.01 0.0125t = 0.719554 T 0 t = 1.187298 T 0 t = 1.776031 T 0 t = 2.056927 T 0 t = 2.373159 T 0 t = 2.715125 T 0 Conclusion We have coupled a higher order phase field model for brittle fracture with a nonlinear thin shell formulation based on a curvilinear surface description. Given a split of the constitutive law into membrane and bending contributions, a split of the elastic energy density has been derived for these terms separately. No spectral decomposition of the strain tensor is required in our formulation. Instead, the surface stretch indicates if there is a contribution to crack evolution or not. We have adopted a thickness integration to capture the asymmetric distribution of volumetric compression and expansion around the mid-plane that occur due to bending. As a consequence, the phase field is constant throughout the thickness and is solely defined on the deforming two-dimensional manifold. A discretization over the thickness or multiple phase fields have thus been avoided by this formulation. The interface between fractured and intact material has been adaptively refined based on the current phase field value. Quadratic LR NURBS have been used for this in the numerical examples. Time discretization is based on a fully implicit generalized-α scheme with adaptive time-stepping, and a monolithic Newton-Raphson procedure is used to solve the discretized coupled system. The examples presented in Sec. 6 include flat membranes and curved shells. For the flat cases, the results qualitatively resemble those presented in the literature. Studying crack propagation on a cylinder indicates the ability of our formulation to capture non-trivial fracture patterns on curved surfaces. It has been observed that the phase field value serves as a suitable indicator for local refinement since only areas along the crack paths are refined. The time step sizes are large if there is no crack evolution and are decreased as soon as the phase field starts evolving. Due to the adaptivity in space and time and the partially analytical thickness integration, the C 1 -continuous solution is achieved within a computationally efficient framework. Looking at the examples in Sec. 6, it does not seem to be necessary to keep a highly resolved mesh in regions of full damage (φ = 0). An adaptive coarsening strategy could be employed, which coarsens the mesh at fully damaged regions. Thus, small elements would only be retained close to the crack tip. A coarsening method for LR NURBS is given in Zimmermann and Sauer (2017). Additionally, stress wave propagation and reflection should be further investigated. Stress wave decay could be modeled by introducing physical viscosity into the system. The corresponding viscous energy and stresses then need to be appropriately split, similar to the energy split outlined in Sec. 4.2.1. The reflection of stress waves at the boundaries could be damped by employing energy absorbing boundary layers. The same could be employed at the interfaces, where different element sizes meet to prevent reflection of stress waves at these LR mesh boundaries. and then updated from iteration step i → i + 1 by x i+1 n+1 = x i n+1 + ∆x i+1 n+1 , x i+1 n+1 =ẋ i n+1 + ∆x i+1 n+1 1 γ ∆t , x i+1 n+1 =ẍ i n+1 + ∆x i+1 n+1 1 β ∆t 2 , φ i+1 n+1 = φ i n+1 + ∆φ i+1 n+1 ,(103) until convergence is achieved. At iteration i we check for the two convergence criteria max f i n+1 f 0 n+1 , f i n+1 f 0 n+1 ≤ tol dyn ,(104) with ... denoting the Euclidean norm and tol dyn = 10 −4 and f f · ∆x ∆φ ≤ tol nrg ,(105) with tol nrg = 10 −25 . B Linearization This section presents the respective elemental contributions for the tangent blocks in Eq. have to be defined. Since we assume the constitutive in-plane response to be fully decoupled from the out-of-plane response, it follows that d αβγδ = e αβγδ = 0. According to Eqs. (49) and (50), the first tangent matrix can be computed based on the contributions ∂τ αβ dil ∂a γδ = K 2 J 2 a αβ a γδ + J 2 − 1 a αβγδ , ∂τ αβ dev ∂a γδ = G 2J I 1 2 a αβ a γδ − I 1 a αβγδ − a αβ A γδ − A αβ a γδ . Based on Eqs. (52) and (53), the tangent matrix f αβγδ can be computed with the contribution ∂ 2Ψ bend (ξ) ∂b αβ ∂b γδ = ξ 2 12 T 3 c A αγ A βδ . Since we consider the fully linearized system in Eq. (100), we also need to linearize the mechanical force vector with respect to the phase field, i.e. ∆ φ f e = k e σφ + k e M φ ∆φ e , with k e σφ := G c g (φ) ∆ x H dA ∆x e ,(112) with ∆ x H := ∆ x max τ ∈[0,t] Ψ + el (x, τ ) ,(113) and ∆ x Ψ + el := τ αβ el,+ a α · N ,β + M αβ 0,+ n · N ;αβ . The linearization off e int with respect to the phase field variables of Ω e reads ∆ φf e int := k e 0 +k e el ∆φ e , The matricesk e 0 andk e el both contribute to the tangent blockK φ in Eq. (100). Figure 1 : 1Phase field profile for the fourth-order theory of Borden et al. (2014). The crack surface Γ = {0} is smoothed out by the function φ(x) in Eq. (31). Figure 3 : 3Refinement strategy Structured mesh: The blue shaded area resembles the support domain of a basis function, that is flagged for refinement. The dashed red lines are then inserted into the parameter domain.(Johannessen et al., 2014) 5.2 Spatial discretization of primary fieldsSubsequently, the finite element approximations of the surface deformation and the phase field are described. It follows the work of Sauer et al. (2014), Sauer et al. (2017), Duong et al. (2017) and Zimmermann et al. (2019). Let n e denote the number of spline basis functions on parametric element Ω e . They are numbered with global indices i 1 , . . . , i ne . The surface representation follows from this as Figure 4 : 4Smoothed jump function (cf. Eq. (90)) used to stabilize the Newton-Raphson solution scheme. Figure 5 : 52D shear test: Specimen geometry, boundary and loading conditions. Fig. 8 Figure 7 : 87shows that at the prescribed deformationū ∈ [0.001018, 0.004474] L 0 , the maximum time step size ∆t max = 0.1 T 0 is used since the crack is not evolving. Thus, the fracture energy stays constant during this time. Since the initial crack is modeled by means of an initial phase field, 2D shear test: Parametric domains of LR meshes at various time steps. Only damage regions are adaptively refined and a coarse mesh is kept in regions of no damage. Figure 9 : 9Dynamic crack branching: (a) Specimen geometry, boundary and loading conditions and (b) initial LR mesh in which the region around the initial crack is refined up to a refinement depth of d = 3. Figure 11 : 11Dynamic crack branching: Final LR meshes as a function of the the loading velocitȳ v. The corresponding crack patterns are illustrated inFig. 10. at t = 2 Figure 12 : 212Dynamic crack branching: Evolution of the phase field on the left and corresponding LR meshes on the right. The loading intensity isv = 2 · 10 −2 L 0 T −1 0 . The final phase field and LR mesh are shown in Figs. 10d and 11d. See also the supplementary movie at https://doi.org/10.5446/42540. Figure 13 : 13Pressurized cylinder: Specimen geometry and loading conditions shown in (a) top view and (b) side view. The dashed line indicates the shell's mid plane. The shell is symmetric across the solid line in (b) which is used to reduce comutational effort. The movement of the two ends is only allowed in the axial direction and not in the radial direction. (101). The linearization of the mechanical force vector f e := f e kin + f e int − f e ext of finite element Ω e with respect to the respective nodal positions x e can be found in the work of Duong et al. (2017). Since we model the pressure as a function of the phase field variable, we need to linearize the external force vector with respect to φ. This linearization of the pressure part f e extp of the external elemental force vector reads ∆ φ f e extp := Ω e N Tp n hN da ∆φ e .(106)For the linearization of the internal force vector the four material tangents c αβγδ := 2 ∂τ αβ ∂a γδ , d αβγδ := ∂τ αβ ∂b γδ , αβ := Jσ αβ and M αβ 0 := JM αβ has been used to map the integrals to the element domain in the reference configuration. According to Eq. (86), the linearization off e with respect to the respective nodal positions x The boundary terms vanish by choosing appropriate boundary conditions for δφ. Boundary conditions for φ are given in Eq. (63). The energy minimization problem now reads δΨ = 0. Since Eq. (58) holds true for all δφ, the Euler-Lagrange equation follows from applying the fundamental lemma of variational calculus, yielding∂Ψ ∂φ − ∂Ψ ∂φ ,α ,α + ∂Ψ ∂φ ;αβ ;αβ = 0 . Table 1 : 12D shear test: Material parameters and imposed load increment ∆ū per time step. Table 2 : 2Dynamic crack branching: Material parameters. Table 3 : 3Pressurized cylinder: Material parameters and imposed pressurep (cf. Eq. (82)). Figure 14: Pressurized cylinder: Crack pattern over time. The stretch reaches up to ≈ 130.49% showing that the proposed formulation can model large deformations. Elements with φ < 0.001 have been removed for visualization. See also the supplementary movie at https://doi.org/10.5446/42541. radial direction, the cylinder ends are refined up to the prescribed refinement level d = 3. The initial mesh consists of 4, 640 elements and 4, 572 control points and the final mesh consists of 35, 672 elements and 34, 756 control points. A uniformly refined mesh would have 131, 072 elements and 128, 777 control points. The surface tension γ(x, t) (cf. Eq. (96)) is visualized inFig. 16. Elements with φ < 0.001 have been removed for visualization. Before the crack reaches the cylinder ends the maximum values are obtained at the crack tips. Small values are obtained behind the crack tip due to the emitted stress waves. The magnitude of the surface tension at the remaining areas is fluctuating due to reflection of stress waves and their following interaction. At the final state inFig. 16, the largest stresses are obtained at the symmetry plane because the largest deformations occur there.Figure 15: Pressurized cylinder: LR meshes in the undeformed configuration during crack branching, before deflection and at the final state. See also the supplementary movie at https://doi.org/10.5446/42541.Figure 16: Pressurized cylinder: Surface tension γ (96) over time. Before the two branches reach the cylinder ends, the maximum values occur at the crack tip. Finally, the maximum values occur at the symmetry plane since the largest deformations occur there. Elements with φ < 0.001 have been removed for visualization.t = 1.187298 T 0 t = 2.056927 T 0 t = 2.715125 T 0 t = 1.187298 T 0 t = 1.776031 T 0 t = 2.373159 T 0 t = 2.715125 T 0 γ(x, t) [E 0 ] This is a part of the Saint Venant-Kirchhoff model, seeDuong et al. (2017). To avoid confusion, we write discrete arrays, such as the shape function array N, in roman font, whereas continuous tensors, such as the normal vector N , are written in italic font. The free nodes refer to the degrees of freedom, which are not given by boundary conditions. T0 refers to a reference time used to obtain a dimensionless formulation, see Sec. 5.7 Note that ρ0 is the surface density and has units [kg/m 2 ]. Appendix A Time integration schemeThe system in Eq. (87) with intermediate quantities and the quantities at time step n + 1x n+1 = x n + ∆tẋ n + 0.5 − β ∆t 2 ẍ n + β∆t 2ẍ n+1 , x n+1 =ẋ n + 1 − γ ∆t ẍ n + γ∆tẍ n+1 ,has to be solved. Here, ∆t = t n+1 −t n refers to the time step. Numerical dissipation is controlled by the parameters γ, β, α f and α m . They are expressed in terms of ρ ∞ ∈ [0, 1], which resembles an algorithmic parameter that corresponds to the spectral radius of the amplification matrix as ∆t → ∞ (seeChung and Hulbert (1993)for further details), i.e.We have found ρ ∞ = 0.5 to be a good choice and have used this in all computations. To solve the nonlinear system of equations in Eq. (87) using the Newton-Raphson procedure, it has to be linearized, i.e.where the tangent matrix blocks are computed fromThe required linearizations of the force vectors are shown in Appendix B. The initial guess for the Newton-Raphson iteration is set tox 0 n+1 = x n + ∆tẋ n + 0.5 − β ∆t 2 ẍ n + β∆t 2 ẍ 0 n+1 , x 0 n+1 =ẋ n ,x 0 n+1 =ẍ n γ − 1 γ , Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements. M Ambati, L De Lorenzis, Computer Methods in Applied Mechanics and Engineering. 312Ambati, M. and De Lorenzis, L. (2016). Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements. Computer Methods in Applied Mechanics and Engineering, 312:351-373. A review on phase-field models of brittle fracture and a new fast hybrid formulation. M Ambati, T Gerasimov, De Lorenzis, L , Computational Mechanics. 552Ambati, M., Gerasimov, T., and De Lorenzis, L. (2015). 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[ "SHOEMAKER-LEVY 9 IMPACT MODELING: I. HIGH-RESOLUTION 3D BOLIDES", "SHOEMAKER-LEVY 9 IMPACT MODELING: I. HIGH-RESOLUTION 3D BOLIDES" ]	[ "I N Apj ", "Press " ]	[]	[]	We have run high-resolution, three-dimensional, hydrodynamic simulations of the impact of comet Shoemaker-Levy 9 into the atmosphere of Jupiter. We find that the energy deposition profile is largely similar to the previous two-dimensional calculations of Mac Low & Zahnle, though perhaps somewhat broader in the range of height over which the energy is deposited. As with similar calculations for impacts into the Venusian atmosphere, there is considerable sensitivity in the results to small changes in the initial conditions, indicating dynamical chaos. We calculated the median depth of energy deposition (the height z at which 50% of the bolide's energy has been released) per run. The mean value among runs is ≈ 70 km below the 1-bar level, for a 1-km diameter impactor of porous ice of density ρ = 0.6 g cm −3 . The standard deviation among these runs is 14 km. We find little evidence of a trend in these results with the resolution of the calculations (up to 57 cells across the impactor radius, or 8.8-m resolution), suggesting that resolutions as low as 16 grid cells across the radius of the bolide may yield good results for this particular quantity.Visualization of the bolide breakup shows that the ice impactors were shredded and/or compressed in a complicated manner but evidently did not fragment into separate, coherent masses, unlike calculations for basalt impactors. The processes that destroy the impactor take place at significantly shallower levels in the atmosphere (∼ −40 km for a 1-km diameter bolide) but the shredded remains have enough inertia to carry them down another scale height or more before they lose their kinetic energy.Comparion basalt-impactor models shows that energy deposition curves for these objects have much less sensitivity to initial conditions than do ice impactors, which may reflect differences in the equation of state for the different kinds of objects, or a scale-dependent breakup phenomenology, with the preferred scale depending on impactor density.Models of impactors covering a ∼ 600-fold range of mass (m) show that larger impactors descend slightly deeper than expected from scaling the intercepted atmospheric column mass by the impactor mass. Instead, the intercepted column mass scales as m 1.2 .	10.1086/504702	[ "https://export.arxiv.org/pdf/astro-ph/0604079v1.pdf" ]	39,720,552	astro-ph/0604079	46c94493b01c99191f320227ef23fecf8cdf151c	SHOEMAKER-LEVY 9 IMPACT MODELING: I. HIGH-RESOLUTION 3D BOLIDES I N Apj Press SHOEMAKER-LEVY 9 IMPACT MODELING: I. HIGH-RESOLUTION 3D BOLIDES (Received 2005 November 7; Revised 2006 March 21; Accepted 2006 March 30) ApJ, in press.arXiv:astro-ph/0604079v1 4 Apr 2006 Preprint typeset using L A T E X style emulateapj v. 6/22/04Subject headings: hydrodynamics -comets: SL9 -planets and satellites: Jupiter We have run high-resolution, three-dimensional, hydrodynamic simulations of the impact of comet Shoemaker-Levy 9 into the atmosphere of Jupiter. We find that the energy deposition profile is largely similar to the previous two-dimensional calculations of Mac Low & Zahnle, though perhaps somewhat broader in the range of height over which the energy is deposited. As with similar calculations for impacts into the Venusian atmosphere, there is considerable sensitivity in the results to small changes in the initial conditions, indicating dynamical chaos. We calculated the median depth of energy deposition (the height z at which 50% of the bolide's energy has been released) per run. The mean value among runs is ≈ 70 km below the 1-bar level, for a 1-km diameter impactor of porous ice of density ρ = 0.6 g cm −3 . The standard deviation among these runs is 14 km. We find little evidence of a trend in these results with the resolution of the calculations (up to 57 cells across the impactor radius, or 8.8-m resolution), suggesting that resolutions as low as 16 grid cells across the radius of the bolide may yield good results for this particular quantity.Visualization of the bolide breakup shows that the ice impactors were shredded and/or compressed in a complicated manner but evidently did not fragment into separate, coherent masses, unlike calculations for basalt impactors. The processes that destroy the impactor take place at significantly shallower levels in the atmosphere (∼ −40 km for a 1-km diameter bolide) but the shredded remains have enough inertia to carry them down another scale height or more before they lose their kinetic energy.Comparion basalt-impactor models shows that energy deposition curves for these objects have much less sensitivity to initial conditions than do ice impactors, which may reflect differences in the equation of state for the different kinds of objects, or a scale-dependent breakup phenomenology, with the preferred scale depending on impactor density.Models of impactors covering a ∼ 600-fold range of mass (m) show that larger impactors descend slightly deeper than expected from scaling the intercepted atmospheric column mass by the impactor mass. Instead, the intercepted column mass scales as m 1.2 . INTRODUCTION For the week starting 16 July 1994, fragments of comet Shoemaker-Levy 9 (SL9) hit Jupiter. Most of the world's telescopes observed the event, collecting an unprecedented volume of imaging, photometry, and spectroscopy that spanned all sensible wavelengths. Papers in collections edited by West & Böhnhardt (1995) and Noll et al. (1996) review the data and early interpretation. Harrington et al. (2004) review the phenomenology, efforts to understand the impact phenomena, and open questions about the impacts. A brief summary of points relevant to this paper follows. The impacts followed the same basic phenomenology. The orbital path of the comet fragments intersected the planetary surface at ∼45 • S latitude, and planetary rotation arranged for a girdle of well-separated impacts there. Each impactor fell into the atmosphere at over 60 km s −1 and an impact angle of about 45 • (Chodas & Yeomans 1996). The ground track of the impactors moved toward the northwest. The bolides crushed, ablated, and decelerated as they fell through the atmosphere, leaving an entry channel filled with superheated gas (e.g., Ahrens et al. 1994a,b;Boslough et al. 1995;Boslough & Gladstone 1997;Crawford 1996;Crawford et al. 1994;Mac Low 1996;Takata et al. 1994;Takata & Ahrens 1997;Shuvalov et al. 1999). This gas rushed back out the entry channel, exiting the atmosphere at an angle and flying ballistically into space. The Hubble Space Telescope (HST) resolved the plumes, and saw several rise to ∼3000 km above the cloud tops (Hammel et al. 1995). Material invisible from Earth rose higher still, as shown by its effects on the magnetosphere (Brecht et al. 2001(Brecht et al. , 1995. The plumes collapsed and re-entered the atmosphere in 20 minutes, heating it and leading to infrared emission so strong that it was dubbed the "main event". In the initial hours after impact, two different systems of expanding rings were seen, one in the visible by HST (Hammel et al. 1995) and the other in the infrared by McGregor et al. (1996). Peculiar patterns remained in the atmosphere, which were spread in latitude over the ensuing weeks (Banfield et al. 1996). Chemistry was just as exciting. Main-event spectra were complex, but only five species were identified. Dozens appeared in the aftermath, and CO and CO 2 are observed to this day, now having crossed the equator into the northern hemisphere Lellouch et al. 2002). Not to be outdone, the magnetosphere responded strongly and in a manner that changed throughout the week of impacts as ionized plume material loaded Jupiter's magnetic field lines. Initially viewed as perfect perturbation experiments, the impacts instead involved complex phenomena at fine spatial and temporal timescales. Models capable of reproducing many of the observed phenomena were too complex by several or-ders of magnitude for the computational power available at the time. Many observers, including most spectroscopists, still have unpublished data that they cannot interpret. We made some headway in explaining phenomena of the images and lightcurves with simplified calculations: extrapolated a published blowout velocity distribution to calculate the fluxes of mass, energy, and momentum on the atmosphere. used those fluxes to drive a twodimensional (2D) vertical-slice model of the atmosphere. The resulting lightcurves are a good match to data from 0.9 -12 µm, and the time-dependent pressure (p) and temperature (T ) grids show phenomena that mimic McGregor's Ring and other effects. Encouraged by this initial success, we have undertaken a study to model the impacts and their aftermath sufficiently to reproduce, in a realistic manner, all of the observed phenomena in the impact, blowout, plume flight, and plume splash phases. By "realistic", we mean that wherever we can eliminate an ad-hoc assumption or an analytic approximation, we do. We no longer use arbitrary initial conditions. Rather, we will use observations and the results of prior models for initialization, and will ultimately produce synthetic images, lightcurves, and spectra derived with radiative transfer from model results. For physical atmospheric effects, our approach is the chaining of hydrodynamic models outlined by Harrington et al. (2004). Our chemical models will be driven by tracer particle histories from the physical models. This paper presents the first results from our impact model. Since the observations of this phase were not as constraining as those of later phases, our primary goal was to produce a data grid with which to initialize subsequent models. Doing this believably required a look at, e.g., resolution convergence and sensitivity to initial conditions. Also, and as with any interesting investigation, there are serendipitous findings that reach beyond this particular set of impacts. Korycansky et al. (2002) and Korycansky & Zahnle (2003) have made 3D calculations of the impact of asteroids into the atmosphere of Venus using ZEUS-MP and its predecessor ZEUS3D. There are a number of similarities between the results of that work and the present study that we note below. Comparable hydrodynamical simulations of the initial phase of the impacts have been previously described by a number of groups Crawford et al. 1994;Gryaznov et al. 1994;Takata et al. 1994;Yabe et al. 1994;Shoemaker et al. 1995;Crawford et al. 1995;Svetsov 1995;Yabe et al. 1995;Crawford 1997;Shuvalov et al. 1997Shuvalov et al. , 1999. Section 2 presents our model. We describe the results of over a dozen runs in section 3, and discuss their implications and future work in section 4. IMPACT MODEL For the calculations described in this paper we employed the ZEUS-MP hydrodynamics code (Norman 2000), which solves the equations for three-dimensional compressible gas flow. We have modified ZEUS-MP (base version 1.0) to include multiple materials. Modifications to the code are described in more detail by Korycansky et al. (2002) and Korycansky & Zahnle (2003). We have not included radiative transfer in ZEUS-MP; the short timescale and high optical depth of the atmosphere below 1 bar, where the main disruption of the impactors and energy deposition takes place, makes it unlikely that radiative transfer would significantly affect the dynamics. That assumption has been tested by Shuvalov et al. (1999), who indeed found that dynamical effects of radiation transfer were insignificant, but that the impacting comets would be strongly heated at large depths. The Jovian atmospheric profile comes from . We also included minor modifications such as a moving grid that tracks the impactor at a variable velocity as it decelerates. Multiple materials were handled by the integration of tracer variables advected in the flow. For non-porous material, the tracer C gives the fraction of mass in the cell that is impactor material. Porosity is tracked by an additional tracer, and is treated with the so-called p − α model for a strengthless, porous solid (Menikoff & Kober 1999). The coordinate system (x 1 , x 2 , x 3 ) is Cartesian and aligned with the bolide's initial velocity such that x 1 is the along-track coordinate, x 2 is horizontal, and x 3 is perpendicular to the others. We relate (x 1 , x 2 , x 3 ) to local Cartesian coordinates (x, y, z) as follows: x = x 2 (1) y = −x 1 sin θ + x 3 cos θ (2) z = x 1 cos θ + x 3 sin θ, (3) where θ is the angle between x 1 and the vertical. Note that local coordinates are not cardinal directions. Fluid velocities in the x 1 , x 2 , and x 3 directions are v 1 , v 2 , and v 3 , respectively. Other quantities that appear in the equations are the density, ρ and the internal energy per unit volume, e. The spatial resolution of the calculations is described by the notation Rn, where n is the number of grid cells across the radius of the body in the high-resolution part of the grid. Away from an inner block of dimensions 4 × 2 × 2 km, the grid spacing increases geometrically (by a factor ∼ 1.04 per grid cell, depending on the overall resolution). The computational grid moves with the impactor and decelerates, keeping the object's front end about 1 km from the front end of the grid so that the object remains in the high-resolution portion as it disrupts. Calculations in the paper were made with resolutions of R16, R32, and R57. We used the Tillotson equation of state, which was formulated for high-velocity impacts (Tillotson 1962;Melosh 1989), although it cannot represent melting or mixed twophase (gas-liquid) regimes. The Tillotson EOS has two regimes, one for cold and/or compressed material, the other for rarefied, hot conditions. For a given mass density of impactor material ρ and internal energy density (energy per unit volume) e, we have E = e/ρ, η = ρ/ρ T , q = η − 1, where ρ T is the density at zero pressure and E is the specific energy. Then, for compressed states (η > 1) and expanded ones (η < 1) where E < E iv , the incipient vaporization energy, the pressure P is the sum of a thermal term and a cold-pressure term: P = P l = a + b 1 + E/(E T η 2 ) e + Aq + Bq 2 1 + e −Kcq ,(4) The parameters a, b, A, B, and E T are described in more detail and listed for common substances by Melosh (1989); K c is described below. For expanded states for which E > E cv , the energy of complete vaporization, P = P h = ae + be 1 + E/(E T η 2 ) + Aq 1 + e −Kcq e −5(1/η−1) 2 .(5) We modify the cold-pressure terms Aq + Bq 2 , and Aq in Eqs. (4) and (5) respectively, by a term of the form 1 + e −Kcq in the denominator, in order to provide a low-density cutoff as recommended by Melosh (1989). We set K c = 1000. A strengthless material is assumed, so negative pressures (tensions) are not sustainable; the pressure cutoff for low densities mimics that effect. For expanded states in which E iv < E < E cv , P is linearly interpolated between P l and P h . Then, on the grid, the resulting Tillotson pressure is weighted by the tracer C. The equation of state parameters for our materials (ice and basalt) are the same as used by Benz & Asphaug (1999) and are listed in table I. (As discussed below, we modeled basalt impactors in order to study the effect of the equation of state on impactor breakup.) For porosity, we use a simple model for a strengthless, porous solid based on the "p − α" formulation (Herrmann 1969;Menikoff & Kober 1999). We distinguish between the solid material density ρ m (and energy e m ) and the same quantities for fluid in the porous void space (ρ m and e m ). Space and material quantities are related by the porosity ǫ, where 0 < ǫ < 1: ρ = (1 − ǫ)ρ m and e = (1 − ǫ)e m . The material pressure p m is provided by the equation of state p m (ρ m , e m ) (the Tillotson EOS described above) and the space pressure is thus p = (1 − ǫ)p m . Initially the porosity is 1−(initial density/nominal density) of the material. During the calculation, porosity is advected with the flow like the material tracer C. At each timestep the advected value of ǫ is then adjusted by comparison with a function ǫ f (p, p c ) = ǫ 0 (1 − p/p c ) 2 that depends on a "crushing pressure", p c , which for ice we set to 10 7 dyne cm −2 . For p > p c , ǫ f = 0. The porosity is given by ǫ = min(ǫ, ǫ f ). During the calculation, porosity can only decrease from its initial value; if porosity is crushed out of a mass element, it does not return even if the pressure drops back below p c . The Tillotson EOS has been used for SL9 calculations by Gryaznov et al. (1994) and Takata et al. (1994). Other equations of state for cometary material used in previous calculations include perfect gas (with adiabatic indices γ = 1.2 and 1.4), (Mac Low 1996), and a "stiffened gas" or Murnaghan EOS with an additional perfect-gas thermal contribution (Mac . A similar stiff equation of state was used by Yabe et al. (1994Yabe et al. ( , 1995. Boslough et al. (1994), Crawford et al. (1994Crawford et al. ( , 1995, and Crawford (1997) used a tabular version of the ANEOS equation of state that took into account melting and vaporization. Shuvalov et al. (1999) used a combination of a non-linear Gruneisen EOS in which the pressure increased roughly as the cube of the density ratio ρ/ρ T combined with a tabular gas EOS for the vaporized component. The most important factors in the various equations of state are probably the relative stiffness dP/dρ and the presence or absence of a dependence of pressure on the internal energy e. A relatively stiff EOS, and one in which P is also a function of e, might be expected to result in calculations that show impactors blowing up at greater altitudes and exhibiting more radial spreading ("pancaking") than otherwise. This may explain some of the differences among the results that have been found from different studies. We will not systematically explore that question in this paper, but in view of the differences we found (discussed below) in results between ice and basalt impactors, we plan to address the issue in future work. Our "standard case" is a 1-km-diameter, porous ice sphere impacting the atmosphere at v = 61.46 km s −1 and θ = 43.09 • . The velocity and impact angle are the means of those of the 21 comet fragments (Chodas & Yeomans 1996) in a frame rotating with Jupiter at the average latitude of the impacts (−44.02 • ). The gravitational acceleration at that latitude (including the J2 and centrifugal terms) is g = 2512 cm s −2 . The bulk density of the bolide is ρ = 0.6 g cm −3 (Asphaug & Benz 1994, 1996Solem 1994Solem , 1995. We have also carried out calculations of impacts with objects of bulk density ρ = 0.1 and 0.92 g cm −3 , the latter corresponding to non-porous solid ice. In addition, we have done calculations with impactors with volumes of 0.2, 3, 40, and 125 times that of the standard case, or diameters of 0.584, 1.44, 3.42, and 5 km. Not all combinations of density, diameters, and resolutions were run. One important result that emerged from the Venusatmosphere calculations was the significant extent to which the results were sensitively dependent on initial conditions in a manner reminiscent of dynamical chaos (Korycansky & Zahnle 2003). Two calculations whose initial conditions (for instance, impact velocity) differed by very small amounts (∼ 0.1%) would develop large divergences in the course of their respective simulations. Korycansky et al. (2000) took the view that the basic process of impactor disruption was due to the growth of Rayleigh-Taylor and Kelvin-Helmholtz instabilities (Field & Ferrara 1995), and it is plausible that the seeds of the perturbations that grow to saturation are irregularities due to the finite resolution on the grid. In the physical case, one would expect analogous irregularities from the inevitable bumpiness of the bolide's surface. While the basic process (fragmentation and ablation) was always the same, there could be significant differences in the results of different runs for impactors of the same gross properties. For example, the diameter of the resulting crater on the surface of Venus might vary by several kilometers (on a scale of ∼ 10 km) depending on the exact details of how the event had unfolded. In the limit where the bolide just reached the surface, different cases might result in anything from a single crater several km in diameter, a group of small craters, or no crater at all. We might expect similar behavior to obtain in this case, and we have attempted to sample the distribution of results by performing several runs of the standard case with small differences in initial impact velocity ∆v of 0.1% of the initial velocity v (≈ 0.06 km s −1 ) or displacements in the cross-track coordinates ∆x 2 , ∆x 3 by one half of a grid cell. For R16 calculations of the 1-km bolide the displacements are 15 m, for R32 they are 7.5 m, and for R57 they are 4.4 m. We use several diagnostics to characterize the calculations. The simplest and most significant is the profile of deposition of impactor kinetic energy dE/dz as a function of height z in the atmosphere, the same quantity as discussed by Mac and Mac Low (1996). The impactor kinetic energy E = E(z) was calculated by integrating the flux of kinetic energy passing through a given height z: E(z) = dt z ρC (v 2 1 + v 2 2 + v 2 3 ) 2 v 1 dx 2 dx 3 .(6) The area integral is taken over all cells whose height z = x 1 cos θ + x 3 sin θ is the desired value. [Note that the projection factors cos θ in the element of area in the z plane (dA = dx 2 dx 3 / cos θ) and the downward velocity (v z = v 1 cos θ) cancel.] The time integral extends to the end of the calculation (typically 8-10 seconds). The density on the grid is weighted by the advected tracer field C that tags material that belongs to the impactor. We computed E(z) for −200 ≤ z ≤ 100 km at 1-km intervals. (For more massive bolides, E(z) was calculated as far down as needed.) Having integrated E(z) for all heights, the deposition profile dE/dz follows by numerical differentiation. For a subset of the runs, we calculated spatially resolved plots of mass flux, similar to those employed by material ρ T (g cm −3 ) a b A (erg cm −3 ) B (erg cm −3 ) E T (erg g −1 ) E iv (erg g −1 ) Ecv (erg g −1 ) ice 0.917 0.3 0.1 9.47 × 10 10 9.47 × 10 10 1.0 × 10 11 7.73 × 10 9 3.04 × 10 10 basalt 2.70 0.5 1.5 2.67 × 10 11 2.67 × 10 11 4.87 × 10 12 4.72 × 10 10 1.82 × 10 11 Korycansky & Zahnle (2004) to study the fragmentation of asteroids in the Venusian atmosphere. Here we hope to correlate events such as fragmentation, as revealed in the mass-flux plot, to specific features (like peaks) in the deposition profile. The time-integrated mass flux µ(z; x, y) at a height z is given by µ(z; x, y) = − ρ(z; x, y)C(z; x, y)v 1 (z; x, y)dt,(7) where z translates to the tilted plane x 1 cos θ + x 3 sin θ in the grid coordinate system. In practice, µ is calculated as the accumulation of a set of integrals at time slices t i , each integrated over the interval t i − ∆t i /2 to t i + ∆t i /2, by assuming that material moves at a constant velocity during that interval and counting all the mass that has crossed or will cross the plane during the interval. Due to the tilt of the grid, the ylocation of the impactor is a function of z; in this case we refer the position of material on the plane to the centerline position defined by x 2 = x 3 = 0. The calculations were performed on a Beowulf cluster using 32 2.4 GHz Opteron processors. The largest (R57) runs took ∼ 3.1 × 10 7 cpu seconds on a grid of 712 × 356 × 356 = 9.0 × 10 7 points. Impactors of 1 km diameter took ∼ 8 − 10 model seconds for the velocity to fall to 0.001× the initial velocity, which was the condition for stopping the calculation. Timesteps were ∼ 5 × 10 −5 s during the initial phases of the impact, increasing to ∼ 3 × 10 −4 s by the end, as the impactor slowed down. High-resolution output was written to disk every 0.1 model seconds, as noted above. The R57 calculations occupied 17 GB of memory and produced ∼ 210 GB of output data. Not all data were saved from all runs. Wall-clock time for an R57 run was about three weeks. RESULTS Our main results are presented in Figs. 1 and 2, which show the profile of kinetic energy deposition (dE/dz) for a number of realizations of the standard case described above. Figs. 1 and 2 show much the same information, plotted in two different ways to emphasize two different points. Each panel shows profiles generated by calculations with very slight differences in initial conditions, as described above. The panels show calculations done at different resolutions: R16, R32, and R57. In Fig. 1, the horizontal scale is logarithmic, to facilitate a comparison to the results of Mac and Crawford (1996). For the most part previous calculations have been 2D axisymmetric calculations at moderately high resolutions (R25) with finite-difference (grid-based) hydrocodes. The exception is the calculation by Takata et al. (1994), a 3D calculation employing the smooth-particle hydrodynamics (SPH) method. Several groups found that a 1-km object penetrated more deeply than we found, to depths well below −100 km, P > 15 bar Crawford et al. 1994;Takata et al. 1994;Crawford et al. 1995). Shuvalov et al. (1999) found different results, partly due to the different density and structure of the objects they model. Some of their calculations were made for an object of ρ=0.22 g cm −3 . They also investigated the impacts of objects of non-uniform density: a dense core (ρ = 1 g cm −3 ) and surrounding shell of low density (ρ = 0.1 g cm −3 ), or a low-density object with high-density inclusions. Their objects had the same diameter (1 km) but were about 1/3 as massive as our standard object. Shuvalov et al. found a rather broad, double-peaked distribution of energy deposition. Our results are most similar to those of Mac and Crawford (1997). The energy-release profile is sharply peaked, though not so strongly as found by Mac Low and Zahnle; a small amount of energy is deposited at levels below 100 km. It is apparent that the sensitivity to initial conditions described above also obtains for these simulations. This is brought out more strongly in Fig. 2, in which the horizontal scale is linear. An important question about simulations of this type is the degree of convergence that they exhibit as a function of numerical parameters such as resolution. In this case convergence means that extrapolation to infinite resolution of a series of models would yield a series of results that converged to the correct answer. Ideally, the limit of resolution-independent results is reached while we are still in the realizable limit of finite resolution. The question in this case is complicated by the sensitivity to initial conditions discussed above, since a degree of scatter in the results is introduced, as can be seen in Fig. 2. The scatter tends to obscure trends in the results as a function of resolution. We must thus compare the results as a function of resolution on a statistical basis. Also included among the runs plotted in Fig. 2 are two calculations (at resolutions R16 and R32) of bolides shaped like the asteroid 4769 Castalia but with the same volume and density as our standard case. These runs are a test of the influence of the bolide's shape on the outcome, the object being in these cases an oblong object with axis ratios 2 : 1 : 0.8 and perhaps a representative shape for non-spherical impactors. The object in these cases was oriented end-on to the direction of motion. The particular model shape was already available for use, having been extensively used in the calculations performed by Korycansky & Zahnle (2003). The results of the runs of Castalia-shaped impactors do not appear to be radically different from spherical ones. Large changes in impactor shape do not seem to affect the outcome more than tiny changes in the initial position or velocity. Presumably, very extreme shapes (e.g., needle-like or plate-shaped objects) would have an effect, but moderately oblong shapes are not a strong influence. For a relatively fragile impactor, it is probably true that the initial shape is quickly "forgotten" as the incoming object is rapidly deformed by aerodynamic forces, and that the same effects that apply to our spherical impactors also apply here. We expect that a sequence of calculations for irregular objects, with small changes in initial conditions, would show a similar degree of scatter in the results. Crawford et al. (1995) also simulated the impact of 2D, nonspherical objects (cylinders with length/diameter ratios of 1:3 and 3:1) and found a signif-icant but not overwhelming dependence on body shape; paradoxically, the 3:1, rod-shaped impactor penetrated the least deeply among the three cases tested. Figure 3 shows various statistical measures of the energy deposition, in particular several different depths that characterize the results. Included in the plot are the mean depth (z = z(dE/dz)dz/E) or the first moment of the energy deposition curve, the mode depth z, (the depth of peak energy deposition), and the depths, z n , at which n% of the bolide's energy has been deposited for n = 90, 50, and 10 (z 50 is the median depth). The mean and median depths are quite similar despite the skewness of the deposition profiles, while the peak energy-deposition depths z are somewhat deeper. The trend (if any) of these measures as a function of resolution is weaker than the amount of scatter, as seen by the standard deviation. In particular the R57 runs give approximately the same results as the ones at lower resolution. Statistical tests (t-and F-tests) applied to the various measures of depth for R16 and R57 find that the differences between them are not statistically significant. However, the variance of the R16 results is 3 − 4× larger than those of R57, and if the same results persisted after ∼ 10 more cases were run, a significant result might emerge. That is, it is possible that the amount of scatter in the results is smaller for high resolution runs. We also note that the results for the Castalia runs do appear to result in systematically slightly shallower depths than the means at R16 and R32. However, only one Castalia run was done for R16 and R32, so the apparent results may not be significant. We also examined the spatially-resolved, integrated fluxes µ(z; x, y) described in the previous section. These provide clues as to exactly how an impact proceeds, in terms of understanding the processes of fragmentation, ablation, and impactor spreading due to aerodynamic forces (pressure gradients). The last process has been denoted "pancaking" (Zahnle 1992;Chyba et al. 1993) and has been modeled semi-analytically and applied to SL9 impacts by several groups (Chevalier & Sarazin 1994;Field & Ferrara 1995;Crawford 1997). We will not make such a model in this paper, but simply discuss the hydrodynamical results. Figure 4 shows the time-integrated flux µ(z; x, y) calculated at various heights z relative to the 1-bar level for the R57 run. The flux is plotted on a logarithmic gray scale for µ > 5 × 10 2 g cm −2 , which emphasizes relatively small values of µ. Values of 5 × 10 4 g cm −2 and above are shown in the deepest shade (black). (For comparison, note that a column 1 km high of ρ = 0.6 g cm −3 has a surface density of 6 × 10 4 g cm −2 .) Only the inner 4 × 4 km region of the projected grid is shown. Due to the zenith angle of the impact (≈ 44 • ) the "footprint" of the impact is elongated in the +y direction in the plots; the effect is most noticeable in the z = 20 km plot, in which the bolide is not yet strongly affected by the atmosphere. The object is torn apart quite high in the atmosphere (approximately between z = −20 and −50 km) compared to the region below -50 km, where most of the energy is deposited. Despite the explosion-like character of the impact, the crushed bolide has enough inertia to carry it down another scale height before it stops and deposits its kinetic energy. The same behavior was noted by Shuvalov et al. (1999) in their calculations. In Fig. 5 we show a similar quantity to µ for the R57 run, namely the column density σ in the (x 1 , x 3 ) plane, or a "side view" of the impactor generated by integrating the density of impactor material in the x 2 direction: σ(x 1 , x 3 ) = ρCdx 2 .(8) Note that σ is not time-integrated; we show particular instants in the calculation when the bolide is passing through z levels that are approximately the same as those shown in Fig. 4. Note also that Fig. 5 is plotted in grid coordinates (x 1 , x 3 ), which are related to z by Eq. 3. Figure 5 shows the compression and disruption of the bolide from a different point of view. The most interesting part of Fig. 5 is the initial compression of the bolide seen at t = 2.7 s, followed by the shredding and sweeping back of material for t > 3 s. As noted above for Fig. 4, the lateral spreading of the impactor during its initial compression is not larger than a factor of 2 or so. A notable feature is the character of the impactor disruption. The impactor is apparently shredded and crushed, but does not seem to fragment into large pieces that separate at significant velocity. Non-axisymmetric filamentary structures develop and then expand into a cloud of material. This behavior is different from what has been seen in calculations of asteroid impacts into the atmosphere of Venus (Korycansky et al. 2002;Korycansky & Zahnle 2003) and inferred from craters on Venus (Korycansky & Zahnle 2004) and the Earth (Passey & Melosh 1980). Other calculations at lower resolution (R32 and R16) show similar behavior. Given that the same code was used for Venus calculations (Korycansky & Zahnle 2003) and the calculations are in many ways very similar, we conclude that the material of the bolide (porous ice vs. rock) and its compressibility must control the character of impactor break-up in these physical situations. To test this idea, we ran an R32 simulation of a 1-km spherical impactor of non-porous basalt (ρ = 2.7 g cm −3 ) with otherwise identical conditions to our standard case. The results are shown in Fig. 6, where we compare an R32 porous-ice calculation (top row) with the basalt impactor (bottom row) at selected heights in the atmosphere. Because the basalt bolide is ∼ 5 times as massive as the ice one, it penetrates more deeply, as reflected in the choice of heights in Fig. 6. Of more interest is the impactor breakup: the basalt object appears to break into several fragments and spread out (pancake) considerably more than does the ice impactor, whose degree of pancaking is rather modest, less than a factor of two in radius. However, to assess the degree to which the pancake model does or does not match the behavior seen in Figs. 4 -6 requires quantitative modeling that we postpone to future work. Fig. 7 shows an additional exploration of the possible outcomes due to differing substances. We ran five R16 impact calculations for 1-km diameter impactors of non-porous ice and porous and non-porous basalt. The differing impactor masses largely account for the variation of average penetration depth per impactor type. However, the differences in the shapes and the level of variation among different trials is unexpected. The difference in outcomes between ice and basalt is marked; the icy impactors show much greater variation in energy deposition (and a much greater overall spread in height) than do the basalt impactors. Differences in initial porosity seem to have little effect compared with the difference in material. Presumably this is due to differences in coefficients in the EOS. We speculate that the greater stiffness of the basalt EOS results in greater pancaking (as in Fig. 6) and (somewhat paradoxically) less variation in the disruption and energy deposition as a result. One way to examine the question is to run models with simplified and artificially Fig. 1, but now with a linear scale on the horizontal axis that emphasizes the differences in energy deposition among the runs. Also included among the plotted runs are two calculations (at resolutions R16 and R32) of impactors shaped like the asteroid 4769 Castalia. Both these runs fall among the main group of runs at each resolution, indicating that initial macroscopic impactor shape does not strongly affect energy-deposition behavior. x, y) as defined in Eq. 7 for the R57 run at various heights z relative to the 1-bar level. The fluxes are plotted on a logarithmic gray scale for values of µ > 5 × 10 2 g cm −2 and the darkest shade (black) corresponds to µ > 5 × 10 4 g cm −2 . Only a grid subsection is shown. stiffened or softened equations of state. Understanding this result would reveal fundamental impact physics and may enable simplified models of atmospheric impacts that do not require extensive hydrodynamic simulation. We hope to address this question in more detail in the future. As noted above, we have also run simulations of the impact of bolides of differing masses corresponding to masses 0.2×, 3×, 40×, and 125× that of the standard case. The corresponding diameters are 0.584, 1.44, 3.42, and 5 km. We ran 5 cases of each diameter at R16 resolution, perturbing the initial positions in x 2 and x 3 by one-half grid cell. The results are shown in Fig. 8, where we plot the median depths of energy deposition z 10 , z 50 , and z 90 in the top panel as a function of bolide mass. The bottom panel shows the same result, but now we plot the corresponding atmosphere columns µ 10,50,90 = ∞ z10,50,90 ρ dz times the initial bolide cross-section A = πr 2 , normalized by the bolide mass m. Least-squares fits for z and µ are: z 10 = −9.50 × 10 −4 (m/g) 0.309 km, z 50 = −5.19 × 10 −3 (m/g) 0.283 km, z 90 = −1.82 × 10 −2 (m/g) 0.257 km, µ 10 A m = 8.16 × 10 −3 (m/g) 0.041 , µ 50 A m = 1.20 × 10 −4 (m/g) 0.199 , µ 90 A m = 1.03 × 10 −4 (m/g) 0.220 . Note that the same sensitivity to initial conditions appears to obtain across a 600-fold range in impactor mass. We expect that impactors would in general penetrate to column depths equivalent to their mass, i.e., µA ∼ m, as was found roughly to be the case by Korycansky & Zahnle (2003) for impacts into the Venusian atmosphere. Thus, we would expect µA ∝ m. As seen in Eq. 9 there is a weak dependence on impactor mass. More massive impactors penetrate somewhat more deeply than would be expected from a strictly proportional relation. x 3 (km) x 1 (km) -Column densities σ in the x 1 − x 3 plane (horizontal and vertical, respectively, for one R57 run. The times and heights roughly correspond to the passage of the bolide through the z planes given in the same panels in Fig. 4. Shading is the same as for Fig. 4, substituting σ for µ. Only a grid subsection is shown. To gain understanding of the SL9 impacts, we have carried out a number of 3D, hydrodynamic simulations of the impact of porous ice comets into the atmosphere of Jupiter. We employed the numerical hydrodynamics code ZEUS-MP, with some modifications to track the comet material (ice), its equation of state, and the degree of porosity, if present. Calculations were carried out at three different resolutions Rn, where n is the number of resolution elements across the bolide radius (for spherical impactors): R16, R32, and R57. We have paid special attention to the profile of energy deposition in the atmosphere, as a measure of how deeply the bolide penetrated and for comparison with previous (mostly 2D) simulations. We carried out several calculations of a "standard case" (a 1km-diameter, porous, ice comet with ρ = 0.6 g cm −3 and initial velocity like that of the SL9 impactors) with tiny variations in initial velocity or shifts in cross-wise position on the computational grid, in order to test for sensitivity of the results to initial conditions, and to sample the distribution of results if the sensitivity were present. Such multiple calculations were carried out at all three of our resolutions to see if there was a convergence trend in the results. Two low and medium resolution calculations were also done of an impactor in the shape of the asteroid 4769 Castalia to see if there were noticeable differences for a non-spherical impactor. Energy-deposition profiles were fairly similar to those found for 2D calculations such as those done by , though they were slightly less sharply peaked. The median depth of energy deposition was ≈ 70 ± 14 km below the 1-bar level, at an atmospheric pressure of ≈10 bars. The aforementioned sensitivity of the results to small changes in initial conditions produced significant variations in the energy deposition profiles, and the error bar just given refers to standard deviation of the individual profiles. Comparing the results of calculations at different resolutions shows very little trend in the results, compared to their scatter. This suggests that, for the purpose of determining the depth of penetration of impactors, relatively low resolution (R16) is sufficient. We have visualized some of the calculations to learn about the impact process and how a bolide responds to aerodynamic x, y) as defined in Eq. 7 for two R32 runs at various heights, z, relative to the 1-bar level. The top row is an ice impactor and the bottom row is an impactor of non-porous basalt (ρ = 2.7 g cm −3 ). Shading is the same as for Fig. 4. Only a grid subsection is shown, but it is larger than that in Fig. 4. forces. The pictures we see are consistent with the "pancake model" of Zahnle (1992) and Chyba et al. (1993). The impactor is flattened quite strongly at early times, but the extent of radial spreading was no more than a factor of two in radius. Shortly thereafter, the impactor develops non-axisymmetric structures and shreds into filamentary structures before coming apart completely. Material is blown back and outward as ablation proceeds until the impactor material expands into a cloud that slows down and deposits its kinetic energy into the atmosphere. The disruption takes place at a considerably shallower depth (at ∼ −40 km) than the peak deposition of energy; the broken-up impactor material has sufficient inertia to be carried downward a significant distance (∼ 1 scale height or more) before being stopped. A set of "low-resolution" (R16) runs of impactors over a 600-fold range in mass (corresponding to diameters 0.584 < d < 5 km) produced median depths of energy deposition ranging from 35 km to ∼ 250 km below the 1-bar level. Scaling the results by the amount of atmospheric mass intercepted by the bolide showed a weak dependence on impactor mass, with more massive bolides penetrating slightly more deeply than predicted by a linear relation between intercepted atmospheric column and bolide mass. Future work will extend these results in a number of directions. We will explore the parameter space of impactor mass and composition. High-resolution calculations will also serve as the basis for new models of the impactor plume and splashback that generated the greater part of the SL9 phenomena ob-served from Earth. The results of these calculations will also serve as input for simplified, semi-analytic, general models of atmospheric impacts for diverse situations such as impacts into the atmospheres of Earth, Venus, and Titan. , dE/dz (erg/cm) Height, z, Relative to 1-bar level (km) FIG. 1.-Energy-deposition profiles for realizations of the impact of a 1-km-diameter ice sphere (ρ = 0.6 g cm −3 ) into Jupiter's atmosphere at 61.4 km s −1 . Calculations at three different resolutions (Rn, see text) are shown. Panel a: R16, initial velocities differ by ∆v = 0.06 km s −1 or initial positions displaced by 0.015 km from the standard case. Panel b: R32, initial velocities differ by ∆v = 0.06 km s −1 or initial positions displaced by 0.0075 km from the standard case. Panel c: R57, initial positions displaced by 0.0044 km from the standard case. , dE/dz (erg/cm) Height, z, Relative to 1-bar level (km) FIG. 2.-Same as FIG. 3 .FIG 3-Measures of energy deposition for the runs shown inFig. 1. Filled squares refer to individual runs whose curves are plotted inFig. 1at resolutions R16, R32, and R57. The horizontal coordinate in each panel is the resolution. Crosses are the means of the runs at a given resolution, and the vertical line extends ± 1 standard deviation of the distribution. The open circles refer to the results of the Castalia-shaped bolides. Panel a: Mean z-value of energy deposition, calculated from the first moments of the energy-deposition profile. Panel b: Mode z, or the depth of maximum energy deposition. Panels c through e, respectively, depth at which 90%, 50%, and 10% the impact kinetic energy has been deposited. . 4.-Time-integrated mass fluxes µ(z; FIG. 5.-Column densities σ in the x 1 − x 3 plane (horizontal and vertical, respectively, for one R57 run. The times and heights roughly correspond to the passage of the bolide through the z planes given in the same panels in Fig. 4. Shading is the same as for Fig. 4, substituting σ for µ. Only a grid subsection is shown. FIG. 6 . 6-Time-integrated mass fluxes µ(z; , dE/dz (erg/cm) z (km) FIG. 7.-Energy-deposition curves for basalt vs. ice and porous vs. non-porous 1-km diameter objects. Five R16 calculations with differences of 0.015 km in initial x 2 and x 3 positions are shown in each panel. Panels a and b are basalt and panels c and d are ice; panels b and d are for impactors of 35% initial porosity. Note the difference in horizontal scale for dE/dz between panels a and b and panels c and d. Panel d shows the same runs those in Figs. 1 and 2. FIG. 8 . 8-Depth of bolide penetration into the Jovian atmosphere as a function of bolide mass. Top panel: Depths of energy deposition z 10 , z 50 and z 90 for five R16 calculations with tiny perturbations to their initial positions. The dashed lines are fits to the results from Eq. 9. Open triangles are z 10 , filled squares are z 50 , and open squares are z 90 . Impactor diameters are 0.584, 1.0, 1.44, 3.42, and 5 km. 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[ "On the converse of Gaschütz' complement theorem", "On the converse of Gaschütz' complement theorem" ]	[ "Benjamin Sambale " ]	[]	[]	Let N be a normal subgroup of a finite group G. Let N ≤ H ≤ G such that N has a complement in H and (\|N \|, \|G : H\|) = 1. If N is abelian, a theorem of Gaschütz asserts that N has a complement in G as well. Brandis has asked whether the commutativity of N can be replaced by some weaker property. We prove that N has a complement in G whenever all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples if Z(N ) ∩ N ′ = 1. For metabelian groups N , the condition Z(N ) ∩ N ′ = 1 implies the existence of complements. Finally, if N is perfect and centerless, then Gaschütz' theorem holds for N if and only if Inn(N ) has a complement in Aut(N ).1 Translation: I did not yet succeed with their [his theorems] generalization to non-abelian extensions. 2 Translation: In particular, it would be of interest to know: is there a bigger class R of groups, than the class of all abelian groups, such that if A ∈ R, then Gaschütz' theorem applies to A. 3 the transliteration Shemetkov is also used in the literature Since this result is not very well-known, we provide a proof for the convenience of the reader. The first lemma generalizes the Schur-Zassenhaus theorem (noting that \|H 1 \| divides \|H\|).Lemma 4. Let N G and H ≤ G such that G = HN . Then there exists H 1 ≤ H such that G = H 1 N and H 1 ∩ N ≤ Φ(H 1 ). In particular, \|H 1 \| and \|G : N \| have the same prime divisors.	null	[ "https://export.arxiv.org/pdf/2303.00254v1.pdf" ]	257,255,500	2303.00254	6e20518352dc645a9d469a1bbc9c63325b84c313	On the converse of Gaschütz' complement theorem 1 Mar 2023 March 2, 2023 Benjamin Sambale On the converse of Gaschütz' complement theorem 1 Mar 2023 March 2, 2023finite groupscomplementsGaschütz' theorem AMS classification: 20D4020E22 Let N be a normal subgroup of a finite group G. Let N ≤ H ≤ G such that N has a complement in H and (\|N \|, \|G : H\|) = 1. If N is abelian, a theorem of Gaschütz asserts that N has a complement in G as well. Brandis has asked whether the commutativity of N can be replaced by some weaker property. We prove that N has a complement in G whenever all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples if Z(N ) ∩ N ′ = 1. For metabelian groups N , the condition Z(N ) ∩ N ′ = 1 implies the existence of complements. Finally, if N is perfect and centerless, then Gaschütz' theorem holds for N if and only if Inn(N ) has a complement in Aut(N ).1 Translation: I did not yet succeed with their [his theorems] generalization to non-abelian extensions. 2 Translation: In particular, it would be of interest to know: is there a bigger class R of groups, than the class of all abelian groups, such that if A ∈ R, then Gaschütz' theorem applies to A. 3 the transliteration Shemetkov is also used in the literature Since this result is not very well-known, we provide a proof for the convenience of the reader. The first lemma generalizes the Schur-Zassenhaus theorem (noting that \|H 1 \| divides \|H\|).Lemma 4. Let N G and H ≤ G such that G = HN . Then there exists H 1 ≤ H such that G = H 1 N and H 1 ∩ N ≤ Φ(H 1 ). In particular, \|H 1 \| and \|G : N \| have the same prime divisors. Introduction It is a difficult problem to classify all finite groups G with a given normal subgroup N and a given quotient G/N . The situation becomes much easier if N has a complement H in G, i. e. G = HN and H ∩ N = 1. Then G is determined by the conjugation action H → Aut(N ) and G ∼ = N ⋊ H. A well-known theorem by Schur asserts that N always has a complement if N is abelian and gcd(\|N \|, \|G : N \|) = 1. Zassenhaus [25,Theorem IV.7.25] observed that the statement holds even without the commutativity of N (now called the Schur-Zassenhaus theorem). Although we are only interested in the existence of complements, we mention that all complements in this situation are conjugate in G by virtue of the Feit-Thompson theorem. In 1952, Gaschütz [6, Satz 1 on p. 99] (see also [11,Hauptsatz III.17.4]) found a way to relax the coprime condition in Schur's theorem as follows. Theorem 1 (Gaschütz). Let N be an abelian normal subgroup of a finite group G. Let N ≤ H ≤ G such that N has a complement in H and gcd(\|N \|, \|G : H\|) = 1. Then N has a complement in G. Unlike Schur's theorem, Theorem 1 does not generalize to non-abelian groups N . The counterexample of smallest order is attributed to Baer, see [20, p. 225]. In modern notation it can be described as a central product G = SL(2, 3) * C 4 (= SmallGroup(48, 33) in the small groups library [5]) where N = Q 8 G has a complement in a Sylow 2-subgroup H = Q 8 * C 4 ≤ G, but not in G (here, Q 8 denotes the quaternion group of order 8 and C 4 is a cyclic group of order 4). A similar counterexample, given by Hofmann [9, p. 32-33] and reproduced in Huppert's book [11,Beispiel I.18.7], has order \|G\| = \|G : H\|\|H : N \|\|N \| = 2 · 3 2 · 3 3 . Finally, a more complicated counterexample of order 2 7 3 2 is outlined in Zassenhaus' book [25, Appendix F, Exercise 5]). We produce more general families of counterexamples in Section 3. Although Gaschütz did obtain some non-abelian variations of his theorem, he confesses: "Ihre Verallgemeinerung auf nichtabelsche Erweiterungen ist mir bisher nicht gelungen." 1 Brandis [2] not only gave a very elementary proof of Theorem 1 (avoiding cohomology), but also replaced abelian groups by solvable groups under further technical assumptions. However, he concludes like Gaschütz with: "Insbesondere wäre es interessant zu wissen: gibt es eine größere Klasse R von Gruppen, als die Klasse der abelschen Gruppen, so daß falls A ∈ R folgt: der Satz von Gaschütz ist für A anwendbar." 2 Following his words we say that Gaschütz' theorem holds for N if for every embedding N ≤ H ≤ G such that N G, gcd(\|N \|, \|G : H\|) = 1 and N has a complement in H, then N has a complement in G. By analogy to the notation of control of fusion/transfer one could say that H controls complements of N in G. Our main theorem generalizes Gaschütz' theorem as follows: Theorem 2. Suppose that all Sylow subgroups of N are abelian. Then Gaschütz' theorem holds for N . The proof of Theorem 2 follows easily from a deep theorem of Šemetkov 3 , which is presented and proved in the next section. We further collect and prove numerous other theorems on the existence of complements in that section. In the last section we construct counterexamples to illustrate the limitations of our results. On the existence of complements Our notation is standard apart from the commutator convention [x, y] := xyx −1 y −1 for elements x, y of a group. The commutator subgroup, the center and the Frattini subgroup of G are denoted by G ′ = [G, G], Z(G) and Φ(G) respectively. For H ≤ G and x ∈ G we write x H = xHx −1 . We will often use the following Theorem 1 implies that an abelian normal subgroup N G has a complement in G if and only if for every Sylow subgroup P/N of G/N , N has a complement in P . This was improved by Šemetkov [ Theorem 3 (Šemetkov). Let N G such that for every prime divisor p of \|G : N \|, N has an abelian Sylow p-subgroup P , and P has a complement in a Sylow p-subgroup of G. Then N has a complement in G. elementary fact: If K is a complement of N in G and N ≤ H ≤ G, then H ∩ K is a complement of N in H.Proof. Choose H 1 ≤ H minimal with respect to inclusion such that G = H 1 N . Note that H 1 ∩ N is normal in H 1 . Let M < H 1 be a maximal subgroup of H 1 . If H 1 ∩ N M , then H 1 = M (H 1 ∩ N ) and we obtain the contradiction G = H 1 N = M N . Hence, H 1 ∩ N is contained in all maximal subgroups of H 1 and it follows that H 1 ∩ N ≤ Φ(H 1 ). In particular, H 1 ∩ N is nilpotent. Since G/N ∼ = H 1 /H 1 ∩ N , the prime divisors of \|G : N \| divide \|H 1 \|. Conversely, let p be a prime divisor of \|H 1 \|. Suppose that p does not divide \|H 1 /H 1 ∩ N \|. Then H 1 ∩ N contains a Sylow p-subgroup P of H 1 . Since H 1 ∩ N is nilpotent, it follows that P is the unique Sylow p-subgroup of H 1 ∩ N H 1 and therefore P H 1 . By the Schur-Zassenhaus theorem, P has a complement K in H 1 . Now K lies in a maximal subgroup M < H 1 , but so does P ≤ H 1 ∩ N ≤ Φ(H 1 ). Hence, H 1 = P K ≤ M , a contradiction. This shows that p divides \|H 1 /H 1 ∩ N \| = \|G : N \|. The next lemma is a weak version of [10, Satz 3.3]. Lemma 5 (Huppert). If G has abelian Sylow p-subgroups, then there exists a characteristic subgroup N G with the following properties: (i) G/N has a normal Sylow p-subgroup. (ii) N has no composition factor of order p. Proof. We argue by induction on \|G\|. If G is abelian, the claim holds for N = 1. Hence, let G ′ = 1. Proof of Theorem 3. In order to argue by induction, we prove a more general statement: Let π be a set of primes such that for every p ∈ π there exists a Sylow p-subgroup P of G such that P ∩ N is abelian and P ∩ N has a complement in P . Then there exists H ≤ G such that G = HN and no prime divisor of \|H ∩ N \| lies in π. The claim follows by letting π be the set of all prime divisors of \|G : N \| and using Lemma 4. We argue by induction on \|G\| + \|π\|. If π = ∅, the claim holds for H = G. Let π = ∅. By induction there exist p ∈ π and H ≤ G such that G = HN and no prime divisor of \|H ∩ N \| lies in π \ {p}. By Lemma 4, we may assume that H ∩ N is nilpotent. Without loss of generality, let p divide \|H ∩ N \|. We choose Sylow p-subgroups P H ≤ P G of H and G respectively. Then P := P H ∩ N is the unique Sylow p-subgroup of H ∩N H. In particular, H ≤ N G (P ). By hypothesis, P N := P G ∩N is abelian and therefore P N ≤ N G (P ). Since \|P N P H \| = \|P N \|\|P H : P \| = \|N \| p \|H : H ∩ N \| p = \|G\| p , (2.1) we conclude that P G = P N P H ≤ N G (P ). Case 1: N G (P ) < G. We apply induction to N N (P ) N G (P ). By hypothesis, the Sylow p-subgroup P N of N N (P ) has a com- plement in the Sylow subgroup P G of N G (P ). For q ∈ π \ {p} we choose Sylow q-subgroups Q H ≤ Q of H and N G (P ) respectively. By hypothesis, Q N := Q ∩ N = Q ∩ N N (P ) is an abelian Sylow subgroup of N N (P ) and Q = Q N Q H as in (2.1). Since \|H ∩ N \| is not divisible by q, Q H is a complement of Q N in Q. By induction, there exists K ≤ N G (P ) such that N G (P ) = N N (P )K and K ∩ N N (P ) = K ∩ N is a π ′ -group. Now the claim follows from G = HN = N G (P )N = KN . Case 2: P G. Let K N be the characteristic subgroup of N provided by Lemma 5. Then \|N/K\| is divisible by p, because P N . Furthermore, P N K/K = 1 is the normal Sylow p-subgroup of N/K. In particular, P N K/K G/K. By hypothesis, P G = P N ⋊ P 1 for some P 1 ≤ P G . Since \|P G K : P N K\| = \|P G : P N K ∩ P G \| = \|P G : P N (K ∩ P G )\| = \|P G : P N \| = \|P 1 \|, P 1 K/K is a complement of P N K/K in P G K/K. By Gaschütz' theorem, P N K/K has a complement L/K in HP N K/K. It follows that LP N = HP N K. Also, L < G since P N K/K = 1. Now we apply induction to K L. For a Sylow p-subgroup P L of L we obtain an abelian Sylow subgroup P K = P L ∩ K of K since K ≤ N . The construction of K reveals O p (KP L ) = O p (K) = K. By another theorem of Gaschütz (see [11, Satz IV.3.8]) , K has a complement in KP L . Consequently, P K also has a complement in P L . For q ∈ π \ {p} we choose Sylow q-subgroups Q H ≤ Q of H and HP N K respectively. Since K ≤ N , Q K = Q ∩ K is an abelian Sylow subgroup of K. Moreover, Q = Q K ⋊ Q H as q ∤ \|H ∩ K\|. Notice that \|L\| q = \|LP N \| q = \|HP N K\| q . By Sylow's theorem, there exists x ∈ HP N K such that x Q = x Q K ⋊ x Q H is a Sylow q-subgroup of L and x Q K is a Sylow subgroup of K. Finally, by induction there exists R ≤ L such that L = RK and R ∩ K is a π ′ -group. We have G = HN = HP N KN = LP N N = RKN = RN and \|R ∩ N : R ∩ K\| = \|R ∩ N : (R ∩ N ) ∩ K\| = \|K(R ∩ N ) : K\| = \|KR ∩ N : K\| = \|L ∩ N : K\|. Since L/K ∩ P N K/K = 1, \|L ∩ N : K\| is not divisible by p. Since (L ∩ N )/K ≤ (N ∩ HP N K)/K = (H ∩ H)P N K/K, the prime divisors of \|L ∩ N : K\| also divide \|H ∩ N \| and therefore, do not lie in π. Consequently, R ∩ N is a π ′ -group and we are done. We are now in a position to proof our main result. Proof of Theorem 2. Let N ≤ H ≤ G such that N G, gcd(\|N \|, \|G : H\|) = 1, and N has a complement K in H. Let p be a prime divisor of \|G : N \|. By Sylow's theorem, a given Sylow p-subgroup Q of K is contained in a Sylow p-subgroup P of H. Then P ∩ N P is a Sylow p-subgroup of N and P = (P ∩ N ) ⋊ Q by comparing orders. By hypothesis, P ∩ N is abelian. If p ∤ \|N \|, then P ∩ N = 1 obviously has a complement in a Sylow p-subgroup of G. On the other hand, if p divides \|N \|, then p ∤ \|G : H\| and P is a Sylow p-subgroup of G. Again P ∩ N has a complement. By Šemetkov's theorem, N has a complement in G. Theorem 6 (Rose). For every finite group N the following assertions are equivalent: (1) Z(N ) = 1 and the inner automorphism group Inn(N ) has a complement in Aut(N ). (2) If N is a normal subgroup of some finite group G, then N has a complement in G. Since Rose's arguments are tailored for infinite groups, we provide a more direct proof. Proof. Suppose that (1) Now assume that (2) is satisfied. By way of contradiction, let Z(N ) = 1 and choose a prime divisor p of \|Z(N )\|. Let x ∈ Z(N ) be of order p and let p n be the maximal order of a p-element in N . Choose C = y ∼ = C p n+1 and define Z : = (x, y p n ) ≤ Z(G × C). We construct the central product G : = (N × C)/Z. Then the map f : N → G, g → (g, 1)Z is a monomorphism. By hypothesis, f (N ) has a complement K ≤ G. By construction, [f (N ), K] = 1 and G = f (N ) × K. Since \|K\| = \|G\|/\|N \| = p n , G does not contain elements of order p n+1 . But (1, y)Z ∈ G does have order p n+1 . This contradiction shows that Z(N ) = 1. Now Inn(N ) ∼ = N has a complement in Aut(G) by hypothesis. Many (non-abelian) simple groups N satisfy Rose's criterion. For instance, all alternating groups apart from the notable exception A 6 . The exceptions among the groups of Lie type were classified in [15]. For centerless perfect groups (i. e. Z(N ) = 1 and N ′ = N ) we will show in Theorem 15 that Rose's criterion is actually necessary to obtain Gaschütz' theorem. A group N is called complete if it satisfies the stronger condition In this case N has a unique normal complement in G (whenever N G). In fact, G = N × C G (N ). Conversely, a theorem of Baer [1,Theorem 1] asserts that N is complete if N always has a normal complement in G whenever N G (see [18,Theorems 7.15,7.17]). Starting with a centerless group G, Wielandt has shown that the automorphism tower G ≤ Aut(G) ≤ Aut(Aut(G)) ≤ . . . terminates in a complete group after finitely many steps (see [12,Theorem 9.10]). If G is non-abelian simple, then already Aut(G) is complete according to a result of Burnside (see [18,Theorem 7.14]). In particular, the symmetric groups S n ∼ = Aut(A n ) for n ≥ 7 are complete (also for n = 3, 4, 5 by different reasons). A large class of complete groups, including some groups of odd order, was constructed in [8] (a paper dedicated to Gaschütz). Proof. (i) Suppose first that N 1 is a normal subgroup of a finite group G such that N 1 has no complement in G. By way of contradiction, suppose that L is a complement of N 1 × . . . × N k inĜ := G × N 2 × . . . × N k . Let K := N 2 . . . N k L ∩ G. Then N 1 K = N 1 . . . N k L ∩ G =Ĝ ∩ G = G and N 1 ∩ K = N 1 ∩ N 2 . . . N k L = 1, because every element ofĜ can be written uniquely as x 1 . . . x k y with x i ∈ N i and y ∈ L. But now K is a complement of N 1 in G. Contradiction. Hence, if Rose's criterion holds for N 1 × . . . × N k , then it holds for N 1 and by symmetry also for N 1 , . . . , N k . Assume conversely that N 1 , . . . , N k fulfill Rose's criterion. Then Z(N 1 × . . . × N k ) = Z(N 1 ) × . . . × Z(N k ) = 1. By the first part of the proof, we may assume that each N i is indecomposable. Since Z(N 1 ) = . . . = Z(N k ) = 1, every automorphism of N 1 ×. . .×N k permutes the N i (see [11,Satz I.12.6]). We may arrange the N i such that N 1 ∼ = . . . ∼ = N k 1 ∼ = N k 1 +1 ∼ = . . . ∼ = N k 1 +k 2 ∼ = . . . . Then Aut(N 1 × . . . × N k ) ∼ = Aut(N k 1 1 ) × . . . × Aut(N ks s ). In order to verify Rose's criterion for N 1 × . . . × N k , we may assume that k 1 = k, i. e. N 1 ∼ = . . . ∼ = N k . In this case we obtain Aut(N k 1 ) ∼ = Aut(N 1 ) ≀ S k . We identify S k with a subgroup of Aut(N k 1 ). By hypothesis, there exists a complement K 1 of Inn(N 1 ) in Aut(N 1 ). It is easy to see that K 1 , S k ∼ = K 1 ≀ S k is a complement of Inn(N k 1 ) in Aut(N k 1 ). (ii) Since every automorphism of N 1 × . . . × N k−1 extends to an automorphism of N , it follows that N 1 is characteristic in N 1 × . . . × N k−1 . Hence, by induction on k, it suffices to consider the case k = 2. Let N ≤ H ≤ G such that N G, gcd(\|N \|, \|G : H\|) = 1 and N has a complement K in H. Then N 1 and N 2 are normal in G, since they are characteristic in N . Moreover, KN 2 is a complement of N 1 in H, because N 1 ∩ KN 2 = N 1 ∩ N ∩ KN 2 = N 1 ∩ (N ∩ K)N 2 = N 1 ∩ N 2 = 1. Since \|N 1 \| is coprime to \|G : H\|, Gaschütz' theorem applied to N 1 yields a complement L of N 1 in G. Therefore,K is a complement of N in G. (iv) Let N ≤ H ≤ G as usual. As in (iii) we find L ≤ G such that G = N L and N ∩ L = M . It suffices to show that M has a complement in L. We do this using Šemetkov's theorem. Let P be a non-trivial Sylow p-subgroup of M . Let Q be a Sylow p-subgroup of a complement of M in N (which exists by hypothesis). By Sylow's theorem, we may assume that Q normalizes P , so that P ⋊ Q is a Sylow p-subgroup of N . Let R be a Sylow p-subgroup of C G (N ). Then QR is a p-subgroup of N C G (N ). Let x = st ∈ P ∩ QR with s ∈ Q and t ∈ R. Then t = s −1 x ∈ N ∩ C G (N ) = Z(N ). Since Z(N ) is a p ′ -group by hypothesis, we obtain t = 1 and x = s ∈ P ∩ Q = 1. This shows that P ∩ QR = 1. Since G/N C G (N ) ≤ Out(N ) and Out(N ) is a p ′ -group, P ⋊ QR is a Sylow p-subgroup of G. Hence, P also has a complement in a Sylow p-subgroup of L. Since P is abelian, Šemetkov's theorem applies to M . Using [11,Satz I.12.6], it is easy to see that N 1 , . . . , N k are characteristic in N = N 1 × . . . × N k if and only if the following holds for all i = j: (i) N i and N j have no common direct factor, (ii) gcd \|N i /N ′ i \|, Z(N j )\| = 1. Concrete examples for Proposition 7(iv) are groups of the form N = P ⋊ Q where P and Q are abelian of order 9 and 12 respectively and \|Z(N )\| = 2 (there are four isomorphism types for N ). Now Proposition 7(ii) applies to N × C 7 (while the other parts do not apply here). Another way to relax the commutativity of N is to consider metabelian groups. Recall that a group G is called metabelian if G/G ′ is abelian, i. e. G ′′ = 1. Theorem 8 (Newman, Yonaha). Let G be a metabelian group such that Z(G) ∩ G ′ = 1. Then G ′ has a complement in G and all such complements are conjugate. Proof. Since the original paper by Yonaha [24] is not widely available and the proof is omitted in Kirtland's book [13], we present a simplification of Yonaha's arguments (which in turn rely on ideas of Newman [16]) for the convenience of the reader. We may assume that G ′ = 1. Let N be a minimal normal subgroup of G contained in G ′ . We argue by induction on \|G ′ : N \|. Since Z(G) ∩ G ′ = 1, there exists x ∈ G \ C G (N ). Consider the map ϕ : N → N , a → [x, a]. For a, b ∈ N we have [x, ab] = [x, a]a[x, b]a −1 = [x, a][x, b], because N ⊆ G ′ is abelian. Thus, ϕ is a homomorphism. Let g ∈ G. Since G/G ′ is abelian, there exists y ∈ G ′ such that gxg −1 = xy. It follows that gϕ(a)g −1 = g[x, a]g −1 = [xy, gag −1 ] = [x, gag −1 ] = ϕ(gag −1 ) for all a ∈ N . This shows that ϕ(N ) is a normal subgroup of G contained in N . By the minimality of N and x / ∈ C G (N ), it follows that ϕ(N ) = N . Therefore, ϕ is injective and C G (x) ∩ N = 1. Case 1: N = G ′ . For every g ∈ G there exists a ∈ G ′ such that [x, g] = ϕ(a) = [x, a], i. e. a −1 g ∈ C G (x) and g ∈ C G (x)G ′ . Consequently, C G (x) is a complement of G ′ in G. Let C ≤ G be another complement of G ′ . Suppose that C < M ≤ G. Then 1 = M ∩ G ′ M . Since G ′ is abelian, also M ∩ G ′ G ′ . Altogether, M ∩ G ′ M G ′ = CG ′ = G. The minimality of N = G ′ implies that M ∩ G ′ = G ′ . Hence, G = CG ′ ≤ M and C is a maximal subgroup of G. Moreover, C ∼ = G/G ′ is abelian. Choose y ∈ C \ C G (G ′ ). Then C = C G (y) by the maximality of C. As above there is an automorphism ψ : N → N , a → [a, y]. Thus, also ϕ • ψ : N → N, a → [x, [a, y]] =: [x, a, y] is an automorphism. In particular, there exists a ∈ N such that [x, a, y] = [x, y −1 ]. But this yields y[a, y] ∈ C G (x) and aya −1 = [a, y]y ∈ y −1 C G (x)y. Furthermore, y ∈ (ya) −1 C G (x)ya. Since C G (x) is abelian, (ya) −1 C G (x)ya ⊆ C G (y) = C. Finally, \|C\| = \|G/G ′ \| = \|C G (x)\| implies that C = (ya) −1 C G (x)ya. Hence, all complements of G ′ are conjugate in G. Case 2: N < G ′ . We first show that Z(G/N )∩G ′ /N = 1. Suppose by way of contradiction that g ∈ G ′ \N and gN ∈ Z(G/N ). Then M := N g G and ϕ extends to a homomorphismφ : M → N , a → [x, a] (note that M ⊆ G ′ is still abelian). Since ϕ is an automorphism, it follows that M = N C M (x). Hence, we may assume that g ∈ C M (x). Then [G, g , x ] = 1 and [ g , x , G] ≤ G ′′ = 1. The 3-subgroups lemma (see [12,Lemma 4.9]) yields [ x , G, g ] = 1 and therefore [G, g ] ≤ Ker(ϕ) = 1. This leads to the contradiction g ∈ Z(G) ∩ G ′ = 1. Now the inductive hypothesis guarantees a complement D/N of G ′ /N in G/N and all such complements are conjugate in G/N (note that G/N is metabelian). As D ′ is normal in D and also normal in the abelian group G ′ , we obtain D ′ DG ′ = G. On the other hand, D/N ∼ = G/G ′ is abelian and D ′ ≤ N . The minimality of N implies that D ′ = 1 or D ′ = N . The first possibility would mean that N ≤ C G (DG ′ ) ∩ G ′ = Z(G) ∩ G ′ = 1. Consequently, D ′ = N and N is a minimal normal subgroup of D since every subgroup of N is normal in G ′ . Moreover, Z(D) ∩ D ′ ≤ Z(G) ∩ G ′ = 1. Hence, Case 1 of the proof applies to D. Thus, D = N ⋊ E and G = G ′ D = G ′ N E = G ′ E with E ∩ G ′ = E ∩ D ∩ G ′ = E ∩ N = 1. Finally, suppose that F is another complement of G ′ in G. Then F N/N is a complement of G ′ /N in G/N . By induction there exists g ∈ G such that gF g −1 N = D. Since gF g −1 ∩ N ≤ g(F ∩ G ′ )g −1 = 1, we see that gF g −1 is a complement of N in D. By Case 1, it follows that gF g −1 is conjugate to E. Surely the proof of Theorem 9 can be adapted to similar situations (using the Schur-Zassenhaus theorem instead of Theorem 8 for instance). G = N N G (K) = M N G (K). For x ∈ M ∩ N G (K) and y ∈ K we have [x, y] ∈ M ∩ K = 1. Hence, x ∈ M ∩ C N (K) = N ′ ∩ Z(N ) = 1, because M is abelian. Therefore, N G (K) is a complement of M in G. Since N/M For the sake of completeness we also address the dual of Rose's theorem which is probably known to experts in cohomology. Theorem 10. For every finite group K = 1 there exist finite groups N G such that G/N ∼ = K and N has no complement in G. Proof. Again it was Gaschütz [7] who proved a stronger statement where N is required to lie in the Frattini subgroup of G (then G is called a Frattini extension of K). The following arguments are inspired by [3,Theorem B.11.8]. (A cohomological proof can be given with Shapiro's lemma, see [19,Proposition 9.76].) Let K = F/R where F is a free group of finite rank and R F . Let P/R ≤ F/R be a subgroup of prime order p (exists since K = 1). By the Nielsen-Schreier theorem, P is free and P/P ′ is free abelian of finite rank. Therefore we find P 1 P with P 1 ≤ R and P/P 1 ∼ = C p 2 . Let Q F be the kernel of the permutation action of F on the cosets F/P 1 . Then Q ≤ P 1 and \|F : Q\| ≤ \|F : P 1 \|! < ∞. Define G := F/Q and N := R/Q. Clearly, G/N ∼ = F/R ∼ = K. Suppose that N has a complement H/Q in G. Then (H ∩ P )/Q is a complement of N in P/Q. Moreover, (H ∩ P )P 1 /P 1 is a complement of R/P 1 in P/P 1 . But this is impossible since P/P 1 ∼ = C p 2 . The situation of Theorem 10 is different for infinite groups: Every group K is a quotient of a free group F . If F splits, then K is a subgroup of F and therefore free by the Nielsen-Schreier theorem. Conversely, by the universal property of free groups, every group extension with a free quotient K splits (including the case K = 1). We use the opportunity to mention a result in the opposite direction by Gaschütz and Eick [4]: Theorem 11 (Gaschütz, Eick). For a finite group N the following assertions are equivalent: (i) There exists a finite group G with N G such that N H < G for all H < G. (ii) There exists a finite group G with N = Φ(G). (iii) Inn(N ) ≤ Φ(Aut(N )). Many more complement theorems can be found in Kirtland's recent book [13]. Some non-existence theorems In the proof of Lemma 5 we have already mentioned [11,Theorem IV.2.2], which implies that Z(G) ∩ G ′ ∩ P ≤ P ′ for every finite group G with Sylow subgroup P . Hence, in the situation of Theorem 2 we have Z(G)∩ G ′ = 1. Our main theorem shows that this is in fact a necessary condition for Gaschütz' theorem. Recall from the introduction that Gaschütz' theorem holds for N if for every embedding N ≤ H ≤ G such that N G, gcd(\|N \|, \|G : H\|) = 1 and N has a complement in H, then N has a complement in G. Theorem 12. Let N be a finite group such that Z(N ) ∩ N ′ = 1. Then for every integer q > 1 coprime to \|N \| there exist groups N ≤ H ≤ G with the following properties: (i) N G and H G. (ii) N has a complement in H, but not in G. (iii) H and N have the same composition factors (up to multiplicities) and G/H is cyclic of order q. In particular, Gaschütz' theorem does not hold for N . Clearly, H = N K. For g = x 1 (x 1 , . . . , x q ) ∈ K ∩ N we must have (x 1 , . . . , x q ) ∈ z and therefore g = 1. Proof. Let 1 = Z = z ≤ Z(N ) ∩ N ′ . Hence, K is a complement of N in H. Suppose by way of contradiction that N has a complement L in G. Note that α is a nilpotent Hall subgroup of G. A theorem of Wielandt asserts that every Hall subgroup of order q is conjugate to α (see [11,Satz III.5.8]; if q is a prime, Sylow's theorem suffices). Since every conjugate of L in G is also a complement of N , we may assume that α ∈ L. It follows that L ∩ H is an α-invariant complement of N in H. For every d ∈ D there exists x ∈ N such that xd ∈ L. Consequently, α(d)d −1 = α(xd)(xd) −1 ∈ L. In particular, (x, x −1 , 1, 1 . . . , 1) ∈ L for all x ∈ N . For x, y ∈ N we compute ([x, y], 1, . . . , 1) = (x, x −1 , 1, . . . , 1)(y, y −1 , 1, . . . , 1)((yx) −1 , yx, 1, . . . , 1) ∈ L. (3.1) Since z ∈ N ′ , we conclude that (z, 1, . . . , 1) ∈ L. But now also z = (z, 1, . . . , 1)α(z, 1, . . . , 1) . . . α q−1 (z, 1, . . . , 1) ∈ L ∩ N. This contradicts L ∩ N = 1. Proof. This follows from Theorem 9. We illustrate that the condition Z(N ) ∩ N ′ = 1 (even Z(N ) = 1) is not sufficient for Gaschütz' theorem in general. A given counterexample N H ≤ G to Gaschütz' theorem can be "blown up" as follows. Let L be a finite group such that gcd(\|L\|, \|G : H\|) = 1 (this is a harmless restriction in the situation of Theorem 12). To an arbitrary homomorphism G → Aut(L), we form the semidirect productsĜ := L ⋊ G,Ĥ := L ⋊ H andN := L ⋊ N . If K is a complement of N in H, then K is also a complement ofN inĤ. Now suppose thatK is a complement ofN inĜ. ThenKL/L a complement ofN /L ∼ = N inĜ/L ∼ = G. Contradiction. Hence,N Ĥ ≤Ĝ is a counterexample to Gaschütz' theorem. The counterexample SL(2, 3) * C 4 mentioned in the introduction lives inside GL (2,5). Therefore, Gaschütz' theorem does not hold for the Frobenius group N = C 2 5 ⋊ Q 8 . Indeed, Z(N ) = 1. In contrast, Gaschütz' theorem does hold the very similar groups C 2 5 ⋊ D 8 and C 2 3 ⋊ Q 8 , because those fulfill Rose's criterion. So we see that the question for an individual group can be very delicate to answer. Other examples arise from our next theorem, which is related to Rose's result as well. Indeed, by the Dedekind law we have N (H ∩ K) = N K ∩ H = H and (H ∩ K) ∩ N = 1. The same argument shows that KM/M is complement of N/M in G/M for every normal subgroup M G contained in N . Since G ′ is characteristic in G, there exists a minimal characteristic subgroup M of G contained in G ′ . Then M is characteristically simple, i. e. a direct product of isomorphic simple groups. By induction there exists N/M G/M with the desired properties. Since an automorphism of G induces an automorphism on G/M , N is characteristic in G. Moreover, G/N ∼ = (G/M )/(N/M ) has a normal Sylow p-subgroup. A composition factor of N or order p must be a composition factor of M . If M is non-abelian, so are all composition factors of M and we are done. Thus, we may assume that M is an elementary abelian p-group. Let P be a Sylow p-subgroup of G containing M . Then P ≤ C G (M ) since P is abelian by hypothesis. If C G (M ) = G, then [11, Satz IV.2.2] (a generalized version of a theorem of Taunt [22, Theorem 4.1]) leads to the contradiction M ≤ Z(G) ∩ G ′ ∩ P ≤ P ′ = 1. Hence, let C G (M ) < G. Then there exists a characteristic subgroup N C G (M ) with the desired properties by induction. Each of M , C G (M ) and N are characteristic in G. Since \|G : C G (M )\| is not divisible by p, the normal Sylow p-subgroup of C G (M )/N is also a normal Sylow subgroup of G/N . holds. Let N G and M := C G (N ) G. Then N ∩ M = Z(N ) = 1 and Inn(N ) ∼ = N ∼ = N M/M G/M ≤ Aut(N ). Hence, there exists K/M ≤ G/M such that G = N K and N M ∩ K = M . It follows that N ∩ K ≤ N ∩ N M ∩ K = N ∩ M = 1. This shows that K is a complement of N in G. ( 1 ' 1) Z(N ) = 1 and Inn(N ) = Aut(N ). The following elementary observation extends the class of groups further (Out(N ) = Aut(N )/Inn(N ) denotes the outer automorphism group of N ).Proposition 7. ( i ) iLet N 1 , . . . , N k be finite groups. Then Rose's criterion holds for N 1 × . . . × N k if and only if it holds for N 1 , . . . , N k . (ii) Let N = N 1 × . . . × N k with characteristic subgroups N 1 , . . . , N k ≤ N . If Gaschütz' theorem holds for N 1 , . . . , N k , then Gaschütz' theorem holds for N . (iii) Let N be a finite group with a characteristic subgroup M such that M fulfills Rose's criterion and Gaschütz' theorem holds for N/M . Then Gaschütz' theorem holds for N . (iv) Let N be a finite group with a characteristic subgroup M such that gcd(\|M \|, \|Z(N )\|\|Out(N )\|) = 1 and all Sylow subgroups of M are abelian. Suppose that M has a complement in N and Gaschütz' theorem holds for N/M . Then Gaschütz' theorem holds for N . Now the canonical map ϕ : L → G/N 1 , x → xN 1 is an isomorphism and we define L N := ϕ −1 (N/N 1 ) and L H := ϕ −1 (H/N 1 ). Since KN 1 /N 1 is a complement of N/N 1 in H/N 1 , also L N has a complement in L H . Moreover, \|L : L H \| = \|G : H\| is coprime to \|L N \| = \|N/N 1 \|. Now Gaschütz' theorem applied to L N ∼ = N/N 1 ∼ = N 2 provides a complement L K of L N in L. Then L K ∩ N ≤ L K ∩ L N = 1 and \|L K \| = \|L : L N \| = \|G : N \|. Therefore, L K is a complement of N in G. (iii) Let N ≤ H = N ⋊ K ≤ G as usual. Since M is characteristic in N ,we have M G and KM/M is a complement of N/M H/M . By Gaschütz' theorem, N/M has a complement L/M in G/M . By Rose's theorem, M has a complementK in L. Now G = LN =KM N =KN and K ∩ N =K ∩ L ∩ N =K ∩ M = 1. For instance, if all non-abelian minimal normal subgroups M 1 , . . . , M n of N fulfill Rose's criterion and if all Sylow subgroups of N/M 1 . . . M n are abelian, then Gaschütz' theorem holds for N by Theorem 2 and Proposition 7. Theorem 9 . 9If N is metabelian and N ′ ∩ Z(N ) = 1, then Gaschütz' theorem holds for N . Proof. Suppose that N ≤ H ≤ G such that N G, gcd(\|N \|, \|G : H\|) = 1 and N has a complement in H. By Theorem 8, M := N ′ has a complement K in N and all such complements are conjugate in N . The Frattini argument implies that is abelian, Gaschütz' theorem applied to N/M ≤ H/M ≤ G/M yields G = N L with N ∩ L = M as usual. As in the proof of Proposition 7, it suffices to show that M has a complement in L. But this is clear, since M has a complement in G. Let α be the automorphism of D := N q = N × . . . × N such that α(x 1 , . . . , x q ) = (x q , x 1 , . . . , x q−1 ) for all (x 1 , . . . , x q ) ∈ D. Then W := D ⋊ α ∼ = N ≀ C q and z := (z, . . . , z) ∈ Z(W ). Hence, we can construct the central product G := (N × W )/ (z, z) ∼ = N * W. We identify N , D and W with their images in G. In this sense, N ∩ W = N ∩ D = z = z −1 . Now H := N D G has the same composition factors as N and G/H ∼ = C q . Consider K := {x 1 (x 1 , . . . , x q ) : (x 1 , . . . , x q ) ∈ D} ≤ H. Corollary 13 . 13Gaschütz' theorem fails for all non-abelian nilpotent groups. Proof. See [11, Satz III.2.6] for instance.Corollary 14. If N is metabelian, then Gaschütz' theorem holds for N if and only if N ′ ∩ Z(N ) = 1. Theorem 15 . 15Let N be a perfect group with trivial center. Then Gaschütz' theorem holds for N if and only if Inn(N ) has a complement in Aut(N ). 21, Theorem 2] as follows (a very similar result for solvable groups appeared in Wright [23, Theorem 2.6]). Another source of examples to Brandis' question comes from a splitting criterion by Rose [17, Corollary 2.3] (obtained earlier by Loonstra [14, Satz 4.3 and Satz 5.1] in a less concise form). AcknowledgmentProposition 7(ii) was found by Scheima Obeidi within the framework of her Master's thesis written under the direction of the author. I appreciate some valuable comments of an anonymous referee. The work is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/4-1). We construct a counterexample similar as in Theorem 12. Since Z(N ) = 1, we will identify N with Inn(N ). Proof. If Inn(N ) has a complement in Aut(N ). then the claim follows from Theorem 6. Now assume conversely that Inn(N ) has no complement in Aut(N ). Let q > 1 be an integer coprime to \|Aut(N )\|Proof. If Inn(N ) has a complement in Aut(N ), then the claim follows from Theorem 6. Now assume conversely that Inn(N ) has no complement in Aut(N ). We construct a counterexample similar as in Theorem 12. Since Z(N ) = 1, we will identify N with Inn(N ). Let q > 1 be an integer coprime to \|Aut(N )\|. . = N Q = N ×, Let α be the automorphism of D := N q = N × . . . x q ∈ N A ≤ H is a complement of N in H. Suppose by way of contradiction that L ≤ G is a complement of N in G. By Wielandt's theorem on nilpotent Hall subgroups, we may assume that α ∈ L. The same computation as in (3.1) shows that D ′ ≤ L. Since N ′ = N by hypothesis. H := N × N Such That Α ; ∈ D. Then W := D ⋊ Α ∼ = N ≀ C ; = N W A, Da G, x q−1 ) for all. Aut(N × D) ∼ = Aut(N ) ≀ S q+1As usual, we identify N , D, W and A with subgroups of G. We show that Gaschütz' theorem fails with respect to N ≤ H ≤ G. Note first that G/H ∼ = α ∼ = C q . As in the proof of Theorem 12. it follows that C := C G (N ) = D α ≤ L. But now L/C is a complement of N C/C ∼ = N in G/C ∼ = Aut(N ). Contradiction× N such that α(x 1 , . . . , x q ) = (x q , x 1 , . . . , x q−1 ) for all (x 1 , . . . , x q ) ∈ D. Then W := D ⋊ α ∼ = N ≀ C q is a subgroup of Aut(N × D) ∼ = Aut(N ) ≀ S q+1 . Since the diagonal subgroup A := (γ, . . . , γ) : γ ∈ Aut(N ) ≤ Aut(N ) q+1 is centralized by α, we can define G := N W A and H := N DA G. As usual, we identify N , D, W and A with subgroups of G. We show that Gaschütz' theorem fails with respect to N ≤ H ≤ G. Note first that G/H ∼ = α ∼ = C q . As in the proof of Theorem 12, it is easy to see that x 1 (x 1 , x 2 , . . . , x q ) : x 1 , . . . , x q ∈ N A ≤ H is a complement of N in H. Suppose by way of contradiction that L ≤ G is a complement of N in G. By Wielandt's theorem on nilpotent Hall subgroups, we may assume that α ∈ L. The same computation as in (3.1) shows that D ′ ≤ L. Since N ′ = N by hypothesis, it follows that C := C G (N ) = D α ≤ L. But now L/C is a complement of N C/C ∼ = N in G/C ∼ = Aut(N ). Contradiction. As promised earlier, Theorem 15 implies that Gaschütz' theorem does not hold for the alternating group A 6. As promised earlier, Theorem 15 implies that Gaschütz' theorem does not hold for the alternating group A 6 . Corollary 16. Let N be a perfect group with trivial center such that Inn(N ) has no complement in Aut(N ). Then for every finite group M , Gaschütz' theorem does not hold for N × M. Corollary 16. Let N be a perfect group with trivial center such that Inn(N ) has no complement in Aut(N ). Then for every finite group M , Gaschütz' theorem does not hold for N × M . Proof. Let N ≤ H ≤ G be the counterexample for N constructed in the proof of Theorem 15 with q coprime to \|M \|. Then N × M ≤ H × M ≤ G × M is a counterexample to Gaschütz' theorem for. Proof. Let N ≤ H ≤ G be the counterexample for N constructed in the proof of Theorem 15 with q coprime to \|M \|. Then N × M ≤ H × M ≤ G × M is a counterexample to Gaschütz' theorem for Let N be a finite group with Z(N ) = 1. Suppose that there exist k ∈ N and an automorphism γ ∈ Aut(N ) such that γ k ∈ Inn(N ) ′ and (δγ) k = 1 for all δ ∈ Inn(N ). Then Gaschütz' theorem does not holds for N. Proposition 17Proposition 17. Let N be a finite group with Z(N ) = 1. Suppose that there exist k ∈ N and an auto- morphism γ ∈ Aut(N ) such that γ k ∈ Inn(N ) ′ and (δγ) k = 1 for all δ ∈ Inn(N ). Then Gaschütz' theorem does not holds for N . Suppose that L is a complement of N in G. Then D ′ ≤ L as shown by a computation as in (3.1). ∈ A ≤ G = N L, Let G = N W A be the group constructed in the proof of Theorem 15 (this does not require N ′ = N ). Since (γ. there exists δ ∈ N such that δ(γ, . . . , γ) ∈ L. It follows that (δγ) k (γ k , . . . , γ k ) ∈ N := (C 2Proof. Let G = N W A be the group constructed in the proof of Theorem 15 (this does not require N ′ = N ). Suppose that L is a complement of N in G. Then D ′ ≤ L as shown by a computation as in (3.1). Since (γ, . . . , γ) ∈ A ≤ G = N L, there exists δ ∈ N such that δ(γ, . . . , γ) ∈ L. It follows that (δγ) k (γ k , . . . , γ k ) ∈ N := (C 2 ⋊ Q 8 ) × C 2 = SmallGroup(144, 187) with \|Z(N )\| = 2. Decide whether or not Gaschütz' theorem holds for N. ⋊ Q 8 ) × C 2 = SmallGroup(144, 187) with \|Z(N )\| = 2. Decide whether or not Gaschütz' theorem holds for N . Absolute retracts in group theory. R Baer, Bull. Amer. Math. Soc. 52R. Baer, Absolute retracts in group theory, Bull. Amer. Math. 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Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden. W Gaschütz, Math. Z. 60W. Gaschütz, Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden, Math. Z. 60 (1954), 274-286. On finite complete groups. B Hartley, D J S Robinson, Arch. Math. (Basel). 35B. Hartley and D. J. S. Robinson, On finite complete groups, Arch. Math. (Basel) 35 (1980), 67-74. K H Hofmann, Zerfällung topologischer Gruppen. 84K. H. Hofmann, Zerfällung topologischer Gruppen, Math. Z. 84 (1964), 16-37. Subnormale Untergruppen und p-Sylowgruppen. B Huppert, Acta Sci. Math. (Szeged). 22B. Huppert, Subnormale Untergruppen und p-Sylowgruppen, Acta Sci. Math. (Szeged) 22 (1961), 46-61. . B Huppert, Endliche I Gruppen, Grundlehren Der Mathematischen Wissenschaften, Springer-Verlag134BerlinB. Huppert, Endliche Gruppen. I, Grundlehren der Mathematischen Wissenschaften, Vol. 134, Springer-Verlag, Berlin, 1967. I M Isaacs, Finite group theory. Providence, RIAmerican Mathematical Society92I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics, Vol. 92, American Mathematical Society, Providence, RI, 2008. Complementation of normal subgroups. J Kirtland, De GruyterBerlinJ. Kirtland, Complementation of normal subgroups, De Gruyter, Berlin, 2017. Das System aller Erweiterungen einer Gruppe. F Loonstra, Arch. Math. (Basel). 12F. Loonstra, Das System aller Erweiterungen einer Gruppe, Arch. Math. (Basel) 12 (1961), 262-279. On the existence of a complement for a finite simple group in its automorphism group. A Lucchini, F Menegazzo, M Morigi, Illinois J. Math. 47A. Lucchini, F. Menegazzo and M. Morigi, On the existence of a complement for a finite simple group in its automorphism group, Illinois J. Math. 47 (2003), 395-418. On a class of metabelian groups. M F Newman, Proc. London Math. Soc. 3M. F. Newman, On a class of metabelian groups, Proc. London Math. Soc. (3) 10 (1960), 354-364. 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[ "Large Scale, Large Margin Classification using Indefinite Similarity Measures", "Large Scale, Large Margin Classification using Indefinite Similarity Measures" ]	[ "Omid Aghazadeh ", "Stefan Carlsson " ]	[]	[]	Despite the success of the popular kernelized support vector machines, they have two major limitations: they are restricted to Positive Semi-Definite (PSD) kernels, and their training complexity scales at least quadratically with the size of the data. Many natural measures of similarity between pairs of samples are not PSD e.g. invariant kernels, and those that are implicitly or explicitly defined by latent variable models. In this paper, we investigate scalable approaches for using indefinite similarity measures in large margin frameworks. In particular we show that a normalization of similarity to a subset of the data points constitutes a representation suitable for linear classifiers. The result is a classifier which is competitive to kernelized SVM in terms of accuracy, despite having better training and test time complexities. Experimental results demonstrate that on CIFAR-10 dataset, the model equipped with similarity measures invariant to rigid and non-rigid deformations, can be made more than 5 times sparser while being more accurate than kernelized SVM using RBF kernels.	null	[ "https://arxiv.org/pdf/1405.6922v1.pdf" ]	774,257	1405.6922	668fd90eabc02206841ed075aef7e69b5d7cadc5	Large Scale, Large Margin Classification using Indefinite Similarity Measures Omid Aghazadeh Stefan Carlsson Large Scale, Large Margin Classification using Indefinite Similarity Measures Despite the success of the popular kernelized support vector machines, they have two major limitations: they are restricted to Positive Semi-Definite (PSD) kernels, and their training complexity scales at least quadratically with the size of the data. Many natural measures of similarity between pairs of samples are not PSD e.g. invariant kernels, and those that are implicitly or explicitly defined by latent variable models. In this paper, we investigate scalable approaches for using indefinite similarity measures in large margin frameworks. In particular we show that a normalization of similarity to a subset of the data points constitutes a representation suitable for linear classifiers. The result is a classifier which is competitive to kernelized SVM in terms of accuracy, despite having better training and test time complexities. Experimental results demonstrate that on CIFAR-10 dataset, the model equipped with similarity measures invariant to rigid and non-rigid deformations, can be made more than 5 times sparser while being more accurate than kernelized SVM using RBF kernels. Introduction Linear support vector machine (SVM) has become the classifier of choice for many large scale classification problems. The main reasons for the success of linear SVM are its max margin property achieved through a convex optimization, a training time linear in the size of the training data, and a testing time independent of it. Although the linear classifier operating on the input space is usually not very flexible, a linear classifier operating on a mapping of the data to a higher dimensional feature space can become arbitrarily complex. Mixtures of linear classifiers has been proposed to increase the non-linearity of linear classifiers [10,1]; which can be seen as feature mappings augmented with non-linear gating functions. The training of these mixture models usually scales bilinearly with respect to the data and the number of mixtures. The drawback is the non-convexity of the optimization procedures, and the need to know the (maximum) number of components beforehand. Kernelized SVM maps the data to a possibly higher dimensional feature space, maintains the convexity, and can become arbitrarily flexible depending on the choice of the kernel function. The use of kernels, however, is limiting. Firstly, kernelized SVM has significantly higher training and test time complexities when compared to linear SVM. As the number of support vectors grows approximately linearly with the training data [22], the training complexity becomes approximately somwehere between O(n 2 ) and O(n 3 ). Testing time complexity scales linearly with the number of support vectors, bounded by O(n). Secondly, the positive (semi) definite (PSD) kernels are sometimes not expressive enough to model various sources of variation in the data. A recent study [21] argues that metric constraints are not necessarily optimal for recognition. For example, in image classification problems, considering kernels as similarity measures, they cannot align exemplars, or model deformations when measuring similarities. As a response to this, invariant kernels were introduced [6] which are generally indef-inite. Indefinite similarity measures plugged in SVM solvers result in non-convex optimizations, unless explicitly made PSD, mainly using eigen decomposition methods [3]. Alternatively, latent variable models have been proposed to address the alignment problem e.g. [9,25]. In these cases, the dependency of the latent variables on the parameters of the model being learnt mainly has two drawbacks: 1) the optimization problem in such cases becomes non-convex, and 2) the cost of training becomes much higher than the case without the latent variables. This paper aims to address these problems using explicit basis expansion. We show that the resulting model: 1) has better training and test time complexities than kernelized SVM models, 2) can make use of indefinite similarity measures without any need for removal of the negative eigenvalues, which requires the expensive eigen decomposition, 3) can make use of multiple similarity measures without losing convexity, and with a cost linear in the number of similarity measures. Our contributions are: 1) proposing and analyzing Basis Expanding SVM (BE-SVM) regarding the aforementioned three properties, and 2) investigating the suitability of particular forms of invariant similarity measures for large scale visual recognition problems. 2 Basis Expanding Support Vector Machine 2.1 Background: SVM Given a dataset D = {(x 1 , y 1 ), . . . , (x n , y n )\|x i ∈ X , y i ∈ {−1, 1}} the SVM based classifiers learn max margin binary classifiers. The SVM classifier is f (x) = w, x ≥ 0 1 . The w is learnt via minimizing 1 2 w, w + C i H (y i , f (x)), where H (y, x) = max(0, 1 − xy) is the Hinge loss. Any positive semi definite (PSD) kernel k : X ×X → R can be associated with a reproducing kernel hilbert space (RKHS) H, and vice versa, that is ψ H (x), ψ H (y) = k(x, y), where ψ H : X → H is the implicitly defined feature mapping associated to H and consequently to k(., .). Representer theorem states that in such a case, ψ H (w) = i γ i k(., x i ) where γ i ∈ R ∀i. For a particular case of k(., .), namely the linear kernel k(x, y) = x · y associated with an Euclidean space, linear SVM classifier is f l (x) = w T x ≥ 0 where w is given by minimizing the primal SVM objective: 1 2 w 2 + C i H (y i , f l (x i )). More generally, given an arbitrary PSD kernel k(., .), the kernelized SVM classifier is f k (x) = i α i k(x, x i ) ≥ 0 where α i s are learnt by minimizing the dual SVM objective: 1 2 α T YKYα − α 1 , 0 ≤ α i ≤ C, α T y = 0 where Y = diag (y). The need for positiveness of k(., .) is evident in the dual SVM objective where the quadratic regularizing term depends on the eigenvalues of K ij = k(x i , x j ). In case of indefinite k(., .)s, the problem becomes non-convex and the inner products need to be re-defined, as there will be no associating RKHS to indefinite similarity measures. Various workarounds for indefinite similarity measures exist, most of which involve expensive eigen decomposition of the gram matrix [3]. A PSD kernel can be learnt from the similarity matrix, with some constraints e.g. being close to the similarity matrix where closeness is usually measured by the Frobenius norm. In case of Frobenius norm, the closed form solution is spectrum clipping, namely setting the negative eigenvalues of the gram matrix to 0 [3]. As pointed out in [4], there is no guarantee that the resulting PSD kernels are optimal for classification. Nevertheless, jointly optimizing for a PSD kernel and the classifier [4] is impractical for large scale scenarios. We do not go into the details of possible re-formulations regarding indefinite similarity measures, but refer the reader to [19,13,3] for more information. Linear and Kernelized SVM have very different properties. Linear SVM has a training cost of O(d x n) and a testing cost of O(d x ) where d x is the dimensionality of x . Kernelized SVM has a training complexity which is O(d k (nn sv )+n 3 sv ) [15] where d k is the cost of evaluating the kernel for one pair of data, and n sv is the number of resulting support vectors. The testing cost of kernelized SVM is O(d k n sv ). Therefore, a significant body of research has been dedicated to reducing the training and test costs of kernelized SVMs by approximating the original problem. Speeding up Kernelized SVM A common approach for approximating the kernelized SVM problem is to restrict the feature mapping of w: ψ H (w) ≈ ψ R (w) = J j=1 β j ψ H (z j ) where J < n. Methods in this direction either learn synthetic samples z j [24] or restrict z j to be on the training data [15]. These methods essentially exploit low rank approximations of the gram matrix K. Low rank approximations of PD K 0, result in speedups in training and testing complexities of kernelized SVM. Methods that learn basis coordinates outside the training data e.g. [24], usually involve intermediate optimization overheads, and thus are prohibitive in large scale scenarios. On the contrary, the Nyström method gives a low rank PSD approximation to K with a very low cost. The Nyström method [23] approximates K using a randomly selected subset of the data: K ≈ K nm K −1 mm K mn(1) where K ab refers to a sub matrix of K = K nn indexed by a = (a 1 , . . . , a n ) T , a i ∈ {0, 1}, and similarly by b. The approximation (1) is derived by defining eigenfunctions of k(., .) as expansions of numerical eigenvectors of K. A consequence is that the data can be embedded in an Euclidean space: K ≈ Ψ T mn Ψ mn , where Ψ mn , the Nyström feature space, is Ψ mn = K − 1 2 mm K mn(2) Methods exist which either explicitly or implicitly exploit this e.g. [14] to reduce both the training and test costs, by restricting the support vectors to be a subset of the bases defined by m. In case of indefinite similarity measures, K − 1 2 mm in (2) will not be real. In the rest of the paper, we refer to an indefinite version of a similarity matrix K withK, and refer to the normalization by K − 1 2 mm with Nyström normalization. In order to get a PSD approximation of an indefiniteK, the indefiniteK mm (1) needs to be made PSD. Spectrum clipping, spectrum flip, spectrum shift, and spectrum square are possible solutions based on eigen decomposition ofK mm . The latter can be achieved without the eigen decomposition step:K T mmKmm 0. If the goal is to find the PSD matrix closest to the original indefiniteK with respect to the reduced basis set m, spectrum clip gives the closed form solution. Therefore, when there are a few negative eigenvalues, the spectrum clip technique gives good low rank approximations toK mm which can be used by (1) to get a low rank PSD approximation toK. However, when there are a considerable number of negative eigenvalues, as it is the case with most of the similarity measures we consider later on in section 2.4, there is no guarantee for the resulting PSD matrix to be optimal for classification. This is true specially when eigenvectors associated with negative eigenvalues contain discriminative information. We experimentally verify in section 3.3 that the negative eigenvalues do contain discriminative information. We seek normalizations that do not assume a PSD K mm , and do not require eigen-decompositions. For example, one can replace K mm in (2) with the covariance of columns of K mn . We experimentally found out that a simple embedding, presented in the next section in (4), is competitive with the Nyström embedding (2) for PSD similarity measures, while outperforming it in case of indefinite ones that we studied. Basis Expanding SVM Basis Expanding SVM (BE-SVM) is a linear SVM classifier equipped with a normalization of the following explicit feature mapφ (x) = [s(b 1 , x), . . . , s(b B , x)] T(3) where B = {b 1 , . . . , b B } is an ordered basis set 2 which is a subset of the training data, and s(., .) is a pairwise similarity measure. The BE-SVM feature space defined by ϕ(x) = 1 E X [ φ − E X [φ] ] (φ(x) − E X [φ])(4) is similar to the Nyström feature space (2) with a different normalization scheme, as pointed out in section 2.2. The centralization ofφ(.) better conditions ϕ(.) for a linear SVM solver, and normalization by the average 2 norm is most useful for combining multiple similarity measures. The BE-SVM classifier is f B (x) = w T ϕ(x) ≥ 0 (5) where w is solved by minimizing the primal BE-SVM objective 1 2 w 2 2 + C i H (y i , f B (x i )) 2(6) An 1 regularizer results in sparser solutions, but with the cost of more expensive optimization than an 2 regularization. Therefore, for large scale scenarios, an 2 regularization, combined with a reduced basis set B, is preferred to an 1 regularizer combined with a larger basis set. Using multiple similarity measures is straightforward in BE-SVM. The concatenated feature map ϕ M (x) = ϕ (1) (x) T , . . . , ϕ (M ) (x) T Indefinite Similarity Measures for Visual Recognition The lack of expressibility of the PSD kernels have been argued before e.g. in [3,4,21]. For example, similarity measures which are not based on vectorial representations of data are most likely to be indefinite. Particularly in computer vision, considering latent information results in lack of a fixed vectorial representation of instances, and therefore similarity measures based on latent information are most likely to be indefinite 3 . A few applications of indefinite similarity measures in computer vision are pointed out below. [6] proposes (indefinite) jitter kernels for building desired invariances in classification problems. [1] uses indefinite pairwise similarity measures with latent positions of objects for clustering. [16] considers deformation models for image matching. [7] defines an indefinite similarity measure based on explicit correspondences between pairs of images for image classification. In this work, we consider similarity measures with latent deformations: s(x i , x j ) = max zi∈Z(xi), zj ∈Z(xj ) K I (φ(x i , z i ), φ(x j , z j )) + R(z i ) + R(z j )(7) where K I (., .) is a similarity measure (potentially a PD kernel), φ(x, z) is a representation of x given the latent variable z, R(z) is a regularization term on the latent variable z, and Z(x) is the set of possible latent variables associated with x. Specifically, when R(.) = 0 and Z(x) involves latent positions, the similarity measure becomes similar to that of [1]. When R(.) = 0 and Z(x) involves latent positions and local deformations, it becomes similar to the zero order model of [16]. Finally, an MRF prior in combination with latent positions and local deformations gives a similarity measure, similar to that of [7]. The proposed similarity measure (7) picks the latent variables which have the maximal (regularized) similarity values K I (., .)s. This is in contrast to [6] where the latent variables were suggested to be those which minimize a metric distance based on the kernel K I (., .). The advantage of a metric based latent variable selection is not so clear, while some works argue against unnecessary restrictions to metrics [21]. Also, if K I (., .) is not PSD, deriving a metric from it is at best expensive. Therefore, the latent variables in (7) are selected according to the similarity values instead of metric distances. Multi Class Classification SVMs are mostly known as binary classifiers. Two popular extensions to the multi-class problems are one-v-res (1vR) and one-v-one (1v1). The two simple extentions have been argued to perform as Training Testing (per sample) Memory Computation Memory Computation Table 1: Complexity Analysis for kernelized SVM and BE-SVM. The number of samples for each of the C classes was assumed to be equal to n C . M is the number of kernels/similarity measures, Md φ is the dimensionality of representations required for evaluating M kernels/similarity measures, and Md K is the cost of evaluating all M kernels/similarity measures. K SVM nMd φ + n 2 C n 2 Md K + n 3 C nMd φ nCMd K BE-SVM nMd φ + nM \|B\| nC\|B\|Md K \|B\|Md φ \|B\|CMd K well as more sophisticated formulations [20]. In particular, [20] concludes that in case of kernelized SVMs, in terms of accuracy they are both competitive, while in terms of training and testing complexities 1v1 is superior. Therefore, we only consider 1v1 approach for kernelized SVM. In case of linear SVMs however, 1v1 results in unnecessary overhead and 1vR is the algorithm of choice. A 1vR BE-SVM can be expected to be both faster and to generalize better than a 1v1 BE-SVM where bases from all classes are used in each of the binary classifiers. In case of 1v1 BE-SVM where only bases from the two classes under consideration are used in each binary classifier, there will be a clear advantage in terms of training complexity. However, due to the reduction in the size of the basis set, the algorithm generalizes less in comparison to a 1vR approach. Therefore, we only consider 1vR formulation for BE-SVM. Table 1 summarizes the memory and computational complexity analysis for 1v1 kernelized SVM and 1vR BE-SVM. Shown are the upper bounds complexities where we have considered n to be the upper bound on n sv . Margin Analysis of Basis Expanding SVM Both kernelized SVM and BE-SVM are max margin classifiers in their feature spaces. The feature space of kernelized SVM ψ H (.) is implicitly defined via the kernel function k(., .) while the feature space of the BE-SVM is explicitly defined via empirical kernel maps. In order to derive the margin as a function of the data, we first need to derive the dual BE-SVM objective, where we assume a non-squared Hinge loss and unnormalized feature mappingsφ(.). Borrowing from the representer theorem and considering the KKT conditions of the primal, one can derive w = i y i β iφ (x i ), and consequently derive the BE-SVM dual objective which is similar to the dual SVM objective but with K ij =φ(x i ) Tφ (x j ). Let S BX refer to the similarity of the data to the bases. We can see that the margin of the BE-SVM, given the optimal dual variables 0 ≤ β i ≤ C, is β T YS T BX S BX Yβ −1 , as opposed to α T YKYα −1 for the kernelized SVM, given the optimal dual variables 0 ≤ α i ≤ C. Furthermore, S T BX S BX is PSD, and that is BE-SVM's workaround for using indefinite similarity measures. We analyze the margin of BE-SVM in case of unnormalized features (φ(.) instead of ϕ(.)) and a non-squared Hinge loss. Given the corresponding dual variables, the margin of the BE-SVM was mentioned to be M BE (β) = β T YS T BX S BX Yβ −1 (8) as opposed to that of the kernelized SVM M K (α) = α T YKYα −1(9) For comparison, the margin of the Nyströmized method is M N (α) = α T YK XB K −1 BB K BX Yα −1(10) BE-SVM vs Kernelized SVM: When s(., .) = k(., .) and all training exemplars are used as bases, the margin of the BE-SVM will be β T YK 2 Yβ −1 . Comparing to the margin of SVM, for the same parameter C and the same kernel, it can be said that the solution (and thus the margin) of BE-SVM is even more derived by large eigenpairs, and even less by small ones. It is straightforward to verify K 2 = i λ 2 i v i v T i . Therefore, the contribution of large eigenpairs, that are {(λ i , v i )\|λ i > 1}, to K 2 is amplified. Similarly, the contribution of small eigenpairs, that are those with λ i < 1, to K 2 is dampened. (c) BE-SVM primal objective Figure 1: Demonstration of kernelized SVM and BE-SVM using two Gaussian RBF kernels with γ 1 = 10, γ 2 = 10 2 and C = 10. 1a is based on equally weighted kernels. 1b is without normalization. 1c is with normalization on 10% of the data randomly selected as bases. 10 fold cross validation accuracy and the number of support vectors are averaged over i = 1 : 20 scenarios based on the same problem but with different spatial noises. The noise model for i th scenario is a zero mean Gaussian with σ i = 10 −2 i. The visualization is on the noiseless data for clarity. Best viewed electronically. BE-SVM vs Nyströmized method: When s(., .) = k(., .) and a subset of training exemplars are used as bases (reduced settings), the resulting margin of BE-SVM is β T YK XB K BX Yβ −1 . Comparing to the margin of the Nyströmized method, we can say that the most of the difference between the Nyströmized method and BE-SVM, is the normalization by K −1 BB . For covariance kernels, that the Nyströmized method is most suitable for, K BB is the covariance of the basis set in the feature space. Therefore, it can be said that the normalization by K −1 BB essentially de-correlates the bases in the feature space. Although this is an appealing property, there is no associating RKHS with indefinite similarity measures and the de-correlation in such cases is non-trivial. In case of covariance kernels, it can be said that BE-SVM assumes un-correlated bases, while bases are always correlated in the feature space. As larger sets of bases usually result in more (non-diagonal) covariances, the un-correlated assumption is more violated with large set of bases. The consequence is that in such cases, that are covariance kernels with large set of bases, BE-SVM can be expected to perform worse than the Nyströmized method. However, for sufficiently small set of bases, or in case of indefinite similarity measures, there is no reason for superiority of the Nyströmized method. In such cases and in practice, BE-SVM is competitive or better than the Nyströmized method. Figure 1 visualizes the use of multiple Gaussian RBF kernels in BE-SVM and kernelized SVM. We point out the following observations. 1) the dual objective of BE-SVM (exact) tends to result in sparser solutions as measured by nonzero support vector coefficient (compare 1a with 1b). We believe the main reason for this to be the modification of the eigenvalues as described in section 2.6. Note however that in order to classify a new sample, its similarity to all training data needs to be evaluated, irrespective of the sparsity of the BE-SVM solution (see equation (13)). In this sense, the BE-SVM dual objective results in completely dense solutions, similar to the primal BE-SVM objective without any basis reduction. However, the solution can be made sparse by construction, by reducing the basis set, similar to the case with the primal BE-SVM objective. We do not demonstrate this here, mainly because our main focus is on the (approximate) primal objective. Demonstration on 2D Toy data 2) due to the definition of the (linear) kernel in BE-SVM (see equation (13)), the solution of the BE-SVM has an inherent bias with respect to the (marginal) distribution of class labels. In other words, the contribution of each class to the norm ofφ(.), and consequently to the value ofK(., .), directly depends on the number of bases from each class. Consequently, the decision boundary of BE-SVM is shifted towards the class with less bases: compare the decision boundaries on the left sides of 1a and 1b. In experiments on CIFAR-10 dataset, as the number of exemplars from different classes are roughly equal, this did not play a crucial role. Related Work There exists a body of work regarding the use of proximity data, similarity, or dissimilarity measures in classification problems. [18] uses similarity to a fixed set of samples as features for a kernel SVM classifier. [11] uses proximities to all the data as features for a linear SVM classifier. [12] uses proximities to all the data as features and proposes a linear program machine based on this representation. In contrast, we use a normalization of the similarity of points to a subset of the data as features for a (fast, approximate) linear SVM classifier. Experiments Dataset and Experimental Setup We present our experimental results on CIFAR-10 dataset [17]. The dataset is comprised of 60,000 tiny 32 × 32 RGB images, 6,000 images for each of the 10 classes involved, divided into 6 folds with inequal distribution of class labels per fold. The first 5 folds are used for training and the 6th fold is used for testing. We use a modified version of the HOG feature [5], described in [9]. Due to the normalization of each of the HOG cells, namely normalizing by gradient/contrast information of the neighboring cells, the HOG cells on border of images are not normalized properly. We believe this to have a negative effect on the results, but as the aim of this paper is not to get the best results possible out of the model, we rely on the consistency of the normalization for all images to address this problem. A possible fix is to up-sample images and ignore the HOG-cells at the boundaries, but we do not provide the results for such fixes. For all the experiments, we center the HOG feature vector and scale feature vectors inversely by the average 2 norm of the centered feature vectors, similar to the normalization of BE-SVM (4). This results in easier selection of parameters C and γ for SVM formulations. Unless stated otherwise, we fix C = 2 and γ = 1 for kernelized SVM with Gaussian RBF kernels, and C = 1 for the rest. We use LibLinear [8] to optimize the primal linear SVM objectives with squared Hinge loss, similar to (6). For kernelized SVM, we use LibSVM [2]. We report multi-class classification results (0-1 loss) on the test set, where we used a 1 v 1 formulation for kernelized SVM, and 1 v all formulation for other methods. Figure 4a shows the performance of linear SVM (H4L and H8L) and kernelized SVM with Gaussian RBF kernel (K4R and K8R) as a function of number of parameters in the models. The number of parameters for linear SVM is the input dimensionality, and for kernelized SVM it is the sum of n sv (d φ + 1) where d φ is the dimensionality of the feature vector the corresponding kernel operates on. The 5 numbers for each model are the results of the model trained on 1, . . . , 5 folds of the training data (each fold contains 10,000 samples). Figure 4b shows the performance kernelized SVM as a function of support vectors when trained on 1, . . . , 5 folds. Except the linear SVM with a HOG cell size of 8 pixels (496 dimensions) which saturates its performance at 4 folds, all models consistently benefit from more training data. Baseline: SVM with Positive Definite Kernels BE-SVM with Invariant Similarity Measures The general form of the invariant similarity measures we consider was given in (7). In particular, we consider rigid and deformable similarity measures where the smallest unit of deformation/translation is a HOG cell. The rigid similarity measure models invariance to translations and is given by K R (x, y) = max z R ∈Z R c∈C φ C (x, c) T φ C (y, c + z R )(11) where is zero for cells outside x (zero-padding). K R (x, y) is the maximal cross correlation between φ(x) and φ(y). Z R = {(z x , z y )\|z x , z y ∈ {−h R , . . . , h R }} The deformable similarity measure allows local deformations (displacements) of each of the HOG cells, in addition to invariance to rigid deformations K L (x, y) = max z R ∈Z R c∈C max z L ∈Z L φ C (x, c) T φ C (y, c + z R + z L )(12) where Z L = {(z x , z y )\|z x , z y ∈ {−h L , . . . , h L }} allows a maximum of h L HOG cell local deformation for each of the HOG cells of y. We consider a maximum deformation of 8 pixels e.g. 2 HOG cells for a HOG cell size of 4 pixels. Regularizing global or local deformations is straightforward in this formulation. However, we did not notice significant improvements for the set of displacements we considered, which is probably related to the small size of the latent set suitable for small images in CIFAR-10. Figure 2a shows the performance of BE-SVM using different similarity measures, when trained on the first fold. It can be seen that the invariant similarity measures improve recognition performance. Particularly, in absence of any other information, modelling rigid deformations (latent positions) seems to be much more beneficial than modelling local deformations. An interesting observation is that aligning the data in higher resolutions is much more crucial: all models (linear SVM, kernelized SVM, and BE-SVM) suffer performace losses when the resolution is increased from a HOG cell size of 8 pixels to 4 pixels. However, BE-SVM achieves significant performance gains by aligning the data in higher resolutions: compare H4L with H4(1,0) and H4(2,0), and H8L with H8(1,0). We tried training linear and kernelized SVM models by jittering the feature vectors, in the same manner that the invariant similarity measures do (11), (12); that is to jitter the HOG cells with zeropadding for cells outside images. This resulted in significant performance losses for both linear SVM and kernelized SVM, while also siginificantly increasing memory requirement and computation times. We believe the reason for this to be the boundary effects; which are also mentioned in previous work e.g. [6]. We also believe that jittering the input images, in combination with some boundary heuristics (see section 3.1), will improve the test performance (while significantly increasing training complexities), but we do not provide experimental results for such cases. Figure 2b shows accuracy of BE-SVM using different similarity measures and different basis selection strategies; for a basis size of B = 10 × 100 exemplars. In the figure, 'Rand' refers to a random selection of the bases, 'Indx' refers to selection of samples according to their indices, 'K KMed' refers to a kernel k-medoids approach based on the similarity measure, and 'Nystrom' refers to selection of bases similar to the 'Indx' approach, but with the Nyström normalization, using a spectrum clip fix for indefinite similarity measures(see section 2.2). The reported results for 'Rand' method is averaged over 5 trials; the variance was not significant. It can be observed that all methods except the 'Nystrom' result in similar performances. We also tried other sophisticated sample selection criteria, but observed similar behaviour. We attribute this to little variation in the quality of exemplars in the CIFAR-10 dataset. Having observed this, for the rest of sub-sampling strategies, we do not average over multiple random basis selection trials, but rather use the deterministic 'Indx' approach. Basis Selection The difference between normalization factors in BE-SVM and Nyström method (see section 2.2) is evident in the figure. The BE-SVM normalization tends to be consistently superior in case of indefinite similarity measures. For PSD kernels (H4L, H8L, H4R, and H8R) , the Nyström normalization tends to be better in lower resolutions (H8) and worse in higher resolutions (H4). We believe the main reason for this is to be lack of significant similarity of bases in higher resolutions in absence of any alignment. In such cases, the low rank assumption of K [23] is violated, and normalization by a diagonally dominant K mm will not capture any useful information. In order to analyze how the performance of BE-SVM depends on the eigenvalues of the similarity measures, we provide the following eigenvalue analysis. We compute the similarity of the bases to themselves -corresponding to K mm in (2) -and perform an eigen-decomposition of the resulting matrix. Table 2 shows the ratio of negative eigenvalues: 'NgRat'= 1 B i [λ i < 0] , and the relative energy of eigenvalues 'NgEng'= i \|λi\| [λi<0] i \|λi\|[λi>0] as a function of various similarity measures for B = 10 × 100 and a HOG cell size of 4. The last two columns, namely 'CorNyst' and 'CorBE' reflect the correlation of the measured entities -'NgRat' and 'NgEng' -to the observed performance of BE-SVM using the Nyström normalization and BE-SVM normalization. We used Pearson's r to measure the extent of linear dependence between the test performances and different normalization schemes. It can be observed that: 1) both normalization schemes have a positive correlation with both the ratio of negative eigenvalues and their relative energy, and 2) BE-SVM normalization correlates more strongly with the observed entities. From this, we conclude that negative eigenvectors contain discriminative information and that BE-SVM's normalization is more suitable for indefinite similarity measures. We also experimented with spectrum flip and spectrum square methods for the Nyström normalization, but they generally provided slightly worse results in comparison to the spectrum clip technique. Multiple Similarity Measures Different similarity measures contain complementary information. Fortunately, BE-SVM can make use of multiple similarity measures by construction. To demonstrate this, using one fold of training data and B = 10 × 50, we greedily -in an incremental way -augmented the similarity measures with the most contributing ones. Using this approach, we found two (ordered) sets of similarity measures with complementary information: 1) a low-resolution set M 1 = {H8R, H8(1, 0), H8(0, 1)}, and 2) a two-resolution set M 2 = {H8R, H4(2, 0), H4(0, 1), H8(1, 0)}. Surprisingly, the two resolution sequence resembles those of the part based models [9], and multi resolution rigid models See text for analysis. [1] in that the information is processed at two levels: a coarser rigid 'root' level and a finer scale deformable level. We then trained BE-SVM models using these similarity measures for various sizes of the basis set, and for various sizes of training data. Figures 4a and 4b show these results, where the BE-SVM models are trained on all 5 folds. The shown number of supporting exemplars (and consequently the number of parameters) for BE-SVM are based on the size of the basis set. It can be seen that using a basis size of B = 10 × 250, the performance of the BE-SVM using more than 3 tworesolution similarity measures surpass that of the kernelized SVM trained on all the data and based on approximately B = 10 × 4000 support vectors. Using low-resolution similarity measures, B = 10 × 500 outperforms kernelized SVMs trained on up to 4 folds of the training data. Furthermore, it can be observed that for the same model complexity, as measured either by the number of supporting exemplars, or by model parametrs, BE-SVM performs better than kernelized SVM. Figure 3 shows the performance of BE-SVM using different similarity measures for various basis sizes and for different training set sizes. It can be observed that using (invariant) indefinite similarity measures can significantly increase the performance of the model: compare the red curve with any other curve with the same line style. For example, using all the training data and a two resolution deformable approach results in 8-10% improvements in accuracy in comparison to the best performing PSD kernel (H8R). Furthermore, the two-resolution approach outperforms the single resolution approach by approximately 3-4% accuracy (compare blue and black curves with the same line style). Measured by model parameters, BE-SVM is roughly 8 times sparser than kernelized SVM for the same accuracy. Measured by supporting exemplars, its sparsity increases roughly to 30. We need to point out that different similarity measures have different complexities e.g. H8(1,0) is more expensive to evaluate than K8R. However, when the bases are shared for different similarity measures, CPU cache can be utilized much more efficiently as there will be less memory access and more (cached) computations. Multiple Kernel Learning with PSD Kernels We tried Multiple Kernel Learning (MKL) for kernelized SVM with PSD kernels. When compared to sophisticated MKL methods, we found the following procedure to give competitive performances, with much less training costs. Defining K C (., .) = αK 1 (., .) + (1 − α)K 2 (., .), our MKL approach consists of performing a line search for an optimal alpha α ∈ {0, .1, . . . , 1} which results in best 5- fold cross validating performance. Using this procedure, linear kernels were found not to contribute anything to Gaussian RBF kernels. The optimal combination for high resolution and low resolution Gaussian RBF kernels (K4R and K8R) resulted in a performance gain of less than 0.5% accuracy in comparison to K8R. We founds this insignificant, and did not report its performance, considering the fact that the number of parameters increases approximately 4 times using this approach. BE-SVM's Normalizations It can be verified that in case of unnormalized featuresφ (m) (.), the corresponding Gram matrix will beK (x i , x j ) = M m=1K (m) (x i , x j ) = M m=1 B b=1 s (m) (b b , x i )s (m) (b b , x j )(13) whereK (m) s are combined with equal weights, the value of each of which depends (locally) on how the similarities of x i and x j correlate with respect to the bases. In the case of normalized features, the centered values of each similarity measure is weighted by While the BE-SVM's normalization of empirical kernel maps is not optimal for discrimination, it can be seen as a reasonable prior for combining different similarity measures. Utilizing such a prior, in combination with linear classifiers and P regularizers, has two important consequences: 1) the centering helps reduce the correlation between dimensions and the scaling helps balance the effect of regularization on different similarity measures, irrespective of their overall norms, and 2) such a scaling directly affects the parameter tuning for learning the linear classifiers: for all the similarity measures (and combinations of similarity measures) with various basis sizes, the same parameter: C = 1 was used to train the classifiers. While cross-validation will still be a better option, cross-validating for different parameters settings -and specially when combining multiple similarity measures -will be very expensive and prohibitive. By using the BE-SVM's normalization, we essentially avoid searching for optimal combining weights for different similarity measures and also tuning for the C parameter of the linear SVM training. (E X [ φ − E X [φ] ]) −2 i.e. In this section, we quantitatively evaluate the normalization suggested for BE-SVM (4), and compare it to a few other combinations. Particularly, we consider various normalizations of the HOG feature vectors, and similarly, various normalization schemes for the empirical kernel mapφ (3). We consider the following normalizations: Figure 5: Performance of BE-SVM for different normalization schemes of the feature vector and the empirical kernel map, and different similarity measures. "F + K (P)" in the legend reflects using F and K normalization schemes for the feature vectors and the empirical kernel maps respectively, which results in the average test performance of P (averaged over the similarity measures). Figure 6: Performance of BE-SVM for different normalization schemes of the feature vector and the empirical kernel map, and different combinations of similarity measures. "F + K (P)" in the legend reflects using F and K normalization schemes for the feature vectors and the empirical kernel maps respectively, which results in the average test performance of P (averaged over the combinations of similarity measures). • No normalization (Unnorm) • Z-Scoring, namely centering and scaling each dimension by the inverse of its standard deviation (Z-Score) • BE-SVM normalization, namely centering and scaling all dimensions by the inverse average 2 norm of the centered vectors (BE-SVM) We report test performances for all combinations of normalizations for the feature vectors and the empirical kernel maps, for two cases: 1) when C = 1, and 2) when the C parameter is crossvalidated from C = {10 −1 , 10 0 , 10 1 }. In both cases, \|B\| = 10 × 100 bases were uniformly subsampled from the first fold of the training set ('Indx' basis selection). Figure 5 shows the performance of BE-SVM in combination with different normalizations of the feature vectors and empirical kernel maps, and for different similarity measures. On top, reported numbers are for C = 1 while on the bottom, C is cross validated. It can be observed that the BE-SVM's normalization works best both for the feature and empirical kernel map normalizations. Although z-scoring is more suitable for linear similarity measures (compare BE-SVM + BE-SVM with Z-SCORE + BE-SVM in H4L, H8L, H4(x,y) and H8(x,y)), overall BE-SVM's normalization of the feature space works better than the alternatives. Particularly, in single similarity measure cases, it seems that normalizing the feature according to the BE-SVM's normalization is more important than normalizing the empirical kernel map. While the cross-validation of the C parameter marginally affects the performance, it does not change the conclusions drawn from the C = 1 case. Figure 6 shows the performance of BE-SVM in combination with different normalizations of the feature vectors and empirical kernel maps, and for different combinations of similarity measures (the sequence of greedily augmented similarity measures M 2 : the set of two resolution similarity measures described in Section 3.5). It can be observed that BE-SVM's normalization of the kernel map is much more important and effective when combining multiple similarity measure (compare to Figure 5) . These observations quantitatively motivate the use of BE-SVM's normalization with the following benefits, at least on the dataset we experimented on: • It removes the need for cross-validation for tuning the C parameter, and mixing weights for different similarity measures. • As the feature vector is centered and properly scaled, the linear SVM solver converges much faster than the unnormalized case, or when C >> 1. • It results in robust learning of BE-SVM which can efficiently combine different similarity measures i.e. RBF kernels (H8R), and linear deformable similarity measures (H4(2,0), H4(0,1), H8(1,0)). Conclusion We analyzed scalable approaches for using indefinite similarity measures in large margin scenarios. We showed that our model based on an explicit basis expansion of the data according to arbitrary similarity measures can result in competitive recognition performances, while scaling better with respect to the size of the data. The model named Basis Expanding SVM was thoroughly analyzed and extensively tested on CIFAR-10 dataset. In this study, we did not explore basis selection strategies, mainly due to the small intra-class variation of the dataset. We expect basis selection strategies to play a crucial role in the performance of the resulting model on more challenging datasets e.g. Pascal VOC or ImageNet. Therefore, an immediate future work is to apply BE-SVM to larger scale and more challenging problems e.g. object detection, in combination with data driven basis selection strategies. T encodes the values of the M similarity measures evaluated on the corresponding bases B (1) , . . . , B (M ) . In this work, we restrict the study to the case that the bases are shared among the M similarity measures: i.e. B (1) = . . . = B (M ) . For most of our experiments, we use HOG cell sizes of 8 and 4, which result in 31 × 1984 dimensional representation of each of the images. Figure 2 : 2Performance of BE-SVM as a function of different similarity measures when trained on the first fold. An H4 (H8) refers to a HOG cell size of 4 (8) pixels. L and R refer to linear and Gaussian RBF kernels respectively, and (h R , h L ) refers to a similarity measure with h R rigid and h L local deformations (11),(12). allows a maximum of h R HOG cells displacements in x, y directions, C = {(x, y)\|x, y ∈ {h 1 , . . . , h H } is the set of indices of h H HOG cells in each direction, and φ C (x, c) is the 31 dimensional HOG cell of x located at position c. φ C (x, c) Figure 3 : 3Performance of BE-SVM using multiple similarity measures for various sizes of the basis set. Results with dotted, dashed, and solid lines represent 1, 3, and 5 folds worth of training data. Figure 4 : 4Performance of BE-SVM vs model parameters for various sizes of the basis set, using multiple similarity measures. Each curve for linear SVM (H4L, H8L) and kernelized SVM (K4R, K8R) represents the result for training on 1, . . . , 5 folds of training data. Each curve for BE-SVM shows the result for training model with a basis set of size B = 10 × {25, 50, 100, 250, 500} when trained on 5 folds of the training data. more global weight is put on (the centered values of) the similarity measures with smaller variances in similarity values. SVM + BE−SVM (65) Unnorm + BE−SVM (64) Z−Score + BE−SVM (59) BE−SVM + Unnorm (56) Z−Score + Unnorm (56) BE−SVM + Z−Score (51) Unnorm + Z−Score (48) Unnorm + Unnorm (47) Z−Score + Z−Score (43) SVM + BE−SVM (66) Unnorm + BE−SVM (65) Z−Score + BE−SVM (60) BE−SVM + Unnorm (58) Z−Score + Unnorm (58) BE−SVM + Z−Score (51) Unnorm + Z−Score (49) Unnorm + Unnorm (47) Z−Score + Z−Score (44) Table 2 : 2Eigenvalue analysis of various similarity measures based on HOG cell size 4. We omit the bias term for the sake of clarity. For the moment assume B is given. 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[ "92¢ /MFlops/s, Ultra-Large-Scale Neural-Network Training on a PIII Cluster", "92¢ /MFlops/s, Ultra-Large-Scale Neural-Network Training on a PIII Cluster" ]	[ "Gordon \nResearch School of Information Sciences and Engineering\nDepartment of Computer Science\nAustralian National University\n0200Canberra, Robert EdwardsAustralia\n", "Jonathan Baxter [email protected] \nAustralian National University\n0200CanberraAustralia\n" ]	[ "Research School of Information Sciences and Engineering\nDepartment of Computer Science\nAustralian National University\n0200Canberra, Robert EdwardsAustralia", "Australian National University\n0200CanberraAustralia" ]	[]	Artificial neural networks with millions of adjustable parameters and a similar number of training examples are a potential solution for difficult, large-scale pattern recognition problems in areas such as speech and face recognition, classification of large volumes of web data, and finance. The bottleneck is that neural network training involves iterative gradient descent and is extremely computationally intensive. In this paper we present a technique for distributed training of Ultra Large Scale Neural Networks 1 (UL-SNN) on Bunyip, a Linux-based cluster of 196 Pentium III processors. To illustrate ULSNN training we describe an experiment in which a neural network with 1.73 million adjustable parameters was trained to recognize machineprinted Japanese characters from a database containing 9 million training patterns. The training runs with a average performance of 163.3 GFlops/s (single precision). With a machine cost of $150,913, this yields a price/performance ratio of 92.4¢ /MFlops/s (single precision). For comparison purposes, training using double precision and the ATLAS DGEMM produces a sustained performance of 70 MFlops/s or $2.16 / MFlop/s (double precision).	10.1109/sc.2000.10031	[ "https://arxiv.org/pdf/1911.05181v1.pdf" ]	417,700	1911.05181	e8c0c7b38759c8fd96c7d2ec64a0d1929b6a55ca	92¢ /MFlops/s, Ultra-Large-Scale Neural-Network Training on a PIII Cluster Gordon Research School of Information Sciences and Engineering Department of Computer Science Australian National University 0200Canberra, Robert EdwardsAustralia Jonathan Baxter [email protected] Australian National University 0200CanberraAustralia 92¢ /MFlops/s, Ultra-Large-Scale Neural-Network Training on a PIII Cluster Bell Price/Performance winner. Student paper award finalist Douglas Aberdeen (corresponding, presenting and student author)neural-networkLinux clustermatrix-multiply Artificial neural networks with millions of adjustable parameters and a similar number of training examples are a potential solution for difficult, large-scale pattern recognition problems in areas such as speech and face recognition, classification of large volumes of web data, and finance. The bottleneck is that neural network training involves iterative gradient descent and is extremely computationally intensive. In this paper we present a technique for distributed training of Ultra Large Scale Neural Networks 1 (UL-SNN) on Bunyip, a Linux-based cluster of 196 Pentium III processors. To illustrate ULSNN training we describe an experiment in which a neural network with 1.73 million adjustable parameters was trained to recognize machineprinted Japanese characters from a database containing 9 million training patterns. The training runs with a average performance of 163.3 GFlops/s (single precision). With a machine cost of $150,913, this yields a price/performance ratio of 92.4¢ /MFlops/s (single precision). For comparison purposes, training using double precision and the ATLAS DGEMM produces a sustained performance of 70 MFlops/s or $2.16 / MFlop/s (double precision). Introduction Artificial neural networks are a class of parametric, nonlinear statistical models that have found wide-spread use in many pattern recognition domains, including speech recognition, character recognition, signal processing, medical diagnosis and finance. The typical network in such an appli-cation has 100-100,000 adjustable parameters and requires a similar number of training patterns in order to generalize well to unseen test data. Provided sufficient training data is available, the accuracy of the network is limited only by its representational power, which in turn is essentially proportional to the number of adjustable parameters. Thus, in domains where large volumes of data can be collectedsuch as speech, face and character recognition, and web page classification -improved accuracy can often be obtained by training a much larger network. In this paper we describe a method for distributed training of Ultra Large Scale Neural Networks (ULSNN), or networks with more than one million adjustable parameters and a similar number of training examples. At its core, the algorithm uses Emmerald, a single-precision (32 bit) general matrix-matrix multiply (SGEMM) based on the Pentium III SIMD Streaming Extensions (SSE), with a peak performance in excess of 1090 MFlops/s on a single 550 MHz Pentium III. The use of single-precision floating point operations is justified by the fact that we have found it sufficient for gradient-based training of ULSNN's. For medium-large scale neural networks as few as 16 bits precision is sufficient [2]. To illustrate the use of our ULSNN training code, we describe an experiment in which a neural network with 1.73 million adjustable parameters is being trained to recognize machine-printed Japanese characters from a database containing 9 million training patterns. The training is running on Bunyip, a 196 processor, Linux-based Intel Pentium III cluster consisting of 98 dual 550 MHz processor PC's, each containing 384 MBytes of RAM, 13 GBytes of hard disk and 3x100 Mb/s fast Ethernet cards. All components in Bunyip are "COTS" (Commodity-Off-The-Shelf), and were sourced from a local PC manufacturer (see http://tux.anu.edu.au/Projects/Beowulf/). Our longest experiment took 56 hours and 52 minutes, requiring a total of 31.2 Peta Flops (10 15 single-precision floating-point operations), with an average performance of 152 GFlops/s (single precision) while under load. With no other user processes running the performance increases to 163.3 GFlops/s which was sustained for a four hour test before returning access to other users. Total memory usage during training was 32.37 GBytes. The total machine cost, including the labor cost in construction, was AUD$253,000, or USD$150,913 at the exchange rate of AUD$1 = .5965¢ USD on the day of the final and largest payment. This gives a final price/performance ratio of USD 92.4¢ /MFlops/s (single precision). For comparison purposes, training using double precision and the ATLAS DGEMM [11] produced a sustained performance of 70 MFlops/s or $2.16 /MFlops/s (double precision). "Bunyip" Hardware Details The machine used for the experiments in this paper is "Bunyip", a 98-node, dual Pentium III Beowulf-class system running Linux kernel 2.2.14. Our main design goals for this machine were to maximise CPU and network performance for the given budget of AUD $250,000 (about USD $149,125). Secondary factors to be balanced into the equation were: amount of memory and disk; reliability; and the overall size of the machine. All dollar figures quoted in the remainder of this paper are US dollars. The Intel Pentium III processors were chosen over Alpha or SPARC processors for price/performance reasons. Dual-CPU systems were preferable as overall cost and size per CPU is lower than single-CPU or quad-CPU systems. Unfortunately, at the time of designing this machine AMD Athlon and Motorola/IBM G4 systems were not available in dual-CPU configurations. We were also keen to use the SSE floating point instructions of the Pentium III range. 550 MHz CPUs were eventually selected as having the best price/performance available in the Pentium III range at that time. For the networking requirements, we decided to go with a commodity solution rather than a proprietary solution. Gigabit ethernet was considered, but deemed too expensive at around $300 per node for the NIC and around $1800 per node for the switch. Instead, a novel arrangement of multiple 100 Mb/s NICs was selected with each node having three NICs which contributed some $65 per node (plus switch costs -see below). The configuration for each node is dual Intel Pentium III 550 CPUs on an EPoX KP6-BS motherboard with 384 MBytes RAM, 13 GByte UDMA66 (IDE) hard disk and three DEC Tulip compatible 100 Mb/s network interfaces, one of which has Wake-On-LAN capability and provision for a Boot ROM. The nodes have no removable media, no video capability and no keyboards. Each node cost $1282. With reference to figure 1, logically the 96 nodes are connected in four groups of 24 nodes arranged as a tetrahedron with a group of nodes at each vertex. Each node in a vertex has its three NICs assigned to one of the three edges emanating from the vertex. Each pair of vertices is connected by a 48-port Hewlett-Packard Procurve 4000 switch (24 ports connecting each way). The switching capacity of the Procurve switches is 3.8 Gb/s. The bi-sectional bandwidth of this arrangement can be determined by looking at the bandwidth between two groups of nodes and the other two groups through 4 switches, giving a total of 15.2 Gb/s. The 48-port switches cost $2386 each. Two server machines, more or less identical to the nodes, with the addition of CD-ROM drives, video cards and keyboards, are each connected to a Netgear 4-port Gigabit switch which is in turn connected to two of the HP Procurve switches via gigabit links. The two server machines also act as connections to the external network. Two hot-spare nodes were also purchased and are used for development and diagnostic work when not required as replacements for broken nodes. Total Cost Single precision general matrix-matrix multiply (SGEMM) Without resorting to the complexities associated with implementing Strassen's algorithm on deep-memory hierarchy machines [9,10], dense matrix-matrix multiplication requires 2M N K floating point operations where A : M × K and B : K × N define the dimensions of the two matrices. Although this complexity is fixed, skillful use of the memory hierarchy can dramatically reduce overheads not directly associated with floating point operations. Memory hierarchy optimization combined with the use of SSE gives Emmerald its performance advantage. Emmerald implements the SGEMM interface of Level-3 BLAS, and so may be used to improve the performance of single-precision libraries based on BLAS (such as LA-PACK [8]). There have been several recent attempts at automatic optimization of GEMM for deep-memory hierarchy machines, most notable are PHiPAC [6] and the more recent ATLAS [11]. ATLAS in particular achieves performance close to vendor optimized commercial GEMMs. Neither ATLAS nor PhiPAC make use if the SSE instructions on the PIII for their implementation of SGEMM. SIMD Parallelisation A SIMD GEMM must aim to minimize the ratio of memory accesses to floating point operations. We employed two core strategies to achieve this: • accumulate results in registers for as long as possible to reduce write backs; Figure 3. L1 blocking for Emmerald: C ← A B where A and B are in L1 and C is accumulated in registers. ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ ¢ £ ¡ £ ¡ £ £ ¡ £ ¡ £ £ ¡ £ ¡ £ £ ¡ £ ¡ £ ¤ ¡ ¤ ¡ ¤ ¤ ¡ ¤ ¡ ¤ ¤ ¡ ¤ ¡ ¤ ¤ ¡ ¤ ¡ ¤ ¥ ¡ ¥ ¥ ¡ ¥ ¥ ¡ ¥ ¦ ¡ ¦ ¦ ¡ ¦ ¦ ¡ ¦ § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ § § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ § ¡ §¨ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡¨ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ buffered k=336 k A' B' B A C m=1 M A' B' C' n=5 N • re-use values in registers as much as possible. In [5] several dot-products were performed in parallel inside the innermost loop of the GEMM. Taking the same approach we found experimentally that 5 dot-products in the inner loop gave the best performance. Figure 2 shows how these 5 dot products utilise SIMD parallelism. Optimizations A number of techniques are used in Emmerald to improve performance. Briefly, they include: • L1 blocking: Emmerald uses matrix blocking [5,6,11] to ensure the inner loop is operating on data in L1 cache. Figure 3 shows the L1 blocking scheme. The block dimensions m and n are determined by the configuration of dot-products in the inner loop (Section 3.2) and k was determined experimentally. • Unrolling: The innermost loop is completely unrolled for all possible lengths of k in L1 cache blocks, taking care to avoid overflowing the instruction cache. • Re-buffering: Since B (Figure 3) is large (336 × 5) compared to A (1 × 336), we deliberately buffer B into L1 cache. While buffering B we re-order its elements to enforce optimal memory access patterns. This has the additional benefit of minimising translation look-aside buffer misses [12]. • Pre-fetching: Values from A are not buffered into L1 cache. We make use of SSE pre-fetch assembler instructions to ensure A values will be in L1 cache when needed. • L2 Blocking: Efficient L2 cache blocking ensures that peak rates can be maintained as long as A, B and C fit into main memory. Emmerald Results The performance of Emmerald was measured by timing matrix multiply calls with size M = N = K = 16 up to 700. The following steps were taken to ensure a conservative performance estimate: • wall clock time on an unloaded machine is used rather than CPU time; • the stride of the matrices, which determines the separation in memory between each row of matrix data, is fixed to 700 rather than the optimal value (the length of the row); • caches are flushed between calls to sgemm(). Timings were performed on a PIII 450MHz running Linux (kernel 2.2.14). Training Neural Networks using SGEMM In this section we describe one-hidden-layer artificial neural networks and, following [3], how to compute the gradient of a neural network's error using matrix-matrix multiplication. We then describe our conjugate-gradient approach to training neural networks. Artificial Neural Networks A one-hidden-layer artificial neural network maps input vectors x = (x 1 , . . . , x ni ) ∈ R ni to output vectors y = (y 1 , . . . , y no ) ∈ R no according to the formula: y i (x) = σ   n h j=1 w ho ij h j (x)   ,(1) where σ : R → R is some squashing function (we use σ = tanh), w ho ij are the adjustable parameters connecting the hidden nodes to the output nodes, and h j (x) is the activation of the j-th hidden node: h j (x) = σ ni k=1 w ih jk x k .(2) In the last expression, w ih jk are the adjustable parameters connecting the input nodes to the nodes in the hidden layer. Given a matrix X of n p training patterns and a matrix T of desired outputs for the patterns in X, X =    x 11 . . . x 1ni . . . . . . . . . x np1 . . . x npni    , T =    t 11 . . . t 1no . . . . . . . . . t npno . . . t npno    , the goal is to find sets of parameters w ho ij and w ih kl minimizing the mean-squared error: E = np i=1 no j=1 [y j (x i ) − t ij ] 2 ,(3) where x i is the i-th row of the data matrix X. Usually (3) is minimized by some form of gradient descent. Computing The Gradient Using Matrix-Matrix Multiply If we write Y for the matrix of outputs y j (x i ), H for the matrix of hidden activations h j (x i ), and W ih and W ho for the parameter matrices w ih kl and w ho ij respectively, then H = σ X * W T ih Y = σ H * W T ho where " * " denotes ordinary matrix multiplication and σ(A) means apply σ elementwize to the components of A. Defining Y ∆ = (I − Y * * Y ) * * (T − Y ) , H ∆ = (I − H * * H) * * (Y ∆ * W ho ) , where " * * " denotes elementwize matrix multiplication, we have ∇ ih E = H T ∆ * X, ∇ oh E = Y T ∆ * H, where ∇ ih E is the gradient of E with respect to the parameters W ih and ∇ ho E is the gradient of E with respect to W ho [3]. Thus, computing the gradient of the error for an artificial neural network can be reduced to a series of ordinary matrix multiplications and elementwize matrix multiplications. For large networks and large numbers of training patterns, the bottleneck is the ordinary matrix multiplications, which we implement using Emmerald's SGEMM routine. In all our experiments we found 32 bits of floating-point precision were enough for training. For neural networks with ≈ 10,000 parameters, as few as 16 bits are sufficient [2]. Armed with the gradient ∇E, we can adjust the parameters W by a small amount in the negative gradient direction W := W − α∇E and hence reduce the error. However, because the gradient computation can be very timeconsuming (a total of 52.2 Tera-floating point operations in our largest experiment), it is more efficient to employ some form of line search to locate a local maximum in the direction ∇E. For the experiments reported in the next section we used the Polak-Ribiére conjugate-gradient descent method [4, §5.5.2] to choose the search direction, combined with an exponential step-size scheme and quadratic interpolation in order to locate a maximum in the search direction. We were also able to speed the search for a maximum by using gradient information to bracket the maximum, since only the sign of the inner product of the gradient with the search direction is required to locate the maximum in that direction, and the sign can be reliably esti-mated with far fewer training patterns than is required to estimate the error. Training Set Parallelism Since the error E and gradient ∇E are additive over the training examples, the simplest way to parallelize the training of a neural network is to partition the training data into disjoint subsets and have each processor compute the error and gradient for its subset. This works particularly well if there are a large number of training patterns so that each processor can work with near-optimal matrix sizes. The communication required is the transmission of the neural network parameters to each slave processor, and the transmission of the error and gradient information back from each slave to a master node which reduces them to a single error or gradient vector. Communication This section discusses the communication costs associated with distributed NN training, arguing that these costs are non-trivial for ULSNNs. A reduce algorithm optimised for Bunyip's topology is also discussed. Communication Costs The inter-process communication costs during network training arise from broadcasting the network parameters to all processes and reducing the network error and gradients from each process to the master process. The parameter broadcasting is cheap, since many copies of the same data is sent to all processes. Broadcasts can take advantage of features such as TCP/IP broadcasting. The reduce process is more difficult with each process generating unique vectors which must be collected and summed by the master process. The time taken to reduce data grows with both the number of parameters and the number of processes. The remaining communication consists of start and stop messages which are insignificant compared to the aforementioned costs. A typical neural network with 100 inputs, 50 hidden layer neurons, and 50 output neurons, requires 7500 parameters, or 30 KBytes of data (single precision), to be sent from every node to the master node. A naive reduction over 194 processes using a 1Gb/s link, such as used in Bunyip, would take 0.05 seconds assuming 100% network utilisation. Our ULSNN with 400 inputs, 480 hidden layer neurons and 3203 output neurons requires 1,729,440 parameters or 6.6 MBytes of data per process which would require 10.1 seconds. There is sufficient memory on each node to occupy both processors for 446 seconds calculating gradients before a reduce operation is required. Consequently the reduce operation would cost at least 2.3% of the avail-able processing time, more if not enough training data is available or the network size is increased. This demonstrates that although communication costs for distributed NN training are minimal for commonly implemented network sizes, ULSNN training must optimise inter-process communication to achieve the best performance. We reduced communication as much as possible by only distributing the neural-network parameters to all the slaves at the very start of training (rather than at each step), and thereafter communicating only the search direction and the amount to step in that direction. One significant reduce operation is required per epoch to send the error gradient vector from each process to the master which then co-ordinates the step size search with the slaves. All communication was done using the LAM implementation of MPI (http://www.mpi.nd.edu/lam). Communicating parameters or directions to all processors required a 6.6 MBytes broadcast operation from the server to each of the 194 processors in the cluster, while reducing the gradient back to the master required 6.6 MBytes of data to be communicated from each processor back to the server. LAM/MPI contains a library reduce operation which uses a simple O(log n) algorithm that distributes the load of the reduce over many processes instead of naively sending 194 gradient vectors to one node [7]. This results in a reduce operation on Bunyip which takes 8.5 seconds over 8 stages. Optimising Reductions There are two problems with existing free implementations of MPI reduce operations. The first is the lack of shared memory protocols on clusters with multi-processor nodes, instead using slow TCP/IP communications between processors on the same motherboard. Secondly, the reduce operation does not take advantage of the topology of the cluster. For example, the best reduce algorithm to use on a ring network might be to send a single vector to each node on the ring in turn, which adds its contribution before passing the vector to the next node. On a star network the best algorithm might be to send each contribution to the central server and sum as they arrive. To decrease the time taken per reduce, we wrote a customized routine utilising shared memory for intra-node communication and MPI non-blocking calls for inter-node communication. This routine is summarised by Figure 5. It is split into 4 stages, each of which takes advantage of an aspect of Bunyip's topology shown in Figure 1. 1. Each node contains two processors, both running an instance of the training process. All 97 nodes (including the server), reduce 6.6 MBytes of data though shared memory between processes, taking 0.18 seconds. The time taken to add the two sets of data together is approximately 0.005 seconds. Each node in group A can open a 100 Mb/s connec- tion to any node in group B via switch 0. Thus all 24 nodes in A can reduce to their B counterparts in parallel. This requires 0.66 seconds. The same trick is used for reducing from group C to D. The reduced data now resides only on the B and D nodes. The total bandwidth for all 96 nodes in this stage is 4.03 Gb/s. 3. Each node contains 3x100 Mb/s NICs. This allows a node to receive data from three other nodes simultaneously provided the TCP/IP routing tables are correctly configured. We split the 24 nodes in each group into 6 sets of 4 nodes. The first of each set (see node BA in Figure 5) is designated as the root and the other three nodes send to it via different NICs. This takes 0.9 seconds achieving a bandwidth of 185 Mb/s into each root node, or 2.22 Gb/s across all 12 root nodes. Overall speedup from optimised reduce Figure 6. The overall training performance speedup exhibited after replacing the MPI library reduce with our optimised reduce against the total number of training patters used. by using the optimised reduce instead of the MPI library reduce, against the total number of training patterns used. In practice our peak performance of 163.3 GFlops/s benefits by roughly 1% from the optimised reduce, however the speedups are much more marked for smaller (and more frequently encountered) data sets. Japanese Optical Character Recognition In this section we describe our distributed application of the matrix-matrix multiply technique of Section 3 used to train an artificial neural network as a classifier for machineprinted Japanese characters. The Problem, Data and Network Architecture Japanese optical character recognition (Japanese OCR) is the process of automatically recognizing machine-printed Japanese documents and converting them to an electronic form. The most difficult aspect of Japanese OCR is correctly classifying individual characters, since there are approximately 4000 characters in common usage. The training data for our neural network consisted of 168,000 scanned, segmented, hand-truthed images of Japanese characters purchased from the CEDAR group at the University of Buffalo. The characters were scanned from a variety of sources, including books, faxes, newspapers and magazines. Figure 7 gives an idea of the varying quality of the character images. Each character in the CEDAR database is represented as a binary image of varying resolution. We down-sampled all the images to a 20 × 20 grey-scale format. The neural network had 400 input nodes, one for each pixel. The database contained examples of 3203 distinct characters, hence the neural-network had 3203 output nodes. The hidden layer Training With reference to equations (1), (2), and (3), the total number of floating point operations required to compute the error E in a neural network is 2 × n p × (n i + n o ) × n h , which equals 32 Tera floating-point operations for the Japanese OCR experiment. A gradient calculation uses n p × (4 × n i × n h + 6 × n h × n o ), or 92 Tera floating-point operations. To assist with load balancing, each slave processor stepped through its training patterns 320 at a time. Between each step the master node was polled to determine whether more steps were required. Once 80% of the total training data had been consumed, the master instructed all slaves to halt computation and return their results (either the error or the gradient). In this way the idle time spent waiting for other slaves to finish was reduced to at most the length of time needed by a single processor to process 320 patterns. With 80% of the data, an error calculation required 26 TFlops and a gradient calculation requires 74 TFlops, or 135 GFlops and 383 GFlops per processor respectively. Results This section describes the classification accuracy achieved; then concentrates on the performance scalability over processors before finishing with peak performance results which result in our claim of a price/performance ratio of 92.4¢ /MFlop/s. Classification Accuracy The network's best classification error on the held-out 6,320 examples is 33%, indicating substantial progress on a difficult problem (an untrained classifier has an error of 1 − 1/3200 = 99.97%). We observed an error rate of 5% on the 40% of the data which contained the most examples of individual characters. Continued training after the 33% error rate was achieved improved the performance on the common characters at the cost of greatly decreased performance on the rare ones. This leads to the conclusion that overfitting is occurring on characters with only one or two examples from the original data set, despite the number of transformations being generated. A more uniform accuracy could be achieved by generating more transforms of rare characters, or preferably, using a greater number of original examples. A very large amount of data is required for two reasons. The first is to avoid overfitting. minimizing the frequency of reduce operations. The maximal patterns test uses 32,000 patterns per processor. All performance values quoted in this paper represent the total flops that contribute to feed forward value and gradient calculations divided by the wall clock time. Implementation specific flops, such as the reduce operations, were not included. Bunyip was under a small load during the performance testing for Figure 8. For a small number of processors, both networks exhibit linear performance scale up, but we observe that for many processors the larger problem scales better despite the increased number of network parameters. This is due to the communication overhead in the small network increasing dramatically as each processor has less data to process before needing to initiate a reduce. The effect would be clearer for a large network (causing long gradient vectors to be reduced) with few training patterns, however this scenario is not usually encountered due to overfitting. Finally we observe that with a large enough data set to fill the memory of every node, we achieve near linear scaling. Price/Performance Ratio Bunyip was dedicated to running the JOCR problem for four hours with 9,360,000 patterns distributed across 196 processors. Bunyip actually consists of 194 processors, however, we co-opted one of the hot-spare nodes (included in the quoted price) to make up the other two processors. Over this four hour period a total of 2.35 PFlops were performed with an average performance of 163.3 GFlops/s. This performance is sustainable indefinitely provided no other processes use the machine. To calculate the price/performance ratio we use the total cost derived in Section 2.1 of USD$150,913, which yields a ratio of 92.4¢ /MFlop/s 2 . Conclusion We have shown how a COTS (Commodity-Off-The-Shelf) Linux Pentium III cluster costing under $151,000 can be used to achieve sustained, Ultra-Large-Scale Neural-Network training at a performance in excess of 160 GFlops/s (single precision), for a price/performance ratio of 92.4¢/MFlop/s. Part of the reason for the strong performance is the use of very large training sets. With the current networking setup, performance degrades significantly with less data per processor, as communication of gradient information starts to dominate over the computation of the gradient. Figure 1 . 1Bunyip architecture Figure 2 . 2Allocation of SSE registers (labelled as xmm[0-7]), showing progression of the dot products which form the innermost loop of the algorithm. Each black circle represents an element in the matrix. Each dashed square represents one floating point value in a SSE register. Thus four dotted squares together form one 128-bit SSE register. Figure 4 shows 4Emmerald's performance compared to AT-LAS and a naive three-loop matrix multiply. The average MFlops/s rate of Emmerald after size 100 is 1.69 times the clock rate of the processor and 2.09 times faster than ATLAS. A peak rate of 890 MFlops/s is achieved when m = n = k = stride = 320. This represents 1.98 times the clock rate. On a PIII 550 MHz (the processors in Bunyip) we achieve a peak of 1090 MFlops/s. The largest tested size was m = n = k = stride = 3696 which ran at 940 MFlops/s at 550 MHz. For more detail see[1]. Figure 4 . 4Performance of Emmerald on a PIII running at 450MHz compared to ATLAS sgemm and a naive 3-loop matrix multiply. Note that ATLAS does not make use of the PIII SSE instructions. Figure 5 . 5The four stages of our customized reduce: Stage 1: SHM intra-node reduce; stage 2: all nodes in group A and C reduce to their counterparts; stage 3: groups B and D reduce to 12 nodes using 3 NICs; stage 4: MPI library reduce to the server node. Figure 7 . 7Example Japanese characters used to train the neural network. was chosen to have 480 nodes. In total, the network had 1.73 million adjustable parameters. 168,000 training examples are not sufficient to avoid overfitting in a network containing 1.73 million adjustable parameters, so we generated synthetic data from the original characters by applying random transformations including line thickening and thinning, shifting, blurring and noise addition. The total number of training examples including the artificial ones was 9,264,000 approximately 5.4 per adjustable network parameter. These were distributed uniformly to 193 of the processors in Bunyip. A further 6320 examples of the CEDAR data set were used for testing purposes. Figure 8 . 8Performance scaling with the number of processors used for training a small network and our large JOCR network with a fixed number of patterns, and the JOCR problem when the total patterns scales with the number of processors. Table 1 . 1Generalisation error decreases as the total number of patterns increases. Table 1 1compares the generalisation accuracy with the total number of training examples used (including transformations of the original 168,000 patterns). Each data point in this graph represents approximately 48 hours training time. Training was halted after 10 epochs result in no classification improvement on the test set. Recalling from Section 5.1 that communication overhead increases with decreasing patterns then the second motivation for large training sets is to reduce such overhead. Figure 8 demonstrates how the performance scales with the number of processors used. The bottom line is the performance versus processors curve for a small network of 400 input nodes, 80 hidden layer nodes, 200 output nodes and a total of 40,960 training patterns. The middle line is our JOCR ULSNN with 163,480 total patterns. The top line is the JOCR network again, however, for this test we allowed the number of patterns to scale with the processors,7.2 Communication Performance Following the convention with integrated circuits, we take ULSNN to mean a neural network with in excess of one million parameters and one million training examples. . The final step is a standard MPI library reduce from 6 B nodes and 6 D nodes to the master process. This is the slowest step in the process taking 3.16 seconds, including the time spent waiting for the the nodes to synchronize since they do not start reducing simultaneously.The overall time taken for the optimised reduce to complete is 4.9 seconds. The actual time saved per reduction is 3.6 seconds. The training performance speedup from this saving varies with the duration of the gradient calculation which depends linearly on the number of training patterns.Figure 6illustrates the expected speedup achieved AcknowledgementsThis project was supported by the Australian Research Council, an Australian National University Major Equipment Grant, and LinuxCare Australia. Thanks are also due to several people who made valuable contributions to the establishment and installation of Bunyip: Peter Christen, Chris Johnson, John Lloyd, Paul McKerras, Peter Strazdins and Andrew Tridgell. Ememrald: A fast matrixmatrix multiply using Intel SIMD technology. D Aberdeen, J Baxter, Research School of Information Science and Engineering, Australian National UniversityTechnical reportD. Aberdeen and J. Baxter. Ememrald: A fast matrix- matrix multiply using Intel SIMD technology. Technical report, Research School of Information Science and En- gineering, Australian National University, August 1999. http://csl.anu.edu.au/∼daa/files/emmerald.ps. Experimental determination of precision requirements for back-propagation training of artificial neural networks. K Asanović, N Morgan, Technical reportThe International Computer Science InstituteK. Asanović and N. Morgan. Experimental determi- nation of precision requirements for back-propagation training of artificial neural networks. Technical report, The International Computer Science Institute, 1991. ftp://ftp.ICSI.Berkeley.EDU/pub/techreports/1991/tr-91- 036.ps.gz. Using PHiPAC to speed Error Back-Propogation learning. J Bilmes, K Asanovic, C.-W Chin, J Demmel, ICASSP. J. Bilmes, K. Asanovic, C.-W. Chin, and J. Demmel. Us- ing PHiPAC to speed Error Back-Propogation learning. In ICASSP, April 1997. Feedforward Neural Network Methodology. T L Fine, SpringerNew YorkT. L. Fine. Feedforward Neural Network Methodology. Springer, New York, 1999. High performance software on Intel Pentium Pro processors or Micro-Ops to Ter-aFLOPS. B Greer, G Henry, Intel. Technical reportB. Greer and G. Henry. High performance software on Intel Pentium Pro processors or Micro-Ops to Ter- aFLOPS. Technical report, Intel, August 1997. http:// www.cs.utk.edu/∼ghenry/sc97/paper.htm. PHiPAC: A portable, high-performace, ANSI C coding methodoloogy and its application to matrix multiply. J Bilmes, K Asanovic, J Demmel, D Lam, C W Chin, J.Bilmes, K.Asanovic, J.Demmel, D.Lam, and C.W.Chin. PHiPAC: A portable, high-performace, ANSI C cod- ing methodoloogy and its application to matrix multiply. Using the exchange rate at the time of writing would yield. Using the exchange rate at the time of writing would yield MFlops/s, however we felt quoting the rate at the time of purchase was a more accurate representation of the cost. 5¢ Usd /, University of TennesseeTechnical report5¢ USD /MFlops/s, however we felt quoting the rate at the time of purchase was a more accurate representation of the cost. Technical report, University of Tennessee, August 1996. http://www.icsi.berkeley.edu/∼bilmes/phipac. . Lam Team, LAM Team. . Lam, mpi source code v6.3.2.Lam/mpi source code v6.3.2. http://www.mpi.nd.edu/lam/download/. Basic Linear Algebra Subroutines. Netlib, Netlib. Basic Linear Algebra Subroutines, November 1998. http://www.netlib.org/blas/index.html. Gaussian elimination is not optimal. V Strassen, Numerische Mathematik. 13V. Strassen. Gaussian elimination is not optimal. Nu- merische Mathematik, 13:354-356, 1969. Tuning strassen's matrix multiplication for memory efficiency. M Thottethodi, S Chatterjee, A R Lebeck, Super Computing. M. Thottethodi, S. Chatterjee, and A. R. Lebeck. Tuning strassen's matrix multiplication for memory efficiency. In Super Computing, 1996. Automatically tuned linear algebra software. R C Whaley, J J Dongarra, Computer Science Department, University of TennesseeTechnical reportR. C. Whaley and J. J. Dongarra. Automatically tuned linear algebra software. Technical report, Com- puter Science Department, University of Tennessee, 1997. http://www.netlib.org/utk/projects/atlas/. Automated empirical optimizations of software and the atlas project. R C Whaley, A Petitet, J J Dongarra, KnoxvilleDept. of Computer Sciences, Univ. of TNTechnical reportR. C. Whaley, A. Petitet, and J. J. Dongarra. Au- tomated empirical optimizations of software and the atlas project. Technical report, Dept. of Com- puter Sciences, Univ. of TN, Knoxville, March 2000. http://www.cs.utk.edu/∼rwhaley/ATLAS/atlas.html.	[]
[ "Numerical-relativity simulation for tidal disruption of white dwarfs by a supermassive black hole", "Numerical-relativity simulation for tidal disruption of white dwarfs by a supermassive black hole" ]	[ "Alan Tsz-Lok ", "Lam \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany\n", "Masaru Shibata \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany\n\nCenter for Gravitational and Quantum-Information Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n", "Kenta Kiuchi \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany\n\nCenter for Gravitational and Quantum-Information Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n" ]	[ "Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany", "Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany", "Center for Gravitational and Quantum-Information Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan", "Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany", "Center for Gravitational and Quantum-Information Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan" ]	[]	We study tidal disruption of white dwarfs in elliptic orbits with the eccentricity of ∼ 1/3-2/3 by a nonspinning supermassive black hole of mass M BH = 10 5 M in fully general relativistic simulations targeting the extreme mass-ratio inspiral leading eventually to tidal disruption. Numerical-relativity simulations are performed by employing a suitable formulation in which the weak self-gravity of white dwarfs is accurately solved. We reconfirm that tidal disruption occurs for white dwarfs of the typical mass of ∼ 0.6M and radius ≈ 1.2 × 10 4 km near the marginally bound orbit around a nonspinning black hole with M BH 4 × 10 5 M . PACS numbers: 04.25.D-, 04.30.-w, 04.40.Dg	10.1103/physrevd.107.043033	[ "https://export.arxiv.org/pdf/2212.10891v2.pdf" ]	254,926,369	2212.10891	d0d0aa82734f31dbef2be49391fc187b150a96f9	Numerical-relativity simulation for tidal disruption of white dwarfs by a supermassive black hole Alan Tsz-Lok Lam Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am Mühlenberg 114476Potsdam-GolmGermany Masaru Shibata Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am Mühlenberg 114476Potsdam-GolmGermany Center for Gravitational and Quantum-Information Physics Yukawa Institute for Theoretical Physics Kyoto University 606-8502KyotoJapan Kenta Kiuchi Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am Mühlenberg 114476Potsdam-GolmGermany Center for Gravitational and Quantum-Information Physics Yukawa Institute for Theoretical Physics Kyoto University 606-8502KyotoJapan Numerical-relativity simulation for tidal disruption of white dwarfs by a supermassive black hole (Dated: April 4, 2023) We study tidal disruption of white dwarfs in elliptic orbits with the eccentricity of ∼ 1/3-2/3 by a nonspinning supermassive black hole of mass M BH = 10 5 M in fully general relativistic simulations targeting the extreme mass-ratio inspiral leading eventually to tidal disruption. Numerical-relativity simulations are performed by employing a suitable formulation in which the weak self-gravity of white dwarfs is accurately solved. We reconfirm that tidal disruption occurs for white dwarfs of the typical mass of ∼ 0.6M and radius ≈ 1.2 × 10 4 km near the marginally bound orbit around a nonspinning black hole with M BH 4 × 10 5 M . PACS numbers: 04.25.D-, 04.30.-w, 04.40.Dg I. INTRODUCTION Tidal disruption of ordinary stars and/or white dwarfs by supermassive black holes has been revealed to be one of the major sources of bright electromagnetic transients (see, e.g., Refs. [1-3]), which have been actively observed in the last decade. In addition, gravitational waves emitted by tidal disruption of white dwarfs closely orbiting supermassive black holes could be observable by Laser Interferometer Space Antenna (LISA) [4]. Electromagnetic signals associated with tidal excitation (e.g., Ref. [5]) or mass stripping (e.g., Refs. [6][7][8][9] for related works) or tidal disruption (e.g., Refs. [10,11]) of white dwarfs can be an important electromagnetic counterpart of gravitational waves. Because the expected event rate is not so high [12] that the signal-to-noise ratio of gravitational waves for the LISA sensitivity is unlikely to be very high, the discovery of the possible electromagnetic counterparts will help extract gravitational waves from the noisy data in the LISA mission. The condition for mass shedding and tidal disruption during the cross encounter of stars with supermassive black holes is often described by the so-called β -parameter defined by β := r t r p ,(1) where r p is the periastron radius for the orbit and r t is the Hill's radius [13] defined by r t := R * M BH M * 1/3 ,(2) with R * the stellar radius, M * the stellar mass, and M BH the mass of the supermassive black hole, respectively. Since the early 1990s (see, e.g., Refs. [14][15][16]), a large number of numerical simulations have been performed in the last three decades (see, e.g., Refs. [17,18] for reviews of the latest works and Refs. [19][20][21][22][23][24] for some of the most advanced works). They have shown that mass stripping can take place at the close encounter if β is larger than about 0.5, and tidal disruption can take place if β 1 for stars in parabolic orbits (see, e.g., Refs. [17,25,26] for Newtonian simulation works, and also early semianalytical work [27,28]). It is also shown that for close orbits around a black hole, the general relativistic effect can significantly reduce the critical value of β for the tidal disruption [23]. Indeed, general relativistic works show that for circular orbits near the innermost stable circular orbit of black holes, the mass shedding can occur even for β ∼ 0.4 [29,30]. However, the previous analyses have been carried out in Newtonian gravity or in relativistic gravity of a black hole with Newtonian (or no) gravity for the companion star or in a tidal approximation with a relativistic tidal potential [29][30][31]. To date, no fully general relativistic (the so-called numericalrelativity) simulation, i.e., a simulation with no approximation except for the finite differencing, has been done for the tidal disruption problem with β 1 (but see Refs. [32][33][34] for a head-on and an off-axis collision). Numerical-relativity simulation is suitable for the tidal disruption problem for the case that the orbit at the tidal disruption is highly general-relativistic. This is particularly the case for tidal disruption of white dwarfs by supermassive black holes because it can occur only for orbits very close to the black-hole horizon. Advantages of the numerical-relativity simulation are: (i) the redistribution of the energy and angular momentum of the star can be followed in a straightforward manner and (ii) we can directly follow the matter motion after the tidal disruption including the subsequent disk formation. In this paper, we present a result of numerical-relativity simulations for tidal disruption of white dwarfs of typical mass (0.6-0.8M ) by a supermassive black hole with relatively low mass (M BH = 10 5 M ) for the first time. For simplicity, the white dwarfs are modeled by the Γ = 5/3 polytropic equation of state. As a first step toward more detailed and systematic studies, we focus on tidal disruption of white dwarfs in mildly elliptic orbits aiming at confirming that our numerical-relativity approach is suitable for reproducing the criteria of tidal disruption, which has been already investigated in many previous works referred to above. The paper is organized as follows. In Sec. II, we describe our formulation for evolving gravitational fields, matter fields, and for providing initial data of a star in elliptic orbits around supermassive black holes. In Sec. III, numerical results are presented paying particular attention to the criterion for tidal disruption. Section IV is devoted to a summary. Throughout this paper, we use the geometrical units of c = 1 = G where c and G are the speed of light and gravitational constant, respectively. The Latin and Greek indices denote the space and spacetime components, respectively. II. BASIC EQUATIONS FOR THE TIME EVOLUTION A. Gravitational field First, we reformulate the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism [35,36] in numerical relativity to a form suitable for the simulation of high-mass ratio binaries, in particular for accurately computing a weak self-gravity of white dwarfs. Throughout this paper, high-mass ratio binaries imply those composed of a very massive black hole of mass M BH 10 5 M and a white dwarf (or an ordinary star) of mass of M * = O(M ) with the radius R * 10 3 km, for which the compactness defined by M * /R * is smaller than 10 −3 . We consider the two-body problem with a compact orbit of the orbital separation r 30M BH . With such setting, the magnitude of the gravitational field generated by the black hole, which is defined by g µν − η µν is, of order M BH /r > 10 −2 . Here g µν and η µν are the spacetime metric and Minkowski metric, respectively. On the other hand, the magnitude of the gravitational field generated by white dwarfs and ordinary stars is of order M * /R * < 10 −3 , which is much smaller than that by the black hole. To accurately preserve the nearly equilibrium state of such stars during their orbits, an accurate computation of the gravitational field by them is required. However, if we simply solve Einstein's equation, a numerical error for the computation of the black-hole gravitational field can significantly affect the gravitational field for the white dwarfs/ordinary stars. To avoid this numerical problem, we separate out the gravitational field into the black hole part and other part, although we still solve fully nonlinear equations. The idea employed here is similar to that of Ref. [33], but we develop a formalism based on the BSSN formalism. In a version of the BSSN formalism [36], the basic equations are written in the form: (∂ t − β k ∂ k )γ i j = −2αÃ i j +γ ik ∂ j β k +γ jk ∂ i β k − 2 3γ i j ∂ k β k ,(3)(∂ t − β l ∂ l )Ã i j = W 2 α R i j − γ i j 3 R k k − D i D j α − γ i j 3 D k D k α +α KÃ i j − 2Ã ikÃ k j +Ã k j ∂ i β k +Ã ki ∂ j β k − 2 3Ã i j ∂ k β k −8π G c 4 αW 2 S i j − 1 3 γ i j S k k ,(4)(∂ t − β l ∂ l )W = W 3 αK − ∂ k β k ,(5)(∂ t − β l ∂ l )K = α Ã i jÃ i j + 1 3 K 2 −W 2 D kD k α − ∂ i W Wγ i j ∂ j α +4π G c 4 α ρ h + S k k ,(6)(∂ t − β l ∂ l )Γ i = −2Ã i j ∂ j α + 2α Γ i jkÃ jk − 2 3γ i j ∂ j K − 8π G c 4γ ik J k − 3 ∂ j W WÃ i j −Γ j ∂ j β i + 2 3Γ i ∂ j β j + 1 3γ ik ∂ k ∂ j β j +γ jk ∂ j ∂ k β i ,(7) where α is the lapse function, β j is the shift vector,γ i j is the conformal three metric defined from the three metric γ i j bỹ γ i j := γ −1/3 γ i j with γ = det(γ i j ), W := γ −1/6 ,Ã i j is the conformal trace-free extrinsic curvature defined from the extrinsic curvature K i j byÃ i j = W 2 (K i j − γ i j K/3) with K := K k k , andΓ i := −∂ jγ i j . ρ h , J i , and S i j are quantities defined from the energy-momentum tensor, T µν , by ρ h = T µν n µ n ν , J i = −T µν n µ γ νi , and S i j = T µν γ µi γ ν j with n µ the timelike unit vector normal to spatial hypersurfaces. In this problem, we employ the so-called puncture gauge [37], in which the evolution equations for α and β i are written as ∂ t α = −2αK,(8)∂ t β i = 3 4 B i ,(9)∂ t B i = ∂ tΓ i − η B B i ,(10) where B i is an auxiliary three-component variable and η B is a constant of order M −1 BH . By introducing a static black-hole solution for the geometric variables, α 0 , β i 0 ,γ 0 i j , W 0 ,Ã 0 i j , and K 0 and by writing all the variables by α = α 0 + α s ,(11)β i = β i 0 + β i s ,(12)γ i j =γ 0 i j +γ s i j ,(13)W = W 0 +W s ,(14)A i j =Ã 0 i j +Ã s i j ,(15)K = K 0 + K s ,(16)Γ i =Γ i 0 +Γ i s ,(17) we then write down the equations for α s , β i s ,γ s i j , W s ,Ã s i j , K s , andΓ i s (these are denoted by a representative variable Q s as follows). Specifically, the evolution equations (4)-(7) and (8)-(10) of the geometrical variables (denoted by a representative variable Q) are schematically written in the form ∂ t Q = F(Q).(18) Then, for the decomposition of Q = Q 0 + Q s with F(Q 0 ) = 0 (under the conditions of ∂ t Q 0 = 0), we write the equation for Q s as ∂ t Q s = F(Q 0 + Q s ) − F(Q 0 ).(19) In numerical simulation, F(Q 0 ) obtained from finite difference is nonzero which contains the truncation error of evolving the stationary background metric numerically. Here, we added the second-term in the right-hand side to explicitly subtract the leading error of evolving the background metric so that the right-hand side of the evolution equation of Q s does not have the zeroth order terms in Q s Any static black-hole solutions can be used for α 0 , β i 0 , · · · , but in the BSSN formalism with the puncture gauge, the metric relaxes to a solution in the limit hypersurface with K 0 = 0. Using such a trumpet-puncture black hole also allows us to construct the initial data in the conformal-thin-sandwich (CTS) formalism [38] (see Sec. II C). Thus, in the present formalism, it is appropriate to employ such a solution. In the nonspinning black hole, the analytic solution is known and is written as [39] α 0 = 1 − 2M BH R + 27M 4 BH 16R 4 ,(20)β i 0 = 3 √ 3M 2 BH 4R 3 x i ,(21)W 0 = r R ,(22)γ 0 i j = δ i j , i.e.,Γ i 0 = 0,(23)A 0 i j = 3 √ 3M 2 BH 4R 3 δ i j − 3 x i x j r 2 ,(24) and K 0 = 0 where R is a function of r determined by [40] r =   2R + M BH + 4R 2 + 4M BH R + 3M 2 BH 4   ×   (4 + 3 √ 2)(2R − 3M BH ) 8R + 6M BH + 3 8R 2 + 8M BH R + 6M 2 BH   1/ √ 2 .(25) We note that r = 0 corresponds to R = 3M BH /2 and the event horizon is located at R = 2M BH (i.e., r ≈ 0.78M BH ) in this solution. B. Hydrodynamics In this paper we model white dwarfs simply by the polytropic equation of state, P = κρ Γ ,(26) where P and ρ are the pressure and rest-mass density, respectively, κ the polytropic constant, and Γ adiabatic index for which we set to be 5/3. For the hydrodynamics, we solve the continuity and Euler equations, ∇ µ (ρu µ ) = 0,(27)∇ µ T µ k = 0,(28) with ∇ µ the covariant derivative with respect to g µν and T µν = (ρ + ρε + P)u µ u ν + Pg µν ,(29) where ε and u µ are the specific internal energy and four velocity, respectively. In this work we do not solve the energy equation, and determine ε simply by ε = κρ Γ−1 /(Γ − 1) which is derived from the condition that the specific entropy is conserved for the fluid elements. The continuity and Euler equations are solved in the same scheme as that used in Refs. [41,42]. The motivation for using the polytropic equation of state comes from the fact that our primary purpose of this paper is to explore the tidal disruption condition for a relatively low value of β < 1 and the formation of shocks by the tidal compression does not play any role. We here focus only on the process of tidal disruption and subsequent short-term evolution of the tidally disrupted material. After the tidal disruption, the fluid is highly elongated and during the long-term evolution of the fluid elements with different specific energy and angular momentum, they collide and shocks are likely to be formed. For such a phase, the shock heating will play an important role. Our plan is to follow this phase by solving the energy equation with a more general equation of state. C. Initial condition First, we describe the formulation employed in this paper for computing the initial data in which white dwarfs are approximately in an equilibrium state in their comoving frame. From Eq. (28), we have ρu µ ∇ µ (hu i ) + ∇ i P = 0,(30) where h is the specific enthalpy defined by h := 1 + ε + P/ρ. To derive Eq. (30), we used Eq. (27). For the isentropic fluid, the first law of thermodynamics is written as ρdh = dP,(31) where dQ denotes the variation of a quantity Q in the fluid rest frame. In the polytropic equations of state employed in this work, we obtain the relation h = dP ρ and ln h = dP ρh .(32) In this situation, Eq. (30) is rewritten to u µ ∇ µ (hu i ) + ∂ i h = 0.(33) Then, we define k µ := u µ /u t . Using this quantity, Eq. (33) is written to u t L k (hu i ) − u t hu µ ∇ i k µ + ∂ i h = 0,(34) where L k denotes the Lie derivative with respect to k µ . The second term of Eq. (34) is written as u t hu µ ∇ i k µ = u t hu µ ∇ i (u µ /u t ) = h∂ i ln u t ,(35) where we used u µ u µ = −1. Thus, Eq. (34) is written to L k (hu i ) + ∂ i (h/u t ) = 0.(36) We consider an initial condition for a system composed of a star of mass M * and radius R * , for which the center is located on the x-axis, around a massive black hole of mass M BH M * and M BH R * which is located at a coordinate origin. We assume that the star predominantly moves toward the y-direction with the identical specific momentum. Thus we set v i := u i /u t = −β i 0 +V i where V i = V δ i y with V being a constant to be determined. Here the term of β i 0 is added to simplify the iteration process for computing quasiequilibrium states. Then, u t is calculated from u t = α 2 − γ i j (v i + β i )(v j + β j ) −1/2 .(37) In the present context, L k (hu i ) can be assumed to be zero for i = y and z, because the star has momentarily translation invariance for the motion toward the yand z-directions. By contrast, with respect to the x-direction, the star receives the force from the massive black hole. Since the radius of the star, R * , is much smaller than the orbital separation, x 0 , and x 0 is larger than the black-hole radius of ∼ M BH , L k (hu i ) for the x-direction can be approximated by ∂ i [A(x − x 0 )] where we take A to be a constant, which should be approximately written as ∼ −M BH /x 2 0 for x 0 > 0. Then, Eq. (33) is integrated to give A(x − x 0 ) + h u t = C,(38) where C is an integration constant. We note that Eq. (38) is not an exact first integral of the Euler equation but can be considered as an approximate one for obtaining an initial condition in which the star is in an approximate equilibrium state. For computing initial conditions, we assume the line elements of the form ds 2 = −(α 2 − β k β k )dt 2 + 2β k dtdx k + ψ 4 δ i j dx i dx j ,(39) where ψ is the conformal factor. Using the Isenberg-Wilson-Mathews formalism [43,44], the basic equations are written as ∆ψ = −2πρ H ψ 5 − ψ −7 8Â i jÂ i j ,(40)∆(αψ) = 2παψ 5 (ρ H + 2S) + 7 8 αψ −7Â i jÂ i j ,(41)∂ iÂ i j = 8πJ j ψ 6 ,(42) where ρ H = ρh(αu t ) 2 − P,(43)S = ρh[(αu t ) 2 − 1] + 3P,(44)J i = ρhαu t u i ,(45) and ∆ is the flat Laplacian.Â i j is defined from the extrinsic curvature,Ã i j , byÂ i j = ψ 6Ã i j and K is set to be zero. Using the CTS decomposition [45,46] with trumpet-puncturê A i j =Â i j 0 + ψ 6 2α (Lβ s ) i j ,(46) Equation (42) is rewritten as ∂ j ∂ j β i s + 1 3 δ i j ∂ j ∂ k β k s = 16παψ 4 J i + (Lβ s ) i j ∂ j ln αψ −6 , (47) where (Lβ s ) i j = δ ik ∂ k β j s + δ jk ∂ k β i s − 2 3 δ i j ∂ k β k s . Note that although there are some works in constructing binary black holes initial data with trumpet-puncture [47][48][49], this is, to our knowledge, the first attempt combining the CTS decomposition and puncture method with the limit (trumpet) hypersurface in constructing quasi-equilibrium initial data in nonvacuum spacetime. We assume that the contribution to the extrinsic curvature from the black hole is negligible because the orbital momentum of the black hole is negligible in this problem, and thus, we set the black hole at rest (however, it is straightforward to take into account the small black-hole motion [50] in our formalism.) For a solution of the initial data, we have to determine the free parameters, A, C, and V . In the polytropic equation of state, we can consider κ as well as the central density ρ c as free parameters. In the following, we first consider that V and rest mass of the star are input parameters and A, C, and κ are parameters to be determined during the iteration process in numerical computation. Our method to adjust κ to a desired value will be described later. To determine these three parameters we need three conditions, for which we choose the following relations. First, we fix the location of the surface of white dwarfs along the x-axis as x = x 1 (referred to as point 1) and x = x 2 (point 2). Typically, we choose x 1 + x 2 = 2x 0 . At the surface, h = 1, and thus, Eq. (38) gives A(x 1 − x 0 ) + 1 u t 1 = A(x 2 − x 0 ) + 1 u t 2 = C,(48) where u t 1 and u t 2 are the values of u t at points 1 and 2. In addition, we fix the rest mass of the star which is defined by m * = d 3 xραψ 6 u t ,(49) where m * is approximately equal to the gravitational mass M * because the star is only weakly self-gravitating. Using the condition (48), the values of C and A are determined, and subsequently, h is determined from Eq. (38). In the polytropic equation of state, the rest-mass density is written as ρ = (h − 1)(Γ − 1) κΓ 1/(Γ−1) ,(50) and thus, from Eq. (49), κ is determined for given values of m * and x 2 − x 1 . Once these free parameters are determined, the rest-mass density are obtained from Eq. (50). For realistic setting, we have to obtain the desired values of the mass of the star and the value of κ. The value of κ is controlled by varying the stellar diameter x 2 − x 1 for a given value of m * . To take into account the effect of the black-hole gravity, we employ the puncture formulation by setting ψ = ψ 0 + φ ,(51)αψ = α 0 ψ 0 + X,(52)β k = β k 0 + β k s ,(53)A i j =Â 0 i j +Â s i j ,(54) where ψ 0 , α 0 , β k 0 , andÂ 0 i j denote the solutions of vacuum Einstein's equation shown already in Sec. II A. Then we numerically solve the equations for φ , X, β k s , andÂ s i j from Eqs. (40), (41), (47), and (46). The initial data is prepared using the octree-mg code [51], an open source multigrid library with an octree adaptive-mesh refinement (AMR) grid, which we modified to support a fourth-order finite-difference elliptic solver. III. NUMERICAL SIMULATION A. Setup The simulation is performed using an AMR algorithm with the equatorial symmetry imposed on the z = 0 (equatorial) plane using the SACRA-TD code (for SACRA see Refs. [41,42]). We prepare two sets of finer domains, one of which comoves with a white dwarf and the other of which is located around the center and covers the massive black hole. Because the radius of the white dwarf, R * , is smaller than the black-hole horizon radius ∼ M BH , we need to prepare more domains for resolving the white dwarf. In addition to these domains, we prepare coarser domains that contain both the finer domains in their inside. All the domains are covered by (2N + 1, 2N + 1, N + 1) grid points for (x, y, z) with N being an even number. Specifically, each domain is labeled by i which runs as 0, 1, 2, · · · , i fix , · · · , i BH , · · · , i max . The grid resolution for the domains with i fix ≤ i ≤ i BH is identical with that with i BH +1 ≤ i ≤ 2i BH − i fix + 1(< i max ), respectively. For 0 ≤ i ≤ i BH , the center of the domain is located at the origin, at which a black hole is present. Strictly speaking, the black hole moves due to the backreaction against the motion of the companion star, but this motion is tiny because of the condition M BH M * . For these domains, the ith level covers a half cubic region of [−L i : L i ] × [−L i : L i ] × [0 : L i ] where L i = N∆x i , ∆x i is the grid spacing for the ith level, and the grid spacing for each level is determined by ∆x i+1 = ∆x i /2 (i = 0, 1, 2, · · · , i BH − 1 and i = i BH + 1, · · · , i max − 1) with ∆x i BH +1 = ∆x i fix and L i BH ∼ 0.8M BH . For the moving domains that cover the white dwarf, the center is chosen to approximately agree with the location of the density maximum. In the present context, the local density maximum is approximately located along a geodesic around the supermassive black hole. The size of the finest domain with i = i max , L max , is chosen so that it is 1.3-1.5R * . We check the convergence of two different models with three grid resolutions as illustrated in Fig. 1. Higher resolution is used for model M8V17 to measure the spin up of the white dwarf more accurately (see Sec. III B). We obtain good convergence for TABLE I. Models considered in this paper and the fate (last column). M7V16a and M7V16b correspond to the models with R * = 8.5 × 10 3 and 7.0 × 10 3 km, respectively. For other models, R * ≈ 10 4 (M * /0.7M ) −1/3 km. r p and r p,A are periastron radius in the present coordinates and the Schwarzschild coordinates, respectively. TD and OC denote tidal disruption and appreciable oscillation of white dwarfs, and NN denotes that no appreciable tidal effect is found. both models, and thus, we employ N = 60 as the standard resolution in this paper. ID V M * (M ) r p /M BH r p,A /M BH J/M BH β B. Numerical results In the present paper we focus on the case that the black-hole mass is M BH = 10 5 M , the white-dwarf mass is M * = 0.6, 0. cases where κ is chosen such that R * = 8.5×10 3 km and R * = 7.0 × 10 3 km. The initial separation is set to be x 0 = 20M BH (it is ≈ 21.01M BH in the Schwarzschild coordinates), and V is chosen to be 0.160, 0.165, 0.170, 0.175, and 0.180 (see Table I With these settings, the white dwarf has an elliptic orbit around the black hole with the periastron at r p ≈ (4.4-10)M BH , and thus, the eccentricity is approximately defined by e = (x 0 − r p )/(x 0 + r p ) is ≈ 1/3-2/3. Here, x 0 (= 20M BH ) and r p are defined in the radial coordinates of the metric of g 0 µν , and thus, the values of e slightly change if we define it in the areal coordinate (Schwarzschild radial coordinate). For the models mentioned above, the value of β is in the range between 0.33 and 0.72 and estimated by we find 0.50 ≤ β ≤ 0.7, and thus, the white dwarf is expected to be strongly perturbed by the black-hole tidal field for M * = 0.6-0.8M . By contrast, for V = 0.180, β < 0.35 with M * = 0.7M , and thus, the tidal force of the black hole is likely to be too weak to perturb the white dwarf. β ≈ 0.59 R * 10 4 km M * 0.7M −1/3 × r p,A 6M BH −1 M BH 10 5 M −2/3 ,(55) For V = 0.170, β ≈ 0.49, 0.44, and 0.40 with M * = 0.6, 0.7, and 0.8M respectively. In these cases, tidal disruption is not very likely to take place but the tidal force from the black hole should induce the stellar oscillation on the white dwarf. Because for Γ = 5/3, the stellar radius depends only weakly on the stellar mass, the presence or absence of the tidal disruption is likely to depend primarily on the value of V (or the specific angular momentum of the white dwarfs) in the present setting. In the following, we will show that our code can reproduce all these expected phenomena. Figure 3 plots the evolution of the maximum density for V = 0.160, 0.165, 0.170, and 0.180 with M * = 0.7M . We note that for M BH = 10 5 M , the orbital period for these parameters are in the range from ≈ 220 s for V = 0.160 to ≈ 250 s for V = 0.180. The figure shows the results expected in the previous paragraphs: For M7V16 and M7V165, the white dwarfs are tidally disrupted while approaching the black hole irrespective of the white-dwarf mass. For M7V17 (β = 0.44), the white dwarf is perturbed by the black hole near the periastron but it is not tidally disrupted. After the close encounter, the white dwarf is in an oscillating state due to the instantaneous tidal force received from the black hole. By contrast, for M7V18, the maximum density is approximately preserved to be constant, suggesting no disruption occurs and the tidal effect is negligible. Note that such tidal field may still perturb the white dwarf and produce detectable electromagnetic or gravitational-wave signal if a sufficient amplitude of oscillation is induced. In Fig. 3, the results of M7V16 (β = 0.65), M7V16a (β = 0.55) and M7V16b (β = 0.45) are also compared. As expected, for the first two models, the white dwarfs are tidally disrupted, while for the most compact white dwarf, the tidal disruption does not occur although it is perturbed significantly by the black-hole tidal force. This illustrates that the β parameter is a good indicator for assessing whether tidal disruption takes place or not irrespective of the white-dwarf radius. Figure 4 shows the evolution of the maximum density when stellar oscillation is induced. For M6V17 (β = 0.49), the white dwarf is significantly elongated by the tidal force from the black hole; the central density is decreased to less than 50% of the original value after passing through the periastron. Associated with the tidal effect, the mass is lost from the white dwarf. However, with the increase of the orbital radius, the central density increases again, resulting in a less massive white dwarf. This is also the case for M8V165 (β = 0.47) and M7V16b (β = 0.45). These results indicate that the critical value of β for the tidal disruption is ∼ 0.50 and the threshold value for exciting a high-amplitude oscillation is β ∼ 0.45. Figure 4 also shows that even for 0.40 β 0.45 an appreciable oscillation is excited by the tidal force. In Fig. 5 and Table I, we summarize the fates of white dwarfs as a result of the tidal interaction. It is found that for β 0.5 tidal disruption takes place and for β 0.4, the white dwarfs are perturbed appreciably by the black-hole tidal field. All these results agree approximately with the expectation from the previous studies. For M7V16, tidal disruption takes place but only a small fraction of the white dwarf matter falls into the black hole because the fluid elements have specific angular momentum large enough to escape capturing by the black hole. Most of the tidally disrupted matter approximately maintains the original elliptic orbit (see Fig. 6) although the matter has an elongated profile. To clarify the eventual matter distribution around the black hole, we will need to follow the matter motion for more than 10 orbits. This topic is one of our major research targets in the future. For 0.4 β 0.5, the white dwarf will be continuously perturbed by the black-hole tidal force whenever it passes through the periastron. In addition the angular momentum is transported during the tidal interaction, and it will lead to the transport of the orbital angular momentum to the white dwarf resulting in a spin-up of it. According to a perturbation study for the stellar encounter, the energy deposition during the tidal interaction in one orbit is written approximately as [52] ∆E tid = f tid M 2 * R * M BH M * 2 R * r p,A 6 = f tid M 2 * R * β 6 .(56) where f tid is a factor of O(0.1), which depends on β and the equation of state. Associated with the energy deposition near the periastron, the angular momentum deposition is also deposited. In one orbit it is approximately estimated by ∆J spin ≈ ∆E tid /Ω p [53] where Ω p = M BH /r 3 p,A , and thus, ∆J spin = f spin M * √ M * R * M BH M * 3/2 R * r p,A 9/2 = f spin M * √ M * R * β 9/2 ,(57) where f spin is a coefficient of the same order of the magnitude of f tid . Because the maximum spin angular momentum of the star is approximately written as M * √ M * R * , we find that ∆J spin can be more than 0.1% of the maximum spin if a white dwarf passes through a close orbit with β 0.4. We approximate the orbit angular momentum J orbit and the spin angular momentum J spin of the white dwarf as where M h := d 3 xψ 6 ρh and the volume average of quantity q is defined as q := 1 M h d 3 xψ 6 ρhq. In such decomposition, the sum of orbital and spin angular momentum equals to the total angular momentum of the white dwarfs. We analyzed the spin angular momentum gain of the white dwarfs for M8V17, and we indeed find ∆J spin /(M * √ M * R * β 9/2 ) ≈ 0.1-0.3 as shown in Fig. 7. Note that the spin up of white dwarf ∆J spin is about 10 −6 of the total angular momentum, and hence, it is not easy to determine ∆J spin accurately. Although we cannot achieve a good convergence in ∆J spin , we are able to obtain a noticeable rise in J spin during the close encounter, which suggests f spin ∼ 0.1-0.3, which is consistent with the above analytic result. For close orbits, the tidal angular-momentum transport can dominate over the orbital angular momentum loss by gravitational-radiation reaction. Assuming that gravitational waves are most efficiently emitted near the periastron at which we may approximate the orbit to be circular, the angular momentum dissipation by gravitational waves in one orbit can be written as [54] ∆J GW ≈ 64π 5 J orbit =M h x u y − y u x ,(58)J spin = d 3 xψ 6 ρh (x − x ) u y − u y − (y − y ) (u x − u x )] ,(59)M 2 BH M 2 * r 2 p,A 1 + 7e 2 8 ,(60) where e denotes the eccentricity. Thus, the ratio of ∆J tid to ∆J GW is written as ∆J spin ∆J GW ≈ 23 f spin r p,A 4M BH −5/2 M BH 10 5 M −3 M * 0.7M −2 × R * 10 4 km 5 1 + 7e 2 8 −1 .(61) Thus it is larger than unity for r p,A 7M BH /c 2 , R * ≈ 10 4 km, M BH = 10 5 M , M * = 0.7M , and f spin = 0.2. This is also the case for the ratio of ∆E tid /∆E GW where ∆E GW is the energy dissipated by gravitational waves in one orbit. Thus, near the tidal disruption orbit, the orbital evolution would be primarily determined not by the gravitational-wave emission but by the tidal effect. To clarify the eventual fate of such a white dwarf, we obviously need a long-term accurate simulation. Such a topic is one of our future targets. We note that both ∆J spin and ∆J GW are much smaller than the orbital angular momentum of order M * M BH r p,A . Thus, the cumulative effect of the tidal angular momentum transport plays an important role just prior to the tidal disruption. By repeated tidal interaction, the spin angular velocity of the white dwarfs is likely to be enhanced up to ∼ M 1/2 BH /r 3/2 p = β 3/2 M 1/2 * /R 3/2 * . In addition, the stellar oscillation for which the oscillation energy is comparable to or larger than the rotational kinetic energy should be excited. As a result, mass loss could be induced, resulting in the increase of the stellar radius and enhancing the importance of tidal interaction. In this type of the system, the tidal disruption is unlikely to take place by one strong impact by the black-hole tidal force but likely to do as a result of a secular increase of the stellar radius (see, e.g., Refs. [9,55] for related studies). IV. SUMMARY We reported a new numerical-relativity code which enables us to explore tidal disruption of white dwarfs by a relatively low-mass supermassive black hole. As a first step toward more detailed future studies, we paid attention to the condition for tidal disruption of white dwarfs with typical mass range in elliptic orbits by a nonspinning supermassive black hole. We showed that our code is capable of determining the condition for the tidal disruption. As expected from previous general relativistic works (e.g., Refs. [23,30]), the tidal disruption takes place for β 0.5 and an appreciable oscillation of the white dwarfs are induced by the black-hole tidal effect for β 0.4 for orbits close to the black hole in the Γ = 5/3 polytropic equation of state. The critical value for the onset of the tidal disruption is smaller than that obtained by Newtonian analysis. For white dwarfs with M * = 0.6M and R * = 1.2 × 10 4 km, β can be larger than 0.4 even for M BH ≈ 4 × 10 5 M if the periastron radius is r p,A = 4M BH . Our result indicates that in such systems with a relatively lowmass (but not intermediate-mass) supermassive black hole for which gravitational waves in the late inspiral phase can be detected by LISA [4], tidal disruption can occur for typical white dwarfs. For spinning black holes with the dimensionless spin parameter of 0.9, r p,A can be smaller than ∼ 1.7M BH [56]. For such black holes, tidal disruption of typical-mass white dwarfs may occur even for M BH ≈ 10 6 M . Investigation of this possibility is a future issue. There are several issues to be explored. The first one is to extend our implementation for spinning black holes. Since no analytic solution is known for the spacetime of spinning black holes on the limit hypersurface, we need to develop a method to provide g 0 µν for employing the formulation introduced in this paper. One straightforward way to prepare such data is just to numerically perform a simulation for a spinning black hole (in vacuum) until the hypersurface reaches the limit hypersurface as a first step, and then, the obtained data are saved and used in the subsequent simulations with white dwarfs. A more subtle issue along this line is to prepare the initial condition. For nonspinning black holes, we can assume that the conformal flatness of the three metric, and as a result, the initial-value equations are composed only of elliptic-type equations with the flat Laplacian. For the spinning black holes, the basic equations are composed of elliptictype equations of complicated Laplacian, and hence, the numerical computation could be more demanding, although in principle it would be still possible to obtain an initial condition. We plan to explore this strategy in the subsequent work. For modeling realistic white dwarfs it is necessary to implement a realistic equation of state. If we assume that the temperature of the white dwarfs is sufficiently low and the pressure is dominated by that of degenerate electrons, it is straightforward to implement this. More challenging issue is to follow the hydrodynamics of tidally disrupted white-dwarf matter for a long term. After the tidal disruption, the matter of the white dwarf is likely to move around the black hole for many orbits. During such orbits, the matter collides each other, and eventually, a compact disk will be formed after the circularization. Such disks are likely to be hot due to the shock heating, and thus, it can be a source of electromagnetic counterparts of the tidal disruption. In the presence of magnetic fields, magnetorotational instability [57] occurs in the disk, and the magnetic fields will be amplified. If the amplified magnetic field eventually penetrates the black hole and if the black hole is appreciably spinning, a jet may be launched through the Blandford-Znajek effect [58]. After the amplification of the magnetic fields, a turbulent state will be developed in the disk and mass ejection could occur by the effective viscosity or magneto-centrifugal force [59]. The ejecta may be a source of electromagnetic signals. One long-term issue is to investigate such scenarios by general relativistic magnetohydrodynamics. FIG. 1 . 1Maximum density as a function of time for model M7V165 (top) with N = 40, 60, 70, and model M8V17 (bottom) with N = 60, 82, 102. We find that a fair convergence is obtained with N = 60. FIG. 2 . 27, and 0.8M . For the polytropic equation of state, the stellar radius, R * , is proportional to M (Γ−2)/(3Γ−4) * for a fixed value of κ. Thus, for Γ = 5/3, the stellar radius depends only weakly on the stellar mass. In the present case we basically choose the value of κ so that R * ≈ 1.0×10 4 (M * /0.7M ) −1/3 km. For M * = 0.7M and V = 0.160, we also prepare two additional Geodesics for V = 0.160, 0.165, 0.170, 0.175 and 0.180 in the coordinates of g 0 µν . Those only for one orbital period started from x = 20M BH and y = z = 0 are plotted. The filled circle at the center represents the black hole with the coordinate radius of its event horizon r ≈ 0.78M BH . ). The corresponding specific angular momentum of the white dwarf is J ≈ 3.7748, 3.8968, 4.0192, 4.142, and 4.2653M BH , and the resulting periastron radius is r p /M BH (r p,A /M BH ) = 4.401(5.456), 5.770 (6.813), 7.030 (8.065), 8.317 (9.346), and 9.681 (10.707) where in the parenthesis the values in the Schwarzschild coordinate, i.e., areal radius (hereafter denoted by r p,A ), are described. In Fig. 2, we plot the geodesics only for one orbital period for V = 0.160, 0.165, 0.170, and 0.180. FIG. 3 . 3where the areal radius r p,A is used for the definition ofβ in this section. For V = 0.160 and 0.165 with M * = 0.6-0.8M , The maximum density as a function of time for V = 0.160, 0.165, 0.170, and 0.180 with M * = 0.7M . The maximum density is normalized by the initial value denoted by ρ 0 . FIG. 4 . 4The maximum density as a function of time for the cases that stellar oscillation occurs (0.5 β 0.4). The red, blue, and green curves show the results with M * = 0.6M , 0.7M and 0.8M , respectively. The maximum density is normalized by the initial value denoted by ρ 0 . FIG . 5. A summary for the fate of the white dwarfs in the plane of M * and β . TD and OC denote that tidal disruption and appreciable oscillation of the white dwarfs are observed after the close encounter of the white dwarfs with the black hole. NN denotes that no appreciable tidal effect is observed. FIG. 6 .FIG. 7 . 67The density profiles of the tidally-disrupted white dwarf for the model V = 0.160 and M * = 0.7M (M7V16). The units of the length scale for the density plots are GM BH /c 2 ≈ 1.48 × 10 5 km. The solid and dashed curves show the time evolution for the location of the maximum density and the elliptic orbit shown in Fig. 2 for V = 0.160 (i.e., geodesic). The length scale of x and y axes is shown in units of M BH . The rescaled change in angular momentum ∆J spin / M * √ M * R * β 9/2 as a function of time for the stellar oscillation scenario (M8V17). This agrees with the analytic expression Eq. (57) if f spin ∼ 0.1-0.3. ACKNOWLEDGMENTSWe thank Kenta Hotokezaka for helpful discussion. This work was in part supported by Grant-in-Aid for Scientific Research (Grant No. JP20H00158) of Japanese MEXT/JSPS. Numerical computations were in part performed on Sakura clusters at Max Planck Computing and Data Facility. Zauderer. K D Alexander, S Van Velzen, A Horesh, B A , 10.1007/s11214-020-00702-warXiv:2006.01159Space Sci. Rev. 216astroph.HEK. D. Alexander, S. van Velzen, A. Horesh, and B. A. 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[ "Anomalous Thermal Transport of SrTiO 3 Driven by Anharmonic Phonon Renormalization", "Anomalous Thermal Transport of SrTiO 3 Driven by Anharmonic Phonon Renormalization" ]	[ "Jian Han ", "Changpeng Lin ", "Ce-Wen Nan ", "Yuan-Hua Lin ", "Ben Xu ", "\nSchool of Materials Science and Engineering\nSchool of Materials Science and Engineering\nTheory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne\nTsinghua University and Graduate School of the China Academy of Engineering Physics\n100088, 1015Beijing, LausannePeople's Republic of China, Switzerland\n", "\nGraduate School of the China Academy of Engineering Physics\nTsinghua University\n100088BeijingPeople's Republic of China\n" ]	[ "School of Materials Science and Engineering\nSchool of Materials Science and Engineering\nTheory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne\nTsinghua University and Graduate School of the China Academy of Engineering Physics\n100088, 1015Beijing, LausannePeople's Republic of China, Switzerland", "Graduate School of the China Academy of Engineering Physics\nTsinghua University\n100088BeijingPeople's Republic of China" ]	[]	SrTiO3 has been extensively investigated owing to its abundant degrees of freedom for modulation. However, the microscopic mechanism of thermal transport especially the relationship between phonon scattering and lattice distortion during the phase transition are missing and unclear. Based on deep-potential molecular dynamics and self-consistent ab initio lattice dynamics, we explore the lattice anharmonicity-induced tetragonal-to-cubic phase transition and explain this anomalous behavior during the phase transition. Our results indicate the significant role of the renormalization of third-order interatomic force constants to second-order terms. Our work provides a robust framework for evaluating the thermal transport properties during structural transformation, benefitting the future design of promising thermal and phononic materials and devices. arXiv:2303.07791v1 [cond-mat.mtrl-sci]	null	[ "https://export.arxiv.org/pdf/2303.07791v1.pdf" ]	257,504,881	2303.07791	74ef8803c7d25b3596566ea2039eec44e5079e58	Anomalous Thermal Transport of SrTiO 3 Driven by Anharmonic Phonon Renormalization 14 Mar 2023 Jian Han Changpeng Lin Ce-Wen Nan Yuan-Hua Lin Ben Xu School of Materials Science and Engineering School of Materials Science and Engineering Theory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne Tsinghua University and Graduate School of the China Academy of Engineering Physics 100088, 1015Beijing, LausannePeople's Republic of China, Switzerland Graduate School of the China Academy of Engineering Physics Tsinghua University 100088BeijingPeople's Republic of China Anomalous Thermal Transport of SrTiO 3 Driven by Anharmonic Phonon Renormalization 14 Mar 2023(Dated: March 15, 2023) SrTiO3 has been extensively investigated owing to its abundant degrees of freedom for modulation. However, the microscopic mechanism of thermal transport especially the relationship between phonon scattering and lattice distortion during the phase transition are missing and unclear. Based on deep-potential molecular dynamics and self-consistent ab initio lattice dynamics, we explore the lattice anharmonicity-induced tetragonal-to-cubic phase transition and explain this anomalous behavior during the phase transition. Our results indicate the significant role of the renormalization of third-order interatomic force constants to second-order terms. Our work provides a robust framework for evaluating the thermal transport properties during structural transformation, benefitting the future design of promising thermal and phononic materials and devices. arXiv:2303.07791v1 [cond-mat.mtrl-sci] Perovskite materials have emerged as one of the most effective material in functional electronics [1][2][3]. However, their thermal conductivities during phase transition remain elusive because of the complex energy landscape and lattice anharmonicity. In particular, the complex patterns of oxygen octahedral distortions that introduce various structural phase transitions and consequently varying degrees of anharmonicity can lead to anomalous thermal transport behavior [4]. For example, SrTiO 3 exhibits antiferrodistortion (AFD) in the low-temperature tetragonal phase, and as the temperature T increases, the AFD amplitude and soft mode frequencies decrease to zero at approximately 105 K, resulting in a phase transition into a cubic phase [5,6]. The thermal conductivity κ of SrTiO 3 exhibits a weak temperature dependence ∼ T −0.4 in the cubic phase and a continuous variation with a bump near the transition point [7,8]. In contrast, lattice-anharmonicity-induced three-phonon interactions generally result in a ∼ T −1 trend in the temperature dependence of the thermal conductivity. In previous studies, the weak temperature dependence was used to be explained by the degeneracy between the transverse optic and the longitudinal acoustic phonon branches [7], nevertheless, this is based on an empirical observation and is not a complete picture. To unveil the mechanism behind the anomalous behavior of κ and its temperature dependence during phase transition, understanding the temperature dependence of lattice dis-tortion and its influence on anharmonicity and thermal conductivity is important, which remains a challenging task. To accurately account for the temperature dependence of the thermal conductivity, Bianco et al. [10] provided a rigorous framework to accurately calculate the influence of anharmonicity and explain the details from the perspective of perturbation. At the lowest perturbation level, 3 rd -and 4 th -order interatomic force constants (IFCs) can introduce renormalizations through tadpole, loop, and bubble diagrams, which result in the frequency shift and broadening of phonon quasiparticles. Approaches based on many-body perturbation theory and frozen phonon method, such as ShengBTE [11] and Phono3Py [12], consider only the broadening from the bubble term. By contrast, the temperature-dependent effective potential (TDEP) method [13,14], self-consistent ab initio lattice dynamics (SCAILD) [15,16] and selfconsistent phonon (SCPH) theory [17] can only additionally consider the frequency shift from the loop diagrams, as the PES is usually expanded around the lattice structure at the ground state. In order to take into account the contributions of the tadpole diagrams in those methods, one needs to also include the temperature dependence of lattice structure, which can be obtained from the NPT (constant particle number, pressure and temperature) ensemble averaging or free energy minimization [10]. A sufficiently broad sampling is necessary in both methods for (a) Comparison of calculated LWTE thermal conductivity with experimental results [7,8]. The LWTE thermal conductivity was calculated using r(T + 95K), Φ (2) (T + 95K) and Φ (3) (T + 95K). (b) Schematic of frozen phonon method [9], SCAILD, and our method combining DP and SCAILD. (c) Diagrammatic description of the relation between finite temperature phonon propagator GS calculated using SCAILD results and the free phonon propagator G0. (d) Diagrammatic description of bare three-phonon and four-phonon vertex and the effective three-phonon vertex at finite temperature. (e) Diagrammatic description of the bubble term to calculate the phonon lifetime in RTA. the lattice structure to converge. However, such a sampling is extremely challenging, since predicting the phase transition process requires an accurate PES from ab initio calculations. In this study, we address this problem by integrating finite-temperature molecular dynamics (MD) simulations with SCAILD. To accurately describe the PES, we trained a deep learning model [18][19][20] based on ab initio calculations for SrTiO 3 . The lattice structure at a certain temperature was determined by MD with a quantum thermal bath (QTB) [21][22][23]. Thereafter, we fitted the effective IFCs via the SCAILD method and obtained their temperature dependence. We solved the linearized Wigner transport equation (LWTE) to calculate the lattice thermal conductivity. Our results agree well with the experimental measurements among a wide temperature range near the phase transition [7,8]. To identify the origin of the thermal conductivity anomaly, we investigated the importance of the effective 2 nd -and 3 rd -IFCs, group velocities, and relaxation times. To demonstrate the importance of the temperature dependence of the lattice structure, we also compared the effective 2 nd -order IFCs with and without the temperature dependence of the lattice structure. Our methodology offers a systematic approach for studying the thermal conductivity and lattice dynamics-related properties during the phase-transition process. We studied the temperature dependence of the lattice structure by considering the NPT ensemble average of the atomic positions and volume in MD simulations as implemented LAMMPS [21] package. Moreover, we incorporated the quantum effect using a quantum thermal bath [22][23][24] method. To describe the interatomic poten-tial energy surface (PES), we used a deep neural network (DNN)-based Deep Potential (DP) model [18,19]. This is consistent with the first-principles results with only 1.21 meV energy error per atom. This model was trained for 1 × 10 6 steps, with more than 2000 labeled data points generated iteratively using a Deep Potential GENerator (DP-GEN) [25,26] platform. The data were labeled via DFT methods [27] from a first-principles perspective. Vienna Ab initio Simulation Package (VASP) [28][29][30][31] was used to perform DFT calculations via the projectoraugmented wave (PAW) method [32]. A revised version of the Perdew-Burke-Ernzerhof [33,34] paraterization of the generalized gradient approximation [35] was applied to describe the exchange-correlation effects of electrons. In all DFT calculations, the plane-wave kinetic energy cutoff was set to 700 eV, and k-spacing was less than 0.14 Å −1 . The temperature dependence of the lattice structure shows that the phase transition temperature in the MD simulation is approximately 200 K, which is higher than the experimental result of 105 K. Such overestimation could be attributed to the limited precision of the PBEsol pseudopotential [36] or the zero-point energy leakage in QTB [37]. Apart from this slight discrepancy, our results capture the phase-transition process and enable us to further delve into the origin of the anomalous thermal conductivity in SrTiO 3 . To investigate the temperature dependency of the 2 ndand 3 rd -order IFCs, we applied the SCAILD method to fit them iteratively with a temperature-dependent lattice structure r of SrTiO 3 . Sampling structures were randomly generated with the following probability: ρ({u a }) ∝ exp − 1 2 uΨ −1 u , where the quantum covari- ance is defined as Ψ ab = ν 2 √ m a m b 1 + 2n ν ω ν ε a ν ε b ν .(1) In the above equation, u a is the displacement of atom i in α direction relative to the finite-temperature equilibrium position r a , a is the shorthand of iα, m a is the atomic mass, ν is the shorthand of qλ, ε a ν is the wavefunction of λ-branch phonon on wavevector q, and n ν is the Bose-Einstein distribution. The interatomic forces of the random structures were calculated using the DP model. We fitted the 2 nd -and 3 rd -order IFCs at the given temperature T with a cutoff radii of 7.0 Å and 3.0 Å, respectively, denoted as Φ (2) and Φ (3) , via least-squares minimization, with the constraints of space-group symmetry, permutation and acoustic sum rules [38,39]. Thereafter, we calculated the thermal conductivity tensor κ αβ as a function of temperature by solving the linearized Wigner equation (LWTE) using the Phono3Py package [12] with a q-mesh of 13 × 13 × 13. The thermal conductivity was subsequently calculated by considering the average of the diagonal part of the thermal conductivity tensor. To compare our results with the experimental measurements, we shifted the phase transition point. For example, the thermal conductivity at 105 K can be evaluated using r, Φ (2) , and Φ (3) at 200 K and the Bose-Einstein distribution at 105 K. In Fig. 1(a), we compare the simulated thermal conductivity with previous experimental measurements [7,8,40]. In our calculations, we capture exactly two features. First, the thermal conductivity demonstrates continuous variation and exhibits a bump at the phase transition point. Second, the temperature dependence in the high-temperature cubic phase agrees well with the experiment and exhibits a tendency of ∼ T −0.4 . Through calculations, we can control the variables to better understand the causes of thermal conductivity anomalies. In the following calculations, we used the relaxation-time approximation (RTA) and calculate the thermal conductivity as follows: κ αβ RTA = 1 N 0 Ω ν C ν v αν v βν τ ν ,(2) where v αν denotes the phonon group velocity, C v is the phonon heat capacity, τ ν is the phonon relaxation time, and N 0 is the total number of q−point in reciprocal space. The relaxation time τ ν = [2Γ ν (ω ν )] −1 can be calculated using the imaginary part Γ ν (ω) of the bubble's self-energy Σ bubble ν (ω) as shown in Fig. 1(e). The bubble self-energy is expressed as follows: Σ bubble ν (ω) = 1 2 ν1,ν2,s=±1 q1+q2+q=0 Φ (3) S (−ν; ν 1 ; ν 2 ) 2 × (n ν1 + n ν2 + 1) sω + ω ν1 + ω ν2 − (n ν1 − n ν2 ) sω + ω ν1 − ω ν2 ,(3) whereΦ S denotes the finite-temperature three-phonon vertex (see Eq. (??)), and Φ (3) S 2 represents the three phonon-phonon (ph-ph) interaction strength. The subscript S denotes the finite-temperature results from SCAILD, and ω = ω + i0 + , where 0 + is a positive infinitesimal. To identify the role of temperature dependence of various physical quantities, we defined the scaled thermal conductivity κ, where the Bose-Einstein distribution n ν is fixed at 300 K. As can be seen in Fig. 2(a) and (b), the orange diamond represents the scaled thermal conductivity with the temperature dependence of other parameters considered explicitly. Such a scaled thermal conductivity can directly demonstrate the influence from the phonon group velocities and lifetimes, without the normal ∼ T −1 dependence owing to an increase in n ν . Therefore, the temperature dependence of κ can recognize the reason for this anomaly in thermal conductivity of SrTiO 3 . To extract the respective contributions to thermal conductivity, we fixed one of the parameters in turn. We find that only at fixed phonon lifetimes, (green diamonds in Fig. 2(a)), κ increases and subsequently decreases with an increase in temperature because of the change in group velocities. Thus, we determine that bump is contributed by the phonon velocity. Additionally, we find that at the phase transition point, higher group velocities result from a lower AFD soft-mode frequency, as shown in Fig. 2(c). The AFD mode frequencies are overestimated in our results; that is, they did not reach zero at the phase transition pint. Such prediction of soft-mode frequency is limited by the computational cost of the phonon shift owing to different order bubble terms [10]. At fixed phonon group velocities, (the blue cubes in Fig. 2(a)), κ increases with the temperature, and this part of the increase counteracts a part of the normal ∼ T −1 dependence of κ. By comparing κ with constant ph-ph interaction strength as shown in Fig. ??, we also find that its the decreasing ph-ph interaction strength contributes to the raise of κ. Accordingly, we confirm that the decreasing ph-ph interaction strength contributes to a weak temperature dependence of κ in the cubic phase, which is in contrast to previous understanding that it can be explained by the degeneracy. The ph-ph interaction strength is mainly determined by the effective 2 nd -and 3 rd -order IFCs. To follow the temperature dependence of the 2 nd -and 3 rd -order effective IFCs on κ, we performed the following procedure: First, we fixed the 2 nd -order IFCs and maintained the temperature dependence of 3 rd -order IFCs. The corresponding κ is shown in Fig. 2(b) using green diamonds. We observe an increasing trend of κ only in the tetragonal phase but not in the cubic phase. This implies that the weak temperature dependence of κ in the cubic phase is not due to the renormalization to the 3 rd -order IFCs. Accordingly, we fix the 3 rd -order effective IFCs and maintain the temperature dependence of 2 nd -order IFCs. The results are shown in Fig. 2(b) by the blue cubic. Similar tendencies to orange diamonds show that the anomalies in κ, including bump and weak temperature dependences are mainly caused by the renormalization to the 2 nd -order IFCs. Therefore, we investigated the ph-ph interaction strength to clarify the reason behind the continuous change in the thermal conductivity at the phase transition point. Fig. 3 shows the number of scattering chan- S } tet to zero could happen only with a continuous decrease in the anisotropy of effective IFCs, especially the 2 nd -order ones. We find that to capture this behaviour, it's necessary to introduce the temperature dependence of the lattice structure. At the lowest perturbation limit, the effective 2 nd -order IFCs expanded around temperature-dependent lattice structure r are Φ (2) ab = Φ (2) 0,ab + c Φ (3) 0,abc δr c + 1 2 cd Φ (4) 0,abcd Ψ cd 0 , (4) where Φ (n) 0 is the bare n th -order IFCs expanded around atomic position r 0 at the ground state, Ψ cd 0 is the temperature-dependent quantum covariance calculated using Φ (2) 0 according to Eq. (1) and δr is the finite temperature atomic displacement. At the lowest perturbation limit, δr is [10]: δr c = r c − r c 0 = − 1 2 abd inv Φ (2) 0 ca Φ (3) 0,abd Ψ bd 0 ,(5) FIG. 4. Temperature dependence of 2 nd -order IFCs Φ (2) . The dark and light symbols are for Φ (2) expanded around r and r0, respectively. r0 was set to the lattice structure at the ground state of the tetragonal phase for T <200 K, and one of the cubic phase for T ≥200 K. We appoint the long axis of tetragonal phase c along z direction. where inv Φ 0 . This renormalization to the 2 nd -order IFCs can therefore be expressed using diagrammatic description, by introducing the tadpole and loop self-energy, denoted as Σ tadpole,0 ν and Σ loop,0 ν , to phonon quasiparticle, as shown in Fig. 1(c). The corresponding finite-temperature phonon propagator is now G S (ω) ≈ G 0 (ω) + G S (ω) = G 0 (ω) + G 0 (ω) Σ loop,0 ν + Σ tadpole,0 ν G 0 (ω), where G 0 (ω) denotes a free phonon propagator. Beyond the lowest perturbation limit, higher-order tadpole and loop diagrams also contribute to effective IFCs. We include the contributions above by expanding the PES around the temperature-dependent lattice structure r using SCAILD, corresponding to the dark blue and orange circles in Fig. 4. In comparison, if the PES were expanded around r 0 , the contribution of the tadpole term is neglected, while the contributions from the loop terms are still maintained, which corresponds to the light blue and orange triangles in Fig. 4. Comparing these two sets of curves, a sharp change in Φ (2) near the phase transition point is found when the tadpole diagrams are neglected, while Φ (2) expanded around r gradually approached the value in the cubic phase with a decreasing degree of anisotropy. We conclude that the tadpole diagram term determines the continuous change in the thermal conductivity at the phase transition point. In summary, we have calculated the thermal conductivity of SrTiO 3 via the SCAILD method with a temperature-dependent lattice structure obtained from DPMD. We explain the origin of the thermal conductivity anomalies in SrTiO 3 . The continuous variation with a bump at the phase transition point is mainly driven by the temperature dependence of the group velocities, whereas the weak temperature-dependent results come from the decreasing ph-ph interaction strength, which is in contrast to the previous understanding that it was explained by the degeneracy of phonon modes. Both phenomena are caused by the renormalization to the effective 2 nd -order IFCs by anharmonicity, where the tadpole self-energy term contributed by 3 rd -order IFCs cannot be neglected. Our study provides a complete prediction and systematic analysis of the anomalous thermal conductivity of SrTiO 3 . The phonon renormalization can also have a significant impact on other transport properties, such as the electrical transport limited by electron-phonon interactions. Therefore, we believe that computational framework adopted in this work promises tremendous potential for predicting the transport behavior of future functional materials. This work was funded by the National Natural Science Foundation of China (Grant Nos. 52072209, 51790494 and 12088101). S1. ACCURACY OF THE DEEP POTENTIAL MODEL The DP model model was trained for 1 × 10 6 steps, with more than 2000 labeled data points generated iteratively using a Deep Potential GENerator (DP-GEN) [1,2] platform. The cutoff radius is 9 Å and the smoothing starts at 2 Å. The initial learning rate was 0.001 and the final learning rate was less than 4 × 10 −8 . This DP model is consistent with the first-principles results with only 1.21 meV energy error per atom. To show the accuracy [3] and (f) phonon lifetime in cubic phase using self-consistent ab initio lattice dynamics (SCAILD) method [4,5] and Phono3Py package [6]. * [email protected] † [email protected] of our model, we compare the calculation results using the DP model and DFT method in NPT molecular dynamics (MD) simulations were performed with periodic boundary conditions and at zero pressure using LAMMPS package [7] with a quantum thermal bath (QTB) [8,9]. The time step was set to 1 fs and the size of supercell was set to 10 × 10 × 10 (5000 atoms). The temperatures were set from 20 K to 400 K with 20 K step. The system reached equilibrium in the first 10 ps. Then the lattice structures were output every 80 fs in the following 160 ps. Because we observe a long axe switching behavior in the MD simulation of SrTiO 3 system based on deep potential (DP) model, which also appeared in previous studies [10], we cannot average the atomic positions directly. Therefore we defined the ordered lattice constant a, b and c, and absolute value of tilting angle θ of oxygen octahedral at each temperature as follows: a(T ), b(T ), c(T ) = ∆t N t t Sort[a(T, t), b(T, t), c(T, t)] θ(T ) = ∆t N t t max [\|θ a (T, t)\| , \|θ b (T, t)\| , \|θ c (T, t)\|] ,(S1) where θ a , θ b and θ c are tilting angles around corresponding axis, a, b and c are lattice constants outputted during MD simulations, T is the temperature and t is the simulation time. κ αα RTA = 1 N 0 Ω ν C ν v α,ν v α ,ν τ ν ,(S4) where ν is the shorthand of qλ, q the wave vector, λ the index of phonon branch, v α,ν the phonon group velocity in direction α, C ν the phonon heat capacity, and τ ν the phonon relaxation time, N 0 the number of unitcells in supercell, Ω the volume of unitcell. The phonon heat capacity is: When the IFCs are expanded around r 0 , there will be a sharp change ofΦ (2) at the phase transition point, as shown in Fig. ??. As a result, there will be also a sharp change in soft C ν = ( ω v ) 2 n ν (n ν + 1) k B T 2 ,(S5) whereΦ (n) 0 is the bare n th -order ph-ph interaction vertex. FIG. 1 . 1FIG. 1. (a) Comparison of calculated LWTE thermal conductivity with experimental results [7, 8]. The LWTE thermal conductivity was calculated using r(T + 95K), Φ (2) (T + 95K) and Φ (3) (T + 95K). (b) Schematic of frozen phonon method [9], SCAILD, and our method combining DP and SCAILD. (c) Diagrammatic description of the relation between finite temperature phonon propagator GS calculated using SCAILD results and the free phonon propagator G0. (d) Diagrammatic description of bare three-phonon and four-phonon vertex and the effective three-phonon vertex at finite temperature. (e) Diagrammatic description of the bubble term to calculate the phonon lifetime in RTA. FIG. 2 . 2(a) The temperature dependence of scaled thermal conductivity κ. For orange diamonds the temperature dependence of v and τ are considered.. For blue cubic, to show the impact of the temperature dependence of v, τ are kept the same. For green diamonds, to show the impact of the temperature dependence of τ and v are kept the same. (b) The temperature dependence of scaled thermal conductivity κ. For orange diamonds, the temperature dependence of r, Φ(3) and Φ(2) are considered. For blue cubic, r and Φ(3) are fixed to show the impact of the temperature dependence of Φ(2) . For green diamonds, r and Φ(2) are fixed to show the impact of the temperature dependence of Φ(3) . (c) The temperature dependence of AFD mode frequencies. FIG. 3 . 3The x-axis is the ph-ph interaction strengths, and the y-axis shows the counts of each value.Φ(3) S is calculated using r, Φ(2) and Φ(3) . The IFCs in 198 K and 199.8 K are interpolated from 180 K and 200 K results. nels with different interaction strengths at different temperatures. Here, {Φ(3) S } tet is the set of ph-ph interaction vertices of scattering channels activated only in the tetragonal phase, and {Φ (3) S } cub is one of the activated scattering channels in the cubic phase. At 180 K, the peaks of {Φ (3) S } cub and {Φ (3) S } tet overlap, and the average value of {Φ (3) S } tet is smaller than that of {Φ (3) S } cub . With an increase in the temperature, the two peaks separate, and the average value of {Φ (3) S } tet gradually approaches zero. Because {Φ (3) S } tet are already zero at the phase-transition point, the sudden change in the scattering channel numbers owing to translational symmetry breaking does not cause a sudden change in the phonon relaxation time, causing the thermal conductivity at the phase transition to change continuously. According to Eq. (??), the continuous change in {Φ (3) (a) and (b) show the on-site terms of Φ (2) for Sr and Ti atom, respectively. Orange symbols indicate the xx and yy component, while the blue symbol indicates the zz component. (c) and (d) show the Frobenius norm of Φ (2) for the on-site term of O atom and nearest Ti-O bond, respectively. Orange symbols show the results of O atom whose nearest Ti-O bond is along the c direction. Blue symbols show the results of O atom whose nearest Ti-O bond is along the a or b direction. is the ca th -component of the inverse of Φ FIG. S1 . S1Comparison of results calculated using the deep potential (DP) model and density functional theory (DFT). (a) energies, (b) atomic forces, (c) viral for all configurations in the training and testing dataset, (d) energy double well of antiferrodistortion (AFD) mode, (e) phonon spectrum in cubic phase using frozen phonon method Fig Fig. S1. FIG. S2. (a)(b) The temperature dependence of average ordered lattice constants and absolute value of tilting angle in the DP model based MD simulations under QTB. (c)(d) The temperature dependence of interpolated lattice constants and tilting angle. Fig. S2(a)(b) shows the temperature dependence of a, b, c and θ, the phase transition temperature in MD simulation is around 200 K. The interpolation was then performed to get the lattice constant and tilting angle at each temperature, which would be used in the further SCAILD calculations. The results were shown in Fig. S2(c)(d). The data used in interpolation are a it , b it , c it and θ it . We choose a it , b it , c it and θ it as follows: (T ) = b it (T ) = a(T )b(T ), T ≤ 100 K a it (T ) = b it (T ) = c it (T ) = 3 a(T )b(T )c(T ). T ≥ 200 K (S3) S3. DETAILS IN CALCULATING SCALED THERMAL CONDUCTIVITY Under relaxation time approximation, the lattice thermal conductivity using linearized Boltzmann transport equation is: S4. THE SCALED THERMAL CONDUCTIVITY USING CONSTANT ph − ph IN-TERACTION STRENGTH Fig. S3 shows the scaled thermal conductivity using constant ph-ph interaction strength 10 −9 eV 2 . FIG. S3. The scaled thermal conductivity using constant ph-ph interaction strength 10 −9 eV 2 . S5. SOFT MODE FREQUENCIES AND RTA THERMAL CONDUCTIVITY US-ING IFCS EXPANDED AROUND r 0 FIG. S4. The soft mode frequency using SCAILD, where the IFCs are expanded around r 0 FIG. S5 . S5The scaled thermal conductivity results where the IFCs are expanded around r 0 . mode frequencies and thermal conductivity, as shown inFig. S4andFig. S5which is not consistent with experimental results.S6. EXPRESSIONS OF TADPOLE AND LOOP SELF-ENERGY TERMSThe lowest order tadpole and loop self-energies are: 0λ 1 ; ν 2 ; −ν 2 ) , Supplementary Information for Anomalous Thermal Transport of SrTiO 3Driven by Anharmonic Phonon Renormalization . B Yan, M Jansen, C Felser, Nature Physics. 9709B. Yan, M. Jansen, and C. 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[ "CONVERGENCE RATES OF INERTIAL SPLITTING SCHEMES FOR NONCONVEX COMPOSITE OPTIMIZATION", "CONVERGENCE RATES OF INERTIAL SPLITTING SCHEMES FOR NONCONVEX COMPOSITE OPTIMIZATION" ]	[ "Patrick R Johnstone \nDepartment of Electrical and Computer Engineering\nUniversity of Illinois at Urbana-Champaign\n\n", "Pierre Moulin [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of Illinois at Urbana-Champaign\n\n" ]	[ "Department of Electrical and Computer Engineering\nUniversity of Illinois at Urbana-Champaign\n", "Department of Electrical and Computer Engineering\nUniversity of Illinois at Urbana-Champaign\n" ]	[]	We study the convergence properties of a general inertial first-order proximal splitting algorithm for solving nonconvex nonsmooth optimization problems. Using the Kurdyka-Łojaziewicz (KL) inequality we establish new convergence rates which apply to several inertial algorithms in the literature. Our basic assumption is that the objective function is semialgebraic, which lends our results broad applicability in the fields of signal processing and machine learning. The convergence rates depend on the exponent of the "desingularizing function" arising in the KL inequality. Depending on this exponent, convergence may be finite, linear, or sublinear and of the form O(k −p ) for p > 1.	10.1109/icassp.2017.7953051	[ "https://export.arxiv.org/pdf/1609.03626v1.pdf" ]	1,879,071	1609.03626	c55cd0cdc382d20edca7c35c531e3c2e1cdf36e7	CONVERGENCE RATES OF INERTIAL SPLITTING SCHEMES FOR NONCONVEX COMPOSITE OPTIMIZATION Patrick R Johnstone Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Pierre Moulin [email protected] Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign CONVERGENCE RATES OF INERTIAL SPLITTING SCHEMES FOR NONCONVEX COMPOSITE OPTIMIZATION Index Terms-Kurdyka-Łojaziewicz InequalityInertial forward- backward splittingheavy-ball methodconvergence ratefirst-order methods We study the convergence properties of a general inertial first-order proximal splitting algorithm for solving nonconvex nonsmooth optimization problems. Using the Kurdyka-Łojaziewicz (KL) inequality we establish new convergence rates which apply to several inertial algorithms in the literature. Our basic assumption is that the objective function is semialgebraic, which lends our results broad applicability in the fields of signal processing and machine learning. The convergence rates depend on the exponent of the "desingularizing function" arising in the KL inequality. Depending on this exponent, convergence may be finite, linear, or sublinear and of the form O(k −p ) for p > 1. INTRODUCTION We are interested in solving the following optimization problem min x∈R n Φ(x) = f (x) + g(x) (1) where g : R n → R ∪ {+∞} is lower semicontinuous (l.s.c.) and f : R n → R is differentiable with Lipschitz continuous gradient. We also assume that Φ is semialgebraic [1], meaning there integers p, q ≥ 0 and polynomial functions Pij, Qij : R n+1 → R such that {(x, y) : y ≥ f (x)} = p ∪ j=1 q ∩ i=1 {z ∈ R n+1 : Pij(z) = 0, Qij(z) < 0}. We make no assumption of convexity. Semialgebraic objective functions in the form of (1) are widespread in machine learning, image processing, compressed sensing, matrix completion, and computer vision [2,3,4,5,6,7,8]. We will list a few examples below. In this paper we focus on the application of Prob. (1) to sparse least-squares and regression. This problem arises when looking for a sparse solution to a set of underdetermined linear equations. Such problems occur in compressed sensing, computer vision, machine learning and many other related fields. Suppose we observe y = Ax+b where b is noise and wish to recover x which is known to be sparse, however the matrix A is "fat" or poorly conditioned. One approach is to solve (1) with f a loss function modeling the noise b and g a regularizer modeling prior knowledge of x, in this case sparsity. The correct choice for f will depend on the noise model and may be nonconvex. Examples of appropriate nonconvex semialgebraic choices for g are the 0 pseudo-norm, and the smoothly clipped absolute deviation (SCAD) [9]. The prevailing convex choice is the 1 norm which is also semialgebraic. SCAD has the advantage over the 1-norm that it leads to nearly unbiased estimates of large coefficients [9]. Furthermore unlike the 0 norm SCAD leads to a solution which is continuous in the data matrix A. Nevertheless 1-based methods continue to be the standard throughout the literature due to convexity and computational simplicity. For Problem (1), first-order methods have been found to be computationally inexpensive, simple to implement, and effective solvers [10]. In this paper we are interested in first order methods of the inertial type, also known as momentum methods. These methods generate the next iterate using more than one previous iterate so as to mimic the inertial dynamics of a model differential equation. In many instances both in theory and in practice, inertial methods have been shown to converge faster than noninertial ones [11]. Furthermore for nonconvex problems it has been observed that using inertia can help the algorithm escape local minima and saddle points that would capture other first-order algorithms [12,Sec 4.1]. A prominent example of the use of inertia in nonconvex optimization is training neural networks, which goes under the name of back propagation with momentum [13]. In convex optimization a prominent example is the heavy ball method [11]. Over the past decade the KL inequality has come to prominence in the optimization community as a powerful tool for studying both convex and nonconvex problems. It is very general, applicable to almost all problems encountered in real applications, and powerful because it allows researchers to precisely understand the local convergence properties of first-order methods. The inequality goes back to [14,15]. In [16,17,18] the KL inequality was used to derive convergence rates of descent-type first order methods. The KL inequality was used to study convex optimization problems in [19,20]. Nonconvex optimization has traditionally been challenging for researchers to study since generally they cannot distinguish a local minimum from a global minimum. Nevertheless, for some applications such as empirical risk minimization in machine learning, finding a good local minimum is all that is required of the optimization solver [21,Sec. 3]. In other problems local minima have been shown to be global minima [22]. Contributions: The main contribution of this paper is to determine for the first time the local convergence rate of a broad family of inertial proximal splitting methods for solving Prob. (1). The family of methods we study includes several algorithms proposed in the literature for which convergence rates are unknown. The family was proposed in [23], where it was proved that the iterates converge to a critical point. However the convergence rate, e.g. how fast the iterates converge, was not determined. In fact in [23], local linear convergence was shown under a partial smoothness assumption. In contrast we do not assume partial smoothness and our results are far more general. We use the KL inequality and show finite, linear, or sublinear convergence, depending on the KL exponent (see Sec. 2). The main inspiration for our work is [18] which studied convergence rates of several noninertial schemes using the KL property. However, the analysis of [18] cannot be applied to inertial methods. Our approach is to extend the framework of [18] to the inertial setting. This is done by proving convergence rates of a multistep Lyapunov potential function which upper bounds the objective function. We also prove convergence rates of the iterates and extend a result of [20,Thm. 3.7 ] to show that our multistep Lyapunov potential has the same KL exponent as the objective function. Finally we include experiments to illustrate the derived convergence rates. Notation: Given a closed set C and point x, define d(x, C) min{ x − c : c ∈ C}. For a sequence {x k } k∈N let ∆ k x k − x k−1 . We say that x k → x * linearly with convergence factor q ∈ (0, 1) if there exists C > 0 such that x k − x * ≤ Cq k . MATHEMATICAL BACKGROUND In this section we give an overview of the relevant mathematical concepts. We use the notion of the limiting subdifferential ∂Φ(x) of a l.s.c. function Φ. For the definition and properties we refer to [1, Sec 2.1]. A necessary (but not sufficient) condition for x to be a minimizer of Φ is 0 ∈ ∂Φ(x). The set of critical points of Φ is crit(Φ) {x : 0 ∈ ∂Φ(x)}. A useful notion is the proximal operator w.r.t. a l.s.c. proper function g, defined as prox g (x) = arg min x ∈R n g(x ) + 1 2 x − x 2 , which is always nonempty. Note that, unlike the convex case, this operator is not necessarily single-valued. Definition A function f : R n → R is said to have the Kurdyka-Lojasiewicz (KL) property at x * ∈ dom ∂f if there exists η ∈ (0, +∞], a neighborhood U of x * , and a continuous and concave function ϕ : [0, η) → R+ such that (i) ϕ(0) = 0, (ii) ϕ is C 1 on (0, η) and for all s ∈ (0, η), ϕ (s) > 0, (iii) for all x ∈ U ∩ {x : f (x * ) < f (x) < f (x * ) + η} the KL inequality holds: ϕ (f (x) − f (x * ))d(0, ∂f (x)) ≥ 1.(2) If f is semialgebraic, then it has the KL property at all points in dom ∂f , and ϕ(t) = c θ t θ for θ ∈ (0, 1]. In the semialgebraic case we will refer to θ as the KL exponent (note that some other papers use 1 − θ [20]). For the special case where f is smooth, (2) can be rewritten as ∇(ϕ • (f (x) − f (x * )) ≥ 1, which shows why ϕ is called a "desingularizing function". The slope of ϕ near the origin encodes information about the "flatness" of the function about a point, thus the KL exponent provides a way to quantify convergence rates of iterative first-order methods. For example the 1D function f (x) = \|x\| p for p ≥ 2 has desingluarizing function ϕ(t) = t 1 p . The larger p, the flatter f is around the origin, and the slower gradient-based methods will converge. In general, functions with smaller exponent θ have slower convergence near a critical point [18]. Thus, determining the KL exponent of an objective function holds the key to assessing convergence rates near critical points. Note that for most prominent optimization problems, determining the KL exponent is an open problem. Nevertheless many important examples have been determined recently, such as least-squares and logistic regression with an 1, 0, or SCAD penalty [20]. A very interesting recent work showed that for convex functions the KL property is equivalent to an error bound condition which is often easier to check in practice [19]. We now precisely state our assumptions on Problem (1), which will be in effect throughout the rest of the paper. Assumption 1. The function Φ : R n → R ∪ {+∞} is semialgebraic, bounded from below, and has desingularizing function ϕ(t) = c θ t θ where c > 0 and θ ∈ (0, 1]. The function g : R n → R is l.s.c., and f : R n → R has Lipschitz continuous gradient with constant L. A FAMILY OF INERTIAL ALGORITHMS We study the family of inertial algorithms proposed in [23]. In what follows s ≥ 1 is an integer, and I = {0, 1, . . . , s − 1}. The algorithm is very general and covers several inertial algorithms proposed in the literature as special cases. For instance the inertial forward-backward method proposed in [12] corresponds to MiFB with s = 1, and b k,0 = 0. The well-known iPiano algorithm also corresponds to this same parameter choice, however the original analysis of this algorithm assumed g was convex [25]. The heavy-ball method is an early and prominent inertial first-order method which also corresponds to this parameter choice when g = 0. The heavy-ball method was originally proposed for strongly convex quadratic problems but was considered in the context of nonconvex problems in [26]. The analysis of [27] applies to MiFB for the special case when s = 1 and a k,0 = b k,0 . However [27] only derived convergence rates of the iterates and not the function values, which are our main interest 1 . Furthermore [27] used a different proof technique to the one used here. This same parameter choice has been considered for convex optimization in [24,28], albeit without the sharp convergence rates derived here. In both the convex and nonconvex settings, employing inertia has been found to improve either the convergence rate or the quality of the obtained local minimum in several studies [12,25,23,24]. Algorithm 1 Multi-step Inertial Forward-Backward splitting (MiFB) Require: x0 ∈ R n , 0 < γ ≤ γ < 1/L. Set x−s = . . . = x−1 = x0, k = 1 repeat Choose 0 < γ ≤ γ k ≤ γ < 1/L, {a k,0 , a k,1 , . . .} ∈ (−1, 1] s , {b k,0 , b k,1 , . . .} ∈ (−1, 1] s . y a,k = x k + i∈I a k,i (x k−i − x k−i−1 ) y b,k = x k + i∈I b k,i (x k−i − x k−i−1 ) x k+1 ∈ prox γ k g (y a,k − γ k ∇f (y b,k )) k = k + 1 until General convergence rates have not been derived for MiFB under nonconvexity and semialgebraicity assumptions. The convergence rate of iPiano has been examined in a limited situation where the KL exponent θ = 1/2 in [20,Thm 5.2]. Note that the primary motivation for studying this framework is its generality -allowing our analysis to cover many special cases from the literature. However the case s = 1 is the most interesting in practice and corresponds to the most prominent inertial algorithms. CONVERGENCE RATE ANALYSIS Throughout the analysis, Assumption 1 is in effect. Before providing our convergence rate analysis, we need a few results from [23]. β k 1 − γ k L − µ − νγ k 2γ k , β lim inf k∈N β k , α k,i sa 2 k,i 2γ k µ + sb 2 k,i L 2 2ν , αi lim sup k∈N α k,i , and z k (x k , x k−1 , . . . , x k−s ) . Define the multi-step Lyapunov function as Ψ(z k ) Φ(x k ) + i∈I s−1 j=i αj ∆ 2 k−i .(3) and δ β − i∈I αi > 0.(4) If the parameters are chosen so that δ > 0 then The assumption that {x k } is bounded is standard in the analysis of algorithms for nonconvex optimization and is guaranteed under ordinary conditions such as coercivity. Since the set of semialgebraic functions is closed under addition, Ψ is semialgebraic [29]. We now give our convergence result. (a) If θ = 1, then x k converges to x * in a finite number of iterations. (i) for all k, Ψ(z k+1 ) ≤ Ψ(z k ) − δ∆ 2 k+1 , (ii) for all k, there is a σ > 0 such that d(0, ∂Ψ(z k )) ≤ σ k j=k+1−s ∆j, (iii) If {x k } is bounded there exists x * ∈ crit(Φ) such that x k → x * and Φ(x k ) → Φ(x * ).(b) If 1 2 ≤ θ < 1, then Φ(x k ) → Φ(x * ) linearly. (c) If 0 < θ < 1/2, then Φ(x k ) − Φ(x * ) = O k 1 2θ−1 . Proof. The starting point is the KL inequality applied to the multistep Lyapunov function defined in (3). Let z * ((x * ) , . . . , (x * ) ) . Suppose Ψ(zK ) = Ψ(z * ) for some K > 0. Then the descent property of Thm. 1(i), along with the fact that Ψ(z k ) → Ψ(z * ), implies that ∆K+1 = 0 and therefore Ψ(z k ) = Ψ(z * ) holds for all k > K. Therefore assume Ψ(z k ) > Ψ(z * ). Now since z k → z * and Ψ(z k ) → Ψ(z * ), there exists k0 > 0 such that for k > k0 (2) holds with f = Ψ. Assume k > k0. Squaring both sides of (2) yields ϕ 2 (Ψ(z k ) − Ψ(z * ))d(0, ∂Ψ(z k )) 2 ≥ 1,(5) Now substituting Thm.1 (ii) into (5) yields σ 2 ϕ 2 (Ψ(z k ) − Ψ(z * ))   k j=k+1−s ∆j   2 ≥ 1. (6) Now   k j=k+1−s ∆j   2 ≤ s k j=k+1−s ∆ 2 j ≤ s δ k j=k+1−s (Ψ(zj−1) − Ψ(zj)) = s δ (Ψ(z k−s ) − Ψ(z k )) , where in the first inequality we have used the fact that ( s i=1 ai) 2 ≤ s n i=1 a 2 i , and in the second inequality we have used Thm. 1(i). Substituting this into (6) yields σ 2 s δ ϕ 2 (Ψ(z k ) − Ψ(z * )) (Ψ(z k−s ) − Ψ(z k )) ≥ 1, from which convergence rates can be derived by extending the arguments in [18,Thm 4]. Proceeding, let r k Ψ(z k ) − Ψ(z * ), and C1 δ σ 2 c 2 s , then using ϕ (t) = ct θ−1 , we get r k−s − r k ≥ C1r 2(1−θ) k .(7) If θ = 1, then the recursion becomes r k−s − r k ≥ C1, ∀k > k0. Since by Theorem 1 (iii), r k converges, this would require C1 = 0, which is a contradiction. Therefore there exists k1 such that r k = 0 for all k > k1. Suppose θ ≥ 1/2, then since r k → 0, there exists k2 such that for all k > k2, r k ≤ 1, and r 2(1−θ) k ≥ r k . Therefore for all k > k2, r k−s − r k ≥ C1r k =⇒ r k ≤ (1 + C1) −1 r k−s ≤ (1 + C1) −p 1 r k 2 ,(8) where p1 (8) holds for all k ≥ k0. k−k 2 s Note that p1 > k−k 2 −s s . Therefore r k → 0 linearly. Note that if θ = 1 2 , 2(1 − θ) = 1 andFinally suppose θ < 1/2. Define φ(t) D 1−2θ t 2θ−1 where D > 0, so φ (t) = −Dt 2θ−2 . Now φ(r k ) − φ(r k−s ) = r k r k−s φ (t)dt = D r k−s r k t 2θ−2 dt. Therefore since r k−s ≥ r k and t 2θ−2 is nonincreasing, φ(r k ) − φ(r k−s ) ≥ D(r k−s − r k )r 2θ−2 k−s . Now we consider two cases. Case 1: suppose 2r 2θ−2 k−s ≥ r 2θ−2 k , then φ(r k ) − φ(r k−s ) ≥ D 2 (r k−s − r k )r 2θ−2 k ≥ C1D 2 .(9) where in the second inequality we have used (7). Case 2: suppose that 2r 2θ−2 k−s < r 2θ−2 k . Now 2θ − 2 < 2θ − 1 < 0, therefore (2θ − 1)/(2θ − 2) > 0, thus r 2θ−1 k > qr 2θ−1 k−s where q = 2 2θ−1 2θ−2 > 1. Thus φ(r k ) − φ(r k−s ) = D 1 − 2θ r 2θ−1 k − r 2θ−1 k−s > D 1 − 2θ (q − 1)r 2θ−1 k−s ≥ D 1 − 2θ (q − 1)r 2θ−1 k 0 C2. (10) Thus putting together (9) and (10) yields φ(r k ) ≥ φ(r k−s ) + C3 where C3 = max(C2, C 1 D 2 ). Therefore φ(r k ) ≥ φ(r k ) − φ(r k−p 2 s ) ≥ p2C3 where p2 k−k 0 s . Therefore r k ≤ 1 − 2θ D 1 2θ−1 (p2C3) 1 2θ−1 ≤ C4 k − s − k0 s 1 2θ−1 . where C4 = C 3 (1−2θ) D 1 2θ−1 . To end the proof, note that Φ(x k ) ≤ Ψ(z k ). In the case where f and g are also convex, we can use parameter choices specified in [24, Thm. 1]. CONVERGENCE RATES OF THE ITERATES The convergence rates of x k − x * can also be quantified. To do so we need another result from [23]. Lemma 3. Recall the notation r k Ψ(z k ) − Ψ(z * ). Let k σ δ (ϕ(r k ) − ϕ(r k+1 )) where σ is defined in Theorem 1 (ii) and δ in (4). Fix κ > 0 so that κ < 2/s. Assume the parameters of MiFB are chosen to so that δ > 0 and {x k } is bounded. Then there exists a k0 > 0 such that for all k > k0 r k > 0 =⇒ ∆ k ≤ κ 2 k−1 j=k−s ∆j + 1 2κ k−1 .(11) Proof. This inequality is proved on page 14 of [23] as part of the proof of [23, Thm 2.2]. We now state our result. (b) If 1 2 ≤ θ < 1, x k → x * linearly. (c) If 0 < θ < 1 2 , x k − x * = O k θ 2θ−1 . Proof. Statement (a) follows trivially from the fact that r k = 0 after finitely many iterations, and therefore ∆ k = 0. We proceed to prove statements (b) and (c). As with Theorem 2 the basic idea is to extend the techniques of [18] to allow for the inertial nature of the algorithm. The starting point is (11). Fix K > k0. Then k≥K ∆ k ≤ κ 2 k≥K k−1 j=k−s ∆j + 1 2κ k≥K k−1 ≤ κs 2 k≥K−s ∆ k + 1 2κ k≥K k−1 Let C = κs 2 and note that 0 < C < 1. Therefore subtracting C k≥K ∆ k from both sides yields k≥K ∆ k ≤ 1 1 − C   C K−1 k=K−s ∆ k + 1 2κ k≥K k−1   . Next note that K−1 k=K−s ∆ k ≤ s δ (Ψ(zK−s−1) − Ψ(zK−1)) 1/2 ≤ s δ √ rK−s−1 Let C C s δ then using k≥K k−1 = σ δ ϕ(rK−1), k≥K ∆ k ≤ 1 1 − C C √ rK−s−1 + σ δ ϕ(rK−1) ≤ 1 1 − C C √ rK−s−1 + σ δ ϕ(rK−s−1) , where in the second inequality we used the fact that r k is nonincreasing and ϕ is a monotonic increasing function. Thus using the triangle inequality and the fact that lim k x k − x * = 0, xK − x * ≤ k≥K ∆ k ≤ 1 1 − C C √ rK−s−1 + σ δ ϕ(rK−s−1) . Hence if r k → 0 linearly, then so does x k − x * , which proves (b). On the other hand if 0 < θ < 1/2, for k sufficiently large we see that x k − x * = O(ϕ(r k−s−1 )), which proves statement (c). KL EXPONENT OF THE LYAPUNOV FUNCTION We now extend the result of [20, Thm 3.7] so that it covers the Lyapunov function defined in (3). Theorem 5. Let s ≥ 1 and consider Ψ (s) (x1, x2, . . . , xs) Φ(x1) + s−1 i=1 ci xi+1 − xi 2 .(12) If Φ has KL exponent θ ∈ (0, 1/2] atx then Ψ (s) has KL exponent θ at [x,x, . . . ,x] . Proof. Before commencing, note that if Φ has desingularizing function ϕ(t) = c θ t θ , the KL inequality (2) can be written in the equivalent "error bound" form: d(0, ∂Φ(x)) 1/α ≥ c (Φ(x) − Φ(x * )) where c c −1 > 0, and α 1 − θ. We now show that this error bound holds for the Lyapunov function in (12). The key is to notice the recursive nature of the Lyapunov function. In particular for all s ≥ 2 Ψ (s) (x 1 s ) = Ψ (s−1) (x s−1 1 ) + cs−1 xs−1 − xs 2 , with Ψ (1) (x 1 1 ) Φ(x1), and x s 1 [x 1 , . . . , x s ] . Since Φ has KL exponent θ atx, Ψ (1) has KL exponent θ atx. We will prove the following inductive step for s ≥ 2: If Ψ (s−1) has KL exponent α (with constant c ) atx s−1 1 , then Ψ (s) has KL exponent α atx s 1 wherē x s 1 [x,x, . . . ,x] wherex is repeated s times. Proceeding, for s ≥ 2 assume x1, x2, . . . , xs are such that xs− xs−1 ≤ 1 and the KL inequality (2) applies to Ψ (s−1) atx s 1 . Then The power of this theorem is that when the KL exponent of the objective function Φ is known, it also applies to the Lyapunov function in (3). This allows us to exactly determine the convergence rate of MiFB via Theorems 2 and 4. ∂Ψ (s) (x 1 s )   ξ s−2 1 ξs−1 0   +   0 cs−1(xs−1 − xs) cs−1(xs − xs−1)   where (ξ s−2 1 , ξs−1) ∈ ∂Ψ (s−1) (x s−2 1 , xs−1). Therefore d(0, ∂Ψ (s) (x s 1 )) 1/α (a) ≥ C1 inf (ξ s−2 1 ,ξ s−1 )∈∂Ψ (s−1) (x s−2 1 ,x s−1 ) ξ c s−1 1/α + ξs−1 + cs−1(xs−1 − xs) 1/α + cs−1(xs − xs−1) 1/α (b) ≥ C1 inf (ξ c s−1 ,ξ s−1 )∈∂Ψ (s−1) (x s−2 1 ,x s−1 ) ξ c s−1 1/α + η1 ξs−1 1/α −η2 cs−1(xs−1 − xs) 1/α + cs−1(xs − xs−1) 1/α (c) ≥ C2 inf (ξ c s−1 ,ξ s−1 )∈∂Ψ (s−1) (x s−2 1 ,x s−1 ) ξ c s−1 1/α + ξs−1 1/α + cs−1c 2 xs − xs−1 1/α (d) ≥ C3 inf (ξ c s−1 ,ξ s−1 )∈∂Ψ (s−1) (x s−2 1 ,x s−1 ) ξ c s−1 ξs−1 1/α + cs−1c 2 xs − xs−1 1/α (e) ≥ C3c Ψ (s−1) (x s−1 1 ) − Ψ (s−1) (x s−1 1 ) + cs−1 2 xs − xs−1 1/α (f ) ≥ C3c Ψ (s−1) (x s−1 1 ) − Ψ (s−1) (x s−1 1 ) + cs−1 2 xs − xs−1 2 = C3c Ψ (s) (x s 1 ) − Ψ (s) (x s 1 ) . NUMERICAL RESULTS One Dimensional Polynomial This simple experiment verifies the convergence rates derived in Theorem 2 for MiFB. Consider the one dimensional function f (x) = \|x\| p for p > 2. Use g(x) = +∞ if \|x\| > 1 and 0 otherwise. The proximal operator is simple projection and f is p(p − 1)-smooth on this set. The function Φ = f + g is semialgebraic with ϕ(t) = pt 1/p , i.e. θ = 1/p. Therefore Theorem 2 predicts O k − p p−2 rates for MiFB, which is verified in Fig. 1 for three parameter choices in the cases p = 4, 18. For simplicity we ignore constants and focus on the sublinear order. For p ≤ 4 this convergence rate is better than that of Nesterov's accelerated method [30], for which only O(1/k 2 ) worst-case rate is known. Faster rates are achievable due to the additional knowledge of the KL exponent. Note a k,: [a k,0 , a k,1 ] and these are log-log plots. SCAD and 1 regularized Least-Squares We solve Prob. (1) with f (x) = 1 2 Ax − b 2 2 and g(x) = n i=1 r(xi) where r is: 1) the SCAD regularizer defined as We choose A ∈ R 500×1000 having i.i.d. N (0, 10 −4 ) entries, and b = Ax0, where x0 ∈ R 1000 has 50 nonzero N (0, 1)distributed entries. For SCAD we use a = 5 and λ = 1 and for the 1 norm we use λ = 0.01. We consider four valid parameter choices. To isolate the effect of inertia, all choices used the same randomly chosen starting point and fixed stepsize, γ k = 0.1/L for SCAD and γ k = 1/L for 1. The inertial parameters were chosen so that δ > 0 (defined in (4)) for SCAD and to satisfy [24, Thm. 1] for the 1 problem. The two figures on the right corroborate Theorem 2 in that all considered parameter choices converge linearly to their limit, which was estimated by using the attained objective function value after 1000 iterations. For the nonconvex SCAD this is a new result. For 1-regularized least squares, inertial methods have been shown to achieve local linear convergence in [24,31] under additional strict complementarity or restricted strong convexity assumptions. However, our analysis, which is based on the KL inequality, does not explicitly require these additional assumptions, as the objective function always has a KL exponent of 1/2 [19, Lemma 10]. Furthermore our result proves global linear convergence, in that the KL inequality (2) holds for all k, implying k0 = 1 in (5) and (8) holds for all k. In addition the two left figures show that the inertial choices appear to provide acceleration relative to the standard non-inertial choice which for SCAD is a new observation. This does not conflict with Theorem 2 which only shows that both non-inertial and inertial methods will converge linearly, however the convergence factor may be different. Estimating the factor is beyond the scope of this paper and we leave it for future work. Finally we mention that FISTA [32] and other Nesterov-accelerated methods [30] are not applicable to SCAD as it is nonconvex. r(xi) =      λ\|xi\| if \|xi\| ≤ λ − \|x i \| 2 −2aλ\|x i \|+λ 2 2(a−1) if λ < \|xi\| ≤ aλ (a+1)λ 2 2 if \|xi\| > aλ,and 2) convergence Note the algorithm as stated leaves open the choice of the parameters {a k,i , b k,i , γ k }. For convergence conditions on the parameters we refer to Section 4 and [24, Thm. 1]. Theorem 1 . 1Fix s ≥ 1 and recall I = {0, 1, . . . , s − 1}. Fix {γ k }, {a k,i } and {b k,i } for k ∈ N and i ∈ I. Fix µ, ν > 0 and define Theorem 2 . 2Assume the parameters of MiFB are chosen such that δ > 0 where δ is defined in (4), thus there exists a critical point x * such that x k → x * . Let θ be the KL exponent of Ψ defined in(3). Theorem 4 . 4Assume {x k } is bounded and the parameters of MiFB are chosen so that δ > 0 where δ is defined in(4). Let θ be the KL exponent of Ψ defined in (3). Then (a) If θ = 1, then x k = x * after finitely many iterations. Fig. 1 : 1(Left) p = 4, (Right) p = 18, Φ * = 0. The dotted line is the slope of the predicted O k i.e. ignoring constants). Fig. 2 : 2(Top Left) Plot of Φ(x k ) for SCAD least-squares. (Top Right) Plot of Φ(x k )−Φ * i with a logarithmic y-axis for SCAD leastsquares. 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[]	[ "P Daskalopoulos ", "N Sesum ", "\nDepartment of Mathematics\nDepartment of Mathematics\nColumbia University\nNew YorkUSA\n", "\nColumbia University\nNew YorkUSA\n" ]	[ "Department of Mathematics\nDepartment of Mathematics\nColumbia University\nNew YorkUSA", "Columbia University\nNew YorkUSA" ]	[]	We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation ∂u ∂t = ∆ log u on R 2 × R. We show that, under the necessary assumption that for every t ∈ R, the solution u(·, t) defines a complete metric of bounded curvature and bounded width, u is a gradient soliton of the form U (x, t) = 2 β (\|x−x 0 \| 2 +δ e 2βt ) , for some x 0 ∈ R 2 and some constants β > 0 and δ > 0.	10.1155/imrn/2006/83610	[ "https://export.arxiv.org/pdf/math/0603525v2.pdf" ]	119,644,055	math/0603525	21111b8566ed1d74aaf6ac16780e27bba0f55de5	23 Mar 2006 P Daskalopoulos N Sesum Department of Mathematics Department of Mathematics Columbia University New YorkUSA Columbia University New YorkUSA 23 Mar 2006arXiv:math/0603525v2 [math.AP] ETERNAL SOLUTIONS TO THE RICCI FLOW ON R 2 We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation ∂u ∂t = ∆ log u on R 2 × R. We show that, under the necessary assumption that for every t ∈ R, the solution u(·, t) defines a complete metric of bounded curvature and bounded width, u is a gradient soliton of the form U (x, t) = 2 β (\|x−x 0 \| 2 +δ e 2βt ) , for some x 0 ∈ R 2 and some constants β > 0 and δ > 0. Introduction We consider eternal solutions of the logarithmic fast diffusion equation (1.1) ∂u ∂t = ∆ log u on R 2 × R. This equation represents the evolution of the conformally flat metric g ij = u I ij under the Ricci Flow ∂g ij ∂t = −2 R ij . The equivalence follows from the observation that the metric g ij = u I ij has scalar curvature R = −(∆ log u)/u and in two dimensions R ij = 1 2 R g ij . Equation (1.1) arises also in physical applications, as a model for long Van-der-Wals interactions in thin films of a fluid spreading on a solid surface, if certain nonlinear fourth order effects are neglected, see [11,2,3]. Our goal in this paper is to provide the classification of eternal solutions of equation (1.1) under the assumption that for every t ∈ R, the solution u(·, t) defines a complete metric of bounded curvature and bounded width. This result is essential in establishing the type II collapsing of complete (maximal) solutions of the Ricci flow (1.1) on R 2 × [0, T ) with finite area R 2 u(x, t) dx < ∞ (c.f. in [9] for the result in the radially symmetric case). For an extensive list of results on the Cauchy problem u t = ∆ log u on R 2 × [0, T ), u(x, 0) = f (x) we refer the reader to [8], [12], [19] and [22]. * : Partially supported by the NSF grants DMS-01-02252, DMS-03-54639 and the EPSRC in the UK. 1 In [10] we introduced the width w of the metric g = u I ij . Let F : R 2 → [0, ∞) denote a proper function F , such that F −1 (a) is compact for every a ∈ [0, ∞). The width of F is defined to be the supremum of the lengths of the level curves of F , namely w(F ) = sup c L{F = c}. The width w of the metric g is defined to be the infimum w(g) = inf F w(F ). We will assume, throughout this paper, that u is smooth, strictly positive and satisfies the following conditions: The width of the metric g(t) = u(·, t) I ij is finite, namely (1.2) w(g(t)) < ∞, ∀t ∈ R. The scalar curvature R satisfies the L ∞ -bound (1.3) R(·, t) L ∞ (R 2 ) < ∞, ∀t ∈ R. Our goal is to prove the following classification result. (1.4) U (x, t) = 2 β (\|x − x 0 \| 2 + δ e 2βt ) for some x 0 ∈ R 2 and some constants β > 0 and δ > 0. Under the additional assumptions that the scalar curvature R is globally bounded on R 2 × R and assumes its maximum at an interior point (x 0 , t 0 ), with −∞ < t 0 < +∞, i.e., R(x 0 , t 0 ) = max (x,t)∈R 2 ×R R(x, t), Theorem 1.1 follows from the result of R. Hamilton on eternal solutions of the Ricci Flow in [15]. However, since in general ∂R/∂t ≥ 0, without this rather restrictive assumption on the maximum curvature, Hamilton's result does not apply. Before we begin with the proof of Theorem 1.1, let us give a few remarks. Remarks: (i) The bounded width assumption (1.2) is necessary. If this condition is not satisfied, then (1.1) admits other solutions, in particular the flat (constant) solutions. (ii) It is shown in [10] that maximal solutions u of the initial value problem u t = ∆ log u on R 2 × [0, T ), u(x, 0) = f (x) which vanish at time T < ∞ satisfy the width bound c (T − t) ≤ w(g(t)) ≤ C (T − t) and the maximum curvature bound c (T − t) −2 ≤ R max (t) ≤ C (T − t) −2 for some constants c > 0 and C < ∞, independent of t. Hence, one may rescale u near t → T and pass to the limit to obtain an eternal solution of equation (1.1) which satisfies the bounds (1.2) and (1.3) (c.f. in [9] for the radially symmetric case). Theorem 1.1 provides then a classification of the limiting solutions. (iii) Since u is strictly positive at all t < ∞, it follows that u(·, t) must have infinite area, i.e., (1.5) R 2 u(x, t) dx = +∞, ∀t ∈ R. Otherwise, if R 2 u(x, t) dx < ∞, for some t < ∞, then by the results in [8] the solution u must vanish at time t + T , with T = (1/4π) R 2 u(x, t) dx, or before. (iv) The proof of Theorem 1.1 only uses that actually u is an ancient solution of equation u t = ∆ log u on R 2 × (−∞, T ), for some T < ∞, such that R = −u t /u ≥ 0. Since, R evolves by R t = ∆ g R + R 2 the strong maximum principle guarantees that R > 0 or R ≡ 0 at all times. Solutions with R ≡ 0 (flat) violate condition (1.2). Hence, R > 0 on R 2 × R. A priori estimates We will establish in this section the asymptotic behavior, as \|x\| → ∞, for any eternal solution of equation (1.1) which satisfies the conditions (1.2)-(1.3). We will show that there exists constants c(t) > 0 and C(t) < ∞ such that (2.1) c(t) \|x\| −2 ≤ u(x, t) ≤ C(t) \|x\| −2 , t ∈ R. This bound is crucial in the proof of Theorem 1.1. We begin with the following lower bound which is a consequence of the results in [22]. (2.2) 1 u(x, t) ≤ O r 2 log 2 r , as r = \|x\| → ∞, ∀t ∈ R. Proof. For t 0 ∈ R fixed, letū denote the maximal solution of the Cauchy problem (2.3)     ū t = ∆ logū, on R 2 × (0, ∞) u(x, 0) = u(x, t 0 ) x ∈ R 2 . It follows by the results of Rodriguez, Vazquez and Esteban in [12] that u satisfies the growth condition (2.4) 1 u(x, t) ≤ O r 2 log 2 r 2t , as r = \|x\| → ∞, ∀t > 0. In particular,ū defines a complete metric. Let us denote byR the curvature of the metricḡ(t) =ū I ij . Claim 2.2. There exists τ > 0 for which sup R 2 \|R(·, t))\| ≤ C(τ ) for t ∈ [0, τ ]. ¿From the claim the proof of the Proposition readily follows by the uniqueness result of Chen and Zhu ( [6]; see also [22]). Indeed, since both u andū define complete metrics with bounded curvature, the uniqueness result in [6] implies that u(x, t) = u(x, t + t 0 ), for t ∈ [0, τ ]. Hence, u satisfies (2.4) which readily implies (2.2), since u is decreasing in time. Proof of Claim 2.2. Sinceḡ(0) = g(t 0 ) and 0 <R(·, 0) = R(·, t 0 ) ≤ C 0 the classical result of Klingenberg (see [14]) implies the injectivity radius bound r 0 = injrad(R 2 ,ḡ(0)) ≥ π C0 . Moreover, (2.5) Volḡ (0) Bḡ (0) (x, r) ≥ V C0 (r), where V C0 (r) is the volume of a ball of radius r in a space form of constant sectional curvature C 0 (see [20] for (2.5)). We will prove the desired curvature bound using (2.5) and Perelman's pseudolocality theorem (Theorem 10.3 in [21]). Let ǫ, δ > 0 be as in the pseudolocality theorem. Choose r 1 < r 0 such that for all r ≤ r 1 , we have V C0 (r) ≥ (1−δ) r 2 . By (2.5), for all x 0 ∈ R 2 we have Volḡ (0) Bḡ (0) (x 0 , r) ≥ (1−δ)r 2 . Since also sup R 2 \|R(·, 0)\| ≤ C 0 , the pseudolocality theorem implies the bound \|R(x, t)\| ≤ (ǫr 1 ) −2 whenever 0 ≤ t ≤ (ǫr 1 ) 2 , dist t (x, x 0 ) < ǫr 1 . Since the previous estimate does not depend on x, we obtain the uniform bound sup R 2 \|R(·, t)\| ≤ (ǫr 1 ) −2 ∀t ∈ [0, (ǫr 1 ) 2 ] finishing the proof of the claim. We will next perform the cylindrical change of coordinates, setting (2.6) v(s, θ, t) = r 2 u(r, θ, t), s = log r where (r, θ) denote polar coordinates. It is then easy to see that the function v satisfies the equation (2.7) v t = (log v) ss + (log v) θθ , for (s, θ, t) ∈ C ∞ × R with C ∞ denoting the infinite cylinder C ∞ = R×[0, 2π]. Notice that the nonnegative curvature condition R ≥ 0 implies that (2.8) ∆ c log v := (log v) ss + (log v) θθ ≤ 0 namely that log v is superharmonic in the cylindrical (s, θ) coordinates. Estimate (2.1) is equivalent to: Lemma 2.3. For every t ∈ R, there exist constants c(t) > 0 and C(t) < ∞ such that (2.9) c(t) ≤ v(s, θ, t) ≤ C(t), (s, θ) ∈ C ∞ . The proof of Lemma 2.3 will be done in several steps. We will first establish the bound from below which only uses that the curvature R ≥ 0 and that the metric is complete. Proposition 2.4. If u(x, t) is a maximal solution of (1.1) that defines a metric of positive curvature, then for every t ∈ R, there exists a constant c(t) > 0 such that (2.10) v(s, θ, t) ≥ c(t), (s, θ) ∈ C ∞ . Proof. Fix a t ∈ R. We will show that (log v) − = max(− log v, 0) is bounded above. We begin by observing that ∆ c (log v) − = (log v) − ss + (log v) − θθ ≥ 0 since ∆ c log v = −R v ≤ 0. Hence, setting V − (s, t) = 2π 0 (log v) − (s, θ, t) dθ the function V − satisfies V − ss ≥ 0, i.e., V − s is increasing in s. It follows that the limit γ = lim s→∞ V − s (s, t) ∈ [0, ∞] exists. If γ > 0, then V − (s, t) > γ 1 s, with γ 1 = γ/2 for s ≥ s 0 sufficiently large, which contradicts the pointwise bound (2.2), which when expressed in terms of v gives v(s, t) ≤ c(t)/s 2 , for s ≥ s 0 sufficiently large. We conclude that γ = 0. Since V − s is increasing, this implies that V − s ≤ 0, for all s ≥ s 0 , i.e. V − (s, t) is decreasing in s and therefore bounded above. We will now derive a pointwise bound on (log v) − . Fix a point (s,θ) in cylindrical coordinates, withs ≥ s 0 + 2π and let B 2π = {(s, θ) : \|s −s\| 2 + \|θ −θ\| 2 ≤ (2π) 2 }. By the mean value inequality for sub-harmonic functions, we obtain (log v) − (s,θ, t) ≤ 1 \|B 2π \| B2π (log v) − (s, θ, t) ds dθ. Since B 2π ⊂ Q = {\|s −s\| ≤ 2π, θ ∈ [−4π, 4π]}, we conclude the bound (log v) − (s,θ, t) ≤ C Q (log v) − (s, θ, t) ds dθ ≤ 4C s+2π s−2π V − (s, t) ds ≤C finishing the proof. The estimate from above on v will be based on the following integral bound. Proposition 2.5. Under the assumptions of Theorem 1.1, for every t ∈ R, we have (2.11) sup s≥0 2π 0 (log v(s, θ, t)) + dθ < ∞. Proof. Fix t ∈ R and define V (s, t) = 2π 0 log v(s, θ, t) dθ. Since ∆ c log v ≤ 0, V satisfies V ss ≤ 0, i.e. V s decreases in s. Set γ = lim s→∞ V s ∈ [−∞, ∞). Claim 2.6. We have γ = 0, i.e. V is increasing in s, and lim s→∞ V (s, t) < ∞. Proof of Claim: We consider the following two cases: Case 1: γ ≥ 0. Then since V s decreases in s, we have V s ≥ 0 for all s, which implies that there is β = lim s→∞ V (s, t) ∈ (−∞, ∞]. If γ > 0, there is s 0 such that for s ≥ s 0 , (2.12) 2π 0 log v(s, θ, t) dθ ≥ γ 1 s, where γ 1 = γ/2 . (1a) If β < ∞, then for s >> 1, V (s, t 0 ) ≤ β, which contradicts (2.12) for big s, unless γ = 0. log v(s,θ, t) ≥ 1 \|B 2π \| B2π log v(s, θ, t) ds dθ. Expressing log v = (log v) + − (log v) − , we observe that (log v) − (s, θ, t) ≤ C(t) + 2 log s, on B 2π from the bound (2.2). Since Q = {\|s −s\| ≤ 1, \|θ −θ\| ≤ π} is contained in B 2π , we obtain B2π (log v) + ≥ s+1 s−1 θ +π θ−π (log v) + = s+1 s−1 2π 0 (log v) + ≥ V (s, t) − C(t) + 2 log s. Combining the above with (2.12) yields the bound (2.13) log v(s,θ, t) ≥ γ 1 s − 2C(t) − 4 log s ≥γs fors >> 1 andγ < γ 1 . We conclude that (2.14) v(s,θ, t) ≥ eγs → ∞, ass → ∞. We will next show that (2.14) actually contradicts our bounded width conditionw(ḡ(t 0 )) < ∞. Indeed, let F : R 2 → [0, ∞) be a proper function on the plane. Denoting by L g {F = c} the length of the c-level curve σ c of F , measured with respect to metric g(t) = u(·, t) I ij , and using the bound (2.13), namely u(x, t)\|x\| 2 = v(log \|x\|, θ, t 0 ) ≥ eγ log \|x\| , we obtain L g (σ c ) = σc √ u dσ c ≥ e min x∈σc ((γ/2−1) log \|x\|) L eucl (σ c ). Ifγ ≥ 2, then denoting by \|x c \| = min x∈σc \|x\| we have L g (σ c ) ≥ e (γ/2−1) log \|xc\| L eucl (σ c ) ≥ e (γ/2−1) log \|xc\| 2π\|x c \| = \|x c \|γ /2 → ∞, as c → ∞ where we have used the fact that the euclidean circle centred at the origin of radius \|x c \| is contained in the region bounded by the curve σ c . If 0 <γ < 2, then α = 1 −γ/2 > 0 and since we may assume that the origin is contained in the interior of the region bounded by the level curve σ c , denoting by \|x c \| = max x∈σc \|x\|, we obtain L g (σ c ) ≥ e −α log \|xc\| L eucl (σ c ) ≥ 1 \|x c \| α \|x c \| = \|x c \| 1−α → ∞ as c → ∞. Here we have used the fact that L eucl (σ c ) ≥ diam eucl (Intσ c ) ≥ max x∈σc \|x\| = \|x c \|. We conclude that So far we have proven that if γ ≥ 0, the only posibility is that γ = 0 and β < ∞. Case 2: γ < 0. Then, there is s 0 such that for s ≥ s 0 , (2.15) 2π 0 log v(s, θ, t)dθ < −\|γ\|s. On the other hand, by (2.2) we have V (s, θ, t) ≥ C(t) s 2 , for s >> 1 which implies −\|γ\|s > 2π 0 log v dθ ≥ 2π log(C(t)) − 4π log s =C(t) − 4π log s, which is not possible for large s. We have shown that the only possibility is that γ = 0 (which means V (s, t) is increasing in s) and β < ∞. This finishes the proof of the Claim. We can now finish the proof of the Proposition. Expressing (2.16)      −∆u = f (x) in Ω, u = 0 on ∂Ω, with f ∈ L 1 (Ω). Then for every δ ∈ (0, 4π) we have Ω e (4π−δ)\|u(x)\| \|\|f \|\| L 1 (Ω) dx ≤ 4π 2 δ (diamΩ) 2 . Proof of Lemma 2.3. The proof follows the ideas of Brezis and Merle in [4]. Set w = log v so that ∆w = −R e w with R denoting the scalar curvature. Fix ǫ ∈ (0, 1). Since R 2 R e w < ∞, there is s 0 such that for all s ≥ s 0 s+2π s−2π 2π 0 R e w dθ dρ < ǫ. Fixs > s 0 and set B 2π (s,θ) = {(s, θ)\| (s−s) 2 +(θ−θ) 2 ≤ (2π) 2 } ⊂ {(s, θ)\| \|s−s\| ≤ 2π, \|θ −θ\| ≤ 2π} = Q(s,θ) so that B2π(s,θ) R e w < ǫ. We shall denote by B 2π the ball B 2π (s,θ) and by Q the cube Q(s,θ). Let w 1 solve problem (2.16) on Ω = B 2π with f = Re w ∈ L 1 (B 2π ) and f L 1 (B2π ) < ǫ. By Theorem 2.7 there exists a constant C > 0 for which B2π e (4π−δ)\|w 1 (x)\| \|\|f \|\| 1 ≤ C δ . Taking δ = 4π − 1 we obtain (2.17) B2π e (4π−δ)\|w 1 (x)\| ǫ ≤ C. Combining (2.17) and Jensen's inequality gives the estimate (2.18) \|\|w 1 \|\| L 1 (B2π ) ≤C(ǫ). The difference w 2 = w − w 1 satisfies ∆w 2 = 0 on B 2π . Hence by the mean value inequality (2.19) \|\|w + 2 \|\| L ∞ (Bπ) ≤ C\|\|w + 2 \|\| L 1 (B2π ) . Since w + 2 ≤ w + + \|w 1 \| combining (2.18) and (2.11) yields \|\|w + 2 \|\| L 1 (B2π) ≤ C. Expressing Re w = R e w2 e w1 and observing that • R is bounded and (2.19) holds on B π • e w1 ∈ L 1 ǫ (B 2π ), where ǫ can be chosen small, so that 1 ǫ > 1, and • −∆w = Re w ≤ C e w2 by standard elliptic estimates we obtain the bound (2.20) \|\|w + \|\| L ∞ (B π/2 ) ≤ C \|\|w + \|\| L 1 (Bπ ) + C \|\|e w1 \|\| L p (Bπ ) ≤C, where p > 1, finishing the proof of the Lemma. Fix τ ∈ R and define the inifinite cylinder Q + (τ ) = {(s, θ, t) : s ≥ 0, θ ∈ [0, 2π], τ − 1 ≤ t ≤ τ }. By Lemma 2.9 (2.21) 0 < c(τ ) ≤ v(s, θ, t) ≤ C(τ ) < ∞, on Q + (τ\|v s \| ≤ C s , \|v ss \| ≤ C s 2 , \|v θ \| ≤ C, \|v θs \| ≤ C s , on Q + (τ ). We will next show that the curvature R(x, t) = −(∆ log u(x, t))/u(x, t) tends to zero, as \|x\| → ∞. We begin by reviewing the Harnack inequality satisfied by the curvature R, shown by R. Hamilton [18] and [16]. In the case of eternal solutions u of (1.1) which define a complete metric, it states as (2.22) ∂ log R ∂t ≥ \|D g log R\| 2 . Since, D g R = u −1 DR, equivalently, this gives the inequality (2.23) ∂R ∂t ≥ \|DR\| 2 R u . Let (x 1 , t 1 ), (x 2 , t 2 ) be any two points in R 2 × R, with t 2 > t 1 . Integrating (2.23) along the path x(t) = x 1 + t−t1 t2−t1 x 2 , also using the bound u(x, t) ≤ C(t 1 )/\|x\| 2 , holding for all t ∈ [t 1 , t 2 ] by (1.2) and the fact that u t ≤ 0, we find the more standard in PDE Harnack inequality (2.24) R(x 2 , t 2 ) ≥ R(x 1 , t 1 ) e −C ( \|x 2 −x 1 \| 2 \|x 1 \| 2 (t 2 −t 1 ) ) . One may now combine (2.24) with Lemmas 2.3 and 2.8 to conclude the following: Proof. For any t ∈ R and r > 0, we denote byR(r, t) the spherical average of R, namelyR (r, t) = 1 \|∂B r \| ∂Br R(x, t) dσ. We first claim that (2.25) lim r→∞R (r, t) = 0, ∀t. Indeed, using the cylindrical coordinates, introduced previously, this claim is equiv- alent to showing that lim s→∞ 2π 0 − (log v) ss (s, θ, t) + (log v) θθ (s, θ, t) v dθ = 0 which is equivalent, using Lemma 2.3, to showing that lim s→∞ 2π 0 −(log v) ss (s, θ, t) dθ = 0. But this readily follows from Lemmas 2.3 and 2.8. Fix t ∈ R. To prove that lim \|x\|→∞ R(x, t) = 0, we use the Harnack inequality (2.24) to show that R(x, t) ≤ C inf y∈∂Br R(y, t + 1) ≤ CR(r, t + 1), ∀x ∈ ∂B r , ∀t ∈ R and use (2.25). Combining the above with classical derivative estimates for linear strictly parabolic equations, gives the following. Proof. For any ρ > 1 we setR(x, t) = R(ρ x, t) and we compute from the evolution equation R t = u −1 ∆R + R 2 of R, that R t = (ρ 2 u) −1 ∆R +R 2 . For τ < T consider the cylinder Q = {(r, t) : 1/2 ≤ \|x\| ≤ 4, τ − 1 ≤ t ≤ τ }. ¿From (2.1) we have 0 < c(τ ) ≤ ρ 2 u(x, t) ≤ C(τ ) < ∞, for all x ∈ Q, henceR satisfies a uniformly parabolic equation in Q. Classical derivative estimates then imply that \|(R) r (x, t)\| ≤ C R L ∞ (Q) for all 1 ≤ \|x\| ≤ 2, τ − 1/2 ≤ t ≤ τ , implying in particular that ρ \|R r (x, τ )\| ≤ C R L ∞ (Qρ) for all ρ ≤ \|x\| ≤ 2 ρ, where Q ρ = {(x, t) : ρ/2 ≤ \|x\| ≤ 4ρ, τ − 1 ≤ t ≤ τ }. The proof now follows from Lemma 2.9. 3. Proof of Theorem 1.1 Most of the computations here are known in the case that u (dx 2 1 + dx 2 2 ) defines a metric on a compact surface (see for example in [7]). However, in the non-compact case an exact account of the boundary terms at infinity should be made. We begin by integrating the Harnack inequality R t ≥ \|DR\| 2 /Ru with respect to the measure dµ = u dx. Since the measure dµ has infinite area, we will intergrate over a fixed ball B ρ . At the end of the proof we will let ρ → ∞. Using also that R t = u −1 ∆R + R 2 we find Bρ ∆R dx + Bρ R 2 u dx ≥ Bρ \|DR\| 2 R dx and by Green's Theorem we conclude (3.1) Bρ \|DR\| 2 R dx − Bρ R 2 udx ≤ ∂Bρ ∂R ∂ν dσ. Next, following Chow ([7]), we consider the vector X = ∇R + R ∇f , where f = − log u is the potential function (defined up to a constant) of the scalar curvature, since it satisfies ∆ g f = R, with ∆ g f = u −1 ∆f denoting the Laplacian with respect to the conformal metric g = u (dx 2 + dy 2 ). As it was observed in [7] X ≡ 0 on Ricci solitons, i.e., Ricci solitons are gradient solitons in the direction of ∇ g f . A direct computation shows Bρ \|X\| 2 R dx = Bρ \|DR\| 2 R dx + 2 Bρ ∇R · ∇f dx + Bρ R \|Df \| 2 dx. Integration by parts implies Bρ ∇R · ∇f dx = − Bρ R ∆f dx + ∂Bρ R ∂f ∂n dσ = − Bρ R 2 u dx + ∂Bρ R ∂f ∂n dσ since ∆f = R u. Hence Bρ \|X\| 2 R dx = Bρ \|DR\| 2 R dx − 2 Bρ R 2 u dx + Bρ R \|Df \| 2 dx + 2 ∂Bρ R ∂f ∂n dσ. (3.2) Combining (3.1) and (3.2) we find that (3.3) Bρ \|X\| 2 R dx ≤ − Bρ R 2 u dx − Bρ R \|Df \| 2 dx + I ρ = −M + I ρ where I ρ = ∂Bρ ∂R ∂n dσ + 2 ∂Bρ R ∂f ∂n dσ. Lemmas 2.8 -2.10 readily imply that (3.4) lim ρ→∞ I ρ = 0. As in [7], we will show next that M ≥ 0 and indeed a complete square which vanishes exactly on Ricci solitons. To this end, we define the matrix M ij = D ij f + D i f D j f − 1 2 (\|Df \| 2 + R u) I ij with I ij denoting the identity matrix. A direct computation shows that M ij = ∇ i ∇ j f − 1 2 ∆ g f g ij , with ∇ i denoting covariant derivatives. It is well known that the Ricci solitons are characterized by the condition M ij = 0, (see in [16]). Claim: (3.5) M := Bρ R 2 u dx − Bρ R \|Df \| 2 dx = 2 Bρ \|M ij \| 2 1 u dx + J ρ where lim ρ→∞ J ρ = 0. To prove the claim we first observe that since ∆f = Ru Bρ R 2 u = Bρ (∆f ) 2 u dx = Bρ D ii f D jj f 1 u dx. Integrating by parts and using again that ∆f = Ru, we find Bρ D ii f D jj f 1 u dx = − Bρ D jii f D j f 1 u dx + Bρ ∆f D j f D j u u 2 dx + ∂Bρ R ∂f ∂n dσ. Integrating by parts once more we find Bρ D jii f D j f 1 u dx = − Bρ \|D ij f \| 2 1 u dx + Bρ D ij f D j f D i u u 2 dx + 1 2 ∂Bρ ∂(\|Df \| 2 ) ∂n 1 u dσ since ∂Bρ D ij f D j f n i 1 u dσ = 1 2 ∂Bρ ∂(\|Df \| 2 ) ∂n 1 u dσ. Combining the above and using that Df = −u −1 Du and ∆f = R u we conclude (3.6) Bρ R 2 u dx = Bρ \|D ij f \| 2 1 u dx + Bρ D ij f D i f D j f 1 u dx − Bρ R \|Df \| 2 dx + J 1 ρ where J 1 ρ = ∂Bρ R ∂f ∂n dσ − 1 2 ∂Bρ ∂(\|Df \| 2 ) ∂n 1 u dσ. Hence (3.7) M = Bρ \|D ij f \| 2 1 u dx + Bρ D ij f D i f D j f 1 u dx − 2 Bρ R \|Df \| 2 dx + J 1 ρ . We will now intergrate \|M ij \| 2 . A direct computation and ∆f = Ru imply Bρ \|M ij \| 2 1 u dx = Bρ \|D ij f \| 2 1 u dx + 2 Bρ D ij f D i f D j f 1 u dx − Bρ R \|Df \| 2 dx + 1 2 Bρ \|Df \| 4 1 u dx − 1 2 Bρ R 2 u dx. (3.8) Combining (3.7) and (3.8) we then find M − 2 Bρ \|M ij \| 2 1 u dx = − Bρ \|D ij f \| 2 1 u dx − 3 Bρ D ij f D i f D j f 1 u dx − Bρ \|Df \| 4 1 u dx + Bρ R 2 u dx + J 1 ρ . Using (3.6) we then conclude that M − 2 Bρ \|M ij \| 2 1 u dx = −2 Bρ D ij f D i f D j f 1 u dx − Bρ \|Df \| 4 1 u dx − Bρ R \|Df \| 2 dx + J 2 ρ . (3.9) where J 2 ρ = ∂Bρ R ∂f ∂n dσ − ∂Bρ ∂(\|Df \| 2 ) ∂n 1 u dσ. We next observe that 2 Bρ D ij f D i f D j f 1 u dx = Bρ D i (\|Df \| 2 ) D i f 1 u and integrate by parts using once more that ∆f = R u and that D i f = −u −1 D i f , to find 2 Bρ D ij f D i f D j f 1 u dx = − Bρ R \|Df \| 2 dx − Bρ \|Df \| 4 1 u dx + J 3 ρ where J 3 ρ = lim ρ→∞ ∂Bρ \|Df \| 2 ∂f ∂n dσ. Combining the above we conclude that M − 2 Bρ \|M ij \| 2 1 u dx = J ρ with J ρ = ∂Bρ R ∂f ∂n dσ − ∂Bρ ∂(\|Df \| 2 ) ∂n + \|Df \| 2 ∂f ∂n 1 u dσ. We will now show that lim ρ→∞ J ρ = 0. Clearly the first term tends to zero, because ∂Bρ \|∂f /∂n\| dσ is bounded Lemma 2.8 and R(x, t) → 0, as \|x\| → ∞, by Lemma 2.9. It remains to show that We first observe that since f = − log u, we have ∂(\|Df \| 2 ) ∂n + \|Df \| 2 ∂f ∂n 1 u = ∂ ∂n \|D log u\| 2 u . Expressing the last term in cylindrical coordinates, setting v(s, θ, t) = r 2 u(r, θ, t), with s = log r, we find ∂Bρ ∂ ∂n \|D log u\| 2 u 1 u dσ = 2π 0 ∂ ∂s [2 − (log v) s ] 2 + [(log v) θ ] 2 v dθ. Further computation shows that ∂ ∂s [2 − (log v) s ] 2 + [(log v) θ ] 2 v = −2 [2 − (log v) s ] (log v) ss v + 2 (log v) θ (log v) θs v + {[2 − (log v) s ] 2 + [(log v) θ ] 2 } (log v) s . By Lemma 2.8 , (log v) θ is bounded as s → ∞, while (log v) s , (log v) θs and (log v) ss tend to zero, as s → ∞. Using also that v is bounded away from zero as s → ∞, we finally conclude that lim s→∞ 2π 0 ∂ ∂s [2 − (log v) s ] 2 + [(log v) θ ] 2 v dθ = 0 implying (3.10) therefore finishing the proof of the claim (3.5). We will now conclude the proof of the Theorem. From This immediately gives that X ≡ 0 and M ij ≡ 0 for all t showing that U is a gradient soliton. It has been shown by L.F. Wu [23] that there are only two types of complete gradient solitons on R 2 , the standard flat metric (R ≡ 0) which is stationary, and the cigar solitons (1.1). The flat solitons violate condition (1.2). Hence, u must be of the form (1.4), finishing the proof of the Theorem. Theorem 1. 1 . 1Assume that u is a positive smooth eternal solution of equation (1.1) which defines a complete metric and satisfies conditions (1.2)-(1.3). Then, u is a gradient soliton of the form R 2 u 2(x, t) dx = +∞. (v) Any eternal solution of equation (1.1) satisfies u t ≥ 0. This is an immediate consequence of the Aronson-Bénilan inequality (or the maximum principle on R = −u t /u), which in the case of a solution on R 2 × [τ, t) states as u t ≤ u/(t − τ ). Letting, τ → −∞, we obtain for an eternal solution the bound u t ≤ 0. (vi) Any eternal solution of equation (1.1) satisfies R > 0. By the previous remark, Proposition 2. 1 . 1Assume that u is a positive smooth eternal solution of equation (1.1) which defines a complete metric and satisfies condition (1.3). Then, ( 1b ) 1bIf β = ∞ and γ > 0 we will derive a contradiction using the boundness ofthe width. The function log v satisfies ∆ c log v = −R v ≤ 0 that is, log v isthe superharmonic function. Fix a point (s,θ) in cylindrical coordinates, withs ≥ s 0 and let B 2π = {(s, θ) : \|s −s\| 2 + \|θ −θ\| 2 ≤ (2π) 2 }. By the mean value inequality for superharmonic functions, we obtain {F = c} ≥ M for any proper function F , which implies that w(g(t)) ≥ M , contradicting our width bound (1.2). (1c) If β = ∞ and γ = 0, we can argue similarly as in (1b), with the only difference that now V (s, t) increases in s and V (s, t) >> M for s >> 1, where M is an arbitrarily large constant. The mean value property applied to log v now shows that log v can be made arbitrarily large for s >> 1. The rest of the argument is the same as in (1b). 2π 0 (( 0log v) + (s, θ, t) dθ = V (s, t) + 2π 0 (log v) − (s, θ, t) dθand using that V (s, t) ≤ β (as shown in Claim 2.6) together with Proposition 2.log v) + (s, θ, t) dθ ≤ β + 2πδ(t) < ∞ as desired.To prove Lemma 2.3, it only remains to show that v(·, t) L ∞ < ∞. The proof of this bound will use the ideas ofBrezis and Merle ([4]), including the following result which we state for the reader's convenience. Theorem 2. 7 ( 7Brezis-Merle). Assume Ω ⊂ R 2 is a bounded domain and let u be a solution of Lemma 2. 9 . 9Under the assumptions of Theorem 1.1 we have lim \|x\|→∞ R(x, t) = 0, ∀t ∈ R. Lemma 2 . 10 . 210Under the assumptions of Theorem 1.1 the radial derivative R r of the curvature satisfies lim \|x\|→∞ \|x\| R r (x, t) = 0, ∀t ∈ R. Acknowledgments. We are grateful to S. Brendle, B. Chow, R. Hamilton, L.Ni, Rafe Mazzeo and F. Pacard for enlightening discussions in the course of this work. 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[ "Poincaré Sphere and Decoherence Problems", "Poincaré Sphere and Decoherence Problems" ]	[ "Y S Kim [email protected] \nCenter for Fundamental Physics\nUniversity of Maryland\n20742College Park, MarylandU.S.A\n" ]	[ "Center for Fundamental Physics\nUniversity of Maryland\n20742College Park, MarylandU.S.A" ]	[]	Henri Poincaré formulated the mathematics of the Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be combined into one mathematical device which can address the internal space-time symmetries of elementary particles, decoherence problems in polarization optics, entropy problems, and Feynman's rest of the universe.	null	[ "https://arxiv.org/pdf/1203.4539v3.pdf" ]	52,066,187	1203.4539	b6e56c13f8bfca750ea0cf63a3b09f88f962fd79	Poincaré Sphere and Decoherence Problems Y S Kim [email protected] Center for Fundamental Physics University of Maryland 20742College Park, MarylandU.S.A Poincaré Sphere and Decoherence Problems Presented at the Fedorov Memorial Symposium: International Conference "Spins and Photonic Beams at Interface," dedicated to the 100th anniversary of F.I.Fedorov (1911-1994) (Minsk, Belarus, 2011) 1 Henri Poincaré formulated the mathematics of the Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be combined into one mathematical device which can address the internal space-time symmetries of elementary particles, decoherence problems in polarization optics, entropy problems, and Feynman's rest of the universe. Introduction It was Henri Poincarś who worked out the mathematics of Lorentz transformations before Einstein and Minkowski. Poincaré's interest in mathematics covers many other areas of physics. Among them, his Poincaré sphere serves as geometrical representation of the Stokes parameters. While there are four Stokes parameters, the traditional Poincaré sphere is based on only three of those parameters [1,2,3]. In this report, we show that the concept of the Poincaré sphere can be extended to accommodate all four Stokes parameters, and show that this extended Poincaré sphere can be used as a representation of the Lorentz group applicable to the four-component Minkowski space. In this way, it is possible to use the Poincarś sphere as a picture of the internal space-time symmetries as defined by Wigner in 1939 [4]. Throughout the paper, we shall use the two-by-two matrix formulation of the Lorentz group [5,6]. The basic advantage is that this representation is the natural language for the two-by-two representation of the four Stokes parameters. Indeed, this two-by-two representation speaks one language applicable to two different branches of physics, as is illustrated in Fig. 1. Let us consider a particle moving along a given direction. We can rotate the system around this direction, and boost along this direction. We can also rotate around and boost along two orthogonal directions perpendicular to the direction of the momentum. These operations can be written in the form of two-by-two matrices. In this way, we can construct a two-by-two representation of the Lorentz group. In this report, we use the same set of two-by-two matrices to perform transformations on the Stokes parameters and the Poincaré sphere. While the original three-dimensional Poincaré sphere is based only on three of the four Stokes parameters, it is possible to extend its geometry to accommodate all four parameters. With these parameters, we can address the degree of polarization between the two orthogonal components of electric field perpendicular to the momentum. In order to deal with this problem, we introduce the decoherence parameter derivable from the traditional degree of polarization. Since this parameter is greater than zero and smaller than one, we can write it as sin χ, and define χ as the decoherence angle. One remarkable aspect of this decoherence parameter is that it is invariant under Lorentz transformations. If the symmetry of the Poincaré sphere is translated into the space-time symmetry, the decoherence parameter remains Lorentz-invariant like the particle mass in the space-time symmetry. Furthermore, to every sin χ, there is cos χ, and their symmetry is well known. Thus we can consider another Poincaré sphere with the same decoherence angle. The second Poincaré sphere could serve as another illustrative example of Feynman's rest of the universe [7,8]. Figure 1: One mathematical instrument for two different branches of physics. As one second-order differential equation takes care of the resonances in the oscillator system and also in the LCR circuit, the Lorentz group can serve as the basic mathematical language for special relativity and modern optics. In Sec. 2, it was noted that Poincaré was the first one to formulate group of Lorentz transformations for the space-time variables. This transformation law can be translated into the language of Naimark's two-by-two Naimark representation. Einstein later derived his E = mc 2 by advancing the concept of the energy-momentum four-vector obeying the same transformation law as that of the space-time coordinates. Thus, the two-by-two representation is possible also for the momentum-energy four-vector. When Einstein was formulating his special relativity, he did not consider internal space-time structures or symmetries of the particles. Also in Sec. 2, it is shown possible to study Wigner's little groups using the two-by-two representation. Wigner's little groups dictate the internal space-time symmetries of elementary particles. They are the subgroups of the Lorentz group whose transformations leave the four-momentum of the given particle invariant [4,9]. In Sec. 3, we first note that the same two-by-two matrices are applicable to two-component Jones vectors. However, the Jones vector cannot address the problem of coherence between two transverse components of the optical beam. This is why the four-component Stokes parameters are needed. The same Lorentz group can be used for the four-component Stokes parameters, and Stokes parameters are like four-vectors, as in the case of energymomentum four vectors. It is noted however, the degree of coherence remains invariant under Lorentz transformations. In Sec. 4, we use the geometry of the Poincaré sphere to describe the Stokes parameters and their transformation properties. It is shown that three different radii are needed to describe them fully. The traditional radius lies between the maximum and minimum radii. It is shown that (S 2 0 − R 2 ) remains lorentz-invariant, where S 0 and R are the maximum radius and the traditional radius respectively. This Lorentz-invariant quantity is dictated by the decoherence parameter. The entropy question is also discussed, and this entropy is shown to be Lorentz-invariant. In Sec. 5, it is shown possible to define another Poincaré sphere in order to deal with variations of the decoherence parameter and the entropy. It is shown that this second Poincaré sphere can serve as another example of Feynman's rest of the universe [7]. In the the Appendix, we shoe how the two-by-two transformation matrix can be translated into the four-by-four matrix applicable to the Missourian four-vector. Poincaré Group, Einstein, and Wigner The Lorentz group starts with a group of four-by-four matrices performing Lorentz transformations on the Minkowskian vector space of (t, z, x, y), leaving the quantity t 2 − z 2 − x 2 − y 2(1) invariant. It is possible to perform this transformation using two-by-two representations [5]. This mathematical aspect is known as the SL(2, c) as the universal covering group for the Lorentz group. In this two-by-two representation, we write the four-vector as a matrix X = t + z x − iy x + iy t − z .(2) Then its determinant is precisely the quantity given in Eq. (1). Thus the Lorentz transformation on this matrix is a determinant-preserving transformation. Let us consider the transformation matrix as G = α β γ δ , G † = α * γ * β * δ * ,(3) with det (G) = 1. This matrix has six independent parameters. The group of these G matrices is known to be locally isomorphic to the group of four-by-four matrices performing Lorentz transformations on the four-vector (t, z, x, y). In other word, for each G matrix there is a corresponding four-by-four Lorentz-transform matrix, as is illustrated in the Appendix. The matrix G is not a unitary matrix, because its Hermitian conjugate is not always its inverse. The group can have a unitary subgroup called SU(2) performing rotations on electron spins. As far as we can see, this G-matrix formalism was first presented by Naimark in 1954 [5]. Thus, we call this formalism the Naimark representation of the Lorentz group. We shall see first that this representation is convenient for studying space-time symmetries of particles. We shall then note that this Naimark representation is the natural language for the Stokes parameters in polarization optics. With this point in mind, we can now consider the transformation X ′ = GXG †(5) Since G is not a unitary matrix, it is not a unitary transformation. In order to tell this difference, we call this "Naimark transformation." This expression can be written explicitly as t ′ + z ′ x ′ − iy ′ x + iy t ′ − z ′ = α β γ δ t + z x − iy x + iy t − z α * γ * β * δ * ,(6) For this transformation, we have to deal with four complex numbers. However, for all practical purposes, we may work with two Hermitian matrices Z(δ) = e iδ/2 0 0 e −iδ/2 , R(δ) = cos(θ/2) − sin(θ/2) sin(θ/2) cos(θ/2) ,(7) and two symmetric matrices B(µ) = e µ/2 0 0 e −µ/2 , S(λ) = cosh(λ/2) sinh(λ/2) sinh(λ/2) cosh(λ/2)(8) The two Hermitian matrices in Eq.(7) lead to rotations around the z and y axes respectively. The symmetric matrices in Eq.(8) perform Lorentz boosts along the z and x directions. Repeated applications of these four matrices will lead to the most general form the of the most general form of the G matrix of Eq.(3) with six independent parameters. For each two-by-two Naimark transformation, there is a four-by-four matrix performing the corresponding Lorentz transformation on the four-component four vector. In the appendix, the four-by-four equivalents are given for the matrices of Eq. (7) and Eq. (8). It was Einstein who defined the energy-momentum four vector, and showed that it also has the same Lorentz-transformation law as the space-time fourvector. We write the energy-momentum four vector as P = E + p z p x − ip y p x + ip y E − p z ,(9) with det (P ) = E 2 − p 2 x − p 2 y − p 2 z ,(10)which means det p = m 2 ,(11) where m is the particle mass. Now Einstein's transformation law can be written as P ′ = GMG † ,(12) or explicitly E ′ + p ′ z p ′ x − ip ′ y p ′ x + ip ′ y E ′ − p ′ z = α β γ δ E + p z p x − ip y p x + ip y E − p z α * γ * β * δ * ,(13) Later in 1939 [4], Wigner was interested in constructing subgroups of the Lorentz group whose transformations leave a given four-momentum invariant, and called these subsets "little groups." Thus, Wigner's little group consists of two-by-two matrices satisfying P = W P W † .(14) This two-by-two W matrix is not an identity matrix, but tells about internal space-time symmetry of the particle with a given energy-momentum fourvector. This aspect was not known when Einstein formulated his special relativity in 1905. If its determinant is a positive number, the P matrix can be brought to the form P = 1 0 0 1 ,(15) corresponding to a massive particle at rest If the determinant is negative, it can be brought to the form P = 1 0 0 −1 ,(16) corresponding to an imaginary-mass particle moving faster than light along the z direction, with its vanishing energy component. If the determinant if zero, we may write P as P = 1 0 0 0 ,(17) corresponding to a massless particle moving along the z direction. For all three of the above cases, the matrix of the form Z(δ) = e iδ/2 0 0 e −iδ/2(18) will satisfy the Wigner condition of Eq. (14). This matrix corresponds to rotations around the z axis, as is shown in the Appendix. For the massive particle with the four-momentum of Eq.(15), the Naimark transformations with the rotation matrix of the form R(θ) = cos(θ/2) − sin(θ/2) sin(θ/2) cos(θ/2) ,(19) also leaves the P matrix of Eq.(15) invariant. Together with the Z(δ) matrix, this rotation matrix lead to the subgroup consisting of unitary subset of the G matrices. The unitary subset of G is SU(2) corresponding to the threedimensional rotation group dictating the spin of the particle [9]. For the massless case, the transformations with the triangular matrix of the form 1 γ 0 1 (20) leaves the momentum matrix of Eq. (17) invariant. The physics of this matrix has a stormy history, and the variable γ leads to gauge transformation applicable to massless particles [10,11]. For a superluminal particle with it imaginary mass, the W matrix of the form S(λ) = cosh(λ/2) sinh(λ/2) sinh(λ/2) cosh(λ/2)(21) will leave the four-momentum of Eq. (16) invariant. This unobservable particle does not appear to have observable internal space-time degrees of freedom. Table 1 summarizes the transformation matrices for Wigner's subgroups for massive, massless, and superluminal transformations. Of course, it is a challenging problem to have one expression for all those three cases, and this problem has been addressed in the literature [12]. Jones Vectors and Stokes Parameters In studying the polarized light propagating along the z direction, the traditional approach is to consider the x and y components of the electric fields. Their amplitude ratio and the phase difference determine the state of polarization. Thus, we can change the polarization either by adjusting the amplitudes, by changing the relative phase, or both. For convenience, we call the optical device which changes amplitudes an "attenuator" and the device which changes the relative phase a "phase shifter." The traditional language for this two-component light is the Jones-matrix formalism which is discussed in standard optics textbooks [13]. In this formalism, the above two components are combined into one column matrix with the exponential form for the sinusoidal function ψ 1 (z, t) ψ 2 (z, t) = a exp {i(kz − ωt + φ 1 )} b exp {i(kz − ωt + φ 2 )} .(22) This column matrix is called the Jones vector. When the beam goes through a medium with different values of indexes of refraction for the x and y directions, we have to apply the matrix e iδ 1 0 0 e iδ 2 = e i(δ 1 +δ 2 )/2 e −iδ/2 0 0 e iδ/2 ,(23) with δ = δ 1 −δ 2 . In measurement processes, the overall phase factor e i(δ 1 +δ 2 )/2 cannot be detected, and can therefore be deleted. The polarization effect of the filter is solely determined by the matrix Z(δ) = e iδ/2 0 0 e −iδ/2 ,(24) which leads to a phase difference of δ between the x and y components. The form of this matrix is given in Eq. (7), which serves as the rotation around the z axis in the Minkowski space and time. Also along the x and y directions, the attenuation coefficients could be different. This will lead to the matrix [14] e −µ 1 0 0 e −µ 2 = e −(µ 1 +µ 2 )/2 e µ/2 0 0 e −µ/2(25) with µ = µ 2 − µ 1 . If µ 1 = 0 and µ 2 = ∞, the above matrix becomes 1 0 0 0 ,(26) which eliminates the y component. This matrix is known as a polarizer in the textbooks [13], and is a special case of the attenuation matrix of Eq.(25). This attenuation matrix tells us that the electric fields are attenuated at two different rates. The exponential factor e −(µ 1 +µ 2 )/2 reduces both components at the same rate and does not affect the state of polarization. The effect of polarization is solely determined by the squeeze matrix [14] B(µ) = e µ/2 0 0 e −µ/2 .(27) This diagonal matrix is given in Eq. (8). In the language of space-time symmetries, this matrix performs a Lorentz boost along the z direction. The polarization axes are not always the x and y axes. For this reason, we need the rotation matrix R(θ) = cos(θ/2) − sin(θ/2) sin(θ/2) cos(θ/2) ,(28) which, according to Eq. which is also given in Eq. (8). In the language of space-time physics, this matrix lead to a Lorentz boost along the x axis. Indeed, the G matrix of Eq. (3) is the most general form of the transformation matrix applicable to the Jones matrix. Each of the above four matrices plays its important role in special relativity, as we discussed in Sec. 2. Their respective roles in optics and particle physics are given in Table 2. However, the Jones matrix alone cannot tell whether the two components are coherent with each other. In order to address this important degree of freedom, we use the coherency matrix [1,2] C = S 11 S 12 S 21 S 22 ,(30) with < ψ * i ψ j >= 1 T T 0 ψ * i (t + τ )ψ j (t)dt,(31) where T is for a sufficiently long time interval, is much larger than τ . Then, those four elements become [15] S 11 =< ψ * 1 ψ 1 >= a 2 , S 12 =< ψ * 1 ψ 2 >= ab e −(σ+iδ) , S 21 =< ψ * 2 ψ 1 >= ab e −(σ−iδ) , S 22 =< ψ * 2 ψ 2 >= b 2 .(32) The diagonal elements are the absolute values of ψ 1 and ψ 2 respectively. The off-diagonal elements could be smaller than the product of ψ 1 and ψ 2 , if the two beams are not completely coherent. The σ parameter specifies the degree of coherency. This coherency matrix is not always real but it is Hermitian. Thus it can be diagonalized by a unitary transformation. If this matrix is normalized so that its trace is one, it becomes a density matrix [7]. If we start with the Jones vector of the form of Eq.(22), the coherency matrix becomes C = a 2 ab e −(σ+iδ) ab e −(σ−iδ) b 2 .(33) We are interested in the symmetry properties of this matrix. Since the transformation matrix applicable to the Jones vector is the two-by-two representation of the Lorentz group, we are particularly interested in the transformation matrices applicable to this coherency matrix. The trace and the determinant of the above coherency matrix are det(C) = (ab) 2 1 − e −2σ , tr(C) = a 2 + b 2 .(34) Since e −σ is always smaller than one, we can introduce an angle χ defined as cos χ = e −σ ,(35) and call it the "decoherence angle." If χ = 0, the decoherence is minimum, and it is maximum when χ = 90 o . We can then write the coherency matrix of Eq.(33) as C = a 2 ab(cos χ)e −iδ ab(cos χ)e iδ b 2 .(36) The degree of polarization is defined as [13] Rotation around z cos(θ/2) − sin(θ/2) sin(θ/2) cos(θ/2) Rotation around y. f = 1 − 4 det(C) (tr(C)) 2 = 1 − 4(ab) 2 sin 2 χ (a 2 + b 2 ) 2 .(37) Squeeze along x and y e η/2 0 0 e −η/2 Boost along z. Squeeze along 45 o cosh(λ/2) sinh(λ/2) sinh(λ/2) cosh(λ/2) Boost along x. (ab) 2 sin 2 χ Determinant (mass) 2 This degree is one if χ = 0. When χ = 90 o , it becomes a 2 − b 2 a 2 + b 2 ,(38) Without loss of generality, we can assume that a is greater than b. If they are equal, this minimum degree of polarization is zero. Under the influence of the Naimark transformation given in Eq.(5), this coherency matrix is transformed as C ′ = G C G † = S ′ 11 S ′ 12 S ′ 21 S ′ 22 = α β γ δ S 11 S 12 S 21 S 22 α * γ * β * δ * .(39) It is more convenient to make the following linear combinations. S 0 = S 11 + S 22 √ 2 , S 3 = S 11 − S 22 √ 2 , S 1 = S 12 + S 21 √ 2 , S 2 = S 12 − S 21 √ 2i .(40) These four parameters are called Stokes parameters, and four-by-four transformations applicable to these parameters are widely known as Mueller matrices [1,3]. However, if the Naimark transformation given in Eq.(39) is translated into the four-by-four Lorentz transformations according to the correspondence given in the Appendix, the Mueller matrices constitute a representation of the Lorentz group. Another interesting aspect of the two-by-two matrix formalism is that the coherency matrix can be formulated in terms of the quarternions [16,17]. The quarternion representation can be translated into rotations in the fourdimensional space. There is a long history between the Lorentz group and the four-dimensional rotation group. It would be interesting to see what the quarternion representation of polarization optics will add to this history between those two similar but different groups. As for earlier applications of the two-by-two representation of the Lorentz group, we note the vector representation by Fedorov [18,19]. Fedorov showed that it is easier to carry our kinematical calculations using his two-by-two representation. For instance, the computation of the Wigner rotation angle is possible in the two-by-two representation [20]. Geometry of the Poincaré Sphere We now have the four-vector (S 0 , S 3 , S 1 , S 2 ), which is Lorentz-transformed like the space-time four-vector (t, z, x, y) or the energy-momentum four vector of Eq.(9). This Stokes four-vector has a three-component subspace (S 3 , S 1 , S 2 ), which is like the three-dimensional Euclidean subspace in the four-dimensional Minkowski space. In this three-dimensional subspace, we can introduce the spherical coordinate system with R = S 2 3 + S 2 1 + S 2 2 S 3 = R cos ξ, S 1 = R(sin ξ) cos δ, S 2 = R(sin ξ) sin δ.(41) The radius R is the radius of the traditional Poincaré sphere, and is R = 1 2 (a 2 − b 2 ) 2 + 4(ab) 2 cos 2 χ.(42) with S 3 = a 2 − b 2 2 .(43) This spherical picture of is traditionally known as the Poincaré sphere [1,2,3] The radius R takes its maximum value S 0 when χ = 0 o . It decreases and reaches its minimum value, S 3 , when χ = 90 o . In terms of R, the degree of polarization given in Eq.(37) is f = R S 0 .(44) This aspect of the radius R is illustrated in Fig. 2. Under the Lorentz transformation, all four Stokes parameters are subject to change. The maximum radius S 0 and the minimum radius S 3 do not remain invariant. The radius R is also subject to change. These three radii are illustrated in Fig. 2. While Lorentz transformations shake up these parameters, is there any quantity remaining invariant? Let us go back to the four-momentum matrix of Eq.(9). Its determinant is m 2 and remains invariant. Likewise, the determinant of the coherency matrix of Eq.36 should also remain invariant. The determinant in this case is S 2 0 − R 2 = (ab) 2 sin 2 χ.(45) This quantity remains invariant. This aspect is shown on the last row of Table 2. While the decoherence parameter is not fundamental is influenced by environment, it plays the same mathematical role as in the particle mass which remains as the most fundamental quantity since Isaac Newton, and even after Einstein. the form ρ(χ) = 1 a 2 + b 2 a 2 (ab)e −iδ (cos χ) (ab)e iδ (cos χ) b 2 ,(46) whose trace is one. This matrix can be diagonalized to ρ(χ) = 1 2 1 + f 0 0 1 − f ,(47) where f is the degree of polarization given in Eq.(37), which can also be written as f = (a 2 + b 2 ) 2 − 4(ab) 2 sin 2 χ a 2 + b 2 .(48) As we noted before, this quantity is one when χ = 0 and takes its minimum value when χ = 90 o . Then, the entropy becomes S = − 1 + f 2 ln 1 + f 2 − 1 − f 2 ln 1 − f 2 .(49) The entropy S becomes zero when χ = 0. When χ = 90 o , the entropy takes its maximum value a 2 a 2 + b 2 ln a 2 + b 2 a 2 + b 2 a 2 + b 2 ln a 2 + b 2 b 2 ,(50) In the special case of a = b, S = − cos 2 χ 2 ln cos 2 χ 2 − sin 2 χ 2 ln sin 2 χ 2 .(51) This entropy is a monotonically increasing function. It is zero when χ = 0. Its maximum value is 2 ln(2) when χ = 90 o . The symmetry between cos χ and sin χ is well known. Let us now consider another Poincaré sphere where cos χ is replaced by sin χ. Then the density matrix and entropy become ρ ′ (χ) = 1 2 1 + f ′ 0 0 1 − f ′ , S ′ = − 1 + f ′ 2 ln 1 + f ′ 2 − 1 − f ′ 2 ln 1 − f ′ 2 ,(52) respectively, with f ′ = (a 2 − b 2 ) 2 + 4(ab) 2 sin 2 χ a 2 + b 2 ,(53) which takes its minimum and maximum values when χ = 0 and χ = 90 o respectively. Indeed, S and S ′ move in opposite directions as χ changes. Thus we are led to consider their addition (S + S ′ ). The question is whether it remains constant. The answer is No. On the other hand, the determinant of the first coherency matrix is (ab) 2 sin 2 χ, and it is (ab) 2 cos χ for the second determinant. Thus the addition of these two determinants (ab) 2 sin 2 χ + (ab) 2 cos χ = (ab) 2 . (54) is independent of the decoherence angle χ. While the determinant of is a Lorentz-invariant quantity, there could be a larger group which will change the value of the determinant, thus the decoherence angle. This question has been addressed in the literature [21]. Then there comes the issue of Feynman's rest of the universe. In his book on statistical mechanics [7], Feynman divides the quantum universe into two systems, namely the world in which we do physics, and the rest of the universe beyond our scope. However, we could gain a better understanding of physics if we can construct a model of the rest of the universe which can be explained in terms of the physical laws applicable to the world which we observe. With this point in mind, Han et al. considered a system of two coupled world [8]. One of those oscillators belong to the world in which we do physics, while the other is in the rest of the universe. This constitute of an illustrative example of Feynman's rest of the universe. In this section, we discussed two Poincaré spheres which are coupled by the addition formula of Eq.(54). One of the spheres talks about the physical world, and the other takes care of the entropy variation of this world through the conservation of Eq.(54). Indeed, these two coupled Poincaré spheres constitute another illustrative example of Feynman's rest of the universe. Concluding Remarks In this report, we noted first that the group of Lorentz transformations can be formulated in terms of two-by-two matrices. This two-by-two formalism can also be used for transformations of the coherency matrix in polarization optics. Thus, this set of four Stokes parameters is like a Minkowskian four-vector under four-by-four Lorentz transformations. The geometry of the Poincaré sphere can be extended to accommodate this four-dimensional transformations. It is shown that the decoherence parameter in the Stokes formalism is invariant under Lorentz transformations, like the particle mass in Einstein's four-vector formalism of the energy and momentum. These matrices appear to be complicated, but it is enough to study the our matrices of Eq.(7) and Eq.(8) enough to cover all the matrices in this group. Thus, we give their four-by-four equivalents in this appendix. 1 : 1Wigner's Little Groups. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. They thus define the internal space-time symmetries of particles. The four-momentum remains invariant under the rotation around it. In addition, they remain invariant under the following transformations. They are different for massive, massless, and superluminal particles. ( 7 ) 7, corresponds to the rotation around the y axis in the space-time symmetry. Among the rotation angles, the angle of 45 o plays an important role in polarization optics. Indeed, if we rotate the squeeze matrix of Eq.(27) by 45 o , we end up with the squeeze matrix Figure 2 : 2Radius of the Poincaré sphere. The radius R takes its maximum value S 0 when the decoherence angle χ is zero. It takes its minimum value S z when χ = 90 o . The degree of polarization is maximum when R = S 0 , and is minimum when R = S z . Table Table 2 : 2Polarization optics and special relativity sharing the same math- ematics. Each matrix has its clear role in both optics and relativity. The determinant of the Stokes or the four-momentum matrix remains invariant under Lorentz transformations. It is interesting to note that the decoher- ence parameter (least fundamental) in optics corresponds to the mass (most fundamental) in particle physics. Polarization Optics Transformation Matrix Particle Symmetry Phase shift δ e δ/2 0 0 e −iδ/2 Rotation around z. Entropy and Feynman's Rest of the UniverseAnother way to measure the lack of coherence is to compute the entropy of the system. The coherency matrix of Eq.(33) leads to the density matrix of AcknowledgmentsFirst of all, I would like to thank Professor Sergei Kilin for inviting me to the International Conference "Spins and Photonic Beams at Interface," honoring Academician F. I. Fedorov. In addition to numerous original contributions in optics, Fedorov wrote a book on two-by-two representations of the Lorentz group based on his own research in this subject. It was quite appropriate for me to present a paper on applications of the Lorentz group to optical science. I would like thank Professors V. A. Dluganovich and M. Glaynskii for bringing to my attention the papers and the book written by Academician Fedorov, as well as their own papers.AppendixIn Sec. 2, we listed four two-by-two matrices whose repeated applications lead to the most general form of the two-by-two matrix G. It is known that every G matrix can be translated into a four-by-four Lorentz transformation matrix through[5,9,15](56) R A M Azzam, I Bashara, Ellipsometry and Polarized Light. North-Holland, AmsterdamR. A. M. Azzam and I. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977). Principles of Optics. M Born, E Wolf, PergamonOxford6th EdM. Born and E. Wolf, Principles of Optics. 6th Ed. (Pergamon, Oxford, 1980). C Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach. New YorkJohn WileyC. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (John Wiley, New York, 1998). On unitary representations of the inhomogeneous Lorentz group. E Wigner, Ann. Math. 40E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40, 149-204 (1939). An English version of this article (translated by F. V. Atkinson) is in the. M A Naimark, American Mathematical Society Translations9Linear Representation of the Lorentz Group, Uspekhi MatM. A. Naimark, Linear Representation of the Lorentz Group, Uspekhi Mat. Nauk 9, No.4(62), 19-93 (1954). An English version of this article (translated by F. V. Atkinson) is in the American Mathematical Society Translations, Series 2. Volume 6, 379-458 (1957). Linear Representations of the Lorentz Group, translated by. M A Naimark, The original book written in Russian was published by Fizmatgiz. Ann Swinfen and O. J. MarstrandMoscowPergamon PressM. A. Naimark, Linear Representations of the Lorentz Group, trans- lated by Ann Swinfen and O. J. Marstrand (Pergamon Press, 1964). The original book written in Russian was published by Fizmatgiz (Moscow, 1958). . R P , Feynman Statistical Mechanics. R. P. Feynman Statistical Mechanics (Benjamin/Cummings, Reading, Massachusetts, 1972). Illustrative example of Feynman's rest of the universe. D Han, Y S Kim, M E Noz, Am. J. Phys. 67D. Han, Y. S. Kim, and M. E. Noz, Illustrative example of Feynman's rest of the universe, Am. J. Phys. 67, 61-66 (1999). Y S Kim, M E , Noz Theory and Applications of the Poincaré Group. Reidel, DordrechtY. S. Kim and M. E. Noz Theory and Applications of the Poincaré Group (Reidel, Dordrecht 1986). E(2)-like little group for massless particles and polarization of neutrinos. D Han, Y S Kim, D Son, Phys. Rev. D. 26D. Han, Y. S. Kim, and D. Son, E(2)-like little group for massless particles and polarization of neutrinos, Phys. Rev. D 26, 3717 -3725 (1982). Space-time geometry of relativistic particles. Y S Kim, E P Wigner, J. Math. Phys. 31Y. S. Kim and E. P. Wigner, Space-time geometry of relativistic par- ticles, J. Math. Phys. 31, 55-60 (1990). One analytic form for four branches of the ABCD matrix. S Başkal, Y S Kim, J. Mod. Opt. 57S. Başkal and Y. S. Kim, One analytic form for four branches of the ABCD matrix, J. Mod. Opt. [57], 1251-1259 (2010). B E A Saleh, M C Teich, Fundamentals of Photonics. Hoboken, New JerseyJohn Wiley2nd EdB. E. A. Saleh and M. C. Teich, Fundamentals of Photonics. 2nd Ed. (John Wiley, Hoboken, New Jersey, 2007). Jones-matrix formalism as a representation of the Lorentz group. D Han, Y S Kim, M E Noz, J. Opt. Soc. Am. A. 14D. Han, Y. S. Kim, and M. E. Noz, Jones-matrix formalism as a rep- resentation of the Lorentz group, J. Opt. Soc. Am. A 14, 2290-2298 (1997). Stokes parameters as a Minkowskian four-vector. D Han, Y S Kim, M E Noz, Phys. Rev. E. 56D. Han, Y.S. Kim, and M. E. Noz, Stokes parameters as a Minkowskian four-vector, Phys. Rev. E 56, 6065-76. (1997). Vector parameterization of the Lorentz group transformations and polar decomposition of Mueller matrices. V A Dlugunovich, Y A Kurochkin, Optics and Spectro. 107V. A. Dlugunovich and Y. A. Kurochkin, Vector parameterization of the Lorentz group transformations and polar decomposition of Mueller matrices, Optics and Spectro. 107, 312-17 (2009). Vectorial Pauli algebraic approach in polarization optics. I. Device and state operators. T Tudor, Optik. 121T. Tudor, Vectorial Pauli algebraic approach in polarization optics. I. Device and state operators, Optik 121, 1226-35 (2010). Vector Parametrization of the Lorentz Group and Relativistic Kinematics. F I Fedorov, Theo. Math. Physics. 2F. I. Fedorov, Vector Parametrization of the Lorentz Group and Rela- tivistic Kinematics, Theo. Math. Physics 2, 248-252 (1970). F I Fedorov, Lorentz Group, Global Science. Moscowin RussianF. I. Fedorov, Lorentz Group (in Russian)(Global Science, Physical- Mathematical Literature, Moscow, 1979). Rotations associated with Lorentz Boosts. S Baskal, Y S Kim, J. Phys. A: Math. Gen. 38S. Baskal and Y. S. Kim, Rotations associated with Lorentz Boosts J. Phys. A: Math. Gen 38 6545-6556 (2005). de Sitter group as a symmetry for optical decoherence. S Başkal, Y S Kim, J. Phys. A. 39S. Başkal and Y. S. Kim, de Sitter group as a symmetry for optical decoherence, J. Phys. A 39, 7775-88 (2006)	[]
[ "WEAK CONVERGENCE OF MONGE-AMPÈRE MEASURES FOR DISCRETE CONVEX MESH FUNCTIONS", "WEAK CONVERGENCE OF MONGE-AMPÈRE MEASURES FOR DISCRETE CONVEX MESH FUNCTIONS" ]	[ "Gerard Awanou " ]	[]	[]	For mesh functions which satisfy a convexity condition at the discrete level, we associate the natural analogue of the Monge-Ampère measure. A discrete Aleksandrov-Bakelman-Pucci's maximum principle is derived. We use it to prove the weak convergence of Monge-Ampère measures for discrete convex mesh functions, converging uniformly on compact subsets, interpolating boundary values of a continuous convex function and with Monge-Ampère masses uniformly bounded. When discrete convex mesh functions converge uniformly on the whole domain and up to the boundary, the associated Monge-Ampère measures weakly converge to the Monge-Ampère measure of the limit function. The analogous result for sequences of convex functions relies on properties of convex functions and their Legendre transform. In this paper we select proofs which carry out to the discrete level. Our result can be used to give alternate proofs of the convergence of some discretizations for the second boundary value problem for the Monge-Ampère equation and was used for a recently proposed discretization of the latter.	10.1007/s10440-021-00400-x	[ "https://arxiv.org/pdf/1910.13870v1.pdf" ]	204,961,207	1910.13870	dc723bce277910ee786391d94f34b91c54812c87	WEAK CONVERGENCE OF MONGE-AMPÈRE MEASURES FOR DISCRETE CONVEX MESH FUNCTIONS 29 Oct 2019 Gerard Awanou WEAK CONVERGENCE OF MONGE-AMPÈRE MEASURES FOR DISCRETE CONVEX MESH FUNCTIONS 29 Oct 2019discrete convex functionsweak convergence of measuresMonge-Ampère measure AMS subject classifications 39A1235J6065N1265M06 For mesh functions which satisfy a convexity condition at the discrete level, we associate the natural analogue of the Monge-Ampère measure. A discrete Aleksandrov-Bakelman-Pucci's maximum principle is derived. We use it to prove the weak convergence of Monge-Ampère measures for discrete convex mesh functions, converging uniformly on compact subsets, interpolating boundary values of a continuous convex function and with Monge-Ampère masses uniformly bounded. When discrete convex mesh functions converge uniformly on the whole domain and up to the boundary, the associated Monge-Ampère measures weakly converge to the Monge-Ampère measure of the limit function. The analogous result for sequences of convex functions relies on properties of convex functions and their Legendre transform. In this paper we select proofs which carry out to the discrete level. Our result can be used to give alternate proofs of the convergence of some discretizations for the second boundary value problem for the Monge-Ampère equation and was used for a recently proposed discretization of the latter. Introduction. Let Ω be a convex bounded domain of R d with boundary ∂Ω and let u be a convex function on Ω. We consider a sequence of mesh functions u h which converges uniformly on compact subsets to u as h → 0. The mesh functions u h are only required to be discrete convex, c.f. Definition 2.1. For a locally integrable function R > 0 on Ω, one associates, through the normal mapping, the R-curvature ω(R, u, .) of the convex function u as a Borel measure. We define the analogous measures ω(R, u h , .) for the mesh functions u h and give conditions under which ω(R, u h , .) weakly converges to ω(R, u, .) as h → 0. The first category of conditions require uniform convergence on Ω of u h to u. The second set of conditions require that the discrete convex mesh functions converge uniformly on compact subsets, interpolate boundary values of a continuous convex function and have Monge-Ampère masses, c.f. Definition 3.3 below, uniformly bounded. The result does not follow immediately from the corresponding result for sequences of convex functions, as most of them rely on properties of convex functions and their Legendre transform. For example [10,Lemma 1.1.3] does not generalize to the discrete setting as it refers to points which may not be mesh points. For another example [9, Proposition 2.6] explicitly uses convexity assumptions. We found proofs which carry out to the discrete level. The result presented in this paper is key to the proof of convergence of a recently proposed discretization for the second boundary value problem for the Monge-Ampère equation [3]. See also [2]. It can also be used, in the case R ≡ 1, to give an alternate proof for the convergence of the discretization proposed by Benamou and Duval in [5]. The method used in [5] does not seem to apply to the discretization we proposed in [3]. For the Dirichlet boundary condition, our result, in the case R ≡ 1, was used in [1] to give a proof of convergence for a discretization, proposed by Benamou and Froese in [6], for a Monge-Ampère equation with right hand side a sum of Dirac masses. We emphasize that the mesh functions u h may not have extensions as convex functions, nor are they nodal convex functions as defined in [16], but only discrete convex mesh functions. As a consequence, the discrete normal mapping at a mesh point x may depend on all grid points. The purpose of the introduction of these notions is not to form the basis of a numerical method, but rather as a theoretical tool for analyzing numerical methods. Because the mesh functions may not have convex extensions, it is not clear that results known for convex functions hold for them. A similar difficulty occurs in other contexts [15,5]. It is also not obvious that properties of the normal mapping also hold for the discrete version of normal mapping we consider. The bulk of the paper consists in verifying that certain arguments we select which are valid for convex functions or valid for the normal mapping are also valid for their discrete versions. The paper is organized as follows. In the next section we collect some notation used throughout the paper and recall the notion of R-Monge-Ampère measure. In section 3, we present our discrete analogues and prove key weak convergence results for our discretization of the normal mapping. The proof of a rather long technical lemma is given in section 4. The paper concludes with the derivation of the discrete Aleksandrov-Bakelman-Pucci's maximum principle. The latter is used in the paper to give weak convergence of Monge-Ampère measures for discrete convex mesh functions interpolating boundary values of a continuous convex function. Preliminaries . For x ∈ Ω and S ⊂ Ω we denote by d(x, S) the distance of x to S. Let h be a small positive parameter and let Z d h = { mh, m ∈ Z d }, denote the orthogonal lattice with mesh length h. We denote by U h the linear space of mesh functions, i.e. real-valued functions defined on Ω ∩ Z d h . For u h ∈ U h and e ∈ Z d h , we define the second order directional difference operator ∆ e u h : Z d h → R, ∆ e u h (x) = u h (x + e) − 2u h (x) + u h (x − e). Let also (r 1 , . . . , r d ) denote the canonical basis of R d . We define Ω h = Ω ∩ Z d h and ∂Ω h = ∂Ω ∩ Z d h . For a function u ∈ C(Ω) its restriction on Ω h is also denoted u by an abuse of notation. DEFINITION 2.1. We say that a mesh function u h is discrete convex if and only if ∆ e u h (x) ≥ 0 for all x ∈ Ω h and e ∈ Z d h for which ∆ e u h (x) is defined. We denote by C h the cone of discrete convex mesh functions. All mesh functions considered in this paper are discrete convex and hence the mention u h ∈ C h will be omitted. For x ∈ Ω recall that d(x, ∂Ω) denote the distance of x to ∂Ω. For a subset S of Ω, diam(S) denotes its diameter. 2.1. R-curvature of convex functions. The material in this subsection is taken from [4,10] to which we refer for proofs. Let Ω be an open subset of R d and let us denote by P(R d ) the set of subsets of R d . DEFINITION 2.2. Let u : Ω → R. The normal mapping of u, or subdifferential of u is the set-valued mapping ∂u : Ω → P(R d ) defined by ∂u(x 0 ) = { p ∈ R d : u(x) ≥ u(x 0 ) + p · (x − x 0 ), for all x ∈ Ω }. (2.1) Let \|E\| denote the Lebesgue measure of the measurable subset E ⊂ Ω. For E ⊂ Ω, we define ∂u(E) = ∪ x∈E ∂u(x).S = { E ⊂ Ω, ∂u(E) is Lebesgue measurable }, is a Borel σ-algebra. Let R be a locally integrable function on Ω such that R > 0. The R-curvature of the convex function u is defined as the set function ω(R, u, E) = ∂u(E) R(p) dp, and can be shown to be a Radon measure on S [10, Theorem 1.1.13]. The set function ω(R, u, .) is also referred to as the R-Monge-Ampère measure associated with the convex function u. We note that there are several equivalent definitions of weak convergence of measures which can be found for example in [ ǫ > 0, there exists h −1 > 0 such that for all h k , 0 < h k < h −1 , we have max x∈K∩Z d h k \|u h k (x) − u(x)\| < ǫ. 3. Discrete normal mapping and weak convergence. For a mesh function u h ∈ C h , the discrete normal mapping of u h at the point x ∈ Ω ∩ Z d h is defined as ∂ h u h (x) = { p ∈ R d , p · e ≥ u h (x) − u h (x − e) ∀e ∈ Z d h such that x − e ∈ Ω ∩ Z d h }. For convenience, we will often omit the mention that we need x−e ∈ Ω∩Z d h in the definition of ∂ h u h (x). For a subset E ⊂ Ω, we define ∂ h u h (E) = ∪ x∈E∩Z d h ∂ h u h (x) , and define the discrete R-curvature of u h as the set function ω(R, u h , E) = ∂ h u h (E) R(p) dp. We prove in Lemma 3.3 below that for E Lebesgue measurable, ∂ h u h (E) is Lebesgue measurable and in Lemma 3.4 below that ω(R, u h , .) defines a Borel measure. Note that for \|E\| sufficiently small and x ∈ E, we have ω(R, u h , E) = ω(R, u h , { x }). We will make the abuse of notation ω(R, u h , x) = ω(R, u h , { x }). We now establish that ω(R, u h , .) does indeed define a Borel measure. LEMMA 3.1. If Ω is bounded, u h ∈ U h and F ⊂ Ω is closed, then ∂ h u h (F ) is also closed. Proof. Recall that ∂ h u h (F ) ⊂ R d . Let {p k } be a sequence in ∂ h u h (F ) which converges to p. We show that p ∈ ∂ h u h (F ). For each k, let x k ∈ F ∩ Z d h such that p k ∈ ∂ h u h (x k ) . Since F is closed and bounded, we may assume that x k converges to x ∈ F . By definition, u h (x k ) − u h (x k − e) ≤ p k · e for all e ∈ Z d h . As a bounded subset of Z d h , F ∩ Z d h is a finite set and so x k = x for k sufficiently large. It follows that u h (x) − u h (x − e) ≤ p k · e for all e ∈ Z d h and hence p ∈ ∂ h u h (F ). DEFINITION 3.1. The discrete Legendre transform of a mesh function u h is the function u * h : R d → R defined by u * h (p) = sup x∈Ω∩Z d h (x · p − u h (x)). As a supremum of affine functions, the discrete Legendre transform is convex and hence is differentiable almost everywhere, c.f. [ p ∈ ∂ h u h (y) if and only if u * h (p) = y · p − u h (y) for y ∈ Ω h . The proof is immediate. The class S h = { E ⊂ Ω, ∂ h u h (E) is Lebesgue measurable },ω(R, u h , E) = x∈E ω(R, u h , x). (3.1) As a consequence ω(R, u h , .) is σ-additive and thus defines a Borel measure. Proof. Since Ω is bounded, the set E is finite. We can therefore write E = { x i , i = 1, . . . , N }, for some integer N . Put ∂ h u h (x i ) = H i . The proof we give is similar to the proof of σ-additivity of the Monge-Ampère measure associated to a convex function [10, Theorem 1.1.13]. The difference is that here the sets H i are not necessarily disjoint but have pairwise intersection of zero measure by Lemma 3.2. We have ∂ h u h (E) = ∪ N i=1 H i = H 1 ∪ (H 2 \ H 1 ) ∪ (H 3 \ (H 2 ∪ H 1 )) ∪ . . . , with the sets on the right hand side disjoints. Moreover H j = [H j ∩ (H j−1 ∪ H j−2 ∪ . . . ∪ H 1 )] ∪ [H j \ (H j−1 ∪ H j−2 ∪ . . . ∪ H 1 )]. But by Lemma 3.2, \|H j ∩ (H j−1 ∪ H j−2 ∪ . . . ∪ H 1 )\| = 0 and hence ω(R, u h , H j ) = ω(R, u h , H j \ (H j−1 ∪ H j−2 ∪ . . . ∪ H 1 ). This implies that ω(R, u h , E) = ω(R, u h , ∪ N i=1 H i ) = N i=1 ω(R, u h , H i ) and proves (3.1). Let E i ⊂ Ω ∩ Z d h be a sequence of sets with E i ∩ E j = ∅ for i = j. Since Ω ∩ Z d h is finite, the union must be finite, i.e. we can find a finite set S such that ∪ ∞ i=1 E i = ∪ l∈S E l and E l = ∅ for l / ∈ S. It follows that ω(R, u h , ∪ ∞ i=1 E i ) = ω(R, u h , ∪ l∈S E l ) = l∈S ω(R, u h , E l ) = ∞ i=1 ω(R, u h , E l ). Next, we prove that ω(R, u h , .) is finite on compact sets. LEMMA 3.5. Let K be a compact set such that K ⊂ Ω. Then ω(R, u h , K) < ∞. Proof. Since K ⊂ Ω we have K ∩ Z d h ⊂ Ω h . By definition for x ∈ Ω h we have x ± hr i ∈ Ω ∩ Z d h for each coordinate direction r i , i = 1, . . . , d. Thus for p ∈ ∂ h u h (x) we have u h (x) − u h (x − hr i ) ≤ hp · r i ≤ u h (x + hr i ) − u h (x). It follows that \|p · r i \| is bounded for each i = 1, . . . , d. Moreover K ∩ Z d h is finite. We conclude that ω(R, u h , K) < ∞. We now prove a weak convergence result for the R-Monge-Ampère measure ω(R, u h , . A k = ∩ n ∪ k≥n A k and lim inf k A k = ∪ n ∩ k≥n A k .h k →0 ∂ h k u h k (K) ⊂ ∂u(K). Proof. Let p ∈ lim sup h k →0 ∂ h k u h k (K) = ∩ n ∪ k≥n ∂ h k u h k (K). Thus for each n, there exists k n ≥ n and x kn ∈ K ∩ Z d h such that p ∈ ∂ h kn u h kn (x kn ). Let x j denote a subsequence of x kn converging to x 0 ∈ K. Since p ∈ ∂ hj u hj (x j ) for all j, u hj (z) ≥ u hj (x j ) + p · (z − x j ), ∀z ∈ Ω ∩ Z d h . (3.2) Next, note that \|u hj (x j ) − u(x 0 )\| ≤ \|u hj (x j ) − u(x j )\| + \|u(x j ) − u(x 0 )\|. By the convergence of x j to x 0 and the uniform convergence of u h to u, we obtain u hj ( x j ) → u(x 0 ) as h j → 0. Similarly u hj (z) → u(z) as h j → 0. Taking pointwise limits in (3.2), we obtain u(z) ≥ u(x 0 ) + p · (z − x 0 ) ∀z ∈ Ω ∩ Z d h . We conclude that p ∈ ∂u(K). The proof of the following lemma is given in section 4. LEMMA 3.7. Assume that u h → u uniformly on compact subsets of Ω, with u convex and continuous. Assume that K is compact and U is open with K ⊂ U ⊂ U ⊂ Ω and that for any sequence h k → 0, a subsequence k j and z kj ∈ Ω h k j with z kj → z 0 ∈ ∂Ω, we have lim inf j→∞ u(z kj ) ≤ lim sup j→∞ u h k j (z kj ). (3.3) Then, up to a set of measure zero, ∂u(K) ⊂ lim inf h k →0 ∂ h k u h k (U ∩ Z d h k ).∂u(K) ⊂ lim inf h k →0 ∂ h k u h k (U ∩ Z d h k ). Proof. By Lemma 3.7, it is enough to show that (3.3) holds. Again, the proof follows from [11,Remark 3 .2]. Put a j = u h k j (z kj ) − u(z kj ) and b j = u(z kj ). Recall that lim sup j (a j + b j ) ≥ lim sup j a j + lim inf j b j . This gives lim sup j→∞ u h k j (z kj ) ≥ lim sup j→∞ (u h k j − u)(z kj ) + lim inf j→∞ u(z kj ) = lim inf j→∞ u(z kj ). The proof is complete. h k →0 ω(R, u h k , K) ≤ ω(R, u, K) (3.4) ω(R, u, U ) ≤ lim inf h k →0 ω(R, u h k , U ). (3.5) The first relation (3.4) follows from Lemma 3.6 as follows. Define B n = ∪ k≥n ∂ h k u h k (K). Let us first assume that B1 R(p)dp < ∞. Since B n is decreasing, we have ω(R, u, K) ≥ lim sup h k →0 ∂ h k u h k (K) R(p)dp = lim n Bn R(p)dp = lim n Bn R(p)dp = inf n Bn R(p)dp ≥ inf n sup k≥n ∂ h k u h k (K)) R(p)dp = lim sup h k →0 ω(R, u h k , K). Next, we show that there exists R > 0 independent of k such that ∂ h k u h k (K) ⊂ B(0, R), where B(0, R) is a ball of center the origin and radius R. This implies that B1 R(p)dp < ∞. If such a R does not exist, for each n ∈ N, ∃x n ∈ K and p n ∈ ∂ h kn u h kn (K) such that \|p n \| > n. Thus u h kn (x) ≥ u h kn (x n ) + p n · (x − x n ), ∀x ∈ Ω. (3.6) Choose δ > 0 such that K δ = { x ∈ Ω, d(x, K) ≤ δ } ⊂ Ω. With x = x n + δp n /\|p n \| in (3.6), we obtain δ\|p n \| ≤ u h kn (x) − u h kn (x n ) But u h → u uniformly on Ω. So there exists a constant M > 0 independent of k such that \|u h kn (x) − u h kn (x n )\| ≤ M for x ∈ Ω. This contradicts \|p n \| → ∞. To prove (3.5) first recall from Lemma 3.5 that ω(R, u, .) is finite on compact sets and hence is a Radon measure. Thus ω(R, u, U ) = sup{ ω(R, u, K), K ⊂ U, K compact } by [8, Theorem 4, section 1.1]. It is therefore enough to show that for K ⊂ U we have ω(R, u, K) ≤ lim inf h k →0 ω(R, u h k , U ). Define C n = ∩ k≥n ∂ h k u h k (U ∩ Z d h k ). We have by Corollary 3.2 ω(R, u, K) ≤ lim inf h k →0 ∂ h k u h k (U∩Z d h k ) R(p)dp = lim n Cn R(p)dp = lim n Cn R(p)dp = sup n Cn R(p)dp ≤ sup n inf k≥n ∂ h k u h k (U∩Z d h k ) R(p)dp = lim inf h k →0 ω(R, u h k , U ). DEFINITION 3.3. We refer to ω(1, v h , Ω h ) as Monge-Ampère mass of the discrete convex function v h . THEOREM 3.4. Assume that u h → u uniformly on compact subsets of Ω, with u convex and continuous. We also assume that ω(1, u h , Ω h ) ≤ C for a constant C independent of h, and u h = g on ∂Ω h where g ∈ C(∂Ω) has an extensiong ∈ C(Ω) which is convex. Then (3.3) holds and as a consequence ω(R, u h , .) tend to ω(R, u, .) weakly. Proof. Let z 0 ∈ ∂Ω, h k → 0, k j a subsequence and z kj ∈ Ω h k j with z kj → z 0 ∈ ∂Ω. Let also x ∈ Ω and x h ∈ Ω h such that x h → x. Part 1 We first show that lim inf j→∞ u(z kj ) ≤ g(z 0 ). Since u h is discrete convex, we have ∆ h u h ≥ 0 where ∆ h u h (x) = d i=1 ∆ ri u h (x)/h 2 is the discrete Laplacian. Let w h denote the solution of the problem ∆ h w h = 0 on Ω h with w h = g on ∂Ω h . We have ∆ h (u h − w h ) ≥ 0 on Ω h with u h − w h = 0 on ∂Ω h .− w h ≤ 0 on Ω h . Since a convex domain is Lipschitz, we can apply the results of [7, section 6.2 ] and claim that w h converges uniformly on compact subsets to the unique viscosity solution of ∆w = 0 on Ω with w = g on ∂Ω. This gives u(x) ≤ w(x) on Ω. But w ∈ C(Ω) [7]. We conclude that lim inf j→∞ u(z kj ) ≤ w(z 0 ) = g(z 0 ). Part 2 Next, we show that g(z 0 ) ≤ lim sup j→∞ u h k j (z kj ). This part of the proof is based on ideas in the proof of [13,Lemma 5.1]. Let ǫ > 0. Since g can be extended to a convex functiong ∈ C(Ω), by [13,Theorem 5.2], the function U defined by U (x) = sup{ L(x), L ≤ g on ∂Ω, L affine }, is in C(Ω) and U = g on ∂Ω. Therefore, there exists an affine function L such that L ≤ g on ∂Ω and L(z 0 ) ≥ g(z 0 ) − ǫ. Let q h = u h − L. Since u h = g on ∂Ω h , we have q h ≥ 0 on ∂Ω h . By the discrete Aleksandrov's maximum principle Lemma 5.2 below, applied to q h on Ω we have (−q h k j (z kj )) d ≤ Cd(z kj , ∂Ω)(diam(Ω)) d−1 ω(1, u h , Ω h ) ≤ Cd(z kj , ∂Ω)ω(1, u h , Ω h ) ≤ C\|\|z kj − z 0 \|\|ω(1, u h , Ω h ). (3.7) By the assumption on the Monge-Ampère masses ω(1, u h , Ω h ) ≤ C with C independent of h. Then (−q h k j (z kj )) d ≤ C\|\|z kj − z 0 \|\|. We conclude that u h k j (z kj ) ≥ L(z kj ) − C\|\|z kj − z 0 \|\| 1 d . This gives lim sup j→∞ u h k j (z kj ) ≥ L(z 0 ) ≥ g(z 0 ) − ǫ. Since ǫ is arbitrary, combined with the result from Part 1, we obtain (3.3). The proof of the last statement is the same as in the proof of Lemma 3.8. 4. Proof of Lemma 3.7. The proof we give here follows the lines of [11,Lemma 3.3]. Not all proofs of weak convergence of Monge-Ampère measures can be adapted to the discrete case. Part 1 We define A = { (x, p), x ∈ K, p ∈ ∂u(x) }. For (x, p) ∈ A, u(z) ≥ u(x)+p·(z −x), ∀z ∈ Ω. We can thus define a mapping v : R d → R by v(z) = sup (x,p)∈A p · (z − x) + u(x), and we have u(z) ≥ v(z) ∀z ∈ Ω. (4.1) For z ∈ K and p ∈ ∂u(z), we have (z, p) ∈ A. And so v(z) ≥ u(z). By (4.1), we get u(z) = v(z) ∀z ∈ K. (4.2) Note that v is defined on R d and not just on Ω. Thus ∂v is defined with respect to R d , i.e. ∀z ∈ R d , ∂v(z) = { p ∈ R d , v(y) ≥ p · (y − z) + v(z), ∀y ∈ R d }. Note also that v takes values in R as Ω is bounded and u bounded on K. Next we prove that ∂v(x) = ∂u(x) ∀x ∈ K. (4.3) Let p ∈ ∂u(x). We have (x, p) ∈ A and for all z ∈ R d , v(z) ≥ u(x) + p · (z − x). By (4.2), u(x) = v(x) and we conclude that p ∈ ∂v(x), i.e. ∂u(x) ⊂ ∂v(x). Let now p ∈ ∂v(x) and x ∈ K. Using (4.1) and (4.2) we obtain for all z ∈ Ω u(z) ≥ v(z) ≥ u(x) + p · (z − x), which proves that p ∈ ∂u(x) and thus we have ∂v(x) ⊂ ∂u(x). This proves (4.3). Part 2 We define W = { p ∈ R d , p ∈ ∂v(x 1 ) ∩ ∂v(x 2 ), for some x 1 , x 2 ∈ R d , x 1 = x 2 }. Since v is convex as the supremum of affine functions, by [10,Lemma 1.1.12], \|W \| = 0. Let K ⊂ Ω be compact and let p ∈ ∂v(K) \ W . By definition of W , there exists a unique x 0 ∈ K such that p ∈ ∂v(x 0 ) and for all x ∈ R d , x = x 0 we have p / ∈ ∂v(x). We claim that v(x) > v(x 0 ) + p · (x − x 0 ), x ∈ R d , x = x 0 . (4.4) Otherwise ∃x 1 ∈ R d , x 1 = x 0 such that v(x 1 ) ≤ v(x 0 ) + p · (x 1 − x 0 ). But then for x ∈ R d , v(x) ≥ v(x 0 ) + p · (x − x 0 ) = v(x 0 ) + p · (x 1 − x 0 ) + p · (x − x 1 ) ≥ v(x 1 ) + p · (x − x 1 ), which gives p ∈ ∂v(x 1 ), a contradiction. Part 3 Recall that K ⊂ U ⊂ U ⊂ Ω and let p ∈ ∂v(K) \ W with p ∈ ∂v(x 0 ), x 0 ∈ K. For k ≥ 1 let δ k = min x∈U∩Z d h k { u h k (x) − p · (x − x 0 ) }, which exists because U ∩ Z d h k is a finite set, and put x k = argmin x∈U∩Z d h k { u h k (x) − p · (x − x 0 ) }. We have u h k (x) ≥ u h k (x k ) + p · (x − x k ), ∀x ∈ U ∩ Z d h k . (4.5) We first prove that x k → x 0 . Let x kj denote a subsequence converging to x ∈ U . We also consider a sequence z j ∈ U ∩ Z d h k j such that z j → x 0 . By the uniform convergence of u h to u and the uniform continuity of u on U , we have u h k j (z j ) → u(x 0 ), and u h k j (x kj ) → u(x). For example \|u h k j (x kj ) − u(x)\| ≤ \|u h k j (x kj ) − u(x kj )\| + \|u(x kj ) − u(x)\|, from which the claim follows. Therefore taking limits in (4.5), we obtain u(x 0 ) ≥ u(x) + p · (x 0 − x).u(x) ≥ v(x) > v(x 0 ) + p · (x − x 0 ) = u(x 0 ) + p · (x − x 0 ) ≥ u(x) + p · (x 0 − x) + p · (x − x 0 ) = u(x). A contradiction. This proves that x k → x 0 . Part 4 We now claim that there exists k 0 such that (4.5) actually holds for all x ∈ Ω∩Z d h k when k ≥ k 0 . Otherwise one can find a subsequence k j and z kj ∈ (Ω \ U ) ∩ Z d h k j such that u h k j (z kj ) < u h k j (x kj ) + p · (z kj − x kj ). (4.7) Since Ω is bounded, up to a subsequence, we may assume that z kj → z 0 ∈ Ω \ U . We show that v(z 0 ) ≤ v(x 0 ) + p · (z 0 − x 0 ). (4.8) Case 1: z 0 ∈ Ω \ U . Using the uniform convergence of u h to u, the uniform continuity of u on U and taking limits in (4.7), we obtain u(z 0 ) ≤ u(x 0 ) + p · (z 0 − x 0 ). By (4.2), u(x 0 ) = v(x 0 ) and by (4.1), v(z 0 ) ≤ u(z 0 ). This gives (4.8). Case 2: z 0 ∈ ∂Ω \ U . Now we have lim sup j→∞ u h k j (z kj ) ≤ v(x 0 ) + p · (z 0 − x 0 ). Note that v is lower semi-continuous as the supremum of affine functions. Using the assumption (3.3) and (4.1), we obtain lim sup j→∞ u h k j (z kj ) ≥ lim inf j→∞ u(z kj ) ≥ lim inf j→∞ v(z kj ) ≥ v(z 0 ). Hence (4.8) also holds in this case. Finally we note that (4.8) contradicts (4.4) and therefore (4.7) cannot hold, i.e. (4.5) actually holds for all x ∈ Ω ∩ Z d h k when k ≥ k 0 . But this means that p ∈ ∪ n ∩ k≥n ∂ 1 h k u h k (U ∩ Z d h k ) and concludes the proof. 5. Discrete Aleksandrov-Bakelman-Pucci's maximum principle. The next lemma is an analogue of [10, Lemma 1.4.1]. LEMMA 5.1. Let v h , w h ∈ U h such that v h ≤ w h on ∂Ω h and v h ≥ w h in Ω h , then ∂ h v h (Ω h ) ⊂ ∂ h w h (Ω h ). Proof. Let p ∈ ∂ h v h (x 0 ), x 0 ∈ Ω h . Then v h (x 0 ) − v h (x 0 − e) ≤ p · e for all e ∈ Z d h such that x 0 − e ∈ Ω ∩ Z d h . Define a = sup e∈Z d h x 0 −e∈Ω∩Z d h { v h (x 0 ) − p · e − w h (x 0 − e) }. We have a ≥ v h (x 0 ) − w h (x 0 ) ≥ 0. Furthermore there exists e 0 such that x 0 − e 0 ∈ Ω ∩ Z d h and a = v h (x 0 ) − p · e 0 − w h (x 0 − e 0 ). Since a ≥ v h (x 0 ) − p · e − w h (x 0 − e), (5.1) we get p · (e − e 0 ) ≥ w h (x 0 − e 0 ) − w h (x 0 − e). We have v h (x 0 − e 0 ) ≥ v h (x 0 ) − p · e 0 = a + w h (x 0 − e 0 ). Hence if a > 0, x 0 −e 0 / ∈ ∂Ω h since v h ≤ w h on ∂Ω h , and p ∈ ∂ h w h (x 0 −e 0 ). If a = 0 we have by (5.1) p · e ≥ v h (x 0 ) − w h (x 0 − e) ≥ w h (x 0 ) − w h (x 0 − e) and p ∈ ∂ h w h (x 0 ). This concludes the proof. The following lemma is a discrete version of the Aleksandrov-Bakelman-Pucci's maximum principle [18,Theorem 8.1]. Analogues can be found in [17] and [14]. LEMMA 5.2. Let u h ∈ C h such that u h ≥ 0 on ∂Ω h . Then for x ∈ Ω h u h (x) ≥ −C(d) diam(Ω) d−1 d(x, ∂Ω)ω(1, u h , Ω h ) 1 d , for a positive constant C(d) which depends only on d. Proof. Assume that there exists x 0 ∈ Ω h such that u h (x 0 ) < 0. We will use below a ball of radius a scalar multiple of −u h (x 0 ). Let F = { v h ∈ C h , v h (x 0 ) ≤ u h (x 0 ) and v h (x) ≤ u h (x) ∀x ∈ ∂Ω h }. Since u h ∈ C h , F = ∅. Define w h (x) = sup v h ∈F v h (x), x ∈ Ω ∩ Z d h . Because u h ∈ F we have u h ≤ w h on Ω ∩ Z d h . This gives in addition w h (x 0 ) = u h (x 0 ) and w h (x) = u h (x) on ∂Ω h . It follows from Lemma 5.1 that for ∂ h u h (Ω h ) ⊃ ∂ h w h (Ω h ) ⊃ ∂ h w h (x 0 ). (5.2) We define E = { p ∈ R d , u h (x 0 ) − p · e ≤ 0 if x 0 − e ∈ ∂Ω h }. We claim that E ⊂ ∂ h w h (x 0 ). Let x ∈ ∂Ω h and put e = x 0 − x. Since u h ≥ 0 on ∂Ω h , for p ∈ E, u h (x 0 ) − p · (x 0 − x) ≤ u h (x). If we define v h (x) = u h (x 0 ) − p · (x 0 − x) we have v h ∈ F . Thus from the definition of F we get u h (x 0 ) − p · (x 0 − x) ≤ w h (x) for all x ∈ Ω ∩ Z d h . But w h (x 0 ) = u h (x 0 ) and therefore for all e such that x 0 − e ∈ Ω ∩ Z d h we have w h (x 0 ) − p · e ≤ w h (x 0 − e). We conclude that p ∈ ∂ h w h (x 0 ). It is not difficult to prove that E is convex. Let x * ∈ ∂Ω h such that \|\|x * − x 0 \|\| = d(x 0 , ∂Ω h ). Since Ω is convex, we have for all x ∈ ∂Ω h (x * − x 0 ) · (x − x 0 ) ≤ \|\|x * − x 0 \|\| 2 . Put z 0 = −u h (x 0 ) d(x 0 , ∂Ω h ) x * − x 0 \|\|x * − x 0 \|\| . We now prove that z 0 ∈ E and that the ball B of center the origin and radius −u h (x 0 )/ diam(Ω) is also contained in E. Let e such that x 0 − e = x ∈ ∂Ω h . We have u h (x 0 ) − z 0 · e = u h (x 0 ) − u h (x 0 ) d(x 0 , ∂Ω h ) (x − x 0 ) · (x * − x 0 ) \|\|x * − x 0 \|\| ≤ u h (x 0 ) − u h (x 0 ) d(x 0 , ∂Ω h ) \|\|x * − x 0 \|\| = u h (x 0 ) − u h (x 0 ) = 0, where we used −u h (x 0 ) ≥ 0 and \|\|x * − x 0 \|\| = d(x 0 , ∂Ω h ). On the other hand if \|\|z\|\| ≤ −u h (x 0 )/ diam(Ω h ) u h (x 0 ) − z · e ≤ u h (x 0 ) + \|\|z\|\| \|\|e\|\| ≤ u h (x 0 ) + \|\|z\|\| diam(Ω h ) ≤ 0. We conclude that E contains the convex hull of B and z 0 which has measure C(d) −u h (x 0 ) diam(Ω h ) d−1 \|\|z 0 \|\|, for a generic constant C(d) which depends only on d. Since E ⊂ ∂ h w h (x 0 ), by (5.2) ω(1, u h , Ω h ) ≥ C(d) (−u h (x 0 )) d (diam(Ω h )) d−1 d(x 0 , ∂Ω h ) . This concludes the proof. THEOREM 2. 3 ([ 10 ] 310Theorem 1.1.13). If u is continuous on Ω, the class DEFINITION 2.4. A sequence µ n of Borel measures converges to a Borel measure µ if and only if µ n (B) → µ(B) for any Borel set B with µ(∂B) = 0. LEMMA 3. 6 . 6Assume that u h → u uniformly on compact subsets of Ω, with u convex and continuous. Then for K ⊂ Ω compact and any sequence h k → 0 lim sup COROLLARY 3. 2 . 2Assume that u h → u uniformly on Ω, with u convex and continuous on Ω. Assume that K is compact and U is open with K ⊂ U ⊂ U ⊂ Ω. Then, up to a set of measure zero, LEMMA 3 . 8 . 38Assume that u h → u uniformly on Ω, with u convex and continuous on Ω.Then ω(R, u h , .) tend to ω(R, u, .) weakly.Proof. By an equivalence criteria of weak convergence of measures, c.f. for example [8, Theorem 1, section 1.9], it is enough to show that for any sequence h k → 0, a compact subset K ⊂ Ω and an open subset U ⊂ U ⊂ Ω, we have lim sup (4. 6 ) 6If x = x 0 , we obtain by (4.1), (4.4), (4.2) and(4.6) Convergence of mesh functions. We will need the following definitions. DEFINITION 2.5. We say that a family of Borel measures µ h converges to a Borel measure µ if for any sequence h k → 0, µ h k weakly converges to µ. DEFINITION 2.6. Let u h ∈ U h for each h > 0. We say that u h converges to a convex function u uniformly on compact subsets of Ω if and only if for each compact set K ⊂ Ω, each sequence h k → 0 and for all8, Theorem 1, section 1.9]. 2.2. If Ω is open, the set of points in R d which belongs to the discrete normal mapping image of more than one point of Ω ∩ Z d h is contained in a set of measure zero. Proof. The proof follows essentially the one of [10, Lemma 1.1.12]. As in the continuous case, it relies on the fact that10, Lemma 1.1.8]. This implies the following LEMMA 3.2. contains the closed sets by Lemma 3.1. Taking into account Lemma 3.2 we obtain. LEMMA 3.3. Assume that Ω is open and bounded. The class S h is a σ-algebra which contains all closed sets of Ω. Therefore if E is a Borel subset of Ω and u h is a mesh function, ∂ h u h (E) is Lebesgue measurable. Proof. The proof is essentially the same as the corresponding one at the continuous level [4, p. 117-118]. LEMMA 3.4. Let Ω be open and bounded. For E ⊂ Ω ∩ Z d h , we have We recall that for a family of sets A k lim sup). Lemmas 3.6-3.8 below are discrete analogues of [10, Lemma 1.2.2 and Lemma 1.2.3]. k By the discrete maximum principle for the discrete Laplacian[12, Theorem 4.77], we have u h Acknowledgments. The author was partially supported by NSF grants DMS-1319640 and DMS-1720276. The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" where part of this work was undertaken. Part of this work was supported by EPSRC grant no EP/K032208/1. Discrete Aleksandrov solutions of the Monge-Ampère equation. G Awanou, G. AWANOU, Discrete Aleksandrov solutions of the Monge-Ampère equation. https://arxiv.org/abs/1408.1729. Computational geometric optics: Monge-Ampère. 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[ "Tau Functions of (n, 1) curves and Soliton Solutions on Non-Zero Constant Backgrounds", "Tau Functions of (n, 1) curves and Soliton Solutions on Non-Zero Constant Backgrounds" ]	[ "Atsushi Nakayashiki " ]	[]	[]	We study the tau function of the KP-hierarchy associated with an (n, 1) curve y n = x − α. If α = 0 the corresponding tau function is 1. On the other hand if α = 0 the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex opertaors to the latter we obtain soliton solutions on non-zero constant backgrounds.	10.1007/s11005-021-01411-3	[ "https://arxiv.org/pdf/2011.10691v1.pdf" ]	227,126,681	2011.10691	1d5c2ad074e14b8af78e941cc4a080fb387bb8e2	Tau Functions of (n, 1) curves and Soliton Solutions on Non-Zero Constant Backgrounds 21 Nov 2020 Atsushi Nakayashiki Tau Functions of (n, 1) curves and Soliton Solutions on Non-Zero Constant Backgrounds 21 Nov 2020 We study the tau function of the KP-hierarchy associated with an (n, 1) curve y n = x − α. If α = 0 the corresponding tau function is 1. On the other hand if α = 0 the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex opertaors to the latter we obtain soliton solutions on non-zero constant backgrounds. Introduction In our previous paper [19] the degeneration of the theta function solution (tau function) of the KP-hierarchy as a result of what we call one step degeneration of an (n, s) curve for n = 2, 3 has been studied. The obtained formula expresses the degenerate tau function as a sum of combinations of exponential functions with tau functions of a lower genus curve. Repeating this process we finally come to the tau function associated with an (n, 1) curve of genus zero. In this paper we derive the explicit formula for the tau function of an (n, 1) curve with an arbitrary n ≥ 2. We use it as a seed solution in the vertex operator construction of solutions of the KP-hierarchy. In the case of n = 2 the solutions obtained in this way are considered as solitons on non-zero constant backgrounds. For n ≥ 2 consider the rational algebraic curve C n defined by y n = x − α,(1.1) which we call an (n, 1) curve. From this curve a solution of the KP-hierarchy (tau function) is constructed as follows. To this end we use the Sato Grassmannian which we denote by UGM (=universal Grassmann manifold) after Sato [23,22]. The Sato Grassmannian is the parameter space of solutions of the KP-hierarchy and is defined as the set of certain subspaces of the vector space V = C((z)) of Laurent series in a variable z. Consider the point ∞ of C n and take the local coordinate z around ∞ such that x = z −n , y = z −1 (1 + O(z)). This choice of the local coordinate is necessary in order to study the degeneration of solutions of the KP-hierarchy associated with an (n, s) curve [19]. Define the vector space V n by V n = ∞ j=0 Cy j . By expanding y j in the coordinate z V n can be considered as a subspace of V . Then it is known that it belongs to UGM (see [18] for example). We denote this point of UGM by V n (z) meaning that it is obtained from V n using the coordinate z. For each point of UGM a solution τ (t) of the KP-hierarchy is constructed, up to constant, in the form of the Schur function expansion as τ (t) = λ ξ λ s λ (t), where the summation is over all partitions, s λ (t) is the Schur function corresponding to λ and ξ λ the Plücker coordinate of the point of UGM. In the present case the Schur function expansion of the tau function corresponding to V n (z) begins from a non-zero constant. We define the tau function τ 0 (t; V n (z)) as that corresponding to V n (z) normalized as τ 0 (t; V n (z)) = 1 + \|λ\|>0 ξ λ s λ (t). ( 1.2) In the case α = 0 we easily have τ 0 (t; V n (z)) = 1. So the actual problem here is to compute τ 0 (t; V n (z)) for α = 0. The result is τ 0 (t; V n (z)) = e where q(t) is a quadratic form in t = t (t 1 , t 2 , t 3 , ...) and L(t) is a linear form in t (see Theorem 3.1). It is difficult to prove such a formula directly from the Schur function expansion. The strategy here to derived the formula (1.3) is to consider a coordinate change of the corresponding wave and the adjoint wave function [4]. This idea comes from the paper [5] by Dubrovin and Natanzon, who first studied the effect of a coordinate change in the construction of solutions of the KP-equation. The Schur function expansion of the tau function (1.3) is also calculated explicitly by normalizing a frame of V n (z) (see Theorem 3.2). We use the Giambelli formula for Plücker coordinates [6,20] in this computation. The second logarithmic derivative of a tau function with respect to t 1 (see (2.2)) gives a solution of the KP-equation (2.3). The solution corresponding to τ 0 (t; V n (z)) is a constant. So (1.3) itself does not look an interesting solution of the KP equation, although it is used to describe the degeneration of theta function solution as mentioned in the beginning. However this is not the case. It plays a role as a seed to create various solutions of the KP-hierarchy. In fact it is well known that soliton solutions of the KP-hierarchy are constructed by applying vertex operators to the trivial solution 1 [4]. Similarly we also get various solutions by applying vertex operators to τ 0 (t; V n (z)). In particular, if n = 2 and α = 0 solutions obtained in this way are considered as solitons with non-zero constant asymptotics (see Corollary 9.3 and Theorem 9.1). Note that for this computation to work it is indispensable to introduce the infinite number of time variables. Finally we mention that in the paper [7] the tau function of the form (1.3) had been studied in relation with Gromov-Witten invariants and the dispertionless KP-hierarchy. In that paper all points of UGM corresponding to tau functions of the n-reduced KPhierarchy, which are expressed in the form of the exponential of a quadratic form, are determined. The point V n (z) is a special family, depending on the parameter α, in them. However such explicit formula as in Theorem 3.1 of the tau function was not derived in [7] as far as the author understands. The paper is organized as follows. After the introduction a brief review on the Sato Grassmannian and the KP-hierarchy are given in section 2. In section 3 the problems and main results are stated. Sections 4 to 7 are devoted to the proofs of main theorems. In section 4 the problem of determining the tau function is reformulated in terms of the wave function. Beginning from the trivial wave function corresponding to the trivial tau function 1, the condition for the tau function which we seek for is formulated by the coordinated changed the wave and the adjoint wave function. The series expansion of the function which appears in the wave function in question is determined in section 5. Based on the results of section 4 and 5 Theorem 3.1 is proved in section 6. In section 7 Theorem 3.2 is proved by computing the Plücker coordinates of V n (z). The generating function of the coefficients of the quadratic form q(t) is computed in section 8. Here the genus zero analogue of the bilinear meromorphic differential [2,3,16] of an (n, s) curve with positive genus plays a crucial role. In section 9 soliton solutions on non-zero constant backgrounds are computed by applying vertex operators to τ 0 (t; V n (z)). In this calculation the result of section 8 is crucial. Sato Grassmannian In this section we beriefly review the theory of the Sato Grassmannian and the KPhierarchy. Let τ (t) be a function of t = t (t 1 , t 2 , ...). The KP-hierarchy is the equation for τ (t) given by Res k=∞ τ (t − s − [k −1 ])(t + s + [k −1 ])e −2 ∞ j=1 s j k j dk 2πi = 0, (2.1) where s = t (s 1 , s 2 , ...), [k] = t (k, k 2 /2, k 3 /3, ...). By expanding in s j , j ≥ 1, (2.1) gives an infinite number of differential equations for τ (t) expressed in Hirota's bilinear form [4]. Set (x, y, t) = (t 1 , t 2 , t 3 ) and, for a solution τ (t) of (2.1) , u(t) = 2 ∂ 2 ∂x 2 log τ (t). (2.2) Then u(t) is a solution of the KP-equation, 3u yy + (−4u t + 6uu x + u xxx ) x = 0. (2. 3) The totality of solutions of the KP-hierarchy constitues a certain infinite dimensional Grassmann manifold called the Sato Grassmannian [23]. Let us recall its definition and fundamental properties. Let V = C((z)) be the vector space of Laurent series in z. Define two subspaces of V by V φ = C[z −1 ], V 0 = zC[[z]]. Then V = V φ ⊕ V 0 . Let π : V −→ V φ be the projection map. Then the Sato Grassmannian which we denote by UGM (=universal Grassmann manifold) after Sato [23,22], is defined as the set of subspaces U of V such that dim(Ker π\| U ) = dim(Coker π\| U ) < ∞. Example 2.1. The subspace V φ belongs to UGM. In this case dim(Ker π\| V φ ) = dim(Coker π\| V φ ) = 0. (2.4) . In general a subspace U defined by U = j≤0 Cξ j , ξ j = z j + ∞ i=1 ξ i,j z i , ξ i,j ∈ C,(2.5) satisfies (2.4) and belongs to UGM. The totality of points of UGM corresponding to such frames is denoted by UGM φ , which forms a cell called the big cell of UGM. A point U of UGM can be specified by its frame, that is, a basis of U . If we associate the infinite column vector (a n ) n∈Z to an element a n z n of V , a frame of U can be expressed by an Z×Z ≤0 matrix ξ = (ξ i,j ) i∈Z,j∈Z ≤0 , where Z ≤0 denotes the set of non-positive integers. In writing the matrix ξ we follow the usual convention that the row numbers increase downward and the column numbers increase rightward. For example the frame ξ = (..., ξ −1 , ξ 0 ) of U given by (2.5) is represented as ξ =             . . . . . . · · · 1 0 · · · 0 1 − − − − − − − − − · · · ξ 1,−1 ξ 1,0 · · · ξ 2,−1 ξ 2,0 . . . . . .             . (2.6) In general it is always possible to take a frame satisfying the following condition; there exists a negative integer l such that ξ i,j = 1 if j < l and i = j 0 if (j < l and i < j) or (j ≥ l and i < l). (2.7) In the sequel we always take a frame which satisfies this condition, although it is not unique. A Maya diagram M = (m j ) ∞ j=0 is a sequence of intergers such that m 0 > m 1 > · · · and m j = −j for all sufficiently large j. For a Maya diagram M = (m j ) ∞ j=0 the corresponding partition is defined by λ(M ) = (j + m j ) ∞ j=0 . By this correspondence the set of Maya diagrams and the set of partitions bijectively correspond to each other. We identify a partition with the corresponding Maya diagram. For a frame ξ and a Maya diagram M = (m j ) ∞ j=0 define the Plücker coordinate of ξ corresponding to M by ξ M = det(ξ m i ,j ) −i,j≤0 . If M corresponds to a partition λ, ξ M is denoted also by ξ λ . Due to the condition (2.7) and the condition of the Maya diagram this infinite determinant can be computed by the finite determinant det(ξ m i ,j ) k≤−i,j≤0 for a sufficiently small k. Define the Schur function s (n) (t) corresponding to the partition (n) by e ∞ n=1 tnκ n = ∞ n=0 s (n) (t)κ n . and the Schur function [13] corresponding to a partition λ = (λ 1 , ..., λ l ) by s λ (t) = det(s (λ i −i+j) (t)) 1≤i,j≤l . We assign weight j to the variable t j . Then s λ (t) becomes homogeneous of weight \|λ\| = λ 1 + · · · + λ l . To a point U of UGM take a frame ξ of U and define the tau function by τ (t; ξ) = λ ξ λ s λ (t), (2.8) where the sum is taken over all partitions. We call it the Schur function expansion of τ (t; ξ). If we change the frame ξ, Plücker coordinates and consequently the tau function are multiplied by a non-zero constant due to the property of a determinant. We call τ (t; ξ), for any frame ξ of U , a tau function of U . So tau functions of a point of UGM differ by non-zero constant multiples to each other. For a frame ξ of the form (2.6), (2.5) ξ (0) = 1 and the Schur function expansion takes the form τ (t; ξ) = 1 + p.w.t, (2.9) where p.w.t. means positive weight terms. For U ∈ UGM φ we denote τ 0 (t; U ) the tau function normalized as in (2.9). We call it the normalized tau function of U . The fundamental theorem of the Sato theory is the following [23] (see also [22,9,14]). The inverse construction from a solution τ (t) of the KP-hierarchy to the point U of UGM is given using the wave function as follows [23,22,9,17]. The wave function Ψ * (t; z) and the adjoint wave function Ψ * (t; z) [4] corresponding to τ (t) are defined by Ψ(t; z) = τ (t − [z]) τ (t) e ∞ i=1 t i z −i . Ψ * (t; z) = τ (t + [z]) τ (t) e − ∞ i=1 t i z −i . (2.10) These functions are solutions of the linear problem associated with the KP-hierarchy [4]. For the inverse construction we use the adjoint wave function. Let Ψ * i (z) be the Laurent series in z defined by (τ (t)Ψ * (t; z)) \| t=(x,0,0,0,...) = ∞ i=0 Ψ * i (z)x i . (2.11) Then U = ∞ i=0 CΨ * i (z). (2.12) By this construction the following property follows. Let U be a point of UGM, τ (t) a tau function corresponding to U and f (z) = e ∞ i=1 a i z i i an invertible formal power series. Then f (z)U belongs to UGM and a tau function corresponding to it is given by e ∞ i=1 a i t i τ (t). (2.13) (n,1) curve and main results In this section the problem is fromulated and main results are stated. Let n ≥ 2 be a positive integer and α a complex number. Consider the rational curve C n defined by y n = x − α j (3.1) which we call (n, 1) curve. The point of infinity of C n corresponds to y = ∞. Take a local coordinate z around ∞ such that x = z −n , y = z −1 (1 − αz n ) 1/n . (3.2) This type of the local coordinate z was used for a general (n, s) curves of genus g ≥ 1 in constructing the multivariate sigma functions [2,3,16] and the quasi-periodic solutions of the n-reduced KP-hierarchy [2,17]. Let V n be the space of meromorphic functions on C n which have a pole only at p ∞ . It is nothing but the vector space generated by y i , i ≥ 0: V n = ∞ i=0 Cy i . By expanding y i in the local coordinate z we consider V n as a subspace of C((z)). We denote this subspace by V n (z) indicating the choice of the local coordinate z. Later we consider a coordinate change. Notice that y i = z −i (1 + O(z)), i ≥ 0. By taking linear cominations of them we have a frame of V n (z) of the form (2.5). Therefore V n (z) ∈ UGM φ . Let τ 0 (t; V n (z)) be the normalized tau function of V n (z). Our main theorem is Theorem 3.1. The following formula holds: τ 0 (t; V n (z)) = e 1 2 (q(t)+L(t)) , q(t) = i,j≥1 q i,j t i t j , L(t) = ∞ i=1 L in t in Here, q i,j = q j,i for any i, j, q i,j = 0 if j = nm for some m ≥ 1 or i + j = 0 mod.n and, for r ≥ 1, s ≥ 0, 1 ≤ p ≤ n − 1, i ≥ 1, q nr−p,ns+p = α r+s pr r + s r − p n r s + p n s . (3.3) L in = − r+s=i,r≥1,s≥0 n−1 p=1 ni (nr − p)(ns + p) q nr−p,ns+p , (3.4) where, for a non-negative integer n, x n = x(x − 1) · · · (x − n + 1) n! , n ≥ 1, x 0 = 1. The Schur function expansion of τ 0 (t; V n (z)) can also be computed. Theorem 3.2. The following expansion holds. τ 0 (t; V n (z)) = 1 + ∞ l=1 r i ,s i ,p i ,q i A l ((r i ), (s i ), (p i ), (q i ))s (nr 1 −p 1 −1,...,nr l −p l −1\|ns 1 +q 1 ,...,ns l +q l ) (t), A l ((r i ), (s i ), (p i ), (q i )) = (−1) l i=1 (ns i +q i +r i ) α l i=1 (r i +s i ) det δ p i ,q j r i + s j − 1 s j s j + p i n r i + s j 1≤i,j≤l . Here the sumation in the second term is over all r i , s i , p i , q i satisfying 1 ≤ p i , q i ≤ n − 1, r i ≥ 1, s i ≥ 0,nr 1 − p 1 > · · · > nr l − p l , ns 1 + q 1 > · · · > ns l + q l , and (m 1 , ..., m l \|m ′ 1 , ..., m ′ l ) is the Frobenius notation of a partition. The proofs of theorems are given in subsequent sections. Coordinate change and wave functions In this section we derive the equation for τ 0 (t; V n (z)) by considering a coordinate change and wave functions. Let us take w = y −1 as another local coordinate around ∞. Then x − α = w −n , y = w −1 . (4.1) By expanding elements of V n in terms of w and identifying the ambient space V of UGM with C((w)), we define the subspace V n (w) of V . Then V n (w) = ∞ i=0 Cw −i = V φ ∈ UGM φ . So τ 0 (t; V n (w)) = 1. Consider the corresponding wave and adjoint wave functions, Ψ(t; w) = e ∞ i=1 t i w −i , Ψ * (t; w) = e − ∞ i=1 t i w −i . (4.2) Here we change the local coordinate from w to z. By (4.1) z and w are connected by z −n − α = w −n . Therefore w = z(1 − αz n ) −1/n . (4.3) Expand (1 − αz n ) i/n = ∞ m=0 a i,m z m ,(4.4) where a i,m = (−α) k i n k , m = nk for some k ≥ 0 0, otherwise. (4.5) We set a i,j = 0 if j < 0 for the sake of convenience. Let us decompose w −i into two parts corresponding to negative and non-negative powers in z, w −i = i−1 m=0 a i,m z −(i−m) + f i (z), f i (z) = ∞ m=i a i,m z m−i . (4.6) Then ∞ i=1 t i w −i = ∞ i=1 t i i−1 m=0 a i,m z −(i−m) + ∞ i=1 t i f i (z). Define the new set of time variables {T i \|i ≥ 1} by T i = ∞ j=i a j,j−i t j . (4.7) Then ∞ i=1 t i w i = ∞ l=1 T l z −l + ∞ i=1 t i f i (z). (4.8) The coefficient matrix (a j,j−i ) i,j≥1 in the right hand side of (4.7) is an upper triangular matrix whose diagonal entries are all one. So it has the inverse which is again an upper triangular matrix with the same property. Therefore t i can be written as t i = ∞ j=1 b i,j T j , (4.9) where b i,j = 0 if j < i and b i,i = 1 for i ≥ 1. Then ∞ i=1 t i f i (z) = ∞ j=1 T j F j (z), (4.10) with F j (z) = ∞ m=0 j i=1 b i,j a i,m+i z m . (4.11) Substituting (4.8) and (4.10) into (4.2) we get Ψ(t; w) = e ∞ i=1 T i F i (z) e ∞ i=1 T j z −j , Ψ * (t; w) = e − ∞ i=1 T i F i (z) e − ∞ i=1 T j z −j . Consider w as a function of z by (4.3), w = w(z). Define new pair of functions of T = (T i ) and z byΨ (T; z) = z 2 w 2 dw dz e ∞ i=1 T i F + i (z) e ∞ i=1 T j z −j , (4.12) Ψ * (T; z) = e − ∞ i=1 T i F + i (z) e − ∞ i=1 T j z −j ,(4.13) where F + i (z) = F i (z) − F i (0). Then Proof. Obviously we have Res w=0 Ψ(t; w)Ψ * (t; w) dw w 2 = 0. Changing the variable from w to z and multiplying by e ∞ i=1 (−T i +T ′ i )F i (0) we get (4.14). Recall the following characterization of the KP-hierarchy in terms of wave functions. Φ(t; z) = (1 + ∞ j=1 Φ j (t)z j )e ∞ i=1 t i z −i , (4.15) Φ * (t; z) = (1 + ∞ j=1 Φ * j (t)z j )e − ∞ i=1 t i z −i ,(4.16) which satisfy Res z=0 Φ(t; z)Φ * (t ′ ; z) dz z 2 = 0, for any t and t ′ . Then there exists a solution τ (t) of the KP-hierarchy, unique up to constant multiples, such that Φ(t; z) and Φ * (t; z) are the corresponding wave and adjoint wave functions. By Proof. We prove the proposition by calculating the expansion ofΨ * (T 1 , 0, 0, ...; z) in T 1 . Let Ψ(T; z) = τ (T − [z]) τ (T) e ∞ i=1 T i z −i , (4.17) Ψ * (T; z) = τ (T + [z]) τ (T) e − ∞ i=1 T i z −i .g(z) = w −1 = z −1 + ∞ m=0 a 1,m+1 z m . Then, by the definition, V n (z) = ∞ i=0 Cg(z) i . On the other handΨ * (T 1 , 0, 0, ...; z) = e −T 1 (z −1 +F + 1 (z)) . We have F 1 (z) = ∞ m=0 a 1,m+1 z m , since b 1,1 = 1. ThereforeΨ * (T 1 , 0, 0, ...; z) = e −T 1 (g(z)−a 1,1 ) . It follows that Span C ∂ i T 1Ψ (T 1 , 0, 0, ...; z)\| T 1 =0 \|i ≥ 0 = ∞ i=0 Cg(z) i = V n (z), which completes the proof of the proposition. Expansion coefficients of F + i (z) In this section we compute the expansion coefficients of F + i (z) which appears in the wave and the adjoint wave functions (4.12), (4.13). Set Then LHS of (5.9) = α r ′ −r r ′ + p n r ′ − r −F + i (z) = ∞ j=1 c i,j z j j .r ′ s=r (−1) s−r r ′ − r r ′ − s = α r ′ −r r ′ + p n r ′ − r r ′ −r s=0 (−1) s r ′ − r s = δ r,r ′ . Proof of Proposition 5. (−1) k r + s k = (−1) s r + s − 1 s . Proof. We have s k=0 (−1) k r + s k = s k=0 (−1) k r + s − 1 k + r + s − 1 k − 1 = s−1 k=0 (−1) k r + s − 1 k + (−1) s r + s − 1 s + s k=1 (−1) k r + s − 1 k − 1 = (−1) s r + s − 1 s . Applying Lemma 5.3 to (5.11) we get (iii) of the proposition. If p = n the right hand side of (5.2) is zero , since which, by (6.1), (6.2), is rewritten as q(T\|S) = ∞ i,j=1 q i,j T i S j , L(T) = ∞ i=1 L i T i ,(6.q(T\|[z]) + 1 2 (q([z]\|[z]) + L([z])) = ∞ i,j=1 c i,j T i z j j . (6.3) This equation is satisfied if we set q(T\|[z]) = ∞ i,j=1 c i,j T i z j j , (6.4) q([z]\|[z]) + L([z]) = 0. (6.5) These equations are solved if we take q i,j = c i,j , (6.6) L k = − i+j=k k ij c i,j . (6.7) Now Theorem 3.1 follows from the following lemma which is used in the proof of Theorem 4.2 in [4]. Lemma 6.1. [4] Let τ i (t), i = 1, 2, be two functions of t = (t 1 , t 2 , ...) such that τ 1 (t + [z]) τ 1 (t) = τ 2 (t + [z]) τ 2 (t) Then τ 2 (t) = cτ 1 (t) for some constant c. Proof. Let f (t) = τ 2 (t)/τ 1 (t). Then the condition is f (t + [z]) = f (t). (6.8) Using f (t + [z]) = e ∞ i=1 z i i ∂ i f (t) = ∞ i=0 z i p i (∂)f (t), (6.9) where ∂ i = ∂/∂t i and∂ = (∂ 1 , ∂ 2 /2, ∂ 3 /3, ...), we have ∞ i=1 z i p i (∂)f (t) = 0. (6.10) Therefore p i (∂)f (t) = 0, i ≥ 1. (6.11) Since p i (∂) = ∂ i i + (terms containing only ∂ j , j < i), we get ∂ i f (t) = 0 for any i ≥ 1. By Corollary 4.5 both τ (T) constructed above and τ 0 (T; V n (z)) saisfy the equation (4.18). Therefore τ (T) = cτ 0 (T; V n (z)) for some constant c by Lemma 6.1. By setting T j = 0 for all j we have c = 1. Then we have (3.3), (3.4) by (6.6), (6.7), Proposition 5.1. Thus Theorem 3.1 is proved. Proof of Theorem 3.2 By the definition of V n (z) the following set of functions give a basis of V n (z), h ni (z) : = z −ni , i ≥ 0, h i (z) : = y i = z −i (1 − αz n ) i n , i ≥ 1, i = 0 mod.n. Then it is obvious that there exists the unique basish i , i ≥ 1 with the propertỹ h i = z −i + O(z) where O(z) denotes an element in zC[[z]]. It is given explicitly as h nr (z) = z −nr , h nr+p (z) = z −nr−p + ∞ k=1 (−1) k α k+r k + r − 1 r r + p n k + r z nk−p , (7.1) where r ≥ 0, 1 ≤ p ≤ n − 1. Proof. It is sufficient to prove thath nr+p (z) is a linear combination of h nr ′ +p (z), r ′ ≥ 0 for each p. We prove it by induction on r. By (4.4), (4.5) h nr+p (z) = z −nr−p + ∞ k=1 a nr+p,nk z n(k−r)−p . For r = 0 we have h p (z) = z −p + ∞ k=1 (−α) k p n k z nk−p =h p (z). Suppose that the assertion is valid until r. Dividing h n(r+1)+p (z) to negative and nonnegative power parts in z we have h n(r+1)+p (z) = z −n(r+1)−p + r+1 l=1 a n(r+1)+p,nl z −n(r+1−l)−p + ∞ k=1 a n(r+1)+p,n(k+r+1) z nk−p . (7.2) We erase the middle trem by subtracting linear combination ofh nk+p (z), 0 ≤ k ≤ r. Set h ′ n(r+1)+p (z) = h n(r+1)+p (z) − r+1 l=1 a n(r+1)+p,nlhn(r+1−l)+p (z). We shall show h ′ n(r+1)+p (z) =h n(r+1)+p (z). Using the assumption of induction we have Then the right hand side of (7.4) h ′ n(r+1)+p (z) = z −n(r+1)−p + ∞ k=1 a n(r+1)+p,n(k+r+1) − r+1 l=1 (−1) k a n(r+1)+p,nl α k+r+1−l × k + r − l r + 1 − l r + 1 − l + p n k + r + 1 − l z nk−p .= (−α) k+r+1 r + 1 + p n r + 1 + k 1 − r l=0 (−1) l k + r + 1 r + 1 − l k − 1 + l l . Lemma 7.2. The following equation holds: 1 − r l=0 (−1) l k + r + 1 r + 1 − l k − 1 + l l = (−1) r+1 k + r r + 1 . Proof. It is sufficient to prove the following identity for polynomials in x: 1 − r l=0 (−1) l x + r + 1 r + 1 − l x − 1 + l l = (−1) r+1 x + r r + 1 . It is easily proved by examining the zeros and the coefficients of x r+1 of both sides. We leave the details to the reader. By Lemma 7.2 we finally have h ′ n(r+1)+p (z) = z −n(r+1)−p + ∞ k=1 (−1) k α k+r+1 r + 1 + p n r + 1 + k k + r r + 1 z nk−p which is equal toh n(r+1)+p (z). This completes the proof of Proposition 7.1. By Proposition 7.1 ξ = [...,h 1 (z),h 0 (z)] is a frame of V n (z) of the form (2.6). To determine the Schur function expansion of τ 0 (t; V n (z)) we have to compute the Plücker coordinates of ξ. The Plücker coordinates corresponding to hook diagrams is easily computed as Lemma 7.3. Let ξ ′ = [..., ξ ′ −2 , ξ ′ −1 , ξ ′ 0 ] be a frame of a point of UGM of the form (2.6), that is, ξ ′ j = z j + ∞ i=1 x i,j z i , j ≤ 0. Then, for i, j ≥ 0, ξ ′ (i\|j) = (−1) j x i+1,−j . By this lemma and Proposition 7.1 we get Corollary 7.4. (i) ξ (i\|j) = 0 if j = 0 mod.n or i + j + 1 = 0 mod.n. (ii) For r ≥ 1, s ≥ 0, 1 ≤ p, q ≤ n − 1, ξ (nr−p−1\|ns+q) = δ p,q (−1) ns+q+r r + s − 1 s s + p n r + s α r+s . Finally let us recall the Giambelli formula for Plücker coordinates. Theorem 7.5. [6,20] Let ξ ′ be the frame of a point of UGM which is the same form as in Lemma 7.3. Then for l ≥ 1, a 1 > · · · > a l ≥ 0, b 1 > · · · > b l ≥ 0, In this section we calculate the generating function of {q i,j } which is used to compute the action of vertex operators on τ 0 (t; V n (z)). Let G(z) = (1 − αz n ) 1 n , H(z 1 , z 2 ) = z 2 G(z 1 ) − z 1 G(z 2 ). (8.1) Since H(z 1 , z 1 ) = 0, H(z 1 , z 2 )/(z 2 − z 1 ) is holomorphic near (0, 0). It has the expansion of the form H(z 1 , z 2 ) z 2 − z 1 = 1 + h.o.t, where h.o.t means terms containing z i 1 z j 2 with i + j ≥ 1. Thus log(H(z 1 , z 2 )/(z 2 − z 1 )) can be expanded to a power series in z 1 , z 2 . Proposition 8.1. The following expansion holds, ∂ z 1 ∂ z 2 log H(z 1 , z 2 ) z 2 − z 1 = ∞ i,j=1 q i,j z i−1 1 z j−1 2 , ∂ z i = ∂/∂z i . (8.2) Remark 8.2. For (p 1 , p 2 ) ∈ C n × C n , p i = (x i , y i ), the bilinear differential d z 1 d z 2 log H(z 1 , z 2 ) = d p 1 d p 2 log(y 1 − y 2 ) = d p 2 n−1 j=0 y j 1 y n−j 2 (x 1 − x 2 )ny n−1 1 dx 1 is the genus zero analogue of that of an (n, s) curve with positive genus which is used in the construction of the multi-variate sigma function [2,3,16]. Thus the proposition is proved. = − n p=1 ∞ i=0 1 ni + p ∞ k,l=0 (−α) k+l −i − p n k i + p n l z n(k+i)+p 1 z n(l−i)−p 2 = − n p=1 ∞ i=0 1 ni + p ∞ s≥i,r≥−i (−α) r+s −i − p n s − i i + p n r + i z ns+p 1 z nr−p 2 . Then ∂ z 1 ∂ z 2 log 1 − z 1 z 2 G(z 2 ) G(z 1 ) + = n p=1 r≥1,Corollary 8.3. (i) e q([z 1 ]\|[z 2 ]) = H(z 1 , z 2 ) z 2 − z 1 .H(z 1 , 0) −z 1 = H(0, z 2 ) z 2 = 1, we get the result. (ii) is obtained by taking the limit z 2 → z 1 in (i). Soliton solution on the non-zero constant background Applying vertex operators to τ 0 (t; V 2 (z)) it is possible to obtain soliton solutions on nonzero backgrounds. In this section we compute them explicitly. Let us consider the vertex operator [4], X(p, q) = e ∞ m=1 tm(p m −q m ) e ∞ m=1 (−p −m +q −m ) ∂m m , ∂ m = ∂ ∂t m . (9.1) Then X(p 1 , q 1 )X(p 2 , q 2 ) = (p 1 − p 2 )(q 1 − q 2 ) (p 1 − q 2 )(q 1 − p 2 ) : X(p 1 , q 1 )X(p 2 , q 2 ) :, (9.2) where the normal ordering symbol : : signifies to move differential operators ∂ n to the right. It then follows the following properties, X(p 1 , q 1 )X(p 2 , q 2 ) = X(p 2 , q 2 )X(p 1 , q 1 ) for any p i , q j , (9.3) X(p 1 , q 1 )X(p 2 , q 2 ) = 0 if p 1 = p 2 or q 1 = q 2 . (9.4) Let M, N be two positive integers, p i , q j , a i,j , 1 ≤ i ≤ M , 1 ≤ j ≤ N complex parameters such that {p i , q j } are all nonzero and mutually distinct. Consider the operator g = e M i=1 N j=1 a i,j X(p i ,q j ) . (9.5) It is well known [4] that, for a solution τ (t) of the KP-hierarchy, gτ (t) is a solution of the KP-hierarchy if it is well defined as a formal power series in t. In particular soliton solutions are obtained by taking τ (t) = 1 [4]. Here we compute gτ 0 (t; V n (z)). l , we assume i 1 < · · · < i l unless otherwise stated. Set, in general, ∆(x 1 , ..., x N ) = i<j (x i − x j ), η(t; z) = ∞ i=1 t i z i .B K = det(b i,k j ) 1≤i,j≤N . Set (κ 1 , ..., κ M +N ) = (q 1 , ..., q N , p 1 , ..., p M ). For K ∈ [M +N ] N define η K (κ) = k∈K η(t; κ k ), η K (κ −1 ) = k∈K η(t; κ −1 k ). ∆ K (κ) = i<j,i,j∈K (κ i − κ j ). To make the result simple we introduce the function τ (n) (g\|t) = ∆(q 1 , ..., q N )e N j=1 η(t;q j ) gτ 0 (t − N j=1 [q −1 j ]; V n (z)),(9.7) which is obviously a solution of the KP-hierarchy. Then Theorem 9.1. Let p i , q j , a i,j are complex parameters such that {p i , q j } are all non-zero and mutually distinct. Then τ (n) (g\|t) =τ (n) (g\|t)τ 0 (t; V n (z)). κ i κ j H(κ −1 i , κ −1 j ) i∈I (1 − ακ −n i ) 1−n n . Proof. Using (9.2)-(9.4) we have g = N j=1 1 + M i=1 a i,j X(p i , q j ) = N l=0 1≤j 1 <···<j l ≤N M i 1 ,...,i l =1 a i 1 ,j 1 · · · a i l ,j l l r<s (p ir − p is )(q jr − q js ) (p ir − q js )(q jr − p is ) × : X(p i 1 , q j 1 ) · · · X(p i l , q j l ) : . Rewriting a i,j in terms ofã i,j using (9.6) and noting that the normal ordering parts are symmetric in {p ir } and {q js } respectively we get ×e η I (p)+η J c (q) τ 0 (t − i∈I [p −1 i ] − j∈J c [q −1 j ]; V n (z)), where q J c = (q j ′ 1 , ..., q j ′ N−l ), p I = (p i 1 , ..., p i l ). By considering the summation in l, I, J as that over the indices (J c , I) we havẽ τ (n) (g\|t) = K∈( [M +N] N ) B K ∆ K (κ)e η K (κ) τ 0 (t − k∈K [κ −1 k ]; V n (z)), Now the theorem follows from (1 − αz j ) 1−n n N i<j H(z i , z j ) z j − z i τ (t; V n (z)). Proof. Using the bilinearity of q(t\|S) and the relation (6.5) we have τ 0 (t − N j=1 [z j ]; V n (z)) = e − N j=1 q(t\|[z j ])+ i≤j q([z i ]\|[z j ]) τ 0 (t; V n (z)) Then the assertion follows from Corollary 8.3. Let u(t) = 2 ∂ 2 ∂x 2 log τ (n) (g\|t) = 2q 1,1 + 2 ∂ 2 ∂x 2 logτ (n) (g\|t) (9.10) be a solution of (2. Remark 9.4. If α = 0 then u(t) given by (9.12) becomes a well known soliton solution [4,15,10,11,21]. If parameters are chosen such thatτ (2) (g\|t) is positive for real varaibales t, the second term of the right hand side of (9.12) decays exponentially as (t 1 , t 2 ) goes to infinity except finite number of directions. Thus u(t) can be considered as a soliton solution on a non-zero constant background if α = 0. Theorem 2. 2 . 2[23] For a frame ξ of a point of UGM τ (t; ξ) is a solution of the KPhierarchy. Conversely for a formal power series solution τ (t) of the KP-hierarchy there exists a unique point U of UGM such that τ (t) is a tau function of U . Lemma 4. 1 . 1The functionsΨ(T; z) andΨ * (T; z) satisfy, for any T = (T i ) and T ′ = (T ′ i ), the following bilinear equation, Res z=0Ψ (T; z)Ψ * (T ′ ; (4.3) ,(4.12),(4.13) it is obvious thatΨ(T; z) andΨ * (T; z) have the expansions of the form (4.15) and (4.16) respectively. Therefore Corollary 4. 3 . 3There exists a unique, up to constant multiples, a solution τ (T) of the KP-hierarchy with the time variables T = (T i ) such that . 4 . 4The point of UGM corresponding to τ (T) in Corollary 4.3 is V n (z). Corollary 4. 5 . 5The tau function τ (T) in Corollary 4.3 is a non-zero constant multiple of τ 0 (T; V n (z)). . 1 . 1The following properties hold.(i) c i,j = c j,i (ii) c i,j = 0 if i + j = 0 mod. n. ( iii) For (i, j) = (ns + p, nr − p) with r ≥ 1, s ≥ 0, 1 ≤ p ≤ n, c i,j = α r+s pr r need to compute (b i,j ) = (a j,j−i ) −1 .Lemma 5.2. The following properties are valid.(i) b i,j = 0 if i = j mod. n.(ii) For r, s ≥ 0, 1 ≤ p ≤ n, b nr+p,ns+p = α s−r s + p n s − r . (5.4)Proof. Since a i,j = 0 for j = 0 mod. n, , s ≥ 0, 1 ≤ p ≤ n. It follows that b i,j = 0 unless i = j mod. n. is sufficient to prove that {b i,j } given by (5.4) satisfy, for any r, r ′ ≥ 0, r ′ s=r a ns+p,n(s−r) b ns+p,nr ′ +p = δ r,r ′ . 1 1Notice that, by(5.3), if c i,j = 0 then there exists l such that l = i, j + l = 0 mod. n.Then i = l = −j mod. n. So c i,j = 0 if −i = j mod.n. This proves (ii). For i = ns + p, j = nr − p, s ≥ 0, r ≥ 1, 1 ≤ p ≤ n, we have, using Lemma 5.2, c ns+p,nr−p = −(nr − p) ns+p l=1 b l,ns+p a l,nr−p+l i,j = 0 if j = 0 mod. n. Thus (iv) of the proposition is proved. Let us prove (i) of the proposition. It is sufficient to prove c ns+p,nr−p = c nr−p,ns+p , r ≥ 1, s ≥ 0, 1 ≤ p ≤ n − 1. Using (5.2) we have c nr−p,ns+p = c n(r−1)+n−p,n(s+1)−(n−p) = α r+s (s + 1)(n − p) r + s r − 1 +We shall find the function τ (T) which satisfies (4.17)and(4.18) in the form 2) where q i,j = q j,i for any i, j, S = (S 1 , S 2 , ...). The equation (4.18) is equivalent to τ (T + [z]) τ (T) = e q(T\|[z])+ 1 2 (q([z]\|[z])+L([z])) 4.5) the coefficient of z nk−p of the right hand side is (−α) k+r+1 r + 1 1 ,...,a l \|b 1 ,...,b l ) = det ξ ′ (a i \|b j ) 1≤i,j≤l . Now Theorem 3.2 follows from Theorem 7.5 and Corollary 7.4. 8 Generating function of q i,j By this expression we see thatqi,j = 0 if i + j = 0 mod. n.A computation shows (ns + p)(nr − p) q ns+p,nr−p . e q([z]\|[z]) = (1 − αz n ) Let us set [M ] = {1, ..., M }. For a nonnegative integer l we denote [M ] l the set of all l-element subsets of [M ]. If I = {i 1 , ..., i l } ∈ [M ] ηFor I (p) = i∈I η(t; p i ), and similarly for η J (q), J ∈ [N ] l . We denote by J c ∈ [N ] N −l the complement of J in [N × (N + M ) matrix B = (b i,K = (k 1 , ..., k N ) ∈ [M +N ] N B K denotes the minor determinant corresponding to the columns specified by K: B I C I (κ)e η I (κ)− i∈I q(t\|[κ −1 i ]) ,(9.9)C I (κ) = i<j,i,j∈I 1 ,...,j l }∈( [N] l ) I={i 1 ,...,i l }∈( [M ] l )B (J c ,I) ∆(q j ′ 1 , ..., q j ′ N−l , p i 1 , ..., p i l ) ∆(q 1 , ..., q N ) × : X(p i 1 , q j 1 ) · · · X(p i l , q j l ) :, where J c = {j ′ 1 , ..., j ′ N −l } ∈ [N ] N −l and (J c , I) = (j ′ 1 , ..., j ′ N −l , i 1 , ..., i l ). On the other hand e N j=1 η(t;q j ) : X(p i 1 , q j 1 ) · · · X(p i l , q j l ) : τ 0 (t − N j=1 [q −1 j ]; V n (z)) = e η I (p)+η J c (q) τ 0 (j 1 ,...,j l )∈( [N] l ) I=(i 1 ,...,i l )∈( [M ] l ) B (J c ,I) ∆(q J c , p I ) Lemma 9. 2 . 2For parameters z i , 1 ≤ i ≤ N , the following equation is valid:τ 0 (t − N j=1 [z j ]; V n (z)) = e − N j=1 q(t\|[z j ]) N j=1 . Letτ(2) (g\|t) be given by (9.9) with n = 2. Thenu(t) = α + 2 ∂ 2 ∂x 2 logτ (2) (g\|t), (9.12)is a solution of the KP equation(2.3). Remark 9. 5 .= 1 . 8 . 518For real κ j , 1 ≤ j ≤ M + N , α, t, the functionτ (n) (g\|t) is non-negative up to overall constant if, for example, κ 1 > · · · > κ M +N > α 1/n ≥ 0 and B I ≥ 0 for allI ∈ [M +N ] N .Here are some examples ofτ(2) (g\|t).Example 9.6. In the case N = M = 1 and B = (1, b) we havẽτ (2) (g\|t) = (1 − ακ −2 1 ) − 1 2 e η(t;κ 1 )−q(t\|[κ −1 1 ]) + b(1 − ακ −2 2 ) − 1 2 e η(t;κ 2 )−q(t\|[κ −1 2 ]) .(9.13) Example 9.7. In the case N = 1, M ≥ 1, B = (b 1 , .., b M +1 ), b 1 In the case N = M AcknowledegementsI would like to thank Saburo Kakei and Yasuhiro Ohta for valuable comments which they gave me at the conference "Varieties of Studies on Non-Linear Waves" held at Kyushu University in November, 2019. I would also like to thank Yasuhiko Yamada for useful comments on the manuscript, and Kanehisa Takasaki for pointing out the reference[7]and invaluable comments related with the dispertionless KP-hierarchy. This work was supported by JSPS KAKENHI Grant Number JP19K03528.Proof of Proposition 8.1.We haveLet us expand the right hand side near (z 1 , z 2 ) = (0, 0) in C 2 assumingFor exampleSince the left hand side of (8.3) is holomorphic at (0, 0), only non-negative powers of z 1 , z 2 should remain in the expansion of the right hand side under the assumption above. DefineThenWe shall show thatq i,j = q i,j . Expand the logarithmic function asWe further expandSubstituting this expression into (8.6) and writing m = ni + p, i ≥ 0, 1 ≤ p ≤ n, we get RHS of (8.6)whereTo study the solution as a function of (x, y, t) := (t 1 , t 2 , t 3 ) the following lemma, which is calculated using Corollary 8.3, is useful. Lemma 9.9. For n = 2 the following equations holds. Sato Grassmannian and degenerate sigma function. J Bernatska, V Enolski, A Nakayashiki, Comm. Math. Phys. 374J. Bernatska, V. Enolski and A. Nakayashiki, Sato Grassmannian and degenerate sigma function, Comm. Math. Phys. 374 (2020) 627-660. Kleinian functions, hyperelliptic Jacobians and applications. V M Buchstaber, V Z Enolski, D V Leykin, Reviews in Math. and Math. Phys. 102Gordon and BreachV. M. Buchstaber, V. Z. Enolski and D. V. Leykin, Kleinian functions, hyperelliptic Jacobians and applications, in Reviews in Math. and Math. Phys. 10, No.2, Gordon and Breach, London, 1997, 1-125. Rational analogue of Abelian functions. V M Buchstaber, V Z Enolski, D V Leykin, Funct. Anal. Appl. 33V. M. Buchstaber, V. Z. Enolski and D. V. Leykin, Rational analogue of Abelian functions, Funct. Anal. Appl. 33 (1999), 83-94 Transformation groups for soliton equations. E Date, M Kashiwara, M Jimbo, T Miwa, Nonlinear Integrable Systems -Classical Theory and Quantum Theory. M. Jimbo and T. MiwaSingaporeE. Date, M. Kashiwara, M. Jimbo, and T. Miwa, Transformation groups for soli- ton equations, in "Nonlinear Integrable Systems -Classical Theory and Quantum Theory", M. Jimbo and T. Miwa (eds.), World Sci., Singapore, 1983, pp.39-119. Real theta function solutions of the Kadomtsev-Petviashvili equation. B Dubrovin, S Natanzon, Math. USSR Izvestiya. 322B. Dubrovin and S. Natanzon, Real theta function solutions of the Kadomtsev- Petviashvili equation, Math. USSR Izvestiya 32-2 (1989) 269-288. Schur function expansions of KP tau functions associated with algebraic curves. (Russian) Uspekhi Mat. V Enolski, J Harnad, Russian Math. Surveys. 664NaukV. Enolski and J. Harnad, Schur function expansions of KP tau functions associated with algebraic curves. (Russian) Uspekhi Mat. Nauk 66 (2011), no. 4(400), 137-178; translation in Russian Math. Surveys 66 (2011), no. 4, 767-807. A n−1 singularities and nKdV hierarchies. A , Mosc. Math. J. 3A. Givental, A n−1 singularities and nKdV hierarchies, Mosc. Math. J., Vol. 3 (2003), 475-505. Solitons and infinite dimensional Lie algebras. M Jimbo, T Miwa, Publ. RIMS, Kyoto Univ. 19M. Jimbo, and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. RIMS, Kyoto Univ., Vol. 19 (1983), 943-1001. Geometric realization of conformal field theory on Riemann surafces. N Kawamoto, Y Namikawa, A Tsuchiya, Y Yamada, Comm. Math. Phys. 116N. Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada, Geometric realization of conformal field theory on Riemann surafces, Comm. Math. Phys. 116 (1988) 247-308. Y Kodama, KP solitons and the Grassmannians. SpringerY. Kodama, KP solitons and the Grassmannians, Springer, 2017. Solitons in two-dimensional shallow water, CBMS-NSF regional conference series in applied mathematics 92. Y Kodama, SIAMY. Kodama, Solitons in two-dimensional shallow water, CBMS-NSF regional confer- ence series in applied mathematics 92, SIAM, 2018. Methods of algebraic geometry in the theory of nonlinear equations. I M Krichever, Russ. Math. Surv. 32I.M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surv., Vol. 32 (1977), 185-213 I G Macdonald, Symmetric Functions and Hall Polynomials. Oxford University Presssecond editionI. G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford University Press, 1995. Algebraic theory of the KP equations, in Perspectives in. M Mulase, Math. Phys., R.Penner and S.T.YauInternational Press CompanyM. Mulase, Algebraic theory of the KP equations, in Perspectives in Math. Phys., R.Penner and S.T.Yau (eds.), International Press Company, (1994), 157-223. D Mumford, Tata lectures on theta II. BirkhauserD. Mumford, Tata lectures on theta II, Birkhauser, 1983. On algebraic expressions of sigma functions for (n, s) curves. A Nakayashiki, Asian J. Math. 14A. Nakayashiki, On algebraic expressions of sigma functions for (n, s) curves, Asian J. Math. 14 (2010), 175-211. Sigma function as a tau function. A Nakayashiki, Int. Math. Res. Not. IMRN. A. Nakayashiki, Sigma function as a tau function, Int. Math. Res. Not. IMRN 2010-3 (2010), 373-394. On reducible degeneration of hyperelliptic curves and soliton solutions. A Nakayashiki, arXiv:1808.06748009, 18 pages. 15A. Nakayashiki, On reducible degeneration of hyperelliptic curves and soliton solu- tions, SIGMA 15 (2019), 009, 18 pages, arXiv:1808.06748. One step degeneration of trigonal curves and mixing solitons and quasi-periodic solutions of the KP equation. A Nakayashiki, arXiv:1911.06524proceedings of Geometric methods in physics XXXVIII. Geometric methods in physics XXXVIIIto appear in theA. Nakayashiki, One step degeneration of trigonal curves and mixing solitons and quasi-periodic solutions of the KP equation, to appear in the proceedings of Geometric methods in physics XXXVIII, arXiv:1911.06524 On the expansion coefficients of KP tau function. A Nakayashiki, S Okada, Y Shigyo, J. Integrable Systems. 27A. Nakayashiki, S. Okada and Y. Shigyo, On the expansion coefficients of KP tau function, J. Integrable Systems 2 (2017), xyx007 S Novikov, S V Manakov, L P Pitaevskii, V E Zakharov, Theory of solitons. New YorkConsultants BureauS.Novikov, S.V. Manakov, L.P. Pitaevskii and V.E. Zakharov, Theory of solitons, Consultants Bureau, New York, 1984. . M Sato, M Noumi, Tokyo, Mathematical Lecture Note. 18Soliton Equations and Universal Grassmann Manifold, Sophia Universityin JapaneseM. Sato and M. Noumi, Soliton Equations and Universal Grassmann Manifold, Sophia University, Tokyo, Mathematical Lecture Note 18, (1984) (in Japanese). Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. M Sato, Y Sato, Nolinear Partial Differential Equations in Applied Sciences. P. D. Lax, H. Fujita and G. StrangNorth-Holland, Amsterdam, and Kinokuniya, TokyoM. Sato and Y. Sato, Soliton equations as dynamical systems on infinite dimen- sional Grassmann manifold, in "Nolinear Partial Differential Equations in Applied Sciences", P. D. Lax, H. Fujita and G. Strang (eds.), North-Holland, Amsterdam, and Kinokuniya, Tokyo, 1982, pp.259-271.	[]
[ "Quantum Antiferromagnets in a Magnetic Field", "Quantum Antiferromagnets in a Magnetic Field" ]	[ "Daniel Loss \nInstitute of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n", "B Normand \nInstitute of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n" ]	[ "Institute of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland", "Institute of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland" ]	[]	Motivated by recent experiments on low-dimensional quantum magnets in applied magnetic fields, we present a theoretical analysis of their properties based on the nonlinear σ model. The spin stiffness and a 1/N expansion are used to map the regimes of spin-gap behavior, predominantly linear magnetization, and spin saturation. A two-parameter renormalization-group study gives the characteristic properties over the entire parameter range. The model is relevant to many systems exhibiting Haldane physics, and is applied here to the two-chain spin ladder compound CuHpCl.	null	[ "https://arxiv.org/pdf/cond-mat/9804151v1.pdf" ]	118,621,935	cond-mat/9804151	cbeb04daad51e5b65aa37ada2f49cf4b771e370b	Quantum Antiferromagnets in a Magnetic Field 15 Apr 1998 (October 20, 2016) Daniel Loss Institute of Physics University of Basel Klingelbergstrasse 82CH-4056BaselSwitzerland B Normand Institute of Physics University of Basel Klingelbergstrasse 82CH-4056BaselSwitzerland Quantum Antiferromagnets in a Magnetic Field 15 Apr 1998 (October 20, 2016) Motivated by recent experiments on low-dimensional quantum magnets in applied magnetic fields, we present a theoretical analysis of their properties based on the nonlinear σ model. The spin stiffness and a 1/N expansion are used to map the regimes of spin-gap behavior, predominantly linear magnetization, and spin saturation. A two-parameter renormalization-group study gives the characteristic properties over the entire parameter range. The model is relevant to many systems exhibiting Haldane physics, and is applied here to the two-chain spin ladder compound CuHpCl. The importance of low-dimensional spin systems in revealing fundamentally new quantum mechanical properties has been recognized since Haldane's conjecture [1] concerning the effects of quantum fluctuations in integerand half-integer-spin antiferromagnetic (AF) chains. We begin with the experimental observation that the magnetization curves of some such materials, thought to be prototypical of the extreme quantum limit, are in fact remarkably classical. As examples, we refer here to the Haldane (S = 1) chain NENP [2], and primarily to the twochain S = 1 2 ladder CuHpCl [3,4], which show regimes of linear magnetization whose gradient is the (classical) Néel susceptibility. With the goal of understanding such behavior, we consider the quantum AF system in an external magnetic field using the nonlinear σ model (NLsM). This widelyapplied treatment is in fact semi-classical, being truly valid only in the limit of large spin S, but has in the past formed the basis for many fundamental deductions concerning the quantum limit. We will demonstrate here its applicability for effectively integer-spin quantum systems in appreciable magnetic fields, and provide justification of this result in terms of the suppression of quantum fluctuations by the field. For the purposes of developing the present theoretical analysis, we will concentrate on the best-characterized sample in recent literature, Cu 2 (1,4diazacycloheptane) 2 Cl 4 (CuHpCl) [3,4], The Hamiltonian for the ladder system in a magnetic field b =gµ B B may be written as (1) where J is the intrachain exchange interaction and J ′ the interchain, or ladder "rung", interaction. The derivation of the NLsM in the presence of a magnetic field is presented in Ref. [5]. For N x -site chains with periodic boundary conditions along x, in the geometry shown in Fig. 1, the resulting model has J x = J, J y = 1 2 J ′ , b = (0, 0, b), and S m,Nx+i = S m,i . There are two key points. First, the spin is written in terms of slowly-varying, orthogonal, staggered and uniform components as S m,i ≃ S[(−1) i+m n m,i + al m,i ], where a is the lattice constant, and the fluctuations l about the staggered configuration must be integrated out subject to the orthogonality constraint n · l = 0. Second, the full Euclidean action in space and inverse temperature, H = i,m=1,2 [JS m,i ·S m,i+1 + J ′ S 1,i ·S 2,i + b · S m,i ],S E = S B + β 0 dτ H, contains in addition a Berry-phase term S B = i S a 2 dτ dx[−al·(n ∧ṅ)] + 4πiS(P 1 + P 2 ). P 1 = P 2 = 1 4π dτ dx(n ∧ṅ)·∂ x n separate to give simply the Pontryagin index on each chain when ∂ y n = 0, as is the case in a ladder of only two chains. The last term in S B is thus i(4πP 1 )2S, demonstrating that the system will have integer-spin characteristics for any value of S, and the topological term may be ignored [6,7]. The action for the quasi-one-dimensional ladder system, in 1+1 Euclidean dimensions denoted by µ, is then S E = 1 2g τ,x [(∂ µ n) 2 − (b 2 − (n · b) 2 ) + 2ib·n ∧ṅ],(2) where g = (2/N y S) J /J x is the bare coupling constant, J = J x + 1 2 J y , and the integral over τ is to upper limit L T = cβ, with c = 2Sa J xJ /h (h → 1) the effective spin-wave velocity. We have left explicit the number N y of chains in the ladder; for the compound under consideration J y > J x and N y = 2, giving an effectively rigid rung coupling. The form of the NLsM in an external field given by the second term in Eq. (2) has been derived previously [5], and its implications considered in the low-field limit. In what follows we will examine its effects for arbitrary fields. To gain initial insight into the effect of the magnetic field, we consider the spin stiffness of the ladder system [8]. Taking the staggered spin configuration to be subject to a twist θ in the plane normal to the applied field ( Fig. 1), we calculate the free energy F (b, θ) to one-loop order in the important out-of-plane spin fluctuations, and deduce the spin stiffness from ρ s = 1 2 cL ∂ 2 F ∂θ 2 θ=0 . ρ s = ρ 0 s 1 − g LL T k 1 k 2 + (b/c) 2 ,(3) where ρ 0 s = c/2g is the classical value and the sum provides both quantum and thermal (through the finite "length") corrections to first order. x y z B FIG. 1. Representation of two-chain ladder system, showing directions of staggered moment, twist and applied field. We restrict the analysis to the low-temperature, or "quantum" case L T > L. Evaluating the summation between spatial limits π/L and π/a, and introducing the "magnetic length" L m as π/L m = b/c, ρ s = ρ 0 s 1 + g 4π ln (a/L) 2 + (a/L m ) 2 (1 + (a/L m ) 2 ) .(4) L in Eq.(4) may be regarded as the correlation length ξ, beyond which segments of the ladder behave independently, and can be computed from ρ s = 0. In the weak-field limit (L m → ∞) one recovers the result [8] ξ 0 = ae 2π/g = ae απS , where α = J x /J. The general solution is ξ(B) = ξ 0 / 1 − (L * m /L m ) 2 , where L * m = a √ e 4π/g − 1 gives the critical field B * at which the correlation length diverges. For fields B < B * , the finite correlation length may be written as ξ(B) = ae απS , whereS = S[1 − g/4π ln(1 − (L * m /L m ) 2 ) ] is a growing value of the effective spin. For B > B * , the field enforces a quasi-long-ranged correlation throughout the system [9], and it is most convenient to write the spin stiffness as ρ s = ρ 0 s [1 − g/4π ln(1 + (L m /a) 2 )], which recovers the bare value as B → ∞. Finally, the divergence of the correlation length at B * corresponds to the closing of the gap ∆ to spin excitations according to ∆ ∝ 1 − (B/B * ) 2 . This situation is summarized in Fig. 2. However, if the B derivative is taken of the free energy in order to compute the magnetization, the resulting terms do not yield the required zero result in the spin gap regime. This indicates the breakdown of the approximation implicit in deriving F (b, θ) that the field be large on the scale of in-plane fluctuation energies \|φ\|. Thus the spin stiffness analysis, while qualitatively revealing of the behavior of the system, is not reliable at low fields. The situation in the weak-field regime may be addressed by a 1/N expansion [10], which is expected to be appropriate in describing spin-gap phases. The staggered spin n is taken to exist in an N -dimensional spin space, in which only the component n z is selected by the field. The relevant parts of the O(N ) action are S E = 1 2g dx(∂ µ n) 2 −b 2 (1 − n 2 z ) − iλ(n 2 − 1),(5) whereb denotes b/c and the constraint that n have unit magnitude is made explicit with the Lagrange multiplier iλ, whose saddle-point value is given by 1 g = (N − 1) k 1 k 2 + iλ + k 1 k 2 + iλ +b 2 ,(6) in which the B-field term is found to appear only at O(1/N ). iλ functions as a mass, or cutoff term in momentum integrations, and is thus an upper lengthscale for cooperative processes in the system, or simply a correlation length (inverse excitation gap). Writing the saddle-point solution as iλ = c 2 π 2 /ξ(B) 2 and carrying out the summation at low T gives an expression analogous to ρ s = 0 emerging from Eq.(4). = − 2b N ensures both that the corresponding magnetization contribution from the k summation terms in F is identically zero to O(1/N ), and that the constant term b 2 + c 2 π 2 /ξ 2 yields 2b(1 − 1/N ). It is clear that the behavior required of a gapped system is returned in the weak-field regime only on making the well-recognized identification (motivated by comparison with renormalization-group results [10]) N → N − 2, and by returning to the physical situation N = 3. Then the magnetization is indeed zero, and the saddle-point solution for ξ(B) becomes precisely that deduced from the spin stiffness analysis, with the same critical field B * . With this replacement caveat we thus obtain from these two approaches a consistent picture of both weak-and strong-field regimes. To quantify the regimes of validity of the foregoing analyses, we consider next a renormalization-group (RG) approach to the NLsM in an applied field. Taking the model in the form S E = 1 2g dx[(∂ µ n) 2 −b 2 (1 − n 2 z ) + 4ibn xṅy ](7) and transforming to variables φ and √ gσ z = n z representing respectively the in-and out-of-plane fluctuations, the latter in a form suitable for perturbative expansion in the coupling constant g, we obtain L E = 1 2g A −b 2 − 1 2 σ z (−∂ 2 µ +b 2 − A)σ z + O(g).(8) A denotes the in-plane terms (∂ µ φ) 2 + 2ibφ, which because φ(τ, x) is assumed to vary slowly can be taken to be a small constant (no fast Fourier modes) in the momentum shell γπ/a < \|k\| < π/a (γ → 1). Performing the integral and expanding in A, the form of Eq.(8) is recovered with coefficients g(a ′ ) andb(a ′ ) 2 given by partial traces. Evaluation of these and differentiation with respect to the flow parameter l = ln(a ′ /a), leads to the coupled RG equations dg dl = g 2 2π 1 1 +β 2 , dβ 2 dl = 2β 2 − g 2 2π ln 1 +β 2 ,(9) in whichβ = a ′b (a ′ ). These new RG equations possess a variety of interesting limiting cases, whose detailed study we defer to a future publication [13]. For the present purposes, we concentrate on the fixed points to obtain a qualitative picture of the RG flow diagram, and on the consequences for the magnetization. (i) Seeking a fixed point by weak-field expansion aroundβ * = 0, we find dg/dl ≃ g 2 /2π and thus d lnβ 2 /dl = 2 − d ln g/dl, which may be solved to yield g 0 g = 1 − g 0 2π l,β =β 0 e l 1 − g 0 2π l 1/2 .(10) The fixed point (g * ,β * ) = (∞, 0) is clearly stable if the flow is stopped at l * = 2π/g 0 . The system will flow to this strong-coupling regime if the starting valueβ 0 is sufficiently small. The lengthscale L * = ae l * at which the flow stops may be compared with the spatial and thermal dimensions L, L T of the system to calculate directly the effects of finite size and temperature [13]. (ii) At strong fields (β → ∞), dg/dl = 0, or g = g 0 , indicating that the coupling is not renormalized, and d lnβ 2 = 2dl, from which it follows thatb(l) =b 0 , i.e. neither is the field. Numerical solution of Eqs. (9) leads to the flow diagram in Fig. 3. The regime (ii) of strong initial field may be termed the weak-coupling situation, where g andb are weakly renormalized to finite values. Here the perturbation theoretic approach is consistent and the weak coupling corresponds to deconfinement of the excitations on the lengthscale L m set by the field. The regime (i) is the strong-coupling condition, of confinement of (gapped) excitations, where the assumption of small g in the derivation proves to be inconsistent. However, in this regime one may deduce the critical lengthscale L * governing the behavior of the system and, as we will show below, that the magnetization is zero (inset Fig. 3). The critical starting field separating the two regions isb * = 0.46. The properties of the two regimes may be illustrated by considering the correlation length in each. ξ is a physical quantity and does not change under the RG flow, meaning that dξ/da ′ = 0, whence ∂g ∂l ∂ξ ∂g + ∂β 2 ∂l ∂ξ ∂β 2 + ξ = 0. (i) At smallβ 2 , −∂ξ/ξ ≃ 2π∂g/g, which has solution ξ 0 = ξe 2π(1/g0−1/g) . Under the reasonable assumption that ξ → a, the bare lattice constant, as the system flows to the strong-coupling limit (1/g 0 → 0), we recover the expected result ξ 0 = ae 2π/g0 (see above) for the finite physical correlation length. (ii) For largeβ 2 , −2∂ξ/ξ ≃ ∂β 2 /β 2 leads to ξ 0 = ξL/a (b 0 invariant) under the RG flow, and we see that for any assumed finite ξ, the bare correlation length ξ 0 is the system size L or L T . We return to the experimental motivation for the above analysis, and compute the magnetization M = ∂F/∂B of the model over the full field range, M = −gµ B N x N y (b/4J) + M d + M f .(12) M contains contributions linear in B from the quadratic term (5), M d from the dynamical term in the trace (cf. (3)) due to fluctuations of n out of the plane perpendicular to B, and M f from in-plane fluctuation termṡ φ. The last give a sawtooth form leading to magnetization steps [13], a finite-size effect which will not be considered further here. While the linear term is always present, we have shown that below a threshold field B * , where the system has a spin gap, it is cancelled by the corresponding correlation-length term. Above an upper threshold B c2 , the magnetization will saturate at the value M s =gµ B SN s (N s = N x N y ), and this effect is not contained in the (large-S) model as applied. Systematic inclusion of an additional total spin constraint is possible [13], but to compress the analysis we will here apply M s as a simple cutoff. The "dynamical" contribution is given consistently to lowest order in the small parameter c/b, and in limit of low T , by the constant M d = 1 2 N xg µ B . Next-order corrections involving the spin excitations have the form M ′ d ∼ B ln(B/J). Specializing to the two-chain ladder material CuHpCl, the exchange constants deduced from the magnetization and susceptibility [3,4] are J ′ = 13.2K and J = 2.4K, whenceJ = 13.3T and J x = 3.6T. Taking the simplest case of constant M d , and the lower critical field B c1 for onset of magnetization where M (B c1 ) = 0, we obtain B c1 =J/g = 6.6T. The saturation field B c2 is given from M (B c2 ) = M s as B c2 = 4S/gJ = 13.3T. These values are in remarkably good agreement with a linear extrapolation of the magnetization data at lowest temperature in Ref. [3], where B c1 = 6.8T and B c2 = 13.7T. Such an extrapolation appears closer to the data than the predictions [11] of a repulsive boson model. We note further that the gradient of the linear magnetization is precisely the classical Néel susceptibility χ ⊥ AF = (gµ B ) 2 /(4Ja 2 ) per unit volume, a result in itself remarkable for an AF in the quantum limit. The computed magnetization is shown in Fig. 4, where the dashed line indicates the validity limit of the calculation. The predicted magnetization shows a surprising degree of accuracy for a model strictly valid only in the limit of large S. We note that the NLsM has recently been used with considerable success also for small systems with small S [12]. Its accuracy in the current context may be ascribed in part to the suppression of quantum fluctuation effects by the field, as shown above, such that in the high-field regime, which covers much of that beyond B c1 , the behavior of a system with no topological term is rather classical. Weak interladder interactions causing the real material to display 3d order at intermediate fields will also enhance classical behavior. Further perspective on the above results is given by computing the critical field B * from the spin stiffness and 1/N treatments. We deduce from L * m that B * = 0.32J/g = 2.1T, a value rather lower than B c1 above, and emphasize that in the regime between these values, as also seen in the RG analysis, the treatment is not reliable. In the RG approach, the magnetization M =g µ B a c ∂F ∂β 0 =g µ B a c ∂β ∂β 0 ∂F ∂β + ∂g ∂β 0 ∂F ∂g ,(13) from which we see in the small-B regime, where ∂g/∂β 0 = 0 and ∂β/∂β 0 = e l 1 − g 0 l/2π, that for any F (b) analytic inb, M (B 0 ) ∼ 1 − g 0 /2π ln(L/a) → 0 as L → L * . Thus the magnetization vanishes in the strongcoupling limit as required. From the numerical solution, the field scaleb * gives B * = 1.6T, in reasonable agreement with the value B * above. We have already shown that the regime approaching the physical B c1 lacks a suitable calculation scheme. Physically, the problem is one of determining the effects of quantum fluctuations of the spin system when the spin gap is weakened by the applied field, and the quantum disordered phase thus made less robust. We have detailed above the lowest-order predictions of the NLsM for the closing of the gap, and expect these to be appropriate when the gap becomes smaller than other energy scales (quantum, thermal and finitesize fluctuations) in the system. The foregoing analysis is not restricted to CuHpCl, but applies also to the S = 1 AF ("Haldane") chain. NENP is considered to be a prototypical case for observation of the Haldane gap but for the complication of a large single-ion anisotropy. The present study predicts again the qualitative features of a gapped regime of zero magnetization, followed by approximately linear behaviour towards saturation (not achieved), as in experiment [2]. We will present elsewhere the quantitative aspect of this problem. There has been considerable recent interest in the possibility of magnetization plateaus in certain systems, and we observe that in the current model these may be expected, for example in S > 1 chains, when the field strength is such that the projected in-plane spin Sn ⊥ is of integer amplitude, leading to a gapped phase. In summary, the nonlinear σ model treatment reproduces well the behavior of quantum antiferromagnets in an external field. For effectively integer-spin systems, meaning those with a trivial topological term, this statement is valid even in the extreme quantum limit of low spin and low dimensionality, because the magnetic field acts to suppress quantum fluctuation effects. We are grateful to the Swiss National Fund for financial support. BN wishes to acknowledge the generosity of the Treubelfonds. FIG. 2 . 2Schematic depiction of behavior of correlation length or spin gap, and of spin stiffness, with applied field.At weak fields (ξ ≪ L m ) one finds ξ ≃ ae 2π/N g (1 + (a 2 /N L 2 m )e 4π/g , a O(1/N ) correction to a result differing from the previous one by a power of 1/N in the ex- FIG. 3 . 3RG flow diagram for g andb. Strong-and weak-coupling regimes are separated by separatrix s. FIG. 4 . 4Computed magnetization of ladder system CuHpCl. . F D M Haldane, Phys. Lett. 93464F. D. M. Haldane, Phys. Lett. 93A, 464 (1983); . Phys. Rev. Lett. 501153Phys. Rev. Lett. 50, 1153 (1983). . Y Ajiro, T Goto, H Kikuchi, T Sakakibara, T Inami, Phys. Rev. Lett. 631424Y. Ajiro, T. Goto, H. Kikuchi, T. Sakakibara, and T. Inami, Phys. Rev. Lett. 63, 1424 (1989). . G Chaboussant, P A Crowell, L P Lévy, O Piovesana, A Madouri, D Mailly, Phys. Rev. B. 553046G. Chaboussant, P. A. Crowell, L. P. Lévy, O. Piovesana, A. Madouri, and D. Mailly, Phys. Rev. B 55, 3046 (1997). . P R Hammar, D H Reich, C Broholm, F Trouw, preprint # cond-mat/9708053P. R. Hammar, D. H. Reich, C. Broholm, and F. Trouw, preprint # cond-mat/9708053. . S Allen, D Loss, Physica A. 23947S. Allen and D. Loss, Physica A, 239, 47 (1997). . D V Kveshchenko, Phys. Rev. B. 50380D. V. Kveshchenko, Phys. Rev. B 50, 380 (1994). . G Sierra, J. Math. Phys. A. 293289G. Sierra, J. Math. Phys. A 29, 3289 (1996). . D Loss, D L Maslov, Phys. Rev. Lett. 74178D. Loss and D. L. Maslov, Phys. Rev. Lett. 74, 178 (1995). Before saturation there is no true long-range order, but the field imposes a quasi-long-range order of the XYmodel type: I Affleck. Phys. Rev. B. 433215Before saturation there is no true long-range order, but the field imposes a quasi-long-range order of the XY - model type: I Affleck, Phys. Rev. B 43, 3215 (1991). A M Polyakov, Gauge Fields and Strings. Harwood, LondonA. M. Polyakov, Gauge Fields and Strings, (Harwood, London, 1986). . C A Hayward, D Poilblanc, L P Lévy, Phys. Rev. B. 5412649C. A. Hayward, D. Poilblanc, and L. P. Lévy, Phys. Rev. B 54, R12649 (1996). . A Chiolero, D Loss, Phys. Rev. Lett. 80169A. Chiolero and D. Loss, Phys. Rev. Lett. 80, 169 (1998). . B Normand, J Kyriakidis, D Loss, in preparationB. Normand, J. Kyriakidis, and D. Loss, in preparation.	[]
[ "GENERALIZED MATRIX SPECTRAL FACTORIZATION WITH SYMMETRY AND APPLICATIONS TO SYMMETRIC QUASI-TIGHT FRAMELETS", "GENERALIZED MATRIX SPECTRAL FACTORIZATION WITH SYMMETRY AND APPLICATIONS TO SYMMETRIC QUASI-TIGHT FRAMELETS" ]	[ "BIN HANChenzhe Diao " ]	[]	[]	Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal and image processing. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for 2 × 2 matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry.2010 Mathematics Subject Classification. 42C40, 42C15, 41A15, 65D07.	10.1016/j.acha.2023.02.002	[ "https://arxiv.org/pdf/2112.01143v1.pdf" ]	244,799,248	2112.01143	018d19dcc84c9ca892bcc92415cecfb67b46103c	GENERALIZED MATRIX SPECTRAL FACTORIZATION WITH SYMMETRY AND APPLICATIONS TO SYMMETRIC QUASI-TIGHT FRAMELETS BIN HANChenzhe Diao GENERALIZED MATRIX SPECTRAL FACTORIZATION WITH SYMMETRY AND APPLICATIONS TO SYMMETRIC QUASI-TIGHT FRAMELETS Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal and image processing. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for 2 × 2 matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry.2010 Mathematics Subject Classification. 42C40, 42C15, 41A15, 65D07. Introduction, Motivations and Main Results Wavelet frames (also called as framelets) are often constructed through their associated filter banks from refinable functions, which further boil down to the fundamental problem of factorizing matrices of Laurent/trigonometric polynomials with various properties. We refer the readers to [1-11, 13, 15, 16, 18-22, 25-29, 31-35] for more details on framelets. As a consequence, appropriately factorizing matrices of Laurent polynomials plays a crucial role in the construction of framelets with desired properties such as symmetry and vanishing moments. Both symmetry and vanishing moments are highly desirable among many key properties of framelets for sparse representations and boundary treatment. This motivates us to study generalized matrix spectral factorization with symmetry. 1.1. Main result on generalized matrix spectral factorization. Let u(z) = k∈Z u(k)z k , z ∈ C\{0} be a Laurent polynomial, where its coefficient sequence u = {u(k)} k∈Z is finitely supported with u(k) ∈ C for all k ∈ Z and u is often called a filter in the literature of engineering. We say that u has symmetry if its coefficient filter u has symmetry: u(z) = z c u(z −1 ) or equivalently, u(k) = u(c − k), ∀ k ∈ Z,(1.1) for some ∈ {−1, 1} and c ∈ Z. For any nonzero Laurent polynomial u having symmetry, we define the following symmetry operator S: Su(z) := u(z) u(z −1 ) , z ∈ C\{0}. (1.2) It is then easy to see that a Laurent polynomial u has symmetry as in (1.1) if and only if Su(z) = z c . In this case, we say that u has symmetry with type z c . For the zero Laurent polynomial 0, we say that it has symmetry with any type. For any t × r matrix P(z) = k∈Z P (k)z k of Laurent polynomials, where P (k) ∈ C t×r for all k ∈ Z, we define the Hermitian conjugate of P via P (z) := P(z −1 ) T = k∈Z P (k) T z −k , z ∈ T := {ζ ∈ C : \|ζ\| = 1}. Let A(z) be a Hermitian 2 × 2 matrix of Laurent polynomials, i.e. A(z) = A (z). Moreover, suppose that all elements of A have symmetry. The problem of generalized matrix spectral factorization with symmetry is to decompose the matrix A(z) into the following form: A(z) = U(z)Diag( 1 , 2 )U (z) with 1 , 2 ∈ {−1, 1} for all z ∈ T,(1.3) where U is some 2 × 2 matrix of Laurent polynomials such that all the entries of U have symmetry. The matrix spectral factorization with symmetry corresponds to (1.3) with 1 = 2 = 1 has been characterized in [21,Theorem 2.3] and [15,Theorems 3.2]. (1.3) without requiring the symmetry property of U for 1 = 2 = 1 is just the well-known standard matrix spectral factorization (also called matrix Fejér-Riesz lemma, e.g., see [24]) in the literature. However, the case 1 = 2 in (1.3) is much more difficult and technical, because we lose the positive semidefinite property A(z) 0 for all z ∈ T and we have to deal with a difference of Hermitian squares instead of sum of Hermitian squares as in [15,21]. For a Laurent polynomial u, we say that u has the difference of (Hermitian) squares (DOS) property with respect to symmetry type z c with ∈ {±1} and c ∈ Z, if there exist Laurent polynomials u 1 and u 2 having symmetry, such that u 1 (z)u 1 (z) − u 2 (z)u 2 (z) = u(z) and Su 1 (z) Su 2 (z) = z c . (1.4) For z 0 ∈ C\{0}, we denote by Z(u, z 0 ) the multiplicity of the zero z 0 of the Laurent polynomial u. We now complete the picture on generalized matrix spectral factorization by stating the following main theorem on the generalized spectral factorization of 2 × 2 matrices of Laurent polynomials. Theorem 1.1. Let A be a 2×2 Hermitian matrix of Laurent polynomials having symmetry and define α(z) := SA 1,2 (z), where A j,k is the (j, k)-entry of A for j, k ∈ {1, 2}. Then there is a 2×2 matrix U (z) of Laurent polynomials with all its entries having symmetry such that A(z) = U(z)Diag(1, −1)U (z) holds (i.e., (1.3) holds with 1 = 1 and 2 = −1) and the symmetry type of U satisfies SU 1,1 (z) SU 2,1 (z) = SU 1,2 (z) SU 2,2 (z) = α(z), (1.5) if and only if the following two conditions hold: (1) det(A(z)) = −d(z)d (z) for some Laurent polynomial d with symmetry. (2) Define p 0 (z) := gcd(A 1,1 (z), A 1,2 (z), A 2,1 (z), A 2,2 (z)) and p(z) := p 0 (z) (z − 1) Z(p 0 ,1) (z + 1) Z(p 0 ,−1) . (1.6) Then p(z) satisfies the DOS (Difference of Squares) condition with respect to type α(z)Sd (z). The technical long proof of Theorem 1.1 will be given in Section 3, where we shall also explain the compatibility condition in (1.5) for the symmetry patterns between the matrices A and U. Writing a Laurent polynomial into sums and differences of Hermitian squares is inevitable for a generalized spectral factorization problem in Theorem 1.1. We now characterize the difference of (Hermitian) squares (DOS) property below whose proof is given in Subsection 3.5. (1) The Laurent polynomial u has real coefficients and u = u; (2) The Laurent polynomial u satisfies exactly one of the following technical conditions: (i) If = 1 and c ∈ 2Z, then there is no condition on u; (ii) If = 1 and c ∈ 2Z + 1, then Z(u, x) ∈ 2Z for all x ∈ (−1, 0); (iii) If = −1 and c ∈ 2Z, then Z(u, x) ∈ 2Z for all x ∈ (−1, 0) ∪ (0, 1); (iv) If = −1 and c ∈ 2Z + 1, then Z(u, x) ∈ 2Z for all x ∈ (0, 1). 1.2. Application of generalized matrix spectral factorization on constructing quasi-tight framelets. Symmetric wavelets and framelets have been extensively studied and applied in applications, e.g., see [2, 5, 9-11, 13, 15, 16, 18, 21-23, 26-36] and references therein. Construction of wavelets and framelets is intrinsically connected with factorizing matrices of Laurent polynomials. In this paper, we are interested in quasi-tight framelets with two generators having the symmetry property. Our investigation is inspired by the recent development of quasi-tight framelets [7,8,[18][19][20] and the importance of (anti-)symmetric wavelet frames in many applications. Using Theorem 1.1, we can obtain a necessary and sufficient condition for the existence of quasi-tight framelets with symmetry. To present our main result on quasi-tight framelets with symmetry, we need several notations. By (l 0 (Z)) r×s we denote the space of all finitely supported matrix-valued sequences/filters on Z, i.e., u ∈ (l 0 (Z)) r×s if u : Z → C r×s has only finitely many nonzero terms. For u = {u(k)} k∈Z ∈ (l 0 (Z)) r×s , its associated matrix Laurent polynomial is u(z) := k∈Z u(k)z k for z ∈ C\{0}. Define u := u(−·) T and then u (z) := [u(z)] = k∈Z u(k) T z −k . For u ∈ (l 0 (Z)) s×r and v ∈ (l 0 (Z)) t×r , their convolution is u * v := j∈Z u(· − j)v(j) and the matrix Laurent polynomial associated with u * v is u(z)v (z). One important feature for a framelet is the orders of vanishing moments of its generators. For any compactly supported ψ ∈ (L 2 (R)) s , we say that ψ has order m vanishing moments if R x j ψ(x)dx = 0, j = 0, . . . , m − 1. We define vm(ψ) := m with m being the largest positive integer satisfying the above identities. For a filter b ∈ l 0 (Z), we say that b has order m vanishing moments if (z − 1) m divides the Laurent polynomial b (z), and we similarly define vm(b) := vm(b) := m with m being the largest positive integer satisfying this. The order of vanishing moments is naturally linked to the concept of the sum rules of a filter. For a sequence a ∈ l 0 (Z), we say that a has n sum rules if (z + 1) n divides the Laurent polynomial a(z). We define sr(a) := sr(a(z)) := n, with n being the largest such integer. For an integer k ∈ Z, we define odd(k) := 1 − (−1) k 2 = 0, if k is even, 1, if k is odd, k ∈ Z. (1.7) As an application of Theorem 1.1, our main result on symmetric quasi-tight framelets with two generators is the following result, whose proof is given in Section 4. Theorem 1.3. Let Θ, a ∈ l 0 (Z) be given filters having symmetry SΘ(z) = 1 and Sa(z) = z c for some c ∈ Z, and Θ = Θ. Let n b be any chosen integer satisfying if and only if the following two conditions hold: (1) det(N a,Θ\|n b (z)) = −d n b (z)d n b (z) for some Laurent polynomial d n b (z) having symmetry. (2) p(z) has the difference of (Hermitian) squares (DOS) property with respect to symmetry type (−1) c+n b z odd(c+n b )−1 Sd n b (z), where p(z) is defined through p 0 (z) := gcd N a,Θ\|n b (z) 1,1 , N a,Θ\|n b (z) 1,2 , N a,Θ\|n b (z) 2,1 , N a,Θ\|n b (z) 2,2 , (1.14) p(z) := p 0 (z) (z − 1) Z(p 0 ,1) (z + 1) Z(p 0 ,−1) . (1.15) Moreover, if Θ(z) = 1, then p 0 (z) = p(z) = 1 and item (2) is automatically satisfied. If in addition a(1) = Θ(1) = 1 and φ ∈ L 2 (R), where φ(ξ) := ∞ j=1 a(e −i2 −j ξ ) for ξ ∈ R, define η := k∈Z Θ(k)φ(· − k) and ψ := 2 k∈Z b (k)φ(2 · −k) (1.16) for = 1, 2, then all η, ψ 1 , ψ 2 ∈ L 2 (R) have symmetry with ψ 1 , ψ 2 having at least order n b vanishing moments, and {η, φ; ψ 1 , ψ 2 } ( 1 , 2 ) forms a compactly supported quasi-tight framelet in L 2 (R), i.e., f = k∈Z f, η(· − k) φ(· − k) + 2 =1 ∞ j=0 k∈Z f, ψ 2 j ;k ψ 2 j ;k , ∀ f ∈ L 2 (R) (1.17) with the above series converging unconditionally in L 2 (R), where ψ 2 j ;k := 2 j/2 ψ (2 j · −k). 1.3. Paper structure. The paper is organized as follows. In Section 2, we briefly review some necessary background on framelets. In particular, we shall explain more on our motivations of considering quasi-tight framelets with symmetry, and some illustrative examples will be provided. In Section 3, we introduce some notations for Laurent polynomials with symmetry first. Next, we consider two special cases of Theorem 1.1 before we prove Theorem 1.1 on the factorization of a 2 × 2 matrix of Laurent polynomials with symmetry. Finally, in Section 4, we use Theorem 1.1 to prove Theorem 1.3, which characterizes quasi-tight framelet filter banks {a; b 1 , b 2 } Θ,(1,−1) with symmetry, and their associated symmetric quasi-tight framelets {η, φ; ψ 1 , ψ 2 } (1,−1) in L 2 (R). Motivation and Examples of Quasi-tight Framelets with Symmetry To further explain our motivations, in this section we shall first review some necessary background on framelets and then provide several illustrative examples of symmetric quasi-tight framelets as an application of Theorems 1.1 and 1.3. 2.1. Introduction to quasi-tight framelets. For φ, ψ 1 , . . . , ψ s ∈ L 2 (R), we say that {φ; ψ 1 , . . . , ψ s } is a framelet (i.e., a wavelet frame) in L 2 (R) if there exist positive constants C 1 , C 2 > 0 such that C 1 f 2 L 2 (R) k∈Z \| f, φ(· − k) \| 2 + s =1 ∞ j=0 k∈Z \| f, ψ 2 j ;k \| 2 C 2 f 2 L 2 (R) , ∀ f ∈ L 2 (R), where ψ 2 j ;k := 2 j/2 ψ (2 j · −k). For φ, η, ψ 1 , . . . , ψ s ∈ L 2 (R) and 1 , . . . , s ∈ {±1}, we say that {η, φ; ψ 1 , . . . , ψ s } ( 1 ,..., s) is a quasi-tight framelet in L 2 (R) if {η; ψ 1 , . . . , ψ s } and {φ; ψ 1 , . . . , ψ s } are framelets in L 2 (R) and f = k∈Z f, η(· − k) φ(· − k) + s =1 ∞ j=0 k∈Z f, ψ 2 j ;k ψ 2 j ;k , ∀ f ∈ L 2 (R),(2.1) with the above series converging unconditionally in L 2 (R). We say that {φ; ψ 1 , . . . , ψ s } ( 1 ,..., s) is a quasi-tight framelet in L 2 (R) if {φ, φ; ψ 1 , . . . , ψ s } ( 1 ,..., s) is a quasi-tight framelet in L 2 (R). By [14,Propositions 4 and 5] or [17,Proposition 4.3], (2.2) implies that {ψ 1 , . . . , ψ s } ( 1 ,..., s) is a homogeneous quasi-tight framelet in L 2 (R), that is, f = s =1 j∈Z k∈Z f, ψ 2 j ;k ψ 2 j ;k , ∀ f ∈ L 2 (R), (2.2) with the above series converging unconditionally in L 2 (R). A quasi-tight framelet {φ; ψ 1 , . . . , ψ s } ( 1 ,..., s) in L 2 (R) with 1 = · · · = s = 1 is often called a tight framelet in the literature. In this case, {ψ 1 , . . . , ψ s } is a homogeneous tight framelet in L 2 (R). In practice, compactly supported wavelets and framelets are highly desired to reduce computational complexity. Quite often, we derive them from a compactly supported refinable function φ, i.e., φ = 2 k∈Z a(k)φ(2 · −k) or equivalently, φ(2ξ) = a(e −iξ ) φ(ξ),(2.3) holds for some finitely supported sequence a ∈ l 0 (Z), where the Fourier transform is defined as f (ξ) := R f (x)e −ixξ dx for f ∈ L 1 (R) and can be naturally extended to square integrable functions and tempered distributions. For a filter a ∈ l 0 (Z) satisfying a(1) = 1 (often called a refinement mask or a low-pass filter in the literature), it is easy to check that (2.3) holds for a compactly supported function/distribution φ by defining φ through its Fourier transform φ(ξ) := ∞ j=1 a(e −i2 −j ξ ), ξ ∈ R. (2.4) From a given refinable function φ ∈ L 2 (R) associated with a low-pass filter a ∈ l 0 (Z), to construct a quasi-tight framelet, we first find an appropriate filter Θ ∈ l 0 (Z) with Θ = Θ and Θ(1) = 1, which is often called a moment correcting filter. Define M a,Θ as in (1.11). Next, we want to find b = (b 1 , . . . , b s ) T ∈ (l 0 (Z)) s×1 and 1 , . . . , s ∈ {±1} such that b 1 (1) = · · · = b s (1) = 0 and M a,Θ (z) = b(z) b(−z) Diag( 1 , . . . , s ) b(z), b(−z) . (2.5) {a; b 1 , . . . , b s } Θ,( 1 ,..., s) satisfying (2.5) is called a quasi-tight framelet filter bank. Define η and ψ as in (1.16) for = 1, . . . , s. If φ ∈ L 2 (R), then {η, φ; ψ 1 , . . . , ψ s } ( 1 ,..., s) is a quasi-tight framelet in L 2 (R) satisfying ψ (0) = 0 for all = 1, . . . , s. This approach of constructing quasi-tight framelets is known as a special case of the Oblique Extension Principle (OEP) in the literature, and such constructed framelets are called OEP-based framelets, see [3,5,6,18] and references therein for details. To construct a symmetric quasi-tight framelet, we require that our low-pass filter a ∈ l 0 (Z) should have symmetry. For the construction of an orthogonal wavelet from φ, its associated refinement mask a ∈ l 0 (Z) must satisfy ( [4]) \|a(z)\| 2 + \|a(−z)\| 2 = 1, ∀z ∈ T. (2.6) As pointed out by Daubechies in [4], there does not exist a compactly supported real-valued (dyadic) orthonormal wavelet with symmetry, except for the Haar wavelet which is discontinuous. Due to the aforementioned restrictions (2.6) on compactly supported (dyadic) real-valued orthogonal wavelets, it is natural to consider tight framelets with symmetry, which only require the following less restrictive condition ( [6,31]): \|a(z)\| 2 + \|a(−z)\| 2 1, ∀z ∈ T. (2.7) The redundancy of tight framelets brings much more freedom for construction and robustness for their performance in applications. To achieve lower computational complexity in the framelet transform, researchers are trying to find tight framelets with small number of generators. Symmetric tight framelet with three generators have been obtained in [2,16,22]. As for tight framelets with two generators and symmetry, the general characterization based on the OEP was given in [ [15,21] that the construction (i.e., (2.5) with s = 2 and 1 = 2 = 1) is related to the problem of splitting a 2 × 2 Hermitian matrix A(z) of Laurent polynomials with symmetry into the form of A(z) = U(z)U (z), z ∈ C \ {0}, (2.8) where U(z) is a 2 × 2 matrix of Laurent polynomials, and all of its elements have symmetry. The necessary and sufficient condition for such a factorization to exist is given in [21,Theorem 2.3] and [15,Theorem 3.2]. Notice that the factorization in (2.8) above is similar to the problem of solving the factorization in (1.3) we stated at the beginning of this paper. The only difference is that (2.8) is a standard spectral factorization, wherein (1.3) we have a generalized spectral factorization of A. However, our generalized matrix spectral factorization problem has some fundamental differences and essential difficulties compared to the standard one. For example, (2.8) implies that A(z) is positive semidefinite for all z ∈ T. We will further explain this issue in later sections. On the other hand, the sparsity of the framelet expansion (2.2) is determined by vanishing moments of the framelet generator ψ. By a simple argument (e.g., see [18, , one has min{vm(b 1 ), . . . , vm(b s )} = min{vm(ψ 1 ), . . . , vm(ψ s )} min sr(a), 1 2 vm Θ(z) − Θ(z 2 )a (z)a(z) , no matter how we choose Θ. Thus we always try to find Θ such that vm (Θ(z) − Θ(z 2 )a (z)a(z)) is as large as possible, and make min{vm(ψ 1 ), . . . , vm(ψ s )} also large. From the above discussions, we see that constructing a tight framelet with symmetry and high order of vanishing moments is not easy. We need to impose an extra condition (2.7) on the low-pass filter a (or stringent conditions on the underlying refinable function φ having stable integer shifts, we refer the readers to [2,3,15,16,22] for more details on this issue), and the moment correcting filter Θ that we choose must make M a,Θ positive semidefinite. This motivates us to consider quasi-tight framelets, which behave almost identical to tight framelets, but have much more flexibility and are much less difficult to construct. As shown in [7,8,19,20], quasi-tight framelets can be constructed from arbitrary compactly supported refinable (vector) functions, with the desired features (e.g., symmetry, a high order of vanishing moments, and high balancing order) being achieved. This makes quasi-tight framelets of interest in both theory and applications. 2.2. Illustrative examples. In this subsection, we provide several examples to illustrate our main results in Theorems 1.1 and 1.3. To check whether the function φ defined in (2.4) belongs to L 2 (R) or not, we shall employ the technical quantity sm(a) (known as the smoothness exponent of a) defined in [12, (4.3)] and [18, (5.6.44)] with p = 2, which can be easily computed by [18,Corollary 5.8.5]. For τ ∈ R, a function φ belongs to the Sobolev space H τ (R) if R \| φ(ξ)\| 2 (1 + \|ξ\| 2 ) τ dξ < ∞. Define sm(φ) := sup{τ ∈ R : φ ∈ H τ (R)}. For a refinable function φ defined in (2.4), the inequality sm(φ) sm(a) holds and we further have sm(φ) = sm(a) if the integer shifts of φ are stable, e.g., see [18,Theorem 6.3.3]. Hence, φ ∈ L 2 (R) if sm(a) > 0. Define δ ∈ l 0 (Z) to be the filter such that δ(0) = 1 and δ(k) = 0 for all k = 0. The first example {a; b 1 , b 2 } δ,(1,−1) of quasi-tight framelet filter banks with symmetry was heuristically obtained in [ a(z) = − 1 16 (z 2 + z −2 ) + 1 4 (z + z −1 ) + 5 8 , b 1 (z) = √ 2 4 z(−z + 2 − z −1 ), b 2 (z) = 1 16 (z 2 − 4z + 6 − 4z −1 + z −2 ), such that sm(a) ≈ 0.8853 > 0, Sa(z) = Sb 2 (z) = 1, Sb 1 (z) = z 2 and vm(b 1 ) = 2, vm(b 2 ) = 4. [18, Example 3.2.2] motivates all later research on quasi-tight framelets in [7,8,19,20] but all such constructed quasi-tight framelets there lack the symmetry property. As an application of our main results Theorems 1.1 and 1.3 in this paper, we now provide a few more examples here. Example 1. Choose Θ = δ and consider a low-pass filter a ∈ l 0 (Z) given by a(z) = − 1 16 (z 2 − 6z + 1)(1 + z) 2 z −2 + 2 √ 2 − 3 32 z −3 (1 + z) 2 (1 − z) 4 . Note that Sa(z) = 1, sr(a) = 2 and vm(Θ(z) − Θ(z 2 )a (z)a(z)) = vm(1 − a (z)a(z)) = 4. Applying Theorem 1.3 with n b = 2, we can construct a quasi-tight framelet filter bank {a; b 1 , b 2 } Θ,(1,−1) with b 1 (z) = (1 − z) 2 2048 (4 − 3 √ 2)(z 3 + z −3 ) − 2 √ 2(z 2 + z −2 ) − (2068 − 1559 √ 2)(z + z −1 ) + 1084 √ 2 , b 2 (z) = (1 − z) 2 2048 (3 √ 2 − 4)(z 3 + z −3 ) + 2 √ 2(z 2 + z −2 ) − (2028 − 1513 √ 2)(z + z −1 ) + 964 √ 2 . Then Sb 1 (z) = Sb 2 (z) = z 2 and vm(b 1 ) = vm(b 2 ) = 2. Define φ through (2.4) and ψ by ψ (ξ) := b (ξ/2) φ(ξ/2) for all ξ ∈ R and = 1, 2. Since sm(a) ≈ 1.0193 > 0, we have φ ∈ L 2 (R). Consequently, we conclude from Theorem 1.3 that {φ; ψ 1 , ψ 2 } (1,−1) is a quasi-tight framelet in L 2 (R) with symmetry and vm(ψ 1 ) = vm(ψ 2 ) = 2. Example 2. Choose Θ = δ. Let φ be a compactly supported refinable function with the associated refinement filter a ∈ l 0 (Z) such that (A) a (B) δ (C) b 1 (D) b 2 (E) φ (F) ψ 1 (G) ψ 2a(z) = 1 1024 (−z 6 + 18z 4 − 32z 3 − 63z 2 + 288z + 604 + 288z −1 − 63z −2 − 32z −3 + 18z −4 − z −6 ). We have Sa(z) = 1, sr(a) = 4, and vm(1 − a a) = 8. Since sm(a) ≈ 1.6821 > 0, φ in (2.4) belongs to L 2 (R). For n b = 4, we can construct a quasi-tight framelet filter bank {a; b 1 , b 2 } Θ,(1,−1) with b 1 (z) = √ 2 32 (−z 4 + 9z 2 − 16z + 9 − z −2 ), b 2 (z) = 1 1024 (z 6 − 18z 4 + 32z 3 + 63z 2 − 288z + 420 − 288z −1 + 63z −2 + 32z −3 − 18z −4 + z −6 ). We have Sb 1 (z) = z 2 , Sb 2 (z) = 1 and vm(b 1 ) = 4, vm(b 2 ) = 8. By letting ψ (ξ) := b (ξ/2) φ(ξ/2) for all ξ ∈ R and = 1, 2, we see that {φ; ψ 1 , ψ 2 } (1,−1) is a quasi-tight framelet in L 2 (R) with symmetry. Example 3. Choose Θ = δ. Let φ be a compactly supported refinable function with the associated refinement filter a ∈ l 0 (Z) such that We have Sa(z) = z, sr(a) = 3, and vm(1 − a a) = 6. Since sm(a) ≈ 1.1543 > 0, φ in (2.4) belongs to L 2 (R). For n b = 3, we can construct a quasi-tight framelet filter bank a(z) = 1 1024 (15z 4 − 63z 3 + 35z 2 + 525z + 525 + 35z −1 − 63z −2 + 15z −3 ). (A) a (B) δ (C) b 1 (D) b 2 (E) φ (F) ψ 1 (G) ψ 2{a; b 1 , b 2 } Θ,(1,−1) with b 1 (z) = 1 1024 (15z 4 − 63z 3 + 385z 2 − 945z + 945 − 385z −1 + 63z −2 − 15z −3 ), b 2 (z) = √ 105 512 (−5z 4 + 21z 3 − 38z 2 + 38z − 21 + 5z −1 ). We have Sb 1 (z) = −z, Sb 2 (z) = −z 3 and vm(b 1 ) = vm(b 2 ) = 3. By letting ψ (ξ) := b (ξ/2) φ(ξ/2) for all ξ ∈ R and = 1, 2, we see that {φ; ψ 1 , ψ 2 } (1,−1) is a quasi-tight framelet in L 2 (R). Example 4. Choose Θ = δ. Let φ be a compactly supported refinable function with the associated refinement filter a ∈ l 0 (Z) such that a(z) = 1 32 (−z 3 + z 2 + 16z + 16 + z −1 − z −2 ). We have Sa(z) = z, sr(a) = 1, and vm(1 − a a) = 4. Since sm(a) ≈ 0.7184 > 0, φ in (2.4) belongs to L 2 (R). For n b = 1, we can construct a quasi-tight framelet filter bank {a; b 1 , b 2 } Θ,(1,−1) with b 1 (z) = √ 2 4096 (z 5 − z 4 − 32z 3 − 32z 2 + 2032z − 2032 + 32z −1 + 32z −2 + z −3 − z −4 ), b 2 (z) = √ 2 4096 (−z 5 + z 4 + 32z 3 + 32z 2 + 1794z − 1794 − 32z −1 − 32z −2 − z −3 + z −4 ). We have Sb 1 (z) = Sb 2 (z) = −z and vm(b 1 ) = vm(b 2 ) = 1. By letting ψ (ξ) := b (ξ/2) φ(ξ/2) for all ξ ∈ R and = 1, 2, we see that {φ; ψ 1 , ψ 2 } (1,−1) is a quasi-tight framelet in L 2 (R) with symmetry. Example 5. Choose Θ(z) = z+z −1 2 . Let φ be a compactly supported refinable function with the associated refinement filter a ∈ l 0 (Z) such that (E), (F) and (G) are the graphs of the refinable function φ and the framelet generators ψ 1 and ψ 2 . We have Sa(z) = 1, SΘ(z) = 1, sr(a) = 2, and vm(Θ − Θ(· 2 )a a) = 4. Since sm(a) ≈ 1, φ in (2.4) belongs to L 2 (R). For n b = 2, we can construct a quasi-tight framelet filter bank a(z) = 1 8 (−z 3 + 3z + 4 + 3z −1 − z −3 ). (A) a (B) δ (C) b 1 (D) b 2 (E) φ (F) ψ 1 (G) ψ 2(A) a (B) δ (C) b 1 (D) b 2 (E) φ (F) ψ 1 (G) ψ 2{a; b 1 , b 2 } Θ,(1,−1) with b 1 (z) = 1 64 z −1 (z − 1) 2 (z 4 + 2z 3 − 12z 2 − 30z − 46 − 30z −1 − 12z −2 + 2z −3 + z −4 ), b 2 (z) = 1 64 z −1 (z − 1) 2 (z 2 − 2z + 4 − 2z −1 + 1)(z 2 + 4z + 8 + 4z −1 + 1). We have Sb 1 (z) = Sb 2 (z) = 1 and vm(b 1 ) = vm(b 2 ) = 2. By letting η(ξ) := Θ(ξ) φ(ξ) and ψ (ξ) := b (ξ/2) φ(ξ/2) for all ξ ∈ R and = 1, 2, we see that {η, φ; ψ 1 , ψ 2 } (1,−1) is a quasi-tight framelet and thus {ψ 1 , ψ 2 } (1,−1) is a homogeneous quasi-tight framelet in L 2 (R) with symmetry. (A) a (B) Θ (C) b 1 (D) b 2 (E) φ (F) η (G) ψ 1 (H) ψ 2 Factorization of Matrices of Laurent Polynomials with Symmetry In this section, we study the factorization problem proposed at the beginning of the paper. Our goal of this section is to prove the main result Theorem 1.1 on the existence of a generalized spectral factorization as in (1.3) with 1 = 1 and 2 = −1. 3.1. Laurent polynomials with symmetry. To prove Theorem 1.1, we need to better understand some properties of Laurent polynomials with symmetry. We first introduce several notations which will be used throughout this paper. For a nonzero Laurent polynomial u(z) = k∈Z u(k)z k , define its lower degree ldeg(u), degree deg(u) and its length len(u) by ldeg(u) := min{k ∈ Z : u(k) = 0}, deg(u) := max{k ∈ Z : u(k) = 0}, len(u) := deg(u) − ldeg(u). If u = 0, then we define len(0) := −∞. Here are some basic facts about the symmetry operator. Proposition 3.1. Suppose u and v are Laurent polynomials with symmetry types u z cu and v z cv respectively. Then the following hold: (1) uv has symmetry with S[uv](z) = u v z cu+cv ; (2) If v divides u, then u/v has symmetry with S[u/v](z) = u v z cu−cv ; (3) u has symmetry with S[u ](z) = [Su] (z) = u z −cu . If u = 0, then S[u ] = Su −1 , c u = ldeg(u) + deg(u) , and odd(len(u)) = odd(c u ), where odd is defined as in (1.7). Proof. Items (1) -(3) are obvious. Note that ldeg(u(· −1 )) = − deg(u). By the definition of S, we have ldeg(u) = c u + ldeg(u(· −1 )) = c u − deg(u) and len(u) = deg(u) − ldeg(u) = c u − 2 ldeg(u). Now all the rest claims follow easily. The symmetry of a Laurent polynomial is related to its multiplicities of roots at ±1. Recall that Z(u, z 0 ) stands for the multiplicity of the root of u at z 0 . We have the following lemma. Lemma 3.2. Let u be a nonzero Laurent polynomial with symmetry type Su(z) = z c for some ∈ {±1} and c ∈ Z. Then = (−1) Z(u,1) and odd(c) = odd(Z(u, 1) + Z(u, −1)). Proof. Write u(z) = (z − 1) Z(u,1) v(z) for some Laurent polynomial v with v(1) = 0. We have z c = u(z) u(z −1 ) = (−z) Z(u,1) v(z) v(z −1 ) . By letting z = 1, we obtain = (−1) Z(u,1) . We can also write u(z) = (z + 1) Z(u,−1) w(z) for a unique Laurent polynomial w with w(−1) = 0. By = (−1) Z(u,1) and the definition of S in (1.2), we have (z + 1) Z(u,−1) w(z) = u(z) = z c u(z −1 ) = (−1) Z(u,1) z c−Z(u,−1) (z + 1) Z(u,−1) w(z −1 ), from which we have w(z) = (−1) Z(u,1) z c−Z(u,−1) w(z −1 ) . Taking z = −1 and noting w(−1) = 0, we conclude that (−1) c−Z(u,−1)+Z(u,1) = 1, i.e., odd(c) = odd(Z(u, 1) + Z(u, −1)). 3.2. Compatibility of matrices of Laurent polynomials with symmetry. We now consider matrices of Laurent polynomials with symmetry. First, we extend the symmetry operator S. For an r × s matrix A := A(z) of Laurent polynomials with symmetry, define SA via [SA(z)] j,k := S(A j,k (z)) for all 1 j r and 1 k s. In this case, we say that the matrix A has symmetry type SA. Next, we discuss how the symmetry property behaves under matrix summations. If A and B are r ×s matrices of Laurent polynomials with symmetry and SA = SB, then it is not difficult to conclude that A ± B have symmetry with S[A + B] = S[A − B] = SA = SB. In this case, we say that the operations A + B and A − B are compatible for preserving symmetry. For an r × s matrix P := P(z) with symmetry, we say that the symmetry type of P is compatible or P has compatible symmetry if SP(z) = Sθ θ θ 1 (z)Sθ θ θ 2 (z) (3.1) holds for some 1 × r and 1 × s row vectors of Laurent polynomials θ θ θ 1 and θ θ θ 2 with symmetry. It is easy to observe from the definition (3.1) that Sθ θ θ 1 (z) gives the symmetry relationship between the rows of P(z), while Sθ θ θ 2 (z) gives the symmetry relationship between the columns of P(z): SP j,l (z) SP k,l (z) = [Sθ θ θ 1 (z)] j [Sθ θ θ 1 (z)] k , ∀l = 1, . . . , s, ∀j, k = 1, . . . , r, (3.2) SP l,j (z) SP l,k (z) = [Sθ θ θ 2 (z)] j [Sθ θ θ 2 (z)] k , ∀l = 1, . . . , s, ∀j, k = 1, . . . , r. (3.3) For an r × s matrix P and an s × t matrix Q of Laurent polynomials with symmetry, we say the multiplication PQ is compatible if SP(z) = Sθ θ θ 1 (z)Sθ θ θ 2 (z), SQ(z) = Sθ θ θ 2 (z)Sθ θ θ 3 (z), for some 1 × r, 1 × s and 1 × t row vectors θ θ θ 1 , θ θ θ 2 and θ θ θ 3 of Laurent polynomials with symmetry. In this case, it is not hard to see that PQ has compatible symmetry with S[PQ](z) = SP(z)SQ(z) = Sθ θ θ 1 (z)Sθ θ θ 3 (z). For an n × n matrix P of Laurent polynomials with compatible symmetry, one can show by induction that det(P) also has symmetry with type S[det(P)](z) = n j=1 ([Sθ θ θ 1 (z)] j [Sθ θ θ 2 (z)] j ) = n j=1 SP j,j (z). (3.4) If we further assume that P is strongly invertible, i.e., det(P) is a nonzero monomial, then P −1 is a matrix of Laurent polynomials with compatible symmetry. To calculate its symmetry type, recall that P −1 = det(P) −1 adj(P). By (3.4), we have [S[adj(P)](z)] j,k = 1 l n,l =k 1 m n,m =j [Sθ θ θ 1 (z)] l [Sθ θ θ 2 (z)] m = S[det(P)](z) [Sθ θ θ 1 (z)] k [Sθ θ θ 2 (z)] j = S[det(P)](z)[Sθ θ θ 1 (z)] k [Sθ θ θ 2 (z)] j . It follows that P −1 has compatible symmetry with S[P −1 ](z) = Sθ θ θ 2 (z)Sθ θ θ 1 (z). (3.5) Consequently, if P, Q and R are n × n, n × m and m × n matrices of Laurent polynomials with symmetry, and if P is strongly invertible, then • the multiplication PQ =: A is compatible implies that the multiplication P −1 A is compatible; • the multiplication RP =: B is compatible implies that the multiplication BP −1 is compatible. Here we discuss some basic properties of compatibility. We summarize them as the following proposition. Proposition 3.3. Let P be an r × s matrix of Laurent polynomials with compatible symmetry satisfying (3.1) for some 1 × r and 1 × s row vectors of Laurent polynomials θ θ θ 1 and θ θ θ 2 with symmetry. (1) Let P l,j be the s × s permutation matrix such that PP l,j switches the l-th and j-th columns of P, and letP l,j be the r × r permutation matrix such thatP k,m P switches the k-th and m-th rows of P. Then PP l,j andP k,m P have compatible symmetry, with S[PP l,j ](z) = Sθ θ θ 1 (z)S[θ θ θ 2 P l,j ](z), S[P k,m P](z) = S[θ θ θ 1 (P k,m ) T ] (z)Sθ θ θ 2 (z). (2) Let D r and D s be r × r and s × s diagonal matrices of Laurent polynomials with symmetry. Then D r P and PD s have compatible symmetry with S[D r P](z) = S[θ θ θ 1 D r ] (z)Sθ θ θ 2 (z), S[PD s ](z) = Sθ θ θ 1 (z)S[θ θ θ 2 D s ](z). (3) Let U be a t × r matrix of Laurent polynomials with symmetry. If there exists k ∈ {1, . . . , s} such that SU j,1 (z)SP 1,k (z) = SU j,2 (z)SP 2,k (z) = · · · = SU j,r (z)SP r,k (z), ∀j = 1, . . . , t, (3.6) then U has compatible symmetry with SU(z) = Sθ θ θ 3 (z)Sθ θ θ 1 (z), where θ θ θ 3 is any 1 × t row vector of Laurent polynomials with symmetry satisfying [Sθ θ θ 3 (z)] j = SU j,1 (z) [Sθ θ θ 1 (z)] 1 for all j = 1, . . . , t. Conversely, if the multiplication UP is compatible, then there must exist k ∈ {1, . . . , s} such that (3.6) holds. (4) Let U be a s × t matrix of Laurent polynomials with symmetry. If there exists l ∈ {1, . . . , r} such that SP l,1 (z)SU 1,m (z) = SP l,2 (z)SU 2,m (z) = · · · = SP l,s (z)SU s,m (z), ∀m = 1, . . . , t, (3.7) then U has compatible symmetry with SU(z) = Sθ θ θ 2 (z)Sθ θ θ 3 (z), where θ θ θ 3 is any 1 × t row vector of Laurent polynomials with symmetry satisfying [Sθ θ θ 3 (z)] m = SU 1,m (z) [Sθ θ θ 2 (z)] 1 for all m = 1, . . . , t. Conversely, if the multiplication PU is compatible, then there must exist m ∈ {1, . . . , s} such that (3.7) holds. Proof. Items (1) and (2) can be easily verified by direct calculation. To prove item (3), by the fact that P has compatible symmetry and using (3.6) and (3.2), we have SU j,q (z) SU j,1 (z) = SP 1,k (z) SP q,k (z) = [Sθ θ θ 1 (z)] 1 [Sθ θ θ 1 (z)] q = [Sθ θ θ 1 (z)] q [Sθ θ θ 1 (z)] 1 , ∀q = 1, . . . , r, j = 1, . . . , t. (3.8) Let θ θ θ 3 be an 1 × t row vector of Laurent polynomials with symmetry satisfying [Sθ θ θ 3 (z)] j = SU j,1 (z) [Sθ θ θ 1 (z)] 1 for all j = 1, . . . , t. It follows from (3.8) that SU j,q (z) = [Sθ θ θ 1 (z)] q [Sθ θ θ 1 (z)] 1 SU j,1 (z) = [Sθ θ θ 1 (z)] q [Sθ θ θ 1 (z)] 1 [Sθ θ θ 3 (z)] j [Sθ θ θ 1 (z)] 1 = [Sθ θ θ 3 (z)] j [Sθ θ θ 1 (z)] q , for all q = 1, . . . , r and j = 1, . . . , t. Hence SU(z) = Sθ θ θ 3 (z)Sθ θ θ 1 (z), that is, U has compatible symmetry and the multiplication UP is compatible. Conversely, suppose the multiplication UP is compatible. This means there exists an 1 × t row vector θ θ θ 3 of Laurent polynomials with symmetry such that SU(z) = Sθ θ θ 3 (z)Sθ θ θ 1 (z). Using (3.2) and (3.3), we see that (3.6) in fact holds for all k = 1, . . . , s. This proves item (3). Finally, item (4) can be proved by using similar arguments as in th proof of item (3). 3.3. Factorization of matrices of Laurent polynomials with symmetry and negative constant determinants. In this subsection, we consider a special case of Theorem 1.1, which is summarized as the following theorem. Theorem 3.4. Let A be a 2 × 2 Hermitian matrix of Laurent polynomials with compatible symmetry and define its symmetry type by SA(z) = 1 α(z) α (z) 1 , where α(z) := SA 1,2 (z). Sup- pose det(A(z)) = C < 0 for all z ∈ T for some negative constant C. Then we can find a ma- trix U(z) = U 1,1 (z) U 1,2 (z) U 2,1 (z) U 2,2 (z) of Laurent polynomials with compatible symmetry such that A(z) = U(z)Diag(1, −1)U (z) holds, and the symmetry type satisfies SU 1,1 (z) SU 2,1 (z) = SU 1,2 (z) SU 2,2 (z) = α(z). (3.9) To prove Theorem 3.4, we need to establish several auxiliary results, which are the foundations for algorithms of constructing a matrix U as in the statement of Theorem 3.4. a 1 (z) := a(z) − b(z)q 1 (z) (3.10) satisfies len(a 1 ) < len(a) and Sa 1 (z) = Sa(z) = Sb(z)Sq 1 (z). Proof. Define M a := deg(a), m a := ldeg(a), M b := deg(b) and m b := ldeg(b). We have a(z) = Ma k=ma a(k)z k and b(z) = M b k=m b b(k)z k . The symmetry properties of a and b yields a(m a ) = a a(M a ), b(m b ) = b b(M b ), M a + m a = c a , m b + M b = c b . Define q 1 (z) := a(M a ) b(M b ) z Ma−M b + a(m a ) b(m b ) z ma−m b . (3.11) We have Sq 1 (z) = q 1 (z) q 1 (z −1 ) = a(Ma) b(M b ) z Ma−M b + a(ma) b(m b ) z ma−m b a(Ma) b(M b ) z M b −Ma + a(ma) b(m b ) z m b −ma = a(Ma) b(M b ) z Ma−M b + a b a(Ma) b(M b ) z (ca−c b )−(Ma−M b ) a(Ma) b(M b ) z M b −Ma + a b a(Ma) b(M b ) z (Ma−M b )−(ca−c b ) = a b z c 1 −c b = Sa(z) Sb(z) . Define a 1 as (3.10), it follows immediately that Sa 1 (z) = Sa(z) = Sb(z)Sq 1 (z). We are left to show that len(a 1 ) < len(a). As len(a) > len (b), we have M a − M b > m a − m b . Thus by the definition of q 1 in (3.11), we have ldeg(q 1 ) = m a − m b and deg(q 1 ) = M a − M b . So ldeg(bq 1 ) = m a and deg(bq 1 ) = M a . Moreover, we obtain from (3.11) that the coefficient of the z Ma term in bq 1 is a(M a ), and the coefficient of the z ma term in bq 1 is a(m a ). Hence we conclude that ldeg(a 1 ) > m a and deg(a 1 ) < M a , which implies len(a 1 ) < len(a). This completes the proof. The next lemma provides a long division algorithm for Laurent polynomials with symmetry. Lemma 3.6. Let a and b be Laurent polynomials with symmetry types Sa(z) = a z ca and Sb(z) = b z c b respectively, for some a , b ∈ {±1} and c a , c b ∈ Z. Suppose b = 0. Consider the following four cases: Case (1): a b = 1, c a − c b ∈ 2Z + 1; Case (2): a b = −1, c a − c b ∈ 2Z + 1; Case (3): a b = 1, c a − c b ∈ 2Z; Case (4): a b = −1, c a − c b ∈ 2Z. Then the following results hold: (i) For cases (1) -(3), we can construct a Laurent polynomial q with symmetry such that r(z) := a(z) − b(z)q(z) (3.12) satisfies len(r) < len(b) and Sr(z) = Sa(z) = Sb(z)Sq(z). (ii) For case (4), we can construct a Laurent polynomial q with symmetry such that the Laurent polynomial r defined in (3.12) satisfies len(r) len(b) and Sr(z) = Sa(z) = Sb(z)Sq (z). Proof. If len(a) < len(b), simply take q := 0 and r := a, and all claims hold. If len(a) len(b), define a 0 := a. Starting from j = 0, if len(a j ) > len(b), then apply Lemma 3.5 to construct a Laurent polynomial q j with symmetry such that a j+1 (z) := a j (z) − b(z)q j (z) satisfies len(a j+1 ) < len(a j ) and Sa j+1 (z) = Sa j (z) = Sb(z)Sq j (z). Note that this process cannot iterate forever: If len(a) = len(b), the process terminates at the first step j = 0 and we set q 0 := 0. Otherwise, the process stops at some step j = K − 1 with K ∈ N when len(a K−1 ) > len(b) len(a K ). In both cases, we have a K (z) = a K−1 (z) − b(z)q K−1 (z) = · · · = a 0 (z) − b(z)(q K−1 (z) + · · · + q 0 (z)). ( 3.13) Moreover, the symmetry types of a j satisfy Sa K (z) = Sa K−1 (z) = · · · = Sa 0 (z) = Sa(z), Sq j (z) = Sa j+1 (z) Sb(z) = Sa(z) Sb(z) , ∀j = 0, . . . , K − 1. Defineq (z) := q K−1 (z) + · · · + q 0 (z). (3. 14) It follows that Sq(z) = Sa(z) Sb(z) and a K (z) = a(z) − b(z)q(z). (3.15) We now consider cases (1)-(4) separately in the following way: • Cases (1) and (2): The condition c a − c b ∈ 2Z + 1 implies that odd(c a ) = odd(c b ). Thus by Proposition 3.1, we have len(a) = len(b). Therefore, we only consider the situation when len(a) > len(b). Define q :=q and r := a K , whereq and a K are defined as (3.14) and (3.13) respectively. By (3.15), we have Sr(z) = Sa K (z) = Sa(z) = Sb(z)Sq(z). Moreover, we have len(r) = len(a K ) len(b). On the other hand, note that Sr(z) = a z ca . It follows from Proposition 3.1 that len(r) = len(b), and thus len(r) < len(b). This proves the claims for cases (1) and (2) in item (i). • Case (3): Defineq and a K as (3.14) and (3.13) respectively. We have len(a K ) len(b) and Sa K (z) = Sa(z) = Sb(z)Sq (z). If len(a K ) < len(b), simply set a :=q and r := a K , and all claims in item (1) Next, note that Sq(z) = z 2M a,K −2M b . As a b = 1, we have a = b . Using Proposition 3.1, direct calculation yields q(z) := a K (M a,K ) b(M b ) z M a,K −M b , r(z) := a K (z) − b(z)q(z).Sb(z)Sq(z) = b z c b +2M a,K −2M b = b z m b +M b +2M a,K −2M b = b z 2M a,K −len(b) = b z 2M a,K −len(a K ) = a z m a,K +M a,K = Sa K (z). Define q :=q +q. We have r(z) = a K (z) − b(z)q(z) = a(z) − b(z)q(z) − b(z)q(z) = a(z) − b(z)q(z), and Sr(z) = Sa K (z) = Sa(z) = Sb(z)Sq (z). This proves all claims for case (3) in item (i). • Case 4: Define q :=q and r := a K , whereq and a K are defined as (3.14) and (3.13) respectively. All claims of item (ii) follows immediately. The proof is now complete. We move on to perform some further analysis on 2 × 2 Hermitian matrices of Laurent polynomials with symmetry. Let A(z) be such a matrix. Then as A(z) = A (z) for all z ∈ T, its symmetry type (1) α(z) = z 2k+1 for some ∈ {±1} and k ∈ Z; (2) α(z) = z 2k for some k ∈ Z. Suppose det(A) = C for some negative constant C. Then there exist a positive integer K and 2 × 2 strongly invertible matrices V (0) , . . . , V (K−1) of Laurent polynomials with compatible symmetry, such that A (K) (z) := V (K−1) (z) · · · V (0) A(z)V (0) (z) · · · V (K−1) (z) satisfies (i) all the multiplications are compatible; (ii) at least one entry of A (K) is 0. Proof. If A has one zero entry, simple take K = 1, V (0) = I 2 and A (1) = A. Here I n stands for the n × n identity matrix. Otherwise, all for entries of A are nonzero, and we claim that This proves (3.17). As a consequence, we have min{len(A 1,1 ), len(A 2,2 )} len(A 2,1 ). (3.19) We now construct a strongly invertible matrix V (0) of Laurent polynomials with compatible symmetry, such that A (1) := V (0) AV (0) satisfies len(A 2,1 ) < len(A 2,1 ) and the multiplications are compatible. We consider the following two cases: • If len(A 1,1 ) len(A 2,1 ), as SA 1,1 (z) = 1 and SA 2,1 (z) = α with α satisfying (1) or (2), we can apply item (i) of Lemma 3.6 to construct a Laurent polynomial Q with symmetry, such that r(z) := A 2,1 (z) − A 1,1 (z)q(z) satisfies len(r) < len(A 1,1 ) and Sr(z) = Sq(z)SA 1,1 (z) = SA 2,1 (z). Define V (0) (z) := 1 0 −q(z) 1 . Then A (1) (z) := V (0) (z)A(z)V (0) (z) = A 1,1 (z) r (z) r(z) A(1) 2,2 (z) . So len(A 2,1 ) = len(r) < len(A 1,1 ) len(A 2,1 ). (3.20) • If len(A 2,2 ) len(A 2,1 ), we apply item (i) of Lemma 3.6 to construct a Laurent polynomial q with symmetry, such that r(z) := A 1,2 (z) − A 2,2 (z)q(z) satisfies len(r) < len(A 2,2 ) and Sr(z) = Sq(z)SA 2,2 (z) = SA 1,2 (z). Define V (0) (z) := 1 −q(z) 0 1 . Then A (1) (z) := V (0) (z)A(z)V (0) (z) = A (1) 1,1 (z) r(z) r (z) A 2,2 (z) . 2,1 ) > . . . , so the iteration terminates within finite steps, say at step j = K − 1, so that the matrix A (K) := V (K−1) A (K−1) V (K−1) = · · · = V (K−1) . . . V (0) AV (0) . . . V (K−1) has one zero entry. Moreover, it can be easily seen from previous discussions that all the multiplications above are compatible, with SA (j) = SA = SV (j) = Sθ θ θ 1 Sθ θ θ 1 , j = 0, . . . , K − 1,(3.22) where A (0) := A and Sθ θ θ 1 (z) := [1, α(z)]. This completes the proof. We now have all the necessary tools to prove Theorem 3.4. Proof of Theorem 3.4 for cases 1-3. We constructively prove Theorem 3.4 for cases 1-3 by the following construction algorithm for U in Theorem 3.4 for cases 1-3 as follows. Algorithm 1. Suppose α(z) = z 2k+1 for some ∈ {±1} and k ∈ Z or α(z) = z 2k for some k ∈ Z. Step 1. Apply Lemma 3.7 to construct 2 × 2 strongly invertible matrices V (0) , . . . , V (K−1) of Laurent polynomials with compatible symmetry such that A (K) (z) := V (K−1) (z) . . . V (0) A(z)V (0) (z) . . . V (K−1) (z) has at least one zero entry, and all above multiplications are compatible. Step 2. Construct the 2 × 2 matrixŨ of Laurent polynomials with compatible symmetry such that A (K) =ŨDiag(1, −1)Ũ . The explicit expression ofŨ is given by the following: (1) If A (K) 1,2 = A (K) 2,1 = 0, then A (K) 1,1 = c 1 and A (K) 2,2 = c 2 for some constants c 1 and c 2 satisfying c 1 c 2 = det(A (K) ) = det(A) = C < 0. Set U :=        Diag( √ c 1 , √ −c 2 ), if c 1 > 0, c 2 < 0, 0 √ −c 1 √ c 2 0 , if c 1 < 0, c 2 > 0; (3.23) (2) If A (K) 1,1 = 0, setŨ := 1 √ 2 A (K) 1,2 A (K) 1,2 1 2 A (K) 2,2 + 1 1 2 A (K) 2,2 − 1 ; (3.24) (3) If A (K) 2,2 = 0, setŨ := 1 √ 2 1 2 A (K) 1,1 + 1 1 2 A (K) 1,1 − 1 A (K) 2,1 A (K) 2,1 . (3.25) Step 3. The matrix U := V (K−1) . . . V (0)Ũ is the one satisfying all claims of Theorem 3.4. Justification of Algorithm 1: By the assumptions of the algorithm, the validity of step 1 is trivial. For step 2: If A Finally, define U as in step 3, it follows immediately that U has compatible symmetry and satisfies A = UDiag(1, −1)U . Moreover, it follows from (3.22) that all the multiplication of U = V (K−1) . . . V (0)Ũ are compatible, and SU = Sθ θ θ 1 Sθ θ θ l where l = 2, 3 or 4. Hence (3.9) holds. This completes the justification of Algorithm 1. Proof of Theorem 3.4 for case 4. This is a harder case, and the construction for U in Theorem 3.4 for case 4 is summarized by the following technical algorithm. Algorithm 2. Suppose α(z) = −z 2k for some k ∈ Z. (S1) Define [−n, n] := fsupp(A 1,1 ). Parametrize Laurent polynomials V(z) = n j=0 V (j)z j , W(z) = n j=0 W (j)z j , where V (0), . . . , V (n), W (0), . . . , W (n) are free parameters. Let Q and R be the Laurent polynomials which are determined uniquely through long division using A 1,1 by (S2) If SŨ 1,1 (z) = z n and SŨ 1,2 = −z n , setŮ 1,1 :=Ũ 1,1 andŮ 1,2 :=Ũ 1,2 . Otherwise, define A 2,1 (z)V(z) + dz n−k W (z) = A 1,1 (z)Q(z) + R(z) with fsupp(R) ⊆ [−n, n − 1],(3.U 1,1 (z) :=Ũ 1,1 (z) − z nŨ 1,1 (z) 2 ,Ů 1,2 (z) :=Ũ 1,2 (z) + z nŨ 1,2 (z) 2 . (3.27) Then the updatedŮ 1,1 andŮ 1,2 are not both 0, and their symmetry types are SŮ 1,1 (z) = −z n and SŮ 1,2 (z) = z n . (S3) There exists λ ∈ R \ {0} such that A 1,1 (z) = λ Ů 1,1 (z)Ů 1,1 (z) −Ů 1,2 (z)Ů 1,2 (z) . (3.28) If λ > 0, define U 1,1 (z) := √ λŮ 1,1 (z), U 1,2 (z) := √ λŮ 1,2 (z). (3.29) Otherwise, define U 1,1 (z) := √ −λŮ 1,2 (z), U 1,2 (z) := √ −λŮ 1,1 (z). (3.30) Now we have SU 1,1 (z) = z n and SU 1,2 (z) = − z n for some ∈ {±1}. (S4) Define U 2,1 (z) := A 2,1 (z)U 1,1 (z) + dz n−k U 1,2 (z) A 1,1 (z) , (3.31) U 2,2 (z) := A 2,1 (z)U 1,2 (z) + dz n−k U 1,1 (z) A 1,1 (z) . (3.32) Then U 2,1 and U 2,2 are well-defined Laurent polynomials with symmetry types SU 2,1 (z) = − z n−2k and SU 2,2 (z) = z n−2k . Set U := U 1,1 U 1,2 U 2,1 U 2,2 . Then U is a desired matrix of Laurent polynomials with compatible symmetry satisfying all claims of Theorem 3.4. Justification of Algorithm 2: For (S1): First note that A 1,1 = 0. Otherwise, we have det(A) = −A 1,2 A 2,1 = −d 2 . This in particular means that A 1,2 is a nonzero monomial, which cannot have symmetry type −z c . Hence, fsupp(A 1,1 ) is a non-empty interval and is symmetric with centre 0. By our choice of R and fsupp(R) ⊆ [−n, n − 1], the condition R = 0 induces a homogeneous linear system of at most 2n equations with 2n + 2 unknowns. Therefore, the space of all solutions, namely the space E, has dimension at least two. In particular, E contains non-trivial solutions, and the Laurent polynomialsŨ 1,1 andŨ 1,2 in (S1) can be defined. For (S2): By our choices ofŨ 1,1 andŨ 1,2 , we have A 2,1 (z)Ũ 1,1 (z) + dz n−kŨ 1,2 (z) = A 1,1 (z)Ũ 2,1 (z),(3.33) whereŨ 2,1 (z) := A 2,1 (z)Ũ 1,1 (z)+dz n−kŨ 1,2 (z) A 1,1 (z) is a well-defined Laurent polynomial. By the symmetry type of A, we have A 2,1 (z) = −z 2k A 2,1 (z). Taking Hermitian conjugates and multiplying z n−2k on both sides of (3.33), we have − z n A 2,1 (z)Ũ 1,1 (z) + dz −kŨ 1,2 (z) = z n−2k A 1,1 (z)Ũ 2,1 (z). (3.34) It follows from (3.33) and (3.34) that A 2,1 (z) Ũ 1,1 (z) − z nŨ 1,1 (z) + dz n−k Ũ 1,2 (z) + z −nŨ 1,2 (z) = A 1,1 (z) Ũ 2,1 (z) + z n−2kŨ 2,1 (z) . (3.35) After (S1), if SŨ 1,1 (z) = z n or SŨ 1,2 (z) = −z n , then either SŨ 1,1 (z) = z 2n SŨ 1,1 (z) or SŨ 1,2 (z) = −z 2n SŨ 1,2 (z). DefineŮ 1,1 andŮ 1,2 as in (3.27), it is trivial thatŮ 1,1 andŮ 1,2 cannot be both zero. Moreover, direct calculation yields SŮ 1,1 = −z n and SŮ 1,2 = z n . For (S3): DefineŮ 2,1 (z) :=Ũ 2,1 (z)+z n−2kŨ 2,1 (z) 2 . Recall that det(A) = −d 2 . It follows that A 1,2 (z) A 2,1 (z)Ů 1,2 (z) + dz n−kŮ 1,1 (z) = A 1,1 (z)A 2,2 (z) + d 2 Ů 1,2 (z) + dz n−kŮ 1,1 (z)A 1,2 (z) =A 1,1 (z)A 2,2 (z)Ů 1,2 (z) + dz n−k dz n−kŮ 1,2 (z) +Ů 1,1 (z)A 1,2 (z) = A 1,1 (z) A 2,2 (z)Ů 1,2 (z) + dz n−kŮ 2,1 (z) . As det(A) = A 1,1 A 2,2 − A 1,2 A 2,1 is a nonzero constant, we see that gcd(A 1,1 , A 1,2 ) = 1. Thus from the above equation, we see that A 1,1 (z) divides A 2,1 (z)Ů 1,2 (z) + dz n−kŮ 1,1 (z). DefineŮ 2,2 (z) := (z) . We have A 2,1 (z)Ů 1,2 (z)+dz n−kŮ 1,1 (z) A 1,1A 2,1 (z)Ů 1,2 (z) + dz n−kŮ 1,1 (z) = A 1,1 (z)Ů 2,2 (z). (3.36) It follows from (3.35) and (3.36) that Ů 2,2 (z) −Ů 1,2 (z) −Ů 2,1 (z)Ů 1,1 (z) A 1,1 (z) A 2,1 (z) = dz n−k Ů 1,1 (z) −Ů 1,2 (z) . (3.37) Multiplying Ů 1,2 ,Ů 1,1 to the left on both sides of (3.37) yields We claim thatŮ 1,1Ů 1,1 −Ů 1,2Ů 1,2 = 0, and thus by what we have got above together with the fact that both A 1,1 andŮ 1,1Ů 1,1 −Ů 1,2Ů 1,2 are Hermitian, we see that (3.28) must hold for some λ ∈ R \ {0}. In fact, by the choices ofŮ 1,1 andŮ 1,2 , either SŮ 1,1 (z) = z n and SŮ 1,2 (z) = −z n , or vice versa. If SŮ 1,1 (z) = z n and SŮ 1,2 (z) = −z n , then by item (1) of Lemma 3.2, we have Z(Ů 1,1 , 1) = Z(Ů 1,1 , 1) ∈ 2Z and Z(Ů 1,2 , 1) = Z(Ů 1,2 , 1) ∈ 2Z + 1. This means Z(Ů 1,1Ů 1,1 , 1) ∈ 4Z and Z(Ů 1,2Ů 1,2 , 1) ∈ 4Z + 2. ThereforeŮ 1,1Ů 1,1 −Ů 1,2Ů 1,2 = 0. The case when SŮ 1,1 = −z n and SŮ 1,2 = z n can be proved similarly. This proves the claim. Now we define U 1,1 and U 1,2 as (3.29) if λ > 0 or as (3.30) if λ < 0. We have SU 1,1 (z) = z n and SU 1,2 (z) = − z n for some ∈ ±{1}, and Ů 2,1 (z)Ů 1,1 (z) −Ů 2,2 (z)Ů 1,2 (z) A 1,1 (z) = Ů 1,1 (z)Ů 1,1 (z) −Ů 1,2 (z)Ů 1,2 (z) A 2,1 (z).A 1,1 (z) = U 1,1 (z)U 1,1 (z) − U 1,2 (z)U 1,2 (z). (3.39) For (S4): By (3.35), (3.36) and our choices of U 1,1 and U 1,2 , one can see that A 1,1 (z) divides A 2,1 (z)U 1,1 (z) + dz n−k U 1,2 (z) and A 2,1 (z)U 1,2 (z) + dz n−k U 1,1 (z). So U 2,1 and U 2,2 are well-defined Laurent polynomials. Since SA 2,1 (z) = −z −2k and SA 1,1 (z) = 1, it is easy to verify that SU 2,1 (z) = − z n−2k and SU 2,2 (z) = z n−2k . Define U := U 1,1 U 1,2 U 2,1 U 2,2 . Observe that (3.31) and (3.32) is equivalent to U 2,2 (z) −U 1,2 (z) −U 2,1 (z) U 1,1 (z) A 1,1 (z) A 2,1 (z) = dz n−k U 1,1 (z) −U 1,2 (z) . (3.40) Multiplying [U 1,1 , U 1,2 ] to the left on both sides of (3.40) and using (3.39) yields det(U(z))A 1,1 (z) = dz n−k A 1,1 (z), and thus det(U(z)) = dz n−k . Multiplying U 1,2 , U 1,1 to the left on both sides of (3.40) yields U 2,1 (z)U 1,1 (z) − U 2,2 (z)U 1,2 (z) A 1,1 (z) = A 1,1 (z)A 2,1 (z). Hence A 2,1 (z) = U 2,1 (z)U 1,1 (z) − U 2,2 (z)U 1,2 (z). (3.41) Multiplying U 2,2 , U 2,1 to the left on both sides of (3.40), together with using (3.41), we have U 2,2 (z)U 2,2 (z) − U 2,1 (z)U 2,1 (z) A 1,1 (z) + A 2,1 (z)A 2,1 (z) =dz n−c det(U(z)) = d 2 = − det(A(z)) = A 1,2 (z)A 2,1 (z) − A 1,1 (z)A 2,2 (z). Since A is Hermitian, we have A 2,1 = A 1,2 . So the above identities imply Hence A 2,2 (z) = U 2,1 (z)U 2,1 (z) − U 2,2 (z)U 2,2 (z). (3.42) Consequently, it follows from (3.39), (3.41) and (3.42) that A = UDiag(1, −1)U . Moreover, we have SU(z) = SU 1,1 (z) SU 1,2 (z) SU 2,1 (z) SU 2,2 (z) = z n − z n − z n−2k z n−2k = Sθ θ θ 1 (z)Sθ θ θ 2 (z), where Sθ θ θ 1 (z) = [ z −n , − z 2k−n ] and Sθ θ θ 2 (z) = [1, −1] . This show that U has compatible symmetry and clearly (1.5) holds. Now the algorithm is fully justified. 3.4. Factorization of matrices of Laurent polynomials with symmetry and coprime entries. We now study the factorization as in Theorem 1.1 for a 2 × 2 Hermitian matrix A of Laurent polynomials with compatible symmetry, without assuming that A has constant determinant. In this subsection, we study the case when all entries of A are coprime, that is, gcd(A 1,1 , A 1,2 , A 2,1 , A 2,2 ) = 1. The main result for this subsection is the following theorem. Proof. If a = 0, then we simply take u = 0 and v = 1. If b = 0, then we take u = 1 and v = 0. Assume now a = 0 and b = 0. Let r := gcd(a, b). By applying the classical Euclidean Algorithm, we can construct Laurent polynomials u 1 and u 2 such that a(z)u 1 (z) + b(z)u 2 (z) = r(z). (3.44) Thus a(z −1 )u 1 (z −1 ) + b(z −1 )u 2 (z −1 ) = r(z −1 ) , which can be rewritten as a(z)u 1 (z −1 ) Sa(z) + b(z)u 2 (z −1 ) Sb(z) = r(z) Sr(z) . u(z) = u 1 (z) + u 1 (z −1 )Sr(z) Sa(z) 2 , v(z) = u 2 (z) + u 2 (z −1 )Sr(z) Sb(z) 2 . (3.46) We claim that the Laurent polynomials u and u defined as (3.46) are the desired ones as required. In fact, note that r = gcd(a, b) has symmetry. Direct calculation yields Su(z) = Sr(z) Sa (z) and Sv(z) = Sr(z) Sb (z) . This implies that u and u have symmetry, and Sa(z)Su(z) = Sb(z)Sv(z) = Sr(z). Let d := gcd(u, v). Then dr divides au + bv = r. Therefore d = 1. The proof is complete. With the Extended Euclidean algorithm developed above, we have the following lemma. Lemma 3.10. Let u 1 and u 2 be nonzero Laurent polynomials with symmetry, and let r := gcd(u 1 , u 2 ). Then one can construct a strongly invertible 2 × 2 matrix P of Laurent polynomials with compatible symmetry such that P(z) u 1 (z) u 2 (z) = r(z) 0 , (3.47) and the above matrix multiplication is compatible. Proof. By Theorem 3.9, we can find coprime Laurent polynomials u and v with symmetry, such that Define P 1 := u v t s . Then det(P 1 ) = 1, which means P 1 is strongly invertible. Moreover, it follows from (3.48) that u 1 (z)u(z) + u 2 (z)v(z) = r(z),P 1 (z) u 1 (z) u 2 (z) = r(z) t(z)u 1 (z) + s(z)u 2 (z) . SP 1 (z) = Su(z) Sv(z) St(z) Ss(z) = Sθ θ θ 1 (z)Sθ θ θ 2 (z), Su 1 (z) Su 2 (z) = Sθ θ θ 2 (z)Sθ θ θ 3 (z), where Sθ θ θ 1 (z) = [Su (z), St (z)], Sθ θ θ 2 (z) = [1, z c ], Sθ θ θ 3 (z) = Su 1 (z). Hence P 1 has compatible symmetry and all multiplications in (3.52) are compatible. Define P 2 := 1 0 −p 1 , where p := tu 1 +su 2 r is a well-defined Laurent polynomial as r = gcd(u 1 , u 2 ) definitely divides tu 1 + su 2 . We see that P 2 is strongly invertible as det(P 2 ) = 1. Furthermore, we have P 2 (z) r(z) t(z)u 1 (z) + s(z)u 2 (z) = r(z) 0 . (3.53) Applying a similar argument as proving that P 1 has compatible symmetry, we see that P 2 has compatible symmetry and all multiplications in (3.53) are compatible. By letting P := P 2 P 1 , we see that P is strongly invertible and (3.47) holds. Moreover, P has compatible symmetry and all multiplications in (3.47) are compatible. The proof is now complete. For a 2 × 2 matrix A of Laurent polynomial, recall its Smith normal form: A(z) = E(z)Diag(d 1 (z), d 2 (z))F(z),(3.54) where E and F are 2 × 2 strongly invertible matrices of Laurent polynomials, d 1 and d 2 are monic polynomials such that d 1 divides d 2 . From the theory of Smith normal form, d j is uniquely determined up to a multiplication by a monomial for j = 1, 2. Thus without loss of generality, we require the polynomials d 1 and d 2 in the factorization (3.54) to have nonzero constant terms. In this case, we call d 1 and d 2 the invariant polynomials of A. A factorization as in (3.54) is obtained by iteratively using long divisions, which might not preserve symmetry structures. As a result, even if A has symmetry, there is not much to say about the symmetry information on E, F, d 1 and d 2 . To resolve this issue, we now introduce a normal form of a 2 × 2 matrix A of Laurent polynomials with compatible symmetry, which will be a backbone of the proof of Theorem 3.8. Proof. If A is the zero-matrix, then we simply take P = Q = I 2 and D the zero matrix will work. If A is nonzero and is diagonal, then we take P = Q = I 2 and D = A. For the other non-trivial cases, some off-diagonal elements of A is nonzero. Here we assume that A 2,1 = 0, and the case for A 1,2 = 0 can be dealt with similarly. Set A (0,0) := A. For j = 0, 1, . . . , perform the following iterative process: Step (i): If A (j,j) 2,1 = 0, construct a strongly invertible 2 × 2 matrix P (j) of Laurent polynomials with compatible symmstry such that A (j+1,j) (z) := P (j) (z)A (j,j) (z) = r (j,j) (z) A (j+1,j) 1,2 (z) 0 A (j+1,j) 2,2 (z) ,(3.P (j) (z) A (j,j) 1,1 (z) A (j,j) 2,1 (z) = r (j,j) (z) 0 . Then P (j) is the desired matrix as required. Moreover, the multiplication in (3.56) is compatible. Step (ii): If A (j+1,j) is a diagonal matrix, i.e., A (j+1,j) 1,2 = 0, we terminate the process. Otherwise, define r (j+1,j) := gcd(r (j,j) , A (j+1,j) 1,2 ) = 0. Apply Lemma 3.10 to find a strongly invertible matrix Q (j) of Laurent polynomials with compatible symmetry, such that Q (j) (z) A (j+1,j) 1,1 (z) A (j+1,j) 1,2 (z) = Q (j) (z) r (j,j) (z) A (j+1,j) 1,2 (z) = r (j+1,j) (z) 0 , It follows that A (j+1,j+1) (z) := A (j+1,j) (z)Q (j) (z) = r (j+1,j) (z) 0 A (j+1,j+1) 2,1 (z) A (j+1,j+1) 2,2 (z) . (3.57) Moreover, item (3) of Proposition 3.3 tells that the multiplication above is compatible. Step (iii) : If A (j+1,j+1) is a diagonal matrix, i.e., A (j+1,j+1) 2,1 = 0, we terminate the process. Otherwise, redefine j := j + 1 and go to step (i). We claim that the above iterative process cannot go on forever, i.e. A (j,j) or A (j+1,j) becomes diagonal for some j ∈ N. For every j ∈ N, we see that A (j+1,j) 1,1 = r (j,j) = gcd(A (j,j) 1,1 , A (j,j) 2,1 ), which divides A (j,j) 1,1 , and similarly A (j+1,j+1) 1,1 divides A (j+1,j) 1,1 . This implies that len(A (j+1,j+1) 1,1 ) len(A (j+1,j) 1,1 ) len(A (j,j) 1,1 ). If A (j,j) 1,1 does not divide A (j,j) 2,1 , we have len(A (j+1,j) 1,1 ) < len(A (j,j) 1,1 ). Similarly if A (j+1,j) 1,1 does not divide A (j+1,j) 1,2 , we have len(A (j+1,j+1) 1,1 ) < len(A (j+1,j) 1,1 ). As the length of the (1, 1)-th entry cannot decrease strictly forever, at some point we must have A Then P (j) has compatible symmetry, (3.56) holds and all multiplications are compatible. Moreover, the matrix A (j+1,j) defined as (3.56 ) is diagonal. If A (j+1,j) 1,1 divides A (j+1,j) 1,2 , then choose Q (j) (z) :=   1 − A (j+1,j) 1,2 (z) A (j+1,j) 1,1 (z) 0 1   . Then Q (j) has compatible symmetry, (3.57) holds and all multiplications are compatible. Moreover, the matrix A (j+1,j+1) defined as (3.56) is diagonal. Therefore, by applying steps (i)-(iii) for finitely many times, we get strongly invertible matrices P and Q of Laurent polynomials with compatible symmetry, such that D := PAQ is a diagonal matrix satisfying (3.55), and all multiplications in (3.55) are compatible. The symmetry of the diagonal elements e 1 and e 2 of D follows trivially from our previous construction steps. Now let d 1 and d 2 be invariant polynomials of A. Then there exist strongly invertible 2×2 matrices E and F such that (3.54) holds. By the definition of invariant polynomials, we have Z(d 1 , z 0 ) Z(d 2 , z 0 ). Without loss of generality, we assume Z(e 1 , z 0 ) Z(e 2 , z 0 ), and prove that Z(d j , z 0 ) = Z(e j , z 0 ) for j = 1, 2. For any z 0 ∈ C \ {0}, write d j (z) = (z − z 0 ) Z(d j ,z 0 ) p j (z) and e j (z) = (z − z 0 ) Z(e j ,z 0 ) q j (z) for j = 1, 2. It follows that A(z) = E(z)Diag (z − z 0 ) Z(d 1 ,z 0 ) , (z − z 0 ) Z(d 2 ,z 0 ) Diag(p 1 (z), p 2 (z))F(z), (3.58) A(z) = P(z) −1 Diag (z − z 0 ) Z(e 1 ,z 0 ) , (z − z 0 ) Z(e 2 ,z 0 ) Diag(q 1 (z), q 2 (z))Q −1 (z). (3.59) It follows that Diag (z − z 0 ) Z(d 1 ,z 0 ) , (z − z 0 ) Z(d 2 ,z 0 ) = W(z)Diag (z − z 0 ) Z(e 1 ,z 0 ) , (z − z 0 ) Z(e 2 ,z 0 ) V(z), (3.60) where W := E −1 P −1 and V := Diag(q 1 , q 2 )Q −1 F −1 Diag(p 1 , p 2 ) −1 . It is trivial that all entries of W and V, as well as det(W) and det(V), are analytic on a neighborhood of z 0 . Moreover, det(W) and det(V) and are nonzero at z 0 . By calculating determinants on both sides of (3.60), we have (z − z 0 ) Z(d 1 ,z 0 )+Z(d 2 ,z 0 ) = (z − z 0 ) Z(e 1 ,z 0 )+Z(e 2 ,z 0 ) det(W(z)) det(V(z)). Note that det(W) and det(V) are analytic in a neighborhood of z 0 , and are nonzero at z 0 . Thus Z(d 1 , z 0 ) + Z(d 2 , z 0 ) = Z(e 1 , z 0 ) + Z(e 2 , z 0 ). (3.61) Furthermore, (3.60) yields (z − z 0 ) Z(d 1 ,z 0 ) = (z − z 0 ) Z(e 1 ,z 0 ) W 1,1 (z)V 1,1 (z) + (z − z 0 ) Z(e 2 ,z 0 )−Z(e 1 ,z 0 ) W 1,2 (z)V 2,1 (z) , which implies Z(e 1 , z 0 ) Z(d 1 , z 0 ). On the other hand, (3.60) is equivalent to Then we can construct 2 × 2 matrices U andÃ of Laurent polynomials with compatible symmetry such that (i)Ã is Hermitian and A = UÃU with all multiplications being compatible; W(z) −1 Diag (z − z 0 ) Z(d 1 ,z 0 ) , (z − z 0 ) Z(d 2 ,z 0 ) V −1 (z) = Diag (z − z 0 ) Z(e 1 ,z 0 ) , (z − z 0 ) Z(e 2 ,z (ii) gcd(Ã 1,1 ,Ã 1,2 ,Ã 2,1 ,Ã 2,2 ) = 1; (iii) det(Ã) = −dd for some nonzero Laurent polynomiald with symmetry; (iv) len(det(Ã)) < len(det(A)) and σ(Ã) σ(A). Proof. By Theorem 3.11, one can construct strongly invertible matrices P and Q of Laurent polynomials with compatible symmetry, such that ( • If z 0 ∈ C \ (T ∪ R): Then z 0 , z −1 0 , z 0 and z 0 −1 are pairwise distinct. Define the Laurent polynomial p(z) := (z − z 0 ) α (z − z −1 0 ) α with symmetry. Then p divides e k , and thus divides the k-th row ofÅ. AsÅ is Hermitian, we see that p divides the k-th column ofÅ. Note that p (z) = z −2α (z − z 0 ) α (z − z 0 −1 ) α , which implies gcd(p, p ) = 1. Hence pp dividesÅ k,k . Define V 1,p (z) := Diag(p(z), 1), V 2,p (z) := Diag(1, p(z)). (3.65) Then we haveÅ (z) = V k,p (z)Ã(z)V k,p (z) (3.66) for some 2 × 2 matrixÃ of Laurent polynomials. By letting U := PV k,p , it follows immediately that A = UÃU . We claim that the matrices U andÃ satisfy all claims of the lemma. Since p has symmetry andÅ has compatible symmetry, we conclude that U andÃ have compatible symmetry, and item (i) clearly holds. Since det(P) = det(Q) = 1, so the Smith normal form ofÅ is the same as the one of A, which is Diag (1, det(A)). This in particular implies gcd(Å 1,1 ,Å 1,2 ,Å 2,1 ,Å 2,2 ) = 1. Therefore, by letting l ∈ {1, 2} \ {k}, we have gcd(Ã 1,1 ,Ã 1,2 ,Ã 2,1 ,Ã 2,2 ) = gcd Å k,k pp ,Å k,l p ,Å l,k p ,Å l,l = 1. This proves item (ii). By det(A) = det(Å) = det(V k,p ) det(Ã) det(V k,p ) = pp det(Ã), we have det(Ã) = det(A) pp = −dd pp . As d has symmetry, we can define β 1 := Z(d, z 0 ) = Z(d, z −1 0 ), β 2 := Z(d, z 0 ) = Z(d, z 0 −1 ), q(z) := (z − z 0 ) β 1 (z − z −1 0 ) β 1 (z −1 − z 0 ) β 2 (z −1 − z 0 −1 ) β 2 . We have α = Z(det(A), z 0 ) = Z(d, z 0 ) + Z(d , z 0 ) = β 1 + β 2 . Hence qq = pp . Moreover, q has symmetry and divides d. Defined := d q . It is trivial thatd is a Laurent polynomial with symmetry, and we have det(Ã) = −dd . This proves item (iii). It is easy to see that len(q) > 0, thus len(d) < len(d) and len(det(Ã)) < len(det(A)). Furthermore, it is trivial that σ(Ã) ⊆ σ(A). Note that Z(det(Ã), z 0 ) = Z det(A) pp , z 0 = Z(det(A), z 0 ) − Z(p, z 0 ) − Z(p , z 0 ) = α − α − 0 = 0, which implies z 0 / ∈ σ(Ã). Hence σ(Ã) σ(A), and item (iv) is verified. • If z 0 ∈ (T ∪ R) \ {0, ±1}: Then z 0 −1 = z 0 . When z 0 ∈ T \ {±1}, we have z 0 −1 = z 0 , so Z(d, z 0 ) = Z(d, z 0 −1 ) = Z(d , z 0 ). When z 0 ∈ R \ {0, ±1}, we have z 0 = z 0 , so Z(d, z 0 ) = Z(d, z −1 0 ) = Z(d, z 0 −1 ) = Z(d , z 0 ). Thus we conclude that Z(d, z 0 ) = Z(d , z 0 ) for this case. It follows that α = Z(det(A), z 0 ) = Z(d, z 0 ) + Z(d , z 0 ) = 2Z(d, z 0 ) ∈ 2Z. (3.67) Define p(z) := z −α/2 (z − z 0 ) α/2 (z − z −1 0 ) α/2 = (z + z −1 − 2 Re(z 0 )) α/2 , if z 0 ∈ T \ {±1}, (z + z −1 − (z 0 + z −1 0 )) α/2 , if z 0 ∈ R \ {0, ±1} . It is easy to check that Sp = 1 and p = p. Given that z 0 = ±1, one can conclude that (z − z 0 ) α (z − z −1 0 ) α divides e k , i.e., pp divides e k . Hence by (3.63), p divides the k-th row of A, p divides the k-th column ofÅ, and pp dividesÅ k,k . Define V j,p as (3.65) for j = 1, 2, and define U := PV k,p . We see that (3.66) holds for some 2 × 2 matrixÃ of Laurent polynomials. Moreover, U andÃ have compatible symmetry, and items (i) and (ii) hold. From (3.67), we see that Z(d, z 0 ) = Z(d, z −1 0 ) = α/2. As d has symmetry, so p divides d. Defined := d/p, thend has symmetry and det(Ã) = −dd . This proves item (iii). As len(p) > 0, we have len(d) < len(d) and thus len(det(Ã)) < len(det(A)). Furthermore, it is trivial that σ(Ã) ⊆ σ(A). Note that Z(det(Ã), z 0 ) = Z(det(A), z 0 ) − Z(p, z 0 ) − Z(p , z 0 ) = α − α/2 − α/2 = 0, which means z 0 / ∈ σ(Ã). Thus σ(Ã) σ(A), and item (iv) is proved. (z). By (3.63), p divides the k-th row ofÅ, p divides the k-th column ofÅ, and pp dividesÅ k,k . Define V j,p as (3.65) for j = 1, 2, and define U := PV k,p . We see that (3.66) holds for some 2 × 2 matrixÃ of Laurent polynomials. Moreover, U andÃ have compatible symmetry, and by using the same arguments as in the previous case, we can prove that items (i) -(iv) hold. The proof is now complete. and (A (j) ) K j=1 of Laurent polynomials with compatible symmetry, and a sequence (d j ) K j=1 of nonzero Laurent polynomials with symmetry, such that • If z 0 ∈ {±1}: Then z 0 = z −1 0 = z 0 = z 0 −1 , which implies Z(d, z 0 ) = Z(d , z 0 −1 ) = Z(d , z 0 ) and thus (3.67) holds. Define p(z) := (z − z 0 ) α/2 . Direct calculation yields Sp(z) = (−z 0 ) α/2 z α/2 and p (z) = (−z 0 z) −α/2 (z − z 0 ) α/2 . Moreover, p(z)p (z) = (−z 0 z) −α/2 (z − z 0 ) α divides e k • A (j) is Hermitian and A (j−1) = U j A (j) U j with all multiplications being compatible for j = 1, . . . , K; • gcd(A (j) 1,1 , A (j) 1,2 , A (j) 2,1 , A (j) 2,2 ) = 1 for all j = 1, . . . , K; • det(A (j) ) = −d (j) d (j) for all j = 0, . . . , K • len(det(A (0) )) > len(det(A (1) )) > · · · > len(det(A (K) )) = 0 and σ(A (0) ) σ(A (1) ) · · · σ(A (K−1) ) = ∅. As a consequence, we have A = A (0) = U 1 · · · U K A (K) U K · · · U 1 . Define B := A (K) and U := U 1 · · · U K . We see that both B and U have compatible symmetry and all multiplications in A = UBU are compatible. Note that det(B) = det(A (K) ) = −dd = det(B) . Thus by len(det(B)) = len(det(A (K) )) = 0, we conclude that det(B) is a negative constant. This proves item (i). Finally, item (ii) is a direct consequence of Theorem 3.4. This completes the proof. 3.5. Difference of (Hermitian) squares property. In this subsection, we prove Theorem 1.2 for the difference of squares (DOS) property of a Laurent polynomial as stated in Section 1. Writing a Laurent polynomial into sums and differences of squares is inevitable for a generalized spectral factorization problem. Therefore it is important to deepen our understandings of the DOS property. We need two lemmas to prove Theorem 1.2. The following lemma demonstrates that the DOS property is preserved under multiplication. Lemma 3.13. Suppose u 1 , u 2 , u 3 and u 4 are Laurent polynomials with symmetry, denote the symmetry types by Su j (z) = j z c j , j ∈ {±1}, c j ∈ Z, j = 1, 2, 3, 4. Suppose Su 1 Su 2 = Su 3 Su 4 . Define u 5 (z) := u 1 (z)u 3 (z) + z c 1 u 2 (z)u 4 (z), u 6 (z) := u 2 (z)u 3 (z) + z c 1 u 1 (z)u 4 (z). (3.68) Then Su 5 (z) = 1 3 z c 1 +c 3 , Su 6 (z) = 2 3 z c 2 +c 3 , Su 5 Su 6 = Su 1 Su 2 = Su 3 Su 4 ,(3. 69) and u 5 u 5 − u 6 u 6 = (u 1 u 1 − u 2 u 2 )(u 3 u 3 − u 4 u 4 ). (3.70) Proof. By Su 1 Su 2 = Su 3 Su 4 , we have 1 2 = 3 4 and c 1 − c 2 = c 3 − c 4 . It follows that S[z c 1 u 2 u 4 ](z) = 2 4 z 2c 1 z −c 2 +c 4 = 1 3 z c 1 +c 3 = S[u 1 u 3 ](z), S[z c 1 u 1 u 4 ](z) = 1 4 z 2c 1 z −c 1 +c 4 = 2 3 z c 2 +c 3 = S[u 2 u 3 ](z). Thus (3.69) follows immediately. Note that (3.68) is equivalent to u 5 u 6 = u 1 u 2 u 2 u 1 u 3 z c 1 u 4 . It follows that u 5 u 5 − u 6 u 6 = u 5 u 6 Diag(1, −1) u 5 u 6 = u 3 z −c 1 u 4 u 1 u 2 u 2 u 1 Diag(1, −1) u 1 u 2 u 2 u 1 u 1 (z) = u 1 (z) Su 1 (z) , u 2 (z) = z c u 2 (z) Su 1 (z) . So for x ∈ R \ {0}, it follows from (1.4) that u(x) = \|u 1 (x)\| 2 − x c \|u 2 (x)\| 2 Su 1 (x) . If = 1 and c ∈ 2Z+1, we have x c < 0 for all x ∈ (−1, 0), and thus Z(u, x) = 2 min{Z(u 1 , x), Z(u 2 , x)} ∈ 2Z. This proves condition (ii), and conditions (iii) and (iv) can be verified similarly. This proves item (2). Sufficiency: Suppose u is a Laurent polynomial satisfying items (1) and (2). By (1.4), for every integer k, we have (z k u 1 (z))(z k u 1 (z)) − u 2 (z)u 2 (z) = u(z) and S[z k u 1 ](z) Su 2 (z) = z c+2k . That is, if u has the DOS property with respect to symmetry type z c , then it has the DOS property with respect to symmetry type z c+2k for all k ∈ Z. As a consequence, it suffices to prove the sufficiency part for c ∈ {0, 1}. Since u is Hermitian and has real coefficients, we conclude that Su = 1 and Z(u, z 0 −1 ) = Z(u , z 0 ) = Z(u, z 0 ). Also Z(u, z 0 ) = Z(u, z −1 0 ) as u has symmetry. It follows that Z(u, z 0 ) = Z(u, z −1 0 ) = Z(u, z 0 −1 ) = Z(u, z 0 ), ∀z 0 ∈ C \ {0}.z −1 (z − z 0 ) 2 , with z 0 ∈ σ ±1 ; Type (b): z −2 (z − z 0 )(z − z −1 0 )(z − z 0 )(z − z 0 −1 ), with z 0 ∈ σ in,up ; Type (c): z −1 (z − z 0 )(z − z 0 ), with z 0 ∈ σ T,up ; Type (d): z −1 (z − z 0 )(z − z −1 0 ) , with z 0 ∈ σ R,up . According to Lemma 3.13, we see that taking products preserves the DOS property of Laurent polynomials. Therefore, we only need to prove that all types of factors above have the DOS property with respect to the symmetry type z c . We discuss them one by one. Type (a): If z 0 ∈ σ ±1 , then z −1 (z − 1) 2 = 0 2 − (z − 1)(z − 1) , z −1 (z + 1) 2 = (z + 1)(z + 1) 2 − 0 2 . Recall that 0 has symmetry of any type. Hence the type (a) factors have the DOS property with respect to symmetry type z c with ∈ {±1} and c ∈ {0, 1}. Type (b): If z 0 ∈ σ in,up , then z −2 (z − z 0 )(z − z −1 0 )(z − z 0 )(z − z 0 −1 ) = \|z 0 \| −1 z −1 (z − z 0 )(z − z 0 ) \|z 0 \| −1 z −1 (z − z 0 )(z − z 0 ) − 0 2 . For the same reason as in the type (a) case, we see that the type (b) factors have the DOS property with respect to symmetry type z c with ∈ {±1} and c ∈ {0, 1}. Type (c): If z 0 ∈ σ T,up , we have v(z) := z −1 (z − z 0 )(z − z 0 ) = z − 2 Re(z 0 ) + z −1 and Re(z 0 ) < 1. We now discuss symmetry conditions (i) -(iv) in item (2) one by one: • For (i), the symmetry type is z 0 = 1. Define v 1 (z) := z − 4 Re(z 0 ) + 2 + z −1 2 2 − 2 Re(z 0 ) , v 2 (z) := z − 2 + z −1 2 2 − 2 Re(z 0 ) . Direct calculation yields v = v 1 v 1 − v 2 v 2 and Sv 1 Sv 2 = 1. • For (ii), the symmetry type is z 1 = z. Define v 1 (z) := z + 1 and v 2 (z) := 2 + 2 Re(z 0 ). It is easy to see that v = v 1 v 1 − v 2 v 2 and Sv 1 Sv 2 = z. • For (iii), the symmetry type is −z 0 = −1. Define v 1 (z) := 1 − Re(z 0 ) 2 (z + 1), v 2 (z) := 1 − Re(z 0 ) 2 (z − 1). It follows that v = v 1 v 1 − v 2 v 2 and Sv 1 Sv 2 = −1. • For (iv), the symmetry type is −z 1 = −z. Define v 1 (z) := 2 − 2 Re(z 0 ) and v 2 (z) := 1 − z −1 . It is straightforward to verify that v = v 1 v 1 − v 2 v 2 and Sv 1 Sv 2 = −z. This finishes the case for type (c). Type (d): If z 0 ∈ σ R,in , in particular, z 0 ∈ (−1, 0) ∪ (0, 1). Let w(z) := z −1 (z − z 0 )(z − z −1 0 ) = z − (z 0 + z −1 0 ) + z −1 . We again consider the following two cases: • Z(u, z 0 ) ∈ 2Z: Since w has symmetry type Sw = 1, we see that Z(u, z −1 0 ) ∈ 2Z, and thus w 2 must divide u. So it suffices to show that w 2 has the DOS property. In fact, since w = w , we have w(z) 2 = w(z)w (z) − 0 2 . Therefore, w 2 has the DOS property with respect to symmetry type z c with ∈ {±1} and c ∈ {0, 1}. • Z(u, z 0 ) ∈ 2Z + 1: We discuss conditions (i) -(iv) individually: -For (i): the symmetry type is 1. Define w 1 (z) := w(z) + 1 4 and w 2 (z) := v(z) − 1 4 . Direct calculation yields w = w 1 w 1 − w 2 w 2 and Sw 1 Sw 2 = 1. -For (ii), the symmetry type is z. Moreover, we have Z(u, z 0 ) ∈ 2Z for all z 0 ∈ (−1, 0) and thus any such points fail the assumption Z(u, z 0 ) ∈ 2Z + 1. So we must have z 0 ∈ (0, 1). Now define w 1 (z) := z + 1 and w 2 (z) := 2 + z 0 + z −1 0 . It is easy to see that w = w 1 w 1 − w 2 w 2 and Sw 1 Sw 2 = z. -For (iii), we have Z(u, z 0 ) ∈ 2Z for all z 0 ∈ (−1, 0) ∪ (0, 1), which cannot happen as we assumed that Z(u, z 0 ) ∈ 2Z + 1. -For (iv), the symmetry type is −z, and we must have z 0 ∈ (−1, 0). Define w 1 (z) := 2 − z 0 − z −1 0 and w 2 (z) := z −1 − 1. It is straightforward to verify that w = w 1 w 1 − w 2 w 2 and Sw 1 Sw 2 = −z. This finishes the proof of the sufficiency part. The theorem is now proved. 3.6. Proof of Theorem 1.1. In previous subsections, we have proved the special cases of Theorem 1.1, namely Theorems 3.4 and 3.8. In this subsection, we are going to complete the proof of Theorem 1.1. Before we do this, we need one last lemma. Furthermore, for every Laurent polynomial d with symmetry satisfying (3.74), there exists k ∈ Z such that Sd(z) = z 2k Su(z) Sv (z) . Proof. The existence of d satisfying (3.74) is a direct consequence of the equivalence between items (1) and (3) in [15, Theorem 2.9]. As both uu and vv satisfy item (1) and hence, item (3) holds for both. Now it is trivial that uu vv satisfies item (3) and hence, item (1) holds for uu vv , which is exactly (3.74). Next, denote Sd(z) = d z c d , Su(z) = u z cu and Sv(z) = v z cv for some d , u , v ∈ {±1} and c d , c u , c v ∈ Z. By Lemma 3.2, we have d = (−1) Z(d,1) = (−1) Z(u,1)−Z(v,1) = u / v . Moreover, by Proposition 3.1, we get odd(c d ) = odd(len(d)) = odd(len(u) − len(v)) = odd(c u − c v ). Therefore, we can find k ∈ Z such that Sd(z) = z 2k Su(z) Sv (z) . Proof of Theorem 1.1. We first prove sufficiency. Assume that items (1) and (2) hold. If det(A) is identically zero, then by Theorem 3.11, there exist strongly invertible 2 × 2 matrices P and Q of Laurent polynomials with compatible symmetry such that A = PDiag(e 1 , 0)Q, where all multiplications are compatible, and e 1 has symmetry. DefineÅ := P −1 AP − , it is trivial thatÅ has compatible symmetry, and all multiplications are compatible. SinceÅ is Hermitian, one can con- Now assume that det(A) is not identically zero. We construct U in the following steps. Step 1. Denote A j,k the (j, k)-th entry of A for 1 j, k 2. Since A = A, we have A 1,1 = A 1,1 . By Lemma 3.14, we have Z(A 1,1 , 1), Z(A 1,1 , −1) ∈ 2Z. Let (3.75) We see thatÃ is a Hermitian matrix of Laurent polynomials with compatible symmetry. Step 2. DefineÃ j,k to be the (j, k)-th entry ofÃ for 1 j, k 2. Definẽ p := gcd(Ã 1,1 ,Ã 1,2 ,Ã 2,1 ,Ã 2,2 ). Hence Z(p, 1) = 0. Similarly, we get Z(p, −1) = 0. Moreover, for z 0 ∈ C \ {0, 1, −1}, we have Z(p, z 0 ) = Z(p 0 , z 0 ), where p 0 = gcd(A 1,1 , A 1,2 , A 2,1 , A 2,2 ). Therefore, we conclude thatp = p, where p is defined as (1.6). According to item (2),p has the DOS property with respect to the symmetry type Sd(z)α(z). By Theorem 1. . (4.9) Now it suffices to find a matrix of Laurent polynomials V = V 1,1 V 1,2 V 2,1 V 2,2 such that N a,Θ\|n b = V Diag(1, −1)V. In this case, by letting b [0] (z) := V 1,1 V 2,1 ,b [1] (z) := V 1,2 V 2,2 ,(4.10) we have N a,Θ\|n b (z) = b [0] (z)b [1] (z) Diag(1, −1) b [0] (z)b [1] (z) , (4.11) which by (4.6) is equivalent to (4.4). By letting b := (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 be such that b(z) = (1 − z) n bb (z), we see that (4.1) holds, and thus {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank. Moreover, we have min{vm(b 1 ), vm(b 2 )} n b . By the discussions above, we summarize the guideline for construction as the following theorem. Theorem 4.1. Let a, Θ ∈ l 0 (Z) be such that Θ = Θ and let b = (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 . Suppose that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank such that min{vm(b 1 ), vm(b 2 )} n b . Then (4.11) holds, where N a,Θ\|n b is defined as (4.9), andb ∈ (l 0 (Z)) 2×1 satisfies b(z) = (1 − z) n bb (z). Conversely, suppose that there exists a matrix of Laurent polynomials V = V 1,1 V 1,2 V 2,1 V 2,2 such that N a,Θ\|n b = V Diag(1, −1)V. Defineb, b ∈ (l 0 (Z)) 2×1 via (4.10) and b(z) := (1 − z) n bb (z) respectively. Then {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank such that min{vm(b 1 ), vm(b 2 )} n b . 4.2. Symmetry property of quasi-tight framelet filter banks. We need to investigate a little further on symmetry property of quasi-tight framelet filter banks. Theorem 4.2. Let a, Θ ∈ l 0 (Z) be such that Θ = Θ and let b = (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 . Suppose that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank such that SΘ(z) = 1, Sa(z) = z c , Sb 1 (z) = 1 z c 1 , Sb 2 (z) = 2 z c 2 , (4.12) for some , 1 , 2 ∈ {±1} and c, c 1 , c 2 ∈ Z. Here Θ, a, b 1 and b 2 are Laurent polynomials associated with Θ, a, b 1 and b 2 respectively. Then c 1 + c, c 2 + c ∈ 2Z. (4.13) Proof. By the definition of a quasi-tight framelet filter bank, we have Our construction of quasi-tight framelet filter banks is based on factorizing the matrix N a,Θ\|n b , whose entries are Laurent polynomials associated with coset sequences. So we need the following result on symmetry property of coset sequences. Lemma 4.3. Let u ∈ l 0 (Z) be such that Su(z) = z c for some ∈ {±1} and c ∈ Z. Then the following statements hold: Θ(z 2 )a(z)a (−z) + b 1 (z)b 1 (−z) − b 2 (z)b 2 (−z) = 0, z ∈ C \ {0}. (1) If c ∈ 2Z, then u [0] and u [1] have symmetry, with Su [0] (z) = z c/2 , Su [1] (z) = z c/2−1 ; (2) If c ∈ 2Z + 1, for all l ∈ Z, define P l (z) : = 1 √ 2 1 z l 1 −z l , l ∈ Z,(4.15) and v 1 (z) v 2 (z) := P l (z) u [0] (z) u [1] (z) = 1 √ 2 u [0] (z) + z l u [1] (z) u [0] (z) − z l u [1] (z) . Then v 1 and v 2 have symmetry, with On the other hand, if v 1 and v 2 are Laurent polynomials with symmetry, then the following statements hold: (3) If Sv 1 (z) = z c+1 and Sv 2 (z) = z c for some ∈ {±1} and c ∈ Z, then u(z) := v 1 (z 2 ) + zv 2 (z 2 ) also has symmetry, with Su(z) = z 2c+2 ; (4) If Sv 1 (z) = z c and Sv 2 (z) = − z c for some ∈ {±1} and c ∈ Z, define P l for all l ∈ Z as (4.15) and w 1 (z) w 2 (z) := P l (z) −1 v 1 (z) v 2 (z) = 1 √ 2 Theorem 1 . 2 . 12Suppose ∈ {±1} and c ∈ Z. A Laurent polynomial u has the DOS property with respect to the symmetry type z c if and only if Figure 1 . 1Example 1: (A),(B),(C) and (D) are the graphs of the filters a, b 1 , b 2 and Θ. (E), (F) and (G) are the graphs of the refinable function φ and the framelet generators ψ 1 and ψ 2 . Figure 2 . 2In Example 2: (A),(B),(C) and (D) are the graphs of the filters a, Θ, b 1 , b 2 . (E), (F) and (G) are the graphs of the refinable function φ and the framelet generators ψ 1 and ψ 2 . Figure 3 . 3In Example 3: (A),(B),(C) and (D) are the graphs of the filters a, Θ, b 1 , b 2 . Figure 4 . 4In Example 4: (A),(B),(C) and (D) are the graphs of the filters a, Θ, b 1 , b 2 . (E), (F) and (G) are the graphs of the refinable function φ and the framelet generators ψ 1 and ψ 2 . Figure 5 . 5In Example 5: (A),(B),(C) and (D) are the graphs of the filters a, Θ, b 1 , b 2 . (E), (F), (G) and (H) are the graphs of the refinable functions φ, η and the framelet generators ψ 1 and ψ 2 . Lemma 3 . 5 . 35Let a and b be Laurent polynomials with symmetry types Sa(z) = a z ca and Sb(z) = b z c b , respectively, for some a , b ∈ {±1} and c a , c b ∈ Z. Suppose b = 0 and len(a) > len(b). Then there exists a Laurent polynomial q 1 with symmetry such that hold. If len(a K ) = len(b), define ldeg(a K ) := m a,K , deg(a K ) := M a,K , ldeg(b) := m b , deg(b) := M b , and define ( 3 . 316) From len(a K ) = len(b), we have M a,k −M b = m a,K −m b . Thus deg(bq) = M b +deg(q) = M a,K and ldeg(bq) = m b + (M a,K − M b ) = m b + (m a,K − m b ) = m a,K . Hence from the definitions ofq and r, it is easy to see that ldeg(r) > m a,K and deg(r) < M a,K . In particular, len(r) < len(a K ) = len(b). possible cases for α(z) = z c : Case 1: = 1, c ∈ 2Z + 1; Case 2: = −1, c ∈ 2Z + 1; Case 3: = 1, c ∈ 2Z; Case 4: = −1, c ∈ 2Z. To prove the main result Theorem 3.4, all the cases 1 -3 can be investigated together, while case 4 need to be dealt with individually. For cases 1 -3, we have the following technical lemma, which plays a key role for constructing a matrix U as required in Theorem 3.4.Lemma 3.7. Let A be a 2 × 2 Hermitian matrix of Laurent polynomials with symmetry type SA = 1 α(z) α (z) 1, where α is one of the following: nonzero Laurent polynomial u, define its support as fsupp(u) := [ldeg(u), deg(u)]. (3.18) As A is Hermitian, it is easy to see that fsupp(A 1,1 A 2,2 ) and fsupp(A 1,2 A 2,1 ) are symmetric intervals with centre 0. Define [−m, m] := fsupp(A 1,1 A 2,2 ), [−n, n] := fsupp(A 1,2 A 2,1 ).Since det(A) = A 1,1 A 2,2 − A 1,2 A 2,1 = C is a constant, we must have m = n. It follows that len(A 1,1 ) + len(A 2,2 ) = len(A 1,1 A 2,2 ) = 2m = 2n = len(A 1,2 A 2,1 ) = len(A 1,2 ) + len(A 2,1 ) = 2 len(A 2,1 ). 1 ) = len(r) < len(A 2,2 ) len(A 2,1 ).(3.21) In both cases above, SV (0) (z) = SA(z) = Sθ θ θ 1 (z)Sθ θ θ 1 (z) where Sθ θ θ 1 (z) :=[1, α(z)]. This shows that the multiplications A (1) = V (0) AV (0) are compatible. Hence we have found a desired V (0) as required.Now if A (1) has a zero entry, we stop here. Otherwise, we repeat the above process to further shrink the length of A (1) 2,1 . In general, if all entries of A (j) are nonzero, we apply the above process to find a strongly invertible matrix V (j) such that A (j+1) := V (j) A (j) V (j) satisfies len(A the multiplications are compatible. Note that len(A 2,1 ) > len(A by SA (K) = SA, we have SA c 2 . Moreover, c 1 and c 2 have opposite signs. DefineŨ as (3.23), direct calculation shows that A (K) =ŨDiag(1, −1)Ũ . Furthermore, since 0 has any types of symmetry, we see thatŨ has compatible symmetry with SŨ = Sθ θ θ 1 Sθ θ θ 2 , where Sθ θ θ 1 := [1, α] and Sθ θ θ 2 := [α, 1]. If A 3.24). Then it is straightforward to verify that A (K) =ŨDiag(1, −1)Ũ and SŨ = Sθ θ θ 1 Sθ θ θ 3 where Sθ θ θ 3 := [α, α]. Lastly, if A (K) 2,2 = 0, it is easy to see that A (K) =ŨDiag(1, −1)Ũ withŨ being defined as (3.25), and SŨ = Sθ θ θ 1 Sθ θ θ 4 where Sθ θ θ 4 := [1, 1]. at least two. Take any nonzero element (t 0 , . . . , t n , s 0 , . . . , s n ) ∈ E and definẽ gcd(A 1,1 , A 2,1 ) = 1. So A 1,1 (z) dividesŮ 1,1 (z)Ů 1,1 (z) −Ů 1,2 (z)Ů 1,2 (z).On the other hand, since fsupp(Ũ 1,1 ) ⊆ [0, n] and fsupp(Ũ 1,2 ) ⊆ [0, n], we see that fsupp(Ů 1,1 ) ⊆ [0, n] and fsupp(Ů 1,2 ) ⊆ [0, n]. Thus fsupp(Ů 1,1Ů 1,1 ) ⊆ [−n, n] and fsupp(Ů 1,2Ů 1,2 ) ⊆ [−n, n]. So fsupp(Ů 1,1Ů 1,1 −Ů 1,2Ů 1,2 ) ⊆ [−n, n] = fsupp(A 1,1 ). Theorem 3 . 8 . 38Let A be a 2 × 2 Hermitian matrix of Laurent polynomials with compatible symmetry. Suppose that(1) gcd(A 1,1 , A 1,2 , A 2,1 , A 2,2 ) = 1; (2) det(A(z)) = −d(z)d (z)for some nonzero Laurent polynomial d(z) with symmetry. Then we can find 2 × 2 matrices V and B of Laurent polynomials with compatible symmetry such that (i) B = B and det(B) = C is a negative constant; (ii) A = VBV where all multiplications are compatible. As a consequence, there exists a 2 × 2 matrix U of Laurent polynomials with compatible symmetry such that A = UDiag(1, −1)U and (1.5) holds. The proof of Theorem 3.8 relies heavily on long division of Laurent polynomials. Note that the symmetry structures are not preserved under long divisions. That is, if a and b are Laurent polynomials with symmetry and b = 0, then we can always do the long division a(z) = b(z)q(z) + r(z),for some Laurent polynomials q and r. However, q and r may not necessarily have any symmetry. Fortunately, we can still establish the following Extended Euclidean Algorithm with symmetry. we have Sa(z)Su(z) = Sb(z)Sv(z) = Sr(z) and gcd(u, v) = 1. 1 (z)Su(z) = Su 2 (z)Sv(z) = Sr(z).(3.49) As u and v are coprime, applying Theorem 3.9 again yields u(z)s(z) − v(z)t(z) = 1, (3.50) for some coprime Laurent polynomials s and t with symmetry satisfying Su(z)Ss(z) = Sv(z)St(z) = 1. (3.51) ( 3 . 52 ) 352Using (3.49) and (3.51), we see thatSu 1 Su 2 = Sv Su = St Ss = z c for some ∈ {±1} and c ∈ Z. This implies that Theorem 3 . 311. (Normal form with compatible symmetry) Let A be a 2 × 2 matrix of Laurent polynomials with compatible symmetry. Then there exist strongly invertible 2 × 2 matrices P and Q of Laurent polynomials with compatible symmetry such that PAQ is a diagonal matrix: D(z) = Diag(e 1 (z), e 2 (z)) := P(z)A(z)Q(z), (3.55) where all multiplications above are compatible, and both e 1 , e 2 have symmetry. Furthermore, if d 1 and d 2 are the invariant polynomials of A, then {Z(e 1 , z 0 ), Z(e 2 , z 0 )} = {Z(d 1 , z 0 ), Z(d 2 , z 0 )} for all z 0 ∈ C \ {0}. = 0 . 056)where r (j,j) := gcd(A In fact, if A (j,j) 1,1 = 0, simply take P (j) apply Lemma 3.10 to find a strongly invertible matrix P (j) of Laurent polynomials with compatible symmetry, such that 0 ) , and by using a similar argument we conclude that Z(d 1 , z 0 ) Z(e 1 , z 0 ). Therefore, Z(d 1 , z 0 ) = Z(e 1 , z 0 ), and thus by (3.61) we have Z(d 2 , z 0 ) = Z(e 2 , z 0 ). This completes the proof. For every square matrix A of Laurent polynomials, we define its spectrum as σ(A) := {z ∈ C \ {0} : det(A(z)) = 0}. (3.62) When det(A) is non-constant, it is clear that σ(A) is non-empty. The following lemma demonstrates how we can shrink the length of det(A) as well as the set σ(A). Lemma 3 . 12 . 312Let A be a 2 × 2 Hermitian matrix of Laurent polynomials with compatible symmetry. Suppose that (1) gcd(A 1,1 , A 1,2 , A 2,1 , A 2,2 ) = 1; (2) det(A) = −dd for some nonzero Laurent polynomial d(z) with symmetry. 3.55) holds, where all multiplications are compatible, and e 1 , e 2 have symmetry. Without loss of generality, we may assume that det(P) = det(Q) = 1, in which case we have e 1 (z)e 2 (z) = det(D(z)) = det(A(z)) = −d(z)d (z). (3.63) Furthermore, item (1) implies that the Smith normal form of A is Diag(1, det(A)). Thus by Theorem 3.11, we have Z(e j , z 0 ) = 0 or Z(e j , z 0 ) = Z(det(A), z 0 ) for all z 0 ∈ C \ {0} and j = 1, 2. Define the Hermitian matrixÅ of Laurent polynomials viå A(z) := P −1 A(z)P − (z) = Diag(e 1 (z), e 2 (z))Q(z)P − (z). (3.64) It is trivial that all multiplications above are compatible. Let z 0 ∈ σ(A). by (3.63), we must have (z − z 0 ) divides e k for some k ∈ {1, 2}. So α := Z(e k , z 0 ) > 0 and our previous discussion yields α = Z(det(A), z 0 ). Moreover, we have Z(e k , z −1 0 ) = α by the symmetry property of e k . There are three possibilities for the location of z 0 . Proof of Theorem 3.8. If det(A) = C is a negative constant, then taking B := A and V := I 2 yields the result. Assume det(A) is not a constant. Set A (0) := A. By Lemma 3.12, we can find two sequences (U j ) K j=1 (u) as (3.71), and defineσ ±1 := σ(u) ∩ {±1}, σ R,in := σ(u) ∩ ((−1, 0) ∪ (0, 1)) , σ T,up := σ(u) ∩ {z ∈ C \ {0} : \|z\| = 1, Im(z) > 0}, σ in,up := σ(u) ∩ {z ∈ C \ {0} : \|z\| < 1, Im(z) > 0}.By (3.72),(3.73) and the fact that u is Hermitian (particularly, fsupp(u) is symmetric about 0), up to multiplication by a real number, u(z) can be split into the product of the following four types of factors: Type (a): Lemma 3 . 15 . 315Suppose that u and v are two Laurent polynomials with symmetry, such that vv divides uu . Then there exists a Laurent polynomial d with symmetry, such that dd = uu vv . (3.74) U has compatible symmetry. Moreover, we haveÅ =ŮDiag(1, −1)Ů , where all multiplications are compatible. Finally, by letting U := PŮ, we see that A = UDiag(1, −1)U holds. Moreover, U has compatible symmetry and satisfies (1.5). This proves the case when det(A) ≡ 0. , −1), Z(A 1,2 , −1) , and define w(z) := (z − 1) α 1 (z + 1) α 2 . Then w divides the first row of A, and w divides the first column of A. Moreover, w(z)w (z) = (−1) α 1 (z − 1) 2α 1 (z + 1) 2α 2 z −α 1 −α 2 , which divides A 1,1 . Define F := Diag(w, 1),Ã := F −1 AF − = choice of α 1 , we have min{Z(Ã 1,1 , 1), Z(Ã 1,2 , 1)} = min {Z(A 1,1 , 1) − 2α 1 , Z(A 1,2 , 1) − α 1 } = 0. A 2, we havep =p, thus Sp = 1. Define A := 1 pÃ , thenÅ is a Hermitian matrix of Laurent polynomials with compatible symmetry, with SÅ = SÃ. Furthermore, we havedet(Å) = det(Ã) p 2 = det(A) det(F) det(F )p 2 = −dd (wp)(wp).(3.77) By Lemma 3.15, there exists a Laurent polynomiald with symmetry such that det(Å) =dd .1 z −1 1 −z −1 = A [0] (z 2 ) + B [0] (z 2 ) A [1] (z 2 ) − B [1] (z 2 ) z 2 A [1] (z 2 ) + B [1] (z 2 ) A [0] (z 2 ) − B [0] (z 2 ) [0] (z) + B [0] (z) A [1] (z) − B [1] (z) z A [1] (z) + B [1] (z) A [0] (z) − B [0] (z) S[Θ(· 2 )aa (−·)](z) = SΘ(z 2 )Sa(z)Sa (−z) = z c (−z) −c = (−1) c , S[b 1 b 1 (−·)](z) = Sb 1 (z)Sb 1 (−z) = 1 z c 1 1 (−z) −c 1 = (−1) c 1 , S[b 2 b 2 (−·)](z) = Sb 2 (z)Sb 2 (−z) = 2 z c 2 2 (−z) −c 2 = (−1) c 2 . We claim that (−1) c = (−1) c 1 = (−1) c 2 , i.e., c, c 1 , c 2 have the same parity, thus (4.13) follows immediately. Assume otherwise. If (−1) c = (−1) c 1 = (−1) c 2 , then we haveS[Θ(· 2 )aa (−·) + b 1 b 1 (−·)] = (−1) c = S[b 2 b 2 (−·)],which contradicts (4.14). Similarly (−1) c = (−1) c 1 = (−1) c 2 or (−1) c = (−1) c 2 = (−1) c 1 cannot happen either. This proves the claim, and finishes the proof of the theorem. 2 +l , Sv 2 (z) = − z c−1 2 +l . By lettingŨ := FȖV, we see thatŨ has compatible symmetry and all multiplications are compatible. Moreover, we have A =pŨDiag(1, −1)Ũ and det(A) = −pp det(Ũ) det(Ũ) .By item (1), we have det(Ũ) det(Ũ) = dd pp . Since det(Ũ) has symmetry, we conclude fromFrom the proof of Theorem 3.4, we can chooseȖ such that SȖ = Sθ θ θ 1 Sθ θ θ with θ θ θ 1 = [1, α] and Sθ θ θ ∈ {[α, 1],[1,1], [α, α]}. DenoteȖ j,k the (j, k)-th entry ofȖ for 1 j, k 2. It follows thatDefine SŨ 1,1 (z) = 1 z k 1 for some 1 ∈ {±1} and k 1 ∈ Z, we conclude thatSincep has the DOS property with respect to symmetry type Sd(z)α(z), it also has the DOS property with respect to symmetry type z 2c−2k 1 Sd(z)α(z). Hence there exist real Laurent polynomials p 1 and p 2 with symmetry such that(z)andwhich further implies thatand Next, we prove item(2). Note that Step 1 in the proof of the sufficiency part does not require any assumptions in item (2), so we can apply that step here. DefineÃ and F as (3.75). ThenÃ is a matrix of real Laurent polynomials with compatible symmetry, and A = FÃF holds. Definep as (3.76). Applying the same argument as in Step 2, one can conclude thatp = p, with p being the Let n b be a positive integer satisfying (1.8). Then we can define the 2 × 2 matrix M a,Θ\|n b of Laurent polynomials as (1.9). More precisely, we have,Next, we are going to factorize M a,Θ\|n b as the following:for someb ∈ (l 0 (Z)) 2×1 . To properly factorize M a,Θ\|n b , we need to introduce the notion of a coset sequence of a filter: For a matrix-valued filter u ∈ (l 0 (Z)) t×r and γ ∈ Z, define the γ-coset sequence of u by Furthermore, (4.6) is equivalent toIt follows from (4.7) that DefineÅ j,k to be the (j, k)-th entry ofÅ for 1 j, k 2. Note that gcd. Step 3. Å 1,1 ,Å 1,2 ,Å 2,1 ,Å 2,2Step 3. DefineÅ j,k to be the (j, k)-th entry ofÅ for 1 j, k 2. Note that gcd(Å 1,1 ,Å 1,2 ,Å 2,1 ,Å 2,2 ) = Step 4. We see thatȂ satisfies all assumptions of Theorem 3.4, and thus we can find a matrix V of real Laurent polynomials with compatible symmetry such thatȂ = VDiag(1, −1)V . Then by our previous discussions, we have same as in (1.6). So it only remains to show thatp has the DOS property with respect to symmetry type Sd(z)α(z). ThusÅ satisfies all assumptions of Theorem 3.8, by which we can find matricesȂ and U of real Laurent polynomials with compatible symmetry, such thatÅ =ȖȂȖ , with all multiplications being compatible and det(Ȃ) = C is a negative constant. As all entries ofÃ have symmetry. by [15, Lemma 2.4], we see thatp also has symmetry. DenoteThusÅ satisfies all assumptions of Theorem 3.8, by which we can find matricesȂ and U of real Laurent polynomials with compatible symmetry, such thatÅ =ȖȂȖ , with all multiplications being compatible and det(Ȃ) = C is a negative constant. Step 4. We see thatȂ satisfies all assumptions of Theorem 3.4, and thus we can find a matrix V of real Laurent polynomials with compatible symmetry such thatȂ = VDiag(1, −1)V . Then by our previous discussions, we have same as in (1.6). So it only remains to show thatp has the DOS property with respect to symmetry type Sd(z)α(z). As all entries ofÃ have symmetry, by [15, Lemma 2.4], we see thatp also has symmetry. Denote Sincep is a greatest common divisor of all entries ofÃ have symmetry, so is z c 0 /2p . Thus without loss of generality, we can redefinẽ p(z) := z c 0 /2p (z), and get Sp = 1. For z 0 ∈ C \ {0}, by the definition ofp and the fact thatÃ is Hermitian. ∈ Z , From the discussions in Step 2, we see that Z(p, 1) = Z(p, −1) = 0, so Lemma 3.2 yields 0 = 1 and c 0 ∈ 2Z. we conclude that Z(p, z 0 −1 ) = Z(p, z 0 ). Therefore, there existc ∈ Z and˜ ∈ C \ {0} such thatp(z) =˜ zcp (z). As∈ Z. From the discussions in Step 2, we see that Z(p, 1) = Z(p, −1) = 0, so Lemma 3.2 yields 0 = 1 and c 0 ∈ 2Z. Sincep is a greatest common divisor of all entries ofÃ have symmetry, so is z c 0 /2p . Thus without loss of generality, we can redefinẽ p(z) := z c 0 /2p (z), and get Sp = 1. For z 0 ∈ C \ {0}, by the definition ofp and the fact thatÃ is Hermitian, we conclude that Z(p, z 0 −1 ) = Z(p, z 0 ). Therefore, there existc ∈ Z and˜ ∈ C \ {0} such thatp(z) =˜ zcp (z). As Note that multiplying˜ −1/2 does not change the symmetry type ofp, so we still have Sp = 1. As a result, we see thatp must have real coefficients. Therefore,p satisfies item (1) of Theorem 1.2. Now defineÅ := 1 pÃ , which is a Hermitian matrix of Laurent polynomials. By Sp = 1, we see that A has compatible symmetry with SÅ = SÃ, and (3.77) holds. By Lemma 3.15, there exists a Laurent polynomiald with symmetry such that det(Å) = −dd , that is item (1) of Theorem 1.1 holds with A and d being replaced byÅ andd respectively. Sp = 1, we know that fsupp(p) is a symmetric interval with center 0, and thusc = 0. Note that z −1 = z for all z ∈ T, in which case we have \|p(z)\| = \|˜ \|\|p(z)\|. Hence˜ ∈ T. If˜ = 1, without loss of generality, we simply redefinep :=˜ −1/2p , and thusp =p. On the other hand, note that gcd(Å 1,1 ,Å 1,2 ,Å 2,1 ,Å 2,2 ) =Sp = 1, we know that fsupp(p) is a symmetric interval with center 0, and thusc = 0. Note that z −1 = z for all z ∈ T, in which case we have \|p(z)\| = \|˜ \|\|p(z)\|. Hence˜ ∈ T. If˜ = 1, without loss of generality, we simply redefinep :=˜ −1/2p , and thusp =p. Note that multiplying˜ −1/2 does not change the symmetry type ofp, so we still have Sp = 1. As a result, we see thatp must have real coefficients. Therefore,p satisfies item (1) of Theorem 1.2. Now defineÅ := 1 pÃ , which is a Hermitian matrix of Laurent polynomials. By Sp = 1, we see that A has compatible symmetry with SÅ = SÃ, and (3.77) holds. By Lemma 3.15, there exists a Laurent polynomiald with symmetry such that det(Å) = −dd , that is item (1) of Theorem 1.1 holds with A and d being replaced byÅ andd respectively. On the other hand, note that gcd(Å 1,1 ,Å 1,2 ,Å 2,1 ,Å 2,2 ) = Using the fact that 1 has the DOS property with respect to any symmetry type, we see that item (2) of Theorem 1.1 holds with A being replaced byÅ. Hence, by the sufficiency part of the theorem that has already been proved, we can find a 2 × 2 matrixŮ of Laurent polynomials with compatible symmetry, such thatÅ =ŮDiag(1, −1)Ů . It follows that UDiag(1, −1)U = A = FÃF =pFÅF =pŨDiag(1, −1)Ũ , whereŨ := FŮ has compatible symmetry. Define Q := adj(Ũ)U andd := det(Ũ). Then Q has compatible symmetry andd has symmetry. Direct calculation yields QDiag(1, −1)Q =p adj(Ũ)ŨDiag(1, −1)Ũ adj(Ũ) =pdd Diag(1, −1Using the fact that 1 has the DOS property with respect to any symmetry type, we see that item (2) of Theorem 1.1 holds with A being replaced byÅ. Hence, by the sufficiency part of the theorem that has already been proved, we can find a 2 × 2 matrixŮ of Laurent polynomials with compatible symmetry, such thatÅ =ŮDiag(1, −1)Ů . It follows that UDiag(1, −1)U = A = FÃF =pFÅF =pŨDiag(1, −1)Ũ , whereŨ := FŮ has compatible symmetry. Define Q := adj(Ũ)U andd := det(Ũ). Then Q has compatible symmetry andd has symmetry. Direct calculation yields QDiag(1, −1)Q =p adj(Ũ)ŨDiag(1, −1)Ũ adj(Ũ) =pdd Diag(1, −1). From the above equation, we getpdd = Q 1,1 Q 1,1 − Q 1,2 Q 1,2 , where Q j,k denotes the (j, k)-th entry of Q for 1 j, k 2. Denote SU 1,2 = 1,2 z c 1,2 with 1,2 ∈ {±1} and c 1,2 ∈. Z. Then SQ. 1z) SQ 1,2 (z) = SU 1,1 (z) SU 1,2 (z) = Sd(zFrom the above equation, we getpdd = Q 1,1 Q 1,1 − Q 1,2 Q 1,2 , where Q j,k denotes the (j, k)-th entry of Q for 1 j, k 2. Denote SU 1,2 = 1,2 z c 1,2 with 1,2 ∈ {±1} and c 1,2 ∈ Z. Then SQ 1,1 (z) SQ 1,2 (z) = SU 1,1 (z) SU 1,2 (z) = Sd(z) 2 (z) = z −2c 1,2 Sd(z)α(z), 2 (z) SU 2,2 (z) = α(z). Therefore,pdd has the DOS property with respect to symmetry type z −2c 1,2 Sd(z)α(z). 1SUIn particular,pdd satisfies itemSU 1,2 (z) SU 2,2 (z) = z −2c 1,2 Sd(z)α(z), 2 (z) SU 2,2 (z) = α(z). Therefore,pdd has the DOS property with respect to symmetry type z −2c 1,2 Sd(z)α(z). In particular,pdd satisfies item (2 . = Z(d, = Z(d, z −1 = , so Z(dd , z 0 ) ∈ 2Z, and odd(Z(p, z 0 )) = odd(Z(pdd. = Z(d, z 0 ), so Z(dd , z 0 ) ∈ 2Z, and odd(Z(p, z 0 )) = odd(Z(pdd , z 0 )) Theorem 1.2,p has the DOS property with respect to symmetry type z −2c 1,2 Sd(z)α(z), and hence also has the DOS property with respect to symmetry type Sd(z)α(z). Consequently, This proves itemConsequently, by Theorem 1.2,p has the DOS property with respect to symmetry type z −2c 1,2 Sd(z)α(z), and hence also has the DOS property with respect to symmetry type Sd(z)α(z). This proves item Symmetric Quasi-tight Framelets with Two Generators In this section, we study quasi-tight framelet filter banks {a. Symmetric Quasi-tight Framelets with Two Generators In this section, we study quasi-tight framelet filter banks {a; Note that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet if and only if (2.5) holds with s = 2, 1 = 1 and 2 = −1. A general guideline for constructing quasi-tight framelets. Let a, Θ ∈ l 0 (Z) be such that Θ = Θ and let b := (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 . Define M a,Θ as (1.11). that is M a,Θ (z) = b(z) b(−z) Diag(1, −1) b(z), b(−zA general guideline for constructing quasi-tight framelets. Let a, Θ ∈ l 0 (Z) be such that Θ = Θ and let b := (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 . Define M a,Θ as (1.11). Note that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet if and only if (2.5) holds with s = 2, 1 = 1 and 2 = −1, that is M a,Θ (z) = b(z) b(−z) Diag(1, −1) b(z), b(−z) . By the choice of n b , we see that A and B are well-defined Laurent polynomials. Furthermore, from the symmetry assumptions of a and Θ, we have SA(z) = 1 and SB(z) = (−1) c+n b . Hence by item (1). A and B as (4.3. of Lemma 4.3, we get SA [0] (z) = 1, SA [1] (z) = z −1 , SB [0] (z) = (−1) c+n b , SB [1] (z) = (−1) c+n b z −1A and B as (4.3). By the choice of n b , we see that A and B are well-defined Laurent polynomials. Furthermore, from the symmetry assumptions of a and Θ, we have SA(z) = 1 and SB(z) = (−1) c+n b . Hence by item (1) of Lemma 4.3, we get SA [0] (z) = 1, SA [1] (z) = z −1 , SB [0] (z) = (−1) c+n b , SB [1] (z) = (−1) c+n b z −1 . Sb 1 (z) = 1 z c 1 and Sb 2 = 2 z c 2 . Since b 1 and b 2 have at least n b order of vanishing moments, there existsb = (b 1 ,b 2 ) T ∈ (l 0 (Z)) 2×1 such that b l (z) = (1 − z). n bb l (z), l = 1, 2Sb 1 (z) = 1 z c 1 and Sb 2 = 2 z c 2 . Since b 1 and b 2 have at least n b order of vanishing moments, there existsb = (b 1 ,b 2 ) T ∈ (l 0 (Z)) 2×1 such that b l (z) = (1 − z) n bb l (z), l = 1, 2. 2 (z) = z −1 . According to Theorem 4.1, {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank with n b order of vanishing moments implies that N a,Θ\|n b = UDiag(1, −1)U . Thus by Theorem 1.1, items (1) and (2) follows immediately. = Su, 1If c+n b ∈ 2Z+1, for any integer l, define P l as (4.15. and define N := P l N a,Θ\|n b P l . By the definition of N a,Θ\|n b and (4.16. we see that N has compatible symmetry with SN == SU 1,2 (z) SU 2,2 (z) = z −1 . According to Theorem 4.1, {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank with n b order of vanishing moments implies that N a,Θ\|n b = UDiag(1, −1)U . Thus by Theo- rem 1.1, items (1) and (2) follows immediately. If c+n b ∈ 2Z+1, for any integer l, define P l as (4.15), and define N := P l N a,Θ\|n b P l . By the definition of N a,Θ\|n b and (4.16), we see that N has compatible symmetry with SN = 1 (z) = SŨ 1,2 (z) SŨ 2,2 (z) = −1. Using the assumption that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank, we have N = P l N a. 2In particular, SŨ 1,1 (z) SŨ. Θ\|n b P l = P l UDiag(1, −1)U P l =ŨDiag(1, −1)ŨIn particular, SŨ 1,1 (z) SŨ 2,1 (z) = SŨ 1,2 (z) SŨ 2,2 (z) = −1. Using the assumption that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank, we have N = P l N a,Θ\|n b P l = P l UDiag(1, −1)U P l =ŨDiag(1, −1)Ũ . Thus the gcd of the 4 entries of N , as well as det(N ), must by the same as those of N a,Θ\|n b up to multiplications by monomials. Therefore, by Theorem 1.1, items (1) and (2) must hold. This finishes the proof of the necessity part. We now prove sufficiency. Suppose items (1) and (2) hold. If c + n b ∈ 2Z, by the definition of N a,Θ\|n b and (4.16), we have SN a,Θ\|n b (z) = polynomials with compatible symmetry. Since P l is strongly invertible, the Smith normal forms of N and N a. Θ\|n b are the same. such that SU 1,1 (z) SU 2,1 (z) = SU 1,2 (z) SU 2,2 (z) = z −1 and N a,Θ\|n b =Since P l is strongly invertible, the Smith normal forms of N and N a,Θ\|n b are the same. Thus the gcd of the 4 entries of N , as well as det(N ), must by the same as those of N a,Θ\|n b up to multiplications by monomials. Therefore, by Theorem 1.1, items (1) and (2) must hold. This finishes the proof of the necessity part. We now prove sufficiency. Suppose items (1) and (2) hold. If c + n b ∈ 2Z, by the definition of N a,Θ\|n b and (4.16), we have SN a,Θ\|n b (z) = polynomials with compatible symmetry, such that SU 1,1 (z) SU 2,1 (z) = SU 1,2 (z) SU 2,2 (z) = z −1 and N a,Θ\|n b = Define b = (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 via b(z) := (1 − z) n bb (z), we can easily verify that b 1 and b 2 have symmetry. Moreover, Theorem 4.1 tells that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank with min{vm(b 1 ), vm(b 2 )} n b . If c+n b ∈ 2Z+1, for any integer l, define P l as (4.15), and define N := P l N a,Θ\|n b P l . By the definition of N a,Θ\|n b and (4.16), it is easy to see that N has compatible symmetry with SN = N and N a,Θ\|n b have the same Smith normal form (up to multiplication by monomials). Thus items (1) and (2) of Theorem 1.1 holds with A = N. Hence by Theorem 1.1, there exists a matrixŨ of Laurent polynomials with compatible symmetry, such that N =ŨDiag(1, −1)Ũ and SŨ 1,1 (z) SŨ 2,1 (z) = SŨ 1,2 (z) SŨ 2,2 (z) = −1. bothb 1 andb 2 have symmetry. Defineb = (b 1 ,b 2 ) T ∈ (l 0 (Z)) 2×1 via b [0]b[1] :=Ũ P − l . By Lemma 4.3, bothb 1 andb 2 have symmetry. Moreover, we see that (4.11) holds. Define b = (b 1 , b 2 ) T ∈ (l 0 (Z)UDiag(1, −1)U . Defineb = (b 1 ,b 2 ) T ∈ (l 0 (Z)) 2×1 via b [0]b[1] := U . According to Lemma 4.3, bothb 1 andb 2 have symmetry. Define b = (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 via b(z) := (1 − z) n bb (z), we can easily verify that b 1 and b 2 have symmetry. Moreover, Theorem 4.1 tells that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank with min{vm(b 1 ), vm(b 2 )} n b . If c+n b ∈ 2Z+1, for any integer l, define P l as (4.15), and define N := P l N a,Θ\|n b P l . By the definition of N a,Θ\|n b and (4.16), it is easy to see that N has compatible symmetry with SN = N and N a,Θ\|n b have the same Smith normal form (up to multiplication by monomials). Thus items (1) and (2) of Theorem 1.1 holds with A = N. Hence by Theorem 1.1, there exists a matrixŨ of Laurent polynomials with compatible symmetry, such that N =ŨDiag(1, −1)Ũ and SŨ 1,1 (z) SŨ 2,1 (z) = SŨ 1,2 (z) SŨ 2,2 (z) = −1. Defineb = (b 1 ,b 2 ) T ∈ (l 0 (Z)) 2×1 via b [0]b[1] :=Ũ P − l . By Lemma 4.3, bothb 1 andb 2 have symmetry. Moreover, we see that (4.11) holds. Define b = (b 1 , b 2 ) T ∈ (l 0 (Z)) 2×1 If in addition a(1) = Θ(1) = 1 and φ ∈ L 2 (R), where φ(ξ) := ∞ j=1 a(e −i2 −j ξ ) for ξ ∈ R, define η, ψ 1 , ψ 2 as (1.16). It follows immediately that φ, η, ψ 1 , ψ s have symmetry, and {η, φ. via b(z) := (1 − z) n bb (z). −1) is a quasi-tight framelet filter bank with min{vm(b 1 ), vm(b 2 )} n b . This finishes the proof of the sufficiency part. ψ 1 , ψ 2 } (1,−1) is a quasi-tight framelet in L 2 (R) by the Oblique Extension Principle. Moreover, ψ 1 and ψ 2 have at least order n b vanishing moments. The proof is now completevia b(z) := (1 − z) n bb (z), it follows from Theorem 4.1 that {a; b 1 , b 2 } Θ,(1,−1) is a quasi-tight framelet filter bank with min{vm(b 1 ), vm(b 2 )} n b . This finishes the proof of the sufficiency part. If in addition a(1) = Θ(1) = 1 and φ ∈ L 2 (R), where φ(ξ) := ∞ j=1 a(e −i2 −j ξ ) for ξ ∈ R, define η, ψ 1 , ψ 2 as (1.16). It follows immediately that φ, η, ψ 1 , ψ s have symmetry, and {η, φ; ψ 1 , ψ 2 } (1,−1) is a quasi-tight framelet in L 2 (R) by the Oblique Extension Principle. Moreover, ψ 1 and ψ 2 have at least order n b vanishing moments. The proof is now complete. and its symmetry type (−1) c+n b z odd(c+n b )−1 Sd n b (z) only differs by a factor of z 2l for some l ∈. Z. 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[ "Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization", "Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization" ]	[ "Agustinus Kristiadi [email protected] \nUniversity of Cambridge\nMunich\n", "Vincent Fortuin [email protected] \nUniversity of Cambridge\nMunich\n" ]	[ "University of Cambridge\nMunich", "University of Cambridge\nMunich" ]	[]	The linearized-Laplace approximation (LLA) has been shown to be effective and efficient in constructing Bayesian neural networks. It is theoretically compelling since it can be seen as a Gaussian process posterior with the mean function given by the neural network's maximum-a-posteriori predictive function and the covariance function induced by the empirical neural tangent kernel. However, while its efficacy has been studied in large-scale tasks like image classification, it has not been studied in sequential decision-making problems like Bayesian optimization where Gaussian processes-with simple mean functions and kernels such as the radial basis function-are the de-facto surrogate models. In this work, we study the usefulness of the LLA in Bayesian optimization and highlight its strong performance and flexibility. However, we also present some pitfalls that might arise and a potential problem with the LLA when the search space is unbounded.	10.48550/arxiv.2304.08309	[ "https://export.arxiv.org/pdf/2304.08309v1.pdf" ]	258,179,126	2304.08309	4f1b462d02a62f7c608b6866738e1ed93623bd40	Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization Agustinus Kristiadi [email protected] University of Cambridge Munich Vincent Fortuin [email protected] University of Cambridge Munich Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization 1-15 Vector Institute Alexander Immer [email protected] ETH Zürich & MPI-IS Tübingen Runa Eschenhagen The linearized-Laplace approximation (LLA) has been shown to be effective and efficient in constructing Bayesian neural networks. It is theoretically compelling since it can be seen as a Gaussian process posterior with the mean function given by the neural network's maximum-a-posteriori predictive function and the covariance function induced by the empirical neural tangent kernel. However, while its efficacy has been studied in large-scale tasks like image classification, it has not been studied in sequential decision-making problems like Bayesian optimization where Gaussian processes-with simple mean functions and kernels such as the radial basis function-are the de-facto surrogate models. In this work, we study the usefulness of the LLA in Bayesian optimization and highlight its strong performance and flexibility. However, we also present some pitfalls that might arise and a potential problem with the LLA when the search space is unbounded. Introduction Bayesian neural networks (BNNs) have been shown to be useful for predictive uncertainty quantification, aiding tasks such as the detection of out-of-distribution data and adversarial examples (Louizos andWelling, 2016, 2017;Kristiadi et al., 2019). Recently, the Laplace approximation (LA, MacKay, 1992a) has emerged to be a compelling practical BNN method and has been successfully deployed in large-scale problems that leverage deep neural nets (NNs), such as image classification and segmentation (Barbano et al., 2022;Miani et al., 2022, etc.), due to its cost-efficiency (Daxberger et al., 2021) and its post-hoc nature. However, its performance in sequential decision-making problems in small-sample regimes-such as active learning, bandits, and Bayesian optimization-has not been studied extensively. Nonetheless, the recently proposed linearized version of the LA (Immer et al., 2021b) has compelling benefits since it can be viewed as a Gaussian process (GP) model-the defacto standard models used in many sequential problems like Bayesian optimization. From this perspective, the LLA is a GP with a posterior mean function given by the maximuma-posteriori (MAP) predictive function of the NN and a covariance function given by the NN's empirical neural tangent kernel at the MAP estimate (Jacot et al., 2018). Just like any GP, the LLA can be tuned via its differentiable marginal likelihood using standard deep learning optimizers without validation data (Immer et al., 2021a(Immer et al., , 2022b. Unlike standard GP models, however, the LLA is much more expressive and accurate due to its NN backbone. In particular, the LLA inherits the inductive biases of the base NN and enables scalable inference due to its post-hoc formulation on top of the MAP-estimated NN. Given its compelling theoretical benefits and its lack of usage in small-sample problems, it is thus interesting to study the performance of the LLA in this regime. In this work, we focus on Bayesian optimization and our empirical findings show that the LLA is competitive or even better than the standard GP baseline, especially when the problem requires a strong inductive bias (e.g., search in the space of images). Nevertheless, the GP view of the LLA also yields a potential pitfall in unbounded domains due to its mean function. We discuss different ideas how the LLA can be improved in the future to mitigate this issue. Linearized Laplace Approximation Let f : X × Θ → Y be a neural network (NN) mapping from the input space X ⊆ R n to the output space Y ⊆ R k under the parameter space Θ ⊆ R d . Given a dataset D := (x i , y i ) m i=1 , the standard way to train f is to find θ MAP ∈ Θ that maximizes the log-posterior function log p(θ \| D). The Laplace approximation (LA) to p(θ \| D) can then be obtained by computing the Hessian H : = −∇ 2 θ log p(θ \| D)\| θ MAP and letting p(θ \| D) ≈ N (θ \| θ MAP , H −1 ) (MacKay, 1992b). The generalized Gauss-Netwon (GGN) matrix is often used to approximate the exact Hessian (Martens, 2020). Since the GGN is the exact Hessian of the linearized network f lin (x; θ) := f (x; θ MAP ) + J(x)(θ − θ MAP ) where J(x) := ∇ θ f (x; θ) \| θ MAP ∈ R k×d is the Jacobian, it is instructive to use f lin to make predictions under the LA, resulting in the linearized-Laplace approximation (LLA, Immer et al., 2021b). Due to its weight-space linearity and Gaussianity, the LLA can be seen as a Gaussian process over the function f (x) directly (Immer et al., 2021b;Rasmussen and Williams, 2005). That is, it defines a prior f ∼ GP(f lin , J(x)J(x ) ) over the function, and its posterior on a test point x * ∈ X is given by the MAP prediction f (x * ; θ MAP ) with uncertainties given by the standard GP posterior variance under the (empirical, at θ MAP ) neural tangent kernel (NTK, Jacot et al., 2018;Lee et al., 2018) k(x 1 , x 2 ) := J(x 1 )J(x 2 ) . In this view, the LLA can be seen as combining the best of both worlds: accurate predictions of a NN and calibrated uncertainties of a GP. A detailed related work is deferred to Appendix C. The Linearized Laplace in Bayesian Optimization We focus on Bayesian optimization problems. We evaluate the LLA on three standard benchmark functions: (i) the Branin function on [−5, 10] × [0, 15] ⊂ R 2 , (ii) the Ackley function on [−32.768, 32.768] 2 ⊂ R 2 , and (iii) the MNIST image generation task minimizing f (x) := x − x 0 on [0, 1] 784 for a fixed x 0 taken from the MNIST training set (Verma et al., 2022). The base NNs are a three-layer ReLU MLP with 50 hidden units each and the LeNet-5 CNN (LeCun et al., 1998), and we uniformly sample the initial training set of size 20. Finally, the acquisition function is the popular and widely-used Expected Improvement (Jones et al., 1998). See Appendix A for the detailed experimental setup. Also refer to (Aerni, 2022) for a comparison of the LLA on sequential decision making prolems. Experiment results The results are shown in Fig. 1. In terms of optimization performance, the LLA is competitive or even better than the (tuned) GP baseline on all benchmark functions. On the high-dimensional MNIST problem, GP-RBF performs similarly (bad) as the random search baseline. Meanwhile, the LLA finds an x with small f (x) in a few iterations. While the MLP-and CNN-based LLA perform similarly in terms of the mean, the CNN-based LLA is considerably less noisy. This indicates that LLA-CNN is more reliable that LLA-MLP in finding the minimizers of the objective function. All in all, the better performance of the LLA can be contributed to the strong predictive performance (in terms of test mean-squared error 1 ) of the base NNs. Moreover, the LLA only induces a small overhead in terms of computation time compared to GP-RBF. This is because, at each iteration, the acquisition function is optimized by backpropagating through the NN and its Jacobian. Nevertheless, we found that the LLA is more memory-efficient than the GP due to the fact that it can leverage mini-batching. Meanwhile, the GP can quickly yield an out-of-memory error (after around 800 iterations). Finally, we compare the two popular methods for tuning LLA: online (Immer et al., 2021a) and post hoc (Kristiadi et al., 2020(Kristiadi et al., , 2021aEschenhagen et al., 2021). We found that post-hoc LLA performs similarly to the online LLA while being cheaper since the marginal likelihood tuning is only being done once after the MAP training (see Fig. 2). This is encouraging because the strong performance of the LLA thus can be obtained at a low cost. Note that, this finding agrees with the conclusion of Aerni (2022). The post-hoc LLA tends to perform at least on-par with the online one, while being computationally cheaper. In Appendix B, we further show empirical evidence regarding the capabilities of the LLA in small-sample regimes. These results further reaffirm the effectiveness of the LLA in Bayesian optimization, and possibly in other sequential problems. Pitfalls We have seen empirically that the LLA can be useful in sequential problems with bounded domains. However, the function-space interpretation of the LLA posterior-MAP-estimated NN as its mean function and GP-NTK as its covariance function-can be a double-edged sword in general. Specifically, Hein et al. (2019) have shown that ReLU networks, the most popular NN architecture, perform pathologically in the extrapolation regime outside of the data in the sense that f ( · ; θ MAP ) is a non-constant linear function, i.e., always increasing or decreasing. Thus, when a ReLU NN is used in the LLA, the LLA will inherit this behavior. As a surrogate function, this ReLU LLA will then behave pathologically under commonlyused acquisition functions α, such as UCB or EI: The acquisition function will always pick x ∈ X that maximizes α under f (x; θ MAP ) and its uncertainty-but when f (x; θ MAP ) is always increasing (or decreasing) outside the data, α will always yield points that are far away from the current data (see Fig. 3 for an illustration). While Kristiadi et al. (2020Kristiadi et al. ( , 2021b have shown that the LLA's uncertainty can be sufficient in counterbalancing the linear growth of the mean function in classification, this mitigation appears to be absent in sequential problems. In Bayesian optimization, for instance, let α( NTK kernel. Then, since outside of the data region, f (x; θ MAP ) and σ(x) is always increasing/decreasing linearly in x (Kristiadi et al., 2020), it is easy to see that α(x) is also increasing/decreasing linearly in x. Since the goal is to obtain a next point x t+1 = arg max x α(x) that maximizes/minimizes α, the proposed x t+1 will always be far away from the current data (infinitely so when X is an unbounded space). 2 While solving this issue is beyond the scope of the present work, possible solutions are (i) better architectural design, e.g., using more suitable activation functions (Meronen et al., 2021), and (ii) designing an acquisition function that takes the behavior induced by the MAP-estimated network into account. For instance, one option for the latter could be to avoid the asymptotic regions of the ReLU network altogether. While this restricts the exploration to be regions near the already gathered data, similar to proximal or trust-region methods (Schulman et al., 2015(Schulman et al., , 2017, it might be worth paying this price to harness the capabilities of neural networks, like strong inductive bias, good predictive accuracy, and scalability. Conclusion We have empirically validated the effectiveness of the linearized Laplace in Bayesian optimization problems: It can outperform standard Gaussian process baselines, especially in tasks where strong inductive biases are beneficial, e.g., when the search space consists of images. The linearized Laplace provides flexibility to the practitioners in incorporating the inductive bias due to its Gaussian-process interpretation: its posterior mean is given by the neural network's MAP predictive function and its posterior mean is induced by the empirical neural tangent kernel of the network. The former implies that the predictive mean of the linearized Laplace directly encodes the inductive bias of the base NN. On the other hand, the aforementioned properties of the linearized Laplace yield a pitfall in Bayesian optimization. Namely, the MAP predictive functions of ReLU networks are known to be pathological: they are always increasing or decreasing outside of the data. Moreover, the induced neural tangent kernel is non-stationary, unlike the commonly-used kernel in Bayesian optimization. They can thus yield pathological behavior in unbounded domains when used in conjunction with standard acquisition functions. The investigation of this issue is an interesting direction for future work. Appendix A. Experiment Details In the following, we describe the experimental setup we used to obtain the Bayesian optimization results presented in the main text. Models For the MLP model, the architecture is n − 50 − 50 − 1. Meanwhile, for the CNN model, the architecture is the LeNet-5 (LeCun et al., 1998) where the two last dense hidden layers are of size 50 and 20. Training For all methods (including the baselines), we initialize the starting dataset by uniformly sampling the search space and obtain the target by evaluating the true objective values on those samples. For both models, we train them for 1000 epochs with full batch ADAM and cosine-annealed learning rate scheduler. The starting learning rate and weight decay pairs are (1 × 10 −1 , 1 × 10 −3 ) and (1 × 10 −3 , 5 × 10 −4 ) for the MLP and CNN models, respectively. For the LLA, the marginal likelihood optimization is done for 10 iterations. For the online LLA, this optimization is done every 50 epochs of the "outer" ADAM optimization loop. For the other hyperparameters for the marginal likelihood optimization, we follow Immer et al. (2021a). Baselines The random search baseline is done by uniformly sampling from the search domain. Meanwhile, for the GP-RBF baseline, we use the standard setup provided by BoTorch (Balandat et al., 2020). In particular, it is tuned by maximizing its marginal likelihood. Appendix B. Additional Experiments B.1. Small-Data Regression Results In Fig. 4 and Fig. 5, we compare the Laplace approximation methods to a deep ensemble of five neural networks on four UCI regression datasets. The models are trained on small increasing subsets from one to 200 data points and evaluated on a held-out validation set consisting of the held-out data points. The mixture of Laplace (MoLA) variants are trained with the same number of components as the deep ensemble. We show results for post-hoc Laplace approximation, which is trained with a fixed prior precision and optimizes it after training (MacKay, 1992a;Eschenhagen et al., 2021), and marglik Laplace approximation, which is trained with Laplace marginal likelihood optimization during training (Immer et al., 2021a). The results show that the online marginal likelihood optimization performs best early during training with the mixture and single Laplace performing similarly well. Further, the methods clearly improve when using the Bayesian posterior predictive in Fig. 5 compared to the MAP prediction in Fig. 4. This is clearly visible by comparing post-hoc MoLA to the Ensemble method (Lakshminarayanan et al., 2017), which is the same model but predicts with the MAP of the individual models instead of their Laplace approximation (Eschenhagen et al., 2021). Even post-hoc Laplace can outperform the Ensemble, despite only using a single model. Figure 5: Test negative log-likelihood (nll) on the same four UCI regression data sets but instead with the Bayesian posterior predictive of the Laplace approximation (i.e. via Bayesian model averaging) in comparison to the same deep ensemble as in the figure above. The online marginal-likelihood optimization methods again clearly outperform the others. "post-hoc" is a Laplace approximation trained with a fixed prior precision and optimizes it after training. Interestingly, the Bayesian predictive helps the unimodal MAP to improve over the deep ensemble and MoLA does so even more. Overall, using the Bayesian predictive tends to improve the performance for all models. repeat this for three runs. The test (predictive) log-likelihood is reported over three runs with standard error. All models are trained for 1000 steps using Adam (Kingma and Ba, 2015) with a learning rate of 1 × 10 −3 decayed to 1 × 10 −5 . The online marginal likelihood variant optimizes every 50 steps for 50 hyperparameter steps and uses early stopping on Immer et al., 2021a). The post-hoc variant optimizes the hyperparameters after training for 1000 steps and uses the last-layer posterior predictive instead of the full-network predictive since it becomes less stable when applied after training without optimizing the prior precision with either cross-validation or the marginal likelihood during training. B.2. Evaluating Joint Predictions on the Neural Testbed Traditionally, the Bayesian deep learning literature focuses on marginal predictions on individual data points (c.f. Appendix C). However, Wen et al. (2021) argue that joint predictive distributions, i.e. predicting a set of labels from a set of inputs, which can capture correlations between predictions on different data points, is much more important for downstream task performance, e.g. in sequential decision making. To facilitate the evaluation of the joint predictive distribution of agents, Osband et al. (2022) propose a simple benchmark of randomly generated classification problems, called the Neural Testbed. While it only consists of small-scale and artificial tasks, the goal is to have a "sanity check" for uncertainty quantification methods in deep learning, to judge their potential on more realistic problems and help guide future research. The authors also show empirically that the performance of the joint predictive is correlated with regret in bandit problems (Gittins, 1979), whereas the marginal predictive performance is not. Recall that the linearized Laplace is a GP: It naturally gives rise to a joint predictive distribution over input points. However, Laplace approximation-based methods have not been evaluated on this benchmark yet. Here, we include preliminary results on the CLASSIFICATION 2D TEST subset of the benchmark 3 . Setup As in the experiments in Appendix B.1, all models are MLPs with fifty hidden units and ReLU activations. We optimize for 100 epochs using Adam with a learning rate of 1 × 10 −3 . We train on CLASSIFICATION 2D TEST, a set of seven 2D binary classification problems. Each method is evaluated on a test set by calculating an approximation to the expected Kullback-Leibler (KL) divergence between the predictive distribution under the method and the true ground-truth data generating process. We normalize the KL divergence to 1.0 for our reference agent which predicts uniform class probabilities. See Osband et al. (2022) for more details. For the post-hoc tuning of the prior precision we optimize the marginal likelihood once after training and use a last-layer Laplace approximation with full covariance. For the online tuning of the prior precision via optimizing the Laplace marginal likelihood, we use a burnin of 20 steps and then take 50 optimization steps every five training steps. Here, we use a Laplace approximation over all weights with a Kronecker factored covariance and a layerwise prior precision. For both variations of the method, we approximate the predictive using the probit approximation (Spiegelhalter and Lauritzen, 1990;MacKay, 1992b). In addition, we tried different predictive approximations, but got comparable results. Results In Table 1, we can see results for a regular MLP, a deep ensemble, and the Laplace approximation. Surprisingly, both variations of the LA perform significantly worse in terms of KL divergence compared to the MLP and the deep ensemble, while the test accuracy is comparable. Online marginal likelihood optimization performs better than post-hoc, but since both are so much worse than the simple baselines, the significance of this difference is unclear. Since we would expect the Laplace approximation to do at least as well as the MLP, we think further investigating the discrepancy in performance is a important step towards applying the Laplace approximation to sequential decision making problems. Also, the results demonstrate the potential pitfalls of expecting Bayesian deep learning techniques to work out-of-the-box in new settings. Immer et al., 2021b), and its associated marginal-likelihood-based model selection capabilities (Immer et al., 2021a). However, we apply these methods to the sequential learning setting, especially Bayesian optimization, which to the best of our knowledge has not been studied before. Bayesian optimization While Bayesian optimization (BO) has been intensely studied (Shahriari et al., 2015;Garnett, 2023, and references therein), the de-facto standard models for this purpose have been Gaussian processes (GPs) (Rasmussen and Williams, 2005;Balandat et al., 2020;Gardner et al., 2018). Indeed, existing work using BNNs as surrogate models for BO is scarce (Snoek et al., 2015;Springenberg et al., 2016;Kim et al., 2021;Rothfuss et al., 2021) and all these approaches use either Bayesian linear regression (Snoek et al., 2015), MCMC (Springenberg et al., 2016), or ensemble-based inference (Kim et al., 2021;Rothfuss et al., 2021). To the best of our knowledge, we are the first to propose Laplace inference for this problem and highlight its performance benefits, low computational cost, and model selection capabilities. Figure 1 : 1Bayesian optimization experiments in terms of objective minimization (top), predictive accuracies of the surrogate models (middle), and the wall-clock time (in seconds) taken to propose a new evaluation point (bottom). The LLA methods outperform the GP on the more complex problems. Figure 2 : 2Post-hoc vs. online LLA in Bayesian optimization, in terms of objective minimization (top), predictive accuracies of the surrogate models (middle), and the wall-clock time (in seconds) taken to propose a new evaluation point (bottom). Figure 3 : 3x) := f (x; θ MAP ) + βσ(x) with β > 0 be the UCB (Garivier and Moulines, 2011) under the LLA posterior, where σ 2 (x) is the GP posterior variance on x under the As a surrogate function, LLA with ReLU activations is (pathologically) always increasing/decreasing far away from the data. This impacts popular acquisition functions like UCB and EI and thus the exploration-exploitation behavior. 2.A similar argument also applies to the expected improvement acquisition function(Jones et al., 1998). SetupFigure 4 : 4All models are MLPs with fifty hidden units and ReLU activation. We train the methods on one to 200 training points, in each turn adding a single data point, and Test negative log-likelihood (nll) for UCI regression datasets with MAP prediction (not Bayesian) of MAP optimized models and model trained with marginal likelihood optimization ("marglik") during training as in (Immer et al., 2021a). All methods were trained on 1 − 200 observation points, where in each turn, one observation point is added. "MoLA" indicates a mixture of Laplace approximations with the same number of components as the deep ensemble. The online marginal-likelihood optimization methods outperform simple MAP with fixed prior precision. Vincent Fortuin, Adrià Garriga-Alonso, Sebastian W Ober, Florian Wenzel, Gunnar Ratsch, Richard E Turner, Mark van der Wilk, and Laurence Aitchison. Bayesian neural network priors revisited. In ICLR, 2022. Jost Tobias Springenberg, Aaron Klein, Stefan Falkner, and Frank Hutter. Bayesian optimization with robust Bayesian neural networks. In NIPS, 2016. Tycho FA van der Ouderaa and Mark van der Wilk. Learning invariant weights in neural networks. In UAI, 2022. Andrew Gordon Wilson and Pavel Izmailov. Bayesian deep learning and a probabilistic perspective of generalization. In NeurIPS, 2020.Yarin Gal and Zoubin Ghahramani. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In ICML, 2016. David J Spiegelhalter and Steffen L Lauritzen. Sequential updating of conditional probabilities on directed graphical structures. Networks, 20(5), 1990. Ekansh Verma, Souradip Chakraborty, and Ryan-Rhys Griffiths. High-dimensional Bayesian op- timization with invariance. In ICML Workshop on Adaptive Experimental Design and Active Learning, 2022. Ziyu Wang, Tongzheng Ren, Jun Zhu, and Bo Zhang. Function space particle optimization for Bayesian neural networks. In ICLR, 2019. Max Welling and Yee W Teh. Bayesian learning via stochastic gradient langevin dynamics. In ICML, 2011. Zheng Wen, Ian Osband, Chao Qin, Xiuyuan Lu, Morteza Ibrahimi, Vikranth Reddy Dwaracherla, Mohammad Asghari, and Benjamin Van Roy. From predictions to decisions: The importance of joint predictive distributions. arXiv preprint arXiv:2107.09224, 2021. © A. Kristiadi, A. Immer, R. Eschenhagen & V. Fortuin. . Available at https://github.com/deepmind/neural_testbed. AcknowledgmentsResources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute. VF was supported by a Branco Weiss Fellowship.Appendix C. Related Work However, since their inception, performing approximate inference in these complex models has remained a lingering challenge. Jospin, Bayesian neural networks Bayesian neural networks promise to combine the expressivity of neural networks with the principled statistical properties of Bayesian inference. NealBayesian neural networks Bayesian neural networks promise to combine the expres- sivity of neural networks with the principled statistical properties of Bayesian inference (MacKay, 1992c; Neal, 1993). However, since their inception, performing approximate in- ference in these complex models has remained a lingering challenge (Jospin et al., 2022). 2021), stochastic weight averaging. Khan, Since exact inference is intractable, approximate inference techniques follow a natural tradeoff between quality and computational cost, from cheap local approximations like Laplace inference. Laplace, 1774Wilson and Izmailovacross ensemble-based methods. Ciosek et al., 2020; He et al.Since exact inference is intractable, approximate inference techniques follow a natural trade- off between quality and computational cost, from cheap local approximations like Laplace inference (Laplace, 1774; MacKay, 1992c; Khan et al., 2019; Daxberger et al., 2021), stochas- tic weight averaging (Izmailov et al., 2018; Maddox et al., 2019), and dropout (Gal and Ghahramani, 2016; Kingma et al., 2015), via variational approximations with different lev- els of complexity (e.g., Graves, 2011; Blundell et al., 2015; Louizos and Welling, 2016; Khan et al., 2018; Osawa et al., 2019), across ensemble-based methods (Lakshminarayanan et al., 2017; Wang et al., 2019; Wilson and Izmailov, 2020; Ciosek et al., 2020; He et al., 2020; D' , Angelo , up to the very expensive but asymptotically correct Markov Chain Monte Carlo (MCMC) approaches. NealWelling and TehD'Angelo et al., 2021; D'Angelo and Fortuin, 2021; Eschenhagen et al., 2021), up to the very expensive but asymptotically correct Markov Chain Monte Carlo (MCMC) approaches (e.g., Neal, 1993; Neal et al., 2011; Welling and Teh, 2011; Garriga-Alonso and Fortuin, 2021; Apart from the challenges relating to approximate inference, recent work has also studied the question of how to choose appropriate priors for BNNs. Izmailov, MacKayImmer et al., 2021a, 2022a,b; Rothfuss et al., 2021, 2022; van der Ouderaa and van der Wilk. In our work, we particularly draw on the recent advances in Laplace inference (Daxberger et al., 2021), particularly linearized LaplaceIzmailov et al., 2021). Apart from the challenges relating to approximate inference, recent work has also studied the question of how to choose appropriate priors for BNNs (e.g., Fortuin et al., 2021, 2022; Nabarro et al., 2022; Sharma et al., 2023; Fortuin, 2022, and references therein) and how to effectively perform model selection in this framework (e.g., Immer et al., 2021a, 2022a,b; Rothfuss et al., 2021, 2022; van der Ouderaa and van der Wilk, 2022; Schwöbel et al., 2022). In our work, we particularly draw on the recent advances in Laplace inference (Daxberger et al., 2021), particularly linearized Laplace (MacKay, 1992b;	[ "https://github.com/deepmind/neural_testbed." ]
[ "BARYONIC DARK MATTER †", "BARYONIC DARK MATTER †" ]	[ "Joseph Silk \nDepartments of Astronomy and Physics, and Center for Particle Astrophysics\nUniversity of California\n94720BerkeleyCA\n" ]	[ "Departments of Astronomy and Physics, and Center for Particle Astrophysics\nUniversity of California\n94720BerkeleyCA" ]	[]	La theorie c'est bien, mais ca n'empeche pas d'exister." Emile CharcotAbstractIn the first two of these lectures, I present the evidence for baryonic dark matter and describe possible forms that it may take. The final lecture discusses formation of baryonic dark matter, and sets the cosmological context. †	10.1007/978-94-011-3410-1_52	[ "https://export.arxiv.org/pdf/astro-ph/9407024v1.pdf" ]	119,035,131	astro-ph/9407024	8422c8e2f4b13783e0d4047b9e6e32b1527e87c0	BARYONIC DARK MATTER † Jul 1994 Joseph Silk Departments of Astronomy and Physics, and Center for Particle Astrophysics University of California 94720BerkeleyCA BARYONIC DARK MATTER † Jul 1994arXiv:astro-ph/9407024v1 7 La theorie c'est bien, mais ca n'empeche pas d'exister." Emile CharcotAbstractIn the first two of these lectures, I present the evidence for baryonic dark matter and describe possible forms that it may take. The final lecture discusses formation of baryonic dark matter, and sets the cosmological context. † INTRODUCTION The nature of the dark matter represents one of the major unsolved problems in astrophysics. In fact, there are really two dark matter problems: the dark matter in the halo and the dark matter that is the predominant contributor to Ω. Baryonic dark matter is a plausible candidate for halo dark matter, but whether it is responsible for Ω is controversial. If Ω = 1, it is most unlikely that baryons predominate, but if Ω ∼ 0.1, the situation is less clear. The uncertainty in the masses of baryonic and non-baryonic dark matter candidates is huge, and should give any experimentalist serious grounds for hesitation before embarking on a finely tuned search for dark matter. This uncertainty has never deterred theorists: on the contrary, it has inspired them to come up with a plethora of candidates. A convenient categorization of dark matter candidates divides the contenders into two regimes, that of particle physics and that of astrophysics. The particle physics-motivated non-baryonic candidates are weakly interacting particles, generally but not invariably massive that are generically classified as WIMPs, for weakly interacting massive particles. The astrophysically motivated baryonic candidates are generally massive and are referred to as MACHOS, for massive astrophysical compact halo objects. MACHOs may of course exist beyond galactic halos, but the evidence is less persuasive. Generally, one can compute the cosmological abundance of WIMPs but there is no proof of their existence. The most compelling WIMP is the lightest supersymmetrical particle, and provides the best candidate for the dark matter that contributes to Ω = 1, if indeed Ω is unity (or even larger). By contrast, MACHO candidates are known to exist, and these lectures will describe the evidence for MACHOs as the dark halo matter. Arguments will be reviewed that both support MACHOs as halo dark matter, and go the additional step of asserting that all dark matter may be baryonic (Lecture 1). These lectures were particularly timely. Shortly after they were given, and before being written up, evidence for MACHO candidate detections was reported in two independent experiments. While these results support the idea that halo dark matter is baryonic, they leave wide open, at present, any inferences about the nature of the MACHOs. Thus in Lecture 2, I describe the possible forms of baryonic halo dark matter that span the range of compact stellar remnants, brown dwarfs and even cold gas clouds. The third lecture sets the cosmological context, and describes how a viable cosmological model may be constructed that consists exclusively of baryonic dark matter. THE EVIDENCE FOR BARYONIC DARK MATTER Primordial Nucleosynthesis The strongest argument that there is a substantial amount of baryonic dark matter, exceeding luminous baryonic matter, comes from primordial nucleosynthesis. The concordance of the predictions of 4 He, 3 He, 2 H and 7 Li abundances, together with the removal of uncertainty in the numbers of neutrino species by the Z 0 decay width and improved measurement of neutron half-life, bounds Ω B . Indeed, the major uncertainty arises from the adopted Hubble constant. The upper limit on Ω B is primarily from 4 He and 7 Li, and the lower bound from 2 H and 3 He. The combined result is [1] Ω B h 2 = 0.0125(±0.0025); 95% confidence limit. Since luminous baryons contribute Ω lum B ≈ 0.007, I conclude that between 30% and 90% of baryons are dark. These bounds contain hidden assumptions about galactic chemical evolution. If halo dark matter consists of stellar remnants, one may have to reevaluate these limits. Indeed, there are already indications from population II abundance studies that lithium may have undergone convective burning at an early stage of galactic chemical evolution [2]. For example, the ratio 6 Li/ 7 Li ≈ 1 20 in HD 84937, an old halo star with [F e/H] = −2.4 [3], implies that old halo stars have certainly destroyed 6 Li by convective burning [4]. Cosmic ray spallation of (α, α)(p, 7 Li) and (p, CN O) (p, 7 Li, ...) undoubtedly produces some 7 [5]. This further complicates use of 7 Li as a tracer of Ω B . If 6 Li was destroyed, it becomes more plausible that convection to a slightly greater depth could have destroyed a substantial amount of 7 Li. The discovery of MACHO candidates (section 2.5) means that the dark halo may consist of stellar relics. The burning of 2 H would generate 3 He that could also be locked up, allowing a large astration, or stellar destruction, factor of 2 H and therefore a lower value of Ω B . If the MACHOs formed from intermediate mass stars, there wouls also be some helium production that would also lower the inferred primordial helium abundance, and therefore also Ω B . The MACHO discovery argues against a large astration factor, since it indicates a value of Ω B that is at least as large as that allowed by the conventional bound. Indeed halo MACHOs, if stellar remnants, favour a value of Ω B that could substantially exceed the canonical bound, by virtue of the lock-up of what may be a high primordial helium abundance. Evidence from Rotation Curves Rotation curves of spiral galaxies show that halos constitutes some 90 percent of the mass of a galaxy and are dark. The local mass-to-luminosity ratio is > ∼ 2000 in the V -band, and > ∼ 64 at K, for edge-on spirals [6]. This precludes ordinary stars from being more than a few percent of the dark halo mass. Even a finely tuned mass function rising below 0.1M ⊙ would produce too much light, if the stars were hydrogen burning (M > 0.08M ⊙ ). Specific alternative baryonic candidates are discussed below. However, a simple argument derived from HI observations of at least some rotation curves suggests that baryonic dark matter is a serious contender for explaining halo dark matter. Consider the curious case of the dwarf galaxy DDO 154. This dwarf spiral has a flat rotation curve that extends to 15 disk scale lengths. The measured HI column density has the same radial profile (N ∝ r −1 ) as the inferred projected dark matter surface density over this range [7]. The HI contributes about 10 percent of the total surface density, and accounts for about 30 percent of the measured rotational velocity. The logical contender for the dark matter is a component that is associated with the halo HI. This points towards baryonic dark matter and possibly even cold gas clouds, as a dominant form of dark matter. To what extent cold gas can exist in substantial amounts in the halo will be discussed below. Another characteristic of DDO 154 and other dwarf spirals is that the rotation curve observations provide unambiguous evidence that the extended dark matter halos have cores of finite extent, typically several kiloparsecs. However halo formation by hierarchical clustering of collisionless dark matter particles in high resolution numerical simulations demonstrates that the density increases to below the resolution limit [8,9]. Any core radius is less than the resolution scale of ∼ 1 kpc. While one can imagine extreme mass loss via supernova-driven winds imprinting a scale on the dark matter halo, a more plausible alternative is simply a dissipative BDM halo [10]. This is the simplest way to generate a halo with a finite core. Dark Halos May Be Flattened One signature of baryonic dissipation is an oblately flattened halo. Collisionless collapse in hierarchical clustering tends to form prolate halos. There are indications that dark halos may be flattened. Studies of polar ring galaxies probe the halo potential along the minor axis and provide evidence of flattening comparable to an E6 galaxy (or 4:10) [11]. Since polar ring galaxies form from mergers, one might expect that isolated galaxies could be more flattened. HI disk warps provide indirect evidence of flattening. The argument here is one of persistence of the nonaxisymmetric instability. An oblate spheroidal halo damps out the warp modes over a time-scale ∼ ǫ −1 t p where ǫ is the flattening (ratio of semiminor to semimajor axes) and t p is the precession time-scale [12]. Survival over the age of the galaxy requires ǫ < ∼ 0.3. Lopsided HI disks with circular velocity fields [13,14,15] also provide evidence for dark baryonic matter. These are not easily understood even if the gas has recently arrived: the simplest explanation for lopsidedness in outer HI disks is that the disk is dark-matter dominated. This matter must be baryonic in order to be in the disk. Stellar kinematics also favor a flattened population II [16,17]. Presumably a gaseous halo that formed contemporaneously with population II would be even more flattened. Simulations of hierarchical clustering in a cold dark matter-dominated universe preferentially form prolate halos [8]. This result is not in agreement with evidence on the distribution of shapes for elliptical galaxies. This may suggest that the simulations are not providing the correct picture since one would expect stars to have formed early in the collapse. A prolonged episode of star formation would result in dissipation and disk formation. The stars should therefore dynamically track the dark matter. Continuity With Population II There are two observationally motivated arguments for believing that dark halos are baryonic. The diskhalo "conspiracy" is the apparent dependence of both the amplitude and shape of the galaxy rotation curve at large galactocentric radii (that is, in the halo-dominated regime) on the luminous content of the galaxy (that is, the disk). Flat rotation curves are formed for normal, spiral galaxies, with disk scale lengths of 3-4 kpc and maximum rotational velocities of 100−200 km s −1 . However, luminous galaxies, with large rotational velocities, generally have declining rotation curves, and dwarf galaxies have rising rotation curves [18]. In the galaxies with flat rotation curves, the disk, which dominates at small radii, and the halo, which dominates at large radii, contribute an approximately equal rotational velocity. Evidently the outer rotation curve, which samples the galaxy halo, is closely coupled to the inner, baryon-dominated galaxy. This situation might naturally arise if the halo is baryonic and formed shortly before the disk formed. Stellar populations in the halo also show a continuity within population II. The stars at large radii and of extremely low metallicity are drawn from a similar narrow range of stellar masses and show similar abundance patterns to population II stars in the inner galaxy. The best nearby laboratories for studying the oldest stars in the halo are globular star clusters. These systems are inferred to contain a substantial fraction of mass in the form of white dwarfs and neutron stars. Perhaps 30 percent of a globular cluster may be in white dwarfs, as determined by dynamical modelling, and at least 1 percent in neutron stars is required to account for the millisecond pulsar population of spun-up neutron stars in binaries. The mass-to-luminosity ratio of an elliptical galaxy (∼ 10h) is several times larger than that of a globular star cluster, and this may be in part due to a larger fraction of white dwarfs. If the initial stellar mass function is sufficiently steep, globular clusters which have short central relaxation times would be depleted in low mass stars relative to an elliptical. However there is little indication that globular clusters with the longest core relaxation times have steeper initial mass functions. While this effect could conceivably account for the higher mass-to-luminosity ratio in ellipticals, if these are low mass star-dominated, the mass-to-light ratio falls far short of that required in dark halos. It is an intriguing coincidence that saturating the lower limit on halo stellar dark matter, where locally M Lv > ∼ 2000, could provide a critical density in the same material, if it were uniformly distributed relative to the luminous density of galaxies. Baryonic dark matter may amount to Ω B ∼ 0.1(h = 0.4) or be as low as Ω B = 0.01(h = 1). It certainly exceeds the stellar contribution, Ω * ≈ 0.007. Galaxy halos coincidentally span the range where this dark matter could be entirely baryonic. The continuity argument suggests that halos are the natural site for the baryonic dark matter. Even galaxy clusters, where gas and stars may dominate the mass, contribute no more than Ω B ≈ 0.1. Globally, only 90 percent of the mass of a spiral galaxy, halo included, is dark. By dark, I mean that this material is not in any identifiable form. Of course, there are white dwarfs and other stellar remnants that cumulatively are dark and add up to a dominant fraction, 80 percent or more, of amn old stellar population. There is dark matter over and above this near the luminous peripheries of spiral galaxies. The unknown dark fraction is not a huge extrapolation from the ∼ 20 percent in globular clusters and ∼ 50 percent in ellipticals that such population synthesis modelling requires. Thus somewhat more extreme pregalactic star formation could have produced the requisite dark matter fraction. Precisely what form the stellar dark matter may take is the subject of section 3. Gravitational Microlensing Experiments Results from two experiments that find strong evidence for the existence of MACHOs were reported in October, 1993 [19,20]. The technique used is gravitational microlensing. If a MACHO passes very close to the line-of-sight from Earth to a distant star, the gravity of the otherwise invisible MACHO causes bending of the starlight and acts as a lens. For halo MACHOs, the star splits into multiple images that are separated by a milliarcsecond, far too small to observe. However, the background star temporarily brightens as the MACHO moves across the line-of-sight in the course of its orbit around the Milky Way halo. The brightening may be by a magnitude or more, which should easily be detectable. The microlensing event has some unique signatures that distinguish it from a variable star. It should be symmetrical in time, achromatic, and should occur only once for a given star. There are two major difficulties with this experiment. First, the microlensing events are very rare. Only about one background star in two million will be microlensed at a given time. Secondly, many stars are intrinsically variable. Studying such rare events may uncover new types of hitherto unknown variable stars. To overcome the low probability of a microlensing event towards the LMC, about 5×10 −7 , the experiments were designed to monitor some ten million stars in the Large Magellanic Cloud. One group, the EROS collaboration of French astrophysicists, utilized a total of more than 300 ESO Schmidt plates taken of the LMC over a 3 year period with red or blue filters. The second group is a US-Australian collaboration that utilizes the 50 inch telescope at Mt. Stromlo, dedicated to the MACHO search, in conjunction with the largest CCD camera in the world built for astronomical use. An analysis of about seven million stars revealed a total of four events that displayed the characteristic microlensing signatures, with event durations of between 20 and 40 days. The duration of the microlensing event directly measures the mass of the MACHO, with some uncertainty because of the unknown transverse velocity of the MACHO across the line-of-sight. The duration of the event is simply the time for the MACHO to cross the Einstein ring radius. The Einstein ring radius is approximately equal to the geometric mean of the Schwarzschild radius of the MACHO and the distance to the MACHO. The MACHO is typically at a halo core radius, which is a sizeable fraction of the distance (55 kpc) to the LMC. Much more data remains to be analyzed by the two groups. The MACHO interpretation, if correct, should result in more events that are distributed according to the expected distribution both of amplifications and of the properties of the background stars. In the meantime, one can speculate about the implications. The four events correspond to MACHO masses of 0.1 to 0.4 M ⊙ with a factor of 3 or so uncertainty. The EROS experimenters are also performing a CCD search that is sensitive to timescales between 30m and 24 h, and therefore to mass scales between 10 −7 to 10 −3 M ⊙ . As yet, no events have been found in this mass range. A third experiment, the OGLE collaboration of Polish and U.S. astronomers [21], has studied 0.7 million stars in the galactic bulge, where there is a higher microlensing probability of detecting disk stars than halo MACHOs towards the LMC. They reported detection of a microlensing event corresponding to a mass of about 0.3 M ⊙ . This approach will eventually provide confirmation of the microlensing technique, since one can predict a minimal expected rate of events from the known disk stellar population. By contrast, the halo would generate no MACHO events if it does not consist of baryonic dark matter. The rate detected appears to be low, perhaps by a factor of three, relative to what the MACHO model of dark halo matter predicts. These conclusions are extremely tentative, and are sensitive to the uncertain experimental efficiency and adopted halo model. The possibility that the detections refer to a thick disk cannot be ruled out if the thick disk contributes about as much as the stellar component of the thin disk. However conventional thick disk observations require at most a six percent contribution in terms of stellar surface density near the sun. These experiments certainly present the strongest evidence to date of dark matter detection. Unless there are perverse types of rare variable stars, MACHOS are likely to constitute a significant fraction of the dark halo. THE POSSIBLE FORMS OF BARYONIC DARK MATTER A Star Formation Primer Star formation is a phenomenological theory. We would like to be able to apply this theory to the formation of BDM. In this section, we review the current status of the theory of star formation. There are three critical ingredients that are essential for understanding how stars form. These are the initial mass function of newly formed stars (IMF), the rate of star formation (SFR), and the star formation effiency (SFE). Initial Mass Function Most of our knowledge of the initial mass function comes from studies of stars in the vicinity of the sun. The IMF is defined as the total number of stars per unit mass ever formed. It is measured per square parsec perpendicular to the galactic plane by counting field and open cluster stars of known distance. Historically, the IMF is approximated by a Salpeter law over 0. 1 M ⊙ to 80 M ⊙ , dn dm ∝ M −1−x ; x = 1.35. Modern data shows that the IMF peaks at 0.3M ⊙ , and declines towards lower masses. At higher masses, the IMF gradually steepens. It has been more accurately approximated as a log normal distribution by Miller and Scalo [22]. The mass range over which one can observe the IMF outside the solar vicinity is limited. In regions of star formation, near infrared imaging has shown that low mass stars are present in numbers consistent with the Miller-Scalo IMF. This of course is essential for the question of how much mass is locked up in stars, and in particular in low luminosity stars. Studies of globular clusters do not find evidence for a turn-over, and the numbers of stars continue to rise to the observational limit of about 0.15M ⊙ . In nearby galaxies, where star formation occurs relatively quiescently, there are also no indications of deviations from the Miller-Scalo IMF at masses down to about ∼ 1M ⊙ . In the Milky Way, observations of the number of Wolf-Rayet stars suggest that the IMF steepens as a function of increasing galactic radius [23]. The budget of ionizing photons, as measured by observations of radio HII regions, has been used to favour a top-heavy IMF in the inner spiral arms [24]. However, in starbursts, regions of intense star formation activity, both modelling and observational indicators suggest that the IMF may vary with time and/or location. Arguments for a top-heavy IMF, weighted towards massive stars, have been summarized by Scalo [25]. The high luminosity per unit of gas mass available to form stars suggests that the IMF is truncated at the low mass end. The best observational evidence is for M82, where spectroscopy of the CO bands near 2µ [26], the high supernova rate and luminosity per unit gas mass [27], and the enhanced ratio of K-band luminosity to mass relative to that in the nucleus [28] all favor a supergiant-dominated starburst in the disk, driven by a top-heavy IMF. A further argument in favour of a top-heavy IMF during galaxy formation comes from the excessive enrichment that is generated, relative to the nucleosynthetic yield in the solar neighbourhood. In galaxy clusters, the high abundance of iron in the intracluster medium, about one-third of the solar iron abundance, has been interpreted as possibly requiring a top-heavy IMF during the formation phase of the observed ellipticals [29,30]. A Miller-Scalo IMF fails by a factor of 10 in providing enough iron. There is also some evidence of a deficiency in very massive stars in at least some starbursts. The absence of massive stars is inferred from the paucity of ionizing photons as measured by the strength of hydrogen recombination lines. The primary theoretical argument for explaining this is due to Wolfire and Cassinelli [31], who note that in metal-rich galactic nuclei, enhanced radiation pressure due to small dust grains is more likely to limit accretion onto massive protostars than in lower metal abundance regions. This would be a primary factor in limiting the upper limit on the IMF to a mass as low as ∼ 30M ⊙ . There is no compelling theory that predicts stellar masses, let alone the IMF. The characteristic stellar mass can be derived by simple dimensional arguments that balance pressure and gravity to have a baryon number of (h c Gm 2 p ) 3 2 , equivalent to 10 57 protons, or to a solar mass. This number is uncertain by at least two orders of magnitude. Indeed, essentially the same argument has been used to derive the brown dwarf mass (0.08M ⊙ ), the Chandrasekhar mass (1.4M ⊙ ), and the maximum mass of a stable star (∼ 100M ⊙ ). A cold, collapsing cloud will realistically form a transient sheet or filament rather than collapse to a point [32]. It radiates freely during the initial collapse, and is unstable to fluctuation growth according to the Jeans criterion. The minimum fragment mass in a cloud at temperature T and surface density µ is M Jeans ≈ c 4 s G 2 µ = 1.6 (T /10K) 2 (µ/150M ⊙ pc −2 ) M ⊙ . A typical value for the surface density of molecular clouds on scales from 0.1 to 30pc is µ = 150M ⊙ pc −2 . The temperature is in the range 10K − 50K. The resulting Jeans mass spans the range observed for molecular cloud cores. Exactly how stars form depends on the continuing evolution and subfragmentation of these cores. Considerable amounts of magnetic flux and specific angular momentum must be lost by the cores on the way to forming stars. Within the more massive cores, large numbers of stars form. Numerical hydrodynamical simulations cannot cope with the dynamical range in density required to study star formation. One has to resort to semi-analytical arguments. The resolution must depend on the initial conditions in the parent cloud. How does the cloud divide itself into stellar mass fragments? The physics of cloud collapse and evolution is complex. It involves fragmentation, coalescence of fragments, accretion by fragments, and binary captures. It is not surprising that the IMF may depend on environment, being different in the nuclei of galaxies, for example, from the IMF in lower density regions. There are indications of gradients of α-nuclei abundances to iron in elliptical galaxies [33], and in the disk of our galaxy [34]. These can be interpreted in terms of IMF variations. However, this invariably is not a unique explanation, as both supernova-driven mass loss from galaxies and accretion of primordial gas into galaxies can modify the abundance gradients. The top-heavy IMF may arise as follows [35]. Interstellar clouds grow by coalescence and then orbit the galaxy. Initially, magnetic support was adequate to provide support against gravitational collapse, with ambipolar diffusion of the field allowing some modest degree of star formation to proceed even in the low mass cores. Because of the limited gas reservoir, one might imagine that predominantly low mass stars are formed in these cores. Spiral density waves provide a non-circular component to the motion that progressively stimulates the aggregation process. After about an orbital time, 10 8 yr or so, many clouds have grown to the point at which they are both Jeans unstable, and magnetically Jeans unstable. The massive clouds now collapse on a free-fall time and form stars of all masses. Evidently, one has two modes of star formation. During the prolonged, quiescent star-forming mode, low mass star formation predominates. Once cloud collapse begins in earnest, both massive and low mass stars form, in the vigorous star-forming mode. Now consider what may happen in the merger of a pair of gas-rich galaxies. The greatly enhanced non-circular cloud motions should drive cloud growth by aggregation on an unprecedented scale [36]. In this situation, the vigorous star-forming model dominates, as the clouds are rapidly driven to the edge of collapse. Hence a top-heavy IMF may arise naturally in regions of intense turbulent motions of clouds as expected in a galaxy merger, and possibly also during the process of galaxy spheroid formation. Star Formation Efficiency Stars form in dense molecular cores that permeate the giant molecular cloud complexes (GMC) [37]. The galactic molecular hydrogen, amounting to about 2 × 10 9 M ⊙ and comparable in mass to the atomic hydrogen, is distributed in ∼ 1000 of these cloud complexes. The overall SFE within the GMCs is a few percent [38], but within the most massive cores the SFE is 30 percent or more [39]., The cores represents a few percent of the mass in the cloud complexes. The overall SFE is about 1 percent in the Milky Way disk. One can understand the low SFE in terms of energy feedback once stars form. Low mass as well as massive protostars are observed to have vigorous bipolar outflows. In addition, low mass protostars are strong x-ray emitters. The enhanced ionization recouples the magnetic field that is undergoing ambipolar diffusion as slow contraction of the cloud cores occurs and low mass stars form. The increased friction between ions and the molecular gas thereby provides an additional magnetic pressure source that resists collapse. The bipolar outflows that are invariably associated with the formation of stars inject a significant amount of momentum and energy into the cold molecular gas, the bulk of which has not yet condensed. A crude measure of efficiency for supra-Jeans mass clouds is obtained as follows [40]. Let typical bipolar outflows be at velocity v out ∼ 100 − 200 km s −1 in molecular clouds of linewidth ∆v ∼ 1 − 3 km s −1 , where characteristic values are used. If momentum is approximately conserved, one would expect that the SFE is ∼ ∆v/v out , or of order one or two percent, over the time-scale over which the flows persist. This might apply over a cloud collapse time-scale, since this is of the same order (∼ 10 6 −10 7 yr at a density n = 10−1000 cm −3 ) as the duration times estimated for many bipolar outflows. Hence feedback from protostellar flows could account for the SFE during the vigorous star-forming phase of a molecular cloud, when gravitational collapse is underway. Star Formation Rate The star formation rate in galaxy disks comparable to the Milky Way is typically in the range 5 − 10 M ⊙ yr −1 . Non-axisymmetric instabilities, such as spiral density waves, are the underlying trigger of star formation, most of which occurs in the spiral arms. These may be driven by a central bar, by the tidal interaction with a companion galaxy, or could even erupt spontaneously as a consequence of the amplification of stochastic noise. Much higher star formation rates are associated with starburst galaxies. Here the SFR may be one or even two orders of magnitude higher, per unit mass in stars. Compelling obervational evidence suggests that many starbursts, and all of the extreme starbursts, are driven by galaxy mergers. The early history of star formation in the Milky Way is inferred to have been relatively quiescent, not differing by more than a factor of 2-3 from the present day SFR [41, 42, 43.] This applies to our galactic disk. One can infer star formation rate histories for nearby spiral galaxies, and similar results are found. The SFR in the late-type spirals (Sd) actually increases slowly with time, whereas the SFR in Sa's and Sb's decreases [44]. A much higher star formation rate per unit mass is inferred for spheroidal stellar populations. The lack of young stars requires all star formation to have terminated at least 6Gyr ago. Population synthesis modelling requires the bulk of the star formation to have occurred in 1Gyr or less. Therefore in spheroids, the SFR was an order of magnitude or more higher than in disks. The Primordial IMF One might expect the IMF to be different for extremely metal-poor stars, if only because many of the processes involved in fragmentation and star formation are sensitive to metallicity. Our best indicator of the primordial IMF comes from examining heavy element abundance ratios in metal-poor stars. From these studies, one can make crude inferences about the IMF of the precursor stars that synthesized the metals [45]. The odd-even pattern of abundances seen in extreme metal-poor stars is identical to that in star formation at the present epoch. The r and s process sites are believed to be massive stars. Hence massive stars (10 − 100M ⊙ ) were present in the first stellar population. Low mass stars were also present. Stars of ∼ 1M ⊙ are found with [Fe/H] < −4. These stars are sufficiently metal-poor that the observed enrichment should have occurred during the first generation of star formation. Theoretical considerations of the fragmentation of primordial clouds result in predictions of minimum fragment masses that differ little from similar predictions for clouds of solar abundance. Opacity-limited fragmentation proceeds as follows. A collapsing cloud is initially transparent to radiation, and cooling regulates the collapse to be approximately isothermal at temperature T. The Jeans mass, proportional to T 3 2 ρ − 1 2 , therefore decreases until the density ρ is sufficiently high that the optical depth across a fragment is appreciable. The ensuing collapse is nearly adiabatic, so that the Jeans mass is proportional to ρ 3 2 (γ− 4 3 ) , with γ ≈ 5 3 . The minimum Jeans scale is about 10 −3 M ⊙ , and is only weakly dependent on metallicity. For a cloud of solar abundance, grain cooling is dominant, whereas for a primordial cloud, molecular hydrogen formation and dissociation control the cooling and fragment mass evolution. In both situations, the effects of finite size of the parent cloud increase the minimum fragment mass scales as fragments can shadow one another and thereby enhance the effective opacity. Other physical effects that are difficult to model but that nevertheless are important include fragment collisions, fragment mergers and accretion of diffuse gas by fragments. The turbulent velocity field induced by asymmetric collapse and by feedback from forming stars will help drive fragment interactions. The general sense of these modifications of the naive, spherically symmetric treatment of opacity-limited fragmentation is to drive the minimum fragment mass up to at least 0.01M ⊙ , and perhaps to 0.1M ⊙ . This could therefore account for the paucity of brown dwarfs in conventional star formation. With regard to primordial star formation, the prospects for BDM being mostly in the form of brown dwarfs are evidently dim. There is no compelling reason that primordial conditions would systematically favor domination by fragments of mass below 0.1M ⊙ . Accretion onto protostellar cores is parametrized, in a simple spherically symmetric situation, by the accretion rate ∼ ∆V 3 /G, where ∆V represents an effective throttle velocity at which inflow occurs. This might be the sound velocity in a quiescent cloud, the turbulent velocity, or the Alfven velocity if magnetic pressure dominates the thermal pressure. If one could imagine an unusually quiescent enviroment, with an accretion rate as low as ∼ 10 −9 M ⊙ yr −1 , it is possible to delay hydrogen ignition and construct brown dwarfs of mass 0.1 or even 0.2M ⊙ [46]. More normal accretion rates are in the range 10 −5 − 10 −3 M ⊙ yr −1 . These lead, for low mass cores, to conventional brown dwarfs, of mass below 0.08M ⊙ for solar and 0.09M ⊙ for primordial composition. It is possible that in a turbulent cloud, where ∆V is enhanced, as well as in a primordial cloud, where inefficient cooling guarantees a high sound speed, the protostellar accretion rates are large. This would provide a possible theoretical justification for a top-heavy IMF in these environments. Halo BDM could conceivably consist of stellar relics if the primordial IMF had very few solar mass stars. Indeed that the primordial IMF was top-heavy is at least as likely as the bottom-heavy option. Several arguments may be adduced to support this possibility [47]. The low dispersion found in the alpha-nuclei relative to iron [34], compared to the large dispersion in [Fe/H] for disk stars, suggests that both the α-nuclei and Fe were mostly produced by massive stars, in contrast to the current epoch IMF that generates Fe from low mass Type I supernovae and α-nuclei from massive stars. The conventional interpretation that only at [Fe/H] < −1 is one dominated by massive star-synthesized alpha-nuclei is probably not tenable in view of recent data [48], which reveals a gradual trend of decreasing α-nuclei with increasing iron abundance. The enhancement with decreasing galactic radius [34] of the alpha-nuclei abundance relative to Fe/H suggests that the primordial IMF in the inner galaxy was systematically top-heavy relative to the solar neighborhood. This latter possibility is also suggested by the analogy between galaxy formation and starbursts. The elevated star formation rate inferred when the old disk and spheroid formed is similar to that encountered in starburst galaxies. The physical mechanism, involving satellite mergers, is common to models of both starbursts and galaxy formation. Modelling of starbursts suggests that a top-heavy IMF is required to account for the observed luminosity, given the available gas supply and a plausible star formation efficiency. One might expect the same situation to have applied when the inner galaxy formed and the bulk of the heavy elements seen in the disk were synthesized. Stellar remnants provide an attractive source of mass to account for the rotation curve in terms of a boosted contribution from the inner disk [49]. It has also been suggested [50] that without a top-heavy IMF at early epochs one would have exhausted the supply of interstellar gas by the present epoch. A top-heavy IMF in the inner galaxy may be required at the present epoch to account for the observed ionizing photon flux [24]. Overproduction of 3 He is avoided with an early IMF that has fewer, by a factor of 2-3, low mass stars than the present-day IMF. One cannot overdo this, otherwise there would be excessive astration of 2 H. If a primordial top-heavy IMF is held responsible for disk and spheroid formation, it is evidently possible to flatten the IMF still further, or even truncate it below ∼ 2M ⊙ , in order to account for halo BDM. The dominant component is most likely to be white dwarfs, since their stellar precursors (< 10M ⊙ ) produce relatively little light or nucleosynthetic contamination compared to more masssive stars. This requires finetuning of the IMF. For example, with an IMF only spanning a range of ∼ 2 − 8M ⊙ , one can avoid excessive CN production, since primordial stars with very low abundance (Z < 10 −4 Z ⊙ ) do not undergo helium flashes and ensuing dredge-up of CN-cycle processed material [51,52]. 4 He production and 2 H destruction cause potential difficulties. Both uncertainties in the primordial abundances and the likelihood of considerable gas recycling offer considerable leeway. It is quite possible that the gas may reside in the halo or outer disk in the form of cold clouds, or else be ejected into the intergalactic medium when the outer BDM halo forms [53]. What Could the (Dark) Matter Be? If the dark matter is baryonic, it makes sense to consider the most reasonable forms that it could take. These are, in order, of decreasing plausibility: Unfortunately, as we have seen, theory is a poor guide. The physical conditions in primordial clouds undoubtedly differ from present-day star-forming clouds. There were no heavy elements, no dust, and, most likely, no significant magnetic fields. However we have so sparse an understanding of how the present-day IMF arises that it is not even possible to infer the sign of any deviation in the primordial IMF from that observed locally. We cannot predict whether the primordial IMF should be biased towards massive or low mass stars. Biased it must be, however, in order to produce sufficiently dark matter. Black holes of mass larger than 10 4 M ⊙ have been recently excluded as a halo dark matter candidate, since otherwise globular clusters [54] and nearby dwarf spheroidal galaxies [55] would be disrupted. Stellar mass objects are preferred, as doing the least injustice to our expectations of what halo dark matter might be. One can distinguish between the various options for stellar mass objects on astrophysical grounds. Brown Dwarfs Not a single brown dwarf is known to exist. Intensive searches for low mass stellar companions of nearby stars by spectroscopy (to detect motions of ∼ 20 m s −1 ) and photometry (to measure shifts of ∼ 0.001 arc-sec/yr) have failed to reveal any candidates below 0.08M ⊙ [56,57]. Searches of binaries have failed to find companions of later spectral types than M 6. Spectroscopy of candidate brown dwarfs that are in the Pleiades, chosen from their location in the H-R diagram, failed to find lithium absorption lines [58]. These stars cannot therefore be brown dwarfs. These failures to find brown dwarfs have not dissuaded theorists from proposing brown dwarfs as BDM candidates. The population II IMF appears to be steep, rising to the detection limit of about 0.15M ⊙ . However the density profile follows that of a de Vaucouleurs law [59]. A halo of brown dwarfs is directly detectable if the IMF of the halo is an extrapolation of the IMF observed in Population II stars. This is because even though one would need to extrapolate the IMF to very low masses, 0.01 − 0.001M ⊙ , given any reasonable slopes, the brown dwarfs just below the main sequence limit are still sufficiently luminous after a Hubble time has elapsed to be detectable via deep star counts in the near-infrared. Recent surveys at 2.2µ suggest that a brown dwarf halo could not be a simple extrapolation of the IMF of Population II stars [60]. However, a strong upturn in the IMF below the main sequence limit of 0.09M ⊙ for primordial brown dwarfs or ∼ 0.08M ⊙ for metal-rich brown dwarfs would not be detectable in the deep counts. Only the gravitational microlensing surveys provide an unambiguous means of searching for these low mass halo objects. The pro-brown dwarf arguments are the following. Brown dwarfs presumably form in considerable numbers, since a fragmenting interstellar cloud is unaware of the minimum mass for hydrogen burning. Cooling flows in galaxy clusters are inferred to undergo mass deposition at a rate of up to ∼ 300M ⊙ yr −1 , and this mass flux cannot end up in stars with a solar neighborhood initial mass function (IMF). Some evidence of star formation is seen, however, for example in the galaxy NGC 1275 at the centre of the Perseus cluster cooling flow [61]. This suggests that the IMF formed in cooling flows may be bottom-heavy, or steeper than the local IMF. Evidence [62] of ∼ 10 11 − 10 12 M ⊙ in cold gas in the cores of cooling flow clusters, based on modelling the x-ray spectrum below 1 keV, may, if confirmed, remove much of the motivation for invoking predominantly low mass star formation in cooling flows. Tentative support for a bottom-heavy IMF comes from a study of several globular cluster luminosity functions. In the mass range ∼ 0.2 − 1M ⊙ , despite large incompleteness corrections, a significantly steeper IMF is found for a few globular clusters, several of which have long core relaxation time-scales [63]. However, the metal abundance in these systems is an order of magnitude lower than that in cooling flows, [F e/H] ≈ −0.5. Halo White Dwarfs White dwarf mergers are believed to result in Type I supernovae. These are luminous and catastrophic events that are powered by the ejection of about 0.6 M ⊙ of radioactive nickel that decays into iron. A dark halo would be detectable were it to generate Type I supernovae at a rate expected for the corresponding number of white dwarfs. However there is some reason to believe that Type I supernovae are subluminous in old stellar populations [64]. A white dwarf halo requires extreme fine-tuning of the primordial IMF. One has to obtain a mass-to-light ratio of > ∼ 2000 in the V-band, as inferred from observations along the minor axes of edge-on spirals [6]. One has also to avoid contamination by ejecta from supernovae. The allowed mass range of the precursor population is 4 − 6 M ⊙ if the stars form in a burst that lasts 2 Gyr, and 2 − 8 M ⊙ if the burst lasts 1 Gyr [53]. If one removes the assumption that the massive star ejecta are recycled in the disk, the constraints on the upper end of the IMF can be relaxed. For example, the gaseous ejecta from the halo could be ejected into the intergalactic medium by supernova-driven winds, which would then give a source for the intracluster iron detected in x-ray observations of rich clusters. The abundances of other elements in the intracluster gas, especially oxygen, will soon be available from ASCA observations, and should help clarify the nature of the parent star population that must have contaminated the intracluster medium early in galactic history. A massive-star origin would result in enhanced oxygen to iron by a factor of 3 or so, as seen for old Population II stars. This would be true for an IMF truncated at the lower end [65]. However a precisely fine-tuned IMF, greatly but not necessarily completely suppressed at both lower and upper ends in order to produce white dwarf halos, would result in a more normal abundance of oxygen relative to iron. Alternatively, the halo gas left over from forming the stellar relic BDM, amounting to as much as 70 percent for a typical return fraction appropriate to a top-heavy IMF, could have condensed into cold gas clouds that remain in the halo, as discussed in the next section. A halo of white dwarfs, with minor components of neutron stars, black holes, and even solar mass stars, as predicted by a top-heavy IMF, is potentially observable via several experiments. If the halo formed less than ∼ 15 Gyr ago, the white dwarfs are sufficiently luminous (L > 10 −6 L ⊙ ) that the nearest ones are observable: for example, a frequency of > ∼ 1/sq deg to m I < 22 is predicted [66]. Perhaps the most dramatic consequence of a white dwarf halo stems from binary mergers. Mergers of close white dwarf pairs formed by tidal capture in the protoclusters where they were formed could produce neutron stars. The required production mechanism must form a substantial number of high galactic latitude pulsars, seen at a distance above the galactic plane > ∼ 1 kpc, and generates pulsar velocities of < ∼ 1000 km s −1 . Neutron stars formed via mergers are plausible candidates for gamma-ray bursters [67]. The existence of high velocity pulsars in the halo is suggested by the observational data. The possibility of their being gamma-ray burst progenitors is to a large degree independent of the theoretical model. Diffuse Gas Clouds The most conservative of assumptions for the nature of dark matter is that it is in the form of diffuse gas. In galaxy halos, this at first sight seems to be completely untenable. Gas at the virial temperature of the Milky Way, ∼ 2 × 10 6 K, would prolifically emit soft x-rays. The diffuse x-ray background allows an x-ray emission measure of at most 0.01 cm −6 pc, corresponding to a halo density at ∼ 10 kpc (the dark halo core radius) of ∼ 10 −3 cm −3 , and therefore to a mass of ∼ 10 8 M ⊙ . With a density profile ρ ∝ r −2 , the diffuse gas mass is only 10 −3 of that required for the dark halo. However several observations suggest that one ought to reexamine the diffuse gas constraints more carefully. The deepest x-ray observations of galaxy clusters indicate that considerable amounts of hot gas may be outside the cluster core. In several clusters, the gas mass amounts to more than 50 percent of the total mass at 3 or 4 Abell radii [68,69,70]. Moreover in the inner cores, where cooling flows are inferred from the x-ray surface brightness profiles, there are indications of x-ray self absorption intrinsic to the cluster. These are best interpreted in terms of > ∼ 10 11 M ⊙ of cold gas, inferred to be in clouds with a covering factor of order unity across the cluster core [71]. High redshift observations of damped Lyman alpha clouds indicate that if the trend observed at z > ∼ 3, where one measures a mass fraction in hydrogen that is roughly equal to that seen in stars at z = 0 [72], continues to z ∼ 5 one may be seeing more cold gas in the form of HI than is in stars at low redshift. With regard to our own galaxy, there may be several times more molecular gas than atomic gas in the disk at a galactocentric distance of ∼ 10kpc [73]. Indeed, molecular gas complexes, excited by HII regions, have been discovered as far out as ∼ 28kpc [74]. There may well be far more colder H 2 present in the outer disk, without accompanying HII regions, than has hitherto been undetected. If this trend were to continue to the outermost disk, at > ∼ 30kpc, one might need to revise the consensus view that the mass in cold gas does not contribute significantly to the rotation velocity. The remarkable case of DDO 154 provides strong testimony for the view that dark matter normally associated with halos may exist at least in part in the form of hitherto undetected cold gas clouds. This dwarf galaxy has one of the best-studied rotation curves, that extends to at least 15 disk scale-lengths. Outside 2 scale-lengths the observed star distribution provides a negligible contribution to the rotation velocity. The HI column density scales as N HI ∝ r −1 to the limit where the rotation curve can be traced. It exactly parallels the dark matter surface density inferred from the rotation curve, and contributes about 10 percent to the required total surface density. It is tempting to infer that the 90 percent shortfall of dark matter is in baryonic form, either H 2 or optically thick HI. It might be physically associated with the observed HI in the disk, or else constitute a flattened halo in which the observed HI clouds are embedded. Hiding a population of cold clouds from detection is possible if the clouds are sufficiently compact so as to only rarely collide. This same condition also guarantees that the cloud surface covering factor is low. It would then be difficult to observe the clouds, either in absorption towards quasars or in emission. There are two difficulties. Clouds passing through the disk would be exposed to the local ionizing radiation field within HII regions and possibly be visible. The overriding question is why such clouds avoid forming stars during a Hubble time. Stabilizing the clouds is possible if pressure support can be maintained. One would need warm cloud cores. It might be possible to achieve this with a modest amount of star formation. Primordial abundances in the cores would also result in higher temperatures. Exotica Baryon dark matter could consist of massive black holes. As noted above, the upper limit on black hole mass is about 10 4 M ⊙ . Precursor supermassive stars in the mass range 100 − 1000M ⊙ implode to form black holes without injecting substantial amounts of enriched material. However, during the precollapse helium-burning phase, there is extensive radiatively-driven mass loss, and considerable amounts of helium are shed. To avoid a discrepancy with primordial nucleosynthesis, one would have to store the ejecta in cold dense clouds that remain in the halo or outer disk. These clouds cannot participate in spheroid and disk star formation, and are not otherwise strongly constrained, as described in the previous section. Nuggets of strange matter, relics of the quark-hadron phase transition, have been proposed as a possible form for dark matter. Such objects may be stable, in certain quark models. However, quark nuggets are likely to have evaporated prior to the nucleosynthesis epoch. The smallest stable objects that might be BDM candidates have masses that can be estimated as follows [75]. These would be made of hydrogen. For a density of solid H 2 of about 0.1g cm −3 , such "snowballs" are gravitationally bound at a temperature of say 30K, corresponding to the CMB temperature at z = 10, if the typical mass exceeds M > ∼ 10 −8 T 30K 3 2 0.1g cm −3 ρ 1 2 M ⊙ Thus the mass range 10 −8 M ⊙ to 10 −3 M ⊙ is the possible range spanned by dark matter snowballs. The central pressure is sufficiently high that if the mass exceeds ∼ 10 −3 M ⊙ , degeneracy is important. More massive objects have higher central density and are smaller. They continue to contract, although at a small rate, and are referred to as brown dwarfs. In summary, I conclude that star formation is a messy problem in nonlinear physics with depressingly many degrees of freedom. These include cloud ionization, metallicity, magnetic field strength, angular momentum, dust grain properties, and possible feedback from forming stars. At least, we can predict the mass of a star, to within an order of magnitude! I have argued that phenomenological arguments provide a useful guide. Unfortunately, a mastery of star formation is critical for understanding the nature of baryonic dark matter. One needs either to prevent star formation from occurring, as is the case if BDM consists of cold clouds or brown dwarfs, or else to fine-tune it, as must be done if BDM is in the form of white dwarfs or black holes. COSMOGONIC IMPLICATIONS Galaxy Morphology The primordial star formation rate is the key to understanding galaxy morphology. The high specific star formation rate inferred during formation of the spheroidal component of a galaxy guarantees that some massive dense stellar subsystems form early in the collapse. These sink deep into the potential well via dynamical friction against the lower density stellar systems that are the prevalent component. The dense star clouds efficiently transfer angular momentum as they spiral into the central regions of the galaxy. In this way, an elliptical galaxy develops that is supported by random stellar motions rather than by systematic rotation. The star formation is completed within 1 or 2 Gyr. In contrast, a disk forms slowly, over several Gyr. The low star formation rate means that the system stays gas-rich. Dissipative cooling controls the rate at which the angular momentum-conserving contraction occurs. Eventually, rotational support halts the collapse process, when the disk has formed. The role of a dark halo is to provide an additional source of mass-collapsing matter against which the gaseous, star-forming component exerts a torque and thereby transfers angular momentum. Evidently, star formation plays a crucial role in determining the various types of galaxies. Galaxy halos are likely to have some BDM, and perhaps to be predominantly BDM. The formation of baryonic dark matter, since it is closely coupled to early star formation, is evidently inseparable from the galaxy morphology issue. An explanation for why spirals predominate in low density regions and ellipticals in dense cluster cores is likely to be related to the problem of BDM. Large-Scale Structure I have hitherto assumed that halo dark matter consists of BDM. This is equivalent to asserting that Ω BDM = 0.03 − 0.07, in accordance with the nucleosynthesis prediction Ω B = 0.015(±0.005)h −2 . However, there is reason to doubt the error bounds on the nucleosynthesis limit. If these are sufficiently relaxed, one is then drawn to consider the case of an open cosmology with Ω = 0.1 − 0.2, in which all of the dark matter is BDM. Could one go the additional step and consider Ω BDM ≈ 1 ? This would grossly violate the nucleosynthesis limits even in non-standard models of inhomogeneous light element production. Such a model almost certainly produces excessive cosmic microwave background fluctuations. Certainly with inflationary initial conditions, a primary motivation for adopting Ω = 1, one has approximately scale-invariant, adiabatic primordial density fluctuations. These are a disaster for δT /T ; nor is δT /T suppressed by reionization on the largest angular scales. However, the Ω = 0.1 − 0.2 cosmology is phenomenologically attractive. It makes the simplest of assumptions: "what you see is what you get." We see baryons, and on scales < 20Mpc, where the observations are most reliable, we measure Ω ∼ 0.1. There is a heavy price to pay for the simplicity. One has to drop inflation, at least in its generic incarnation, and one has to abandon the hypothesis of primordial scale-invariant curvature fluctuations. The result is a model that is ugly but simple. A low Ω universe, containing only baryons, must be seeded by primordial isocurvature fluctations. These are equivalent to primordial spatial variations in the specific entropy or in the baryon number. There is no accepted theory for the origin of such fluctuations. However, one might anticipate that some models of baryogenesis, for which there is not a universally accepted theory, and which provide nB nγ , are also capable of producing ∆ nB nγ . Indeed, there are such models in the literature [76,77]. However there are essentially no predictions for the fluctuation spectrum, which accordingly is treated phenomenologically, as a power-law of arbitrary slope and normalization. Primordial entropy perturbations δs are defined as perturbations in the number of photons per baryon, so that δs = δ T 3 n , whence δs s = 3 4 δρ γ ρ γ − δρ B ρ B . Here, δρ γ is the perturbation in radiation density and δρ B is the perturbation in baryon density. Requiring that there be no net curvature perturbation, the isocurvature mode being orthogonal to the adiabatic or curvature mode, then leads one to write δρ γ + δρ B = 0. In the late time, matter-dominated limit, one obtains δT T = 1 3 δs. This is valid on scales larger than the horizon at last scattering, and shows that one can map out the intrinsic entropy fluctuations on sufficiently large angular scales (greater than a few degrees). In the absence of a predicted spectrum, one adopts a power-law form for the primordial entropy fluctuations, with power spectrum P (k) ≡ \|δ k \| 2 ∝ k n , where δ k is the Fourier amplitude, δρ ρ = δ k exp(ik.x) d 3 k. The rms fluctuations (δρ/ρ) 2 are equal to \|δ k \| 2 . An empirical fit to the COBE DMR data over spherical harmonics (l = 2 − 10) finds n ≈ 1.1 but with large uncertainty, ∆n ≈ ±0.5. Comparison with the Tenerife data (l ≈ 18) suggests that n ≈ 1.5, as does analysis of the second year DMR data. If confirmed, this would favour a non-inflationary primordial fluctuation spectrum as expected in a low Ω universe. Primordial Density Fluctuation Power Spectrum An empirical fit to the matter fluctuations can be performed using the power spectrum derived from various redshift surveys. Over scales of 10 − 50Mpc, the linear regime of power is effectively probed, albeit with uncertainties that depend on the inevitable distortions involved in transforming from redshift to threedimensional space. An empirical fit requires n ≈ −1, with an uncertainity of about ∆n ≈ ±0.5. The more negative values of n result in excessive CMB temperature fluctuations on scales of order 10 degrees, where reionization is ineffective. A compromise value is n ≈ −0.5 for the primordial power law index. In terms of the invariant mass M associated with comoving wavenumber k, the corresponding mass spectrum is δρ/ρ ∝ M − n+3 6 ∝ M −0.4 for n = −0.5. Hence in contrast to the scale-invariant inflationary spectrum (n ≈ 1), which is only logarithmically divergent, with n ef f ≈ n − 4, on scales smaller than that of the horizon at matter-radiation equality, roughly a galactic mass, the isocurvature spectrum is strongly divergent towards high redshift. Early formation of small galaxies is inevitable in this model. Star formation, and the associated supernovae, must result in production of an ionizing photon flux that is capable of at least partially reionizing the intergalactic medium. Even with a small efficiency of ionizing photon production, recoupling of the CMB is likely to be almost inevitable at z > ∼ 100. This has two notable effects. Radiation drag inhibits growth of matter fluctuations on sub-horizon scales. Rescattering of the CMB smooths out the associated temperature fluctuations. To produce the large-scale structure, as characterized by the correlation amplitude on 10Mpc, the suppressed growth implies that one needs a larger initial amplitude for the primordial fluctuations than would be the case were early reionization (z > 100) not to have occurred. This has interesting consequences for the generation of large-scale peculiar velocity fields. The baryonic dark matter power spectrum has a generic large-scale peak that corresponds to the maximum Jeans mass scale. This is approximately equal to 110 0.1/Ωh 2 Mpc. The sound speed prior to recombination is c s = dp dρ 1 2 = dp r d(ρ m + ρ r ) 1 2 = c √ 3 1 + 3 4 ρ m ρ r − 1 2 ∝ (1 + z) 1 2 at ρ m > ρ r . The Jeans length l J ∼ c s t ∝ (1 + z) −1 , and therefore the comoving Jeans length is constant. After recombination, the temperature abruptly drops to 3000 K and the sound speed correspondingly declines. The Jeans mass prior to recombination was 2 × 10 18 0.1 Ωh 2 2 M ⊙ ; after recombination, it drops to 5 × 10 5 0.1 Ωh 2 1 2 M ⊙ . Since the sound speed prior to recombination is about 10 4 times larger than that after recombination, any pressure fluctuations propagating as sound waves are greatly amplified over scales much larger than the post-recombination Jeans length. One consequence is the occurrence of dramatic oscillations in the matter transfer function (Figure 1) that are eventually quenched by radiation drag, in a universe where reionization occurs at z < 1000. It is unclear whether the corresponding oscillations in the galaxy correlation function, the amplitude of which depends sensitively on the model for small-scale nonlinearity, would be observable. A large-scale coherent velocity field is another consequence of this model [78]. Large-scale bulk flows are more directly computable, being insensitive to any non-linear corrections for the range of n of interest, and measurable. The velocity correlation function is shown in Figure 2. The large-scale matter distribution is normalised to give unit variance in mass fluctuations averaged over a sphere of radius 8h −1 Mpc, where the luminous galaxy counts have unit variance. δT T on Intermediate and Small Angular Scales The initial expectation for δT T in a low Ω universe is that the Jeans mass peak would have a substantial effect. On scales below the horizon at recombination, corresponding to several hundred Mpc, gravity would amplify the primordial entropy fluctuations. Associated adiabatic fluctuations are generated by gravityinduced velocity fields. This should lead to δT T ∼ υ c ∼ L t δρ ρ . However the first-order Doppler fluctuations are erased by rescattering of the CMB photons. The probability of rescattering is to t n e σ T cdt = 0.04hΩ B Ω With Ω B ∼ Ω o ∼ 0.1, primary fluctuations are erased if reionization occurs at z > ∼ 50, over angular scales of up to ∼ 10 degrees. However, temperature fluctuations are regenerated on the last scattering surface. Only in second order, δT T ∼ ( v c )( δρ ρ ), do the fluctations add in quadrature, the first-order fluctuations (∼ v c ) self-cancelling. While any surviving first-order fluctuations would be on degree scales, and correspond to the primary Doppler peaks, the second order, regenerated fluctuations are on arc-minute scales. These are a unique signature of BDM on these scales, since the primary last scattering surface has a thickness of about 5 arc-minutes, and in the canonical CDM model, one expects no primary fluctuations on smaller scales. The best current limit on δT T over small angular scales is that from the Australia Telescope Compact Array. Over a beam of 0 ′ .9, the rms temperature fluctuations are less than 9 × 10 −6 . This allows a small area of BDM parameter space, with Ω B ∼ 0.1, n ∼ −0.5, and h ∼ 0.8 [79]. With the Hubble constant as low as h = 0.5, excessive fluctuations are generated on arcminute scales. where T 4 ≡ T 10 4 K is the temperature of the intergalactic gas. The COBE FIRAS experiment sets an upper limit, y < 2.5 × 10 −5 . The Compton y Constraint This is sufficient to exclude models in which reionization occurred at z > ∼ 800, with h ∼ 0.8, Ω B ∼ Ω 0 ∼ 0.1 [80]. A BDM Scenario Consider the following model for a cosmology dominated by BDM. Take Ω ∼ 0.1 ∼ Ω B . This alone requires early structure formation, even clusters of galaxies forming at z > ∼ 10. Galaxies form much earlier. With primordial entropy fluctuations allowed as possible seeds, adiabatic fluctuations being observationally excluded, one infers a linear fluctuation distribution described by δρ ρ ∝ M − 1 2 − n 6 t 2 3 , z > ∼ Ω −1 − 1, 0 > ∼ n > ∼ − 0.5. Nonlinearity occurs on the Jeans mass scale, ∼ 10 6 M ⊙ , as early as z ∼ 1100. What happens next is pure speculation. One scenario is the following. The primordial clouds, of mass comparable to globular star clusters, collapse, and fragment into stars by z ∼ 500. Ionizing photons from the first massive stars ensure that Compton drag forces will initially inhibit further gas collapse and star formation. However once z > ∼ 200, the cloud contraction is sufficiently shorter than the Hubble time that star formation resumes. Supernovae drive gas outflows that will soon disrupt star formation in the low mass clouds. Only later, by z ∼ 30, when sufficiently massive potential wells have developed that can efficiently retain the ejection from supernova-driven winds, will galaxy formation begin in earnest. The baryonic dark matter consists in part of the compact remnants of early massive star formation, and also, at least in the intergalactic medium, of diffuse gas. CONCLUSIONS The BDM hypothesis provides a reasonably complete description both of dark halos and of dark matter that is more broadly distributed. It leads to five unique predictions. Three of these are related to CMB temperature fluctuations. a. Secondary fluctuations at a level δT T ∼ 10 −5 are predicted on arc-minute scales because of the early reionization that the BDM model requires in order to erase the primary Doppler peaks. b. Secondary Doppler peaks δT /T ∼ 10 −5 are generated on degree scales. Their location depends on Ω. c. On large angular scales, > ∼ 10 degrees, curvature effects dominate the predicted temperature anisotropies. In addition to the usual Sachs-Wolfe anisotropies, δT T = 1 3 φ LS , from the last scattering surface, there is the integrated effect of time-varying potentials along the line-of-sight, δT T = ∂φ ∂t dt. The curvature scale ( c H0 )(1 − Ω 0 ) − 1 2 becomes less than the particle horizon scale, 2c H0 Ω 0 at Ω 0 < 0.85. Hence one expects a suppression of the low-order multipoles relative to the higher order multipoles, over scales l > ∼ Ω −1 . The detailed shape of the predicted low multipole power spectrum is dominated by the primordial entropy fluctuations and the details of how the fluctuation spectrum is defined. d. Compton y-distortions of the CMB spectrum are inevitable in a BDM universe at a level y ∼ 10 −5 because of the early reionization. e. A peak in the matter power spectrum is inevitable at 100−300Mpc, corresponding to the maximum Jeans mass in the early universe. This could manifest itself as a source of a systematic, coherent large-scale flow that is discrepant in direction with the CMB dipole, and possibly is aligned with the CMB quadrupole (if such a quadrupole is indeed measured). Perhaps the very large-scale flow (∼ 800 km s −1 ) inferred from the dipole moment of a sample of Abell clusters at a distance of ∼ 150 h −1 Mpc [81], if confirmed, would be best explained in such a model. The major weakness in the BDM model arises from our poor understanding of star formation in extreme environments, such as that of protogalaxies. This applies equally whether we wish to account for a level Ω B ≈ 0.02 that suffices to account for dark halos and to satisfy the primordial nucleosynthesis constraint, or aim for the grander goal of Ω B ≈ 0.1 in the cosmological setting. The luminous regions of galaxies provide Ω * ≈ 0.007 in the form of known types of stars and gas. There is not a unique prescription for arriving at this value of Ω * . This is true regardless of whether the universal Ω is 0.1 or 1. Thus it seems eminently plausible that BDM both does exist and should exist, and dominate the known luminous matter content of the universe. Whether BDM accounts for all of the matter in the universe is more problematical and controversial. Certainly, the trend towards a high H 0 pushes one towards a low Ω universe, as does the most reliable and systematic-free, that is to say, the most local, of the large-scale structure data. Should BDM provide the resolution to both the large-scale and the small-scale dark matter problems, it is encouraging to note that we must be on the verge of detecting its elusive signature. BDM is (barely) alive and well. Figure 1. The matter transfer function for baryonic dark matter-dominated cosmological models [82]. For each combination of Ω and h, results are shown for several epochs of reionization. Figure 2. The velocity correlation function for baryonic dark matter-dominated cosmological models [82]. For Ω = 0.2 and h = 1, results are shown with normalization σ 8 = 1 for reionization at z = 1000 compared to no reionization, and for n = −1 and n = 0.. For comparison, also shown is the velocity correlation function for unbiased CDM. a. Stellar mass objects, from 10 −3 M ⊙ to 10 3 M ⊙ . These could be brown dwarfs (10 −3 to ∼ 0.08M ⊙ ), white dwarfs (0.4 − 1.4M ⊙ ), neutron stars (0.4 − 2M ⊙ ), stellar relic black holes from ordinary massive stars (∼ 2 − 10M ⊙ ), or black holes that formed from supermassive stars (∼ 100 − 10 4 M ⊙ ).b. Diffuse dense clouds of cold hydrogen, c. Exotica, including primordial black holes and nuggets of strange matter. A significant spectral distortion of the CMB blackbody arises if reionization occurs very early. Compton scattering not only erases angular fluctuations, but transfers energy from the hotter electrons to the CMB photons, ∆hν hν ∼ kT mec 2 . 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[ "Volumetric Bias in Segmentation and Reconstruction: Secrets and Solutions", "Volumetric Bias in Segmentation and Reconstruction: Secrets and Solutions" ]	[ "Yuri Boykov \nComputer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada\n", "Hossam Isack \nComputer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada\n", "Carl Olsson \nComputer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada\n", "Ismail Ben Ayed [email protected] \nComputer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada\n" ]	[ "Computer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada", "Computer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada", "Computer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada", "Computer Science UWO\nCentre for Math. Sciences\nMedical Biophysics UWO\nLund University\nCanada, Sweden, Canada" ]	[]	Many standard optimization methods for segmentation and reconstruction compute ML model estimates for appearance or geometry of segments, e.g. Zhu-Yuille [21], Torr [18], Chan-Vese [6], GrabCut [16], Delong et al. [8].We observe that the standard likelihood term in these formulations corresponds to a generalized probabilistic K-means energy. In learning it is well known that this energy has a strong bias to clusters of equal size, which can be expressed as a penalty for KL divergence from a uniform distribution of cardinalities [10]. However, this volumetric bias has been mostly ignored in computer vision. We demonstrate significant artifacts in standard segmentation and reconstruction methods due to this bias. Moreover, we propose binary and multi-label optimization techniques that either (a) remove this bias or (b) replace it by a KL divergence term for any given target volume distribution. Our general ideas apply to many continuous or discrete energy formulations in segmentation, stereo, and other reconstruction problems.	10.1109/iccv.2015.206	[ "https://arxiv.org/pdf/1505.00218v1.pdf" ]	1,233,136	1505.00218	3e2f630410f1de9f5821a03fecc727dfc9f7562c	Volumetric Bias in Segmentation and Reconstruction: Secrets and Solutions Yuri Boykov Computer Science UWO Centre for Math. Sciences Medical Biophysics UWO Lund University Canada, Sweden, Canada Hossam Isack Computer Science UWO Centre for Math. Sciences Medical Biophysics UWO Lund University Canada, Sweden, Canada Carl Olsson Computer Science UWO Centre for Math. Sciences Medical Biophysics UWO Lund University Canada, Sweden, Canada Ismail Ben Ayed [email protected] Computer Science UWO Centre for Math. Sciences Medical Biophysics UWO Lund University Canada, Sweden, Canada Volumetric Bias in Segmentation and Reconstruction: Secrets and Solutions Y. Boykov, H. Isack, C. Olsson, I.B. Ayed, arXiv:xxxx, May 2015 p.1 Many standard optimization methods for segmentation and reconstruction compute ML model estimates for appearance or geometry of segments, e.g. Zhu-Yuille [21], Torr [18], Chan-Vese [6], GrabCut [16], Delong et al. [8].We observe that the standard likelihood term in these formulations corresponds to a generalized probabilistic K-means energy. In learning it is well known that this energy has a strong bias to clusters of equal size, which can be expressed as a penalty for KL divergence from a uniform distribution of cardinalities [10]. However, this volumetric bias has been mostly ignored in computer vision. We demonstrate significant artifacts in standard segmentation and reconstruction methods due to this bias. Moreover, we propose binary and multi-label optimization techniques that either (a) remove this bias or (b) replace it by a KL divergence term for any given target volume distribution. Our general ideas apply to many continuous or discrete energy formulations in segmentation, stereo, and other reconstruction problems. Introduction Most problems in computer vision are ill-posed and optimization of regularization functionals is critical for the area. In the last decades the community developed many practical energy functionals and efficient methods for optimizing them. This paper analyses a widely used general class of segmentation energies motivated by Bayesian analysis, discrete graphical models (e.g. MRF/CRF), information theory (e.g. MDL) , or continuous geometric formulations. Typical examples in this class of energies include a log-likelihood term for models P k assigned to image segments S k E(S, P ) = − K k=1 p∈S k log P k (I p ), where, for simplicity, we focus on a discrete formulation with data I p for a finite set of pixels/features p ∈ Ω and seg- Secrets (1) Solutions (6), (9)(10) (a) GrabCut [16] with unbiased data term (10) (b) plane fitting [18,8,1] with unbiased data term (10) (c) Chan-Vese [6] + [7] with target volumes (6) Figure 1. Left: segmentation and stereo reconstruction with standard likelihoods or probabilistic K-means energy E(S, P ) in (1) has bias to equal size segments (2). Right: (a-b) corrections due to unbiased data termÊ(S, P ) in (9,10) or (c) weighted likelihoods EW (S, P ) in (6) biased to proper target volumes, see (5). ments S k = {p ∈ Ω\|S p = k} defined by variables/labels S p ∈ N indicating the segment index assigned to p. In different vision problems models P k could represent Gaussian intensity models [6], color histograms [2], GMM [21,16], or geometric models [18,8,1] like lines, planes, homographies, or fundamental matrices. p.2 Depending on application, the energies combine likelihoods (1), a.k.a. data term, with different regularization potentials for segments S k . One of the most standard regularizers is the Potts potential, as in the following energy E P otts (S, P ) = − K k=1 p∈S k log P k (I p ) + λ · \|\|∂S\|\|, where \|\|∂S\|\| is the number of label discontinuities between neighboring points p on a given neighborhood graph or the length of the segmentation boundary in the image grid [3]. Another common regularizer is sparsity or label cost for each model P k with non-zero support [18,21,1,8], e.g. E sp (S, P ) = − K k=1 p∈S k log P k (I p ) + γ · k [S k = ∅]. In general, energies often combine likelihoods (1) with multiple different regularizers at the same time. This paper demonstrates practically significant bias to equal size segments in standard energies when models P = {P k } are treated as variables jointly estimated with segmentation S = {S k }. This problem comes from likelihood term (1), which we interpret as probabilistic K-means energy carefully analyzed in [10] from an information theoretic point of view. In particular, [10] decomposes energy (1) as 1 E(S, P ) c = K k=1 \|S k \| · KL(I k \|P k ) + \|Ω\| · (H(S\|I) − H(S)) where KL(I k \|P k ) is KL divergence for model P k and the true distribution 2 of data I k = {I p \| p ∈ S k } in segment k. Conditional entropy H(S\|I) penalizes "non-deterministic" segmentation if variables S p are not completely determined by intensities I p . The last term is negative entropy of segmentation variables −H(S), which can be seen as KL divergence − H(S) c = KL(S\|U ) := K k=1 \|S k \| \|Ω\| · ln \|S k \|/\|Ω\| 1/K(2) between the volume distribution for segmentation S V S := \|S 1 \| \|Ω\| , \|S 2 \| \|Ω\| , . . . , \|S K \| \|Ω\|(3) and a uniform distribution U = { 1 K , ..., 1 K }. Thus, this term represents volumetric bias to equal size segments S k . Its minimum is achieved for cardinalities \|S k \| = Ω K . 1 Symbol c = represents equality up to an additive constant. 2 The decomposition above applies to either discrete or continuous probability models (e.g. histogram vs. Gaussian). The continuous case relies on Monte-Carlo estimation of the integrals over "true" data density. Contributions Our experiments demonstrate that volumetric bias in probabilistic K-means energy (1) leads to practically significant artifacts for problems in computer vision, where this term is widely used for model fitting in combination with different regularizers, e.g. [21,18,6,16,8]. Section 2 proposes several ways to address this bias. First, we show how to remove the volumetric bias. This could be achieved by adding extra term \|Ω\|·H(S) to any energy with likelihoods (1) exactly compensating for the bias. We discuss several efficient optimization techniques applicable to this high-order energy term in continuous and/or discrete formulations: iterative bound optimization, exact optimization for binary discrete problems, and approximate optimization for multi-label problems using α-expansion [5]. It is not too surprising that there are efficient solvers for the proposed correction term since H(S) is a concave cardinality function, which is known to be submodular for binary problems [14]. Such terms have been addressed previously, in a different context, in the vision literature [11,17]. Second, we show that the volumetric bias to uniform distribution could be replaced by a bias to any given target distribution of cardinalities W = {w 1 , w 2 , ..., w K }.(4) In particular, introducing weights w k for log-likelihoods in (1) replaces bias KL(S\|U ) as in (2) by divergence between segment volumes and desired target distribution W KL(S\|W ) = K k=1 \|S k \| \|Ω\| · ln \|S k \|/\|Ω\| w k .(5) Our experiments in supervised or unsupervised segmentation and in stereo reconstruction demonstrate that both approaches to managing volumetric bias in (1) can significantly improve the robustness of many energy-based methods for computer vision. Log-likelihood energy formulations This section has two goals. First, we present weighted likelihood energy E W (S, P ) in (6) and show in (8) that its volumetric bias is defined by KL(S\|W ). Standard data term E(P, S) in (1) is a special case with W = U . Then, we present another modification of likelihood energyÊ(S, P ) in (9) and prove that it does not have volumetric bias. Note that [10] also discussed unbiased energyÊ. The analysis ofÊ below is needed for completeness and to devise optimization for problems in vision where likelihoods are only a part of the objective function. Weighted likelihoods: Consider energy E W (S, P ) := − K k=1 p∈S k log(w k · P k (I p )),(6) which could be motivated by a Bayesian interpretation [8] where weights W explicitly come from a volumetric prior. It is easy to see that E W (S, P ) = E(S, P ) − K k=1 \|S k \| · log w k (7) = E(S, P ) + \|Ω\| · H(S\|W ) where H(S\|W ) is a cross entropy between distributions V S and W . As discussed in the introduction, the analysis of probabilistic K-means energy E(S, P ) in [10] implies that E W (S, P ) c = K k=1 \|S k \| · KL(I k \|P k ) + \|Ω\| · H(S\|I) − \|Ω\| · H(S) + \|Ω\| · H(S\|W ). Combining two terms in the second line gives E W (S, P ) c = K k=1 \|S k \| · KL(I k \|P k ) + \|Ω\| · H(S\|I) + \|Ω\| · KL(S\|W ).(8) In case of given weights W equation (8) implies that weighted likelihood term (6) has bias to the target volume distribution represented by KL divergence (5). Note that optimization of weighted likelihood term (6) presents no extra difficulty for regularization methods in vision. Fixed weights W contribute unary potentials for segmentation variables S p , see (7), which are trivial for standard discrete or continuous optimization methods. Nevertheless, examples in Sec. 3 show that indirect minimization of KL divergence (5) substantially improves the results in applications if (approximate) target volumes W are known. Unbiased data term: If weights W are treated as unknown parameters in likelihood energy (6) they can be optimized out. In this case decomposition (8) implies that the corresponding energy has no volumetric bias: E(S, P ) := min W E W (S, P ) (9) = K k=1 \|S k \| · KL(I k \|P k ) + \|Ω\| · H(S\|I). Weights V S in (3) are ML estimate of W that minimize (8) by achieving KL(S\|W ) = 0. Putting optimal weights W = V S into (7) confirms that volumetrically unbiased data term (9) is a combination of standard likelihoods (1) with a high-order correction term H(S): This standard fact is easy to check: function −z log z (blue curve) is concave and its 1st-order approximation at zt = w k t = \|S k t \|/\|Ω\| (red line) is a tight upper-bound or surrogate function [13]. Note that unbiased data termÊ(S, P ) should be used with caution in applications where allowed models P k are highly descriptive. In particular, this applies to Zhu&Yuille [21] and GrabCut [16] where probability models are histograms or GMM. According to (9), optimization of model P k will over-fit to data, i.e. KL(I k \|P k ) will be reduced to zero for arbitrary I k = {I p \| p ∈ S k }. Thus, highly descriptive models reduceÊ(S, P ) to conditional entropy H(S\|I), which only encourages consistent labeling for points of the same color. While this could be useful in segmentation, see bin consistency in [17], trivial solution S 0 = Ω becomes good for energyÊ(S, P ). Thus, bias to equal size segments in standard likelihoods (1) is important for histogram or GMM fitting methods [21,16]. E(S, P ) = E(S, P ) − K k=1 \|S k \| · log \|S k \| \|Ω\| = E(S, P ) + \|Ω\| · H(S).(10) Many techniques with unbiased data termÊ(S, P ) avoid trivial solutions. Over-fitting is not a problem for simple models, e.g. Gaussians [6], lines, homographies [18,8]. Label cost could be used to limit model complexity. Trivial solutions could also be removed by specialized regional terms added to the energy [17]. Indirectly, optimization methods that stop at a local minimum help as well. Bound optimization for (9-10): One local optimization approach forÊ(S, P ) uses iterative minimization of weights W for E W (S, P ). According to (8) the optimal weights at any current solution S t are W t = { \|S 1 t \| \|Ω\| , ..., \|S K t \| \|Ω\| } since they minimize KL(S t \|W ) . The al-gorithm iteratively optimizes E Wt (S, P ) over P, S and resets to energy E Wt+1 (S, P ) at each step until convergence. This block-coordinate descent can be seen as bound optimization [13]. Indeed, see Figure 2, at any given S t energy E Wt (S, P ) is an upper bound forÊ(S, P ), that iŝ E(S, P ) ≤ E Wt (S, P ) ∀Ŝ E(S t , P ) = E Wt (S t , P ). This bound optimization approach toÊ(S, P ) is a trivial modification for any standard optimization algorithm for energies with unary likelihood term E W (S, P ) in (6). High-order optimization for entropy in (9-10): Alternatively, optimization of unbiased termÊ(S, P ) could be based on equation (10). Since term E(S, P ) is unary for S the only issue is optimization of high-order entropy H(S). The entropy is a combination of terms −z log z for z = \|S k \|/\|Ω\|. Each of these is a concave function of cardinality, which are known to be submodular [14]. As explained below, entropy is amenable to efficient discrete optimization techniques both in binary (Sec.3.2) and multi-label cases (Sec.3.2-3.3). Optimization of concave cardinality functions was previously proposed in vision for label consistency [11], bin consistency [17], and other applications. Below, we discuss similar optimization methods in the context of entropy. We use a polygonal approximation with triangle functions as illustrated in Figure 3. Each triangle function is the minimum of two affine cardinality functions, yielding an approximation of the type − \|S k \| \|Ω\| log \|S k \| \|Ω\| ≈ l min a L l \|S k \|, a U l \|S k \| + b U l .(11) Optimization of each "triangle" term in this summation can be done as follows. Cardinality functions like a L l \|S k \| and a U l \|S k \| + b U l are unary. Evaluation of their minimum can be done with an auxiliary variable y l ∈ {0, 1} as in min y l y l (a L l \|S k \|) +ȳ l (a U l \|S k \| + b U l )(12) which is a pairwise energy. Indeed, consider binary segmentation problems S p ∈ {0, 1}. Since \|S k \| = p∈Ω S p , if k = 1 p∈Ω (1 − S p ), if k = 0(13) (12) breaks into submodular 3 pairwise terms for y l and S p . Thus, each "triangle" energy (12) can be globally optimized with graph cuts [12]. For more general multi-label problems S p ∈ N energy terms (12) can be iteratively optimized via binary graph-cut moves like α-expansion [5]. Indeed, let 3 Depending on k, may need to switch y l andȳ l . variables x p ∈ {0, 1} represent α-expansion from a current solution S t = {S k t } to a new solution S. Since \|S k \| = p∈Ω x p , if k = α p∈S k t (1 − x p ), if k = α(14) (12) also reduces to submodular pairwise terms for y l , x p . The presented high-order optimization approach makes stronger moves than the simpler bound optimization method in the previous sub-section. However, both methods use block coordinate descent iterating optimization of S and P with no quality guarantees. The next section shows examples with different optimization methods. Examples This sections considers several representative examples of computer vision problems where regularization energy uses likelihood term (1) with re-estimated models P k . We empirically demonstrate bias to segments of the same size (2) and show advantages of different modifications of the data term proposed in the previous section. Segmentation with target volumes In this section we consider a biomedical example with K = 3 segments: S 1 background, S 2 liver, S 3 substructure inside liver (blood vessels or cancer), see Fig.4. The energy combines standard data term E(S, P ) from (1), boundary length \|\|∂S\|\|, an inclusion constraint S 3 ⊂ S 2 , and a penalty for L 2 distance between the background segment and a given shape template T , as follows E(S, P ) + λ\|\|∂S\|\| + [S 3 ⊂ S 2 ] + β\|\|S 1 − T \|\| L2 . (15) For fixed models P k this energy can be globally minimized over S as described in [7]. In this example intensity likelihood models P k are histograms treated as unknown parameters and estimated using block-coordinate descent for variables S and P . Figure 4 compares optimization of (15) in (b) with optimization of a modified energy replacing standard likelihoods E(S, P ) with a weighted data term in (6) E W (S, P ) + λ\|\|∂S\|\| + [S 3 ⊂ S 2 ] + β\|\|S 1 − T \|\| L2 (16) for fixed weights W set from specific target volumes (c-d). The teaser in Figure 1(c) demonstrates a similar example for separating a kidney from a liver based on Gaussian models P k , as in Chan-Vese [6], instead of histograms. Standard likelihoods E(P, S) in (15) show equal-size bias, which is corrected by weighted likelihoods E W (P, S) in (16) with approximate target volumes W = {0.05, 0.95}. Segmentation without volumetric bias We demonstrate in different applications a practically significant effect of removing the volumetric bias, i.e., using our functionalÊ(S, P ). We first report comprehensive comparisons of binary segmentations on the GrabCut data set [16], which consists of 50 color images with groundtruth segmentations and user-provided bounding boxes 4 . We compared three energies: high-order energyÊ(S, P ) (10), standard likelihoods E(S, P ) (1), which was used in the well-known GrabCut algorithm [16], and E W (S, P ) (6), which constrains the solution with true target volumes (i.e., those computed from ground truth). The appearance models in each energy were based on histograms encoded by 16 bins per channel, and the image data is based color specified in RGB coordinates. For each energy, we added a standard contrast-sensitive regularization term [16,2]: λ p,q∈N α pq [S p = Sq], where α pq denote standard pairwise weights determined by color contrast and spatial distance between neighboring pixels p and q [16,2]. N is the set neighboring pixels in a 8-connected grid. We further evaluated two different optimization schemes for high-order energyÊ(S, P ): (i) bound optimization and (ii) high-order optimization of concave cardinality potential H(S) using polygonal approximations; see Sec.2 for details. Each energy is optimized by alternating two iterative steps: (i) fixing the appearance histogram models and optimizing the energy w.r.t S using graph cut [4]; and (ii) fixing segmentation S and updating the histograms from current solution. For all methods we used the same appearance model initialization based on a user-provided box 5 . The error is evaluated as the percentage of mis-classified pixels with respect to the ground truth. Table 1 reports the best average error over λ ∈ [1 . . . 30] for each method. As expected, using the true target volumes yields the lowest error. The second best performance was obtained byÊ(S, P ) with high-order optimization; removing the volumetric bias substantially improves the performance of standards loglikelihoods reducing the error by 6%. The bound optimization obtains only a small improvement as it is more likely to get stuck in weak local minima. We further show representative examples for λ = 16 in the last two rows of Table 1, which illustrate clearly the effect of both equal-size bias in (1) and the corrections we proposed in (10) and (6). It is worth noting that the error we obtained for standard likelihoods (the last column in Table 1) is significantly higher than the 8% error previously reported in the literature, e.g., [19]. The lower error in [19] is based on a different (more recent) set of tighter bounding boxes [19], where the size of the ground-truth segment is roughly half the size of the box. Therefore, the equal-size bias inÊ(S, P ) (10) for this particular set of boxes has an effect similar to the effect of true target volumes W in E W (S, P ) (6) (the first column in Table 1), which significantly improves the performance of standard likelihoods (the last column). In practice, both 50/50 boxes and true W are equally unrealistic assumptions that require knowledge of the ground truth. Fig. 5 depicts a different application, where we segment a magnetic resonance image (MRI) of the brain into multiple regions (K > 2). Here we introduce an extension of E(S, P ) using a positive factor γ that weighs the contribu- tion of entropy against the other terms: E γ (S, P ) = E(S, P ) + γ\|Ω\|H(S).(17) This energy could be written as K k=1 \|S k \|KL(I k \|P k ) + \|Ω\|H(S\|I) + (γ − 1)\|Ω\|H(S)) using the high-order decomposition of likelihoods E(S, P ) from [10] presented in the intro. Thus, the bias introduced by H(S) has two cases: γ ≤ 1 (volumetric equality bias) and γ ≥ 1 (volumetric disparity bias), as discussed below. We used the Chan-Vese data term [6], which assumes the appearance models in E(S, P ) are Gaussian distributions: − log P k (I p ) c = (I p − µ k ) 2 /2σ 2 , with µ k the mean of intensities within segment S k and σ is fixed for all segments. We further added a standard total-variation term [20] that encourages boundary smoothness. The solution is sought following the bound optimization strategy we discussed earlier; See Fig. 2. The algorithm alternates between two iterative steps: (i) optimizing a bound ofÊ γ (S, P ) w.r.t segmentation S via a continuous convexrelaxation technique [20] while model parameters are fixed, and (ii) fix segmentation S and update parameters µ k and w k using current solution. We set the initial number of models to 5 and fixed λ = 0.1 and σ = 0.05. We run the method for γ = 0, γ = 1 and γ = 3. Fig. 5 displays the results using colors encoded by the region means obtained at convergence. Column (a) demonstrates the equal-size bias for γ = 0; notice that the yellow, red and brown components have approximately the same size. Setting γ = 1 in (b) removed this bias, yielding much larger discrepancies in size between these components. In (c) we show that using large weight γ in energy (17) has a sparsity effect; it reduced the number of distinct segments/labels from 5 to 3. At the same time, for γ > 1, this energy introduces disparity bias; notice the gap between the volumes of orange and brown segments has increased compared to γ = 1 in (b), where there was no volumetric bias. This disparity bias is opposite to the equality bias for γ < 1 in (a). Geometric model fitting Energy minimization methods for geometric model fitting problems have recently gained popularity due to [9]. Similarly to segmentation these methods are often driven by a maximum likelihood based data term measuring model fit to the particular feature. The theory presented in Section 2 applies to these problems as well and they therefore exhibit the same kind of volumetric bias. Figures 1 (b) shows a simple homography estimation example. Here we captured two images of a scene with two planes and tried to fit homographies to these (the right image with results is shown in Figure 1). For this image pair γ < 1 γ = 1 γ > 1 image equality bias no bias disparity bias (a) (b) (c) Figure 5. Segmentation using energy (17) combined with a standard total-variation regularization [20]. We used the Chan-Vese model [6] as appearance term and bound optimization to compute a local minimum of the energy (See Fig. 2). At each iteration, the bound is optimized w.r.t segmentation using the convex-relaxation technique in [20]. Initial number of models: 5. λ = 0.1, σ = 0.05. Upper row (from left to right): image data and the results for γ = 0, γ = 1 and γ = 3. Lower row: histograms of the number of assignments to each label and the entropies obtained at convergence. SIFT [15] generated 3376 matches on the larger plane (paper and floor) and 135 matches on the smaller plane (book). For a pair of matching points I p = {x p , y p } we use the log likelihood costs p∈S k − log w k · P H k ,Σ k (I p ) ,(18) where P H k ,Σ k (I p ) = 1 (2π) 2 √ \|Σ k \| e − 1 2 d H k ,Σ k (xp,yp) 2 and d H k ,Σ k is the symmetric mahalanobis transfer distance. The solution to the left in Figure 1 (b) was generated by optimizing over homographies and covariances while keeping the priors fixed and equal (w 1 = w 2 = 0.5). The volume bias makes the smaller plane (blue points) grab points from the larger plane. For comparison Figure 1 (b) also shows the result obtained when reestimating w 1 and w 2 . Note that the two algorithms were started with the same homographies and covariances. Figure 6 shows an independently computed 3D reconstruction using the same matches as for the homography experiment. Multi Model Fitting Recently discrete energy minimization formulations have been shown to be effective for geometric model fitting tasks [9,8]. These methods effectively handle regularization terms needed to produce visually appealing results. The where V (S) = (p,g)∈N V pq (S p , S q ) is a smoothness term and L(S) is a label cost preventing over fitting by penalizing the number of labels. The data term D(S, W, Θ) = − k Sp=k log(w k ) + P (m p \|Θ k ) (20) consists of log-likelihoods for the observed measurements m p , given the model parameters Θ. Typically the prior distributions w k are ignored (which is equivalent to letting all w k be equal) hence resulting in a bias to equal partitioning. Because of the smoothness and label cost terms the bias is not as evident in practical model fitting applications as in k-means, but as we shall see it is still present. Multi model fitting with variable priors presents an additional challenge. The PEARL (Propose, Expand And Reestimate Labels) paradigm [9] naturally introduces and removes models during optimization. However, when reestimating priors, a model k that is not in the current labeling will have w k = 0 giving an infinite log-likelihood penalty. Therefore a simple alternating approach (see bound optimization in Sec.2) will be unable to add new models to the solution. For sets of small cardinality it can further be seen that the entropy bound in Figure 2 will become prohibitively large since the derivative of the entropy function is unbounded (when approaching w = 0). Instead we use α-expansion moves with higher order interactions to handle the entropy term, as described in Section 2. Figure 7 shows the result of a synthetic line fitting experiment. Here we randomly sampled points from four lines with different probabilities, added noise with σ = 0.025 and added outliers. We used energy (19) without smoothness and with label cost h times the number of labels (excluding the outlier label). The model parameters Θ consist of line location and orientation. We treated the noise level for each line as known. Although the volume bias seems to manifest itself more clearly when the variance is reestimated, it is also present when only the means are estimated. Using random sampling we generated 200 line proposals to be used by both methods (fixed and variable W). Figure 7 (c), (d) and (e) show the results with fixed W for three different strengths of label cost. Both the label cost and the entropy term want to remove models with few assigned points. However, the label cost does not favor any assignment when it is not strong enough to remove a model. Therefore it cannot counter the volume bias of the standard data term favoring more assignments to weaker models. In the line fitting experiment of Figure 7 we varied the strength of the label cost (three settings shown in (c), (d) and (e)) without being able to correctly find all the 4 lines. Reestimation of W in Figure 7 (f) resulted in a better solution. Figures 8 and 9 show the results of a homography estimation problem with the smoothness term V (S). For the smoothness term we followed [9] and created edges using a Delauney triangulation with weights e −d 2 /5 2 , where d is the distance between the points. For the label costs we used h = 100 with fixed W and h = 5 with variable W . We fixed the model variance to 5 2 (pixels 2 ). The two solutions are displayed in Figure 8 and Figure 9 shows a histogram of the number of assigned points to each model (black corresponds to the outlier label). Even though smoothness and label costs mask it somewhat, the bias to equal volume can be seen here as well. Conclusions We demonstrated significant artifacts in standard segmentation and reconstruction methods due to bias to equal size segments in standard likelihoods (1) following from the general information theoretic analysis [10]. We proposed binary and multi-label optimization methods that either (a) remove this bias or (b) replace it by a KL divergence term for any given target volume distribution. Our general ideas apply to many continuous or discrete problem formulations. these examples in details. Figure 2 . 2(Entropy -bound optimization) According to(7,10) energy EW t (S, P ) is a bound forÊ(S, P ) since cross entropy H(S\|Wt) is a bound for entropy H(S) with equality at S = St. Figure 3 . 3(Entropy -high order optimization) (a) polygonal approximation for −z log z. (b) "triangle" functions decomposition. 25} Figure 4 . 25}4for W = U (c) for W = {0.04, 0.96} (d) for W = {0.75, 0.Equal volumes bias KL(S\|U ) versus target volumes bias KL(S\|W ). Grey histogram is a distribution of intensities for the ground truth liver segment including normal liver tissue (the main mode), blood vessels (the small mode on the right), and cancer tissue (the left mode). (a) Initial (normalized) histograms for two liver parts. Initial segmentation shows which histogram has larger value for each pixel's intensity. (b) The result of optimizing energy (15). The solid blue and green histograms at the bottom row are for intensities at the corresponding segments. (c-d) The results of optimizing energy (16) for fixed weights W set for specific target volumes. Figure 6 . 63D-Geometry of the book scene inFigure 1 (b). typical objective functions are of the type E(S, W, Θ) = V (S) + D(S, W, Θ) + L(S), Figure 7 . 7Line fitting: (a) data generated from three lines, (b) data with outliers, (c) fixed W and h = 100, (d) fixed W and h = 200, (e) fixed W and h = 300, (f) variable W and h = 5. Figure 8 . 8Homography fitting: fixed (left) and variable W (right). Figure 9 . 9Histogram of the number of assignments to each label (model) inFigure 8. Fixed W (left) and variable W (right). Table 1. Comparisons on the GrabCut data set.Energy E W (S, P ) (6) true target volumes WÊ (S, P ) (10) high-order optimizationÊ (S, P ) (9) bound optimization E(S, P ) (1) standard likelihoods Overall Error (50 images) 5.29% 7.87% 13.41% 14.41% Examples error: 4.75% error: 2.29% error: 6.85% error: 4.95% error: 9.64% error: 41.20% error: 14.69% error: 40.88% http://research.microsoft.com/en-us/um/cambridge/projects /visionimagevideoediting/segmentation/grabcut.htm5 The data set comes with two boxes enclosing the foreground segment for each image. We used the outer bounding box to restrict the image domain and the inner box to compute initial appearance models. On the Detection of Multiple Object Instances using Hough Transforms. O Barinova, V Lempitsky, P Kohli, IEEE conference on Computer Vision and Pattern Recognition (CVPR). O. Barinova, V. Lempitsky, and P. Kohli. On the Detection of Multiple Object Instances using Hough Transforms. 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In International Conference on Computer Vision (ICCV), 2009. 1, 4 Fast Approximate Energy Minization with Label Costs. A Delong, A Osokin, H Isack, Y Boykov, International Journal of Computer Vision (IJCV). 961A. Delong, A. Osokin, H. Isack, and Y. Boykov. Fast Ap- proximate Energy Minization with Label Costs. Interna- tional Journal of Computer Vision (IJCV), 96(1):1-27, Jan- uary 2012. 1, 2, 3, 7 Energy-based Geometric Multi-Model Fitting. H N Isack, Y Boykov, International Journal of Computer Vision (IJCV). 972H. N. Isack and Y. Boykov. Energy-based Geometric Multi- Model Fitting. International Journal of Computer Vision (IJCV), 97(2):123-147, April 2012. 6, 7, 8 An Information-Theoretic Analysis of Hard and Soft Assignment Methods for Clustering. M Kearns, Y Mansour, A Ng, Thirteenth Conference on Uncertainty in Artificial Intelligence (UAI). M. Kearns, Y. Mansour, and A. Ng. An Information- Theoretic Analysis of Hard and Soft Assignment Methods for Clustering. 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[ "Quantum Clock Synchronization with a Single Qudit", "Quantum Clock Synchronization with a Single Qudit" ]	[ "Armin Tavakoli \nPhysics Department\nStockholm University\nS-10691StockholmSweden\n", "Adán Cabello \nDepartamento de Física Aplicada II\nUniversidad de Sevilla\nE-41012SevillaSpain\n", "Marek Żukowski \nInstytut Fizyki Teoretycznej i Astrofizyki\nUniwersytet Gdański\nPL-80-952GdańskPoland\n", "Mohamed Bourennane \nPhysics Department\nStockholm University\nS-10691StockholmSweden\n" ]	[ "Physics Department\nStockholm University\nS-10691StockholmSweden", "Departamento de Física Aplicada II\nUniversidad de Sevilla\nE-41012SevillaSpain", "Instytut Fizyki Teoretycznej i Astrofizyki\nUniwersytet Gdański\nPL-80-952GdańskPoland", "Physics Department\nStockholm University\nS-10691StockholmSweden" ]	[]	Clock synchronization for nonfaulty processes in multiprocess networks is indispensable for a variety of technologies. A reliable system must be able to resynchronize the nonfaulty processes upon some components failing causing the distribution of incorrect or conflicting information in the network. The task of synchronizing such networks is related to Byzantine agreement (BA), which can classically be solved using recursive algorithms if and only if less than one-third of the processes are faulty. Here we introduce a nonrecursive quantum algorithm, based on a quantum solution of the detectable BA, which achieves clock synchronization in the presence of arbitrary many faulty processes by using only a single quantum system. I n many multiprocess networks, including data transfer networks, telecommunications networks, the global positioning system, and long baseline interferometry, the individual processes need to have clocks that must be synchronized with one another 1,2 . To this purpose, individual processes' clocks must periodically be resynchronized. This motivates the need for clock synchronization algorithms which work despite the faulty behavior by some of the processes. Faulty behavior can occur due to a variety of causes, including crashing, transmission failure, and distribution of incorrect or inconsistent information in the network 3 . A clock synchronization algorithm should achieve the following tasks: C1) For any given instant, the time of all nonfaulty processes' clocks must be the same. This is necessary, but not sufficient, since simply stopping all clocks at zero satisfies C1. We therefore need to assume that a process' logical clock also keeps the rate of its corresponding physical clock. In addition, synchronizing may cause further errors, so we require that: C2) There is a small bound on the amount that a process' clock is changed during synchronization 4 .Reliable clock synchronization algorithms can be complicated. To simplify the problem we shall work under the following assumptions 4 : A1) Initially, all clocks are synchronized to the same value. Physical clocks typically do not keep perfect time but drift with respect to one another. This motivates the following assumption: A2) All nonfaulty processes' clocks run at one second in clock time per second in real time. A general problem arises from the clocks continuously changing during the synchronization procedure. Unless the synchronization algorithm is very fast, this will cause problems. This motivates our last assumption: A3) A nonfaulty process can read the time difference between the clock of another process and its own.A method to achieve synchronization is to use interactive consistency algorithms (ICAs) in which all nonfaulty processes reach a mutual agreement about all the clocks 4 . An ICA should satisfy that, for every process p: (1) Any two nonfaulty processes obtain the same value of process p's clock, even if p is faulty. (2) If p is nonfaulty, then every nonfaulty process obtains the value of p's clock. The synchronization problem can classically be solved using recursive algorithms if and only if less than one-third of the clock are faulty.The conditions for an ICAs are similar to the ones of the problem of Byzantine Agreement (BA) in the case of which: (i) All nonfaulty processes obtain the same value and (ii) if process p is nonfaulty, then all nonfaulty processes obtain the value it sends 4,5 . Nevertheless, it has been shown that even quantum methods cannot solve the BA if one-third or more of processes are faulty 6 .However, for most applications, including clock synchronization, it is sufficient to consider a scenario called detectable Byzantine agreement (DBA) or detectable broadcast 7,8 . In this case, conditions (i) and (ii) are replaced with: (i9) either all nonfaulty processes obtain the same value or all abort, and (ii9) if process p is nonfaulty, then either every nonfaulty process obtains the same value or aborts. By ''abort'' we mean treating the value as undefined and exiting the protocol.Classical ICAs can only achieve DBA if less than one-third of the processes are faulty 4 and agreement is achieved by majority voting using a recursive algorithm, called OM(n), where n is the number of faulty processes. The OM(n) algorithm works as follows. We label the processes as P k , with k 5 1, 2, …, m. If n 5 0, then P 1 OPEN SUBJECT AREAS: QUANTUM INFORMATION SINGLE PHOTONS AND QUANTUM EFFECTS	10.1038/srep07982	null	9,861,364	1501.05758	680c670b7ec065a8d0c65e6530b2266f74d46a41	Quantum Clock Synchronization with a Single Qudit Published 23 January 2015 Armin Tavakoli Physics Department Stockholm University S-10691StockholmSweden Adán Cabello Departamento de Física Aplicada II Universidad de Sevilla E-41012SevillaSpain Marek Żukowski Instytut Fizyki Teoretycznej i Astrofizyki Uniwersytet Gdański PL-80-952GdańskPoland Mohamed Bourennane Physics Department Stockholm University S-10691StockholmSweden Quantum Clock Synchronization with a Single Qudit Published 23 January 201510.1038/srep07982Received 10 November 2014 Accepted 24 December 2014Correspondence and requests for materials should be addressed to M.B. ([email protected]. se) SCIENTIFIC REPORTS \| 5 : 7982 \| Clock synchronization for nonfaulty processes in multiprocess networks is indispensable for a variety of technologies. A reliable system must be able to resynchronize the nonfaulty processes upon some components failing causing the distribution of incorrect or conflicting information in the network. The task of synchronizing such networks is related to Byzantine agreement (BA), which can classically be solved using recursive algorithms if and only if less than one-third of the processes are faulty. Here we introduce a nonrecursive quantum algorithm, based on a quantum solution of the detectable BA, which achieves clock synchronization in the presence of arbitrary many faulty processes by using only a single quantum system. I n many multiprocess networks, including data transfer networks, telecommunications networks, the global positioning system, and long baseline interferometry, the individual processes need to have clocks that must be synchronized with one another 1,2 . To this purpose, individual processes' clocks must periodically be resynchronized. This motivates the need for clock synchronization algorithms which work despite the faulty behavior by some of the processes. Faulty behavior can occur due to a variety of causes, including crashing, transmission failure, and distribution of incorrect or inconsistent information in the network 3 . A clock synchronization algorithm should achieve the following tasks: C1) For any given instant, the time of all nonfaulty processes' clocks must be the same. This is necessary, but not sufficient, since simply stopping all clocks at zero satisfies C1. We therefore need to assume that a process' logical clock also keeps the rate of its corresponding physical clock. In addition, synchronizing may cause further errors, so we require that: C2) There is a small bound on the amount that a process' clock is changed during synchronization 4 .Reliable clock synchronization algorithms can be complicated. To simplify the problem we shall work under the following assumptions 4 : A1) Initially, all clocks are synchronized to the same value. Physical clocks typically do not keep perfect time but drift with respect to one another. This motivates the following assumption: A2) All nonfaulty processes' clocks run at one second in clock time per second in real time. A general problem arises from the clocks continuously changing during the synchronization procedure. Unless the synchronization algorithm is very fast, this will cause problems. This motivates our last assumption: A3) A nonfaulty process can read the time difference between the clock of another process and its own.A method to achieve synchronization is to use interactive consistency algorithms (ICAs) in which all nonfaulty processes reach a mutual agreement about all the clocks 4 . An ICA should satisfy that, for every process p: (1) Any two nonfaulty processes obtain the same value of process p's clock, even if p is faulty. (2) If p is nonfaulty, then every nonfaulty process obtains the value of p's clock. The synchronization problem can classically be solved using recursive algorithms if and only if less than one-third of the clock are faulty.The conditions for an ICAs are similar to the ones of the problem of Byzantine Agreement (BA) in the case of which: (i) All nonfaulty processes obtain the same value and (ii) if process p is nonfaulty, then all nonfaulty processes obtain the value it sends 4,5 . Nevertheless, it has been shown that even quantum methods cannot solve the BA if one-third or more of processes are faulty 6 .However, for most applications, including clock synchronization, it is sufficient to consider a scenario called detectable Byzantine agreement (DBA) or detectable broadcast 7,8 . In this case, conditions (i) and (ii) are replaced with: (i9) either all nonfaulty processes obtain the same value or all abort, and (ii9) if process p is nonfaulty, then either every nonfaulty process obtains the same value or aborts. By ''abort'' we mean treating the value as undefined and exiting the protocol.Classical ICAs can only achieve DBA if less than one-third of the processes are faulty 4 and agreement is achieved by majority voting using a recursive algorithm, called OM(n), where n is the number of faulty processes. The OM(n) algorithm works as follows. We label the processes as P k , with k 5 1, 2, …, m. If n 5 0, then P 1 OPEN SUBJECT AREAS: QUANTUM INFORMATION SINGLE PHOTONS AND QUANTUM EFFECTS distributes its value to every other process. Every process uses the value received from P 1 and, in case no value is obtained, uses 0. If n . 0, then P 1 distributes its value to every other process. For k 5 2, …, m, let x k denote the value obtained by P k from P 1 . If P k receives no message, then let x k 5 0. P k acts as P 1 in algorithm OM(n 2 1) by distributing x k to the remaining m 2 2 processes. For every k and for all j ? k, let x j be the value received by P k from P j using OM(n 2 1), and in case no value was received x j 5 0. P k decides on the value obtained from the median of (x 1 , …, x m ). Thus, OM(n) requires O(m n11 ) transmitted messages to solve the task. The DBA is an example of a communication task for which quantum resources can provide a solution, while classical tools cannot. Nevertheless, the special case of DBA in a three process network with one faulty process, has been solved using quantum methods based on three-qutrit singlet states 7,9 , four-qubit entangled states 10 , and three 8 or two 12 pairwise quantum key distribution (QKD) channels, and experimentally demonstrated using four photon-polarization entangled state 11 . Interestingly, later works have shown that there are quantum solutions for certain communication complexity problems and secret sharing tasks which do not require entanglement, but, instead, sequential communication of a single quantum system 13,14 . These protocols have been shown to be much more resistant to noise and imperfections, and significantly more scalable than protocols based on entanglement. In this article, we introduce a quantum algorithm that solves the DBA and achieves clock synchronization in the presence of an arbitrary number of faulty processes, with only one single round of message passing per process independently of the number of faulty processes, utilizing only a single quantum system. In order to solve the DBA problem, the m processes need to share data in the form of lists l k , of numbers subject to specific correlations, and the distribution must be such that the list l k held by process P k is known only by P k . Quantum mechanics provides methods to generate and securely distribute such data, here we shall seek for one which is simple, efficient, and easily extendible to an arbitrary number of processes. We assume that all processes can communicate with one another with oral messages by pairwise authenticated error-free classical channels and pairwise authenticated quantum channels. Correlated lists and their use The initial stage of the quantum protocol is to distribute lists l k , for k 5 1, …, m, each of them available only to process P k . All lists have to be of the same length L and are required to satisfy the property that if N 5 0 (or 1) is at position j in l 1 , then 0 (respectively, 1) is at position j in lists l k for k 5 2, …, m (i.e., they are perfectly correlated). However, if N g {2, …, m 2 1} is at position j in l 1 , then the sum of numbers at positions j in lists l k for k 5 2, …, m equals m 2 N, and all elements in these lists are either 0 or 1. Given an N, all the possible combinations of binary numbers satisfying the condition are uniformly probable. Note that, on one hand, P 1 has information about at which positions the lists of all other processes the values are perfectly correlated, and at which positions they are random bits, with the property that their sum is anticorrelated with the value, N $ 1, in l k . On the other hand, the holder of one the lists l k , with k 5 2, …, m, has no information whatsoever on whether the lists are correlated at a given position or not. Once the processes have these lists, they can use them to achieve mutual agreement and solve the DBA by applying the algorithmic part of the protocol, which we shall call QB(n, m). The special case, QB(1, 3), reproduces the protocol in 11 . (1) P 1 sends bit-valued messages to all processes. The message sent to process P k will be denoted by m 1,k . Together with each message, P 1 sends a list l 1,k of all of the positions in l 1 in which the value m 1,k appears. If P 1 is nonfaulty all lists and messages are identical. The full information which P k receives from P 1 will be denoted by {m 1,k , l 1,k }. (2) The receiving processes P k analyze (singlehandedly) the obtained lists and messages. If the analysis of P k shows that l 1,k is of appropriate length (i.e., about L/m) and {m 1,k , l 1,k } is consistent with l k at all positions, then if P k is nonfaulty, it conveys {m 1,k , l 1,k } to all other processes P k?1 . A faulty process sends a flipped bit value of the message with whatever list it chooses. The full information which P j receives from P k will be denoted by {m k,j , l k,j }. A nonfaulty P k will also decide on the final bit value it adopts V k . This is m 1,k , unless messages from the other processes force it to decide that P 1 is faulty. However, if {m 1,k , l 1,k } is not consistent with l k , then P k immediately ascertains that P 1 is faulty and relays to other processes neither 0 nor 1 but H, meaning ''I have received inconsistent data.'' (3) Once all messages have been exchanged between P 2 , …, P m , each process considers the obtained data and acts according to the instructions in Table 1. The overall aim is, if P 1 is nonfaulty, to have the same value of V k for all nonfaulty processes, or all of them aborting. Quantum protocol for distributing lists l k All processes are equipped with devices which can unitarily transform qudits. In addition, P 1 has a source of single qudits of dimension m and the last process, P m , has additionally a measurement device. The protocol runs as follows (for an illustration, see Fig. 1): (I) P 1 prepares the state y 0 j i~1ffiffiffiffi m p X m{1 j~0 j j i:ð1Þ (II) P 1 randomly chooses the ''encoding basis'' from m different options U 0 ,...,U m21 and labels the choice c 1 . Having chosen the c 1 'st encoding basis, process P 1 applies the following unitary transformation to the qudit: U c1 ¼ j0ih0j þ X mÀ1 k¼1 v c1 jkihkj;ð2Þ where v~e i 2p m . From the interferometric point of view, applying U c 1 introduces a phase-shift of 22pc 1 /m in the first beam. (III) After that, P 1 randomly chooses a value N 1 in the set {0, 1, …, m 2 1} and encodes N 1 , by applying the following unitary transformation: U N 1 ð Þ~X m{1 j~0 v jN1 j j i j h j:ð3Þ Afterwards, the qudit is sent to P 2 . (IV) P 2 , in the same manner as P 1 , choses a c 2 g{0,...,m21} and applies the unitary U c 2 corresponding to choice of encoding basis. (V) Next, P 2 randomly chooses a value N 2 in the set {0, 1}. If N 2 5 0, no action is taken, i.e., P 2 applies the transformation U N 2~0 ð Þ~. If N 2 5 1, then P 2 applies U(N 2 5 1) and then sends the qudit to P 3 . and, if X m k~1 N k~0 , modulo m, then P m k~1 U N k ð Þ~:ð5Þ Whenever this condition is not satisfied, the final state of the system is orthogonal to jy 0 ae and will therefore never be an outcome of P m 's measurement. Clock synchronization Now, we will show how to apply our method for solving the DBA to achieve fault tolerant clock synchronization. However, a problem arises from clocks ticking during the synchronization procedure. This is solved by exploiting assumption A3: Instead of sending a number, the processes send their clock differences to each other. In the classical case, we achieve clock synchronization by running the algorithm OM(1) m times, sending clock differences instead of the binary values, and analogously for OM(n) 4 . In analogy with the classical case, the processes send clock differences also in the quantum case, exploiting the fact that the clock differences can be decomposed into binary strings up to arbitrary accuracy agreed upon in advance. We run QB(n, m) m times in such a way that for each run a new processes takes the roll of P 1 in QB(n, m). More explicitly, P y reads the clock difference D xy between its own clock and the clock of P x . If P y is nonfaulty it will relay D xy to P z but if P y is a faulty process, it can arbitrarily change D xy before sending it. If P y relays the value obtained from P x to P z , then P z knows the time difference between P x and P y . Also, since QB(n, m) is ran m times, P z will also obtain D yz from P y and thus P z knows that P y is claiming that the time difference between P x and P z is D xy 1 D yz , which can then be compared to D xz obtained directly from P x . Comparison with the other solutions The correlated lists needed for achieving DBA can be distributed by other means than with the single-qudit protocol. Successful distribution can be achieved by the process P m sharing a QKD channel with every other process. P m uses a QKD protocol, e.g., BB84 17 to distribute numbers such that (1) P m and P 1 share a string All QKD channels except that shared between P 1 and P m transmit bit values. In order to transmit elements of {0, …, m 2 1} to P 1 , the numbers must be encoded into qlog 2 m ð Þr qubits. One additional requirement that has to be made for solving the DBA using the QKD distributed lists is that P m is not required to convey any lists. This is necessary since P m has full knowledge about the lists of all other processes and therefore easily could cheat. Instead, P m may announce the message it received from P 1 , and if any inconsistency is noted by P 2 , …, P m21 , then P m will change its final value if the other processes convince P m of them being nonfaulty. There are also other proposed solutions to the DBA considering three processes where one is faulty. The first one, proposed in Ref. 7, relies on the three qutrit entangled Aharonov state. The goal is to distribute lists given by all permutations of the elements of the set {0, 1, 2}, i.e., (0-1-2, 0-2-1, 1-0-2, 1-2-0, 2-0-1, and 2-1-0). Table 1 \| Once P k receives all messages and lists from all other processes, it will study the obtained lists and messages and compare to its own list l k . Depending on the consistency between the obtained and private data, P k will act according to table below. Notation m j,k ,l j,k È É %l k means that m j,k and l j,k are found to be consistent with l k whereas 6% means ''inconsistent with.'' The symbol H means ''I have received inconsistent data.'' By M k we denote some nonempty subset of {1, …, m} \ {k} local analysis of all data received by P k decision of P k on the value V k (iia) Vj[N m \ k f g, m j,k ,l j,k È É %l k and all messages are equal V k 5 m 1,k , no faulty process (iib) Vj[N m \ k f g, m j,k ,l j,k È É %l k and not all messages are equal as P 1 is faulty, V k 5 abort (iic) Vj[M k , m j,k ,l j,k È É 6%l k and Vj 6 [M k , m j,k ,l j,k È É %l k V k 5 m j,k , for j 6 [ M k ,k m j i~1 ffiffiffiffiffi ffi m! p X i~s Sm ð Þ {1 ð Þ N s Sm ð Þ ð Þ i 1 , . . . ,i m j i ,ð6Þ where i~i 1 , . . . ,i n f g , S m 5 {0, …, m 2 1} and N(s(S m )) is the parity of the permutation of S m . Already for the simplest case of m 5 3, this approach requires the preparation of a very complex state which, to our knowledge, has not yet been experimentally realized. However, for the three process case, it has been pointed out in Ref. 12 that the distribution of the lists can be realized without the state (6), by utilizing two separated QKD channels. With small modification for the m process setting, distribution of the lists is achieved with m 2 1 QKD channels. However, to encode the entire space provided by S m , the QKD requires qlog 2 m ð Þr qubits. If the efficiency of a detector g is not perfect and the QKD is performed with single qubits using von Neuman measurements, successful distribution occurs only with probability g m{1 ð Þ qlog 2 m ð Þr . Typically, the classical part of the protocol in Ref. 7 and its possible generalizations scale rapidly with the number of processes. It is required that m! different types of lists are distributed. However, a solution to the three party DBA exploiting four-qubit entanglement provides a simpler classical part of the protocol: the number of different lists is lowered from six to four 11 . The general m process protocol presented in this paper generalizes the protocol in Ref. 11 and requires 2 m21 different types of lists. As emphasized earlier, the distribution of the required lists can be achieved both with single-qudit and with m 2 1 QKD channels. Using QKD channels, only one channel needs to transmit all elements in S m while the remaining m 2 2 channels only transmit bit values. In the presence of nonperfect detectors, successful distribution occurs with probability g m{2zqlog 2 m ð Þr . However, in the single-qudit approach only one single detection is needed and, therefore, successful distribution of the lists occur with probability g independently of m. The single-qudit protocol is highly scalable, both in terms of success probability with inefficient detectors and requirements on the classical lists. Conclusions We have presented a single-qudit protocol which provides an efficient solution to an important multiparty communication problem: It solves DBA and achieves clock synchronization in the presence of arbitrary many faulty clocks. In principle, our quantum algorithm is not limited to the case of clock synchronization, it can with small adaptation be used for other tasks requiring oral message interactive consistency. Interestingly, our algorithm works by transmitting a single qudit among the parties rather than by distributing a quantum entangled state among them. This makes the protocol much more practical, as single qudits can be experimentally realized easily in many ways. For example, using unbiased multiport beamsplitters 15 or time-bin 16 . Compared to schemes based on several QKD channels, the single-qubit protocol is more scalable and robust against detection inefficiencies. This results shows that single-qudit quantum information protocols are interesting beyond QKD 18,19 and random number generation 20,21 , and should stimulate experimental implementations and further research in quantum information protocols. (VI) P 3 , …, P m consecutively repeat the same procedure as P 2 with independent choices of basis and encoding their respective random values N 3 , …, N m . (VII) In addition, P m measures the qudit using a device which distinguishes the state jy 0 ae from any set of states orthogonal to it.www.nature.com/scientificreports SCIENTIFIC REPORTS \| 5 : 7982 \| DOI: 10.1038/srep07982(VIII) If P m obtains jy 0 ae, then the processes consecutively reveal their encoding bases (but not their values N k ) in reverse order: First P m and last P 1 . If it turns out that the sum of the basis choices modulo m equals zero, then the run is treated as a valid distribution of the numbers N k at the same position in the private lists l k .The protocol distributes the numbers in the required way because all the unitary operators are diagonal and, therefore, commute. . ( 4 ) 4m [ 0, . . . ,m{1 f g .(2) For every l 5 2, …, m 2 1, P m and P l share a string K lNone of P 2 , …, P m21 have any information about a particular list element of any other process. (5) Whenever P 1 receives an element k j 1,m §2, P 1 has no information on the bit value of k j l,m for l 5 2, …, m, and whenever P m~p for all l 5 2, …, m. Figure 1 \| 1as the other P j 's are faulty(iid) Vj[M k , m j,k ,l j,k È É %l k and \Vj 6 [M k V k 5 m 1,k , although P 1 could be faulty(iie)Vj[M k , m j,k ,l j,k È É %l k , but with unequal messages, and H from Vj 6 [M k V k 5 abort, at least P 1 is faulty Scheme of the quantum protocol for the distribution of the correlated lists. P 1 prepares a uniform d-level superposition state, makes a choice of basis and encoding, and forwards the qudit to P 2 which applies a choice a basis and encoding and forwards the qudit to P 3 . Processes P 3 , …, P m act in analogy with P 2 . Finally P m projects the state onto the initial state prepared by P 1 and, if the outcome is 1, the processes reveal their bases and, if all bases are the same, the round is treated as valid. SCIENTIFIC REPORTS \| 5 : 7982 \| DOI: 10.1038/srep07982 AcknowledgmentsThis project was supported by the Swedish Research Council, ADOPT, the Project No. FIS2011-29400 (MINECO, Spain) with FEDER funds, the FQXi large grant project ''The Nature of Information in Sequential Quantum Measurements,'' MNiSW Grant No. IdP2011 000361 (Ideas Plus) and Foundation for Polish Science TEAM project co-financed by the EU European Regional Development Fund.Author contributionsA.C., M.Z. and M.B. proposed and initiated the project. 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[ "HEAT: Hyperedge Attention Networks", "HEAT: Hyperedge Attention Networks" ]	[ "Dobrik Georgiev ", "Marc Brockschmidt ", "Miltiadis Allamanis [email protected] ", "\nDepartment of Computer Science and Technology\nUniversity of Cambridge\nUK\n", "\nMicrosoft Research\nCambridgeUK\n", "\nMicrosoft Research\nCambridgeUK\n" ]	[ "Department of Computer Science and Technology\nUniversity of Cambridge\nUK", "Microsoft Research\nCambridgeUK", "Microsoft Research\nCambridgeUK" ]	[]	Learning from structured data is a core machine learning task. Commonly, such data is represented as graphs, which normally only consider (typed) binary relationships between pairs of nodes. This is a substantial limitation for many domains with highly-structured data. One important such domain is source code, where hypergraph-based representations can better capture the semantically rich and structured nature of code.In this work, we present HEAT, a neural model capable of representing typed and qualified hypergraphs, where each hyperedge explicitly qualifies how participating nodes contribute. It can be viewed as a generalization of both message passing neural networks and Transformers. We evaluate HEAT on knowledge base completion and on bug detection and repair using a novel hypergraph representation of programs. In both settings, it outperforms strong baselines, indicating its power and generality.	null	[ "https://export.arxiv.org/pdf/2201.12113v2.pdf" ]	246,411,285	2201.12113	7428f9b16a82839e2cb6e6c7a77c1ffeab898813	HEAT: Hyperedge Attention Networks Dobrik Georgiev Marc Brockschmidt Miltiadis Allamanis [email protected] Department of Computer Science and Technology University of Cambridge UK Microsoft Research CambridgeUK Microsoft Research CambridgeUK HEAT: Hyperedge Attention Networks Published in Transactions on Machine Learning Research (09/2022) Reviewed on OpenReview: https: // openreview. net/ forum? id= gCmQK6McbR Learning from structured data is a core machine learning task. Commonly, such data is represented as graphs, which normally only consider (typed) binary relationships between pairs of nodes. This is a substantial limitation for many domains with highly-structured data. One important such domain is source code, where hypergraph-based representations can better capture the semantically rich and structured nature of code.In this work, we present HEAT, a neural model capable of representing typed and qualified hypergraphs, where each hyperedge explicitly qualifies how participating nodes contribute. It can be viewed as a generalization of both message passing neural networks and Transformers. We evaluate HEAT on knowledge base completion and on bug detection and repair using a novel hypergraph representation of programs. In both settings, it outperforms strong baselines, indicating its power and generality. Introduction Large parts of human knowledge can be formally represented as sets of relations between entities, allowing for mechanical reasoning over it. Common examples of this view are knowledge graphs representing our environment, databases representing business details, and first-order formulas describing mathematical insights. Such structured data hence regularly appears as input to machine learning (ML) systems. In practice, this very generic framework is not easy to handle in ML models. One issue is that the set of relations is not necessarily known beforehand, and that the precise structure of a relation is not easily fixed. As an example, consider studied(person:P , institution:I, major:M ), encoding the fact that a person P studied at institution I, majoring in M . However, if the institution is unknown, we may want to just consider studied(person:P , major:M ), or we may need to handle case of people double-majoring, using studied(person:P , major:M 2 , major:M 1 ). Existing ML approaches usually cast such data as hypergraphs, and extend the field of graph learning approaches to this setting, but struggle with its generality. Some solutions require to know the set of relations beforehand (fixing their arity, and assigning fixed meanings to each parameter), while others abstract the set of entities to a simple set and forego the use of the name of the relation and its parameters. Figure 1: Representation of relation ρ(τ 1 :n 1 , τ 2 :n 2 , τ 3 :n 3 ) as typed and qualified hyperedge e, i.e. a relation of type ρ, and the qualified participation of the nodes n 1 as τ 1 , n 2 as τ 2 , and n 3 as τ 3 in e. HEAT operates on hypergraphs with such hyperedges. One form of data that can profit from being modeled as a hypergraph is program source code. Existing approaches model code either as simple sequence of tokens (Hindle et al., 2012) or as a graph (Allamanis et al., 2018b;Hellendoorn et al., 2020). While the former can easily leverage successful techniques from the NLP domain, it is not possible to include additional domain-specific knowledge (such as the flow of data) in the input. On the other hand, graph models are able to take some of this additional information, but struggle to represent more complex relationships that require hyperedges. In this work, we propose a new architecture, HEAT (HyperEdge ATtention), that is able to handle an open set of relations that may appear in several variants. To this end, we combine the idea of message passing schemes that follow the graph structure with the flexibility and representational power of Transformer architectures. Concretely, our model handles arbitrary relations by presenting each one as a separate sequence in a Transformer, where an idea akin to standard positional encodings is used to represent how each entity participates in the relation. The Transformer output is then used to update the representations of participating entities. We illustrate the success of this technique in two very different settings: learning to infer additional relations in knowledge graphs, and learning to find and repair bugs in programs. In both cases, our model shows improvements over strong state-of-the-art methods. Concretely, we (a) define HEAT as a novel hypergraph neural network architecture (Sec. 2), (b) define a novel representation of programs as hypergraphs (Sec. 3), (c) evaluate HEAT on the tasks of detecting and repairing bugs (Sec. 4.1) and link prediction in knowledge graphs (Sec. 4.2). Our implementation of the HEAT model is available on the heat branch of https://github.com/microsoft/ neurips21-self-supervised-bug-detection-and-repair/tree/heat. This includes code for the extraction of hypergraph representations of Python code as discussed in Sec. 3. The HEAT Model In this work, we are interested in representing typed and qualified relations of the form relName(parName1:n 1 , parName2:n 2 , ...) where relName represents the name of a relation, n 1 , n 2 , . . . are entities participating in the relation, with parName1, . . . describes (qualifies) their role in the relation. Formally, such relations can be represented as typed and qualified hypergraphs: we consider a set of nodes (entities) N = {n 1 , . . .} and a set of hyperedges H = {e 1 , . . .}. Each hyperedge e = (ρ, {(τ 1 , n 1 ) , . . . , (τ k , n k )}) describes a named relationship of type ρ among nodes n 1 . . . n k where the role of node n k within e is qualified using τ k . Fig. 1 illustrates the form of such a hyperedge. Note that this is in contrast to traditional hypergraphs where nodes participate in a hyperedge without any qualifiers. Instead, typed and qualified graphs can accurately represent a large range of domains maintaining valuable information. Background Message passing neural networks (MPNN) (Gilmer et al., 2017) operate on sets of nodes N and sets of (typed) edges E, where each edge (n i , τ i , n j ) of type τ connects a pair of nodes. In MPNNs, each node n ∈ N is associated with representations h (t) n that are computed incrementally. The initial h (0) n stems from input features, but subsequent representations are computed by exchanging information between nodes. Optionally, each edge e may also be associated with a state/representation h (t) e that is also computed incrementally. Concretely, each edge gives rise to a "message" using a learnable function f m : m (t) (ni,τ,nj ) = f m h (t) ni , τ, h (t) nj , h (t) e . To update the representation of a node n, all incoming messages are aggregated with a permutation-invariant function Agg(·) into a single update, i.e. u (t) nj = Agg m (t) (ni,τ,nj ) \| (n i , τ, n j ) ∈ E . (1) Agg is commonly implemented as summation or max-pooling, though attention-based variants exist (Veličković et al., 2018). Finally, each node's representation is updated using its original representation and the aggregated messages. Many different update mechanisms have been considered, ranging from just using the aggregated messages (Kipf & Welling, 2017) to gated update functions . Transformers (Vaswani et al., 2017) learn representations of sets of elements N without explicit edges. Instead, they consider all pairwise interactions and use a learned attention mechanism to identify particularly important pairs. Transformer layers are split into two sublayers. First, multihead attention (MHA) uses the representations of all N to compute an "update" u n for each n ∈ N , i.e. u (t) n1 , . . . , u (t) n k = MHA h (t) n1 , . . . , h (t) n k .(2) Internally, MHA uses an attention mechanism to determine the relative importance of pairs of entities and to compute the updated representations. Note that MHA treats its input as a (multi-)set, and hence is permutation-invariant. To provide ordering information, a common approach is to use positional encodings, i.e. to extend each representation with explicit information about the position. In practice, this means that MHA operates on h (t) n1 ⊕ p 1 , . . . , h(t) n k ⊕ p k , where p i provides information about the position of n i , and ⊕ is a combination operation (commonly element-wise addition). The second Transformer sublayer is used to combine the updated representations with residual information and add computational depth to the model. This is implemented as q (t) ni = LN h (t) ni + u (t) ni (3) h (t+1) ni = LN q (t) ni + FFN q (t) ni ,(4) where LN is layer normalisation (Ba et al., 2016) and FFN is a feedforward neural network (commonly with one large intermediate hidden layer). HEAT: An Attention-Based Hypergraph Neural Network We now present HEAT, a Transformer-based message passing hypergraph neural network that learns to represent typed and qualified hypergraphs. Overall, we follow the core idea of the standard message passing paradigm: our aim is to update the representation of each entity using messages arising from its relations to other entities. Following Battaglia et al. (2018), we also explicitly consider representations of each hyperedge, based on their type and the representation of adjacent entities. Intuitively, we want each entity to "receive" one message per hyperedge it participates in, reflecting both its qualifier (i.e. role) in that hyperedge, as well as the relation to other participating entities. h (t) e h (t) n1 r (t) τ1 h (t) n2 r (t) τ2 h (t) n3 r (t) τ3 ⊕ ⊕ ⊕ Multihead Attention m (t) e m (t) e→n1 m (t) e→n2 m (t) e→n3 (a) Message computation for hyperedge e from Fig. 1. The previous node states h (t) n i are combined with their respective qualifier embeddings r (t) τ i . Then, a multihead attention mechanism computes the messages m To compute these messages, we borrow ideas from the Transformer architecture. We view each qualified hyperedge as a set of entities, with each entity being associated with a position (its qualifier in the hyperedge), and then use multihead attention -as in Eq. 2 -to compute one message per involved entity: m (t) e , m (t) e→n1 , . . . , m (t) e→n k = MHA h (t) e , r (t) τ1 ⊕ h (t) n1 , . . . , r (t) τ k ⊕ h (t) n k .(5) Here, h (t) e is the current representation of the hyperedge, h (t) ni is the current representation of node n i , r (t) τi is the representation (embedding) of the qualifier τ i which denotes the role of n i in e, and ⊕ combines two vectors. Fig. 2a illustrates this. In this work, we consider element-wise addition for ⊕ but other operators, such as vector concatenation would be possible. We can then compute a single update u (t) ni for node n i by aggregating the relevant messages computed for the hyperedges it occurs in, i.e. u (t) ni = Agg m (t) e→ni \| e = (ρ, {. . . , (τ i , n i ), . . .}) ∈ H . This is equivalent to the aggregation of messages in MPNNs (cf. Eq. 1). We then compute an updated entity representation h (t+1) ni following the Transformer update as described in Eqs. 3, 4. This is illustrated in Fig. 2b. The core difference to the Transformer case is the aggregation Agg(·) used to combine all messages (as in a message passing network). Agg(·) can be any permutation invariant function, such as elementwise-sum and max, or even an another transformer (see Sec. 4.1 for experimental evaluation of these variants). Finally, the hyperedge states h (t) e are also updated as in Eqs. 3, 4 using the messages m (t) e computed as in Eq. 5, and no aggregation is required because there is only a single message for each hyperedge e. Initial node states h (0) ni are initialised with node-level information (as in GNNs). Qualifier embeddings r τi (resp. initial hyperedge states h (0) e ) are obtained by breaking the name of τ i (resp. ρ e ) into subtokens (e.g., parName is split into par and name or foo_bar2 into foo, bar, and 2), embedding these through a (learnable) vocabulary embedding matrix, and then sum-pooling. Then for each HEAT layer r (t) τi is computed through a linear learnable layer of the sum-pooled embedding, i.e. r (t) τi = W (t) τ r τi + b (t) τ . Generalising Transformers Note that if the hypergraph is made up of a single hyperedge of the form Seq(pos1:n 1 , pos2:n 2 , ...) then HEAT degenerates to the standard transformer of Vaswani et al. (2017) with positional encodings over the sequence n 1 , n 2 , . . . at each layer. In the case of such sequence positions, we simply use the sinusoidal positional encodings of Vaswani et al. (2017) as position embedding, rather than using a learnable embedding. Thus HEAT can be thought to generalise transformers from sets (and sequences) to richer structures. Computational Considerations A naïve implementation of a HEAT message passing step requires one sample per hyperedge to be fed into a multihead attention layer, as each hyperedge of k nodes gives rise to a sequence of length k + 1, holding the edge and all node representations. To enable efficient batching, this would require to pad all such sequences to the longest sequence present in a batch. However, the length of these sequences may vary wildly. On the other hand, processing each hyperedge separately would not make use of parallel computation in modern GPUs. To resolve this, we use two tricks. First, we consider "microbatches" with a pre-defined set of sequence lengths {16, 64, 256, 768, 1024}, minimising wastage from padding. Second, we "pack" several shorter sequences into a single sequence, and then use appropriate attention masks in the MHA computation to avoid interactions. For example, a hyperedge of size 9 and a hyperedge of size 7 can be joined together to yield a sequence of length 16. Appx. A details the packing algorithm used. This process is performed in-CPU during the minibatch preparation. Representing Code as Hypergraphs Program source code is a highly structured object that can be augmented with rich relations obtained from program analyses. Traditionally, source code is represented in machine learning either as a sequence of tokens (Hindle et al., 2012), a tree (Yin & Neubig, 2017), or as a graph (Allamanis et al., 2018b;Hellendoorn et al., 2020). Graph-based representations (subsuming tree-based ones) commonly consider pairwise relationships among entities in the code, whereas token-level ones simply consume a token sequence. In this section, we present a novel hypergraph representation of code that retains the best of both representations. Our program hypergraph construction is derived from the graph construction presented by Allamanis et al. (2021), but uses hyperedges to produce a more informative and compact representation. The set of nodes in the generated hypergraphs include tokens, expressions, abstract syntax tree (AST) nodes, and symbols. However, in contrast to prior work, we do not only consider pairwise relationships, but instead use a typed and qualified hypergraph (Fig. 1). We detail the used hyperedges and their advantages next. Fig. 3 in Appx. B illustrates some of the considered relations on a small synthetic snippet. Tokens For the token sequence t 1 , t 2 , ..., t L of a snippet of source code, we create the relation Tokens(p1:t 1 , p2:t 2 , ...). This is the entire information used by token-level Transformer models, considering all-to-all relations among tokens. Note that in standard graph-based code representations, the token sequence is usually represented using a chain of NextToken binary edges, meaning that long-distance relationships are hard to discover for models consuming such graphs. For very long token sequences, which may cause memory issues, we "chunk" the sequence into overlapping segments of length L, similar to windowed, sparse attention approaches. We use L = 512 in the experiments. Within HEAT and specifically for the Tokens(·) hyperedges, to reduce the used parameters and to allow arbitrarily long sequences, the qualifier embeddings r p1 , r p2 , r p3 , ... are computed using the fixed sinusoidal embeddings of Vaswani et al. (2017) instead of learnable embeddings. AST We represent the program's abstract syntax tree using AstNode relations, with qualifiers corresponding to the names of children of each node. For example, an AST node of a binary operation is represented as AstNode(node:n BinOp , left:n left , op:n op , right:n right ). Similarly, a AstNode(node:n IfStmt , cond:n c , then:n t , else:n e ) represents an if statement node n IfStmt . In cases where the children have the same qualifier and are ordered, (e.g. the sequential statements within a block) we create numbered relations, i.e. AstNode(node:n Block , s1:n 1 , s2:n 2 , ...). For Python, we use as qualifier names those used in libCST. In contrast, most graph-based approaches use Child edges to connect a parent node to all of its children, and thus lose information about the specific role of child nodes. As a consequence, this means that left and right children of non-commutative operations can not easily be distinguished. Allamanis et al. (2021) attempts to rectify this using a NextSibling edge type, but still is not able to make use of the known role of each child. Control and Data Flow To indicate that program execution can flow from any of the AST nodes p 1 , p 2 , ... to one of the AST nodes s 1 , s 2 , ..., we use the relation CtrlF(prev:n p1 , prev:n p2 , ..., succ:n s1 , succ:n s2 ). For example, if c: p1 else: p2; s1 would yield CtrlF(prev:p1, prev:p2, succ:s1). Similarly, we use relations MayRead and MayWrite to represent dataflow, where the previous location at which a symbol may have been read (written to) are connected to the succeeding locations. Note that these relations compactly represent data and control flow consolidating N -to-M relations (e.g. N states may lead to one of M states) into a single relation, which would otherwise require N · M edges in a standard graph. Symbols We use the relation Symbol(sym:n s , occ:n 1 , occ:n 2 , ..., may_last_use:n u1 , may_last_use:n u2 , ... ) to connect all nodes n 1 , n 2 , ... referring to a symbol (e.g. variable) to a fresh node n s , introduced for each occurring symbol. We annotate within this relation the nodes n u1 , n u2 , ... that are the potential last uses of the symbol within the code snippet. Functions A challenge in representing source code is how to handle calls to (potentially user-defined) functions. As it is usually not feasible to include the source code of all called functions in the model input (for computational and memory reasons), appropriate abstractions need to be used. For a call (invocation) of a function foo(arg1, ..., argN) defined by def foo(par1, ..., parN): ..., we introduce the relation foo(rval:n, par1:n arg1 , ..., par3:n argN ) where n is the invocation expression node and n arg1 , . . . , n argN are the nodes representing each of the arguments. Hence, we generate one relation symbol for each defined function, and match nodes representing arguments to the formal parameter names as qualifiers. This representation allows to naturally handle the case of variable numbers of parameters and arbitrary functions. Syntactic sugar and operators are converted in the same way, using the name of the corresponding built-in function. For example, in Python, a in b is converted into the relation __contains__(self:n b , item:n a ) and a -= b is converted into __isub__(self:n a , other:n b ) following the reference Python data model. Finally, we use the relation Returns(fn:n f , from:n 1 , from:n 2 , ...) to connect all possible return points for a function with the AST node n f for the function definition. Similarly, a Yields(·) is defined for generator functions. Evaluation We evaluate HEAT on two tasks from the literature: bug detection and repair (Allamanis et al., 2021) and knowledge base completion (Galkin et al., 2020). We implemented it as a Py-Torch (Paszke et al., 2019) Module, available on the heat branch of https://github.com/microsoft/ neurips21-self-supervised-bug-detection-and-repair/tree/heat. HEAT for Bug Detection & Repair We evaluate HEAT on the bug localisation and detection task of Allamanis et al. (2021) in the supervised setting. This is a hard task that requires combining ambiguous information with reasoning capabilities able to detect bugs in real-life source code. (1) we adapt the graph construction from programs to produce hypergraphs as discussed in Sec. 3, and (2) use HEAT to compute entity representations from the generated graphs, rather than GNNs or GREAT. Dataset We use the code of Allamanis et al. (2021) to generate a dataset of randomly inserted bugs to train and evaluate a neural network in a supervised fashion. Consequently, we obtain a new variant of the "Random Bugs" test dataset, consisting of ∼ 760k graphs. We additionally re-extract the PyPIBugs dataset with the provided script, generating hypergraphs as consumed by HEAT, and graphs generated by the baseline models. However, since the PyPIBugs dataset is provided in the form of GitHub URLs referring to the buggy commit SHAs, some of them have been removed from GitHub and thus our PyPIBugs dataset contains 2354 samples, 20 less than the one used by Allamanis et al. (2021). Model Architecture We modify the architecture of Allamanis et al. (2021) to use 6 HEAT layers with hidden dimension of 256, 8 heads, feed-forward (FFN in Eq. 4) hidden layer of 2048, and dropout rate of 0.1. As discussed above, our datasets differ slightly from the data used by Allamanis et al. (2021), and we re-evaluated their released code on our new datasets. We found this rerun to perform notably better than what was originally reported in the paper. In private communication, the authors explained that their public code included a small change to the model architecture compared to the paper: the subnetworks used for selecting a program repair rule are now shallow MLPs (rather than inner products), which increases performance across the board. Our HEAT extension follows the code release, and hence we use max-pooled subtoken embeddings to initialise node embeddings, a pointer network-like submodel to select which part of the program to repair, and a shallow MLPs to select the repair rules. We also re-use the PyBugLab supervised objective, which is composed of two parts: a PointerNet-style objective requiring to identify the graph node corresponding to a program bug, and a ranking objective requiring to select a fixing program rewrite at the selected location. Results We show the results of our experiments in Table 1, where "Loc." refers to the accuracy in identifying the buggy location in an input program, "Repair" to the accuracy in determining the correct fix given the buggy location, and "Joint" to solving both tasks together. The results indicate that HEAT improves performance on both considered datasets, improving the joint localisation and repair accuracy by ∼ 10% over the two well-tuned baselines. In particular, we observe that HEAT substantially improves over GREAT (Hellendoorn et al., 2020), which also adapts the Transformer architecture to include relational information. However, GREAT eschews explicit modelling of edges, and instead uses (binary) relations between tokens to bias the attention weights in the MHA computation. We observe a less pronounced gain over GNNs, which we believe is due to the simpler information flow across long distances and the clearer way of encoding structural information in HEAT. (see Sec. 3) We note that Allamanis et al. (2021) showed that their models also outperform fine-tuned variants of the cuBERT (Kanade et al., 2020) model, which stems from self-supervised pre-training using masking objectives. Consequently, we believe that HEAT outperforms such large models as well, though a comparison to recent very large models, adapted to the task (Chen et al., 2021;Austin et al., 2021;Li et al., 2022) is left to future work. Variations and Ablations To understand the importance of different components of HEAT, we experiment with five ablations and variations, shown on the bottom of Table 1. First, we study the importance of using multi-head attention to compute messages. In particular, we are interested in determining whether considering interactions between different nodes participating in a hyperedge is necessary. To this end, we consider an alternative scheme in which we first compute a hyperedge representation using aggregation of all adjacent nodes, using their qualifier information, i.e. In our experiments, we use a Deep Set (Zaheer et al., 2017) model to implement Agg . We then compute messages for each node n i using a single linear layer W , i.e. q (t) e = Agg h (t) e , r τ1 ⊕ h (t) n1 , r τ2 ⊕ h (t) n2 , ... .m (t) e→ni = ReLU W · [r τi ⊕ h (t) ni , q (t) e ] . The results indicate that this model variant is still stronger than the GNN and GREAT architectures, but that HEAT profits from explicitly considering the relationships of nodes participating in a hyperedge. Next, we analyse the importance of using qualifier information in our model. To this end, we consider a variant of HEAT in which we remove from Eq. 5 the qualifier information r τi , i.e. to m (t) e , m (t) e→n1 , m (t) e→n2 , ... = MHA h (t) e , h (t) n1 , h (t) n2 , ... . The results clearly indicate that the qualifier-as-position scheme used in HEAT is crucial for good performance. In particular, it indicates that the qualifier information contained in the data is very valuable, and emphasises the importance of considering qualified hypergraphs. We also considered using a more expressive aggregation mechanism for messages, replacing the max pooling we use to implement Agg. Specifically, we use a multi-head attention between the current node state (as queries in the attention mechanism) and the messages computed for all adjacent hyperedges (appearing as keys and values). This is reminiscent of graph attention networks (Veličković et al., 2018), which use an attention mechanism to determine the respective importance of binary edges when aggregating messages. In our experiments, this performed slightly worse than the aggregation using a simple max, while being substantially more memoryand compute-intensive. Next, we analyse whether the representational capacity is improved by the feedforward network (Eq. 4) in the node and edge update. Removing these substantially reduces the number of parameters of the model. The results indicate that these intermediate, per-representation computation steps add valuable capacity to the model. This is especially apparent on the PyPIBugs dataset. Another ablation we consider is a model variant in which we do not use evolving edge states, but instead re-use h (0) e on all HEAT layers. We observe a similar or slightly reduced performance on the joint localisation and repair. This suggests that updating the edge state provides valuable representational capacity to the model. Edge states, can be seen as the [CLS] token in traditional transformers, providing "scratch space" for storing intermediate information for each hyperedge. Finally, we assess the importance of the underlying data representation, investigating whether our representation of programs as hypergraphs (differing from the PyBugLab baseline) alone explains the performance improvement. For this, we convert the generated hyperedges to (binary) edges and use HEAT to learn from these (binary) graphs. This experiment intents to disentagle the effects of the different data representation from the effects of HEAT's architecture. The results, shown in Table 2, indicate that even on binarised hypergraphs, HEAT outperforms the baseline GNN and GREAT models (especially on accuracy of choosing program repairs), even though large parts of its architecture are not properly utilised by the data. However, HEAT also takes advantage of the more expressive and compact data representation. HEAT for Knowledge Graph Completion Knowledge graphs (KGs) can be accurately represented as typed and qualified hypergraphs. We now focus on link prediction over a hyper-relational KG, which can be viewed as completing a knowledge base by additional likely facts. Dataset Following the discovery of test leaks and design flaws by Galkin et al. (2020) in common benchmark datasets such as WikiPeople (Guan et al., 2019) and JF17K (Wen et al., 2016) we chose one of the variations of the new WD50K dataset presented there -WD50K (100). It is derived from Wikidata, containing 31k statements, all of which use some qualifier. To model a qualified triple statement of the form (s, r, o, Q) with Q a set of qualifier/entity pairs (qr i , qv i ), we create a single hyperedge with special qualifiers src (resp.) obj for the source and object of the relation, i.e. (r, {(src, s) , (obj, o)} ∪ Q). Using the example of Fig. 1 of Galkin et al. (2020), the fact that Einstein studied mathematics at ETH Zurich in his undergraduate is expressed as the hyperedge: EducatedAt(src:n Einstein , obj:n ETH Zurich , degree:n Bachelor , major:n Mathematics ). In the released WD50K dataset, raw Wikidata identifiers (e.g. P69, Q937, etc.) are used to refer to entities and relation names. We enrich the dataset by additionally retrieving natural language information for these entities from Wikidata (e.g. replacing P69 by "educated at") and allow the models to consume this information, to encourage similar treatment of similarly named relationships and entities. Model Architecture We modified the open-source release of StarE by Galkin et al. (2020), replacing the GNN-based StarE encoder by HEAT. We used a single layer of HEAT with embedding size of 100 and a single layer of the Transformer used for calculating the final predictions 1 . Apart from some training parameters (see below), hyperparameters (dropout, etc.) remain as documented in Appendix C of Galkin et al. (2020). Since our goal is to evaluate the effectiveness of HEAT, the remainder of the model is identical to the original implementation of Galkin et al. (2020): queries for a relation j are represented as the concatenation of the entity embeddings, relation embedding, and qualifier embeddings, as shown in Fig. 3 of Galkin et al. (2020). These are then passed through a Transformer block and a fully-connected layer to obtain the probability over each entity being a possible object of the relation. To process the additional natural language information about relations and entities we extracted (see above), we build a vocabulary using byte pair encoding (BPE), and then embed the individual tokens and use sum pooling to obtain initial node and relation representations. We evaluated variations of both the original StarE model and our HEAT-based variant using this information, see below. Training and evaluation Training is performed as in Galkin et al. (2020) using binary cross entropy with label smoothing. In this link prediction task, a matching score is calculated for all possible relation objects given source, relation and qualifier-entity pairs (Dettmers et al., 2018, p.3). We trained our model for 1k epochs with a learning rate of 0.0004 and batch size of 512. Hyperparameters were fine-tuned manually, using the provided validation set in the StarE implementation. For direct comparison with Table 3 of Galkin et al. (2020) we report mean reciprocal rank (MRR) and hits at 1 and 10 (H@1, H@10) matching the original evaluation setting. We train and evaluate all models using 5 random seeds and report standard deviations. Table 3 shows the results of our evaluation on WD50K (100). We reran the original StarE implementation, both to validate our setup and to obtain standard deviations. First, we observe that using HEAT instead of the original GNN-based StarE encoder improves results, without any further changes to the architecture. We expect that recent orthogonal work improving StarE by Yu & Yang (2021) would similarly improve with our model. Next, we consider the influence of using natural language information extracted from Wikidata. We note that the original StarE encoder does not profit from this information. In contrast, the HEAT-based model slightly improves results, even though most words in the extracted data are extremely sparse. Finally, we evaluate the less expressive message aggregation scheme of max pooling (as discussed in Sec. 4.1). Here, we see that its performance is only marginally worse. Galkin et al. (2020). Standard deviations are obtained over 5 different seeds. Results Model MRR H@1 H@10 StarE ( Hy-Transformer (Yu & Yang, 2021) 0.699 0.637 0.812 † Rerun of Galkin et al. (2020) implementation with 5 seeds. * Current state-of-the-art. Related Work We review closely related techniques for learning on hypergraphs, and then briefly discuss some particularly relevant work from the application areas of learning on code and knowledge bases. Hypergraph learning We broadly classify hypergraph learning into three approaches: hyperedge expansion, spectral methods, and spatial (message passing-based) methods. Compared to HEAT, none of the existing methods can directly work on typed and qualified hypergraphs. In the first class, hypergraphs are transformed into (binary) graphs and then a standard GNN is applied on the resulting graph. This approach is taken by Agarwal et al. (2006), who use clique and star expansion (representing each hyperedge as a full or a star graph respectively), Yadati et al. (2019), who represent each hyperedge by a simple weighted edge whose endpoints can be further connected with weighted edges to mediator nodes and Yang et al. (2020), who create a node for each pair of incident nodes and hyperedges and connect those stemming from the same node or hyperedge. These approaches can re-use existing GNN architectures, but at the cost of a significantly increased number of edges. The next class of methods aims at generalising the concepts of graph Laplacians and spectral convolutions to the domain of hypergraphs. Feng et al. (2019) use the observations that nodes in the same hyperedge should not differ much in embedding to define a normalised hypergraph Laplacian. A similar approach has been employed by Fu et al. (2019), who utilise the hypergraph p-Laplacian. Bai et al. (2021) extend Feng et al. (2019 where the node-edge incidence matrix is weighted via an attention mechanism. Such methods are usually applied to transductive learning tasks and either do not fully support typed or qualified hyperedges or are limited to a fixed number of hyperedge types or nodes in a hyperedge. This makes them inapplicable in our setting. The final class of methods generalises the concept of neural message passing to hypergraphs. HyperSAGNN (Zhang et al., 2020) is a self-attention based neural network, capable of handling variable-sized hyperedges, but has been developed for the purposes of hyperedge prediction and is not directly applicable for node classification. HyperSAGE (Arya et al., 2020) performs convolution in two steps: nodes to hyperedges and hyperedges to nodes, where the aggregation during the node and hyperedge message passing step is a powermean function, but it suffers from poor parallelisation and other practical issues (Huang & Yang, 2021, p. 2). Chien et al. (2021) propose AllSet, a message passing scheme based on representing hyperedges as sets and aggregations using permutation-invariant functions. AllSet can provably subsume a substantial number of previous hypergraph convolution methods. Chien et al. (2021) propose two implementations of AllSet, using DeepSets (Zaheer et al., 2017) and Transformers (Vaswani et al., 2017). This work is most similar to ours, but does not consider the setting of qualified hyperedges. A natural consequence is that the model then computes a single "message" per hyperedge, whereas HEAT computes different messages for each participating node (which only is required when nodes play different roles in the hyperedge). In Sec. 4.1, we consider two relevant ablations. These show that the use of qualifier information is crucial for good results in the PyBugLab setting, and that our message computation based on multihead attention is stronger than a DeepSet-based alternative. To the best of our knowledge, we are the first to focus on processing typed and qualified hyperedges: two hyperedges with the same element set can be different if their type does not match and elements in a hyperedge can have different qualifiers (roles). Furthermore, our architecture is not restricted to a fixed number of hyperedge types/qualifier and can generalise to types/qualifiers unseen during train time. Knowledge Graph Completion Knowledge graph (KG) completion has emerged due to the incompleteness of KGs (Ji et al., 2021). Embedding-based models (Bordes et al., 2013;Schlichtkrull et al., 2018;Shi & Weninger, 2017) first learn a low-dimensional embedding and then use it to calculate scores based on these embeddings and rank the top k candidates. HEAT is a variation of an embedding-based model, but we focus on hyper-relational KGs. Other, non-embedding-based approaches also exist, e.g. reinforcement learning (Xiong et al., 2017) or rule-based ones (Rocktäschel & Riedel, 2017). Since we do not use such techniques, we omit discussing them here and refer the reader to Ji et al. (2021, §IV.A). Similarly to hypergraph learning, other works model hyper-relational KGs by simplifying qualified relations to simpler representations: Wen et al. (2016) use clique expansion, which can be costly, (Fatemi et al., 2020) represent hyper-relational facts as n-ary relations, but do not have explicit source/object of the relation and instead of considering the qualifier of an entity as in HEAT, their model only considers the position of an entity within the relation; Guan et al. (2019) break n-ary facts into n + 2 qualifier/entity pairs 2 , making qualifier/entity pairs indistinguishable to "standard" (s, rel, o) triples. These expansion-based approaches cannot leverage semantic information such as the interaction of different qualifier/entity pairs (Yu & Yang, 2021). Closest to our work is StarE (Galkin et al., 2020). StarE uses a GNN-like convolution, consisting of several steps (cf. equations (5)-(7) of Galkin et al., 2020) on hyper-relational graphs to calculate updated entity embeddings, which are then fed through a Transformer module. In Sec. 4 we show that HEAT outperforms their GNN-based encoder. Learning on Code Over the last decade, machine learning has been applied to code on a variety of tasks (Allamanis et al., 2018a). A central theme in this research area is the representation of source code. Traditionally, token-level representations have been used (Hindle et al., 2012), but Allamanis et al. (2018b) showed that graph-level approaches can leverage additional semantic information for substantial improvements. Subsequently, Fernandes et al. (2019); Hellendoorn et al. (2020) showed that combining token-level and graphlevel approaches yields best results. By using a relational transformer model over the code tokens Hellendoorn et al. (2020) overcomes the inability of GNN models to handle long-range interactions well, while allowing to make use of additional semantic information expressed as graph edges over the tokens. However, token-based representations do not provide unambiguous locations for annotating semantic information (i.e. edges) for non-token units such as expressions (e.g. a+b or a+b+c). Additionally, all these approaches have been limited to standard (binary) edges, making the resulting graphs large and/or imprecise (see Sec. 3). Our experiments with HEAT show that a suitable representation as typed and qualified hypergraph further improves over the combination of token-level and binary graph-level approaches. Conclusions & Discussion We introduced HEAT, a neural network architecture that operates on typed and qualified hypergraphs. To model such hypergraphs, HEAT combines the idea of message passing with the representational power of Transformers. Furthermore, we showed how to convert program source code into such hypergraphs. Our experiments show that HEAT is able to learn well from these highly structured hypergraphs, outperforming strong recent baselines on their own datasets. A core insight in our work is to apply the power of Transformers to several, overlapping sets of relations at the same time. This allows to concurrently model sequences and graph-structured data. We believe that this opens up exciting opportunities for future work in the joint handling of natural language and knowledge bases. A Packing Hyperedges for HEAT Hyperedges in HEAT have a variable size, i.e. a variable number of nodes that participate in each hyperedge. For each hyperedge HEAT uses a multihead attention to compute the outgoing messages m ei→nj . However, multihead attention has a quadratic computational and memory cost with respect to the size of the hyperedge. A naïve solution would require us to pad all hyperedges up to a maximum size. However, this would be wasteful. On the other hand, we need to batch the computation across multiple hyperedges for efficient GPU utilisation. To achieve a good trade-off between large batch sizes (a lot of elements) and minimal padding, we consider a small set "microbatches". To pack the hyperedges optimally, we would solve the following integer linear program (ILP) min 1..k j L 2 j · y j − i l 2 i s.t. y j ∈ {0, 1}, x ij ∈ {0, 1}, ∀i, j 1..k j x ij = 1, ∀i 1..n i x ij s i ≤ y j L j , ∀j, where y j is a binary variable indicating whether "bucket" j of size L j (i.e. belonging in the microbatch of size L j ) is used (i.e. contains any elements). Then, l j is the width of the jth hyperedge. x ij are binary variables for i = 1..n and j = 1..k indicating if the element i is in bucket j and s i the width of bucket i. The objective creates as few buckets as possible to minimise the wasted (quadratic) space/time needed for multihead attention over variable-sized sequences. The first constraint requires that each element is assigned to exactly one bucket and the second constraint that each used bucket is filled up to its capacity. However, the complexity of this ILP is prohibitive. Instead, we resort in using a greedy algorithm to select the "microbatches" used. This is detailed in Algorithm 1. Sample Hyperedges (Relations) •Tokens(p1:n 1 , p2:n 2 , p3:n 3 , p4:n 4 , p5:n 5 , p6:n 7 , p7:n 8 , p8:n 9 , p9:n 10 , p10:n 11 , p11:n 12 , p12:n 13 , · · · ) •AstNode(node:n 19 , test:n 6 , body:n 17 ) •AstNode(node:n 17 , value:n 16 , target:n 9 ) •foo(rval:n 12 , fzz:n 9 ) Assuming that foo is defined as def foo(fzz): ... •__getattribute__(rval:n 23 , self:n 20 , name:n 22 ) •MayRead(prev:n 4 , prev:n 13 , succ:n 25 ) •CtrlF(prev:n 6 , prev:n 16 , succ:n 23 ) •Symbol(sym:n x , occ:n 4 , occ:n 9 , occ:n 13 , may_last_use:n 25 ) Figure 4: A full hypergraph sample for the snippet disc Hyperedges are denoted as shaded (blue) boxes. Best viewed on screen. The Tokens(·) hyperedge is omitted for clarity but the sequence of tokens is placed in the box (right). B Code as Hypergraph Example The state update for each node n in the hypergraph. Apart from Agg this matches a standard Transformer layer ofVaswani et al. (2017). Figure 2 : 2The HEAT Architecture. For this, we built on the open-source release of PyBugLab of Allamanis et al. (2021), making two changes: GNN) † (Galkin et al., 2020) 0.654±0.002 0.586±0.002 0.777±0.002 StarE (GNN) -with NL information 0.653±0.003 0.586±0.003 0.774±0.005 StarE (HEAT) -Agg max, no NL information 0.666±0.003 0.605±0.004 0.779±0.002 StarE (HEAT) -Agg CrossAtt, no NL information 0.666±0.003 0.599±0.004 0.787±0.003 StarE (HEAT) -Agg CrossAtt, with NL information 0.667±0.003 0.601±0.003 0.789±0.001 Figure 3 : 3Sample relations for the synthetic snippet shown on the top. The AST nodes and tokens are wrapped in boxes and numbered appropriately in a preorder fashion. Only a few samples of the relations (mapped to hyperedges) are shown below. Fig. 3 3shows a synthetic code snippet with all the token and AST nodes enclosed in boxes. Some sample relations are also shown. Finally,Fig. 4shows a full hypergraph for the following code snippet Table 1 : 1Evaluation Results on Supervised Bug Detection and Repair on supervised PyBugLab.Random Bugs PyPIBugs Joint Loc. Repair Joint Loc. Repair Table 2 : 2Results on the Random Bugs dataset when applying HEAT on a graph dataset representing edges as (binary) edges.Joint Localisation & Repair Localisation Repair Table 3 : 3Link prediction results on WD50K (100) of Algorithm 1 Greedy Hyperedge Packing into Microbatches buckets ← [ ] for h in sortedDescendingByWidth(hyperedges) do wasAdded ← False for bucket in buckets do if bucket.remainingSize ≥ h.width then bucket.add(h) wasAdded ← True break if not wasAdded then bucketSize ← smallestFittingMicrobatchWidth(h.width) newBucket ← createBucket(bucketSize) newBucket.add(e) buckets.append(newBucket) return GroupBucketsToMicrobatches(buckets) if 1 is_foo 2 ( 3 x 4 ) 5 6 : 7 19 [INDENT] 8 x 9 = 10 foo 11 ( 12 x 13 ) 14 15 16 17 19 [DEDENT] 18 19 y 20 . 21 bar 22 23 ( 24 x 25 ) 26 27 Figure 3ofGalkin et al. (2020), bottom rectangle. +2 for the source and object qualifiers Higher order learning with graphs. 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[ "arXiv:hep-ph/9809229v1 2 Sep 1998 THE QUARK-ANTIQUARK WILSON LOOP FORMALISM IN THE NRQCD POWER COUNTING SCHEME", "arXiv:hep-ph/9809229v1 2 Sep 1998 THE QUARK-ANTIQUARK WILSON LOOP FORMALISM IN THE NRQCD POWER COUNTING SCHEME" ]	[ "Antonio Vairo \nInstitut für Hochenergiephysik\nÖsterreichische Akademie der Wissenschaften Nikolsdorfergasse 18A-1050ViennaAustria\n" ]	[ "Institut für Hochenergiephysik\nÖsterreichische Akademie der Wissenschaften Nikolsdorfergasse 18A-1050ViennaAustria" ]	[]	The quark-antiquark interaction from the NRQCD Lagrangian is studied in the Wilson loop formalism.	10.1142/9789812793713_0032	[ "https://export.arxiv.org/pdf/hep-ph/9809229v1.pdf" ]	15,659,727	hep-ph/9809229	16ca085e04f84c7d8377a8d02d7541bbcd38b69c	arXiv:hep-ph/9809229v1 2 Sep 1998 THE QUARK-ANTIQUARK WILSON LOOP FORMALISM IN THE NRQCD POWER COUNTING SCHEME Antonio Vairo Institut für Hochenergiephysik Österreichische Akademie der Wissenschaften Nikolsdorfergasse 18A-1050ViennaAustria arXiv:hep-ph/9809229v1 2 Sep 1998 THE QUARK-ANTIQUARK WILSON LOOP FORMALISM IN THE NRQCD POWER COUNTING SCHEME The quark-antiquark interaction from the NRQCD Lagrangian is studied in the Wilson loop formalism. The NRQCD Lagrangian Large radii quarkonia (i.e. excited heavy mesons likebb andcc) cannot be described in terms of perturbative QCD with the addition of leading nonperturbative effects encoded into local condensates. 1 This happens since in this situation the gluonic correlation length cannot be considered infinitely large with respect to the other scales of the system. A successful description needs therefore a systematic inclusion of non-local condensates. A solution is provided by the so-called Wilson loop formalism. The non-local quantities are field strength insertions in the Wilson loop made up by the quark trajectories. [2][3][4] While a full relativistic QCD formulation in this formalism is still lacking, 5 such a formulation is possible for heavy quark bound states which are essentially described by non relativistic dynamics. These systems are characterized by a dynamical adimensional parameter, the quark velocity v, which is small and allows a classification of the energy scales of the problem in hard (∼ m), soft (∼ mv) and ultrasoft (∼ mv 2 ). Moreover, this provides a power counting scheme for the operators in the Lagrangian. The relation between v and the QCD parameters is unknown (for infinitely heavy quarks v coincides with α s , for realistic quarks it is the result of perturbative and nonperturbative effects) but irrelevant once the power counting scheme works. Due to the heavy quark mass m, at a scale µ between m and mv the physics is still dominated by perturbative effects. Therefore, in order to describe heavy quark bound states, it is possible to substitute the QCD Lagrangian with an effective non relativistic Lagrangian via perturbative matching at that scale. The new Lagrangian is simpler, since the hard degrees of freedom have been integrated out explicitly, but equivalent to the QCD one at a given order in α s and v. This effective Lagrangian is known as the NRQCD Lagrangian. 6 At order O(v 4 ) the NRQCD Lagrangian describing a bound state between a quark of mass m 1 and an antiquark of mass m 2 is 7,8 L = Q † 1 iD 0 + c (1) 2 D 2 2m 1 + c (1) 4 D 4 8m 3 1 + c (1) F g σ · B 2m 1 + c (1) D g D·E − E·D 8m 2 1 +ic (1) S g σ·(D×E − E×D) 8m 2 1 Q 1 + antiquark terms (1 ↔ 2) + d 1 m 1 m 2 Q † 1 Q 2 Q † 2 Q 1 + d 2 m 1 m 2 Q † 1 σQ 2 Q † 2 σQ 1 + d 3 m 1 m 2 Q † 1 T a Q 2 Q † 2 T a Q 1 + d 4 m 1 m 2 Q † 1 T a σQ 2 Q † 2 T a σQ 1 , (1) where Q j are the heavy quark fields. The coefficients c (j) 2 , c (j) 4 , .. . are evaluated at the matching scale µ for a particle of mass m j . They encode the ultraviolet regime of QCD order by order in α s . The explicit expressions and a numerical discussion can be found in. 9 The power counting rules 6 for the operators of Eq. (1) are Q ∼ (mv) 3/2 , D ∼ mv, gA 0 ∼ mv 2 , gA ∼ mv 3 , gE ∼ m 2 v 3 and gB ∼ m 2 v 4 . Four quark operators which are apparently of order v 3 are actually suppressed by additional powers in α s in the matching coefficients and the octet contributions by an additional power in v 2 on singlet states. Therefore in the following we will neglect these contributions with the exception of a term which mixes under RG transformation with the chromomagnetic operator contribution to the spin-spin potential. 10 We will call the corresponding matching coefficient d. The Wilson Loop Formalism The use of the Wilson loop formalism on the Lagrangian (1) was first suggested in. 10 Let us sketch the derivation of the heavy quark potential. The 4-point gauge invariant Green function G associated with the Lagrangian (1) is defined as G(x 1 , y 1 , x 2 , y 2 ) = 0\|Q † 2 (x 2 )φ(x 2 , x 1 )Q 1 (x 1 )Q † 1 (y 1 )φ(y 1 , y 2 )Q 2 (y 2 )\|0 , where φ(x 2 , x 1 ) ≡ exp −ig 1 0 ds (x 2 − x 1 ) µ A µ (x 1 + s(x 2 − x 1 ) ) is a Schwinger line added to ensure gauge invariance. After integrating out the heavy quark fields, G can be expressed as a quantum-mechanical path integral over the quark trajectories 5 : G(x 1 , y 1 , x 2 , y 2 ) = x 1 y 1 Dz 1 Dp 1 x 2 y 2 Dz 2 Dp 2 exp    i T /2 −T /2 dt 2 j=1 p j · z j − m j − c (j) 2 p 2 j 2m j +c (j) 4 p 4 j 8m 3 j 1 N c Tr P T s exp −ig Γ dz µ A µ (z) + i T /2 −T /2 dz 0j c (j) F g σ · B 2m j +c (j) D g D·E − E·D 8m 2 j + ic (j) S g σ·(D×E − E×D) 8m 2 j × exp i m 1 m 2 T /2 −T /2 dt g 2 d T a(1) σ (1) T a(2) σ (2) δ 3 (z 1 − z 2 ) ≡ x 1 y 1 Dz 1 Dp 1 x 2 y 2 Dz 2 Dp 2 exp    i T /2 −T /2 dt 2 j=1 p j · z j − m j − p 2 j 2m j + p 4 j 8m 3 j − i T /2 −T /2 dt U    , where the bracket means the Yang-Mills average over the gauge fields, Γ is the Wilson loop made up by the quark trajectories z 1 and z 2 and the endpoints Schwinger strings and y 0 2 = y 0 1 ≡ −T /2, x 0 2 = x 0 1 ≡ T /2. Reparameterization invariance 7 fixes c 2 = c 4 = 1. Assuming that the limit exists, we define the heavy quark-antiquark potential V as lim T →∞ T /2 −T /2 dt U. For a discussion on the existence of a quark-antiquark potential we refer to. 11,12 Expanding in v (following the power counting given in the previous section) or in the inverse of the mass we get V (r) = lim T →∞ i log W (Γ) T (2) + S (1) · L (1) m 2 1 + S (2) · L (2) m 2 2 2c + F V ′ 1 (r) + c + S V ′ 0 (r) 2r + S (1) · L (2) + S (2) · L (1) m 1 m 2 c + F V ′ 2 (r) r + S (1) · L (1) m 2 1 − S (2) · L (2) m 2 2 2c − F V ′ 1 (r) + c − S V ′ 0 (r) 2r + S (1) · L (2) − S (2) · L (1) m 1 m 2 c − F V ′ 2 (r) r + 1 8   c (1) D m 2 1 + c (2) D m 2 2   (∆V 0 (r) + ∆V E a (r)) + 1 8   c (1) F m 2 1 + c (2) F m 2 2   ∆V B a (r) + c (1) F c (2) F m 1 m 2 S (1) ·rS (2) ·r r 2 − S (1) ·S (2) 3 V 3 (r) + S (1) ·S (2) 3m 1 m 2 c (1) F c (2) F V 4 (r) − 48πα s C F d δ 3 (r) . W (Γ) ≡ P exp −ig Γ dz µ A µ (z)∆V E a (r) = 2 lim T →∞ T 0 dt E(0, 0)E(0, t) − E(0, 0) E(0, t) , ∆V B a (r) = 2 lim T →∞ T 0 dt B(0, 0)B(0, t) , r k r V ′ 1 (r) = ǫ ijk lim T →∞ T 0 dt t B i (0, 0)E j (0, t) , r k r V ′ 2 (r) = 1 2 ǫ ijk lim T →∞ T 0 dt t B i (0, 0)E j (r, t) , r i r j r 2 − δ ij 3 V 3 (r) = 2 lim T →∞ T 0 dt B i (0, 0)B j (r, t) − δ ij 3 B(0, 0)B(r, t) , V 4 (r) = 2 lim T →∞ T 0 dt B(0, 0)B(r, t) , where ≡ W (Γ 0 ) / W (Γ 0 ) . Conclusions Two comments in order to conclude. The explicit expression for the potential (2) in terms of field strength insertions in a static Wilson loop is suitable for direct lattice 13 and analytic 14 evaluations. This makes the obtained result of particular importance. Different vacuum models can be easily compared on the heavy quark potential predictions, once the Wilson loop average has been evaluated. The O(v 2 ) leading order NRQCD Lagrangian L = Q † 1 (iD 0 + ∂ 2 /2m 1 )Q 1 + antiquark part does not contribute to (2) only with a static potential (with the exception of the perturbative contribution if evaluated in Coulomb gauge). Since the corresponding Wilson loop P exp −ig Γ dz 0 A 0 (z) is a function of the non-static loop Γ, its expansion produces in general velocity dependent terms as well. 3,13,14 This is not surprising since the power counting scheme of NRQCD has to be considered as a leading order power counting scheme. An exact value in v cannot be assigned to each term of the effective Lagrangian at least before the soft and ultrasoft degrees of freedom have been disentangled. 12 An extensive analysis of the topic discussed here can be found in, 9 with particular emphasis on the relevance of the matching procedure in order to have a consistent potential in the perturbative regime. F is the non-static Wilson loop. The expansion of it around the static Wilson loop W (Γ 0 ) (Γ 0 is a r × T rectangle) gives the static potential V 0 = lim T →∞ i log W (Γ 0 ) /T plus velocity (non-spin) dependent terms. 3 S (j) and L (j) are the spin and orbital angular momentum operators of the particle j. The matching coefficients are defined as 2c ± F,S ≡ c ,S . The "potentials" V 1 , V 2 , ... are scale dependent gauge field averages of electric and magnetic field strength insertions in the static Wilson loop: . F J Yndurain, Nucl. Phys. Proc. Suppl. B. 64433F. J. Yndurain, Nucl. Phys. Proc. Suppl. B 64, 433 (1998). . L S Brown, W I Weisberger, Phys. Rev. D. 203239L. S. Brown and W. I. Weisberger, Phys. Rev. D 20, 3239 (1979); . E Eichten, F Feinberg, Phys. Rev. D. 232724E. Eichten and F. Feinberg, Phys. Rev. D 23 (1981) 2724; M E , No. 207Peskin in Proceeding of the 11th SLAC Institute. P. Mc Donough151SLAC ReportM. E. Peskin in Proceeding of the 11th SLAC Institute, SLAC Report No. 207, 151, edited by P. 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Proc. Supp. B. N. Brambilla, J. Soto and A. Vairo in preparation; N. Brambilla and A. Vairo in Proceedings of QCD'98, edited by S. Narison, to appear in Nucl. Phys. Proc. Supp. B. . Y Chen, Y Kuang, R J Oakes, Phys. Rev. D. 52264Y. Chen, Y. Kuang and R. J. Oakes, Phys. Rev. D 52, 264 (1995). G P Lepage, these Proceedings. G. P. Lepage in these Proceedings. . A Pineda, J Soto, Nucl. Phys. Proc. Suppl. B. 64428A. Pineda and J. Soto, Nucl. Phys. Proc. Suppl. B 64, 428 (1998); J Soto, these Proceedings. J. Soto in these Proceedings. . G S Bali, K Schilling, A Wachter, Phys. Rev. D. 562566G. S. Bali, K. Schilling and A. Wachter, Phys. Rev. D 56, 2566 (1997). . N Brambilla, A Vairo, Phys. Rev. D. 553974N. Brambilla and A. Vairo, Phys. Rev. D 55, 3974 (1997).	[]
[ "Common Origin of µ−τ and CP Breaking in Neutrino Seesaw, Baryon Asymmetry, and Hidden Flavor Symmetry", "Common Origin of µ−τ and CP Breaking in Neutrino Seesaw, Baryon Asymmetry, and Hidden Flavor Symmetry" ]	[ "Hong-Jian He [email protected]†[email protected] \nInstitute of Modern Physics and Center for High Energy Physics\nTsinghua University\n100084BeijingChina\n\nCenter for High Energy Physics\nPeking University\n100871BeijingChina\n\nKavli Institute for Theoretical Physics China\nChinese Academy of Sciences\n100190BeijingChina\n", "Fu-Rong Yin \nInstitute of Modern Physics and Center for High Energy Physics\nTsinghua University\n100084BeijingChina\n" ]	[ "Institute of Modern Physics and Center for High Energy Physics\nTsinghua University\n100084BeijingChina", "Center for High Energy Physics\nPeking University\n100871BeijingChina", "Kavli Institute for Theoretical Physics China\nChinese Academy of Sciences\n100190BeijingChina", "Institute of Modern Physics and Center for High Energy Physics\nTsinghua University\n100084BeijingChina" ]	[]	We conjecture that all CP violations (both Dirac and Majorana types) arise from a common origin in neutrino seesaw. With this conceptually attractive and simple conjecture, we deduce that µ−τ breaking shares the common origin with all CP violations. We study the common origin of µ−τ and CP breaking in the Dirac mass matrix of seesaw Lagrangian (with right-handed neutrinos being µ−τ blind), which uniquely leads to inverted mass-ordering of light neutrinos. We then predict a very different correlation between the two small µ−τ breaking observables θ 13 − 0 • and θ 23 − 45 • , which can saturate the present experimental upper limit on θ 13 . This will be tested against our previous normal mass-ordering scheme by the on-going oscillation experiments. We also analyze the correlations of θ 13 with Jarlskog invariant and neutrinoless ββ-decay observable. From the common origin of CP and µ−τ breaking in the neutrino seesaw, we establish a direct link between the low energy CP violations and the cosmological CP violation for baryon asymmetry. With these we further predict a lower bound on θ 13 , supporting the on-going probes of θ 13 at Daya Bay, Double Chooz and RENO experiments. Finally, we analyze the general model-independent Z 2 ⊗ Z 2 symmetry structure of the light neutrino sector, and map it into the seesaw sector, where one of the Z 2 's corresponds to the µ−τ symmetry Z µτ 2 and another the hidden symmetry Z s 2 (revealed in our previous work) which dictates the solar mixing angle θ 12 . We derive the physical consequences of this Z s 2 and its possible partial violation in the presence of µ−τ breaking (without or with neutrino seesaw), regarding the θ 12 determination and the correlation between µ−τ breaking observables.PACS numbers: 14.60. Pq, 12.15.Ff, 13.15.+g, 13.40.Em Phys. Rev. D 84 (2011) 033009 and arXiv:1104.2654 *	10.1103/physrevd.84.033009	[ "https://arxiv.org/pdf/1104.2654v4.pdf" ]	56,276,831	1104.2654	9dc8cca1e35ff9eb40884748e0a83869d58d897e	Common Origin of µ−τ and CP Breaking in Neutrino Seesaw, Baryon Asymmetry, and Hidden Flavor Symmetry 17 Apr 2012 Hong-Jian He [email protected]†[email protected] Institute of Modern Physics and Center for High Energy Physics Tsinghua University 100084BeijingChina Center for High Energy Physics Peking University 100871BeijingChina Kavli Institute for Theoretical Physics China Chinese Academy of Sciences 100190BeijingChina Fu-Rong Yin Institute of Modern Physics and Center for High Energy Physics Tsinghua University 100084BeijingChina Common Origin of µ−τ and CP Breaking in Neutrino Seesaw, Baryon Asymmetry, and Hidden Flavor Symmetry 17 Apr 2012 We conjecture that all CP violations (both Dirac and Majorana types) arise from a common origin in neutrino seesaw. With this conceptually attractive and simple conjecture, we deduce that µ−τ breaking shares the common origin with all CP violations. We study the common origin of µ−τ and CP breaking in the Dirac mass matrix of seesaw Lagrangian (with right-handed neutrinos being µ−τ blind), which uniquely leads to inverted mass-ordering of light neutrinos. We then predict a very different correlation between the two small µ−τ breaking observables θ 13 − 0 • and θ 23 − 45 • , which can saturate the present experimental upper limit on θ 13 . This will be tested against our previous normal mass-ordering scheme by the on-going oscillation experiments. We also analyze the correlations of θ 13 with Jarlskog invariant and neutrinoless ββ-decay observable. From the common origin of CP and µ−τ breaking in the neutrino seesaw, we establish a direct link between the low energy CP violations and the cosmological CP violation for baryon asymmetry. With these we further predict a lower bound on θ 13 , supporting the on-going probes of θ 13 at Daya Bay, Double Chooz and RENO experiments. Finally, we analyze the general model-independent Z 2 ⊗ Z 2 symmetry structure of the light neutrino sector, and map it into the seesaw sector, where one of the Z 2 's corresponds to the µ−τ symmetry Z µτ 2 and another the hidden symmetry Z s 2 (revealed in our previous work) which dictates the solar mixing angle θ 12 . We derive the physical consequences of this Z s 2 and its possible partial violation in the presence of µ−τ breaking (without or with neutrino seesaw), regarding the θ 12 determination and the correlation between µ−τ breaking observables.PACS numbers: 14.60. Pq, 12.15.Ff, 13.15.+g, 13.40.Em Phys. Rev. D 84 (2011) 033009 and arXiv:1104.2654 * Introduction We conjecture that all CP violations (both Dirac and Majorana types) arise from a common origin in neutrino seesaw. With this conceptually attractive and simple conjecture, we deduce that µ−τ breaking shares the common origin with all CP violations, since the µ−τ symmetric limit enforces vanishing mixing angle θ 13 and thus Dirac CP conservation. In a recent work [1], we studied the common origin of soft µ−τ and CP breaking in the neutrino seesaw, which is uniquely formulated in the dimension-3 Majorana mass term of singlet right-handed neutrinos. This formulation predicts the normal mass ordering (NMO) for light neutrinos. In this work, we study in parallel a different realization of the common origin of µ−τ and CP breaking in the "µ−τ blind seesaw", where the right-handed neutrinos are singlet under the µ−τ transformation. We then find the Dirac mass-matrix to be the unique place for the common origin of µ−τ and CP breaking in the µ−τ blind seesaw. Since the Dirac mass-matrix arises from Yukawa interactions with Higgs boson(s), this can also provide an interesting possibility of realizing spontaneous CP violation with CP phases originating from the vacuum expectation values of Higgs fields. Different from our previous construction [1], we reveal that the common origin of µ−τ and CP breaking in the Dirac mass-matrix uniquely leads to the inverted mass-ordering (IMO) of light neutrinos and thus different neutrino phenomenology. Hence, the present mechanism can be distinguished from the previous one [1] by the on-going and upcoming experiments on the neutrino oscillations [2] and neutrinoless double-beta decays [3]. The oscillation data from solar and atmospheric neutrinos, and from the terrestrial neutrino beams produced in the reactor and accelerator experiments, have measured two mass-squared differences ∆m 2 31 , ∆m 2 21 and two large mixing angles (θ 12 , θ 23 ) to good accuracy [4] [5]. The two compelling features are [4] [5]: (i) the atmospheric neutrino mixing angle θ 23 has only small deviations from its maximal value of θ 23 = 45 • ; (ii) the reactor neutrino mixing angle θ 13 is found to be small, having its allowed range still consistent with θ 13 = 0 • at 90%C.L. Hence, the pattern of (θ 23 , θ 13 ) = (45 • , 0 • ) is strongly supported by the experimental data as a good zeroth order approximation. It is important to note that this pattern corresponds to the µ − τ symmetry and Dirac CP conservation in the neutrino sector, where the µ−τ symmetry is determined by both values of (θ 23 , θ 13 ) = (45 • , 0 • ) and the Dirac CP conservation is due to θ 13 = 0 • . On the theory ground, it is natural and tempting to expect a common origin for all CP-violations, although the Dirac and Majorana CP-violations appear differently in the light neutrino mass-matrix of the low energy effective theory. Given such a common origin for two kinds of CP-violations, then they must vanish together in the µ−τ symmetric limit. For the µ−τ blind seesaw, we can uniquely formulate this common breaking in the Dirac mass matrix, leading to distinct neutrino phenomenology. With such a conceptually attractive and simple construction of the common breaking of two discrete symmetries, we can predict the µ−τ breaking at low energies and derive quantitative correlations between the two small deviations, θ 23 − 45 • and θ 13 − 0 • , very different from that of the previous NMO scheme [1]. Our predicted range of θ 13 can saturate its present experimental upper limit. The improved measurements of θ 23 will come from the Minos [7] and T2K [8] experiments, etc, while θ 13 will be more accurately probed by the on-going reactor experiments, Daya Bay [10] [11], Double Chooz [12], and RENO [13], as well as the accelerator experiments T2K [8], NOνA [14] and LENA [15], etc. We further derive the observed baryon asymmetry via leptogenesis at seesaw scale, and analyze the correlation between the leptogenesis and the low energy neutrino observables in the present IMO scheme. Especially, we deduce a lower bound on the reactor neutrino mixing angle θ 13 1 • , and demonstrate that most of the predicted parameter space will be probed by the on-going Double Chooz, Daya Bay, and RENO reactor experiments. Finally, we will analyze the most general Z 2 ⊗ Z 2 symmetry structure of the light neutrino sector, and map it into the seesaw sector, where one of the Z 2 's is the µ−τ symmetry Z µτ 2 and another the hidden symmetry Z s 2 (revealed in our recent work [1] for the NMO scheme), which dictates the solar mixing angle θ 12 . We derive the physical consequences of the Z s 2 for the most general light neutrino mass-matrix (without seesaw) and for the seesaw models (with different µ−τ breaking mechanisms). In particular, we analyze the partial violation of Z s 2 in the presence of µ−τ breaking for the µ−τ blind seesaw, which leads to a modified new correlation between the µ−τ breaking observables, very different from that of Ref. [1]. The determination of θ 12 is systematically studied for the current IMO scheme and the partial violation of Z s 2 will be clarified. We organize this paper as follows. In Sec. 2 we present a unique construction for the common origin of the µ−τ and CP breakings in the neutrino seesaw with µ−τ blind right-handed neutrinos. Then, we give in Sec. 3 a model-independent reconstruction of light neutrino mass-matrix under inverted mass-ordering and with small µ − τ and CP violations at low energies. In Sec. 4. 1, we explicitly derive the low energy µ−τ and CP violation observables from the common breaking in the Dirac mass-matrix of the µ−τ blind seesaw. These include the two small deviations for the mixing angles θ 23 − 45 • and θ 13 − 0 • , the Jarlskog invariant for CP-violations, and the M ee element for neutrinoless double-beta decays. In Sec. 4. 2 we study the cosmological CP violation via leptogenesis in our model, this can generate the observed baryon asymmetry of the universe. Using all the existing data from neutrino oscillations and the observed baryon asymmetry [16,17], we derive the direct link between the cosmological CP-violation and the low energy Jarlskog invariant J . We further predict a lower bound on the reactor mixing angle θ 13 , and deduce a nonzero Jarlskog invariant J with negative range. We also establish a lower limit on the leptogenesis scale for producing the observed baryon asymmetry. In Sec. 5, we analyze the determination of solar mixing angle θ 12 and its relation to the hidden symmetry Z s 2 in the light neutrino sector (without seesaw) and in the seesaw sector (with two different realizations of µ − τ breaking). Finally, conclusions are summarized in the last section 6. Common Origin of µ−τ and CP Breaking from Neutrino Seesaw with Inverted Ordering The current global fit of neutrino data [4] for the three mixing angles and two mass-squared differences is summarized in Table-1. We note a striking pattern of the mixing angles, where the atmospheric angle θ 23 has its central value slightly below the maximal mixing [18] of 45 • and the reactor angle θ 13 slightly above 0 • . So the neutrino data support two small deviations θ 23 − 45 • and θ 23 − 0 • of the same order, symmetries emerge, i.e., the µ−τ symmetry [19] and the Dirac CP conservation in the neutrino sector. − 7.0 • < (θ 23 − 45 • ) < 5.5 • , 0 • (θ 13 − 0 • ) < 9.5 • ,(2. It is clear that the µ − τ symmetry and the associated Dirac CP-invariance are well supported by all neutrino data as a good zeroth order approximation, and have to appear in any viable theory for neutrino mass-generation. We also note that the θ 13 = 0 • limit does not remove the possible low energy Majorana CP-phases, but since the Majorana CP-violation comes from a common origin with the Dirac CP-violation in our theory construction (cf. below), it has to vanish as the Dirac CP-violation goes to zero in the µ−τ symmetric limit. Table 1: Updated global analysis [4] of solar, atmospheric, reactor and accelerator neutrino data for three-neutrino oscillations, where the AGSS09 solar fluxes and the modified Gallium capture cross-section [20] are used. In our theory construction, we conjecture that all CP violations (both Dirac and Majorana types) have a common origin and thus they must share the common origin with the µ−τ breaking. For the neutrino seesaw with heavy right-handed neutrinos blind to the µ−τ symmetry, this common origin can only come from the Dirac mass-term. In the following, we first consider the minimal neutrino seesaw Lagrangian with exact µ−τ and CP invariance, from which we will derive the seesaw massmatrix for the light neutrinos. Diagonalizing this zeroth order mass-matrix we predict the inverted mass-ordering of light neutrinos and deduce the mixing angles, (θ 23 , θ 13 ) 0 = (45 • , 0 • ) , as well as a formula for the solar angle θ 12 . Then we will construct the common origin for the µ−τ and CP breaking in the Dirac mass-matrix. Finally, we systematically expand the small µ−τ and CP breaking effects in the seesaw mass-matrix to the first nontrivial order. µ−τ and CP Symmetries of Neutrino Seesaw with Inverted Ordering The right-handed neutrinos are singlets under the standard model gauge group, and thus can be Majorana fields with large masses. This naturally realizes the seesaw mechanism [21] which provides the simplest explanation for the small masses of light neutrinos. For simplicity, we consider the Lagrangian for the minimal neutrino seesaw [22,23], with two right-handed singlet Majorana neutrinos besides the standard model (SM) particle content, L ss = − L Y ℓ Φℓ R − L Y ν Φ N + 1 2 N T M R CN + h.c. = − ℓ L M ℓ ℓ R − ν L m D N + 1 2 N T M R CN + h.c. + (interactions) ,(2.2) where L represents three left-handed neutrino-lepton weak doublets, ℓ = (e, µ, τ ) T denotes charged leptons, ν L = (ν e , ν µ , ν τ ) T is the light flavor neutrinos, and N = (N 1 , N 2 ) T contains two heavy right-handed singlet neutrinos. The lepton Dirac-mass-matrix M ℓ = v Y ℓ / √ 2 and the neutrino Dirac-mass-matrix m D = v √ 2 Y ν arise from the Yukawa interactions after spontaneous electroweak symmetry breaking, Φ = (0, v √ 2 ) T = 0 , and the Majorana mass-term for M R is a gauge-singlet. We can regard this minimal seesaw Lagrangian in Eq. (2.2) as an effective theory of the general three-neutrino seesaw where the right-handed singlet N 3 is much heavier than the other two (N 1 , N 2 ) and thus can be integrated out at the mass-scales of (N 1 , N 2 ), leading to Eq. (2.2). As a result, the minimal seesaw generically predicts a massless light neutrino [22]; this is always a good approximation as long as one of the light neutrinos has a negligible mass in comparison with the other two (even if not exactly massless). Extension to the three-neutrino seesaw will be discussed in Sec. 4. 3. Let us integrate out the heavy neutrinos (N 1 , N 2 ) in (2.2) and derive the seesaw formula for the 3 × 3 symmetric Majorana mass-matrix of the light neutrinos, M ν ≃ m D M −1 R m T D ,(2.3) where m D is the 3 × 2 Dirac mass-matrix, and M R is the 2 × 2 Majorana mass-matrix. The di- agonalization of M ν is achieved by unitary rotation matrix U ν via U T ν M ν U ν = D ν with D ν = diag(m 1 , m 2 , m 3 ) . The Lagrangian (2.2) is defined to respect both the µ−τ and CP symmetries. Under the µ−τ symmetry Z µτ 2 , we have the transformation, ν µ ↔ pν τ , where p = ± denotes the even/odd parity assignments of the light neutrinos under Z µτ 2 . Since the µ − τ symmetry has been tested at low energy via mixing angles of light neutrinos, it is logically possible that the right-handed heavy Majorana neutrinos in the seesaw Lagrangian (2.2) are singlets under Z µτ 2 (called "µ − τ blind"), which is actually the simplest realization of µ−τ symmetry in the neutrino seesaw. In this work we consider that the right-handed Majorana neutrinos N to be µ−τ blind, i.e., both (N 1 , N 2 ) are the singlets under Z µτ 2 , and thus can be first rotated into their mass-eigenbasis without affecting the µ−τ symmetric structure of the Dirac mass-matrix m D . So, in the mass-eigenbasis of (N 1 , N 2 ), we have M R = diag(M 1 , M 2 ) . Under the µ−τ and CP symmetries, the Dirac mass-matrix m D is real and obeys the invariance equation, G T ν m D = m D , (2.4) with G ν =   1 0 0 0 0 p 0 p 0   . (2.5) Next, we note that due to the large mass-splitting of µ and τ leptons, the lepton sector can exhibit, in general, a different flavor symmetry G ℓ from the µ−τ symmetry Z µτ 2 in the neutrino sector. The two symmetries Z µτ 2 and G ℓ could originate from spontaneous breaking of a larger flavor symmetry G F [24]. Under the transformation of left-handed leptons F ℓ ∈ G ℓ , we have the invariance equation of lepton mass-matrix, F † ℓ M ℓ M † ℓ F ℓ = M ℓ M † ℓ . As we will show in Sec. 4. 2, we are free to choose an equivalent representation d ℓ = U † ℓ F ℓ U ℓ of G ℓ from the start under which the left-handed leptons are in their mass-eigenbasis, where U ℓ is the transformation matrix diagonalizing the lepton mass-matrix, U † ℓ M ℓ M † ℓ U ℓ = D 2 ℓ with D ℓ = diag(m e , m µ , m τ ) . This means that in the lepton mass-eigenbasis, the conventional Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix V [28] in the leptonic charged current (an analog of the CKM matrix [29] in the quark sector) is fixed by the transformation U ν of neutrino mass-diagonalization, V = U ν . We can further rotate the right-handed leptons into their mass-eigenbasis, without affecting the PMNS matrix, except making the lepton-mass-term diagonal in the seesaw Lagrangian (2.2), i.e., M ℓ = diag(m e , m µ , m τ ) . Under the µ − τ and CP symmetries, we find the Dirac mass-matrix m D to have the following form, 6) with all elements being real, and σ 1 ≡ m 0 M 1 , σ 2 ≡ m 0 M 2 . As will be shown shortly, the parameter m 0 is defined at the seesaw scale and equals the nonzero mass-eigenvalue of the light neutrinos at zeroth-order under the µ−τ symmetric limit. In (2.6) we have also defined four dimensionless parameters, m D =    āā ′ bc bc     =     σ 1 a σ 2 a ′ σ 1 b σ 2 c σ 1 b σ 2 c     ,(2.(a, b) ≡ (ā,b) √ m 0 M 1 , (a ′ , c) ≡ (ā ′ ,c) √ m 0 M 2 . (2.7) Then, we find it convenient to define a dimensionless Dirac matrix, m D ≡ m D ( m 0 M R ) − 1 2 =     a a ′ b c b c     . (2.8) Substituting the above into the seesaw equation (2.3), we derive the µ−τ and CP symmetric massmatrix for light neutrinos, M ν ≃ m D M −1 R m T D = m 0 m D m T D = m 0      a 2 + a ′2 ab + a ′ c ab + a ′ c b 2 + c 2 b 2 + c 2 b 2 + c 2      , (2.9) which we call the zeroth order mass-matrix. In the next subsection we will further include the small µ−τ and CP breaking effect. Note that from (2.9), we have det(M ν ) = 0 , which generally holds in any minimal seesaw. Diagonalizing the mass-matrix (2.9), we derive the mass-eigenvalues and mixing angles at zeroth order, m 1,2 = m 0 2 (a 2 + a ′2 + 2b 2 + 2c 2 ) ∓ [(a 2 + a ′2 ) − 2(b 2 + c 2 )] 2 + 8(ab + a ′ c) 2 , (2.10a) m 3 = 0 , (2.10b) tan 2θ 12 = 2 √ 2\|ab + a ′ c\| \|a 2 + a ′2 − 2(b 2 + c 2 )\| , θ 23 = 45 • , θ 13 = 0 • , (2.10c) where we have made all mass-eigenvalues positive and the mixing angles (θ 12 , θ 13 , θ 23 ) within the range 0, π 2 by properly defining the rotation matrix. (As shown in Table-1, the solar angle θ 12 is most precisely measured and its 3σ range is below 37.7 • , so we always have 2θ 12 < π 2 and tan 2θ 12 > 0 .) The mixing angles (θ 23 , θ 13 ) = (45 • , 0 • ) are direct consequence of the µ−τ symmetry, but this symmetry does not fix θ 12 . Eqs. (2.10a)-(2.10b) show that the mass-spectrum of light neutrinos falls into the "inverted mass-ordering" (IMO), m 2 m 1 ≫ m 3 . Table-1 shows that the ratio of two mass-squared differences, ∆m 2 21 \|∆m 2 31 \| ≪ 1 . Since for the minimal seesaw model with IMO, the equation det(M ν ) = 0 leads to m 3 = 0 , so the above ratio requires the approximate degeneracy m 1 ≃ m 2 to be a good zeroth order approximation as enforced by the neutrino oscillation data. So, we will realize the exact degeneracy m 1 = m 2 for the µ−τ and CP symmetric mass-matrix (2.9), by imposing the relations for Eq. (2.10a), (a 2 + a ′2 ) − 2(b 2 + c 2 ) = 0 , ab + a ′ c = 0 . (2.11) As will be shown in the next subsection, including the common origin of µ−τ and CP breaking in the neutrino seesaw can produce small non-degeneracy between m 1 and m 2 at the next-to-leading order (NLO). Since the mass-parameter m 0 is introduced in (2.7) for defining the dimensionless parameters (a, b, c), we can now fix m 0 by defining 12) as the zeroth order mass-eigenvalue of light neutrinos, under the normalization condition, (a 2 + a ′2 ) + 2(b 2 + c 2 ) = 2 . m 0 ≡ m 1 = m 2 ,(2. (2.13) Combining this relation to Eq. (2.11), we can deduce, (2.14) where we see that three of the four parameters, (a, a ′ , c), can all be solved in terms of b . The last equation in (2.14) is not independent, but it helps to fix a relative sign. We note that in (2.9) the µ−τ symmetric seesaw mass-matrix M ν contains five parameters, the mass-parameter m 0 and the four dimensionless parameters (a, b, c, a ′ ). The inverted mass-spectrum have imposed a LO condition m 1 = m 2 , which results in two constraints in (2.11), and the normalization condition m 0 ≡ m 1 in (2.12) leads to the third constraint (2.13). In consequence, we end up with only two independent parameters, m 0 and b . a 2 = 2c 2 = 1 − 2b 2 , a ′2 = 2b 2 , c 2 = 1 2 − b 2 , a ′ c = −ab , We note that under the condition of (2.11), the mixing angle θ 12 given by (2.10c) has no definition at the zeroth order (the µ−τ symmetric limit) due to the vanishing numerator and denominator in the formula of tan 2θ 12 . But including the small µ − τ breaking effect will generate the nonzero expression of θ 12 at the NLO even though its final formula does not depend on the µ−τ breaking parameter (cf. Sec. 2. 2). As we will show in Sec. 2. 2, the µ − τ breaking arises from deviation in the element c of m D , so we can apply the l ′ Hôpital rule to the expression of tan 2θ 12 by taking the first-order derivatives on its numerator/denominator respect to c and deduce, 15) which is consistent with (4.5) of Sec. 4. 1 from the explicit NLO analysis. For the case with µ − τ breaking arising from deviation in the element b of m D , we can apply the l ′ Hôpital rule again to infer the formula, 16) which is the inverse of (2.15). As will be shown in Sec. 5. 2, the different forms of µ−τ breaking will affect the determination of the solar mixing angle θ 12 . But it is worth to note that the expression of θ 12 is fixed by the µ−τ symmetric m D as in (2.15) or (2.16), and does not explicitly depend on the µ−τ breaking parameter. We will systematically analyze these features in Sec. 5 and clarify the difference from our previous construction [1]. tan 2θ 12 = \|a ′ \| √ 2 \|c\| = \|ā ′ \| √ 2 \|c\| ,(2.tan 2θ 12 = \|a\| √ 2 \|b\| = √ 2\|c\| \|a ′ \| ,(2. Common Origin of µ−τ and CP Breaking in the µ−τ Blind Seesaw In this subsection, we will construct a unique breaking term providing a common origin for both µ−τ and CP breaking. From this we will further derive predictions of the common µ−τ and CP breaking for the low energy light neutrino mass-matrix, by treating the small breaking as perturbation up to the first nontrivial order (Sec. 4). We will analyze the seesaw-scale leptogenesis and its correlations with the low energy observables in Sec. 4. 2. As we have explained, the µ − τ symmetry serves as a good zeroth order flavor symmetry of the neutrino sector, which predicts θ 13 = 0 and thus the Dirac CP-conservation. Hence, the µ−τ symmetry breaking is generically small, and must generate all Dirac CP-violations at the same time. On the theory ground, it is natural and tempting to expect a common origin for all CP-violations, even though the Dirac and Majorana CP-violations appear differently in the light neutrino massmatrix of the low energy effective theory. For the two kinds of CP-violations arising from a common origin, then they must vanish together in the µ−τ symmetric limit. Different from our previous study [1], we consider the heavy right-handed neutrinos to be µ−τ blind in the neutrino seesaw. Thus the Majorana mass-matrix M R of the right-handed neutrinos must be µ−τ singlet. Hence, we deduce that the unique common origin of the µ−τ and CP breaking must arise from the Dirac mass-matrix of the seesaw Lagrangian (2.2). For the minimal seesaw, the most general form of m D is 17) where the scaling factors σ 1 ≡ m 0 M 1 and σ 2 ≡ m 0 M 2 are real mass-parameters as defined in m D =     a a ′ b 1 c 1 b 2 c 2     =     σ 1 a σ 2 a ′ σ 1 b 1 σ 2 c 1 σ 1 b 2 σ 2 c 2     ,(2. Eq. (2.6). The six elements of m D can be complex in general. But there are three rephasing degrees of freedom for the left-handed lepton-doublets. So we can always rotate the three elements in the first column of m D to be all real, hence the remaining CP phases (associated with the µ−τ breaking) have to appear in the elements c 1 and c 2 because a ′ cannot break µ−τ symmetry and thus should be real. We have conjectured that all CP violations arise from a common origin, which then must originate from the µ − τ breaking; so we can formulate such a common origin as a single phase in either c 1 or c 2 in the minimal construction, where the other two elements in the second column of m D should be real. Hence, we present a unique minimal construction to formulate the common origin of µ−τ and CP breaking in the Dirac mass-matrix m D as follows, 18) where the dimensionless parameters −1 < ζ ′ < 1, 0 ζ < 1 , and the CP-phase angle ω ∈ [0, 2π) . m D =     σ 1 a σ 2 a ′ σ 1 b σ 2 c(1 − ζ ′ ) σ 1 b σ 2 c(1 − ζe iω )     ,(2. Here we have set b 1 = b 2 ≡ b since (b 1 , b 2 ) are already made real and thus cannot serves as the common source of the µ−τ and CP breaking. Inspecting (2.18) we see that, for any nonzero ζ and ω, the µ−τ and CP symmetries are broken by the common source of ζe iω . We could also absorb the real parameter ζ ′ into c by defining c ′ ≡ c(1 − ζ ′ ) . Thus we have, m D =     σ 1 a σ 2 a ′ σ 1 b σ 2 c ′ σ 1 b σ 2 c ′ (1 − ζ ′′ e iω ′ )     , (2.19) with ζ ′′ e iω ′ = ζe iω − ζ ′ 1 − ζ ′ . (2.20) Given the ranges of (ζ, ζ ′ ) as defined above, we see that the corresponding new parameter ζ ′′ of the µ−τ breaking has a much larger range, including values within 1 \|ζ ′′ \| 3 (when \|ζ\|, \|ζ ′ \| 0.6 for instance), which are beyond the perturbative expansion. We find that if enforce \|ζ ′′ \| < 1 , the parameter-space of (2.19) becomes smaller than (2.18) and insufficient for making the model fully viable. This means that our formulation of (2.18) is more general and has larger parameter-space for making theoretical predictions. Hence, we will apply (2.18) for the physical analyses below. We note another formulation of such a breaking in the Dirac mass-matrix m D , 21) which is connected to (2.18) by a µ−τ transformation for the light neutrinos ν = (ν e , ν µ , ν τ ) into 5). Accordingly, the mass-matrix (2.21) transforms as, 22) which goes back to (2.18). So the two different formulations (2.18) and (2.21) just cause the µ−τ asymmetric parts in the seesaw mass-matrix M ν = m D M −1 R m T D to differ by an overall minus sign. As we will comment further in Sec. 4. 1, this does not affect our predictions for the physical observables and their correlations. So we only need to focus on the formulation (2.18) for the rest of our analysis. m D =     σ 1 a σ 2 a ′ σ 1 b σ 2 c(1 − ζe iω ) σ 1 b σ 2 c(1 − ζ ′ )     ,(2.ν ′ = (ν e , ν τ , ν µ ) , via ν = G ν ν ′ , with G ν [p = 1] defined in Eq. (2.m D → m ′ D = G T ν m D = m D ,(2. We may also first rotate the three elements in the second column of (2.17) to be real and then formulate the common origin of µ−τ and CP breaking as follows, m D =     σ 1 a σ 2 a ′ σ 1 b(1 − ζ ′ ) σ 2 c σ 1 b(1 − ζe iω ) σ 2 c     . (2.23) As will be clarified in Sec. 5, this will lead to the determination of solar mixing angle θ 12 as in (2. 16), in contrast to (2.18) which predicts a different θ 12 as in (2.15). Here θ 12 is explicitly fixed by the µ−τ and CP symmetric parameters of m D in either case. But, we find the predictions for all other µ−τ and CP breaking observables and their correlations to remain the same as those from the construction in (2.18). Finally, it is interesting to note that for an extended Higgs sector (consisting of two Higgs doublets or more) we can generate all CP-phases in the Dirac mass-matrix m D via spontaneous CP violation [30], which is beyond the current scope and will be elaborated elsewhere [31]. Perturbative Expansion for µ−τ and CP Breaking Let us first consider the 3 × 3 mass-matrix M ν light neutrinos, which can be generally presented as, 24) where the zeroth order matrix M (0) ν corresponds to vanishing µ−τ breaking with ζ i = 0 , and the NLO mass-matrix δM M ν =    A B 1 B 2 C 1 D C 2    ≡    A 0 B 0 B 0 C 0 D 0 C 0    +    δA δB 1 δB 2 δC 1 δD δC 2    ≡ M (0) ν + δM ν = M (0) ν + δM (1) ν + O(ζ 2 i ) ,(2. (1) ν includes the µ−τ breaking to the first nontrivial order. We find it useful to further decompose δM (1) ν into the µ−τ symmetric and anti-symmetric parts, δM (1) ν ≡ δM s ν + δM a ν ≡    δA δB s δB s δC s δD δC s    +    0 δB a −δB a δC a 0 −δC a   , (2.25) with δB s ≡ 1 2 (δB 1 + δB 2 ) , δB a ≡ 1 2 (δB 1 − δB 2 ) , (2.26a) δC s ≡ 1 2 (δC 1 + δC 2 ) , δC a ≡ 1 2 (δC 1 − δC 2 ) . (2.26b) This decomposition is actually unique. From our construction in the previous subsection, the µ−τ and CP breaking Dirac mass-matrix m D as well as the Majorana mass-matrix M R is uniquely parameterized as follows, 27) with σ 1,2 ≡ m 0 M 1,2 and m D =     σ 1 a σ 2 a ′ σ 1 b σ 2 c 1 σ 1 b σ 2 c 2     , M R = diag(M 1 , M 2 ) ,(2.c 1 = c 1 − ζ ′ , c 2 = c 1 − ζe iω . (2.28) Thus, we can explicitly derive the seesaw mass-matrix for light neutrinos, M ν = m 0      a 2 + a ′2 ab + a ′ c 1 ab + a ′ c 2 b 2 + c 2 1 b 2 + c 1 c 2 b 2 + c 2 2      . (2.29) Since the neutrino data require the µ−τ breaking to be small, we can further expand M ν in terms of small breaking parameter ζ as, M ν ≡ M (0) ν + δM ν = M (0) ν + δM (1) ν + O(ζ 2 ) ,(2.30) with M (0) ν = m 0      a 2 + a ′2 ab + a ′ c ab + a ′ c b 2 + c 2 b 2 + c 2 b 2 + c 2      = m 0    1 0 0 1 2 1 2 1 2    , (2.31a) δM (1) ν = m 0      0 −a ′ c ζ ′ −a ′ c ζe iω −2 c 2 ζ ′ −c 2 (ζ ′ + ζe iω ) −2c 2 ζe iω      , (2.31b) where we have used the solution (2.14) for the second step of (2.31a) and the µ−τ breaking expression A 0 = m 0 (a 2 + a ′2 ) = m 0 , (2.32a) B 0 = m 0 (ab + a ′ c) = 0 , (2.32b) C 0 = D 0 = m 0 (b 2 + c 2 ) = 1 2 m 0 , (2.32c) and δA = 0 , δD = − m 0 c 2 (ζ ′ + ζe iω ) , δB s = − 1 2 m 0 a ′ c(ζ ′ + ζe iω ) , δC s = − m 0 c 2 (ζ ′ + ζe iω ) , δB a = − 1 2 m 0 a ′ c(ζ ′ − ζe iω ) , δC a = − m 0 c 2 (ζ ′ − ζe iω ) . (2.33) Note that from (2.33) we can compute the ratio, 34) where in the last step we have used the resolution (2.14). It is interesting to note that the ratio (2.34) of the µ−τ asymmetric parts in the light neutrino mass-matrix M ν only depends on the µ−τ symmetric elements of the Dirac mass-matrix m D . This ratio just corresponds to the determination of the solar angle θ 12 in (2.15) and will be further confirmed later by the full NLO analysis of Sec. 4. 1. δB a δC a = a ′ 2c = − b a ,(2. Inverted Ordering: Reconstructing Light Neutrino Mass Matrix with µ−τ and CP Violations at Low Energy In this section, we give the model-independent reconstruction of the Majorana mass-matrix for light neutrinos under inverted mass-ordering (IMO), in terms of the low energy observables (masseigenvalues, mixings angles and CP phases). We expand this reconstruction by experimentally welljustified small parameters up to the next-to-leading order (NLO). Applying this reconstruction formulation to our model will allow us to systematically derive the physical predictions for the correlations among the low energy observables as well as for the link to the baryon asymmetry via leptogensis at the seesaw scale. Notation Setup and Model-Independent Reconstruction Let us consider the general 3×3 symmetric and complex Majorana mass-matrix for the light neutrinos, M ν ≡     m ee m eµ m eτ m µµ m µτ m τ τ     ≡     A B 1 B 2 C 1 D C 2     . (3.1) In the mass-eigenbasis of charged leptons, the neutrino mass-matrix M ν can be diagonalized by a unitary transformation V (= U ν ) , i.e., V T M ν V = D ν ≡ diag(m 1 , m 2 , m 3 ) , and thus we can write the reconstruction equation, M ν = V * D ν V † . (3.2) The mixing matrix V can be generally expressed as a product of three unitary matrices including a CKM-type mixing matrix U plus two diagonal rephasing matrices U ′ and U ′′ , V ≡ U ′′ U U ′ , (3.3a) U ≡    c s c x −s s c x −s x e iδ D s s c a − c s s a s x e −iδ D c s c a + s s s a s x e −iδ D −s a c x s s s a + c s c a s x e −iδ D c s s a − s s c a s x e −iδ D c a c x   , (3.3b) U ′ ≡ diag(e iφ 1 , e iφ 2 , e iφ 3 ) , U ′′ ≡ diag(e iα 1 , e iα 2 , e iα 3 ) , (3.3c) where δ D is the Dirac CP-phase. For notational convenience, we have denoted the three neutrino mixing angles of the PMNS matrix as, (θ 12 , θ 23 , θ 13 ) ≡ (θ s , θ a , θ x ) , by following Ref. [23]. We will further use the notations, (s s , s a , s x ) ≡ (sin θ s , sin θ a , sin θ x ) and (c s , c a , c x ) ≡ (cos θ s , cos θ a , cos θ x ) . For the diagonal rephasing matrix U ′ , only two of its three Majorana phases are measurable (such as φ 3 − φ 1 and φ 2 − φ 1 ) after extracting an overall phase factor. The matrix U ′′ contains another three phases which associate with the flavor-eigenbasis of light neutrinos and are needed for the consistency of diagonalizing a given mass-matrix M ν . For convenience we define the rephased mass-eigenvalues D ν ≡ U ′ * D ν U ′ † ≡ ( m 1 , m 2 , m 3 ) = (m 1 e −i2φ 1 , m 2 e −i2φ 2 , m 3 e −i2φ 3 ) , so the reconstruction equation (3.2) becomes, M ν = V ′ * D ν V ′ † , ( V ′ ≡ U ′′ U ) . (3.4) Thus, we can fully reconstruct all elements of M ν in terms of the rephased mass-eigenvalues ( m 1 , m 2 , m 3 ), the mixing angles (θ s , θ a , θ x ), the Dirac phase δ D , and the rephasing phases α i (which do not appear in physical PMNS mixing matrix), m ee = e −i2α 1 c 2 s c 2 x m 1 + s 2 s c 2 x m 2 + s 2 x e −2iδ D m 3 , (3.5a) m µµ = e −i2α 2 (s s c a − c s s a s x e iδ D ) 2 m 1 + (c s c a + s s s a s x e iδ D ) 2 m 2 + s 2 a c 2 x m 3 , (3.5b) m τ τ = e −i2α 3 (s s s a + c s c a s x e iδ D ) 2 m 1 + (c s s a − s s c a s x e iδ D ) 2 m 2 + c 2 a c 2 x m 3 , (3.5c) m eµ = e −i(α 1 +α 2 ) c s c x (s s c a −c s s a s x e iδ D ) m 1 −s s c x (c s c a + s s s a s x e iδ D ) m 2 +s a s x c x e −iδ D m 3 , (3.5d) m eτ = e −i(α 1 +α 3 ) c s c x (s s s a +c s c a s x e iδ D ) m 1 −s s c x (c s s a −s s c a s x e iδ D ) m 2 −c a s x c x e −iδ D m 3 , (3.5e) m µτ = e −i(α 2 +α 3 ) (s s c a − c s s a s x e iδ D )(s s s a + c s c a s x e iδ D ) m 1 + (c s c a + s s s a s x e iδ D )(c s s a − s s c a s x e iδ D ) m 2 − s a c a c 2 x m 3 , (3.5f) where among the Majorana phases φ 1,2,3 (hidden in the mass-parameters m 1,2,3 ) only two are independent because an overall phase factor of U ′ can be absorbed into the diagonal rephasingmatrix U ′′ . For the case with a vanishing mass-eigenvalue (such as m 3 = 0 in our present model), only one independent phase combination, say e i(φ 2 −φ 1 ) , will survive. If we impose µ−τ symmetry on the light neutrino mass-matrix M ν , we can deduce [1], (θ a , θ x ) 0 = (45 • , 0 • ) , α 20 = α 30 . (3.6) The solar mixing angle θ s is independent of the µ−τ symmetry and is thus left undetermined. To predict θ s , we will uncover a new flavor symmetry beyond the Z µτ 2 (cf. Sec. 5). Reconstruction of Light Neutrino Mass Matrix with Inverted Ordering Now we are ready to apply the above general reconstruction formalism to the inverted mass-ordering (IMO), m 2 m 1 ≫ m 3 , with m 3 = 0 (as predicted by the present minimal seesaw model), in contrast to our previous model which predicts the normal mass-ordering (NMO) [1]. We introduce a small mass-ratio for light neutrinos, y ′ ≡ m 2 2 − m 2 1 m 2 1 = ∆m 2 21 ∆m 2 13 = 0.029 − 0.036 ≪ 1 ,(3.7) as constrained by the neutrino data at 90% C.L. (Table-1). So it is sufficient to make perturbative expansion in y ′ up to its linear order. Thus, at the zeroth order of y ′ , we have equal mass-eigenvalues, m 10 = m 20 = m 0 . Under the y ′ -expansion up to next-to-leading order (NLO), m i = m 0 + δm i , we have y ′ ≃ 2(δm 2 − δm 1 ) m 1 = 2(m 2 − m 1 ) m 1 . (3.8) We can define another small ratio z ≡ δm 1 m 1 = O(y ′ ) , and deduce, δm 1 = z m 1 , δm 2 = z + y ′ 2 m 1 ,(3.9) where m 1 = ∆m 2 13 is fixed by the neutrino data, and m 0 = m 1 − δm 1 = (1 − z)m 1 ≃ ∆m 2 13 . Next, we consider the mixing angles and CP-phases. Since the neutrino oscillation data strongly support the µ−τ symmetry as a good approximate symmetry (3.6), we can define the small deviations from the general µ−τ symmetric solution (3.6), δ a ≡ θ a − π 4 , δ x ≡ θ x − 0 ,(3.10) which characterize the µ−τ symmetry breaking. From the data in Table- For our analysis we will systematically expand the small parameters (δ a , δ x , y ′ , z) up to their linear order. For the Majorana CP-phases, φ 3 drops due to m 3 = 0 ; we also remove an overall redundant Majorana phase φ 1 (from U ′ ) into the redefinition of α j (in U ′′ ). So, the remaining independent Majorana phase is only φ ,ᾱ j ≡ α j + φ 1 , (j = 1, 2, 3) , (3.12a) φ ≡ φ 2 − φ 1 = φ 0 + δφ . (3.12b) The expansion up to the NLO for our current reconstruction analysis will include (δᾱ 1 , δᾱ 2 , δᾱ 3 , δφ) . The solar angle θ s (≡ θ 12 ) is independent of the µ−τ breaking and thus receives no NLO correction. Furthermore, we note that the Dirac phase e iδ D is always associated with the small mixing parameter s x (≃ δ x ) , so it only appears at the NLO and thus receive no more correction at this order of expansion. Finally, we give a summary of all relevant NLO parameters in our reconstruction analysis, ( y ′ , z, δ a , δ x , δᾱ 1 , δᾱ 2 , δᾱ 3 , δφ) , (3.13) Each of them is defined as the difference between its full value and zeroth-order value under the µ−τ symmetric limit. In Sec. 4 we will derive these deviations from our seesaw model for the common origin of µ−τ and CP breaking, and analyze their correlations. Making the perturbative expansion of (3.13) under the inverted mass-ordering, we first deduce the LO form of the light neutrino mass-matrix (3.1), m (0) ee ≡ A 0 = m 0 e −2iᾱ 10 c 2 s + s 2 s e −i2φ 0 , (3.14a) m (0) eµ = m (0) eτ ≡ B 0 = 1 √ 2 m 0 s s c s e −i(ᾱ 10 +ᾱ 20 ) 1 − e −i2φ 0 , (3.14b) m (0) µµ = m (0) τ τ ≡ C 0 = 1 2 m 0 e −2iᾱ 20 s 2 s + c 2 s e −2iφ 0 = D 0 , (3.14c) where we have also matched to our notation of M (0) ν in (2.24). Then, we derive elements of the NLO mass-matrix δM (1) ν from (3.5), δm (1) ee ≡ δA = m 0 e −i2ᾱ 10 z + s 2 s 2 y ′ − i2(s 2 s δφ + δᾱ 1 ) , (3.15a) δm (1) eµ ≡ δB 1 = m 0 √ 2 e −i(ᾱ 10 +ᾱ 20 ) − c s s s 2 y ′ − e iδ D δ x + i2c s s s δφ , (3.15b) δm (1) eτ ≡ δB 2 = m 0 √ 2 e −i(ᾱ 10 +ᾱ 20 ) − c s s s 2 y ′ + e iδ D δ x + i2c s s s δφ , (3.15c) δm (1) µµ ≡ δC 1 = m 0 e −i2ᾱ 20 z 2 + c 2 s 4 y ′ − δ a − i(c 2 s δφ + δᾱ 2 ) , (3.15d) δm (1) τ τ ≡ δC 2 = m 0 e −i2ᾱ 20 z 2 + c 2 s 4 y ′ + δ a − i(c 2 s δφ + δᾱ 3 ) , (3.15e) δm (1) µτ ≡ δD = m 0 e −i2ᾱ 20 z 2 + c 2 s 4 y ′ − i 2 (2c 2 s δφ + δᾱ 2 + δᾱ 3 ) , (3.15f) where we have matched to our notation of δM (1) ν as defined in (2.24). In the above formulas, we have used the µ−τ symmetric relations for the LO parameters, (θ a0 , θ x0 ) = ( π 4 , 0) andᾱ 20 =ᾱ 30 , as well as m 3 ≡ 0 . From (2.25), we can uniquely decompose the elements of δM (1) ν in (3.15) as the µ−τ symmetric and anti-symmetric parts, δM (1) ν ≡ δM s ν + δM a ν , with their elements given by, δB s ≡ δB 1 + δB 2 2 = m 0 √ 2 e −i(ᾱ 10 +ᾱ 20 ) − c s s s 2 y ′ + i2c s s s δφ , δB a ≡ δB 1 − δB 2 2 = − m 0 √ 2 e −i(ᾱ 10 +ᾱ 20 ) e iδ D δ x , δC s ≡ δC 1 + δC 2 2 = m 0 e −i2ᾱ 20 z 2 + c 2 s 4 y ′ − i 2 (2c 2 s δφ + δα 2 + δα 3 ) = δD , δC a ≡ δC 1 − δC 2 2 = − m 0 e −i2ᾱ 20 δ a + i 2 (δα 2 − δα 3 ) . (3.16) With these, we will be ready to apply the above reconstruction formulas (3.14), (3.15) and (3.16) to match with (2.24) in our seesaw model at the LO and NLO, respectively. We will systematically solve these matching conditions in the next section, which allows us to connect the seesaw parameters to the low energy neutrino observables and deduce our theoretical predictions. For matching the seesaw predictions to our reconstruction formalism, we note that the latter was presented at the low energy scale so far. We need to connect the low energy neutrino parameters to the model predictions at the seesaw scale, where the possible renormalization group (RG) running effects should be taken into account in principle. Such RG effects were extensively discussed in the literature [25], and can be straightforwardly applied to the present analysis. Below the seesaw scale, heavy right-handed neutrinos can be integrated out from the effective theory and the seesaw mass-eigenvalues m j (j = 1, 2, 3) for light neutrinos obey the approximate one-loop RG equation (RGE) [25], 17) to good accuracy [26], where t = ln(µ/µ 0 ) with µ the renormalization scale. For the SM, the coupling-parameter α ≃ −3g 2 2 + 6y 2 t + λ , with (g 2 , y t , λ) denoting the SU (2) L weak gauge coupling, the top Yukawa coupling and Higgs self-coupling, respectively. Hence, we can deduce the running mass-parameter m j from scale µ 0 to µ , dm j dt = α 16π 2 m j ,(3.m j (µ) = χ(µ, µ 0 ) m j (µ 0 ) ≃ exp 1 16π 2 t 0 α(t ′ ) dt ′ m j (µ 0 ) ,(3.18) with t = ln(µ/µ 0 ) . In the present analysis we will choose, (µ 0 , µ) = (M Z , M 1 ) , with Z boson mass M Z representing the weak scale and the heavy neutrino-mass M 1 characterizing the seesaw scale. Consider the minimal neutrino seesaw with inverted mass-spectrum, m 2 m 1 ≫ m 3 = 0 . We note that the zero-eigenvalue m 3 and the mass ratio y ′ do not depend on the RG running scale µ . So we can derive the running of the two nonzero mass-parameters from weak scale to seesaw scale, m 1 ≡ m 1 (M 1 ) = χ 1 m 1 (M Z ), (3.19a) m 2 ≡ m 2 (M 1 ) = χ 1 m 2 (M Z ) = 1 + y ′ m 1 , (3.19b) with χ 1 ≡ χ(M 1 , M Z ) . In Sec. 4, we will compute the RG running factor χ 1 ≡ χ(M 1 , M Z ) numerically, which depends on the inputs of initial values for α 2 = g 2 2 /(4π) , y t and the Higgs boson mass M H , via the combination α defined above. Using the electroweak precision data [17,27], Other running effects due to the leptonic mixing angles and CP-phases are all negligible for the present study since their RGEs contain only flavor-dependent terms and are all suppressed by y 2 τ = O(10 −4 ) at least [25]. For the analyses below (Sec. 4), we will first evolve the mass-parameters from the seesaw scale M 1 down to the low energy scale for neutrino oscillations, and then match them with those in our reconstruction formalism. Including such RG effects just requires to replace the light masseigenvalues ( m 1 , m 2 ) at seesaw scale M 1 by the corresponding (m 1 , m 2 ) at low energy, and vice versa. Predictions of Common µ−τ and CP Breaking with Inverted Ordering In this section we apply the reconstruction formalism (including the RG running effects) in Sec. 3. 2 to our common µ−τ and CP breaking seesaw in Sec. 2. 3. Then, we systematically derive the predictions for the low energy neutrino observables. This includes the nontrivial correlation between two small µ−τ breaking parameters δ x (≡ θ 13 − 0) and δ a ≡ θ 23 − π 4 . Furthermore, we study the correlations of θ 23 − 45 • and θ 13 with Jarlskog invariant J and neutrinoless ββ-decay observable M ee . Finally, we study the matter-antimatter asymmetry (baryon asymmetry) via leptogenesis in the µ−τ blind seesaw, and establish the direct link with low energy neutrino observables. Furthermore, we will derive a nontrivial lower bound on the reactor mixing angle, θ 13 1 • , and restrict the Jarlskog invariant into a negative range, −0.037 J −0.0035 . Predicting Correlations of Low Energy Neutrino Observables Both µ − τ and CP violations arise from a common origin in the seesaw Lagrangian of our model, which is characterized by the breaking parameter ζe iω and shows up at the NLO of our perturbative expansion. Hence, in the light neutrino mass-matrix, the small µ − τ breaking parameters (δ a , δ x ) together with all CP-phases are controlled by ζ and ω . In the following, we will use the reconstruction formalism (Sec. 3. 2) under IMO for diagonalizing the light neutrino mass-matrix at the NLO. Then, we will further derive quantitative predictions for these low energy observables and their correlations. We first inspect the reconstructed LO mass-matrix M (0) ν in (3.14). Matching (3.14) with our model prediction (2.31a) at the same order, we find the solutions, α 10 =ᾱ 20 = φ 0 = 0 , (4.1a) m 10 = m 20 = m 0 , m 3 = 0 , (4.1b) a 2 = 2c 2 = 1 − 2b 2 , a ′2 = 2b 2 , c 2 = 1 2 − b 2 , a ′ c = −ab , (4.1c) which is also consistent with Eq. (2.14). Here all the LO CP-phases (ᾱ 10 ,ᾱ 20 , φ 0 ) = 0 because the original CP-violation in the seesaw Lagrangian vanishes in the ζ = 0 limit (Sec. 2. 2). Then, we analyze the NLO light neutrino mass-matrix δM (1) ν , as given by (2.25) of our model and by the reconstruction formula (3.15). We match the two sets of equations at the low energy for the µ−τ symmetric elements, δA = 0 = m 0 z + s 2 s 2 y ′ − i2(s 2 s δφ + δᾱ 1 ) , (4.2a) δB s = − m 0 2 a ′ c(ζ ′ + ζe iω ) = m 0 √ 2 − c s s s 2 y ′ + i2c s s s δφ , (4.2b) δC s = − m 0 c 2 (ζ ′ + ζe iω ) = m 0 2 z + c 2 s 2 y ′ − i(2c 2 s δφ + δᾱ 2 + δᾱ 3 ) = δD, (4.2c) and for µ−τ anti-symmetric elements, δB a = − m 0 2 a ′ c(ζ ′ − ζe iω ) = − m 0 √ 2 e iδ D δ x , (4.3a) δC a = −m 0 c 2 (ζ ′ − ζe iω ) = −m 0 δ a + i 2 (δᾱ 2 − δᾱ 3 ) , (4.3b) where using Eq. (3.19) we have run the mass-parameter m 0 from the seesaw scale down to the corresponding m 0 at low energy for the left-hand-sides of Eqs. (4.2) and (4.3). From the µ−τ symmetric Eqs. (4.2a)-(4.2b), we can infer six independent conditions for the real and imaginary parts of (δA, δB s , δC s ) , respectively, z = − s 2 s 2 y ′ , (4.4a) δᾱ 1 = −s 2 s δφ , (4.4b) c s s s √ 2 y ′ = a ′ c ζ ′ + ζ cos ω , (4.4c) 2 √ 2 c s s s δφ = −a ′ c ζ sin ω , (4.4d) z 2 + c 2 s 4 y ′ = −c 2 ζ ′ + ζ cos ω , (4.4e) − 1 2 (2c 2 s δφ + δᾱ 2 + δᾱ 3 ) = −c 2 ζ sin ω . (4.4f) Thus, with the aid of (4.4a) we take the ratio of (4.4c) and (4.4e), and derive (4.5) which coincides with (2.15) in Sec. 2. 1. Using Eq. (4.5), we deduce from Eq. (4.1c), tan 2θ s = − a ′ √ 2 c = √ 2 b a ,a = p a cos 2θ s , b = p a 1 √ 2 sin 2θ s , (4.6a) a ′ = p a ′ sin 2θ s , c = −p a ′ 1 √ 2 cos 2θ s , (4.6b) with p a , p a ′ = ± denoting the signs of (a, a ′ ). Here we see that the four dimensionless LO parameters (a, a ′ , b, c) in the Dirac mass-matrix (2.18) are fixed by the solar mixing angles θ s , since the conditions in (4.1c) make three of them non-independent. Finally, we further resolve (4.4) and derive the NLO parameters, y ′ = −2 cos 2θ s ζ ′ + ζ cos ω , (4.7a) z = s 2 s cos 2θ s ζ ′ + ζ cos ω , (4.7b) δᾱ 1 = − 1 2 s 2 s (c 2 s − s 2 s ) ζ sin ω , (4.7c) δφ = 1 2 (c 2 s − s 2 s ) ζ sin ω , (4.7d) δᾱ 2 + δᾱ 3 = s 2 s (s 2 s − c 2 s ) ζ sin ω . (4.7e) It is interesting to note that the present model predicts a generically small Majorana CP-phase angle at low energy, φ = δφ = O(ζ) , in contrast to our soft breaking model [1] where the low energy Majorana CP-phase angle (φ 23 ) is not suppressed. Next, we analyze the µ−τ anti-symmetric equations (4.3a)-(4.3b) for δM (1) ν . With (4.6), we can deduce from (4.3a)-(4.3b), 1 2 sin 2θ s cos 2θ s (ζ ′ − ζe iω ) = −e iδ D δ x , (4.8a) 1 2 cos 2 2θ s ζ ′ − ζe iω = δ a + i 2 (δᾱ 2 − δᾱ 3 ) , (4.8b) which decompose into cos δ D δ x = − 1 2 sin 2θ s cos 2θ s ζ ′ − ζ cos ω , (4.9a) sin δ D δ x = 1 2 sin 2θ s cos 2θ s (ζ sin ω) , (4.9b) δ a = 1 2 cos 2 2θ s ζ ′ − ζ cos ω , (4.9c) δᾱ 2 − δᾱ 3 = − cos 2 2θ s (ζ sin ω) . (4.9d) Thus the Dirac CP-phase angle δ D can be derived from the ratio of (4.9a) and (4.9b), tan δ D = ζ sin ω ζ cos ω − ζ ′ = δᾱ 2 − δᾱ 3 2δ a . (4.10) With Eqs. (4.7a), (4.10) and (4.9), we finally deduce, (4.11) and thus ζ ′ + ζ cos ω = − 1 2 cos 2θ s y ′ , −ζ sin ω = 2 tan δ D cos 2 2θ s δ a ,cos δ D δ x = − sin 2θ s 4 y ′ + 4 cos 2θ s ζ ′ = sin 2θ s 4 y ′ + 4 cos 2θ s cos ω ζ , (4.12a) δ a = cos 2θ s 4 y ′ + 4 cos 2θ s ζ ′ = − cos 2θ s 4 y ′ + 4 cos 2θ s cos ω ζ , (4.12b) δᾱ 2 − δᾱ 3 = 2 tan δ D δ a . (4.12c) From Eqs. (4.12a) and (4.12b), we derive a nontrivial correlation between the low energy µ − τ breaking observables δ a and δ x , (4.13) This shows that at the NLO the two small µ−τ breaking parameters are proportional to each other, δ a = − cot 2θ s cos δ D δ x .δ x ∝ δ a . Because of \| cos δ D \| 1 , we can infer from Eq. (4.13) a generic lower bound on δ x , for any nonzero δ a , δ x \|δ a \| tan 2θ s ,(4.14) where we have δ x ≡ θ 13 ∈ [0, π 2 ] in our convention. It is worth to note that our previous soft breaking model [1] also predicted a correlation and a lower bound, δ a = − cot θ s cos δ D δ x , (Prediction of Ref. [1]), (4.15a) ⇒ δ x \|δ a \| tan θ s , (4.15b) where the quantitative difference from the present predictions is that we have the coefficient cot 2θ s in Eq. (4.13) as compared to cot θ s in Eq. (4.15a). In fact, this is a profound difference. From the present oscillation data in Table-1, we observe that the deviation of the solar angle θ s (≡ θ 12 ) from its maximal mixing value is relatively small, 16) and this limit only relaxes slightly at 99% C.L., 7.8 • < 45 • − θ 12 < 12.9 • . Hence, we see that the range of the deviation 45 • − θ 12 is at the same level as the two other small deviations θ 23 − 45 • and θ 13 − 0 • shown in Eq. (2.1). So, we can define a new naturally small quantity, 17) and make expansion for δ s as well. Then, we immediately observe a qualitative difference between cot 2θ s ≃ 2δ s ≪ 1 in (4.13) and cot θ s ≃ 1 + 2δ s 1 in (4.15a). Hence, we can rewrite the two correlations (4.13) and (4.15a) in the well expanded form, 9.0 • < 45 • − θ 12 < 12.2 • , (at 90% C.L.),(4.δ s ≡ π 4 − θ s ,(4.δ a ≃ −2 cos δ D (δ s δ x ) ≪ δ x , (Current Prediction), (4.18a) δ a ≃ − cos δ D δ x = O(δ x ) , (Prediction of Ref. [1]). (4.18b) Two comments are in order. First, we deduce from (4.18) the following patterns of the three mixing angles, (4.19b) where for the current model Eq. (4.19a) predicts a nearly maximal atmospheric angle θ 23 ≃ π 4 ; while for the soft-breaking model [1], Eq. (4.19b) allows all three deviations to be comparable. Second, for each given nonzero δ a = θ 23 − π 4 , we can deduce the lower limits on δ x = θ 13 from (4.18), (θ 12 , θ 23 , θ 13 ) = π 4 − δ s , π 4 − O(δ s δ x ), δ x , (in the current model), (4.19a) (θ 12 , θ 23 , θ 13 ) = π 4 − δ s , π 4 − δ a , δ x , (in the model of Ref. [1]),δ x \|δ a \| 2δ s ≫ \|δ a \| , (Current Prediction), (4.20a) δ x \|δ a \| , (Prediction of Ref. [1]). (4.20b) Given the 99% C.L. range of 7.8 • < δ s < 12.2 • , we derive the lower limit from (4.14) or (4.20a) for the present model, θ 13 (3.6 ∼ 2.1)\|θ 23 − 45 • \| ,(4.21) which allows θ 13 to easily saturate its current upper limit. As another illustration, taking the current "best fit" values (θ 12 , θ 23 ) = (34.5 • , 42.8 • ) as in Table-1, we derive from (4.14) or (4.20a) the lower limits θ 13 6 • for the present model, and θ 13 1.5 • for Ref. [1]. Hence, in contrast with Ref. [1], the present model favors a larger θ 13 , and can saturate its current upper limit, as will be demonstrated in Fig. 2 below. In the following, we systematically analyze the predicted parameter space and correlations in the present model (with inverted mass-ordering). We will find these to be very different from that in our soft breaking model (with normal mass-ordering) [1]. So, the present model can be tested against that in Ref. [1] by the on-going and upcoming neutrino experiments. Using the neutrino data for θ s and (∆m 2 21 , ∆m 2 13 ) ( Table-1), and scanning the Dirac CP phaseangle δ D ∈ [0 • , 360 • ) , we can plot the two µ−τ breaking mixing angles, θ 13 (≡ δ x ) and θ 23 − 45 • (≡ δ a ), from (4.12a)-(4.12b) and (4.13), as functions of the theory parameter ζ cos ω and δ D . Our findings are depicted in Fig. 1(a)-(d) with the experimental inputs varied within 90% C.L. ranges and with ζ cos ω ∈ [−0.6, 0.6] in the natural perturbative region. Here we find that the theory prediction of θ 23 − 45 • lies in the range, 22) which is within the current experimental bounds. On the other hand the predicted θ 13 can saturate the current experimental limits, and has distinct distributions. Table-1. From the theory relations (4.12a)-(4.12b), we can further explore the correlation between the two µ−τ breaking mixing angles θ 13 and θ 23 − 45 • . This is displayed in Fig. 2, where we have varied the measured parameters within their 90% C.L. ranges, and input the Dirac-phase angle δ D ∈ [0, 2π) as well as \|ζ ′ \| 0.6 . The current 90% C.L. limits on θ 13 are shown by the shaded region (yellow), while the θ 13 sensitivities of the on-going Double Chooz [12], RENO [13] and Daya Bay [10] [32]. − 4 • θ 23 − 45 • 4 • ,(4. Inspecting Fig. 2, we find that the sharp edges on the two sides of the allowed parameter space are essentially determined by the lower bound given in (4.14), δ x \|δ a \| tan 2θ s , where the current data require, 2.2 tan 2θ s 3.1 at 90% C.L. (Table-1 (Table-1) and this feature is quite robust [6]. From Furthermore, as shown in Fig. 2, detecting a nonzero θ 13 3 • will strongly favor a nonzero θ 23 −45 • . Hence, we further encourage the improved measurements of θ 23 by Minos [7] and T2K [8], as well as future neutrino factory and super-beam facility [33,34]. Note that our previous soft breaking model [1] predicted a lower bound δ x \|δ a \| tan θ s with the slope 0.64 tan θ s 0.73 at 90% C.L., which is about 3.4 − 4.2 times smaller than the present model. This means that given the same nonzero deviation of θ 23 − 45 • , the current model will place a much stronger lower bound on θ 13 , higher than that in Ref. [1] by a factor of 3.4 − 4.2. Hence, the prediction of Fig. 2 is really encouraging for the upcoming neutrino oscillation experiments, which will probe the µ−τ violating observables θ 13 −0 • and θ 23 −45 • to much higher precision. Then, we analyze our model predictions for the low energy CP-violation (via Jarlskog invariant J) and the neutrinoless double-beta decays (via the element \|m ee \| of M ν ). From our theory construction in Sec. 2. 2, the original CP-phase e iω in the Dirac mass-matrix of seesaw Lagrangian is the common source of both low energy Dirac and Majorana CP-violations via the phase angles δ D and δφ . The Dirac CP-violation is characterized by the Jarlskog invariant J [35] in the light neutrino sector with nonzero CP-phase δ D and can be measured by the long baseline neutrino oscillation experiments. On the other hand, the neutrinoless double-beta decay observable \|m ee \| contains both δ D and Majorana CP-phase δφ . We can express the Jarlskog invariant J as follows [35], J ≡ 1 8 sin 2θ s sin 2θ a sin 2θ x cos θ x sin δ D = δ x 4 sin 2θ s sin δ D + O(δ 2 x , δ 2 a ) ,(4.23) where as defined earlier, δ x ≡ θ x and δ a ≡ θ a − π 4 . The solutions (4.12a)-(4.12b) leads to the correlation (4.13). We can input the neutrino data for mixing angles, (θ s , δ x ) ≡ (θ 12 , θ 13 ) , and mass-ratio, y ′ ≡ ∆m 2 21 /∆m 2 13 , as well as scanning the model-parameter ζ ′ in its perturbative range \|ζ ′ \| 0.6 . We then study the neutrinoless double-beta decays. Our present model predicts the inverted mass-ordering (IMO) with m 3 = 0 , so from (3.5a) we can derive the mass-matrix element \|m ee \| for neutrinoless double-beta decays, We plot the correlation between θ 13 and the Jarlskog invariant J in Fig. 3(a), and the neutrinoless ββ-decay observable M ee is depicted in Fig. 3 (b). For the analysis of Fig. 3(a), we have used Eq. (4.12a) where we vary the model-parameter ζ ′ ∈ [−0.6, 0.6] in its perturbative range. We scan all other measured parameters within their 90%C.L. ranges. The shaded region (yellow) in Fig. 3 is allowed by the neutrino data at 90% C.L. Fig. 3(a) shows that any nonzero J will lead to a lower bound on θ 13 due to δ x 4\|J\|/ sin 2θ s as inferred from Eq. (4.23). Combining the current upper limit θ 13 < 9.5 • (shaded region in yellow) with our parameter space in Fig. 3 which can be probed by the on-going neutrinoless double beta decay experiments [3]. M ee ≡ \|m ee \| = V * ej 2 m j = m 1 c 2 x c 2 s + s 2 s 1 + y ′ e −i2φ ≃ m 1 1 + 1 2 s 2 s y ′ − δ 2 x − 2s 2 s c 2 s δφ 2 ,(4. Before concluding this subsection, we compare our prediction ( work [36]. In Ref. [36], using a charged lepton perturbation, Friedberg and Lee derived a very interesting prediction, cos 2θ 23 = tan 2 θ 13 , leading to π 4 − θ 23 ≃ 1 2 θ 2 13 ≪ θ 13 ,(4.26) which does not contain CP phase and predicts a nearly maximal θ 23 . For comparison, we rewrite our predictions (4.18a)-(4.18b) in the same notations, π 4 − θ 23 ≃ 2 cos δ D π 4 − θ 12 θ 13 ≪ θ 13 , (Current Prediction), (4.27a) π 4 − θ 23 ≃ cos δ D θ 13 = O(θ 13 ) , (Prediction of Ref. [1]), (4.27b) where our correlations explicitly contain the CP-phase angle δ D . Moreover, our present model predicts a deviation π 4 − θ 23 to be significantly smaller than θ 13 as in (4.27a), due to the suppression of π 4 − θ 12 = 0.16 − 0.21 at 90% C.L. But, taking cos δ D = O(1) , we see that the right-hand-side of (4.27a) is larger than that of (4.26) by a factor of 4( π 4 − θ 12 )/θ 13 = (36.0 − 48.8 • )/θ 13 at 90% C.L., which is clearly bigger than one. On the other hand, our previous soft breaking model [1] predicts the two small µ−τ breaking observabes to be of the same order, π 4 − θ 23 = O(θ 13 ) , as in (4.27b). Hence, the predictions by Friedberg-Lee [36] and by us differ in a nontrivial and interesting way, which strongly motivate the on-going and future neutrino experiments for tests and resolution. Baryon Asymmetry from µ−τ Blind Seesaw and Direct Link to Low Energy In this subsection, we study the predictions of our µ−τ blind seesaw model for cosmological baryon asymmetry (matter-antimatter asymmetry) via thermal leptogenesis [37,38]. We build up the direct link between leptogenesis CP-asymmetry and the low energy Dirac CP-phase, and further predict the low energy leptonic Jarlskog invariant J [35]. Imposing the WAMP data on the baryon asymmetry [16], we predict a negative Jarlskog invariant, J < 0 , and derive a lower bound on the reactor mixing angle, θ 13 1 • . We also analyze the correlations of the leptogenesis scale with the low energy observables such as the Jarlskog-invariant J and neutrinoless ββ-decay parameter M ee [3]. We further deduce a lower bound on the leptogenesis scale for producing the observed baryon asymmetry. Our universe is exclusively dominated by matter rather than antimatter. The asymmetry of baryon-anti-baryon density n B − n B (≃ n B ) relative to the photon density n γ is measured to be a tiny nonzero ratio [16], η B ≡ n B − n B n γ = (6.19 ± 0.15) × 10 −10 . (4.28) The SM fails to generate the observed baryon asymmetry because of the too small CP-violations from CKM matrix and the lack of sufficiently strong first-order electroweak phase transition [39], which violate Sakharov's condition for baryongenesis [40]. It is important that the seesaw extension of the SM allows the thermal leptogenesis [37] with CP-violations originating from the neutrino sector and the lepton-number asymmetry produced during out-of-equilibrium decays of heavy Majorana neutrino N j into the lepton-Higgs pair ℓH and its CP-conjugatelH * . Then, the lepton asymmetry can be partially converted to a baryon asymmetry via the nonperturbative electroweak sphaleron [41] interactions which violate B + L [42] but preserve B − L [43,44] , η B = ξ f N f B−L = − ξ f N f L , where ξ is the fraction of B −L asymmetry converted to baryon asymmetry via sphaleron process [43] and ξ = 28/79 for the SM. The dilution factor f = N rec γ /N * γ = 2387/86 is computed by considering standard photon production from the onset of leptogenesis till recombination [44]. The effect of the heavier right-handed neutrino (N 2 ) decays will be washed out in the thermal equilibrium, only the lightest one (N 1 ) can effectively generate the net lepton asymmetry for M 1 ≪ M 2 . (In the numerical analysis below, we will consider the parameter space with M 2 /M 1 5 , to ensure the full washout of lepton asymmetry from N 2 -decays.) Thus, the net lepton asymmetry N f L is deduced as [44], N f L = 3 4 κ f ǫ 1 . Hence, we can derive the final baryon asymmetry, (4.29) where d ≡ 3ξ/(4f ) ≃ 0.96×10 −2 , and the factor κ f measures the efficiency of out-of-equilibrium N 1decays. The κ f is determined by solving the Boltzmann equation numerically [44,45]. In practice, useful analytical formulas for κ f can be inferred by fitting the numerical solution of the Boltzmann equation. We find it convenient to use the following fitting formula of κ f [45], formulas than (4.30) to the exact solution of κ f in the literature [44] agree with each other quite well for the relevant range of m 1 .) The CP asymmetry parameter ǫ 1 is defined as η B = − 3 ξ 4f κ f ǫ 1 = −d κ f ǫ 1 ,κ −1 f ≃ m 1 0.55×10 −3 eV 1.16 + 3.3×10 −3 eV m 1 ,(4.ǫ 1 ≡ Γ[N 1 → ℓH] − Γ[N 1 → ℓH * ] Γ[N 1 → ℓH] + Γ[N 1 → ℓH * ] = 1 4πv 2 F M 2 M 1 ℑm [(m † D m D ) 12 ] 2 (m † D m D ) 11 ,(4.31) where v denotes the vacuum expectation value of the SM Higgs boson. As we constructed in Sec. 2. 2, the Dirac mass-matrix m D is complex and provides the common origin of the µ−τ and CP breaking; the complexity of m D causes the difference between the decay widths Γ[N 1 → ℓH] and Γ[N 1 → ℓH * ] , and thus a nonzero CP asymmetry ǫ 1 = 0 . For the SM, the function F (x) in (4.31) takes the form, F (x) ≡ x 1 − (1 + x 2 ) ln 1 + x 2 x 2 + 1 1 − x 2 = − 3 2x + O 1 x 3 , (for x ≫ 1 ) .(4.32) For our numerical analysis of the thermal leptogenesis, the mass ratio M 2 /M 1 ≫ 1 and thus the above expanded formula of F (x) holds with good accuracy. Then, we proceed to compute the matrix elements, (m † D m D ) 11 = m 0 M 1 a 2 + 2b 2 = m 0 M 1 , (4.33a) (m † D m D ) 12 = − m 0 M 1 M 2 bc ζ ′ + ζe iω . (4.33b) So we can deduce the effective mass-parameter m 1 as introduced below (4.30), 34) and the imaginary part, where the RG running factor χ 1 = χ(M 1 , m Z ) is defined in Eqs. (3.18)-(3.19). Using Eq. (4.34) together with the neutrino data (Table-1), we find that the light neutrino mass-parameter m 1 lies in the 3σ range, 0.046 < m 1 /χ 1 < 0.053 eV, where the RG factor χ 1 ≃ 1.3 − 1.4 is evaluated numerically, as explained around the end of Sec. 3. 2. So,in Eq. (4.30) the second term on the right-hand-side is negligible and κ f is thus dominated by the first term. m 1 = m 0 ≃ χ 1 ∆m 2 13 ,(4.ℑm [(m † D m D ) 12 ] 2 = − 1 2 m 2 0 M 1 M 2 y ′ sin 2θ s sin δ D δ x ,(4. With these and from (4.31), we derive the CP asymmetry parameter ǫ 1 as follows, ǫ 1 ≃ 3y ′ m 0 M 1 16πv 2 sin 2θ s sin δ D δ x .(4.36) Finally, inspecting Eqs. (4.29), (4.31) and (4.32), we can derive, η B M 1 = −d κ f 3y ′ m 0 16πv 2 sin 2θ s sin δ D δ x .(4.37) Since the WMAP measurement (4.28) finds the baryon asymmetry η B > 0 , so we can infer the constraint, sin δ D < 0 , which restricts the Dirac phase angle, δ D ∈ (π, 2π) . Then, from Eq. (4.37) we compute the ratio η B /M 1 for any nonzero sin δ D , where we vary all measured quantities within their 90% C.L. ranges. Since 0 < \| sin δ D \| 1 , we can deduce a robust numerical upper bound, η B M 1 < 1.8 × 10 −23 GeV −1 .(4.38) Inspecting (4.37) we can also reexpress the leptogenesis scale M 1 in terms of baryon asymmetry η B and other physical observables, M 1 = −16πv 2 η B 3d κ f m 0 y ′ sin 2θ s sin δ D δ x . (4.39) With the data of η B from (4.28), we can plot, in Fig. 4, the leptogenesis scale M 1 as a function of Dirac CP-phase δ D , where all experimentally measured quantities are scanned within their 90% C.L. range (with 1500 samples). Fig. 4 reveals a robust lower bound on M 1 , M 1 > 3.5 × 10 13 GeV . With the above constraint on the parameter space from realizing successful thermal leptogenesis, we can rederive the correlation between θ 13 and θ 23 − 45 • , as shown in the new Fig. 6, which should be compared with Fig. 2 in Sec. 4. 1 (without requiring leptogensis). We note that the realization of successful thermal leptogenesis puts a general lower bound on the mixing angle θ 13 , should be compared to Fig. 3 where leptogenesis is not required. We see that due to the constraint from the observed baryon asymmetry, the parameter space of J > 0 is forbidden in Fig. 7(a). On the other hand, the constrained range for M ee in Fig. 7(b) is almost the same as Fig. 3(b), since Eq. (4.24) shows that the observable M ee has rather weak dependence on small NLO parameters δ x (= θ 13 ) and δφ via their squared terms. Thus, from Fig. 7(a)-(b), we infer the following constraints on J and M ee , We further analyze the correlations of the neutrinoless ββ-decay observable M ee with the Jarlskog invariant J and the light neutrino mass m 1 (≃ m 2 ) , in Fig. 8(a-b) and Fig. 8(c-d), respectively. The two left plots in Fig. 8(a) and ( shows that our model predictions are located at the upper boundaries of the whole parameter space, giving rise to the largest allowed M ee . This is very distinctive and highly testable. Furthermore, in with the reactor angle θ 13 in Fig. 9(a), and with the Jarlskog invariant J in Fig. 9 at, M 1 ∝ 1/\|J\| . This behavior is impressively reflected in Fig. 9(b), as expected. In addition, the relation, M 1 ∝ (δ x sin δ D ) −1 θ −1 13 , nicely explains the lower arched edge in Fig. 9(a). θ 13 1 • ,(4.−0.037 J − 0.0035 ,(4. Extension to General Three-Neutrino Seesaw In this subsection, we analyze the extension to the general neutrino seesaw with three right-handed neutrinos N ′ = (N 1 , N 2 , N 3 ) T , where N ′ is µ−τ blind. Then, in the µ−τ and CP symmetric limit, the mass-matrices m D and M R are extended to 3 × 3 matrices, m D =    āā ′ā′′ bcd bcd     ≡     σ 1 a σ 2 a ′ σ 3 a ′′ σ 1 b σ 2 c σ 3 d σ 1 b σ 2 c σ 3 d     , M R = diag(M 1 , M 2 , M 3 ) ,(4.M ν = m 0     a 2 +a ′2 +a ′′2 ab + a ′ c + a ′′ d ab + a ′ c + a ′′ d b 2 +c 2 +d 2 b 2 +c 2 +d 2 b 2 +c 2 +d 2     ≡   A B s B s C s C s C s   ,(4.45) from which we deduce the mass-eigenvalues and mixing angles, m 1,2 = 1 2 (A + 2C s ) ∓ (A − 2C s ) 2 + 8B 2 s = m 0 2 (a 2 + a ′2 + a ′′2 + 2b 2 + 2c 2 + 2d 2 ) ∓ [(a 2 + a ′2 + a ′′2 ) − 2(b 2 + c 2 + d 2 )] 2 + 8(ab + a ′ c + a ′′ d) 2 , (4.46a) m 3 = C s − C s = 0 , (4.46b) tan 2θ 12 = 2 √ 2B s A − 2C s = 2 √ 2\|ab + a ′ c + a ′′ d\| \|a 2 +a ′2 +a ′′2 −2(b 2 +c 2 +d 2 )\| , (4.46c) θ 23 = 45 • , θ 13 = 0 • , (4.46d) where the mass-spectrum remains the inverted mass-ordering (IMO). The third mass-eigenvalue m 3 vanishes because our µ − τ blind seesaw (4.44) predicts the seesaw mass-matrix (4.45) with its 23element equal to the 22-element and 33-element. This is also a general feature of any µ−τ symmetric IMO scheme at the LO, as to be shown in (5.5) of Sec. 5. 1. Furthermore, we will demonstrate shortly that the third mass-eigenvalue m 3 = 0 actually holds up to the NLO after including the µ−τ and CP breaking in our analysis. So this resembles very much the minimal seesaw we studied earlier. Similar to Eqs. (2.11) and (2.13) in Sec. 2. 1, we can realize the IMO at the LO of three-neutrino seesaw, m 1 = m 2 = m 0 , which leads to the three extended conditions, (a 2 + a ′2 + a ′′2 ) + 2(b 2 + c 2 + d 2 ) = 2 , (4.47a) (a 2 + a ′2 + a ′′2 ) − 2(b 2 + c 2 + d 2 ) = 0 , (4.47b) ab + a ′ c + a ′′ d = 0 . (4.47c) With these we deduce from (4.45) the generic LO seesaw mass-matrix for the IMO, (4.48) which is the same as the LO mass-matrix (2.31a) we derived earlier for the minimal seesaw. Hence, despite that the LO mass-matrix M in the final form of (4.48) which is parameter-free except an overall mass-scale. As a result of the IMO conditions (4.47), we note that the solar angle formula (4.46c) gives tan 2θ 12 = 0 0 at the LO, which is now undetermined. So, the θ 12 has to be derived from the NLO contributions related to µ−τ breaking terms. Before getting into detail, it is convenient to infer θ 12 by using the l ′ Hôpital rule, similar to what we did in Sec. 2. 1 for the minimal seesaw. Thus we have, M (0) ν = m 0      a 2 +a ′2 +a ′′2 ab+a ′ c+a ′′ d ab+a ′ c+a ′′ d b 2 +c 2 +d 2 b 2 +c 2 +d 2 b 2 +c 2 +d 2      = m 0    1 0 0 1 2 1 2 1 2   ,tan 2θ 12 = \|a\| √ 2 \|b\| , (4.49a) for µ−τ breaking arising from the deviation in the element b of m D , or tan 2θ 12 = \|a ′ \| √ 2 \|c\| ,(4.m D =     σ 1 a σ 2 a ′ σ 3 a ′′ σ 1 b σ 2 c 1 σ 3 d σ 1 b σ 2 c 2 σ 3 d     , (4.50a) c 1 = c 1 − ζ ′ , c 2 = c 1 − ζe iω . (4.50b) Thus we can deduce the NLO part of the seesaw mass-matrix M ν = M (0) ν +δM (1) ν for light neutrinos, δM (1) ν = m 0      0 −a ′ c ζ ′ −a ′ c ζe iω −2c 2 ζ ′ −c 2 (ζ ′ +ζe iω ) −2c 2 ζe iω      ,(4.and φ ′ ≡ φ 3 − φ 1 = φ ′ 0 + δφ ′ . Note that the LO phases vanish, α i0 = φ 0 = φ ′ 0 = 0 . So the NLO elements of M ν are reconstructed as follows, Then, with the NLO µ−τ symmetric parts from (2.33) and (4.53), we deduce the solar angle θ 12 , δA = m 0 z + s 2 s 2 y ′ − i2(s 2 s δφ + δᾱ 1 ) , (4.53a) δB s = m 0 2 √ 2 sin 2θ s − 1 2 y ′ + i2δφ , (4.53b) δC s = m 0 2 z + z ′ + c 2 s 2 y ′ − i(2c 2 s δφ + δα 2 + δα 3 ) , (4.53c) δD = m 0 2 z − z ′ + c 2 s 2 y ′ − i(2c 2 s δφ + δα 2 + δα 3 ) , (4.53d) δB a = − m 0 √ 2 e iδ D δ x , (4.53e) δC a = −m 0 δ a + i 2 (δα 2 − δα 3 ) ,(4.tan 2θ s = − a ′ √ 2 c ,(4.54) which coincides with Eq. (4.5) as we derived earlier for the minimal seesaw. Next, connecting the µ−τ anti-symmetric parts in (2.33) and (4.53) gives, m 0 2 a ′ c(ζ ′ − ζe iω ) = − m 0 √ 2 e iδ D δ x , (4.55a) −m 0 c 2 (ζ ′ − ζe iω ) = −m 0 δ a + i 2 (δᾱ 2 − δᾱ 3 ) ,(4.55b) from which we arrive at cos δ D δ x = a ′ c √ 2 ζ ′ − ζ cos ω , (4.56a) sin δ D δ x = − a ′ c √ 2 (ζ sin ω) , (4.56b) δ a = c 2 ζ ′ − ζ cos ω , (4.56c) δᾱ 2 − δᾱ 3 = −2c 2 (ζ sin ω) . (4.56d) Here for the left-hand-sides of (4.55a)-(4.55b) we have used the Eq. (3.19) to evolve the overall massparameter m 0 from seesaw scale down to the corresponding m 0 at low energy. Finally, using Eqs. (4.54), (4.56a) and (4.56c), we derive the key correlation between two low energy µ−τ breaking observables δ a and δ x , δ a = − cot 2θ s cos δ D δ x ,(4.57) which coincides with (4.13) as we derived earlier for the minimal seesaw. We note that it is also possible to construct the common origin of µ−τ and CP breaking in the element d of m D , instead of the element c . Then we can rewrite the Dirac mass-matrix (4.50) as m D =     σ 1 a σ 2 a ′ σ 3 a ′′ σ 1 b σ 2 c σ 3 d 1 σ 1 b σ 2 c σ 3 d 2     , (4.58a) d 1 = d 1 − ζ ′ , d 2 = d 1 − ζe iω . (4.58b) This results in the following NLO seesaw mass-matrix, δM (1) ν = m 0      0 −a ′′ d ζ ′ −a ′′ d ζe iω −2d 2 ζ ′ −d 2 (ζ ′ +ζe iω ) −2d 2 ζe iω      ,(4.59) from which we derive the solar angle, 60) and the reconstruction conditions, tan 2θ s = − a ′′ √ 2 d ,(4.cos δ D δ x = a ′′ d √ 2 ζ ′ − ζ cos ω , (4.61a) sin δ D δ x = − a ′′ d √ 2 (ζ sin ω) , (4.61b) δ a = d 2 ζ ′ − ζ cos ω ,(4. 61c) δᾱ 2 − δᾱ 3 = −2d 2 (ζ sin ω) . (4.61d) So, from Eqs. (4.60), (4.61a) and (4.61c), we can readily derive the correlation between two µ − τ breaking observables, δ a = − cot 2θ s cos δ D δ x , (4.62) which coincides with (4.57). In summary, the general three-neutrino seesaw (with right-handed neutrinos being µ−τ blind) still predicts the inverted mass-ordering (IMO) for light neutrinos [cf. Eqs. (4.46a)-(4.46b)]. Despite that the LO conditions (4.47) for the IMO contains two new parameters (a ′′ , d), the LO seesaw mass-matrix (4.48) is shown to take the same form as in the minimal seesaw. Furthermore, the NLO µ−τ and CP breaking part of our seesaw mass-matrix (4.51) or (4.59) exhibits the same structure as in the minimal seesaw. This makes our final physical prediction of the correlation (4.57) or (4.62) coincides with (4.13). Hidden Symmetry and Dictation of Solar Mixing Angle So far, by analyzing the µ−τ symmetry and its breaking, we have studied the atmospheric mixing angle θ 23 and the reactor mixing angle θ 13 in great detail. As shown in Table-1, the solar mixing angle θ 12 is best measured [49,50] among the three mixing angles. In this section we will clarify the connection between µ−τ breaking and the determination of the solar mixing angle θ 12 for both inverted mass-ordering (IMO) (cf. Sec. 2) and normal mass-ordering (NMO) [1]. Then, we analyze the general model-independent Z 2 ⊗ Z 2 symmetry structure in the light neutrino sector, and map it into the seesaw sector, where one of the Z 2 symmetries corresponds to the µ−τ symmetry Z µτ 2 and another the hidden symmetry Z s 2 (which we revealed in [1] for the NMO of light neutrinos and is supposed to dictate θ 12 ). We will further derive the general consequences of this Z s 2 and its possible violation in the presence of µ−τ breaking for cases either with or without neutrino seesaw, regarding the θ 12 determination. µ−τ Breaking versus θ 12 Determination: Inverted Mass-Ordering In Ref. [1] we proved that the solar mixing angle θ 12 (≡ θ s ) is not affected by the soft µ−τ breaking from the neutrino seesaw, and we revealed a hidden symmetry Z s 2 for both the seesaw Lagrangian and the light neutrino mass-matrix which dictates θ s , where the normal mass-ordering (NMO) is realized. In this subsection, we generally analyze mass-eigenvalues and mixing angles for the µ−τ symmetric mass-matrix of light neutrinos under the inverted mass-ordering (IMO). Then we explain why the µ−τ breaking is invoked for the θ s determination and why the hidden symmetry Z s 2 will be violated. The µ−τ blind seesaw constructed in Sec. 2 belongs to an explicit realization of the IMO scheme. Let us start with the general µ−τ symmetric mass-matrix for light neutrinos, (5.1a) which can be diagonalized as follows [1,53], M (s) ν =   A B s B s C s D C s   ,m 1,2 = 1 2 [A + (C s +D)] ∓ [A − (C s +D)] 2 + 8B 2 s , (5.2a) m 3 = C s − D , (5.2b) tan 2θ s = 2 √ 2 B s A − (C s +D) , θ a = 45 • , θ x = 0 • . (5.2c) Substituting (5.2c) into (5.2a), we arrive at m 1,2 = 1 2 {[A + (C s +D)] ∓ \|A − (C s +D)\| sec 2θ s } . (5.3) For the IMO scheme, we have the mass-spectrum m 2 m 1 ≫ m 3 , where a small m 3 = 0 is also generally allowed for the analysis below. So we can derive, for the general IMO scheme, (5.4) where in the last step we have used the neutrino data (Table-1 For the µ−τ blind seesaw defined in Sec. 2. 1, we find that the light neutrino mass-spectrum must be inverted ordering, as given in Eqs. (2.10a)-(2.10b). So, following the consistency with neutrino data (5.4) and matching the reconstruction formalism (5.5) for the IMO scheme, we can explicitly realize the degeneracy m 1 = m 2 at the LO by imposing the condition (2.11) on the elements of m D . A − (C s +D) A + (C s +D) = m 2 − m 1 m 2 + m 1 cos 2θ s ≃ ∆m 2 21 4∆m 2 13 cos 2θ s = (2.1 − 3.8) × 10 −3 , (Here m 3 = 0 is an outcome of the minimal seesaw.) Thus, as expected, we find a problem for the θ s determination in the µ−τ symmetric limit, tan 2θ s = 2 √ 2\|ab + a ′ c\| \|a 2 + a ′2 − 2(b 2 + c 2 )\| = 0 0 , (5.7) which is just an explicit realization of our above general IMO analysis [cf. (5.5)]. Hence, it is clear that θ s must be inferred from the NLO formula (5.6), where the NLO elements will be predicted by a given model, e.g., by the first four expressions in Eq. (2.33) in the µ−τ blind seesaw with all NLO corrections arising from the µ−τ breaking [51]. Thus the explicit expression of θ s from such underlying models will depend on how the µ − τ breaking is constructed. This is contrary to the neutrino seesaw with normal mass-ordering (NMO) of light neutrinos as studied in Ref. [1], where we find that the formula of tan 2θ s [cf. (5.2c) above] is well defined in the µ−τ symmetric limit. As we noted in Sec. 2. 1, the structure 0 0 in Eq. (5.7) allows us to use the l ′ Hôpital rule on (5.7) by taking the first derivatives on both its numerator and denominator. We need to decide for which parameter in (5.7) constructions of µ−τ breaking for m D lead to two different θ s formulas above. This is an essential difference from the soft µ−τ breaking model in Ref. =          \|a ′ \| √ 2 \|c\| , (µ−τ breaking in c) , \|a\| √ 2 \|b\| , (µ−τ breaking in b) ,(5. [1], where θ s is dictated by the hidden symmetry Z s 2 under which the soft µ−τ breaking term in M R is an exact singlet. In the next subsections we will analyze the general model-independent Z 2 ⊗ Z 2 symmetry in the light neutrino sector, and then map it into the seesaw sector. This allows us to explore, at a deeper level, the Z s 2 symmetry and its possible partial violation under the µ−τ breaking in a unified way, concerning θ s determination. Z s 2 Symmetry under General µ−τ Breaking and General Determination of θ 12 This subsection consists of two parts. In Sec. 5. 2. 1, we analyze the general model-independent Z 2 ⊗Z 2 symmetry structure of the light neutrino sector, in both the mass and flavor eigenbases. We will show that, in the flavor eigenbasis of light neutrinos, one of the Z 2 's is the Z µτ 2 symmetry which predicts the mixing angles (θ 23 , θ 13 ) = (45 • , 0 • ) , and another is the Z s 2 symmetry which generally dictates the solar angle θ 12 by its group parameter (allowing deviations from the conventional tri-bimaximal mixing ansatz). With general µ − τ breaking parameters, we will derive a nontrivial correlation between the two µ − τ breaking observables which is necessary for holding the Z s 2 symmetry. In Sec. 5. 2. 2, we will further analyze the general µ−τ breaking in the light neutrino mass-matrix M ν and derive a nontrivial consistency condition to hold the Z s 2 symmetry. From this condition and using the general reconstruction formalism of Sec. 3. 1, we will deduce the same correlation between the µ−τ breaking observables, for both the normal mass-ordering and inverted mass-ordering of light neutrinos (without approximating the lightest neutrino mass to zero) [52]. Z s 2 Symmetry for General Determination of Solar Angle θ 12 Let us inspect the flavor symmetries in the lepton and neutrino sectors. In general, the lepton and neutrino sectors are expected to obey different flavor symmetries. After spontaneous symmetry breaking, the residual symmetry groups for the lepton and neutrino mass-matrices may be denoted as G ℓ and G ν , respectively. Consider the symmetry transformations F j ∈ G ℓ and G j ∈ G ν for left-handed leptons and neutrinos. Thus the mass-matrices of leptons (M ℓ ) and light neutrinos (M ν ) will satisfy the invariance equations [57], F † j M ℓ M † ℓ F j = M ℓ M † ℓ , G T j M ν G j = M ν . (5.9) The above mass-matrices can be diagonalized by unitary rotations for left-handed leptons and neutrinos, U † ℓ M ℓ M † ℓ U ℓ = D ℓ ≡ diag m 2 e , m 2 µ , m 2 τ , U T ν M ν U ν = D ν ≡ diag (m 1 , m 2 , m 3 ) . (5.10) Then, combining the invariance equations (5.9) and diagonalization equations (5.10) result in (5.11) where d ℓ and d ν are diagonal phase-matrices obeying d † ℓ d ℓ = I 3 and d 2 ν = I 3 (with I 3 the 3 × 3 unit matrix), which require d ℓ = diag(e iγ 1 , e iγ 2 , e iγ 3 ) and d ν = diag(±1, ±1, ±1) . So, up to an overall phase factor, the {d (j) ℓ } forms the generic Abelian group U (1) ⊗ U (1) = G ℓ for leptons, and {d (j) ν } has only two independent d ν , 12) forming the generic discrete group Z 2 ⊗ Z 2 = G ν for neutrinos. From Eq. (5.11) the following consistency solutions are deduced, (5.13) This proves that {F j } and {d ν } the "kernel representations", with which the equivalent "flavor representations" {F j } and {G j } can be generated as in (5.13) via the disgonalization matrices U ℓ and U ν , respectively. Hence, we are free to choose an equivalent lepton symmetry group representation {F j } = {d (j) ℓ } with U ℓ = I 3 , and accordingly, rewrite the representation of neutrinos symmetry group, (5.14) with V = U † ℓ U ν = U ν equal to the physical PMNS mixing matrix as defined in Eq. (3.3) of Sec. 3[54]. Let us rewrite the PMNS matrix (3.3) U † ℓ F † j M ℓ M † ℓ F j U ℓ = d † ℓ D ℓ d ℓ = D ℓ , U T ν G T j M ν G j U ν = d T ν D ν d ν = D ν ,d (1) ν = diag(1, 1, −1) , d (2) ν = diag(−1, 1, 1) ,(5.F j = U ℓ d (j) ℓ U † ℓ , G j = U ν d (j) ν U † ν .G j = V d (j) ν V † ,, V = U ′′ U U ′ = V ′ U ′ , with V ′ ≡ U ′′ U as introduced in Eq. (3.4). So we see that the Majorana phase-matrix U ′ cancels in G j , (5.15) According to the most general reconstruction formulation in Sec. 3. 1, we can expand the matrix V ′ to NLO in terms of the small parameters, (δ a , δ x , δα i ) , where (δ a , δ x ) characterizes the low energy µ−τ breaking and the CP-angle δα i arises from the phase matrix U ′′ (which is not directly observable and only needed for the consistency of diagonalizing the mass matrix M ν ). There is no need to expand the Dirac CP-phase e iδ D itself since it is always associated with the small µ − τ breaking parameter δ x . So, under this expansion we derive G j = V ′ d (j) ν V ′ † .V ′ = V s + δV ′ , (5.16) with V s =     c s −s s 0 ss √ 2 cs √ 2 − 1 √ 2 ss √ 2 cs √ 2 1 √ 2     , (5.17a) δV ′ =      ic s δα 1 is s δα 1 −δ x e −iδ D − ssδa+csδxe iδ D +issδα 2 √ 2 −csδa+ssδxe iδ D −icsδα 2 √ 2 − δa−iδα 2 √ 2 ssδa+csδxe iδ D −issδα 3 √ 2 csδa−ssδxe iδ D −icsδα 3 √ 2 − δa+iδα 3 √ 2      . (5.17b) Let us first consider the µ−τ symmetric limit with V ′ = V s . So substituting V s into Eq. (5.15) we deduce, G µτ ≡ G 1 = V s d (1) ν V † s =   1 0 0 0 0 1 0 1 0   ,(5.18) which, as expected, just gives the Z µτ 2 symmetry-transformation matrix G ν for light neutrinos as we explicitly constructed in (2.5) earlier for the seesaw Lagrangian (2.2). Next, we derive the symmetry-transformation matrix G 0 s corresponding to d (2) ν of (5.12) in the µ−τ symmetric limit with ( δV ′ = 0 ), (5.19b) which is symmetric since V s and d (j) ν are real. (For the same reason G µτ is also symmetric.) In the last step, for convenience we have defined, 20) with k (or equivalently, tan θ s ) serving as the group parameter of Z s 2 , tan θ s = k , (5.21) where we can always choose the convention of θ s ∈ 0, π 2 such that, tan θ s = k 0 . Noting (d (j) ν ) 2 = I 3 and using the relation G 0 G 0 s = V s d (2) ν V † s =    s 2 s −c 2 s − √ 2 s s c s − √ 2 s s c s c 2 s −s 2 s c 2 s    (5.19a) = 1 1 + k 2    k 2 −1 − √ 2k − √ 2k 1 −k 2 1   ,(s s , c s ) = (k, 1) √ 1 + k 2 ,(5.s = V s d(2) ν V † s , we can readily verify (G 0 s ) 2 = I 3 and thus indeed G 0 s ∈ Z s 2 . Hence, the solar angle θ s is dictated by the group parameter k of the 3-dimensional representation of the hidden symmetry Z s 2 [55]. We stress that the G 0 s in (5.19b), as the 3d representation of Z s 2 , is uniquely fixed by the µ−τ symmetric matrix V s ; we call Z s 2 a hidden symmetry since it generally exists for any µ−τ symmetric neutrino mass-matrix M i.e., any µ−τ symmetric neutrino sector must automatically contain the hidden Z s 2 symmetry which dictates the solar angle θ s as in (5.21). As pointed out in Ref. [1], a particular choice of k = 1 √ 2 gives the conventional tri-bimaximal ansatz [56] tan θ s = 1 √ 2 (θ s ≃ 35.3 • ), but other choices of the group parameter k allow deviations from the conventional tri-bimaximal mixing, e.g., we can make a very simple choice of k = 2 3 , leading to tan θ s = 2 3 (θ s ≃ 33.7 • ), which agrees to the neutrino data equally well (cf. Table-1 in Sec. 2) or even better (cf. Table-2 in "Note Added in Proof"). The Z s 2 itself, as the minimal hidden symmetry for θ s , is not restrictive enough to fix its group parameter k . But, extending the Z µτ 2 ⊗ Z s 2 symmetry into a larger simple group can fix a particular k value and thus the solar angle θ s . As we demonstrated in Sec. 6.3 of Ref. [1], a simple example is to enlarge Z µτ 2 ⊗ Z s 2 to the permutation group S 4 [57], under which we can infer k = 1 √ 2 , corresponding to the tri-bimaximal mixing θ s = arctan 1 √ 2 . Then, we examine how such a Z s 2 symmetry could possibly survive after including general µ−τ breaking terms in V ′ = V s + δV ′ . Expanding the small µ−τ breaking parameters up to NLO, we can derive the symmetry-transformation matrix G s corresponding to d (2) ν of (5.12), G s ≡ G 2 = V ′ d (2) ν V ′ † = V s d (2) ν V † s + (V s d (2) ν δV ′ † + δV ′ d (2) ν V † s ) ≡ G 0 s + δG s ,(5.] =      0 − s 2s δa+2c 2 s cos δ D δx √ 2 s 2s δa+2c 2 s cos δ D δx √ 2 − s 2s δa+2c 2 s cos δ D δx √ 2 −2s 2 s δ a −s 2s cos δ D δ x 0 s 2s δa+2c 2 s cos δ D δx √ 2 0 2s 2 s δ a +s 2s cos δ D δ x      , (5.23a) Im[δG s ] =      0 s 2s (δα 1 −δα 2 )−2c 2 s sin δ D δx √ 2 s 2s (δα 1 −δα 3 )+2c 2 s sin δ D δx √ 2 − s 2s (δα 1 −δα 2 )−2c 2 s sin δ D δx √ 2 0 s 2 s (δα 2 −δα 3 )+s 2s sin δ D δ x − s 2s (δα 1 −δα 3 )+2c 2 s sin δ D δx √ 2 −s 2 s (δα 2 −δα 3 )−s 2s sin δ D δ x 0      , (5.23b) where s 2s ≡ sin 2θ s . Because the symmetry transformation G s ∈ Z s 2 , we have the condition G 2 s = I 3 . Then, expanding this up to the NLO, we have verified the consistency condition, (5.24) Requiring that the Z s 2 symmetry persists under µ−τ breaking, i.e., the form of G s remains unaffected by the µ−τ violation, we have the condition, {G 0 s , δG s } = 0 .G s = G 0 s , or, δG s = 0 . (5.25) Thus, with (5.23a)-(5.23b), we can derive the following solutions, (5.26) from the real part condition Re[δG s ] = 0 , and ; but now it is re-derived by requiring that the Z s 2 symmetry persists in the presence of general low energy µ − τ breaking. In addition, the above Eq. (5.27b) also coincides with Eq. (4.12b) of Ref. δ a δ x = − cot θ s cos δ D ,2δα 1 = δα 2 + δα 3 , (5.27a) δα 2 − δα 3 = −2 cot θ s sin δ D δ x = 2 tan δ D δ a ,(5. [1]. As we will demonstrate in the next subsection, the Z s 2 symmetry is independent of the soft µ−τ breaking in the seesaw model of Ref. [1]. We note that in the current construction of common µ−τ and CP breaking with seesaw mechanism (Sec. 2. 2), such a Z s 2 symmetry is not fully respected, hence the correlation (5.26) no longer holds and we have predicted a modified correlation (4.13), which can be tested against (5.26) by the on-going and upcoming neutrino oscillation experiments. To summarize, as we have demonstrated above from general low energy reconstruction formulation, the transformations G µτ = G 1 and G s = G 2 in the µ−τ symmetric limit correspond to the discrete groups Z µτ 2 ⊗ Z s 2 , which are equivalent to and originate from the generic symmetry Z 2 ⊗ Z 2 in the neutrino mass-eigenbasis because they are connected by the similarity transformations via (5.13). The µ − τ symmetry Z µτ 2 has been known before, and the hidden symmetry Z s 2 (as the minimal group dictating the solar angle θ s ) was revealed by Ref. [1] in the context of neutrino seesaw. In this work, we further find that requiring the symmetry Z s 2 to persist in the presence of most general µ−τ breaking terms will predict a new correlation (5.26) between the small µ−τ breaking parameters (δ a , δ x ) . As we will prove below, the Z s 2 symmetry is respected by a class of soft µ−τ breaking seesaw models in Ref. [1], but is partially violated in the present µ−τ breaking seesaw model (Sec. 2. 2). Z s 2 Symmetry and Neutrino Mass-Matrix with General µ−τ Breaking In this subsection, we directly analyze the generally reconstructed light neutrino mass-matrix M ν under the hidden symmetry Z s 2 and the determination of solar angle θ s . The mass-matrix (3.1) can be uniquely decomposed into the µ−τ symmetric and anti-symmetric parts, M ν = M (s) ν + δM (a) ν ,(5.28) with M (s) ν =   A B s B s C s D C s   , δM (a) ν =   0 δB a −δB a δC a 0 −δC a   , (5.29a) B s ≡ 1 2 (B 1 + B 2 ) , C s ≡ 1 2 (C 1 + C 2 ) , (5.29b) δB a ≡ 1 2 (B 1 − B 2 ) , δC a ≡ 1 2 (C 1 − C 2 ) , (5.29c) where we generally allow m 1 m 2 m 3 = 0 . Then, from (5.9), the invariance equation of M ν under G s corresponds to G † s (M (s) ν + δM (a) ν )G s = M (s) ν + δM (a) ν ,(5.30) which uniquely gives, G † s M (s) ν G s = M (s) ν , (5.31a) G † s δM (a) ν G s = δM (a) ν . (5.31b) Note that two possibilities may exist: (i). The Z s 2 symmetry is a full symmetry of the light neutrino mass-matrix M ν if both (5.31a) and (5.31b) hold. (ii). The Z s 2 symmetry is a partial symmetry of M ν if the µ−τ anti-symmetric part M (a) ν breaks (5.31b). We can prove that the Z s 2 is always a symmetry of the µ−τ symmetric part M (s) ν and generally holds (5.31a). Substituting (5.16) into (3.4) and noting that the decomposition (5.28) is unique, we can reconstruct the µ−τ symmetric and anti-symmetric parts of M ν , respectively, M (s) ν = V * s D ν V † s , (5.32a) δM (a) ν = V * s D ν δV ′ † + δV ′ * D ν V † s + δV ′ * D ν δV ′ † = V * s D ν δV ′ † + δV ′ * D ν V † s + O(δ 2 j ) ,(5.32b) where δ j denotes all possible NLO parameters under consideration (such as δ x , δ a and y ′ , etc). This shows that the µ − τ symmetric part M (s) ν is diagonalized by V s . Hence, the corresponding Z s 2 transformation matrix is just G 0 s , as given by (5.19). The G 0 s must be the symmetry of M (s) ν and thus always holds the invariance equation (5.31a). This proves that the solar mixing angle θ s (as contained in the rotation matrix V s and symmetry transformation matrix G 0 s ) is generally dictated by the Z s 2 symmetry, independent of any specific model. On the other hand, the validity of (5.31b) is highly nontrivial because the requirement of G s = G 0 s [cf. (5.25)] does not generally hold under µ−τ breaking, and it has to be checked case by case. As we will prove in Sec. 5. 3, the µ−τ anti-symmetric part M (a) ν will break Z s 2 in the current µ−τ blind seesaw (Sec. 2), while it preserves Z s 2 in the soft µ−τ breaking seesaw of Ref. [1]. Using the expression of G s [Eqs. (5.19a) and (5.25)], we can derive the solution from (5.31a) for the µ−τ symmetric part, tan 2θ s = 2 √ 2 B s A − (C s +D) , (5.33) and another solution from (5.31b) for the µ−τ anti-symmetric part, 34) which further leads to, tan θ s = − √ 2 δB a δC a ,(5.tan 2θ s = − 2 √ 2 δB a δC a δC 2 a − 2δB 2 a . (5.35) Hence, if the Z s 2 would be a full symmetry of M ν (including its µ−τ breaking part), the two solutions (5.33) and (5.35) for the solar angle θ s must be identical, leading to a nontrivial consistency condition, tan 2θ s = 2 √ 2 B s A − (C s + D) − 2 √ 2 δB a δC a δC 2 a − 2δB 2 a . (5.36) An explicit counter example to this condition will be given in Sec. 5. 3. 2. In the following, we apply the most general reconstruction formalism (Sec. 3. 1) to compute the µ−τ symmetric and anti-symmetric parts of light neutrino mass-matrix M ν = M (s) ν + M (a) ν . With these, we will explicitly verify Eq. (5.33) by using the elements of µ − τ symmetric M Reconstruction Analysis for General Normal Mass-Ordering Scheme Eq. (3.5) reconstructs all the elements of M ν in terms of three mass-eigenvalues, three mixing angles and relevant CP-phases. The normal mass-ordering (NMO) has the spectrum m 1 < m 2 ≪ m 3 , so we can define the small ratios, (5.38) with α 10 = α 20 = α 30 ≡ α 0 , α 30 + φ 30 = nπ , and the NLO elements in δM ν , δA = e −i2α 0 e −i2φ 10 c 2 s y 1 + e −i2φ 20 s 2 s y 2 m 30 , (5.39a) y 1 ≡ m 1 m 3 , y 2 ≡ m 2 m 3 , y 3 ≡ m 3 −m 30 m 3 . (5.37) M (0) ν = m 30   0 0 0 1 2 − 1 2 1 2   ,δB s = 1 2 √ 2 e −i2α 0 e −i2φ 10 y 1 − e −i2φ 20 y 2 sin 2θ s m 30 , (5.39b) δC s + δD = e −i2α 0 e −i2φ 10 s 2 s y 1 + e −i2φ 20 c 2 s y 2 m 30 , (5.39c) δB a = 1 √ 2 e −iδ D δ x m 30 , (5.39d) δC a = 1 2 [2δ a − i (δα 2 − δα 3 )] m 30 . (5.39e) From (5.38), we have A (0) s = B (0) s = C (0) s + D (0) = 0 . Thus, using the µ−τ symmetric NLO elements (5.39a)-(5.39c), we can compute the ratio, 2 √ 2 B s A s − (C s +D) = (e −i2φ 10 y 1 − e −i2φ 20 y 2 ) sin 2θ s (e −i2φ 10 y 1 − e −i2φ 20 y 2 )(c 2 s − s 2 s ) = tan 2θ s ,(5.40) which explicitly verifies our Eq. (5.33) [as generally derived from the invariance equation (5.31a) under Z s 2 ] for the current NMO scheme. This is an explicit proof up to NLO that for a general NMO scheme the µ−τ symmetric mass-matrix M (s) ν = M (0) ν + δM (s) ν does hold the Z s 2 symmetry. Then, using the µ−τ anti-symmetric elements (5.39d)-(5.39e), we derive the ratio, (5.41) where in the last step we have used Eq. (5.34) under the assumption that Z s 2 symmetry also holds for the µ−τ anti-symmetric mass-matrix M (a) ν , i.e., the validity of the invariance equation (5.31b). Analyzing the real and imaginary parts of (5.41), we deduce two relations, − √ 2 δB a δC a = − e −iδ D δ x δ a − i 2 (δα 2 −δα 3 ) = tan θ s ,δ a = −δ x cot θ s cos δ D , (5.42a) δα 2 − δα 3 = 2 tan δ D δ a . (5.42b) These are in perfect agreement with (5.26) and (5.27b), which are generally derived under a single assumption that the Z s 2 symmetry persists in the presence of µ−τ breaking. But, as will be shown in Sec. 5. 3. 2, this assumption does not generally hold, and the current µ−τ blind seesaw (Sec. 2. 2) provides a nontrivial counter example. Reconstruction Analysis for General Inverted Mass-Ordering Scheme For the inverted mass-ordering (IMO), the light neutrinos have the spectrum m 2 m 1 ≫ m 3 , so we can define the small ratios, z 1 ≡ m 1 − m 0 m 1 , z 2 ≡ m 2 − m 0 m 1 , z 3 ≡ m 3 m 1 ,(5.43) where we have z 1 = z and z 2 ≃ z + 1 2 y ′ in connection to the NLO parameters (y ′ , z) introduced in Eqs. (3.7)-(3.9) of Sec. 3. 2. Then we have the independent NLO parameters for the IMO analysis, (z 1 , z 2 , z 3 , δ a , δ x , δα i , δφ i ). Expanding them perturbatively, we derive the LO form of the symmetric mass-matrix M ν , (5.44) with α 10 = α 20 = α 30 = α 0 , φ 10 = φ 20 = −α 0 , and the NLO elements of M ν , M (0) ν = m 0   1 0 0 1 2 1 2 1 2   ,δA = m 0 c 2 s z 1 + s 2 s z 2 − i2(c 2 s δφ 1 + s 2 s δφ 2 + δα 1 ) , (5.45a) δB s = 1 2 √ 2 m 0 sin 2θ s [z 1 − z 2 − i2 (δφ 1 − δφ 2 )] , (5.45b) δC s + δD = m 0 s 2 s z 1 + c 2 s z 2 − i 2s 2 s δφ 1 + 2c 2 s δφ 2 + δα 2 + δα 3 , (5.45c) δB a = − 1 √ 2 m 0 e iδ D δ x , (5.45d) δC a = −m 0 δ a + i 2 (δα 2 − δα 3 ) . (5.45e) From (5.44), we have B (0) s = 0 and A (0) s − (C(0) s +D (0) ) = 0 . So using the µ−τ symmetric NLO elements (5.45a)-(5.45c), we can compute the ratio, 2 √ 2 B s A s − (C s + D) = sin 2θ s [z 1 − z 2 − i2 (δφ 1 − δφ 2 )] cos 2θ s [z 1 − z 2 − i2 (δφ 1 − δφ 2 )] − i (2δα 1 − δα 2 − δα 3 ) , (5.46) from which we deduce the consistent solution, under Z s 2 ] for the current IMO scheme. Also the above solution (5.47b) exactly coincide with the general Eq. (5.27). The above is an explicit proof up to NLO that for a general IMO scheme the µ−τ symmetric mass-matrix M (s) 2 √ 2 B s A s − (C s +D) = tan 2θ s ,(5.ν = M (0) ν + δM (s) ν does hold the Z s 2 symmetry. Then, with the µ−τ anti-symmetric elements (5.45d)-(5.45e), we further evaluate the ratio, (5.48) where in the last step we have applied (5.34) under the assumption that the µ−τ anti-symmetric massmatrix M (a) ν also respects the Z s 2 symmetry, i.e., the invariance equation (5.31b) holds. Inspecting the real and imaginary parts of (5.48), we deduce the following, ν . But, as we will prove in Sec. 5. 3. 2, the above assumption is not generally true and for the µ−τ blind seesaw with IMO (Sec. 2. 2) the Z s 2 symmetry is violated by δM (a) ν . − √ 2 δB a δC a = − e iδ D δ x δ a + i 2 (δα 2 − δα 3 ) = tan θ s ,δ a = −δ x cot θ s cos δ D ,(5.δα 2 − δα 3 = 2 tan δ D δ a ,(5. So far we have explicitly proven the relations (5.26) and (5.27) for general NMO and IMO schemes via the general model-independent reconstruction formalism (Sec. 3. 1), where the only assumption is that the Z s 2 symmetry fully persists in the presence of µ − τ breaking. In the next subsection, we will map the Z µτ 2 ⊗ Z s 2 symmetry into the neutrino seesaw Lagrangian, and demonstrate that the hidden Z s 2 symmetry is a full symmetry of our soft µ−τ breaking model in Ref. [1] where the physical prediction (5.26) holds; while for the current µ−τ blind seesaw model the Z s 2 is only a partial symmetry (respected by the µ−τ symmetric part M (s) ν ), and is violated by the µ−τ anti-symmetric part δM (a) ν , leading to our prediction of the modified new correlation (4.13) in Sec. 4. 1,in contrast to (5.49a) or (5.26). Mapping Z 2 ⊗ Z 2 Hidden Symmetry into Neutrino Seesaw Consider the general seesaw Lagrangian in the form of (2.2) with two or three right-handed neutrinos. After spontaneous electroweak symmetry breaking, consider the invariance of (2.2) under the residual symmetry transformations, 50) where G j is 3-dimensional unitary matrix, and G R j is 2 × 2 or 3 × 3 matrix (depending on two or three right-handed neutrinos invoked in the neutrino seesaw). Accordingly, we have the following invariance equations for the Dirac and Majorana neutrino mass-matrices, ν L → G j ν L , N → G R j N ,(5.G T j m D G R j = m D , (5.51a) G R j T M R G R j = M R , (5.51b) from which we deduce the invariance equation for the seesaw mass-matrix of light neutrinos, G T j M ν G j = M ν ,(5.52) where M ν = m D M −1 R m T D . Let us diagonalize the Majorana mass-matrices M ν and M R as follows, (5.53) in which D ν = diag(m 1 , m 2 , m 3 ) and D R = diag(M 1 , · · ·, M n ) with n = 2 for the minimal seesaw or n = 3 for three-neutrino-seesaw. Thus, from (5.51)-(5.53), we can express G j and G R j as, U T ν M ν U ν = D ν , U T R M R U R = D R ,G j = U ν d (j) ν U † ν , G R j = U R d (j) R U † R ,(5.54) where kernel representation {d (j) ν } is given in (5.12), and corresponds to the product group Z µτ 2 ⊗ Z s 2 via the equivalent flavor representation {G j } for the light neutrino sector. For d (j) R in (5.54), we give its nontrivial forms, d (1) R = diag(−1, 1) , (for minimal seesaw), (5.55a) d (1) R = diag(1, 1, −1) , d(2) R = diag(−1, 1, 1) , (for 3-neutrino-seesaw), (5.55b) where d R forms a Z ′ 2 symmetry for right-handed neutrinos in the minimal seesaw, and {d (1) R , d(2) R } form a product group Z ′µτ 2 ⊗ Z ′s 2 for right-handed neutrinos in the three-neutrino-seesaw. The trivial case with d (j) R equal to unity matrix is not listed here which corresponds to the singlet representation G R j = I . Since the low energy oscillation data do not directly enforce a Z ′µτ 2 symmetry for heavy right-handed neutrinos, we find two possibilities when mapping the Z µτ 2 to the seesaw sector: (i). the right-handed neutrinos have correspondence with the light neutrinos in each fermion family and transform simultaneously with the light neutrinos under the Z µτ 2 to ensure the invariance equation (5.51a); this means Z ′µτ 2 = Z µτ 2 . (ii). the right-handed neutrinos are singlet of the usual Z µτ 2 symmetry (called "µ−τ blind"), so the extra symmetry Z ′µτ 2 in the N sector is fully independent of the Z µτ 2 for light neutrinos; this means that under Z µτ 2 the invariance equation (5.51a) has G 1 ∈ Z µτ 2 for light neutrinos and G R = I for right-handed neutrinos. As generally shown in Sec. 5. 2, the Z s 2 symmetry dictates the solar angle θ s for light neutrinos. The extra group Z ′s 2 in the right-handed neutrino sector also has two possibilities: one is Z ′s 2 = Z s 2 , and another is for the right-handed neutrinos being singlet of the Z s 2 symmetry with G R s = I . Neutrino Seesaw with Common Soft µ−τ and CP Breaking In Ref. [1], we studied the common soft µ−τ and CP breaking in the minimal neutrino seesaw, where the right-handed neutrinos N = (N µ , N τ ) T obeying the same Z µτ 2 (= Z ′µτ 2 ) at the LO, and small soft µ − τ breaking is uniquely constructed in M R at the NLO. In the µ − τ symmetric limit, we inferred that the diagonalization matrix U R is a 2 × 2 orthogonal rotation with its rotation angle θ R ≡ θ R 23 = π 4 [1] , as expected. Thus, inputting (5.55a) for d (1) R , we deduce from (5.54), G R µτ = 0 1 1 0 ,(5.56) which is just the Z µτ 2 transformation matrix for right-handed neutrinos. With the two right-handed neutrinos N = (N µ , N τ ) T shown above, there is no rotation angle θ R 12 and also no corresponding Z ′s 2 symmetry. So the right-handed neutrinos can only belong to the singlet representation G R s = I 2 under Z s 2 symmetry, with d (2) R = I 2 . In our soft µ−τ breaking model [1], the Dirac mass-matrix, 57) exhibits the exact Z µτ 2 symmetry, so it should obey the hidden Z s 2 as well, (5.58) where G s = G 0 s is given by (5.19) and G R s = I 2 . This further leads to the invariance equation for the seesaw mass-matrix of light neutrinos, (5.59) where M ν = m D M −1 R m T D , and the invariance equation for M R is trivial here since G R s = I 2 . m D =   a a b c c b   ,(5.G T s m D G R s = m D ,G T s M ν G s = M ν , [Given the form of G s = G 0 s as constructed in (5.19), we can also explicitly verify the equations (5.58) and (5.59).] Hence, the group parameter k of Z s 2 and the corresponding solar angle θ s via Eq. (5.21) are fully fixed by the elements of the µ−τ symmetric m D , and is independent of the soft µ−τ breaking in M R (which is the Z s 2 singlet). This is a general proof based on group theory, without relying on making any expansion of the µ−τ breaking terms in M R . As can be explicitly solved from Eq. (5.58) above, we have [1], tan θ s = \|k\| = √ 2\|a\| \|b + c\| . (5.60) As another nontrivial check, we inspect the consistency condition (5.36). With the form of M ν in Ref. [1], we explicitly verify that (5.36) indeed holds, We note that this Z s 2 symmetry has a nice geometric interpretation. The two vectors, u 1 = (a, b, c) T and u 2 = (a, c, b) T , in the Dirac mass-matrix m D = (u 1 , u 2 ) , determine a plane S, obeying the plane-equation, (5.62) where the parameter k is given in (5.21). As shown in Ref. tan 2θ s = 2 √ 2 B s A − (C s +D) = − 2 √ 2 δB a δC a δC 2 a − 2δB 2 a = 2 √ 2 a(b + c) 2a 2 − (b + c) 2 ,(5.x − k √ 2 (y + z) = 0 , [1], the 3-dimensional representation G s is just the reflection transformation respect to the plane S. For the case of three-neutrino-seesaw, the µ−τ symmetric Dirac mass is extended to a 3×3 matrix, m ′ D =    a ′ a a b ′ b c b ′ c b    = (u 0 , u 1 , u 2 ) . (5.63) Thus, to hold m ′ D invariant under the Z s 2 symmetry, we just need to require its first column u 0 = (a ′ , b ′ , b ′ ) T to lie in the S plane, i.e., a ′ √ 2 b ′ = √ 2 a b + c = k ,(5.64) where k = tan θ s as in (5.21). This means that the Dirac mass matrix (5.63) only contains one more independent parameter than that of the minimal seesaw; furthermore, m ′ D is rank-2 and thus det M ν = (det m ′ D ) 2 (det M R ) −1 = 0 always holds, as in the minimal seesaw. µ−τ Blind Seesaw with Common µ−τ and CP Breaking As constructed in Sec. 2, the µ − τ blind seesaw defines the right-handed neutrinos N as singlet of Z µτ 2 symmetry. This means that we must have the Z µτ 2 transformation matrix G R µτ = I 2 and d µτ R = d (2) R = I 2 . Consider the general Dirac and Majorana mass-matrices in the minimal seesaw, m D =   ãã ′ b 1c1 b 2c2   , M R = M 11 M 12 M 12 M 22 . (5.65) The Majorana mass-matrix M R can be diagonalized by the unitary rotation U R , U T R M R U R = M R ≡ diag(M 1 , M 2 ) ,(5.66) Then we can derive the seesaw mass-matrix for light neutrinos, M ν ≃ m D M −1 R m T D = m D M −1 R m T D ,(5.67) where m D = m D U R takes the form as in (2.17). For the µ−τ blind seesaw with N being Z µτ 2 singlet, we can always start with the mass-eigenbasis of N with M R = diag(M 1 , M 2 ) , which means that the rotation U R becomes automatically diagonal and real, U R = I 2 . Then, the extra symmetry Z ′ 2 of M R must be independent of the Z µτ 2 of light neutrinos, i.e., Z ′ 2 = Z µτ 2 . So the natural choice is Z ′ 2 = Z s 2 . The Z ′ 2 can have a nontrivial d s R = d (1) R = diag(−1, 1) as in (5.55a). Thus, the corresponding symmetry transformation for M R is G R s = U R d s R U † R = d (1) R = diag(−1, 1) . (5.68) There is also a singlet representation of Z ′ 2 , corresponding to d R = I 2 . Then, let us inspect the possible Z s 2 symmetry for the Dirac mass-matrix by including the µ−τ breaking effects [cf. (2.17) G T s m D G R s = m D , G R s T M R G R s = M R ,(5.69) which will become, in the mass-eigenbasis of right-handed neutrinos, G T s m D d s R = m D , d s T R M R d s R = M R .(5.G T s m D d s R = m D ,(5.71) where m D ≡ m D ( m 0 M R ) − 1 2 . [The µ−τ symmetric form of m D was given in Eq. (2.8).] Using the notation m D , we can reexpress the seesaw mass-matrix, M ν = m 0 m D m T D . So we can further deduce the invariance equations under G s and d s R , respectively, G T s m D m T D G s = m D m T D , d s T R m T D m D d s R = m T D m D . (5.72) Next, we inspect the two equations in (5.72) to check the validity of the Z s 2 symmetry after embedding the µ − τ breaking into m D [such as those constructed in (2.18) for instance]. From (5.72), we will explicitly prove that the G s is a symmetry only for the µ − τ symmetric part of M ν ∝ m D m T D ; while d s R is violated by the µ − τ breaking terms in m T D m D . Hence, the Z s 2 symmetry is only a partial symmetry of the light neutrinos, valid for the µ−τ symmetric part M (s) ν . We can write down the mass-matrix m D with the most general µ−τ breaking, m D =    a a ′ b 1 c 1 b 2 c 2    =    a a ′ b c b c    +     0 0 − δb 1 +δb 2 2 − δc 1 +δc 2 2 − δb 1 +δb 2 2 − δc 1 +δc 2 2     +     0 0 − δb 1 −δb 2 2 − δc 1 −δc 2 2 + δb 1 −δb 2 2 + δc 1 −δc 2 2     = m (0) D + δm (s) D + δm (a) D = m (s) D + δm (a) D (5.73) where b 1 ≡ b − δb 1 , b 2 ≡ b − δb 2 , c 1 ≡ c − δc 1 , and c 2 ≡ c − δc 2 . For the symmetric mass-matrix product, m D m T D = M ν / m 0 ≡ M ν , we compute, up to the NLO, in m D or m D : one is for δb 1 = δb 2 = 0 and (δc 1 , δc 2 ) = c(ζ ′ , ζe iω ) , which corresponds to m D in (2.18); and another is for δc 1 = δc 2 = 0 and (δb 1 , δb 2 ) = b(ζ ′ , ζe iω ) , which corresponds to m D in (2.23). As we pointed out earlier, the invariance of the product (5.74) under G s ∈ Z s 2 [cf. (5.72)] would be justified so long as our general consistency condition (5.36) could hold. So, with (5.74) we can explicitly compute tan 2θ s from the two expressions in (5.36) including the µ−τ symmetric and anti-symmetric mass-matrix elements, respectively. We thus arrive at tan 2θ (s) (5.75b) for δb 1 = δb 2 = 0, and tan 2θ (s) (5.74) does respect the Z s 2 symmetry, and its invariance equation (5.31a) leads to the correct solution (5.33) and thus (5.75a) for the solar angle θ s . Substituting (5.21) into (5.75a) or (5.76a), we derive the m D m T D =    1 0 0 1 2 1 2 1 2    − (δb 1 + δb 2 )    0 a 2 a 2 b b b    − (δc 1 + δc 2 )     0 a ′ 2 a ′ 2 c c c     −(δb 1 − δb 2 )     0 a 2 − a 2 b 0 −b     − (δc 1 − δc 2 )     0 a ′ 2 − a ′ 2 c 0 −c     ≡ M (0) ν + δM (s) ν + δM (a) ν = M (s) ν + δMs = 2 √ 2 B s A − (C s +D) = − a ′ √ 2 c , (5.75a) tan 2θ (a) s = − 2 √ 2 δB a δC a δC 2 a − 2δB 2 a = 2 √ 2 a ′ c a ′2 −2c 2 = tan 4θ (s) s ,s = 2 √ 2 B s A − (C s +D) = − a √ 2 b , (5.76a) tan 2θ (a) s = − 2 √ 2 δB a δC a δC 2 a − 2δB 2 a = 2 √ 2 ab a 2 −2b 2 = tan 4θ (s) s ,(5.equation, k 2 + 2 r 0 k − 1 = 0 , with r 0 ≡ a ′ √ 2c corresponding to (5.75a), or r 0 ≡ a √ 2b corresponding to (5.76a). So we can fix the Z s 2 group parameter k in terms of the ratio of seesaw mass-parameters in m D , k = −1 ± 1 + r 2 0 . (5.77) Finally, we compute the other symmetric product m T D m D , up to the NLO, m T D m D = 1 0 0 1 − (δb 1 + δb 2 ) 2b c c 0 − (δc 1 + δc 2 ) 0 b b 2c . (5.78) The last two matrices of (5.78) arise from the µ−τ breaking, which make m T D m D non-diagonal at the NLO, and thus explicitly violate the second invariance equation of (5.72). This violation of Z s 2 does not directly lead to observable effect at low energies since the seesaw mass-matrix M ν for light neutrinos is given by the first product m D m T D in Eq. (5.74). Also, we could choose to assign the right-handed neutrinos to be singlet under the Z s 2 from the light neutrinos, i.e., d R = I 2 , then the invariance equation for m D m T D becomes trivial. But the first invariance equation in (5.72) under G s ∈ Z s 2 is still broken by the µ − τ anti-symmetric mass-matrix δM From the analyses above, we conclude that the hidden symmetry Z s 2 is a partial symmetry of the present model, respected by the µ−τ symmetric part M (s) ν of the light neutrino mass-matrix, and thus determines the solar angle θ s as in Eqs. (5.75a) and (5.77 Conclusion In this work, we have studied the common origin of µ−τ breaking and CP violations in the neutrino seesaw with right-handed Majorana neutrinos being µ−τ blind. The oscillation data strongly support we have further generated the observed matter-antimatter asymmetry (the baryon asymmetry) from thermal leptogenesis in the µ − τ blind seesaw. Under the successful leptogenesis, we derived the constrained correlation between θ 23 − 45 • and θ 13 − 0 • , as presented in Fig. 6. This figure predicts a lower bound on the key mixing angle, θ 13 1 • , which will be explored soon by the on-going reactor neutrino experiments at Daya Bay [10], Double-Chooz [12] and RENO [13]. Finally, we have studied the determination of solar mixing angle θ 12 and its connection to a hidden flavor symmetry Z s 2 and its possible breaking in Sec. 5. The general model-independent Z 2 ⊗ Z 2 symmetry structure of light neutrino sector was analyzed in Sec. 5.2.1. We first reconstructed the 3-dimensional representation G 0 s for Z s 2 group in the µ − τ symmetric limit as in Eq. (5.19). We proved that hidden symmetry Z s 2 holds for any µ−τ symmetric mass-matrix M ν of light neutrinos and determines the solar angle θ 12 via its group parameter, k = tan θ 12 , as in Eq. (5.21). Then, requiring that Z s 2 persists in the presence of general µ−τ breaking, i.e., G s = G 0 s as in (5.25), we deduce a unique correlation equation (5.26) which strikingly coincides with Eq. (4.15a), as predicted by our soft µ−τ breaking seesaw [1]. In Sec. 5.2.2, we further analyzed the validity of Z s 2 symmetry from general model-independent reconstructions of light neutrino mass-matrix M ν . We derived the general consistency condition (5.36) for the validity of Z s 2 symmetry in the presence of all possible µ−τ breakings. Under this condition, we derived the nontrivial correlation (5.42a) or (5.49a) between the two µ − τ breaking observables θ 23 − 45 • and θ 13 − 0 • , which agrees to Eq. (5.26) as derived earlier from pure group theory approach. We stress that the agreement between (5.26) [or (5.42a)] and the prediction (4.15a) from our soft µ−τ breaking seesaw is not a coincidence. As we explained in Sec. 5.3.1, the true reason lies in the fact that the soft µ − τ breaking is uniquely embedded in the right-handed Majorana mass-matrix M R which is a singlet of the Z s 2 group and thus does not violate Z s 2 . On the other hand, for the µ−τ blind seesaw, the µ−τ breaking is solely confined in the Dirac mass-matrix m D which would have nontrivial transformation (5.70) or (5.72) if Z s 2 could actually hold. As we have verified in Sec. 5.3.2, the invariance equation (5.72) Minos reported 62 e-like events above an estimated background of 49 events, and favors a nonzero θ 13 at 1.5σ level. The resultant confidence interval yields, 0 sin 2 2θ 13 < 0. 12 (0.19) at 90% C.L. for NMO (IMO) with δ D = 0; and the best-fit value is sin 2 2θ 13 = 0.04 (0.08) for NMO (IMO). On the other hand, the T2K experiment observed 6 e-like events with an estimated background of 1.5 events, indicating a nonzero θ 13 at 2.5σ level. This gives the 90% C.L. limits, 0.03 (0.04) < sin 2 2θ 13 < 0.28 (0.34) for NMO (IMO) with δ D = 0; and the best-fit value is sin 2 2θ 13 = 0.11 (0.14) for NMO (IMO). These new data indicate a relatively large θ 13 mixing angle, Shortly afterwards, a new global analysis of oscillation data has been performed [60] to include the latest T2K and Minos data. With this we can update our [60] by including the latest data from Minos [59] and T2K [58] long-baseline accelerator experiments. (Using the new reactor fluxes will slightly shift the mixing angles θ 12 and θ 13 a bit as shown in [60].) With Table-2, we have systematically updated our numerical analyses in Sec. 4. We find that the predictions of Fig. 2, Fig. 6 and Fig. 4 exhibit more constrained parameter space in an interesting way, while the other figures remain largely the same as before. For comparison, we use the updates in Table-2 and replot Figs. 2, 6, 4 as new Fig. 10, Fig. 11 and Fig. 12, respectively. In Figs. 10 and 11, we see that the updated 90% C.L. constraint on θ 13 (yellow area) just picks up the central region of our predicted theory parameter-space. Comparing the two plots in Figs. 10-11, we see that imposing successful leptogensis in Fig. 11 makes the parameter space more centered along the two wings, and the region around θ 23 ∼ 45 • is clearly disfavored. Since the new global fit of Table-2 gives the 90% C.L. limits, −7.5 • < θ 23 − 45 • < 2.9 • , with a central value θ 23 − 45 • = −4.6 • , it is clear that the left-wing of the theory parameter-space is more favored over the right-wing. Furthermore, imposing the θ 23 and θ 13 limits from Table-2 on our parameter-space in Fig. 11, we deduce the allowed range at 90% C.L., −4.8 • < θ 23 − 45 • < 2.9 • , which is shifted towards the negative side by about 1 • as compared to Eq. Table-2, with 1200 samples. Then, Fig. 12 shows that the predicted parameter region in the M 1 − δ D plane is much more centered along the two edges in Fig. 4, and a high leptogenesis scale M 1 > 10 15 GeV is strongly excluded except for the tiny regions of the CP-angle δ D very close to 180 • and 360 • . Finally, as a comparison, we further analyze the prediction from our Eq. (5.26) [Sec. 5.2] which we derived for light neutrinos alone (without invoking seesaw) and under the assumption of an exact Z s 2 symmetry. As we already proved in Sec. 5.3, this Z s 2 only holds in a class of models including our soft µ−τ and CP breaking model in Ref. [1], but can be violated in other class of models including the current µ − τ blind seesaw model. Hence, the Z s 2 symmetry cannot generally hold in a model-independent way. With Eq. (5.26), we plot the correlation between the µ − τ breaking parameters θ 13 and θ 23 − 45 • in Fig. 13(a)-(b). In plot-(a) we show the correlation by using the global fit in Fig. 13 shows that our predicted parameter space can easily saturate the current upper limit on θ 13 , and thus accommodates a relatively large θ 13 as indicated by the new data from T2K [58] and Minos [59]. The prediction of Fig. 13 differs from the above Fig. 10 significantly, because the coefficient in Eq. (4.18a) [corresponding to Fig. 10] has a nontrivial suppression factor relative to that of Eq. (4.18b) or Eq. (5.26) [corresponding to Fig. 13]. Furthermore, we note that the above Fig. 13 should also be compared to our previous Fig. 2 in Ref. [1] because the correlation (5.26) applies to both of them. But there are large differences between these two figures, the major reason is that we input the parameter θ 23 − 45 • in Fig. 13 according to the oscillation data (Table-1 or Table-2) and without invoking seesaw, while the θ 23 − 45 • in the Fig. 2 of Ref. [1] was derived as a function of fundamental µ − τ and CP breaking parameters in the seesaw Lagrangian which were scanned within their theoretically allowed ranges. This also leads to a stronger upper limit of θ 13 6 • in Ref. [1]. Note Added-2 : After the publication of this paper in Phys. Rev. D 84 (2011) 033009, Daya Bay and RENO collaborations announced new measurements of nonzero θ 13 on March 8, 2012 [61] and April 8, 2012 [62], respectively. Daya Bay experiment made a 5.2σ discovery of nonzero θ 13 [61], sin 2 2θ 13 = 0.092 ± 0.016(stat) ± 0.005(syst); and RENO found a nonzero θ 13 at 4.9σ level [62], sin 2 2θ 13 = 0.113 ± 0.013(stat) ± 0.019(syst) . These give the following 3σ ranges of nonzero θ 13 , Daya Bay: 5.7 • < θ 13 (8.8 • ) < 11.1 • , (6.2a) RENO: 5.9 • < θ 13 (9.8 • ) < 12.6 • , (6.2b) where the numbers in the parentheses θ 13 = 8.8 • and θ 13 = 9.8 • correspond to the central values. Then, we can re-plot the Fig. 10, Fig. 11 and Fig. 13(b) as the new Fig. 14, Fig. 15 and Fig. 16, respectively. In these new plots, we have scanned the experimental inputs within 3σ ranges. For the successful leptogenesis, we find that the lower bound on the leptogenesis scale M 1 becomes, M 1 > 2 × 10 13 , at 3σ level. The successful leptogenesis in Fig. 15 further requires, θ 13 1 • . To compare with our predictions, we have displayed the 3σ range of θ 13 from the new Daya Bay measurement [61] in the green shaded region. Furthermore, we show the 3σ lower and upper limits of θ 13 from the new RENO data [62] by the horizontal red-lines. The horizontal black dashed-lines in each plot denote the 3σ limits from the global fit [60]. From Figs. 14-15, we see that the new limits from Daya Bay [61] and RENO [62] experiments nicely pick up the central regions of our predicted parameter space of θ 13 . [61] are shown as the green shaded region; and the 3σ limits of the new RENO data [62] are depicted by the horizontal red-lines. The horizontal black dashed-lines denote the 3σ limits of the global fit [60]. Figure 15: Update of Fig. 11, with 2000 samples. The experimental inputs are scanned within 3σ ranges. The 3σ ranges of the new Daya Bay data [61] are shown as the green shaded region; and the 3σ limits of the new RENO data [62] are depicted by the horizontal red-lines. The horizontal black dashed-lines denote the 3σ limits of the global fit [60]. Finally, Fig. 16 shows the correlation of θ 13 and θ 23 − 45 • as predicted by our Eq. (5.26) [Sec. 5.2] under the assumption of an exact Z s 2 symmetry and without invoking seesaw. We have scanned the 3σ ranges of θ 12 in (5.26). We see that the new data of Daya Bay [61] and RENO [62] pick up the upper parts of our predicted parameter space of θ 13 . [61] are shown as the green shaded region; and the 3σ limits of the new RENO data [62] are depicted by the horizontal red-lines. The horizontal black dashed-lines denote the 3σ limits of the global fit [60]. .L., with the best fitted values, (θ 23 − 45 • ) = −2.2 • and (θ 13 − 0 • ) = 5.1 • . This justifies a fairly good zeroth order approximation, θ 23 = 45 • and θ 13 = 0 • , under which two exact discrete ( M Z ) = 29.57 ± 0.02 , m t = 173.1 ± 1.4 GeV, and the Higgs-mass range 115 M H 149 GeV [90% C.L.] for the SM, we find the running factor χ(M 1 , M Z ) ≃ 1.3 − 1.4 for M 1 = 10 13 − 10 16 GeV. Figure 1 : 1Predictions of θ 13 and θ 23 − 45 • as functions of the µ − τ breaking parameter ζ cos ω and CP breaking parameter δ D . The experimental inputs are scanned within 90% C.L. ranges and the Dirac phase angle δ D ∈ [0, 2π) , with 1500 samples. The shaded region (yellow) denotes the 90% C.L. limits on θ 13 and θ 23 − 45 • , from Figure 2 : 2) and the lower limit tan 2θ s = 2.2 just corresponds to the slopes of the sharp edges which are nearly straight lines. Hence, for any measured nonzero value of θ 23 − 45 • = 0 , theFig. 2imposes a lower bound on θ 13 , which will be tested by the reactor experiments such as Daya Bay, RENO and Double Chooz. The current oscillation data favor Correlation between θ 13 and θ 23 − 45 • , based on Eqs. (4.12a)-(4.12b), where the experimental inputs are scanned within 90% C.L. ranges and the Dirac phase angle δ D ∈ [0 • , 360 • ) , with 1500 samples. The sensitivities of Double Chooz[12], RENO[13] and Daya Bay[10] experiments to θ 13 are shown by the three horizontal (red) solid lines at 90% C.L., as 5.0 • , 4.1 • and 2.9 • (from top to bottom). The Daya Bay's future sensitivity (2.15 • ) is shown by the horizontal dashed (red) line. the central value of θ 23 to be smaller than 45 • Fig. 2 , 2we see that taking the current central value of θ 23 − 45 • = −2.2 • (Table-1), the lower bound on θ 13 is already very close to the sensitivity of Double Chooz experiment; and a minor deviation of θ 23 − 45 • = −1.4 • will push θ 13 up to the sensitivity of Daya Bay experiment. Hence, the Daya Bay, RENO and Double Chooz reactor experiments hold great potential to discover a nonzero θ 13 . 24) where in the last step we have expanded δ x and δφ to the second order since y ′ = O(10 −2 ) is relatively small as constrained by the current data [cf.(3.7)]. Eq.(4.24) shows that the neutrinoless ββ-decay observable M ee only contains the second orders of the µ−τ breaking quantity δ x (= θ 13 ) and the Majorana CP-phase angle δφ . Hence, M ee is less sensitive to the µ − τ breaking and Majorana CP-violation at low energies. Figure 3 : 3Correlations of θ 13 (in degree) with the Jarlskog invariant J [plot-(a)] and with the neutrinoless ββ-decay observable M ee [plot-(b)]. Each plot has computed 1500 samples. The shaded region (yellow) is allowed by the current data at 90% C.L. Figure 4 : 430) with m 1 ≡ (m † D m D ) 11 /M 1 , and m D ≡ m D U R with U R being the rotation matrix diagonalizing the mass-matrix M R of right-handed neutrinos. In the present µ − τ blind seesaw, it is natural to set the right-handed neutrinos in their mass-eigenbasis from the start, M R = diag(M 1 , M 2 ) , as we defined in Sec. 2. 1. So we have U R = I with I the unit matrix, and thus m D = m D . Leptogenesis scale M 1 is plotted as a function of Dirac CP-phase angle δ D , where the seven years of WMAP measurement (4.28) is imposed. All experimental inputs are scanned within their 90% C.L. ranges, with 1500 samples. Figure 5 : 5Seesaw scale M 1 and M 2 as functions of the elements (ā,b) and (ā ′ ,c) in the Dirac massmatrix m D , where the shaded regions correspond to the natural perturbative region (ā,b,ā ′ ,c) ∈ [1, 300] GeV, and 600 samples are generated in each plot. This puts an upper bound, M 1 3.5 × 10 15 GeV from plot-(b), and M 2 1.7 × 10 15 GeV from plot-(c). . (2.7) and (4.6), we connect the seesaw scale (M 1 , M 2 ) to the elements of the Dirac mass-matrix m D , Dirac mass-parameters (ā,b,ā ′ ,c) arise from the Yukawa interactions, (ā,b,ā ′ ,c) = (y a , y b , y a ′ , y c )v/ √ 2 . So we can plot M 1 as a function of the magnitude of the Dirac mass-parameter \|ā\| or \|b\| in Fig. 5(a)-(b), and M 2 as a function of the magnitude of the Dirac mass-parameter \|ā ′ \| or \|c\| in Fig. 5(c)-(d), where we have varied the measured quantities in their 90% C.L. ranges. We note that the Yukawa couplings (y a , y b , y a ′ , y c ) cannot be too small (to avoid excessive fine-tuning) or too large (to keep valid perturbation). So, we will take the Dirac mass-parameters (ā,b,ā ′ ,c) in the natural range [1, 300] GeV, corresponding to the Yukawa couplings y j no smaller than O(10 −2 ) and no larger than O(y t ), where y t = √ 2m t /v ≃ 1 is the top-quark Yukawa coupling in the SM. This natural perturbative range of (ā,b,ā ′ ,c) is indicated by the shaded area in Fig. 5(a)-(d), which results in an upper limit on the seesaw scale (M 1 , M 2 ) due to the perturbativity requirement. From Fig. 5(b) we infer an upper bound M 1 3.5 × 10 15 GeV, while Fig. 5(c) requires M 2 1.67 × 10 15 GeV. For the above construction of natural thermal Leptogenesis we consider the parameters space M 2 /M 1 5 , so with the upper bound of Fig. 5(c) we further deduce a stronger limit M 1 3.3 × 10 14 GeV. Figure 6 : 642) even for the region around θ 23 = 45 • .Under successful leptogenesis, the correlations of θ 13 with the Jarlskog invariant J and the neutrinoless double beta decay observable M ee are plotted inFig. 7(a) and (b), respectively. This Correlation between θ 13 and θ 23 − 45 • , where all the inputs are the same asFig. 2, except requiring successful leptogenesis in the present analysis, with 1500 samples. Figure 7 : 7c) show the correlations of M ee with J and with m 1 after imposing the leptogenesis. For the two right plots in Fig. 8(b)(d), we have replotted the same model-predictions as in the two corresponding left plots of Fig. 8(a)(c) (all in blue color). For comparison, we have further plotted, in Fig. 8(b)(d) with green color, the model-independent parameter space of M ee [cf. (4.24)] versus J [cf. (4.23)] or m 1 (= ∆m 2 13 ) , for the IMO scheme with m 3 ≃ 0 , where the relevant observables are varied within their 90% C.L. ranges and δ D ∈ (0, 2π]. This comparison Correlations of θ 13 with Jarlskog invariant J in plot-(a) and with neutrinoless double beta decay observable M ee in plot-(b), where all inputs are the same as Fig. 3, except requiring the successful leptogenesis in the present figure, with 1500 samples for each plot. Fig. 8 ( 8b)(d), we have compared our predictions with the sensitivities of the future neutrinoless ββdecay experiments CUORE (CU)[46] and Majorana[47]/GERDA III[48] (M/G), which are depicted by the horizontal dashed lines at 15 meV (black) and 20 meV (red), respectively.The leptogenesis scale M 1 can be determined from the baryon asymmetry η B , the reactor angle θ 13 , the Dirac phase sin δ D and other neutrino observables as in Eq.(4.39). Since the low energy parameter J in Eq. (4.23) is also predicted as a function of θ 13 and sin δ D , so it will correlate with the leptogenesis scale M 1 . Hence, we can plot the correlations of the leptogenesis scale M 1 Figure 8 : 8(b). InspectingEqs.(4.23) and(4.39), we deduce, J ∝ δ x sin δ D and M 1 ∝ (δ x sin δ D ) −1 , from which we arrive Upper plots (a)-(b) show the correlations between the neutrinoless ββ-decay observable M ee and the Jarlskog invariant J with successful leptogenesis. Lower plots (c)-(d) depict the correlations between M ee and light neutrino mass m 1 (≃ m 2 ) with successful leptogenesis. All experimental inputs are varied within 90% C.L. ranges, for 1500 samples. The background (green) regions in plots (b) and (d) represent the model-independent parameter space of the IMO scheme with m 3 ≃ 0 . The horizontal dashed lines in (b) and (d) depict the sensitivities of the future neutrinoless ββ-decay experiments CUORE (CU)[46] and Majorana[47]/GERDA III[48] (M/G), at 15 meV (black) and 20 meV (red), respectively. Figure 9 : 944) with σ 1 ≡ m 0 M 1 , σ 2 ≡ m 0 M 2 , and σ 3 ≡ m 0 M 3 , where the µ − τ blind right-handed neutrinos N ′ can always be rotated into their mass-eigenbasis without affecting the structure of Correlations of leptogenesis scale M 1 with the reactor mixing angle θ 13 in plot-(a), and with the low energy Jarlskog invariant J in plot-(b). Each plot contains 1500 samples. m D . Thus, we rederive the µ−τ and CP symmetric seesaw mass-matrix for the light neutrinos, new parameters (a ′′ , d) at the beginning, the realization of IMO eliminates them all and reduces M (0) ν to the universal LO mass-matrix as shown 49b) for µ−τ breaking arising from the deviation in the element c of m D , or tan 2θ 12 = \|a ′′ \| √ 2 \|d\| , (4.49c) for µ−τ breaking arising from the deviation in the element d of m D . As noted in Sec. 2. 2, we can always rotate the first column in m D to be all real by rephasing. For the convenience of comparison with the minimal neutrino seesaw, we will thus formulate the common origin of µ−τ and CP breaking in the element c of m D . It is possible to construct such a breaking in the element d of m D , but this does not affect our physical conclusions as will be clarified below, after Eq.(4.57). [Since we are constructing a common origin of µ−τ and CP breaking from a single source in m D , we do not consider this breaking to occur in both c and d elements of m D at the same time.] So, we build the Dirac mass-matrix m D with the common µ−τ and CP breaking in the following form, ( 4 . 451) gives Eq. (2.33) with the equality δC s = δD , we deduce z ′ = 4.53d), and thus m 3 = 0 holds up to the NLO. Hence, we have shown that our model with the general three-neutrino seesaw under IMO does share the essential feature of m 3 = 0 with the minimal seesaw. ) to estimate the allowed range of this ratio at 90%C.L. Literally, Eq. (5.4) shows a fine-tuned cancellation between the mass-matrix elements A and (C s + D) down to the level of 10 −3 . As will be clear in Sec. 5. 2. 2 by using the general reconstruction formalism for the IMO scheme, we find that the LO form of the µ−τ symmetric mass-matrix M(0) ν predicts the exact relations [cf. Eq. (5.44)], A (0) − (C (0) s + D (0) s − D (0) = 0 , which ensures m 1 = m 2 and m 3 = 0 at the LO. So, the small ratio (5.4) naturally arises from the NLO elements [δA − (δC s + δD)] = 0 , and thus there is no real fine-tuning in(5.4). This also means that at the LO the solar angle θ s is undetermined from the formula (5.2c), tan 2θ s = 0 0 , and the real determination of θ s is given by the NLO elements of M will explicitly verify in the next subsection for the general IMO scheme [cf. Eqs. (5.46)-(5.47a)]. the derivatives should be taken. There are only two possible choices, either c or b, since the µ−τ breaking under the µ−τ blind seesaw could appear in either c or b element of m D , as we explicitly constructed in Eqs. (2.18) and (2.23). Thus, applying the l ′ Hôpital rule to (5.7) we have tan 2θ s as expected, gives finite expressions for θ s , depending only on the LO parameters of the Dirac mass-matrix m D . This also agrees to Eqs. (2.15)-(2.16) in Sec. 2. 1. But Eq. (5.8) shows that θ s does depend on how the µ−τ breaking is built in the seesaw Lagrangian, and the two different ℓ } are just connected by the similarity transformations, and are thus two equivalent representations of the same group G ℓ ; similarly, {G j } and {d (j) ν } are two equivalent representations of the same group G ν . We may call the representation {d (j) ℓ } and {d (j) 27b) from the imaginary part condition Im[δG s ] = 0 , where in the last step of (5.27b) we have made use of (5.26) for simplification. Note that the correlation (5.26) precisely agrees to what derived from our soft breaking model in Eq. (4.12a) of Ref. [1] further derive physical consequences of the consistency condition (5.36) by using the elements of µ angle θ s . The last equality in(5.61) can be derived also from the solution (5.60) above, they are all consistent. Hence, the Z s 2 is a full symmetry of the seesaw sector and the light neutrino mass-matrix M ν in this soft µ−τ breaking model. sum of the first three matrices and δM (a) ν equals the sum of the last two matrices. For deriving the LO matrix M (0) ν in (5.74) we have used the relations (2.14) for the IMO scheme. There exist two basic realizations for the common breaking of µ−τ and CP symmetries s 76b)for δc 1 = δc 2 = 0. The above explicitly demonstrates the inequality θ , and thus proves the violation of the consistency condition(5.36). This is because the µ−τ anti-symmetric massbreaks the Z s 2 symmetry. Hence, Z s 2 is not a full symmetry of the mass-matrix M ν . Nevertheless, we find that the µ − τ symmetric part M , as shown by Eq. (5.75) or (5.76) above. Fig. 7 ( 7a) further constrains the Jarlskog invariant J into the negative range, −0.037 J −0.0035 , while Fig. 7(b) predicts the range of neutrinoless ββ-decay observable, 45.5 meV M ee 50.7 meV, which can be probed by the on-going neutrinoless ββ-decay experiments [3]. A lower bound on the leptogenesis scale M 1 is inferred from Fig. 4, M 1 > 3.5 × 10 13 GeV, and is given in Eq. (4.40). The correlations of the leptogenesis scale M 1 with the reactor angle θ 13 and the Jarlskog invariant J are analyzed in Fig. 9(a)-(b). hold only for the µ−τ symmetric part of the light neutrino mass-matrix M ν , and is partially violated by its µ−τ antisymmetric part [cf. Eq. (5.74)]. In consequence, we found: (i) the solar mixing angle θ 12 is dictated by the group parameter k of the hidden symmetry Z s 2 acting on the µ−τ symmetric mass-matrix M (s) ν [cf. Eqs. (5.75a) and (5.77)]; (ii) the consistency condition (5.36) no longer holds, and we predicted a modified new correlation (4.13), which can be experimentally distinguished from Eq. (4.15a) as predicted by our soft µ−τ breaking seesaw [1]. In contrast to our previous prediction (4.15a), Fig. 6 points to an important feature of the new correlation (4.13) by showing a more rapid increase of θ 13 as a function of θ 23 − 45 • [cf. also (4.20a)]; this allows θ 13 to saturate the current experimental upper limit, and confines the deviation θ 23 − 45 • into a more restrictive range, −4 • θ 23 − 45 • 4 • at 90%C.L., as in Eq. (4.22). These distinctive predictions of the present µ−τ blind seesaw can be systematically tested against those of our previous soft µ−τ breaking seesaw [1], by the on-going and upcoming neutrino experiments. Note Added in Proof : After the submission of this paper to arXiv:1104.2654 on April 14, 2011, two long-baseline accelerator experiments newly announced evidences for θ 13 via the ν µ → ν e appearance channel, one by the T2K Collaboration [58] on June 14, 2011 and another by the Minos Collaboration [59] on June 24, 2011. inverted mass-ordering (IMO) and favors a naturally larger θ 13 even for a rather small deviation ofθ 23 − 45 • , as shown in Eq. (4.20a) and our Fig. 2 (Sec. 4.1) or Fig. 6 (Sec. 4.2). Figure 10 :Figure 11 :Figure 12 : 101112Update of Fig. 2 (Sec. 4.1) by using the improved global fit in Table-2, with 2000 samples. The shaded region (yellow) shows the updated constraint on θ 13 at 90% C.L. Update of Fig. 6 (Sec. 4.2) by using the improved global fit in Table-2, with 2000 samples. The shaded region (yellow) shows the updated constraint on θ 13 at 90% C.L. Update of Fig. 4 (Sec. 4.2) by using the improved global fit in Figure 13 : 13Correlation of θ 13 and θ 23 − 45 • as predicted by our Eq. (5.26) [Sec. 5.2] without seesaw and under the assumption of exact Z s 2 symmetry. Plot-(a) shows the correlation by using the global fit in Table-1, while plot-(b) depicts the correlation under the improved global fit in Table-2, with 2000 samples in each plot. The shaded regions (yellow) give the allowed 90% C.L. ranges by the corresponding global fit. Figure 14 : 14Update of Fig. 10, with 2000 samples. The experimental inputs are scanned within 3σ ranges. The 3σ ranges of the new Daya Bay data Figure 16 : 16Correlation of θ 13 and θ 23 − 45 • as predicted by our Eq. (5.26) without seesaw and under the assumption of exact Z s 2 symmetry, with 2000 samples. The experimental inputs are scanned within 3σ ranges. The 3σ ranges of the new Daya Bay data experiments are depicted by the three horizontal (red) lines at 90% C.L., as 5.0 • , 4.1 • and 2.9 • (from top to bottom), based on three years of data-taking. The horizontal dashed (red) line represents Daya Bay's future sensitivity (2.15 • ) with six years of running 53f ) 53fwhere we note that the Majorana phase δφ ′ does not appear at the NLO because it is always suppressed by another NLO parameter z ′ =m 3 m 1 . Moreover, since the µ−τ and CP breaking matrix 49a) and (2.18) in Sec. 2. 2]. This means to hold the invariance equations in (5.51), 70 ) 70Since M R and d R are both diagonal, the invariance equation for M R always holds. So we can rewrite the above invariance equation for m D as, ). This also agrees to the result (2.15) [Sec. 2. 1] or (4.5) [Sec. 4. 1] which we derived earlier. As a final remark, we stress that the violation of the hidden Z s 2 symmetry by the µ−τ anti-symmetric mass-matrix δM has an important physical impact: it predicts a modified new correlation (4.13), and can be experimentally distinguished from Eq. (5.26) as predicted before by our soft µ−τ breaking of neutrino seesaw [1].(a) ν = m 0 δM (a) ν in (5.74) µ−τ symmetry as a good approximate symmetry in the light neutrino sector, leading to the zeroth order pattern, (θ 23 , θ 13 ) = (45 • , 0 • ) . Hence the µ−τ breakings, together with the associated CP violations, are generically small. For the µ−τ blind seesaw, we have convincingly formulated their common origin into Dirac mass matrix m D (Sec. 2. 2), leading to the unique inverted mass-ordering (IMO) of light neutrinos and distinct neutrino phenomenology. This is parallel to our previous work [1] where the common origin of µ−τ and CP breaking arises from the Majorana mass matrix of the singlet right-handed neutrinos and uniquely leads to the normal mass-ordering (NMO) of light neutrinos.In Sec. 3, we gave the model-independent reconstruction of low energy µ − τ and CP breakings with inverted neutrino mass-spectrum. With this we derived various predictions of the µ−τ blind neutrino seesaw in Sec. 4. In particular, we deduced a modified new correlation(4.13) between the two small µ − τ breaking observables θ 23 − 45 • and θ 13 − 0 • , as depicted inFig. 2and is very different from that in Ref.[1]. Eq. (4.13) is shown to also hold for the general three-neutrino seesaw in Sec. 4. 3. This correlation can be experimentally tested against Eq. (4.15a) as deduced from our soft µ−τ breaking seesaw mechanism [1]. As shown inFig. 2 and Fig. 6, our predicted range of θ 13 can saturate its present experimental upper bound. Imposing the current upper limit on θ 13 , we derived a restrictive range of the deviation, −4 • θ 23 − 45 • 4 • at 90%C.L., in Eq.(4.22). In Sec. 4. 2, T2K : 5 :.0 • < θ 13 (9.7 • ) < 16.0 • , (for NMO), (6.1a)5.8 • < θ 13 (11.0 • ) < 17.8 • ,at 90% C.L., where the central values are shown in the parentheses. We would like to point out that the new data from T2K and Minos further support our theory predictions which give the unique(for IMO); (6.1b) Minos: 0 • θ 13 (5.8 • ) < 10.1 • , (for NMO), (6.1c) 0 • θ 13 (8.2 • ) < 12.9 • , (for IMO); (6.1d) Table - 1 -accordingly, and translate the improvements [60] into the new Table 2. • − 35.4 • 31.0 • − 36.4 • 32.6 • − 34.7 • 30.6 • − 36.8 • • − 47.9 • 36.3 • − 51.3 • 38.6 • − 45.0 • 35.7 • − 53.1 • • − 10.4 • 3.5 • − 11.6 • 6.5 • − 9.6 • 1.8 • − 12.1 •Parameters Best Fit 90% C.L. 99% C.L. 1σ Limits 3σ Limits ∆m 2 21 (10 −5 eV 2 ) 7.58 7.15 − 7.94 7.07 − 8.09 7.32 − 7.80 6.99 − 8.18 ∆m 2 13 (10 −3 eV 2 ) 2.35 2.20 − 2.55 2.10 − 2.63 2.26 − 2.47 2.06 − 2.67 θ 12 33.6 • 32.0 θ 23 40.4 • 37.5 θ 13 8.3 • 5.09 Table 2 : 2The updated global analysis Table - 1 -, while in plot-(b) we depict the correlation under the improved global fit inTable-2. Each plots contains 2000 samples. The shaded regions (yellow) display the allowed 90% C.L. parameter space by the corresponding global fit. Note that Eq. (5.26) holds for both normal mass ordering and inverted mass ordering of light neutrinos. AcknowledgmentsWe thank C. S. Lam, Rabindra N. Mohapatra, Werner Rodejohann and Alexei Yu. Smirnov for useful discussions on this subject, and Eligio Lisi for discussing the updated global fit[60]in connection with the new data from T2K[58]and Minos[59]. We are grateful to Yi-Fang Wang, Kam-Biu Luk and Jun Cao for discussing Daya Bay experiment[10]. We also wish to thank Profs. T. D. Lee and R.Friedberg for valuable correspondence and discussions on the comparison of θ 13 predictions in our study and their work[36], which we showed at the end of Secis, d(1) ℓ = diag(1, η, η 2 ) ∈ Z 3 with η = exp i 2π 3 , and Z 3 is a subgroup of the general symmetry G ℓ = U (1) ⊗ U (1) of lepton mass-matrix. The invariance of lepton mass-matrix M ℓ M † ℓ under the transformation F 1 = d(1) ℓ automatically ensures left-handed leptons in their mass-diagonal basis.[55] The 3-dimensional representation(5.19b) of Z s 2 was derived in Eq. (6.26) of Ref.[1] for our soft µ−τ breaking seesaw model where the group parameter k (re-denoted as k ′ here) is related to the current k of (5.19b) by a simple notational conversion, k ′ ≡ − √ 2/k . we have the independent NLO parameters for the NMO analysis, (y 1 , y 2 , z, δ a , δ x , δα i , δφ i ). we have the independent NLO parameters for the NMO analysis, (y 1 , y 2 , z, δ a , δ x , δα i , δφ i ) . Expanding them perturbatively, we derive the LO form of the µ−τ symmetric mass-matrix M ν , References. arXiv:1001.0940JCAP. 1] S.-F. Ge, H.-J. He, F.-R. Yin0517Expanding them perturbatively, we derive the LO form of the µ−τ symmetric mass-matrix M ν , References [1] S.-F. Ge, H.-J. He, F.-R. Yin, JCAP 05, 017 (2010) [arXiv:1001.0940]. 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[ "Analytic Isomorphisms of compressed local algebras ", "Analytic Isomorphisms of compressed local algebras " ]	[ "J Elias ", "M E Rossi " ]	[]	[]	In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Fröberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see[6]. This unexpected result has been extended to compressed level K-algebras of socle degree 3 in[4]. In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s ≤ 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.Compressed algebras and Inverse Systemis the formal power series ring and K is an algebraically closed field of characteristic zero. We denote by m = M/I the maximal ideal of A where M = (x 1 , . . . , x n ) and let Soc(A) = 0 : m be the socle of A. Throughout this paper we denote by s the socle degree of A, that is the maximum integer j such that m j = 0. The type of A is t := dim K Soc(A).Let A be of initial degree v, that is I ⊆ M v \ M v+1 and socle degree s. The socle type E = E(A) = (0, . . . , e v−1 , e v , . . . , e s , 0, 0, . . . ) of A is the sequence E of natural numbers where e i := dim K ((0 : m) ∩ m i /(0 : m) ∩ m i+1 ).Clearly e s > 0 and e j = 0 for j > s, but other conditions are necessary on E for being permissible for an Artin K-algebra of given initial degree v and socle degree s (see[10]). Essentially the conditions require room for generators of I of valuation v and assure that there are not generators of valuation v − 1.A level algebra A of socle degree s (say also s-level), type t and embedding dimension n is an Artin quotient of R whose socle is concentrated in a single degree (i.e. Soc(A) = m s ) and satisfying dim K Soc(A) = t. Hence e j = 0 for j = s and e s = t. The Artin algebra is Gorenstein if t = 1.It make sense to consider maximum length Artin algebras of given socle type E and they can be characterized in terms of their Hilbert function. The Hilbert vector of A is denoted byis the Hilbert function of A. By its definition, the Hilbert function of A coincides with the Hilbert function of the corresponding associated graded ring gr m (A) = ⊕ i≥0 m i /m i+1 . We say that the Hilbert function HF (A) is maximal in the class of Artin level algebras of given embedding dimension and socle type, if for each integer i, h i (A) ≥ h i (A ′ ) for any other Artin algebra A ′ in the same class. The existence of a maximal HF (A) was shown for graded algebras by A. Iarrobino [10]. In the general case by Fröberg and Laksov [9], by Emsalem [8], by A. Iarrobino and the first author in [5] in the local case. * 2010 Mathematics Subject Classification. Primary 13H10; Secondary 13H15; 14C05	10.1090/s0002-9939-2014-12313-6	[ "https://arxiv.org/pdf/1207.6919v1.pdf" ]	119,733,195	1207.6919	7403b05a3859fc30f4df598eb143d247a7acaa98	Analytic Isomorphisms of compressed local algebras * 30 Jul 2012 J Elias M E Rossi Analytic Isomorphisms of compressed local algebras * 30 Jul 2012and Phrases: Artin Gorenstein local ringsInverse systemHilbert functionsIsomorphism classes † Partially supported by In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Fröberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see[6]. This unexpected result has been extended to compressed level K-algebras of socle degree 3 in[4]. In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s ≤ 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.Compressed algebras and Inverse Systemis the formal power series ring and K is an algebraically closed field of characteristic zero. We denote by m = M/I the maximal ideal of A where M = (x 1 , . . . , x n ) and let Soc(A) = 0 : m be the socle of A. Throughout this paper we denote by s the socle degree of A, that is the maximum integer j such that m j = 0. The type of A is t := dim K Soc(A).Let A be of initial degree v, that is I ⊆ M v \ M v+1 and socle degree s. The socle type E = E(A) = (0, . . . , e v−1 , e v , . . . , e s , 0, 0, . . . ) of A is the sequence E of natural numbers where e i := dim K ((0 : m) ∩ m i /(0 : m) ∩ m i+1 ).Clearly e s > 0 and e j = 0 for j > s, but other conditions are necessary on E for being permissible for an Artin K-algebra of given initial degree v and socle degree s (see[10]). Essentially the conditions require room for generators of I of valuation v and assure that there are not generators of valuation v − 1.A level algebra A of socle degree s (say also s-level), type t and embedding dimension n is an Artin quotient of R whose socle is concentrated in a single degree (i.e. Soc(A) = m s ) and satisfying dim K Soc(A) = t. Hence e j = 0 for j = s and e s = t. The Artin algebra is Gorenstein if t = 1.It make sense to consider maximum length Artin algebras of given socle type E and they can be characterized in terms of their Hilbert function. The Hilbert vector of A is denoted byis the Hilbert function of A. By its definition, the Hilbert function of A coincides with the Hilbert function of the corresponding associated graded ring gr m (A) = ⊕ i≥0 m i /m i+1 . We say that the Hilbert function HF (A) is maximal in the class of Artin level algebras of given embedding dimension and socle type, if for each integer i, h i (A) ≥ h i (A ′ ) for any other Artin algebra A ′ in the same class. The existence of a maximal HF (A) was shown for graded algebras by A. Iarrobino [10]. In the general case by Fröberg and Laksov [9], by Emsalem [8], by A. Iarrobino and the first author in [5] in the local case. * 2010 Mathematics Subject Classification. Primary 13H10; Secondary 13H15; 14C05 Definition 1.1. An Artin algebra A = R/I of socle type E is compressed if and only if it has maximal length e(A) = dim K A among Artin quotients of R having socle type E and embedding dimension n. The maximality of the Hilbert function characterizes compressed algebras as follows. If A is an Artin algebra (graded or local) of socle type E, it is known that for i ≥ 0, h i (A) ≤ min{dim K R i , e i dim K R 0 + e i+1 dim K R 1 + · · · + e s dim K R s−i }. Accordingly with [10], Definition 2.4. B, we can rephrase the previous definition in terms of the Hilbert function. Definition 1.2. A local (or graded) K-algebra A of socle degree s, socle type E and initial degree v is compressed if h i (A) =    s u=i e u (dim K R u−i ) if i ≥ v dim K R i otherwise In particular a s-level algebra A of type t and embedding dimension n is compressed if h i (A) = min{ n + i − 1 i , t n + s − i − 1 s − i }. If t = 1 and the above equality holds then A is called an extremal Gorenstein algebra or also compressed Gorenstein algebra. It is clear that compressed algebras impose several restrictive numerical conditions on the socle sequence E (see [10], Definition 2.2). For instance if v is the initial degree of A, then e v−1 = max{0, dim K R v−1 − u≥v (e u dim K R u−(v−1) )}.(1) If s ≥ 2(v − 1), then it is easy to see that e v−1 = 0 because dim K R s−(v−1) ≥ dim K R v−1 . This is the case if A is Gorenstein. Y.H. Cho and I. Iarrobino proved that the family of graded compressed level quotients of P = K[x 1 , . . . , x n ] is parameterized by an open dense subset of the Grassmanian of codimension c quotients of P s and has dimension c(dim P s − c) (see [3]). Boij studied the minimal free resolutions for compressed graded level algebras [1], Stanley and Hibi have studied connections of these algebras with combinatorics. Accordingly with Emsalem [8,Pag. 408] we give the following definition. Notice that, as a consequence, A is canonically graded if and only if A is analytically isomorphic to a standard graded K-algebra. Remark that a local K-algebra can be considered graded in a non-standard way, but not canonically graded. For instance, consider A = K[[t 2 , t 3 ]]; in this case A is not analytically isomorphic to gr m (A) ∼ = K[x, y]/(y 2 ) because A is reduced and gr m (A) is not reduced, but we remark that A ∼ = K[x, y]/(y 2 − x 3 ) is graded setting deg(x) = 2 and deg(y) = 3. Actually it is very rare that a local ring is isomorphic to its associated graded ring. If this is the case, several can be the applications, see for example [2], [6], [7]. If A is an Artin local (or graded) ring, the dual module V = Hom(A, K) can be identified with a maximal length submodule of the polynomial ring P = K[y 1 , . . . , y n ] which is closed under partial derivation. We recall the main facts and establish notation concerning the inverse system of Macaulay in the study of Artin local rings (A, m). The reader should refer to [8] and [11] for an extended treatment. It is known that P has an R-module structure under the following action • : R × P −→ P (f, g) → f • g = f (∂ y1 , . . . , ∂ yn )(g) where ∂ yi denotes the partial derivative with respect to y i . If we denote by x α = x α1 1 · · · x αn n and y β = y β1 1 · · · y βn n then x α • y β =    β! (β−α)! y β−α if β i ≥ α i for i = 1, · · · , n 0 otherwise where β! (β−α)! = n i=1 βi! (βi−αi)! . If S ⊆ P, then its annihilator in R is Ann R (S) = {f ∈ R : f • g = 0 for all g ∈ S}. The R-submodule generated by S in P is closed under partial derivation, then Ann R (S) is an ideal of R. Conversely, for any ideal I ⊂ R we define the following R-submodule of P : I ⊥ := {g ∈ P \| f, g = 0 ∀f ∈ I } = {g ∈ P \| I • g = 0}. Starting from an Artin local algebra R/I with socle-degree s, then I ⊥ is generated by polynomials of degree ≤ s. If S is generated by a sequence G := G 1 , . . . , G t of polynomials of P, then we will write Ann R (G) and A G = R/Ann R (G) where Ann R (G) = ∩ t j=1 Ann R (G j ). If A G is compressed, then we may assume that each A Gj is an extremal Gorenstein algebra. Notice that even if each A Gj is canonically graded, then A G is not necessarily canonically graded because it could not exists an uniform analytic isomorphism (see Example 3.5). F. S. Macaulay in [12] proved that there exists a one-to-one correspondence between ideals I ⊆ R such that R/I is an Artin local ring and R-submodules M of P which are finitely generated, see also Emsalem [8, Section B, Proposition 2] and Iarrobino [11,Lemma 1.2]. In this correspondence, Artin K-algebras A = R/I of embedding dimension n, socle degree s and type t correspond to R-submodules of P generated by t polynomials G 1 , . . . , G t of degree d i (depending on the socle type) such that the leading forms [4]). G 1 [d 1 ], . . . , G t [d t ] of degree d i (write G i = G i [d i ] + The following result was proved in [10, Proposition 3.7 and Corollary 3.8]. Proposition 1.4. The compressed local algebra A whose dual module is generated by G 1 , . . . , G t of degrees d 1 , . . . , d t has a compressed associated graded ring gr m (A) whose dual module is generated by G 1 [d 1 ], . . . , G t [d t ] , the leading forms of G 1 , . . . , G t . Conversely if gr m (A) is compressed, then A is compressed and E(A) = E(gr m (A)). It is possible to compute the Hilbert function of A = R/I via the inverse system. We define the following K-vector space: (I ⊥ ) i := I ⊥ ∩ P ≤i + P <i P <i .(2) Then it is known that h i (R/I) = dim K (I ⊥ ) i .(3) Notice that in the graded setting A = R/Ann(G), then h i (R/I) = dim K (I ⊥ ) i = dim K ∂ s−i G(4) where ∂ s−i denotes the partial derivatives of order s − i. We will translate the above equality in terms of the rank of suitable matrices associated to G. Let Ω = {ω i } be the canonical basis of R/M s+1 as a K-vector space consisting of the standard monomials x α ordered by the deg-lex order with x 1 > · · · > x n and, then the dual basis with respect to the action • is the basis Ω * = {ω * i } of P ≤j where (x α ) * = 1 α! y α , in fact ω * i (ω j ) = ω j , ω * i = δ ij , where δ ij = 0 if i = j and δ ii = 1. Given a form G of degree s and an integer q ≤ s, we denote by ∆ q (G) the n−1+s−q n−1 × n−1+q n−1 matrix whose columns are the coordinates of ∂ i (G), \|i\| = q, with respect (x L ) * = 1 L! y L , \|L\| = s − q (i and L are n-uples of integers). We will denote by (L, i) the corresponding position in the matrix ∆ q (G). In the following L + i denotes the sum in N n . Proposition 1.5. Let G ∈ P = K[y 1 , · · · , y n ] be a form of degree s. Then h s−i (A G ) = rank(∆ i (G)) ≤ min{ n − 1 + s − i n − 1 , n − 1 + i n − 1 } for i = 0, · · · , s. The equality holds if and only if A G is compressed. Proof. It is straight consequence of (2) and (3), and the definition of the matrix ∆ i (G) of size n−1+i n−1 × n−1+s−i n−1 . Proposition 1.6. Let G = \|j\|=s β j 1 j! y j be a degree s homogeneous polynomial. Given an integer q, the (L, i)-component of ∆ q (G), \|L\| = s − q, \|i\| = q, is ∆ q (G) (L,i) = β L+i Proof. Let us consider the derivative ∂ i (G) = \|j\|=s, j≥i β j 1 (j − i)! y j−i \|i\| = q; j ≥ i means that j w ≥ i w for w = 1, · · · , n. If we write L = j − i, then ∂ i (G) = \|L\|=s−q β L+i 1 L! y L From this we get the claim. It is easy to deduce the following corollary which gives via (4) an alternative proof of the fact that a graded Gorenstein algebra A G has symmetric Hilbert function. Corollary 1.7. Let G be a form of P of degree s Given an integer i ≤ s, then ∆ i (G) = t ∆ s−i (G) where t denotes the transpose matrix. With the previous notation, let A G be a graded level algebra We can define for all integer i ≤ s, ∆ i (G) =    ∆ i (G 1 ) . . . ∆ i (G t )    (5) which is a t n−1+s−i n−1 × n−1+i n−1 matrix. We get the following result. Proposition 1.8. Let A = A G be a compressed s-level local algebra of type t. Then for every i = 1, . . . , s h i (A) = rank(∆ i (G[s]) = min{ n − 1 + i n − 1 , t n − 1 + s − i n − 1 }. Proof Automorphisms of Artin local algebras and associated matrices Given a K-algebra C, we will denote by Aut(C) the group of the automorphisms of C as a K-algebra and by Aut K (C) as a K-vector space. The automorphisms of R as a K-algebra are well known. They act as replacement of x i by z i , i = 1, · · · , n, such that M = (x 1 , . . . , x n ) = (z 1 , . . . , z n ). Actually, since M s+1 ⊆ I, we are interested in the automorphisms of R/M s+1 of K-algebras induced by the projection π : R −→ R/M s+1 . Clearly Aut(R/M s+1 ) ⊆ Aut K (R/M s+1 ) . For all p ≥ 1, I p denotes the identity matrix of order n+p−1 p . For any ϕ ∈ Aut K (R/M s+1 ) we may associate a matrix M (ϕ) with respect to the basis Ω of size r = dim K (R/M s+1 ) = n+s s . Given I and J ideals of R such that M s+1 ⊂ I, J, there exists an isomorphism of K-algebras ϕ : R/I → R/J if and only if ϕ is canonically induced by a K-algebra automorphism of R/M s+1 sending I/M s+1 to J/M s+1 . In particular ϕ is an isomorphism of K-vector spaces. Passing to Hom( , K), ϕ corresponds to ϕ * : J ⊥ → I ⊥ where t M (ϕ) is the matrix associated to ϕ * with respect to the basis Ω * of P ≤s . We denote by R the subgroup of Aut K (P ≤s ) (automorphisms of P ≤s as a K-vector space) represented by the matrices t M (ϕ) of Gl r (K) with ϕ ∈ Aut(R/M s+1 ). By Emsalem, [8, Proposition 15], the classification, up to analytic isomorphism, of the Artin local Kalgebras of multiplicity d, socle degree s and embedding dimension n is equivalent to the classification, up to the action of R, of the K-vector subspaces of P ≤s of dimension d, stable by derivations and containing P ≤1 = K[y 1 , . . . , y n ] ≤1 . Our goal is to translate the study of analytic isomorphisms of Artin level K-algebras A = R/I of socle s and type t in terms of the corresponding dual polynomials of degree s in P in an effective computational framework. This section is quite technical and the idea is the generalization of the method used in [6]. The main goal is Corollary 2.3 and, in view of it, the machinery could be presented assuming s ≤ 4. Actually the presentation does not improve substantially and one loses the control of the general case that could be useful for further investigations. Let F = F 1 , . . . , F t , respectively G = G 1 , . . . , G t , be polynomials of degree s. Let ϕ ∈ Aut(R/M s+1 ), from the previous facts we have ϕ(A F ) = A G if and only if (ϕ * ) −1 ( F R ) = G R .(6)If F i = b i1 w * 1 + . . . b ir w * r ∈ P ≤s , then we will denote the row vector of the coefficients of the polynomial with respect to the basis Ω * by [F i ] Ω * = (b i1 , . . . , b ir ). If there exists ϕ ∈ Aut(R/M s+1 ) such that [G i ] Ω * M (ϕ) = [F i ] Ω * , for every i = 1, . . . , t, then ϕ(A F ) = A G(7) Let s be a positive integer, the aim of this section is to provide a structure of the matrix M (ϕ) associated to special K-algebra isomorphisms ϕ ∈ Aut(R/M s+1 ). Let ϕ s−p be an automorphism of R/M s+1 such that ϕ s−p = Id modulo m p+1 , with 1 ≤ p ≤ s, that is ϕ s−p (x j ) = x j + \|i\|=p+1 a j i x i + higher terms(8) for j = 1, . . . , n and a j i ∈ K for each n-uple i such that \|i\| = p + 1. In the following we will denote a := (a 1 i , \|i\| = p + 1; · · · ; a n i , \|i\| = p + 1) ∈ K n( n+p n−1 ) . The matrix associated to ϕ s−p , say M (ϕ s−p ), is an element of Gl r (K), r = n+s s+1 , with respect to the basis Ω of R/M s+1 . We write M (ϕ s−p ) = (B i,j ) 0≤i,j≤s where B i,j is a n+i−1 i × n+j−1 j matrix of the coefficients of monomials of degree i appearing in ϕ(x j ) where j = (j 1 , . . . , j n ) such that \|j\| = j. It is easy to verify that: B i,j =            0, 0 ≤ i < j ≤ s, or j = 1, i = 1, · · · , s, I i , i = j = 0, · · · , s, 0, j = s − p, · · · , s − 1, i = j + 1, · · · , s, and (i, j) = (s, s − p). The matrix M (ϕ s−p ) has the following structure M (ϕ s−p ) =                    1 0 · · · 0 0 0 0 0 0 I 1 0 0 0 0 0 . . . 0 0 I 2 0 0 0 0 . .                   The entries of B p+1,1 , B p+2,2 , . . . , B s,s−p are linear forms in the variables a j i , with \|i\| = p+1, j = 1, · · · , n. We are mainly interested in B s,s−p which is a n+s−1 s × n+s−p−1 s−p matrix whose columns correspond to x W with \|W \| = s − p and the rows correspond to the coefficients of x L with \|L\| = s in ϕ(x W ). One has ϕ(x W ) = x W + n j=1 w j ( \|i\|=p+1 a j i x W −δj +i ) + . . . where W = (w 1 , . . . , w n ) ∈ N n with \|W \| = s − p, here δ j is the n-uple with 0-entries but 1 in position j. We remark that \|W − δ j + i\| = s. Then the entry of B s,s−p corresponding to the L row, \|L\| = s, and W column, \|W \| = s − p, is (B s,s−t ) L,W = W −δj +i=L w j a j i .(9) Let F, G be polynomials of degree s of P and let ϕ s−p be a K-algebra isomorphism of type (8) sending A F to A G . We denote by F [j] (respectively G[j]) the homogeneous component of degree j of F (respectively of G), that is F = F [s] + F [s − 1] + . . . (G = G[s] + G[s − 1] + . . . ). By (7) we have [G] Ω * M (ϕ s−p ) = [F ] Ω * ,(10) in particular we deduce [F [j]] Ω * =    [G[s − p]] Ω * + [G[s]] Ω * B s,s−p , j = s − p, [G[j]] Ω * , j = s − p + 1, · · · , s.(11) We ) with the set of indexes (j, i), j = 1, · · · , n, \|i\| = p+1, corresponding to the entries of a = (a 1 i , \|i\| = p + 1; · · · ; a n i , \|i\| = p + 1) ∈ K n( n+p n−1 ) . For every i = 1, · · · , n, we denote S i p the set of monomials x α of degree p such that x α ∈ x i (x i , · · · , x n ) p−1 , hence #(S i p ) = p−1+n−i p−1 . By definition S 1 p ∪ · · · ∪ S n p is the set of monomials of degree p and the last p − 2 + n − i p − 1 elements of S i p correspond to x i x i+1 S i+1 p ,(13)For instance S 1 3 = {x 3 1 , x 2 1 x 2 , x 2 1 x 3 , x 1 x 2 2 , x 1 x 2 x 3 , x 1 x 2 3 }, S 2 3 = {x 3 2 , x 2 2 x 3 , x 2 x 2 3 }, S 3 3 = {x 3 3 }. We write log(x α ) = α for all α ∈ N n . Lemma 2.2. The matrix M [s−p] (G[s]) has the following upper-diagonal structure M [s−p] (G[s]) =        M 1 * · · · * * 0 M 2 · · · * * . . . . . . . . . . . . . . . 0 0 0 M n−1 * 0 0 0 0 M n        where M j is a matrix of size s−p−1+n−j s−p−1 × n+p n−1 , j = 1, · · · , n, defined as follows: the entries of M j are the entries of M [s−p] (G[s]) corresponding to the rows W ∈ log(S j s−p ) and columns (j, i), \|i\| = p + 1. We label the entries of M j with respect to these multi-indexes. Then it holds: (i) for all W = (w 1 , · · · , w n ) ∈ log(S 1 s−p ) and i, \|i\| = p + 1, We may generalize the previous facts to a sequence G = G 1 , . . . , G t of polynomials of degree s of P. Let ϕ s−p be a K-algebra isomorphism of type (8) sending A F to A G where F = F 1 , . . . , F t . In particular we assume that, as in (10), w 1 ∆ p+1 (G[s]) (W −δ1,i) = M 1(W,(1,i)) , (ii) for all j = 1, · · · , n − 1, W ∈ log(S j+1 s−p ),[G r ] Ω * M (ϕ s−p ) = [F r ] Ω * , for every r = 1, . . . , t. We deduce the analogous of (11) and we restrict our interest to . . . M [s−p] (G t [s])    (14) which is a t n−1+s−p n−1 × n n+p n−1 matrix, we get τ ([G[s]] Ω * B ⊕t s,s−p ) = M [s−p] (G[s]) τ a.(15) The matrix M [s−p] (G[s]) has the same shape of M [s−p] (G[s]), already described in Lemma 2.2 and its blocks correspond to suitable submatrices of (∆ p+1 (G[s])) (see (5)). Hence we have an analogous to (11) for the level case [F r [j]] Ω * =    [G r [s − p]] Ω * + a τ (M [s−p] (G[s])), j = s − p, [G r [j] ] Ω * , j = s − p + 1, · · · , s. for all r = 1, . . . , t. Compressed Gorenstein algebras Let A = A G be an compressed Artin Gorenstein local K-algebra A of socle degree s. We recall that, by Proposition 1.8, if G = G[s] + G[s − 1] + . . . , then h i (A) = rank(∆ i (G[s]) = min{ n − 1 + i n − 1 , n − 1 + s − i n − 1 } for every i = 1, . . . , s. The main result of [6] shows that if s ≤ 3 then A is canonically graded. Let assume s = 4, then the Hilbert function is {1, n, n+1 2 , n, 1}. Because A G [4] is an extremal Gorenstein algebra with the same Hilbert function of A, we may assume G = G[4] + G [3]. In fact P 1 , P 2 ⊆< G[4] > R because of (2) and, as a consequence, it easy to see that < G [4] + G[3] > R =< G[4] + G[3] + G[2] + ... > R . So we have to prove that, however we fix G [3], there exists an automorphism ϕ ∈ Aut(R/M 5 ) such that A G ≃ A G [4] . We consider for every j = 1, . . . , n (11) and (12) we get [F [3]] Ω * = [G [3]] Ω * + a τ (M [3] (G [4])) ϕ(x j ) = x j + \|i\|=2 a j i x i + higher terms If A F = ϕ −1 3 (A G ), then from [F [4]] Ω * = [G [4]] Ω * where a = (a 1 i , . . . , a n i ). By Proposition 1.8 and Corollary 2.3, we know that the matrix M [3] (G [4]) has maximal rank and it coincides with the number of the rows, so there exists a solution a ∈ K n of (17) such that F [3] = 0 and F [4] = G [4]. Let A be a local K-algebra of embedding dimension n and socle degree s. By looking at the dual module, it is clear that if s ≤ 2, then A is graded because the dual module can be generated by homogeneous polynomials of degree at least two. The aim is now to list the local compressed algebras of embedding dimension n, socle degree s and socle type E = (0, . . . , e v−1 , e v , . . . , e s , 0, 0, . . . ) which are canonically graded. Examples will prove that the following result cannot be extended to higher socle degrees. Proof. Since a local ring with Hilbert function {1, n, t} is always graded (the dual module can be generated by quadratic forms), we may assume s ≥ 3. If s = 3 and A is level compressed, then A is canonically graded by [4]. If A is not necessarily level, but compressed, then by (1) the socle type is {0, 0, e 2 , e 3 } and the Hilbert function is {1, n, h 2 , e 3 } where h 2 = min{dim K R 2 , e 2 + e 3 n}. Because gr m (A) = P/I * has embedding dimension n, then P ≤1 ⊆ (I * ) ⊥ . Then we may assume that in any system of coordinates I ⊥ is generated by e 2 quadratic forms and e 3 polynomials G 1 , . . . , G e3 of degree 3. Then the result follows because R/ Ann R (G 1 , . . . , G e3 ) is a 3-level compressed algebra of type e 3 and hence canonically graded. Assume s = 4 and e 4 = 1. We recall that if A is Gorenstein, then the result follows by Theorem 3.1. Since A is compressed, then by (1) the socle type is (0, 0, 0, e 3 , 1). This means that I ⊥ is generated by e 3 polynomial of degree 3 and one polynomial of degree 4. Similarly to the above part, because P ≤2 ⊆ (I * ) ⊥ , I ⊥ can be generated by e 3 forms of degree 3 and one polynomial of degree 4. As before the problem is reduced to the Gorenstein case with s = 4 and the result follows. Assume s = 4 and n = 2. If e 4 = 1, then we are in case (2). If e 4 > 1, because A is compressed, the possible socle types are: E i = (0, 0, 0, 0, i) with i = 2, · · · , 5 and since A is compressed, the corresponding Hilbert function is {1, 2, 3, 4, i}. In each case A is graded because the Hilbert function forces the dual module to be generated by forms of degree four. The following example shows that Theorem 3.1 fails if A is Gorenstein of socle degree s = 4, but not compressed, i.e. the Hilbert function is not maximal. Example 3.3. Let A be an Artin Gorenstein local K-algebra with Hilbert function HF A = {1, 2, 2, 2, 1}. The local ring is called almost stretched and a classification can be found in [7]. In this case A is isomorphic to one and only one of the following rings : (a) A = R/I with I = (x 4 1 , x 2 2 ) ⊆ R = K[[x 1 , x 2 ]] , and I ⊥ = y 3 1 y 2 . In this case A is canonically graded, (b) A = R/I with I = (x 4 1 , −x 3 1 + x 2 2 ) ⊆ R = K[[x 1 , x 2 ]] , and I ⊥ = y 3 1 y 2 + y 3 2 . The associated graded ring is of type (a) and it is not isomorphic to R/I. Hence A is not canonically graded. (c) A = R/I with I = (x 2 1 + x 2 2 , x 4 2 ) ⊆ R = K[[x 1 , x 2 ]] , and I ⊥ = y 1 y 2 (y 2 1 − y 2 2 ) . In this case A is graded. The following example shows that Theorem 3.1 cannot be extended to extremal Gorenstein algebras of socle degree s = 5. I = (x 4 1 , x 3 2 − 2x 3 1 x 2 ) ⊂ R = K[[x 1 , x 2 ]]. The quotient A = R/I is an extremal Gorenstein algebra with HF A = {1, 2, 3, 3, 2, 1}, I * = (x 4 1 , x 3 2 ) and I ⊥ = y 3 1 y 2 2 + y 4 2 . We will prove that I is not isomorphic to I * . Assume that there exists an analytic isomorphism ϕ of R mapping I into I * . It is easy to see that Jacobian matrix of ϕ is diagonal because (I * ) ⊥ =< y 3 1 y 2 2 > . We perform the computations modulo (x 1 , x 2 ) 5 , so we only have to consider the following coefficients of ϕ    ϕ(x 1 ) = ax 1 + . . . ϕ(x 2 ) = bx 2 + ix 2 1 + jx 1 x 2 + kx 2 2 + . . . where a, b are units, i, j, k ∈ K. After the isomorphism x 1 → 1/ax 1 , x 2 → 1/bx 2 , we may assume a = b = 1. Then we have I * = ϕ(I) = (x 4 1 , x 3 2 − 2x 3 1 x 2 + 3ix 2 1 x 2 2 + 3jx 1 x 3 2 + 3kx 4 2 ) modulo (x 1 , x 2 ) 5 . Hence there exist α ∈ K, β ∈ R such that x 3 2 − 2x 3 1 x 2 + 3ix 2 1 x 2 2 + 3jx 1 x 3 2 + 3kx 4 2 = αx 4 1 + βx 3 2 modulo (x 1 , x 2 ) 5 . From this equality we deduce α = 0 and 2x 3 1 x 2 = x 2 2 (x 2 + 3ix 2 1 + 3jx 1 x 2 + 3kx 2 2 − βx 2 ) modulo (x 1 , x 2 ) 5 , a contradiction, so I is not isomorphic to I * . It is interesting to notice that we can get the same conclusion by following the line of the proof of Theorem 3.1. Let ϕ as above sending I into I * . If we denote by (z i ) i=1,...,6 the coordinates of an homogeneous form G [5] of degree 5 in y 1 , y 2 with respect Ω * , then by Lemma 2.1, the matrix M [4] (G [5]) (s = 5, p = 1) has the following shape       4z 1 4z 2 4z 3 0 0 0 3z 2 3z 3 3z 4 z 1 z 2 z 3 2z 3 2z 4 2z 5 2z 2 2z 3 2z 4 z 4 z 5 z 6 3z 3 3z 4 3z 5 0 0 0 4z 4 4z 5 4z 6       In our case G[5] = y 3 1 y 2 2 , so all z i are zero but z 3 = 12, hence the above matrix has rank 4 and it has not maximal rank accordingly with Corollary 2.3.. Since all the rows are not zero except the last one, it is easy to see that F [4] = y 4 2 is not in the image of M [4] (G [5]), as (11) requires. The following example shows that Theorem 3.2 cannot be extended to compressed type 2 level algebras of socle degree s = 4. Then A = R/I is a compressed level algebra with socle degree 4, type 2 and Hilbert function HF A = {1, 3, 6, 6, 2}. We prove that A is not canonically graded. We know that I * = Ann(G 1 [4], G 2 [4]) and we prove that A and gr m (A) are not isomorphic as Kalgebras. Let ϕ an analytic isomorphism sending I to I * , then it is easy to see that ϕ = I 3 modulo (x 1 , x 2 , x 3 ) 2 . Following the approach of this paper, we compute the matrix M [3] (G 1 [4], G 2 [4]) of size 20 × 18 and, accordingly with (11), we show that y 3 3 is not in the image of M [3] (G 1 [4], G 2 [4]). Let F 1 [4], F 2 [4] be two homogeneous forms of degree 4 of R = K[y 1 , y 2 , y 3 ]. We denote by (z j i ) i=1,...,15 the coordinates of F j [4] with respect the basis Ω * , j = 1, 2. Then the 20 × 18 matrix M [3] (F 1 [4], F 2 [4]) has the following shape, see (14),                          Definition 1. 3 . 3A local algebra (A, m) is canonically graded if there exists a K-algebra isomorphism between A and its associated graded ring gr m (A). lower terms . . . ) are linearly independent. If A = A G where G := G 1 , . . . , G t is s-level local algebra of type t (d i = s for all i = 1, . . . , t), then the associated graded ring gr m (A) is not necessarily level and in general dim K Soc(gr m (A)) ≥ t. Actually the graded K-algebra Q = P/Ann P (G[s]) (where G[s] := G 1 [s], . . . , G t [s]) is a graded s-level algebra of type t and it is the unique quotient of gr m (A) with socle degree s and type t. In particular gr m (A) is level if and only if gr m (A) ≃ Q (see . By Proposition 1.4 we know that gr m (A) is level compressed of socle degree s and type t. Since gr m (A) is level if and only if gr m (A) ≃ Q = P/Ann(G[s]), the result follows by Proposition 1.5. are going to study [G[s]] Ω * B s,s−p . Let [α i ] be the vector of the coordinates of G[s] w.r.t. Ω * , i.e. of [G[s]] Ω * B s,s−p are bi-homogeneous forms in the components of [α i ] and a = (a 1 i , . . . , a n i ) such that \|i\| = p + 1 of bi-degree (1, 1). Hence there exists a matrix M [s−p] (G[s]) of size n−in the K[α i ] such that τ ([α i ]B s,s−p ) = M [s−p] (G[s]) τ a (12) where τ a denotes the transpose of the row-vector a. We are going to describe the entries of M [s−p] (G[s]). We label the columns of M [s−p] (G[s] Lemma 2. 1 . 1The entry of M [s−p] (G[s]) corresponding to the W -row, \|W \| = s − p, and column (j, i) ∈ {1, · · · , n} × {i; \|i\| = p + 1} is M [s−p] (G[s]) W,(j,i) = w j α W −δj +i . Proof. Given W , \|W \| = s − p, the coordinate of [G[s]] Ω * B s,s−p with respect (x W ) * is, by (9), ([G[s]] Ω * B s,s−p ) (x W ) * = \|L\|=s (B s,s−p ) W,L α α W −δj +i a j i so the entry of M [s−p] (G[s]) corresponding to the W row, \|W \| = s − p, and column (j, i) ∈ {1, · · · , n} × {i; \|i\| = p + 1} is w j α W −δj +i . The goal is now to present a structure of M [s−p] (G[s]) in terms of G[s]. In particular we will prove that the rank of M [s−p] (G[s]) can be expressed in terms of the Hilbert function of A G[s] . We need further notations. M j+1,(W,(j+1, * )) = w j+1 M j,(L,(j, * )) with L = δ j + W − δ j+1 , Proof. First we prove that M [s−p] (G[s] has the upper-diagonal structure as in the claim. Since the entry of M [s−p] (G[s]) corresponding to the W row, \|W \| = s − p, and column(j, i) ∈ {1, · · · , n} × {i; \|i\| = p + 1} is w j α W −δj +i , this entry is zero if W ∈ log(S t s−p ) and j < t ≤ n. (i) Notice that the set of multi-indexes L = W − δ 1 , \|W \| = s − p,agrees with the set of multi-indexes of degree s − p − 1, see (13). Hence by Proposition 1.6 and Proposition 2.1 we havew 1 ∆ p+1 (G[s]) (W −δ1,i) = w 1 α W −δ1+i = M 1(W,(1,i)) .(ii) If \|i\| = p + 1 and L = δ j + W − δ j+1 , then, Proposition 2.1, M j+1,(W,(j+1, * )) = w j+1 α W −δj+1+i = w j+1 α L−δj+i = w j+1 M j,(L,(j, * )) .As a consequence of (i) and (ii), the matrix ∆ p+1 (G[s]) is strongly involved in the computation of the rank of M [s−p] (G[s].Corollary 2.3. If s ≤ 4 then rank (M [s−p] (G[s])) is maximal if and only if rank (∆ p+1 (G[s])) is maximal. Proof. Notice that M [s−p] (G[s]) has an upper-diagonal structure where the rows of the diagonal blocks M j are a subset of the rows of the first block matrix M 1 . Let us assume that the number of rows of M 1 is not bigger than the number of columns of M 1 , as a consequence the same holds for M j with j > 1. Then we can compute the rank of M [s−p] (G[s]) by rows, so rank (M [s−p] (G[s])) is maximal if and only if rank (∆ p+1(G[s])) is maximal. Since M 1 is a s−pwe get the result. This inequality is equivalent to n + s − p − 2 ≤ n + p, i.e. s ≤ 2p + 2, since p ≥ 1 we get that s ≤ 4.Example 3.4 shows that Corollary 2.3 fails for s = 5. gluing t times the matrix B s,s−p and where [G[s]] Ω * is the row ([G r [s]] Ω * : r = 1, . . . , t). Accordingly with (12), it is defined the matrix M [s−p] (G r [s]) of size n−entries depending on [G[s]] Ω * such that t ([G r [s]] Ω * B s,s−p ) = M [s−p] (G r [s]) Theorem 3. 1 . 1Let A be an extremal Artin Gorenstein local K-algebra. If s ≤ 4 then A is canonically graded. Proof. Let A be a extremal Artin Gorenstein local K-algebra of socle degree s ≥ 2 and embedding dimension n. Then A = A G with G ∈ P = K[y 1 , . . . , y n ] a polynomial of degree s and gr m (A) = P/ Ann(G[s]) is an extremal Gorenstein graded algebra of socle degree s ≥ 2 and embedding dimension n (see Proposition 1.4). Theorem 3 . 2 . 32Let A be a compressed Artin K-algebra of embedding dimension n, socle degree s and socle type E. Then A is canonically graded in the following cases: (1) s ≤ 3, (2) s = 4 and e 4 = 1, (3) s = 4 and n = 2. Example 3 . 4 . 34Let us consider the ideal .. . . . . . 0 . . . . . . . . . . . . . . . 0 B p+1,1 0 . . . I s−p 0 0 . . . 0 . . . B p+2,2 0 0 I s−p+1 0 . . . 0 . . . · · · . . . . . . 0 . . . 0 0 B s,1 B s,2 . . . B s,s−p 0 . . . I s It is enough to specialize the matrix to our case for proving that y 3 3 is not in the image of M[3](G 1[4], G 2[4]).Remark 3.6. Let C s,t be the family of level Artin algebras of socle degree s and type t. This family can be parameterized by a non-empty open Zariski subset I s,t of the affine space of t-uples of degree s polynomials of P .The previous examples suggested that if s, t are not in the hypothesis of the last Theorem, then a generic element F ∈ I s,t defines a non-canonically graded level Artin algebra R/Ann R ( F ) of socle degree s and type t. Betti numbers of compressed level algebras. M Boij, J. Pure Appl. Algebra. 1342M. Boij, Betti numbers of compressed level algebras, J. Pure Appl. Algebra 134 (1999), no. 2, 111-131. Poincare' series and deformations of Gorenstein local algebras. G Casnati, J Elias, R Notari, M E Rossi, DOInumber:10.1080/00927872.2011.636643Comm. in Algebra. 40G. Casnati, J. Elias, R. Notari, M.E. Rossi, Poincare' series and deformations of Gorenstein local algebras, Comm. in Algebra, 40, Issue 3, DOI number: 10.1080/00927872.2011.636643 (2012). Hilbert Functions and Level Algebras. Y H Cho, A Iarrobino, Journal of Algebra. 241Y.H. Cho and A. Iarrobino, Hilbert Functions and Level Algebras, Journal of Algebra. 241 (2001), 745-758. Short Artinian level local k-algebras. A De Stefani, to appear in Comm. in AlgebraA. De Stefani, Short Artinian level local k-algebras, to appear in Comm. in Algebra. The Hilbert function of a Cohen-Macaulay local algebra: extremal Gorenstein algebras. J Elias, A Iarrobino, J. Alg. 110J. Elias and A. Iarrobino, The Hilbert function of a Cohen-Macaulay local algebra: extremal Gorenstein algebras, J. Alg. 110 (1987), 344-356. Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system. J Elias, M E Rossi, Trans. Am. Math. Soc. 3649J. Elias and M.E. Rossi, Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system, Trans. Am. Math. Soc., 364 (2012), no. 9, 4589-4604. Isomorphism classes of certain Artinian Gorenstein Algebras, Algebra and Representation Theory. J Elias, G Valla, DOI10.1007/s10468-009-9196-8J. Elias and G. Valla, Isomorphism classes of certain Artinian Gorenstein Algebras, Algebra and Representation Theory, DOI 10.1007/s10468-009-9196-8. Géométrie des pointsépais. J Emsalem, Bull. Soc. Math. France. 1064J. Emsalem, Géométrie des pointsépais, Bull. Soc. Math. France 106 (1978), no. 4, 399-416. Compressed algebras. R Froberg, D Laksov, Springer Verlag 1092R. Froberg and D. Laksov, Compressed algebras, L.N.M. Springer Verlag 1092 (1984), 121-151. Compressed algebras: Artin algebras having given socle degrees and maximal length. A Iarrobino, Trans. Amer. Math. Soc. 2851A. Iarrobino, Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 (1984), no. 1, 337-378. Associated graded algebra of a Gorenstein Artin algebra. Mem. Amer. Math. Soc. 107514115, Associated graded algebra of a Gorenstein Artin algebra, Mem. Amer. Math. Soc. 107 (1994), no. 514, viii+115. Juan Elias Departament d'Àlgebra i Geometria Universitat de Barcelona Gran Via 585. F S Macaulay, 8007Barcelona, Spain e-mailCambridge UniversityThe algebraic theory of modular systems. [email protected]. Macaulay, The algebraic theory of modular systems, Cambridge University, 1916. Juan Elias Departament d'Àlgebra i Geometria Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain e-mail: [email protected]	[]
[ "Anomaly detection in high-energy physics using a quantum autoencoder", "Anomaly detection in high-energy physics using a quantum autoencoder" ]	[ "Vishal S Ngairangbam \nPhysical Research Laboratory\nAhmedabad -380009GujaratIndia\n\nDiscipline of Physics\nIndian Institute of Technology\nGandhinagar -382424PalajGujaratIndia\n", "Michael Spannowsky \nInstitute for Particle Physics Phenomenology\nDurham University\nDH1 3LEDurhamUnited Kingdom\n", "Michihisa Takeuchi \nDepartment of Physics\nOsaka University\n560-0043OsakaJapan\n" ]	[ "Physical Research Laboratory\nAhmedabad -380009GujaratIndia", "Discipline of Physics\nIndian Institute of Technology\nGandhinagar -382424PalajGujaratIndia", "Institute for Particle Physics Phenomenology\nDurham University\nDH1 3LEDurhamUnited Kingdom", "Department of Physics\nOsaka University\n560-0043OsakaJapan" ]	[]	The lack of evidence for new interactions and particles at the Large Hadron Collider has motivated the high-energy physics community to explore model-agnostic data-analysis approaches to search for new physics. Autoencoders are unsupervised machine learning models based on artificial neural networks, capable of learning background distributions. We study quantum autoencoders based on variational quantum circuits for the problem of anomaly detection at the LHC. For a QCD tt background and resonant heavy Higgs signals, we find that a simple quantum autoencoder outperforms classical autoencoders for the same inputs and trains very efficiently. Moreover, this performance is reproducible on present quantum devices. This shows that quantum autoencoders are good candidates for analysing high-energy physics data in future LHC runs. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] 1 By classical, we mean any machine learning algorithm that leverages only discrete bit computations, while by quantum, we imply a computation that uses the properties of quantum mechanics and qubits, even if they are simulated on classical hardware.	10.1103/physrevd.105.095004	[ "https://arxiv.org/pdf/2112.04958v3.pdf" ]	245,006,312	2112.04958	f19974421f94a4f4994a2185b921e30d1809e4a5	Anomaly detection in high-energy physics using a quantum autoencoder Vishal S Ngairangbam Physical Research Laboratory Ahmedabad -380009GujaratIndia Discipline of Physics Indian Institute of Technology Gandhinagar -382424PalajGujaratIndia Michael Spannowsky Institute for Particle Physics Phenomenology Durham University DH1 3LEDurhamUnited Kingdom Michihisa Takeuchi Department of Physics Osaka University 560-0043OsakaJapan Anomaly detection in high-energy physics using a quantum autoencoder May 20, 2022, OU-HET-1125, IPPP/21/54 The lack of evidence for new interactions and particles at the Large Hadron Collider has motivated the high-energy physics community to explore model-agnostic data-analysis approaches to search for new physics. Autoencoders are unsupervised machine learning models based on artificial neural networks, capable of learning background distributions. We study quantum autoencoders based on variational quantum circuits for the problem of anomaly detection at the LHC. For a QCD tt background and resonant heavy Higgs signals, we find that a simple quantum autoencoder outperforms classical autoencoders for the same inputs and trains very efficiently. Moreover, this performance is reproducible on present quantum devices. This shows that quantum autoencoders are good candidates for analysing high-energy physics data in future LHC runs. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] 1 By classical, we mean any machine learning algorithm that leverages only discrete bit computations, while by quantum, we imply a computation that uses the properties of quantum mechanics and qubits, even if they are simulated on classical hardware. I. INTRODUCTION In the absence of a confirmed new physics signal and in the presence of a plethora of new physics scenarios that could hide in the copiously produced LHC collision events, unbiased event reconstruction and classification methods [1][2][3][4] have become a major research focus of the high-energy physics community. Unsupervised machine learning models [5][6][7][8], popularly used as anomaly-detection methods [9][10][11][12][13][14][15], are trained on Standard Model processes and should indicate if a collision event is irreconcilable with the kinematic features of events predicted by the Standard Model. One of the most popular neural network-based approach are autoencoders [16]. Autoencoders consist of an encoder step that compresses the input features into a latent representation with reduced dimensionality. Subsequently, the latent representation is decoded into an output of the same dimensionality as the input feature space. The entire network is then trained to minimise the reconstruction error. The latent space acts as an information bottleneck, and its dimension is a hyperparameter of the network. The assumption is that the minimal dimension of the latent space for which the input features can still be reconstructed corresponds to the intrinsic dimension of the input data, here Standard Model induced background processes. However, the trained autoencoder would poorly reconstruct any unknown new-physics process with a higher intrinsic dimension. If the signal is kinematically sufficiently different from the background samples, the loss or reconstruction error will be larger for signal than for background events. Such autoencoders can be augmented with convolutional neural networks [17,18], graph neural networks [19,20] or recurrent neural networks [21,22] on its outset, making it a very flexible anomaly detection method for a vast number of use cases. With the advent of widely available noisy intermediate-scale quantum computers (NISQ) [23] the interest in quantum algorithms applied to high-energy physics problems has spurred. Today's quantum computers have a respectable quantum volume and can perform highly non-trivial computations. This technical development has resulted in a community-wide effort [24,25] exploring the applications of quantum computers for studying quantum physics in general and in particular, the application to challenges in the theoretical description of particle physics. Some recent studies in the direction of LHC physics include evaluating Feynman loop integrals [26], simulating parton showers [27] and structure [28], quantum algorithm for evaluating helicity amplitudes [29], and simulating quantum field theories [30][31][32][33][34][35]. An interesting application of quantum computers is the nascent field of quantum machine learning-leveraging the power of quantum devices for machine learning tasks, with the capability of classical 1 machine learning algorithms for various applications at the LHC already recognised, it is only natural to explore whether quantum machine learning (QML) can improve the classical algorithms [36][37][38][39][40][41][42][43]. This work explores the feasibility and potential advantages of using quantum autoencoders (QAE) for anomaly detection. Most quantum algorithms consist of a quantum state, encoded through qubits, which evolves through the application of a unitary operator. The necessary compression and expansion of data in the encoding and decoding steps are manifestly non-unitary, which has to be addressed by the QAE using entanglement operations and reference states which disallow information to flow from the encoder to the decoder. To this end, a QAE should, in principle, be able to perform tasks ordinarily accomplished by a classical autoencoder (CAE) based on deep neural networks (DNN). The ability of DNNs are known to scale with data [44], and large datasets are necessary to bring out their better performance over other machine-learning algorithms. Interestingly, we find that a quantum autoencoder, augmented using quantum gradient descent [39,45] for its training, is much less dependent on the number of training samples and reaches optimal reconstruction performance with minuscule training datasets. Since the use of quantum gradient descent is a relatively new way of improving the convergence speed and reliability of the quantum network training, we provide a detailed introduction in Appendix A. Moreover, compared to CAEs, which use the same input variables as the QAE, QAEs have better anomaly detection capabilities for the two benchmark processes we use in our study. This better performance is particularly interesting as the CAE has O(1000) parameters compared to just O(10) for the QAE. The study indicates the possibility to study quantum latent representations of high-energy collisions, in analogy to classical autoencoders [19,[46][47][48]. Our results indicate that quantum autoencoders could be advantageous in anomaly detection tasks in the NISQ era. The rest of the paper is organised as follows. In section II, we present an introduction to classical autoencoders based on deep neural networks. We then describe the basic ideas of quantum machine learning and a quantum autoencoder in section III. The details of the data simulation, network architecture, and training are described in section IV. We present the performance of a quantum autoencoder compared to a classical autoencoder in section V. We conclude in section VI. II. CLASSICAL AUTOENCODERS Encoder Decoder Encoder Decoder FIG. 1: Schematic representation of a simple dense classical autoencoder (left) and a quantum autoencoder (right) for a four dimensional input space and a two dimensional latent space. To induce an information bottleneck in quantum unitary evolutions, we throw away states \|β i (trash states) at the encoder output (green lines), which are replaced by reference states \|βi (shown in orange lines ), containing no information of the input \|xj . The mechanism can be better understood by dividing the Hilbert space of the complete system into three parts: HA the subspace formed by the qubits that are fed to the decoder, HB the subspace of the qubits that are discarded after encoding, and H B the subspace where a fixed reference state (initialised as \|0 ⊗ dim H B ) unacted by the encoder is fed to the decoder. SWAP gates can achieve the exchange of states denoted by black lines. Autoencoders are neural networks utilised in various applications of unsupervised learning. They learn to map input vectors x to a compressed latent vector z via an encoder. This latent vector feeds into a decoder that reconstructs the inputs. Denoting the encoder and decoder networks as E(Θ E , x) and D(Θ D , z) with Θ E and Θ D denoting the learnable parameters of the respective network, we have z = E(Θ E , x) ,x = D(Θ D , z) ,(1) wherex denote the reconstructed output vector. The whole network is trained via gradient descent to reduce a faithful distance L, between the reconstructed outputx and the input vector x. For instance L can be the root-mean-squareerror (RMSE), L(x,x) = i=n i=1 (x i − x i ) 2 n ,(2) wherex i and x i are the i th component of the reconstructed and input vectors respectively, and n is their dimension. A faithful encoding should have an optimal latent dimension k < n, with k being the intrinsic dimension of the data set. This dimensionality reduction is crucial in many applications of autoencoders, which otherwise learns trivial mappings to reconstruct the output vectorsx. Unsupervised learning deals with learning probability distributions, and properly trained autoencoders are excellent for many applications. A dense CAE for a four feature input and two-dimensional latent space is shown in figure 1. The encoder and the decoders are also enclosed in red and blue boxes, respectively. One popular usage of autoencoders in collider physics is anomaly detection. In various scenarios at the LHC, the background processes' contributions are orders of magnitude larger than most viable signals. However, a plethora of possible signal scenarios exist that could be realised in nature, making it unlikely that the signal-specific reconstruction techniques of supervised learning methods comprehensively cover all possible scenarios. This motivates unsupervised anomaly detection techniques, wherein a statistical model learns the probability distribution of the background to classify any data not belonging to it as anomalous (signal) data. Using an autoencoder as an anomaly detector, we train it to reconstruct the background data faithfully. Many signals have a higher intrinsic dimension 2 than background data due to their increased complexity. Hence, they incur higher reconstruction losses. Thus, the loss function can be used as a discriminant to look for anomalous events. III. QUANTUM AUTOENCODERS Quantum machine learning broadly deals with extending classical machine learning problems to the quantum domain with variational quantum circuits [50]. We can divide these circuits into three blocks: a state preparation that encodes classical inputs into quantum states, a unitary evolution circuit that evolves the input states, and a measurement and post-processing part that measures the evolved state and processing the obtained observables further. For this discussion, we will always work in the computational basis with the basis vectors {\|0 , \|1 } denoting the eigen states of the Pauli Z operatorσ z for each qubit. There are many examples of state preparation in literature [51], which has their own merits in various applications. We prepare the states using angle encoding, which encodes real-valued observables φ j as rotation angles along the x-axis of the Bloch sphere \|Φ = n i=1 R x (φ j ) \|0 = n j=1 cos φ j 2 \|0 − i sin φ j 2 \|1 ,(3) where R x = e −i φ j 2σ x denote the rotation matrix. The number of qubits required n, is same as the dimensions of the input vector. A parametrised unitary circuit U(Θ), with Θ denoting the set of parameters, evolves the prepared state \|Φ to a final state \|Ψ , \|Ψ = U(Θ) \|Φ .(4) The final measurement step involves the measurement of an observable on the final state \|Ψ . Since measurements in quantum mechanics are inherently probabilistic, we measure multiple times (called shots) to get an accurate result. In order to do that, we need quantum hardware that can prepare a large number of pure identical input states \|Φ for each data point. After defining a cost function, the parameters Θ can be trained and updated using an optimisation method. To better capture the geometry of the underlying Hilbert space and to achieve a faster training of the quantum network, 3 we will use quantum gradient descent [45], where the direction of steepest descent is evaluated according to the Fubini-Study metric [52,53]. The general idea is to make the optimisation procedure aware of the weight space's underlying quantum geometry, which improves the speed and reliability of finding the global minimum of the loss function. A brief outline of quantum gradient descent is given in Appendix A. While we have not discussed the specific form of the parametrised unitary operation U(Θ), it is important to note that one of the major advantages of quantum computation is due to its ability to produce entangled states, a phenomenon absent in devices based on classical bits. The prepared input state is separable into the component qubits, and a product of unitaries acting on single-qubit states will not entangle the subsystems. The CNOT gate is a standard two-qubit gate, which will be used in our circuit to entangle the subsystems. A. Quantum autoencoders on variational circuits Quantum autoencoders based on variational circuit models have been proposed for quantum data compression [54]. In our work, we want to learn the parameters of such a network to compress the background data efficiently. Along the same principles as anomaly detection on classical autoencoders, we expect that the compression and subsequent reconstruction will work poorly on data with different characteristics to the background. A quantum autoencoder, in analogy to the classical autoencoders has an encoder circuit which evolves the input state \|Φ to a latent state \|χ via a unitary transformation U(Θ), and then reconstructs the input state, via its hermitian conjugate \|Φ = U † (Θ)\|χ . However, note that since unitary transformations are probability conserving and act on spaces having identical dimensions, there is no data compression in such a setup. In order to have data compression, some qubits at the initial encoding \|χ are discarded and replaced by freshly prepared reference states. Such a setup for a four feature input and two dimensional latent space if shown in figure 1. The unitary operators output identical number of qubits, however at the encoder step, two of its outputs (shown by green lines) are replaced by freshly prepared reference states (shown in orange lines), devoid of any information of the input states. We describe the basics of quantum autoencoding in the following, mainly based on the discussion of quantum autoencoders for data compression from ref. [54]. Quantum anomaly detection of simulated quantum states has been investigated in ref [55]. To the best of our knowledge, our study is the first to explore anomaly detection of classical inputs via a quantum autoencoder. The main difference between existing studies and ours is that the input states for the former are inherently quantum mechanical. In contrast, the choice of input embedding of the classical numbers in our case determines the nature of the quantum state. We will use angular encoding, where the quantum states are separable into the constituent qubits. We will, however, be extensively using CNOT gates in the unitary evolution which will entangle the different qubits. Let us denote the Hilbert space containing the input states by H. For describing a quantum autoencoder, it is convenient to expand H as the product of three subspaces, H = H A ⊗ H B ⊗ H B ,(5) with subspace H A denoting the space of qubits fed into the decoder from the encoder, and H B denoting the space corresponding to the ones that are re-initialised, and H B denoting the Hilbert space containing the reference state. In the following, we will denote states belonging to any subspace with suffixes while the full set will have no suffix. For example, \|a AB ∈ H A ⊗ H B , \|κ ∈ H, \|b B ∈ H B etc. We will use the same convention for operators acting on the various subspaces. Since we entangle the separable input qubits in the subspaces H A ⊗ H B via U AB (Θ), the latent state \|χ AB ∈ H A ⊗ H B , in general, is not seperable. The input of the larger composite system including the reference state is \|Φ AB ⊗ \|β B , with \|β B denoting a freshly prepared reference state (initialised as \|0 ⊗ dim H B ) not acted on by the unitary U AB . The process of encoding can be therefore written as, \|χ AB ⊗ \|β B = (U AB (Θ) ⊗ I B ) \|Φ AB ⊗ \|β B ,(6) where I B denotes the identity operator on H B . Explicitly, the dimensions of the subspaces H A , H B , and H B are 2 N lat , 2 N trash , and 2 N trash , respectively, where N lat is the number of qubits passed to the decoder directly from the encoder, while N trash are the ones that are discarded. Swapping the B and B , gives the input to the decoder as \|χ = I A ⊗ V BB \|χ AB ⊗ \|β B ,(7) where V BB indicates a unitary that performs the swap operation, 4 and I A is the identity operator on H A . The output 4 For instance swapping the state of two qubits in the basis {\|00 , \|01 , \|10 , \|11 }, can be implemented via the unitary matrix V BB =     1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1     . of the decoder can now be written as \|Ψ = U † AB (Θ) ⊗ I B \|χ ,(8) with I B being the identity operator on H B . The decoding, therefore, takes the swapped latent state \|χ , and the unitary U † AB evolves it with no information from the encoder in the subspace H B . The reconstruction efficiency of the autoencoder can be quantified in terms of the fidelity between the input and output states in the subspace H A ⊗ H B , which quantifies their similarity. For two quantum states \|ψ and \|φ , it is defined as F (\|φ , \|ψ ) = F (\|ψ , \|φ ) = \| φ\|ψ \| 2 . For normalized states, we have 0 ≤ F ≤ 1, with F = 1 only when \|φ and \|ψ are exactly identical. We can write the fidelity of the complete system as F (\|Φ AB ⊗ \|β B , \|Ψ ) = F (\|Φ AB ⊗ \|β B , U † AB V BB U AB \|Φ AB ⊗ \|β B ) , where we have implicitly assumed that the unitary operators are extended to the whole space via a direct product with the identity operator on the subspace it does not act on, for notational compactness. Noting that U AB \|Φ AB = \|χ AB , we can write this as, F (\|Φ AB ⊗ \|β B , \|Ψ ) = F (\|χ AB ⊗ \|β B , V BB \|χ AB ⊗ \|β B ) . Writing the swapped state as V BB \|χ AB ⊗ \|β B = \|χ AB ⊗ \|β B , we have F (\|Φ AB ⊗ \|β B , \|Ψ ) = F (\|χ AB ⊗ \|β B , \|χ AB ⊗ \|β B ) .(9) Since we are interested in the wave functions belonging to the subspace H A ⊗ H B , we trace over B to get the required fidelity. However, a perfect fidelity between the input and outputs of the AB system can be achieved when the complete information of the input state passes to the decoder, i.e. U AB \|Φ AB = \|Φ c A ⊗ \|β B .(10) The state \|Φ c A denotes a compressed form of \|Φ AB , i.e it should contain the information of the AB system in the input, while \|β B is equivalent to the reference state, with no information of the input. If the B and B systems are identical during the swap operation, the entire circuit reduces to the identity map. The output of the B system, hereby referred to as the trash state, is itself the determining factor of the output state fidelity. The output of the B system can be obtained after tracing over the A system as:ρ B = Tr A {\|χ χ\| AB } and the required fidelity of the B system is F (\|β B ,ρ B ). A perfect reconstruction of the input is possible only when the trash state fidelity F (\|β B ,ρ B ) = 1. Thus a quantum autoencoder can be trained by maximising the trash state fidelity instead of the output fidelity, which has the advantage of reducing the resource requirements during training. Although, the output fidelity obtained by tracing over the B system is numerically not equal to the trash state fidelity, we can use the latter in anomaly detection as well, since it is a faithful measurement of the output fidelity. Thus, unlike vanilla classical autoencoders, we can reduce the execution and training of QAEs into the encoder circuit for anomaly detection. The above discussions have focused on the underlying principles behind a quantum autoencoding process on single input states. As stated before, we need to prepare identical input-states for each data point and repeat the unitary evolution and measurement to get a useful estimate of the fidelity, evident also from the use of density operators to express the output state. Referring to the ensemble of the input states as {p i , \|Φ i AB }, we obtain for the cost function C(Θ) = − i p i F (\|β B ,ρ B ) ,(11) where the negative sign converts the optimisation process into minimising the cost function. It is important to note that the ensemble should not be taken as being analogous to the batch training in classical neural networks, as it is required for the accurate prediction of the network output even when testing the autoencoder network. The fidelity between two qubits at the encoder output and the reference states is measured via a SWAP test. IV. ANALYSIS SETUP A. Data simulation To show the prowess of the quantum autoencoders, we study two processes with distinctive features: a QCD continuum background of top pair production taking possible signal signatures of resonant heavy Higgs decaying to a pair of top quarks, and invisible Z decays into neutrinos with a likely signal of the 125 GeV Higgs decaying to two dark matter particles. As we shall see in the following sections, the relative performance of QAEs over CAEs show parallels in these two different signatures, pointing towards an advantage of QAEs over CAEs not governed by the specific details of the final state. Resonant Higgs signal over continuum tt background The first background and signal samples used in our analysis consist of the QCD tt continuum production, pp → tt, and the scalar resonance production pp → H → tt, respectively. The background and the signal events are generated with a centre-of-mass energy of 14 TeV, as expected during future LHC runs. Each top decays to a bottom quark and a W boson, and we focus on the decay of the W 's into muons exclusively. We consider four different masses of the scalar resonance, m H = 1.0, 1.5, 2.0, and 2.5 TeV. All events are generated with MadGraph5 aMC@NLO [56], and showered and hadronization is performed by Pythia8 [57]. Delphes3 [58] is utilized for the detector simulation, where the jets are clustered using FastJet [59]. We generate about 30k events for the background samples, while for each signal sample, we generate about 15k. The background events are divided into 10k training, 5k validation and 15k testing samples. For the object reconstruction, a standard jet definition using the anti-k t algorithm [60] with the jet radius R = 0.5 is used. For the signal bottom jets, the output from Delphes 3 is used and require p b T > 30 GeV. For isolated leptons, we requires p l T > 30 GeV and its isolation criteria with R = 0.5. We extracted four variables {p b1 T , p l1 T , p l2 T , / E T } for our analysis, keeping in mind the limitations of current devices. To conserve the aperiodic topology of these variables in the angle embedding (given in eq. 6) we fix the range of each variable to [0, 1000] by adding two points 5 and map the whole dataset to a range [0, π] via the MinMaxScaler implemented in scikit-learn [61]. The two added points are then removed from the dataset. This maps each feature's minimum and maximum to two distinct angles separated by a finite distance due to the selection criteria. Invisible Higgs signal over invisible Z background To test the anomaly detection capabilities of QAEs in a different scenario, we study invisible decays of a Z boson produced with two jets originating from QCD vertices. As a possible signal, we take the production of the 125 GeV Higgs boson and two jets originating from Electroweak vertices, decaying to two scalar dark matter particles. The generation is carried out in the same manner as in the previous case, including the definition of jets. We demand that we have at least two reconstructed jets with p T > 30 GeV, and the events have a missing transverse momentum / E T > 30 GeV. For the background, we have 30k events divided into 10k training, 5k validation, and 15k test events, while for the signal, we have 15k test events. We extract six variables to train the QAE and the CAE. They are the absolute separation in pseudorapidity between the two jets \|∆η jj \|, the invariant mass of the dijet system m jj and the sum of transverse energies H η C T = \|ηi\|<η C E i T , within four ranges of pseudorapidity η C ∈ {1.0, 1.5, 2.0, 2.5}. The mapping to conserve the aperiodic topology of these variables in the angular embedding is done by increasing their range on the higher side. B. Network architecture and training The QAE was implemented and trained using Pennylane [62]. As stated before, we train and test the QAE model with only the encoder circuit. After the input features are embedded as the rotation angle of the x-axis in the Bloch sphere, the unitary evolution U(Θ) consists of two stages. In the first step, each qubit is rotated by an angle θ i in the y-axis of the Bloch sphere. The values of these angles are to be optimized via gradient descent. After this, we apply the CNOT gate to all the possible pairs of qubits, with the ordering determined by the explicit number of the qubit. This circuit is shown in figure 2 for a four qubit input QAE with two-qubit latent representation. It is given by, U AB = C 23 ⊗ C 13 ⊗ C 12 ⊗ C 23 ⊗ C 03 ⊗ C 02 ⊗ C 01 ⊗ R 0 y (θ 0 ) ⊗ R 1 y (θ 1 ) ⊗ R 2 y (θ 2 ) ⊗ R 3 y (θ 3 ) , where C ij is the CNOT operation acting on the composite space of two qubits i and j, and R i y (θ i ) is the rotation of a single qubit i about the y-axis of the Bloch sphere. Note that the expression does not contain the operations of the SWAP test, which will be explained in the following paragraphs. The training proceeds to find the optimal values for θ i . The number of qubits discarded at the encoder, the size of the trash-state, fixes the latent dimension 6 via N lat = N in − N trash , with N lat the latent dimension, N in the size of the input state, and N trash the number of discarded qubits. The reference state \|β B , has the same number of qubits N trash , and it is initialized to be \|β B = \|0 ⊗N trash . We measure the fidelity between the trash-stateρ B and the reference state \|β B via a SWAP test [63]. It is a way to measure the fidelity between two multi-qubit states. For any two states \|φ and \|ψ with the same dimensions, the fidelity F (\|φ , \|ψ ) can be measured as the output of an ancillary qubit \|a anc after the following operation, H anc ⊗ I (c-SWAP) H anc ⊗ I \|0 anc ⊗ \|φ ⊗ \|ψ ,(12) where H anc is the Hadamard gate acting on the ancillary qubit, and c-SWAP is the controlled swap operation between the states \|φ and \|ψ controlled by the ancillary qubit. Thus the total number of qubits required for a fixed N in and N trash is N in + N trash + 1. Due to the limitation of current quantum devices we limit the input feature to four, and scan over the possible latent dimensions. The peak shifts towards the right in analogy to the CAE, however the shift is not as pronounced. With a single training sample, the network is not able to converge completely while for anything greater than 10, the increase in training size has practically no effect. The quantum network is trained by minimising the cost function (c.f eq. 11) with quantum gradient descent for the one, two and three-dimensional latent spaces. We train these instances for different training sizes of 1, 10, 100, 1000, and 10000 events to study the dependence of the QAE's performance on the size of the training data. We update the weights for each data sample, with 5000 shots in all training scenarios. For training sizes greater than or equal to 100, we train the networks for 50 epochs. In comparison, for sample sizes 1 and 10, we train the QAE for 500 and 200 epochs, respectively. To benchmark the performance of a QAE on a quantum computer, we train a QAE with the two inputs p l1 T and p b1 T with quantum-gradient descent on Pennylane, and compare the test performance with the simulation and the IBM-Q. For running on the IBM-Q, we build and implement the test circuits in Qiskit [64]. We also train classical autoencoders using Keras-v2.4.0 [65] with Tensorflow-v2.4.1 [66] for the same input features, for comparison. The encoder is a dense network mapping the input space to a latent dimension of N lat ∈ {1, 2, 3}, and has three hidden layers with 20, 15, and 10 nodes. The hidden layers have ReLU activations while the latent output has Linear activation. The decoder has a symmetric configuration to the encoder. The networks are trained with Adam [67] optimiser with a learning-rate of 10 −3 to minimise the root-mean-squared error between the input vector x and the reconstructed vectorx. For the CAEs, we found that training with single data per update (technically batch size=1) has a volatile validation loss per epoch, with slow convergence. Therefore, we choose a batch size of 64 to train the CAEs. 7 We train the QAE with analogous architecture for a six-dimensional input for the second scenario for a twodimensional latent space in a similar fashion for all training sizes. For the CAE keeping the number of nodes and layers identical to the previous case for six-dimensional input and output vectors, we perform a hyperparameter scan, the details of which is given in Appendix C. All results shown in the next section for this scenario is for the best performing hyperparameters. V. RESULTS Results of the various training scenarios are presented in this section. We present a detailed investigation of the QAE and CAE's properties for the tt background scenario in Sections V A to V D. The lessons learnt from these analyses, particularly the training size independence and the relative performance, are then tested for the invisible Z background in Section V E. A. Dependence of test reconstruction efficiency on the number of training samples The distribution of the loss function of the independent background test samples for different training sizes of the CAE is shown in figure 3. Although training with a single data point is inherently inaccurate, we perform such an exercise as a sanity check of the CAE's comparison to a QAE. The test distribution shifts towards the left as one increases the training size, thereby signifying increased reconstruction efficiency. For training sizes of up to 10 2 , the limited statistics will produce a very high statistical uncertainty. Since it is not the main emphasis of our present work, we do not comment any further. Looking at the distribution across different latent dimensions for 10 3 and 10 4 training samples, one can see the impact of the information bottleneck. For a singular latent dimension, the passed information is already available from 10 3 samples, and hence the loss distribution is very close to the one trained on 10 4 . This relative separation increases as we go to higher latent dimensions, denoting the higher information passed to the decoder to reconstruct the input, which is exploited with higher training samples. For an analogous comparison with the quantum fidelity, we define the cosine similarity between the input vector x and the reconstructed vectorx as, cos α = x.x \|x\| \|x\| ,(13) where the dot product is done with a Euclidean signature. The distribution of the cosine similarity shown in figure 4, shows similar features to the loss function's distribution, with efficient reconstruction possible only when the train size is at least 10 3 . We have seen that CAEs cannot be trained with limited statistics to reconstruct the statistically independent test dataset. From the distribution of the test sample's fidelity in figure 5, we see that QAEs are much more effective in learning from small data samples. Although training with a single data point has not reached the optimal reconstruction efficiency, it is obtained with ten sample events. Unlike CAEs, see figs. 3 and 4, the test fidelity distribution for all latent spaces are identical for training sizes greater than or equal to ten. The independence of the sample size is particularly important in LHC searches where the background cross section is small. This particularly interesting feature may be due to the interplay of an enhancement of statistics via the uncertainty of quantum measurements and the relatively simple circuits employed in our QAE circuit. For a single input point and assuming that we have hardware capable of building exact copies, a finite number of measurement processes always introduces a non-zero uncertainty in the network output. This uncertainty can act as additional information in the quantum gradient minimisation, which is performed after the measurement process, increasing the convergence for smaller data samples. Moreover, existing studies [68,69] show the advantage of quantum machine learning over classical approaches. Additionally, the use of quantum gradient descent [39] makes the loss landscape more convex, thereby speeding up convergence. B. Classification Performance We compare the QAE and the CAE's performance for the four-dimensional input feature space. The metrics used in this presentation bear similarity to those used in a supervised framework. It also assumes that a randomly chosen event is equally likely to be either the background or the signal. This assumption is not sound in the context of LHC searches or in an anomaly detection technique since the background's cross-section is orders of magnitude larger than that of the signal. Nevertheless, they are handy when comparing different classifiers. For each value of m H , we plot the Receiver-Operator-Characteristics (ROC) curve between the signal acceptance and the background rejection in figure 6, for the networks trained with 10k samples. The ROC curve is obtained by evaluating the signal acceptance as a function of the background rejection since both are functions of the threshold T 0 applied on the loss function. The signal acceptance S ∈ [0, 1] quantifies the fraction of accepted signal events when one puts a threshold T 0 on the variable x, while the background rejection¯ B ∈ [0, 1] measures the fraction of rejected background events for the threshold T 0 . An outline of how the ROC curve is obtained is given in Appendix B. The black dotted lines denote the performance of a random classifier with no knowledge of either the signal or the background, and the lines further away from it indicate better performance than those in its vicinity. The performance reduces with increasing latent dimensions for CAEs and QAEs, with the highest background rejection coming for a singular latent dimension. Comparing the QAEs and the CAEs (dotted vs solid lines for each colour), we find that QAEs perform better than CAEs consistently in all latent dimensions and the different values of m H . This better performance may be a universal property of QAEs. However, as our analysis is a proof-of-concept, an in-depth exploration of the properties of QAEs in general and anomaly detection at colliders, in particular, is needed to affirm this observation. C. Anomaly detection We now explore the performance of the autoencoders in a semi-realistic search scenario. When we scale the normalization of the signal and the background by their respective probability of occurrence, i.e. their respective cross-sections, we are essentially in an anomaly detection scenario since the background is orders of magnitude larger than the signal. The performance of the autoencoders can then be quantified in terms of statistical significance as a function of the threshold applied on the loss. For the background, we scale the cross-section obtained from Madgraph by a global k factor of 1.8 [70], while for all the signal masses, we fix a reference value of 10 fb. The yield is then calculated as N p = p σ p L E p (T 0 ) , where p is the baseline selection efficiency, σ p the cross-section, and E p (T 0 ) the efficiency at a threshold T 0 of the loss distribution, for a process p, while L is the integrated luminosity which we take to be 3000 fb −1 . Since it is natural to use the best classifier in a search, we evaluate the significance of the autoencoders with one latent dimension, trained on 10k samples. We apply the threshold for the QAE and the CAE on the quantum trash state fidelity and the RMSE loss, respectively. We use (1 − Fidelity) for the QAE to make the signal-rich regions same in both scenarios. RMSE loss is chosen over the cosine similarity since the former was found to have a higher performance. The significance N S / √ N B for each of the signal masses as a function of the threshold T 0 is shown in figure 7. We fix the threshold range so that there are enough background test statistics in the least background like bin. Looking at the peak of the significance, we note that QAEs outperform CAEs, which is only natural from the preceding discussions. However, an interesting development is the relative performance for the different masses. Even though the ROCs indicated higher discrimination with increasing mass, the significance increases for m H = 1.0 TeV to 1.5 TeV and decreases for higher masses. Since we have fixed a fiducial cross-section for each signal mass, it plays no role in this irregularity. The trend arises via an interplay between the higher discrimination by the autoencoder output and the decrease in baseline efficiency with increasing mass m H . The decreasing selection efficiency is due to the isolation criteria of the jets and the leptons, which would be naturally boosted when we go to higher resonant masses m H , thereby becoming more collimated. D. Benchmarking on a quantum device We now compare the performance of the quantum simulator and the actual quantum hardware. Since there is a limitation on the available number of qubits, we limit the feature space in two dimension, which consists of {p b1 T , p l1 T }. For our QAE setup, in addition to the two qubits for embedding the input features, one qubit for the reference state and another ancilliary qubit for the SWAP test are needed. We use the simpler version of the quantum circuit shown in figure 2, which is implemented and trained using PennyLane. To compare the performance, we use the same circuit with the same optimized parameters both for PennyLane and for the IBM-Q belem backend. Accessing the IBM hardware was done through Qiskit. In figure 8, we show the fidelity distributions for the background and the signal samples for our QAE circuit with the optimized circuit parameters computed by the simulator in Pennylane and in the actual quantum device of IBM-Q belem backend. The plot shows the shape of the distribution (denoted by the width of the shaded region) in the y-axis for each bins of size 0.1 in the x-axis (plotted at each bin center). The lines at each ending denote the range of the data of the y-axis. Since IBM-Q does not have a shallow implementation of the CSWAP operation, the fidelity distributions are smeared toward 0.5, and it is especially worse around 1. One of the advantages of using the SWAP test is to reduce the number of qubits for the evaluations of the fidelity during the optimization process. For example, to check the performance of the current circuit, directly measuring the fidelity between the reference state and the output for the second qubit would be enough. It can be achieved by the simple Pauli z measurements. The correlation of the fidelities obtained by Pennylane and by IBM-Q belem, based on the SWAP test and on the Pauli z measurement are shown in the right panel as the violin plots, in blue and in orange, respectively. The correlation is better for the Pauli z measurements for the same circuit part with the identical input parameters. It suggests that the decoherence effects from a deeper circuit obscure the performance. In figure 9, we show the ROC curves based on the fidelity distributions for the background and the signal samples evaluated by Pennylane simulator in the left panel. The central panel shows the ROC curves based on the fidelities evaluated by the SWAP test, while the right panel shows those by the second qubit Pauli z measurements, for the same IBM-Q device of belem backend. As one can see, the performances based on the Pauli z measurements on the IBM-Q device follow those obtained by the Pennylane simulator. The AUCs for them are also essentially the same. Thus, the deficit in the performance with the SWAP test is due to the too deep circuit realization for the CSWAP operation in the IBM-Q device. Therefore, the realisation of a CSWAP operation with a shallow circuit is necessary. To check the efficacy of quantum hardware for the four input QAE, we evaluate the trash state fidelity of a QAE with four-dimensional input features. Due to hardware limitations discussed above, we estimate it without the SWAP test for a single trash qubit giving us a three-dimensional latent representation. The correlation between the Pennylane evaluated fidelity and the output from IBM-Q lagos, shown in figure 8, displays a good agreement between the simulation and the hardware. E. Comparative training efficiency and performance for pp → Z(νν)jj background We have seen that a QAE trains efficiently and performs better than a CAE in a hypothetical resonant signal scenario. To gauge how these important behaviours carry over to a different process, we study the training size dependence and performance of a QAE and CAE for an invisible background (and signal), detailed in the last paragraph of Section IV A for a two-dimensional latent space. Note that all the results for the CAE are for the best model chosen after a hyperparameter scan described in Appendix C. The loss distribution of the test dataset for the background for different sizes of training data and their ROC curve for the case of 10k training samples are shown in figure 10. The characteristics are similar to the previous scenario, giving further evidence that the training efficiency of the QAE is not limited to a specific kind of process. Moreover, from the ROC and the AUC value, we see that the QAE also performs better than the CAE. This superior performance is particularly noteworthy given that the CAE's hyperparameters has been chosen after a hyperspace scan restricted to a fixed width and depth. VI. CONCLUSION The lack of evidence for new interactions and particles at the Large Hadron Collider has motivated the high-energy physics community to explore model-agnostic data-driven approaches to search for new physics. Machine-learning anomaly detection methods, such as autoencoders, have shown to be a powerful and flexible tool to search for outliers in data. Autoencoders learn the kinematic features of the background data by training the network to minimise the reconstruction error between input features and neural network output. As the kinematic characteristics of the signal are different to the background, the reconstruction error for the signal is expected to be larger, allowing signal events to be identified as anomalous. Although quantum architecture capable of processing huge volumes of data is not yet feasible, noisy-intermediate scale devices could have very real applications at the Large Hadron Collider in the near future. With the origin of the collisions being quantum-mechanical, a quantum autoencoder could, in principle, learn quantum correlations in the data that a bit based autoencoder fails to see. We have shown that quantum-autoencoders based on variational quantum circuits have potential applications as anomaly detectors at the Large Hadron Collider. Our analysis shows that for the scenario we consider, i.e. the same set of input variables, quantum autoencoders outperform dense classical autoencoders based on artificial neural networks, asserting that quantum autoencoders can indeed go beyond their classical counterparts. They are very judicious with data and converge with very small training samples. This independence opens up the possibility of training quantum autoencoders on small control samples, thereby opening up data-driven approaches to inherently rare processes. f S (T 0 ) = T0 −∞ dx p S (x) ,f B (T 0 ) = ∞ T0 dx p B (x) , where we have assumed that the signal rich regions are on the lower side of the variable x. The ROC curve is then obtained by expressing the signal acceptance as a function of the background rejection as T 0 =f −1 B (¯ B ) =⇒ S = f S (T 0 ) = f S (f −1 B (¯ B )) . The ROC curve therefore shows the function S (¯ B ) without any reference to the threshold T 0 , which is implicitly assumed in the evaluation of the dependent ( S ) and the independent (¯ B ) quantities. The variable x can be any physical observable or the output of a neural network model. For the studies conducted here, it is the RMSE loss for the CAE, and the fidelity for the QAE. The details of the hyperparameter scan of classical autoencoder with six-dimensional inputs and outputs are given in this appendix. We use the RandomSearch algorithm implemented in KerasTuner [74] for the scan. The number of nodes in the hidden layers of the encoder is kept fixed to 20, 15, and 10. With a (fixed) two-dimensional latent space, we use a symmetric decoder setup. Once the skeleton of the architecture is fixed, we scan over the activation function of the layers, L1 regularisation and L2 regularisation of the weights, the dropout value between two successive layers, and the training's learning rate and batch size. Their respective values along with the best one chosen for the final training are given in table I. The best value of the hyperparameters are from thousand trials trained for hundred epochs, and the training is terminated if the validation loss does not improve for ten epochs (implemented as the EarlyStopping callback during training). We do not vary the width or the depth to compare the capabilities of CAEs with at least some degree of comparability to the simple QAE used in the study. Increasing the width and depth will undoubtedly increase the expressive power of a CAE, which is not the objective of the current study. Networks like Convolutional or graph autoencoders acting on low-level high dimensional data will undoubtedly perform better than currently executable QAEs. However, existing quantum resources cannot process such high dimensional data. FIG. 2 : 2The figure shows a Quantum autoencoder circuit for a four qubit input and two latent qubits. The inputs are already embedded in qi (by the input embedding circuit), which are then rotated by tunable angles θi in the y-direction of the Bloch sphere by Ry(θi) gates (shown in purple boxes). Each pair of these qubits are entangled via CNOT gates(shown with blue lines). For the trash training, we need a two-dimensional reference state denoted by ti qubits and an ancillary qubit a0. FIG. 3 :FIG. 4 :FIG. 5 : 345Loss distribution of the test background samples (15k) for different sizes of training dataset. We can see that the distribution shifts significantly towards the left (direction with lower loss) as one increases the training data size, which reflects that there is noticeable increase in learning with larger data samples. Cosine similarity (analogous to quantum fidelity) distribution of the test background samples (15k) for different sizes of training dataset of the CAE. Quantum Fidelity distribution of the test background samples (15k) for different sizes of training dataset for the QAE. FIG. 6 : 6ROC curve between signal acceptance vs background rejection for Quantum Autoencoder (QAE) and Classical Autoencoder (CAE) for various values of mH and different latent dimensions for a training datasize of 10k samples. The trend across latent dimensions is same for both QAE and CAE with QAEs performing better in all cases. FIG. 7 : 7Significance as a function of the threshold T0 on the fidelity and root-mean-square-error (RMSE) of the QAE and the CAE, respectively, for each of the signal scenarios and singular latent dimension trained on 10k samples. To keep the signal rich region on the right side for both, we have used (1 − Fidelity) for the QAE. We fix the cross-section of all signals to 10 fb, and evaluate the yields at an integrated luminosity of 3000 fb −1 . FIG. 8 : 8The correlation between the fidelity values obtained by Pennylane and by the IBM-Q backends. On the left we show the comparison of a 2-1-2 QAE, where we directly measure the trash state (orange) and with a SWAP test employing a CSWAP gate. We find that the shallower implementation without the CSWAP gate has lesser decoherence effects, and hence better agreement with the simulation. The correlation with the direct measurement for the 4-3-4 case is shown on the right..0TeV, AUC=0.89 mH=1.5TeV, AUC=0.94 mH=2.0TeV, AUC=0.95 mH=2.5TeV, with CSWAP) mH=1.0TeV, AUC=0.68 mH=1.5TeV, AUC=0.78 mH=2.0TeV, AUC=0.81 mH=2.5TeV, without CSWAP) mH=1.0TeV, AUC=0.89 mH=1.5TeV, AUC=0.94 mH=2.0TeV, AUC=0.95 mH=2.5TeV, AUC=0.96FIG. 9: ROC curves based on the fidelity distributions. Those evaluated by the Pennylane simulator (left panel), by the quantum device IBM-Q belem backend with the SWAP test (central panel), and with the second qubit measurements (right panel) are shown. FIG. 10: The test distribution of the invisible Z background scenario for different training sizes of a CAE (left) and a QAE (center) for a two-dimensional latent representation, and their respective ROC curve (right) for the training done with 10k events. Similar to the previous case, the QAE has converged with much smaller datasets than the CAE. Moreover, the QAE performs relatively better than the CAE for the particular signal.Training Size=10k CAE AUC 0.7000 QAE AUC 0.7524 Appendix C: Details of hyperparameter scanSl. no. Hyper Parameter Value Space Best value 1. Activation function tanh, ReLu, Sigmoid, Linear ReLu 2. L1 Regularisation 0,0.1,0.01,0.001,0.0001 0 3. L2 Regularisation 0,0.1,0.01,0.001,0.0001 0 4. Dropout 0,0.1,0.2,0.3 0 5. Learning Rate 0.01,0.001,0.0003 0.0003 6. Batch Size 32,64,128,256,512,1024 64 TABLE I : IThe table shows the different values of the hyperparameters and their best values after the scan. It has been found in ref[49] that Convolutional Autoencoders cannot detect signals of lower intrinsic dimensions. While Quantum Autoencoders could alleviate this issue, we do not study their properties for lower dimensional signals. The study aims to validate their workings on similar scenarios where classical autoencoders work. 3 See[39] for a brief presentation of the Fubini-Study metric and a comparison of natural and quantum gradient descent for the training of classical and quantum networks. It was shown that quantum gradient descent improves the training of a variational quantum circuit significantly. Events with the variables lying above 1000 GeV are very rare and excluded in our case. In a realistic analysis, the upper bound can be determined from the data. In our discussion, we will use the number of latent qubits as the latent dimension, although the Hilbert space would have 2 N lat dimensions. The network performs one update per epoch for training with a number of samples less than 64. These training sizes are too small for a CAE to have any good learning capability. Hence, we do not try to modify the batch sizes or interpret the test distributions. Appendix A: Quantum Gradient DescentWe discuss the basic idea behind quantum gradient descent[45]in this appendix. The general idea is to make the optimisation procedure aware of the underlying quantum geometry of the weight space. Denoting any generic weight vector by Θ, we have the vanilla gradient descent update as,where L is a well-behaved loss function. This expression implicitly assumes that l 2 distances correctly describe the underlying geometry of the weight space, placing all directions in the weight space on an equal footing. In reality, however, the geometry of the weight space can be much more complicated, and such a straightforward update rule may not converge to the optimal point. Therefore, to have an idea of the underlying geometry, we modify eq. A1 with the metric tensor G,to get the Natural Gradient descent[71]. Note that Natural Gradient descent gives the usual gradient descent (eq. A1) for a Euclidean metric G = I. 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[ "Evolving Robust Neural Architectures to Defend from Adversarial Attacks", "Evolving Robust Neural Architectures to Defend from Adversarial Attacks" ]	[ "Shashank Kotyan [email protected] \nDepartment of Informatics\nKyushu University\nJapan\n", "Danilo Vasconcellos Vargas [email protected] \nDepartment of Informatics\nKyushu University\nJapan\n" ]	[ "Department of Informatics\nKyushu University\nJapan", "Department of Informatics\nKyushu University\nJapan" ]	[]	Neural networks are prone to misclassify slightly modified input images. Recently, many defences have been proposed, but none have improved the robustness of neural networks consistently. Here, we propose to use adversarial attacks as a function evaluation to search for neural architectures that can resist such attacks automatically. Experiments on neural architecture search algorithms from the literature show that although accurate, they are not able to find robust architectures. A significant reason for this lies in their limited search space. By creating a novel neural architecture search with options for dense layers to connect with convolution layers and vice-versa as well as the addition of concatenation layers in the search, we were able to evolve an architecture that is inherently accurate on adversarial samples. Interestingly, this inherent robustness of the evolved architecture rivals state-ofthe-art defences such as adversarial training while being trained only on the non-adversarial samples. Moreover, the evolved architecture makes use of some peculiar traits which might be useful for developing even more robust ones. Thus, the results here confirm that more robust architectures exist as well as opens up a new realm of feasibilities for the development and exploration of neural networks.	10.1145/3377929.3389962	[ "https://arxiv.org/pdf/1906.11667v3.pdf" ]	195,699,904	1906.11667	21f197e08105f99d2b3fd1aa16fbf38feb4bea65	Evolving Robust Neural Architectures to Defend from Adversarial Attacks Shashank Kotyan [email protected] Department of Informatics Kyushu University Japan Danilo Vasconcellos Vargas [email protected] Department of Informatics Kyushu University Japan Evolving Robust Neural Architectures to Defend from Adversarial Attacks Neural networks are prone to misclassify slightly modified input images. Recently, many defences have been proposed, but none have improved the robustness of neural networks consistently. Here, we propose to use adversarial attacks as a function evaluation to search for neural architectures that can resist such attacks automatically. Experiments on neural architecture search algorithms from the literature show that although accurate, they are not able to find robust architectures. A significant reason for this lies in their limited search space. By creating a novel neural architecture search with options for dense layers to connect with convolution layers and vice-versa as well as the addition of concatenation layers in the search, we were able to evolve an architecture that is inherently accurate on adversarial samples. Interestingly, this inherent robustness of the evolved architecture rivals state-ofthe-art defences such as adversarial training while being trained only on the non-adversarial samples. Moreover, the evolved architecture makes use of some peculiar traits which might be useful for developing even more robust ones. Thus, the results here confirm that more robust architectures exist as well as opens up a new realm of feasibilities for the development and exploration of neural networks. Introduction Neural Architecture Search (NAS) and adversarial samples have rarely appeared together. Regarding adversarial samples, they were discovered in 2013 when neural networks were shown to behave strangely for nearly the same images [Szegedy, 2014]. Afterwards, a series of vulnerabilities were found in [Moosavi-Dezfooli et al., 2017;Su et al., 2019]. Such adversarial attacks can also be easily applied to real-world scenarios which confer a big problem for current deep neural networks' applications. Currently, there is not any known learning algorithm or procedure that can defend against adversarial attacks consistently. Regarding NAS, the automatic design of architectures has been of broad interest for many years. The aim is to develop methods that do not need specialists in order to be applied to a different application. This would confer not only generality but also easy of use. Most of the algorithms for NAS are either based on reinforcement learning [Pham et al., 2018; or evolutionary computation [Real et al., 2017;Miikkulainen et al., 2019]. On the one hand, in reinforcement learning approaches, architectures are created from a sequence of actions which are afterwards rewarded proportionally to the crafted architecture's accuracy. On the other hand, in evolutionary computation based methods, small changes in the architecture (mutations) and recombinations (crossover) are used to create new architectures. All architectures evolved are evaluated based on their accuracy. Some of the best architectures based on this accuracy are chosen to continue to the next generation. Here we propose the use of NAS to tackle the robustness issues exposed by adversarial samples. In other words, architecture search will be employed not only to find accurate neural networks but also robust ones. This is based on the principle that robustness of neural networks can be assessed by using accuracy on adversarial samples as an evaluation function. We hypothesise that if there is a solution in a given architecture search space, the search algorithm would be able to find it. This is not only a blind search for a cure. The best architectures found should also hint which structures and procedures provide robustness for neural networks. Therefore, it would be possible to use the results of the search to understand further how to improve the representation of models as well as design yet more robust ones. Adversarial Machine Learning Adversarial machine learning is a constrained optimisation problem. Let f (x) ∈ [[1..N ]] be the output of a machine learning algorithm in multi-label classification setting. Here, x ∈ R k is the input of the algorithm for the input of size k and N is the number of classes in which x can be classified. In Image Classification problem k = m × n × 3 where m × n is the the size of the image. Adversarial samples x can be thus defined as follows: x = x + x such that f (x ) = f (x)(1) in which x ∈ R k is a small perturbation added to the input. Therefore, adversarial machine learning can be defined as an arXiv:1906.11667v3 [cs.NE] 16 Jul 2020 optimization problem 1 : minimize x g(x + x ) c subject to x ≤ th (2) where th is a pre-defined threshold value and g() c is the softlabel or confidence for the correct class c such that f (x) = argmax g(x) Moreover, attacks can be divided according to the function optimised. In this way, there are L 0 (limited number of pixels attacked), L 1 , L 2 and L ∞ (limited amount of variation in each pixel) types of attacks. There are many types of attacks as well as their improvements. Universal perturbation types of attacks were shown possible in which a single perturbation added to most of the samples is capable of fooling a neural network in most of the cases [Moosavi-Dezfooli et al., 2017]. Moreover, extreme attacks such as only modifying one pixel (L 0 = 1) called one-pixel attack is also shown to be surprisingly effective [Su et al., 2019;Vargas and Kotyan, 2019]. Most of these attacks canbe easily transferred to real scenarios by using printed out versions of them [Kurakin et al., 2016]. Moreover, carefully crafted glasses [Sharif et al., 2016] or even general 3D adversarial objects are also capable of causing misclassification [Athalye and Sutskever, 2018]. Regarding understanding the phenomenon, it is argued in [Goodfellow et al., 2014] that neural networks' linearity is one of the main reasons. Another recent investigation proposes the conflicting saliency added by adversarial samples as the reason for misclassification [Vargas and Su, 2019]. Many defensive systems were proposed to mitigate some of the problems. However, current solutions are still far from solving the problems. Defensive distillation [Papernot et al., 2016] uses a smaller neural network to learn the content from the original one; however, it was shown not to be robust enough [Carlini and Wagner, 2017]. The addition of adversarial samples to the training dataset, called adversarial training, was also proposed [Goodfellow et al., 2014;Huang et al., 2015;Madry et al., 2018]. However, adversarial training has a strong bias in the type of adversarial samples used and is still vulnerable to attacks Many recent variations of defences were proposed which are carefully analysed, and many of their shortcomings explained in [Athalye et al., 2018;Uesato et al., 2018]. In this article, different from previous approaches, we aim to tackle the robustness problems of neural networks by automatically searching for inherent robust architectures. Neural Architecture Search There are three components to a neural architecture search: search space, search strategy and performance estimation. A search space substantially limits the representation of the architecture in a given space. A search strategy must be employed to search for architectures in a defined search space. Some widely used search strategies for NAS are: Random Search, Bayesian Optimization, Evolutionary Methods, Reinforcement Learning, and Gradient Based Methods. Finally, 1 Here the definition will only concern untargeted attacks but a similar optimization problem can be defined for targeted attacks a performance estimation (usually error rate) is required to evaluate the explored architectures. Currently, most of the current NAS suffer from high computational cost while searching in a relatively small search space [Lee et al., 2018;Lima and Pozo, 2019].It is already shown in [Yu and Kim, 2018] that, if there is a possibility of fitness approximation at small search spaces, we could evolve algorithms in an ample search space. Moreover, many architecture searches focus primarily on the hyper-parameter search while using architecture search spaces around previously hand-crafted architecture [Lee et al., 2018;He et al., 2019] such as DenseNet which are proved to be vulnerable to adversarial attacks [Vargas and Kotyan, 2019]. Therefore, for finding robust architectures, it is crucial to expand the search space beyond the current NAS. SMASH [Brock et al., 2017] uses a neural network to generate the weights of the primary model. The main strength of this approach lies in preventing high computational cost, which is incurred in other searches. However, this comes at the cost of not being able of tweaking hyperparameters which affect weights like initialisers and regularisers. Deep Architect [Negrinho and Gordon, 2017] follows a hierarchical approach using various search algorithms such as Monte Carlo Tree Search (MCTS) and Sequential Model based Global Optimization (SMBO). Searching for Robust Architectures A robust evaluation (defined in Section 4.1) and search algorithm must be defined to search for robust architectures. The search algorithm may be a NAS provided that some modifications are made (Section 4.2). However, to allow for a more extensive search space, which is better suited to the problem, we also propose the Robust Architecture Search (Section 5). Robustness Evaluation Adversarial accuracy may seem like a natural evaluation function for assessing neural networks' robustness. However, there are many types of perturbations possible; each will result in a different type of robustness assessment and evolution. For example, let us suppose an evaluation with th = 5 is chosen, robust networks against th = 5 might be developed. At the same time, nothing can be said for other th and attack types (different L). Therefore, th plays a role but the different types of L 0 , L 1 , L 2 and L ∞ completely change the type of robustness, such as wide perturbations (L ∞ ), punctual perturbations (L 0 ) and a mix of both (L 1 and L 2 ). To avoid creating neural networks that are only robust against one type of robustness and at the same time to allow robustness to slowly build-up from any partial robustness, a set of adversarial attacks for varying th and L are necessary. To evaluate the robustness of architectures in varying th and L while at the same time keeping computational cost low, we use here a transferable type of attack. In other words, adversarial samples previously found by attacking other methods are stored and used as possible adversarial samples to the current model under evaluation. This solves the problem that most of the attacks are usually slow to be put inside a loop which can make the search for architectures too expensive. Figure 1: Illustration of the proposed RAS structure with three subpopulations. Model Attack Optimiser L 0 Attack L ∞ Attack Total th = 1 th = 3 th = 5 th = 10 th = 1 th = 3 th = 5 th = 10 Table 1 shows a summary of the number of images used from each type of attack, totalling 2812 adversarial samples. Samples were generated using the model agnostic dual quality assessment [Vargas and Kotyan, 2019]. Specifically, we use the adversarial samples from two types of attacks (L 0 and L ∞ attacks) with two optimization algorithms (Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [Hansen et al., 2003] and Differential Evolution (DE) [Storn and Price, 1997]). We use CIFAR-10 dataset [Krizhevsky et al., 2009] to generate the adversarial samples. We attacked traditional architectures such as ResNet [He et al., 2016] and CapsNet [Sabour et al., 2017]. We also attacked some state-of-artdefences such as Adversarial Training (AT) [Madry et al., 2018] and Feature Squeezing (FS) [Xu et al., 2017] defending ResNet. The evaluation procedure consists of calculating the amount of successful adversarial samples divided by the total of possible adversarial samples. This also avoids problems with different amount of perturbation necessary for attacks to succeed, which could cause incomparable results. Robust Search Conversion of existing NAS By changing the fitness function (in the case of evolutionary computation based NAS) or the reward function (in the case of reinforcement learning based NAS), it is possible to create robust search versions of NAS algorithms. In other words, it is possible to convert the search for accuracy into the search for robustness and accuracy. Here we use SMASH and DeepArchitect for the tests. The reason for the choice lies in the difference between the methods and availability of the code. Both methods have their evaluation function modified to contain not only accuracy but also robustness (Section 4.1). Robust Architecture Search (RAS) Here, we propose an evolutionary algorithm to search for robust architectures called Robust Architecture Search (RAS) 2 . It makes sense to focus here on search spaces that allow for unusual layer types and their combinations to happen, which is more vast than the current traditional search spaces. The motivation to consider this vast search space is that some of the most robust architectures might contain these unusual combinations which are not yet found or deeply explored. RAS Overview RAS works by creating three initial populations (layer, block and model populations). Every generation, the model population have each of its members modified five times by mutations. The modified members are added to the population as new members. Here we propose a utility evaluation in which layer and block populations are evaluated by the number of models (architectures) using them. Models are evaluated by their accuracy and attack resilience (accuracy on adversarial samples). All blocks and layers which are not used by any of the current members of the model population are removed at the end step of each generation. More- Child Population Model 1_1 Model 1_2 Model 2_1 Model 2_2 Model 25_1 Model 25_2 Mutations Model i Model c Individual Fitness Evaluation Closest Individual to Model i based on Spectrum Better Fitness Individual Step 1 Step 2 Step 3 Step 4 Description of Population For such a vast search space to be more efficiently searched, we propose to use three subpopulations, allowing for the reuse of blocks and layers. Specifically, the layers consist of: Layer Population: Raw layers (convolutional and fully connected) which make up the blocks. Block Population: Blocks which are a combination of layers. Model Population: A population of architectures which consists of interconnected blocks. Figure 1 illustrates the architecture. The initial population consists of 25 random architectures which contain U (2, 5) blocks made up of U (2, 5) layers, in which U (a, b) is a uniform random distribution with minimum a and maximum b values. The possible available parameters for the layers are as follows: for convolutional layers, filter size might be 8, 16, 32 or 64, stride size may be 1 or 2 and kernel size is either 1, 3 or 5; for fully connected layers, the unit size may assume the values of 64, 128, 256 or 512. All the layers use Rectified Linear Unit (ReLU) as an activation function and are followed by a batch-normalisation layer. Mutation Operators Regarding the mutation operators used to evolve the architecture, they can be divided into layer, block and model mutations which can only be applied to the respective layer, block and model populations' individuals. The following paragraphs define the possible mutations. Layer Mutation Layer mutations are of the following types: a) Change Kernel: Changes the kernel size of the convolution layer, b) Change Filter: Changes the filter size of the convolution layer, c) Change Units: Changes the unit size of the fully connected layer, d) Swap Layer: Chosen layer is swapped with a random layer from the layer population. Block Mutation Block mutation change a single block in the block population. The possibilities are: e) Add Layer: A random layer is added to a chosen random block, f) Remove Layer: A random layer is removed from a chosen random block, g) Add Layer Connection: A random connection between two layers from the chosen random block is added, h) Remove Layer Connection: A random connection between the two layers from the chosen random block is removed, i) Swap Block: Chosen block is swapped with a random block from the population. Model Mutation Model mutation modify a given architecture. The possible model mutations are: j) Add Block: A random block is added to the model, k) Remove Block: A random block is removed from the model, l) Add Block connection: A random connection between the two blocks is added, m) Remove Block connection: A random connection between the two blocks is removed. All mutations add a new member to the population instead of substituting the previous one. In this manner, if nothing is done, the population of layers and blocks may explode, increasing the number of lesser quality layers and blocks. This would cause the probability of choosing functional layers and blocks to decrease. To avoid this, when the layer or block population exceeds 100 individuals, the only layer/block mutation available is swap layers/blocks. Objective (Fitness) Function Fitness of an individual of the model population is measured using the final validation accuracy of the model after training for a maximum of 200 epochs with early stopping if accuracy or validation accuracy do not change more than 0.001 in the span of 15 epochs. Regarding the fitness calculation, the fitness is calculated as the accuracy of the model plus the robustness of the model (Fitness = Accuracy + Robustness). The Accuracy of the architecture is calculated after the model is trained for 50 epochs over the whole set of samples (50000 samples) of the CIFAR-10's training dataset for every 10 generation(s) or over 1000 random samples of the CIFAR-10 training dataset for all other generations. This allows an efficient evolution to happen in which blocks and layers evolve at a faster rate without interfering with the architecture's accuracy. Using entire dataset subjects to evolving the architecture to have better accuracy and using a subset of the dataset evolves the layers and blocks of the architecture at a faster rate. The Robustness of the architecture is calculated using accuracy on adversarial samples as described in Section 4.1. Spectrum-based Niching Scheme To keep a high amount of diversity while searching in a vast search space by using a novel algorithm described below also shown in Figure 2. This niching scheme uses the idea of Spectrum-based niching from [Vargas and Murata, 2017] but explores a different approach to it. First, all the initial population is converted into a cluster population such that each individual in the initial population is a cluster representative. Then we create two child individuals for each cluster representative by randomly applying five mutation operators on cluster representative. We then find the closest cluster representative to the child individual using spectrum described below. If the fitness of the child individual is better than the closest cluster representative than the child individual becomes the new cluster representative, and the old cluster representative is removed from the population and the generation. The process is completed for all the individuals in a cluster population. We are hence evolving a generation of the evolution. Here, we use the spectrum as a histogram containing the features: Number of Blocks, Number of Total Layers, Number of Block Connections, Number of Total Layer Connections, Number of Dense Layers, Number of Convolution Layers, Number of Dense to Dense Connections, Number of Dense to Convolution Connections, Number of Convolution to Dense Connections, and Number of Convolution to Convolution Connections. By using this Spectrum-based niching scheme, we aim to achieve an open-ended evolution, preventing the evolution from converging to a single robust architecture. Preserving diversity in the population ensures that the exploration rate remains relatively high, allowing us to find different architectures even after many evolution steps. For the vast search space of architectures, this property is especially important, allowing the algorithm to traverse the vast search space efficiently. Experiments on RAS and Converted NAS Architecture Search Testing ER ER on Adversarial Samples DeepArchitect* 25% 75% SMASH* 23% 82% Ours 18% 42% Table 2: Error Rate (ER) on both the testing dataset and adversarial samples when the evaluation function has both accuracies on the testing data and accuracy on the adversarial samples. *Both DeepArchitect and SMASH had their evaluation function modified to be the sum of accuracy on the testing and adversarial samples. Here, experiments are conducted on both the proposed RAS and converted versions of DeepArchitect and SMASH. The objective is to achieve the highest robustness possible using different types of architecture search algorithms and compare their result and effectiveness. Initially, DeepArchitect and Smash found architectures which had an error rate of 11% and 4% respectively when the fitness is only based on the neural network's testing accuracy. However, when the accuracy on adversarial samples is included in the evaluation function, the final error rate increases to 25% and 23% respectively (Table 2). This may also indicate that poisoning the dataset might cause a substantial decrease in accuracy for the architectures found by SMASH and DeepArchitect. In the case of RAS, even with a more extensive search space, an error rate of 18% is achieved. Regarding the robustness of the architectures found, Table 2 shows that the final architecture found by DeepArchitect and SMASH were very susceptible to attacks, with error rate on adversarial samples of 75% and 82% respectively. Despite the inclusion of the R (measured accuracy on adversarial samples) on the evaluation function, the architectures were still unable to find a robust architecture. This might be a consequence of the relatively small search space used and more focused initialisation procedures. Moreover, the proposed method (RAS) finds an architecture which has an error rate of only 42% on adversarial samples. Note, however, that in the case of the evolved architecture, this is an inherent property of the architecture found. The architecture is inherently robust without any kind of specialised training or defence such as adversarial training (i.e., the architecture was only trained on the training dataset). The addition of defences should increase its robustness further. Analyzing RAS In this section, we will evaluate the proposed architecture regarding its evolution quality and how subpopulations behave throughout the process. Figure 3 shows how the mean accuracy of the architectures evolved increases over time. The pattern of behaviour is typical of evolutionary algorithms, showing that evolution is happening as expected. In Figure 4, the overall characteristics of the evolved architectures throughout the generations are shown. The average number of blocks and the connections between them increase over the generations. However, the average number of layers never reaches the same complexity as the initial models. The number of layers decreases steeply initially while slowly increasing afterwards. Therefore, the overall behaviour is that blocks become smaller and numerous. A consequence of this is that the number of connections becomes proportional to the number of connections between blocks and therefore exhibit similar behaviour. The average number of layers per block and the average number of connections shows little change, varying only 0.5 and 0.16 respectively. Notice that the average number of layers increases but the average number of layers per block continues to decrease albeit slowly. Consequently, blocks tend to degenerate into a few layers, resulting in around three layers per block from the first average number of 3.2 layers per block. Lastly, the average number of connections in a block is kept more or less the same, with the mean varying throughout only from 2.1 to 2.26. The behaviour described above might suggest that it is hard to create big reusable blocks. This seems to be supported by both the decrease of complexity observed as well as the increase in the number of blocks. 8 Analyzing the Final Architecture: Searching for the Key to Inherent Robustness RAS found an architecture that possesses inherent robustness capable of rivalling current defences. To investigate the reason behind this robustness, we can take a more in-depth look at the architecture found. Figure 5 show(s) some peculiarities from the evolved architecture: multiple bottlenecks, projections into high-dimensional space and paths with different constraints. Multiple Bottlenecks and Projections into High-Dimensional Space The first peculiarity is the use of Dense layers in-between Convolutional ones. This might seem like a bottleneck similar to the ones used in variational autoencoders. However, it is the opposite of a bottleneck ( Figure 5); it is a projection in high-dimensional space. The evolved architecture uses mostly a low number of filters while, in some parts of it, high-dimensional projections exist. In the whole architecture, four Dense layers in-between Convolutional ones were used, and all of the projects into higher dimensional space. This follows directly from Cover's Theorem which states that projecting into high dimensional space makes a training set linearly separable [Cover, 1965]. Paths with Different Constraints The second peculiarity is the use of multiple paths with the different number of filters and output sizes after high-dimensional projections. Notice how the number of filters differs in each of the Convolutional layers in these paths. This means there are different constraints over the learning in each of these paths, which should foster different types of features. Therefore, this is a multi-bottleneck structure forcing the learning of different sets of features which are now easily constructed from the previous high-dimensional projection. Conclusions Automatic search for robust architectures is proposed as a paradigm for developing and researching robust models. This paradigm is based on using adversarial attacks together with error rate as evaluation functions in NAS. Experiments on using this paradigm with some of the current NAS had poor results. This was justified by the small search space used by current methods. Here, we propose the RAS method, which has a broader search space, including concatenation, connections between dense to convolutional layer and viceversa. Results with RAS showed that inherently robust architectures do indeed exist. In fact, the evolved architecture achieved robust results comparable with state-of-the-art defences while not having any specialised training or defence. In other words, the evolved architecture is inherently robust. Such inherent robustness could increase if adversarial training, or other types of defence, or a combination of them are employed together with it. Moreover, investigating the reasons behind such robustness have shown that some peculiar traits are present. The evolved architecture has overall a low number of filters and many bottlenecks. Multiple projections into high-dimensional space are also present to possibly facilitate the separation of features (Cover's Theorem). It also uses multiple paths with different constraints after the high-dimensional projection, which should, consequently, cause a diverse set of features to be learned by the network. Thus, in the search space of neural networks, more robust architectures do exist, and more research is required to find and fully document them as well as their features. Figure 2 : 2Illustration of the proposed Evolutionary Strategy. In this strategy, Step 2, 3, and 4 are repeated for all the individuals in the child population.Step 1 is repeated when all the individuals in the child population have been evaluated against a cluster individual. over, architectures compete with similar ones in their subpopulation, such that only the fittest of each subpopulation survives. Figure 3 : 3Accuracy improvement over the generations. Figure 4 : 4The overall distribution of the architectures found in each generation. The connections from the input layer and the softmax layer are always present and, therefore, they are omitted in the calculation. Figure 5 : 5Two fragments of the evolved architecture which has peculiar traits. Table 1 : 1The number of samples used from each type of black-box attack to compose the 2812 adversarial samples. Based on the principle of the transferability of adversarial samples, these adversarial samples are used as a fast attack for the robustness evaluation of architectures. Details of the attacks as well as the motivation for using a model-agnostic (black-box) dual quality (L0 and L∞) assessment are explained in detail at[Vargas and Kotyan, 2019]. Code is available at http://bit.ly/RobustArchitectureSearch AcknowledgmentsThis work was supported by JST, ACT-I Grant Number JP-50243 and JSPS KAKENHI Grant Number JP20241216. 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[ "Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance", "Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance" ]	[ "Marc Hallin [email protected] \nLIDAM/ISBA\nECARES\nUniversité libre de Bruxelles Avenue F.D\nRoosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium\n", "Gilles Mordant [email protected] \nLIDAM/ISBA\nECARES\nUniversité libre de Bruxelles Avenue F.D\nRoosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium\n", "Johan Segers [email protected] \nLIDAM/ISBA\nECARES\nUniversité libre de Bruxelles Avenue F.D\nRoosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium\n" ]	[ "LIDAM/ISBA\nECARES\nUniversité libre de Bruxelles Avenue F.D\nRoosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium", "LIDAM/ISBA\nECARES\nUniversité libre de Bruxelles Avenue F.D\nRoosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium", "LIDAM/ISBA\nECARES\nUniversité libre de Bruxelles Avenue F.D\nRoosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium" ]	[]	Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. This includes the important problem of testing for multivariate normality with unspecified mean vector and covariance matrix and, more generally, testing for elliptical symmetry with given standard radial density and unspecified location and scatter parameters. The calculation of test statistics boils down to solving the well-studied semidiscrete optimal transport problem. Exact critical values can be computed for some important particular cases, such as null hypotheses of ellipticity with given standard radial density and unspecified location and scatter; else, approximate critical values are obtained via parametric bootstrap. Consistency is established, based on a result on the convergence to zero, uniformly over certain families of distributions, of the empirical Wasserstein distance-a novel result of independent interest. A simulation study establishes the practical feasibility and excellent performance of the proposed tests.	10.1214/21-ejs1816	[ "https://arxiv.org/pdf/2003.06684v1.pdf" ]	212,725,185	2003.06684	2cdaec78444ed2efa36d8345df5ff3c4ccb4c6d2	Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance Marc Hallin [email protected] LIDAM/ISBA ECARES Université libre de Bruxelles Avenue F.D Roosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium Gilles Mordant [email protected] LIDAM/ISBA ECARES Université libre de Bruxelles Avenue F.D Roosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium Johan Segers [email protected] LIDAM/ISBA ECARES Université libre de Bruxelles Avenue F.D Roosevelt 42, UCLouvain Voie du Roman Pays 20/L1.04.011050, B-1348Brussels, Louvain-la-NeuveBelgium, Belgium Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance and phrases: CopulaElliptical distributionGoodness-of-fitMultivariate normalityOptimal transportSemi-discrete problemSkew-t distributionWasserstein distance Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. This includes the important problem of testing for multivariate normality with unspecified mean vector and covariance matrix and, more generally, testing for elliptical symmetry with given standard radial density and unspecified location and scatter parameters. The calculation of test statistics boils down to solving the well-studied semidiscrete optimal transport problem. Exact critical values can be computed for some important particular cases, such as null hypotheses of ellipticity with given standard radial density and unspecified location and scatter; else, approximate critical values are obtained via parametric bootstrap. Consistency is established, based on a result on the convergence to zero, uniformly over certain families of distributions, of the empirical Wasserstein distance-a novel result of independent interest. A simulation study establishes the practical feasibility and excellent performance of the proposed tests. Introduction Wasserstein distances are metrics on spaces of probability measures with certain finite moments. They measure the distance between two such distributions by the minimal cost needed to move probability mass in order to transform one distribution into the other one. Wasserstein distances have a long history and continue to attract interest from diverse fields in statistics, machine learning and computer science, in particular image analysis; see for instance the monographs and reviews by Santambrogio (2015), Peyré and Cuturi (2019), and Panaretos and Zemel (2019). A natural application of any meaningful distance between distributions is to the goodness-of-fit (GoF) problem-namely, the problem of testing the null hypothesis that a sample comes from a population with fully specified distribution P 0 or with unspecified distribution within some postulated parametric model M. GoF problems certainly are among the most fundamental and classical ones in statistical inference. Typically, GoF tests are based on some distance between the empirical distribution P n and the null distribution P 0 or an estimated distribution in the model M. The most popular ones are the Cramér-von Mises (Cramér, 1928;von Mises, 1928) and Kolmogorov-Smirnov (Kolmogorov, 1933;Smirnov, 1939) tests, involving distances between the cumulative distribution function of P 0 and the empirical one. Originally defined for univariate distributions only, they have been extended to the multivariate case, for instance in Khmaladze (2016), who proposes a test that has nearly all properties one could wish for, including asymptotic distribution-freeness, but whose implementation is computationally quite heavy and quickly gets intractable. Many other distances have been considered in this context, though. Among them, distances between densities (after kernel smoothing) have attracted much interest, starting with Bickel and Rosenblatt (1973) in the univariate case. Bakshaev and Rudzkis (2015) recently proposed a multivariate extension; the choice of a bandwidth matrix, however, dramatically affects the outcome of the resulting testing procedure. Fan (1997) considers a distance between characteristic functions, which accommodates arbitrary dimensions; the idea is appealing but the estimation of the integrals involved in the distance seems tricky. McAssey (2013) proposes a heuristic test that relies on a comparison of the empirical Mahalanobis distance with a simulated one under the null. Still in a multivariate setting, Ebner, Henze and Yukich (2018) define a distance based on sums of powers of weighted volumes of kth nearest neighbour spheres. The use of the Wasserstein distance for GoF testing has been considered mostly for univariate distributions (Munk and Czado, 1998;del Barrio et al., 1999;del Barrio et al., 2000;del Barrio, Giné and Utzet, 2005). For the multivariate case, available methods are restricted to discrete distributions (Sommerfeld and Munk, 2018) and Gaussian ones (Rippl, Munk and Sturm, 2016). Indeed, serious difficulties, both computational and theoretical, hinder the development of Wasserstein GoF tests for general multivariate continuous distributions, particularly in the case of composite null hypotheses. Composite null hypotheses are generally more realistic than simple ones. Of particular practical importance is the case of location-scale families: tests of multivariate Gaussianity, tests of elliptical symmetry with given standard radial density, etc., belong to that type. Although the asymptotic null distribution of empirical processes with estimated parameters is well known (van der Vaart, 1998, Theorem 19.23), the actual exploitation of that theory in GoF testing remains problematic because of the difficulty of computing critical values. The aim of this paper is to explore the potential of the Wasserstein distance for GoF tests of simple (consisting of one fully specified distribution) and composite (consisting of a parametric family of distributions) null hypotheses involving continuous multivariate distributions. The tests we are proposing are based on the Wasserstein distance between P n and the distribution P 0 in the case of a simple null hypothesis and on the Wasserstein distance between P n and a model-based distribution estimate in the case of a composite null hypothesis. They are computationally feasible, have the correct size, and enjoy good power properties in comparison with other tests available in the literature. We concentrate on the continuous case, i.e., the distributions under the null hypothesis are absolutely continuous with respect to the Lebesgue measure on R d . The test statistic involves the Wasserstein distance between P n , which is discrete, and a distribution from the null hypothesis to be tested, which is continuous. Calculating such a distance requires solving the so-called semidiscrete transportation problem, an active area of research in computer science. In case of a simple null hypothesis, the null distribution of the test statistic does not depend on unknown parameters. Exact critical values can be calculated with arbitrary precision via a Monte Carlo procedure, by simulating from the null distribution and computing empirical quantiles. Exact critical values can also be computed for Wasserstein tests for the GoF of a location-scatter family of elliptical distributions with known radial distribution. We handle the presence of unknown nuisance parameters by using empirically standardized data. An important and well-studied special case is that of testing for multivariate normality. Out of the many available tests in the literature, we select the ones of Royston (1982), Henze and Zirkler (1990) and Rizzo and Székely (2016) as benchmark for our Wasserstein test. For general parametric models, we rely on the bootstrap to calculate critical values. The question whether the method has the correct size under the null hypothesis remains open. A proof of that property would require knowledge of non-degenerate limit distributions of the empirical Wasserstein distance-a hard and long-standing open problem, which we briefly review in Section 1.2. Monte Carlo experiments, however, suggest that our tests have the correct asymptotic size. In all cases, we show that our Wasserstein GoF tests are consistent against fixed alternatives, that is, the null hypothesis under such alternatives is rejected with probability tending to one. For the general parametric case, this property relies on the uniform consistency in probability of the empirical distribution with respect to the Wasserstein distance, uniformly over adequate classes of probability measures. To the best of our knowledge, this result, which is of independent interest, is new in the literature. Measure transportation has attracted much interest in the recent statistical literature. Carlier et al. (2016), Chernozhukov et al. (2017 and del Barrio et al. (2018) propose measure transportation-based concepts of multivariate ranks, signs, and quantiles. These notions have been successfully applied by Shi, Drton and Han (2019), Deb and Sen (2019), and Ghosal and Sen (2019) in the construction of distribution-free tests in a multivariate context, by Hallin, La Vecchia and Liu (2019) for R-estimation of VARMA models with unspecified innovation densities. The outline of the paper is as follows. In the remainder of this introduction, we introduce the Wasserstein distance (Section 1.1), review the asymptotic theory of empirical Wasserstein distance (Section 1.2), and provide some information on the computational methods for the semi-discrete problem underlying the implementation of the Wasserstein GoF tests (Section 1.3). In Section 2, we give a formal description of the GoF test procedure for simple null hypotheses. Section 3 addresses the composite null hypothesis that the unknown distribution belongs to an elliptical family with unknown location and scatter (covariance) parameters and known radial distribution; the multivariate normal family is an important special case. Composite null hypotheses covering general parametric models are treated in Section 4. In Section 5, we conduct a simulation study to assess the finite-sample performance of the Wasserstein tests in comparison to other GoF tests, both for simple and composite null hypotheses. In Appendix A, the convergence of the empirical Wasserstein distance uniformly over certain classes of underlying distributions is stated and proved. In Appendix B, the algorithms we are using in the computation of critical values are listed and explained. Wasserstein distance Let P(R d ) be the set of Borel probability measures on R d and let P p (R d ) be the subset of such measures with a finite moment of order p ∈ [1, ∞). For P, Q ∈ P(R d ), let Γ(P, Q) be the set of probability measures γ on R d ×R d with marginals P and Q, i.e., such that γ (B × R d ) = P(B) and γ(R d × B) = Q(B) for Borel sets B ⊆ R d . The p-Wasserstein distance W p (P, Q) between P, Q ∈ P p (R d ) is W p (P, Q) := inf γ∈Γ(P,Q) R d ×R d x − y p dγ(x, y) 1/p , with · the Euclidean norm. In terms of random variables X ∼ P and Y ∼ Q, the p-Wasserstein distance is the smallest value of {E( X − Y p )} 1/p over all possible couplings (X, Y ) ∼ γ. The p-Wasserstein distance W p defines a metric on P p (R d ) which, when endowed with the Wasserstein distance W p , is a complete separable metric space (Villani, 2009, Theorem 6.18 and the bibliographical notes). Convergence in the W p metric is equivalent to weak convergence plus convergence of moments of order p; see for instance Bickel and Freedman (1981, Lemmas 8.1 and 8.3) and Villani (2009, Theorem 6.9). It is thus quite natural to consider W p in the construction of GoF tests for multivariate distributions. For univariate distributions P and Q with distribution functions F and G, the p-Wasserstein distance boils down to the L p -distance W p (P, Q) = 1 0 F −1 (u) − G −1 (u) p du 1/p between the respective quantile functions F −1 and G −1 . This representation considerably facilitates both the computation of the distance and the asymptotic theory of its empirical versions. Also, the optimal transport plan mapping X ∼ P to Y ∼ Q is immediate: if F has no atoms, then Y := G −1 • F (X) ∼ Q, while monotonicity of G −1 • F implies the optimality of the coupling (X, Y ). Asymptotic theory Let X n = (X 1 , . . . , X n ) be an independent random sample from P ∈ P(R d ). Its distribution as a random vector in (R d ) n is the n-fold product P n of P with itself. Let L n : (R d ) n → P(R d ) map x n = (x 1 , . . . , x n ) ∈ (R d ) n to the discrete probability measure L n (x n ) := n −1 n i=1 δ xi , with δ x the Dirac measure at x. The empirical distribution of the sample is P n := L n (X n ) = n −1 n i=1 δ Xi . We study its distribution as a random element in P(R d ). The Wasserstein distance between the empirical distribution L n (x n ) and a probability measure P ∈ P p (R d ) is the value at x n ∈ (R d ) n of the map W p (L n , P) : x n ∈ (R d ) n → W p (L n (x n ), P) ∈ [0, ∞). Consider the distribution of this map under P n , i.e., for an independent random sample of size n from P. In perhaps more familiar notation, the random variable of interest is the empirical Wasserstein distance W p ( P n , P). According to Bickel and Freedman (1981, Lemma 8.4), if P ∈ P p (R d ), the empirical distribution is strongly consistent in the Wasserstein distance: for an i.i.d. sequence X 1 , X 2 , . . . with common distribution P, we have W p ( P n , P) → 0 almost surely as n → ∞. The corresponding consistency rates have been studied intensively; see Panaretos and Zemel (2019, Section 3.3) for a review. If P is nondegenerate, then E[W p ( P n , P)] is at least of the order n −1/2 , and if P is absolutely continuous, which is the case of interest here, the convergence rate cannot be faster than n −1/d . Actually, the rate can be arbitrarily slow (Bobkov and Ledoux, 2019, Theorem 3.3). Precise rates under additional moment assumptions are given for instance in Fournier and Guillin (2015). Asymptotic distribution results for the empirical Wasserstein distance in dimension d ≥ 2 are, however, surprisingly scarce. The one-dimensional case is well-studied thanks to the link to empirical quantile processes, see for instance del Barrio, Giné and Utzet (2005). Also for discrete distributions, non-degenerate limit distributions are known (Sommerfeld and Munk, 2018;Tameling, Sommerfeld and Munk, 2019). For multivariate Gaussian distributions, a central limit theorem for the empirical Wasserstein between the true normal distribution and the one with estimated parameters is given in Rippl, Munk and Sturm (2016). Although interesting and useful for GoF testing (see Section 5.1 below), this result does not cover the case of the empirical distribution P n . Important steps have been taken recently by del Barrio and Loubes (2019) who, quite remarkably, manage to obtain some asymptotic results under alternatives. For general P, Q ∈ P 4+δ (R d ) for some δ > 0, they establish a central limit theorem for n 1/2 W 2 ( P n , Q) − E{W 2 ( P n , Q)} . Unfortunately, if Q = P, which is the case of interest here, the asymptotic variance is zero, meaning that the random fluctuations of W 2 ( P n , P) around its mean are of order less than n −1/2 . Moreover, as mentioned above, E{W p ( P n , P)} may converge to zero at a slower rate than n −1/2 . The crucial problem of the limiting distribution of the empirical Wasserstein distance under the null so far remains an important and difficult open problem, which apparently precludes the implementation of multivariate analogues of the existing one-dimensional procedures. The most recent progress perhaps has been booked in Goldfeld and Kato (2020), who obtain a central limit theorem for the empirical W 1 -distance after smoothing the empirical and true distributions with a Gaussian kernel. To construct critical values of Wasserstein GoF tests of general parametric models, we will propose in Section 4 the use of the parametric bootstrap. In general, proving consistency of the parametric bootstrap typically requires having non-degenerate limit distributions under contiguous alternatives of the statistic of interest (Beran, 1997;Capanu, 2019). As the above review shows, such results are still beyond the horizon. Computational issues Important numerical developments have taken place recently in the area of measure transportation and, more particularly, in the computation of the 2-Wasserstein distance between a discrete and a continuous distribution, the so-called semi-discrete optimal transportation problem; see for instance Leclaire and Rabin (2019) and the references therein. The efficiency and high accuracy of the algorithms developed by Mérigot (2011), Lévy (2015, or Kitagawa, Mérigot and Thibert (2017) make it possible to simulate from the exact null distribution of empirical Wasserstein distances. Moreover, Kitagawa, Mérigot and Thibert (2017) establish, under certain assumptions, the convergence of their algorithm. This opens the door for the implementation, based on simulated critical values, of the Wasserstein distance-based GoF tests in dimension d ≥ 1 for which asymptotic critical values remain unavailable. Most algorithms to date rely on the dual formulation of the problem, assuming that the source continuous probability measure P admits a density f w.r.t. the Lebesgue measure on R d . We follow Santambrogio (2015, Section 6.4.2) for a brief exposition. In line with the set-up relying on the empirical measure, we work with the quadratic cost function (p = 2) and a discrete measure P n = n −1 n i=1 δ xi over n distinct atoms x 1 , . . . , x n ∈ R d , each of mass 1/n. The semi-discrete problem requires constructing a power diagram or Laguerre-Voronoi diagram, partitioning R d into power cells V ψ (i) := x ∈ R d : 1 2 x − x i 2 − ψ i ≤ 1 2 x − x j 2 − ψ j , ∀j = 1, . . . , n for i = 1, . . . , n, defined in terms of a vector ψ = (ψ 1 , . . . , ψ n ) ∈ R n . Each power cell V ψ (i) corresponds to a set of linear constraints and, therefore, is a convex polyhedron. The dual to the problem of minimizing the expected transportation cost R d ×R d 1 2 x − y 2 dγ(x, y) over the couplings γ ∈ Γ(P n , P) is then the maximisation, with respect to the vector ψ, of the objective function F (ψ) := 1 n n i=1 ψ i + n i=1 V ψ (i) 1 2 x − x i 2 − ψ i f (x) dx. The function ψ → F (ψ) is differentiable. Setting its partial derivatives to zero yields the equations V ψ (i) f (x) dx = 1 n , i = 1, . . . , n, specifying that each power cell V ψ (i) should receive mass P n ({x i }) = n −1 under P. The optimal transport plan from P to P n then consists in mapping all points in the interior of V ψ (i) to x i . The dual formulation above is the basis for the multi-scale algorithm developed in Mérigot (2011) based on the method for solving constrained least-squares assignment problems in Aurenhammer, Hoffmann and Aronov (1998). For the Monte Carlo simulation experiments, we use the implementation of that algorithm in the function semidiscrete provided by the R package transport (Schuhmacher et al., 2019). Further improvements of the multi-scale algorithm are introduced in Lévy (2015) and Kitagawa, Mérigot and Thibert (2017). Recently, stochastic algorithms in Genevay et al. (2016) and Leclaire and Rabin (2019) are claimed to perform even better. To the best of our knowledge, implementations of these algorithms are not yet available in R (R Core Team, 2018), the language in which we programmed the simulation experiments. Our aim in this paper is to demonstrate the feasability of goodness-of-fit tests for multivariate distributions based on the Wasserstein distance. Advances in computational methods and software implementations can only strengthen that case. Wasserstein GoF tests for simple null hypotheses Let X n = (X 1 , . . . , X n ) be an independent random sample from some unknown distribution P ∈ P(R d ). For some given fixed P 0 ∈ P p (R d ), consider testing the simple null hypothesis H n 0 : P = P 0 against H n 1 : P = P 0 based on the observations X n . Note that P, under the alternative, is not required to have finite moments of order p. Consider the test statistic T n := W p p ( P n , P 0 ), the pth power of the p-Wasserstein distance between the empirical distribution P n = n −1 n i=1 δ Xi and the distribution P 0 specified by the null hypothesis. Having bounded support, P n trivially belongs to P p (R d ), so that T n is well-defined. Actual computation of T n amounts to solving the semi-discrete optimal transport problem, as reviewed in Section 1.3 for p = 2. In the simulations of Section 5, we therefore limit ourselves to p = 2; the theory developed in this section, however, is developed for general p ≥ 1. For 0 < α < 1, the test φ n P0 we are proposing has the form φ n P0 = 1 if T n > c(α, n, P 0 ), 0 otherwise,(1) with critical value c(α, n, P 0 ) := inf c > 0 : P n 0 [T n > c] ≤ α(2) where P n 0 stands for n-fold product measure of P 0 on (R d ) n , that is, the distribution under H n 0 of the observation X n . By construction, the exact size of the GoF test in (1) is P n 0 T n > c(α, n, P 0 ) ≤ α, with equality if the law of T n under P n 0 is continuous. The risk of a false rejection is thus bounded by the nominal size α, and often equal to it. Although the critical level c(α, n, P 0 ) cannot be calculated analytically, its value can be approximated to any desired degree of precision via Monte Carlo simulation. To this end, draw a large number N , say, of independent random samples of size n from P 0 and compute the test statistic for each such sample. The empirical (1 − α) quantile of the N simulated test statistics thus obtained is then a consistent and asymptotically normal estimator of c(α, n, P 0 ) as N → ∞ provided that the distribution of T n has a continuous and positive density at c(α, n, P 0 ). The approximation error is of the order N −1/2 and can be made arbitrarily small by choosing N sufficiently large. The null distribution of T n depends on P 0 , so that c(α, n, P 0 ) needs to be calculated for each P 0 separately. Under the alternative hypothesis, the following proposition establishes that the test is rejecting the null with probability tending to one, i.e., is consistent against any fixed alternative P = P 0 . Proposition 1 (Consistency). For every P 0 ∈ P p (R d ), the test φ n P0 is consistent against any P ∈ P(R d ) with P = P 0 : lim n→∞ P n [φ n P0 = 1] = 1 for any α > 0. Proof. Fix P 0 ∈ P p (R d ). For any α > 0, the critical value c(α, n, P 0 ) tends to zero as n → ∞. Indeed, by Bickel and Freedman (1981, Lemma 8.4), we have T n → 0 in P n 0 -probability and thus lim n→∞ P n 0 [T n > ε] = 0 for any ε > 0. It follows that, for every α > 0 and every ε > 0, we have c(α, n, P 0 ) ≤ ε for all sufficiently large n. Let P ∈ P(R d ) with P = P 0 . We consider two cases according as P has finite moments of order p or not. First, suppose that P ∈ P p (R d ). Still by Bickel and Freedman (1981, Lemma 8.4), we have W p ( P n , P) → 0 in P n -probability as n → ∞. The triangle inequality for the metric W p yields W p ( P n , P 0 ) − W p (P, P 0 ) ≤ W p ( P n , P) → 0, n → ∞ in P n -probability. Hence T n = W p p ( P n , P 0 ) → W p p (P, P 0 ) in P n -probability as n → ∞. But W p p (P, P 0 ) > 0 since P, P 0 ∈ P p (R d ) and P = P 0 by assumption. It follows that lim n→∞ P n [T n > c(α, n, P 0 )] = 1, as required. Second, suppose that P ∈ P(R d ) \ P p (R d ). Let δ 0 denote the Dirac measure at 0 ∈ R d . Since W p is a metric, the triangle inequality implies W p ( P n , P 0 ) ≥ W p ( P n , δ 0 ) − W p (P 0 , δ 0 ). Now, W p (P 0 , δ 0 ) is a constant and W p p ( P n , δ 0 ) = n −1 n i=1 X i p . As the expectation of X 1 2 under P is infinite, the law of large numbers implies that W p p ( P n , δ 0 ) → ∞ in P n -probability as n → ∞. But the same then is true for T n and thus lim n→∞ P n [T n > c(α, n, P 0 )] = 1, as required. Wasserstein GoF tests for elliptical families The distribution P ∈ P(R d ) of a d-dimensional random vector Z with den- sity f is called spherical with radial density f rad if f (z) is of the form f rad ( z ) for z ∈ R d where ∞ 0 f rad (r) dr = 1. The radial density f rad is called stan- dard if ∞ 0 r 2 f rad (r) dr = d. The distribution P is then in P 2 (R d )-denote it by P f rad -and Z has mean zero and covariance matrix I d . The distribution P ∈ P(R d ) of a d-dimensional random vector X is called elliptical with standard radial density f rad if there exist µ ∈ R d and a full- rank d × d matrix A such that the distribution of A −1 (X − µ) is spherical with radial density f rad satisfying r 2 f rad (r)dr = d; the distribution P then is in P 2 (R d ) and X has mean µ and covariance matrix Σ = AA . We refer to Cambanis, Huang and Simons (1981) or Fang, Kotz and Ng (1990) for details. Let E(f rad ) denote the family of d-variate elliptical distributions with standard radial density f rad . Such families are indexed by a location vector µ ∈ R d and a positive definite d × d covariance matrix Σ; the choices µ = 0 and Σ = I d yield the spherical P f rad . Common examples of elliptical families are the multivariate normal family, with f rad the density of the root of a χ 2 d variable, and the multivariate Student t distribution with ν > 2 degrees of freedom, where f rad is the density of the root of a rescaled Fisher F (d, ν) variable. In general, elliptical distributions are not subject to moment constraints (Σ := AA is then a scatter rather than a covariance matrix), but here we intend to use the Wasserstein distance of order p = 2 and therefore restrict to elliptical families with finite second-order moments. Given an i.i.d. sample X 1 , . . . , X n from some unspecified P ∈ P 2 (R d ), we wish to test the null hypothesis that P is elliptical with specified standard radial density f rad , namely, H n 0 : P ∈ E(f rad ) against H n 1 : P ∈ E(f rad ). (3) The location vector µ and the covariance matrix Σ of P are unknown nuisance parameters. In contrast to Section 2, the null hypothesis is thus a composite one. Our testing strategy is to compute residuals of the form Z n,i :=Â −1 n (X i −μ n ), i = 1, . . . , n,(4) yielding an empirical distribution PẐ n := n −1 n i=1 δẐ n,i . The test statistic we propose is T E(f rad ),n := W 2 2 ( PẐ n , P f rad ). If the null distribution of (Ẑ n,1 , . . . ,Ẑ n,n ) does not depend on the unkown µ and Σ, then we can define critical values for T E(f rad ),n as if µ = 0 and Σ = I d . As in Section 2, such critical values can then be approximated with any desired accuracy via Monte Carlo random sampling from P f rad . In the sequel, we letμ n = n −1 n i=1 X i = X n and choose forÂ n the Cholesky triangle L n,X ∈ R d×d of the empirical covariance matrix S n,X := 1 n − 1 n i=1 (X i −μ n )(X i −μ n ) . Recall that for every symmetric positive definite matrix S ∈ R d×d , there exists a unique lower triangular matrix L ∈ R d×d with positive diagonal elements, called Cholesky triangle, producing the Cholesky decomposition S = LL (Golub and Van Loan, 1996, Theorem 4.2.5). If Σ is invertible, then S n,X is invertible with probability tending to one; even more, for an i.i.d. sequence X 1 , X 2 , . . . from P ∈ E(f rad ), with probability one, the matrix S n,X is invertible for all n large enough depending on the sample. On the event that S n,X is invertible, the residuals (4) are thuŝ Z n,i = L −1 n,X (X i − X n ), i = 1, . . . , n.(5) For completeness, on the event that S n,X is not invertible, we setẐ n,i = 0 for i = 1, . . . , n, although a precise definition is immaterial for the results to follow. Let us show that the joint distribution of the vector of residuals computed in this way does not depend on the unknown µ or Σ. The key is the following elementary property. Lemma 1. Let x 1 , . . . , x n ∈ (R d ) n have mean vector x n = n −1 n i=1 x i and covariance matrix S n,x = (n − 1) −1 n i=1 (x i − x n )(x i − x n ) ∈ R d×d . Let µ ∈ R d and let L ∈ R d×d be lower triangular with positive diagonal elements. Put z i := L −1 (x i − µ), i = 1, . . . , n; with obvious notation, similarly define z n and S n,z . Then, S n,x is invertible if and only S n,z is, in which case their Cholesky factors L n,x and L n,z are related by L n,x = LL n,z , which implies L −1 n,x (x i − x n ) = L −1 n,z (z i − z n ), i = 1, . . . , n. Proof. We have x i = µ + Lz i for all i = 1, . . . , n, whence x n = µ + Lz n and S n,x = LS n,z L . Since L is invertible, S n,x is invertible if and only if S n,z is. Suppose they both are, and let L n,x and L n,z denote their Cholesky factors. The matrix LL n,z is lower triangular, has positive diagonal elements, and satisfies LL n,z (LL n,z ) = LS n,z L = S n,x . By the uniqueness of the Cholesky decomposition, L n,x = LL n,z . Finally, L −1 n,x (x i −x n ) = (LL n,z ) −1 {(µ+Lz i )−(µ+Lz n )} = L −1 n,z (z i −z n ), i = 1, . . . , n. Proposition 2. Let X 1 , . . . , X n be an i.i.d. sample from P ∈ E(f rad ) with mean µ and full-rank covariance Σ. The joint distribution ofẐ n,i in (5) for i = 1, . . . , n does not depend on µ nor Σ. Proof. Let L be the Cholesky factor of Σ. In view of Lemma 1, it is sufficient to show that Z i = L −1 (X i −µ), for i = 1, . . . , n, is an independent random sample of the spherical distribution P f rad with mean zero, covariance identity, and standard radial density f rad . By the assumption on P, there exists an invertible A ∈ R d×d with AA = Σ such that X i = µ + Aζ i for i = 1, . . . , n, where ζ 1 , . . . , ζ n is an independent random sample from P f rad . Then, Z i = L −1 Aζ i for all i = 1, . . . , n, where the matrix L −1 A is orthogonal: indeed, (L −1 A)(L −1 A) = L −1 AA (L ) −1 = L −1 Σ(L ) −1 = L −1 LL (L ) −1 = I d . It thus follows from sphericity that the common distribution of Z 1 , . . . , Z n is the same as that of ζ 1 , . . . , ζ n , that is, P f rad . For the hypothesis testing problem (3), we propose the test φ n E(f rad ) := 1 if T E(f rad ),n > c E (α, n, f rad ), 0 otherwise, at level α ∈ (0, 1) and with critical value c E (α, n, f rad ) = inf c > 0 : P n f rad [T E(f rad ),n > c] ≤ α .(6) The probability in (6) is calculated under the spherical distribution with radial density f rad , which is free of nuisances. By Proposition 2, the size of the test is at most α: for all P ∈ E(f rad ), P n [T E(f rad ),n > c E (α, n, f rad )] = P n f rad [T E(f rad ),n > c E (α, n, f rad )] ≤ α. The size of the test is equal to α if the null distribution of T E(f rad ),n is continuous at the critical value. In practice, calculation of the critical value is implemented by Monte Carlo simulation, see Algorithm 2 in Appendix B. The consistency of the test follows from a large of law numbers in 2-Wasserstein distance for the empirical distribution of the residuals defined in (5). Proposition 3. Let P ∈ P 2 (R d ) have mean vector µ ∈ R d and invertible covariance matrix Σ ∈ R d×d with Cholesky triangle L ∈ R d×d . Let X 1 , X 2 , . . . be a sequence of i.i.d. random vectors with common distribution P. ForẐ n,i as in (5), we have W 2 2 ( PẐ n , Q 0 ) → 0 almost surely as n → ∞, where PẐ n := n −1 n i=1 δẐ n,i and Q 0 ∈ P 2 (R d ) is the distribution of L −1 (X 1 − µ). Proof. The random vectors Z i = L −1 (X i − µ) for i = 1, 2, . . . form an i.i.d. sequence with common distribution Q 0 . By the strong law of large numbers, X n → µ and S n,X → Σ a.s. as n → ∞. With probability one, S n,X is invertible for n large enough (depending on the sample) and admits a unique Cholesky factor L n,X . The map that sends a positive definite symmetric matrix to its Cholesky triangle is differentiable (Smith, 1995) and thus continuous. It follows that L n,X → L and L −1 n,X → L −1 a.s. as n → ∞. Let P Z n := n −1 n i=1 δ Zi . By the triangle inequality for the Wasserstein distance, W 2 ( PẐ n , Q 0 ) ≤ W 2 ( PẐ n , P Z n ) + W 2 ( P Z n , Q 0 ). We already know that W 2 ( P Z n , Q 0 ) → 0 almost surely as n → ∞ (Bickel and Freedman, 1981, Lemma 8.4). It remains to show that W 2 ( PẐ n , P Z n ) → 0 in probability as n → ∞. Consider the coupling of PẐ n and P Z n via the discrete uniform distribution on the pairs (Ẑ n,i , Z i ) for i = 1, . . . , n. From the definition of the Wasserstein distance, we have W 2 2 ( PẐ n , P Z n ) ≤ 1 n n i=1 Ẑ n,i − Z i 2 . For each i = 1, . . . , n, the identity X i = µ + LZ i yields Ẑ n,i − Z i = L −1 n,X (µ + LZ i − X n ) − Z i ≤ L −1 n,X µ − X n + L −1 n,X L − I d Z i featuring the matrix norm on R d×d associated with the Euclidean norm on R d . It follows that W 2 2 ( PẐ n , P Z n ) ≤ 2 L −1 n,X 2 µ − X n 2 + 2 L −1 n,X L − I d 2 1 n n i=1 Z i 2 . The right-hand side converges to zero almost surely as n → ∞ in view of the law of large numbers for n −1 n i=1 Z i 2 , (7), and (8). The result follows. Proposition 4 (Consistency). The sequence of tests φ n E(f rad ) is consistent against any P ∈ P 2 (R d ) \ E(f rad ) with positive definite covariance matrix: lim n→∞ P n φ n E(f rad ) = 1 = 1 for every α > 0. Proof. Let α > 0 and ε > 0. By Proposition 3, lim n→∞ P n f rad [T E(f rad ),n > ε] = 0 and thus c E (α, n, f rad ) ≤ ε for all sufficiently large n (depending on α and ε). It follows that lim n→∞ c E (α, n, f rad ) = 0. Let P be as in the statement. It is sufficient to show that there exists ε > 0 such that lim n→∞ P n [T E(f rad ),n > ε] = 1. By Proposition 3, we have W 2 ( PẐ n , Q 0 ) → 0 in P n -probability, n → ∞, with Q 0 as in the statement of that proposition. By the triangle inequality, W 2 ( PẐ n , P f rad ) ≥ W 2 (P f rad , Q 0 ) − W 2 ( PẐ n , Q 0 ). By assumption, Q 0 = P f rad and thus W 2 (P f rad , Q 0 ) > 0 since otherwise P ∈ E(P 0 ). For ε > 0 less than W 2 2 (P f rad , Q 0 ), we obtain lim n→∞ P n [T E(f rad ),n > ε] = 1, as required. Wasserstein GoF tests for general parametric families Extending the scope of Section 3, consider the problem of testing whether the unknown common distribution P of a sample of observations belongs to some parametric family M := P θ : θ ∈ Θ of distributions on R d where the parameter space Θ is some metric space and the map θ → P θ is assumed to be one-toone and continuous in a sense to be specified. Given an independent random sample X n = (X 1 , . . . , X n ) from some unknown P ∈ P(R d ), the goodness-of-fit problem is about testing H n 0 : P ∈ M against H n 1 : P / ∈ M. Assume that every P θ ∈ M has a finite moment of order p ∈ [1, ∞), that is, M ⊆ P p (R d ). The test statistic we propose is T M,n := W p p ( P n , Pθ n ) whereθ n = θ n (X n ) is some consistent (under H n 0 ) estimator sequence of the true parameter θ. The distribution of X n under H n 0 in (9) being P n θ for some θ ∈ Θ, we would like to take c M (α, n, θ) = inf{c > 0 : P n θ [T M,n > c] ≤ α}(10) as the critical value of a test with nominal size α ∈ (0, 1). This choice is infeasible, however, since the true parameter θ is unknown. Therefore, we propose to replace it by the bootstrapped quantity c M (α, n,θ n ), yielding the test φ n M := 1 if T M,n > c M (α, n,θ n ), 0 otherwise,(11) rejecting H n 0 whenever T M,n exceeds c M (α, n,θ n ). Given the parameter estimateθ n , the proposed critical value can be approximated by resampling from the estimated distribution Pθ n . The idea is as follows and is given in more detail in Appendix B, in particular Algorithm 3: 1. generate a large number B of samples X * n,b = (X * 1,b , . . . , X * n,b ) ∈ (R d ) n , say, for b = 1, . . . , B, of size n from Pθ n ; 2. letting P * n,b = n −1 n i=1 δ X * i,b denote the empirical distribution of the bootstrap sample number b, compute (a) the parameter estimateθ * n,b = θ n (X * n,b ), and (b) the test statistic T * M,n,b = W 2 2 ( P * n,b , Pθ * n,b ); 3. compute the empirical quantile c M,B (α, n,θ n ) = inf c > 0 : B −1 B b=1 I(T * M,n,b > c) ≤ α . As B → ∞ and since, conditionally on the data, the Monte Carlo approximation c M,B (α, n,θ n ) converges to the true quantile c M (α, n,θ n ) of the distribution of T M,n under P n θn , provided the latter distribution has a positive and continuous density at the stated limit point. The rate of convergence in probability is O(1/ √ B). In what follows, we assume we can compute c M (α, n,θ n ) to any desired degree of accuracy. By construction, we have ∀θ ∈ Θ, P n θ T M,n > c M (α, n, θ) ≤ α, that is, if we could use the critical value at the true parameter θ, the risk of a false rejection of the null hypothesis would be bounded by α; it would be even equal to α if the distribution of T M,n is continuous at c M (α, n, θ). But as the true parameter θ is unknown, we use the estimated oneθ n instead, so that the risk of a false rejection is P n θ T M,n > c M (α, n,θ n ) . The question remains open whether under the null hypothesis the actual size of the test indeed converges to α. To prove this would require non-degenerate limit distribution theory for W p p ( P n , P θ ), not only for fixed θ ∈ Θ, but even for sequences θ n converging to θ at certain rates. As discussed in Section 1.2, such limit results are still beyond the horizon. In the simulation study, however, we check that the proposed bootstrap method indeed produces a test with approximately the right size. Nevertheless, against a fixed alternative, the consistency of the test (11) based on the parametric bootstrap can be established theoretically. The key is the uniform convergence in probability of the empirical Wasserstein distance treated in Appendix A. For the parameter estimatorθ n , we need to assume weak consistency locally uniformly in θ: if ρ denotes the metric on Θ and if K(Θ) denotes the collection of compact subsets of Θ, we will require that ∀ε > 0, ∀K ∈ K(Θ), lim n→∞ sup θ∈K P n θ ρ(θ n , θ) > ε = 0. As illustrated in Remark 1 below, this condition is satisfied, for instance, for moment estimators of a Euclidean parameter under a uniform integrability condition. Proposition 5 (Consistency). Let M = {P θ : θ ∈ Θ} ⊆ P p (R d ), p ∈ [1, ∞) , be a model indexed by a metric space (Θ, ρ). Assume that the following conditions are satisfied: (a) the map Θ → P p (R d ) : θ → P θ is one-to-one and W p -continuous; (b)θ n is weakly consistent locally uniformly in θ ∈ Θ, i.e., (12) holds. Then, the following properties hold: (i) T M,n → 0 in P n θ -probability locally uniformly in θ ∈ Θ, i.e., (iii) for every P ∈ P(R d ) \ M such that there exists K ∈ K(Θ) with P n θ n ∈ K → 1 as n → ∞, we have lim n→∞ P n φ n M = 1 = 1. Proof. (i) By the triangle inequality, it follows that T 1/p M,n = W p ( P n , Pθ n ) ≤ W p ( P n , P θ ) + W p (P θ , Pθ n )(13) for all θ ∈ Θ. For compact K ⊆ Θ, it is then sufficient to show that each of the W p -distances on the right-hand side of (13) converges to 0 in P n θ -probability uniformly in θ ∈ K. First, since K is compact and θ → P θ is W p -continuous, the set M K := {P θ : θ ∈ K} is compact in P p (R d ) equipped with the W p -distance. By Bickel and Freedman (1981, Lemma 8.3(b)) or Villani (2009, Definition 6.8(b) and Theorem 6.9) and a subsequence argument, it follows that x → x p is uniformly integrable with respect to M K , i.e., lim r→∞ sup θ∈K x >r x p dP θ (x) = 0. Corollary 1 then implies that W p ( P n , P θ ) → 0 in P n θ -probability as n → ∞, uniformly in θ ∈ K. Second, as K is compact and θ → P θ is W p -continuous, there exists, for every scalar ε > 0, a scalar δ = δ(ε) > 0 such that 1 ∀θ ∈ K, ∀θ ∈ Θ, ρ(θ, θ ) ≤ δ =⇒ W p (P θ , P θ ) ≤ ε. It follows that ∀θ ∈ K, P n θ W p (P θ , Pθ n ) > ε ≤ P n θ ρ(θ,θ n ) > δ . By condition (b), the latter probability converges to 0 as n → ∞ uniformly in θ ∈ K. (ii) Fix α > 0, ε > 0, and K ∈ K(Θ). By (i), there exists an integer n(ε) ≥ 1 such that ∀n ≥ n(ε), ∀θ ∈ K, P n θ T M,n > ε ≤ α. By definition of the critical values, also c M (α, n, θ) ≤ ε for all n ≥ n(ε) and θ ∈ K. (iii) Let P and K be as in the statement. Put c n = sup θ∈K c M,n (α, n, θ). We have P n φ n M = 1 ≥ P T M,n > c M (α, n,θ n ),θ n ∈ K ≥ P T M,n > c n ,θ n ∈ K . In view of (ii), we have c n → 0 as n → ∞, so that it is sufficient to show that there exists ε > 0, depending on P and M, such that lim n→∞ P n T M,n > ε = 1. Consider two cases, P ∈ P p (R d ) \ M and P ∈ P(R d ) \ P p (R d ), according as P has a finite moment of order p or not. First, suppose that P ∈ P p (R d ) \ M. We have W p (P, P θ ) > 0 for every θ ∈ Θ while the map θ → W p (P, P θ ) is continuous. As K is compact, η := 1 This is a slight generalization of the well-known property that a continuous function on a compact set is uniformly compact. As a proof, fix ε > 0 and consider for each θ ∈ K a scalar δ(θ) > 0 such that for all θ ∈ Θ with ρ(θ, θ ) ≤ δ(θ) we have Wp(P θ , P θ ) ≤ ε/2. Cover K by open balls with centers θ ∈ K and radii δ(θ)/2. By compactness, extract a finite cover with centers θ 1 , . . . , θm ∈ K. Put δ = min j δ(θ j )/2. For every θ ∈ K and θ ∈ Θ with ρ(θ, θ ) ≤ δ, there exists j = 1, . . . , m such that ρ(θ, θ j ) < δ(θ j )/2 and then also ρ(θ , θ j ) < δ(θ j ). By the triangle inequality, Wp(P θ , P θ ) ≤ Wp(P θ j , P θ ) + Wp(P θ j , P θ ) ≤ ε. inf W p (P, P θ ) : θ ∈ K > 0. On the event {θ n ∈ K}, the triangle inequality implies T 1/p M,n = W p ( P n , Pθ n ) ≥ W p (P, Pθ n ) − W p ( P n , P) ≥ η − W p ( P n , P). We obtain that P n φ n M = 1 ≥ P T 1/p M,n > c 1/p n ,θ n ∈ K ≥ P W p ( P n , P) < η − c 1/p n ,θ n ∈ K . As η > 0 and lim n→∞ c n = 0, the latter probability converges to one by the assumption made on K and the fact that W p ( P n , P) → 0 in P n -probability as n → ∞. Second, suppose that P ∈ P(R d ) \ P p (R d ). Let δ 0 be the Dirac measure at 0 ∈ R d . Since θ → W p (P θ , δ 0 ) is continuous, s = sup θ∈K W p (P θ , δ 0 ) is finite. By the triangle inequality, on the event {θ n ∈ K}, T 1/p M,n = W p ( P n , Pθ n ) ≥ W p ( P n , δ 0 ) − W p (Pθ n , δ 0 ) ≥ W p ( P n , δ 0 ) − s. Moreover, W p p ( P n , δ 0 ) = n −1 n i=1 X i p diverges to x p dP(x) = ∞ in P nprobability by the weak law of large numbers. It follows that lim n→∞ P n T M,n > c n ,θ n ∈ K = 1. Remark 1 (Uniform consistency). Under a mild moment condition, the uniform consistency condition (b) in Proposition 5 is satisfied for method of moment estimators-call them moment estimators-of a Euclidean parameter θ ∈ Θ ⊆ R k . In the method of moments, an estimatorθ n of θ is obtained by solving (with respect to θ) the equations 1 n n i=1 f j (X i ) = E θ [f j (X)], j = 1, . . . , k, for some given k-tuple f := (f 1 , . . . , f k ) of functions such that m : θ → E θ [f (X)] is a homeomorphism between Θ and m(Θ); see, for instance, van der Vaart (1998, Chapter 4). The consistency ofθ n = m −1 (n −1 n i=1 f (X i )) uniformly in θ ∈ K for any compact K ⊆ Θ then follows from the uniform consistency over K of n −1 n i=1 f (X i ) as an estimator of E θ [f (X)] for such θ. By van der Vaart and Wellner (1996, Proposition A.5.1), a sufficient condition for the latter is that the functions f j are P θ -uniformly integrable for θ ∈ K, i.e., lim M →∞ sup θ∈K E θ \|f j (X)\| I{\|f j (X)\| > M } = 0, j = 1, . . . , k. Since I{\|f j (X)\| > M } ≤ \|f j (X)\| η /M η for η > 0, a further sufficient condition is that there exists η > 0 such that sup θ∈K E θ [\|f j (X)\| 1+η ] < ∞ for j = 1, . . . , k. Remark 2 (Parameter estimate under the alternative). In Proposition 5(iii), the condition that there exists a compact K ⊆ Θ such that lim n→∞ P n [θ n ∈ K] = 1 holds, for instance, when Θ is locally compact andθ n is consistent for a pseudoparameter θ(P) ∈ Θ. This is the case for the moment estimators of Remark 1 when Θ ⊆ R k is open and f is P-integrable with f (x) dP(x) ∈ m(Θ). Remark 3 (Location-scale parameters). Let p = 2 and consider a parametric model M = {Q ψ : ψ ∈ Ψ} ⊆ P 2 (R d ) where ψ = (µ, σ, θ) ∈ R d × (0, ∞) d × Θ, such that, for X i = (X 11 , . . . , X id ) ∼ Q ψ ∈ M, we have µ j = E[X ij ] and σ j = var(X ij ) for all j = 1, . . . , d. The range Θ of θ is supposed not to depend on the location-scale parameter vectors µ and σ. For instance, θ could be a vector of shape parameters for the marginal distributions and/or determine the copula of Q ψ . Then, we can simplify the procedure by employing estimated residuals of the formẐ n,i = (Ẑ n,i1 , . . . ,Ẑ n,id ) witĥ Z n,ij = (X ij − X n,j )/s n,j,X , i = 1, . . . , n, j = 1, . . . , d, where X n,j and s n,j,X are the empirical means and standard deviations, respectively, of X 1j , . . . , X nj . The joint distribution of these estimated residuals depends only on θ but not on (µ, σ). Indeed, we have X ij = µ j + σ j Z ij where the distribution of Z i = (Z i1 , . . . , Z id ) is P θ , which is defined as Q ψ with ψ = ((0, . . . , 0), (1, . . . , 1), θ), that is, µ j = 0 and σ j = 1 for all j = 1, . . . , d. In obvious notation, we haveẐ n,ij = (Z ij − Z n,j )/s n,j,Z . Letθ n denote a strongly consistent estimator of θ that depends on the data only throughẐ n,1 , . . . ,Ẑ n,n . Consider the empirical distributions PẐ n := n −1 n i=1 δẐ n,i and P Z n := n −1 n i=1 δ Zi . To test the hypothesis H n 0 : P ∈ M, we propose the location-scale adjusted statistic T ls M,n := W 2 2 PẐ n , Pθ n . Its distribution still depends on θ but no longer on µ or σ. Critical values can thus be computed as if µ j = 0 and σ j = 1 for all j = 1, . . . , d. For a test of size α ∈ (0, 1), we reject the null hypothesis as soon as the test statistic exceeds the critical value c ls M (α, n,θ n ) where c ls M (α, n, θ) := inf c ≥ 0 : P n θ T ls M,n > c ≤ α , θ ∈ Θ.(16) In practice, critical values are calculated by a parametric bootstrap procedure as before. The advantage of the estimated residuals (14) is that the critical values are a function of θ only rather than a function of ψ = (µ, σ, θ), which greatly simplifies their computation. Under the null hypothesis, we have T ls M,n → 0 almost surely as n → ∞ since it is bounded by a multiple of 1 n n i=1 d j=1 (Ẑ n,ij − Z ij ) 2 + W 2 2 P Z n , P θ + W 2 2 P θ , Pθ n where each of the three terms converges to zero almost surely. Under an alternative P ∈ P 2 (R d ) \ M such thatθ n remains in a compact set with probability tending to one, T ls M,n remains bounded away from zero and the test is consistent by an argument similar to the proof of Proposition 5. Finite-sample performance of GoF tests This section is devoted to a numerical assessment of the finite-sample performance of the Wasserstein-based GoF tests introduced in the previous sections and we compare them, whenever possible, with other tests. The case of a simple null hypothesis (Section 2) is treated in Section 5.1. The performances of various tests for multivariate normality, which is a special case of the hypothesis of elliptical symmetry considered in Section 3, are compared in Section 5.2, along with an illustration involving a Student t distribution with known degrees of freedom. Section 5.3 considers, in line with Remark 3, the more general composite null hypothesis of a parametric family indexed by marginal location and scale along with a copula parameter θ. Numerical results support the validity of the bootstrap-based calculation of critical values. To the best of our knowledge, no GoF test is available in the literature for such cases except for the method described by Khmaladze (2016), the numerical implementation of which, however, remains unsettled. Throughout, we consider the Wasserstein distance of order p = 2. The level α of the tests is set to 5%, the sample size is n = 200, and the number of replicates considered in the estimation of power curves is 1000. We rely on the R package transport (Schuhmacher et al., 2019), which is why we restrict ourselves to dimension d = 2. As explained in Section 1.3, stochastic algorithms have recently been proposed to solve the semi-discrete problem in higher dimensions, but these are not yet implemented in R. Simple null hypotheses The setting is as in Section 2: given an independent random sample X 1 , . . . , X n from some unknown P ∈ P(R d ), we consider testing the simple null hypothesis H n 0 : P = P 0 , where P 0 ∈ P 2 (R d ) is fully specified. Other GoF tests Two other goodness-of-fit tests will be used as benchmarks. Rippl, Munk and Sturm (2016) consider the fully specified Gaussian null hypothesis H n 0 : P = N d (µ 0 , Σ 0 ) with given mean and covariance. Recall that the squared 2-Wasserstein distance between two d-variate Gaussian distributions is W 2 2 N d (µ 1 , Σ 1 ), N d (µ 2 , Σ 2 ) = µ 1 − µ 2 2 + tr Σ 1 + Σ 2 − 2(Σ 1/2 1 Σ 2 Σ 1/2 1 ) 1/2 . The Rippl-Munk-Sturm test statistic is W 2 2 (N d (X n , S n,X ), N d (µ 0 , Σ 0 )) , with X n and S n,X the sample mean and sample covariance matrix, respectively. This test is sensitive to changes in the parameters of the Gaussian distribution but not to other types of alternatives. Calculation of the test statistic is straightforward. To compute critical values, we relied on a Monte Carlo approximation, drawing many samples from the Gaussian null distribution and taking the empirical quantile of the resulting test statistics. Khmaladze (2016) constructs empirical processes in such a way that they are asymptotically distribution-free, which facilitates their use for hypothesis testing. A special case of the construction is as follows. Let the d-variate cumulative distribution function (cdf) F be absolutely continuous with joint density f , marginal densities f 1 , . . . , f d , and copula density c. Define l(x) = {c(F 1 (x 1 ), . . . , F d (x d ))} 1/2 , x ∈ R d , with F 1 , . . . , F d the marginal cdfs of F . The d-variate cdf G(x) = d j=1 F j (x j ) has the same margins as F , but coupled via the independence copula. Letting κ(x) = (−∞,x] l(y) f (y) dy and κ = l(y) f (y) dy, it follows from Corollary 4 in Khmaladze (2016) that the empirical process v F,n (x) = 1 √ n n i=1 l(X i )I(X i ≤ x) − κ(x) − G(x) − κ(x) 1 − κ 1 √ n n i=1 l(X i ) − κ based on an independent random sample X 1 , . . . , X n from F converges weakly to a G-Brownian bridge, i.e., has the same weak limit as the ordinary empirical process v G,n (x) = 1 √ n n i=1 I(Y i ≤ x) − G(x) based on an independent random sample Y 1 , . . . , Y n from G. The asymptotic distribution of a test statistic based onṽ F,n which is invariant with respect to coordinate-wise continuous monotone increasing transformations is thus the same as if F (or G) were the uniform distribution on [0, 1] d . This includes the Kolmogorov-Smirnov type statistic sup x∈R d ṽ F,n (x) , which (with F the cdf of P 0 ) we are considering below for comparison with our Wasserstein-based test. In case F has independent margins, F and G coincide and the procedure reduces to a classical Kolmogorov-Smirnov test. To ensure that the test has the right size at finite sample size, we calculate critical values by Monte Carlo approximation rather than relying on the asymptotic theory. Results In Figure 1, we assess the performance of the GoF tests of H n 0 : P = P 0 where P 0 = N 2 (0, I 2 ) is a centered bivariate Gaussian with identity covariance matrix. The alternatives P in panels (a)-(f) are as follows: (a) P = N 2 µ µ , I 2 with location shift µ along the main diagonal (rejection frequencies plotted against µ ∈ R); (b) P = N 2 0, σ 2 0 0 σ 2 (rejection frequencies plotted against σ 2 > 0); (c) P = N 2 0, 1 ρ ρ 1 with correlation ρ (rejection frequencies plotted against ρ ∈ (−1, 1)); (d) P has standard normal margins but Gumbel copula with parameter θ (rejection frequencies plotted against θ ∈ [1, ∞)); (e) P has standard Gaussian margins but a bivariate Student t copula with ν = 4 degrees of freedom and correlation parameter ρ (rejection frequencies plotted against ρ ∈ (−1, 1)); 2 (f) P is the "banana-shaped" Gaussian mixture described in Appendix C (rejection frequencies plotted against the mixing weight p ∈ (−1, 1)). 3 The Gumbel and Student t copula simulations in (d) and (e) were implemented from the R package copula (Hofert et al., 2018). Inspection of Figure 1 indicates that the Khmaladze test, as a rule, is uniformly outperformed by the Rippl-Munk-Sturm and Wasserstein tests. The Rippl-Munk-Sturm test, of course, does relatively well under the Gaussian alternatives of panels (a)-(c) where, however, the Wasserstein test is almost as powerful (while its validity, contrary to that of the Rippl-Munk-Sturm test, extends largely beyond the Gaussian null hypothesis). Against the non-Gaussian alternatives in panels (d)-(f), the Wasserstein test has higher power than the Rippl-Munk-Sturm and Khmaladze tests, with the exception of the Gumbel copula alternative in panel (d), where the Rippl-Munk-Sturm and Wasserstein tests perform equally well. For the "banana mixture" of panel (f), the Rippl-Munk-Sturm test fails to capture the change in distribution. Figures 2 and 3 are dealing with non-Gaussian simple null distributions P 0 , so that the Rippl-Munk-Sturm test no longer applies. In Figure 2, the null distribution is the mixture of Gaussians P 0 = 0.5 N 2 (0, I 2 ) + 0.5 N 2 3 0 , I 2 . The alternatives in both panels are (a) P = 0.5 N 2 (0, I 2 ) + 0.5 N 2 3+δ 0 , I (rejection frequencies plotted against the location shift δ ∈ R); Empirical powers of various GoF tests for the simple Gaussian null hypothesis H n 0 : P = N2(0, I2). Three tests are considered: the Wasserstein-2 distance (Section 2), the Rippl-Munk-Sturm test (Rippl, Munk and Sturm, 2016), and the Khmaladze Kolmogorov-Smirnov type test (Khmaladze, 2016), see Section 5.1.1. The alternatives P in panels (a)-(f) are described in Section 5.1.2 (note that in (e), P is not Gaussian even when ρ = 0). (b) P 0 = λ N 2 (0, I 2 ) + (1 − λ) N 2 3 0 , I 2 (rejection frequencies plotted against the mixing weight λ ∈ [0, 1]). The Wasserstein test uniformly outperforms the Khmaladze one. In Figure 3, P 0 has standard Gaussian margins and a Gumbel copula with parameter θ = 1.7. The alternative P is of the same form but with another value θ = 1.7 of the copula parameter θ ∈ [1, ∞). Again, the Wasserstein test yields uniformly higher empirical power. Elliptical families If the radial density f rad is the density of the root of a chi-square random variable with d degrees of freedom, the elliptical family E(f rad ) corresponds to the Gaussian family. The null hypothesis in (3) then is that P is multivariate Gaussian with unknown mean vector and positive definite covariance matrix. Testing multivariate normality is a well-studied problem for which many tests have been put forward. As benchmarks, we will consider here the tests proposed in Royston (1982), Henze and Zirkler (1990), and Rizzo and Székely (2016). Royston's test is a generalisation of the well-known Shapiro-Wilks test. The Henze-Zirkler test statistic is an integrated weighted squared distance between the characteristic function under the null and its empirical counterpart. Interestingly, Ramdas, García Trillos and Cuturi (2017) showed that the Wasserstein distance and the energy distance of Rizzo and Székely (2016) are connected, as the so-called entropy-penalized Wasserstein distance interpolates between them two. We borrowed the implementation of these tests from the R package MVN (Korkmaz, Goksuluk and Zararsiz, 2014). The test by Rippl, Munk and Sturm (2016) tests (see Section 5.1.1) for the simple null hypothesis H n 0 : P = P0 with P0 a bivariate distribution with standard Gaussian margins and Gumbel copula with parameter θ = 1.7; rejection frequencies are plotted against the copula parameter θ. (2016) considered in Section 5.1 does not apply here, since it only can handle fully specified Gaussian distributions. The alternatives in the two panels of Figure 4 are (a) P with standard normal margins and a Gumbel copula with parameter θ ranging over [1, ∞); (b) P with independent margins, one of which is standard normal while the other one is Student t with ν > 0 degrees of freedom. Inspection of Figure 4 reveals that the Wasserstein test has the highest power against the copula alternative in panel (a), while Royston's test has no power at all. For the Student t alternative in panel (b), Royston's test comes out as most sensitive, but the Wasserstein and energy tests (Rizzo and Székely, 2016) performe quite well too. In Figure 5, we consider the bivariate Student (ν = 12 degrees of freedom) elliptical family, with radial density f rad the density of the root of a rescaled Fisher F (d, 12) variable. Figure 5 provides a plot of rejection frequencies under bivariate skew-t alternatives (Azzalini, 2014) with marginal skewness parameters α 1 and α 2 . Simulations were based on the function rmst from the R package sn (Azzalini, 2020). In principle, the empirical process approach in Khmaladze (2016) leads to test statistics that are asymptotically distribution-free, but their numerical implementation involves a number of multiple integrals, the computation of which remains problematic. Empirical power curves of various tests of the hypothesis that P is bivariate Gaussian with unknown mean vector and covariance matrix. The Wasserstein test in Section 3 is compared to three other multivariate normality tests mentioned in Section 5.2. In (a), the alternative P has a Gumbel copula with parameter θ; rejection frequencies are plotted against θ ∈ [1, ∞). In (b), one of the marginals of P is a Student t distribution with ν degrees of freedom; rejection frequencies are plotted against ν > 0. General parametric families We now turn to the more general example of a non-elliptical parametric model M where the parametric bootstrap procedure described in Section 4 nevertheless applies. In the notation of Remark 3, let M = {Q ψ : ψ ∈ Ψ} consist of the bivariate distributions with Gaussian marginals and an Ali-Mikhail-Haq (AMH) copula, yielding a five-dimensional parameter vector ψ = (µ 1 , σ 1 , µ 2 , σ 2 , θ) where µ 1 , µ 2 ∈ R and σ 1 , σ 2 ∈ (0, ∞) are marginal location and scale parameters, and θ ∈ Θ = [−1, 1] is the AMH copula parameter. We applied the method involving the location-scale reduction described in Remark 3. Following Genest, Ghoudi and Rivest (1995), the copula parameter θ was estimated via a rank-based maximum pseudo-likelihood estimator. Obviously, the componentwise ranks of the data and those of the residuals in (14) coincide, so thatθ n , as required, depends on the data only through the residuals. We first checked the validity of the parametric bootstrap procedure of Section 4. To do so, we simulated 1 000 independent random samples of size n = 200 from P ∈ M with θ = 0.7. For each sample, we calculated the test statistic T ls M,n in (15) and checked whether or not it exceeds the bootstrapped critical value c ls M (α, n,θ n ) for α equal to multiples of 5%. The critical value function θ → c ls M (α, n, θ) in (16) was pre-computed by Algorithm 3, or rather a Fig 5. Empirical power of the Wasserstein test in Section 3 for the hypothesis that P is bivariate Student t with ν = 12 degrees and unknown mean vector and covariance matrix. The alternatives P are bivariate skew-t with skewness parameters α1 and α2. variation thereof taking into account the estimated residuals in Eq. (14). The points in Figure 6(a) show the empirical type I errors as a function of α. The diagonal line fits the points well, lending support to the validity of the parametric bootstrap method (if not proving it). Figure 6(b) similarly displays the rejection frequencies of the Wasserstein test under an alternative P whose copula belongs to the Frank family with parameter η. If η = 0, the Frank copula reduces to the independence copula, which is a member of the AMH family too. Again, the approach in Khmaladze (2016) in principle also applies, but its actual implementation is intricate and remains unsettled. x >r x p dP(x) = 0. Then, lim n→∞ sup P∈M E P W p p ( P n , P) = 0. The condition on M is equivalent to the one that the closure of M in the metric space (P p (R d ), W p ) is compact. This follows from Prohorov's theorem and the characterization of W p -convergence in Bickel and Freedman (1981, Lemma 8.3) or Villani (2009, Theorem 6.9). The convergence rate of E P {W p p ( P n , P)} has been studied intensively; see, for instance, Fournier and Guillin (2015, Theorem 1). However, those rates require the existence of moments of order q higher than p. Proof of Theorem 1. The following smoothing argument is inspired by the proof of Theorem 1.1 in Horowitz and Karandikar (1994). Let U σ denote the Lebesgueuniform distribution on the ball {x ∈ R d : x ≤ σ} in R d with radius σ ∈ (0, ∞) and centered at the origin. Denoting by * the convolution of probability measures, we have, for any Q ∈ P p (R d ), W p (Q * U σ , Q) ≤ σ. Indeed, if X and Y are independent random vectors with distributions Q and U σ , respectively, then (X + Y, X) is a coupling of Q * U σ and Q, so that W p p (Q * U σ , Q) ≤ E[ Y p ] ≤ σ p . By the triangle inequality, it follows that W p ( P n , P) ≤ 2σ + W p ( P n * U σ , P * U σ ). Taking expectations and using the elementary inequality (a + b) p ≤ 2 p−1 (a p + b p ) for p ≥ 1, a ≥ 0, and b ≥ 0, we obtain E P W p p ( P n , P) ≤ 2 p−1 2 p σ p + E W p p ( P n * U σ , P * U σ ) . If we can show that ∀σ > 0, lim n→∞ sup P∈M E W p p ( P n * U σ , P * U σ ) = 0,(17) then it will follow that ∀σ > 0, lim sup n→∞ sup P∈M E P W p p ( P n , P) ≤ 2 2p−1 σ p . But then, this latter lim sup is actually a lim and is equal to zero, as required. Let us proceed to show (17). Fix σ > 0 for the remainder of the proof. Let f σ denote the density function of U σ . The measures P n * U σ and P * U σ are absolutely continuous too and have density functions x → n −1 n i=1 f σ (x − X i ) and x → R d f σ (x − y)dP(y) , respectively. The Wasserstein distance can be controlled by weighted total variation (Villani, 2009, Theorem 6.15): W p p ( P n * U σ , P * U σ ) ≤ 2 p−1 R d x p d\| P n * U σ − P * U σ \|(x) = 2 p−1 R d x p 1 n n i=1 f σ (x − X i ) − R d f σ (x − y) dP(y) dx. Take expectations and apply Fubini's theorem to see that E P W p p ( P n * U σ , P * U σ ) ≤ 2 p−1 R d x p g n (x; P) dx (18) where g n (x; P) = E P 1 n n i=1 f σ (x − X i ) − R d f σ (x − y) dP(y) . Let r > σ and split the integral in (18) according to whether x > r or x ≤ r. Note that f σ (u) = f σ (0) if y ≤ σ and f σ (u) = 0 otherwise. For any P ∈ P(R d ) and any x ∈ R d , we have, by the Cauchy-Schwarz inequality, g n (x; P) ≤ n −1/2 f σ (0). It follows that lim n→∞ sup P∈P(R d ) x ≤r x p g n;P (x) dx = 0. But then, in view of (18), we have lim sup n→∞ sup P∈M E P W p p ( P n * U σ , P * U σ ) ≤ lim sup n→∞ sup P∈M 2 p−1 x >r x p g n (x; P) dx. By the triangle inequality, we also have, for all n, g n (x; P) ≤ 2 R d f σ (x − y)dP(y). Applying Fubini's theorem once more, we find that x >r x p g n (x; P) dx ≤ 2 x >r x p y∈R d f σ (x − y) dP(y) dx = 2 y∈R d x >r x p f σ (x − y) dx dP(y) = 2 y∈R d u+y >r u + y p f σ (u) du dP(y). Since f σ (u) = 0 whenever u > σ and since r > σ, we have u+y >r u + y p f σ (u) du ≤ 2 p−1 (σ p + y p ) if y > r − σ, 0 otherwise. Choosing r > 2σ, we get that y > σ for all y in the non-zero branch above, and thus, for all n, x >r x p g n (x; P) dx ≤ 2 p+1 y >r−σ y p dP(y). It follows that, for every r > σ, lim sup n→∞ sup P∈M E P W p p ( P n * U σ , P * U σ ) ≤ 2 2p sup P∈M y >r−σ y p dP(y). The left-hand side does not depend on r. The condition on M implies that the right-hand side converges to zero as r → ∞. It follows that the left-hand side must be equal to zero. But this is exactly (17), as required. The proof is complete. Corollary 1. For M as in Theorem 1, we have ∀ε > 0, lim n→∞ sup P∈M P n W p p (L n , P) > ε = 0, i.e., W p p ( P n , P) → 0 in probability as n → ∞, uniformly in P ∈ M. Proof. By Markov's inequality, for every ε > 0 and every P ∈ P p (R d ), we have P n W p (L n , P) > ε ≤ ε −p (R d ) n W p p (L n , P) dP n . In view of Theorem 1, the integral converges to zero uniformly in P ∈ M. Appendix B: Algorithms for the computation of critical values Our test statistics involve the Wasserstein distance between an empirical measure and a continuous one. Their calculation requires solving a semi-discrete optimal transport problem (Section 1.3), for which we relied on the function semidiscrete in the R package transport (Schuhmacher et al., 2019), which implements the method of Mérigot (2011). The method starts from a discretization of the source density. The quality of approximation can be set by choosing a sufficiently fine mesh and selecting the tolerance parameter to a low value. The meshes considered here consisted of approximately 10 5 cells. Below we provide pseudo-code algorithms to sketch the main steps in the actual computation of the critical values. We start with the case of a simple null hypothesis (Algorithm 1), then turn to elliptical families with a given generator (Algorithm 2) and finally propose the bootstrap procedure for a general parametric family (Algorithm 3). The empirical distribution associated with a sample X = (X 1 , . . . , X n ) ∈ (R d ) n is denoted by P n (X). The largest integer not larger than a scalar x ∈ R is denoted by x . In Algorithm 3, we first compute c M (α, n, θ) for θ in a finite mesh Θ 1 ⊆ Θ. From these values, we reconstruct the function θ → c M (α, n, θ) by smoothing. It is into the resulting function that we plug in the actual estimateθ n . Further, we restrict the bootstrap parameter estimatesθ * n,b to be in another finite mesh Θ 2 ∈ Θ, because calculation of the bootstrapped test statistics T * M,n,b requires a preliminary discretization of the density associated toθ * n,b in order to solve the corresponding semi-discrete optimal transport problem. The first loop in Algorithm 3 is discretizing the densities of P θ for θ ∈ Θ 2 . The second loop is calculating c M (α, n, θ) for θ ∈ Θ 1 by drawing B samples of size n from P θ . The final step of the algorithm consists of reconstructing the function θ → c M (α, n, θ) by smoothing. This smoothing step is illustrated in Figure 7 for the five-parameter bivariate Gaussian-AMH model in Section 5.3, applying the location-scale reduction in Remark 3. The quality of the approximate critical thresholds is ensured by choosing a large enough number of Monte Carlo replications N (Algorithms 1 and 2) or bootstrap replicates B (Algorithm 3). In the simulation experiments, we chose N between 3 000 and 10 000 depending on the time required, while B = 1 000. Algorithm 1: Computation of c(α, n, P 0 ) in Eq. (2) Input: • A mesh that supports the source density f associated to P 0 Appendix C: A banana-shaped distribution The "banana-shaped" distribution in Section 5.1 and Figure 1 of three Gaussian components. Figure 8 shows a scatterplot for p = 0.35 of a random sample of size n = 500 from this distribution. ∀ε > 0 , 0∀K ) the critical values c M (α, n, θ) tend to zero uniformly in θ, i.e., lim n→∞ sup θ∈K c M (α, n, θ) = 0 ∀α > 0, ∀K ∈ K(Θ); Fig 1. Empirical powers of various GoF tests for the simple Gaussian null hypothesis H n 0 : P = N2(0, I2). Three tests are considered: the Wasserstein-2 distance (Section 2), the Rippl-Munk-Sturm test (Rippl, Munk and Sturm, 2016), and the Khmaladze Kolmogorov-Smirnov type test (Khmaladze, 2016), see Section 5.1.1. The alternatives P in panels (a)-(f) are described in Section 5.1.2 (note that in (e), P is not Gaussian even when ρ = 0). Fig 2 . 2Empirical powers of the Wasserstein andKhmaladze (2016) tests for the simple null hypothesis H n 0 : P = P0 with P0 an equal-weights mixture of N2(0, I2) and N2 3 0 , I2 . In panel (a), the alternative P is an equal-weights mixture of N2(0, I2) and N2 3+δ 0 , I2 ; rejection frequencies are plotted against δ ∈ [−1, 1]. In panel (b), the alternative P is a mixture of the same two components, but with weights λ ∈ (0, 1) and (1 − λ); rejection frequencies are plotted against λ ∈ [0.25, 0.75]. Fig 3 . 3Empirical powers of the Wasserstein and Khmaladze Fig 4. Empirical power curves of various tests of the hypothesis that P is bivariate Gaussian with unknown mean vector and covariance matrix. The Wasserstein test in Section 3 is compared to three other multivariate normality tests mentioned in Section 5.2. In (a), the alternative P has a Gumbel copula with parameter θ; rejection frequencies are plotted against θ ∈ [1, ∞). In (b), one of the marginals of P is a Student t distribution with ν degrees of freedom; rejection frequencies are plotted against ν > 0. Fig 6 . 6Wasserstein test for H n 0 : P ∈ M with M the family of bivariate distributions with Gaussian margins with unknown location-scale parameters and Ali-Mikhail-Haq (AMH) copula with unknown parameter θ ∈ [−1, 1] (Section 5.3). Test statistic and critical values computed based on estimated residuals and parametric bootstrap as in Remark 3. Panel (a) shows real versus nominal type I errors α based on 1 000 samples of size n = 200 drawn from P ∈ M with θ = 0.8. Panel (b) shows the power against alternatives P with Gaussian marginals and Frank copula with parameter η ≥ 0; if η = 0, the Frank copula is the independence one, which is part of the AMH family too.Theorem 1. Let M ⊆ P p (R d ) be such that lim r→∞ sup P∈M • 4 TFig 7 . 47A number of replications N • A sample size n • A level α Output: An approximation of c(α, n, P 0 )1 T ← [0, . . . , 0] ∈ R N // Initialization 2 for i = 1 to N do 3 X ← rand(n, P 0 ) // Generation of sample of size n from P 0 [i] ← W 2 2 ( Pn(X), P 0 ) 5 sort(T ) 6 c(α, n, P 0 ) ← T [ (1 − α)N ]// Empirical quantile (order statistic) Illustration of the last step in Algorithm 3 for the bivariate five-parameter Gaussian-AMH model in Section 5.3 using the location-scale reduction in Remark 3. The function θ → c ls M (α, n, θ) (in red) is constructed by smoothing Monte Carlo estimates (circles) of c ls M (α, n, θ) for θ ∈ Θ1 ⊆ Θ = [−1, 1], with α = 0.05, n = 200 and B = 1 000 samples per point. The smoother is a 6th-degree polynomial fitted by ordinary least squares. Fig 8 . 8Scatterplot of a sample of size 500 from the "banana-shaped" mixture (19). Note that P is not Gaussian, even for ρ = 0. 3 The mixture is constructed so that the first and second moments of P remain close to those of P 0 . Appendix A: Uniform convergence of the empirical Wasserstein distanceThe aim of this appendix is to establish the convergence to zero in probability, uniformly in the underlying distribution P ∈ M, of the empirical Wasserstein distance W p ( P n , P) when M ⊆ P p (R d ) has a compact W p -closure. Actually, Theorem 1 establishes the stronger result that the convergence to zero holds uniformly in the p-th mean. The Markov inequality then implies (Corollary 1) the desired uniform convergence in probability. The notation is that of Section 1.2, with E P standing for expectation under an independent random sample from P.Algorithm 2: Computation of c E (α, n, f rad ) in Eq.(6)Input:• A mesh that supports the source density f associated to P 0 .// Generation of sample of size n from P f rad 4Ẑ ← standardize(X, method = "Cholesky") // Residuals as in(5)Algorithm 3: Computation of θ → c M (α, n, θ) in(10)Input:• A finite set Θ 1 ⊆ Θ of values of θ at which to calculate c M (α, n, θ) initially• A larger finite set Θ 2 ⊆ Θ into which to force the bootstrapped estimatesθ * n,b• A number of bootstrap replications B• A sample size n9 ColumnSort(T ) // For each θ ∈ Θ 1 , sort T [ · , θ] 10 for θ in Θ 1 do 11 c M (α, n, θ) ← T [ (1 − α)N , θ] 12 c M (α, n, · ) ← Smooth (θ, c M (α, n, θ)) : θ ∈ Θ 1 // function θ → c M (α, n, θ) Minkowski-type theorems and least-squares clustering. F Aurenhammer, F Hoffmann, B Aronov, Algorithmica. 20Aurenhammer, F., Hoffmann, F. and Aronov, B. (1998). 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[ "QURG: Question Rewriting Guided Context-Dependent Text-to-SQL Semantic Parsing", "QURG: Question Rewriting Guided Context-Dependent Text-to-SQL Semantic Parsing" ]	[ "Linzheng Chai \nState Key Lab of Software Development Environment\nBeihang University\nBeijingChina\n", "Dongling Xiao \nTencent Cloud Xiaowei\n\n", "Jian Yang \nState Key Lab of Software Development Environment\nBeihang University\nBeijingChina\n", "Liqun Yang \nState Key Lab of Software Development Environment\nBeihang University\nBeijingChina\n", "Qian-Wen Zhang \nTencent Cloud Xiaowei\n\n", "Yunbo Cao [email protected] \nTencent Cloud Xiaowei\n\n", "Zhoujun Li \nState Key Lab of Software Development Environment\nBeihang University\nBeijingChina\n", "Zhao Yan \nTencent Cloud Xiaowei\n\n" ]	[ "State Key Lab of Software Development Environment\nBeihang University\nBeijingChina", "Tencent Cloud Xiaowei\n", "State Key Lab of Software Development Environment\nBeihang University\nBeijingChina", "State Key Lab of Software Development Environment\nBeihang University\nBeijingChina", "Tencent Cloud Xiaowei\n", "Tencent Cloud Xiaowei\n", "State Key Lab of Software Development Environment\nBeihang University\nBeijingChina", "Tencent Cloud Xiaowei\n" ]	[]	Context-dependent Text-to-SQL aims to translate multi-turn natural language questions into SQL queries. Despite various methods have exploited context-dependence information implicitly for contextual SQL parsing, there are few attempts to explicitly address the dependencies between current question and question context. This paper presents QURG, a novel QUestion Rewriting Guided approach to help the models achieve adequate contextual understanding. Specifically, we first train a question rewriting model to complete the current question based on question context, and convert them into a rewriting edit matrix. We further design a two-stream matrix encoder to jointly model the rewriting relations between question and context, and the schema linking relations between natural language and structured schema. Experimental results show that QURG significantly improves the performances on two large-scale contextdependent datasets SParC and CoSQL, especially for hard and long-turn questions. * Corresponding author. CAR_DATA Id MPG Horsepower cylinders ... ... MakeId CAR_NAMES CAR_MAKES Model Make MakeId : How about with the max MPG? : Show its make! SELECT Id FROM CARS_DATA ORDER BY Horsepower DESC LIMIT 1 SELECT Id FROM CARS_DATA ORDER BY MPG DESC LIMIT 1 SELECT T1.Make FROM CAR_NAMES AS T1 JOIN CARS_DATA AS T2 ON T1.MakeId = T2.Id ORDER BY T2.MPG DESC LIMIT 1	10.48550/arxiv.2305.06655	[ "https://export.arxiv.org/pdf/2305.06655v2.pdf" ]	258,615,209	2305.06655	29bce162bd5b83cbc4c00732d03844b33b8a6007	QURG: Question Rewriting Guided Context-Dependent Text-to-SQL Semantic Parsing Linzheng Chai State Key Lab of Software Development Environment Beihang University BeijingChina Dongling Xiao Tencent Cloud Xiaowei Jian Yang State Key Lab of Software Development Environment Beihang University BeijingChina Liqun Yang State Key Lab of Software Development Environment Beihang University BeijingChina Qian-Wen Zhang Tencent Cloud Xiaowei Yunbo Cao [email protected] Tencent Cloud Xiaowei Zhoujun Li State Key Lab of Software Development Environment Beihang University BeijingChina Zhao Yan Tencent Cloud Xiaowei QURG: Question Rewriting Guided Context-Dependent Text-to-SQL Semantic Parsing Context-dependent Text-to-SQL aims to translate multi-turn natural language questions into SQL queries. Despite various methods have exploited context-dependence information implicitly for contextual SQL parsing, there are few attempts to explicitly address the dependencies between current question and question context. This paper presents QURG, a novel QUestion Rewriting Guided approach to help the models achieve adequate contextual understanding. Specifically, we first train a question rewriting model to complete the current question based on question context, and convert them into a rewriting edit matrix. We further design a two-stream matrix encoder to jointly model the rewriting relations between question and context, and the schema linking relations between natural language and structured schema. Experimental results show that QURG significantly improves the performances on two large-scale contextdependent datasets SParC and CoSQL, especially for hard and long-turn questions. * Corresponding author. CAR_DATA Id MPG Horsepower cylinders ... ... MakeId CAR_NAMES CAR_MAKES Model Make MakeId : How about with the max MPG? : Show its make! SELECT Id FROM CARS_DATA ORDER BY Horsepower DESC LIMIT 1 SELECT Id FROM CARS_DATA ORDER BY MPG DESC LIMIT 1 SELECT T1.Make FROM CAR_NAMES AS T1 JOIN CARS_DATA AS T2 ON T1.MakeId = T2.Id ORDER BY T2.MPG DESC LIMIT 1 Introduction The past decade has witnessed increasing attention on text-to-SQL semantic parsing task, which aims to map natural language questions to SQL queries. Previously, works have mainly concentrated on the context-independent text-to-SQL task (Zhong et al., 2017;Yu et al., 2018), which translates single questions to SQL queries. The key to solving context-independent text-to-SQL is to model the relationships between questions and schema. Recent works have made great progress (Wang et al., 2020;Lin et al., 2020b; by employing schema linking mechanism which aligns schema items to entity references in the questions. With the extensive demand for interactive systems, the context-dependent text-to-SQL task which translates multi-turn questions to SQL : What is id of the car with the max MPG? : What is make of the car with the max MPG? Figure 1: An example of the context-dependent Textto-SQL task with the phenomenon of co-reference and omission. u rw t denotes the rewritten question of the current question u t at t-th conversation turn. queries has attracted more attention. Compared with the context-independent text-to-SQL task, the context-dependent text-to-SQL task faces more challenges, that not only need to consider the relationship between natural language questions and the schema, but also the relationship between the current question and question context. In a multiturn scenario, as shown in Figure 1, current questions may contain two contextual phenomena: coreference and omission which are heavily associated with the historical context, meanwhile the question context may also contain information irrelevant to current questions. Thus models are required to selectively leverage contextual information to correctly address the user's intention of the current questions. Previous works (Zheng et al., 2022;Yu et al., 2021b;Scholak et al., 2021b) on context-dependent text-to-SQL typically model the context dependencies in a simple way that feeds the concatenation of the current question, question context and schema into a neural networks encoder. To exploit contextdependence information, Hui et al. (2021) propose a dynamic relation decay mechanism to model the dynamic relationships between schema and question as conversation proceeds. Several works di- Figure 2: An example of rewriting edit matrix. Given rewritten question u rw t , we convert it to relations between current question u t and question context u <t . rectly leverage historical generated SQL (Zhang et al., 2019;Zheng et al., 2022) or track interaction states associated with historical SQL (Cai and Wan, 2020; to enhance the current SQL parsing. Furthermore, Yu et al. (2021b) and introduce task-adaptive pre-trained language models and auxiliary training tasks on question and context to help models achieve adequate contextual understanding. However, these works neglect the explicit guidance on resolving contextual dependency, that SQL parsing and context understanding are coupled for model training. The question rewriting (QR) task is to convert the current question into a self-contained question that can be understood without contextual information, and has been widely explored to represent multi-turn utterances currently (Su et al., 2019a;Pan et al., 2019;Elgohary et al., 2019a). is the first attempt at contextdependent text-to-SQL task by question rewriting, where a question rewrite model first explicitly completes the dialogue context, and then a contextindependent Text-to-SQL parser follows. However, this approach relies on in-domain QR annotations and complex algorithms to obtain the rewritten question data. To address the above limitations, we propose QURG, a novel QUestion Rewriting Guided approach, which consists of three steps: 1) rewriting the current question into self-contained question and further converting it into a rewriting edit matrix; 2) jointly representing the rewriting matrix, multi-turn questions, and schema; 3) decoding the SQL queries. Firstly, we train and evaluate the QR model on the out-of-domain dataset CA-NARD (Elgohary et al., 2019a) and initialize the QR model with a pre-trained sequence generator for more precise rewritten questions. Secondly, inspired by (Liu et al., 2020b), we propose to integrate rewritten results into the text-to-SQL task in the form of a rewriting relation matrix between question and context. We observed that directly replacing or concatenating original input with rewritten question may mislead the model for correctly SQL parsing. The reason is the unavoidable noise in rewritten questions and some questions are semantically complete and do not need to be rewritten. The rewriting edit matrix denotes the relations between question and context words, these relations could clearly guide the model in solving longrange dependencies. Taking figure 2 as an example, the rewritten question can be produced from original question through a series of edit operations in rewriting matrix (substitute "one" in the current question with "cities" in context, and insert "arriving flights" before "?"). Furthermore, we propose a two-stream relation matrix encoder based on the relation-aware Transformer (RAT) (Shaw et al., 2018) to jointly model the rewriting relation features between the current question and the context, and the schema linking relation features between multi-turn question and database schema. Finally, we aggregate the representations from the two relation matrix encoders to generate current SQL queries. We evaluate our proposed QURG on two largescale cross-domain context-dependent benchmarks: SParC (Yu et al., 2019b) and CoSQL (Yu et al., 2019a). We summarize the contributions of this work as follows: • We present a novel context-dependent text-to-SQL framework QURG that explicitly guides models to resolve contextual dependencies. • Our framework incorporates rewritten questions in a novel way that explicitly represents multiturn questions through rewriting relation matrix and two-stream relation matrix encoder. • Experimental results show that QURG achieves comparable performance to recent state-of-theart works on two context-dependent text-to-SQL datasets. Besides, we further explore different approaches to incorporate rewritten questions into text-to-SQL to access the advantages of QURG. 2 Related Work Text-to-SQL The text-to-SQL task aims to map natural language questions to database-related SQL queries. Spi-der (Yu et al., 2018) is a widely evaluated crossdomain context-independent dataset and numerous works have shown that modeling the relation between question and schema can effectively improve performance on Spider. Shaw et al. (2018) adopt relation-aware Transformer (RAT) (Shaw et al., 2018) to encode the relational position for sentence representations, which has been widely transferred to text-to-SQL works (Wang et al., 2020;Lin et al., 2020b;Scholak et al., 2021a;Yu et al., 2021a) to encode the schema-linking relations between natural language questions and structured database schema. further improve relations modeling by line graph neural networks. In face of the co-reference and omission in multiturn questions, context-dependent text-to-SQL task is more challenging. Several works (Zhang et al., 2019;Zheng et al., 2022) utilize previously generated SQL queries to resolve long-range dependency and improve the parsing accuracy. Cai and Wan (2020) and Hui et al. (2021) use graph neural network to jointly encode multiturn questions and schema. Inspired by the success of pre-trained models (Raffel et al., 2020;?;Yang et al., 2020Yang et al., , 2022, Yu et al. (2021b) and propose auxiliary state switch prediction tasks to model multi-turn question relations. Scholak et al. (2021b) simply constrain the auto-regressive decoders of super large pre-trained language models T5-3B. propose a dual learning method to produce rewritten question data with in-domain QR annotations and directly use rewritten questions to generate SQL queries. In this work, we propose to adopt the rewriting matrix to explicitly model the relationships between the current question and context, and enhance the ability to capture long-range contextual dependencies. Question Rewriting (QR) Question rewriting is to complete the co-reference and omission in the current question with historical context information, and help models understand multi-turn questions (Elgohary et al., 2019b). Most works (Pan et al., 2019;Su et al., 2019b;Liu et al., 2021) conduct experiments on QR task as a sequence generation task with the copy mechanism. While Liu et al. (2020b) and Hao et al. (2021) formulate the task as a semantic segmentation task and sequence-tagging task respectively. Moreover, QR is widely applied to downstream tasks like conversational question answer (CQA) (Vakulenko et al., 2021;Anantha et al., 2021;Kim et al., 2021;Yang et al., 2019;Bai et al., 2023), conversational retrieval (Dalton et al., 2020;Yu et al., 2020) and context-dependent text-to-SQL . Differently, we convert rewritten questions to rewriting relation matrix between question and context, and propose a two-stream relation matrix encoder to jointly model the rewriting and schema linking relations for context-dependent text-to-SQL parsing. Furthermore, we Preliminaries In this section, we first formalize the contextdependent Text-to-SQL task, and then we introduce the relation-aware Transformer (RAT) (Wang et al., 2020), which is widely adopted to encode relations between sequence elements in text-to-SQL tasks, and which we use to build our two-stream encoder. Task Formulation The context-dependent text-to-SQL task is to generate the SQL query y t given current user question u t , historical question context u <t = {u 1 , u 2 , . . . , u t−1 } , and database schema S = T , C , which consists of a series of tables T = {t 1 , ..., t \|T \| } and columns C = {c 1 , ..., c \|C\| }. Relation-Aware Transformer (RAT) The relation-aware transformer is an extension of the vanilla transformer (Vaswani et al., 2017). RAT can integrate the pre-defined relation features by adding relation embedding to the self-attention mechanism of the vanilla transformer. The vanilla Transformer is a model architecture which consists of a stack of multi-head selfattention layers, which has been widely used for tasks that process sequence inputs. Given input em- bedding sequence X = {x i } n i=1 where x i ∈ R dx , each Transformer layer transform the input element x i into y i with H heads as follows: e (h) ij = x i W (h) Q x j W (h) K d z /H (1) α (h) ij = Softmax j e (h) ij (2) z (h) i = n j=1 α (h) ij x j W (h) V (3) z i = Concat(z (1) i , ..., z (H) i ) (4) y i = LayerNorm(x i + z i ) (5) y i = LayerNorm( y i + FC(ReLU( y i ))) (6) where h denotes the h-th head, a (h) ij is the attention weights, Concat(·) is a concatenate operation, FC(·) is a full-connected layer, LayerNorm(·) is layer normalization, ReLU(·) is the activation function and W (h) Q ,W (h) K ,W (h) V are learnable projec- tion parameters. Compared to the vanilla transformer, RAT integrates learnable relation embedding into the selfattention module to bias model toward pre-defined relational information as: e (h) ij = x i W (h) Q x j W (h) K + r K ij d z /H (7) z (h) i = n j=1 α (h) ij x j W (h) V + r V ij (8) where r ij is the pre-defined relation embedding between input elements x i and x j . Figure 3 illustrates our framework of QURG. It contains three parts: 1) Question rewriting model which is employed to obtain rewritten question u rw t from current question u t and context u <t . 2) Rewriting matrix generator which converts rewritten question to word-level relation matrix between u t and u <t . 3) SQL parser with two-stream encoder which can effectively integrate rewrite matrix for solving context dependencies. Methodology Question Rewriting Model Following Lin et al. (2020a) and Kim et al. (2021), we employ a pre-trained T5-base sequence generator (Raffel et al., 2020) as our QR model. Due to the lack of QR annotations on the text-to-SQL task, we directly use the out-of-domain QR dataset CA-NARD (Elgohary et al., 2019a) for QR model training and evaluation. Specifically, given the current user question u t and historical context u <t , we train QR models to produce rewritten question u rw t as: P(u rw t \|{[history], u t−1 , [query], u t }), where [history] and [query] are special symbols to distinguish the context input and current question. Rewriting Matrix Construction Instead of feeding the rewritten question directly into the text-to-SQL model, we further convert the rewritten question into rewriting matrix which contains the key information to resolve context dependencies in current question. Following Liu et al. (2020b), we adopt a heuristic method to construct the bi-directional rewriting matrix R rw ∈ R (\|Xutter\|×\|Xutter\|) , where X utter is the concatenate of historical context u <t and current question u t . Firstly, we compare u t with u rw t to find the Longest Common Subsequence (LCS), for each word in u rw t , if it is not in LCS, we will tag it as ADD. Similarly, for each word in u t , if it is not in LCS, we tag it as DEL. After tagging, we merge consecutive words of the same tag to obtain ADD span and DEL span. Secondly, we traverse the ADD span in u rw t , if it appears in the historical context u <t and corresponds to DEL in the same position in u t , then we consider it as a "substitute" operation between original question u t and historical context u <t , while there is no corresponding DEL in u t , we consider it as a "insert" operation. Furthermore, if the ADD span does not appear in the context u <t , we ignore it because these spans correspond to some unimportant word in most cases. Taking figure 2 as an example, give current question u t : "Which one has the most?" and rewritten question u rw t : "Which city has the most arriving flights?", the longest common sequence is "which has the most?". We tag "one" in u t as DEL, and tag "city" and "arriving flights" in u rw t as ADD. Since the two ADD spans "city" and "arriving flights" all appear in context u <t , the relation between "one" in u t and "cities" in u <t is "substitute", the relation between "?" in u t and "arriving flights" in u <t is "insert". Note that the rewriting matrix shown in Figure 2 is uni-directional (u <t → u t ), when encoding later, question u t is concatenated with context u <t , so we extend the uni-directional rewriting matrix to bi-directional rewriting matrix R rw following the relation types between u t and u <t as shown in Table 1 (An example are described in Appendix A). Through the above method, we can associate the existing omission and co-reference in the current question u t with the historical context u <t , retaining context-dependencies in the form of a rewriting matrix, while ignoring trivial information or noise in the rewritten question u rw t . QURG: SQL Parser with Rewriting Matrix Our QURG model is an extension of RAT-SQL (Wang et al., 2020) following the common encoderdecoder architecture, which consists of three modules, as shown in Figure 3: 1) Pre-trained Language Model (PLM) encoder which jointly transforms u t u t u t 1 _ u 2 u 1 t1 t2 c 1 c 2 Current Question Current Question Question Context Question Context Database Schema ( Type of x i Type of x j Relation type Description question u t , context u <t and schema S into embedding as X u , X ctx and X sc respectively; 2) Twostream relation matrix input encoder which further encodes element embedding with pre-defined pairwise relation features as H; 3) Grammar-based decoder which generates SQL query corresponding to the current question. Question u t Context u <t Q-C-INS Insert x j before x i Q-C-SUB Substitute x j for x i NONE None operation Context u <t Question u t C-Q-INS Insert x i before x j C-Q-SUB Substitute x i for x j NONE None operation Pre-trained Language Model Encoder We concatenate the current question u t , context u <t and schema S as the input sequence of pretrained language models: X ={[CLS], u t , [SEP], u t−1 , ..., u 1 , [SEP], t 1 , , t 2 , ..., t \|T \| , [SEP], c 1 , c 2 , ...c \|C\| , [SEP]}. Following , we randomly shuffle the order of tables and columns in different mini-batches to alleviate the risk of over-fitting. Moreover, since each table name or column name may consist of multiple words, we use the average of the beginning and ending hidden vector as the schema element representation. Finally, the joint embedding vector of X is represented as X = Concat(X u ; X ctx ; X sc ). Two-stream Relation Matrix Encoder This module contains two streams of relation matrix encoders: Schema Linking matrix R link encoder and Rewriting matrix R rw encoder. Firstly, the schema linking aids the model with aligning column/table references in the question and context to the corresponding schema columns/tables id. The schema linking relation matrix R link is borrowed from RATSQL (Wang et al., 2020) which builds relations between natural language and schema elements. Through the schema linking method, we can get schema linking matrix R link ∈ R (\|X\|×\|X\|) . Then, the schema linking matrix R link encoder takes joint embeddings of current question X u , context X ctx and schema word X sc as input and applies L link stacked RAT layers to produce contextual representation H link u , H link ctx and H link sc respectively: H link (0) = Concat (X u ; X ctx ; X sc ) (9) H link (l) = RAT (l) H link (l−1) , R link(10) where l ∈ [1, L link ] denote the index of the l-th RAT layer. Similarly, the rewriting matrix R rw encoder takes the joint embeddings of the current question X u and context X ctx as input and applies L rw stacked RAT layers to get the rewriting enhanced representations H rw u , H rw ctx of question and context respectively: H rw (0) = Concat (X u ; X ctx )(11) H rw (l) = RAT (l) H rw (l−1) , R rw where l ∈ [1, L rw ] denote the index of the l-th RAT layer. Finally, we aggregate the representations of the two-stream encoder as: H = Concat H link u +H rw u ; H link ctx +H rw ctx ; H link sc Grammar-based Decoder We follow Wang et al. (2020) and , using a grammar-based syntactic neural decoder that generates the target SQL action sequence in the depth-first-search order of the abstract syntax tree (AST). We use single-layer LSTM (Hochreiter and Schmidhuber, 1997) as the auto-regressive decoder. At each step, the decoder predicts the probability of actions and uses the pointer mechanism to predict the probability of table/column id. We refer the reader to (Yin and Neubig, 2018) for details. Experiments In this section, we describe the experimental setups and evaluate the effectiveness of our proposed QURG. We compare QURG with previous works and conduct several ablation experiments. We also compare our method with the other two approaches of incorporating rewritten questions into text-to-SQL, to further verify the advantages of our QURG. In addition, the experimental details of the QR model are described in appendix B. Experimental Setup Datasets We train our QURG model on two large-scale cross-domain context-dependent textto-SQL datasets, SparC (Yu et al., 2019b) and CoSQL (Yu et al., 2019a). The details of those datasets are organized in Table 2. Evaluation Metrics For evaluation, we employ two main metrics on both SParC and CoSQL datasets: Question match (QM) accuracy and Interaction match (IM) accuracy. Specifically, for QM, if all clauses in a predicted SQL are exactly matching those of the target SQL, the matching score is 1.0, otherwise, the score is 0.0. For IM, if all the predicted SQL in interaction is correct, the interaction match score is 1.0, otherwise the score is 0.0. Implementation Details For Text-to-SQL tasks, we use ELECTRA (Clark et al., 2020) as our pretrained language model for all experiments. We set the learning rate to 1e-4, batch size to 32, and the maximum gradient norm to 10. The number of training epochs is 300 and 320 for SParC and CoSQL respectively. The numbers of RAT layers L link = 8 for schema linking matrix encoder and L rw = 4 for rewriting matrix encoder respectively. During inference, we set the beam size to 5 for SQL parsing. Models are trained with 8 NVIDIA V100 GPU cards. Experimental Results As shown in Table 3 and 5, we compare the performances of QURG with previous works on the development set of SParC and CoSQL datasets. QURG achieves comparable performance to previous state-of-the-art methods, including SCoRE (Yu et al., 2021b), RAT-SQL-TC and HIE-SQL (Zheng et al., 2022) which effectively promote performances by using task-adaptive pretrained language models. Besides, our QURG outperforms DELTA which directly uses rewritten questions to predict SQL queries. In terms of IM accuracy on the CoSQL, QURG also surpasses PICARD (Scholak et al., 2021b) which is based on the super large pre-trained models T5-3B (Raffel et al., 2020). To further study the advantages of QURG on contextual understanding, as shown in Table 4, we evaluate the performances of the different question turns on SparC and CoSQL, and compare our QURG with previous powerful methods. As the number of turns increases, the difficulty of the task increases because models need to resolve the coreference and omission with longer dependencies. Besides, our QURG can achieve more improvements as the interaction turn increase. Furthermore, we evaluate the performance of QURG on the different difficulty levels of target SQL as shown in the right of Table 4, we observe that our QURG consistently achieves comparable performances to recent state-of-the-art works. Ablation Study As shown in Table 6, we conduct several ablation studies to evaluate the contribution of rewriting matrix integration for our QURG. To explore the effects of the rewriting matrix (−R rw ), we set all the relation types in the rewriting matrix to NONE and keep the model structure unchanged. It degrades the model performances on both datasets by 1.0%-2.9% which confirms that rewriting matrix can effectively improve SQL parsing ability through enhanced context understanding. Then we further remove the whole rewriting matrix (Yu et al., 2019b) and CoSQL dataset (Yu et al., 2019a). encoder (−Enc rw ) to verify the effect of the additional encoder parameters on question u t and context u <t , we observe that the additional parameters slightly degrade the performance on SParC, while slightly improving on CoSQL (−Enc rw → −R rw ), which indicates that the additional parameters have little effect on the improvements of QURG. Moreover, we explore the effects of different approaches to integrating rewritten questions into text-to-SQL tasks, as shown in Table 7: 1) "ONLY" indicates only using rewritten questions u rw t to generate SQL queries, discarding the original question u t and context u <t ; 2) "CONCAT" indicates concatenating original question u t , context u <t with rewrite questions u rw t , treating u rw t as additional information. As shown in Table 7 and Figure 5, ONLY feeding rewritten questions into text-to-SQL models results in a substantial performance drop, since the QR model is trained on out-of-domain data, the rewritten questions may contain a lot of noise. For "CONCAT", question and context are retained to meet the potential noise in rewritten questions, while it is still not ideal and causes performance to degraded against QURG, especially with the increase of turns, the performance of CON-CAT decreases more significantly. Not all questions need to be rewritten, for the questions without coreferences or omissions, CONCAT will introduce redundant information or noise. Compared with the ONLY and CONCAT, QURG is more competitive by leveraging the rewriting matrix, which effectively preserves the rewriting information without introducing redundant information (if the question does not need to be rewritten, the rewriting matrix degrades into nondistinctive bias). Case Study In figure 4 , we offer some cases from the development set of SparC, in which QURG generates SQL queries correctly, while baseline (without rewriting matrix encoder −Enc rw ) fails. In the first case, the "car makers from France" is omitted in the current question, the baseline model resolves the omitted subject "car maker" while ignoring the condition "from France", our QURG utilizes the complete "Insert" relation between u 4 and context u <4 to understand u 4 and generated correct SQL query. In the second case, "results" in current question u 3 refers to "names, country and ages" in context u <3 , "their" refers to "singers" in u 1 . The baseline model failed to build the relation between "results" and "age", and our QURG successfully incorporates this relation to model and products correct SQL query. Conclusions We propose QURG, a novel context-dependent textto-SQL framework that utilizes question rewriting to resolve long-distance dependencies between the current question and historical context. Firstly, QURG adopts a pre-trained sequence generator to produce rewritten questions and further converts them to the rewriting relation matrix between question and context. Secondly, QURG employs a twostream matrix encoder to incorporate the rewriting edit relations into a text-to-SQL model to enhance contextual understanding. Experimental results show that our QURG achieves comparable performance with recent state-of-the-art works on two large-scale context-dependent text-to-SQL datasets SParC and CoSQL. (Hui et al., 2021) 45.7 19.5 DELTA 51.7 21.5 SCoRE (Yu et al., 2021b) 52.1 22.0 PICARD † (Scholak et al., 2021b) 56.9 24.2 HIE-SQL (Zheng et al., 2022) 56.4 28.7 QURG (Ours) 56.6 26.6 A Example of Bi-directional Rewriting Matrix The method to extend the uni-directional rewriting matrix to the bi-directional rewriting matrix is illustrated in Figure 6. When encoding rewriting matrix in two-stream encoder, question u t is concatenated with context u <t . Given question context u <t = {x 1 , x 2 , ..., x 5 } and current question u t = {x 6 , x 7 , x 8 }, we first generate uni-directional rewriting matrix, which contain three type relations: "None", "Substitute", and "Insert". Then we extend relations to bidirectional: "substitute" is extended to "Q-C-SUB" and "C-Q-SUB", "insert" is extended to "Q-C-INS" and "C-Q-INS". We train and evaluate the QR model on the CA-NARD dataset (Elgohary et al., 2019a) which consists of 31K, 3K, and 5K QA pairs for training, development, and test sets, respectively. The semantic-incomplete questions in CANARD are generated by rewriting a subset of the original questions in QuAC (Choi et al., 2018). B.2 Implementation Details We adopt a pre-trained T5-base model to initialize our QR model, the learning rate is 1e-4, the batch size is 256, the number of training epochs is 20 and we use AdamW optimizer with linear warmup scheduler. Model are trained with 8 NVIDIA V100 GPU cards. B.3 Metrics And Results We employ automatic metrics ROUGE to evaluate the QR model, where ROUGE n (R m ) mea-sures the n-gram overlapping between the rewritten questions and the golden ones, ROUGE L (R L ) measures the longest matching sequence between them. As shown in B.4 Rewriting Matrix Evaluation To verify that rewriting edit matrix holds the key rewritten information from rewritten question, we compare the restored rewritten question from rewriting matrix (denoted as RES-RQ) with rewritten question generated by QR model (denoted as GEN-RQ) and evaluate it with ROUGE. As shown in Table 9, we take the GEN-RQ as the target and evaluate the difference between the target and RES-RQ. On both datasets, RES-RQ and GEN-RQ maintain a high level of consistency. This indicates that the rewriting matrix effectively stores the key information of question rewriting. Datasets Train Dev R 1 R 2 R L R 1 R 2 R L SPARC 91.7 81.6 90.0 91.8 82.0 90.3 COSQL 89.0 81.5 87.9 88.4 80.6 87.2 Table 9: Evaluation results of restoring rewritten question from rewriting matrix. Figure 3 : 3Illustration of QURG framework: Left: Question rewriting module which produces rewritten questions u rw t and rewriting matrix R rw . Right: PLM encoder and our proposed two-stream relation matrix encoders. Figure 4 : 4Baseline: SELECT COUNT() FROM car_makers QURG: SELECT COUNT() FROM car_makers JOIN countries WHERE CountryName = "value" Baseline: SELECT Name, Country FROM singer ORDER BY Age DESC QURG: SELECT Name, Country, Age FROM singer ORDER BY Age DESC : Show all the different car makers? : What are the names of the countries they belong to? : Show just the car makers from France ! : How many are there? : Show names of all singers. : For each of them, also show country and age. : List the results in descending order of their age. Case studies on SParC dataset. Baseline denotes the QURG model without rewriting matrix encoder. Figure 5 : 5Detailed results on different question turns for models ONLY, CONCAT and QURG. Figure 6 : 6An example of bi-directional rewriting matrix for two-stream matrix relation encoder. arXiv:2305.06655v2 [cs.CL] 16 May 2023Substitute Insert None w h i c h name all cities that have destination arriving flights o n e h a s t h e m o s t ? Which one has the most ? Question Context Current Question Rewritten Question Which city has the most arriving flights ? Name all cities that have destination airports. </s> Order them by number of arriving flights. Substitute In se rt u t ut u rw u t ut Table / Column) /Pre-trained Language Model t1 x ut 1 _ x ut X u X ctx Hu rw rw Hctx X sc x u1 x x t2 x c1 x c2 x u2 u t 1 _ u 1 u rw Question Rewriter Aggregator Attn Grammar-based Decoder rw L Schema Linking Matrix R Encoder link link L Xu Xctx ; [ ] Xu Xctx Xsc ; ; [ ] Rewritten Question Rewriting Matrix rw R Hu link link Hctx link H SQL sc Question Context Current Question Rewriting Matrix R Encoder rw Substitute Insert None u t u t [history] [query] Table 1 : 1Relation types between question and context. Relation edges exist from source token x i ∈ X utter to target token x j ∈ X utter if the pair meets one of the descriptions listed in the table, where X utter is token set of u t and u <t . Table 2 : 2Detailed statistics for SParC dataset In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers), pages 464-468. 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SParC (→) Turn 1 Turn 2 Turn 3 Turn 4 / Easy Medium Hard Extra Models (↓) # 422 # 422 # 270 # 89 # 483 # 441 # 145 # 134 EditSQL a 62.2 45.1 36.1 19.3 / 68.8 40.6 26.9 12.8 IGSQL b 63.2 50.8 39.0 26.1 / 70.9 45.4 29.0 18.8 R 2 SQL c 67.7 55.3 45.7 33.0 / 75.5 51.5 35.2 21.8 RAT-SQL+TC d 75.4 64.0 54.4 40.9 / - - - - QURG (Ours) 75.4 66.1 53.7 44.3 / 80.1 64.4 43.4 35.1 CoSQL (→) Turn 1 Turn 2 Turn 3 Turn 4 Turn> 4 Easy Medium Hard Extra Models (↓) # 293 # 285 # 244 # 114 # 71 # 417 # 320 # 163 # 107 EditSQL a 50.0 36.7 34.8 43.0 23.9 62.7 29.4 22.8 9.3 IGSQL b 53.1 42.6 39.3 43.0 31.0 66.3 35.6 26.4 10.3 IST-SQL e 56.2 41.0 41.0 41.2 26.8 66.0 36.2 27.8 10.3 SCoRE f 60.8 53.0 47.5 49.1 32.4 - - - - QURG (Ours) 64.5 55.4 55.7 50.0 42.3 77.2 50.0 40.5 20.6 Table 4 : 4Detailed question match accuracy (QM) results in different interaction turns and SQL difficulties on the development set of SParC and CoSQL datasets. Results of a (Zhang et al., 2019), b (Cai and Wan, 2020), c (Wang et al., 2021), d (Li et al., 2021), e (Wang et al., 2021) and f (Yu et al., 2021b) are from the original paper.Models CoSQL QM IM EditSQL (Zhang et al., 2019) 39.9 12.3 GAZP (Zhong et al., 2020) 42.0 12.3 IGSQL (Cai and Wan, 2020) 44.1 15.8 RichContext (Liu et al., 2020a) 41.0 14.0 IST-SQL (Wang et al., 2021) 44.4 14.7 R 2 SQL Table 5 : 5Performances on the development set of CoSQL dataset. Model with † mark is based on the super large T5-3B (Raffel et al., 2020) pre-trained model.Table 6: Ablation studies for the components of QURG. Note that −Enc rw is also the baseline without the integration of rewritten questions u <t .Models SParC CoSQL QM IM QM IM QURG 64.9 46.5 56.6 26.6 −R rw 62.6 43.6 55.6 25.2 −Enc rw 63.4 44.7 55.0 24.5 Models SParC CoSQL QM IM QM IM ONLY 52.4 29.4 45.9 15.0 CONCAT 62.3 41.2 53.0 20.8 QURG 64.9 46.5 56.6 26.6 Table 7 : 7Studies on different approaches to inject rewritten question into context-dependent text-to-SQL. Table 8 , 8our QR model gets better results against all baselines on the CANARD dataset.Models R 1 R 2 R L Copy † 72.7 54.9 68.5 Pronoun Pub † 73.1 63.7 73.9 L-Ptr-GEN † 78.9 62.9 74.9 RUN † 79.1 61.2 74.7 T5-base (Ours) 81.3 70.1 78.4 Table 8 : 8Evaluation results of the QR models on CA-NARD dataset. †: Results are from(Liu et al., 2020b) Models SParCQM IMEditSQL(Zhang et al., 2019)47.2 29.5 GAZP(Zhong et al., 2020)48.9 29.7 IGSQL(Cai and Wan, 2020)50.7 32.5 RichContext (Liu et al., 2020a) 52.6 29.9 IST-SQL 47.6 29.9 R 2 SQL(Hui et al., 2021)54.1 35.2 DELTA 58.6 35.6 SCoRE(Yu et al., 2021b)62.2 42.5 HIE-SQL(Zheng et al., 2022)QURG (Ours)64.9 46.5Table 3: Performances on the development set of SParC dataset. The models with mark employ task adaptive pre-trained language models. Open-domain question answering goes conversational via question rewriting. Raviteja Anantha, Svitlana Vakulenko, Zhucheng Tu, Shayne Longpre, Stephen Pulman, Srinivas Chappidi, ACL. 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[ "Serial Contrastive Knowledge Distillation for Continual Few-shot Relation Extraction", "Serial Contrastive Knowledge Distillation for Continual Few-shot Relation Extraction" ]	[ "Xinyi Wang \nState Key Laboratory for Novel Software Technology\nNanjing University\nChina\n", "Zitao Wang \nState Key Laboratory for Novel Software Technology\nNanjing University\nChina\n", "Wei Hu \nState Key Laboratory for Novel Software Technology\nNanjing University\nChina\n\nNational Institute of Healthcare Data Science\nNanjing University\nChina\n" ]	[ "State Key Laboratory for Novel Software Technology\nNanjing University\nChina", "State Key Laboratory for Novel Software Technology\nNanjing University\nChina", "State Key Laboratory for Novel Software Technology\nNanjing University\nChina", "National Institute of Healthcare Data Science\nNanjing University\nChina" ]	[]	Continual few-shot relation extraction (RE) aims to continuously train a model for new relations with few labeled training data, of which the major challenges are the catastrophic forgetting of old relations and the overfitting caused by data sparsity. In this paper, we propose a new model, namely SCKD, to accomplish the continual few-shot RE task. Specifically, we design serial knowledge distillation to preserve the prior knowledge from previous models and conduct contrastive learning with pseudo samples to keep the representations of samples in different relations sufficiently distinguishable. Our experiments on two benchmark datasets validate the effectiveness of SCKD for continual few-shot RE and its superiority in knowledge transfer and memory utilization over state-of-the-art models.	10.48550/arxiv.2305.06616	[ "https://export.arxiv.org/pdf/2305.06616v1.pdf" ]	258,615,627	2305.06616	ea3162754bb2698e4b2fde0debd0541c333f14ad	Serial Contrastive Knowledge Distillation for Continual Few-shot Relation Extraction Xinyi Wang State Key Laboratory for Novel Software Technology Nanjing University China Zitao Wang State Key Laboratory for Novel Software Technology Nanjing University China Wei Hu State Key Laboratory for Novel Software Technology Nanjing University China National Institute of Healthcare Data Science Nanjing University China Serial Contrastive Knowledge Distillation for Continual Few-shot Relation Extraction Continual few-shot relation extraction (RE) aims to continuously train a model for new relations with few labeled training data, of which the major challenges are the catastrophic forgetting of old relations and the overfitting caused by data sparsity. In this paper, we propose a new model, namely SCKD, to accomplish the continual few-shot RE task. Specifically, we design serial knowledge distillation to preserve the prior knowledge from previous models and conduct contrastive learning with pseudo samples to keep the representations of samples in different relations sufficiently distinguishable. Our experiments on two benchmark datasets validate the effectiveness of SCKD for continual few-shot RE and its superiority in knowledge transfer and memory utilization over state-of-the-art models. Introduction Relation extraction (RE) aims to recognize the semantic relations between entities in texts, which is widely applied in many downstream tasks such as language understanding and knowledge graph construction. Conventional studies (Zeng et al., 2014;Heist and Paulheim, 2017;Zhang et al., 2018) mainly assume a fixed pre-defined relation set and train on a fixed dataset. However, they cannot work well with the new relations that continue emerging in some real-world scenarios of RE. Continual RE (Wang et al., 2019;Han et al., 2020;Wu et al., 2021) was proposed as a new paradigm to solve this situation, which applies the idea of continual learning (Parisi et al., 2019) to the field of RE. Compared with conventional RE, continual RE is more challenging. It requires the model to learn emerging relations while maintaining a stable and accurate classification of old relations, i.e., the socalled catastrophic forgetting problem (Thrun and Mitchell, 1995;French, 1999), which refers to the * Corresponding author severe loss of prior knowledge during the model is learning new tasks. Recent continual learning works leverage the regularization-based models, the architecture-based models, and the memorybased models to alleviate catastrophic forgetting. Several studies (Wang et al., 2019; have shown that the memory-based models are more promising for NLP tasks, and a number of memory-based continual RE models (Cui et al., 2021;Zhao et al., 2022;Hu et al., 2022;Zhang et al., 2022) have made significant progress. In real life, the shortage of labeled training data for relations is an unavoidable problem, especially severe in emerging relations. Therefore, the continual few-shot RE paradigm (Qin and Joty, 2022) was proposed to simulate real human learning scenarios, where new knowledge can be acquired from a small number of new samples. As illustrated in Figure 1, the continual few-shot RE paradigm expects the model to continuously learn new relations through abundant training data only for the first task, but through sparse training data for all subsequent tasks. Thus, the model needs to identify the growing relations well with few labeled data for them while retaining the knowledge on old relations without re-training from scratch. As relations grow, the confusion about relation representations leads to catastrophic forgetting. In continual fewshot RE, catastrophic forgetting becomes more severe since the few samples of new relations may not be representative for these relations. The possibility of confusion between relation representations greatly increases. Since the emerging relations are few-shot, the problem of overfitting becomes another key challenge in the continual few-shot RE task. The overfitting for samples in few-shot tasks aggravates the model's forgetting of prior knowledge as well. Existing few-shot learning works (Fan et al., 2019;Gao et al., 2019a;Obamuyide and Vlachos, 2019;Geng et al., 2020) are worthy of reference by continual few-shot RE models to ensure good generalization. Inspired by knowledge distillation (Hinton et al., 2015) to transfer knowledge well and contrastive learning (Wu et al., 2018) to constrain representations explicitly, we propose SCKD, a model built with serial contrastive knowledge distillation for continual few-shot RE. Through it, we tackle the aforementioned two key challenges. First, how to alleviate the problem of catastrophic forgetting? SCKD follows the memory-based methods for continual learning and preserves a few typical samples from previous tasks. Furthermore, we present serial knowledge distillation to preserve the prior knowledge from previous models and conduct contrastive learning to keep the representations of samples in different relations sufficiently distinguishable. Second, how to mitigate the negative impact of overfitting caused by sparse samples? We leverage bidirectional data augmentation between memory and current tasks to obtain more samples for few-shot relations. The pseudo samples generated in serial contrastive knowledge distillation can help prevent overfitting as well. In summary, our main contributions are twofold: • We propose SCKD, a novel model built with serial contrastive knowledge distillation for resolving the continual few-shot RE task. With the proposed serial knowledge distillation and contrastive learning with pseudo samples, our SCKD can take full advantage of memory and effectively alleviate the problems of catastrophic forgetting and overfitting under considerably few memorized samples. • We perform extensive experiments on two benchmark datasets FewRel (Han et al., 2018) and TACRED (Zhang et al., 2017). The results demonstrate the superiority of SCKD over the state-of-the-art continual (few-shot) RE models. Furthermore, the proposed data augmentation, serial knowledge distillation, and contrastive learning all contribute to performance improvement. Related Work In this section, we review related work on continual RE and few-shot RE. Continual RE. The goal of continual learning is to accomplish new tasks sequentially without catastrophically forgetting the acquired knowledge from previous tasks. For continual RE, RP-CRE (Cui et al., 2021) refines sample embeddings for prediction with the generated relation prototypes from memory. However, its relation prototype calculation is sensitive to typical samples. CRL (Zhao et al., 2022) introduces supervised contrastive learning and knowledge distillation to generate sample representations when replaying memory. It narrows the representations of samples belonging to the same relation through supervised contrastive learning but fails to keep the representations of samples in different relations far away to avoid confusion. Besides, knowledge distillation between prototypes calculated by averaging sample representations may lose some features of specific samples. CRECL (Hu et al., 2022) contrasts a given sample with all the candidate relation prototypes stored in memory by a contrastive network. It faces the same problem as RP-CRE on typical samples for computing relation prototypes. Conducting contrastive learning only with relation prototypes may not guarantee the differences between sample representations belonging to different relations. KIP-Framework (Zhang et al., 2022) generates knowledge-infused relation prototypes to leverage the relational knowledge from pre-trained language models with prompt tuning. Compared with other models, KIP-Framework needs extra knowledge such as relation descriptions, and its overall procedure is more time-consuming. All these works rely on plenty of training data for learning new relations and large memory for retaining prior knowledge. In contrast, our model only needs a few training samples to learn new relations well through bidirectional data augmentation and the generated pseudo samples from relation prototypes. Furthermore, our model can avoid catastrophic forgetting under limited memory through serial knowledge distillation and contrastive learning. As far as we know, ERDA (Qin and Joty, 2022) is the only work addressing continual few-shot RE. It imposes relational constraints in the embedding space and generates new training data from unlabeled text. However, our model does not need to import extra data like ERDA. Instead, it generates pseudo samples from relation prototypes and augments training data by modifying original samples to alleviate the overfitting problem. Few-shot RE. Few-shot learning aims to leverage only a few novel samples to adapt the model for solving tasks. For few-shot RE, its goal is to enable the model to quickly learn the characteristics of relations with very few samples, so as to accurately classify these relations. At present, there are two main lines of work: (1) The metric learning methods (Fan et al., 2019;Gao et al., 2019a) use various metric functions (e.g., the Euclidean or Cosine distance) learned from prior knowledge to map the input into a subspace so that they can distinguish similar and dissimilar sample pairs easily to assign the relation labels. (2) The meta-learning methods (Obamuyide and Vlachos, 2019;Geng et al., 2020) learn general relation classification experience from the meta-training stage and leverage the experience to quickly converge on specific relation extraction during the meta-testing stage. In this paper, our problem setting is different from the above few-shot RE works, as we expect the model to continuously learn new few-shot relations instead of conducting the few-shot relation learning just once. Furthermore, these few-shot RE works do not have the capacity for continual learning. Methodology Task Definition The objective of RE is to identify the relations between entity mentions in sentences. Continual RE aims to accomplish a sequence of J RE tasks {T 1 , T 2 , . . . , T J }, where each task T j has its own dataset D j and relation set R j . The relation sets of different tasks are disjoint. Once finishing T j , D j is no longer available for future learning, and the model is assessed on all previous tasks {T 1 , . . . , T j } for identifyingR j = j i=1 R i . Also, the trained model serves as the base model for the subsequent task T j+1 . In real-world scenarios, labeled training data for new tasks are often limited. Therefore, we define the continual few-shot RE task in this paper, where only the first task T 1 has abundant data for model training and the subsequent tasks are all few-shot. Let N be the relation number of each few-shot task and K be the sample number of each relation, the task can be called N -way-K-shot. A continual fewshot RE model is expected to perform well on all historical few-shot and non-few-shot tasks. Our Framework Algorithm 1 shows the end-to-end training for task T j , with the model Φ j−1 previously trained. Following the memory-based methods for continual learning (Lopez-Paz and Ranzato, 2017;Chaudhry et al., 2019), we use a memoryM j−1 to preserve a few samples in all previous tasks {T 1 , . . . , T j−1 }. 1. Initialization (Line 1). The current model Φ j inherits the parameters of Φ j−1 , except for Φ 1 randomly initialized. We adapt Φ j on D j to learn the knowledge of new relations in T k . 2. Prototype generation (Lines 2-6). Inspired by (Han et al., 2020;Cui et al., 2021), we apply the k-means algorithm to select L typical samples from D j for every relation r ∈ R j , which constitute a memory M r . The memory for the current task is M j = r∈R j M r , and the overall memory for all observed relations until now isM j =M j−1 ∪ M j . Then, we generate a prototype p r for each r ∈R j . 3. Data augmentation (Line 7). To cope with the scarcity of samples, we conduct bidirectional data augmentation between D j andM j . By measuring the similarity between entities in samples, we generate an augmented dataset D * j and an augmented memoryM * j by mutual replacement between similar entities. Serial Contrastive Knowledge Distillation (Lines 8-10). We construct a set of pseudo samples based on the prototype set. Then, we carry out serial contrastive knowledge distillation with the pseudo samples on D * j and oñ M * j , respectively, making the sample representations in different relations distinguishable and preserve the prior knowledge for identifying the relations in previous tasks well. We detail the procedure in the subsections below. Initialization for New Task To adapt the model for the new task T j , we perform a simple multi-classification task on dataset D j . Specifically, for a sample x in T j , we use special tokens [E 1 ] and [E 2 ] to denote the start positions of Algorithm 1: Training procedure for T j Input: Φ j−1 ,R j−1 ,M j−1 , D j , R j Output: Φ j ,M j 1 initialize Φ j from Φ j−1 , and adapt it on D j ; 2M j ←M j−1 ; 3 foreach r ∈ R j do 4 pick L samples in D j , and add intoM j ; 5R j ←R j−1 ∪ R j ; 6 generate prototype setP j based onM j ; 7 generate augmented dataset D * j and memorỹ M * j by mutual replacement; 8 generate pseudo sample setS j based onP j ; 9 update Φ j by serial contrast. knowl. distill. on D * j ,S j ; // re-train current task 10 update Φ j by serial contrast. knowl. distill. onM * j ,S j ; // memory replay two entities in x, respectively. Then, we obtain the representations of special tokens using the BERT encoder (Devlin et al., 2019). Next, the feature of sample x, denoted by f x , is defined as the concatenation of token representations of [ E 1 ] and [E 2 ]. We obtain the hidden representation h x of x as h x = LN(W Dropout(f x ) + b),(1) where W ∈ R d×2h and b ∈ R d are two trainable parameters. d is the dimension of hidden layers. h is the dimension of BERT hidden representations. LN(·) is the layer normalization operation. Finally, based on h x , we use the linear softmax classifier to predict the relation label. The classification loss, L csf , is defined as L csf = − 1 \|D j \| x∈D j \|R j \| r=1 y x,r · log P x,r ,(2) where y x,r ∈ {0, 1} indicates whether x's true label is r. P x,r denotes the r-th entry in x's probability distribution calculated by the classifier. Prototype Generation After the initial adaption above, we pick L typical samples for each relation r ∈ R j to form memory M r . We leverage the k-means algorithm upon the hidden representations of r's samples, where the number of clusters equals the number of samples that need to be stored for representing r. Then, in each cluster, the sample closest to the centroid is chosen as one typical sample. To obtain the prototype p r for r, we average the hidden representations of L typical samples in M r : p r = 1 L x∈Mr h x .(3) The prototype setP j stores the prototypes of all relations inR j , i.e.,P j = ∪ r∈R j {p r }. Bidirectional Data Augmentation For a sample x in D j orM j , the token representations of [E 1 ] and [E 2 ] generated by BERT are used as the representations of corresponding entities. We obtain the entity representations from all samples and calculate the cosine similarity between the representations of any two different entities. Once the similarity exceeds a threshold τ , we replace each of the two entities in the original sample with the other entity. Our intuition is that one certain entity in a sentence is replaced by its close entity with everything else unchanged, the relation represented by the sentence is unlikely to change much. For example, "The route crosses the Minnesota River at the Cedar Avenue Bridge." and "The route crosses the River MNR at the Cedar Avenue Bridge." have the same relation "crosses". We assign the same relation label to the new samples as their original samples and store them together as the augmented dataset D * j and the augmented memoryM * j . Serial Contrastive Knowledge Distillation Knowledge distillation (Hinton et al., 2015;Cao et al., 2020) has demonstrated its effectiveness in transferring knowledge. In this paper, we propose a serial contrastive knowledge distillation method to leverage the knowledge from the previous RE model to guide the training of the current model. The procedure of serial contrastive knowledge distillation is depicted in Figure 2. We detail it below. Feature distillation. In this step, we expect the encoder of the current model to extract similar features with the previous model. For a sample x, let f j−1 x and f j x be x's features extracted by the previous model Φ j−1 and the current model Φ j , respectively. We propose a feature distillation loss to enforce the extracted features unbiased towards new relations: L fd = 1 \|M * j \| x∈M * j 1 − (f j−1 x ) f j x . (4) Prob. Prob. Pseudo sample set Classifier BERT Dropout Current model Previous model Φ ! Φ !"# ! " Dataset & memory " " * " * % " $%& ' ! "'( ! "'( ! "'( ! " BERT Dropout Classifier Pseudo samples generation. We attempt to construct pseudo samples for all the observed relations, which are used in the next hidden contrastive distillation step. Specifically, we assume the sample representations of relations follow the Gaussian distribution with the corresponding prototypes as their average values. The construction of pseudo samples is based on prototype setP j , and one pseudo sample for r can be constructed as follows: s r = p r + η · δ r ,(5) where η ∼ N (0, 1) is a standard Gaussian noise, and δ r is the root of the diagonal covariance based on the hidden representations of all r's samples when r first appears in the relation set of one task. The diagonal covariance consists of the variance in each dimension, which can describe the differences in each dimension of the sample representations belonging to that relation. We multiply the Gaussian noise with the root of the diagonal covariance and add the result to the prototype representation for generating pseudo samples. In this way, the generated samples can more closely match the real samples of the relation rather than random. We repeat the above operation n times for each relation in {T 1 , . . . , T j } and store the constructed pseudo samples in the pseudo sample setS j . Hidden contrastive distillation. In this step, we expect the current model to obtain similar hidden representations with the previous model. We also want to keep the hidden representations of samples in different relations distinguishable. First, we consider the distillation between sample hidden representations. We feed a sample x's feature f j x into the dropout layers of the previous model Φ j−1 and the current model Φ j to obtain the hidden representations, which are denoted by h j−1 x and h j x , respectively. Then, we formulate the representation distillation loss as follows: L rd = 1 \|M * j \| x∈M * j 1 − (h j−1 x ) h j x . (6) Moreover, based on the previously-constructed pseudo samples and the real samples from the training data, we conduct contrastive learning to make the hidden representations of samples for different relations as distinct as possible, which can enhance the knowledge distillation. To achieve this, we mine hard positives and hard negatives with previous representations while contrasting them with current representations, which can ensure that the current model can obtain similar representations as the previous model. We put forward a distillation triplet loss function: L dtr = 1 \|M * j \| x∈M * j max 0, \|\|h j x − z + max \|\| 2 − \|\|h j x − z − min \|\| 2 ,(7) where z + max and z − min are selected through h j−1 x . z + max is the representation farthest from h j−1 x in all sample representations that belong to the same relation with x, and z − min is the representation nearest from h j−1 x in all sample representations that belong to the different relations with x. Overall, the loss function for hidden contrastive distillation is defined as We propose a prediction distillation loss function: L hcd = L rd + L dtr .(8)L pd = − 1 \|M * j \| x∈M * j \|R j−1 \| r=1 c j−1 x,r log c j x,r ,(9)c j−1 x,r = exp o j−1 x,r T \|R j−1 \| l=1 exp o j−1 x,l T , c j x,r = exp o j x,r T \|R j−1 \| l=1 exp o j x,l T ,(10) where T is the temperature scalar. This prediction distillation loss encourages the predictions of the current model on previous relations to match the soft labels by the previous model. The total distillation loss consists of the above three losses: L dst = α · L fd + β · L hcd + γ · L pd ,(11) where α, β and γ are adjustment coefficients. We optimize the classification loss and distillation loss with multi-task learning. Therefore, the final loss is L = λ 1 · L csf + λ 2 · L dst ,(12) where λ 1 and λ 2 are also adjustment coefficients. Experiments In this section, we assess the proposed SCKD and report our results. The datasets and source code for SCKD are accessible from GitHub. 1 Experiment Setup Datasets. Our experiments are conducted on the following two benchmark RE datasets: • FewRel (Han et al., 2018) is a popular dataset for few-shot RE containing 100 relations and 70,000 samples in total. Following (Qin and Joty, 2022), we adopt the version of 80 relations and split them into 8 tasks, where each task contains 10 relations (10-way). The first task T 1 has 100 samples per relation while the subsequent tasks T 2 , . . . , T 8 are all few-shot. We conduct 5-shot and 10-shot experiments. • TACRED (Zhang et al., 2017) is a large-scale RE dataset with 42 relations and 106,264 samples from Newswire and Web documents. Following (Qin and Joty, 2022), we filter out 1 https://github.com/nju-websoft/SCKD "no_relation" and divide the remaining 41 relations into 8 tasks. The first task T 1 has 6 relations and 100 samples per relation. All the other tasks have 5 relations (5-way), and we carry out 5-shot and 10-shot experiments. Evaluation metrics. We measure average accuracy in our experiments. At task T j , it can be calculated as ACC j = 1 j j i=1 ACC j,i , where ACC j,i denotes the accuracy (i.e., the number of correctlylabeled samples divided by all samples) on the test set of task T i after training the model on task T j . We repeat the experiments six times using random seeds, and report means and standard deviations. Competing models. We compare SCKD against two baselines: The finetuning model trains the RE model only with the training data of the current task while inheriting the parameters of the model trained on the previous task. It serves as the lower bound. The joint-training model stores all samples from previous tasks in memory and uses all the memorized data to train the re-initialized model for the current task. It can be regarded as the upper bound. We also compare SCKD with four recent opensource models for continual RE: RP-CRE (Cui et al., 2021), CRL (Zhao et al., 2022), CRECL (Hu et al., 2022), and ERDA (Qin and Joty, 2022). Since RP-CRE, CRL, and CRECL do not investigate the few-shot scenario while ERDA reported its results under the "loose" evaluation which picks no more than 10 negative labels from the observed labels, we re-run these models using their source code and report the new results. KIP-Framework (Zhang et al., 2022) has not released its source code, thus we cannot re-run it for comparison. Implementation details. We develop our SCKD based on PyTorch 1.7.1 and Huggingface's Transformers 2.11.0 (Wolf et al., 2020). See Appendix A for the selected hyperparameter values. For a fair comparison, we set the random seeds of the experiments identical to those in (Qin and Joty, 2022), so that the task sequence is exactly the same. We employ the "strict" evaluation method proposed in (Cui et al., 2021), which chooses the whole observed relation labels as negative labels for evaluation. We stipulate that the memory can only store one sample for each relation (L = 1) when running all models. Table 1 lists the result comparison on the 10-way-5-shot setting on the FewRel dataset and the 5-way-5-shot setting on the TACRED dataset. We have the following findings: (1) Our proposed SCKD performs significantly better than the competing models on all tasks. After learning all tasks, SCKD outperforms the second-best model CRECL by 2.99% and 6.09% on FewRel and TACRED, respectively. Results and Analyses Main Results (2) Regarding the two baselines, the finetuning model leads to rapid drops in average accuracy due to severe overfitting and catastrophic forgetting. The joint-training model may not always be the upper bound (e.g., T 2 to T 5 on FewRel) due to the extremely imbalanced data distribution. Besides, after learning the final task of FewRel, SCKD can achieve close results to the joint-training model with considerably few memorized samples. (3) ERDA performs worst among the four com-peting models. This is because the extra training data from the unlabeled Wikipedia corpus for data augmentation may contain errors and noise, which makes the model unable to fit the emerging relations well. (4) RP-CRE, CRL, and CRECL can effectively acquire knowledge from new relations without catastrophic forgetting of prior knowledge. However, their performance is all affected by the limited memory size, since they all need more memorized samples for each relation to generate more representative relation prototypes. See Appendix B.3 for the 10-way-10-shot results on FewRel and 5-way-10-shot results on TACRED. Ablation Study We conduct an ablation study to validate the effectiveness of each module. Specifically, for "w/o distillation", we disable the serial contrastive knowledge distillation module. For "w/o augmentation", we use the original (not augmented) dataset and memory. For "w/o both", we update the model via the simple re-training on memory. From Table 2, we obtain several findings: (1) The average accuracy at each task reduces when we disable any modules, showing their usefulness. (2) If we remove the serial contrastive knowledge distillation module, the results drop drastically, which shows that knowledge distillation and contrastive learning can alleviate catastrophic forgetting and overfitting. Furthermore, we conduct a fine-grained ablation study to investigate serial contrastive knowledge distillation. We disable L fd , L rd , L dtr , L pd in the FewRel Table 3: Fine-grained ablation study on serial contrastive knowledge distillation. model update, to assess their influence. Table 3 shows the results, and we have several findings: (1) The results decline if we remove any losses, which demonstrates that each loss contributes to the overall performance. T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8SCKDFewRel T * 2 T * 3 T * 4 T * 5 T * 6 T * 7 T * 8SCKD (2) The drops caused by disabling the distillation triplet loss L dtr are most obvious since SCKD cannot keep the hidden representations of samples in different relations sufficiently distinguishable without contrastive learning. Comparison with Few-shot RE Models We compare SCKD with classic few-shot RE models provided in (Gao et al., 2019b). For a fair comparison, the few-shot RE models treat the training and test sets of previous tasks as the support and query sets for training, respectively. The training set of the current task serves as the support set for testing. We test our model and the few-shot models using the accuracy on the test set of current task. Table 4 presents the results, and we observe that SCKD is always superior to GNN (CNN), Proto (CNN), Proto (BERT), and BERT-PAIR, as it conducts contrastive learning with pseudo samples on the few-shot tasks, which maximizes the distance between the representations of different relations. Knowledge Transfer Capability Backward transfer (BWT) measures how well the continual learning model can handle catastrophic forgetting. The BWT of accuracy after finishing all tasks is defined as follows: Figure 3 shows the BWT of SCKD and the competing models. Due to the overwriting of learned knowledge, BWT is always negative. The performance drops of SCKD are the lowest, showing its effectiveness in alleviating catastrophic forgetting. See Appendix B.1 for the 10-shot results on FewRel and TACRED. BWT = 1 J − 1 J−1 i=1 ACC J,i − ACC i,i . (13) Sample Representation Discrimination To investigate the effects on discriminating sample representations, we use t-SNE (van der Maaten and Hinton, 2008) to visualize the sample representations of six selected relations after the training of CRECL and SCKD. From Figure 4, we see that, compared to CRECL, SCKD can make the representations of samples in different relations more distinguishable. For example, the two relations, "spouse" and "follows", with close sample representations in CRECL can be clearly separated by SCKD, which shows that SCKD has a better ability to maintain the differences between relations. L = 2 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 Influence of Memory Size For the memory-based continual RE models, memory size has an important impact on performance. Due to the limited samples in the few-shot scenario, the models only store one sample per relation (L = 1) in the previous experiments. In this experiment, we conduct experiments on the 10-way-10-shot setting of FewRel with different memory sizes (L = 2, 3). We choose this setting because it ensures that the memorized data only occupy a small fraction of all samples. The comparison results are shown in Table 5, and we can see that: (1) With memory size growing, all the models perform better, confirming that memory size is a key factor that affects continual learning. (2) SCKD maintains the best performance with different memory sizes, which demonstrates the effectiveness of SCKD in leveraging the memory for continual few-shot RE. See Appendix B.2 for the results on TACRED. Conclusion In this paper, we propose SCKD for continual fewshot RE. To alleviate the problems of catastrophic forgetting and overfitting, we design the serial contrastive knowledge distillation, making prior knowledge from previous models sufficiently preserved while the representations of samples in different relations remain distinguishable. Our experiments on FewRel and TACRED validate the effectiveness of SCKD for continual few-shot RE and its superiority in knowledge transfer and memory utilization. For future work, we plan to investigate how to apply the serial contrastive knowledge distillation to other classification-based continual few-shot learning tasks. Limitations The work presented here has a few limitations: (1) The proposed model belongs to the memory-based methods for continual learning, which requires a memory that costs extra storage. In some extremely storage-sensitive cases, there may be restrictions on the usage of our model. (2) The proposed model has currently been evaluated under the RE setting. It is better to transfer it to other continual few-shot learning settings (e.g., event detection and even image classification) for a comprehensive study. We run all the experiments on an X86 server with two Intel Xeon Gold 6326 CPUs, 512 GB memory, four NVIDIA RTX A6000 GPU cards, and Ubuntu 20.04 LTS. The training procedure is optimized with Adam. Following the convention, we conduct a grid search to choose the hyperparameter values. Specifically, the search space of important hyperparameters is as follows: For all the competing models ERDA (Qin and Joty, 2022), RP-CRE (Cui et al., 2021), CRL (Zhao et al., 2022) and CRECL (Hu et al., 2022), we just assign the same memory size as ours, and retain other hyperparameter settings reported in their original papers. Figure 5 presents the 10-way-10-shot BWT results on FewRel and the 5-way-10-shot BWT results on TACRED. From this figure as well as Figure 3 in the main text, we can observe that: (1) SCKD achieves the best BWT scores again under this different shot setting. (2) Compare with the competing models, the performance of SCKD declines lowest, which shows that SCKD alleviates catastrophic forgetting effectively. (a) FewRel (10-way-10-shot) Backward transfer Figure 5: Results of BWT on FewRel (10-way-10-shot) and TACRED (5-way-10-shot). B More Experimental Results B.1 Knowledge Transfer Capability B.2 Influence of Memory Size To enrich the experimental results on the influence of memory size, we also conduct an experiment on TACRED with different memory sizes and show the results in Table 7. Based on these results and the results listed in Table 5 of the main text, we can find that: SCKD maintains the best performance with different memory sizes not only on FewRel but also on TACRED. This demonstrates that our model is effective and versatile in making good use of memory. Table 8 shows the 10-way-10-shot results on the FewRel dataset and the 5-way-10-shot results on the TACRED dataset. Based on these results and the experimental results on memory size listed in Table 5 of the main text, we have the following findings: (1) Compared with the competing models, our model still performs best. It gains a significant Table 8: Result comparison on FewRel (10-way-10-shot) and TACRED (5-way-10-shot). Means ± stds are reported. L = 2 T 1 T 2 T 3 T 4 T 5 T 6 T 7T B.3 Results with Different Shots accuracy improvement over the second-best model by 3.65% on FewRel and 5.79% on TACRED at last. (2) Our model achieves a close performance with L = 1 (62.98% on FewRel and 52.11% on TACRED) to the competing models with L = 2. This demonstrates that our model can make better use of memory. Figure 2 : 2Procedure of serial contrastive knowledge distillation. Prediction distillation. In this step, we expect the classifier of the current model to predict similar probability distributions with the classifier of the previous model on the previous relation set.For a sample x's hidden representation h j x , the output logits of the previous model are o j−1 x = o j−1 x,1 , . . . , o j−1 x,\|R j−1 \| while the logits of the current model are o j x = o j x,1 , . . . , o j x,\|R j−1 \| , . . . , o j x,\|R j \| . Figure 3 :Figure 4 34Results of BWT on FewRel and TACRED. movement spouse located on terrain feature main subject follows sport : t-SNE plot of sample representations belonging to six relations from FewRel (10-way-5-shot) Continually learn new relations. . .Data for movement Data for crosses Data for performer Data for sport Data for follows Data for father Task 1 Task 2 Task i Learn relations: crosses, movement Learn relations: sport, performer Learn relations: father, follows . . . . . . . . . Figure 1: Continual few-shot RE paradigm ±2.18 28.19 ±1.51 22.46 ±0.64 17.89 ±0.92 14.39 ±0.91 12.61 ±0.65 10.68 ±0.64 Joint-train 94.87 ±0.27 80.83 ±3.79 74.41 ±2.32 71.73 ±0.85 70.12 ±2.55 67.37 ±1.62 65.67 ±1.75 64.48 ±0.45 Joint-train 87.93 ±0.68 78.02 ±1.51 72.84 ±1.38 68.23 ±5.21 63.42 ±4.98 62.01 ±3.89 59.62 ±2.33 57.63 ±1.41FewRel T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 Finetune 94.32 ±0.21 43.54 RP-CRE 93.97 ±0.64 76.05 ±2.36 71.36 ±2.83 69.32 ±3.98 64.95 ±3.09 61.99 ±2.09 60.59 ±1.87 59.57 ±1.13 CRL 94.68 ±0.33 80.73 ±2.91 73.82 ±2.77 70.26 ±3.18 66.62 ±2.74 63.28 ±2.49 60.96 ±2.63 59.27 ±1.32 CRECL 93.93 ±0.22 82.55 ±6.95 74.13 ±3.59 69.33 ±3.87 66.51 ±4.05 64.60 ±1.92 62.97 ±1.46 59.99 ±0.65 ERDA 92.43 ±0.32 64.52 ±2.11 50.31 ±3.32 44.92 ±3.77 39.75 ±3.34 36.36 ±3.12 34.34 ±1.83 31.96 ±1.91 SCKD 94.77 ±0.35 82.83 ±2.61 76.21 ±1.61 72.19 ±1.33 70.61 ±2.24 67.15 ±1.96 64.86 ±1.35 62.98 ±0.88 TACRED T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 Finetune 87.97 ±0.53 25.81 ±4.57 19.65 ±4.75 18.38 ±1.25 15.68 ±1.31 11.88 ±2.61 10.77 ±2.49 9.69 ±2.26 RP-CRE 87.32 ±1.76 74.90 ±6.13 67.88 ±4.31 60.02 ±5.37 53.26 ±4.67 50.72 ±7.62 46.21 ±5.29 44.48 ±3.74 CRL 88.32 ±1.26 76.30 ±7.48 69.76 ±5.89 61.93 ±2.55 54.68 ±3.12 50.92 ±4.45 47.00 ±3.78 44.27 ±2.51 CRECL 87.09 ±2.50 78.09 ±5.74 61.93 ±4.89 55.60 ±5.78 53.42 ±2.99 51.91 ±2.95 47.55 ±3.38 45.53 ±1.96 ERDA 81.88 ±1.97 53.68 ±6.31 40.36 ±3.35 36.17 ±3.65 30.14 ±3.96 22.61 ±3.13 22.29 ±1.32 19.42 ±2.31 SCKD 88.42 ±0.83 79.35 ±4.13 70.61 ±3.16 66.78 ±4.29 60.47 ±3.05 58.05 ±3.84 54.41 ±3.47 52.11 ±3.15 Table 1 : 1/o dst. 94.67 82.47 74.13 68.59 66.31 63.43 61.36 58.96 w/o aug. 94.77 82.56 75.78 71.75 70.37 66.87 64.39 62.51 w/o both 94.63 82.39 73.96 68.14 65.97 62.92 60.62 58.41Result comparison on FewRel (10-way-5-shot) and TACRED (5-way-5-shot). Means ± stds are reported. FewRel T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 SCKD 94.77 82.83 76.21 72.19 70.61 67.15 64.86 62.98 wTACRED T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 SCKD 88.42 79.35 70.61 66.78 60.47 58.05 54.41 52.11 w/o dst. 88.38 77.12 66.95 61.64 56.25 53.39 48.09 46.52 w/o aug. 88.35 79.16 70.08 66.32 60.15 57.73 54.04 51.79 w/o both 88.12 76.48 65.45 60.99 55.79 52.46 47.31 45.79 Table 2 : 2Ablation study on modules. Table 4 : 4Result comparison with few-shot RE models. Table 5 : 5Results w.r.t. memory size on FewRel (10- way-10-shot). 1 . 1The search range for the dropout ratio is [0.2, 0.6] with a step size of 0.1. 2. The search range for the threshold τ is [0.80, 0.99] with a step size of 0.01. 3. The search range for the number of pseudo samples per relation is [5, 20] with a step size of 5. 4. The search range for α, β, γ and λ 1 , λ 2 is [0.1, 1] with a step size of 0.1. The selection is illustrated in Table 6. Dim. of BERT representations 768 Dim. of hidden representations 768 Threshold τ for augmentation 0.95 No. of pseudo samples per relationHyperparameters Values Batch size 16 Dropout ratio 0.5 Gradient accumulation steps 4 Learning rate for the encoder 0.00001 Learning rate for the dropout layer 0.00001 Learning rate for the classifier 0.001 10 Temperature scalar 0.08 α, β, γ 0.5, 1.0, 0.5 λ 1 , λ 2 1, 1 Table 6 : 6Hyperparameter setting in our model. 8 RP-CRE 86.42 78.69 70.41 62.73 58.37 55.79 52.55 50.43 CRL 86.71 77.48 68.02 61.65 59.18 56.55 53.45 52.18 CRECL 84.58 74.83 66.80 57.57 56.58 55.26 52.26 52.01 ERDA 79.69 54.06 40.40 34.41 33.34 29.47 28.43 26.51 SCKD 88.27 79.07 71.11 64.88 62.14 58.91 56.41 54.84L = 3 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 RP-CRE 87.19 78.98 70.57 63.25 60.68 57.24 55.78 51.89 CRL 87.01 79.35 69.94 62.96 61.01 58.72 56.61 53.76 CRECL 86.06 76.93 68.39 62.83 60.11 59.78 56.81 55.96 ERDA 80.75 55.13 44.63 37.29 34.53 32.37 31.13 29.20 SCKD 88.59 80.47 74.26 66.56 64.85 61.78 59.34 56.74 Table 7 : 7Results w.r.t. memory size on TACRED (5way-10-shot). Finetune 95.67 ±0.28 46.64 ±2.22 29.68 ±1.98 22.41 ±1.48 18.47 ±0.58 14.84 ±0.99 13.02 ±0.59 11.23 ±0.72 Joint-train 95.82 ±0.37 87.17 ±5.11 80.73 ±5.95 77.75 ±5.33 76.77 ±3.74 74.26 ±2.14 72.96 ±1.81 71.57 ±0.39 RP-CRE 95.19 ±0.21 79.21 ±6.35 74.72 ±4.18 71.39 ±5.11 67.62 ±3.83 64.43 ±2.72 63.08 ±2.59 61.46 ±1.19 CRL 95.01 ±0.31 82.08 ±6.91 79.52 ±4.85 75.48 ±4.91 69.41 ±3.05 66.49 ±2.23 64.86 ±1.45 62.95 ±0.59 CRECL 95.63 ±0.28 83.81 ±3.69 78.06 ±5.91 71.28 ±4.54 68.32 ±3.52 66.76 ±3.84 64.95 ±1.40 63.01 ±1.62 ERDA 92.68 ±0.57 66.59 ±8.29 56.33 ±6.23 48.62 ±5.96 40.51 ±2.22 37.21 ±2.25 36.39 ±3.17 33.51 ±1.47 SCKD 95.45 ±0.34 86.64 ±4.72 80.06 ±6.73 76.02 ±5.96 73.82 ±4.33 70.57 ±3.22 68.34 ±2.34 66.66 ±0.75 Finetune 85.84 ±1.95 25.63 ±3.75 21.49 ±4.63 17.45 ±2.05 14.32 ±1.95 13.14 ±3.01 11.34 ±2.59 9.21 ±1.59 Joint-train 86.56 ±1.12 80.14 ±2.17 74.67 ±2.86 70.31 ±2.79 70.04 ±2.96 67.31 ±2.19 65.42 ±2.03 61.59 ±1.19 RP-CRE 86.68 ±1.72 78.43 ±4.25 69.43 ±6.22 60.71 ±4.34 55.84 ±5.28 51.17 ±4.24 47.27 ±3.49 47.16 ±1.88 CRL 87.81 ±0.39 77.68 ±7.89 63.31 ±7.77 56.51 ±2.82 53.21 ±2.01 52.42 ±4.02 48.54 ±4.19 46.46 ±3.73 CRECL 83.88 ±1.68 73.45 ±2.85 59.24 ±5.55 53.51 ±5.04 49.27 ±3.24 47.41 ±2.85 45.15 ±3.61 44.33 ±2.48 ERDA 79.37 ±0.95 51.28 ±5.67 36.97 ±4.95 29.39 ±5.07 27.80 ±4.23 25.18 ±3.29 24.47 ±1.22 22.37 ±3.92 SCKD 88.84 ±1.51 78.64 ±5.03 70.08 ±3.17 64.27 ±2.99 61.73 ±2.82 58.19 ±3.95 55.91 ±2.79 52.95 ±3.14FewRel T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 TACRED T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 AcknowledgmentsThis work was supported by the National Natural Science Foundation of China (No. 62272219) and the Collaborative Innovation Center of Novel Software Technology & Industrialization. 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[ "Finite element grad grad complexes and elasticity complexes on cuboid meshes", "Finite element grad grad complexes and elasticity complexes on cuboid meshes" ]	[ "Jun Hu [email protected] \nPeking University\nPeking University\nPeking University\n\n", "Yizhou Liang \nPeking University\nPeking University\nPeking University\n\n", "Ting Lin [email protected] \nPeking University\nPeking University\nPeking University\n\n" ]	[ "Peking University\nPeking University\nPeking University\n", "Peking University\nPeking University\nPeking University\n", "Peking University\nPeking University\nPeking University\n" ]	[]	This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an H 2 conforming finite element space, an H(curl; S) conforming finite element space, an H(div; T) conforming finite element space and an L 2 finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element elasticity complex, a vector H 1 conforming space and an H(curl curl T ; S) conforming space are constructed. Combining with an existing H(div; S) ∩ H(div div; S) element and H(div; S) element, respectively, these finite element spaces form two different finite element elasticity complexes. The exactness of all the finite element complexes is proved.	10.48550/arxiv.2302.03783	[ "https://export.arxiv.org/pdf/2302.03783v1.pdf" ]	256,662,280	2302.03783	d160e3794bb637749362be88a07d88ed07bc79a6	Finite element grad grad complexes and elasticity complexes on cuboid meshes Jun Hu [email protected] Peking University Peking University Peking University Yizhou Liang Peking University Peking University Peking University Ting Lin [email protected] Peking University Peking University Peking University Finite element grad grad complexes and elasticity complexes on cuboid meshes 1 This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an H 2 conforming finite element space, an H(curl; S) conforming finite element space, an H(div; T) conforming finite element space and an L 2 finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element elasticity complex, a vector H 1 conforming space and an H(curl curl T ; S) conforming space are constructed. Combining with an existing H(div; S) ∩ H(div div; S) element and H(div; S) element, respectively, these finite element spaces form two different finite element elasticity complexes. The exactness of all the finite element complexes is proved. Introduction Differential complexes have been an important tool in the study and design of finite element methods [4,5,6]. The most canonical differential complex is the de Rham complex, and it plays an important role in the finite element study of electromagnetism and fluid dynamics. In this paper, we focus on the construction of finite elements for another two differential complexes, i.e., the so-called gradgrad complex [2,21] (1.2) Here T and S denote the spaces of traceless and symmetric matrices in three dimensions, respectively, the operators curl and div act row-wise on the matrix-valued functions, and the operator curl T acts column-wise on the matrix-valued functions. Here P 1 is the space of linear function and the rigid motion space RM := {a + b × x : a, b ∈ R 3 }. The finite element discretization of the gradgrad complex (1.1) is related to the linearized Einstein-Bianchi system [22]. The first finite element gradgrad complex on tetrahedral grids was constructed in [15], and those finite element spaces can be used to solve the linearized Einstein-Bianchi system within the mixed form. Recently, several discrete divdiv complexes were constructed [9,16,17], and the associated finite element spaces can be used to discretize the linearized Einstein-Bianchi system within the dual formulation introduced in [22]. The elasticity complex (1.2) plays an important role in the theoretical and numerical analysis of linear elasticity problems, cf. [1,3]. It can be derived from the composition of de Rham complexes in the so-called Bernstein-Gelfand-Gelfand (BGG) construction [2]. By the BGG construction, a two-dimensional finite element elasticity complex has been constructed in [11]. On the Clough-Tocher split in two dimensions, a finite element elasticity complex was proposed in [13]; On the Alfeld split in three dimensions, a finite element elasticity complex was proposed in [12]. Recently, a finite element elasticity complex on a general tetrahedral mesh was constructed in [10] in which the H(div, Ω; S) finite element is the Hu-Zhang element for the symmetric stress tensor [14,19,20]. In this paper, we construct new families of finite element complexes of these two complexes on cuboid meshes. For each complex, two families of discrete complexes with different local and global regularity have been constructed. For the finite element gradgrad complex, the H 2 finite element is the three-dimensional Bogner-Fox-Schmit (BFS) element, whose restriction on each face of each element is a two-dimensional BFS element. The H(curl; S) conforming finite element space constructed in this paper is shown to possess higher regularity. In fact, the finite element space is also H(curl curl T ; S) conforming, and each component is H 1 conforming. As a result, this finite element is also used in the construction of the finite element elasticity complexes. For the H(div; T) conforming finite element space, the degrees of freedom defined on each diagonal component are actually those of the Lagrange element. As a result, the diagonal components admit H 1 regularity. The regularity seems necessary if the degrees of freedom on each component are considered separately. The finite element complexes formed by the above spaces are shown to be exact in a contractible domain. Further, an H(curl; S) element and an H(div; T) element with reduced regularity are constructed. In this case, the H(curl; S) element is not H(curl curl T ; S) conforming, and the degrees of freedom for diagonal components of H(div; T) element are coupled. For the finite element elasticity complex, the H 1 conforming element is newly constructed, and the H(curl curl T ; S) element comes from the construction of the above finite element gradgrad complex. There are two different choices of H(div; S) elements, one is H(div; S) conforming element from in [18], the other is the H(div; S) ∩ H(div div; S) element from [17]. Combined with the cuboid Brezzi-Douglas-Marini [7] and DG element, respectively, two finite element elasticity complexes are formed and proved to be exact. The rest of the paper is organized as follows: Section 2 introduces the notation. Section 3 constructs two families of finite element gradgrad complexes with different regularity on cuboid meshes. Section 4 designs two families of the finite element elasticity complexes. Notations Let Ω be a contractible domain with Lipschitz boundary, which can be partitioned into cuboid grids. Denote by M the space of (3 × 3) matrices, and S and T the space of symmetric and traceless matrices, respectively. For convenience, the components of vectors and matrices are all indexed by x, y, and z. For example, v =   v x v y v z   (2.1) and σ =   σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz   . (2.2) For matrices, the operators curl and div are applied on each row, while curl T means applying the curl operator on each column, that is, curl T σ = (curl σ T ) T for σ ∈ M. The symmetric part of matrix σ is denoted as sym σ = 1 2 (σ + σ T ), and the traceless part is denoted as dev σ = σ − 1 3 (σ xx + σ yy + σ zz ). The mesh considered in this paper will be a cuboid mesh T h . For edge e of an element K, the subscript will be imposed to indicate its direction. For example, e x represents an edge parallel to the x-direction. Similar notations will be adopted for the faces of elements, for example, F yz represents a face normal to the x-direction, see Figure 1. Set Q k 1 ,k 2 ,k 3 (x, y, z) := span{x l 1 y l 2 z l 3 : 0 ≤ l i ≤ k i }. (2.3) Unless otherwise specified, Q k 1 ,k 2 ,k 3 (x, y, z) will be shorten as Q k 1 ,k 2 ,k 3 , and the polynomial spaces on edges and faces Q k (x) and Q k 1 ,k 2 (x, y) are defined similarly. The corresponding space is defined as a null space when one of the index k i is negative. Two finite element spaces on the cuboid mesh will be used without extra definition: the Lagrange finite element space L k,k,k = {u ∈ C 0 (Ω); u\| K ∈ Q k,k,k , K ∈ T h }, and the discontinuous finite element space DG k 1 ,k 2 ,k 3 = {u ∈ L 2 (Ω); u\| K ∈ Q k 1 ,k 2 ,k 3 , K ∈ T h }. For unisolvency, the proof is considered on the reference element T = [0, 1] 3 . To prove the exactness, it is necessary to count the dimension of the constructed finite element spaces. Denote by V the number of vertices, E the number of edges, F the number of faces, and T the number of cells. Henceforth, the vector-valued Sobolev spaces are given by H 1 and L 2 . Discrete gradgrad complex This section considers four types of finite element spaces: H 2 conforming space U h , H(curl, Ω; S) conforming space Σ h , H(div, Ω; T) conforming space Ξ h and L 2 (Ω) space Q h . These finite element spaces will be used to construct P 1 ⊂ −→ U h grad grad −→ Σ h curl −→ Ξ h div −→ Q h −→ 0, (3.1) a discrete subcomplex of the following continuous gradgrad complex, P 1 ⊂ −→ H 2 (Ω) grad grad −→ H(curl, Ω; S) curl −→ H(div, Ω; T) div −→ L 2 (Ω) −→ 0, (3.2) where P 1 is the space of polynomials of degree ≤ 1 (dim = 4), the spaces H(curl, Ω; S) := {v ∈ L 2 (Ω; S) \| curl v ∈ L 2 (Ω; M)},(3.3) and H(div, Ω; T) := {v ∈ L 2 (Ω; T) \| div v ∈ L 2 (Ω, R 3 )}. (3.4) Local version: the polynomial complex In this subsection, the local version of the finite element subcomplex is constructed with the following form: P 1 ⊂ −→ U T grad grad −→ Σ T curl −→ Ξ T div −→ Q T −→ 0, (3.5) for k ≥ 3, where the local spaces U T := Q k,k,k , Σ T :=   Q k−2,k,k Q k−1,k−1,k Q k−1,k,k−1 Q k−1,k−1,k Q k,k−2,k Q k,k−1,k−1 Q k−1,k,k−1 Q k,k−1,k−1 Q k,k,k−2   , Ξ T :=   Q k−1,k−1,k−1 Q k−2,k,k−1 Q k−2,k−1,k Q k,k−2,k−1 Q k−1,k−1,k−1 Q k−1,k−2,k Q k,k−1,k−2 Q k−1,k,k−2 Q k−1,k−1,k−1   , and Q T :=   Q k−2,k−1,k−1 Q k−1,k−2,k−1 Q k−1,k−1,k−2   . Clearly, (3.5) is a complex for the choice of local function spaces, the dimension counting is currently admitted and this will be postponed to the global discrete complex. Proof. It suffices to show the discrete divergence operator is surjective, and the kernel of the discrete curl operator is the image of grad grad. First, for q ∈ Q T , define v ∈ Ξ T such that v xy (x, y, z) = y 0 q x ds, and similarly define v yz , v zx . The other components of v are set as zero. Then it holds that div v = q, which proves div Ξ T = Q T . Second, suppose that σ ∈ Σ T such that curl σ = 0, then there exists φ = [φ x , φ y , φ z ] ∈ [Q k−1,k,k , Q k,k−1,k , Q k,k,k−1 ], such that σ = grad φ. The symmetric property of matrix σ implies that ∂ ∂y φ x = ∂ ∂x φ y . Together with the other off-diagonal entries, this leads to that curl φ = 0, which implies there exists u ∈ Q k,k,k such that φ = grad u. Hence, it holds that σ = grad grad u. This is, grad grad U T = {σ ∈ Σ T : curl σ = 0}. A combination of these two facts and the dimension counting indicates the exactness. The above proposition, together its intermediate results, is useful to prove the exactness of the global finite element complexes. H 2 conforming finite element space The H 2 conforming finite element space U h is a generalization of Bogner-Fox-Schmit (BFS) element spaces [8,23] in three dimensions, where the shape function space is Q k,k,k for k ≥ 3. Given u ∈ Q k,k,k , the degrees of freedom are defined as follows: 1. The function value and partial derivatives of u at each vertex x of T , It will be shown that the set of degrees of freedom is unisolvent with respect to the shape function space Q k,k,k , and that the resulting finite element space is a subspace of H 2 (Ω). u(x), ∂ ∂x u(x), ∂ ∂y u(x), ∂ ∂z u(x), ∂ 2 ∂x∂y u(x), ∂ 2 ∂x∂z u(x), ∂ 2 ∂y∂z u(x), ∂ 3 ∂x∂y∂z u(x). Proposition 3.2. Suppose k ≥ 3, then the above set of degrees of freedom is unisolvent with respect to the shape function space Q k,k,k , and the finite element space U h is H 2 conforming. Proof. The dimension of the shape function space is (k +1) 3 , which is equal to the number of the total degrees of freedom is (k − 3) 3 + 12 × (k − 3) 2 + 36 × (k − 3) + 64 = (k + 1) 3 . Hence it suffices to prove that if u ∈ Q k,k,k vanishes at all degrees of freedom (3.6), (3.7), (3.8), (3.9), then u = 0. Since for each face F with the normal vector n, u\| F , ∂u ∂n \| F ∈ Q k,k vanishes for all the degrees of freedom of the two-dimensional BFS element. Hence, it follows from the unisolvency of the two-dimensional BFS element that u = ∂u ∂n = 0 on the faces of the element T . Therefore, on T = [0, 1] 3 , the standard argument yields that u = x 2 (1 − x) 2 y 2 (1 − y) 2 z 2 (1 − z) 2 u 1 for some u 1 ∈ Q k−4,k−4,k−4 (x, y, z). The degrees of freedom inside element T make sure that u 1 = 0. This proves the unisolvency, while the H 2 continuity is implied by the previous argument. Note that the dimension of the space U h is as follows: dim U h = 8V + 4(k − 3)E + 2(k − 3) 2 F + (k − 3) 3 T . (3.10) H(curl; S) conforming finite element space This subsection considers the construction of an H(curl; S) conforming finite element space Σ h on T h . For this element, the shape function space on T is taken as   σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz   ∈   Q k−2,k,k Q k−1,k−1,k Q k−1,k,k−1 Q k−1,k−1,k Q k,k−2,k Q k,k−1,k−1 Q k−1,k,k−1 Q k,k−1,k−1 Q k,k,k−2   =: Σ T . (3.11) The following lemma is useful to prove the H(curl; S) conformity of a symmetric matrixvalued piecewise polynomial on the cuboid mesh T h . Lemma 3.1 (A sufficient condition for H(curl; S) conformity). If a symmetric matrix-valued piecewise polynomial σ ∈ S satisfies that 1. σ xx is single-valued across all the faces F xy and F xz of T h , and a corresponding condition holds for σ yy and σ zz , with the index changing cyclicly. 2. All the off-diagonal components, say σ xy , are single-valued across all the faces of T h . Then σ is in H(curl, Ω; S). Proof. It follows from a straightforward argument. For σ ∈ Σ h , due to its symmetry, the degrees of freedom will be separated into six parts as follows: σ xx , σ yy , σ zz , σ xy = σ yx , σ xz = σ zx , σ yz = σ zy . In what follows, only the degrees of freedom of σ xx and σ xy will be specified, since those of the remaining four components can be similarly defined via cyclic permutation. The degrees of freedom of σ xx are defined as follows: 1. The moments of σ xx and its normal derivatives on each edge e x of T , ex σ xx p dx, ex ∂ ∂y σ xx p dx, ex ∂ ∂z σ xx p dx, ex ∂ 2 ∂y∂z σ xx p dx for p ∈ Q k−2 (x). (3.12) 2. The moments of σ xx and its normal derivative on each face F xy of T , Fxy σ xx p dxdy, Fxy ∂ ∂z σ xx p dxdy for p ∈ Q k−2,k−4 (x, y). (3.13) 3. The moments of σ xx and its normal derivative on each face F xz of T , Fxz σ xx p dxdz, Fxz ∂ ∂y σ xx p dxdz for p ∈ Q k−2,k−4 (x, z). (3.14) 4. The moments of σ xx inside T , T σ xx p dxdydz for p ∈ Q k−2,k−4,k−4 (x, y, z). (3.15) The degrees of freedom of σ yy , σ zz can be defined in a similar way as those of σ xx by cyclic permutation. The degrees of freedom of σ xy are defined as follows: 1. The value of σ xy and its partial derivative along z-direction, at each vertex x of T , σ xy (x), ∂ ∂z σ xy (x). (3.16) 2. The moments of σ xy and its normal derivative on each e x of T , ex σ xy p dx, ex ∂ ∂z σ xy p dx for p ∈ Q k−3 (x). (3.17) 3. The moments of σ xy and its normal derivative on each e y of T , ey σ xy p dy, ey ∂ ∂z σ xy p dy for p ∈ Q k−3 (y). 5. The moments of σ xy and its normal derivatives on each face F xy of T , Fxy σ xy p dxdy, Fxy ∂ ∂z σ xy p dxdy for p ∈ Q k−3,k−3 (x, y). (3.20) 6. The moments of σ xy on each face F xz of T , Fxz σ xy p dxdz for p ∈ Q k−3,k−4 (x, z). (3.21) 7. The moments of σ xy on each face F yz of T , Fyz σ xy p dydz for p ∈ Q k−3,k−4 (y, z). 8. The moments of σ xy inside the element T , T σ xy p dxdydz for p ∈ Q k−3,k−3,k−4 (x, y, z). (3.23) The degrees of freedom of σ yz and σ zx are similar those of σ xy , by cyclic permutation. The following proposition shows that the set of degrees of freedom is unisolvent and the resulting finite element space is H(curl; S) conforming. Proposition 3.3. When k ≥ 3, the degrees of freedom defined in (3.12)-(3.23) are unisolvent for the shape function space Σ T in (3.11), and the resulting finite element space Σ h is H(curl; S) conforming. Proof. The unisolvency of diagonal entries and off-diagonal entries will be separately considered. It suffices to show the unisolvency of σ xx and σ xy . Unisolvency of σ xx The number of degrees of freedom defined for σ xx is (k − 1)(k − 3) 2 + 8(k − 1)(k − 3) + 16(k − 1) = (k − 1)(k − 3 + 4) 2 = (k − 1)(k + 1) 2 , (3.24) which is equal to the dimension of Q k−2,k,k . It remains to show that if σ xx ∈ Q k−2,k,k vanishes for these degrees of freedom, then σ xx = 0. Since σ xx \| ex , ∂σ xx ∂y \| ex , ∂σ xx ∂z \| ex , ∂ 2 σ xx ∂y∂z \| ex ∈ Q k−2 (x), the degrees of freedom in (3.12) imply that they vanish on e x . When k = 3, consider any face, say F x , normal to x-direction. The restriction σ xx \| Fx of σ xx on such a face is in Q k,k (y, z), and vanishes at each degree of freedom of the two-dimensional BFS element, hence it can be concluded that σ xx = 0. Now consider the case k ≥ 4. Since σ xx \| Fxy and ∂ ∂z σ xx \| Fxy are in Q k−2,k−4 , the degrees of freedom in (3.13) show that σ xx , ∂σxx ∂z vanish on F xy . A similar argument shows that u, ∂σxx ∂y vanish on F xz . It follows that σ xx = y 2 (1 − y) 2 z 2 (1 − z) 2 σ 1 with σ 1 ∈ Q k−2,k−4,k−4 . Finally, the degrees of freedom in (3.15) imply that σ xx = 0. Unisolvency of σ xy Now it turns to show the unisolvency of σ xy . The number of the degrees of freedom in (3.16)-(3.23) is 16 + 16(k − 2) + 4(k − 3) + 4(k − 2) 2 + 4(k − 2)(k − 3) + (k − 2) 2 (k − 3) = k 2 (k + 1), (3.25) which is equal to the dimension of the space Q k−1,k−1,k . It suffices to show that for any polynomial σ xy ∈ Q k−1,k−1,k , it vanishes at all the degrees of freedom if and only if σ xy = 0 on element T . It follows from the degrees of freedom defined in (3.16), (3.17), (3.18) that σ xy and ∂σxy ∂z vanish on all e x , e y . When k ≥ 4, it follows from (3.16) and (3.19) that σ xy vanishes on all the edges e z . It then follows from (3.20) that σ xy and ∂σxy ∂z vanish on face F xy , and from (3.21) and (3.22) that σ xy vanishes on the faces F yz and F xz . It indicates that σ xy = z 2 (1 − z) 2 x(1 − x)y(1 − y)σ 2 , with σ 2 ∈ Q k−3,k−3,k−4 . Finally, by (3.23) it indicates that σ xy = 0. When k = 3, a similar argument shows that σ xy = z 2 (1 − z) 2 σ 2 for some polynomial σ 2 . Since σ xy ∈ Q 2,2,3 it can be concluded that σ xy = 0. The H(curl; S) conformity of σ is already implied by the previous proof, and Lemma 3.1. The dimension of Σ h is dim Σ h =[4(k − 1)E + 4(k − 1)(k − 3)F + 3(k − 1)(k − 3) 2 T ] + [6V + 4(k − 2)E + (k − 3)E + 2(k − 2)(2k − 5)F + 3(k − 2) 2 (k − 3)T ]. H(div; T) conforming finite element space This subsection considers the construction of an H(div; T) conforming space Ξ h . For this element, the shape function space on T is   τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz   ∈   Q k−1,k−1,k−1 Q k−2,k,k−1 Q k−2,k−1,k Q k,k−2,k−1 Q k−1,k−1,k−1 Q k−1,k−2,k Q k,k−1,k−2 Q k−1,k,k−2 Q k−1,k−1,k−1   =: Ξ T . (3.26) Like those of the space Σ T , the degrees of freedom of τ are componentwisely defined. Only the construction of the first row τ x is given, since the degrees of freedom of the remaining components can be defined in a similar way. The degrees of freedom on τ xx are defined as follows: 1. The value of τ xx at each vertex x of T , namely, τ xx (x). 2. The moments of τ xx on each edge e of T , e τ xx p dl for p ∈ Q k−3 . (3.27) 3. The moments of τ xx on each face F of T , F τ xx p ds for p ∈ Q k−3,k−3 . (3.28) 4. The moments of τ xx inside T , T τ xx p dxdydzf or for p ∈ Q k−3,k−3,k−3 (x, y, z). (3.29) The degrees of freedom of τ yy and τ zz can be similarly defined, by a cyclic permutation. Remark 3.1. Notice that since the matrix-valued piecewise polynomial τ is traceless, only two of components τ xx , τ yy and τ zz should be given. The degrees of freedom of τ xy are defined as follows: 1. The moments of τ xy and ∂ ∂y τ xy on each edge e x of T , ex τ xy p dx, ex ∂ ∂y τ xy p dx for p ∈ Q k−2 (x). (3.30) 2. The moments of τ xy on each face F xy of T , Fxy τ xy p dxdy for p ∈ Q k−2,k−4 (x, y). (3.31) 3. The moments of τ xy and ∂ ∂y τ xy on each face F xz of T , Fxz τ xy p dxdz, Fxz ∂ ∂y τ xy p dxdy for p ∈ Q k−2,k−3 (x, z). (3.32) 4. The moments of τ xy inside the element T , T τ xy p dxdy for p ∈ Q k−2,k−4,k−3 (x, y, z). (3.33) The degrees of freedom of the remaining five components can be similarly defined via permutation. The next proposition shows the unisolvency of the degrees of freedom with respect to the shape function spasce, and H(div; T) conformity of the corresponding finite element space. Proposition 3.4. Suppose k ≥ 3, the degrees of freedom defined above are unisolvent for the shape function defined in (3.26), and the resulting finite element space Ξ h is H(div; T) conforming. Proof. It suffices to prove that τ xx and τ xy are unisolvent, since the proof of the remaining components is similar. For the diagonal part, it is the classical Lagrange element. Unisolvency of τ xy Next, it suffices to prove the unisolvency of τ xy . The number of degrees of freedom of τ xy is equal to the dimension of the shape function space Q k−2,k,k−1 , that is, 8(k−1)+2(k−1)(k−3)+4(k−1)(k−2)+(k−1)(k−3)(k−2) = (k−1)k(k+1) = dim Q k−2,k,k−1 . (3.34) For any τ xy ∈ Q k−2,k,k−1 , it can be shown that it vanishes at all degrees of freedom if and only if τ xy = 0. = y 2 (1 − y) 2 z(1 − z)τ 1 with τ 1 ∈ Q k−2,k−4,k−3 . Then (3.33) implies that τ xy = 0. This completes the proof. 2. When k = 3: From (3.30), τ xy and ∂ ∂y τ xy vanish at all edges e x . Then on F xz , it follows that τ xy = z(1 − z)τ 1 for some τ 1 ∈ Q 1,0 (x, z). It follows from (3.32) that τ xy vanishes on F xz . Similarly it can deduced that ∂ ∂y τ xy vanishes on F xz . Hence τ xy = y 2 (1 − y) 2 τ 1 for some polynomial τ 1 and since τ ∈ Q 2,3,1 (x, y, z), yielding that τ xy = 0, which completes the proof. The H(div; T) conformity comes from the fact that τ xy is single-valued on the faces F xz , F xy (and the similar results on the remaining off-diagonal entries holds) and the diagonal entries (e.g. τ xx ) is continuous. The dimension of Ξ h is dim Ξ h = [2V + 2(k − 2)E + 2(k − 2) 2 F + 2(k − 2) 3 T ] + [4(k − 1)E + 2(k − 1)(k − 3)F + 4(k − 1)(k − 2)F + 6(k − 1)(k − 2)(k − 3)T ]. (3.35) L 2 finite element space This subsection considers an L 2 conforming vector-valued finite element space Q h . For this element, the shape function on T is   q x q y q z   ∈   Q k−2,k−1,k−1 Q k−1,k−2,k−1 Q k−1,k−1,k−2   =: Q T . (3.36) Given q ∈ Q h , the degrees of freedom q x are defined as follows: 1. The moments of q x on each edge e x of T , ex q x p dx for p ∈ Q k−2 (x). The degrees of freedom of q y and q z can be defined similarly as those of q x by permutation. Proposition 3.5. Suppose that k ≥ 3, the degrees of freedom defined above are unisolvent with respect to the shape function space (3.36). Proof. Here consider the unisolvency of q x only. The number of degrees of freedom is equal to the dimension of the shape function space Q k−2,k−1,k−1 , namely, 4(k − 1) + 2(k − 1)(k − 2) + 2(k − 1)(k − 2) + (k − 1)(k − 2) 2 = (k − 1)k 2 . (3.41) Hence it suffices to prove that for q x ∈ Q k−2,k−1,k−1 , vanishing at all the degrees of freedom defined above, then q x = 0. Since q x \| ex ∈ Q k−2 the degrees of freedom in (3.37) indicates that q x \| ex = 0. Restricting q x on F xy , it can be deduced that q x \| Fxy = q 1 y(1 − y) for some q 1 ∈ Q k−2,k−3 (x, y). Hence by (3.38), q x \| Fxy = 0. Similarly by (3.39), q x \| Fxz = 0. Hence q x = q 2 y(1 − y)z(1 − z) for some q 2 ∈ Q k−2,k−3,k−3 (x, y, z), which leads to q x = 0 by (3.40). The dimension of Q h is dim Q h = (k − 1)E + 2(k − 1)(k − 2)F + 3(k − 1)(k − 2) 2 T . (3.42) Finite element complex and its exactness This section proves that the following finite element sequence P 1 ⊂ −→ U h grad grad −→ Σ h curl −→ Ξ h div −→ Q h −→ 0 (3.43) is an exact complex. Since there hold the following dimensions of these spaces U h , Σ h , Ξ h and Q h defined in the previous subsections, Then the following result holds, indicating that the discrete div operator is surjective for the above two bubble function spaces. The proof is similar to that in the local version, see Section 3.1. dim U h =8V + 4(k − 3)E + 2(k − 3) 2 F + (k − 3) 3 T , dim Σ h =[4(k − 1)E + 4(k − 1)(k − 3)F + 3(k − 1)(k − 3) 2 T ], + [6V + 4(k − 2)E + (k − 3)E + 2(k − 2) 2 F + 2(k − 2)(k − 3)F + 3(k − 2) 2 (k − 3)T ], dim Ξ h =[2V + 2(k − 2)E + 2(k − 2) 2 F + 2(k − 2) 3 T ] + [4(k − 1)E + 2(k − 1)(k − 3)F + 4(k − 1)(k − 2)F + 6(k − 1)(k − 2)(k − 3)T ], dim Q h =(k − 1)E + 2(k − 1)(k − 2)F + 3(k − 1)(k − 2) 2 T , this leads to dim P 1 − dim U h + dim Σ h − dim Ξ h + dim Q h = 4 − 4V + 4E − 4F + 4TLemma 3.2. It holds that divΞ(T ) =Q(T ) when k ≥ 3. Proof. It is easy to see that divΞ(T ) ⊂Q(T ). For any q ∈Q(T ), there exists τ 0 ∈ H 1 (T ; T) with vanishing trace such that div τ 0 = q. Then define τ ∈Ξ(T ) such that while the remaining components can be similarly defined. As a result, τ ∈Ξ(T ). Then for any p = [p x , p y , T τ xx p = T (τ 0 ) xx p, ∀p ∈ Q k−3,k−3,k−3 (T ),(3.p z ] T with p x ∈ Q k−2,k−3,k−3 , p y ∈ Q k−3,k−2,k−3 , p z ∈ Q k−3,k−3,k−2 , it follows that T div τ · p = T τ : ∇p = T τ 0 : ∇p = T div τ 0 · p = T q · p. It follows from the definition of τ that div τ is of the form div τ =   y(1 − y)z(1 − z)γ x x(1 − x)z(1 − z)γ y x(1 − x)y(1 − y)γ z   , where γ x ∈ Q k−2,k−3,k−3 and γ y ∈ Q k−3,k−2,k−3 , γ z ∈ Q k−3,k−3,k−2 are in the corresponding space. Hence div τ = q. Proposition 3.6. The discrete divergence operator div : Ξ h → Q h is surjective. Proof. The proof is divided into two steps. In the first step, for a given q ∈ Q h , a function τ ∈ Ξ h will be constructed such that (div τ − q)\| T ∈Q(T ) for each T ∈ T h . The construction is as follows: First, let τ xx = τ yy = τ zz = 0. Second, for each e x of T , let ex τ xy p dx = 0 for p ∈ Q k−2 (x), This shows that the degrees of freedom (3.37) vanish for div τ − q. By (3.47), (3.50), (3.51), an integration by parts yields, for any p ∈ Q k−2,k−3 (x, z) that Fxz ( ∂ ∂y τ xy + ∂ ∂z τ xz − q x )p dxdz = Fxz ( ∂ ∂y τ xy − q x )p dxdz + ex 1 τ xz pdx − ex 0 τ xz pdx − Fxz τ xz ∂ ∂z p dxdz =0. (3.53) It indicates the degrees of freedom in (3.38) and (3.39) vanish for div τ − q. Now it suffices to show the construction is of mean zero inside the element T , which is directly from the following calculation T ( ∂ ∂y τ xy + ∂ ∂z τ xz − q x ) dxdydz ( F + xz τ xy dxdz − F − xz τ xy dxdz) + ( F + xy τ xz dxdy − F − xy τ xz dxdy) − T q x dxdydz =0. (3.54) By Lemma 3.2, there existsτ ∈ Ξ h , such that divτ = q − div τ . Therefore, τ +τ ∈ Ξ h and div(τ +τ ) = q. Hence the proof of the proposition is complete. To show U h grad grad −→ Σ h curl −→ Ξ h is exact, it suffices to prove that ker curl = im grad grad on the discrete level for those finite element spaces. Proposition 3.7. Suppose σ ∈ Σ h such that curl σ = 0, then σ = grad grad u for some u ∈ U h . That is, ker curl = im grad grad on the discrete level. Proof. Since curl σ = 0, there exists u ∈ H 2 (Ω) such that grad grad u = σ, from (3.2). Since σ\| T ∈ Σ h is a symmetric matrix-valued polynomial for each element T ∈ T , u\| T is a polynomial of degree k for all variables x, y, z. Combining with u ∈ H 2 , it follows that u ∈ C 1 (Ω). To show that u ∈ U h , it suffices to check the higher order continuity across the internal vertices, edges, and faces of T h . It follows from ∂ 3 ∂x∂y∂z u = ∂ ∂z σ xy , ∂ 2 ∂x∂y u = σ xy , ∂ 2 ∂y∂z u = σ yz and ∂ 2 ∂x∂z u = σ xz , A discrete complex with reduced regularity In this section, a new H(curl; S) conforming space and a new H(div; T) conforming space with reduced regularity will be constructed. H(curl; S) conforming finite element space with reduced regularity This subsection considers an H(curl; S) conforming finite element space Σ h with reduced regularity. For this element, the shape function space on T is   σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz   ∈   Q k−2,k,k Q k−1,k−1,k Q k−1,k,k−1 Q k−1,k−1,k Q k,k−2,k Q k,k−1,k−1 Q k−1,k,k−1 Q k,k−1,k−1 Q k,k,k−2   =: Σ T . (3.55) For σ ∈ Σ h , due to its symmetry, the degrees of freedom will be separated into six parts, σ xx , σ yy , σ zz , σ xy = σ yx , σ xz = σ zx , σ yz = σ zy . In the follows, only the degrees of freedom of σ xx and σ xy will be specified, since those of the remaining four components are similarly defined via cyclic permutation of the index. The degrees of freedom of σ xx are defined as follows: 1. The moments of σ xx on each edge e x of T , ex σ xx p dx, for p ∈ Q k−2 (x). Compared to the version provided in Section 3.3, the reduced element relaxes some partial regularity on edges. The degrees of freedom of σ xy are defined as follows: 1. The value and partial derivative along z-direction of σ xy at each vertex x of T , Compared to the finite element in Section 3.4, the major difference here is in the fifth set of the degrees of freedom. σ xy (x), ∂ ∂z σ xy (x). The next proposition shows the unisolvency of the degrees of freedom with respect to the shape function spaces (3.55), and the corresponding finite element space is H(curl; S) conforming. Proposition 3.8. The above degrees of freedom are unisolvent with respect to the shape function space Σ T in (3.55), and the resulting finite element space Σ h is H(curl; S) conforming. Proof. The unisolvency of σ xx is classical. For the component σ xy , the proof is similar as that of σ xy of Σ h . The H(curl, S) conformity is implied in the previous proof. The dimension of Σ h is dim Σ h = [(k − 1)E + 2(k − 1) 2 F + 3(k − 1) 3 T ] + [6V + 4(k − 2)E + (k − 3)E + (k − 2) 2 F + 2(k − 2)(k − 3)F + 3(k − 2) 2 (k − 1)T ]. H(div; T) conforming finite element space with reduced regularity This subsection considers a new H(div; T) conforming finite element space Ξ h with reduced regularity. The shape function space on T is as follows:   τ xx τ xy τ xz τ yx τ xy τ yz τ zx τ zy τ zz   ∈   Q k−1,k−1,k−1 Q k−2,k,k−1 Q k−2,k−1,k Q k,k−2,k−1 Q k−1,k−1,k−1 Q k−1,k−2,k Q k,k−1,k−2 Q k−1,k,k−2 Q k−1,k−1,k−1   = Ξ T . (3.68) The following bubble space on T is needed for the following construction: B div,k;T (T ) = (τ xx , τ yy , τ zz ) ∈ Q k−1,k−1,k−1 × Q k−1,k−1,k−1 × Q k−1,k−1,k−1 : τ xx vanishes on F yz , τ yy vanishes on F xz , τ zz vanishes on F xy , and τ xx + τ yy + τ zz = 0 . Proof. Since x(1 − x)\|τ xx , y(1 − y)\|τ yy , z(1 − z)\|τ zz , and τ xx + τ yy + τ zz = 0, there exist p ∈ Q k−3,k−1,k−1 and q ∈ Q k−1,k−3,k−1 such that x(1 − x)p(x, y, 0) + y(1 − y)q(x, y, 0) = 0 (3.71) and x(1 − x)p(x, y, 1) + y(1 − y)q(x, y, 1) = 0. (3.72) It follows that there exist f (x, y) ∈ Q k−3,k−3 (x, y) and g(x, y) ∈ Q k−3,k−3 (x, y) such that p(x, y, 0) = y(1 − y)f (x, y), q(x, y, 0) = −x(1 − x)f (x, y) and p(x, y, 1) = y(1 − y)g(x, y), q(x, y, 1) = −x(1 − x)g(x, y) for f, g ∈ Q k−3,k−3 (x, y). Regard p as a polynomial of z, of degree ≤ k − 1 and coefficients in Q k−3,k−1 (x, y). Since the values of p at z = 0 and z = 1 are given as above, additional values of p(z i ) at k − 2 points z i , i = 1, 2, · · · , k − 2, uniquely determine p. Therefore, the dimension of B div,k;T (T ) is 2(k − 2) 2 k + 2(k − 2) 2 = 2(k − 2) 2 (k + 1). (3.73) Based on the above observations, now it is ready to define the degrees of freedom. The degrees of freedom of (τ xx , τ yy , τ zz ) are defined as follows: 1. The values τ xx (x), τ yy (x), τ zz (x) at each vertex x of element T . 4. The following moments inside the element T , T (τ xx ξ xx + τ yy ξ yy + τ zz ξ zz )dxdydz for (ξ xx , ξ yy , ξ zz ) ∈ B div,k;T (T ). (3.76) The degrees of freedom on τ xy are defined as follows: 1. The moments of τ xy each face F xz of T , Fxz τ xy p dxdz, for p ∈ Q k−2,k−1 (x, z). The degrees of freedom for the remaining off-diagonal components can be similarly defined. The unisolvency and desired conformity of the finite element is stated in the following proposition. Proposition 3.9. The above degrees of freedom are unisolvent for the shape function Ξ T , defined in (3.68). Moreover, the resulting finite element space Ξ h is H(div; T) conforming. Proof. Unisolvency of (τ xx , τ yy , τ zz ). For this case, the degrees of freedom of these three diagonal components are coupled. Note that the dimension of the shape function space of (τ xx , τ yy , τ zz ) is 2k 3 . While the total number of degrees of freedom is 16 + 24(k − 2) + 6(k − 2) 2 + 2(k − 2) 2 (k + 1) = 2k 3 , equals to the dimension of (τ xx , τ yy , τ zz ). Hence, it suffices to show that if (τ xx , τ yy , τ zz ) belonging to {(τ xx , τ yy , τ zz ) ∈ Q k−1,k−1,k−1 × Q k−1,k−1,k−1 × Q k−1,k−1,k−1 : τ xx + τ yy + τ zz = 0} vanishes for all the DOFs, then τ xx = τ yy = τ zz = 0. By (3.74), (3.75), it holds that τ xx vanishes on face F yz , τ yy vanishes on face F xz and τ zz vanishes on face F xy . Therefore, this yields (τ xx , τ yy , τ zz ) ∈ B div,k;T (T ). At end, the fourth set of DOFs in (3.76) indicates that τ xx = τ yy = τ zz = 0. The unisolvency of τ xy is straightforward and hence omitted. Next, consider the H(div; T) conformity of the resulting spacee. This is from (1) τ xx is single-valued on F yz , and (2) τ xy is single-valued on F xz , and the other conditions obtained via a cyclic permutation. The dimension of Ξ h is dim Ξ h = [2V + 2(k − 2)E + (k − 2) 2 F + 2(k − 2) 2 (k + 1)T ] + [2(k − 1)kF + 6(k − 1) 2 kT ]. (3.79) A new finite element complex and its exactness For this case, the finite element subspace Q h of L 2 (Ω, R 3 ) is the space of discontinuous piecewise polynomials. For this element, the shape function space on T is   q x q y q z   ∈   DG k−2,k−1,k−1 DG k−1,k−2,k−1 DG k−1,k−1,k−2   ,(3.80) This subsubsection aims at proving the following finite element sequence P 1 ⊂ −→ U h grad grad −→ Σ h curl −→ Ξ h div −→ Q h −→ 0 (3.81) is exact. Since the dimensions of the finite element space U h , Σ h , Ξ h and Q h are as follows: dim U h =8V + 4(k − 3)E + 2(k − 3) 2 F + (k − 3) 3 T , dim Σ h =[(k − 1)E + 2(k − 1) 2 F + 3(k − 1) 3 T ] + [6V + 4(k − 2)E + (k − 3)E + (k − 2) 2 F + 2(k − 2)(k − 3)F + 3(k − 2) 2 (k − 1)T ]. dim Ξ h =[2V + 2(k − 2)E + (k − 2) 2 F + 2(k − 2) 2 (k + 1)T ] + [2(k − 1)kF + 6(k − 1) 2 kT ], dim Q h =3(k − 1)k 2 T , this leads to dim U h − dim Σ h + dim Ξ h − dim Q h =4(V − E + F − T ) =4 = dim P 1 ,(3.82) by Euler's formula. Proposition 3.10. Suppose that k ≥ 3, the discrete divergence operator div : Ξ h → Q h is surjective. Proof. Given q = (q x , q y , q z ) T ∈ Q h , set τ xy by τ xy (x, y, z) = y 0 q x ds, and similarly define τ zx and τ yz . The remaining entries are set zero. Clearly, τ is H(div; T) conforming, and belongs to the space Ξ h . In addition, div τ = q. Proposition 3.11. For k ≥ 3, if σ ∈ Σ h satisfies that curl σ = 0, then there exists u ∈ U h such that grad grad u = σ. Proof. The proof is similar to that in Proposition 3.7. Let a piecewise polynomial function u ∈ H 2 such that grad grad u = σ. Again, it suffices to check that ∂ 2 ∂y∂z u is single-valued on edge e x . This directly comes from the fact σ yz is single-valued on e x . Theorem 3.2. The finite element sequence in (3.81) is an exact complex, provided k ≥ 3. Proof. By dimension counting in (3.82), the result follows from Proposition 3.10 and Proposition 3.11. Discrete elasticity complex This section considers four type of finite element spaces: H 1 (Ω) conforming space V h , H(curl curl T ; S) conforming space Φ h , H(div; S) conforming space Γ h and L 2 (Ω) space Z h . These finite element spaces will be used to construct RM ⊆ −→ X h sym grad −→ Φ h curl curl T −→ Γ h div −→ Z h −→ 0, (4.1) a discrete subcomplex on the cuboid grids of the following continuous elasticity complex, The following vector identity will be used in the following proof. RM ⊆ −→ H 1 Ω; R 3 sym grad −→ H(curl curl T , Ω; S) curl curl T −→ H(div, Ω; S) div −→ L 2 (Ω; R 3 ) −→ 0. (4.2) Here RM = {a + b × x : a, b ∈ R 3 } is Local version: the polynomial complex In this subsection, the local version of the finite element complex is constructed with the following form, (where T = [0, 1] 3 ) RM ⊆ −→ X T sym grad −→ Φ T curl curl T −→ Γ T div −→ Z T −→ 0, (4.5) where k ≥ 2, and X T =   Q k,k+1,k+1 Q k+1,k,k+1 Q k+1,k+1,k   , Φ T =   Q k−1,k+1,k+1 Q k,k,k+1 Q k,k+1,k Q k,k,k+1 Q k+1,k−1,k+1 Q k+1,k,k Q k,k+1,k Q k+1,k,k Q k+1,k+1,k−1   , Γ T =   Q k+1,k−1,k−1 Q k,k,k−1 Q k,k−1,k Q k,k,k−1 Q k−1,k+1,k−1 Q k−1,k,k Q k,k−1,k Q k−1,k,k Q k−1,k−1,k+1   , and Z T =   Q k,k−1,k−1 Q k−1,k,k−1 Q k−1,k−1,k   . Clearly, the above sequence is a complex. The following proposition indicates that the complex is exact. Proof. By the choice of the shape function spaces, the polynomial sequence is a complex. It remains to prove its exactness. First, for q ∈ Z T , define v ∈ Γ T such that v xx (x, y, z) = x 0 q x ds. The components v yy and v zz are similarly defined, and the offdiagonal components (e.g. v xy ) are defined as zero. Clearly, it holds that div v = q. Second, suppose that σ ∈ Φ T such that curl curl T σ = 0. Then there exists a vectorvalued polynomial a such that curl T σ = grad a. Since curl T σ is a traceless matrix, therefore grad a is traceless, which implies div a = 0. Therefore there exists a vectorvalued polynomial u such that a = curl u. It then follows from (4.4) that curl T σ = grad curl u = 2 curl T sym grad u. By the local exactness of the gradgrad complex, there exists a polynomial φ such that σ − 2 sym grad u = grad grad φ, therefore σ = sym grad(2u + grad φ). A combination of these two results and the dimension counting (admitted here, and will be shown in (4.26), implies the exactness. H 1 conforming space This subsection considers the construction of an H 1 conforming finite element vectorvalued space X h on T h . For this element, the shape function space on T is as follows:   u x u y u z   ∈   Q k,k+1,k+1 Q k+1,k,k+1 Q k+1,k+1,k   := X T . (4.6) The degrees of freedom of u x are defined as follows: 1. The function value and the following partial derivatives of u x , at each vertex x of T , u x (x), ∂ ∂y u x (x), ∂ ∂z u x (x), ∂ 2 ∂y∂z u x (x). (4.7) 2. The moments of function itself and partial derivatives of u x on each edge e x of T , ex u x p dx, ex ∂ ∂y u x p dx, ex ∂ ∂z u x p dx, ex ∂ 2 ∂y∂z u x p dx for p ∈ Q k−2 (x).Fxy ∂ ∂z u x p dxdy for p ∈ Q k−2,k−3 (x, y),(4. 12) Fxz u x p dxdz, Fxz ∂ ∂y u x p dxdz for p ∈ Q k−2,k−3 (x, z). (4.13) 6. The moment of u x inside the element T , T u x p dxdydz for p ∈ Q k−2,k−3,k−3 (x, y, z). (4.14) The degrees of freedom of the remaining components (u y and u z ) can be similarly defined, by a cyclic permutation on the index. Proposition 4.2. When k ≥ 2, then the degrees of freedom defined above are unisolvent with respect to the shape function space (4.6), and the resulting finite element space X h is H 1 conforming. Proof. The number of degrees of freedom defined for u x is 4 + 16(k − 1) + 16(k − 2) + 2(k − 2) 2 + 4(k − 1)(k − 2) + (k − 1)(k − 2) 2 = (k + 1)(k + 2) 2 , which is equal to the dimension of Q k,k+1,k+1 . It then suffices to show that if u x ∈ Q k,k+1,k+1 vanishes at the degrees of freedom defined above, then u x = 0. It follows from (4.7) and (4.8) that u x , ∂ ∂y u x , ∂ ∂z u x , ∂ 2 ∂y∂z u x vanish at all vertices and edges e x . Then it follows from (4.9) and (4.10) that u x , ∂ ∂z u x vanish on edges e y and u x , ∂ ∂y u x vanish on edges e z . Therefore, on any face, say F yz , the restriction u x \| Fyz can be expressed as u x \| Fyz = y 2 (1 − y) 2 z 2 (1 − z) 2 u 1 , u 1 ∈ Q k−3,k−3 (y, z). Due to (4.11), it is easy to see that u x vanishes on F yz . Similarly, u x vanishes on all faces and ∂ ∂z u x vanishes on face F xy , ∂ ∂y u x vanishes on face F xz , by (4.12) and (4.13). Hence u x = x(1 − x)y 2 (1 − y) 2 z 2 (1 − z) 2 u 2 for some u 2 ∈ Q k−2,k−3,k−3 and consequently is zero, by (4.14). The dimension of X h is dim X h = 12V + [4(k − 1) + 4(k − 2)]E + [(k − 2) 2 + 4(k − 1)(k − 2)]F + 3(k − 1)(k − 2) 2 T . (4.15) Remark 4.1. As it will be shown later, X h is actually H 1 (curl) conforming. H(curl curl T ; S) conforming space The discrete H(curl curl T ; S) conforming space Φ h (of degree k) is taken exactly as the proposed H(curl; S) conforming space Σ h of degree k + 1, see Section 3.3. The following lemma shows that the proposed Σ h has H(curl curl T ; S) regularity. • The diagonal component σ xx is single-valued across e x of T h , σ xx and ∂σxx ∂n is singlevalued across F xy and F xz of T h . A corresponding condition holds for σ yy and σ zz . • The off-diagonal component σ xy is continuous on T h , ∂ ∂z σ xy is single-valued across F xy of T h . A corresponding condition holds for σ xz and σ yz . then σ ∈ H(curl curl T ; S). Proof. Set v = curl curl T σ, then a direct calculation yields that v xx = ∂ 2 σ yy ∂z 2 + ∂ 2 σ zz ∂y 2 − 2 ∂ 2 σ yz ∂y∂z , v xy = ∂ 2 σ xy ∂z 2 + ∂ 2 σ zz ∂x∂y − ∂ 2 σ xz ∂y∂z − ∂ 2 σ yz ∂x∂z . By assumption it holds that T ∂ 2 σ yy ∂z 2 p = T ∂ 2 p ∂z 2 σ yy + Fxy [[ ∂σ yy ∂z ]]p + Fxy [[σ yy ]] ∂p ∂z = T ∂ 2 p ∂z 2 σ yy . T ∂ 2 σ yz ∂y∂z p = T ∂ 2 p ∂y∂z σ yz + Fxy [[σ yz ]]p + Fxz [[σ yz ]]p + Ex [[σ yz ]]p = T ∂ 2 p ∂y∂z σ yz . The other components are calculated similarly. As a result, it can be concluded that curl curl T σ ∈ L 2 . From this lemma and the proof of Section 3.3, one can easily derive the following proposition. H(div; S) conforming space The H(div; S) conforming space Γ h is taken as the H(div div; S) conforming space in [17]. The shape function spaces of Γ h on T is taken as   σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz   ∈   Q k+1,k−1,k−1 Q k,k,k−1 Q k,k−1,k Q k,k,k−1 Q k−1,k+1,k−1 Q k−1,k,k Q k,k−1,k Q k−1,k,k Q k−1,k−1,k+1   =: Γ T . (4.16) The degrees of freedom on σ xx is defined as follows: 1. The moments of σ xx and ∂ ∂x σ xx on the face F yz , Fyz σ xx p dydz Fyz ∂ ∂x σ xx p dydz for p ∈ Q k−1,k−1 (y, z). (4.17) 2. The moments of σ xx inside the element T , T σ xx for p ∈ Q k−3,k−1,k−1 (x, y, z). The degrees of freedom on σ xy is defined as follows: 1. The moments of σ xy on each edge e z of T , ez σ xy p dz for p ∈ Q k−1 (z). (4.19) 2. The moments of σ xy on each face F xz and F yz of T , Fxz σ xy p dxdz for p ∈ Q k−2,k−1 (x, z), Fyz σ xy p dydz for p ∈ Q k−2,k−1 (y, z). (4.20) 3. The moments of σ xy inside the element T , T σ xy p dxdydz for p ∈ Q k−2,k−2,k−1 (x, y, z). The proof of unisolvency and H(div; S) conformity is classical, and actually shown in [17]. The dimension of Θ h is kE + [2k 2 + 2k(k − 1)]F + [3k 2 (k − 2) + 3k(k − 1) 2 ]T . L 2 conforming space This section constructs an L 2 finite element space Z h . For this element, the shape function space on T is   q x q y q z   ∈   Q k,k−1,k−1 Q k−1,k,k−1 Q k−1,k−1,k   =: Z T . (4.22) The degrees of freedom of q x are defined as follows: 1. The moment of q x on each face F yz of T , Fyz q x p dydz for p ∈ Q k−1,k−1 (y, z). The degrees of freedom of q y and q z can be similarly defined as those of q x , by a cyclic permutation. The verification of the unisolvency of Z h is straightforward and hence omitted. The dimension of Z h is dim Z h = k 2 F + 3(k − 1)k 2 T . Finite element complex and its exactness This section proves that the following finite element sequence RM ⊆ −→ X h sym grad −→ Φ h curl curl T −→ Γ h div −→ Z h −→ 0, (4.25) is an exact complex. Since there hold the following dimensions of these spaces X h , Φ h , Γ h , Z h defined in the previous subsections, dim X h =12V + [4(k − 1) + 4(k − 2)]E + [(k − 2) 2 + 4(k − 1)(k − 2)]F + 3(k − 1)(k − 2) 2 T , dim Φ h =[4kE + 4k(k − 2)F + 3k(k − 2) 2 T ] + [6V + 4(k − 1)E + (k − 2)E + 2(k − 1) 2 F + 2(k − 1)(k − 2)F + 3(k − 1) 2 (k − 2)T ], dim Γ h =kE + [2k 2 + 2k(k − 1)]F + [3k 2 (k − 2) + 3k(k − 1) 2 ]T , dim Z h =k 2 F + 3(k − 1)k 2 T , this leads to dim X h − dim Φ h + dim Γ h − dim Z h =6(V − E + F − T ) =6 = dim RM (4.26) by Euler's formula. The following result holds, indicating that the discrete divergence operator is surjective. Proposition 4.4. The discrete divergence operator: div : Γ h → Z h is surjective. Proof. The argument is similar to that in [18]. Consider the symmetric matrix-valued piecewise polynomial function σ such that σ xx = Set σ xy = σ xz = σ yz = 0. Clearly, it holds that σ\| T ∈ Φ T , it then suffices to check the continuity. By such a construction, τ xx is continuous on each face F yz . It directly follows from the facts that ∂ ∂x σ xx = q x and q x is continuous when crossing F yz , which completes the proof. The following proposition shows that the kernel of the discrete curl curl T operator is just the image of sym grad. Proposition 4.5. If σ ∈ Φ h satisfies that curl curl T σ = 0, then there exists v ∈ X h such that σ = sym grad v. Proof. It follows from (4.2) that curl curl T σ = 0 implies that there exists a function u ∈ H 1 (Ω) such that (sym grad u) = σ. It follows from the local discrete complex that each u\| K is a polynomial and u\| K ∈ X K , for each element K. Since u ∈ H 1 (Ω), u is continuous when crossing the internal faces. Additionally, since Φ h is H(curl) conforming by Proposition 3.3 , which implies curl sym grad u ∈ L 2 (Ω; M). By (4.3) curl sym grad u = 1 2 (grad curl u) T , it shows that curl u ∈ H 1 (Ω). Since curl u is a piecewise polynomial, it then implies that curl u is continuous. It follows from both [curl u] x = ∂ ∂y u z − ∂ ∂z u y and [sym grad u] yz = ∂ ∂y u z + ∂ ∂z u y are continuous that so are ∂ ∂y u z and ∂ ∂z u y . Similarly, it holds that ∂ ∂y u x , ∂ ∂z u x , ∂ ∂x u y , ∂ ∂x u z are continuous. It remains to show that ∂ 2 ∂y∂z u x are single-valued at each vertex of x, which follows from a simple calculation ∂ 2 ∂y∂z u x = 1 2 [ ∂ ∂y σ xz + ∂ ∂z σ xy − ∂ ∂x σ yz ]. Since the three terms on the right hand side are single-valued at each vertex, by the degrees of freedom, ∂ 2 ∂y∂z u x is also single-valued. A finite element complex with reduced regularity This section considers a finite element complex, which reduces some additional regularity. The goal of this subsection is to construct This subsubsection introduces an H(div; S) conforming element space Γ h from [18], with reduced regularity. The shape function space on T is taken as   σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz   ∈   Q k+1,k−1,k−1 Q k,k,k−1 Q k,k−1,k Q k,k,k−1 Q k−1,k+1,k−1 Q k−1,k,k Q k,k−1,k Q k−1,k,k Q k−1,k−1,k+1   = Γ T . RM ⊆ −→ X h sym grad −→ Φ h curl curl T −→ Γ h div −→ Z h −→ 0,(4. A finite element complex with regularity and its exactness Take Z h as the vector-valued discontinuous element space, and the following relationship holds, dim X h =12V + [4(k − 1) + 4(k − 2)]E + [(k − 2) 2 + 4(k − 1)(k − 2)]F + 3(k − 1)(k − 2) 2 T . dim Φ h =[4kE + 4k(k − 2)F + 3k(k − 2) 2 T ] + [6V + 4(k − 1)E + (k − 2)E + 2(k − 1) 2 F + 2(k − 1)(k − 2)F + 3(k − 1) 2 (k − 2)T ] dim Γ h =kE + [k 2 + 2k(k − 1)]F + [3k 3 + 3k(k − 1) 2 ]T . dim Z h =3(k + 1)k 2 T . A calculation yields that dim X h − dim Φ h + dim Γ h − dim Z h =6(V − E + F − T ) =6 = dim RM. (4.36) Proposition 4.7. When k ≥ 2, it holds that div Γ h = Z h . Proof. The proof is similar to that of Proposition 4.4. Theorem 4.2. When k ≥ 2, the finite element sequence (4.28) is an exact complex. Proof. A combination of dimension counting (4.36), Propositions 4.5 and 4.7 completes the proof. H(curl curl T , Ω; S) curl curl T −→ H(div, Ω; S) div −→ L 2 (Ω) −→ 0. Figure 1 : 1An illustration of notations. Proposition 3 . 1 . 31The polynomial sequence (3.5) is an exact complex. The moments of u and its normal derivatives on each edge e, say e x , of T , degrees of freedom defined on e y and e z are similarly defined, by a cyclic permutation.1 3. The moments of u and its normal derivatives on each face F , say F yz , of T , degrees of freedom defined on the other faces F xz and F xy , by a cyclic permutation.4. The moments of u inside the element T , T up dxdydz for p ∈ Q k−4,k−4,k−4 (x, y, z).(3.9) The moments of σ xy on each e z of T , ez σ xy p dx for p ∈ Q k−4 (z).(3.19) The moments of q x on each face F xy of T , Fxy q x p dxdy for p ∈ Q k−2,k−3 (x, y).(3.38)3. The moments of q x on each face F xz of T ,Fxz q x p dxdz for p ∈ Q k−2,k−3 (x, z).(3.39) 4. The moments of q x inside the element T , T q x p dx for p ∈ Q k−2,k−3,k−3 (x, y, z). (3.40) 's formula. Before proving the exactness of the sequence in (3.43), two auxiliary spaces (so-called bubble function spaces) on each element T ∈ T h are introduced: Ξ(T ) := {τ ∈ Ξ T vanishes for the DOFs defined on the boundary ∂T }, Q(T ) := {q ∈ Q T vanishes for the DOFs defined on the boundary ∂T and T q = 0}. that the degrees of freedom of u defined at vertices are single-valued by the definition of Σ h . A similar argument shows the continuity of the following moments ex ∂ 2 ∂y∂z up dx = ex σ yz pdx for any p ∈ Q k−4 (x). Hence it holds that u ∈ U h . A combination of the dimension counting (3.44), Proposition 3.6 and Proposition 3.7 indicates the following result. Theorem 3 . 1 . 31The finite element sequence in (3.43) is an exact complex. The moments of σ xx on each face F xy of T , Fxy σ xx p dxdy, for p ∈ Q The moments of σ xx on each face F xz of T , Fxz σ xx p dxdz, for p ∈ Q k−2,k−2 (x, z).(3.58)4. The moments of σ xx inside the element T , T σ xx p dxdydz for p ∈ Q k−2,k−2,k−2 (x, y, z).(3.59) The moments of σ xy on each edge e x of T , ex σ xy p dx, ex ∂ ∂z σ xy for p ∈ Q k−3 (x). (3.61) 3. The moments of σ xy on each edge e y of T , ey σ xy p dy, ey ∂ ∂z σ xy for p ∈ Q k−3 (y). (3.62) 4. The moments of σ xy on each edge e z of T , ez σ xy p dx for p ∈ Q k−4 (z). (3.63) 5. The moments of σ xy on each face F xy of T , Fxy σ xy p dxdy, for p ∈ Q k−3,k−3 (x, y).(3.64) 6. The moments of σ xy on each face F xz of T , Fxz σ xy p dxdz for p ∈ Q The moments of σ xy on each face F yz of T , Fyz σ xy p dydz for p ∈ Q k−3,k−4 (y, z). (3.66) 8. The moments of σ xy inside the element T , T σ xy p dxdydz for p ∈ Q k−3,k−3,k−2 (x, y, z). (3.67) 2 . 2The moments of τ xx , τ yy , τ zz on each edge e of element T , e τ xx p dl, e τ yy p dl, e τ zz p dl for p ∈ Q k−3 .(3.74)3. The moments of τ xx on each face F yz of element T , Fyz τ xx p ds for p ∈ Q k−3,k−3 (y, z).(3.75)A similar set of degrees of freedom can be defined for τ yy and τ zz by cyclic permuatation of the index. The moments of τ xy inside the element T , T τ xy p dxdy for p ∈ Q k−2,k−2,k−1 (x, y, z).(3.78) the rigid motion space (dim = 6), the spaces H(curl curl T , Ω; S) := {u ∈ L 2 (Ω; S); curl curl T u ∈ L 2 (Ω; S)} and H(div, Ω; S) := {σ ∈ L 2 (Ω; S) : div σ ∈ L 2 (Ω)}. Proof. A straightforward calculation yields that curl T grad u = grad curl u. curl u) T . Proposition 4 . 1 . 41The polynomial sequence (4.5) is an exact complex. The moments of function itself and partial derivatives of u x on each edge e y , e z of T ,ey u x p dx, ey ∂ ∂z u x p dx, for p ∈ Q k−3 x p dx, for p ∈ Q k−3 (z).(4.10) 4. The moments of u x on each face F yz , Fyz u x p dydz for p ∈ Q k−3,k−3 (y, z). (4.11) 5. The moments of function itself and partial derivatives of u x on F xy and F xz of T , Fxy u x p dxdy, Lemma 4.2 (H(curl curl T ; S) conformity). If a symmetric matrix-valued piecewise polynomial σ satisfies that Proposition 4 . 3 . 43When k ≥ 3, the space Φ h is H(curl curl T ; S) conforming. The moment of q x inside T , T q x p dxdydz for p ∈ Q k−2,k−1,k−1 (x, y, z). Theorem 4. 1 . 1When k ≥ 2, the finite element sequence (4.25) is an exact complex. Proof. A combination of Proposition 4.4, Proposition 4.5, and the dimension counting (4.26) completes the proof. 1 . 1When k ≥ 4: From (3.30), τ xy and ∂τxy ∂y vanish at all edges e x . From (3.31) and (3.32), τ xy vanishes on face F xy , F xz , Similarly, one can show that ∂τxy ∂y vanishes on F xz . It follows that τ xy Fourth, for each face F xz , which is normal to the y−direction, letThe remaining degrees of freedom of τ xy can be treated as zero. The component of τ xz can be similiarly defined. Next, it can be verified that (div τ x − q x )\| T ∈Q T .47) and ex ∂ ∂y τ xy p dx = 1 2 ex q x p dx for p ∈ Q k−2 (x). (3.48) Third, for two faces F + xz and F − xz , let F + xz τ xy dxdz − F − xz τ xy dxdz = 1 2 T q x . (3.49) Fxz ∂ ∂y τ xy p dxdy = Fxz q x p dxdy for p ∈ Q k−2,k−3 (x, z). (3.50) Last, for each face F xy , set Fxy τ xy p dxdy = 0 for p ∈ Q k−2,k−4 (x, y). (3.51) It follows from (3.47) and (3.48) that ex ( ∂ ∂y τ xy + ∂ ∂z τ xz − q)p dxdz = 0 for p ∈ Q k−2 (x). (3.52) 28 ) 284.7.1 H(div; S) conforming space with reduced regularity This means x → y, y → z, z → x. The degrees of freedom on σ xx is defined as follows:1. The moments of σ xx on each face F yz of T , Fyz σ xx p dydz for p ∈ Q k−1,k−1 (y, z).(4.30)2. 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[ "Four infinite families of ternary cyclic codes with a square-root-like lower bound", "Four infinite families of ternary cyclic codes with a square-root-like lower bound" ]	[ "Tingfang Chen ", "Cunsheng Ding [email protected] ", "Chengju Li [email protected] ", "Zhonghua Sun [email protected] ", "C ", "C Li ", "Z ", "\nDepartment of Computer Science and Engineering\nDepartment of Computer Science and Engineering\nThe Hong Kong University of Science and Technology\nClear Water BayKowloon, Hong KongChina\n", "\nis with the Shanghai Key Laboratory of Trustworthy Computing\nThe Hong Kong University of Science and Technology\nClear Water BayKowloon, Hong KongChina\n", "\nand also with the National Mobile Communications Research Laboratory\nEast China Normal University\n200062ShanghaiChina\n", "\nSchool of Mathematics\nSoutheast University\nChina\n", "\nHefei University of Technology\n230601HefeiAnhuiChina\n" ]	[ "Department of Computer Science and Engineering\nDepartment of Computer Science and Engineering\nThe Hong Kong University of Science and Technology\nClear Water BayKowloon, Hong KongChina", "is with the Shanghai Key Laboratory of Trustworthy Computing\nThe Hong Kong University of Science and Technology\nClear Water BayKowloon, Hong KongChina", "and also with the National Mobile Communications Research Laboratory\nEast China Normal University\n200062ShanghaiChina", "School of Mathematics\nSoutheast University\nChina", "Hefei University of Technology\n230601HefeiAnhuiChina" ]	[]	Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, and the arXiv paper arXiv:2301.06446, the objectives of this paper are the construction and analyses of four infinite families of ternary cyclic codes with length n = 3 m − 1 for odd m and dimension k ∈ {n/2, (n + 2)/2} whose minimum distances have a square-root-like lower bound. Their DRAFT 2 duals have parameters [n, k ⊥ , d ⊥ ], where k ⊥ ∈ {n/2, (n − 2)/2} and d ⊥ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes.	10.48550/arxiv.2303.06849	[ "https://export.arxiv.org/pdf/2303.06849v2.pdf" ]	257,496,359	2303.06849	d62e23f6b190b969ee1cdf07aac48a327c51b398	Four infinite families of ternary cyclic codes with a square-root-like lower bound 2 May 2023 May 3, 2023 Tingfang Chen Cunsheng Ding [email protected] Chengju Li [email protected] Zhonghua Sun [email protected] C C Li Z Department of Computer Science and Engineering Department of Computer Science and Engineering The Hong Kong University of Science and Technology Clear Water BayKowloon, Hong KongChina is with the Shanghai Key Laboratory of Trustworthy Computing The Hong Kong University of Science and Technology Clear Water BayKowloon, Hong KongChina and also with the National Mobile Communications Research Laboratory East China Normal University 200062ShanghaiChina School of Mathematics Southeast University China Hefei University of Technology 230601HefeiAnhuiChina Four infinite families of ternary cyclic codes with a square-root-like lower bound 2 May 2023 May 3, 20231 C. Ding's research was supported by The Hong Kong Research Grants Council, Proj. No. 16302121. C. Li's research was supported by the National Natural Science Foundation of China (12071138) and Shanghai Natural Science Foundation (22ZR1419600). Z. Sun's research was supported by The National Natural Science Foundation of China under Grant Number 62002093. T. Chen is with theIndex Terms Cyclic code, linear code Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, and the arXiv paper arXiv:2301.06446, the objectives of this paper are the construction and analyses of four infinite families of ternary cyclic codes with length n = 3 m − 1 for odd m and dimension k ∈ {n/2, (n + 2)/2} whose minimum distances have a square-root-like lower bound. Their DRAFT 2 duals have parameters [n, k ⊥ , d ⊥ ], where k ⊥ ∈ {n/2, (n − 2)/2} and d ⊥ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes. INTRODUCTION In this paper, we assume that the reader is familiar with the basics of linear codes and cyclic codes over finite fields [1], [3] and do not introduce them here. Let C be an [n, k, d] linear code over F q . C is said to be distance-optimal if there does not exist an [n, k, d ′ ≥ d + 1] linear code over F q . A code is called an optimal code if it is distance-optimal or meets a bound for linear codes. If a lower bound on the minimum weight d of C is close to √ n, we then call it a square-root-like lower bound. Cyclic codes have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms [3]. They are also important in theory, as they are closely related to algebra, algebraic geometry, algebraic function fields, algebraic number theory, association schemes, combinatorics, finite fields, finite geometry, graph theory, and group theory. There are a lot of references on cyclic codes over finite fields. However, no infinite family of ternary cyclic codes with parameters [n, k ∈ {n/2, (n ± 2)/2}, d ≥ √ n] is seen in the literature. It is very difficult to design such an infinite family of cyclic codes, as the dimension is about half of the length of each code in the family. These are the motivations of this paper. Recently, several infinite families of binary cyclic codes with parameters [n, (n±1)/2, d] were constructed, where the minimum distance d has a lower bound which is quite close to √ n [7], [5], [6]. Inspired by the recent results in [7], [5], [6], in this paper we construct four infinite families of ternary cyclic codes with parameters [n, k, d], where k ∈ {n/2, (n ± 2)/2} and the minimum distance d has a lower bound which is quite close to √ n. Their duals have parameters [n, k ⊥ , d ⊥ ], where k ⊥ ∈ {n/2, (n ± 2)/2} and the minimum distance d ⊥ also has a lower bound which is quite close to √ n. These four families of codes and their duals contain distance-optimal cyclic codes. The authors are not aware of such an infinite family of ternary cyclic codes in the literature. The rest of this paper is organized as follows. In Section 2, we give a general construction of ternary cyclic codes. In Section 3, we settle the dimensions of the four infinite families of ternary cyclic codes. In Sections 4-7, we develop lower bounds on their minimum distances. In Section 8, we investigate the parameters of their dual codes. In Section 9, we conclude this paper and make some remarks. A GENERAL CONSTRUCTION OF TERNARY CYCLIC CODES Let Z n = {0, 1, 2, . . . , n − 1} be the ring of integers modulo n. For any integer b, let b mod n denote the unique b 0 ∈ Z n such that b ≡ b 0 (mod n). For any s ∈ Z n , the 3-cyclotomic coset of s modulo n is defined by C (3,n) s = {s, s3, s3 2 , . . . , s3 ls−1 } mod n ⊆ Z n , where l s is the smallest positive integer such that s ≡ s3 ls (mod n). For an integer i with 0 ≤ i ≤ 3 m − 1, let i = i m−1 3 m−1 + i m−2 3 m−2 + · · · + i 1 3 + i 0 be the 3-adic expansion of i, where i j ∈ {0, 1, 2} for 0 ≤ j ≤ m−1. For any i with 0 ≤ i ≤ n−1, define w 3 (i) = m−1 j=0 i j , which is called the 3-weight of i. From now on, we fix n = 3 m − 1 and let α denote a primitive element of F 3 m , where m ≥ 2. Define a polynomial g (i 1 ,i 2 ,m) (x) = 1≤j≤n−1 w 3 (j)≡i 1 or i 2 (mod 4) (x − α j ),(1) where i 1 and i 2 are a pair of distinct elements in the set {0, 1, 2, 3}. It is easily seen that g (i 1 ,i 2 ,m) (x) ∈ F 3 [x], as ω 3 (j) is a constant on each 3-cyclotomic coset C (3,n) s . Let C (i 1 ,i 2 ,m) May 3, 2023 DRAFT denote the ternary cyclic code of length n = 3 m − 1 with generator polynomial g (i 1 ,i 2 ,m) (x). Denote T (i 1 ,i 2 ,m) = {1 ≤ j ≤ n − 1 : w 3 (j) ≡ i 1 or i 2 (mod 4)}. It is clear that T (i 1 ,i 2 ,m) is the defining set of C (i 1 ,i 2 ,m) with respect to the n-th primitive root of unity α. Notice that the codes C (0,2,m) and C (1,3,m) have very poor minimum distances when m = 3. These codes may not be interesting. In this paper, we study C (0,3,m) , C (1,2,m) , C (0,1,m) and C (2,3,m) only for odd m, as the parameters of these four codes for some even m are not the best known. We will also study the parameters of the duals of these ternary cyclic codes. THE DIMENSIONS OF THE TERNARY CODES C (i 1 ,i 2 ,m) In this section, we always assume that m is odd and we mainly determine the dimensions of the ternary codes C (0,3,m) , C (1,2,m) , C (0,1,m) and C (2,3,m) . To this end, we compute the cardinalities of the sets T i = {1 ≤ j ≤ n − 1 : w 3 (j) ≡ i (mod 4)}, where i ∈ {0, 1, 2, 3}. Clearly, we see that T 0 ∪ T 2 = {1 ≤ j ≤ n − 1 : w 3 (j) ≡ 0 (mod 2)}, T 1 ∪ T 3 = {1 ≤ j ≤ n − 1 : w 3 (j) ≡ 1 (mod 2)}, and both are disjoint unions. It is easy to see that w 3 (j) ≡ j (mod 2). Then we have \|T 0 ∪ T 2 \| = \|T 0 \| + \|T 2 \| = (3 m − 3)/2, \|T 1 ∪ T 3 \| = \|T 1 \| + \|T 3 \| = (3 m − 1)/2.(2) We begin to determine \|T i \| for i ∈ {0, 1, 2, 3}. Consider the polynomial (1 + x + x 2 ) m ∈ Z[x]. It is easy to see the coefficient of x t in (1 + x + x 2 ) m is equal to k 1 , k 2 ≥0 k 1 +2k 2 =t m k 1 m − k 1 k 2 ,(3) May 3, 2023 DRAFT where t = 0, 1, · · · , 2m. Let ω = e 2π √ −1 4 ∈ C be a 4-th primitive root of unity. Then ω 2 + 1 = 0. Taking x = ω, it then follows from (3) that ω m = (1 + ω + ω 2 ) m = s 0 + s 1 ω + s 2 ω 2 + s 3 ω 3 = s 0 − s 2 + (s 1 − s 3 )ω, where s i = 0≤t≤2m t≡i (mod 4) k 1 , k 2 ≥0 k 1 +2k 2 =t m k 1 m − k 1 k 2 for i ∈ {0, 1, 2, 3}. Then we have s 0 = s 2 and s 1 − s 3 =        1 if m ≡ 1 (mod 4), −1 if m ≡ 3 (mod 4). It is easily deduced from the definitions of T i that \|T 0 \| = s 0 − 1 = s 2 − 1 = \|T 2 \|, \|T 1 \| = s 1 , \|T 3 \| = s 3 . Then by (2) we have        \|T 0 \| = \|T 1 \| − 1 = \|T 2 \| = \|T 3 \| = (3 m − 3)/4 if m ≡ 1 (mod 4), \|T 0 \| = \|T 1 \| = \|T 2 \| = \|T 3 \| − 1 = (3 m − 3)/4 if m ≡ 3 (mod 4).(4) When m ≡ 3 (mod 4) ≥ 3 is odd, the dimensions of C (i 1 ,i 2 ,m) are treated in the following theorem. Theorem 3.1. Let m ≡ 3 (mod 4) ≥ 3. Then we have the following. • The ternary cyclic code C (0,3,m) has length n = 3 m − 1 and dimension k = n/2, and the ternary cyclic code C (1,2,m) has length n = 3 m − 1 and dimension k = (n + 2)/2. • The ternary cyclic code C (2,3,m) has length n = 3 m − 1 and dimension k = n/2, and the ternary cyclic code C (0,1,m) has length n = 3 m − 1 and dimension k = (n + 2)/2. May 3, 2023 DRAFT Proof: Note that T (i 1 ,i 2 ,m) = T i 1 ∪ T i 2 is a disjoint union and dim(C (i 1 ,i 2 ,m) ) = n − \|T (i 1 ,i 2 ,m) \| = n − \|T i 1 \| − \|T i 2 \|. The desired conclusions then follow from (4) directly. When m ≡ 1 (mod 4) ≥ 5 is odd, the dimensions of C (i 1 ,i 2 ,m) are given in the following theorem whose proof is similar to that of Theorem 3.1 and is omitted here. THE TERNARY CODE C (0,3,m) In this section, we develop a lower bound on minimum distance of the ternary cyclic code 3,m) . When m = 3, the parameters of the code C (0, 3,3) are documented in the following example and are the parameters of the best known linear codes [4]. This code is also an optimal cyclic code [2, Table A.92]. This fact motivates us to investigate the parameters of the code C (0,C (0,3,m) for odd m.x 13 + 2x 11 + x 10 + x 8 + x 6 + x 4 + 2x 3 + 1. The following two lemmas on the defining set T (0,3,m) are necessary to derive the lower bound on minimum distance of C (0,3,m) . Lemma 4.2. Let m ≡ 1 (mod 4) ≥ 5. Let v = (3 (m+1)/2 − 1)/2 and δ = (3 (m−1)/2 + 5)/2. Then gcd(v, n) = 1. Define T (0,3,m) (v) = {vi mod n : i ∈ T (0,3,m) }. May 3, 2023 DRAFT Then {1, 2, . . . , δ − 1} ⊂ T (0,3,m) (v). Proof: Since v = (3 (m+1)/2 − 1)/2, we have gcd(2v, n) = gcd(3 (m+1)/2 − 1, 3 m − 1) = 3 gcd((m+1)/2,m) − 1 = 2. It then follows that gcd(v, n) = 1. We now prove the second desired conclusion. It can be easily verified that v −1 mod n = (3 m − 1)/2 + 3 (m+1)/2 + 1. Consider now any positive integer a with a ≤ δ − 1 = (3 (m−1)/2 + 3)/2 < 3 (m−1)/2 − 1. When a is even, we have av −1 mod n = 3 (m+1)/2 a + a. It then follows that w 3 (av −1 mod n) = 2w 3 (a). Notice that w 3 (a) ≡ a ≡ 0 (mod 2). Consequently, w 3 (av −1 mod n) ≡ 0 (mod 4). When a is odd and a ≤ δ − 1, we have 1 ≤ a ≤ δ − 2 = (3 (m−1)/2 + 1)/2. It is easily verified that av −1 mod n = ((3 m − 1)/2 + (3 (m+1)/2 + 1)a) mod n = (3 (m−1)/2 + 3 (m+1)/2 3 (m−1)/2 − 1 2 + a + 3 (m−1)/2 − 1 2 + a) mod n.(5) The rest of the proof can be divided into the following two cases by noting that a is odd. 1) Let 1 ≤ a ≤ 3 (m−1)/2 −3 2 be odd. It follows from (5) that w 3 (av −1 mod n) = 1 + 2w 3 ((3 (m−1)/2 − 1)/2 + a). May 3, 2023 DRAFT Note that (3 (m−1)/2 − 1)/2 + a is odd. Then we have w 3 ((3 (m−1)/2 − 1)/2 + a) ≡ 1 (mod 2). It then follows that w 3 (av −1 mod n) ≡ 3 (mod 4). 2) Let a = 3 (m−1)/2 +1 2 . It follows from (5) that w 3 (av −1 mod n) = w 3 ((3 (m−1)/2 + (3 (m+1)/2 + 1)3 (m−1)/2 ) mod n) = w 3 (2 · 3 (m−1)/2 + 1) ≡ 3 (mod 4). The desired second conclusion then follows. Lemma 4.3. Let m ≡ 3 (mod 4) > 3. Let v = (3 (m+1)/2 + 1)/2 and δ = (3 (m−1)/2 + 7)/2. Then gcd(v, n) = 1. Define T (0,3,m) (v) = {vi mod n : i ∈ T (0,3,m) }. Then {1, 2, . . . , δ − 1} ⊂ T (0,3,m) (v). Proof: Since v = (3 (m+1)/2 + 1)/2 and m/ gcd((m + 1)/2, m) = 1 is odd, we have gcd(2v, n) = gcd(3 (m+1)/2 + 1, 3 m − 1) = 2. It then follows that gcd(v, n) = 1. We now prove the desired second conclusion. It can be easily verified that v −1 mod n = (3 m − 1)/2 + 3 (m+1)/2 − 1. Consider now any positive a with a ≤ δ − 1 = (3 (m−1)/2 + 5)/2 < 3 (m−1)/2 − 1. When a is even, we have av −1 mod n = 3 (m+1)/2 a − a = 3 (m+1)/2 (a − 1) + 3 (m+1)/2 − a. May 3, 2023 DRAFT It then follows that w 3 (av −1 mod n) = w 3 (a − 1) + w 3 (3 (m+1)/2 − 1 − (a − 1)) = w 3 (3 (m+1)/2 − 1) = m + 1. Consequently, w 3 (av −1 mod n) ≡ 0 (mod 4). When a is odd and a ≤ δ − 1, we have 1 ≤ a ≤ δ − 2 = (3 (m−1)/2 + 3)/2. It is easily verified that av −1 mod n = ((3 m − 1)/2 + (3 (m+1)/2 − 1)a) mod n = (3 (m−1)/2 + 3 (m+1)/2 3 (m−1)/2 − 1 2 + a + 3 (m−1)/2 − 1 2 − a) mod n.(6) The rest of the proof can be divided into the following two cases. 1) Let 1 ≤ a ≤ 3 (m−1)/2 −1 2 be odd. It follows from (6) that av −1 mod n = 3 (m−1)/2 + 3 (m+1)/2 3 (m−1)/2 − 1 2 + a + 3 (m−1)/2 − 1 − 3 (m−1)/2 − 1 2 + a . Note that (3 (m−1)/2 − 1)/2 + a ≤ 3 (m−1)/2 − 1, then we have w 3 (av −1 mod n) = 1 + w 3 ((3 (m−1)/2 − 1)/2 + a) + w 3 (3 (m−1)/2 − 1 − ((3 (m−1)/2 − 1)/2 + a)) = 1 + w 3 (3 (m−1)/2 − 1) = m ≡ 3 (mod 4). 2) Let a = 3 (m−1)/2 +3 2 . It follows from (6) that w 3 (av −1 mod n) = w 3 ((3 (m+1)/2 (3 (m−1)/2 + 1) + 3 (m−1)/2 − 2) mod n) = w 3 (3 (m+1)/2 + 3 (m−1)/2 − 1) = m ≡ 3 (mod 4). May 3, 2023 DRAFT The desired second conclusion then follows. The following theorem gives two lower bounds on the minimum distance of C (0,3,m) , which are obtained from Lemmas 4.2 and 4.3 directly by employing the BCH bound on cyclic codes. The dimension of the code follows from Theorems 3.1 and 3.2. Theorem 4.4. Let m ≥ 5 be odd. Then the ternary cyclic code C (0,3,m) has parameters [n, k, d] with d ≥      3 (m−1)/2 +7 2 if m ≡ 3 (mod 4), 3 (m−1)/2 +5 2 if m ≡ 1 (mod 4) and k =      n 2 if m ≡ 3 (mod 4), n+2 2 if m ≡ 1 (mod 4). Let δ max (v) denote the largest value ℓ such that {a, (a + 1) mod n, . . . , (a + ℓ − 1) mod n} ⊂ T (0,3,m) (v) for some integer a with 0 ≤ a ≤ n − 1. Define δ max = max{δ max (v) : 1 ≤ v ≤ n − 1, gcd(v, n) = 1}. Then we have the following results on m and corresponding δ max by Magma: (m, δ max ) = (3, 5), (5, 11), (7,19), (9, 43). When m = 9, we have The parameters of the code C (1,2,3) are documented in the following example and are the parameters of the best linear codes known [4]. This code is an optimal cyclic code [2, Table A.92]. Example 5.1. Let m = 3. Then C (1,2,m) has parameters [26, 14, 7] and generator polynomial x 12 + x 11 + 2x 10 + x 9 + 2x 8 + 2x 7 + x 6 + x 5 + x 4 + 2x 3 + x 2 + x + 1. The following two lemmas on the defining set T (1,2,m) are necessary to derive the lower bound on the minimum distance of C (1,2,m) . It then follows that gcd(v, n) = 1. We now prove the desired second conclusion. It can be easily verified that v −1 mod n = (3 m − 1)/2 − 3 (m+1)/2 + 3. Consider now any positive a with a ≤ δ − 1 = (3 (m−1)/2 + 11)/2 < 3 (m−1)/2 − 1. When a is even, we have av −1 mod n = n − 3 (m+1)/2 a + 3a, It then follows that w 3 (av −1 mod n) = w 3 (3 m−1 av −1 mod n) = w 3 (a − 1) + w 3 (3 (m+1)/2 − 1 − (a − 1)) = w 3 (3 (m+1)/2 − 1) = m + 1. Consequently, w 3 (av −1 mod n) ≡ 2 (mod 4). When a is odd and a ≤ δ − 1, we have 1 ≤ a ≤ δ − 2 = (3 (m−1)/2 + 9)/2. It is easily verified = 3 m−1 + 3 (m−1)/2 3 (m−1)/2 − 1 2 − a + 3 (m−1)/2 − 1 2 + a.(7) Clearly, w 3 (av −1 mod n) = w 3 (3 m−1 av −1 mod n). The rest of the proof can be divided into the following four cases. 1) Let 1 ≤ a ≤ 3 (m−1)/2 −1 2 be odd. It follows from (7) that 3 m−1 av −1 mod n = 3 m−1 + 3 (m−1)/2 3 (m−1)/2 − 1 2 − a + 3 (m−1)/2 − 1 − 3 (m−1)/2 − 1 2 − a . Then we have w 3 (av −1 mod n) = w 3 (3 m−1 av −1 mod n) = 1 + w 3 ((3 (m−1)/2 − 1)/2 − a) + w 3 (3 (m−1)/2 − 1 − ((3 (m−1)/2 − 1)/2 − a)) = 1 + w 3 (3 (m−1)/2 − 1) = m ≡ 1 (mod 4). 2) Let a = 3 (m−1)/2 +1 2 . It follows from (7) that w 3 (av −1 mod n) = w 3 (3 m−1 ) ≡ 1 (mod 4). 3) Let a = 3 (m−1)/2 +5 2 . It follows from (7) that w 3 (av −1 mod n) = w 3 (3 m−1 − 3 (m+1)/2 + 3 (m−1)/2 + 2) = m ≡ 1 (mod 4). 4) Let a = 3 (m−1)/2 +9 2 . It follows from (7) that w 3 (av −1 mod n) = w 3 (3 m−1 − 3 (m+1)/2 − 3 (m−1)/2 + 3 + 1) = m ≡ 1 (mod 4). The desired second conclusion then follows. It then follows that gcd(v, n) = 1. We now prove the desired second conclusion. It can be easily verified that v −1 mod n = (3 m − 1)/2 − 3 (m+1)/2 − 3. Consider now any positive a with a ≤ δ − 1 = (3 (m−1)/2 + 9)/2 < 3 (m−1)/2 − 1. When a is even, we have av −1 mod n = n − 3 (m+1)/2 a − 3a, and 3 m−1 av −1 mod n = n − 3 (m−1)/2 a − a = 3 (m−1)/2 (3 (m+1)/2 − 1 − a) + 3 (m−1)/2 − 1 − a. It then follows that w 3 (av −1 mod n) = w 3 (3 m−1 av −1 mod n) = w 3 (3 (m+1)/2 − 1 − a) + w 3 (3 (m−1)/2 − 1 − a) = m + 1 − w 3 (a) + m − 1 − w 3 (a) = 2m − 2w 3 (a). Notice that w 3 (a) ≡ a ≡ 0 (mod 2). Consequently, w 3 (av −1 mod n) ≡ 2m ≡ 2 (mod 4). Clearly, w 3 (av −1 mod n) = w 3 (3 m−1 av −1 mod n). The rest of the proof can be divided into the following three cases. 1) Let 1 ≤ a ≤ 3 (m−1)/2 −1 2 be odd. It follows from (8) that w 3 (av −1 mod n) = 1 + 2w 3 ((3 (m−1)/2 − 1)/2 − a). Note that (3 (m−1)/2 − 1)/2 − a is even, we have w 3 ((3 (m−1)/2 − 1)/2 − a) ≡ 0 (mod 2). It then follows that w 3 (av −1 mod n) ≡ 1 (mod 4). 2) Let a = 3 (m−1)/2 +3 2 . It follows from (8) that w 3 (av −1 mod n) = w 3 (3 m−1 − 2 · 3 (m−1)/2 − 2) = 2m − 5 ≡ 1 (mod 4). 3) Let a = 3 (m−1)/2 +7 2 . It follows from (8) that w 3 (av −1 mod n) = w 3 (3 m−1 − 3 (m+1)/2 − 3 (m−1)/2 − 3 − 1) = 2m − 5 ≡ 1 (mod 4). The desired second conclusion then follows. The following theorem gives two lower bounds on the minimum distance of C (1,2,m) , which are obtained from Lemmas 5.2 and 5.3 directly by employing the BCH bound on cyclic codes. The dimension of the code follows from Theorems 3.1 and 3.2. The parameters of the code C (0,1,3) are documented in the following example and are the parameters of the best linear codes known [4]. This code is an optimal cyclic code [2, Table A.92]. Example 6.1. Let m = 3. Then C (0,1,m) has parameters [26,14,7] and generator polynomial x 12 + x 11 + x 10 + 2x 9 + x 8 + x 7 + x 6 + 2x 5 + 2x 4 + x 3 + 2x 2 + x + 1. The following two lemmas on the defining set T (0,1,m) are necessary to develop the lower bound on the minimum distance of C (0,1,m) . It then follows that gcd(v, n) = 1. We now prove the desired second conclusion. It can be easily verified that v −1 mod n = (3 m − 1)/2 − 3 (m+1)/2 + 3. Consider now any positive a with a ≤ δ − 1 = (3 (m−1)/2 + 11)/2 < 3 (m−1)/2 − 1. When a is even, we have It then follows that w 3 ((n − a)v −1 mod n) = w 3 (3 m−1 (n − a)v −1 mod n) = w 3 (a − 1) + w 3 (3 (m−1)/2 − 1 − (a − 1)) = w 3 (3 (m−1)/2 − 1) = m − 1. Consequently, be odd. It follows from (9) that w 3 ((n − a)v −1 mod n) ≡ 0 (mod 4).3 m−1 (n − a)v −1 mod n = 3 m−1 + 3 (m−1)/2 3 (m−1)/2 − 1 − 3 (m−1)/2 − 1 2 − a + 3 (m−1)/2 − 1 2 − a. Note that (3 (m−1)/2 − 1)/2 − a ≤ 3 (m−1)/2 − 1, then we have w 3 (av −1 mod n) = 1 + w 3 (3 (m−1)/2 − 1 − ((3 (m−1)/2 − 1)/2 − a)) + w 3 ((3 (m−1)/2 − 1)/2 − a) = 1 + w 3 (3 (m−1)/2 − 1) = m ≡ 1 (mod 4). 2) Let a = 3 (m−1)/2 +1 2 . It follows from (9) that 1 (mod 4). w 3 ((n − a)v −1 mod n) = w 3 (2 · 3 m−1 − 1) = 2m − 1 ≡3) Let a = 3 (m−1)/2 +5 2 . It follows from (9) that w 3 ((n − a)v −1 mod n) = w 3 (2 · 3 m−1 + 2 · 3 (m−1)/2 − 3) = m ≡ 1 (mod 4). 4) Let a = 3 (m−1)/2 +9 2 . It follows from (9) that w 3 ((n − a)v −1 mod n) = w 3 (2 · 3 m−1 + 3 (m+1)/2 + 3 (m−1)/2 − 3 − 2) = m ≡ 1 (mod 4). The desired second conclusion then follows. It then follows that gcd(v, n) = 1. We now prove the desired second conclusion. It can be easily verified that v −1 mod n = (3 m − 1)/2 − 3 (m+1)/2 − 3. Consider now any positive a with a ≤ δ − 1 = (3 (m−1)/2 + 9)/2 < 3 (m−1)/2 − 1. When a is even, we have (n − a)v −1 mod n = 3 (m+1)/2 a + 3a, and 3 m−1 (n − a)v −1 mod n = 3 (m−1)/2 a + a. It then follows that w 3 ((n − a)v −1 mod n) = w 3 (3 m−1 (n − a)v −1 mod n) = 2w 3 (a). Notice that w 3 (a) ≡ a ≡ 0 (mod 2). Consequently, It then follows that w 3 ((n − a)v −1 mod n) ≡ 1 (mod 4). w 3 ((n − a)v −1 mod n) ≡ 0 (mod 4). 2) Let a = 3 (m−1)/2 +3 2 . It follows from (10) that 1 (mod 4). w 3 ((n − a)v −1 mod n) = w 3 (3 m−1 + (3 (m−1)/2 + 1) 2 ) = 5 ≡ 3) Let a = 3 (m−1)/2 +7 2 . It follows from (10) that w 3 ((n − a)v −1 mod n) = w 3 (3 m−1 + (3 (m−1)/2 + 1)(3 (m−1)/2 + 3)) = w 3 (2 · 3 m−1 + 3 (m+1)/2 + 3 (m−1)/2 + 3) = 5 ≡ 1 (mod 4). The desired second conclusion then follows. The following theorem gives two lower bounds on the minimum distance of C (0,1,m) , which are obtained from Lemmas 6.2 and 6.3 directly by employing the BCH bound on cyclic codes. The dimension of the code follows from Theorems 3.1 and 3.2. THE TERNARY CYCLIC CODES C (2,3,m) The parameters of the code C (2,3,3) are documented in the following example and are the parameters of the best linear codes known [4]. This code is an optimal cyclic code [2, Table A.92]. Example 7.1. Let m = 3. Then C (2,3,m) has parameters [26, 13, 8] and generator polynomial x 13 + 2x 10 + x 9 + x 7 + x 5 + x 3 + 2x 2 + 1. The following two lemmas on the defining set T (2,3,m) are necessary to develop the lower bound on the minimum distance of C (2,3,m) . Their proofs are very similar to those of Lemmas 6.2 and 6.3 and omitted here. The following theorem gives two lower bounds on the minimum distance of C (2,3,m) , which are obtained from Lemmas 7.2 and 7.3 directly by employing the BCH bound on cyclic codes. if m ≡ 1 (mod 4). THE DUAL CODES OF THESE TERNARY CYCLIC CODES In this section, we investigate the parameters of the dual codes of C (0,3,m) , C (1,2,m) , C (0, 1,m) and C (2,3,m) for odd m. A. The ternary cyclic codes C ⊥ (0,3,m) The parameters of the code C ⊥ (0, 3,3) are documented in the following example and are the parameters of the best linear codes known [4]. This code is an optimal cyclic code [2, Table A.92]. x 13 + x 10 + 2x 9 + x 6 + 2x 4 + x 3 + 2x 2 + 2. Proof: Note that (x − 1)g (0,3,m) (x)g (1,2,m) (x) = x n − 1. The complement code C c (0,3,m) of C (0,3,m) has generator polynomial (x − 1)g (1,2,m) (x) and is a subcode of C (1,2,m) with dim(C c (0,3,m) ) = dim(C (1,2,m) ) − 1. It is well-known that C c (0,3,m) and C ⊥ (0,3,m) have the same parameters [3]. Note that 0 is contained in the defining set of C c (0,3,m) . The desired conclusions then follow from Theorem 5.4. B. The ternary cyclic codes C ⊥ (1,2,m) The parameters of the code C ⊥ (1,2,3) are documented in the following example and are the parameters of the best linear codes known [4]. This code is an optimal cyclic code [2, if m ≡ 1 (mod 4). Proof: Note that (x − 1)g (0,3,m) (x)g (1,2,m) (x) = x n − 1. May 3, 2023 DRAFT The complement code C c (1,2,m) of C (1,2,m) has generator polynomial (x − 1)g (0,3,m) (x) and is a subcode of C (0,3,m) . The desired conclusions then similarly follow from Theorem 4.4. C. The ternary cyclic codes C ⊥ (0,1,m) The parameters of the code C ⊥ (0,1,3) are documented in the following example and are the parameters of the best linear codes known [4]. This code is an optimal cyclic code [2, The parameters of the code C ⊥ (2,3,3) are documented in the following example and are the parameters of the best linear codes known [4]. This code is an optimal cyclic code [2, The binary codes in [5] and the ternary codes C (i 1 ,i 2 ,m) treated in this paper can certainly be generalised to cyclic codes C (i 1 ,i 2 ,q,m) over F q for any prime power q. But it is amazing that these codes C (i 1 ,i 2 ,q,m) over F q may have very bad parameters for other q. For example, we have the following numerical results. • C (2,3,5,3) In addition, the development of the lower bounds on the minimum distances of these codes depends on the specific value of q. The authors do not see a uniform way for developing the lower bounds on the binary cyclic codes in [5] and the lower bounds on the ternary cyclic codes in this paper. This explains why only the ternary case was treated in this paper and the binary case was considered in [5] separately. Theorem 3. 2 . 2Let m ≡ 1 (mod 4) ≥ 5. Then we have the following.• The ternary cyclic code C (0,3,m) has length n = 3 m − 1 and dimension k = (n + 2)/2, and the ternary cyclic code C (1,2,m) has length n = 3 m − 1 and dimension k = n/2.• The ternary cyclic code C (2,3,m) has length n = 3 m − 1 and dimension k = (n + 2)/2, and the ternary cyclic code C (0,1,m) has length n = 3 m − 1 and dimension k = n/2. Example 4. 1 . 1Let m = 3. Then C (0,3,m) has parameters [26, 13, 8] and generator polynomial Lemma 5. 2 . 2Let m ≡ 1 (mod 4) > 1. Let v = (3 (m−1)/2 + 1)/2 and δ = (3 (m−1)/2 + 13)/2.Then gcd(v, n) = 1. DefineT (1,2,m) (v) = {vi mod n : i ∈ T (1,2,m) }.Then {1, 2, . . . , δ − 1} ⊂ T (1,2,m) (v). Proof: Since v = (3 (m−1)/2 + 1)/2 and m/ gcd((m − 1)/2, m) = 1 is odd, we have gcd(2v, n) = gcd(3 (m−1)/2 + 1, 3 m − 1) = 2. ( 3 3(m+1)/2 − a) + a − 1. that av −1 mod n = ((3 m − 1)/2 − (3 (m−1)/2 − 1)3a) mod n, and 3 m−1 av −1 mod n = (3 m − 1)/2 − 3 (m−1)/2 a + a Lemma 5. 3 . 3Let m ≡ 3 (mod 4) > 3. Let v = (3 (m−1)/2 − 1)/2 and δ = (3 (m−1)/2 + 11)/2. Then gcd(v, n) = 1. Define T (1,2,m) (v) = {vi mod n : i ∈ T (1,2,m) }. Then {1, 2, . . . , δ − 1} ⊂ T (1,2,m) (v). Proof: Since v = (3 (m−1)/2 − 1)/2, we have gcd(2v, n) = gcd(3 a is odd and a ≤ δ − 1, we have 1 ≤ a ≤ δ − 2 = (3 (m−1)/2 + 7)/2. It is easily verified that av −1 mod n = (3 m − 1)/2 − (3 (m−1)/2 + 1)3a mod n, and 3 m−1 av −1 mod n = (3 m − 1)/2 − (3 (m−1)/2 + 1)a = 3 m−1 + 3 (m−1)/2 3 (m−1)/2 − 1 2 − a + 3 (m−1)/2 − 1 2 − a. Theorem 5. 4 . 4Let m ≥ 5 be odd. Then the ternary cyclic code C (1,2,m) has parameters [n, Lemma 6. 2 . 2Let m ≡ 1 (mod 4) > 1. Let v = (3 (m−1)/2 + 1)/2 and δ = (3 (m−1)/2 + 13)/2.Then gcd(v, n) = 1. DefineT (0,1,m) (v) = {vi mod n : i ∈ T (0,1,m) }.Then {n − (δ − 1), . . . , n − 2, n − 1} ⊂ T (0,1,m) (v). Proof: Since v = (3 (m−1)/2 + 1)/2 and m/ gcd((m − 1)/2, m) = 1 is odd, we have gcd(2v, n) = gcd(3 (m−1)/2 + 1, 3 m − 1) = 2. (n − a)v −1 mod n = 3 (m+1)/2 a − 3a, a is odd and a ≤ δ − 1, we have 1 ≤ a ≤ δ − 2 = (3 (m−1)/2 + 9)/2. It is easily verified that (n − a)v −1 mod n = (3 m − 1)/2 + (3 (m−1)/2 − 1)3a mod n, and 3 m−1 (n − a)v −1 mod n =(3 m − 1)w 3 ((n − a)v −1 mod n) = w 3 (3 m−1 (n − a)v −1 mod n). The rest of the proof can be divided into the following four cases. 1) Let 1 ≤ a ≤ 3 (m−1)/2 −1 2 Lemma 6. 3 . 3Let m ≡ 3 (mod 4) > 3. Let v = (3 (m−1)/2 − 1)/2 and δ = (3 (m−1)/2 + 11)/2. Then gcd(v, n) = 1. Define T (0,1,m) (v) = {vi mod n : i ∈ T (0,1,m) }. Then {n − (δ − 1), . . . , n − 2, n − 1} ⊂ T (0,1,m) (v). Proof: Since v = (3 (m−1)/2 − 1)/2, we have gcd(2v, n) = gcd(3 (3 m − 1 1a is odd and a ≤ δ − 1, we have 1 ≤ a ≤ δ − 2 = (3 (m−1)/2 + 7)/2. It is easily verified that (n − a)v −1 mod n = ((3 m − 1)/2 + (3 (m−1)/2 + 1)3a) mod n, and 3 m−1 (n − a)v −1 mod n = w 3 ((n − a)v −1 mod n) = w 3 (3 m−1 (n − a)v −1 mod n). The rest of the proof can be divided into the following three cases. 1) Let 1 ≤ a ≤ 3 (m−1)/2 −1 2 be odd. It follows from (10) that w 3 ((n − a)v −1 mod n) = 1 + 2w 3 ((3 (m−1)/2 − 1)/2 + a). Theorem 6. 4 . 4Let m ≥ 3 be odd. Then the ternary cyclic code C (0,1,m) has parameters [n, Lemma 7. 2 . 2Let m ≡ 1 (mod 4) > 1. Let v = (3 (m+1)/2 − 1)/2 and δ = (3 (m−1)/2 + 5)/2. Then gcd(v, n) = 1. Define T (2,3,m) (v) = {vi mod n : i ∈ T (2,3,m) }. Then {n − (δ − 1), . . . , n − 2, n − 1} ⊂ T (2,3,m) (v).Lemma 7.3. Let m ≡ 3 (mod 4) > 3. Let v = (3 (m+1)/2 + 1)/2 and δ = (3 (m−1)/2 + 7)/2. Then gcd(v, n) = 1. Define T (2,3,m) (v) = {vi mod n : i ∈ T (2,3,m) }. Then {n − (δ − 1), . . . , n − 2, n − 1} ⊂ T (2,3,m) (v). Example 8. 1 . 1Let m = 3. Then the dual code C ⊥ (0,3,m) has parameters [26, 13, 8] and generator polynomial Theorem 8. 2 . 2Let m ≥ 3 be odd. Then the ternary cyclic code C ⊥ (0,3,m) has parameters [n, m ≡ 1 (mod 4), n 2 if m ≡ 3 (mod 4). The proof is similar to that of Theorem 8.2 and omitted here.9. SUMMARY AND CONCLUDING REMARKSThe main contributions of this paper are the construction and analysis of the four infinite families of ternary cyclic codes with parameters[3 m − 1, k ∈ {(3 m − 1)/2, (3 m + 1)/2}, d]for odd m, where d has a square-root-like lower bound. Their dual codes have parameters[3 m − 1, k ⊥ ∈ {(3 m − 1)/2, (3 m − 3)/2}, d ⊥ ]for odd m, where d ⊥ also has a square-root-like lower bound. When m = 3, these codes are among the best linear codes known and are optimal ternary cyclic codes. The authors are not aware of any ternary cyclic code that is better than a code in the four families of codes and their duals of length 3 m − 1 for odd m. These ternary codes presented in this paper are interesting in the sense they are the first infinite families of ternary codes codes with parameters of the form [n, k ∈ {n/2, (n ± 2)/2}, d], where d has a square-root-like lower bound. Table A . A92]. Example 8.3. Let m = 3. Then the dual code C ⊥ (1,2,m) has parameters [26, 12, 9] and generator polynomialx 14 + 2x 13 + 2x 11 + 2x 10 + 2x 9 + x 8 + 2x 7 + x 6 + 2x 5 + x 4 + x 3 + x 2 + x + 2.Theorem 8.4. Let m ≥ 3 be odd. Then the ternary cyclic code C ⊥ (1,2,m) has parameters [n, k, d]with d ≥      3 (m−1)/2 +9 2 if m ≡ 3 (mod 4), 3 (m−1)/2 +7 2 if m ≡ 1 (mod 4) and k =      n−2 2 if m ≡ 3 (mod 4), n 2 Table A .92]. AExample 8.5. Let m = 3. Then the dual code C ⊥ (0,1,m) has parameters[26, 12, 9] and generator polynomial x 14 + 2x 13 + 2x 12 + 2x 11 + 2x 10 + x 9 + 2x 8 + x 7 + 2x 6 + x 5 + x 4 + x 3 + x + 2.Theorem 8.6. Let m ≥ 3 be odd. Then the ternary cyclic code C ⊥ (0,1,m) has parameters [n, k, d]Proof: The proof is similar to that of Theorem 8.2 and omitted here.with d ≥      3 (m−1)/2 +9 2 if m ≡ 3 (mod 4), 3 (m−1)/2 +7 2 if m ≡ 1 (mod 4) and k =      n−2 2 if m ≡ 3 (mod 4), n 2 if m ≡ 1 (mod 4). D. The ternary cyclic codes C ⊥ (2,3,m) Table A .92]. AExample 8.7. Let m = 3. Then the dual code C ⊥ (2,3,m) has parameters [26, 13, 8] and generator polynomialx 13 + x 11 + 2x 10 + x 9 + 2x 7 + x 4 + 2x 3 + 2.Theorem 8.8. Let m ≥ 3 be odd. Then the ternary cyclic code C ⊥ (2,3,m) has parameters [n, k, d]with d ≥      has parameters [124, 62, 3] and C ⊥ (2,3,5,3) has parameters [124, 62, 3].• C (0,1,5,3) has parameters [124, 63, 3] and C ⊥ (0,1,5,3) has parameters [124, 61, 4]. • C (0,3,5,3) has parameters [124, 63, 3] and C ⊥ (0,3,5,3) has parameters [124, 61, 4]. • C (1,2,5,3) has parameters [124, 62, 3] and C ⊥ (1,2,5,3) has parameters [124, 62, 3]. • C (1,3,5,3) has parameters [124, 62, 2] and C ⊥ (1,3,5,3) has parameters [124, 62, 2]. • C (0,2,5,3) has parameters [124, 63, 2] and C ⊥ (0,2,5,3) has parameters [124, 61, 4]. May 3, 2023DRAFT Open problems on cyclic codes. P Charpin, Handbook of Coding Theory. V. S. Pless and W. C. HuffmanAmsterdam, The NetherlandsElsevier1P. Charpin, "Open problems on cyclic codes," in Handbook of Coding Theory, vol. 1, V. S. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier, 1998, pp. 963-1063. C Ding, Codes from Difference Sets. SingaporeWorld ScientificC. Ding, Codes from Difference Sets. Singapore: World Scientific, 2018. W C Huffman, V Pless, Fundamentals Error-Correcting Codes. Cambridge, U.K.Cambridge Univ. PressW. C. Huffman and V. Pless, Fundamentals Error-Correcting Codes. Cambridge, U.K.: Cambridge Univ. Press, 2003. Bounds on the minimum distance of linear codes and quantum codes. M , M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at http://www.codetables.de, accessed on 2023-1-1. Five infinite families of binary cyclic codes and their related codes with good parameters. H Liu, C Li, C Ding, arXiv:2301.06446H. Liu, C. Li, C. Ding, "Five infinite families of binary cyclic codes and their related codes with good parameters," arXiv:2301.06446. Parameters of several families of binary duadic codes and their related codes. H Liu, C Li, H Qian, arXiv:2302.13461H. Liu, C. Li, H. Qian, "Parameters of several families of binary duadic codes and their related codes," arXiv:2302.13461. Binary [n, (n + 1)/2] cyclic codes with good minimum distances. C Tang, C Ding, IEEE Trans. Inf. Theory. 6812C. Tang, C. Ding, "Binary [n, (n + 1)/2] cyclic codes with good minimum distances," IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022.	[]
[ "Monte Carlo generator ELRADGEN 2.0 for simulation of radiative events in elastic ep-scattering of polarized particles", "Monte Carlo generator ELRADGEN 2.0 for simulation of radiative events in elastic ep-scattering of polarized particles" ]	[ "I Akushevich ", "O F Filoti ", "A Ilyichev ", "N Shumeiko ", "\nDuke University\n27708DurhamNCUSA\n", "\nNational Center of Particle and High Energy Physics\nUniversity of New Hampshire\n03824, 220040Durham, MinskNHUSA, Belarus\n" ]	[ "Duke University\n27708DurhamNCUSA", "National Center of Particle and High Energy Physics\nUniversity of New Hampshire\n03824, 220040Durham, MinskNHUSA, Belarus" ]	[]	The structure and algorithms of the Monte-Carlo generator ELRADGEN 2.0 designed to simulate radiative events in polarized ep-scattering are presented. The full set of analytical expressions for the QED radiative corrections is presented and discussed in detail. Algorithmic improvements implemented to provide faster simulation of hard real photon events are described. Numerical tests show high quality of generation of photonic variables and radiatively corrected cross section. The comparison of the elastic radiative tail simulated within the kinematical conditions of the BLAST experiment at MIT BATES shows a good agreement with experimental data.	10.1016/j.cpc.2012.01.015	[ "https://arxiv.org/pdf/1104.0039v1.pdf" ]	37,673,552	1104.0039	7651547cad431d2550d90c250d86236c023fda4b	Monte Carlo generator ELRADGEN 2.0 for simulation of radiative events in elastic ep-scattering of polarized particles 31 Mar 2011 I Akushevich O F Filoti A Ilyichev N Shumeiko Duke University 27708DurhamNCUSA National Center of Particle and High Energy Physics University of New Hampshire 03824, 220040Durham, MinskNHUSA, Belarus Monte Carlo generator ELRADGEN 2.0 for simulation of radiative events in elastic ep-scattering of polarized particles 31 Mar 2011 The structure and algorithms of the Monte-Carlo generator ELRADGEN 2.0 designed to simulate radiative events in polarized ep-scattering are presented. The full set of analytical expressions for the QED radiative corrections is presented and discussed in detail. Algorithmic improvements implemented to provide faster simulation of hard real photon events are described. Numerical tests show high quality of generation of photonic variables and radiatively corrected cross section. The comparison of the elastic radiative tail simulated within the kinematical conditions of the BLAST experiment at MIT BATES shows a good agreement with experimental data. Introduction The exclusive photon production in lepton-nucleon scattering is the routine experimental tool in investigating the hadronic structure. Depending on the design of experiments, the measurements of this process can give an access to the generalized parton distributions [1,2] or the generalized polarizabilities [3,4]. In some cases the exclusive photon production appears as a background effect to inelastic [5,6] or elastic [7] lepton nucleon scattering. The last scenario, i.e. the situation when the events with the real photon emission accompany the elastic electron-proton scattering is the most advanced due to the infrared problem, therefore it will be in our main focus. The set of processes contributed to the observed cross section in the next order of perturbation theory is referred to as the lowest order radiative corrections (RC). The basic contribution to the lowest order RC appears from the square of amplitude that only includes real photon emission from the lepton leg. This contribution contains the so-called large logarithm (i.e., the logarithm of the lepton mass) and normally is only held in the lowest order RC. In practice of data analysis, RC are calculated theoretically or their contribu-tion to the observed cross sections (or asymmetries) are minimized by experimental methods. Due to finite detector resolution, a complete removal of the events with radiated hard photon(s) by pure experimental methods is not possible. Furthermore, the contributions of additional virtual particles and soft photon emission cannot be removed in principle. The theoretical calculation provides with analytical expressions included the contributions of loops and photon emission which are infrared free after the procedure of the cancellation of the infrared divergence. The contribution of the hard photon radiation is presented in the form of integrals over photon phase space. Partly the integration is performed analytically without additional simplifying assumptions or assumptions on specific functional forms describing hadronic structure. The pioneering approach for RC calculation in inelastic processes was suggested by Mo and Tsai in their seminal paper in 1969 [8]. They also developed the peaking approximation allowing for analytical estimating integrals over photon angles. The approximation is used in many data analysis, e.g. , in ref. [9] the electromagnetic RC in elastic ep-scattering was calculated in peaking approximation with taking into account the one-photon emission both from lepton and hadron legs. The Mo and Tsai approach requires involvement of the artificial parameter ∆ separating the integration region over photon energy on parts with soft and hard photon contributions. To cancel infrared divergence analytically only leading terms in the expansion of the soft photon contribution over reciprocal of the photon energy are kept. As a result, the final expressions contain undesired dependence on the artificial parameter. Bardin and Shumeiko developed the approach [10] for exact separation and cancellation of infrared divergence when the final expressions for RC were completely free from any artificial parameters like ∆. Using this approach, RC in polarized elastic ep-scattering withing QED theory has been calculated in refs. [11,12]. Basing on these calculations the FORTRAN code MASCARAD has been developed and successfully used for data processing of the relevant parity conservation experiments [13,14]. Other approaches were also used for RC calculation in elastic ep-scattering. Thus, the total lowest order RC (both to lepton and hadron legs) was also calculated in [7] with soft photon approximation and the method of electron structure functions suggested in the work [15] was also applied for estimation of RC to elastic ep scattering [16,17]. The use of realistic detector geometry requires essentially complicated integration over the real photon phase space. As a result, the researchers come to the necessity of using the Monte Carlo technique which constitutes a complementary approach to the theoretical calculations of RC using respective codes such as MASCARAD. The Monte Carlo generators for simulation of radiative events have been developed for many specific processes and intensively used in data analysis. Thus, the Monte Carlo generator RADGEN [18] for simulation of radiative events in inclusive deep inelastic scattering has been developed on the basis of the FORTRAN code POLRAD [19]. The Monte Carlo generator MERADGEN [20] for simulation of radiative events in Møller scattering appeared on the base of FORTRAN code MERA [21]. In this paper we present and describe in detail the latest version 2.0 of the Monte Carlo generator ELRADGEN. The prototype of the code [22] dealt with the simulation of real hard photon emission as a background effect in the unpolarized elastic electron-proton scattering. The present version 2.0 is extended on the initial polarized particles: longitudinally polarized electron and arbitrary polarized proton. The theoretical background for the developments is presented in ref. [11]. The paper is organized as follows. Section 2 describes the kinematics of the investigated process and the generation method. The different contributions to the lowest order RC and the multi-soft photon emission are presented and discussed in Section 3. The brief structure of the code and the input-output datafiles are described in Section 4. Test runs, comparison with MASCARAD, and numerical comparison of the simulated and measured cross sections of the radiative tail from elastic peak in the BLAST experiment are presented in Section 5. Conclusions and final remarks are given in Section 6. The four-momenta reconstruction formulae, explicit expressions for the lepton and target polarization vectors, some lengthy formulae for RC, and test outputs are given in Appendices. Kinematics and Method of Generation The lowest order (Born) ( Fig.1 (a)) as well as the additional virtual particle ( Fig.1 (b,c)) contributions to the polarized elastic lepton-nucleon scattering e(k 1 , ξ L ) + p(p 1 , η) −→ e ′ (k 2 ) + p ′ (p 2 )(1) (k 2 1 = k 2 2 = m 2 , p 2 1 = p 2 2 = M 2 ) can be described by the following three variables: Q 2 = −q 2 = −(k 1 − k 2 ) 2 , S = 2k 1 p 1 , φ,(2) where φ is the azimuthal angle between the scattering plane (k 1 , k 2 ) and the ground level. The Lab system is used with OZ axis along the beam direction and plane OZX parallel to the ground level. The explicit expressions of polarization vectors (ξ L and η) and four-momenta reconstructed in the lab system are presented in Appendix A. The description of the phase space of the radiative process ( Fig.1(d,e)) e(k 1 , ξ L ) + p(p 1 , η) −→ e ′ (k 2 ) + p ′ (p 2 ) + γ(k),(3) (k 2 = 0) requires three new kinematic variables: a virtual proton transfer momentum squared t = −(k 1 − k 2 − k) 2 , the inelasticity v = (p 2 + k) 2 − M 2 , and the azimuthal angle φ k between the planes (q, k) and (k 1 , k 2 ). This set of variables defines the four-momenta of all final particles. The simulation of radiative events requires an additional definition of the lowest bound of the photon energy (or another respective quantity, inelasticity v min in our case) separating the photon phase space into the region of soft and hard photons. Only hard photons need to be simulated while soft photons cannot be simulated because of the infrared divergence. The observed cross section can be presented in terms of two positively definite parts: σ obs = σ rad (v min ) + σ BSV (v min ).(4) The first term, σ rad (v min ), describes the cross section with an additional hard photon emitted, and the second, σ BSV (v min ), contains the contributions of the Born cross section, soft-photon emission, and virtual corrections. Here and later we define σ ≡ dσ/dQ 2 dφ. Note that σ obs does not depend on v min while terms σ BSV and σ rad do. The strategy for simulation of one event can be defined in a standard way [22]: • For the fixed initial energy, Q 2 , the angle φ, and the missing mass square resolution v min , the two positively-definite contributions to the observed (radiative-corrected) cross section σ obs , σ rad (v min ), and σ BSV (v min ) are calculated separately. • The corresponding channel of scattering (i.e., BSV or radiative process) is simulated for this event in accordance with partial contributions of these two positive parts into the total cross section. More specifically, the channel of scattering is simulated in accordance with the Bernoulli trial where the probability of "success" (i.e., radiative channel) is calculated as a ratio of the radiative part of the cross section to the total cross section. • For the radiative event the kinematic variables t, v and φ k are simulated in accordance with their calculated distributions. The distributions of v and φ k are conditional (e.g. , v is simulated conditionally on t, and φ k is simulated conditionally on t and v). The explicit expressions for the probability densities of these variables are defined by eqs. 15. • The four-momenta of all final particles in a required reference frame are calculated. The initial values of Q 2 (and φ) can be non-fixed but externally simulated according to a probability distribution (for example, the Born cross section). If the Q 2 distribution is simulated over the Born cross section, then the realistic observed Q 2 distribution is calculated as sum of weights computed as ratios of the total and Born cross sections for each simulated event. If the observed cross section is used for the simulation of Q 2 , then reweighting is not required. 3 Explicit expressions for σ rad (v min ) and σ BSV (v min ) The analytical expressions for the lowest order RC on which the ELRAD-GEN is based, were obtained in ref. [11] (see eqs. (50) and (51)). The result for the observed cross section can be formally outlined as σ obs = (1 + δ)σ 0 + C dv v [σ R (v) − σ 0 ], where C is a kinematic coefficient proportional to α and the quantity σ R (v) is proportional to the bremsstrahlung cross section (σ R (0) = σ 0 ). This expression does not reproduce the form of eq. (4), because the term with the integral is not positively definite and the term with σ R (v) cannot be separated because it is singular for v → 0. Instead, the following transformation of this term was used: dv v (σ R (v) − σ 0 ) = dv v σ R (v)θ(v − v min ) − dv v σ 0 θ(v − v min ) + dv v [σ R (v) − σ 0 ]θ(v min − v).(5) The first term in (5) represents the contribution of hard photons, i.e., with inelasticity above v min . This term is positively definite and it is used as σ rad (v min ) in (4). Its structure and explicit expressions are discussed in Section 3.2. The second term admits the analytic integration resulting in correction δ add (v min ). This term (as well as the third term in the eq. (5) discussed in Section 3.3) contributes to the σ BSV (v min ) that represents the part of the observed cross section not contained in the contributions of radiated photons with inelasticity above v min . BSV cross section The BSV -part of observed cross section includes the Born cross section ( Fig.1 (a)), loop effects ( Fig.1 (b,c)) and the contribution of soft photons. The latter is restricted by the inelasticity value v < v min : σ BSV (v min ) = (1 + δ V R + δ l vac + δ h vac )e δ inf σ 0 + δ add (v min )σ 0 +σ add R (v min ).(6) The Born contribution to the cross section reads: σ 0 = α 2 S 2 Q 4 4 i=1 θ B i F i (Q 2 ).(7) The kinematic coefficients θ B are presented in Appendix B. The structure functions F i are the squared combinations of the electric and magnetic elastic form factors: F 1 (Q 2 ) = 4τ p M 2 G 2 M (Q 2 ), F 2 (Q 2 ) = 4M 2 G 2 E (Q 2 ) + τ p G 2 M (Q 2 ) 1 + τ p , F 3 (Q 2 ) = −2M 2 G E (Q 2 )G M (Q 2 ), F 4 (Q 2 ) = −M 2 G M (Q 2 ) G E (Q 2 ) − G M (Q 2 ) 1 + τ p (8) with τ p = Q 2 /4M 2 . The factorizing corrections in the first term of (6) describe the effects of loops and soft-photon emission. The correction δ inf comes from the emission of soft photons, the δ V R appears as a result of an infrared cancellation of real ( Fig.1 (d,e)) and virtual ( Fig.1 (c)) photon contribution. The explicit expressions for them are: where Li 2 is the Spence function. δ inf = α π log Q 2 m 2 − 1 log v 2 max S(S − Q 2 ) , δ V R = α π 3 2 log Q 2 m 2 − 2 − 1 2 log 2 S S − Q 2 + Li 2 1 − M 2 Q 2 S(S − Q 2 ) − π 2 6 ,(9)t 1 t 2 t' 1 t' 2 v 1 (t) v 1 (t) v 1 '(t) v 1 '(t) v 1 (t)=v min v max The effect of vacuum polarization by leptons (hadrons) depicted on Fig.1 (b) is described by δ l vac (δ h vac ) . The explicit expression for δ l vac is defined by eq. (21) of ref. [5] while the fit for δ h vac has been taken from [23]. The term δ add (v min )σ 0 in the R.H.S. of (6) contains the correction coming from the second term in (5). δ add (v min ) = − 2α π log Q 2 m 2 − 1 log v max v min .(10) The last term in (6) is discussed in Section 3.3. Bremsstrahlung cross section Since the structure functions depend only on t, and therefore integrals over other variables (i.e., v and φ k ) can be evaluated analytically or numerically with high precision, a reasonable sequence of integration variables is chosen such that integration over t is external. This approach allows us to speed up the generation of radiative events. The radiative photon phase space for t-and v-variables are presented on Fig. 2. It is separated into hard and soft photon emission by the line v = v min . The cross section of hard-photon bremsstrahlung is σ rad (v min ) = − α 3 4πS 2 t 2 t 1 dt 4 i=1 F i (t) t 2 θ R i (v 1 , v max ).(11) The quantities θ R i (v 1 , v max ) result from the integration over inelasticity v. Their arguments correspond to the limits of integration: θ R i (v 1 , v max ) = k i j=1 vmax v 1 dvR j−3 θ R ij (v).(12) Here R = Q 2 + v − t, and the upper sum limits are defined as k i = (3, 3, 4, 5). Accordingly, the quantities θ R ij (v) result from the integration over φ k : θ R ij (v) = 2π 0 dφ k θ R ij (v, φ k ).(13) The set of quantities θ R is defined in Appendix B. Kinematical bounds are defined as v 1 = max{ (t − Q 2 )( √ t − √ 4M 2 + t) 2 √ t , (t − Q 2 )( √ t + √ 4M 2 + t) 2 √ t , v min }, v max = 2Q 2 (S 2 − 4M 2 m 2 − Q 2 (S + m 2 + M 2 )) Q 2 (S + 2m 2 ) + Q 2 (S 2 − 4M 2 m 2 )(Q 2 + 4m 2 ) ≈ S − Q 2 − M 2 Q 2 S , t 1,2 = 2M 2 Q 2 + v max Q 2 + v max ∓ (Q 2 + v max ) 2 + 4M 2 Q 2 2(M 2 + v max ) .(14) The probability distributions used for simulation of the photonic variables are obtained using (11) and (13): ρ(t) = 1 N t 4 i=1 F i (t) t 2 θ R i (v 1 , v max ), N t = ρ(v\|t) = 1 N v 4 i=1 k j j=1 F i (t)θ R ij (v)R j−3 , N v = 4 i=1 F i (t)θ R i (v 1 , v max ), ρ(φ k \|v, t) = 4 i=1 k j j=1 F i (t)θ R ij (v, φ k )R j−3 N φ k , N φ k = 4 i=1 k j j=1 F i (t)θ R i (v)R j−3 . (15) 3.3 Contribution of σ add R (v min ) The contribution of σ add R (v min ) can be presented as an integral over the softphoton region in Fig. 2: σ add R (v min ) = − α 3 4πS 2 t ′ 2 t ′ 1 dt v min v ′ 1 dv 4 i=1 k j j=2 R j−3 θ R ij (v) F i (t) t 2 + 1 R 2 θ R i1 (v) F i (t) t 2 − 4θ B i F IR (v) F i (Q 2 ) Q 4 .(16) The limits of integration over variables t and v read: v ′ 1 = max{ (t − Q 2 )( √ t − √ 4M 2 + t) 2 √ t , (t − Q 2 )( √ t + √ 4M 2 + t) 2 √ t }, t ′ 1,2 = 2M 2 Q 2 + v min Q 2 + v min ∓ (Q 2 + v min ) 2 + 4M 2 Q 2 2(M 2 + v min ) .(17) The infrared divergences could occur in the limit v ′ 1 → 0 (i.e. at t → Q 2 ) in the terms containing R −2 . However, one can see that σ add R (v min ) is infrared-free. Indeed, taking into account Eqs. v ′ 1 → 0 we have lim v→0 θ R i1 (v) F i (t) t 2 = 4θ B i F IR (0) F i (Q 2 ) Q 4 .(18) This cancels one degree of R. The second degree of R cancels because the integration region is collapsed into a point within this limit. 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 The structure of the program The set of files included in the package ELRADGEN 2.0 contains three FORTRAN files (elradgen.f, run.f, test.f), six INCLUDE files (const.inc, grid.inc, output.inc, par.inc, pol.inc, test.inc), two data files (rnd.dat, test.dat), and one Makefile. No installation is required for this code. The elradgen.f is a source code of the Monte Carlo generator ELRADGEN 2.0. It contains the set of functions and subroutines for simulation of a single event. The loop over simulated events as well as initialization of constants requires coding in the external program. Two versions of such external programs are given in files run.f and test.f. The file run.f is a typical external program for simulation of an event with fixed Q 2 and φ. The file test.f is designed to run several tests discussed below. The structure of the code is illustrated in Fig. 3: • program main is a sample of an external program that invokes ELRAD-GEN (in our case it is the program included in run.f and test.f); • elrad init defines all constants (such as beam energy and polarization degrees) which are necessary for generation; • grid init prepares the grids for generation of photonic kinematic variables; • elradgen is a main subroutine governing the simulation of an event; • urand is a generator of uniformly distributed random numbers; • fsib calculates the Born cross section; • fsigt invokes one of the subroutines fsigtan or fsigtnu to calculate the cross section dσ/dt; • fsigtan calculates the analytical cross section dσ/dt for unpolarized scattering; • fsigtnu calculates the cross section dσ/dt with numerical integration over variable v for polarized scattering; • fsigtv calculates the analytical cross sections dσ/(dtdv) and dσ/(dtdvdφ k ); • ffpro is a model for elastic form factors. The six INCLUDE files are: • const.inc includes all necessary constants, e.g. , the fine electromagnetic constant, the proton and lepton masses; • grid.inc includes nets of bins for simulation of the three photonic variables; • output.inc contains variables governing the form of output as discussed below • pol.inc includes quantities which describe the polarization state (defined in Appendix A.1); • test.inc includes variables and nets of bins required for test run; • par.inc includes variables required for calculation of σ add R (v min ). The file rnd.dat includes an initial integer for the flat generator urand, and test.dat is an example of output data for a test run (when test.f is used as an external program); the results of different test are presented in Appendix C. The commands "make" or "make test" need to be run for creating the executable file for the simulation or for the test runs, respectively. Input-output data Input data in ELRADGEN 2.0 are set up in program main of run.f or test.f. Majority of them are transferred to the main program through parameters in the subroutine elradgen. They are: • ebeam is an energy of electron beam; • q2 is a virtual photon momentum squared Q 2 ; • phi is an azimuthal angle between the scattering plane and the ground level; • vvmin is a missing mass square resolution v min for separation of radiatively corrected cross section into radiative and BSV parts; • vcut is a cut-off quantity v cut that allows to exclude the simulation of hard photons above v cut . The last variable provides the opportunity to exclude simulation of events with inelasticity above a predetermined level. This could be convenient when simulation is performed for experimental design, when hard real photon are removed from experimental data by putting a cut on the missing mass of the undetectable particle. The quantities describing the polarization characteristics of beam and target (defined in Appendix A.1) are transferred to the code through the common block pol containing four variables: i,ii) plrun and pnrun, the polarization degrees of the lepton beam P L and target P N , iii) thetapn, the angle θ η between 3-vectors of the target polarization η and initial lepton momentum k 1 , and iv) phipn, the angle φ η between OZX and (k 1 , η) planes. One additional variable itest governs the form of the output. If itest = 0, all output information is printed to the file test.dat. If itest = 0, the output data are collected in two common blocks of the file output.inc: common/variables/tgen,vgen,phigen,weight,ich and common/vectors/vprad,phrad Here tgen, vgen, and phigen are the generated photonic variables t, v, and φ k , respectively, weight is a ratio of the observable cross section to the Born one, variable ich shows whether the scattering channel is radiative (ich = 1) or BSV (ich = 0). The quantities vprad = p 2 − p 1 and phrad := k are fourmomenta of virtual and real photons defined in the Lab system. For BSV events, vgen = 0, tgen = Q 2 , φ k = 0, phrad = 0 and vprad = k 1 −k 2 . Numerical tests and comparison with experimental data Below we describe three types of numerical experiments allowing: i) to crosscheck some key distributions and parameter estimates in ELRADGEN, ii) to investigate issues related to a possible dependence of simulated cross sections on v min , and iii) to perform a comparison with data collected in the BLAST experiment. Tests implemented in ELRADGEN There are five tests implemented in the program. The first three deal with checking to which extent the simulated distributions on photonic variables t, v, and φ k correspond to analytical probability distributions given in eq. (15). The sharp peaks in the t-distribution coming from the collinear singularities, i.e., from the kinematical region where the real photon is emitted along either the initial or the final lepton. After integration over the inelasticity v, these two singularities are situated near t = Q 2 and are only slightly different. The peaks on the plots of the v-distributions correspond to the collinear singularities as well. Since the variable t is external, the v-distribution is conditional on t, and therefore only one peak corresponding to either the initial or the final electron appears for each v-distribution. Finally, the φ k -distributions show that most of the photons are emitted in the scattering plane. For generation of the three types of distributions, one has to set itest = 1, itest = 2, or itest = 3, in the file test.f, and then type "make test" and "./test.exe". In Appendix C, the test outputs for t, v, and φ k generation with P = 1, E Lab b =4 GeV, θ η = 48 0 , 20 bins for the histogramming and 10 8 radiative events are presented. After generating all photonic variables for one radiative event, ELRADGEN reconstructs the four-momenta of the final particles. To make sure that the vectors are constructed properly, the next test corresponding to itest := 4 is implemented. This test allows to perform the numerical comparison of the generated variables t, v, and φ k with the value of these variables reconstructed from four-momenta of the particles. This test also reconstructs the mass of the real photon that has to be equal to zero. The test with itest := 5 provides us with the comparison of the unpolarized cross section integrated over v analytically and numerically. v min -dependence and comparison with MASCARAD The Monte Carlo generator ELRADGEN 2.0 was developed on the basis of the FORTRAN code MASCARAD, therefore the agreement of outputs of both programs with the same input parameters has to be demonstrated as a primary test. Here we restrict our crosscheck to the JLab kinematic conditions without cuts on inelasticity v and focus on the comparison of the ratio of the radiatively corrected cross section to the Born one. Define components of the cross sections as: Table 1 The v min -dependence of the ratios of radiative, BSV, and observable contributions to the unpolarized (P L P N ≡ 0) electron-proton cross section to the Born cross section for JLab kinematic conditions (E beam = 4 GeV and Q 2 = 3 GeV 2 ): a (b) presents the results of analytical (numerical) integration over v in ELRADGEN, while c shows the results of the calculation using MASCARAD [11]. Table 2 v min -dependence of the radiative, BSV, and observable contribution to electronproton scattering with longitudinally polarized target (θ η = 0) for a different spin orientation in the Born units and comparison with MASCARAD [11] for JLab kinematic conditions (E beam = 4 GeV and Q 2 = 3 GeV 2 ). v min σ L rad /σ L 0 σ L BSV /σ L 0 weight = σ L obs /σ L 0 GeV 2 ELRADGEN ELRADGEN ELRADGEN MASCARAD P L P N 1 -1 1 -1 1 -1 1 -1 1σ L,T a (ξ L , η L,T ) = σ u a + P L P N σ p a (ξ L , η L,T ),(19) where a = 0, BSV, rad, obs First, we consider the unpolarized scattering for which an option with analytical integration over v is available. Table 1 presents the results of the analytical and numerical integration for the BVS-and radiative contributions (i.e., σ BSV (v min ) and σ rad (v min )) to the observed cross section, as well as the results obtained using MASCARAD. Each of these contributions changes essentially by decreasing v min from 1 to 10 −4 GeV 2 , while the observable cross sections barely change for both analytical and numerical integration over v. Table 3 v min -dependence of the radiative, BSV and observable contribution to electronproton scattering with transversely polarized target (θ η = π/2, φ = φ η ) for different spin orientation in the Born units and comparison with MASCARAD [11] for JLab kinematic conditions (E beam = 4 GeV and Q 2 = 3 GeV 2 ). The similar behavior of the radiative and BSV parts takes place for the polarized case. This is illustrated in Tables 2 and 3. The observable cross sections change by no more than 1%. Results from the BLAST data Analyzing the ∆-excitation region in ep-scattering, it is necessary to extract the contribution of real hard-photon emission that accompanies the elastic ep-scattering (so-called elastic radiative tail) and cannot be removed from the data by any experimental cuts. The main radiative photons are emitted by the electron leg (see Fig.1 (d,e)), because their contributions include the logarithm of the electron mass. These radiative events are spin-dependent, and therefore affect not only the cross section, but other extracted quantities in the ∆region as well, e.g. asymmetries, spin-correlation parameters, spin-structure functions, etc. The BLAST experiment was designed to study spin-dependent electron scattering off protons and deuterons with small systematic uncertainties [25]. The experiment used a longitudinally polarized, an intense electron beam and isotropically pure highly-polarized internal targets of hydrogen and deuterium from an atomic beam source. For extraction of the elastic radiative tail contribution, the new version 2.0 of Monte Carlo generator ELRADGEN has been applied. This generator was incorporated into the BLAST Monte Carlo event generator [26], where longitudinally polarized electrons at an energy of 850 MeV and at a polarization factor of 65%, were scattered off a highlypolarized hydrogen internal gas target (P N ∼ 80%), with the average target In order to estimate the contribution of the elastic radiative tail to the ∆excitation region, the results of the Monte Carlo simulations were normalized to the data elastic peak, as shown in Fig. 5. In this figure, the normalized yields correspond to the outgoing electrons detected in each sector of the BLAST detector (inclusive scattering). The radiative tail obtained from the above normalization is subtracted from the measured yields (radiative corrections), and then the left and right asymmetries are extracted. For comparison, in Fig. 6 we show the left and right asymmetries with and without the radiative corrections. The asymmetry from ELRADGEN alone is also shown (dotted line) in order to see the radiative tail effect to the overall asymmetries and its spin dependence. Conclusion In this paper, we presented a new version of Monte Carlo generator ELRAD-GEN for simulation of real-photon events within elastic lepton nucleon scattering for longitudinally polarized lepton and arbitrary polarized target. Following the absolute necessity of both accuracy and quickness for our program, we have developed the fast and highly precise code using analytical integration wherever it was possible. The developed program has a broad spectrum of applications in data analysis of various experimental designs on polarized ep-scattering, including the measurements of the generalized parton distributions, the generalized polarizabilities, and the evaluation of spin asymmetries in elastic scattering. Also, it can be used as a generator of the "Born" process in DVCS measurements and of the radiative tail from the elastic peak in DIS. The most significant application of the generator is in the experiments with the complex detector geometry. The set of numerical tests of the presented version of this code proved its high quality. First, a good agreement with FORTRAN code MASCARAD [11] was found. Second, no dependence on the missing mass square resolution was found. Third, the distributions of the generated radiative events are found to be in accordance with the corresponding probability densities. Fourth, a good agreement with the radiative tail from the elastic peak measured in the BLAST experiment was demonstrated. Several additional steps allowing to make the simulation even faster are planned. They include implementation of analytical integration over variable v of the hard-photon emission contribution with the longitudinally polarized target and utilizing the look-up table option for faster simulation of radiative events with transverse component of the target polarization vector. The spectrum of applications of the presented code could be extended after a certain substantive upgrade in several directions including the development of this generator for transferred polarization from lepton beam to recoil proton [28], and for involving this generator into the measurement of the electromagnetic form-factors of the proton in elastic scattering with unpolarized [29] and polarized targets [30,31]. Inclusion of electroweak effects will provide the generalization for the investigation of electroweak corrections in experiments on axial form factors of the nucleon [32] and parity violation elastic scattering [33]. This generator can be included in data analysis of experiments with the measurement of unpolarized and spin-flip generalized polarizabilities in virtual Compton scattering [3,4]. Appendix A.1 Polarization vector definitions As it was mentioned above we assume that the electron beam has a longitudinal polarization. Therefore its polarization vector has a form [19]: ξ L = 1 √ λ s S m k 1 − 2mp 1 . (A.1) The target polarization vector in the Lab. system can be decomposed into longitudinal η L = 1 √ λ s 2Mk 1 − S M p 1 (A.2) and transverse η T components as it is depicted in Fig. 7 η = cos(θ η )η L + sin(θ η )η T , (A.3) where θ η is the angle between 3-vectors k 1 and η. Transverse η T component can be presented as: η T = cos(φ − φ η )η t + sin(φ − φ η )η ⊥ , (A.4) where φ η is the angle between (k 1 , η) and OZX planes, and η t = (4m 2 M 2 + 2Q 2 M 2 − SX)k 1 + λ s k 2 − (SQ 2 + 2m 2 S x )p 1 √ λλ s , η ⊥ = 0, k 2 × k 1 \|k 2 \|\|k 1 \| sin θ = (0, sin φ, − cos φ, 0). (A.5) Here X = 2k 2 p 1 , S x = S − X, λ = SXQ 2 − M 2 Q 4 − m 2 λ q , λ q = S 2 x + 4Q 2 M 2 . (A.6) It should be noted, that for the BSV process the variable X is fixed by Q 2 and S: X = S − Q 2 , while for the radiative one, the variable X (as well as S x and λ q ) depends on inelasticity: X = S − Q 2 − v, S x = Q 2 + v, λ q = (Q 2 + v) + 4Q 2 M 2 . (A.7) As it follows from (A.9) the normal to scattering plane component of η satisfies the equations: k 1 η ⊥ = k 2 η ⊥ = p 1 η ⊥ = 0, kη ⊥ = −p 2 η ⊥ = sin φ k √ λ 3 λ q . (A.8) Appendix A.2 Four-momenta reconstruction After generation of photonic variable t, v and φ k the four-momenta of final proton p 2 = (p (0) 2 , p(1)2 , p(2)2 , p(3) 2 ), lepton k 2 = (k (0) 2 , k(1)2 , k(2) 2 , k 2 ) and real photon k = (k (0) , k (1) , k (2) , k (3) ) in the LAB system read: p (1) 2 = √ λ 3 (λ 1 cos φ cos φ k − λ q S sin φ sin φ k ) + λ 2 √ λ 4 cos φ λ q S , p(2)2 = √ λ 3 ( λ q S cos φ sin φ k + λ 1 sin φ cos φ k ) + λ 2 √ λ 4 sin φ λ q S , p(3)2 = λ 1 λ 2 − 4M 2 √ λ 3 λ 4 cos φ k 2λ q MS , p (0) 2 = t + 2M 2 2M , k(1)2 = √ λ 4 cos φ S , k(2)2 = √ λ 4 sin φ S , k(3)2 = S 2 − λ 1 2MS , k (0) 2 = S − Q 2 − v 2M , k (1) = √ λ 3 ( λ q S sin φ sin φ k − λ 1 cos φ cos φ k ) + (λ q − λ 2 ) √ λ 4 cos φ λ q S , k (2) = − √ λ 3 ( λ q S cos φ sin φ k + λ 1 sin φ cos φ k ) + (λ q − λ 2 ) √ λ 4 sin φ λ q S , k (3) = λ 1 (λ q − λ 2 ) + 4M 2 √ λ 3 λ 4 cos φ k 2λ q MS , k (0) = Q 2 + v − t 2M , (A.9) where λ 1 = S(Q 2 + v) + 2M 2 Q 2 , λ 2 = t(Q 2 + v) + 2M 2 (Q 2 + t), λ 3 = tv(Q 2 − t + v) − M 2 (Q 2 − t) 2 , λ 4 = Q 2 S(v max − v). (A.10) Appendix B Explicit expressions for the kinematic quantities θ The kinematic coefficients θ B i appear as a convolution of the leptonic tensor L B µν with corresponding hadronic structures w µν 1 = −g µν , w µν 2 = p µ p ν M 2 , w µν 3 = −iP N ǫ µνλσ q λ η σ M , w µν 4 = iP N ǫ µνλσ q λ p σ ηq M 3 (B.1) and read θ B 1 = 1 2 L B µν w µν 1 = Q 2 , θ B 2 = 1 2 L B µν w µν 2 = 1 2M 2 (S(S − Q 2 ) − M 2 Q 2 ), θ B 3 = 1 2 L B µν w µν 3 = P L P N 2m M (qη k 2 ξ − ξη Q 2 ), θ B 4 = 1 2 L B µν w µν 4 = P L P N mQ 2 qη M 3 (2p 1 ξ − k 2 ξ). (B.2) Here P L and P N define the degree of the lepton and target polarization respectively and the explicit expressions for polarized vectors of the scattering particles ξ and η can be found in Appendix A.1. The quantities θ i (v 1 , v 2 ) appear as a convolution of the leptonic tensor that is responsible for the real photon emission: L R µν = − 1 2 Tr[(k 2 + m)Γ µα (1 − P Lξ γ 5 )(k 1 + m)Γ αν ], Γ µα = k 1α kk 1 − k 2α kk 2 γ µ − γ µk γ α 2kk 1 − γ αk γ µ 2kk 2 , Γ αν = k 1α kk 1 − k 2α kk 2 γ ν − γ αk γ ν 2kk 1 − γ νk γ α 2kk 2 (B.3) with hadronic structures presented in eq.(B.1). As a result θ R i (v 1 , v 2 ) = − 1 4π v 2 v 1 dv λ q 2π 0 dφ k L R µν w µν i (q → q − k) = k i j=1 v 2 v 1 dvR j−3 θ R ij (v) = k i j=1 v 2 v 1 dv 2π 0 dφ k R j−3 θ R ij (v, φ k ), (B.4) where R and k i are defined after eq. (12). Appendix B.1 Quantities θ ij (v) and θ ij (v, φ k ) Here, we combine explicit expressions for θ R ij (v) and θ R ij (v, φ k ) quantities calculated for polarized scattering [5,19,34] and present the explicit expressions θ R i (v 1 , v 2 ) calculated for unpolarized scattering only. Both types of quantities θ R ij (v) and θ R ij (v, φ k ) for i = 1, 2, 3 take the similar form θ R 11 = 4Q 2 F IR , θ R 12 = 4τ F IR , θ R 13 = −4F − 2τ 2 F d , θ R 21 = 2(SX − M 2 Q 2 )F IR /M 2 , θ R 22 = (2m 2 S p F 2− + S p S x F 1+ + 2(S x − 2M 2 τ )F IR − τ S 2 p F d )/2M 2 , θ R 23 = (4M 2 F + (2M 2 τ − S x )τ F d − S p F 1+ )/2M 2 , θ R 31 = P L P N 8m M (ηq k 2 ξ − Q 2 ξη)F IR , θ R 32 = −P L P N 2m M (2ηq (τ k 2 ξF d − 2F ξ IR ) + Q 2 ηK(F ξ 2+ − F ξ −2 − 2F ξ d ) +4ξητ F IR − 4m 2 k 2 ξ(2F η d − F η 2+ )) θ R 33 = P L P N 2m M (ηK τ (2F ξ d + F ξ 2− − F ξ 2+ ) − 2 k 2 ξ τ F η d − 4m 2 F ξη d − 6F ξη IR +Q 2 (F ξη 2+ − F ξη 2− )), θ R 34 = P L P N 2mτ M ( 2F ξη d + F ξη 2+ − F ξη 2− ) , (B.5) where τ = (t − Q 2 )/R and the four-vector K = k 1 + k 2 . For i = 4 we have where θ 41 = P L P N 4m M 2 ( 2 ξp Q 2 − S x ξk 2 ) F IR , θ 42 = P L P N m M 2 (2 ( S p − 2S x ) F ξ IR + 2 k 2 ξ τ S x F d + 8 ξp τ F IR +S p ( Q 2 m F ξ 2+ − Q 2 F ξ 2− ) − 4m 2 ( k 2 ξ (2F d − F 2+ ) + S p (F ξ 2+ − F ξ d )), θ 43 = P L P N m M 2 (( Q 2 − τ S p ) F ξ 2− − ( Q 2 m − τ S p ) F ξ 2+ + 2 k 2 ξ τ F d +6F ξ IR − 2F ξ d τ S p ), θ 44 = −P L P N mτ M 2 ( 2F ξ d − F ξ 2− + F ξ 2+ ) . (B.7) The quantities θ η 4j are calculated as: θ η 4j = θ 4j (F all → F η all , F ξ all → F ξη all ). (B.8) The upper indices in F all appears in the following way: 2F ξ 2+ = (2F 1+ + τ F 2− )s ξ + F 2+ r ξ , 2F η 2+ = (2F 1+ + τ F 2− )s η + F 2+ r η + 4 sin φ k d η R λ 3 λ q F 2+ , 2F ξ 2− = (2F d + F 2+ )τ s ξ + F 2− r ξ , 2F ξ d = F 1+ s ξ + F d r ξ , 2F η d = F 1+ s η + F d r η + 4 sin φ k d η R λ 3 λ q F d , 4F ξη 2+ = (2F 1+ + τ F 2− )(r η s ξ + s η r ξ ) + F 2+ (r η r ξ + τ 2 s η s ξ ) +4(2F + F d τ 2 )s η s ξ + 8 sin φ k d η R λ 3 λ q F ξ 2+ , 4F ξη 2− = (2F d + F 2+ )(r η s ξ + s η r ξ ) + F 2− (r η r ξ + τ 2 s η s ξ ) + 4τ F 1+ s η s ξ + 8 sin φ k d η R λ 3 λ q F ξ 2− , 4F ξη d = F 1+ (r η s ξ + s η r ξ ) + F d (r η r ξ + τ 2 s η s ξ ) + 4F s η s ξ + 8 sin φ k d η R λ 3 λ q F ξ d . (B.9) The quantities s {ξ,η} = a {ξ,η} + b {ξ,η} , r {ξ,η} = τ (a {ξ,η} − b {ξ,η} ) + 2c {ξ,η} (B.10) are combinations of coefficients of polarization vectors ξ and η expansion over basis (see Appendix A.1) ξ = 2(a ξ k 1 + b ξ k 2 + c ξ p), η = 2(a η k 1 + b η k 2 + c η p + d η η ⊥ ). (B.11) We note that the scalar products from (B.5,B.6,B.7) are also calculated in terms of the polarization vector coefficients: ηq = −Q 2 (a η − b η ) + S x c η , ηK = (Q 2 + 4m 2 )(a η + b η ) + S p c η , k 2 ξ = Q 2 m a ξ + 2m 2 b ξ + Xc ξ , ξp = Sa ξ + Xb ξ + 2M 2 c ξ , 1 2 ξη = 2m 2 (a ξ a η + b ξ b η ) + 2M 2 c ξ c η + Q 2 m (a ξ b η + b ξ a η ) +S(a ξ c η + c ξ a η ) + X(b ξ c η + c ξ b η ). (B.12) With the exception of the contribution proportional to d η all dependencies of θ ij on the photonic variable φ k are included in the quantities F , however in both cases we have: F IR = m 2 F 2+ − Q 2 F d (B.13) So for θ ij (v, φ k ) the quantities F read F d (v, φ k ) = F (v, φ k ) z 1 z 2 , F 1+ (v, φ k ) = F (v, φ k ) 1 z 1 + 1 z 2 , F 2± (v, φ k ) = F (v, φ k ) 1 z 2 2 ± 1 z 2 1 , F (v, φ k ) = 1 2π λ q . (B.14) Here z 1 = 2kk 1 R = 1 λ q (Q 2 S p + τ (SS x + 2M 2 Q 2 ) − 2M λ z cos φ k ), z 2 = 2kk 2 R = 1 λ q (Q 2 S p + τ (XS x − 2M 2 Q 2 ) − 2M λ z cos φ k ), (B.15) and λ z = (τ − τ min )(τ max − τ )λ, τ max/min = S x ± λ q 2M 2 , S p = S + X = 2S − Q 2 − v. (B.16) The following equalities define the functions F for θ R ij (v): F (v) = λ −1/2 q , F d (v) = τ −1 (C −1/2 2 (τ ) − C −1/2 1 (τ )), F 1+ (v) = C −1/2 2 (τ ) + C −1/2 1 (τ ), F 2± (v) = B 2 (τ )C −3/2 2 (τ ) ∓ B 1 (τ )C −3/2 1 (τ ), (B.17) where B 1,2 (τ ) = − 1 2 ( λ q τ ± S p (S x τ + 2Q 2 ) ) , C 1 (τ ) = (Sτ + Q 2 ) 2 + 4m 2 (Q 2 + τ S x − τ 2 M 2 ), C 2 (τ ) = (Xτ − Q 2 ) 2 + 4m 2 (Q 2 + τ S x − τ 2 M 2 ). (B.18) We note that F d has a 0/0-like uncertainty for τ = 0 (inside the integration region). It leads to difficulties in numerical integration, so another form is used also F d (v) = S p (τ S x + 2Q 2 ) C 1/2 1 (τ )C 1/2 2 (τ )(C 1/2 1 (τ ) + C 1/2 2 (τ )) . (B.19) Appendix B.2 Quantities θ R i (v 1 , v 2 ) As it was mentioned above for the unpolarized scattering, the integration over v is performed analytically resulting in: θ R 1 (v 1 , v 2 ) = −4I F + 4tI −2 2+ − 2(Q 4 + t 2 )I −2 d , θ R 2 (v 1 , v 2 ) = 1 2M 2 (4M 2 I F − t(I 0 1+ + 2I 0 d ) + t(2S − t)I −1 1+ + 4I 0 21 −4(2S − t)I −1 21 + 4(S 2 − t(M 2 + S))I −2 2+ +t(Q 2 + 4S − 3t)I −1 d +[t(Q 2 t − (2S − t) 2 ) + 2M 2 (t 2 + Q 4 )]I −2 d ), (B.20) where I F = v 2 v 1 dvF (v) = log   Q 2 + v 2 + (Q 2 + v 2 ) 2 + 4M 2 Q 2 Q 2 + v 1 + (Q 2 + v 1 ) 2 + 4M 2 Q 2   , I 0 1+ = v 2 v 1 dvF 1+ (v) = (Q 2 − t) S Q 4 ∆L 1 − S − t t 2 ∆L 2 + ∆ 1 1 Q 4 + ∆ 1 2 t 2 ,I −1 1+ = v 2 v 1 dv R F 1+ (v) = 1 Q 2 ∆L 1 + 1 t ∆L 2 , I 0 21 = 1 2 m 2 v 2 v 1 dv(F 2+ (v) − F 2− (v)) = 1 2 (Q 2 − t) S 2 Q 4 ∆ 0 1 , I −1 21 = 1 2 m 2 v 2 v 1 dv R (F 2+ (v) − F 2− (v)) = S 2Q 2 ∆ 0 1 , I −2 2+ = m 2 dv R 2 F 2+ (v) = 1 2(Q 2 − t) ∆ 0 1 − Q 2 t ∆ 0 2 , I 0 d = v 2 v 1 dvF d (v) = (Q 2 − t) S 2 Q 6 ∆L 1 − (S − t) 2 t 3 ∆L 2 + 2 S Q 6 ∆ 1 1 +2 S − t t 3 ∆ 1 2 + 1 2(Q 2 − t) ∆ 2 1 Q 6 − ∆ 2 2 t 3 , I −1 d = v 2 v 1 dv R F d (v) = S Q 4 ∆L 1 + S − t t 2 ∆L 2 + 1 Q 2 − t ∆ 1 1 Q 4 − ∆ 1 2 t 2 , I −2 d = v 2 v 1 dv R 2 F d (v) = 1 Q 2 − t 1 Q 2 ∆L 1 − 1 t ∆L 2 . (B.21) Here ∆ 2 i = \|D i (v 2 )\|D i (v 2 ) − \|D i (v 1 )\|D i (v 1 ), ∆ 1 i = \|D i (v 2 )\| − \|D i (v 1 )\|, ∆ 0 i = D i (v 2 )/D 2 i+3 (v 2 ) − D i (v 1 )/D 2 i+3 (v 1 ), (B.22) ∆L 1 = log 2m 2 t(Q 2 − t + 2v 2 ) + Q 2 D 1 (v 2 ) + D 4 (v 2 ) √ 4m 2 t + Q 4 2m 2 t(Q 2 − t + 2v 1 ) + Q 2 D 1 (v 1 ) + D 4 (v 1 ) √ 4m 2 t + Q 4 , ∆L 2 = log 2m 2 √ t(Q 2 − t + 2v 2 ) + √ tD 2 (v 2 ) + D 5 (v 2 ) √ 4m 2 + t 2m 2 √ t(Q 2 − t + 2v 1 ) + √ tD 2 (v 1 ) + D 5 (v 1 ) √ 4m 2 + t and D 1 (v) = (t − Q 2 )(S − Q 2 ) + Q 2 v, D 2 (v) = S(Q 2 − t) + tv, D 3 (v) = vt(v − t + Q 2 ) − M 2 (Q 2 − t) 2 , D 4 (v) = D 2 1 (v) + 4m 2 D 3 (v), D 5 (v) = D 2 2 (v) + 4m 2 D 3 (v). (B.23) Appendix C Test output Here, we present the results of the test as test.dat output file corresponding to: 1) itest := 1 -the generation of ρ(t) distribution and comparison with the analytical cross section corresponding to the first formula in (15) (here and below invariants v, t are in GeV 2 ) itest=1 t generation rgen is generated probability rcalc is calculated probability Ebeam=.850 GeV Q2=.200 GeV2 vmin=0.10E-01 GeV2 vcut=0.00 GeV2 PLPN=-1.00 lepton polarization times nucleon polarization thetapn=48.0 degrees angle between target polarization vector and beam momentum in Lab system phipn=0.00 degrees athimutal angle in Lab system number of bins 20 number of radiative events 100000000 initial random number 12 bin t rgen rcalc rgen/rcalc GeV2 GeV(-2) 1 0. 3) itest := 3 -the generation of ρ(φ k ) distribution and comparison with the analytical cross section corresponding to the third formula in (15) itest=3 phik generation rgen is generated probability rcalc is calculated probability Ebeam=.850 GeV Q2=.200 GeV2 vmin=0.10E-01 GeV2 vcut=0.00 GeV2 PLPN=-1.00 lepton polarization times nucleon polarization thetapn=48.0 degrees angle between target polarization vector and beam momentum in Lab system phipn=0.00 degrees athimutal angle in Lab system number of bins 20 number of radiative events 100000000 initial random number 12 t= 0.5218 GeV2 v= 0.7962 GeV2 bin phik rgen rcalc rgen/rcalc rad rad(-1) GeV2 vmin=0.10E-01 GeV2 vcut=0.00 GeV2 PLPN=-1.00 beam polarization times target polarization thetapn=48.0 degrees angle between target polarization vector and beam momentum in Lab system phipn=0. GeV2 reconstructed v from 4-vector v=0.579509 GeV2 generated v itest=5 comparison of the analytical and numerical integration over t rnum is the numerically integrated cross section ran is the analytically integrated cross section Ebeam=4.00 GeV Q2=3.00 GeV2 vmin=0.10E-01 GeV2 vcut=.200 GeV2 PLPN= 0.00 beam polarization times target polarization bin t rnum ran rnum/ran GeV*2 nbarnGeV*(-4)rad(- Fig. 1 . 1Feynman graphs contributing to radiatively corrected cross sections of elastic lepton-nucleus scattering: Born (a), additional virtual particles (b,c) and real photon emission (d,e) contributions. Fig. 2 . 2The region of integration over v-and t-variables for JLab kinematics (Q 2 = 3 GeV 2 , S = 7.5 GeV 2 ). The line v = v min splits it into hard (solid lines) and soft (dashed lines) real photon regions. (13), (B.2), (B.5), and (B.6), in the limit Fig. 3 . 3The structure of the program ELRADGEN 2.0 4 The structure of the program and input-output data Fig. 4 . 4Histogram (points) and corresponding probability densities (solid lines) for variables describing the exclusive real hard photon production in polarized electron proton scattering at JLab kinematic conditions (E beam = 4 GeV, Q 2 = 3 GeV 2 , ) for transverse polarized proton (θ η = 90 0 ) with φ = φ η , P L P N = −1 and v min = 10 −2 GeV 2 . Fig. 4 4presents the t-, v-, and φ k -distributions calculated numerically and generated by ELRADGEN under JLab kinematic conditions for a transverse polarized target. The theoretical and simulated distributions of this case (as well as for unpolarized and longitudinally polarized targets) are almost identical. Fig. 5 .Fig. 6 . 56Normalized yields as a function of the invariant mass, W [GeV] over 0.08 < Q 2 < 0.38 GeV 2 . The dots show the BLAST ABS hydrogen data corrected for the background contributions, and the solid line represents the Monte Carlo simulations with radiative effects (ELRADGEN 2.0). The effect of the radiative contributions to the asymmetry in the ∆-excitation region. The left (left) and right (right) asymmetries are shown with (dots) and without (squares) radiative corrections (RC), for 0.08 < Q 2 < 0.38 GeV 2 . Monte Carlo simulations using the MAID 2003 model [27] (straight line) and ELRADGEN (dotted line) are shown for comparison. spin direction oriented at 48.84 • to the left of the beam direction. Fig. 7. 3-vectors decomposition in the LAB system R 41 = 41ηq θ 41 /M, θ R 42 = (ηq θ 42 − θ η 41 )/M, θ R 43 = (ηq θ 43 − θ η 42 )/M, θ R 44 = (ηq θ 44 − θ η 43 )/M, θ R 45 = − θ η 44 /M, (B.6) 00 degrees athimutal angle in Lab system number of bins 20 number of radiative events 5 initial random number 12 ---------------------------------------- itest := 2 -the generation of ρ(v) distribution and comparison with the analytical cross section corresponding to the second formula in(15) 4237E-01 1.624 1.786 0.9093 2 0.9197E-01 1.619 1.589 1.019 3 0.1453 2.325 2.274 1.023 4 0.1969 12.14 35.10 0.3460 5 0.2351 1.518 1.666 0.9108 6 0.2893 0.3042 0.3153 0.9647 7 0.3411 0.1113 0.1134 0.9816 8 0.3923 0.5102E-01 0.5163E-01 0.9883 9 0.4433 0.2666E-01 0.2688E-01 0.9917 10 0.4943 0.1518E-01 0.1526E-01 0.9945 11 0.5449 0.9216E-02 0.9259E-02 0.9954 12 0.5960 0.5872E-02 0.5865E-02 1.001 13 0.6466 0.3893E-02 0.3872E-02 1.006 14 0.6973 0.2577E-02 0.2632E-02 0.9791 15 0.7479 0.1822E-02 0.1833E-02 0.9937 16 0.7983 0.1278E-02 0.1304E-02 0.9801 17 0.8490 0.9465E-03 0.9396E-03 1.007 18 0.9003 0.6965E-03 0.6806E-03 1.023 19 0.9509 0.4965E-03 0.4945E-03 1.004 20 0.9973 0.2696E-03 0.3558E-03 0.7577 2) itest=2 v generation rgen is generated probability rcalc is calculated probability Ebeam=.850 GeV Q2=.200 GeV2 vmin=0.10E-01 GeV2 vcut=0.00 GeV*2 PLPN=-1.00 lepton polarization times nucleon polarization thetapn=48.0 degrees angle between target polarization vector and beam momentum in Lab system phipn=0.00 degrees athimutal angle in Lab system number of bins 20 number of radiative events 100000000 initial random number 12 t= 0.5218 GeV2 bin v rgen rcalc rgen/rcalc GeV2 GeV(-2) 1 0.6260 0.1660 0.1657 1.002 2 0.6599 0.1931 0.1933 0.9990 3 0.6937 0.2265 0.2266 0.9998 4 0.7275 0.2676 0.2677 0.9996 5 0.7613 0.3206 0.3205 1.000 6 0.7952 0.3904 0.3913 0.9978 7 0.8291 0.4927 0.4921 1.001 8 0.8630 0.6478 0.6486 0.9987 9 0.8971 0.9284 0.9278 1.001 10 0.9316 1.579 1.580 0.9993 11 0.9698 6.013 6.043 0.9951 12 0.9886 13.45 16.44 0.8179 13 1.029 1.759 1.760 0.9992 14 1.064 0.9512 0.9502 1.001 15 1.098 0.6343 0.6361 0.9973 16 1.132 0.4694 0.4686 1.002 17 1.166 0.3649 0.3645 1.001 18 1.199 0.2937 0.2942 0.9985 19 1.233 0.2435 0.2436 0.9999 20 1.267 0.2071 0.2053 1.008 i=1 t 2 t 1 dt F i (t) t 2 θ R i (v 1 , v max ), AcknowledgmentsThe authors would like to acknowledge useful discussion with E.Tomasi-Gustafsson. 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[ "Rainbow path and color degree in edge colored graphs", "Rainbow path and color degree in edge colored graphs" ]	[ "Anita Das \nE-Comm Research Lab Education & Research\nInfosys Limited Bangalore\nIndia\n", "S V Subrahmanya \nE-Comm Research Lab Education & Research\nInfosys Limited Bangalore\nIndia\n", "P Suresh \nE-Comm Research Lab Education & Research\nInfosys Limited Bangalore\nIndia\n" ]	[ "E-Comm Research Lab Education & Research\nInfosys Limited Bangalore\nIndia", "E-Comm Research Lab Education & Research\nInfosys Limited Bangalore\nIndia", "E-Comm Research Lab Education & Research\nInfosys Limited Bangalore\nIndia" ]	[]	Let G be an edge colored graph. A rainbow path in G is a path in which all the edges are colored with distinct colors. Let d c (v) be the color degree of a vertex v in G, i.e. the number of distinct colors present on the edges incident on the vertex v. Let t be the maximum length of a rainbow path in G. Chen and Li(2005)showed that if d c k (k 8), for every vertex v of G, then t 3k 5 + 1. Unfortunately, the proof by Chen and Li is very long and comes to about 23 pages in the journal version. Chen and Li states in their paper that it was conjectured by Akira Saito, that t 2k 3 . They also state in their paper that they believe t k − c for some constant c.In this note, we give a short proof to show that t 3k 5 , using an entirely different method. Our proof is only about 2 pages long. The draw-back is that our bound is less by 1, than the bound given by Chen and Li. We hope that the new approach adopted in this paper would eventually lead to the settlement of the conjectures by Saito and/or Chen and Li.	10.37236/3770	[ "http://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i1p46/pdf" ]	3,959,503	1312.5067	c22e6e515fd623b8dd9d88581a2106f75060b3da	Rainbow path and color degree in edge colored graphs Anita Das E-Comm Research Lab Education & Research Infosys Limited Bangalore India S V Subrahmanya E-Comm Research Lab Education & Research Infosys Limited Bangalore India P Suresh E-Comm Research Lab Education & Research Infosys Limited Bangalore India Rainbow path and color degree in edge colored graphs Submitted: Oct 1, 2013; Accepted: Feb 12, 2014; Published: Feb 28, 2014 Mathematics Subject Classifications: 05C15, 05C38edge colored graphsrainbow pathcolor degree Let G be an edge colored graph. A rainbow path in G is a path in which all the edges are colored with distinct colors. Let d c (v) be the color degree of a vertex v in G, i.e. the number of distinct colors present on the edges incident on the vertex v. Let t be the maximum length of a rainbow path in G. Chen and Li(2005)showed that if d c k (k 8), for every vertex v of G, then t 3k 5 + 1. Unfortunately, the proof by Chen and Li is very long and comes to about 23 pages in the journal version. Chen and Li states in their paper that it was conjectured by Akira Saito, that t 2k 3 . They also state in their paper that they believe t k − c for some constant c.In this note, we give a short proof to show that t 3k 5 , using an entirely different method. Our proof is only about 2 pages long. The draw-back is that our bound is less by 1, than the bound given by Chen and Li. We hope that the new approach adopted in this paper would eventually lead to the settlement of the conjectures by Saito and/or Chen and Li. Introduction Given a graph G = (V, E), a map c : E → N (N is the set of non-negative integers) is called an edge coloring of G. A graph G with such a coloring c is called an edge colored graph. We denote the color of an edge e ∈ E(G) by color(e). For a vertex v of G, the color neighborhood CN (v) of v is defined as the set {color(e)\|e is incident on v} and the color degree of v, denoted by d c (v) is defined to be d c (v) = \|CN (v)\|. A path in an edge colored graph with no two edges sharing the same color is called a rainbow path. Similarly, a cycle in an edge colored graph is called a rainbow cycle if no two edges of the cycle share the same color. A survey on rainbow paths, cycles and other rainbow sub-graphs can be found in [5]. Several theorems and conjectures on rainbow cycles can be found in a paper by Akbari, Etesami, Mahini and Mahmoody in [1]. Let t denote the length of the maximum length rainbow path in G. In [2], Chen and Li studied the maximum length rainbow path problem in edge-colored graphs and proved that if G is an edge colored graph with d c (v) k (k 8), for every vertex v of G, then G has a rainbow path of length at least 3k 5 + 1. Chen and Li state in their paper that it was conjectured by Akira Saito, that t 2k 3 . They also state in their paper that they believe t k − c for some constant c, after showing an example where the rainbow path cannot be more than k − 1. In this note, we give a short proof to show that t 3k 5 , using an entirely different method. Our proof is only about 2 pages long. The draw-back is that our bound is less by 1, than the bound given by Chen and Li. We hope that the new approach adopted in this paper would eventually lead to the settlement of the conjectures by Saito and/or Chen and Li. Preliminaries All graphs considered in this paper are finite, simple and undirected. A graph is a tuple (V, E), where V is a finite set of vertices and E is the set of edges. For a graph G, we use V (G) and E(G) to denote its vertex set and edge set, respectively. The neighborhood N (v) of a vertex v is the set of vertices adjacent to v but not including v. The degree of a vertex v is d v = \|N (v)\|. A path is a non-empty graph P = (V, E) of the form V = {p 1 , p 2 , . . . , p k } and E = {(p 1 , p 2 ), (p 2 , p 3 ), . . . , (p k−1 , p k )}, which we usually denote by the sequence {p 1 , p 2 , . . . , p k }. The length of a path is its number of edges. If P = {p 1 , p 2 , . . . , p k } is a path, then the graph C with V (C) = V (P ) and E(C) = P ∪ {(p k , p 1 )} is a cycle, and \|E(C)\| is the length of C. We represent this cycle by the cyclic sequence of its vertices, for example C = {p 1 , p 2 , . . . , p k , p 1 }. Proof of the main results Let G be an edge colored graph with d c (v) k, for every vertex v of G and t be the maximum length rainbow path in G. Let C denote the set of colors used in the edge coloring of G. The following lemma ensures a rainbow path of length k+1 2 starting from any vertex in an edge colored graph. Proof. Let P be a rainbow path in G of maximum length, say t. Thus P contains t + 1 vertices, t edges and hence t distinct colors. Let P = {x = u 0 , u 1 , . . . , u t = y}, U = the electronic journal of combinatorics 21(1) (2014), #P1.46 {color(u i , u i+1 ), 0 i t − 1) and U c = C \ U . Select a subset E(x) of edges incident on x as follows: Let (x, u 1 ) ∈ E(x). Now, select k − 1 more edges incident on x and add to E(x) so that all the k edges in E(x) have different colors. Clearly it is possible to do this, since d c (x) k. Let N P (x) = {u i : 1 i t and (x, u i ) ∈ E(x)}, N P c (x) = {a ∈ V (G) \ V (P ) : (x, a) ∈ E(x)}. Claim. If z ∈ N P c (x), then color(x, z) ∈ U − {color(x, u 1 )}. Suppose that color(x, z) ∈ U c . As z / ∈ V (P ), P = {z, x = u 0 , u 1 , . . . , u t = y} is a path. Moreover, P is a rainbow path as color(x, z) ∈ U c . Now P is a rainbow path in G of length t + 1. This is a contradiction to the fact that t is the length of the maximum length rainbow path in G. Also it is obvious that color(x, z) = color(x, u 1 ). Hence the claim is true. From the above claim we infer that N P C (x)\| \|U \|−1 = t−1. So, \|N P (x)\| k−(t−1). But, number of vertices in P excluding x is t. So, k − (t − 1) t. Hence, t k+1 2 . Since t is an integer, we have t k+1 2 . The following lemma ensures that if the maximum length of a rainbow path is small enough, then we can convert the maximum rainbow path into a rainbow cycle by some simple modifications. Lemma 2.2. Let G be an edge colored graph and d c (v) k, for every v ∈ V (G). Let t be the length of the maximum length rainbow path in G. If t < 3 5 k , then G contains a rainbow cycle of length (t + 1). Proof. Assume for contradiction that there is no rainbow cycle of length t + 1 in G. Let P = {u 0 (= x), u 1 , u 2 , . . . , u t (= y)} be a rainbow path of length t in G. Let U = {color(u i , u i+1 ), 0 i t − 1} and U c = C \ U , where C is the set of colors used to color the edges of G. Clearly \|U \| = t. Let T x = {u i : 0 i t, (x, u i ) ∈ E(G) and color(x, u i ) ∈ U c } and let T y = {u i : 0 i t, (y, u i ) ∈ E(G) and color(y, u i ) ∈ U c }. First note that, \|{(x, z) ∈ E(G) : color(x, z) ∈ U c }\| k−t. Moreover, if (x, z) ∈ E(G) with color(x, z) ∈ U c , then z ∈ V (P ), i.e., z = u i for some 1 i t, since otherwise we would have a rainbow path of length t + 1 in G. It follows that \|T x \| k − t. By a similar argument, we get \|T y \| k − t. Note that u 0 , u 1 / ∈ T x since u 0 = x and color(x, u 1 ) ∈ U . Also, u t / ∈ T x , since if (x, u t ) is an edge and is colored using a color from U c , then we already have a t + 1 length rainbow cycle, contrary to the assumption. So, we can write T x = {u i : 2 i t − 1 and color(x, u i ) ∈ U c }. By similar reasoning, we can write, T y = {u i : 1 i t − 2 and color(y, u i ) ∈ U c }. Define M x = {u j : u j+1 ∈ T x }. Observation 1. \|M x \| = \|T x \| k − t. Claim 1. M x ∩ T y = ∅. If possible suppose M x ∩ T y = ∅. Now, \|M x \| + \|T y \| t − 1, as both M x ⊂ V (P ) and T y ⊂ V (P ) and number of vertices on P excluding x and y is t − 1. (Note that, x, y / ∈ M x and x, y / ∈ T y .) As \|M x \| k − t and \|T y \| k − t and M x ∩ T y = ∅ by assumption, we have k − t + k − t t − 1. That is, 2k 3t − 1. So, t 2k+1 3 . This is a contradiction to the fact that t < 3 5 k . Hence Claim 1 is true. Claim 2. If u i ∈ M x ∩ T y , then color(y, u i ) = color(x, u i+1 ). Suppose Claim 2 is false. That is, ∃u i ∈ M x ∩T y such that color(y, u i ) = color(x, u i+1 ). Now consider the cycle: CL = {x, u 1 , . . . , u i , y, u t , u t−1 , . . . , u i+1 , x}. Clearly CL is a rainbow cycle, as color(y, u i ) = color(x, u i+1 ), color(y, u i ) ∈ U c and color(x, u i+1 ) ∈ U c . Note that the length of CL is t + 1, as we removed exactly one edge, namely (u i , u i+1 ) from P and added two new edges, namely (y, u i ) and (x, u i+1 ) to CL. So, the length of CL is t − 1 + 2 = t + 1, contradiction to the assumption. Hence, we can infer that if u i ∈ M x ∩ T y , then color(y, u i ) = color(x, u i+1 ). Let S y = {v ∈ V (P ) − (M x ∪ {y, u t−1 }) : color(y, v) ∈ U }}. Observation 2. \|M x \| + \|T y \| + \|S y \| − \|M x ∩ T y \| t − 1. Proof: This is because S y is disjoint from M x ∪T y and S y ∪M x ∪T y ⊆ V (P )−{y, u t−1 }. (Note that y(= u t ) and u t−1 do not appear in M x , T y or S y .) We partition the set M x ∩ T y as follows. Let u i ∈ M x ∩ T y . If color(u i , u i+1 ) appears in one of the edges incident on y, then u i ∈ A otherwise u i ∈ B. Observation 3. \|T y \| k − t + \|B\|. Proof: To see this first note that there are at least k edges of different colors incident on y (as by assumption, color degree of y is at least k)and at most t − \|B\| of them can get the colors from U , since \|B\| colors in U do not appear on the edges incident on y, by the definition of B. So, at least k − t + \|B\| of the edges incident on y have colors from U c , and clearly any w, such that (y, w) is an edge, colored by a color in U c has to be on P , since otherwise we have a longer rainbow path. It follows that \|T y \| k − t + \|B\|. Claim 3. If u i ∈ A, then the edge incident on y with color color(u i , u i+1 ) has its other end point on the rainbow path P . That is, if w is such that (y, w) is an edge and color(u i , u i+1 ) = color(y, w), then w ∈ V (P ). Suppose Claim 3 is false. Let (y, w) ∈ E(G) with color(y, w) = color(u i , u i+1 ) and w / ∈ V (P ). Now, consider the path: P = {w, y, u t−1 , u t−2 , . . . , u i+1 , x(= u 0 ), u 1 , . . . , u i }. Clearly P is a rainbow path as color(u i , u i+1 ) = color(y, w), the edge (u i , u i+1 ) / ∈ E(P ) and color(u i+1 , x) ∈ U c , since u i ∈ M x . Note that, the length of P is t + 1. This is a contradiction to the fact that t is the maximum length rainbow path in G. Hence Claim 3 is true. Now, partition A as follows: if u i ∈ A, then by the above claim the edge incident on y with the color color(u i , u i+1 ) has its other end point say w, on P . If w ∈ M x , then let w) is an edge and color(y, w) = color(u i , u i+1 ) ∈ U . Since u i ∈ A 2 ⊂ M x , we have i < t − 1 and thus color(u i , u i+1 ) = color(y, u t−1 ). Therefore w(u i ) cannot be y or u t−1 , for any u i ∈ A 2 . It follows that {w(u i ) : u i ∈ A 2 } ⊆ S y , and therefore we have \|S y \| \|A 2 \|. Recall that, for each u i ∈ A 1 , there is a unique vertex w = w(u i ) such that (y, w) is an edge with color(u i , u i+1 ) = color(y, w). Moreover, w ∈ M x , by the definition of A 1 and A 1 ∪ {w(u i ) : u i ∈ A 1 } ⊆ M x . Note that w(u i ) is uniquely defined for u i since it is the end point of the edge incident on y colored with the color of the edge (u i , u i+1 ). Moreover, A 1 ∩ {w(u i ) : u i ∈ A 1 } = ∅, since A 1 contains vertices which are end points of edges from y, colored by the colors in U c whereas each w(u i ) is the end point of some edge from y which is colored by a color in U . It follows that 2\|A 1 \| \|M x \|. That is, \|A 1 \| \|Mx\| 2 , as required. u i ∈ A 1 , else u i ∈ A 2 . Observation 4. \|M x ∩ T y \| = \|A\| + \|B\| = \|A 1 \| + \|A 2 \| + \|B\|. Observation 5. \|S y \| \|A 2 \|. To see this, recall that S y = {v ∈ V (P ) − (M x ∪ {y, u t−1 }) : color(y, v) ∈ U }. By definition of A 2 , for each u i ∈ A 2 there exists a unique vertex w = w(u i ) ∈ V (P ) − M x such that (y, Now, substituting k − t + \|B\| for \|T y \| (by Observation 3), \|A 2 \| for \|S y \| (by Observation 5), and \|A 1 \| + \|A 2 \| + \|B\| = \|M x ∩ T y \| (by Observation 4) in the inequality of Observation 2, and simplifying we get \|M x \| + k − t − \|A 1 \| t − 1. Now using \|A 1 \| \|M x \|/2 (Claim 4) and and simplifying we get \|Mx\| 2 + k − t t − 1. Recall that \|M x \| k − t (Observation 1). Substituting and simplifying we get, t 3k+2 5 . It follows that t 3 5 k , contradicting the initial assumption. Hence the Lemma is true. F i = {z ∈ V (CL c ) : (u i , z) ∈ E(G)}. Claim 1. \|F i \| 2k 5 . Moreover, for z ∈ F i , color(u i , z) ∈ U . First part follows from the fact that the color degree of u i is at least k as d c (u i ) k and there are at most 3k 5 vertices in CL. If possible suppose color(u i , z) ∈ U c . Now consider the path P = {z, u i , u i+1 , u i+2 , . . . , u t , u 0 , . . . , u i−1 }. Clearly, P is a rainbow path as color(u i , z) ∈ U c and {u i , u i+1 , u i+2 , . . . , u t , u 0 , . . . , u i−1 } is already a rainbow path being a part of the rainbow cycle CL. Note that, the length of P is t + 1. This is a contradiction to the assumption that t is the maximum length rainbow path in G. Hence Claim 1 is true. Let G = (V , E ), where V (G ) = V (G) and E (G ) = E(G) \ {e ∈ E(G) : color(e) ∈ U }. Clearly, in G there is no edge between V (CL) to V (CL c ), since by Claim 1, every such edge is colored by a color in U . Consider the induced subgraph on V (CL c ) in G . Suppose not. Let z ∈ F 0 be such that color(u 0 , z) ∈ U 0 . Without loss of generality assume that color(u 0 , z) ∈ U 1 . Then consider the path P * = (u k 5 , . . . , u t , u 0 , z), which is clearly a rainbow path, since the edge of CL with its color equal to color(u 0 , z) is not there in this path. Also the length of P * is t+1− k 5 . By Observation 1, G has minimum color degree at least 2k 5 , and therefore by Lemma 2.1, G has a rainbow path of length at least k 5 starting from the vertex z, let us call this path P . Clearly concatenating the path P with P * we get a rainbow path since colors used in P * belong to U whereas the colors used in P belong to U c . Moreover, the length of this rainbow path is at least t + 1, a contradiction, to the assumption that t is the length of the maximum rainbow path in G. Let G = G [V (CL c )]. Let d c (v) k for every v ∈ V (G ). Observation 1. k 2k 5 . Proof: Clearly k k − \|U \| = k − (t + 1) k − 3k 5 2k 5 . Consider the following subset U 0 of U , defined by U 0 = U 1 ∪ U 2 , where U 1 = color(u i , u i+1 ) : 0 i k 5 , U 2 = color(u i , u i+1 ) : (t + 1) − k 5 i t − 1 ∪ {color(u t , u 0 )}. Now we complete the proof as follows: In view of Claim 2, and Claim 1, we know that \|F 0 \| \|U − U 0 \|. But \|U − U 0 \| (since we can assume k 5: for smaller values of k, the Theorem is trivially true). This is a contradiction to the first part of Claim 1. Lemma 2 . 1 . 21Let G be an edge colored graph and d c (v) k, for every v ∈ V (G). Then, given any vertex x in G there exists a rainbow path of length at least k+1 2 starting from x. the electronic journal of combinatorics 21(1) (2014), #P1.46 Claim 4 . 4\|A 1 \| \|Mx\| 2 .the electronic journal of combinatorics 21(1) (2014), #P1.46 Theorem 2 . 3 . 23Let G be an edge colored graph and d c (v) k, for every v ∈ V (G). If t is the maximum length of a rainbow path in G, then t 3k 5 . Proof. If possible suppose t < 3k 5 . By Lemma 2.2, G contains a rainbow cycle of length t + 1. Let CL be this cycle. Note that, CL contains (t + 1) vertices and (t + 1) edges. Now, t + 13k 5 . Let CL = {u 0 , u 1 , . . . , u t , u 0 } and V (CL c ) = V (G) \ V (CL). Let U = {color(e) : e ∈ E(CL)} and U c = C \ U , where C is the set of colors used to color the edges of G. Let the electronic journal of combinatorics 21(1) (2014), #P1.46Claim 2. {color(u 0 , z) : z ∈ F 0 } ∩ U 0 = ∅. On rainbow cycles in edge colored complete graphs. S Akbari, O Etesami, H Mahini, M Mahmoody, Australasian Journal of Combinatorics. 37S. Akbari, O. Etesami, H. Mahini, M. Mahmoody. On rainbow cycles in edge colored complete graphs. Australasian Journal of Combinatorics 37, 33-42, 2007. Long heterochromatic paths in edge-colored graphs. H Chen, X Li, The Electronic Journal of Combinatorics. 1233H. Chen and X. Li. Long heterochromatic paths in edge-colored graphs. The Electronic Journal of Combinatorics. 12 (2005), #R33. On rainbow cycles and paths. H Gebauer, F Mousset, arXiv:1207.0840H. Gebauer, F. Mousset. On rainbow cycles and paths. (2012). arXiv:1207.0840 Rainbow and orthogonal paths in factorizations of K n. A Gyárás, M Mhalla, Journal of Cominatorial Designs. 183A. Gyárás, M. Mhalla. Rainbow and orthogonal paths in factorizations of K n . Journal of Cominatorial Designs, 18(3):167-176, 2010. Monochromatic and heterochromatic subgraphs in edge-colored graphs -a survey. M Kano, X Li, #P1.46the electronic journal of combinatorics. 241Graphs Combin.M. Kano, X. Li. Monochromatic and heterochromatic subgraphs in edge-colored graphs -a survey. Graphs Combin. 24 (2008), 237 -263. the electronic journal of combinatorics 21(1) (2014), #P1.46	[]
[ "King's College London Deformed general relativity", "King's College London Deformed general relativity" ]	[ "ProfRhiannon Mairi Cuttell \nTheoretical Particle Physics and Cosmology Department of Physics\nKING'S COLLEGE LONDON\n\n", "Sakellariadou \nTheoretical Particle Physics and Cosmology Department of Physics\nKING'S COLLEGE LONDON\n\n" ]	[ "Theoretical Particle Physics and Cosmology Department of Physics\nKING'S COLLEGE LONDON\n", "Theoretical Particle Physics and Cosmology Department of Physics\nKING'S COLLEGE LONDON\n" ]	[]	School of Natural and Mathematical SciencesDepartment of PhysicsDoctor of PhilosophyDeformed general relativity by Rhiannon CuttellIn this thesis, I investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. The specific deformation of general covariance is predicted by some studies into loop quantum cosmology.I firstly find the minimally-deformed model for a scalar-tensor theory, thereby establishing a classical reference point, and investigate the cosmological effects of a non-minimal coupled scalar field. By treating the deformation perturbatively, I derive the deformed gravitational action which includes the nearest order of curvature corrections. Then working more generally, I derive the deformed scalar-tensor constraint to all orders and I find that the momenta and spatial derivatives from gravity and matter must combine in a very specific form. It suggests that the deformation should be equally affected by matter field derivatives as it is by gravitational curvature. Finally, I derive the deformed gravitational action to all orders, and find how intrinsic and extrinsic curvatures differently affect the deformation.The deformation seems to be required to satisfy a non-linear equation usually found in fluid mechanics. iv	null	[ "https://arxiv.org/pdf/1910.01699v1.pdf" ]	203,736,494	1910.01699	2de81d6db245f96a244492f7bbd76c9d351a761a	King's College London Deformed general relativity 1st April 2019 3 Oct 2019 ProfRhiannon Mairi Cuttell Theoretical Particle Physics and Cosmology Department of Physics KING'S COLLEGE LONDON Sakellariadou Theoretical Particle Physics and Cosmology Department of Physics KING'S COLLEGE LONDON King's College London Deformed general relativity 1st April 2019 3 Oct 2019Doctoral Thesis Author: School of Natural and Mathematical SciencesDepartment of PhysicsDoctor of PhilosophyDeformed general relativity by Rhiannon CuttellIn this thesis, I investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. The specific deformation of general covariance is predicted by some studies into loop quantum cosmology.I firstly find the minimally-deformed model for a scalar-tensor theory, thereby establishing a classical reference point, and investigate the cosmological effects of a non-minimal coupled scalar field. By treating the deformation perturbatively, I derive the deformed gravitational action which includes the nearest order of curvature corrections. Then working more generally, I derive the deformed scalar-tensor constraint to all orders and I find that the momenta and spatial derivatives from gravity and matter must combine in a very specific form. It suggests that the deformation should be equally affected by matter field derivatives as it is by gravitational curvature. Finally, I derive the deformed gravitational action to all orders, and find how intrinsic and extrinsic curvatures differently affect the deformation.The deformation seems to be required to satisfy a non-linear equation usually found in fluid mechanics. iv List of Symbols g ab space-time metric q ab spatial metric q determinant of q ab ∂ q logarithmic partial derivative with respect to q Q abcd symmetric combination of two spatial metrics p ab momentum canonically conjugate to q ab v ab normal derivative of q ab P traceless part of p ab squared w traceless part of v ab squared K ab extrinsic curvature tensor K standard extrinsic curvature contraction L Y X Lie derivative of X with respect to Y a H Hubble expansion rate R Ricci curvature scalar N lapse function N a shift vector n a future-pointing vector normal to the spatial manifold S action L Lagrangian H total Hamiltonian C Hamiltonian constraint D a diffeomorphism constraint ∂ a partial derivative with respect to the coordinate x a ∇ a covariant derivative with respect to the coordinate x a w ρ cosmological equation of state β deformation function ψ scalar field ∂ ψ partial derivative with respect to ψ π momentum canonically conjugate to ψ ν normal derivative of ψ ϕ minimally coupled scalar field ω gravitational coupling φ I constraints δ ab cd symmetric combination of two Kronecker delta functions π • ratio of a circle's circumference to its diameter Introduction In this thesis I investigate deformed general relativity, which is a semi-classical model attempting to capture the leading effects of a correction to general relativity predicted in some studies of loop quantum gravity. It uses the methods of canonical gravity but with space-time covariance deformed by a phase-space function. By assuming a general deformation, I find the general models which are consistent with it, demonstrating multiple routes which can be taken to find them. Before going into more depth on this, I must first discuss the motivations for this investigation. The need for a theory of quantum gravity It is known that matter fields are quantised due to the remarkable agreement of experimental results with quantum field theory [1][2][3]. There have been some attempts to allow for classical gravity to couple to quantum fields at a fundamental level [4,5], and some interesting phenomena have been discovered from considering effective models of quantum fields on a curved space-time [6][7][8]. However, it is generally expected that gravity must be quantised too [9,10]. The gravitational field, like all other fields, therefore must be quantized, or else the logical structure of quantum field theory must be profoundly altered, or both. [11,B. DeWitt] Besides gravity being known to couple to quantum fields, there are known limitations to the current common understanding. General relativity predicts its own demise due to singularities arising in the equations describing black holes and the very early universe [12]. They are known to exist due to robust experimental observations supporting the existence of black holes [13] and supporting an early universe which closely matches what is predicted of a hot big bang [14]. These phenomena exist at the intersection of general relativity and quantum mechanics since they involve both massive systems and small scales. It seems they cannot be fully understood without a framework which consistently bridges the gap. As a precedent for the singularity problem, classical mechanics could not sufficiently account for experimental results showing that atoms contained small, massive nuclei orbited by electrons (the Nagaoka-Rutherford model). This is due to accelerating point charges (electric field singularities) being known to emit radiation as per the Landau formula, and therefore an electron orbit should radiatively decay, causing atoms to be unstable. However, the development of quantum mechanics resolved this by introducing discrete and stationary orbitals in the Bohr model. The hope is that quantising gravity will similarly cure it of some of its pathologies. One might not want to jettison all that is good about general relativity in pursuit of a quantised theory. The key underlying idea, equivalence of all frames, is considered a philosophically and aesthetically satisfying aspect. Conversely, the requirement in the orthodox interpretation of quantum mechanics for an external observer is considered troubling, hence why Einstein spent much of the latter part of his career challenging it [15]. One crucial sticking point in reconciling general relativity and quantum mechanics is the problem of time [16,17]. In quantum mechanics time is a fixed external parameter, in general relativity it is internal to the system and is not uniquely defined. These are seemingly incommensurable differences, and to bridge the gap requires significant compromise. The solution in canonical gravity for reconciling the two is to split space-time at the formal level, but include symmetry requirements so that the full general covariance is kept implicitly [10,18,19]. One is left with a description of a spatial slice evolving through time rather than one of a static and eternal bulk. These methods are often required for numerically simulating general relativity due to the necessity of specifying a time coordinate when setting up an evolution simulation. This introduces on each spatial manifold a conserved quantity or 'constraint' given by φ I → 0 for each dimension of time and space, analogous to a generalisation of the conservation of energy and momentum. These constraints form an algebra which contains important information about the geometric nature of space-time, and is of the form {φ I , φ J } = f K IJ φ K [10,20]. This is a Lie algebroid which describes the relationships between the constraints and generates transformations between different choices of coordinates [21,22]. The important {C, C} part of this algebra ensures that the spatial manifold evolving through time is equivalent to a stack of spatial manifolds embedded in a geometric spacetime manifold. In this more general case of gravitation in interaction with other fields, [the equation 1 ] not only guarantees the embeddability of the 3-geometries in a spacetime but also ensures that these additional fields evolve consistently within this space-time. [23,C. Teitelboim] This part of the algebra is what I am going to consider to be deformed, but where does this hypothesis come from? Loop quantum gravity Though there are several candidates for a theory of quantum gravity, I am going to only consider loop quantum gravity [24,25]. There are other somewhat related theories which also deal directly with quantising gravity, such as: causal dynamical triangulations [26]; causal set theory [27]; group field theory [28]; and asymptotically safe gravity [29]. The main alternative candidate is string theory and its variants, which prioritises bringing gravity into the established framework for quantum particles in order to create a unified theory [30,31]. Loop quantum gravity focuses on maintaining some key concepts from general relativity such as background independence and local dynamics throughout the process of combining gravity and quantum mechanics. It describes space-time as not being a continuous manifold, but instead being a network of nodes connected by ordered links with quantum numbers for geometrical quantities such as volume. Such a network is not merely embedded in space but is space itself. As such, due to the quantisation of geometry, one cannot shrink the length of a link between nodes to being infinitesimal as in the classical case. If general relativity is truly the classical limit of loop quantum gravity, then there should be a semi-classical limit where the dynamics are well approximated by general relativity with minor quantum corrections. These should become larger at small scales and in regions of high curvature. A closely related theory is loop quantum cosmology, which uses concepts and techniques from loop quantum gravity and applies them directly at the cosmological level by using midi-superspace models [32,33]. That is, by quantising a universe which already has certain symmetries assumed such as isotropy to simplify the process. There has been some progress towards proving that loop quantum gravity can be symmetry-reduced to loop quantum cosmology, but as yet this has not been shown definitively [34,35]. For models of loop quantum cosmology to be self-consistent and anomaly-free while including some of the interesting effects from the discrete geometry, it seems that the algebra of constraints must be deformed. Specifically, some of the structure functions become more dependent on the phase space variables through a deformation function [36][37][38][39][40][41][42]. Deforming rather than breaking the algebra in principle maintains general covariance but the transformations between different choices of coordinates become highly non-linear [43]. It becomes less clear to what extent one can still interpret space-time geometrically, at least in terms of classical notions of geometry. f K IJ (q) → β(q, p)f K IJ (q) However, there is ambiguity in the correct choice of variables used for loop quantum gravity. The results cited in the previous paragraph are for real variables for which there has been significant difficulty including matter and local degrees of freedom [44]. The main alternative, self-dual variables, have had some positive results for including those degrees of freedom without deforming the constraint algebra [45], but might not have the desired quality of resolving curvature singularities [46]. Interesting predictions coming from loop quantum gravity include: a bouncing universe [47]; black hole singularity resolution and transition to white holes [48]; and signature change of the effective metric [41]. Some of these predictions are closely associated with a deformation of classical symmetries in regions of high energy density. Why study deformed general relativity? Deformed general relativity builds directly from the idea that the constraint algebra is deformed [49]. It is constructed by taking the deformed constraint algebra, and finding a corresponding model which includes local degrees of freedom a priori. This can be done because, if one starts from an algebra and makes some reasonable assumptions, one can deduce the general form of all the constraints [21,50]. This should provide a more intuitive understanding of how the deformation affects dynamics and may provide a guide for how to include the problematic degrees of freedom when working with real variables in loop quantum gravity. The constraint algebra is important because, as said previously, it closely relates to the structure of space-time [23]. Quantum geometry will behave differently to classical geometry, and deformed general relativity attempts to capture some of the effects in a semiclassical model which is more amenable to phenomenological investigations. Phenomenological models which are comparable to deformed general relativity, such as deformed special relativity [51] and rainbow gravity [52], struggle to go beyond describing individual particles coupled to an energy-dependent metric. They can suffer from a breakdown of causality [53], or find it difficult to describe multi-particle states [54]. Deformed general relativity does not suffer from these problems by construction. Overview of this thesis The main focus of this thesis is to investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. I review important concepts and methodology in chapter 2. In chapter 3, I find the minimally- The research chapters 3 and 4 are adapted from the previously published papers [55] and [56], respectively. The other research chapters, 5 and 6, were recently submitted for publication [57,58] Wider impact This study is directly motivated by the prediction of a deformed constraint algebra appearing in loop quantum cosmology [36][37][38][39][40][41][42]. As such it should provide insight into the lingering questions of how matter and local degrees of freedom need to be incorporated into the motivating theory in the presence of a deformation, and how spatial and time derivatives are differently affected. There are also potentially wider implications for this study. For example, it has been shown that taking the deformed constraint algebra to the flat-space limit gives a deformed version of the Poincaré algebra, which leads to a modified dispersion relation [46,59]. This might indicate something such as a variable speed of light or an observer-independent energy scale. In this respect it is similar to the phenomenological models of deformed special relativity [51] and rainbow gravity [52]. The deformation might indicate a non-commutative character to geometry [60,61] although apparently not a multifractional one [62]. It might represent a variable dimensionality of space-time and a running of the spectral dimension [63]. The deformation function may change sign, as suggested in the motivating studies [41]. This makes the hyperbolic equations become elliptical and implies a phase transition from classical Lorentzian spacetime to an effectively Euclidean quantum regime [22,64]. It therefore may be a potential mechanism for the Hartle-Hawking no-boundary proposal [65]. Methodology In this thesis I am primarily building on preceding work done by others [21,49,50] and elaborating on previously published material [55,56]. Space-time decomposition Quantum mechanics naturally works in the canonical or Hamiltonian framework. The canonical framework takes variables defined at a certain time and evolves them through time. That evolution defines a canonical momentum for each variable. To make general relativity more amenable to quantum mechanics, one must likewise make a distinction between the time dimension and the spatial dimensions. So I foliate the bulk space-time manifold M into a stack of labelled spatial hypersurfaces, Σ t . I assume it is globally hyperbolic, so topologically M = Σ × R [10,18,19]. A future-pointing vector normal to the spatial hypersurface Σ t is defined such that g ab n a n b = −1. The spatial slices Σ t are themselves Riemannian manifolds with an induced metric q ab = g ab + n a n b , such that q ab n b = 0. The spatial metric has an inverse defined as q ab = g ab + n a n b , so that q b a := q ac q bc = δ b a + n a n b acts as a spatial 1 projection tensor. If the spatial foliation, and therefore the spatial coordinates, are arbitrary, the timeevolution vector field t a cannot be uniquely determined by the time function t. One can project it into its normal and spatial components, defining the lapse function N = −n a t a , and the spatial shift vector N a = q a b t b . Therefore, t a = N n a + N a . Since the coordinates are arbitrary, it is convenient to take the normal to the spatial surface as the time-like direction for defining velocities rather than using the time-vector itself. Therefore, v ab := L n q ab = 1 N q ab − 2∇ (a N b) , ν I := L n ψ I = 1 N ψ I − N a ∂ a ψ I ,(2.1) whereẊ := L t X, and the extrinsic curvature of the spatial slice is related to this by K ab = 1 2 v ab . Canonical formalism I take a general first-order action for a model with dynamical fields ψ I , and non-dynamical fields λ I , S = d 4 x L (ψ I , ∂ a ψ I , λ I ) , (2.2) where ∂ a ψ I := ∂ψ I ∂x a =: ψ I,a . Varying the action with respect to each field, fixing the variation at the boundaries, and imposing the principle of least action, δS δψ I ≈ 0, δS δλ I ≈ 0,(2.3) gives the Euler-Lagrange equations of motion, 0 ≈ ∂L ∂ψ I − ∂ a ∂L ∂(∂ a ψ I ) , (2.4a) 0 ≈ ∂L ∂λ I . (2.4b) The approximation symbol is used to indicate something that is true in the dynamical regime, or 'on-shell', rather than something that is true kinematically, or 'off-shell'. The non-dynamical fields λ I can be seen to produce constraints on the system given by (2.4b), they are also known as Lagrange multipliers. Making a space-time decomposition as in section 2.1, one can define the canonical momenta of each field, π I ψ := δS δψ I = ∂L ∂ψ I , π I λ := δS δλ I = ∂L ∂λ I . (2.5) Since L does not depend onλ I , one can see that π I λ ≈ 0 are primary constraints on the system. If the matrix ∂ 2 L ∂ψ I ∂ψ J is non-degenerate, then the above equation can be inverted to findψ I =ψ I (ψ J , π J ψ , λ J ), and so one can replace the time derivatives in the action. Making a Legendre transform to find the Hamiltonian associated to this action, H = dtd 3 x Iψ I π I ψ + I µ λ I π I λ − S,(2.6) where µ λ I is a coefficient which acts like a Lagrange multiplier. The Poisson bracket of a quantity with the Hamiltonian equals the time derivative of that quantity on-shell, F ≈ {F, H} = d 3 x I δF δψ I (x) δH δπ I ψ (x) + I δF δλ I (x) δH δπ I λ (x) − (F ↔ H) ,(2.7) and if F ≈ 0 should be true at all times, thenḞ ≈ 0 must also be true [20]. Therefore, evaluating {π I λ , H} either gives back a function of the primary constraints π J λ , produces a secondary constraint φ I (ψ J , π J ψ , λ J ) ≈ 0, or gives a specific form for the coefficients of the constraints µ I . The equations (2.4b) appear here as secondary constraints. I repeat the process with {φ I , H} until I have found all the constraints on the system, at which point there is no need to differentiate between primary and secondary constraints, and I have found the generalised Hamiltonian, H = dtd 3 x Iψ I π I ψ + I µ I φ I − S ≈ H. (2.8) The set of constraints has a Poisson bracket structure {φ I , φ J } = f K IJ φ K + α IJ , α IJ / ∈ {φ K },(2.9) and if α IJ = 0 then some of φ I are what are called 'second-class' constraints, in which case some of the coefficients µ I are uniquely determined. If α IJ = 0 then all of φ I are 'firstclass', in which case the constraints not only restrict the values of the dynamical fields, but also generate gauge transformations [10,20]. This is because, in general the evolution (2.7) will depend on µ I . For an undetermined µ I to influence the mathematics but not the physical observables, a change of its value must correspond to a gauge transformation generated by the relevant first-class constraint. For classical general relativity, the action does not depend onṄ orṄ a (up to boundary terms) and is only linearly dependent on N and N a . 2 As such, there are primary constraints given by π N and π N a , which generate secondary constraints known as the Hamiltonian constraint and diffeomorphism constraint respectively, C := δH δN = {H, π N } , D a := δH δN a = H, π N a ,(2.10) which are all first-class constraints. This means that N and N a are gauge functions which do not affect the observables, and therefore the spatial slicing does not affect the dynamics. The theory is background independent and the constraints generate gauge transformations 3 , {F, C[N ]} = N L n F, {F, D a [N a ]} = L N F. (2.11) The Hamiltonian can be rewritten as a sum of the constraints up to a boundary term, H = dtd 3 x N C + N a D a + µ N π N + µ a N π N a . (2.12) Considering the Poisson bracket structure of these constraints, given by (2.9) with φ I ∈ {C, D a }, one finds that they form a Lie algebroid 4 [22], 2 Or rather, it is only linearly dependent on N and N a when velocities are represented by normal derivatives (2.1). 3 The square brackets indicates the constraint is 'smeared' over the spatial surface using the function in the brackets, e.g. C[N ] = d 3 xN (x)C(x). 4 'Algebroid' refers to the fact that some of the structure coefficients f K IJ are phase space functions D a [N a 1 ], D b [N b 2 ] = D a L N 2 N a 1 , (2.13a) C[N 1 ], D a [N a 2 ] = C L N 2 N 1 , (2.13b) C[N 1 ], C[N 2 ] = D a q ab (N 1 ∂ b N 2 − ∂ b N 1 N 2 ) . (2.13c) where (N 1 , N a 1 ) and (N 2 , N a 2 ) each represent the lapse and shift of two different hypersurface transformations. As interpreted in ref. [23], (2.13a) shows that D a is the generator of spatial morphisms, (2.13b) shows that C is a scalar density of weight one (as defined in appendix B) and (2.13c) specifies the form of C such that it ensures the embeddability of the spatial slices in space-time geometry. Choice of variables Classical canonical general relativity can be formulated equivalently using different variables. There is geometrodynamics, which uses the spatial metric and its canonical momentum (q ab , p cd ), the latter of which is directly related to extrinsic curvature, p ab = ω 2 √ q K ab − Kq ab ,(2.14) where q := det q ab and ω is the gravitational coupling. An alternative is connection dynamics, which uses the Ashtekar-Barbero connection and densitised triads (A I a , E b J ), where capital letters signify internal indices rather than coordinate indices [66,67]. This can be related to geometrodynamics by using the equations [10], q δ IJ = q ab E a I E b J , (2.15a) A I a = Γ I a + γ BI K I a , (2.15b) Γ I a = 1 2 √ q q bc IJK E b J ∇ a E c K √ q , (2.15c) K I a = 1 √ q δ IJ K ab E b J , (2.15d) where γ BI is the Barbero-Immirzi parameter and IJK is the covariant Levi-Civita tensor. The exact value of γ BI should not affect the dynamics [68]. The other alternative I mention here is loop dynamics, which uses holonomies of the connec- When each set of variables is quantised, they are no longer equivalent, for example the value of γ BI does now affect the dynamics [46,69]. For complex γ BI , care has to be taken to make sure the classical limit is real general relativity, rather than complex general relativity. Significantly, quantising loop variables (loop quantum gravity) discretises geometry, and so cannot be taken to be infinitesimal [25, p. 105]. In this work, I choose to use metric variables to build a semi-classical model of gravity. This is because the comparison to other modified gravity models should be clearer, and there is no ambiguity arising from γ BI . Higher order models of gravity In four dimensions, the Einstein-Hilbert action for general relativity is given by S = ω 2 d 4 x √ −g (4) R. (2.16) where ω = 1/8π • G is the gravitational coupling and g := det g ab . The integrand is the four dimensional Ricci curvature scalar which is contracted from the Riemann curvature tensor (4) R := (4) R a bac g bc . For any Riemannian manifold, this is defined using the commutator of two covariant derivatives of an arbitrary vector, ∇ c ∇ d A a − ∇ d ∇ c A a = R a bcd A b . (2.17) There are many reasons why theoretical physicists seek to find models of gravity which go beyond the Einstein-Hilbert action. For instance, mysteries known as dark matter [70] and dark energy [71] may originate with gravity behaving differently than expected rather than being due to unknown dark substances [72]. The indication that there was a period of inflationary expansion in the early universe has also caused a search for relevant models [73,74]. Moreover, the classical equations of gravity predict their own demise in extraordinary circumstances such as in a black hole or at a hot big bang. A theory of gravity that solves these problems to which classical general relativity is the low-curvature, largescale limit may have a semi-classical regime where corrections appear, at leading orders, similar to these theories of modified gravity [73,75,76]. One way of attempting to find alternative models of gravity is by constructing actions from higher order combinations of the Riemann tensor, so you instead have the general action S = ω 2 d 4 x √ −gF (4) R a bcd . (2.18) To bring this in line with the space-time split, I replace the determinant, g = −N 2 q. The Riemann tensor must be decomposed by projecting it along its normal and tangential components relative to the spatial slice, q e a q f b q g c q h d (4) R ef gh = 1 4 v ac v bd − 1 4 v ad v bc + (3) R abcd , (2.19a) q e a q f b q g c n h (4) R ef gh = 1 2 ∇ a v bc − 1 2 ∇ b v ac , (2.19b) q e a n f q g b n h (4) R ef gh = − 1 2 L n v ab + 1 4 q bc v ac v bd + 1 N ∇ (a ∇ b) N. (2.19c) These identities are respectively known as the Gauss equation, the Codazzi equation, and the Ricci equation [10,77]. All other projections vanish due to the tensor's antisymmetry. As can be seen from (2.19c), there are second order time derivatives included in the Riemann tensor. Including second order time derivatives in an action is problematic because it may introduce the Ostrogradsky instability [78]. To demonstrate what this means, I take a one dimensional model action, S = dtL (q,q,q) ,(2.20) I cannot find the associated Hamiltonian when there are time derivatives higher than second order, and the Euler-Lagrange equations may involve fourth order time derivatives, 0 ≈ ∂L ∂q − d dt ∂L ∂q + d 2 dt 2 ∂L ∂q , (2.21) if ∂ 2 L ∂q 2 = 0. So I must introduce an additional variable to absorb the higher order terms. The Ostrogradsky method [79] is to replaceq with an independent variable v. S = dt {L (q, v,v) + ψ (v −q)} ,(2.22) however, I instead do this slightly differently for reasons which will be apparent later. Following the method used in ref. [77,80] and using variables like in ref. [81], I instead replaceq with an auxiliary variable a, S = dt {L (q,q, a) + ψ (q − a)} , (2.23) and integrate by parts to move the second order time derivative to the Lagrange multiplier ψ, promoting it to a dynamical variable, S = dt L (q,q, a) −qψ − ψa ,(2.H = dt qp +ψπ + µ a π a − S, = dt {−pπ + µ a π a − L (q, π, a) + ψa} , (2.26) where µ a is a Lagrange multiplier. The equation of motion for a produces the secondary constraint φ = ∂L ∂a − ψ ≈ 0. Finding {φ, H} ≈ 0 produces an equation for µ a and therefore φ is a second-class constraint and a is uniquely determined. The constraint can be solved for a (q, ψ, π) as long as ∂ 2 L ∂a 2 = 0 and this can be substituted into the Hamiltonian without incident, in which case I find, H = dt {−pπ − L (q, ψ, π) + ψ a (q, ψ, π)} (2.27) which is only linear in p. This means that the energy is unbounded from below and above, and so the model may be unstable [79]. For specific models of this kind rather than this simple example, I can find a well behaved Hamiltonian if there are sufficient restrictions on the values that ψ can take [81]. If I do have a well behaved Hamiltonian, it is clear that the higher order derivative action L(q,q,q) contains an additional degree of freedom, which has been absorbed by ψ. Non-minimally coupled scalar from F (4) R gravity In ref. [77,80], it was shown how to find the Hamiltonian form of any F (4) R a bcd action. The Riemann tensor is split into its normal and tangential components (2.19), and auxiliary tensors are introduced as in (2.23). The tensor which is the Lagrange multiplier of (2.19c) becomes dynamical by integrating by parts. This turns the action into being first order in time derivatives, and therefore one can find the associated Hamiltonian. This field contains the additional degrees of freedom allowed by the higher order derivatives. To include tensor contractions such as (4) R ab (4) R ab and (4) R abcd (4) R abcd produces several additional degrees of freedom, and requires considering spatial derivatives of velocity or momenta because of (2.19b). For the sake of simplicity, in this chapter and throughout the thesis, I will only consider models which are comparable with F (4) R . So the action is given by, S = ω 2 dtd 3 xN √ q F (ρ) + ψ (4) R − ρ . (2.28) I decompose the Ricci scalar using (2.19), R = R + q ab L n v ab + 1 4 v 2 − 3 2 v ab v ab − 2 N ∆N, R = (3) R,(4) where ∆ := q ab ∇ a ∇ b . Then integrate the action (2.28) by parts to move the second order time derivative to ψ, S = ω 2 dtd 3 xN √ q F (ρ) + ψ R − K − 2 N ∆N − ρ − νv ,(2.30) where q := det q ab , ν := L n ψ, and K := v 2 − v ab v ab /4 is the standard extrinsic curvature contraction. The conjugate momenta are, p ab := δS δq ab = 1 N δS δv ab = ω 2 √ q ψ 2 v cd Q abcd − q ab q cd − νq ab , (2.31a) π := δS δψ = 1 N δS δν = −ω 2 √ q v, (2.31b) where Q abcd := q a(c q d)b for convenience. I can invert these to find, v ab = 2 ω √ q 2 ψ p T ab − q ab π , ν = 2 3ω √ q (ψπ − pH = d 3 x q ab p ab +ψπ + µ ρ π ρ + µ N π N + µ N a π a N − S, = d 3 x N C + N a D a + µ ρ π ρ + µ N π N + µ N a π a N ,(2.34) with the corresponding Hamiltonian constraint, C := δH δN = 2 ω √ q 1 ψ P − 1 3 pπ + ψ 6 π 2 + ω √ q 2 ψρ − ψR − F (ρ) + 2∆ψ ,(2.35) where P := p T ab p ab T . Finding {π ρ , H} gives a secondary constraint, φ ρ = ω 2 N √ q ψ − F (ρ) ≈ 0, (2.36) which is second-class. It can be solved to find ρ(ψ) as long as F = 0, in which case we can find the Hamiltonian constraint in terms of only the metric and the scalar field ψ. This leaves me with a term depending on ψ which acts like a scalar field potential, U geo (ψ) = ω 2 ψρ (ψ) − F ρ (ψ) = ω 2 ψ F −1 (ψ) − F F −1 (ψ) ,(2.37) which I call the geometric scalar potential. As I will further elaborate in section 3, this scalar-tensor model I have derived from an F (4) R model of gravitation is equivalent to letting the gravitational coupling in the Einstein-Hilbert action become dynamical, ω → ωψ. So models of gravity that have an action which is an arbitrary function of the spacetime curvature scalar (4) R can be converted into a scalar-tensor theory in the Hamiltonian formalism. The structure of general covariance underlying general relativity should be preserved in these models, though they do contain an additional degree of freedom. Deformed constraint algebra As previously mentioned in section 1.2, loop quantum cosmology predicts that the symmetries of general relativity should be deformed in a specific way in the semi-classical limit [36][37][38][39][40][41][42]. This appears from incorporating loop variables in a mini-superspace model, but specifying that all anomalies α IJ in (2.9) vanish while allowing counter-terms to deform the classical form of the algebra. This ensures that the constraints are first-class, retaining the gauge invariance of the theory and of the arbitrariness of the lapse and shift. If anomalous terms were to appear in the constraint algebra, then the gauge invariance would be broken and the constraints could only be solved at all times for specific N or N a . This means that there would a privileged frame of reference, and therefore no general covariance. In the referenced studies, it is strongly indicated that the bracket of two Hamiltonian constraints (2.13c) is deformed by a phase space function β, {C[N 1 ], C[N 2 ]} = D a [βq ab (N 1 ∂ b N 2 − ∂ b N 1 N 2 )]. (2.38) This has not been shown generally, but has been shown for several models independently. There are no anomalies in the constraint algebra, so a form of general covariance is preserved. However, it may be that the interpretation of a spatial manifold evolving with time being equivalent to a foliation of space-time (also known as 'embeddability') is no longer valid. These deformations only appear to be necessary for models when the Barbero-Immirzi parameter γ BI is real. For self-dual models, when γ BI = ±i, this deformation does not appear necessary [45]. However, self-dual variables are not desirable in other ways. They do not seem to resolve curvature singularities as hoped, and obtaining the correct classical limit is non-trivial [46]. Because of this, even though I use metric variables in this work, considering β = 1 and ensuring the correct classical limit means there should be relevance to the models of loop quantum cosmology with real γ BI . Derivation of the distribution equation From the constraint algebra, I am able to find the specific form of the Hamiltonian constraint C for a given deformation β. The diffeomorphism constraint D a is not affected when the deformation is a weightless scalar 5 and so is completely determined as shown in appendix B. With D a and β as inputs, I can find C by manipulating (2.38). Firstly, I must find the unsmeared form of the deformed algebra. At this point I do not need to specify my canonical variables, and leave them merely as (q I , p I ), 0 = {C[N 1 ], C[N 2 ]} − D a [βq ab (N 1 ∂ b N 2 − ∂ b N 1 N 2 )], (2.39a) = d 3 z I δC[N 1 ] δq I (z) δC[N 2 ] δp I (z) − (D a βN 1 ∂ a N 2 ) z − (N 1 ↔ N 2 ) . (2.39b) Take the functional derivatives with respect to N 1 (x) and N 2 (y), 0 = I d 3 z δC(x) δq I (z) δC(y) δp I (z) − (D a β∂ a ) x δ (x, y) − (x ↔ y) ,(2.40) where δ(x, y) is the three dimensional Dirac delta distribution 6 . If I note that I will only consider constraints without spatial derivatives on momenta, this simplifies, 0 = I δC(x) δq I (y) ∂C ∂p I y − (βD a ∂ a ) x δ (x, y) − (x ↔ y) . (2.41) For when I wish to derive the action instead of the constraint, I can transform the equation by noting that, δC[N ] δq I = − δL[N ] δq I , N v I = δC[N ] δp I ,(2.42) where v I := L n q I and the Lagrangian is here defined such that S = dtd 3 xN L. I substitute these into (2.39b), then take the functional derivatives to remove N 1 and N 2 , 0 = I δL(x) δq I (y) v I (y) + (βD a ∂ a ) x δ (x, y) − (x ↔ y) . (2.43) To find a useful form for this, I need to use a specific form for the diffeomorphism constraint. Because it depends on momenta, I must replace them using, p I := δS δq I = 1 N δL[N ] δv I ,(2. Order of the deformed action and constraint I can determine the relationship between the order of the deformation function and the order of the associated constraint (or action) by comparing orders of momenta (or velocity). Hamiltonian route As an example, take the distribution equation (2.41) with only a scalar field, so that I can consider orders of π in a way analogous to dimensional analysis. This equation must be satisfied independently at each order of momenta, so I isolate the coefficient of 0 = δC(x) δψ(y) ∂C ∂π y − (βπ∂ a ψ∂ a ) x δ (x, y) − (x ↔ y) ,(2.π n , 0 = n C m=1 m ∂C (n−m+1) ∂∆ C (m) − β (n−1) ,(2.48) where I have expanded the constraint and deformation, C = n C m=0 C (m) π m , β = n β m=0 β (m) π m . (2.49) The highest order contribution to (2.48) comes when m = n C and n − m + 1 = n C , in which case n = 2n C − 1. This is the highest order at which β won't automatically be constrained to vanish, so I find its highest order of momenta to be n β = 2n C − 2. However, this result does not take into account the fact that the combined order of momenta and spatial derivatives may be restricted. If this is the case (as is found in chapter 5), then the highest order contribution to the (2.48) will be when n − m + 1 = n C − 2, in which case I find the relation 2n C − n β = 4. (2.50) I see that a deformed second order constraint only requires considering a zeroth order deformation as I do in chapter 3, but a fourth order constraint requires considering a fourth order deformation. I consider the constraint to general order in chapter 5. Note that this relation suggests there are higher order deformations which allow for constraints given by finite order polynomials. Lagrangian route Consider the distribution equation (2.43) with only a scalar field, 0 = δL(x) δψ(y) ν(y) + β ∂L ∂ν ∂ a ψ∂ a x δ(x, y) − (x ↔ y) ,(2.51) where I have used the diffeomorphism constraint (B.6) and the momentum definition (2.45). Let me consider a simplified model to match the derivative orders for the deformation and the derivative orders for the Lagrangian in a way analogous to dimensional analysis. First order time derivatives are given by ν and two orders of spatial derivatives are given by ∆. I can collect terms in the distribution equation of the same order of time derivatives as they are linearly independent. Schematically, the distribution equation is given by, 0 = ∂L ∂∆ ν + ∂L ∂ν β,(2.52) and expanding the Lagrangian and deformation in powers of ν, L = n L m=0 L (m) ν m , β = n β m=0 β (m) ν m , (2.53) the coefficient of ν n is then given by, 0 = ∂L (n−1) ∂∆ + n β m=0 (n − m + 1) L (n−m+1) β (m) . (2.54) I can relabel and rearrange to find a schematic solution for the highest order of L appearing here, L (n) = −1 nβ (0) ∂L (n−2) ∂∆ + n β m=1 (n − m) β (m) L (n−m) . (2.55) I can see that if n β > 0, then this equation is recursive and n L → ∞ because there is no natural cut-off, suggesting that L is required to be non-polynomial. If I wish to truncate the action at some order, then it must be treated as an perturbative approximation. I consider a perturbative fourth order action in chapter 4, and the completely general action in chapter 6. Cosmology Since the main motivations for this study centre around cosmological implications of the deformed constraint algebra, I need to lay out how I find the cosmological dynamics of a model. I restrict to an isotropic and homogeneous space, using the Friedmann-Lemaître-Robertson-Walker metric (FLRW), q ab = a 2 (t)Σ ab , a = (det q ab ) 1/6 N a = 0, (2.56) where Σ ab is time-independent and describes a three dimensional spatial slice with constant curvature k. When space is flat, k = 0, this is given by Σ ab = δ ab . The normal derivative of the spatial metric is given by, v ab = 2 N aȧΣ ab , ∴ K = 6ȧ 2 a 2 N 2 =: 6 N 2 H 2 , (2.57) where H is the Hubble expansion rate, and the Ricci curvature scalar is given by, R = 6k a 2 . (2.58) When using canonical coordinates, the metric momentum is given by p ab =p Σ ab ,p = det p ab 1/3 , (2.59) which changes the metric's commutation relation, q ab (x), p cd (y) = δ cd ab (x)δ(x, y) → a(x),p(y) = δ(x, y) 6a(x) , (2.60) where δ cd ab := δ c (a δ d b) . The spatial derivatives of matter fields vanish, ∂ a ψ I = 0. One may couple a perfect fluid to the metric by including the energy density ρ in the constraint or the action [82], C ⊃ a 3 ρ, L ⊃ −a 3 ρ,(2.61) which must satisfy the continuity equation, ρ + 3Hρ (1 + w ρ ) = 0, (2.62) where w ρ is the perfect fluid's cosmological equation of state, the ratio of the pressure density to the energy density. For investigations into whether there are implications for the hypothesised inflationary period in the very early universe, I must define what is considered to be a period of inflation. The simple definition is when the finite scale factor is both expanding and accelerating, a > 0 andä > 0. As said above, loop quantum cosmology with real variables seems to predict a big bounce instead of a big bang or crunch. In this thesis, I take the very literal interpretation of this (as found in ref. [83]) and define a bounce as a turning point for a finite scale factor, a > 0, a = 0 andä > 0. This definition may be usable, but it is not ideal. If a bounce does indeed happen when β < 0, as predicted in the literature, then this is when the effective metric signature is Euclidean, whenȧ may be a complex number. Ideally, I would like to extract cosmological observables such as the primordial scalar index to find phenemenological constraints [84]. However, to calculate the power spectra of primordial fluctuations would require adapting the cosmological perturbation theory formalism to ensure it is valid for deformed covariance, something which would probably be highly non-trivial. Unfortunately, there was not enough time to investigate this. Second order scalar-tensor model and the classical limit In this chapter, I derive the general form of a minimally-deformed, non-minimally-coupled scalar-tensor model which includes up to two orders in momenta or time derivatives. This allows me to demonstrate that the higher order gravity model derived in section 2.4.1 does not deform the constraint algebra or general covariance, and therefore show how the deformed models derived in subsequent chapters are distinct. For those later chapters, this minimally-deformed model provides a useful reference point. This chapter is adapted from work I previously published in ref. [55]. I find the form of the model by deriving restrictions on the constraint using (2.41) and then transform to find the action. It would be completely equivalent to derive the action first, because the minimally deformed case maintains a linear relationship between velocities and momenta, meaning that the transformation between the action and constraint is trivial. After finding the constraint and action, I look at some of the cosmological implications in section 3.3, especially the interesting influence of the non-minimal coupling of the scalar field. I use the structure of the scalar-tensor constraint which is a parameterisation of F ( (4) R), (2.35), to guide the structure of my general ansatz for a spatial metric coupled to several scalar fields. I include spatially covariant terms up to second order in momenta or spatial derivatives, and ignore terms linear in momenta, C = C ∅ + C (R) R + C (p 2 ) abcd p ab p cd + C (pπ I ) pπ I + C (ψ I ψ J ) ∂ a ψ I ∂ a ψ J + C (ψ I ) ∆ψ I + C (π I π J ) π I π J ,(3.1) with summation over I and J implied. I have included C (ψ I ψ J ) because it appears in the constraint for minimally coupled scalar fields [10, p. 62]. I aimed to define the most general ansatz for a scalar-tensor constraint containing up to two orders in derivatives which is covariant under general spatial diffeomorphisms, as well as under time reversal, and preserves spatial parity. Each coefficient is potentially a function of q and ψ I , allowing for non-minimal coupling. The spatial indices of C (p 2 ) abcd only represent different combinations of the metric. The zeroth order term might include terms such as scalar field potentials or perfect fluids, and it behaves as a generalised potential C ∅ = √ q U (q, ψ I ). Solving the distribution equation 0 = δC 0 (x) δq ab (y) 2p cd C (p 2 ) abcd + πq ab C (pπ) y + δC 0 (x) δψ(y) p C (pπ) + 2π C (π 2 ) y + 2β ∂ b p ab + Γ a bc p bc − β∂ a ψ π x ∂ a(x) δ(x, y) − (x ↔ y) , (3.2) where C 0 is the part of the constraint without momenta. From here there are two routes to solution, by focusing on either the p ab and π components. I must do both to find all consistency conditions on the coefficients of the Hamiltonian constraint. p ab sector To proceed to the metric momentum sector, I take (3.2) and find the functional derivative with respect to p ab (z), 0 = 2 δC 0 (x) δq cd (y) C (p 2 ) abcd (y) + δC 0 (x) δψ(y) C (pπ) ab (y) δ(z, y) + 2β(x) δ c (a ∂ b) x δ(z, x) + Γ c ab (x)δ(z, x) ∂ c(x) δ(x, y) − (x ↔ y) ,(3.3) where I explicitly show the coordinate of the partial derivative as ∂ a(y) := ∂ ∂y a because the distinction is important when integrating by parts. I then proceed by moving derivatives away from δ(z, y) terms and discarding total derivatives, 0 = 2 δC 0 (x) δq cd (y) C (p 2 ) abcd (y) + δC 0 (x) δψ(y) C (pπ) ab (y) + 2∂ c(y) βδ c (a ∂ b) y δ(y, x) − 2 βΓ c ab ∂ c y δ(y, x) δ(z, y) − (x ↔ y) , (3.4) which can be rewritten as, 0 = A ab (x, y)δ(z, y) − A ab (y, x)δ(z, x). (3.5) Integrating over y, I find that part of the equation can be combined into a tensor dependent only on x, 0 = A ab (x, z) − δ(z, x) d 3 yA ab (y, x), = A ab (x, z) − δ(z, x)A ab (x), where A ab (x) = d 3 yA ab (y, x) . (3.6) Substituting in the definition of A ab (x, z) then relabelling, 0 = 2 δC 0 (x) δq cd (y) C (p 2 ) abcd (y) + δC 0 (x) δψ(y) C (pπ) ab (y) + 2∂ c(y) βδ c (a ∂ b) y δ(y, x) − 2 (βΓ c ab ∂ c ) y δ(y, x) − A ab (x)δ(y, x). (3.7) Multiplying by an arbitrary test tensor θ ab (y), then integrating by parts over y, I get 0 = θ ab (· · · ) ab + ∂ c θ ab 2C (p 2 ) abde ∂C 0 ∂q de,c + 4∂ d C (p 2 ) abef ∂C 0 ∂q ef,cd + C (pπ) ab ∂C 0 ∂ψ ,c +2∂ d C (pπ) ab ∂C 0 ∂ψ ,cd + 2δ c (a ∂ b) β + 2βΓ c ab + ∂ cd θ ab 2C (p 2 ) abef ∂C 0 ∂q ef,cd + C (pπ) ab ∂C 0 ∂ψ ,cd + 2βδ cd ab ,(3.8) where I do not need to consider the zeroth derivative terms because they do not produce restrictions on the form of the constraint. Since θ ab is arbitrary beyond the symmetry of its indices, each unique contraction of it forms a linearly independent equation. To calculate the derivatives of C 0 , I must use the decomposition of the Riemann tensor (A.6) and the second covariant derivative of the metric variation expressed in terms of partial derivatives (A.9). This gives, ∂C 0 ∂ψ ,ab = C (ψ ) q ab , ∂C 0 ∂ψ ,a = 2C (ψ 2 ) ∂ a ψ − C (ψ ) Γ a , ∂C 0 ∂q ab,cd = C (R) Φ abcd , ∂C 0 ∂q ab,c = C (ψ ) 1 2 q ab ∂ c ψ − q c(a ∂ b) ψ − C (R) Φ def g Γ c f g δ ab de + 4δ (a (d Γ b) e)(f δ c g) ,(3.9) where Φ abcd = Q abcd − q ab q cd as found in (A.8). Note that Q abcd := q a(c q d)b and δ ab de := δ (a d δ b) e . I evaluate the coefficient of ∂ dc θ ab and find the linearly independent components, q ab ∂ 2 θ ab : 0 = −2C (R) 2C (p 2 ) + C (p 2 x) + C (ψ ) C (pπ) , (3.10a) ∂ ab θ ab : 0 = C (R) C (p 2 x) + β, (3.10b) where I have decomposed the constraint coefficient C (p 2 ) abcd = q ab q cd C (p 2 ) + Q abcd C (p 2 x) . Then evaluating similarly for ∂ c θ ab , q ab ∂ c ψ∂ c θ ab : 0 = 2 C (ψ 2 ) + C (ψ ) ∂ ψ C (pπ) + C (ψ ) − 8C (R) ∂ ψ C (p 2 ) + C (ψ ) − 4C (R) ∂ ψ C (p 2 x) , (3.11a) ∂ b ψ∂ a θ ab : 0 = −C (ψ ) + 2C (R) ∂ ψ C (p 2 x) + ∂ ψ β, (3.11b) X b ∂ a θ ab : 0 = C (R) (1 + 2∂ q ) C (p 2 x) + ∂ q β, (3.11c) X c q ab ∂ c θ ab : 0 = −2C (R) (1 + 4∂ q ) 2C (p 2 ) + C (p 2 x) (3.11d) + C (ψ ) (1 + 4∂ q ) C (pπ) , (3.11e) where ∂ ψ := ∂ ∂ψ , ∂ q := ∂ ∂ log q and X a := q bc ∂ a q bc . Note that the equations for ∂ c q ab ∂ c θ ab , ∂ a q bc ∂ c θ ab and q ab ∂ d q cd ∂ c θ ab are not included because they are identical to (3.10). Using (3.10b) to solve for C (p 2 x) , then substituting it into (3.11c), I find, ∂ log C (R) ∂ log q = 1 2 1 + ∂ log β ∂ log q , (3.12) which is solved by C (R) (q, ψ) = f (ψ) q \|β (q, ψ)\|, where f (ψ) is some unknown function. If I solve (3.10) for C (p 2 ) and C (p 2 x) , then substitute them into (3.11e), I find a similar equation to the one above for C (R) , and therefore C (ψ ) (q, ψ) = f (ψ ) (ψ) q \|β (q, ψ)\|. Taking (3.11b) then substituting in for C (p 2 x) , C (R) and C (ψ ) , I find that f (ψ ) (ψ) = −2∂ ψ f (ψ), C (R) = f q \|β\|, C (ψ ) = −2∂ ψ f q \|β\|, (3.13a) C (p 2 x) = −σ β f \|β\| q , C (p 2 ) = σ β 2f \|β\| q − ∂ ψ f 2f C (pπ) , (3.13b) where σ β := sgn(β), which is all the conditions which can be obtained from the metric momentum sector of the distribution equation. The remaining conditions must be found in the scalar momentum sector. π sector Similar to subsection 3.1.1 above, I take the functional derivative of (3.2) with respect to π(z), 0 = δC 0 (x) δq ab (y) C (pπ) ab (y) + 2 δC 0 (x) δψ(y) C (π 2 ) (y) δ(z, y) − (β∂ a ψ∂ a ) x δ(x, y)δ(z, x) − (x ↔ y) , (3.14) then exchange terms to find the coefficient of δ(z, y), 0 = δC 0 (x) δq ab (y) C (pπ) ab (y) + 2 δC 0 (x) δψ(y) C (π 2 ) (y) + (β∂ a ψ∂ a ) y δ(y, x) δ(z, y) − (x ↔ y) , (3.15) which can be rewritten as, 0 = A(x, y)δ(z, y) − A(y, x)δ(z, x), (3.16a) 0 = A(x, z) − δ(z, x) d 3 yA(y, x), (3.16b) = A(x, z) − δ(z, x)A(x), where A(x) = d 3 yA (y, x) , (3.16c) leading to 0 = δC 0 (x) δq ab (y) C (pπ) ab (y) + 2 δC 0 (x) δψ(y) C (π 2 ) (y) + (β∂ a ψ∂ a ) y δ(y, x) − A(x)δ(y, x). (3.17) Multiplying by an arbitrary test function η(y), then integrating by parts over y, I get 0 = η (· · · ) + ∂ ab η C (pπ) cd ∂C 0 ∂q cd,ab + 2C (π 2 ) ∂C 0 ∂ψ ,ab + ∂ a η C (pπ) bc ∂C 0 ∂q bc,a + 2∂ b C (pπ) cd ∂C 0 ∂q cd,ab + 2C (π 2 ) ∂C 0 ∂ψ ,a + 4∂ b C (π 2 ) ∂C 0 ∂ψ ,ab − β∂ a ψ . (3.18) I then substitute in (3.9) to find the linearly independent conditions, ∂ 2 η : 0 = C (R) C (pπ) − C (ψ ) C (π 2 ) , (3.19a) ∂ a ψ∂ a η : 0 = 1 2 C (ψ ) − 4C (R) ∂ ψ C (pπ) + 4 C (ψ 2 ) + C (ψ ) ∂ ψ C (π 2 ) − β, (3.19b) X a ∂ a η : 0 = C (R) (1 + 4∂ q ) C (pπ) − C (ψ ) (1 + 4∂ q ) C (π 2 ) . (3.19c) Note that there is another condition from ∂ b q ab ∂ a η, but it is identical to (3.19a). I can solve (3.19a) for C (pπ) = C (ψ ) C (π 2 ) /C (R) , and then substitute into (3.19b) to find, 0 = C (π 2 ) C (ψ 2 ) − ∂ ψ C (ψ ) + C (ψ ) C (R) ∂ ψ C (R) + C (ψ ) 8 − β 4 ,(3.20) which I can solve for C (π 2 ) , and is the same conclusion I get from (3.11a) (though I did not explicitly write it above because it is simpler to write it here). The condition (3.19c) is solved when I substitute in all my results so far, C (π 2 ) = σ β 4 \|β\| q C (ψ 2 ) q \|β\| + 2f − 3f 2 2f −1 , (3.21a) C (pπ) = −σ β f 2f \|β\| q C (ψ 2 ) q \|β\| + 2f − 3f 2 2f −1 , (3.21b) and if I collect all of the coefficients, I find the Hamiltonian constraint, C = q \|β\| f R − 2f ∆ψ + C (ψ 2 ) ∂ a ψ∂ a ψ + C ∅ + σ β \|β\| q    1 f p 2 6 − P + 1 4 π − f f p 2 C (ψ 2 ) q \|β\| + 2f − 3f 2 2f −1    ,(3.22) so the freedom in any (3+1) dimensional scalar-tensor theory with time symmetry and minimally deformed general covariance comes down to the choice of f (ψ), β (q, ψ), C (ψ 2 ) (q, ψ) and the zeroth order term C ∅ (q, ψ). It is convenient to make a redefinition, C (ψ 2 ) = g (q, ψ) q \|β\|, where I have made the scalar weight and expected dependence on β explicit. It is worth remembering that this is an assumption, and that g could be a function of β. It is also convenient to treat the zeroth order term as a general potential, and to extract the scalar density, C ∅ = √ q U (q, ψ). I find the effective Lagrangian associated with this Hamiltonian constraint by performing a Legendre transformation, L = q \|β\| f K β − R + f νv β + 2∆ψ + g + 2f ν 2 β −g ∂ a ψ∂ a ψ − U \|β\| .L cov = q \|β\| −f (4,β) R − g + 2f ∂ (4,β) µ ψ ∂ µ (4,β) ψ − √ q U, (3.24) where the deformed four dimensional Ricci scalar and partial derivative are given by, (4,β) R = R + σ β \|β\| q ab L n v ab \|β\| + 1 4β v 2 − 3 4β v ab v ab − 2∆ \|β\| N \|β\| N , (3.25a) ∂ (4,β) µ ψ ∂ µ (4,β) ψ = ∂ a ψ ∂ a ψ − 1 β ν 2 . (3.25b) If this is compared to (2.29), I see that the deformation seems to have transformed the effective lapse function N → \|β\| N , and transformed the effective normalisation of the normal vector to g µν n µ n ν = −σ β . Here is where I see the effective signature change which comes from the deformation. It is useful to take the Lagrangian in covariant form and use it to redefine the coupling functions so that minimal coupling is when the functions are equal to unity, f = − 1 2 ω R and g = − 1 2 ω ψ + ω R , L cov = 1 2 q \|β\| ω R (ψ) (4,β) R − ω ψ (q, ψ) ∂ (4,β) µ ψ ∂ µ (4,β) ψ − √ q U (q, ψ) ,(3.26) so the effective forms of the constraint and Lagrangian are given by, L = 1 2 q \|β\| ω R R − K β − ω R νv β + 2∆ψ + ω ψ ν 2 β − ω ψ + 2ω R ∂ a ψ∂ a ψ − √ q U, (3.27a) C = q \|β\| 2σ β qω R P − p 2 6 − ω R 2 R + σ β 2q π − ω R ω R p 2 ω ψ + 3ω 2 R 2ω R −1 +ω R ∆ψ + ω ψ 2 + ω R ∂ a ψ∂ a ψ + √ q U, (3.27b) which is the main result of this section in its most useful form. Since I have non-minimal coupling, I am working in the Jordan frame. I can get to the Einstein frame by making a specific conformal transformation which absorbs the coupling ω R by setting q ab = ω Rqab and N = ω −1/2 RÑ , L = 1 2 q \|β\| R −K β + ω ψ ω R + 3ω 2 R 2ω 2 R ν 2 β −q ab ∂ a ψ∂ b ψ − q U ω 2 R , (3.28) where variables with tildes are Einstein-frame quantities. So the Einstein frame couplings are given byω R = 1,ω ψ = ω ψ ω R + 3ω 2 R /2 /ω 2 R , and the potential byŨ = U/ω 2 R . When the term 'Einstein frame' is used elsewhere in the literature, it often refers to an action which is transformed further so that the effective scalar coupling is also unity. I can make this transformation to a minimally coupled scalar ϕ by solving the differential equation, ∂ϕ ∂ψ = ω ψ ω R + 3 2 ∂ ψ ω R ω R 2 , (3.29) for example, when ω ψ = 0, this is solved by ϕ (ψ) = 3 2 log ω R (ψ) sgn(∂ ψ log ω R (ψ)). Multiple scalar fields Consider the case of multiple scalar fields. I start from the distribution equation as before, but label the scalar field variables with an index. Proceeding like in section 3.1.1 by taking functional derivatives with respect to p ab and then integrating by parts with test function θ ab , I obtain the conditions, ∂ ab θ ab : 0 = C (R) C (p 2 x) + β, (3.30a) q ab ∂ 2 θ ab : 0 = −2C (R) 2C (p 2 ) + C (p 2 x) + I C (ψ I ) C (pπ I ) , (3.30b) X b ∂ a θ ab : 0 = C (R) (1 + 2∂ q ) C (p 2 x) + ∂ q β, (3.30c) ∂ b ψ I ∂ a θ ab : 0 = C (ψ I ) − 2C (R) ∂ ψ I C (p 2 x) − ∂ ψ I β, (3.30d) q ab ∂ c ψ I ∂ c θ ab : 0 = C (ψ I ) − 8C (R) ∂ ψ I C (p 2 ) + C (ψ I ) − 4C (R) ∂ ψ I C (p 2 x) +2 C (ψ 2 I ) + C (ψ I ) ∂ ψ I C (pπ I ) + J =I C (ψ I ψ J ) + C (ψ J ) ∂ ψ I C (pπ J ) . (3.30e) I note that there are other independent terms, but they do not produce any extra conditions. Likewise, if I follow the route taken in section 3.1.2, taking the functional derivative with respect to π I then integrating by parts with test function η I , I find the conditions, ∂ 2 η I : 0 = C (R) C (pπ I ) − C (ψ I ) C (π 2 I ) − 1 2 J =I C (ψ J ) C (π I π J ) , (3.31a) X a ∂ a η I : 0 = C (R) (1 + 4∂ q ) C (pπ I ) − C (ψ I ) (1 + 4∂ q ) C (π 2 I ) − 1 2 J =I C (ψ J ) (1 + 4∂ q ) C (π I π J ) , (3.31b) ∂ a ψ I ∂ a η I : 0 = 1 2 C (ψ I ) − 4C (R) ∂ ψ I C (pπ I ) + 4 C (ψ 2 I ) + C (ψ I ) ∂ ψ I C (π 2 I ) + J =I C (ψ I ψ J ) + 2C (ψ J ) ∂ ψ I C (π I π J ) − β, (3.31c) ∂ a ψ J =I ∂ a η I : 0 = 1 2 C (ψ J ) − 2C (R) ∂ ψ J C (pπ I ) + 2 C (ψ I ψ J ) + 2C (ψ I ) ∂ ψ J C (π 2 I ) +2 C (ψ 2 J ) + C (ψ J ) ∂ ψ J C (π I π J ) + K =I,J C (ψ J ψ K ) + 2C (ψ K ) ∂ J C (π I π K ) , (3.31d) and similar to above, there are other independent terms which do no produce any unique conditions. To solve this system of equations I must make assumptions, in particular about the relationship between the scalar fields. One choice might be to assume an O (N ) symmetry, where the coupling and deformation would only depend on the absolute value of the scalar field multiplet \|ψ\| = I ψ 2 I , and relationships between the C (ψ I ψ J ) coefficients could be assumed. However, I instead choose to take one non-minimally coupled field (ψ, π ψ ) and one minimally coupled field (ϕ, π ϕ ) with no cross-terms in the spatial derivative sector, C (ϕ ψ ) = 0. The minimally coupled field only appears in terms other than the potential U (q, ψ, ϕ) through the deformation function β(q, ψ, ϕ). For example, C (R) = C (R) (q, ψ, β). Solving (3.30a) and (3.30c) gives me, C (R) = f (ψ) q \|β (q, ψ, ϕ)\|, C (p 2 x) = −1 f (ψ) \|β (q, ψ, ϕ)\| q , (3.32) as before. Substituting these into (3.30b) and (3.30d) gives me, 33) and the remaining conditions are, C (ψ ) = −2f q \|β\|, C (ϕ ) = 0, C (p 2 ) = σ β 2f \|β\| q − f 2f C (pπ ψ ) ,(3.C (pπ ψ ) = −σ β f 2f \|β\| q C (ψ 2 ) q \|β\| + 2f − 3f 2 2f −1 (3.34a) C (πϕπ ψ ) = −∂ ϕ β ∂ ψ f 4C (ϕ 2 )          2C (ψ 2 ) √ q\|β\| 1 − ∂ log C (ψ 2 ) ∂ log β + 2f − 3f 2 2f C (ψ 2 ) √ q\|β\| + 2f − 3f 2 2f 2          , (3.34b) C (π 2 ϕ ) = β 4C (ϕ 2 ) , C (pπϕ) = − f f C (πϕπ ψ ) . (3.34c) I note that the constraint is significantly simpler if I assume C (ϕ 2 ) = g ϕ (ψ) q \|β\| and C (ψ 2 ) = g ψ (ψ) q \|β\|, where g ϕ and g ψ are arbitrary functions. In this case the whole Hamiltonian constraint is 35) and the associated Lagrangian density is C = q \|β\| f R − 2f ∆ψ + g ϕ ∂ a ϕ∂ a ϕ + g ψ ∂ a ψ∂ a ψ + √ q U + σ β \|β\| q    π 2 ϕ 4g ϕ + 1 f p 2 6 − P + π ψ − f f p π ψ − f f p − f ∂ϕβ βgϕ π ϕ 4 g ψ + 2f − 3f 2 2f    ,(3.L = q \|β\| f K β − R + f ν ψ v β + 2∆ψ + ĝ ψ h + 3f 2 2f ν 2 ψ β −g ψ ∂ a ψ∂ a ψ + g ϕ hβ ν 2 ϕ − g ϕ ∂ a ϕ∂ a ϕ + f ∂ ϕ β hβ ν ϕ ν ψ − √ q U, (3.36a) g ψ = g ψ + 2f − 3f 2 2f , h = 1 − f 2 ∂ ϕ β 2 4g ϕĝψ β 2 . (3.36b) If β does not depend on ϕ, then this can be simplified greatly, in which case the effective and covariant forms of the Lagrangian are given by, L = 1 2 q \|β\| ω R R − K β − ω R ν ψ v β + 2∆ψ + ω ϕ ν 2 ϕ β − ∂ a ϕ∂ a ϕ + ω ψ ν 2 ψ β − ω ψ + 2ω R ∂ a ψ∂ a ψ − √ q U, (3.37a) L cov = 1 2 q \|β\| ω R (4,β) R − ω ψ ∂ (4,β) µ ψ∂ µ (4,β) ψ − ω ϕ ∂ (4,β) µ ϕ∂ µ (4,β) ϕ − √ q U, (3.37b) where ω R = −2f , ω ψ = 2 (g ψ + 2f ), ω ϕ = 2g ϕ . Therefore, when I assume that the minimally coupled scalar field can also be considered to be minimally coupled to the deformation function, I find that the action simplifies to the expected form. It would be interesting to see what effects appear for scalar field multiplets, especially for non-Abelian symmetries, but that is beyond the scope of this study. Instead, I now turn to studying the cosmological dynamics of my results. Cosmology To find the cosmological dynamics, I restrict to a flat, homogeneous, and isotropic metric in proper time (N = 1). I also assume that β does not depend on the minimally coupled scalar field ϕ for the sake of simplicity. From (3.37), I find the Friedmann equation, which can be written in two equivalent forms, H ω R H + ω Rψ = 1 3 ω ψ 2ψ 2 + ω ϕ 2φ 2 + σ β \|β\| U , (3.38a) ω R H + 1 2 ω Rψ 2 = 1 3 1 2 ω R ω ψ + 3 2 ω 2 R ψ 2 + ω R ω ϕ 2φ 2 + σ β ω R \|β\| U . (3.38b) From (3.38b) I see that ω R ω ψ + 3ω 2 R /2 ≥ 0 and ω R ω ϕ ≥ 0 are necessary when U → 0 to ensure real-valued fields. If I compare this condition to the Einstein frame Lagrangian (3.28), I can see that it is also the condition which follows from insisting that the scalar field ψ is not ghost-like in that frame. Similarly, I see that σ β ω R > 0 is necessary wheṅ ψ,φ → 0. For the reasonable assumption that the minimally coupled field ϕ does not affect the deformation function β, the only way that field is modified is through a variable maximum phase speed c 2 ϕ = β. Due to this minimal modification, it does not produce any of the cosmological phenomena I am interested in (bounce, inflation) through any novel mechanism. Therefore, I will ignore this field for the rest of the chapter. I find the equations of motion by varying the Lagrangian (3.37) with respect to the fields. For the simple undeformed case β = 1 the equations are given by, ω R ω ψ + 3 2 ω 2 R ψ = −3ψH ω R ω ψ + ω 2 R − ω R ∂ ψ U + 3 2 ω R ω R H 2 − 1 2ψ 2 ω R ω ψ + 3 2 ω R ω ψ + 3ω R ω R + 3 2 ω R 1 + a 3 ∂ ∂a U, (3.39a) ω R ω ψ + 3 2 ω 2 R ä a = − 1 2 H 2 ω R ω ψ + 3ω 2 R + ω ψ 2 1 + a 3 ∂ ∂a U − 1 4ψ 2 ω 2 ψ + 2ω ψ ω R − ω ψ ω R + 1 2 ω R ω ψψ H − ω R 2 ∂ ψ U, (3.39b) where I can see from the equations of motion that the model breaks down if ω R ω ψ + 3ω 2 R /2 → 0 because it will tend to cause \|ψ\| → ∞ and \|ä\| → ∞. Bounce I will address the question of whether there are conditions under which there can be a big bounce as defined in section 2.8. I find in chapter 4 (and in ref. [56]) that a deformation function which depends on curvature terms can generate a bounce. Elsewhere in the literature on loop quantum cosmology the bounce happens in a regime when β < 0 because the terms depending on curvature or energy density overpower the zeroth order terms [40,41]. However, I am not including derivatives in the deformation here so the effect would have to come from the non-minimal coupling of the scalar field or the zeroth order deformation. I takeȧ = 0 for finite a, include a deformation and I ignore the minimally coupled field for simplicity. From the Friedmann equation (3.38) I find, 0 = ω ψ 2ψ 2 + σ β \|β\| U,(3.40) which implies that σ β ω ψ < 0 for a bounce because otherwise the equation cannot balance for U > 0 and ψ ∈ R. Substituting (3.40) into the full equation of motion for the scale factor, and demanding thatä > 0 to make it a turning point, I find the following conditions, σ β ω ψ < 0, (3.41a) ω R ω ψ + 3 2 ω 2 R > 0, (3.41b) σ β \|β\| ω ψ + 2ω R U − σ β ω R 2ω ψ ∂ ψ \|β\|ω ψ U + aβ 6 ∂ ∂a ω ψ U \|β\| > 0, (3.41c) from which I can determine what the coupling functions, deformation and potential must be for a bounce. For example, if I look at the minimally coupled case, when ω R = ω ψ = 1, and assume that U > 0, I can see that the conditions are given by, σ β < 0, ∂ log \|β\| −1/2 U ∂ log a < −6. (3.42) Since I must have β → 1 in the classical limit and σ β < 0 at the moment of the bounce, then β must change sign at some point. Therefore, a universe which bounces purely due to a zeroth order deformation must have effective signature change. Another example is obtained by assuming scale independence and choosing β = 1 and U > 0. In this case the bounce conditions become, ω ψ < 0, ω ψ ω R + 3 2 ω 2 R > 0, ω ψ + 2ω R − 1 2 ω R ∂ ψ log (ω ψ U ) > 0,(3.43) which I can use to find a model which bounces purely due to a scale-independent nonminimally coupled scalar. I present this model in subsection 3.3.5. Inflation Now consider the inflationary dynamics. For simplicity I assume that inflation will come from a scenario similar to slow-roll inflation with possible enhancements coming from the non-minimal coupling or the deformation. The conditions for slow-roll inflation are, ψ 2 U, ψ ψ H , Ḣ H 2 ,(3.44) assuming the couplings, potential and deformation are scale independent and the deformation is positive, I get the following slow roll equations, H β 1/2 U 3ω R , (3.45a) ψ − β 1/2 U 3ω R   ∂ ψ log U β 1/2 ω 2 R ω ψ ω R + ω 2 R ω 2 R + β ω R 2βω R   , (3.45b) and define the slow-roll parameters, := −Ḣ H 2 , η := −Ḧ HḢ , ζ := −ψ ψH ,(3.46) which, under slow-roll conditions are given by, ∂ ψ log β 1/2 U ω R ∂ ψ log U β 1/2 ω 2 R 2 ω ψ ω R + ω 2 R ω 2 R + β ω R 2βω R (3.47a) η   ∂ ψ log U β 1/2 ω 2 R ω ψ ω R + ω 2 R ω 2 R + β ω R 2βω R   ∂ ψ log + 2 , (3.47b) ζ ∂ ψ   ∂ ψ log U β 1/2 ω 2 R ω ψ ω R + ω 2 R ω 2 R + β ω R 2βω R   + , (3.47c) where a prime indicates a partial derivative with respect to ψ, i.e. β = ∂ ψ β. The slow-roll regime ends when the absolute value of any of these three parameters approaches unity. Defining N to mean the number of e-folds from the end of inflation, a (t) = a end e −N (t) , I find that, N = − t t end dtH = − ψ ψ end dψ Ḣ ψ ,(3.48) and using the slow-roll approximation, N ψ ψ end dψ ω ψ ω R + ω 2 R ω 2 R + β ω R 2βω R ∂ ψ log U β 1/2 ω 2 R ,(3.49) which can be solved once I specify the form of the couplings, deformation and potential. I cannot find equations for observables such as the spectral index n s because it would require investigating how the cosmological perturbation theory is modified in the presence of non-minimal coupling and deformed general covariance. Beyond this, it is difficult to make general statements about the dynamics unless I restrict to a given model, so I will now consider some models and discuss their specific dynamics. Geometric scalar model As demonstrated in the previous chapter, section 2.4.1, the geometric scalar model comes from parameterising F (4) R gravity so that the additional degree of freedom of the scalar curvature is instead embodied in a non-minimally coupled scalar field ψ [77,80]. Its couplings are given by ω R = ψ and ω ψ = 0. This model is a special case of the Brans-Dicke model, which has ω ψ = ω 0 /ψ, when the Dicke coupling constant ω 0 vanishes. I can add in a minimally coupled scalar field with ω ϕ = 1 and thereby see the effect of this scalar-tensor gravity on the matter sector. However, I set ω ϕ = 0 because it does not significantly affect my results. The effective action for this model is given by, L geo = 1 2 q \|β\| ψ R − K β − ν ψ v β − 2∆ψ − √ q U (ψ) , (3.50a) U (ψ) = ψ 2 F −1 (ψ) − 1 2 F F −1 (ψ) , (3.50b) where F refers to the F (4) R function which has been parameterised. The equations of motion when β → 1 are given by, H ψH +ψ = 1 3 U, (3.51a) a a = −H 2 + 1 3 ∂U ∂ψ , (3.51b) ψ = −2ψH + ψH 2 + 1 + a 3 ∂ ∂a − 2ψ 3 ∂ ∂ψ U, (3.51c) from which I can see that the scalar field has very different dynamics compared to minimally coupled scalars. This reflects its origin as a geometric degree of freedom rather than a purely matter field. Looking at inflation, the geometric scalar model with a potential corresponding to the Starobinsky model, can indeed cause inflation through a slow-roll of the scalar field down its potential. The non-minimal coupling of the scalar to the metric also causes the scale factor to oscillate unusually, however. It is interesting to compare in Fig. 3.1 the scale factor in the Jordan frame, a, and the conformally transformed scale factor in the Einstein frame,ã = a √ ω R . F (4) R = (4) R + 1 2M 2 (4) R 2 → U = M 2 4 (ψ − 1) 2 ,(3. Assuming ψ > 1 during inflation, the slow-roll parameters (3.47) are given by, ψ + 1 (ψ − 1) 2 , η −2 ψ 2 − 1 , ζ 1 ψ − 1 ,(3.53) so the slow-roll regime of inflation ends at ψ ≈ 3 when → 1. The equation for the number of e-folds of inflation in the slow-roll regime (3.49) is given by N 1 2 ψ − ψ end − log ψ ψ end . (a) (b) Figure 3.2: A contour plot of ω R ω ψ + 3ω 2 R /2 for the non-minimally enhanced scalar model is shown in (a). In (b), the red region is when the metric becomes ghost-like (when ω R < 0). In both, the white regions are forbidden because it is where ω R ω ψ + 3ω 2 R /2 < 0, implying imaginary fields. The green region is the region of well-behaved evolution. Non-minimally enhanced scalar model Unlike the geometric scalar model considered above, the non-minimally enhanced scalar model (NES) from [85], takes a scalar field from the matter sector and introduces a nonminimal coupling rather than extracting a degree of freedom from the gravity sector. The coupling functions are given by ω R = 1 + ξψ 2 , ω ψ = 1 and ω ϕ = 0. The strength of the quadratic non-minimal coupling is determined by the constant ξ. The deformed effective Lagrangian for this model is given by, The equations of motion for this model when it is undeformed are given by, L NES = q \|β\| 1 2 1 + ξψ 2 R − K β + 1 2 ν 2 ψ β − ∂ a ψ∂ a ψ −2ξ ψν ψ v 2β + ψ∆ψ + ∂ a ψ∂ a ψ − √ q U (ψ) .1 + ξψ 2 H 2 + 2ξψψH = 1 3 1 2ψ 2 + U , (3.55a) 1 + (1 + 6ξ) ξψ 2 ä a = −1 2 H 2 1 + (1 + 12ξ) ξψ 2 − 1 + 4ξ 4ψ 2 + ξψψH + 1 2 1 + a 3 ∂ ∂a U + ξψ∂ ψ U, (3.55b) 1 + (1 + 6ξ) ξψ 2 ψ = −3ψH 1 + (1 + 4ξ) ξψ 2 − 1 + ξψ 2 ∂ ψ U + 3ξψ 1 + ξψ 2 H 2 − 1 + 4ξ 2ψ 2 + U + a 3 ∂U ∂a . (3.55c) and I proceed to use them to consider this model's inflationary dynamics. For a power-law potential U = λ n \|ψ\| n and ξ > 0, the slow-roll parameter which reaches unity first is at ψ end ±n 2 + n (6 − n) ξ . The number of e-folds from the end of inflation is given by, N NES (ψ) ψ ψ end dϕ ϕ 1 + (1 + 4ξ) ξϕ 2 (1 + ξϕ 2 ) (n + (n − 4) ξϕ 2 ) ,(3.56) and if I specify that n = 4, I find term in the scalar equation behaves like N NES 1 + 4ξ 8 ψ 2 − 1 + 1 2 log 1 + 12ξ (1 + 4ξ) (1 + ξψ 2 ) ,(3.U ψ = −λψ 2 2 (1 + 6ξ) + λ (1 + 3ξ) ξ (1 + 6ξ) 2 log 1 + (1 + 6ξ) ξψ 2 ,(3.58) which is not bounded from below when ξ > 0 and λ > 0 and is therefore unstable. More generally, there are local maxima in the effective potential at ψ = ± n ξ (4 − n) , so for ξ > 0 the model is stable for bare potentials which are of quartic order or higher. Bouncing scalar model L BS,cov = √ q cos ψ 2 (4) R − 1 + b cos ψ 2 (1 + b) ∂ µ ψ∂ µ ψ − U , (3.59a) L BS = √ q 2 cos ψ (R − K) + sin ψ (νv + 2∆ψ) + 1 + b cos ψ 1 + b ν 2 + (2 + b) cos ψ − 1 1 + b ∂ a ψ∂ a ψ − 2U . (3.59b) As confirmed by numerically evolving the equations of motion, I know from the bouncing conditions (3.41) that this model will bounce when b > 1 because then there is a value of ψ for which ω ψ < 0. As I show in Fig. 3.4, the collapsing universe excites the scalar field so much that it 'tunnels' through to another minima of the potential. The bounce happens when the field becomes momentarily ghost-like, when ω ψ < 0. I can construct other models which produce a bounce purely through non-minimal coupling by having any U (ψ) with multiple minima and couplings of the approximate form ω ∼ 1 − U . However, to ensure the scalar does not attempt to tunnel through the potential to infinity and thereby not prevent collapse, the coupling functions must become negative only for values of ψ between stable minima. For example, for the Z 2 potential U (ψ) = λ ψ 2 − 1 2 , couplings which are guaranteed to produce a bounce are ω R (ψ) = ω ψ (ψ) = 1 − e − when λ > 1. Summary In I presented a model which produces a cosmological bounce purely through non-minimal coupling of a periodic scalar field to gravity. I also provided the general method of producing similar models without a periodic symmetry. I did not consider in detail the effect that the deformation has on the cosmological dynamics. However, I did show that a big bounce which is purely due to a zeroth order deformation necessarily involves effective signature change. Perhaps most importantly, I have established the minimally-deformed low-curvature limit that the subsequent chapters refer to. Fourth order perturbative gravitational action As I showed in section 2.7.2, the deformed action doesn't seem to naturally have a cut-off for higher powers of derivatives, and it must either be considered completely in general or treated perturbatively as a polynomial expansion. In this chapter I will treat it perturbatively in order to find the lowest order corrections which are non-trivial. This chapter is mostly adapted from a previously published paper [56]. Solving the action's distribution equation The general deformed action must satisfy the distribution equation (2.43), I restrict to the case when there is only a metric field, for which the diffeomorphism constraint is given by (B.11), 0 = δL(x) δq ab (y) v ab (y) + I δL(x) δψ I (y) ν I (y) + (βD a ∂ a ) x δ (x, y) − (x ↔ y) .D a = −2∇ b p ab = −2 δ a (b ∂ c) + Γ a bc ∂L ∂v bc . (4.2) Firstly, I integrate (4.1) by parts to move spatial derivatives from L and onto the delta functions. I discard the surface term and find, 0 = δL(x) δq ab (y) v ab (y) − 2 β ∂L ∂v bc Γ a bc ∂ a x δ(x, y) + 2 ∂L ∂v ab ∂ b x [(β∂ a ) x δ(x, y)] − (x ↔ y) ,(4.3) from this I take the functional derivative with respect to v ab (z) (after relabelling the other indices), I move the derivative from δ(x, z) and exchange some terms using the (x ↔ y) symmetry to find it in the form, 0 = δL(x) δq ab (y) δ(y, z) + δ∂L(x) δq cd (y)∂v ab (x) v cd (y) +2 ∂ ∂v ab ∂ d β ∂L ∂v cd − β ∂L ∂v de Γ c de ∂ c + ∂ ∂v ab β ∂L ∂v cd ∂ cd x δ(x, y) δ(x, z) + 2 ∂β ,d ∂v ab,e ∂L ∂v cd x ∂ c(x) δ(x, y)∂ d(x) δ(x, z) − (x ↔ y) .0 = A ab (x, y)δ(y, z) − A ab (y, x)δ(x, z),(4.5) where, Integrating over y, I find that part of the equation can be combined into a tensor dependent only on x, A ab (x, y) = δL(x) δq ab (y) − v cd (x) δ∂L(y) δq cd (x)∂v ab (y) + 2 ∂ ∂v ab β ∂L ∂v de Γ c de − ∂ d β ∂L ∂v cd ∂ c − ∂ ∂v ab β ∂L ∂v cd ∂ cd + ∂ e ∂β ,0 = A ab (x, z) − δ(z, x) d 3 yA ab (y, x), = A ab (x, z) − δ(z, x)A ab (x), where A ab (x) = d 3 yA ab (y, x) . (4.7) Substituting in the definition of A ab (x, z) then relabelling, 0 = δL(x) δq ab (y) − v cd (x) δ∂L(y) δq cd (x)∂v ab (y) + 2 ∂ ∂v ab β ∂L ∂v de Γ c de − ∂ d β ∂L ∂v cd ∂ c − ∂ ∂v ab β ∂L ∂v cd ∂ cd + ∂ e ∂β ,d ∂v ab,e ∂L ∂v cd ∂ c y δ(y, x) − A ab (x)δ(x, y). (4.8) To find this in terms of one independent variable, I multiply by the test tensor θ ab (y) and integrate by parts over y, Then collecting derivatives of θ ab , 0 = ∂L ∂q ab θ ab + ∂L ∂q ab,c ∂ c θ ab + ∂L ∂q ab,cd ∂ cd θ ab − v cd ∂ 2 L ∂q cd ∂v ab θ ab + v cd ∂ e ∂ 2 L ∂q cd,e ∂v ab θ ab − v cd ∂ ef ∂ 2 L ∂q cd,ef ∂v ab θ ab + 2∂ c θ ab ∂ ∂v ab ∂ d β ∂L ∂v cd − β ∂L ∂v de Γ c de − θ ab ∂ e ∂β ,0 = θ ab (· · · ) ab + ∂ c θ ab ∂L ∂q ab,c + v de ∂ 2 L ∂q de,c ∂v ab − 2v ef ∂ d ∂ 2 L ∂q ef,cd ∂v ab +2 ∂ ∂v ab ∂ d β ∂L ∂v cd − β ∂L ∂v de Γ c de − 4∂ d ∂ ∂v ab β ∂L ∂v cd + 2∂ e ∂β ,d ∂v ab,c ∂L ∂v de + ∂ cd θ ab ∂L ∂q ab,cd − v ef ∂ 2 L ∂q ef,cd ∂v ab − 2 ∂ ∂v ab β ∂L ∂v cd + 2 ∂β ,e ∂v ab,(c ∂L ∂v d)e ,(4.10) where I have discarded the terms containing θ ab without derivatives, because they do not provide any restrictions on the form of the action. This is simplified by noting that ∂ c and ∂ ∂v ab commute, and that ∂β ,e ∂v ab,c = δ c e ∂β ∂v ab . Therefore, the solution is given by, 0 = θ ab (· · · ) ab + ∂ c θ ab ∂L ∂q ab,c + v de ∂ 2 L ∂q de,c ∂v ab − 2v ef ∂ d ∂ 2 L ∂q ef,cd ∂v ab −2Γ c de ∂ ∂v ab β ∂L ∂v de − 2∂ d β ∂ 2 L ∂v ab ∂v cd − 4β∂ d ∂ 2 L ∂v ab ∂v cd −2 ∂β ∂v ab ∂ d ∂L ∂v cd + ∂ cd θ ab ∂L ∂q ab,cd − v ef ∂ 2 L ∂q ef,cd ∂v ab − 2β ∂ 2 L ∂v ab ∂v cd . (4.11) At this point I need to make some assumptions about the form of the action before I can use this equation to restrict its form. Finding the conditions on the action w := Q abcd v T ab v T cd = v T ab v ab T . Substituting these variables into (4.11), the resulting equation contains a series of unique tensor combinations. The test tensor θ ab is completely arbitrary so the coefficient of each unique tensor contraction with it must independently vanish if the whole equation is to be satisfied. Firstly, I focus on the terms depending on the second order derivative ∂ cd θ ab . I evaluate each individual term in appendix C. Substituting (C.3) into (4.11), I find the following independent conditions, q ab ∂ 2 θ ab : 0 = ∂L ∂R − 2v 3 ∂ 2 L ∂R∂v + 2β ∂ 2 L ∂v 2 − 2 3 ∂L ∂w , (4.12a) Q abcd ∂ cd θ ab : 0 = ∂L ∂R − 4β ∂L ∂w , (4.12b) q ab v cd T ∂ cd θ ab : 0 = ∂ 2 L ∂R∂v + 4β ∂ 2 L ∂w∂v , (4.12c) v ab T ∂ 2 θ ab : 0 = v 3 ∂ 2 L ∂R∂w − β ∂ 2 L ∂v∂w , (4.12d) v ab T v cd T ∂ cd θ ab : 0 = ∂ 2 L ∂R∂w + 4β ∂ 2 L ∂w 2 . (4.12e) Before I analyse these equations, I will find the conditions from the first order derivative part of (4.11). There are many complicated tensor combinations that need to be considered, so for convenience I define X a := q bc ∂ a q bc and Y a := q bc ∂ c q ab . I evaluate the individual terms in appendix C. When I substitute the results (C.4) into (4.11), I once again find a series of unique tensor combinations with their own coefficient which vanishes independently. Most of these conditions are the same as those found in (4.12) so I won't bother duplicating them again here. However, I do find the following new conditions, X a ∂ b θ ab : 0 = ∂L ∂R − 4 (∂ q β + 2β∂ q ) ∂L ∂w , (4.13a) q ab X c ∂ c θ ab : 0 = −1 2 ∂L ∂R + v 3 (4∂ q − 1) ∂ 2 L ∂v∂R + ∂β ∂v (1 − 2∂ q ) ∂L ∂v + (β − 2∂ q β − 4β∂ q ) ∂ 2 L ∂v 2 − 2 3 ∂L ∂w , (4.13b) v ab T X c ∂ c θ ab : 0 = v 3 (4∂ q − 1) ∂ 2 L ∂w∂R + ∂β ∂w (1 − 2∂ q ) ∂L ∂v + (β − 2∂ q β − 4β∂ q ) ∂ 2 L ∂v∂w , (4.13c) q ab v cd T X d ∂ c θ ab : 0 = (1 − 2∂ q ) ∂ 2 L ∂v∂R − 4 (∂ q β + 2β∂ q ) ∂ 2 L ∂v∂w − 4 ∂β ∂v ∂ q ∂L ∂w , (4.13d) v ab T v cd T X d ∂ c θ ab : 0 = (1 − 2∂ q ) ∂ 2 L ∂w∂R − 4 (∂ q β + 2β∂ q ) ∂ 2 L ∂w 2 − 4 ∂β ∂w ∂ q ∂L ∂w , (4.13e) q ab v cd T Y d ∂ c θ ab : 0 = 2β ∂ 2 L ∂v∂w + ∂β ∂v ∂L ∂w , (4.13f) v ab T v cd T Y d ∂ c θ ab : 0 = 2β ∂ 2 L ∂w 2 + ∂β ∂w ∂L ∂w , (4.13g) ∂ a F ∂ b θ ab : 0 = ∂β ∂F + 2β ∂ ∂F ∂L ∂w , (4.13h) q ab ∂ c F ∂ c θ ab : 0 = 2v 3 ∂ 3 L ∂F ∂v∂R − ∂β ∂v ∂ 2 L ∂F ∂v − ∂β ∂F + 2β ∂ ∂F ∂ 2 L ∂v 2 − 2 3 ∂L ∂w , (4.13i) v ab T ∂ c F ∂ c θ ab : 0 = 2v 3 ∂ 3 L ∂F ∂w∂R − ∂β ∂w ∂ 2 L ∂F ∂v − ∂β ∂F + 2β ∂ ∂F ∂ 2 L ∂v∂w , (4.13j) q ab v cd T ∂ d F ∂ c θ ab : 0 = 1 2 ∂ 3 L ∂F ∂v∂R + ∂β ∂v ∂ 2 L ∂F ∂w + ∂β ∂F + 2β ∂ ∂F ∂ 2 L ∂v∂w , (4.13k) v ab T v cd T ∂ d F ∂ c θ ab : 0 = 1 2 ∂ 3 L ∂F ∂w∂R + ∂β ∂w ∂ 2 L ∂F ∂w + ∂β ∂F + 2β ∂ ∂F ∂ 2 L ∂w 2 ,(4. Evaluating the fourth order perturbative action For this section, I construct an ansatz for the action and deformation that is explicit in being a perturbative expansion. For each time derivative above the classical solution, I include the small parameter ε, and consider up to O ε 2 . I consider two orders because in models of loop quantum cosmology which have deformed covariance, the holonomy corrections to the action expand into even powers of time derivatives [39,42]. Therefore, considering a fourth order action and a second order deformation should include the nearest higher-order terms in an expansion of those holonomy functions. Therefore I write, L = L 0 + L (v) v + L (w) w + L (v 2 ) v 2 + ε L (vw) vw + L (v 3 ) v 3 + ε 2 L (w 2 ) w 2 + L (v 2 w) v 2 w + L (v 4 ) v 4 + O(ε 3 ), (4.14a) β = β 0 + εβ (v) v + ε 2 β (v 2 ) v 2 + β (w) w + O(ε 3 ), (4.14b) where each coefficient is potentially a function of q and R. I take the condition from Q abcd ∂ cd θ ab , (4.12b) and truncate to O(ε 2 ). Separating different powers of v and w, it gives the following conditions for the Lagrangian coefficients, ε 2 w 2 : ∂ R L (w 2 ) = 0, ε 2 v 2 w : ∂ R L (v 2 w) = 0, ε 2 v 4 : ∂ R L (v 4 ) = 0, εvw : ∂ R L (vw) = 0, εv 3 : ∂ R L (v 3 ) = 0, (4.15a) w : ∂ R L (w) = 4ε 2 β (w) L (w) + 2β 0 L (w 2 ) , v 2 : ∂ R L (v 2 ) = 4ε 2 β (v 2 ) L (w) + β (v) L (vw) + β 0 L (v 2 w) , v : ∂ R L (v) = 4ε β (v) L (w) + β 0 L (vw) . (4.15b) So from the five conditions in (4.15a), one can see that terms with three or four time derivatives must not contain any spatial derivatives. From the three conditions in (4.15b), one can see that including R in these coefficients requires including a factor of ε for every combined derivative order above two. Therefore, the spatial derivatives must be treated equally with time derivatives when one is performing a perturbative expansion, as expected. So I can now further expand the ansatz to include explicit factors of R, L = L ∅ + L (v) v + L (w) w + L (v 2 ) v 2 + L (R) R + ε L (vw) vw + L (v 3 ) v 3 +L (vR) vR + ε 2 L (w 2 ) w 2 + L (v 2 w) v 2 w + L (v 4 ) v 4 + L (wR) wR +L (v 2 R) v 2 R + L (R 2 ) R 2 + O(ε 3 ), (4.16a) β = β ∅ + εβ (v) v + ε 2 β (v 2 ) v 2 + β (w) w + β (R) R + O(ε 3 ), (4.16b) where each coefficient is potentially a function of q. I now substitute this ansatz into the conditions found for the action so that its form can be progressively restricted. Looking once again at the condition from Q abcd ∂ cd θ ab (4.12b), one finds it is satisfied by the following solutions, ∅ : L (w) = L (R) 4β ∅ , (4.17a) εv : L (vw) = 1 4β 2 ∅ β ∅ L (vR) − 4β (v) L (R) , (4.17b) ε 2 R : L (wR) = 1 4β 2 ∅ 2β ∅ L (R 2 ) − 2β (R) L (R) , (4.17c) ε 2 v 2 : L (v 2 w) = 1 4β 3 ∅ β 2 ∅ L (v 2 R) − β ∅ β (v) L (vR) + β 2 (v) − β ∅ β (v 2 ) L (R) , (4.17d) ε 2 w : L (w 2 ) = 1 32β 3 ∅ 2β ∅ L (R 2 ) − β (R) + 4β ∅ β (w) L (R) , (4.17e) and then looking at the condition from v ab T ∂ 2 θ ab , (4.12d), ε : L (vR) = β (v) L (R) β ∅ , (4.18a) ε 2 v : L (v 2 R) = 1 6β 2 ∅ 2β ∅ L (R 2 ) + 6β ∅ β (v 2 ) − β (R) L (R) , (4.18b) where (4.18a) and (4.17b) combine to give L (vw) = 0. Then looking at the condition from q ab v cd T ∂ cd θ ab , (4.12c) ε : β (v) = 0, (4.19a) ε 2 v : L (R 2 ) = L (R) 2β ∅ β (R) − 3β ∅ β (v 2 ) , (4.19b) one can see that L (vR) = 0 and therefore all the third order terms all vanish. Looking at the condition from q ab ∂ 2 θ ab , (4.12a), ∅ : L (v 2 ) = −L (R) 6β ∅ , (4.20a) εv : L (v 3 ) = 0, (4.20b) ε 2 w : β (v 2 ) = −2 3 β (w) , (4.20c) ε 2 v 2 : L (v 4 ) = −β (w) L (R) 36β 2 ∅ , (4.20d) and then from X a ∂ b θ ab , (4.13a), ∅ : L (R) = f q \|β ∅ \|, (4.21a) ε 2 R : β (w) = b − β (R) 4β ∅ , (4.21b) where f and b arise as integration constants. From q ab X c ∂ c θ ab , (4.13b), εv : L (v) = ξ √ q,(4.22) where ξ is also an integration constant. Finally, the condition from ∂ a R∂ b θ ab , (4.13h), means that To make sure the classical limit of the result matches the action found in chapter 3, I set f = ω/2, and replace the normal derivatives with the standard extrinsic curvature ε : b = 0.contraction K = v 2 6 − w 4 . Therefore, the fourth order perturbative gravitational action is given by, L = L ∅ + ξv √ q + ω 2 q \|β ∅ \| R − K β ∅ − ε 2 β (R) 4β ∅ R + K β ∅ 2 + O ε 3 ,(4.24) with the associated deformation β = β ∅ + ε 2 β (R) R + K β ∅ + O ε 3 . (4.25) So the remaining freedom in the action comes down to the constants ξ and ω, the functions β ∅ and β (R) . There is also a term which doesn't affect the kinematic structure and acts like a generalised notion of a potential, so can be rewritten as L ∅ (q) = − √ q U (q). Cosmology In this section I find the cosmological implications of the nearest order corrections coming from the deformation to general covariance. Since it is a perturbative expansion, the results when the corrections become large should be taken to be indicative rather than predictive. I restrict to a flat FLRW metric as in section 2.8, L = −a 3 U (a) − 3σ ∅ ωa 3 N 2 \|β ∅ \| H 2 1 + 3ε 2 β 2 2N 2 β ∅ H 2 + O ε 3 ,(4.26) where a is the scale factor, H =ȧ/a is the Hubble expansion rate, σ ∅ := sgn(β ∅ ), and β 2 = β (R) /β ∅ is the coefficient of K in the deformation. I couple this to matter with energy density ρ and pressure density P = w ρ ρ. I Legendre transform the effective Lagrangian to find the Hamiltonian. Imposing the Hamiltonian constraint C ≈ 0 gives us 1 N 2 H 2 1 + 9ε 2 β 2 2β ∅ N 2 H 2 = σ ∅ 3ω \|β ∅ \| U,(4.27) which can be solved to find the modified Friedmann equation, 1 N 2 H 2 = 2σ ∅ \|β ∅ \| 3ω(1 + α) U,(4.28) where the correction factor is α := 1 + 6ε 2 β 2 ω \|β ∅ \| U .σ ∅ \|β ∅ \| 6α U 2 + ∂ log U ∂ log a + 2 ∂ log N ∂ log a + 1 2 ∂ log β ∅ ∂ log a +2 α − 1 α + 1 1 + ∂ log N ∂ log a − 1 2 ∂ ∂ log a log β 2 β ∅ . (4.30) If I take a perfect fluid, then U = ρ, where ρ is the fluid's energy density, which satisfies the continuity equationρ + 3Hρ(1 + w ρ ) = 0. (4.31) where w ρ is the perfect fluid's equation of state. Note that there are corrections to the matter sector due to the modified constraint algebra [86,87], as shown for scalar fields in other chapters. However, these have not been included here, as it is not known how the deformation would affect a perfect fluid. Since ε is a small parameter, it can be used to expand Eq. (4.28), 1 N 2 H 2 = σ ∅ \|β ∅ \| 3ω ρ 1 + 3ε 2 β 2 ω \|β ∅ \| ρ + O ε 3 ,(4.32) and expanding the bracket in Eq. (4.30) to first order, it can be seen thatä/a > 0 when w ρ < w a , where w a = −1 3 1 − 1 2 ∂ log β ∅ ∂ log a + 6ε 2 β 2 ω \|β ∅ \| ρ 1 − 1 2 ∂ ∂ log a log β 2 β ∅ ,(4.33) having set N = 1, so this is applicable for cosmic time. When β 2 < 0, the modified Friedmann equation (4.32) suggests a big bounce rather than a big bang at high energy density, sinceȧ → 0 when a > 0 andä > 0 is possible when ρ → ρ c where ρ c = ω \|β ∅ \| 6ε 2 \|β 2 \| . (4.34) This requires either ρ c to be constant, or for it to diverge at a slower rate than ρ as a → 0. Let me emphasise that the bounce is found considering only holonomy corrections manifesting as higher-order powers of of second-order derivatives and not considering ignoring β(a,ȧ) = f (a)Σ(a,ȧ) ∂ 2 ∂ȧ 2 γ ∅ (a,ȧ) sin[γ BI µ(a)ȧ] γ BI µ(a) 2 , (4.35) where γ BI is the Barbero now becomes β = f √ γ ∅ cos 2γ BI γ ∅ ♦a δ H , (4.36) The "old dynamics" or "µ 0 scheme" corresponds to δ = 1, and the favoured "improved dynamics" or "μ scheme" corresponds to δ = 0 [88,89]. In the semi-classical regime, H √ ♦ 1, so I can Taylor expand this equation for the correction function to get β = f √ γ ∅ − 2γ 2 BI ♦a 2δ f (γ ∅ ) 3/2 H 2 + O ♦ 2 . (4.37) The way that γ ∅ is defined is that it multiplies the background gravitational term in the Hamiltonian constraint relative to the classical form. Since I am assuming γ ∅ = γ ∅ (a), I can isolate it by taking the Lagrangian (4.26) and setting β 2 = 0. If I then Legendre transform to find a Hamiltonian expressed in terms of the momentum of the scale factor, I find that it is proportional to \|β ∅ \|. Thus, I conclude that γ ∅ = \|β ∅ \| when γ ∅ is just a function of the scale factor. Using this to compare (4.37) to (4.25), β = β ∅ + 6ε 2 β 2 H 2 + O ε 3 ,(4.38) I find that f = σ ∅ \|β ∅ \| 3/4 , and therefore f = σ ∅ γ 3/2 ∅ . From this, I can now deduce the form of the coefficient for the higher-order corrections, ε 2 β 2 = −σ ∅ 3 γ 2 BI ♦a 2δ γ 3 ∅ . (4.39) The exact form of γ ∅ (a) is uncertain, and the possible forms that have been found also contain quantisation ambiguities. The form given by Bojowald in Ref. [90] is γ ∅ = 3r 1−l 2l (r + 1) l+2 − \|r − 1\| l+2 l + 2 − r (r + 1) l+1 − sgn(r − 1)\|r − 1\| l+1 l + 1 , (4.40) where l ∈ (0, 1), r = a 2 /a 2 and a is the characteristic scale of the inverse-volume corrections, related to the discreteness scale. I will only use the asymptotic expansions of this function, namely γ ∅ ≈          1 + (2 − l)(1 − l) 10 a a −4 , if a a 3 1 + l a a 2(2−l) , if a a (4.41) and even then I will only take γ ∅ ≈ 1 for a a , since the correction quickly becomes vanishingly small. I replace the area gap with a dimensionless parameter♦ = ♦ω which is of order unity. The modified Friedmann equation (4.32) is now given by 42) which I need to compare for different types of matter. First of all I will consider a perfect fluid, and then I will consider a scalar field with a power-law potential. H 2 = σ ∅ γ ∅ 3ω ρ 1 − σ ∅ γ 2 BI♦ 3ω 2 a 2δ γ 2 ∅ ρ ,(4. Perfect fluid H 2 ∝ a −3(1+wρ) 1 − γ 2 BI♦ 3ω 2 ρ 0 a Θ , (4.44) where Θ depends on which regime of (4.41) we are in, namely which is also required to match the classical limit. The reason this is required is because ρ needs to diverge faster than ρ c as a → 0 in order for there to be a bounce. This will happen when w ρ > w b , where Θ =      2δ − 3(1 + w ρ ), if a a , 2δ + 4(2 − l) − 3(1 + w ρ ), if a a ,w b =      − 1 + 2 3 δ, if a a − 1 + 2 3 δ + 4 3 (2 − l), if a a (4.46) which means that, if the bounce does not happen in the a a regime, the inverse-volume corrections make the bounce less likely to happen. If I use the favoured value of δ = 0, and assume l = 1, then w b = 1/3 and so w ρ still needs to be greater than that found for radiation in order for there to be a bounce. A possible candidate for this would be a massless (or kinetic-dominated) scalar field, where w ρ = 1. Another aspect to investigate is whether the conditions for inflation are modified. Taking (4.33), I see that acceleration happens when w ρ < w a , where Holonomy-type corrections increase the range since Θ ≤ 0, and so may inverse-volume corrections. However, the latter also seems to include a cut-off when the last term of Eq. (4.47) in the a a regime dominates. Since a bounce requiresȧ = 0 andä > 0, the condition w b < w ρ < w a must be satisfied and so it must happen before the cut-off dominates if it is to happen at all. w a =          − 1 3 + 2γ 2 BI♦ 9ω 2 (1 − δ)ρ 0 a Θ , if a a 1 − 2l 3 − 2γ 2 BI♦ ω 2 a 4(2−l) 1 + δ − l (1 + l) 2 ρ 0 a Θ , if a a Scalar field I now investigate the effects that the inverse-volume and holonomy corrections can have when I couple gravity to an undeformed scalar field. In this case, the energy and pressure densities are given by ρ = 1 2φ 2 + U (ϕ), P = 1 2φ 2 − U (ϕ),(4.H 2 = σ ∅ γ ∅ 3ω U 1 − σ ∅ γ 2 BI♦ 3ω 2 a 2δ γ 2 ∅ U . (4.50b) If I substitute (4.50b) into (4.50a), take the derivative with respect to time and substitute in (4.50b) and (4.50a) again, I find ϕ U = 1 3 η,φ 2 2U = 1 3 , (4.51) where the slow-roll parameters are η := 1 1 − ς ω γ ∅ U U − (1 − 2ς) + χ − δς , (4.52a) := 1 1 − ς ω 2γ ∅ U U 2 , (4.52b) whereλ > 0 and n/2 ∈ N, then the number of e-folds before the end of inflation is χ := 1 − 3ς 2 ∂ log γ ∅ ∂ log a (4.52c) ς := γ 2 BI♦ 3ω 2 a 2δ γ 2 ∅ U,N (ϕ) = − ϕ ϕ end dϕ Ḣ ϕ = ϕ ϕ end dϕ γ ∅ U ωU 1 − γ 2 BI♦ 3ω 2 a 2δ γ 2 ∅ U .N (ϕ) = 1 2nω ϕ 2 − ϕ 2 end − γ 2 BI♦λ 3n 2 (n + 2)ω 1+ n 2 ϕ 2+n − ϕ 2+n end . (4.56) If I take the approximation that slow-roll inflation is valid beyond the regime specified by (4.53), then I can calculate a value for the maximum number of e-folds by starting inflation at the big bounce, which is applicable only for the region where ρ is below a critical value, N max = 1 2n    3n γ 2 BI♦λ 2 n − ϕ 2 end ω    − γ 2 BI♦λ 3n 2 (n + 2)    3n γ 2 BI♦λ 1+ 2 n − ϕ 2 end ω 1+ n 2    ,(4.ϕ +φ 3γ ∅ ω 1 2φ 2 + U 1 − γ 2 BI♦ 3ω 2 a 2δ γ 2 ∅ 1 2φ 2 + U + U = 0.1 − γ 2 BI♦ 3ω 2 1 2φ 2 + λ n ϕ n > 0,ϕ ≈ − nλω 3 ϕ n 2 −1 1 − γ 2 BI♦ λ 3nω 2 ϕ n −1/2 , (4.62) where the term in the bracket is the correction to the classical solution. Looking at The condition for acceleration for the case I am considering here is w ρ < w a = −1 3 1 − 2γ 2 BI♦ 3ω 2 1 2φ 2 + λ n ϕ n (4.63) where we can define the effective equation of state as w ρ = P (ϕ)/ρ(ϕ) using (4.48). I plot in Fig. 4.2 this region on the phase space of the scalar field to see how accelerated expansion can happen in a wider range than in the classical case. In order to be able to solve the equations and make plots, I have neglected non-zero values of δ and non-unity values of γ ∅ . It may be that in these cases the big bounce and inflation are no longer inevitable, as was found for the perfect fluid. Discussion In this chapter, I calculated the general conditions on a deformed action which has been formed from the variables (q, v, w, R). I then found the nearest-order curvature corrections coming from the deformation by solving these conditions for a fourth order action. I found that these corrections can act as a repulsive gravitational effect which may produce a big bounce. When coupling gravity to a perfect fluid, the effects that the quantum corrections have depend on the equation of state, but inflation and a big bounce are possible. I coupled deformed gravity to an undeformed scalar in this preliminary investigation into higher order curvature corrections. I investigated slow-roll inflation and a big bounce in the presence of this scalar field. In chapter 5, I find that scalar fields must be deformed in much the same way as the metric. Therefore, these results might be interesting on some level, but cannot be taken too literally. Unfortunately, there was simply not enough time to research the fully deformed cases, hence why this material remains. Deformed scalar-tensor constraint to all orders In this chapter I find the most general gravitational constraint which satisfies the deformed constraint algebra. To find the constraint is easier than finding the action, so I also include a non-minimally coupled scalar field in order to find the most general deformed scalartensor constraint. This material has not been previously published. As stated in chapter 2, I am not looking for models with degrees of freedom beyond a simple scalar-tensor model. Since actions which contain Riemann tensor squared contractions introduce additional tensor degrees of freedom [77], I automatically do not consider such terms here. This means I only need to expand the constraint using variables which are tensor contractions containing up to two orders of spatial derivatives or up to two in momenta. It also means I do not need to consider spatial derivatives of momenta in the constraint. Therefore, for a metric tensor field q ab , p cd and a scalar field (ψ, π), I expand the constraint into the following variables, q = det q ab , p = q ab p ab , P = Q abcd p ab T p cd T , R, ψ, π, ∆ := q ab ∇ a ∇ b ψ = ∂ 2 ψ − q ab Γ c ab ∂ c ψ, γ := q ab ∇ a ψ∇ b ψ = ∂ a ψ∂ a ψ, (5.1) where p ab T := p ab − 1 3 pq ab is the traceless part of the metric momentum. Therefore, I start with the constraint given by C = C(q, p, P, R, ψ, π, ∆, γ). I must solve the distribution equation again to find the equations which restrict the form of the constraint. The calculations in this chapter generalise those presented in chapter 3 where the minimally deformed scalar-tensor constraint was regained from the constraint algebra. Solving the distribution equation Starting from (2.41), I have the general distribution equation for a Hamiltonian constraint, without derivatives of the momenta, which depends on a metric tensor and a scalar field, 0 = δC(x) δq ab (y) ∂C ∂p ab y + δC(x) δψ(y) ∂C ∂π y − (βD a ∂ a ) x δ (x, y) − (x ↔ y) . (5.2) To solve this I will take the functional derivative with respect to a momentum variable, manipulate a few steps and then integrate with a test tensor to find several equations which the constraint must satisfy. Since I have two fields, I must do this procedure twice. The first route I consider will be where I take the derivative with respect to the metric momentum. p ab route Starting from the distribution equation (5.2), relabel indices, then take the functional derivative with respect to p ab (z), 0 = δC(x) δq cd (y) ∂ 2 C ∂p ab ∂p cd y δ(z, y) + δ∂C(x) δq cd (y)∂p ab (x) ∂C ∂p cd y δ(z, x) + δC(x) δψ(y) ∂ 2 C ∂p ab ∂π y δ(z, y) + δ∂C(x) δψ(y)∂p ab (x) ∂C ∂π y δ(z, x) − ∂ c(x) δ(x, y) ∂(βD c ) ∂p ab + β ∂D c ∂p ab ,d ∂ d x δ(z, x) − (x ↔ y) . (5.3) Move derivatives and discard surface terms so that it is reorganised into the form, 0 = A ab (x, y)δ(z, y) − A ab (y, x)δ(z, x),(5.4) where, A ab (x, y) = δC(x) δq cd (y) ∂ 2 C ∂p ab ∂p cd y − δ∂C(y) δq cd (x)∂p ab (y) ∂C ∂p cd x + δC(x) δψ(y) ∂ 2 C ∂p ab ∂π y − δ∂C(y) δψ(x)∂p ab (y) ∂C ∂π x + ∂(βD c ) ∂p ab ∂ c y δ(y, x) − ∂ d(y)    β ∂D c ∂p ab ,d ∂ c y δ(y, x)    . (5.5) If I take (5.4) and integrate over y, I can find A ab (x, y) in terms of a function dependent on only a single independent variable, 0 = A ab (x, z) − A ab (x)δ(z, x), where, A ab (x) = d 3 yA ab (y, x). (5.6) I then multiply this by an arbitrary, symmetric test tensor θ ab (z), integrate over z, and separate out different orders of derivatives of θ ab , 0 = θ ab (· · · ) ab + ∂ c θ ab ∂C ∂q ef,c ∂ 2 C ∂p ab ∂p ef + 2 ∂ 2 C ∂q ef,cd ∂ d ∂ 2 C ∂p ab p ef + ∂C ∂p ef ∂ 2 C ∂q ef,c ∂p ab − 2 ∂C ∂p ef ∂ d ∂ 2 C ∂q ef,cd ∂p ab + ∂C ∂ψ ,c ∂ 2 C ∂p ab ∂π +2 ∂C ∂ψ ,cd ∂ d ∂ 2 C ∂p ab ∂π + ∂C ∂π ∂ 2 C ∂ψ ,c ∂p ab − 2 ∂C ∂π ∂ d ∂ 2 C ∂ψ ,cd ∂p ab − ∂(βD c ) ∂p ab − ∂ d β ∂D d ∂p ab ,c + ∂ cd θ ab ∂C ∂q ef,cd ∂ 2 C ∂p ab ∂p ef − ∂C ∂p ef ∂ 2 C ∂q ef,cd ∂p ab + ∂C ∂ψ ,cd ∂ 2 C ∂p ab ∂π − ∂C ∂π ∂ 2 C ∂ψ ,cd ∂p ab − β ∂D c ∂p ab ,d . (5.7) As done in previous chapters, I disregard the term zeroth order derivative of θ ab because it does not provide useful information. Before I can attempt to interpret this equation, I must first separate out all the different tensor combinations that there are. Because θ ab is arbitrary, the coefficients of each unique tensor combination must vanish independently. When I substitute in C = C(q, p, P, R, ψ, π, ∆, γ), there are many complicated tensor combinations that need to be considered, so for convenience I define X a := q bc ∂ a q bc . I evaluate each term in the ∂ cd θ ab bracket, and write them in (D.2), in appendix D. So the linearly independent terms depending on ∂ cd θ ab produce the following conditions, ∂ ab θ ab : 0 = ∂C ∂R ∂C ∂P + β, (5.8a) q ab ∂ 2 θ ab : 0 = ∂C ∂p ∂ 2 C ∂p∂R − ∂C ∂R ∂ 2 C ∂p 2 + 1 3 ∂C ∂P + 1 2 ∂C ∂∆ ∂ 2 C ∂π∂p − 1 2 ∂C ∂π ∂ 2 C ∂p∂∆ , (5.8b) q ab p cd T ∂ cd θ ab : 0 = ∂C ∂R ∂ 2 C ∂p∂P − ∂C ∂P ∂ 2 C ∂p∂R , (5.8c) p T ab ∂ 2 θ ab : 0 = ∂C ∂p ∂ 2 C ∂P∂R − ∂C ∂R ∂ 2 C ∂p∂P + 1 2 ∂C ∂∆ ∂ 2 C ∂π∂P − 1 2 ∂C ∂π ∂ 2 C ∂P∂∆ , (5.8d) p T ab p cd T ∂ cd θ ab : 0 = ∂C ∂R ∂ 2 C ∂P 2 − ∂C ∂P ∂ 2 C ∂P∂R . (5.8e) I then evaluate each term in the ∂ c θ ab bracket of (5.7) and write them in (D.3). There are many unique terms which should be considered here, but in this case most of these are already solved by a constraint which satisfies (5.8). So the equations containing new information are, ∂ a ψ∂ b θ ab : 0 = 2 ∂C ∂R ∂ ψ − ∂C ∂∆ ∂C ∂P + ∂ ψ β, (5.9a) q ab ∂ c ψ∂ c θ ab : 0 = 1 2 ∂C ∂∆ − 4 ∂C ∂R ∂ ψ ∂ 2 C ∂p 2 + 1 3 ∂C ∂P + 1 2 ∂C ∂∆ ∂C ∂P + ∂C ∂p 1 2 ∂ 2 C ∂p∂∆ + 4∂ ψ ∂ 2 C ∂p∂R + 2 ∂C ∂γ + ∂C ∂∆ ∂ ψ ∂ 2 C ∂π∂p + 2 ∂C ∂π ∂ 2 C ∂p∂γ − ∂ ψ ∂ 2 C ∂p∂∆ − π ∂β ∂p , (5.9b) p T ab ∂ c ψ∂ c θ ab : 0 = 1 2 ∂C ∂∆ − 4 ∂C ∂R ∂ 2 C ∂p∂P + ∂C ∂p 1 2 ∂ 2 C ∂P∂∆ + 4∂ ψ ∂ 2 C ∂P∂R +2 ∂C ∂γ + ∂C ∂∆ ∂ ψ ∂ 2 C ∂π∂P + 2 ∂C ∂π ∂ 2 C ∂P∂γ − ∂ ψ ∂ 2 C ∂P∂∆ − π ∂β ∂P , (5.9c) q ab p cd T ∂ d ψ∂ c θ ab : 0 = 2 ∂C ∂R ∂ ψ − ∂C ∂∆ ∂ 2 C ∂p∂P − ∂C ∂P 2∂ ψ ∂ 2 C ∂p∂R + ∂ 2 C ∂p∂∆ (5.9d) p T ab p cd T ∂ d ψ∂ c θ ab : 0 = 2 ∂C ∂R ∂ ψ − ∂C ∂∆ ∂ 2 C ∂P 2 − ∂C ∂P 2∂ ψ ∂ 2 C ∂P∂R + ∂ 2 C ∂P∂∆ (5.9e) X a ∂ b θ ab : 0 = ∂C ∂R (1 + 2∂ q ) ∂C ∂P + ∂ q β, (5.9f) q ab X c ∂ c θ ab : 0 = ∂C ∂p (4∂ q − 1) ∂ 2 C ∂p∂R − ∂C ∂R (4∂ q + 1) ∂ 2 C ∂p 2 + 1 3 ∂C ∂P + 1 2 ∂C ∂π (1 − 4∂ q ) ∂ 2 C ∂∆∂p + 1 2 ∂C ∂∆ (1 + 4∂ q ) ∂ 2 C ∂π∂p − 1 3 p ∂β ∂p , (5.9g) p T ab X c ∂ c θ ab : 0 = ∂C ∂p (4∂ q − 1) ∂ 2 C ∂P∂R − ∂C ∂R (4∂ q + 1) ∂ 2 C ∂P∂p + 1 2 ∂C ∂π (1 − 4∂ q ) ∂ 2 C ∂P∂∆ + 1 2 ∂C ∂∆ (1 + 4∂ q ) ∂ 2 C ∂P∂π − 1 3 p ∂β ∂P , (5.9h) q ab p cd T X d ∂ c θ ab : 0 = ∂C ∂R (1 + 2∂ q ) ∂ 2 C ∂p∂P + ∂C ∂P (1 − 2∂ q ) ∂ 2 C ∂p∂R , (5.9i) p T ab p cd T X d ∂ c θ ab : 0 = ∂C ∂R (1 + 2∂ q ) ∂ 2 C ∂P 2 + ∂C ∂P (1 − 2∂ q ) ∂ 2 C ∂P∂R , (5.9j) ∂ a F ∂ b θ ab : 0 = 2 ∂C ∂R ∂ 2 C ∂F ∂P + ∂β ∂F , (5.9k) q ab ∂ c F ∂ c θ ab : 0 = 2 ∂C ∂p ∂ 3 C ∂F ∂p∂R − 2 ∂C ∂R ∂ ∂F ∂ 2 C ∂p 2 + 1 3 ∂C ∂P + ∂C ∂∆ ∂ 3 C ∂F ∂p∂π − ∂C ∂π ∂ 3 C ∂F ∂p∂∆ + 1 3 δ p F ∂β ∂p , (5.9l) p T ab ∂ c F ∂ c θ ab : 0 = 2 ∂C ∂p ∂ 3 C ∂F ∂P∂R − 2 ∂C ∂R ∂ 3 C ∂F ∂p∂P + ∂C ∂∆ ∂ 3 C ∂F ∂P∂π − ∂C ∂π ∂ 3 C ∂F ∂P∂∆ + 1 3 δ p F ∂β ∂P , (5.9m) q ab p cd T ∂ d F ∂ c θ ab : 0 = ∂C ∂R ∂ 3 C ∂F ∂p∂P − ∂C ∂P ∂ 3 C ∂F ∂p∂R , (5.9n) p T ab p cd T ∂ c F θ ab : 0 = ∂C ∂R ∂ 3 C ∂F ∂P 2 − ∂C ∂P ∂ 3 C ∂F ∂P∂R , (5.9o) where F ∈ {p, P, R, ∆, γ}. These conditions strongly restrict the form of the constraint, but before I attempt to consolidate them I must find the conditions coming from the scalar field. π route Similar to the calculation using the metric momentum, I return to the distribution equation (5.2) and take the functional derivative with respect to π(z), 0 = δC(x) δq ab (y) ∂ 2 C ∂π∂p ab y δ(z, y) + δ∂C(x) δq ab (y)∂π(x) ∂C ∂p ab y δ(z, x) + δC(x) δψ(y) ∂ 2 C ∂π 2 y δ(z, y) + δ∂C(x) δψ(y)∂π(x) ∂C ∂π y δ(z, x) − δ(z, x) ∂(βD a ) ∂π ∂ a x δ(x, y) − (x ↔ y) ,(5.10) which can be rewritten as, 0 = A(x, y)δ(z, y) − A(y, x)δ(z, x),(5.11) where, A(x, y) = δC(x) δq ab (y) ∂ 2 C ∂π∂p ab y − δ∂C(y) δq ab (x)∂π(y) ∂C ∂p ab x + δC(x) δψ(y) ∂ 2 C ∂π 2 y − δ∂C(y) δψ(x)∂π(y) ∂C ∂π x + ∂(βD a ) ∂π ∂ a y δ(y, x),(5.12) and similar to above, (5.11) can be solved to find 0 = A(x, z) − A(x)δ(x, z). Multiply this by a test scalar field η(z) and integrate over z, 0 = η (· · · ) + ∂ a η ∂C ∂q cd,a ∂ 2 C ∂π∂p cd + 2 ∂C ∂q cd,ab ∂ b ∂ 2 C ∂π∂p cd + ∂C ∂p cd ∂ 2 C ∂q cd,a ∂π −2 ∂C ∂p cd ∂ 2 C ∂q cd,ab ∂π + ∂C ∂ψ ,a ∂ 2 C ∂π 2 + 2 ∂C ∂ψ ,ab ∂ b ∂ 2 C ∂π 2 + ∂C ∂π ∂ 2 C ∂ψ ,a ∂π −2 ∂C ∂π ∂ b ∂ 2 C ∂ψ ,ab ∂π − ∂(βD a ) ∂π + ∂ ab η ∂C ∂q cd,ab ∂ 2 C ∂π∂p cd − ∂C ∂p cd ∂ 2 C ∂q cd,ab ∂π + ∂C ∂ψ ,ab ∂ 2 C ∂π 2 − ∂C ∂π ∂ 2 C ∂ψ ,ab ∂π . (5.13) I evaluate each of the terms for ∂ ab η, and write them in (D.4). From these, I find the independent equations, ∂ 2 η : 0 = ∂C ∂p ∂ 2 C ∂π∂R − ∂C ∂R ∂ 2 C ∂π∂p + 1 2 ∂C ∂∆ ∂ 2 C ∂π 2 − 1 2 ∂C ∂π ∂ 2 C ∂∆∂π , (5.14a) p ab T ∂ ab η : 0 = ∂C ∂R ∂ 2 C ∂π∂P − ∂C ∂P ∂ 2 C ∂π∂R . (5.14b) Then, I evaluate all the terms for ∂ a η, and write them in (D.5). Therefore, ignoring terms solved by (5.14), the equations I get from ∂ a η are, ∂ a ψ∂ a η : 0 = 1 2 ∂C ∂∆ − 4 ∂C ∂R ∂ ψ ∂ 2 C ∂π∂p + ∂C ∂p 1 2 ∂ 2 C ∂∆∂π + 4∂ ψ ∂ 2 C ∂R∂π +2 ∂C ∂γ + ∂C ∂∆ ∂ ψ ∂ 2 C ∂π 2 + 2 ∂C ∂π ∂ 2 C ∂γ∂π − ∂ ψ ∂ 2 C ∂∆∂π − β + π ∂β ∂π , (5.15a) p ab T ∂ b ψ∂ a η : 0 = ∂C ∂R ∂ ψ − 1 2 ∂C ∂∆ ∂ 2 C ∂π∂P − ∂C ∂P ∂ ψ ∂ 2 C ∂π∂R + 1 2 ∂ 2 C ∂π∂∆ , (5.15b) X a ∂ a η : 0 = ∂C ∂p (4∂ q − 1) ∂ 2 C ∂π∂R − ∂C ∂R (4∂ q + 1) ∂ 2 C ∂π∂p + 1 2 ∂C ∂∆ (1 + 4∂ q ) ∂ 2 C ∂π 2 + 1 2 ∂C ∂π (1 − 4∂ q ) ∂ 2 C ∂π∂∆ − 1 3 p ∂β ∂π , (5.15c) p ab T X b ∂ a η : 0 = ∂C ∂R (1 + 2∂ q ) ∂ 2 C ∂π∂P + ∂C ∂P (1 − 2∂ q ) ∂ 2 C ∂π∂R , (5.15d) ∂ a F ∂ a η : 0 = ∂C ∂p ∂ 3 C ∂F ∂π∂R − ∂C ∂R ∂ 3 C ∂F ∂π∂R + 1 2 ∂C ∂∆ ∂ 3 C ∂F ∂π 2 − 1 2 ∂C ∂π ∂ 3 C ∂F ∂π∂∆ + 1 6 δ p F ∂β ∂π , (5.15e) p ab T ∂ b F ∂ a η : 0 = ∂C ∂R ∂ 3 C ∂F ∂π∂P − ∂C ∂P ∂ 3 C ∂F ∂π∂R , (5.15f) where F ∈ {p, P, R, ∆, γ}. Now that I have all of the conditions restricting the form of the constraint, I can move on to consolidating and interpreting them. Solving for the constraint Now I have the full list of equations, I seek to find the restrictions on the form of C they impose. Firstly, I use the condition from ∂ ab θ ab , (5.8a) to find ∂C ∂R = −β ∂C ∂P −1 ,(5.16) which I substitute into the equation from p T ab p cd T ∂ cd θ ab , (5.8e), 0 = ∂C ∂R ∂ 2 C ∂P 2 − ∂C ∂P ∂ 2 C ∂P∂R = −2β ∂C ∂P −1 ∂ 2 C ∂P 2 + ∂β ∂P = β ∂ ∂P log β ∂C ∂P −2 ,(5.17) and because β → 1 in the classical limit and so cannot vanish generally, I find that, β = b 1 ∂C ∂P 2 , where ∂b 1 ∂P = 0. (5.18) Substituting this back into (5.16) gives me ∂C ∂R = −b 1 ∂C ∂P , and from this I can find the first restriction on the form of the constraint, C(q, p, P, R, ψ, π, ∆, γ) = C 1 (q, p, ψ, π, ∆, γ, χ 1 ), where χ 1 := P − R 0 b 1 (q, p, x, ψ, π, ∆, ψ)dx. (5.19) Substituting this into the condition from ∂ a F ∂ b θ ab , (5.9k), gives 0 = ∂b 1 ∂F ∂C 1 ∂χ 1 2 , for F ∈ {p, P, R, π, ∆, γ}, (5.20) and therefore b 1 must only be a function of q and ψ. Substituting this into (5.19) leads to χ 1 = P − b 1 R. Turning to the condition from X a ∂ b θ ab , (5.9f), I find 0 = ∂C 1 ∂χ 1 2 (∂ q − 1) b 1 ,(5.21) which is solved by b 1 (q, ψ) = q b 2 (ψ). This is as expected because it means both terms in χ 1 have a density weight of two. From this I see that the condition coming from ∂ a ψ∂ b θ ab , (5.9a), gives 0 = ∂C 1 ∂χ 1 qb 2 ∂C 1 ∂χ 1 − ∂C 1 ∂∆ (5.22) which provides further restrictions on the form of the constraint, C = C 2 (q, p, ψ, π, γ, χ 2 ), χ 2 := P − q b 2 R − b 2 ∆ . (5.23) Look at the condition from p T ab ∂ 2 θ ab , (5.8d), 0 = ∂C ∂p ∂ 2 C ∂P∂R − ∂C ∂R ∂ 2 C ∂p∂P + 1 2 ∂C ∂∆ ∂ 2 C ∂π∂P − 1 2 ∂C ∂π ∂ 2 C ∂P∂∆ , = qb 2 − ∂C 2 ∂p ∂ 2 C 2 ∂χ 2 2 + ∂C 2 ∂χ 2 ∂ 2 C ∂p∂χ 2 + b 2 2b 2 ∂C 2 ∂χ 2 ∂ 2 C 2 ∂π∂χ 2 − ∂C 2 ∂π ∂ 2 C 2 ∂χ 2 2 = qb 2 ∂C 2 ∂χ 2 ∂C 2 ∂p + b 2 2b 2 ∂C 2 ∂π ∂ ∂χ 2 log ∂C 2 ∂p + b 2 2b 2 ∂C 2 ∂π ∂C 2 ∂χ 2 −1 ,(5.24) and because b 2 is a non-zero constant in the classical limit, this can be integrated to find ∂C 2 ∂p + b 2 2b 2 ∂C 2 ∂π = g 1 (q, p, ψ, π, γ) ∂C 2 ∂χ 2 , (5.25) where g 1 is a unknown function arising as an integration constant, and needs to be determined. This provides a further restriction on the form of the constraint, C = C 3 (q, ψ, γ, Π, χ 3 ) , Π := π − b 2 2b 2 p,χ 3 := P − q b 2 R − b 2 ∆ + p 0 g 1 q, x, ψ, Π + b 2 2b 2 x, γ dx. (5.26) Substituting this into the condition from ∂ 2 η, (5.14a), gives 0 = q b 2 ∂C 3 ∂χ 3 2 ∂ ∂π g 1 (q, p, ψ, π, γ) ,(5.27) and therefore, χ 3 = P − q b 2 R − b 2 ∆ + p 0 g 1 (q, x, ψ, γ) dx. (5.28) Evaluating the condition from q ab ∂ 2 θ ab , (5.8b), gives 0 = 1 3 qb 2 ∂C 3 ∂χ 3 2 1 + 3 ∂ ∂p g 1 (q, p, ψ, γ) ,(5.29) which can be integrated to find g 1 = g 2 (q, ψ, γ) − p/3 and therefore (5.28) becomes, χ 3 = P − 1 6 p 2 + g 2 p − q b 2 R − b 2 ∆ . (5.30) Then look at the condition from p T ab X c ∂ c θ ab , (5.9h), from which can be found 0 = 2qb 2 ∂C 3 ∂χ 3 ∂ 2 C 3 ∂χ 2 3 (2∂ q − 1) g 2 ,(5.31) which can be solved by, g 2 (q, ψ, γ) = √ q g 3 (ψ, γ) if we assume that ∂ 2 C 3 ∂χ 2 3 = 0 generally, which is true for any deformation dependent on curvature ∂β ∂χ 3 = 0. This is what is expected for the density weight of each term in χ 3 to match. I now look at the condition for p T ab ∂ c γ∂ c θ ab , which is (5.9m) with F = γ, 0 = q 3/2 b 2 ∂g 3 ∂γ ∂C 3 ∂χ 3 ∂ 2 C 3 ∂χ 2 3 , (5.32) which is true when g 3 = g 3 (ψ). At this point it gets harder to progress further as I have done so far. To review, I have restricted the constraint and deformation to the forms, C (q, p, P, R, ψ, π, ∆, γ) = C 3 (q, ψ, Π, γ, χ 3 ) , β = q b 2 (ψ) ∂C 3 ∂χ 3 2 , Π = π − b 2 2b 2 p, χ 3 = P − 1 6 p 2 + p √ q g 3 (ψ) − q b 2 R − b 2 ∆ ,(5.33) which satisfies all the conditions in (5.8), (5.14), (5.15) and (5.9) apart from the conditions for q ab ∂ c ψ∂ c θ ab , (5.9b), and ∂ a ψ∂ a η, (5.15a). As it stands, these conditions are not easy to solve. Solving the fourth order constraint to inform the general case To break this impasse, I use a test ansatz for the constraint which contains up to four orders in momenta, C 3 → C 0 + C (Π) Π + C (Π 2 ) Π 2 + C (Π 3 ) Π 3 + C (Π 4 ) Π 4 + C (χ) χ 3 + C (χ 2 ) χ 2 3 + C (Πχ) Πχ 3 + C (Π 2 χ) Π 2 χ 3 ,(5.34) where each coefficient is an unknown function to be determined dependent on q, ψ and γ. There is an asymmetric term included in χ 3 determined by the function g 3 (ψ), so I do not restrict myself to only even orders of momenta, unlike section 3. Substituting this into (5.15a), I can separate out the multiplier of each unique combination of variables as an independent equation. For each of the terms which are the multipliers of 5 or 6 orders of momenta, I find a condition specifying that the constraint coefficients for terms 3 or 4 orders of momenta must not depend on γ, e.g. ∂ ∂γ C (χ 2 ) = 0, ∂ ∂γ C (Π 3 ) = 0. Since γ depends on two spatial derivatives, I see that each term in the constraint must not depend on a higher order of spatial derivatives than it does momenta. If I include higher orders of spatial derivatives in the ansatz, I quickly find them ruled out in a similar fashion. Therefore, I use this information to further expand my ansatz, C 3 → C ∅ + C (γ) γ + C (γ 2 ) γ 2 + C (Π) Π + C (Πγ) Πγ + C (Π 2 ) Π 2 + C (Π 2 γ) Π 2 γ + C (Π 3 ) Π 3 + C (Π 4 ) Π 4 + C (χ) χ 3 + C (χγ) χ 3 γ + C (χ 2 ) χ 2 3 + C (Πχ) Πχ 3 + C (Π 2 χ) Π 2 χ 3 ,(5.35) where each coefficient is now an unknown function of q and ψ. One can find all the necessary conditions from (5.15a), for which the solution also satisfies (5.9b). I will show a route which can taken to progressively restrict C. The condition coming from P 2 is solved if C (Π 2 χ) = 1 2 C (χ 2 ) C (χγ) 2qb 2 C (χ 2 ) + 7b 2 2 8b 2 2 − b 2 b 2 −1 ,(5.36) the condition from γ 2 is solved by, C (Π 2 γ) = 1 4 C (χγ) 2C (γ 2 ) b 2 C (χγ) + 7b 2 2 8b 2 2 − b 2 b 2 −1 ,(5.37) the condition from γP is solved by, C (γ 2 ) = C 2 (χγ) 4C (χ 2 ) ,(5.38) the condition from π 4 is solved by, C (Π 4 ) = 1 16 C (χ 2 ) C (χγ) 2qb 2 C (χ 2 ) + 7b 2 2 8b 2 2 − b 2 b 2 −1 ,(5.39) and all the other conditions coming from four momenta are solved. Turning to the third order, the condition from πP is solved by, 40) and the condition from πγ is solved by, 41) and the condition from π 3 is solved by, C (Π 3 ) = 1 12 C (Πχ) 1 qb 2 3C (χγ) 2C (χ 2 ) − 2C (Πγ) C (Πχ) + 7b 2 2 8b 2 2 − b 2 b 2 − √ qC (χ 2 ) 4g 3 − 3g 3 b 2 b 2 C (χγ) 2qb 2 C (χ 2 ) + 7b 2 2 8b 2 2 − b 2 b 2 −2 ,(5.C (Πγ) = C (Πχ) C (χγ) 2C (χ 2 ) ,(5.C (Πχ) = −1 2 √ qC (χ 2 ) 4g 3 − 3g 3 b 2 b 2 C (χγ) 2qb 2 C (χ 2 ) + 7b 2 2 8b 2 2 − b 2 b 2 −1 ,(5.42) which completes all the terms from third order. The only new condition coming from second order is solved by, 43) and the only new condition coming from first order is solved by, C (Π 2 ) = 1 4 C (Πχ) 1 qb 2 C (χγ) C (χ 2 ) − C (γ) C (χ) + 7b 2 2 8b 2 2 − b 2 b 2 + 1 16 qC (χ 2 ) 4g 3 − 3g 3 b 2 b 2 2 C (χγ) 2qb 2 C (χ 2 ) + 7b 2 2 8b 2 2 − b 2 b 2 −2 ,(5.C (Π) = −1 4 √ qC (χ) 4g 3 − 3g 3 b 2 b 2 1 qb 2 C (χγ) C (χ 2 ) − C (γ) C (χ) + 7b 2 2 8b 2 2 − b 2 b 2 × C (χγ) 2qb 2 C (χ 2 ) + 7b 2 2 8b 2 2 − b 2 b 2 −2 ,(5.44) and from the zeroth order, C (χγ) = 2C (γ) C (χ 2 ) C (χ) ,(5.45) When all of these terms are combined, I find the solution for the fourth order constraint, C = C ∅ + C (χ) χ 3 + Π (Π − Ξ) 4Ω + C (γ) C (χ) γ + C (χ 2 ) χ 3 + Π (Π − Ξ) 4Ω + C (γ) C (χ) γ 2 , (5.46) where Ω = C (γ) b 2 qC (χ) + 7b 2 2 8b 2 2 − b 2 b 2 , Ξ = √ q 4g 3 − 3g 3 b 2 b 2 . (5.47) If this solution is generalised to all orders, (5.48) one can check that it satisfies all the conditions from (5.8), (5.14), (5.15) and (5.9). It is possible that directly generalising from the fourth order constraint rather than continuing to work generally means that this is not the most general solution. However, at least I now know a form of the constraint which can solve all the conditions. Now that I have a form for the general constraint, I seek to compare it to the low-curvature limit, when C → χ 4 C χ + C ∅ , and match terms with that found previously (3.27) in chapter 3 and [55]. I find that, C = C 4 (q, ψ, χ 4 ) , χ 4 = P − 1 6 p 2 + √ qpg 3 − q b 2 R − b 2 ∆ + Π (Π − Ξ) 4Ω + C (γ) C (χ) γ,b 2 = σ β ω 2 R 4 , σ β := sgn(β) = sgn(β ∅ ). C χ = 2σ β ω R \|β ∅ \| q , C γ = q \|β ∅ \| ω ψ 2 + ω R ,(5.49) For convenience, I redefine the function determining the asymmetry, g 3 = ξ/2, and I expand the constraint in terms of the weightless (or 'de-densitised') scalar R := χ 4 /q. This means that the general form of the deformed constraint is given by, C = C (q, ψ, R) , β = σ β q ∂C ∂R 2 , (5.50a) R := 2σ β qω R P − 1 6 p 2 − ω R 2 R + ω R ∆ψ + ω ψ 2 + ω R ∂ a ψ∂ a ψ, + σ β ω R ω ψ ω R + 3 2 ω 2 R 1 2q π − ω R ω R p 2 + ξ √ q ω ψ ω R p + 3ω R 2ω R π − ξ ξ π − ω R ω R p . (5.50b) It is probably more appropriate to see the deformation function itself as the driver of deformations to the constraint, so I rearrange (5.50a), ∂C ∂R = q \|β\|,(5.51) which can be integrated to find, C = R 0 q \|β(q, ψ, r)\|dr + C ∅ (q, ψ). (5.52) From either form of the general solution (5.50a) or (5.52), one can now understand the meaning of (2.50), which relates the order of the constraint and the deformation, 2n C − n β = 4. The differential form (5.50a) is like n β = 2 (n C − 2), and the integral form (5.52) is like n C = 2 + n β /2.β = β ∅ (1 + β 2 R) n → C = C ∅ +          2 q \|β ∅ \| (n + 2) β 2 sgn(1 + β 2 R) \|1 + β 2 R\| n+2 2 − 1 , n = −2, q \|β ∅ \| β 2 sgn(1 + β 2 R) log \|1 + β 2 R\| , n = −2, C ∅ + q \|β ∅ \| R + nβ 2 4 R 2 + · · · . (5.53) β = β ∅ e β 2 R → C = C ∅ + 2 q \|β ∅ \| β 2 e β 2 R/2 − 1 C ∅ + q \|β ∅ \| R + β 2 4 R 2 + · · · , (5.54) β = β ∅ sech 2 (β 2 R) → C = C ∅ + q \|β ∅ \| β 2 gd (β 2 R) , C ∅ + q \|β ∅ \| R − β 2 2 6 R 3 + · · · ,(5.55) where gd(x) := x 0 dt sech(t) is the Gudermannian function. Most other deformation functions would need to be integrated numerically to find the constraint. As can be seen from the small R expansions, it would be possible to constrain β ∅ and β 2 phenomenologically but the asymptotic behaviour of β would be difficult to determine. The simplest constraint that can be expressed as a polynomial of R that contains higher orders than the classical solution is given by, 56) which is equivalent to the fourth order constraint found in (5.46). β = β ∅ (1 + β 2 R) 2 → C = C ∅ + q \|β ∅ \| R + β 2 2 R 2 ,(5. Looking back at the constraint algebra D a [βq ab (N ∂ b M − ∂ b N M )], = dxD a βq ab (N ∂ b M − ∂ b N M ) = dxD a σ β ∂C ∂C 2 (N ∂ b M − ∂ b N M ) = dxD a σ β (N ∂ bM − ∂ bNM ), = D a [σ β q ab (N ∂ bM − ∂ bNM )],(5.58) which I can combine to show the that the following two equations are equivalent, {C[N ], C[M ]} = D a [βq ab (N ∂ b M − ∂ b N M )], (5.59a) {C[σ ∂CN ],C[σ ∂CM ]} = D a [σ β q ab (N ∂ bM − ∂ bNM )]. (5.59b) The two σ ∂C on the left side should cancel out, but they are included here to show the limit to the redefinition of the lapse functions. While it may seem like I have regained the undeformed constraint algebra up to the sign σ β with a simple transformation, it shouldn't be taken to mean that this is actually the algebra of constraints. That is, the above equation doesn't ensure thatC ≈ 0 instead of C ≈ 0 when on-shell. The surfaces in phase space described byC = 0 and C = 0 are different in general. Cosmology I restrict to an isotropic and homogeneous space to find the background cosmological dynamics, following the definitions in section 2.8. Writing the constraint as C = C(a, ψ, R) where R = R(a, ψ,p, π), the equations of motion are given by, a N = 1 6a ∂R ∂p ∂C ∂R ,ṗ N = −1 6a ∂C ∂a + ∂R ∂a ∂C ∂R , ψ N = ∂R ∂π ∂C ∂R ,π N = − ∂C ∂ψ − ∂R ∂ψ ∂C ∂R ,(5.60) into which I can substitute ∂C ∂R = a 3 \|β\|. When I assume minimal coupling (ω R = 0, ω ψ = 0) and time-symmetry (ξ = 0), the equations of motion become, R → −3σ βp 2 ω R a 2 − 3kω R a 2 + σ β π 2 2ω ψ a 6 , a N = −σ βp ω R \|β\|,ψ N = σ β π ω ψ a 3 \|β\|,π N = − ∂C ∂ψ , p N = −1 6a ∂C ∂a − a \|β\| σ βp 2 ω R a 2 + kω R a 2 − σ β π 2 2ω ψ a 6 . (5.61) To find the Friedmann equation, find the equation for H 2 /N 2 , and substitute in for R, H 2 N 2 = \|β\|p 2 ω 2 R = β −R 3ω R − k a 2 + σ β π 2 6ω R ω ψ a 6 ,(5.62) and when the constraint is solved, C ≈ 0, then R can be found in terms of C ∅ . Cosmology with a perfect fluid I here find the deformed Friedmann equations for various forms of the deformation. For simplicity, I ignore the scalar field and include a perfect fluid C ∅ = a 3 ρ(a). From the deformation function β = β ∅ (1 + β 2 R) n , solving the constraint (5.53) gives R =              σ 2 β 2 σ 2 − (n + 2) σ 2 β 2 ρ 2 \|β ∅ \| n+2 2 − 1 β 2 , n = −2, σ 2 β 2 exp −σ 2 β 2 ρ \|β ∅ \| − 1 β 2 , n = −2,(5.63) where σ ∅ := sgn(β ∅ ) and σ 2 := sgn(1 + β 2 R). When I simplify by assuming σ 2 = 1, the Friedmann equation is given by, The singularities for 0 > n > −2 appear to be similar to sudden future singularities characterised in [83,91]. However, the singularities here might instead be called sudden 'past' singularities as they happen when a is small (but non-zero) and ρ is large. Moreover, they happen for any perfect fluid with w > −1, i.e. including matter and radiation. H 2 N 2 =            β ∅ 3ω R β 2 1 − 1 − ρ ρ c (n) 2 n+2 − kβ ∅ a 2 1 − ρ ρ c (n) 2n n+2 , n = −2, β ∅ 3ω R β 2 1 − exp −β 2 ρ \|β ∅ \| − kβ ∅ a 2 exp 2β 2 ρ \|β ∅ \| n = −2, (5.64) (a) β = β∅ (1 + β2R) n (b) β ∼ exp R For the deformation function β = β ∅ exp (β 2 R) from (5.54), solving the constraint gives, 65) and the Friedmann equation is given by, R = 2 β 2 log 1 − β 2 ρ 2 \|β ∅ \| ,(5.H 2 N 2 = −2β ∅ 3ω R β 2 log 1 − β 2 ρ 2 \|β ∅ \| − kβ 2 a 2 1 − β 2 ρ 2 \|β ∅ \| 2 . (5.66) and a critical density appears for ρ → 2 \|β ∅ \| β 2 . For the deformation function β = β ∅ sech 2 (β 2 R) from (5.55), solving the constraint gives, R = −1 β 2 gd −1 β 2 ρ \|β ∅ \| . (5.67) Substituting this back into the deformation function gives, 5.68) and the Friedmann equation is given by β = β ∅ cos 2 β 2 ρ \|β ∅ \| (H 2 N 2 = β ∅ 3ω R β 2 gd −1 β 2 ρ \|β ∅ \| − kβ ∅ a 2 cos 2 β 2 ρ \|β ∅ \| . (5.69) where there is a critical density 1 , ρ → π • \|β ∅ \| 2β 2 . These exponential-type deformation functions that I consider all predict a upper limit on energy density. To illustrate this, I plot the modified Friedmann equations for these functions in Fig. 5.1(b). Cosmology with a minimally coupled scalar field Since the metric and scalar kinetic terms must combine into one quantity, R, a deformation function should not affect the relative structure between fields. To illustrate this, take a free scalar field (without a potential) which is minimally coupled to gravity, and assume no perfect fluid component. This means that the generalised potential term C ∅ will vanish, in which case solving the constraint, C ≈ 0, merely implies R = 0. Consequently, since the deformation function β is a function of R, the only deformation remaining will be the zeroth order term β = β ∅ (q, ψ). Combining the equations of motion (5.61) allows me to find the Friedmann equation, H 2 N 2 = ω ψψ 2 6ω R N 2 − kβ ∅ a 2 ,(5.70) that is, the minimally-deformed case. For β = β ∅ , it is required that R must not vanish, which itself requires that C ∅ must be non-zero. Therefore, for the dynamics to depend on a deformation which is a function of curvature, there must be a non-zero potential term which acts as a background against which the fields are deformed. Deformation correspondence As discussed in the perturbative action chapter 4, the form of the deformation used in the literature which includes holonomy effects is given by the cosine of the extrinsic curvature [40][41][42]. Of particular importance to this is that the deformation vanishes and changes sign for high values of extrinsic curvature. Since the extrinsic curvature is proportional to the Hubble expansion rate, write the deformation (4.36) here as, In Fig. 5.2, I plot β(h) and h(r) in the region \|h\| ≤ π • /2. After making the transformation, I find β(r). Note that, unlike for h, β does not vanish for finite r. So it seems that a deformation which vanishes for finite extrinsic curvature does not necessarily vanish for finite intrinsic curvature or metric momenta (at least not in the isotropic and homogeneous Returning to the solution for the constraint, (5.52), reducing it to depending on only a andp gives β = β ∅ cos (β k H) .(a) β (h) (b) h(r) (c) β (r) (d) C k (r)C = −6a ω R p 0 dp σ β p \|β (a, p )\| + C ∅ (a) ,(5.73) and transforming fromp to r as defined in (5.72), while making the assumptions σ β = 1, N = 1, β ∅ = 1, and β k ∼ constant, this becomes If instead of the extrinsic curvature itself, the deformation is a cosine of the standard extrinsic curvature contraction, β = cos β k K ∼ cos h 2 , it still cannot be transformed analytically. However, it does match the function β(h) = 1−4π −2 • h 4 well, as I have plotted C = −6ω R a 3 β 2 k C k (r) + C ∅ (a in Fig. 5.3. However, numerically finding the constraint for these two deformations, then considering the low R limit, I see that C ∼ R 2 + C ∅ . Therefore, this deformation can be ruled out if C ∼ R + C ∅ is known to be the low curvature limit of the Hamiltonian constraint. Considering the function β(h) = 1−4π −2 • h 2 in Fig. 5.2, transforming from h to K and from r to R to R, we can see the correspondence between different limits of the deformation function, β (K, 0) = 1 − β 2 K, → β (0, R) = 1 1 + β 2 R . (5.75) This is what I found in chapter 6, where the general form of this particular deformation is actually the product of these two limits. However, for non-linear deformation functions, β(K, R) cannot be determined so easily from β(K, 0) and β(0, R). That being said, given β(R), the dependence on K could be found by simply solving and evolving the equations of motion. Discussion In this chapter, I have found the general form that a deformed constraint can take for non-minimally coupled scalar-tensor variables. The momenta and spatial derivatives for all fields must maintain the same relative structure in how they appear compared to the minimally-deformed constraint. This means that the constraint is a function of the fields and the general kinetic term R. The freedom within this kinetic term comes down to the coupling functions. While a lapse function transformation can apparently take the constraint algebra back to the undeformed form, this seems to be merely a cosmetic change as it does not in fact alter the Hamiltonian constraint itself. Deformed gravitational action to all orders As shown in section 2.7, the deformed action must be calculated either perturbatively, as has been done in chapter 4, or completely generally. It appears that this is because it does not permit a closed polynomial solution when the deformation depends on curvature. In this chapter I attempt this general calculation. This material has been subsequently published in ref. [57]. and so I can solve up to a sign, σ L := sgn ∂L ∂w , ∂L ∂w = σ L λ 1 β . (6.3) Then, from Q abcd ∂ cd θ ab , (4.12b), I find ∂L ∂R = 4β ∂L ∂w = 4σ L σ β \|λ 1 β\|,(6.4) where σ β := sgn(β(q, v, w, R)). If I then compare the second derivative of the action, ∂ 2 L ∂w∂R , using both equations, I find a nonlinear partial differential equation for the deformation function, 0 = ∂β ∂R + 4β ∂β ∂w , (6.5) which is the same form as Burgers' equation for a fluid with vanishing viscosity [92]. However, before I attempt to interpret this, I will find further restrictions on the action and deformation. I now seek to find how the trace of the metric's normal derivative, v, appears. Take the condition for v ab T ∂ 2 θ ab , (4.12d) 0 = v 3 ∂ 2 L ∂R∂w − β ∂ 2 L ∂v∂w = σ L 2 λ 1 β 4v 3 ∂β ∂w + ∂β ∂v (6.6) which I can solve to find that β = β (q,w, R), wherew = w −2v 2 /3. So in the deformation, the trace v must always be paired with the traceless tensor squared w like this. I can see that this is related to the standard extrinsic curvature contraction byw = −4K. To find how the trace appears in the action, I look at the condition from q ab ∂ 2 θ ab , (4.12a), 0 = ∂L ∂R − 2v 3 ∂ 2 L ∂v∂R + 2β ∂ 2 L ∂v 2 − 2 3 ∂L ∂w (6.7) inputting my solutions so far, I can solve for the second derivative with respect to the trace, ∂ 2 L ∂v 2 = −4σ L 3 λ 1 β 1 − v 2 ∂β ∂v . (6.8) I integrate over v to find the first derivative, ∂L ∂v = −4vσ L 3 λ 1 β + ξ 1 (q, w, R) = −4v 3 ∂L ∂w + ξ 1 (q, w, R). (6.9) To make sure that the solutions (6.3), (6.4) and (6.9) match for the second derivatives ∂ 2 L ∂v∂R and ∂ 2 L ∂v∂w , I find that ξ 1 = ξ 1 (q). Therefore, from this I can see that the action should have the metric normal derivatives appear in the combined formw apart from a single linear term L ⊃ vξ 1 (q). I now just have to see what conditions there are on how the metric determinant appears in the action. First I have the condition from X a ∂ b θ ab , (4.13a), 0 = ∂L ∂R − 4 (∂ q β + 2β∂ q ) ∂L ∂w , = 4σ L σ β \|λ 1 β\| 1 − ∂ q λ 1 λ 1 , ∴ λ 1 (q) = qλ 2 ,(6.10) and second I have the condition from v ab T X c ∂ c θ ab , (4.13c), 0 = v 3 (4∂ q − 1) ∂ 2 L ∂w∂R + ∂β ∂w (1 − 2∂ q ) ∂L ∂v + (β − 2∂ q β − 4β∂ q ) ∂ 2 L ∂v∂w , = ∂β ∂w (ξ 1 − 2∂ q ξ 1 ) , ∴ ξ 1 (q) = ξ 2 √ q,(6.11) and both these results show that my action will indeed have the correct density weight when β → 1, that is L ∝ √ q. All the remaining conditions from the distribution equation that have not been explicitly referenced are solved by what I have found so far, so to make progress I must now attempt to consolidate my equations to find an explicit form for the action. If I integrate (6.3), I find L = σ L \|qλ 2 \| w 0 dx \|β(q, x, R)\| + f 1 (q, v, R),(6.12) and then if I match the derivative of this with respect to v with (6.9), I find the v part of the second term, f 1 (q, v, R) = vξ 2 √ q + f 2 (q, R). (6.13) If I then match the derivative of (6.12) with respect to R with (6.4), I see that ∂L ∂R = 4σ L σ β \|qλ 2 β\| = ∂f 2 ∂R − σ L 2 \|qλ 2 \| w 0 σ β dx \|β(q, x, R)\| 3/2 ∂ ∂R β(q, x, R) (6.14) and using (6.5) to change the derivative of β, 4σ L σ β \|qλ 2 β\| = ∂f 2 ∂R + 2σ L \|qλ 2 \| w 0 dx \|β(q, x, R)\| ∂ ∂x β(q, x, R),(6.15) and so I can change the integration variable, 4σ L σ β \|qλ 2 β\| = ∂f 2 ∂R + 2σ L \|qλ 2 \| β(q,w,R) β(q,0,R) db \|b\| ,(6.16) the upper integration limit cancels with the left hand side of the equality, and therefore ∂f 2 ∂R = 4σ L sgn(β(q, 0, R)) \|qλ 2 β(q, 0, R)\|. (6.17) Then integrating this over R, f 2 (q, R) = 4σ L \|qλ 2 \| R 0 sgn(β(q, 0, r)) \|β(q, 0, r)\|dr + f 3 (q),(6.L = σ L \|qλ 2 \| w 0 dx \|β(q, x, R)\| + 4 R 0 sgn(β(q, 0, r)) \|β(q, 0, r)\|dr Now, I test this with a zeroth order deformation so I can match terms with my previous results. Using β = β ∅ (q), + vξ 2 √ q + f 3 (q).L = σ L \|qλ 2 \| w \|β ∅ \| + 4R sgn(β ∅ ) \|β ∅ \| + vξ 2 √ q + f 3 (q),(6.20) comparing this to (4.24) and usingw = −4K leads to σ L = σ β , \|λ 2 \| = ω 8 , ξ 2 = ξ, f 3 = − √ qV (q),(6.21) and therefore, the full solution is given by, 22) and the deformation function must satisfy the non-linear partial differential equation, L = ωσ β √ q 2 R 0 sgn(β(q, 0, r)) \|β(q, 0, r)\|dr − K 0 dk \|β(q, k, R)\| + √ q (vξ − V (q)) ,(6.0 = ∂β ∂R − β ∂β ∂K . (6.23) By performing a Legendre transform, I can see that the Hamiltonian constraint associated with this action is given by, C = ωσ β √ q 2 K 0 dk \|β(q, k, R)\| − 2K \|β(q, K, R)\| − R 0 sgn(β(q, 0, r)) \|β(q, 0, r)\|dr + √ q V,(6.dR dK = −1 β (q, K, R) (6.27) and because β is constant along the trajectories, they are a straight line in the (K, R) plane. I must have an 'initial' condition in order to solve the equation, and because R is here the analogue of −t in (6.25) I define the initial function when R = 0, given by β(q, K, 0) =: α(q, K). Since there are trajectories along which β is constant, I can use α to solve for R(K) along those curves, given an initial value K 0 , R = K 0 − K α(K 0 ) . (6.28) Reorganising to get, K 0 = K + Rα(K 0 ), and then substituting into β, this leads to the implicit relation, β(q, K, R) = α (q, K + Rβ(q, K, R)) . (6.29) I invoke the implicit function theorem to calculate the derivatives of β, ∂β ∂K = α 1 − Rα , ∂β ∂R = −βα 1 − Rα ,(6.30) which show that a discontinuity develops when Rα → 1. This is the point where the characteristic trajectories along which β is constant converge to form a caustic. Beyond this point, β seems to become a multi-valued function. An analytic solution to β only exists when α is linear, α = α 1 (q) + α 2 (q)K, β = α 1 (q) + α 2 (q)K 1 − α 2 (q)R ,(6.31) and when α 2 (q) is small, I can expand β into a series, β α 1 + α 2 (K + α 1 R) ∞ n=0 R n α n 2 ,(6.32) and by comparing this to the perturbative deformation found previously, (4.25), I can see This cross section appears to match what was found in section 5.4.3 when I attempted to find the correspondence between β(K, 0) and β(R). It would seem that β(0, R) should be a non-vanishing function of the shape as shown in Fig. 5.2(c). When the inviscid Burgers' equation is being simulated in the context of fluid dynamics, a choice must be made on how to model the shock wave [92]. The direct continuation of the equation means that the density function u becomes multi-valued, and the physical intepretation of it as a density breaks down. The alternative is to propagate the shock wave as a singular object, which requires a modification to the equations. Considering my case of the deformation function, allowing a shock wave to propagate does not seem to make sense. It might require being able to interpret β as a density function and the space of (K, R) to be interpreted as a medium. Whether or not the shock wave remains singular or becomes multi-valued, the most probable interpretation is that it represents a disconnection between different branches of curvature configurations. That is, for a universe to transition from one side of the discontinuity to the other may require taking an indirect path through the phase space. Linear deformation If I take the analytic solution for the deformation function when its initial condition is linear (6.31), I can substitute it into the general form for the gravitational action (6.22). If I assume I am in a region where 1 − α 2 R > 0, I get the solution, L = ω √ q α 2 sgn 1 + α 2 K α 1 \|α 1 \| − \|α 1 +α 2 K\| \|1−α 2 R\| + √ q (vξ −V ) , (6.33) and expanding in series for small α 2 when I am in a region where \|α 1 \| \|α 2 K\|, L = ω 2 q \|α 1 \| R − K α 1 − α 2 4 R + K α 1 2 + O α 3 2 + √ q (vξ − V ) ,(6.34) which matches exactly the fourth order perturbative action I found previously (4.24). The Hamiltonian constraint associated with the non-perturbative action can be found from (6.24), and then I can solve for K when the constraint vanishes (as long as I specify that it must be finite in the limit α 2 → 0), K = 2 ω sgn(α 1 ) \|α 1 \|V 1 − α 2 V 2ω \|α 1 \| − α 1 R 1 − α 2 V ω \|α 1 \| −2 ,(6.35) and if I restrict to the FLRW metric and a perfect fluid as in section 2.8, I find the modified Friedmann equation, H 2 N 2 = sgn(α 1 ) \|α 1 \| 3ω ρ 1 − α 2 ρ 2ω \|α 1 \| − α 1 k a 2 1 − α 2 ρ ω \|α 1 \| −2 . (6.36) There is a correction term similar to that found for the fourth order perturbative action which suggests there could be a bounce when ρ → 2ω \|α 1 \|/α 2 . However, there is also an additional factor which causes H to diverge when ρ → ω \|α 1 \|/α 2 , which is before that potential bounce. This is directly comparable to the modified Friedmann equation found for the deformation function β(R) = β ∅ (1 + β 2 R) −1 , (5.64) investigated in section 5.4.1, with α 1 = β ∅ and α 2 = ωβ 2 /2. As is found here, those results suggested a sudden singularity where H diverges when a and ρ remain finite. Discussion I have found the general form of the deformed gravitation action when considering tensor combinations of derivatives up to second order. The way in which the deformation, and thereby the action, depends on the extrinsic and intrinsic curvature was found to be highly non-linear. Curiously, its form matches an equation found in fluid dynamics. The meaning of this comparison is far from clear. For different initial functions, I numerically solved for the deformation function until a discontinuity formed. The meaning of this discontinuity is not clear, but might manifest as a barrier across which paths through phase space cannot cross. Chapter 7 Conclusions I have attempted to thoroughly investigate the effects that a quantum-motivated deformation to the hypersurface deformation algebra of general relativity has in the semi-classical limit. Starting from the algebra, I have shown how to regain a deformed gravitational action or a deformed scalar-tensor constraint. Finding the minimally-deformed version of a non-minimally coupled scalar-tensor model, I was able to establish the classical low-curvature reference point. I was able to show how the higher-order curvature terms arising from a deformation are qualitatively different from conventional higher-order terms which can absorbed by a non-minimally coupled scalar field. I also investigated some of the interesting effects which non-minimal coupling has on cosmology. As a first step towards including higher-order curvature terms coming from a deformation, I derived the fourth order gravitational action perturbatively. The nearest order corrections demonstrate a change in the relative structure between time and space since the higher order curvature terms appear with a different sign. I investigated the cosmological implications of the higher order terms, albeit while using the assumption that the action found perturbatively could be extended beyond the perturbative regime. In attempting to find the deformed scalar-tensor constraint to any order, I was able to show how the momenta and spatial derivatives maintain the same relative kinetic structure. Interestingly, the way the scalar field and gravitational kinetic terms combine must also be unchanged. That is to say that higher order gravitational terms are necessarily accompanied by higher order scalar terms of the same form. The main consequence of this seems to be that a potential term (in a general sense) must be present for a deformation of the kinetic terms to affect the dynamics. By testing different deformation functions, I was able to show what kinds of cosmological effects should be expected. Interestingly, the deformations which cause a big bounce seem to be required to vanish, but are not required to change sign. For the final chapter, I derived the general deformed gravitational action. The way the deformation function is differently affected by extrinsic and intrinsic curvature (or, equivalently, by time and space derivatives) was found to be similar to a differential equation which usually appears in fluid mechanics. Discontinuities in the deformation function seem to be inevitable, but the interpretation of what they mean is not clear. By checking the nearest order perturbative corrections, I was able to validate the perturbative action derived in an earlier chapter. One of the original motivations of this study was to provide insight into the problem of incorporating spatial derivatives, local degrees of freedom and matter fields into models of loop quantum cosmology which deform space-time covariance. From my results, it would seem that the problem comes from considering the kinetic terms as separable, or as differently deformed. The kinetic term, when constructed with canonical variables, cannot have its internal structure deformed beyond a sign. The deformation can only be a function of the combined term, which means that matter field derivatives deform the space-time covariance in a similar way to curvature. This may strike at the heart of the way the loop quantisation project, which attempts to first find a quantum theory of gravity, typically adds in matter as an afterthought. That being said, there are important caveats to this work which must be kept in mind. The fact that I used metric variables rather than the preferred connection or loop variables might limit the applicability of my results when comparing to the motivating theory. Moreover, the deformation of the constraint algebra is only predicted for real values of γ BI . I also only considered combinations of derivatives or momenta that were a maximum of two orders, when higher order combinations and higher order derivatives are likely to appear in true quantum corrections. As said in the introduction, 1, there are potentially wider implications for this study. The deformation can lead to a modified dispersion relation, possibly indicating a variable speed of light or an invariant energy scale. It might be related to non-classical geometric qualities such a non-commutativity or scale-dependent dimensionality. In the literature, it is indicated that the deformation function may change sign, implying a transition from a Lorentzian to a Euclidean geometry at high densities. In such a way, it might be a potential mechanism for the Hartle-Hawking no-boundary proposal. Appendix A Decomposing the curvature In our calculations, we need to decompose the three dimensional Riemann curvature frequently, so we collect the relevant identities in this appendix. The Riemann tensor is defined as the commutator of two covariant derivatives of a vector ∇ c ∇ d A a − ∇ d ∇ c A a = R a bcd A b , (A.1) and can be given in terms of the Christoffel symbols, The two equations we need most are the derivative of the Ricci scalar with respect to the first and second spatial derivative of the metric, and we can find these from combining the above equations, ∂R ∂(∂ d ∂ c q ab ) = ∂(∇ h ∇ g q ef ) ∂(∂ d ∂ c q ab ) ∂R ∂(∇ h ∇ g q ef ) = δ d h δ c g δ ab ef Φ ef gh = Φ abcd ∴ ∂R ∂q ab,cd = Φ abcd = Q abcd − q ab q cd , (A.10a) ∂R ∂(∂ c q ab ) = ∂(∇ h ∇ g q ef ) ∂(∂ c q ab ) ∂R ∂(∇ h ∇ g q ef ) = −Γ c gh δ ab ef − 4δ (a (e Γ b) f )(g δ c h) Φ ef gh , ∴ ∂R ∂q ab,c = 3 2 Q abde ∂ c q de − Q edc(a ∂ b) q de + q ab Y c − 2q c(b Y a) − 1 2 q ab X c + q c(b X a) , (A.10b) where X a := q bc ∂ a q bc and Y a := q bc ∂ (c q b)a = ∂ b q ba . w Ψ ∈ R transforms under the change x a → x a , Ψ b 1 ...b i a 1 ...a j = det ∂x c ∂x c w Ψ Ψ b 1 ...b i a 1 ...a j ∂x b 1 ∂x b 1 · · · ∂x b i ∂x b i ∂x a 1 ∂x a 1 · · · ∂x a j ∂x a j , (B.3) and one can 'de-densitise' to find a tensor 1 by multiplying it by q −w Ψ /2 , because √ q is a scalar density of weight one [10, p.276]. The integration measure d 3 x has a weight of −1, so for an integral to be appropriately tensorial, the integrand must have a weight of +1, e.g. d 3 x √ q. Since making a Legendre transformation requires using the term d 3 xψ π for a conjugate pair (ψ, π), when the variable ψ is of weight w ψ , the momentum π is of weight 1 − w ψ . B.1 Diffeomorphism constraint for a scalar field I consider a scalar field (ψ, π). Take (B.2) with F = ψ, {ψ(x), D a [N a ]} = d 3 yN a (y) δD a (y) δπ(x) , = N a ∂D a ∂π − ∂ b N a ∂D a ∂π ,b + ∂ bc N a ∂D a ∂π ,bc + . . . Checking what result I get for F = π merely produces the same equations and therefore the diffeomorphism constraint for a scalar field is given by, D a = π∂ a ψ. (B.6) I considered up to second order spatial derivatives here as a demonstration, but no diffeomorphism constraint goes beyond first order, so I will not bother with them for further equations below. = N a ∂D a ∂π − ∂ b ∂D a ∂π ,b + ∂ bc ∂D a ∂π ,bc + ∂ b N a − ∂D a ∂π ,b + 2∂ c ∂D a ∂π , B.2 Diffeomorphism constraint for a vector I consider a weightless contravariant vector (A a , P b ). Take (B.2) with F = A a , {A a (x), D b [N b ]} = d 3 yN b (y) δD b (y) δP a (x) , = N b ∂D b ∂P a − ∂ c N b ∂D b ∂P a,c + . . . = N b ∂D b ∂P a − ∂ c ∂D b ∂P a,c + ∂ c N b − ∂D b ∂P a,c + . . . (B.7a) L N A a = N b ∂ b A a − A b ∂ b N a ,( B.4 Diffeomorphism constraint for a tensor density For the general case of a tensor density with n covariant indices, m contravariant indices and weight w Ψ , Ψ b 1 ···bm a 1 ···an , Π c 1 ···cn d 1 ···dm where the canonical momentum has weight 1 − w Ψ , the associated diffeomorphism constraint is given by, × (∂ c − Y c ) ∂L ∂v + 2 v cd T ∂ d + q cd ∂ e v T de − v cd T Y d − W c ∂L ∂w . (C.4h) B need for a theory of quantum gravity . . . . . . . . . . . . . . . . . . . 1 1.2 Loop quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Why study deformed general relativity? . . . . . . . . . . . . . . . . . . . . 5 1.4 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Wider impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -time decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Canonical formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Choice of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Higher order models of gravity . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 Non-minimally coupled scalar from F (4) R gravity . . . . . . . . . . 16 2.5 Deformed constraint algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Derivation of the distribution equation . . . . . . . . . . . . . . . . . . . . . 20 2.7 Order of the deformed action and constraint . . . . . . . . . . . . . . . . . . 21 2.7.1 Hamiltonian route . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7.2 Lagrangian route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.8 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Second order scalar-tensor model and the classical limit 26 3.1 Solving the distribution equation . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.1 p ab sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 π sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Multiple scalar fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1 Bounce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.2 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.3 Geometric scalar model . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.4 Non-minimally enhanced scalar model . . . . . . . . . . . . . . . . . 44 3.3.5 Bouncing scalar model . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 Fourth order perturbative gravitational action 49 4.1 Solving the action's distribution equation . . . . . . . . . . . . . . . . . . . 49 vi 4.2 Finding the conditions on the action . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Evaluating the fourth order perturbative action . . . . . . . . . . . . . . . . 54 4.4 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.1 Linking the β function to LQC . . . . . . . . . . . . . . . . . . . . . 60 4.4.2 Perfect fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.3 Scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5 Deformed scalar-tensor constraint to all orders 70 5.1 Solving the distribution equation . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1.1 p ab route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1.2 π route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Solving for the constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.1 Solving the fourth order constraint to inform the general case . . . . 80 5.3 Looking back at the constraint algebra . . . . . . . . . . . . . . . . . . . . . 84 5.4 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.4.1 Cosmology with a perfect fluid . . . . . . . . . . . . . . . . . . . . . 86 5.4.2 Cosmology with a minimally coupled scalar field . . . . . . . . . . . 88 5.4.3 Deformation correspondence . . . . . . . . . . . . . . . . . . . . . . . 89 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6 Deformed gravitational action to all orders 93 6.1 Solving for the deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Linear deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The general diffeomorphism constraint 109 B.1 Diffeomorphism constraint for a scalar field . . . . . . . . . . . . . . . . . . 110 B.2 Diffeomorphism constraint for a vector . . . . . . . . . . . . . . . . . . . . . 111 B.3 Diffeomorphism constraint for a tensor . . . . . . . . . . . . . . . . . . . . . 111 B.4 Diffeomorphism constraint for a tensor density . . . . . . . . . . . . . . . . 112 C Fourth order perturbative gravitational action: Extras 113 D Deformed scalar-tensor constraint to all orders: Extras 116 vii List of Figures 3.1 Geometric scalar inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Non-minimally enhanced scalar model parameter-space . . . . . . . . . . . . 44 3.3 Non-minimally enhanced scalar inflation . . . . . . . . . . . . . . . . . . . . 46 3.4 Bouncing scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Phase space trajectories for undeformed scalar coupled to 4th order deformed gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Accelerating regions of phase space for undeformed scalar coupled to 4th order deformed gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1 The Friedmann equation for various deformation functions β(R) . . . . . . 87 5.2 Matching the Hubble expansion and canonical momentum for a cosine deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.1 Numerically solved deformation with initial function α = tanh ωK . . . . . . 99 6.2 Numerically solved deformation with initial function α = cos (ωK) . . . . . 100 viii tion and gravitational flux (h [A], F I [E]). Classically, h [A] is given by the path-ordered exponential of the connection integrated along a curve and F I [E] is the flux of the densitised triad through a surface that the curve intersects. If is taken to be infinitesimal, one can easily relate loop dynamics and connection dynamics because then h = 1 + A(˙ ) + O(\| \| 2 ) [25, p. 21]. I substitute into the distribution equation (2.41) my ansatz for a second order constraint (3.1), the diffeomorphism constraint from (B.6) and (B.11), and a zeroth order deformation β (q, ψ), parts at the level of the action does not affect the dynamics because it only eliminates boundary terms. This allows me to find the effective form of the Lagrangian, with a space-time decomposition and without second order time derivatives. I can also do this in the opposite direction to find the covariant form of the above effective Lagrangian, For the parameterisation of F ( (4) R) given in section 2.4.1, ω R = ωψ, and the transformation is given by ψ (ϕ) ∝ e ϕ √ 2/3 as long as ψ > 0. Figure 3 . 1 : 31Inflation from the geometric scalar model version of the Starobinsky model through slow-roll of the non-minimally coupled scalar field. For the scale factor, I compare the Jordan and Einstein frames because the coupling causes the former to oscillate unusually. Initial conditions, a = 1, ψ = 20,ψ = 0, M = 1. negative values of ξ, there are values of ψ which are forbidden if I am to keep my variables real, shown in Fig. 3.2. 57) and the presence of ξ in the dominant first term shows how the non-minimal coupling enhances the amount of inflation. If I compare this result to numerical solutions inFig. 3.3, I see this effect.The slow-roll approximation works less well as ξ increases. I can see this when I look atFig. 3.3(b) where I compare the slow-roll approximation to when I numerically determine the end of inflation, i.e. when = −Ḣ/H 2 = 1.I must be wary when dealing with this model, because the coupling can produce an effective potential which is not bounded from below. If I substitute the Friedmann equation (3.55a) into (3.55b) and (3.55c) I can find effective potential terms. These terms are those which do not vanish when all time derivatives are set to zero, and I can infer what bare potential they effectively behave like. If the bare potential is U = λψ 2 /2, then the effective potential Figure 3 . 3 : 33For the non-minimally enhanced scalar model with U = ψ 4 /4, (a) shows numerical solutions of inflation for different coupling strengths. Initial conditions, ψ = 20,ψ = 0, H > 0. In (b), N for ψ = 20 is compared for the numerical solutions (red crosses) and the analytical solution in the slow-roll approximation (3.57) (blue line). Figure 3 . 4 : 34As I said in subsection 3.3.1, I have taken the bounce conditions and constructed a model which bounces purely from the non-minimal coupling. This model consists of a nonminimally coupled scalar with periodic symmetry. My couplings are given by ω R = cos ψ and ω ψ = 1 + b cos ψ 1 + b , where b is some real constant, and for simplicity I ignore deformations and the minimally coupled scalar field. The bouncing scalar model Lagrangian in Cosmological bounce generated by non-minimally coupled scalar field with b = 2 and U = sin 2 (ψ/2). Initial conditions, ψ = 0,ψ = 1/25, H < 0 covariant and effective forms are given by, this chapter I have presented my calculation of the most general action for a second-order non-minimally coupled scalar-tensor model which satisfies a minimally deformed general covariance. I presented a similar calculation which involves multiple scalar fields. I showed how the magnitude of the deformation can be removed by a transformation of the lapse function, but the sign of the deformation and the associated effective signature change cannot be removed. I explored the background dynamics of the action, in particular showing the conditions required for either a big bounce or a period of slow-roll inflation. By specifying the free functions I showed how to regain well-known models from my general action. In particular I discussed the geometric scalar model, which is a parameterisation of F (4) R gravity and related to the Brans-Dicke model; and I discussed the non-minimally enhanced scalar model of a conventional scalar field with quadratic non-minimal coupling to the curvature. Firstly , I solve the distribution equation for the deformed gravitational action in section 4.1. Then I specify the variables used to construct the action and thereby find the conditions restricting its form in section 4.2. Afterwards, I progressively restrict the action when it is perturbatively expanded to fourth order in derivatives section 4.3. Finally, I investigate the cosmological consequences of the results in section 4.4. Firstly , the variables used for the action and deformation must be determined. I am considering only the spatial metric field q ab and its normal derivative v ab , and for simplicity I am only considering tensor contractions which contain up to second order in derivatives, as previously stated in section 2.4.1. The only covariant quantities I can form up to second order in derivatives from the spatial metric are the determinant q = det q ab and the Ricci curvature scalar R. The normal derivative can be split into its trace and traceless components, v ab = v T ab + 1 3 vq ab , so it can form scalars from the trace v and a variety of contractions of the traceless tensor v T ab . However, to second order I only need to consider point on the remaining equations don't provide any new conditions on the Lagrangian coefficients. to the effective Lagrangian, and varying it with respect to the scale factor, I find the Euler-Lagrange equation of motion. When I substitute in Eq. -Immirzi parameter, γ ∅ is the function which contains information about inverse-volume corrections, Σ(a,ȧ) depends on the form of γ ∅ , and f (a) is left unspecified. I just consider the case where γ ∅ = γ ∅ (a), in which case Σ = 1/ 2 √ γ ∅ and µ = a δ−1 √ γ ∅ ♦. The constant ♦ is usually interpreted as being the "area gap" derived in loop quantum gravity. I leave δ unspecified for now, because different quantisations of loop quantum cosmology give it equal to different values in the range [0, 1]. Equation (4.35) I consider the simple case of a perfect fluid. Solving the continuity equation (2.62) gives us the energy density as a function of the scale factor, ρ(a) = ρ 0 a −3(1+wρ) .(4.43) To investigate whether there can be a big bounce, I insert this into Eq. (4.42), which becomes of the form simply ignored the constant coefficients for a a . Whether a bounce happens depends on whether H → 0 when a > 0, which would happen if the higher-order correction in the modified Friedmann equation became dominant for small values of a, i.e. if Θ < 0, range of values of w ρ which can cause accelerated expansion is indeed modified. like to investigate how these semi-classical effects affect the number of e-folds of the scale factor during inflation. The number of e-folds before the end of inflation N (ϕ) is defined by a(ϕ) = a end e −N (ϕ) , where remove the explicit dependence on a from the integral by setting δ = 0 and γ ∅ = remove the explicit scale-factor dependence of the equation by setting δ = 0 and γ ∅ = 1 (the same assumptions as I used to find N ). Then substituting in the power-+ λϕ n−1 = 0, (4.60) andφ are complex. I use this equation to plot phase space trajectories in Fig. 4.1.I can find the slow-roll attractor solution for \|φϕ 1−n Fig. 4 . 41(b) and 4.1(d), I conclude that the attractor solutions diverge from a linear relationship as they approach the boundary. a) Full phase space for U (ϕ) = λϕ 2 /2 (b) Attractor solution for U (ϕ) = λϕ 2 /2 (c) Full phase space for U (ϕ) = λϕ 4 /4 (d) Attractor solution for U (ϕ) = λϕ 4 /4 Figure 4 . 1 : 41Line integral convolution plots showing trajectories in phase space for a scalar field with potential λϕ n /n with holonomy corrections. The hue at each point indicates the magnitude of the vector (φ,φ), with blue indicating low values. The trajectories do not extend outside of the region (4.61). The attractor solution (the trajectory approached by a wide range of inital conditions) is well approximated by (4.62), corresponding to slow-roll inflation. I useλ = (8π • ) (4−n)/2 ,♦ = √ 3γ BI /4, δ = 0, γ ∅ = 1. Plots are in Planck units, ω = 1/8π • (a) Accelerating values of wρ for U (ϕ) = λϕ 2 /2 (b) Accelerating values of wρ for U (ϕ) = λϕ 4 /4 Figure 4 . 2 : 42Contour plots showing the region in scalar phase space satisfying the condition for accelerated expansion when holonomy corrections are included (4.63). The dashed line indicates the classical acceleration condition w a = −1/3 and the dotted line indicates the bounce boundary. The white line indicates the slow-roll solution (4.62). The contours indicate the value of w ρ by their colour, and the most blue contour is for w ρ ≈ 0.2. I usẽ λ = (8π • ) (4−n)/2 ,♦ = √ 3γ BI /4, δ = 0, γ ∅ = 1. Plots are in Planck units, ω = 1/8π • . From the integral form of the solution (5.52), I can now check a few examples of what constraint corresponds to certain deformations. Here are a few examples of easily integrable functions with the appropriate limit, this deformed constraint to mean anything, it must not reduce to the undeformed constraint through a simple transformation. If I write the constraint as a function of the undeformed vacuum constraintC = √ q R, I see that the deformation in the constraint algebra can be absorbed by a redefinition of the lapse functions, {C[N ], C[M ]} = dxdyN (x)M (y){C(x), C(y)}, σ ∂CN x σ ∂CM y {C(x),C(y)}, (5.57c) = {C[σ ∂CN ],C[σ ∂CM ]}, (5.57d) whereN := N ∂C/∂C ,M := M ∂C/∂C and σ ∂C := sgn(∂C/∂C), because the lapse functions should remain positive. The other side of the equality, Figure 5 . 1 : 51Behaviour of the Friedmann equation for various deformation functions β(R) when k = 0. where ρ c (n) = 2 \|β ∅ \| β 2 (n + 2) . To see the behaviour of the modified Friedmann equation for different values of n, look at Fig. 5.1(a). For n > 0, the Hubble rate vanishes as the universe approaches the critical energy density, this indicates that a collapsing universe reaches a turning point at which point the repulsive effect causes a bounce. For 0 > n > −2, there appears a sudden singularity in H at finite ρ (therefore finite a). In the ρ → ∞ limit, H 2 ∼ e 2ρ when n = −2 and H 2 ∼ ρ 2n n+2 when n < −2. to find C(R) and β(R) associated with this deformation of form β(K). To do so, I need to find the relationship between the Hubble parameter H =ȧ/a and the momentum p, and thereby infer the form of β(R). Then, using (5.52) I can find the constraint C(R).So, using the equations of motion (5.61), I findh = r \|cos h\|, where, h := β k H, r := − N σ βp ω R a β k \|β ∅ \|,(5.72)this is an implicit equation which cannot be solved analytically for h(r), and so must be solved numerically.For the general relation h = r \|β(h)\|, there are similar β functions which can be transformed analytically. One example is β(h) = 1 − 4π −2 • h 2 , which also has the same limits of β(0) = 1 and β(h → ±π • /2) = 0, and can be transformed to find β(r) = 1 + 4π −2 • r 2 −1 . Figure 5 . 2 : 52Plot showing the process of starting from a deformation β(h) (a), transforming h(r) (b), finding the new form of the deformation β(r) (c), and finding the kinetic part of the constraint C k (r) (d). I include the function β = 1 − 4π −2 • h 2 (blue dashed line) because it has the same limits as β = cos h (red solid line) for the region \|h\| ≤ π • /2 but the transformation can be done analytically case). In this respect, it matches the dynamics found for exponential-form deformations in Fig. 5.1. integrate the solution for β(r) found for when β = cos(h). I plot the function C k (r) inFig. 5.2(d). Figure 5 . 3 : 53Plot showing transformations for the deformations given by β(h) = cos h 2 (red solid line) and β(h) = 1 − 4π −2 • h 4 (blue dashed line). I have shown how to obtain the cosmological equations of motion, and given a few simple examples of how they are modified. For some deformation functions, a upper bound on energy density appears, which probably generates a cosmological bounce. For other deformation functions, a sudden singularity in the expansion appears when the deformation diverges for high densities. I have shown that deformations to the field dynamics requires a background general potential against which the deformation must be contrasted. Using the cosmological equations of motion, I made contact with the holonomy-generated deformation which is a cosine of the extrinsic curvature. Through this, I have demonstrated how the relationship of momenta and extrinsic curvature becomes non-linear with a nontrivial deformation. It seems that when the deformation produces an upper bound on extrinsic curvature, there does not seem to be an upper bound on intrinsic curvature or momenta. Figure 6 . 1 : 61Numerically solved deformation function for initial function α = tanh (ωK). The numerical evolution breaks for R > ω because a discontinuity has developed. The initial function is indicated by the black line. The plots are in ω = 1 units.the correspondence α 1 = β ∅ and α 2 = ε 2 β (R) /β ∅ = ε 2 β 2 .For other initial functions, I must numerically solve the deformation. As a test, inFig. 6.1, I numerically solve for β when α = tanh (ωK). I see that, as R increases, the positive gradient in K intensifies to form a discontinuity, and softens as R decreases.I have also numerically solved for the deformation when the initial function is given by α = cos (ωK), shown inFig. 6.2. This function is motivated by loop quantum cosmology models with holonomy corrections[40][41][42]. As with the tanh numerical solution inFig. 6.1, I see the positive gradient intensify and the negative gradient soften. I could not evolve the equations past the formation of the shock wave so I cannot say for certain whether a periodicity emerges in R, but I can compare the cross sections for β inFig. 6.2(d). Figure 6 . 2 : 62Numerically solved deformation function with an initial function α = cos (ωK) and periodic boundary conditions. The numerical evolution breaks for \|R\| > ω because discontinuities have developed. The initial function is indicated by the black line. The plots are in ω = 1 units. a bcd = ∂ c Γ a db − ∂ d Γ a cb + Γ a ce Γ e db − Γ a de Γ e cb , (A.2)which are given byΓ a bc = q ad ∂ (b q c)d − 1 2 ∂ a q bc , (A.3)The variation of the Riemann tensor is given by the Palatini equation,δR a bcd = ∇ c δΓ a db − ∇ d δΓ a cb , (A.4)where the variation of the connection isδΓ a bc = q ad ∇ (b δq c)− q cd δ ef ab − q ef δ cd ab , (A.8b) Φ abcd := q ef Φ abcd ef = Q abcd − q ab q cd . (A.8c)To decompose the Riemann tensor in terms of partial derivatives, use this formula for decomposing the second covariant derivative of the variation of the metric, ∇ d ∇ c δq ab = ∂ d ∂ c δq ab + ∂ g δq ef −Γ g dc δ ef ab − 4δ bc + ∂ bc N a ∂D a ∂π ,bc + . . . ,(B.4a) L N ψ = N a ∂ a ψ, (B.4b)comparing these two equations, one can easily see that∂D a ∂π = ∂ a ψ, ∂D a ∂π ,b = 0, ∂D a ∂π ,bc = 0. (B.5) B.7b)looking at the derivative of N a , I can see that ∂D b ∂Pa,c = δ a b A c , and substituting this backinto the equation I find, ∂D b ∂Pa = δ a b ∂ c A c + ∂ b A a .If I check with F = P a I find the same equations, leading us to the diffeomorphism constraintD a = P b ∂ a A b + ∂ b (P a A a ) . (B.8)B.3 Diffeomorphism constraint for a tensorI consider a rank-2 tensor defined on a three dimensional spatial manifold q ab , p cd . I use the example of the metric, but our result is general. Test (B.2) using F = q ab ,{q ab (x), D c [N c ]} = d 3 yN c (y) δD c (y) δp ab (x) , = N c ∂D c ∂p ab − ∂ d N c ∂D c ∂p ab ,d + . . . = N c ∂D c ∂p ab − ∂ d ∂D c ∂p ab ,d + ∂ d N c − ∂D c ∂p ab ,d + . . . (B.9a) L N q ab = N c ∂ c q ab + 2q c(b ∂ a) N c , (B.9b)looking at the derivative of N a , , and substituting this back into the equation I find, ∂Dc ∂p ab = ∂ c q ab − 2∂ (a q b)c . If I check with F = p ab I find the same equations, leading us to the diffeomorphism constraint D a = p bc ∂ a q bc − 2∂ (c q b)a p bc , (B.10) and for the specific example of the metric, this reduces to D a = −2q ab ∇ c p bc . (B.11) D a = Π b 1 ···bn c 1 ···cm ∂ a Ψ c 1 ···cm b 1 ···bn − w Ψ ∂ a Π b 1 ···bn c 1 ···cm Ψ c 1 ···cm b 1 ···bn − n ∂ (b 1 Ψ c 1 ···cm b 2 ···bn)a Π b 1 ···bn c 1 ···cm + m ∂ (c 1 Π b 1 ···bn c 2 ···cm)a Ψ c 1 ···cm b 1 ···bn . deformed gravitational action to all orders, and find how intrinsic and extrinsic curvatures differently affect the deformation. I identify some of the cosmological consequences for the significant results of each chapter. There are several research questions which I attempt to answer in this thesis. How are the form of the deformation function and the form of the model related? In particular, what is the deformed scalar-tensor Hamiltonian and what is the deformed gravitational Lagrangian, using either perturbative or non-perturbative methods? How do they relate to the classical limit and to each other? How can matter fields be incorporated in deformed models? How does the deformation function depend on curvature, and is it different for intrinsic and extrinsic curvatures?deformed model for a scalar-tensor theory, establishing a classical reference point. Then in chapter 4, I derive the deformed gravitational action which includes the lowest non-trivial order of perturbative curvature corrections coming from the deformation. In chapter 5, I derive the deformed scalar-tensor constraint to all orders and I find that the momenta and space derivatives must combine in a specific form. Finally, in chapter 6, I find the By this point, I have accumulated all conditions on the form of the Lagrangian for my choice of variables. The next step is to try and consolidate them.13l) where F ∈ {v, w, R}. Take the equations (4.12) and(4.13), which solve the distribution equation for the gravitational action when I expand it in terms of the variables (q, v, w, R), and see what can be deduced about the action when it is treated non-perturbatively.Start with the equation for ∂ a F ∂ b θ ab where F ∈ {v, w, R}, (4.13h), this can be rewrittenas 0 = β ∂L ∂w 2 ∂ ∂F log β ∂L ∂w 2 , (6.1) which implies that β ∂L ∂w 2 = λ 1 (q), (6.2) 18 ) 18which means that finally I have my solution for the general action, 24 ) 246.1 Solving for the deformationThe nonlinear partial differential equation for the deformation function is an unexpected result, and invites a comparison to a very different area of physics. I can compare it to Burgers' equation for nonlinear diffusion, [92], (where u is a density function), and see that the deformation equation is very similar to the limit of vanishing viscosity η → 0. This equation is not trivial to solve because it can develop discontinuities where the equation breaks down, termed 'shock waves'. Returning to my own equation (6.23), I analyse its characteristics. It implies that there are trajectories parameterised by s given by along which β is constant. These trajectories have gradients given by,∂u ∂t + u ∂u ∂x = η ∂ 2 u ∂x 2 , (6.25) dq ds = 0, dR ds = 1, dK ds = −β (q, K, R) , (6.26) the equation referenced in the quote as the same as (2.13c) by 'spatial', I mean tangential to the spatial manifold See appendix B for information about weight.6 Defined such that δq I (x) δq I (y) = δ(x, y). It is non-zero when x a = y a , behaves as a scalar with respect to its first argument and as a scalar density with respect to its second argument. (a) (b) Chapter 4. Fourth order perturbative gravitational action where π• ≈ 3.14. a tensor is a tensor density of weight zero, which I sometimes also call weightless. If something is called a tensor density without any reference to its weight, it is probably of weight one. Appendix BThe general diffeomorphism constraint I start from the assumption that the equal-time slices of our foliation are internally diffeomorphism covariant. That is to say that spatial transformations and distortions are not deformed by the deformation of the constraint algebra. As such, the Hamiltonian constraint is susceptible to deformation and the diffeomorphism constraint is not. Therefore I need to consider what form the diffeomorphism constraint has. In the hyperspace deformation algebra (2.13), the diffeomorphism constraint forms a closed sub-algebra,(B.1)This equation shows that the diffeomorphism constraint is the generator of spatial diffeomorphisms (hence the name),Evaluating each term in the ∂ cd θ ab bracket of (4.11), by substituting in the variablesand using the equations derived for decomposing R in appendix A,(C.3c)Evaluating each term in the ∂ c θ ab bracket of (4.11),Appendix DDeformed scalar-tensor constraint to all orders: ExtrasUse the following definitions for convenience,Evaluating each term in the ∂ cd θ ab bracket of (5.7), ∂C ∂q ef,cdEvaluating each term in the ∂ c θ ab bracket of (5.7),Appendix D. Deformed scalar-tensor constraint to all orders: ExtrasEvaluating each term in the ∂ cd η ab bracket of (5.13), ∂C ∂q cd,abEvaluating each term in the ∂ c η ab bracket of (5.13),pX a + ∂ a ψ β + π ∂β ∂π . (D.5g) Measurement of the electromagnetic coupling at large momentum transfer. 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[ "Study the effect of scratching depth and ceramic-metal ratio on the scratch behavior of NbC/Nb Ceramic/Metal nano-laminates using molecular dynamics simulation and machine learning", "Study the effect of scratching depth and ceramic-metal ratio on the scratch behavior of NbC/Nb Ceramic/Metal nano-laminates using molecular dynamics simulation and machine learning" ]	[ "Mesbah Uddin \n2 590 Garden Road60115DeKalbIL\n", "Iman Salehinia ", "\nDepartment of Mechanical Engineering\nNorthern Illinois University\n\n" ]	[ "2 590 Garden Road60115DeKalbIL", "Department of Mechanical Engineering\nNorthern Illinois University\n" ]	[]	Developing a new class of coating materials is necessary to meet the increasing demands of energy and defense-related technologies, aerospace engineering, and harsh environmental conditions. Functional-based coatings, such as ceramic-metal nanolaminates, have gained popularity due to their ability to be customized according to specific requirements. To design and develop advanced coatings with the necessary functionalities, it is crucial to understand the effects of various parameters on the mechanical and tribological properties of these coatings. In this study, we investigate the impact of penetration depth, individual layer thickness, and ceramic-metal ratio on the mechanical and tribological properties of ceramic-metal nanolaminates, particularly NbC/Nb. Our findings reveal that the thickness of the individual metallic and ceramic layers significantly affects the coatings' properties. However, some models exhibited punctures on the top ceramic layer, which altered the scratching behavior and reduced the impact of layer thickness on it. This is because the top ceramic layer's thickness is too low, and the indenter can easily puncture it instead of pushing the ceramic atoms. The minimum thickness required to resist indentation is called the critical thickness, which depends on the indentation size and penetration depth. In the latter part of this paper, we employed 1 Corresponding author, Email: [email protected] machine learning to reduce computational costs, and the model predicts the friction coefficient with an R-squared value of 0.958.	null	[ "https://export.arxiv.org/pdf/2305.03698v2.pdf" ]	258,546,896	2305.03698	c33a99bb53403e40fe80cb10be66e7ced1bb8877	Study the effect of scratching depth and ceramic-metal ratio on the scratch behavior of NbC/Nb Ceramic/Metal nano-laminates using molecular dynamics simulation and machine learning Mesbah Uddin 2 590 Garden Road60115DeKalbIL Iman Salehinia Department of Mechanical Engineering Northern Illinois University Study the effect of scratching depth and ceramic-metal ratio on the scratch behavior of NbC/Nb Ceramic/Metal nano-laminates using molecular dynamics simulation and machine learning Scratch behaviorceramic/metal nano-laminate coatingfriction coefficientmaterial removalmetal thicknessmolecular dynamicsmachine learning Developing a new class of coating materials is necessary to meet the increasing demands of energy and defense-related technologies, aerospace engineering, and harsh environmental conditions. Functional-based coatings, such as ceramic-metal nanolaminates, have gained popularity due to their ability to be customized according to specific requirements. To design and develop advanced coatings with the necessary functionalities, it is crucial to understand the effects of various parameters on the mechanical and tribological properties of these coatings. In this study, we investigate the impact of penetration depth, individual layer thickness, and ceramic-metal ratio on the mechanical and tribological properties of ceramic-metal nanolaminates, particularly NbC/Nb. Our findings reveal that the thickness of the individual metallic and ceramic layers significantly affects the coatings' properties. However, some models exhibited punctures on the top ceramic layer, which altered the scratching behavior and reduced the impact of layer thickness on it. This is because the top ceramic layer's thickness is too low, and the indenter can easily puncture it instead of pushing the ceramic atoms. The minimum thickness required to resist indentation is called the critical thickness, which depends on the indentation size and penetration depth. In the latter part of this paper, we employed 1 Corresponding author, Email: [email protected] machine learning to reduce computational costs, and the model predicts the friction coefficient with an R-squared value of 0.958. INTRODUCTION A functionally based coating material is a type of composite material that has varying compositions in different areas depending on the performance requirements. Ceramic-metal nanolaminates are a promising option for this type of coating due to their unique properties. By changing the combination of different geometric parameters of ceramic-metal nanolaminates, we can create coatings with different functionalities. These materials offer exceptional properties such as high strength, hardness, and wear resistance [1][2][3][4]. They can also be used as advanced materials for various applications, including engine parts, medical implants, the automotive industry, aerospace, and defense industry. Scratching of CMLs can occur due to external forces acting on the material, such as during manufacturing or application, or due to wear and tear during service. The scratching process involves the interaction of the tip and the surface, resulting in deformation and material removal. Scratching can lead to various surface defects, including cracks, delamination, and wear. Understanding the scratching behavior of CMNs is important for improving their performance in practical applications. In recent years, computation tools have been used for different studies [5][6][7][8][9][10][11][12][13][14][15], i.e., molecular dynamics (MD) simulations have emerged as a powerful tool for investigating the scratching behavior of CMNs at the atomic scale . The scratching behavior of CMLs is influenced by several factors. The effect of those various factors such as interface type [41,45], scratching speed [45,46,47,48], scratching direction [45,49], temperature [45,50], indenter size [51,52], shape [45,53], scratching depth [41,49,52,54,55] on the mechanical and tribological behavior have been investigated. These studies have reported a signification effect of these parameters on the scratching behavior. The current study on scratching behavior of ceramic/metal multilayer materials at the atomistic size scale has received limited attention, particularly with regards to the effect of scratching depth as most of the previous studies are done in nanoscale. Understanding the impact of scratching depth on the mechanical response of these materials is crucial for realistic interpretation of computational results in materials design, particularly in applications beyond the nanoscale. However, the high computational cost associated with molecular dynamics atomistic simulations often causes researchers to overlook the effects of penetration depth, leading to an incomplete understanding of the mechanical behavior of materials at different scales. Furthermore, the impact of design parameters, such as individual layer thickness, on scratching behavior has not been thoroughly explored. Given that both individual layer thickness and ceramic-metal ratio significantly influence the mechanical behavior of nanolaminates, it is crucial to control these factors during fabrication for tribological applications. In a previous study, we investigated the impact of individual layer thickness on the scratching behavior of NbC/Nb multilayers. However, the scratching only penetrated 3nm, indicating that only the first few layers of ceramic and a small section of the metallic layer were affected. Therefore, this study did not provide a comprehensive understanding of the effects of individual layer thickness. This academic paper builds upon our prior research and employs molecular dynamics atomistic simulations to investigate how scratch depth impacts the scratching behavior of NbC/Nb multilayer samples with different metal thicknesses. In addition, the individual layer thickness dependency on the scratching behavior was captured. In addition, molecular dynamics simulation cost is very expensive and time consuming. Recently machine learning has become a popular method to predict different features [63]. In the last section of this paper, a machine-learning approach has been used to predict the friction coefficient of different models of ceramic-metal nanolaminate. The models were subjected to periodic boundary conditions on their side faces, with the top surface left unrestricted and the bottom surface fixed to prevent any rigid movements of the model while undergoing indentation. A rigid spherical nano-indenter was used with a set penetration depth of 7 nm to ensure adequate penetration of the metallic layers. One should note that for 7 nm penetration depth, the indenter radius cannot be 5 nm, therefore, the indenter radius was kept at 10 nm. The indenter speed was set to 100 m/s for nano-indentation and 250 m/s for nano-scratching, with a scratching length of 20 MODELING nm. Prior to equilibration, an energy minimization using conjugate gradient method was applied. Dynamic relaxation was then used to equilibrate the models at 10 K and 0 pressure using the NPT (isothermal-isobaric) ensemble for 10 ps. The simulation used a microcanonical ensemble during nano-indentation and nano-scratching, while temperature was maintained at 10 K to better observe deformation mechanisms. RESULTS AND DISCUSSION The effect of penetration depth on the scratching behavior of NbC/Nb laminates was discussed here. Figure 1 shows the removed material volume for different models. It is shown that both CMNLs models produce a high volume of removed material at higher penetration depth. This is in contrast to our study at 3 nm penetration depth where the simulations only showed material removal for the model with the thinnest metallic layers, i.e., 22. In addition, the CMNL model with thicker metallic layers displayed the highest removed material volume at 7 nm penetration depth, followed by the CMNLI and NbC single crystal models. To comprehend this behavior, it is necessary to examine the atomic snapshot during scratching. Figure 14 presents the perspective and front views of the scratched surface at the end of a 20 nm scratching length for both models at 7 nm penetration depth. Figure 15 (a, b). However, the rate of increase in scratching load is more pronounced than the increasing rate of the normal load. This behavior is in good agreement with reported data [52] and it is linked to the fact that within the increase of penetration c) d) Puncture Puncture a) b) depth, both normal and transverse area increase but the increase of transverse contact area is more rapid. However, the normal load of CMNLII, 2826 at higher penetration depth is continuously decreasing and then it became saturated. This is mainly the indenter directly contacts with soft Nb metal due to the puncture at the top NbC (Figure 14), therefore, decreasing the normal force. However, it is noticeable that the normal load is very high compared to the scratching load for both models. Therefore, the normal load dominates the friction properties when the penetration depth is deeper, leading to a higher friction coefficient for the CMNLII model with the thickest metallic layer. To understand the effect of the metal-ceramic ratio on the scratching behavior more detailed, more simulations were carried out for multilayers having the same bi-layer thickness but with different individual layer thicknesses. Figure 6 shows the removed Molecular dynamics simulation cost is very expensive and time consuming. Therefore, we aimed to predict the frictional coefficient using a machine-learning approach based on a dataset of 35 unique simulation models. If we could generate scratching load, normal load and friction coefficient by molecular dynamics simulation, that will allow us to reduce the computational cost and we can study any different model. identified important features, further investigation is needed to understand the underlying mechanisms that govern the relationship between these features and the frictional coefficient. Despite these limitations, our study demonstrates the potential of using machine learning to gain insights into material design and frictional behavior. Thus, we can reduce the computational cost of molecular dynamics simulation. In future study, we will improve our model and predict scratching, normal, friction coefficient and will validate different models data obtained from the molecular dynamics simulation. CONCLUSION Molecular dynamics atomistic simulations were performed to investigate the scratching behavior of NbC/Nb ceramic/metal nanolaminates with various metal thicknesses. Three models with different metal thicknesses were considered and their tribological behavior was compared to that of NbC and Nb single crystals. Nano-scratching was simulated by penetrating and moving a spherical indenter into the CMNLs. The results showed a significant effect of the metal thickness on the mechanical response of the models. The material removal rates of the CMNL models were lower than those for NbC and Nb single crystals proving the improving effect of the alternating metallic and ceramic layers on the scratch behavior of the hybrid materials. The scratching response of the models was linked to the underlying deformation mechanisms during the scratching process. The strain hardening of the metallic layers and their compliance was introduced as the significant factors affecting the scratching behavior of the models. LAMMPS software was utilized to conduct molecular dynamics simulations with the second nearest neighbor modified embedded atom method (2NN-MEAM) interatomic potentials for NbC and Nb. These potentials have been previously employed by Salehinia et al. to examine the deformation mechanisms of NbC/Nb under uniform compression and nano-indentation. Additionally, the potential was able to accurately reproduce various physical properties of metals and ceramics, including lattice parameters, enthalpy of formation, elastic properties, surface energies, and interface energies. The interface of NbC/Nb was found to adopt the Baker-Nutting orientation relationship (OR), where the (001) planes of Nb and NbC single crystals form the interface, and the [100]crystallographic direction of NbC is parallel to the [110] direction of Nb[59]. Fig. 1 - 1The simulation cell used for nanoindentation and scratching. The layers are numbered 1-4 from top to bottom. Only the thickness of layer 2 was varied. The utilized multilayer samples in this study consisting of alternating layers of ceramic and metal, with a minimum of four layers and a fixed 2 nm thickness for the ceramic layers and 6 nm thickness for the bottom metal layer to replicate the behavior of ceramic/metal nanolaminates, with the bottom metal layer acting as an elastic foundation for the layers above. The width of each multilayer was chosen to minimize coherency strains and boundary effects, and periodic boundary conditions were applied to the side faces. The thickness of the first metallic layer was varied to investigate its effect on the mechanical and tribological behavior of the samples. The top ceramic and metal layer thickness were varied and used in this study. For ease of visualization, all layers except the top ceramic layer have been removed. The images demonstrate that the indenter punctured the top ceramic layer for both models, which is caused by the larger attack angle, defined as the acute angle between the tangent of the semi-sphere indenter at the contact radius and the model's top surface, increasing with penetration depth. Once the ceramic layer is punctured, atoms easily pile up around the indenter, regardless of the metal-ceramic ratio, resulting in a large amount of material removal in both models. Figure 1 : 1Comparison of material removed volume at different penetration depth Figure 2 : 2Perspective (a, c) and front view (b, d) of scratched surface at end of the scratching distance for CMNLI, 2226 (a, c) and CMNLII, 2826 (b, d). Atoms are colored based on the width for perspective view. Figure 3 3compares the scratching load, normal load, and friction coefficient at different penetration depths. It can be seen from Figure 15 that all tribological properties are significantly affected by the penetration depth. The scratching load and normal load of Model 22 become increasingly higher when the scratch is performed at deeper penetration depth, as shown in Figure 3 : 3The variations of a) scratching load, b) normal load and c) friction coefficient at different model As pointed out in the introduction, the majority of the available literature has reported the scratching behavior of materials for extremely low penetration depth. To understand the scratching behaviors at high penetration depth, CMNL models were also compared with the single crystal Nb and NbC. Fig. 4 4shows the variation of the friction coefficient with the scratching distance for all the models including multilayers and single crystals at the 7nm penetration depth. The NbC single crystal exhibits the lowest friction coefficient. However, CMNLII model with the thickest metallic layer shows the highest friction properties, which also contradicts our previous study at 3nm penetration depth. To explain this behavior, one the scratching and normal load are observed.Fig. 5a and bshow the variation of the scratching (friction) load and the normal load with the scratching distance, respectively, for all models at higher penetration depth. The scratching load and normal load for NbC single crystal are still the Figure 4 : 4The variations of friction coefficient of different model at penetration depth 7 nm. Figure 5 : 5The variations of a) scratching load, and b) normal load of different model at penetration depth 7 nm. material for each model. The contibution of individual ceramic and mettalic layers on total removed volume was also counted. It is shown that the removed volume are almost the same for all models except CMNL,2826 and CMNL,1926. That indicates the low dependency of pile-up atoms to the metal-ceramic thickness ratio at higher penetration depth. When the penetration depth is higher, all models produce a high amount of material removal, no matter the metal/ ceramic ratio. However, the CMNL,2826 and CMNL, 1926show a very high amount of removed volume of material than the other models. Also, while only ceramic layer contributes to the total removed volume for other models, both metallic and ceramic layer contribute to the material removed volume for CMNL,1926, CMNL, 2826 models. Fig. 6 -Figure 7 : 67Removed volume of the different model at the scratching length 20 nm To understand this observed behavior, the atomic snapshot at the end of scratching was observed for each model. Figure 7 shows the perspective view of the top ceramic layer for CMNL,1926, CMNL, 2826 and CMNL,3726. For the sake of better comparison, only those models are kept which have a puncture during the scratching. Here, only CMNL,1926 and CMNL,2826 show the puncture at top ceramic layer during the scratching, but it is not seen in the other models. The reason for being punctured in only CMNL,1926 and CMNL,2826 models is the low thickness of the top ceramic layer. The thicknesses of the top ceramic layer in the CMNL,1926, CMNL, 2826 are too thin that the indenter can easily puncture the top ceramic during the indentation. Once the puncture initiates in the ceramic layer, it is very easy for both the top ceramic and metallic atoms to be piled up, creating a significant amount of pile-up atoms. Therefore, some contribution to the material removed volume from the metallic layer are seen for these two models as shown in Figure 18. With the increase of ceramic layer thickness, the number of ceramic atoms become enough to resist the indenter to puncture the top ceramic layer. The least thickness of the ceramic layer that can resist the indenter making a puncture can be defined as a critical thickness. For this specific study, the critical thickness is between 2 and 3 nm for indenter radius 10 nm at 7 nm penetration depth. This critical thickness depends on the indenter radius and penetration depth. For a constant indenter size, the critical thickness increases with the increase of penetration depth while it decreases with the indenter radius. Perspective view of scratched surface at end of the scratching distance showing puncture. a) CMNL,1926, b) CMNL,2826, c) CMNL, 3726. Figure 8 Figure 8 : 88shows the variation of scratching load, normal load, and friction coefficient with the scratching distance. The obtained results show similar behavior to our study at penetration depth 7 nm. The scratching load and normal load are still highest for CMNL models with the thinnest metallic layer as like our previous study. As the rate of variations of the normal load is more rapid, the normal load dominates the friction properties, leading to a higher friction coefficient for the model with the thickest The variations of a) scratching load, b) normal load and c) Figure 9 : 9The comparison of friction coefficient (Molecular Dynamics Simulation vs Machine Learning)Figure 9suggests that our machine learning model could accurately predict the frictional coefficient, capturing the underlying relationships between the input features and the target variable. The model identified several important features that contributed to predicting the frictional coefficient. These features were primarily related to the simulation models' depth, radius, and geometry. By leveraging machine learning, we can optimize the material design and identify the proper material strength, leading to more efficient and effective systems in various applications. However, there are some limitations to our approach. Firstly, our dataset is relatively small, with only 35 simulation models. This may limit the model's ability to generalize to a broader range of scenarios. Secondly, although our model has Figure Captions List Fig. 1 - List1The simulation cell used for nanoindentation and scratching. The layers are numbered 1-4 from top to bottom. Only the thickness of layer 2 was varied. Fig. 2 - 2Load-displacement curves for CMNL models. Table 1 - 1The details of the considered models.Layer 1 Layer 2 Layer 3 Layer 4 Number of atoms Material Ceramic Metal Ceramic Metal CMNLI, 2226 (22) 2 nm 2 nm 2 nm 6 nm 648584 highest followed by CMNLI,2226, CMNLII,2826, and Nb single crystal which is good agreement with our previous study []. The variation in tangential force observed in CMNL (Ceramic Matrix Nanocomposite) models can be attributed to the Nb layer situated beneath the top ceramic layer, given that the top layer remains constant across all models. It has been noted that the C2/M2 models exhibit a thinner metallic layer which leads to a higher resistance against dislocation propagation, thus increasing the friction force compared to the C2/M8 model. This can be attributed to the greater difficulty in nucleating dislocations within the metallic layer of C2/M2 models. In contrast, the C2/M8 model showcases dislocations with greater freedom of movement within the metallic layer, owing to the greater distance between opposing interfaces, resulting in a lower friction force. These mechanisms contribute to the difference in friction force between the two models.2226, d-3nm 2226, d-7nm 2826, d-7nm 2826, d-3nm 0 1000 2000 3000 4000 5000 6000 0 50 100 150 200 Normal Load, nN Scractching Distance, (Å) 2226,d-3nm 2226,d-7nm 2826,d-7nm 2826,d-3nm strength for various applications. The dataset was divided into training, validation, and testing sets, with 23 models (70%) used for training, five models (15%) for validation, and the remaining seven models (15%) for testing. We generated features from the simulated model's intended radius, depth, and geometry to build our predictive model. After training the model, we obtained a validation Mean Squared Error (MSE) of approximately 0.00098, indicating that the model performed well on the validation data. When evaluating the model on the test data, we found the following scores:By leveraging machine learning, we sought to enhance our understanding of material design and identify the proper material Mean Squared Error: 0.0030 Root Mean Squared Error: 0.055 Mean Absolute Error: 0.045 R-squared: 0.958 ACKNOWLEDGMENTWe would like to thank the Center for Research Computing & Data at Northern IllinoisUniversity for their support and providing us the access to their computer cluster, GAEA. 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[ "Error correction and extraction in request dialogs", "Error correction and extraction in request dialogs" ]	[ "Stefan Constantin stefan.constantin\|[email protected] \nInstitute for Anthropomatics and Robotics\nKarlsruhe Institute of Technology\n\n", "Alex Waibel \nInstitute for Anthropomatics and Robotics\nKarlsruhe Institute of Technology\n\n" ]	[ "Institute for Anthropomatics and Robotics\nKarlsruhe Institute of Technology\n", "Institute for Anthropomatics and Robotics\nKarlsruhe Institute of Technology\n" ]	[]	We propose a dialog system utility component that gets the two last utterances of a user and can detect whether the last utterance is an error correction of the second last utterance. If yes, it corrects the second last utterance according to the error correction in the last utterance. In addition, the proposed component outputs the extracted pairs of reparandum and repair entity. This component offers two advantages, learning the concept of corrections to avoid collecting corrections for every new domain and extracting reparandum and repair pairs, which offers the possibility to learn out of it.Related WorkStudies have been conducted in the area of interactive repair dialog. In(Suhm et al., 1996)a multimodal approach is used.	null	[ "https://export.arxiv.org/pdf/2004.04243v3.pdf" ]	215,548,425	2004.04243	9b4c1bb84df6b21cb2bf31cbb6b31f94f0af16b2	Error correction and extraction in request dialogs Stefan Constantin stefan.constantin\|[email protected] Institute for Anthropomatics and Robotics Karlsruhe Institute of Technology Alex Waibel Institute for Anthropomatics and Robotics Karlsruhe Institute of Technology Error correction and extraction in request dialogs We propose a dialog system utility component that gets the two last utterances of a user and can detect whether the last utterance is an error correction of the second last utterance. If yes, it corrects the second last utterance according to the error correction in the last utterance. In addition, the proposed component outputs the extracted pairs of reparandum and repair entity. This component offers two advantages, learning the concept of corrections to avoid collecting corrections for every new domain and extracting reparandum and repair pairs, which offers the possibility to learn out of it.Related WorkStudies have been conducted in the area of interactive repair dialog. In(Suhm et al., 1996)a multimodal approach is used. For the error correction one sequence labeling and two sequence to sequence approaches are presented. For the error correction detection these three error correction approaches can also be used and in addition, we present a sequence classification approach. One error correction detection and one error correction approach can be combined to a pipeline or the error correction approaches can be trained and used end-to-end to avoid two components. We modified the EPIC-KITCHENS-100 dataset to evaluate the approaches for correcting entity phrases in request dialogs. For error correction detection and correction, we got an accuracy of 96.40 % on synthetic validation data and an accuracy of 77.85 % on human-created real-world test data. Introduction Errors and ambiguities are difficult to avoid in a dialog. Corrections allow to recover from errors and to disambiguate ambiguities. For example, a household robot gets the request "Put the cleaned spoons into the cutlery drawer", but the robot does not know which one of the drawers is the cutlery drawer. It can choose one of the drawers and puts the spoons there. If its choice is wrong, the user must correct the robot, e. g. "No, into the drawer right of the sink". Alternatively, the robot can ask which one of the drawers is the cutlery drawer. The clarification response of the user, e. g. "It's the drawer right of the sink", is also a correction because the response disambiguates the ambiguity. Another type of correction occurs when the user changes their mind, e. g. "I changed my mind, the forks", or when the system misunderstands the user request (e. g. because of automatic speech recognition or natural language understanding errors). All these correction types can be processed in the same manner and therefore we propose a component that gets a request and a correction and outputs a corrected request. To get this corrected request, the phrases in the correction phrase replace their corresponding phrases in the request. In this work, we restrict on entity phrases like "drawer right of the sink". To replace other phrases like verb phrases is out of scope for this work. The request "Put the cleaned spoons into the cutlery drawer" with its correction "No, into the drawer right of the sink" is converted to "Put the cleaned spoons into the drawer right of the sink". Such a component has two advantages compared to handling the corrections in the actual dialog component. First, it reduces the amount of required training data for the actual dialog component because corrections will not need to be learned if there is an open-domain correction component. Second, this kind of correction component can be extended so that it outputs the extracted pairs of reparandum and repair entity. In our example there is one pair: "cutlery drawer" and "drawer right of the sink". These entity pairs can be used, for example, for learning in a life-long learning component of a dialog system to reduce the need for correction in future dialogs, e. g. the robot can learn which one of the drawers is the cutlery drawer. wrong phrases and respeak or spell the correct phrase, or choose from alternatives in the n-best list of the automatic speech recognition component, or use handwriting to write the correct phrase. These error strategies are improved in (Suhm and Waibel, 1997) by considering the context. In (Suhm et al., 1999(Suhm et al., , 2001 the previous approaches are evaluated in more detail in a dictation system with real users. Different human strategies for error correction are presented in (Gieselmann, 2006). Sagawa et al. (2004) propose an error handling component based on correction grammars. These correction grammars have the advantage that they can be used domain-independently. However, they need a grammar based dialog system. An error correction detection module and strategies to handle the detected errors are proposed by Griol and Molina (2016). The corrected request must be handled by the Spoken Language Understanding component. That means, for every domain the Spoken Language Understanding component must be adapted to the possible corrections. Kraljevski and Hirschfeld (2017) propose a domain-independent correction detection by checking the speech for hyperarticulation. Other features than hyperarticulation are not used. In (Béchet and Favre, 2013), a system is presented that detects errors in automatic speech recognition transcripts and asks the user for a correction. There are also studies that research automatic error correction without user interaction. In (Xie et al., 2016) a character-based approach to correct language errors is used. They used a characterbased approach to avoid out-of-vocabulary words because of orthographic errors. In (Weng et al., 2020), the authors used a multi-task setup to correct the automatic speech recognition outputs and do the natural language understanding. The task of request correction presented in the introduction is related to the task of disfluency removal. In disfluency removal, there are the reparandum (which entity should be replaced), the interruption point (where the correction begins), the interregnum (which phrase is the signal phrase for the correction), and the repair phrase (the correct entity) (Shriberg, 1994). In Figure 1, a disfluent utterance annotated with this terminology is depicted. spoon into the drawer reparandum interruption pt. uh interregnum sink repair Figure 1: disfluent utterance annotated with repair terminology A lot of work has been conducted for disfluency removal (Cho et al., 2014;Dong et al., 2019;Wang et al., 2016;Jamshid Lou et al., 2018). In all these works, it is assumed that it is enough to delete tokens of the disfluent utterance to get a fluent utterance. A disfluent utterance with the copy and delete labels is depicted in Figure 2. spoons into the drawer uh sink C C C D D C Figure 2: disfluent utterance labeled with copy and delete labels However, in the task of corrections, longdistance replacements can occur. That means, that between the reparandum and the repair are words that are important and must not be deleted. Such a long-distance replacement is depicted in Figure 3. Dataset Our dataset is based on the annotations in natural language of the EPIC-KITCHENS-100 dataset (Damen et al., 2020(Damen et al., , 2022. The EPIC-KITCHENS-100 dataset comprises 100 hours of recordings of actions in a kitchen environment. An example annotation of such an action is "put pizza slice into container" and the corresponding verb is "put-into" and the corresponding entities are "slice:pizza" and "container". Annotations in this dataset have one verb and zero to six entities. The verb, the corresponding verb class, the entities and the corresponding entity classes are explicitly saved to every annotation. The order of the entities and the corresponding entity classes is the same as in the annotation. If the verb has a preposition, the verb is saved including its preposition. The words of the entities are represented in a hierarchy. The most general word of the hierarchy is left and the words are more specialized the further to the right of the hierarchy. The words of each hierarchy are separated by a colon. There are 67 218 annotations in the training and 9669 annotations in the validation dataset of the EPIC-KITCHENS-100 dataset. There is no test dataset. Some annotations occur multiple times, because different recordings of the 100 hours recordings have the same annotation. By considering only the unique annotations, 15 968 annotations are in the training and 3835 annotations are in the validation dataset. For our dataset, we used only the annotations that have one or two entities. We excluded the annotations with no entities because we need at least an entity that can be corrected. Annotations including more than two entities amount only to less than 1.15 % of all annotations and therefore we decided to exclude them because of dataset balancing reasons. The verb classes of the EPIC-KITCHENS-100 datasets are imbalanced. To get a better balance in the validation dataset, we removed annotations of verb classes that occur very often from the validation dataset. We wanted a more balanced dataset to evaluate whether the model gets along with very different verb classes. We calculated the number of desired remaining annotations of a verb class, called r, by dividing the number of annotations, called a, by 100, but we determined a minimal number of remaining annotations of verb classes: 2 for one entity annotations (r = max(2, a/100)) and 4 for two entity annotations (r = max(4, a/100)). In some cases, there are less than the desired remaining annotations of a verb class in the EPIC-KITCHENS-100 dataset. We then used the possible number. We chose the values for minimal examples to get a nearly balanced dataset: 142 annotations with one entity and 122 annotations with two entities. To get the annotations of a verb class, we chose the verbs occurring in a verb class equally distributed. In total, we have 264 annotations in the reduced validation dataset. The number of unique annotations in respect to the verb class before and after the reduction are depicted in Figure 4. In the EPIC-KITCHENS-100 dataset, the training and validation datasets are similar: all 78 verb classes of the validation dataset occur in the training dataset and 346 of the 372 first level word of the entity hierarchies of the validation dataset occur in the training dataset. Because of this, we decided to reduce the training dataset to have more difference To use these annotations for training and evaluating the error correction detection and correction component, we had to add corrections to the annotations. For the training and validation dataset, we generated the corrections synthetically. There are three options for the entitiy replacement: the first entity should be replaced, the second entity should be replaced, or both entities should be replaced. We drew uniformly distributed which of these three options should be applied. If both entities should be corrected, we drew uniformly distributed in which order they should be corrected. For the training and validation dataset, we had 8 and 6, respectively, templates to introduce the correction phrase, followed by the corrected entities. An entity could be replaced by an entity that occurs in an annotation of the same verb class in the same position. An example for one corrected entity is "Be so kind and pick the oregano" for the request and "it's the chilli" for the correction and an example for two corrected entities is "Could you put the tin in the Cupboard?" for the request and "no the olives in the Fridge" for the correction. For the test dataset, we had nine human data collectors who could freely write the corrections, they only knew what entities should be replaced with what other entities (but were allowed to use synonyms for the other entities) and whether the correction should be a correction to a wrong action of the robot, a clarification, or a correction because the user changed their mind (equally distributed). We added 19 and 14 templates before the narration to increase the variety of the natural language of the training and validation dataset, respectively. In the EPIC-KITCHENS-100 dataset, the articles of the entities are missing, therefore we added a "the" before the entities. For the test dataset, we used the narrations of our validation dataset and let the same nine annotators that created the corrections for our test dataset paraphrase them. The test dataset is more challenging than the validation dataset because it differs even more from the training dataset. The nine data collectors were told to use a large variety of natural language. We used the 4822 annotations of the reduced training dataset to generate with the different data augmentations 52 357 request and correction pairs for the error correction training dataset. The error correction validation dataset has 264 request and correction pairs and the error correction test dataset has 960 request and correction pairs. To train and evaluate the error correction detection, we need examples where the last utterance is no correction. To achieve this, the second last and the last utterance are made of all the requests of the error correction data. The requests were shuffled for the last utterance. This approach doubled the number of examples to the correction examples, that means, we have 104 714 pairs in the error correction detection and error correction training dataset, 528 pairs in the error correction detection and error correction validation dataset, and 1920 pairs in the error correction detection and error correction test dataset. The target for the error correction datasets is the corrected request and the reparandum repair pairs and the target for the error correction detection and error correction dataset depends whether the source has a request and correction pair or a request and request pair. In the first case, there is an error correction and the target is the same as in the error correction datasets, in the second case, the target is to copy both requests. There is a further dataset, the error correction detection dataset. The sources are the same as in the error correction detection and error correction dataset but the target is the binary value whether there is a correction or not. We created the described datasets in different forms for the different approaches. For the sequence labeling approach, we labeled the source tokens with different labels, see Figure 5 and Section 4 for an explanation of the labels. For the sequence to sequence approach with generative token generation, we created source and target pairs, see Figure 6. For the sequence to sequence approach with generation by copying source tokens, we added the order of copy operations. Additionally, the separator tokens that are needed in the target will be inserted to the source, see Figure 7. would C it C be C possible C to C wash C the C Models For the error correction and extraction, we developed three different approaches. The first approach is a sequence labeling approach, the second approach is a sequence to sequence approach where the output tokens are sampled from a fixed vocabulary, and the third approach is a sequence to sequence approach where output tokens are copied from the source tokens. For the sequence labeling approach, every word is labeled with one of the following labels: C (copy), D (delete), R1 (entity 1 potentially to be replaced), R2 (entity 2 potentially to be replaced), S1 (entity to replace entity 1), or S2 (entity to replace entity 2). For the correction target, the S1 and S2 labeled entities are used to replace the R1 and R2 labeled entities, respectively. For the extraction target, the output is the pairs R1 and S1 as well as R2 and S2 if there is a replacement available for the first or second entity, respectively. In Figure 8, an example request and correction pair is labeled and both targets are given. For the sequence labeling, we propose finetuning the cased BERT large model (24 Transformer encoder blocks, hidden size of 1024, 16 self-attention heads, and 340 million parameters) (Devlin et al., 2019). For the sequence to sequence approach where the output tokens are sampled from a fixed vocabulary, we propose fine-tuning a T5 large model (Raffel et al., 2020). The T5 model is a pre-trained Transformer network (Vaswani et al., 2017) and the T5 large model has the following properties: 24 Transformer encoder blocks, 24 Transformer decoder blocks, hidden size of 1024 (in-and output) and 4096 (inner-layer), 16 self-attention heads, 737 million parameters. The probability distribution over the fixed vocabulary V can be calculated in the following way: P generate (V ) = sof tmax(dec T · W generate ) where dec is the output of the Transformer decoder and W generate ∈ R hidden size decoder×vocabulary size is a learnable matrix. We call this T5 model T5 generate. In the corrected request there are only tokens of the input sequence. To utilize this property, we developed a pointer network model (Vinyals et al., 2015) with the T5 large model that calculates which input token has the highest probability to be copied to the output sequence. This is our third approach. The probability distribution over the input sequence tokens V can be calculated in the following way: P copy (V ) = sof tmax(dec T · enc T ) where dec is the output of the Transformer decoder and enc ∈ R source input length×embedding size . To utilize the knowledge of the pre-trained model, we feed the source input token with the highest probability into the encoder instead of the position of the source input token. That means, that in the generation stage the copy mechanism is only used, otherwise it is like a normal T5 model. To be able to output the separators, we add this to the source, so that they can also be copied. We call this modificated T5 model T5 copy. To decide whether an utterance is a correction for the previous request command, the described three approaches can also be used. If all output labels of the sequence labeling approach are C, no error correction is detected, otherwise there is an error correction. The sequence to sequence approaches detect an error correction if the source and the target without the separators are not equal, otherwise there is no error correction. In the T5 copy approach, the source for the comparison is the original source and not the source with the inserted separators. In addition to these three approaches, a sequence classification can also be used for the error correction detection. For the sequence classification, we propose to fine-tune the cased BERT large model (24 Transformer encoder blocks, hidden size of 1024, 16 self-attention heads, and 340 million parameters) (Devlin et al., 2019). Implementation We used the HuggingFace (Wolf et al., 2020) Pytorch (Paszke et al., 2019) BERT and T5 models for our implementations of the models described in Section 4 and published our implementations and request \|\| correction put the milk into the shelf no the soja milk into the left shelf labels C R1 R1 R2 R2 R2 D S1 S1 S1 S2 S2 S2 S2 corrected request put the soja milk into the left shelf pairs of reparandum and repair entity milk → soja milk -into the shelf → into the left shelf Evaluation In this section, we will first evaluate the different error correction detection component approaches described in Section 4. After that, the error correction component approaches described in Section 4 are evaluated. Third, we will compare whether it is better to separate the error correction detection and error correction in separate components and use a pipeline approach or whether an end-to-end approach is better. For all evaluations, we used the datasets described in Section 3. We fine-tuned the sequence classification and labeling approaches one epoch with the following hyperparameters: AdamW optimizer (Loshchilov and Hutter, 2019) with learning rate of 2 · 10 −5 , batch size of 32 and maximum input length of 128. The T5 generate and T5 copy models were finetuned one epoch with the following hyperparameters: Adam optimizer (Kingma and Ba, 2015) with learning rate of 2.5 · 10 −4 , batch size of 24 and a maximum input length of 12; in the embedding layer, the first two encoder blocks were frozen. The results of the error correction detection components are depicted in Table 1. Accuracy means how many examples were classified correctly, precision is how many of the positive classified examples are really positive, recall how many of the positive examples are found by the component and the F 1 -score is the harmonic mean of the precision and recall. We calculated the precision, recall and F 1 -score for the case that detecting corrections were the positive examples and for the case that detecting no corrections were the positive examples to get better insights in the quality of the differently trained models. The sequence classification approach was trained with the error correction detection dataset and the other approaches were trained with the error correction detection and error cor-1 https://github.com/msc42/ seq2seq-transformer https://github.com/ msc42/seq-labeling-and-classification rection dataset. The best approach is the sequence labeling approach (if all words have the copy label C, it is no error correction, otherwise it is an error correction). It has an accuracy of 100 % for the validation and 88.49 % for the test dataset. The recall for detecting no corrections is 99.90 % and the precision 81.34 % (F 1 -score 89.67 %) in the test dataset. That means, if there is no correction, the component detects it in most of the cases and make no unnecessary correction. This is a good property, because it is better not detecting a correction than correcting something which is already correct. The error correction detection and error correction component should improve the overall system and not make it worse. Nevertheless, the results for detecting corrections with a recall of 77.08 % and a precision of 99.87 % (F 1 -score 87.01 %) in the test dataset are good. In some cases where the component fails, it is really difficult to detect the correction like in "Kindly turn off the heat on the oven \| Please turn off the water tap on the oven". The classification approach has similar results to the sequence labeling approach: 100 % accuracy for the validation dataset and 87.86 % for the test dataset. This approach also prefers detecting no corrections over corrections. The T5 generate approach is worse. It has an accuracy of 98.67 % on the validation dataset and an accuracy of 84.01 % on the test dataset. The worst results are from the T5 copy approach (71.78 % and 77.45 % validation and test dataset accuracy, respectively). The results of the error correction components are depicted in Table 2. We evaluated the error correction with the metric accuracy. The correction is correct if the predicted correction and the reference correction are the same. The extraction of the reparandum and repair pairs is correct if the predicted pairs are equal to the reference pairs. The order and entities that map to themselves are ignored. Both are correct if the correction as well as the extraction are correct. For this evaluation the error correction datasets are used. On the validation dataset, the sequence labeling approach that Table 3: evaluation results of the error correction detection and error correction (metric accuracy), the end-to-end (E2E) models were trained on the error correction detection and error correction dataset and the other models were trained on the error correction dataset, "and" means that the error correction detection was done by the best error correction detection model (sequence labeling) and the error correction detection by the model mentioned after the "and" if a correction was detected was trained on the error correction detection and error correction datasets has the best overall accuracy (95.08 %). The accuracy for the correction is 96.59 % and for the extraction 95.08 %. On the test dataset, the T5 generate approach that is trained on the error correction dataset has the best accuracy (71.98 %). In general, all approaches trained on the error correction detection and error correction dataset have a higher accuracy on the validation dataset and all approaches trained on the error correction dataset have a higher accuracy on the test dataset. The T5 copy extraction could be optimized by bookkeeping the order of copy operations, stopping after finishing the correction and use the bookkeeping to reconstruct the reparandum and repair pairs. We relinquished this optimization because the correction results were much worse and we did not see any sense in further optimizations that will only lead to minimal improvements. The results of the error correction detection and error correction components are depicted in Table 3. We used the same metric accuracy as in the error correction evaluation. For the error correction detection in the pipeline approach, we used the best error correction detection model evaluated in this section. It is the sequence labeling approach where no correction is in the example if all labels are C. After the error correction detection, the error correction will occur. We evaluated all three approaches described in Section 4 in their version trained on the error correction detection and error correction dataset and their version trained on the error correction dataset. In the end-to-end setting, a component executes the error correction detection and the error correction in one run. The best approach is the pipeline approach with the T5 generate approach only trained on the error correction dataset with an accuracy of 96.40 % on the validation and 77.85 % accuracy on the test dataset. The evaluation results show that the test dataset is more challenging than the validation dataset. The nine data collectors were able to introduce even more variety of natural language than the validation dataset has. Conclusions and Further Work The proposed error correction detection and error correction component shows high potential. For the validation dataset, we got very good results: in 96.40 % of the cases, we could detect whether there is a correction or not and if there is a cor-rection, it outputs a correct corrected request and could extract correctly the reparandum and repair pairs. The results for the human-generated realworld data with 77.85 % shows that the proposed component is learning the concept of corrections and can be developed to be used as an upstream component to avoid the need for collecting data for request corrections for every new domain. In addition, the extraction of the pairs of reparandum and repair entity can be used for learning in a life-long learning component of a dialog system to reduce the need for correction in future dialogs. In future work, the training dataset could be extended to a bigger variety of natural language which will enable the model to learn the concept of corrections better and to get better results on human-generated real-world data. The mentioned life-long learning component could also be part of future work and the classification of correction types could improve the performance of such a lifelong learning component. To improve the accuracy, architectures that have a better NER performance than our used BERT model, like the architecture proposed by (Baevski et al., 2019), could be used. A further future research goal is to be able to correct all phrases and not only entity phrases. Figure 3 : 3request and correction phrase annotated with repair terminology Figure 4 : 4unique annotations in respect to the verb class before and after the reduction of the EPIC-KITCHENS-100 validation dataset between them. We removed the verb classes of the 49 less frequent occurring verb classes (in total 98 verb classes are in the training dataset) from the training dataset and removed all entities from the training dataset when its first part was also in the validation dataset. That means, if bowl:washing:up was in the validation dataset, an annotation with bowl:salad in the training dataset was removed. After the reduction 4822 annotations were left in the training dataset. Figure 6 :Figure 7 : 67sequence to sequence with fixed vocabulary data example source file: Would it be possible to wash the table ? \| no the Wok instead of the table . sequence to sequence with copy source token approach data example Figure 8 8Figure 8: error correction example source file: Would it be possible to wash the table ? \| no the Wok instead of the table . target file: Would it be possible to wash the Wok ? \| table -> Woktable R1 ? C \| D no D the D wok S1 instead D of D the D table D . D Figure 5: sequence labeling data example Table 2 : 2evaluation results of the error correction (metric accuracy), the end-to-end (E2E) models were trained on the error correction detection and error correction dataset and the other models were trained on the error correction datasetmodel(s) validation dataset test dataset correction extraction both correction extraction both detection and seq. labeling 98.11 % 97.35 % 97.35 % 67.12 % 68.42 % 66.75 % detection and E2E seq. labeling 98.30 % 97.54 % 97.54 % 69.62 % 71.76 % 69.31 % E2E seq. labeling 98.30 % 97.54 % 97.54 % 69.58 % 71.77 % 69.27 % detection and T5 generate 96.40 % 98.67 % 96.40 % 78.48 % 79.89 % 77.85 % detection and E2E T5 generate 98.11 % 98.30 % 98.11 % 68.32 % 68.79 % 67.64 % E2E T5 generate 97.54 % 97.16 % 96.40 % 68.07 % 68.49 % 66.88 % detection and T5 copy 75.19 % 94.13 % 75.00 % 68.47 % 72.75 % 66.96 % detection and E2E T5 copy 85.42 % 96.59 % 84.66 % 63.31 % 66.81 % 62.12 % E2E T5 copy 69.70 % 78.03 % 56.25 % 55.00 % 58.91 % 47.40 % AcknowledgementsThis work has been supported by the German Federal Ministry of Education and Research (BMBF) under the project OML (01IS18040A). 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[ "Composition of Terrestrial Exoplanet Atmospheres from Meteorite Outgassing Experiments", "Composition of Terrestrial Exoplanet Atmospheres from Meteorite Outgassing Experiments" ]	[ "Maggie A Thompson \nDepartment of Astronomy and Astrophysics\nUniversity of California Santa Cruz\n95064Santa CruzCA\n", "Myriam Telus \nEarth and Planetary Sciences\nUniversity of California Santa Cruz\n95064Santa CruzCA\n", "Laura Schaefer \nGeological Sciences\nSchool of Earth, Energy, and Environmental Sciences\nStanford University\n94305StanfordCA\n", "Jonathan J Fortney \nDepartment of Astronomy and Astrophysics\nUniversity of California Santa Cruz\n95064Santa CruzCA\n", "Toyanath Joshi \nDepartment of Physics\nUniversity of California Santa Cruz\n95064Santa CruzCA\n", "David Lederman \nDepartment of Physics\nUniversity of California Santa Cruz\n95064Santa CruzCA\n" ]	[ "Department of Astronomy and Astrophysics\nUniversity of California Santa Cruz\n95064Santa CruzCA", "Earth and Planetary Sciences\nUniversity of California Santa Cruz\n95064Santa CruzCA", "Geological Sciences\nSchool of Earth, Energy, and Environmental Sciences\nStanford University\n94305StanfordCA", "Department of Astronomy and Astrophysics\nUniversity of California Santa Cruz\n95064Santa CruzCA", "Department of Physics\nUniversity of California Santa Cruz\n95064Santa CruzCA", "Department of Physics\nUniversity of California Santa Cruz\n95064Santa CruzCA" ]	[]	Terrestrial exoplanets likely form initial atmospheres through outgassing during and after accretion, although there is currently no first-principles understanding of how to connect a planet's bulk composition to its early atmospheric properties. Important insights into this connection can be gained by assaying meteorites, representative samples of planetary building blocks. We perform laboratory outgassing experiments that use a mass spectrometer to measure the abundances of volatiles released when meteorite samples are heated to 1200 • C. We find that outgassing from three carbonaceous chondrite samples consistently produce H 2 O-rich (averaged ∼66 %) atmospheres but with significant amounts of CO (∼18 %) and CO 2 (∼15 %) as well as smaller quantities of H 2 and H 2 S (up to 1%). These results provide experimental constraints on the initial chemical composition in theoretical models of terrestrial planet atmospheres, supplying abundances for principal gas species as a function of temperature.	10.1038/s41550-021-01338-8	[ "https://arxiv.org/pdf/2104.08360v1.pdf" ]	233,295,957	2104.08360	7c85f7c116c099595da4b41570e82e4af943e074	Composition of Terrestrial Exoplanet Atmospheres from Meteorite Outgassing Experiments April 20, 2021 16 Apr 2021 Maggie A Thompson Department of Astronomy and Astrophysics University of California Santa Cruz 95064Santa CruzCA Myriam Telus Earth and Planetary Sciences University of California Santa Cruz 95064Santa CruzCA Laura Schaefer Geological Sciences School of Earth, Energy, and Environmental Sciences Stanford University 94305StanfordCA Jonathan J Fortney Department of Astronomy and Astrophysics University of California Santa Cruz 95064Santa CruzCA Toyanath Joshi Department of Physics University of California Santa Cruz 95064Santa CruzCA David Lederman Department of Physics University of California Santa Cruz 95064Santa CruzCA Composition of Terrestrial Exoplanet Atmospheres from Meteorite Outgassing Experiments April 20, 2021 16 Apr 2021Email: [email protected] 1 Terrestrial exoplanets likely form initial atmospheres through outgassing during and after accretion, although there is currently no first-principles understanding of how to connect a planet's bulk composition to its early atmospheric properties. Important insights into this connection can be gained by assaying meteorites, representative samples of planetary building blocks. We perform laboratory outgassing experiments that use a mass spectrometer to measure the abundances of volatiles released when meteorite samples are heated to 1200 • C. We find that outgassing from three carbonaceous chondrite samples consistently produce H 2 O-rich (averaged ∼66 %) atmospheres but with significant amounts of CO (∼18 %) and CO 2 (∼15 %) as well as smaller quantities of H 2 and H 2 S (up to 1%). These results provide experimental constraints on the initial chemical composition in theoretical models of terrestrial planet atmospheres, supplying abundances for principal gas species as a function of temperature. We are at the dawn of an exciting technological era in astronomy with new large-aperture telescopes and advanced instrumentation, both in space and on the ground, leading to major advances in exoplanet characterization. To optimize the use of these new facilities, we need suitable theoretical models to obtain a better understanding of the diversity of exoplanet atmospheres. Statistical studies using NASA's Kepler mission data suggest that terrestrial and other low-mass planets are common around G, K and M stars [1,2,3]. Given the large number of current and anticipated low-mass exoplanet discoveries, the next phase in exoplanet science is to characterize the physics and chemistry of their atmospheres. For the foreseeable future, Solar System meteorites provide the only direct samples that can be rigorously studied in the laboratory to gain insight into the initial atmospheric compositions of these planets. Although gas giant planets, like Jupiter and Saturn, form primary atmospheres by capturing gases from the stellar nebula, atmosphere formation for low-mass planets is more complicated. While nebular ingassing can contribute to early atmosphere formation if a protoplanet accretes enough mass before the gas disk dissipates, several factors can result in loss of nebular volatiles early in the planet's history [4,5,6]. For instance, terrestrial planets' inability to retain significant primary atmospheres can be due to low planetary mass, large impactors and high EUV and X-ray flux from young host stars [7]. Instead, low-mass planets likely have secondary atmospheres which form via outgassing of volatiles during and after planetary accretion [8]. The Solar System's terrestrial planets are believed to have formed by accretion of planetesimals that have compositions similar to chondritic meteorites, which are a likely source of atmospheric volatiles for such planets [9,7]. As a result, an important step towards establishing the connection between terrestrial planets' bulk compositions and their atmospheres is to directly measure the outgassed volatiles from meteorites. While meteorites come in a wide variety with a range of volatile contents, they can be classified into three main types: chondrites, achondrites and irons. Chondrites are stony meteorites that come from undifferentiated planetesimals composed of aggregate material from the protoplanetary disk, while achondrites and iron meteorites have melted and derive from partially or fully differentiated planetesimals. Both chondrites and achondrites likely contributed to forming the Sun's terrestrial planets [7]. Our study focuses on CM-type carbonaceous chondrites because their compositions provide a strong match to that of the solar photosphere, second only to CI-type chondrites, so they serve as representative samples of the bulk composition of material in the solar nebula during planet formation. While planet formation alters planetesimals through various thermal and differentiation processes, carbonaceous chondrite-like material was a likely source of volatiles for the terrestrial planets, and CM chondrites are among the most volatile-rich of remnant materials, making these samples well-suited for studying initial outgassed atmospheres [10,11]. Planetary outgassing has been modeled both for the Solar System's terrestrial planets and for some low-mass exoplanets. Many prior studies have focused on outgassing during a planet's magma ocean phase and Earth's early degassing history, with several models investigating the outgassing composition from meteorites (e.g., [12,7,13,14,15]). Currently, however, there is limited experimental data to constrain these models and, in particular, none to inform meteorite outgassing studies. Prior experiments that heated meteorites were limited in several key ways due to restrictions in the number and type of samples used, the temperatures to which the samples where heated, and the number of gas species that were accurately measured (e.g., [16,17,18,19,20]), while this study measures many of the dominant outgassing species across a temperature range relevant for terrestrial planet atmospheres for multiple meteorite samples. In addition, some studies shocked samples prior to analyzing their volatile species so they do not properly simulate outgassing conditions from bulk chondritic material (e.g., [21]), while other studies focused on either a small subset of volatiles or trace metals and other volatile elements that are not major constituents of rocky planet atmospheres (e.g., [22,16,20], see "Comparison with Prior Studies" in Methods). As a result, these studies are unsuitable for validating outgassing models for terrestrial planets. Therefore, to inform the initial composition of terrestrial planet atmospheres, we have designed an experimental procedure to directly measure by mass spectrometry a large set of the major outgassed species (e.g., H 2 O, CO 2 ) from diverse meteorites over a wide temperature range. For this study, three chondrites are analyzed: Murchison, Jbilet Winselwan and Aguas Zarcas. Murchison was observed to fall in Australia in 1969 [23]; Jbilet Winselwan was collected in Western Sahara in 2013 [24]; and Aguas Zarcas fell in Costa Rica in 2019 [25]. We minimized terrestrial contamination and weathering effects by ensuring that none of our samples have fusion crust, which is altered during atmospheric entry; by using two fall samples which are minimally altered by terrestrial contamination; and by significantly reducing the majority of adsorbed species on the samples (see Methods). Our experimental set-up consists of a furnace connected to a residual gas analyzer (RGA) mass spectrometer and a vacuum system. This system heats samples at controlled rates (up to 1200 • C) in a high-vacuum environment (∼ 10 −4 Pa (10 −9 bar) at lower temperatures and ∼ 10 −3 Pa (10 −8 bar) at higher temperatures) and measures the partial pressures of up to 10 volatile species made up of hydrogen, carbon, nitrogen, oxygen and sulfur. For each of our experiments, a sample was heated in an open crucible (and thereby open to mass loss) from 200 to 1200 • C (475-1475 K) at a rate of 3.3 • C/minute and the partial pressures of the outgassing molecular species were continuously monitored using the RGA. The results from these experiments are expressed in three major forms. The first two are the instantaneous partial pressures (p i , for species i) and mole fractions (χ i = p i p Total , for species i where p Total = i p i , and see Methods (e.g., Equation 3) for elemental mole fractions) of outgassed volatiles as a function of temperature. The third is relative abundances of outgassed volatiles from each sample, reported as partial pressures normalized to the total pressure of released gases summed over temperature, P i,Total = T p i T p Total , for species i, and for elemental abundances P j,Total = T p j j ( T p j ) , for element j (i.e., H, C, O, N, and S). The RGA measures the partial pressures of 10 selected species according to their molecular masses, assuming species are singly ionized (i.e., the mass-to-charge ratio equals the molecular mass): 2 amu (H 2 ), 12 amu (carbon), 14 amu (nitrogen), etc. The signals for each of the species tracked during the outgassing experiments have been corrected for ion fragments and the possibility of terrestrial atmospheric adsorption and contamination using a set of linear equations. This approach also accounts for the background signal (Extended Data Figure 1). In addition, since the masses of several molecules overlap, we developed a method to address these degeneracies (see Methods). An alternative approach to correct for ion fragments using a least squares regression produces generally similar results (see Methods, Supplementary Table 3 and Extended Data Figure 2). Results Tables 1 and 2 report the relative abundances of outgassed volatile species and elements from the three chondrites. We analyzed samples of Jbilet Winselwan twice under identical conditions to test the reproducibility of our experimental procedure, so its final reported relative abundances are given by the mean and 95% confidence interval of the mean of the two analyses (see Methods and Extended Data Figure 3). As shown in Table 1, we find that H 2 O has the largest relative abundance (∼66±11 %) for all the meteorite samples followed by CO (∼18±8 %), CO 2 (∼15±5 %), and H 2 (∼1±1 %) (see Extended Data Figure 4). The signal at 34 amu has a lower relative abundance while the signals at 12, 14, 16, 32, and 40 amu along with N 2 have relative abundances that are effectively 0 (see Methods). In terms of the elemental abundances in Table 2, hydrogen has the highest concentration (∼48±5 %) followed by oxygen (∼41±2 %), carbon (∼12±4 %), and sulfur (∼0.03±0.02 %), averaged across all three samples (uncertainties reported as the 95% confidence interval). We expect the three CM chondrite samples to have similar outgassed abundances given their similar bulk compositions which are within 20 mg/g for most volatile elements (Supplementary Table 2). Our experimental results confirm this prediction with the relative abundances for each species across the three samples being within 3σ of each other. Figure 1 shows the mole fractions of the measured volatile species as a function of temperature from each of the three meteorite samples. Several differences to note between the three meteorites are that Murchison has larger water but smaller CO outgassed abundances compared to Winselwan and Aguas Zarcas. In addition, Aguas Zarcas has a larger CO abundance but smaller CO 2 abundance than Murchison and Winselwan. Figure 2 shows the relative abundance ratios for the three samples as a function of temperature, which can inform the chemistry of the initial atmospheres. The mean outgassed carbon-to-oxygen, hydrogen-tocarbon, sulfur-to-oxygen and hydrogen-to-oxygen ratios summed over temperature for the three chondrite samples are 0.29±0.08, 4.15±1.80, 0.0008±0.0006, and 1.18±0.18, respec-tively (uncertainties reported as the 95% confidence interval), with abundance ratios for the three chondrites within 2σ of each other. The average C/O, H/C, S/O and H/O ratios for the initial bulk CM chondrite composition are 0.06±0.01, 7.09±1.18, 0.04±0.01, and 0.45±0.01, respectively [26,27]. These initial bulk elemental abundances represent the volatile outgassing inventory for a planet that is outgassing predominantly CM chondrite-like material ( Table 2). By comparing the outgassed and bulk CM chondrite abundance ratios, we find that, aside from the S/O ratios, the outgassed and bulk ratios are within an order of magnitude of each other. The C/O and H/O outgassed ratios are larger than the bulk ratios, which is likely due to the fact that many of the phases that host carbon and hydrogen (e.g., phyllosilicates, organics and carbonates) readily break down upon heating whereas a significant portion of the oxygen is in phases that do not easily decompose (e.g., forsterite and CaO). On the other hand, the H/C and S/O ratios are smaller than the bulk ratios, with the largest difference between the outgassed and bulk S/O ratios. This may be due to the fact that models predict that S 2 and SO 2 should also outgas in this temperature range but we only measure H 2 S due to the RGA's 10-species limit. There are several major similarities between the results from our experiments ( Figure 3 (b, d)) and those from the equilibrium calculations ( Figure 3 (a, c)). For instance, water is the dominant outgassed species over almost the entire temperature range for both experimental and theoretical methods, with the average mole fraction being 0.6 and 0.4 for the experiments and theoretical models, respectively. In addition, CO and CO 2 constitute significant fractions of the vapor phase over the temperature range. In particular, for the Murchison sample, the experimental and theoretical outgassing trends for CO 2 and CO generally match with CO 2 outgassing more at the lowest (∼300 • C) and highest temperatures (∼1100 • C), and CO outgassing more at intermediate temperatures (∼800-1000 • C). In addition to the experimental results, we calculated 'equilibrium-adjusted' abundances using the equilibrium model to re-compute gas speciation based on the experimental abundances at intervals of 50 • C (dashed curves Figure 3 (b, d)). Generally, the 'equilibrium-adjusted' experimental H 2 O and CO 2 abundances provide a better match to the equilibrium model results, and the CO abundance provides a better match at higher temperatures. On the other hand, the other significant outgassing species (H 2 and H 2 S) did not exhibit an improved match compared to the experimental results, although the equilibrium-adjusted H 2 abundance is much larger compared to the experimental H 2 . Although our experiments monitor signals at 12 amu (carbon), 14 amu (nitrogen), 16 amu (CH 4 /O) and 40 amu (Ar), once we correct for ion fragments and atmosphere adsorption, we do not detect significant amounts of these species, also matching chemical equilibrium calculations (see Methods). There are also significant differences between the experimental and theoretical results. For instance, although H 2 S mole fractions reach similar maxima of ∼1E-3, the peak is displaced from ∼600 • C in the theoretical calculations to ∼950 • C in the experiments. This offset may be due to the fact that, in carbonaceous chondrites, sulfur can occur as gypsum (a sulfate mineral, CaSO 4 H 2 O) which breaks down at 700 • C and the corresponding phase change for sulfur gas may be kinetically inhibited, causing it to outgas at higher temperatures [28]. Preliminary X-ray diffraction (XRD) analyses carried out in our lab indicate that gypsum may be breaking down during the experiments (see Methods section on solid phases). However, the equilibrium models do not show gypsum being in a solid phase, which may explain the mismatch in outgassing trends of H 2 S. In addition, iron sulfide (FeS) and tochilinite (2Fe 0.9 S1.67Fe(OH) 2 ), which are known to be in carbonaceous chondrites, may decompose and contribute to the outgassed H 2 S [29, 30]. The 'equilibrium-adjusted' H 2 S abundances are much lower than both the model and experimental results, further supporting H 2 S production being kinetically inhibited. Another significant difference is the experiments' absence of N 2 gas which does not match equilibrium models that show outgassing over the entire temperature range. The primary reason the experimental N 2 abundance is negligible is due to the atmospheric adsorption correction we apply to account for the possibility of contaminated N 2 gas from Earth's atmosphere being adsorbed by the samples (see Methods). Without this correction, the measured N 2 outgassing varies from moderate levels consistent with theoretical models (i.e., average mole fractions of ∼5E-2) to negligible amounts, depending on the sample being investigated. Several other differences between experimental outgassing and equilibrium model results lack complete explanations. For example, the second most abundant volatile species predicted to outgas over most temperatures in equilibrium models is H 2 with maxima mole fractions of ∼0.4, but our experimental results indicate an order of magnitude lower average H 2 mole fractions near ∼0.03. While the cause for much lower experimental H 2 abundance is uncertain, it may be due to the fact that our experiments do not allow sufficient time for some gas-gas reactions to take place that could raise the H 2 abundance, as equilibrium is reached between H 2 O and H 2 . The 'equilibrium-adjusted' H 2 abundances are larger than the experimental results, pointing to a likely disequilibrium for gas phase reactions in the experiments. By comparing gas-gas reaction rates to the experiment's vacuum pumping rate, we conclude that these species likely do not have sufficient time to reach chemical equilibrium (see Methods). In addition, the H 2 background signals are large relative to the samples' signals, so over-subtraction of the background signal could also explain the lower abundance. Finally, our experiments do not detect a significant amount of outgassed species with mass number 32 amu after correcting for the possibility of contaminated terrestrial atmospheric adsorption of O 2 (see Methods). In the chemical equilibrium calculations, O 2 should only begin outgassing significantly at the highest temperatures (∼1100 • C). As described further in Methods, an alternative data analysis technique using least-squares fitting produces nonzero yields of CH 4 , which is not predicted to outgas based on equilibrium models (Extended Data Figure 2, Supplementary Table 3). It is possible that our original analysis (see Equations 4-15 in Methods) applies an overly conservative correction to CH 4 's abundance, and further investigation is required to confirm if CH 4 is indeed outgassing from the samples. Oxygen fugacity (f O 2 ) represents the chemical potential of oxygen in a system which affects its gas chemistry and may be equated to the partial pressure of oxygen in a gas phase under low pressure and near-ideal gas conditions such as present in these experiments. Our instrument set-up does not allow equilibrium to be achieved, especially at lower temperatures, because many gas species do not have sufficient time to interact since their mean collision time is either comparable to or longer than the vacuum system's evacuation rate. Therefore, it is not surprising that the experimental f O 2 curves from H 2 O/H 2 and CO 2 /CO do not agree in magnitude, except at the highest temperatures, revealing that the gas phase is likely not in equilibrium (see Figure 4). Discussion The results from our outgassing experiments have several important implications for the initial atmospheric chemistry of low-mass exoplanets. While terrestrial planets experience a diversity of conditions during planet formation, these experiments represent an empiricallydetermined boundary condition of the less-theoretically-explored lower temperature/pressure paths that could be used to test outgassing models. Our experimental conditions approximately simulate the initial heating phase during planet formation, revealing the initial volatile species that would outgas assuming the bulk composition of material being outgassed is CM chondrite-like. As such low-mass planets form their initial atmospheres via outgassing during accretion, H 2 O-rich steam atmospheres form. These atmospheres will also likely contain significant amounts of CO and CO 2 and smaller amounts of H 2 and H 2 S. Until now, several common assumptions used for low-mass exoplanet atmospheric modeling include ad hoc abundances such as H 2 O-only or CO 2 -only, solar abundances (dominated by H 2 and He), or atmospheric compositions of Solar System planets [31,32,33,34,35]. Our outgassing experiments suggest initial atmospheres may differ significantly from many of the common assumptions in atmosphere models, and provide the basis for more refined future models of terrestrial planets' initial atmospheres (see "Comparison with Prior Studies" in Methods). In proposing that our results be used as initial conditions for terrestrial exoplanet atmosphere models, we note that while our experiments cover a wide range of temperatures, they were conducted in a low-pressure environment. Schaefer & Fegley 2010 [14] suggest that varying pressure does not have a significant effect on the major gas composition of outgassed atmospheres from CM chondrite material. Chemical equilibrium or kinetics calculations can determine how these sets of initial compositions from outgassing experiments evolve as the pressure varies within a planet's atmosphere. Theoretical models of terrestrial planet atmosphere formation usually lack experimental constraints. Based on the results from our experiments, the most direct way to improve such models of outgassing from CM chondritic-like building blocks would be to assume initial abundances at planetary surface boundaries averaging approximately 66% H 2 O, 18% CO and 15% CO 2 over a temperature range similar to that used in this study. Depending on the capabilities of a particular model, temperature dependencies for species abundances could also be incorporated using our experimental results. Additional improvements could include the Langmuir coefficients for evaporation (and condensation) for the major outgassing species measured in our experiments to properly simulate chemical kinetics effects. Determining these coefficients often requires conducting Langmuir evaporation experiments [36]. In addition, reaction rates between the solid and gas phases relevant for chondrite outgassing would improve our understanding of when chemical equilibrium conditions are acceptable and when chemical kinetics effects are important. Subsequent experiments will determine whether other species predicted by chemical equilibrium models also outgas significantly. In addition, future experiments will measure volatiles from a wider variety of meteorites including ordinary and enstatite chondrites. Tables Table 1: Total mass of volatile species released (in g) and relative abundances (in %) of outgassed species summed over temperature for the three CM chondrite samples. Abundances are in partial pressures normalized to the total pressure of all released gases measured summed over temperature and are reported as percentages. The species corresponding to each mass number is in parentheses. The uncertainties reported for Jbilet Winselwan are the 95% confidence intervals of the means. Since some species have overlapping mass numbers (e.g., S and O 2 ), we provide a detailed description of the calculations made in determining these relative abundances in Methods. Murchison Methods Sample Preparation & Experimental Procedure CM chondrite samples were powdered with an agate mortar and pestle and sieved so that only material between 20 and 106 µm in diameter was analyzed to ensure homogeneous samples. Powdered samples were stored in a desiccator under vacuum to minimize terrestrial contamination. For each heating experiment, ∼3 mg of powdered sample was evenly distributed into a 6.5 × 4.0 mm 2 alumina crucible, as shown in Supplementary Figure 1. This sample size was chosen because larger sample sizes saturate the RGA. Prior to assaying each sample, we first heated an empty small crucible and the larger 50 × 20 × 20 mm 3 combustion boat in the tube furnace to allow impurities, particles in the tube and adsorbed volatiles to degas which otherwise could interfere with our measurements. The heating procedure to bake-out the tube and sample containers consisted of five steps: (1) heating from room temperature to 200 • C over 40 minutes, (2) holding at 200 • C for 30 minutes, (3) heating from 200 • C to 1200 • C over 5 hours, (4) holding at 1200 • C for 5 hours, and (5) cooling the system to room temperature over 5 hours. After one of the bake-outs, we calibrated the background signal by performing a similar heating procedure on empty sample containers except for step (4) in which the time at 1200 • C is reduced to 10 minutes. The RGA mass spectrometer used in this study operates inside the vacuum chamber and ionizes gas molecules according to their molecular masses (up to 100 atomic mass units (amu)) and measures their partial pressures. Since an RGA is commonly used for detecting low-levels of contamination in vacuum systems, its sensitivity to trace amounts of gas makes it ideal for carrying out this study [37]. The experimental procedure for each sample is identical to that used to determine the background signal. We chose the heating rate of 3.3 • C/min because it is similar to those in prior studies using mass spectrometers to monitor released gases from meteorite samples (e.g., [17], [38]). We also note that the combination of very low pressure and high vacuum pumping rate precludes many gas-gas reactions or volatile phase changes in the experimental system. At the high-temperature end of the experiments, some gas-gas reaction rates approach or exceed the vacuum pumping rate suggesting that some gas species, but not all, may approach equilibrium. Data Calibration: Ion Fragmentation, Terrestrial Atmospheric Adsorption & Background Subtraction Corrections The RGA's ionizer can cause different types of ions to be produced from a single species of gas molecule due to processes such as molecular fragmentation [37]. The mass spectrum of each molecule has contributions from all ion fragments formed from that molecule and they define the molecule's fragmentation pattern. In residual gas analysis, standard fragment patterns of common atoms and molecules are well established. Supplementary Table 1 explains, for a given gas species whose mass number is analyzed during the experiments, the percentage each of its known ion fragments contributes to the intensity relative to the major peak due to that gas species itself. These fragment patterns were determined from the NIST Mass Spectrometry Data Center [39]. To correct for ion fragments for each species, we subtract its partial pressure from the partial pressures of other species that contribute to its mass signal weighted by the percentage of the other species's contribution (see . For example, the partial pressure of H 2 is given by: p H 2 = p 2amu − (0.02 * p H 2 O ).(1) We also correct for the possibility of terrestrial atmospheric adsorption onto the samples. We assume that the signal at 40 amu is due entirely to atmospheric argon adsorbed onto the samples (see the section on degeneracies below). Given the composition of Earth's atmosphere (78% N 2 , 21% O 2 , 1% Ar), we determine the amount of atmospheric N 2 and O 2 . After correcting for ion fragments, we subtract the atmospheric N 2 and O 2 contributions from the signals due to N 2 and 32 amu (see Equations 8,12 and 14). We also subtract the ion fragments of atmospheric N 2 and O 2 from the signals at 14 and 16 amu (see Equations 6 and 9). The step-wise heating procedure allows us to disentangle terrestrial weathering and contamination from the actual volatile composition of our samples [40]. In the heating procedure, we hold each sample at 200 • C for 30 minutes which helps eliminate any adsorbed water or nitrogen that is not intrinsic to the sample. Although we conduct each heating experiment under high-vacuum conditions (∼ 10 −4 Pa), slight contamination may still be possible. Therefore, to properly calibrate the background signal, we conducted an additional experiment following the same procedure used for the empty sample containers (see Extended Data Figure 1). The partial pressures during this background measurement are corrected for ion fragments and terrestrial atmospheric adsorption and then serve as the background pressures which are subtracted from the ion fragment-corrected and atmospheric adsorption-corrected partial pressures during sample heating to determine the fully calibrated (i.e., ion fragmentcorrected, atmospheric adsorption-corrected and background-subtracted) partial pressures (i.e., for species i, p i = p i,heatingp i,background ). The total background pressure averages ∼6E-4 Pa, and the dominant background species, H 2 O, has an average partial pressure of ∼5E-4 Pa, both of which are ∼1.5 times lower than their corresponding sample values. The total pressure of the system at each temperature step is given by: p Total = i p i,heating − i p i,background(2) Supplementary Figure 2 shows the variations in total pressure with temperature for the samples. To calculate the mole fraction (χ i ) of a gas species (atomic or molecular) at each temperature step, we divide its background-subtracted and ion fragment-corrected partial pressure by the total pressure, χ i = p i /p Total . For the elemental mole fractions of hydrogen, carbon, oxygen, nitrogen, and sulfur at each temperature step, we sum the mole fraction of each gas species containing the element of interest multiplied by the number of atoms of that element in the species, and divide by a normalization factor. For example, for hydrogen: χ H = 2χ H 2 O + 4χ CH 4 + 2χ H 2 + 2χ H 2 S Norm(3) where Norm is the normalization factor that ensures that the elemental mole fractions sum to unity and is given by χ H + χ C + χ O + χ N + χ S . The reported relative abundance of a given species i is its partial pressure summed over temperature and normalized to the total pressure of the released gases also summed over temperature (P i, Total = T p i / T p Total ), and is expressed as a percent (see Extended Data Figure 4). The relative abundance of a given element j is its partial pressure, determined the same way as Equation 3 except using partial pressures instead of mole fractions, summed over temperature and normalized to the sum of the pressures of all elements measured in the experiments also summed over temperature (P j, Total = T p j / T ( j p j )). Calculations to Determine Gas Species' Partial Pressures Several of the mass numbers analyzed for this study correspond to multiple gas species (e.g., 28 amu corresponds to CO and N 2 ). Bulk composition measurements of the samples, measurements of the other masses, and melting/evaporation temperatures for the different species allow us to disentangle which species dominate the signal and, in some cases, distinguish between different gas species' signals that correspond to the same mass number. Equations 4-15 below show the calculations to determine the partial pressures of different volatile species by accounting for ionization fragmentation, disentangling some of the species with overlapping mass numbers and correcting for atmospheric adsorption: p H 2 = p 2 amu − 0.02p H 2 O (4) p H 2 O = 1.04p 18 amu (5) p CH 4 = 1.25p 16 amu − 0.10p CO 2 − 0.015p H 2 O − 4.96p 40 amu(6) p N 2 , pre-atmosphere correction = (p 14 amu − 0.21p CH 4 )/0.14 (7) p N 2 = 1.14p N 2 , pre-atmosphere correction − (83.96p 40 amu ) (8) p N = p 14 amu − 0.21p CH 4 − 0.14p N 2(9) p CO = 1.07(p 28 amu − p N 2 , pre-atmosphere correction ) − 0.10p CO 2 (10) p C = p 12 amu − 0.09p CO 2 − 0.05p CO − 0.04p CH 4(11) p Ar = p 40 amu (All due to atmospheric adsorption) (14) p CO 2 = 1.29p 44 amu(15) In Equation 6, we account for the fact that the signal at 16 amu can be due to ion fragments of CO 2 and H 2 O. We also account for the fact that contaminated O 2 due to atmospheric adsorption has an ion fragment at 16 amu (see "16 and 32 amu" section in "Degeneracies" section). Although 16 amu can also be due to ion fragments of CO, we do not account for them because 16 amu only contributes 2% to CO. Equations 7-10 explain how we disentangle the signals due to CO and N 2 given that they have the same mass number (28 amu). In Equation 7, we first account for the fact that the signal at 14 amu can be due to ion fragments of CH 4 . We assume that the remaining signal at 14 amu is due entirely to atomic nitrogen which is an ion fragment of N 2 , and we use it to determine the partial pressure of N 2 . In Equation 8, we account for contaminated N 2 due to atmospheric adsorption by assuming all of the 40 amu signal is due to terrestrial atmospheric argon and using its signal and the known composition of Earth's atmosphere to determine the amount of contaminated N 2 . In Equation 9, we determine the partial pressure of atomic nitrogen which is zero since we assume all of it was an ion fragment of N 2 . In Equation 10, we determine the signal due to CO by subtracting the total amount of N 2 from the signal at 28 amu and also accounting for the fact that this signal can be an ion fragment of CO 2 . Equation 11 determines the partial pressure of atomic carbon and accounts for the various ion fragments at 12 amu. The signal at 32 amu can be due to either O 2 or atomic sulfur. Equation 12 accounts for the contamination from atmospheric adsorbed O 2 , and once this correction is applied the signal at 32 amu becomes negligible. Although the signal at 32 amu can be an ion fragment of H 2 S, we do not account for it since we are not certain that this signal is due to sulfur or O 2 . Equation 13 determines the partial pressure of H 2 S, and Equation 14 assumes that the signal at 40 amu is entirely due to terrestrial atmospheric argon. Finally, Equation 15 determines the partial pressure of CO 2 . Equations 5, 6, 8, 10, 12, 13, and 15 determine the partial pressures of molecules subject to fragmentation. An additional factor that should be taken into account when correcting for ion fragments is adding fragments back to those species that are subject to molecular fragmentation. While this may cause a slight over-correction, it does not significantly affect the results, as the average difference between the relative abundances summed over temperature with or without adding fragments back in is ∼1%. The differences between the relative abundances summed over temperature with or without adding fragments back in are also within the uncertainties (expressed as 95% confidence intervals of the means) for each species and atomic abundance. As examples, to determine H 2 O and CH 4 's partial pressures, we add back the contributions from their fragments: Reproducibility of Experimental Results In order to test the reproducibility of our experiment and to confirm that it precisely measures the outgassed species from various samples, we analyzed samples of Jbilet Winselwan twice under identical conditions. Jbilet Winselwan's final reported relative abundances are given by the mean and the 95% confidence interval of the mean calculated from a t-distribution of the two trials (see Table 1 in main article). As Extended Data Figure 4 illustrates, the relative abundances of the three most abundant outgassed species, H 2 O, CO and CO 2 , between the two experiments agree with each other within 6 % with 95% confidence intervals less than 35%. The other species' abundances between the two experiments have variations of up to ∼3 % and 95 % confidence intervals less than 16 %. As Tables 1 and 2 in the main article illustrate, the species with the largest uncertainties are CO 2 , H 2 O, and the total amount of H (95 % confidence intervals of ∼34 %, 34 %, and 36 %, respectively). These large confidence intervals are due to the the small sample size. All other species have confidence intervals less than 18 % with difference between the two measurements less than 3 %. Calculating Oxygen Fugacity Although our experiments simulate a non-equilibrium open system, we can compare our results to what is expected at equilibrium. For determining the oxygen fugacity of the system as shown in Figure 4 H 2 O = H 2 + 0.5O 2 (18) log 10 (K 1 ) = −12794 T + 2.7768 (19) f O 2 = (K 1 χ H 2 O χ H 2 ) 2(20) Similarly, to calculate f O 2 from CO and CO 2 we use Equations 21-23: CO 2 = CO + 0.5O 2 (21) log 10 (K 2 ) = −14787 T + 4.5472 (22) f O 2 = (K 2 χ CO 2 χ CO ) 2(23) K 1 and K 2 are the equilibrium constants that are functions of temperature T (in Kelvin) and taken from the IVTANTHERMO database (see [41] for details). Figure 4 Comparison with Model Assumptions A key distinction between our experimental results and equilibrium model calculations is that our experiments simulate initial (or instantaneous) outgassing compositions, not the longterm outgassing abundances once equilibrium has been achieved. Also, in our experiments the meteorite composition changes as the temperature increases and volatiles are removed whereas the equilibrium calculations assume a closed system in which the volatiles are not removed. Nevertheless, the preliminary outgassed abundances determined experimentally may have important implications for the subsequent evolution of outgassed atmospheres that eventually achieve chemical equilibrium, as initial outgassed species' abundances control what is available to subsequently evolve within an atmosphere. For example, our experiments find that H 2 S outgasses at higher temperatures than predicted in equilibrium models, which means that if a planet does not reach 900-1000 • C, H 2 S may not have a significant atmospheric abundance. The experimental results for this work are the instantaneous outgassing compositions because those are the more appropriate ones to compare to thermochemical equilibrium models rather than the cumulative outgassing compositions. Both the instantaneous measurements and equilibrium model results represent contained assemblages, although the composition in the experiments is evolving. On the other hand, the cumulative outgassed abundances do not represent such an assemblage, since the gases are removed at each measurement and do not react with material outgassed at higher temperatures. However, the cumulative outgassed compositions are a useful way to determine the extent to which volatiles have been released, so we have calculated the cumulative outgassing composition. Supplementary Figure 4 illustrates the cumulative outgassing compositions as a function of temperature for each of the three chondrite samples as well as the average of the three samples. We see a leveling-off behavior at high temperatures for nearly all of the outgassing species across each of the three CM chondrite samples. Some of this leveling-off behavior is due to the fact that a volatile's outgassing is decreasing (e.g., H 2 S), but there could still be other gas species being produced that we do not track in these experiments. Compared to the other volatiles measured, CO 2 and H 2 's cumulative outgassing trends do not level off as significantly at the higher temperatures. Least Squares Regression Technique for Ion Fragments and Species Degeneracies Our chosen technique to correct for ion fragmentation and, when possible, break the degeneracies between volatile species that have the same mass number involves making logical assumptions regarding which gas species likely dominates a given mass signal and using a set of arithmetic corrections (see "Calculations to Determine Gas Species' Partial Pressures" section above). However, a non-linear least squares regression is an alternate method to account for ion fragments and disentangle gas species with overlapping mass numbers. This technique involves performing a least squares regression on the normalized mass spectrum library (Supplementary Table 1, column 4) and constraining the outgassed abundances to be positive and less than appropriate upper bounds. For all of the species, the upper bounds are twice the maximum value of that species mass number. Extended Data Figure 2 shows the fully-calibrated outgassing abundances calculated using this method and Supplementary Table 3 compares the average partial pressures for each gas species calculated using our original analysis to those from the least squares technique. In order to calculate the uncertainties on the fitted parameters from the least squares analysis, we ran a Monte Carlo simulation on a sparser data array. The least squares analysis results are consistent with the original results within 2σ for most species. While the results for the most dominant outgassing species (i.e., H 2 O, CO, CO 2 ) are similar to our original analysis, the least squares result finds non-negligible amounts of methane and atomic sulfur. Outgassing of these species is not predicted from chemical equilibrium calculations, which suggests that this technique may not fully account for ion fragments compared to our original calculations or, in the case of sulfur, they are leftover fragments from species we are not currently measuring in our experiments (e.g., S 2 , SO 2 , etc.). Degeneracies between Gas Species and Mass Numbers For mass numbers that could correspond to multiple volatile species, we describe in the following subsections additional details on how we either determined which species dominates the signal or disentangled multiple species' signals. 16 and 32 amu: 16 amu is the molecular weight of CH 4 and the atomic weight of oxygen. As shown in Supplementary Table 1, mass number 16 is affected by ionization fragmentation, being fragments of CO 2 , CO, H 2 O and O 2 . Ion fragments at 16 amu contribute to 22% of O 2 's signal (assuming the signal at 16 amu is due to atomic oxygen), 9.8% of CO 2 's signal, 2.2% of CO's signal, and 1.5% of H 2 O's signal. We assume that the majority of the signal at 16 amu outgassing from the samples is due to CH 4 because prior meteorite ablation studies have detected small amounts CH 4 from carbonaceous chondrites and atomic oxygen is not expected to outgas significantly [43]. 32 amu is the atomic weight of sulfur and the molecular weight of O 2 and methanol (CH 3 OH). If mass number 32 amu is due to atomic sulfur, then it can be an ion fragment of hydrogen sulfide (H 2 S), contributing to 45% of H 2 S's signal. According to theoretical calculations, atomic sulfur is not predicted to outgas significantly and O 2 is only predicted to begin outgassing around 1100 • C. As indicated below, our data does not allow a definitive determination of which species dominate the signals at 16 and 32 amu. For the 32 amu signal, we correct for the possibility of atmospheric adsorption of O 2 onto the samples by assuming the 40 amu signal is entirely due to atmospheric argon and using the known ratio of O 2 /Ar in Earth's atmosphere to subtract the atmospheric contribution to the signal at 32 amu (Equation 12). After correcting for atmospheric adsorption, we do not detect a significant outgassing signal at 32 amu for any of the samples. The signal at 32 amu is likely not due to methanol because it is not predicted to outgas significantly across the entire temperature range. For the 16 amu signal, even after correcting for ion fragments of CO 2 and H 2 O, the signal is still significant at lower temperatures (up to ∼600 • C) for all three samples. However, since O 2 has an ion fragment at 16 amu, we also have to account for the possibility of atmospheric adsorption of O 2 's ion fragment at 16 amu so we subtract 22 % of the atmospheric adsorbed O 2 from the 16 amu signal. When we apply this correction, the signal at 16 amu becomes negligible across all temperatures for all samples. If the signal at 32 amu is predominantly due to sulfur not O 2 , then the signal at 16 amu would be significant and likely due to CH 4 . However, because we cannot definitively resolve which species dominate the signals at 32 and 16 amu, we must conservatively apply the atmospheric adsorption correction for both masses. In order to disentangle CO and N 2 , as described in the next subsection, we assume the ion fragment of CO at 16 amu is negligible, which is reasonable because it only contributes to 2 % of CO's signal. If we assume the signal at 16 amu is due entirely to atomic oxygen resulting from ion fragments of O 2 , we can use it to disentangle the abundances of sulfur and O 2 . To calculate O 2 's signal from the 16 amu signal, we first correct for the fact that 16 amu can contribute to ion fragments of CO 2 and H 2 O (we assume the ion fragment of CO at 16 amu is negligible), and then use the remaining signal to calculate the abundance of O 2 , knowing that 16 amu contributes to 22% of O 2 's mass spectrum (Equation 24). We then calculate the signal due to sulfur, by subtracting O 2 's signal from the 32 amu signal and correcting for ion fragmentation of H 2 S (Equation 25). Finally, we correct for atmospheric adsorption of O 2 (Equation 26). The resulting signal due to 16 amu is given by Equation 27. Once we disentangle the signals from O 2 and sulfur, we find that sulfur and O 2 abundances are negligible at all temperatures including higher temperatures where predictions indicate that O 2 should begin outgassing (Supplementary Figure 3). In addition when the sulfur and O 2 components are separated, the abundance at 16 amu is also negligible. Ultimately, further work is required to determine which species dominate the signals at 16 and 32 amu. p O 2 , pre-atmosphere correction = 1.22((1.25p 16 amu − 0.10p CO 2 − 0.02p H 2 O )/0.22) (24) p S = p 32 amu − p O 2 , pre-atmosphere correction − 0.45p H 2 S (25) p O 2 = p O 2 , pre-atmosphere correction − (22.53p 40amu )(26)p CH 4 = 1.25p 16 amu − 0.10p CO 2 − 0.02p H 2 O − (0.22p O 2 )(27) 28 amu: This is the molecular weight of CO, N 2 and ethylene (C 2 H 4 ). Mass number 28 amu can also be an ion fragment of CO 2 , contributing to 10% of CO 2 's signal (Supplementary Table 1). To disentangle the signals due to CO and N 2 , we assume the signal at 14 amu is predominantly due to ion fragments of N 2 which is valid because atomic nitrogen is not expected to outgas. We correct for the fact that 14 amu is also an ion fragment of CH 4 , and then we use the resulting signal at 14 amu to calculate the signal due to N 2 , knowing that 14 amu contributes to 14 % of N 2 's mass spectrum (Equation 7). We then determine the signal due to CO by subtracting the N 2 signal from the 28 amu signal and correcting for ion fragmentation of CO 2 (Equation 10). We correct for atmospheric adsorption of N 2 by assuming the signal at 40 amu is entirely due to atmospheric argon and using the known ratio of N 2 /Ar in Earth's atmosphere to subtract the atmospheric contribution to N 2 (Equation 8). Even after disentangling the signal at 28 amu into the contributions from N 2 and CO and correcting for the effects of ionization fragmentation, the abundance of CO is still very high, being the second most abundant species (Figure 1 and Extended Data Figure 4). After correcting for atmospheric adsorption, N 2 's outgassed abundance is not significant for any of the three samples. Since CM chondrites have a higher bulk abundance of oxygen (432 mg/g) compared to hydrogen (14 mg/g) while prior theoretical and experimental studies do not predict significant amounts of C 2 H 4 to outgas, the 28 amu signal is more likely to be CO than C 2 H 4 (see Supplementary Table 2). Further investigation is required to definitively rule out C 2 H 4 contributing to the signal at 28 amu, so our experimental results should be considered an upper limit on the CO abundances. As Figure 3 illustrates, this result agrees fairly well with chemical equilibrium calculations. The fact that the oxygen fugacity calculated from the abundance ratios of CO 2 and CO is lower than f O 2 under chemical equilibrium at lower temperatures suggests that there is more CO than CO 2 in our experiments than would be expected if at equilibrium. Once our experiment reached higher temperatures (∼900 • C), the experimental f O 2 determined by CO 2 /CO matches the theoretical chemical equilibrium value. 40 amu: This is the atomic weight of argon and the molecular weight of sodium hydroxide (NaOH), potassium hydride (KH), and methyl cyanide (CH 3 CN). Mass number 40 should not contribute to the signals of any other gas species due to the ionization fragmentation process. In terms of the average bulk composition of CM chondrites, oxygen has the highest abundance (432 mg/g) followed by carbon (22 mg/g), hydrogen (14 mg/g) and finally sodium (4.1 mg/g), nitrogen (1.52 mg/g) and potassium (0.4 mg/g) (Supplementary Table 2). Although Argon has an even smaller bulk abundance than these species, it is relatively abundant in Earth's atmosphere ([Ar]/[O 2 ] for air is 0.05). Atmospheric 40 Ar is known to contaminate prior meteorite experiments (e.g., [44]). In addition, NaOH, KH, and CH 3 CN are not predicted to outgas significantly from CM chondrites at these temperatures. Therefore, we conclude that the 40 amu signal is due to atmospheric Ar. As described earlier, we use this signal to determine the atmospheric contributions of N 2 and O 2 . Future investigation is required to determine if any of the 40 amu signal is due to outgassing from the samples rather than atmospheric adsorption of Ar. Preliminary X-ray diffraction (XRD) analyses were performed on the sample residues and unheated samples. For each XRD measurement, the ∼3 mg sample was spread in a thin layer over a silicon sample holder and continuously rotated 360 • for two hours while data was collected, covering angles 0 to 70 • . Comparing the solid phases from the equilibrium calculations to what we detect in the samples from our preliminary XRD analysis, we find that almost all of these phases may be present in the unheated samples and the post-heated residues but, for the post-heated residues, most of the phases have reduced signals despite the unheated and post-heated sample masses being nearly the same. Notable exceptions include Ca 3 (PO 4 ) 2 and Co which were not definitively detected in the unheated samples and post-heated residues, and CaAl 2 Si 2 O 8 and Na 8 Al 6 Si 6 O 24 C l2 which were not detected in most of the post-heated residues. For example, troilite (FeS) is present in the unheated samples but has a much weaker signal in the post-heated residues, matching the equilibrium calculations that have FeS being a stable phase up until ∼775 • C. Our XRD analysis suggests that gypsum (CaSO 4 (H 2 O) 2 ) may be breaking down during the experiments. However, in the equilibrium calculations, gypsum is never stable, and this difference may be due to an issue with the data for gypsum that is used in the equilibrium models or uncertainties in the bulk composition used for the equilibrium calculations. Further XRD analyses are required to confirm these preliminary results. Solid Phases Outgassed Gas Species' Masses The average molar mass of the mixture across 200 to 1200 • C of volatile species i is determined by the equation:M = i M i * T χ i , where M i is the molar mass of species i. To calculate the mass fraction of a species, w i : w i = 100 * (( T χ i ) * M i )/M .(28) To determine the outgassed mass of a certain element or species (Mass i ), the mass fraction is multiplied by the total outgassed mass (Mass Total ) which is determined by measuring the mass change of the sample before and after heating: Mass i = w i × Mass Total .(29) Each sample was weighed before and after heating to determine the mass loss of each volatile species as a result of outgassing. The total gas released during the experiments based on mass loss measurements is similar between Murchison and Winselwan but higher for Aguas Zarcas. For all three chondrites, the mass released is mostly in H 2 O, CO, and CO 2 . Comparing the initial bulk abundance of an element for CM chondrites to the outgassed abundance informs the degree to which the samples have outgassed relative to complete vaporization of the samples ( Table 2 in main article). On average, the samples have higher outgassed abundances of hydrogen and carbon but lower outgassed abundances of oxygen, nitrogen and sulfur compared to the initial bulk abundances. These differences between the initial bulk abundances for an average CM chondrite composition and the outgassed abundances suggests that the samples have not outgassed fully relative to complete vaporization and could also reflect heterogeneities in the meteorite samples themselves. Comparison with Prior Studies Planetary outgassing has been modeled both for the Solar System's terrestrial planets and for some low-mass exoplanets. For instance, studies find that Earth's early degassing produced a steam atmosphere during planetary accretion and a reducing atmosphere of H 2 and/or CH 4 near the end of accretion (e.g., [12,18,45,46] [12,46]). Outgassing models for a planet's magma ocean phase suggest that the degassed atmospheric composition depends on the concentration of volatiles in the accreted body and the pressure at which degassing occurs [7,13]. Many of these studies assume chemical equilibrium conditions and lack experimental data to validate some of their assumptions. For example, Gaillard & Scaillet 2014 considered outgassed species composed of a limited set of elements (H, C, O, S, Fe) to investigate volcanic outgassing of basaltic material on planetary atmospheres. However, they do not include other potentially important elements (e.g., F, Na, Cl, K) nor do they apply experimental data to validate using a simplified set of elements. Finally, prior research used chemical equilibrium calculations assuming meteorite abundances to determine planet atmospheric compositions [47,14,48]. These planetary outgassing models have been applied to low-mass exoplanet atmosphere studies to help interpret current observational data (e.g., [49,50]). As noted above, there is limited experimental data to inform these theoretical outgassing models and, in particular, none to fully inform meteorite outgassing work. Prior meteorite heating experiments have used a variety of instrumental techniques including mass spectrometry, infrared spectroscopy and shock devolatilization (e.g., [16,17,51,18,21,52,22,19,20]). However, studies that heated meteorites were limited in several key ways due to restrictions in the number and type of samples used, the temperatures to which the samples where heated, and the number of gas species that were accurately measured. For example, some prior studies focused on the contribution from meteorites on impact-induced atmosphere formation which often involved shocking samples prior to analyzing their volatile contents, and therefore do not properly simulate conditions expected for outgassing from a planet (e.g., [16,18,21]). In addition, these experiments only measured a small subset of volatile species, namely H 2 O and CO 2 ([16, 21]). It is important to note that the prior studies that focused on shock-induced devolatilization experiments did not continuously monitor the composition of degassed species and focused on higher pressures (10 −4 − 10 4 bars) than those in our experiments (∼ 10 −8 bars). These prior works cannot be quantitatively compared to theoretical outgassing models because they either monitored the evolving composition of only a few gas species as a function of temperature or instead inferred loss of volatiles by comparing samples before and after heating. Other studies focused on trace metals (e.g., Co, Zn, In) and moderately volatile and volatile elements (e.g., Se, Ga, As), which are not major constituents of the atmospheres of temperate rocky planets ([20, 22] [49]. In summary, our results provide a comprehensive experimental comparison to prior theoretical chemical equilibrium models [47,14] that aim to study the outgassing compositions of chondritic meteorites and their implications for terrestrial planets' early atmospheres. Additional experiments on a wider range of chondritic meteorites, including ordinary and enstatite chondrites, will allow for a more complete comparison with prior theoretical work and will reveal more insight into the possible atmospheric composition of early Earth as well as various exoplanets. Data Availability: The data that support the findings of this study and corresponding plots in the paper are available from https://github.com/maggieapril3/CMChondritesOutgassingData or from the corresponding author upon request. Figures 1-4 Supplementary Figures 2-4 have associated raw data that is available from https://github.com/maggieapril3/CMChondritesOutgassingData or from the corresponding author. The thermochemical equilibrium models used in Figures 3 and 4 are available from L.S. upon request. and Extended Data Figures 1-5 and Code Availability: The code used to calibrate and analyze the data used in this study is also available from https://github.com/maggieapril3/CMChondritesOutgassingData. The boat is inserted into an alumina tube to the center of the furnace that can reach temperatures up to 1200 • C. The furnace is connected to a turbomolecular pump which maintains the entire system at a high-level vacuum, and to a residual gas analyzer which measures the partial pressures of up to 10 species continuously throughout the experiment. A thermocouple inside the tube measures the temperature as a function of time. The thermocouple is placed within 50 mm of the sample containers and both are within the furnace's 13 cm hotspot to ensure accurate temperature measurements. Supplementary Information Supplementary Table 1: Table of the mass spectrum for each gas species in our experiments. For the ten masses measured during the experiments, several could correspond to gas molecules that, when ionized by the RGA, produce fragments that contribute to the signal of other masses measured. For each of the gas species that correspond to one of the ten measured masses, we used the mass spectrum from National Institute of Standards and Technology (NIST)'s Mass Spectrometry Data Center to determine the possible ion fragments [1]. We also include an additional ion fragment for water at 2 amu from MKS [2]. For each species in the table, we list the mass numbers of its known ion fragments and their signal intensities relative to the major peak due to the gas species itself (i.e., the signal percentage of the gas species is 100%). In the last column, we provide the normalized signal intensities (i.e., all of the signals sum to 100%). We only list ion fragments whose mass numbers correspond to those measured in our outgassing experiments. For atoms and species that either do not suffer from ion fragments or whose fragments correspond to masses that we do not measure, we assume all of its signal is concentrated at its mass number. [ 15 ] 15Herbort, O., Woitke, P., Helling, C. & Zerkle, A. The atmospheres of rocky exoplanets i. outgassing of common rock and the stability of liquid water. Astronomy & Astrophysics 636, 1-19 (2020). [16] Court, R. W. & Sephton, M. A. Meteorite ablation products and their contribution to the atmospheres of terrestrial planets: An experimental study using pyrolysis-ftir. Geochimica et Cosmochimica Acta 73, 3512-3521 (2009). [17] Gooding, J. L. & Muenow, D. W. Experimental vaporization of the holbrook chondrite. Meteoritics 12, 401-408 (1977). [18] Lange, M. A. & Ahrens, T. J. The evolution of an impact-generated atmosphere. Icarus 51, 96-120 (1982). [19] Burgess, R., Wright, I. P. & Pillinger, C. T. Determination of sulphur-bearing components in c1 and c2 carbonaceous chondrites by stepped combustion. Meteoritics 26, 55-64 (1991). [20] Springmann, A. et al. Thermal alteration of labile elements in carbonaceous chondrites. Icarus 324, 104-119 (2019). [21] Tyburczy, J. A., Frisch, B. & Ahrens, T. J. Shock-induced volatile loss from a carbonaceous chondrite: implications for planetary accretion. Earth and Planetary Science Letters 80, 201-207 (1986).[22] Ikramuddin, M., Binz, C. M. & Lipschutz, M. E. Thermal metamorphism of primitive meteorites iii. ten trace elements in krymka l3 chondrite heated at 400-1000 c. Geochimica et Cosmochimica Acta 41, 393-401 (1977). [23] Krinov, E. L. Fall of murchison stone meteorite shower, australia. The Meteoritical Bulletin (Meteoritics) 5, 2 (1969). [24] Ruzicka, A., Grossman, J., Bouvier, A., Herd, C. D. K. & Agee, C. B. The meteoritical bulletin, no 102. Meteoritics and Planetary Science 50 (2015). [25] Gattacceca, J., McCubbin, F. M., Bouvier, A., & Grossman, J. N. Meteoritical bulletin no. 108 Meteoritics and Planetary Science 55 (2020). [26] Nittler, L. R. et al. Bulk element compositions of meteorites: a guide for interpreting remote-sensing geochemical measurements of planets and asteroids. Antarctic Meteorite Research 17, 231-251 (2004). [27] Alexander, C. M. O'D. et al. The Provenances of Asteroids, and Their Contributions to the Volatile Inventories of the Terrestrial Planets. Science 337, 721 (2012). [28] O'Brien, W. J. & Nielsen, J. P. Decomposition of gypsum investment in the presence of carbon. Journal of Dental Research 38, 541-547 (1959). [29] Zhao, S., Jiang, J. & Zheng, J. Thermal analysis on the kinetics of thermal decomposition of FeS. Journal of Chongqing University (2011). Author Information M. Thompson performed the outgassing experiments and data analysis and wrote the manuscript. M. Thompson and M. Telus conceived the research. D.L., T.J., M. Telus and M. Thompson collaborated to configure the experimental set-up. M. Telus provided the meteorite samples used in the experiments and imparted essential guidance on the data analysis and interpretation of the results. D.L. provided the laboratory equipment for the experiments and helpful suggestions for the data analysis. T.J. helped with preparing the experiments and maintaining the instruments. L.S. provided the chemical equilibrium models and greatly contributed to interpreting the results and their implications. J.F. gave important insight into the scope of this work and its implications for exoplanet atmospheres. All authors contributed to editing the manuscript.The authors declare no competing interests. Figure 1 :Figure 2 :Figure 3 : 123Mole fractions of the measured species outgassed as a function of temperature for each chondrite sample. The results are for 3 mg samples of (a) Murchison, (b) and (c) Jbilet Winselwan, and (d) Aguas Zarcas. We analyzed two 3 mg samples of Jbilet Winselwan under identical conditions to test reproducibility and show the results in (b) and (c). H 2 has the largest variation between the two experiments with Jbilet Winselwan. Across the three samples, some species exhibit major variations in their relative abundances over specific temperature intervals. For instance, CO and CO 2 's abundances increase around 650 -750 • C. Although the mole fraction of H 2 S varies considerably over the entire heating range, it peaks near ∼900-1000 • C and then decreases at higher temperatures for all three chondrites. For most samples, there is a prominent increase in H 2 's abundance near ∼1100 • C. Ratios of mole fractions of outgassed bulk elements hydrogen, carbon, oxygen, and sulfur as a function of temperature for the three chondrite samples. From top to bottom the ratios are: carbon/oxygen, hydrogen/carbon, sulfur/oxygen and hydrogen/oxygen. Blue, purple and orange curves represent elements outgassed from Murchison, Winselwan, and Aguas Zarcas, respectively. Comparison between equilibrium calculations (left) and experimental results (right) under the same pressure and temperature conditions.Figures (a)and (b) illustrate outgassing abundances calculated assuming chemical equilibrium for an average CM chondrite bulk composition at 1E-3 Pa (a) and experimental outgassing results for the average of the three CM chondrite samples measured at 1E-3 Pa (b). In (b), each species' curve is dominated by the sample that has the most abundant amount of that species at a given temperature.Figures (c) and (d)show the results for outgassing from a Murchison composition using chemical equilibrium calculations (c) and experimental outgassing results from the Murchison sample (d). The dashed curves in (b) and (d) show 'equilibrium-adjusted' experimental abundances in which the equilibrium model was used to recalculate gas speciation using the experimental abundances at intervals of 50 • C. The mass (in amu) of each species is in parentheses. See Extended DataFigure 5for other volatile species that theoretically degas with mole fractions above 1 × 10 −4 according to chemical equilibrium calculations but are not measured in the experiments. Figure 4 : 4Oxygen fugacities relative to the quartz-fayalite-magnetite (QFM) buffer from theory and experiments. Oxygen fugacity of an average bulk CM chondrite composition as a function of temperature from chemical equilibrium calculations (black curve, labeled Theory) and the oxygen fugacity of the average of the three CM chondrites measured experimentally (blue and orange curves, labeled Experiment Figure 2 :Figure 3 : 23Murchison Measurement Corrected for Ion Fragments and Atmospheric Adsorption andBackground-Subtracted (i.e., data from (d) subtracted from data from (c)).Extended DataFigure 1: Data Calibration Steps. Each figure illustrates the partial pressures (bars) for the molecular species measured from 200 • C to 1200 • C. Each sample's data is calibrated by first correcting for ion fragments and atmospheric adsorption and then background subtracting. Results of analyzing ion fragments using a non-linear least squares regression. The outgassing abundances in (a) are for the Murchison sample with the panel on the right side showing the average standard deviation determined from the Monte Carlo simulation for each of the species measured. The abundances in (b) are the average of the three CM chondrites. Comparison between the yields of major volatiles released from Jbilet Winselwan samples during two identical experiments. The mole fraction summed over temperature for each volatile species is normalized to the total mole fraction of released gases summed over temperature and expressed as a percentage. The uncertainty on the mean relative abundance for each volatile species is the 95% confidence interval of the mean. The volatile yields are fairly reproducible between the two experiments, especially for the most dominant outgassed species (H 2 O, CO, CO 2 ). Figure 4 :Figure 5 : 45Comparison between the yields of major volatiles released from the samples. The mole fraction summed over temperature for each volatile species is normalized to the total mole fraction of released gases summed over temperature and expressed as a percentage. The data for Winselwan is the mean of the two individual experiments conducted with the uncertainty reported as the 95% confidence interval of the mean (see Methods and Extended DataFigure 3). The mean relative abundance of all three samples for each volatile species is also shown with the uncertainty reported as the 95% confidence interval of the mean. All three samples have similar outgassing abundances for the most dominant outgassing species (H 2 O, CO, and CO 2 ). While H 2 and H 2 S have larger variations up to an order of magnitude, the relative abundances for each species across the three samples are within 2σ of each other. Additional Outgassing Species from Chemical Equilibrium Calculations Outgassing abundances for additional species not measured in the experiments calculated assuming chemical equilibrium for Murchison (a) and an average CM chondrite bulk composition (b) at 1E-3 Pa. The outgassing of H 2 O is also shown as a reference. p H 2 O 2= (1.0p 18 amu ) + (0.02p 18 amu ) + (0.02p 18 mu ) = 1.04p 18 amu (16) p CH 4 = ((1.0p 16 amu ) + (0.21p 16 amu ) + (0.04p 16 mu )) − (0.10p CO 2 ) − (0.02p H 2 O ) − (4.96p 40 amu ) = 1.25p 16 amu − (0.10p CO 2 ) − (0.02p H 2 O ) − (4.96p 40 amu )(17) Supplementary Figure 1 : 1Schematic of Instrument Set-Up. Each powdered sample is placed inside a small alumina crucible which itself is placed inside a alumina mini combustion boat. 3 :Figure 2 :Figure 4 : 324Comparison of primary algebraic data analysis and Monte Carlo non-linear least squares (MC) data analysis for Murchison. The second and third columns show the average partial pressure (in bars) for each species. The fourth column shows the standard deviation of the average partial pressure for each species analyzed using the Monte Carlo technique. These partial pressures are corrected for ion fragments and atmospheric adsorption but have not been background subtracted. Total pressure of measured volatiles released from the samples as a function of temperature. Variations in total pressure with temperature suggest that the amount of outgassing varies throughout the experiment. The average difference between the maximum and minimum total pressure is 6E-9 bars. Most samples show an increase in total pressure near 400• C. Average of 3 CM Chondrite Samples with the Signal at 32 amu Separated into Sulfur and O 2 Components Supplementary Figure 3: Comparison between original results and results of separating the 32 amu signal into sulfur and O 2 components. Figure (a) shows the outgassing abundances in which the signal at 32 amu is not separated into the sulfur and O 2 components (i.e., Figure 3 (b)). Figure (b) shows the results of separating the signal at 32 amu into its sulfur and O 2 abundances. Cumulative Outgassing Abundances. The cumulative outgassing trends for samples of (a) Murchison, (b) Jbilet Winselwan, (c) Aguas Zarcas, and (d) the average of the three CM chondrite samples. ). Two abundance ratios were used to calculate f O 2 : H 2 O/H 2 (blue curve) and CO 2 /CO (orange curve). We cannot determine f O 2 directly from the O 2 abundance because after correcting for terrestrial atmospheric adsorption its abundance goes to zero (see Methods). Table 2 : 2Relative outgassed atomic abundances (in %) summed over temperature of hydrogen, carbon, oxygen, nitrogen and sulfur for the three samples. As inTable 1, abundances are in partial pressure normalized to the total pressure of all released gases measured and are reported as percentages. The uncertainties reported for Jbilet Winselwan are the 95% confidence intervals on the means. Comparison between the initial (pre-degassing) normalized atomic abundances for average CM chondrites and the outgassed normalized atomic abundances are shown, both reported as percentages (bottom two rows). These atomic abundances are normalized to the sum of the elements measured in the experiments, i.e., H, C, O, N, S. The uncertainties for the pre-degassing normalized atomic abundances are (1σ) standard deviations. The uncertainties for the outgassed quantities are expressed as the 95% confidence intervals of the means. The initial bulk atomic abundances of CM chondrites come from the literature[26, 27].Sample Total H Total C Total O Total N Total S Murchison 50.20 9.82 39.96 0.0 0.02 Jbilet Winselwan 44.95±35.69 13.39±17.86 41.62±17.97 0.0 0.04±0.14 Aguas Zarcas 47.83 12.04 40.09 0.0 0.04 Initial* Bulk CM Abundance 28.81±0.44 4.07±0.67 64.37±0.12 0.18±0.05 2.58±0.70 Outgassed CM Abundance 47.66±5.33 11.75±3.65 40.55±1.88 0.0 0.03±0.02 *Pre-degassing ). The major factors controlling speciation during Earth's early degassing included the water content of accreting planetesimals as well as temperature and pressure conditions during the atmosphere's degassing history. For the Zahnle et al. 1988 model of Earth's steam atmosphere during accretion, water is the only atmospheric species considered, while the Hashimoto et al. 2007 model of Earth's reducing atmosphere assumed accretion of only a specific type of chondritic material with varying amounts of water ( ). As a result, prior studies are unsuitable for validating outgassing models for low-mass planets. To fill this gap in the understanding of meteorite outgassing compositions, we designed an experimental procedure to analyze the abundances of a wide range of degassed components: H 2 , C, N, CH 4 /O, H 2 O, CO, N 2 , S/O 2 , H 2 S, Ar, CO 2 , which informs the initial compositions of outgassed atmospheres assuming the outgassing material is CM chondrite-like. Comparing our results to other prior meteorite heating experiments, we find that our detection of significant outgassing of H 2 S from Murchison (beginning at 800 • C) is consistent with the stepped combustion experiments of Murchison from Burgess et al. 1991 that found the highest outgassing yield of sulfur occurring at 800 • C [19]. Court & Sephton 2009 rapidly heated CM2 chondrites to 1000 • C and using FTIR found that outgassed H 2 O and CO 2 yields relative to the initial sample masses were ∼9 % and 5 %, respectively; they did not detect significant amounts of CO and CH 4 . These H 2 O and CO 2 yields are similar to those measured in our experiments, which reached higher temperatures over a much longer period of time: ∼9 % for H 2 O and ∼6 % for CO 2 , where both of these values are determined by taking the outgassed mass of the volatile species divided by the initial sample mass [16]. Mbarek & Kempton 2016 [49] used the theoretical outgassing composition of chondritic meteorites from Schaefer & Fegley 2010 [14] as their initial condition and then performed Gibbs free energy minimization to determine what condensate cloud species may form in super-Earth atmospheres. This work explores similar atmospheric temperatures to those measured in our experiments (∼350-1500 K) and finds that the C/O and H/O ratios have a strong influence on cloud chemistry in exoplanet atmospheres. They claim that if a planet's bulk composition is made of CM chondrite-like material, its outgassed atmosphere will have C/O and H/O ratios of 0.18 and 1.39, respectively [49]. From our experimental outgassing abundances, we find similar C/O and H/O ratios, 0.29±0.08 and 1.18±0.18, respectively. Our experimental C/O ratio is between Mbarek & Kempton's values for CM, CI and CV chondrites, whereas our H/O ratio is closest to their values for CM and CI chondrites. They predict that the atmospheres of super-Earth exoplanets with bulk compositions similar to CM chondrites may form KCl and ZnS clouds, but slightly more oxidizing conditions (e.g., CV chondrites) may hinder the formation of cloud condensates Gas Species Mass Number (amu) % of Signal Relative to Major Peak Normalized % of Signal H SupplementaryTable 2: Previously determined average bulk composition of CM chondrites and Murchison from literature a [3], b[4], c[5]. The uncertainties are the 1σ standard deviations.2 : 2 100 100 C: 12 100 100 N: 14 100 100 CH 4 : 16 100 79.99 14 20.7 16.56 12 4.31 3.448 O: 16 100 100 H 2 O: 18 100 96.62 16 1.5 1.449 2 2 1.932 CO: 28 100 93.11 16 2.2 2.048 12 5.2 4.842 N 2 : 28 100 87.72 14 14 12.28 S: 32 100 100 O 2 : 32 100 81.97 16 22 18.03 H 2 S: 34 100 68.97 32 45 31.03 Ar: 40 100 100 CO 2 : 44 100 77.58 28 10.2 7.913 16 9.8 7.603 12 8.9 6.905 Element Average CM Chondrite Murchison H 11.5±0.18 a mg/g 10.7±0.002 a mg/g C 19.5±3.24 a mg/g 20.8 a mg/g N 996.5±280 a µg/g 1050 a µg/g O 412.0±0.75 a,b mg/g 410 b mg/g S 33±9.0 b mg/g 14 c mg/g K 400 b µg/g 280 c µg/g Na 4.1 b mg/g 4.2 c mg/g Supplementary Table AcknowledgementsWe thank A. K. Skemer for his helpful insights and K. Kim for performing the preliminary XRD experiments. M. Thompson acknowledges support from the ARCS Foundation Scholarship. M. Telus is supported by NASA EW grant 80NSSC18K0498 and NASA ECA grant 80NSSC20K1078. T. J. was supported from U. C. 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[ "Solution manifolds of differential systems with discrete state-dependent delays are almost graphs", "Solution manifolds of differential systems with discrete state-dependent delays are almost graphs" ]	[ "Tibor Krisztin [email protected]@math.uni-giessen.de \nELKH-SZTE Analysis and Applications Research Group Bolyai Institute\nUniversity of Szeged\nMathematisches Institut Universität Gießen\n\n", "Hans-Otto Walther \nELKH-SZTE Analysis and Applications Research Group Bolyai Institute\nUniversity of Szeged\nMathematisches Institut Universität Gießen\n\n" ]	[ "ELKH-SZTE Analysis and Applications Research Group Bolyai Institute\nUniversity of Szeged\nMathematisches Institut Universität Gießen\n", "ELKH-SZTE Analysis and Applications Research Group Bolyai Institute\nUniversity of Szeged\nMathematisches Institut Universität Gießen\n" ]	[]	We show that for a system x ′ (t) = g(x(t − d1(Lxt)), . . . , x(t − d k (Lxt))) of n differential equations with k discrete state-dependent delays the solution manifold, on which solution operators are differentiable, is nearly as simple as a graph over a closed subspace in C 1 ([−r, 0], R n ). The map L is continuous and linear from C([−r, 0], R n ) onto a finite-dimensional vectorspace, and g as well as the delay functions dκ are assumed to be continuously differentiable.	10.3934/dcds.2023036	[ "https://export.arxiv.org/pdf/2208.06491v1.pdf" ]	251,564,639	2208.06491	40e87d696d671e7a87a489bb837b6f59f7ebae28	Solution manifolds of differential systems with discrete state-dependent delays are almost graphs Aug 2022 Tibor Krisztin [email protected]@math.uni-giessen.de ELKH-SZTE Analysis and Applications Research Group Bolyai Institute University of Szeged Mathematisches Institut Universität Gießen Hans-Otto Walther ELKH-SZTE Analysis and Applications Research Group Bolyai Institute University of Szeged Mathematisches Institut Universität Gießen Solution manifolds of differential systems with discrete state-dependent delays are almost graphs Aug 2022Delay differential equationstate-dependent delaysolution mani- foldalmost graph 2020 AMS Subject Classification: Primary: 34K4334K1934K05; Secondary: 58D25 We show that for a system x ′ (t) = g(x(t − d1(Lxt)), . . . , x(t − d k (Lxt))) of n differential equations with k discrete state-dependent delays the solution manifold, on which solution operators are differentiable, is nearly as simple as a graph over a closed subspace in C 1 ([−r, 0], R n ). The map L is continuous and linear from C([−r, 0], R n ) onto a finite-dimensional vectorspace, and g as well as the delay functions dκ are assumed to be continuously differentiable. Introduction For an integer n > 0 and a real number r > 0 let C n = C([−r, 0], R n ) and C 1 n = C 1 ([−r, 0], R n ) denote the Banach spaces of continuous and continuously differentiable maps [−r, 0] → R n , respectively, with the norms given by \|φ\| C = max −r≤t≤0 \|φ(t)\| and \|φ\| = \|φ\| C + \|φ ′ \| C , for a chosen norm on R n . If a differential equation with state-dependent delay, like for example the equation x ′ (t) = g(x(t − ∆)), ∆ = d(x(t))(1) with real functions g : R → R and d : R → [0, r], is written in the general form x ′ (t) = f (x t )(2) of an autonomous delay differential equation, with a map f : U → R n defined on an open subset of C 1 n , and x t (s) = x(t + s), −r ≤ s ≤ 0, then the associated solution manifold is the set X f = {φ ∈ U : φ ′ (0) = f (φ)}. If f is continuously differentiable and satisfies the extension property that (e) each derivative Df (φ) : C 1 n → R n , φ ∈ U , continues to a linear map D e f (φ) : C n → R n and the map U × C n ∋ (φ, χ) → D e f (φ)χ) ∈ R n is continuous and if X f is non-empty then X f is a continuously differentiable submanifold of codimension n in C 1 n , and the initial value problem for initial data in X f is well-posed, with each solution operator continuously differentiable [10,4]. The extension property (e) is a version of the notion of being almost Fréchet differentiable which was introduced for maps on domains U ⊂ C n by Mallet-Paret, Nussbaum, and Paraskevopoulos [9]. Let us recall that in contrast to theory for differential equations with constant time lags [3,2] the initial value problem for equations with state-dependent delays is in general not well-posed for initial data in open subsets of the space C n . The present paper about solution manifolds continues work which started with [5,13]. Under a boundedness condition on the extended derivatives D e f , or under a condition on f which generalizes delays being bounded away from zero, solution manifolds are graphs which can be written as X f = {χ + α(χ) : χ ∈ dom} with a continuously differentiable map α from an open subset dom of the closed subspace X 0 = {φ ∈ C 1 n : φ ′ (0) = 0} into a complementary space Q ⊂ C 1 n ,X = {ζ + α(ζ) ∈ E : ζ ∈ dom}, α(ζ) = 0 on dom ∩ X, α(ζ) ∈ E \ H on dom \ X, and the map H ⊃ dom ∋ ζ → ζ + α(ζ) ∈ X is a diffeomorphism onto X. An example of an almost graph in the plane is the unit circle without (0, 1) tr , for H = R × {0} and α the inverse of the stereographic projection [14, Section 1]. A property stronger than being an almost graph is existence of an almost graph diffeomorphism as introduced in [ A(X ∩ O) = H ∩ A(O) and if A leaves the points of (X ∩ O) ∩ H fixed. In [14,Section 1] it is shown that the existence of an almost graph diffeomorphism with respect to X, H, O implies that X ∩ O is an almost graph over H. The main results of [14], Theorems 3.5 and 4.8, imply that for a class of systems more general than those studied in [13] the associated solution manifolds carry a finite atlas of manifold charts whose domains are almost graphs over X 0 . The size of the atlas is determined precisely by the zerosets of the delays. If for example the delay function d in Eq. (1) is non-constant and has zeros then the atlas found in [14] consists of exactly 2 manifold charts. Let us now recall the systems x ′ (t) = g(x(t − d 1 (Lx t )), . . . , x(t − d k (Lx t )))(3) 1 as in the definition of a continuously differentiable submanifold introduced in [14]. In Eq. (3) the delays are given by compositions of a continuous linear map L : C n → F onto a finite-dimensional normed vectorspace over the field R, with continuously differentiable delay functions d κ : W → [0, r], W ⊂ F open and κ ∈ {1, . . . , k}. The nonlinearity g is a continuously differentiable map from an open subset V ⊂ R nk into R n . The notation for the argument of g in Eq. (3) is an abbreviation for the column vector y ∈ R nk with components y ι = x ν (t − d κ (Lx t )) ( = (x t ) ν (−d κ (Lx t )) ) for ι = (κ − 1)n + ν with κ ∈ {1, . . . , k} and ν ∈ {1, . . . , n}. With regard to the form of the delays in Eq. (3) one may think of Lφ as an approximation of φ ∈ C n in the subspace F ⊂ C n , and view d κ (L·) as a substitute for a more general delay functional defined on an open subset of C n . In order that Eq. (3) makes sense it is assumed that (V) there exist φ ∈ C 1 n with Lφ ∈ W and (φ(−d 1 (Lφ)), . . . , φ(−d k (Lφ))) ∈ V . (in notation as described above for φ = x t ). With φ = (φ(−d 1 (Lφ)), . . . , φ(−d k (Lφ))) we get that U = {φ ∈ C 1 n : Lφ ∈ W and φ ∈ V } is non-empty, and Eq. (3) takes the form of Eq. (2) with f : U → R n given by f (φ) = g( φ). According to [14, Propositions 2.1 and 2.3] the set U is open, f is continuously differentiable with property (e), and X f = ∅, so that X f is a continuously differentiable submanifold of codimension n in C 1 n . In the present paper we consider system (3) under the above conditions and construct an almost graph diffeomorphism with respect to the whole solution manifold X f , to the subspace X 0 , and to the open neighbourhood O = U of X f in C 1 n . The result is stated as Theorem 3.5 below. Let us emphasize that the finite atlas result from [14] holds true under further hypotheses on g or on the delay functions d κ whereas the construction in Sections 2-3 below requires nothing beyond smoothness as stated above. This discrepancy reflects the fact that in the proof of [14, Theorem 5.1], as well as in the proofs of [13, Theorems 2.4 and 5.1], the invariance property A(φ) = φ of a diffeomorphism A from X f onto an open subset of X 0 is established and exploited. The approach in the present paper, which originated in the case study [15], proceeds without recourse to the said invariance property. The Introduction of [14] lists a wide variety of special cases of system (3). Therefore, by Theorem 3.5, the solution manifolds associated with many familiar delay differential systems with state-dependent delays are almost graphs over a closed subspace of C 1 n . It is an open problem whether this is true for the solution manifold of the general equation (2) with a continuously differentiable f : U → R n , defined on an open U ⊂ C 1 n , satisfying the extension property (e). There are differential equations with discrete delays so that the delay functions are not of the form δ κ (Lφ) with a continuous linear map L : C n → F into a finite dimensional vectorspace F . For example, threshold delays and transmission delays are implicitly defined (see e.g. [1,7,8,11,12,4,6]), and the corresponding delay functions σ(φ) are defined on an open subset of C 1 n in order to write the system in the form (2) with the required properties. It would be interesting to describe the solution manifolds for systems with threshold and transmission delays as well. Finite-dimensional vectorspaces are always equipped with the canonical topology which makes them topological vectorspaces. Notation, conventions, preliminaries. For subsets A ⊂ B of a topological space T we say A is open in B if A is On R nk we use a norm which satisfies \|y j \| ≤ \|y\| ≤ nk ι=1 \|y ι \| for all y ∈ R nk , j ∈ {1, . . . , nk}. In case V = R nk the expression dist(v, R nk \ V ) = min y∈R nk \V \|y − v\| defines a continuous function V → (0, ∞). An upper index as in (x 1 , . . . , x N ) tr ∈ R N denotes the transpose of the row vector (x 1 , . . . , x N ). Vectors in R N which occur as argument of a map are always written as row vectors. The vectors of the canonical basis of R N are denoted by e ν , ν ∈ {1, . . . , N }; e νµ = 1 for ν = µ and e νµ = 0 for ν = µ, ν and µ in {1, . . . , N }. Derivatives and partial derivatives of a map at a given argument are continuous linear maps, indicated by a capital D. In case of real functions on domains in R and in R N , φ ′ (t) = Dφ(t)1 and ∂ ν g(x) = D ν g(x)1, respectively. For n = 1 we abbreviate C = C 1 and C 1 = C 1 1 . We define continuous bilinear products C × R n → C n and R n × C n → C n by (φ · q) ν = q ν φ ∈ C and (q · φ) ν = q ν φ ν ∈ C for ν = 1, . . . , n. Obviously q · φ = n ν=1 q ν (φ ν · e ν ) for all q ∈ R n , φ ∈ C n . The maps C 1 n × W ∋ (ψ, η) → ψ ν (−d κ (η)) ∈ R, ν ∈ {1, . . . ,U ∋ φ → φ ∈ V ⊂ R nk is continuously differentiable. The inclusion C 1 n ∋ φ → φ ∈ C n , differentiation ∂ : C 1 n ∋ φ → φ ′ ∈ C n , and evaluation ev t : C ∋ φ → φ(t) ∈ R at t ∈ [−r, 0] are continuous linear maps. In the sequel a diffeomorphism is a continuously differentiable injective map with open image whose inverse is continuously differentiable. Preparations The following lemma is a version of [13, Proposition 4.1] which includes smallness in C of the function found. Lemma 2.1 Let λ : C → F be a continuous linear map into a finite-dimensional real vectorspace F , and let ǫ > 0 be given. There exists ψ ∈ C 1 with λψ = 0, ψ ′ (0) = 1, and \|ψ\| C < ǫ. Proof. 1. Proof that there exists a complementary space K ⊂ C 1 for λ −1 (0) in C. We have λC 1 = λC 1 (with dim λC 1 ≤ dim λC < ∞) ⊃ λC (with C 1 dense in C) ⊃ λC 1 , hence λC 1 = λC. Set K = k j=1 Rψ j with preimages ψ j ∈ C 1 of a basis of λC and verify C = λ −1 (0) ⊕ K. The projection P : C → C along K ⊂ C 1 onto λ −1 (0) maps C 1 into λ −1 (0) ∩ C 1 . Choose a sequence (φ m ) ∞ 1 in C 1 with φ ′ m (0) = 1 for all m ∈ N and \|φ m \| C → 0 as m → ∞. Then \|(id − P )φ m \| C → 0 as m → ∞. As K is finite-dimensional we also get \|(id − P )φ m \| → 0 as m → ∞. It follows that ev 0 ∂(id − P )φ m → 0 as m → ∞. Using this and P φ m ∈ C 1 we get 1 = φ ′ m (0) = ev 0 ∂φ m = ev 0 ∂(P φ m + (id − P )φ m ) = ev 0 ∂P φ m + ev 0 ∂(id − P )φ m for all m ∈ N, which yields ev 0 ∂P φ m → 1 as m → ∞. For m so large that ev 0 ∂P φ m = 0 the functions ψ m = 1 ev 0 ∂P φ m P φ m belong to C 1 and satisfy λψ m = 0 and ψ ′ m (0) = ev 0 ∂ψ m = 1. For m sufficiently large we also obtain \|ψ m \| C < ǫ. The next proposition provides functions in C 1 which after multiplication with the unit vectors e ν ∈ R n yield bases of subspaces which are complementary for X 0 and depend on φ ∈ U via v = φ ∈ V ⊂ R nk . For φ ∈ X f the subspace is complementary also for the tangent space T φ X f . We omit proofs of these facts as they will not be used in the sequel. The almost graph diffeomorphism associated with X f , X 0 , and U which will be constructed in the next section is composed of translations by vectors in the complementary spaces just mentioned. Recall the map L : C n → F from Eq. (3). H ν : V → C 1 so that for all v ∈ V , L(H ν (v) · e ν ) = 0, (H ν (v)) ′ (0) = 1, \|H ν (v)\| C ≤ h(v), and for each µ ∈ {1, . . . , nk}, \|D µ H ν (v)1\| C ≤ h(v). Proof. 1. There is a sequence of non-empty open subsets V j1 , V j2 , V j of V , with j ∈ N, such that ∞ j=1 V j = V, and for every j ∈ N, V j1 ⊂⊂ V j2 ⊂⊂ V j and V j ⊂⊂ V j+1,1 . With V 02 = ∅ we have that for each integer j ≥ 1, (V j+1 \ V j2 ) ∩ (V j \ V j−1,2 ) = V j \ V j2 while for integers j ≥ 1 and k ≥ j + 2, (V k \ V k−1,2 ) ∩ (V j \ V j−1,2 ) = V j \ V k−1,2 ⊂ V j \ V j+1,2 = ∅. 2. For every j ∈ N choose a continuously differentiable function a j : R nk → [0, 1] with a j (v) = 1 on V j1 , a j (v) = 0 on R nk \ V j2 . For every j ∈ N choose an upper bound A j > 1 + nk µ=1 max v∈R nk \|D µ a j (v)1\| = max v∈Vj \|a j (v)\| + nk µ=1 max v∈Vj \|D µ a j (v)1\| so that the sequence (A j ) ∞ 1 in [1, ∞) is increasing. The sequence (h j ) ∞ 1 given by h j = min v∈Vj h(v) > 0 is nonincreasing. We have h j 2A j ≤ h j for all j ∈ N, and the sequence (h j /2A j ) ∞ j=1 is decreasing. 3. For each j ∈ N apply Lemma 2.1 to λ : C → F given by λφ = L(φ · e ν ), and to ǫ = h j /2A j . This yields a sequence of functions ψ ν,j ∈ C 1 , j ∈ N, which satisfy L(ψ ν,j · e ν ) = 0, (ψ ν,j ) ′ (0) = 1, \|ψ ν,j \| C < h j 2A j . The maps H ν,j : V j \ V j−1,2 → C 1 , j ∈ N, given by H ν,j (v) = a j (v)ψ ν,j + (1 − a j (v))ψ ν,j+1 are continuously differentiable and satisfy L(H ν,j (v) · e ν ) = a j (v)L(ψ ν,j · e ν ) + (1 − a j (v))L(ψ ν,j+1 · e ν ) = 0, (H ν,j (v)) ′ (0) = a j (v)ψ ′ ν,j (0) + (1 − a j (v))ψ ′ ν,j+1 (0) = 1, \|H ν,j (v)\| C ≤ a j (v)\|ψ ν,j \| C + (1 − a j (v))\|ψ ν,j+1 \| C ≤ a j (v)h j + (1 − a j (v))h j+1 ≤ a j (v)h j + (1 − a j (v))h j = h j ≤ h(v) for every j ∈ N and all v ∈ V j \ V j−1,2 . Moreover, for such j and v, and for every µ ∈ {1, . . . , kn}, D µ H ν,j (v)1 = (D µ a j (v)1) ψ ν,j − (D µ a j (v)1) ψ ν,j+1 , hence \|D µ H ν,j (v)1\| C ≤ \|D µ a j (v)1\|(\|ψ ν,j \| C + \|ψ ν,j+1 \| C ) ≤ A j h j 2A j + h j+1 2A j+1 ≤ A j 2 h j 2A j = h j ≤ h(v). 4. It remains to show that the maps H ν,j , j ∈ N, define a map H ν : V → C 1 . This follows from Part 1 of the proof provided H ν,j+1 and H ν,j coincide on the intersection V j \V j2 of their domains, for every j ∈ N. For j ∈ N and v ∈ V j \V j2 we have H ν,j+1 (v) = a j+1 (v)ψ ν,j+1 + (1 − a j+1 (v))ψ ν,j+2 = ψ ν,j+1 due to a j+1 (v) = 1 on V j+1,1 ⊃ V j ⊃ V j \ V j2 , and H ν,j (v) = a j (v)ψ ν,j + (1 − a j (v))ψ ν,j+1 = ψ ν,j+1 due to a j (v) = 0 on R nk \ V j2 ⊃ V j \ V j2 . The almost graph diffeomorphism The function h V : V → (0, ∞) given by 2 (1 + max ι=1,...,nk;ν=1,...,n \|∂ ι g ν (v)\| + max ν=1,...,n \|g ν (v)\|) . h(v) = min{1, dist(v, R nk \ V )} 2(nk) in case V = R nk and h(v) = 1 2(nk) 2 (1 + max ι=1,...,nk;ν=1,...,n \|∂ ι g ν (v)\| + max ν=1,...,n \|g ν (v)\|) for V = R nk is continuous. For h = h V choose functions H ν : V → C 1 , ν ∈ {1, . . . , n}, according to Proposition 2.2 and define H : V → C 1 n by H(v) = n ν=1 H ν (v) · e ν , or equivalently, (H(v)) ν = H ν (v) for ν = 1, . . . , n. The map A : U → C 1 n given by A(φ) = φ − g(v) · H(v) = φ − n ν=1 g ν (v)(H ν (v) · e ν ) with v = φ is continuously differentiable and satisfies A(X f ) ⊂ X 0 as for every φ ∈ X f we have (A(φ)) ′ (0) = φ ′ (0) − (g(v) · H(v)) ′ (0) (with v = φ) = φ ′ (0) − n ν=1 g ν (v)(H ν (v) · e ν ) ′ (0) = φ ′ (0) − n ν=1 g ν (v) (H ν (v)) ′ (0) e ν = φ ′ (0) − g(v) = φ ′ (0) − g( φ) = 0 (with φ ∈ X f ), which means A(φ) ∈ X 0 . From the above lines it is also obtained that if φ ∈ U and A(φ) ∈ X 0 , then 0 = (A(φ)) ′ (0) = φ ′ (0) − g( φ), that is φ ∈ X f . We also have A(φ) = φ on X 0 ∩ X f since φ ∈ X 0 ∩ X f yields 0 = φ ′ (0) = g( φ), hence A(φ) = φ − g( φ) · H( φ) = φ. For A to be an almost graph diffeomorphism associated with the submanifold X f ⊂ C 1 n , with the open set U ⊂ C 1 n , and with the closed subspace X 0 ⊂ C 1 n , it remains to prove that A is a diffeomorphism onto an open subset of C 1 n . Observe that due to L(H ν (v) · e ν ) = 0 we have LA(φ) = Lφ − L n ν=1 g ν (v)(H ν (v) · e ν ) = Lφ(4) for every φ ∈ U . Next we examine the relation between v = φ, φ ∈ U , and y = χ for χ = A(φ). Let η = Lχ = Lφ ∈ W . For ι = (κ − 1)n + ν with κ ∈ {1, . . . , k} and ν ∈ {1, . . . , n}, y ι = χ ι = A(φ) ι = [φ ν − (g(v) · H(v)) ν ](−d κ (η)) = φ ι − (g ν (v)(H ν (v))(−d κ (η)) = v ι − g ν (v)(H ν (v))(−d κ (η)) = S ι (η, v) with the continuously differentiable map S : W × V → R nk given by S(η, v) = v − R(η, v) and R ι (η, v) = g ν (v)(H ν (v))(−d κ (η)) = ev −dκ(η) (g ν (v)H ν (v)) for ι = (κ − 1)n + ν with κ ∈ {1, . . . , k} and ν ∈ {1, . . . , n}. Proposition 3.1 (i) For all (η, v) ∈ W × V , \|D 2 R(η, v)\| Lc(R nk ,R nk ) ≤ 1 2 . (ii) In case V = R nk , \|R(η, v) < dist(v, R nk \ V ) 2 for all (η, v) ∈ W × V . Proof. 1. On assertion (i). Let η ∈ W be given and define R η : V → R nk by R η (v) = R(η, v). For every v ∈ V and for all y ∈ R nk with \|y\| ≤ 1 we get \|D 2 R(η, v)y\| = \|DR η (v)y\| ≤ nk ι=1 nk j=1 \|∂ j R η,ι (v) · y j \| ≤ nk ι=1 nk j=1 \|∂ j R η,ι (v)\|, and for j ∈ {1, . . . , nk} and ι = (κ − 1)n + ν with κ ∈ {1, . . . , k} and ν ∈ {1, . . . , n}, by the chain rule, \|∂ j R η,ι (v)\| = \|D j R η,ι (v)1\| = \|ev −dκ(η) (D j g ν (v)(1) · H ν (v) + g ν (v) · D j H ν (v)1) \| = \|D j g ν (v)(1) · (H ν (v))(−d κ (η)) + g ν (v) · (D j H ν (v)1)(−d κ (η))\| ≤ (\|∂ j g ν (v)\| + \|g ν (v)\|)h(v) (with Proposition 2.2). It follows that 2. On assertion (ii). Assume V = R nk . For all (η, v) ∈ W × V and for each ι = (κ − 1)n + ν with κ ∈ {1, . . . , k} and ν ∈ {1, . . . , n} we have \|D 2 R(η, v)\| Lc(R nk ,R nk ) = sup \|y\|≤1 \|D 2 R(η, v)y\| ≤ (nk) 2 [ max\|R ι (η, v)\| = \|g ν (v)(H ν (v))(−d κ (η))\| ≤ \|g ν (v)\|\|H ν (v)\| C ≤ \|g ν (v)\|h(v). Hence \|R(η, v)\| ≤ nk ι=1 \|R ι (η, v)\| ≤ h(v) nk ι=1 max ν=1,...,n \|g ν (v)\| ≤ h(v) nk max ν=1,...,n \|g ν (v)\| < dist(v, R nk \ V ) 2 . It is convenient to introduce the continuously differentiable maps S η : V ∋ v → S(η, v) ∈ R nk , η ∈ W . Proposition 3.2 (i) The set ∪ η∈W {η} × S η (V ) ⊂ F × R nk is open. (ii) Each map S η : V → R nk , η ∈ W , is a diffeomorphism onto the open set S η (V ) = S({η} × V ) ⊂ R nk . (iii) The map ∪ η∈W {η} × S η (V ) ∋ (η, y) → S −1 η (y) ∈ V is continuously differ- entiable. Proof. 1. On assertion (i). Let η 0 ∈ W and y 0 = S η0 (v 0 ) = S(η 0 , v 0 ) with v 0 ∈ V be given. Choose a closed ball V 0 ⊂ V with center v 0 and radius ǫ > 0, and an open neighbourhood W 0 of η 0 in W such that for all η ∈ W 0 , \|R(η, v 0 ) − R(η 0 , v 0 )\| < ǫ 8 . For η ∈ W 0 and v, v 1 in V 0 the Mean Value Theorem in combination with Proposition 3.1 yield \|R(η, v) − R(η, v 1 )\| ≤ 1 2 \|v − v 1 \|. Let Y ⊂ R nk denote the open ball with center y 0 and radius ǫ/8. Let η ∈ W 0 and y ∈ Y be given and consider the map V 0 ∋ v → y + R(η, v) ∈ R nk , which is a contraction. For each v ∈ V 0 , \|y + R(η, v) − v 0 \| = \|y + R(η, v) − (y 0 + R(η 0 , v 0 )\| ≤ \|y − y 0 \| + \|R(η, v) − R(η, v 0 )\| + \|R(η, v 0 ) − R(η 0 , v 0 )\| ≤ ǫ 8 + 1 2 \|v − v 0 \| + ǫ 8 ≤ 3ǫ 4 ≤ ǫ, and we see that the previous contraction has range in V 0 . Consequently there is a fixed point v = y + R(η, v) ∈ V 0 ⊂ V. Hence y = v − R(η, v) = S η (v). It follows that W 0 × Y ⊂ ∪ η∈W {η} × S η (V ), which yields the assertion. 2. On assertion (ii). Let η ∈ W be given. It follows from assertion (i) that the set S η (V ) ⊂ R nk is open. 2.1. Proof that S η is injective in case V = R nk . Let v,ṽ in V be given with S η (v) = S η (ṽ). Then v −ṽ = R(η, v) − R(η,ṽ). Without loss of generality, dist(ṽ, R nk \V ) ≤ dist(v, R nk \V ). Using Proposition 3.1 (ii) we infer \|ṽ−v\| = \|R(η,ṽ)−R(η, v)\| < 1 2 (dist(ṽ, R nk \V )+dist(v, R nk \V )) ≤ dist(v, R nk \V ). It follows that the line segment v + [0, 1](ṽ − v) belongs to V , and the Mean Value Theorem in combination with Proposition 3.1 (i) yields \|v −ṽ\| = \|R(η, v) − R(η,ṽ)\| ≤ 1 2 \|v −ṽ\|, which gives us v =ṽ. 2.2. The proof of injectivity of S η for V = R nk is simpler due to convexity. From the estimates \|id − DS η (v)\| Lc(R nk ,R nk ) = \|D 2 R(η, v)\| Lc(R nk ,R nk ) ≤ 1 2 < 1 for v ∈ V we infer that each DS η (v), v ∈ V , is an isomorphism. Therefore the Inverse Mapping Theorem applies and yields that S −1 η is given by continuously differentiable maps on neighbourhoods of the values y ∈ S η (V ). On assertion (iii). For every η ∈ W and y = S η (v) with v ∈ V we have that v = S −1 η (y) satisfies y − (v − R(η, v)) = 0, or equivalently, F (η, y, v) = 0 for the continuously differentiable map F : (∪ η∈W {η} × S η (V )) × V → R nk given by F (η, y, v) = y − (v − R(η, v)). Because of the estimates \|D 3 F (η, y, v) − id\| Lc(R nk ,R nk ) = \|D 2 R(η, v)\| Lc(R nk ,R nk ) ≤ 1 2 < 1 for η ∈ W, y ∈ S η (V ), and v ∈ V, each map D 3 F (η, y, v) is an isomorphism. The Implicit Function Theorem applies and yields that the map ∪ η∈W {η} × S η (V ) ∋ (η, y) → S −1 η (y) ∈ R nk is locally given by continuously differentiable maps. Using Proposition 3.2 (i) and continuity we obtain that the set Proof. Let φ ∈ U be given and set χ = A(φ). Then Lχ = Lφ ∈ W , and y = χ satisfies y = S η (v) for v = φ and η = Lφ. Hence χ ∈ S η (V ). It follows that A(φ) = χ ∈ O, and we obtain A(U ) ⊂ O. With φ, χ, y, v, η as before, v = S −1 η (y), and thereby, Proof. 1. In order to obtain B(O) ⊂ U let χ ∈ O be given and set φ = B(χ), η = Lχ, y = χ. As Lφ = Lχ ∈ W we may consider φ ∈ R nk . In order to show φ ∈ U we need to verify φ ∈ V . From χ ∈ O we have y = S η (v) for some v ∈ V . For every ι = (κ − 1)n + ν with κ ∈ {1, . . . , k} and ν ∈ {1, . . . , n}, φ ι = φ ν (−d κ (Lφ)) = χ ν (−d κ (Lφ)) + g ν (S −1 η (y))(H ν (S −1 η (y)))(−d κ (Lφ)) = χ ν (−d κ (Lχ)) + g ν (v)(H ν (v))(−d κ (Lχ)) = y ι + g ν (v)(H ν (v))(−d κ (η)) = y ι + R ι (η, v) = S ι (η, v) + R ι (η, v) = v ι , which yields φ = v ∈ V . 2. For χ, φ, η, y, v as in Part 1 of the proof we saw that φ = v. It follows that A(B(χ)) = A(φ) = φ − g( φ) · H( φ) = [χ + g(S −1 η (y)) · H(S −1 η (y))] − g(v) · H(v) = χ (with v = S −1 η (y)). B(A(φ)) = B(χ) = χ + g(v) · H(v) = [φ − g(v) · H(v)] + g(v) · H(v) = φ. Theorem 3.5 The map A is an almost graph diffeomorphism with respect to the continuously differentiable submanifold X f , to the closed subspace X 0 of codimension n in C 1 n , and to the open subset U ⊃ X f of C 1 n , and the solution manifold X f is an almost graph over X 0 . open with respect to the relative topology on B. Analogously for A closed in B. The relation A ⊂⊂ B for open subsets of T means that the closure A of A is compact and contained in B. Proposition 2. 2 2Let a continuous function h : V → (0, ∞) and ν ∈ {1, . . . , n} be given. Then there exists a continuously differentiable map j=1,...,nk;ν=1,...,n \|∂ j g ν (v)\| + max ν=1,...,n \|g ν (v)\|]h(v) ≤ 1 2 (see the choice of h). O = {χ ∈ C 1 n : Lχ ∈ W and χ ∈ S η (V ) for η = Lχ} is open. The map B : O → C 1 n given byB(χ) = χ + g(v) · H(v) for v = S −1 η (y), y = χ, η = Lχis continuously differentiable. Analogously to Eq. (4) we have LB(χ) = Lχ for every χ ∈ O. Proposition 3.3 A(U ) ⊂ O and B(A(φ)) = φ for all φ ∈ U . Proposition 3. 4 4B(O) ⊂ U and A(B(χ)) = χ for all χ ∈ O. An example of the form (1) with d : R → (0, r] in[13, Section 3] shows that in general solution manifolds do not admit any graph representation. However,[13, Theorem 5.1] says that for a class of systems with discrete state-dependent delays all of which are strictly positive the solution manifolds are nearly as simple as a graph, namely, they are almost graphs over X 0 in the sense of the following definition from[14, Section 1]: A continuously differentiable submanifold X of a Banach space E is called an almost graph over a closed subspace H ⊂ E if H has a closed complementary subspace in E and if there is a continuously differentiable map α : H ⊃ dom → E, dom an open subset of H, such thatsee the proof of [5, Lemma 1] and [13, Theorem 2.4], respectively. 14, Section 1]: For E, X, H as above, and for an open set O ⊂ E, a diffeomorphism A : O → E onto an open subset of E is called an almost graph diffeomorphism with respect to X, H, and O, if 1 n}, κ ∈ {1, . . . , k}, are continuously differentiable, compare Part 2.1 of the proof of [14, Proposition 2.1]. 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[ "A fully hybrid integrated Erbium-based laser", "A fully hybrid integrated Erbium-based laser" ]	[ "Yang Liu \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Zheru Qiu \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Xinru Ji \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Andrea Bancora \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Grigory Lihachev \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Johann Riemensberger \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Rui Ning Wang \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Andrey Voloshin \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n", "Tobias J Kippenberg \nInstitute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n\nCenter for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland\n" ]	[ "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Institute of Physics\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland", "Center for Quantum Science and Engineering\nSwiss Federal Institute of Technology Lausanne (EPFL)\nCH-1015LausanneSwitzerland" ]	[]	Erbium-doped fiber lasers [1-3] exhibit high coherence and low noise as required for applications in fiber optic sensing[4], gyroscopes, Li-DAR, and optical frequency metrology[5]. Endowing Erbium-based gain in photonic integrated circuits can provide a basis for miniaturizing lownoise fiber lasers to chip-scale form factor, and enable large-volume applications. Yet, while major progress has been made in the last decade on integrated lasers based on silicon photonics with III-V gain media[6][7][8][9][10][11][12][13][14], the integration of Erbium lasers on chip has been compounded by large laser linewidth. Recent advances in photonic integrated circuit-based highpower Erbium-doped amplifiers make a new class of rare-earth-ion-based lasers possible[15]. Here, we demonstrate a fully integrated chip-scale Erbium laser that achieves high power, narrow linewidth, frequency agility and the integration of a III-V pump laser. The laser circuit is based on an Erbium-implanted ultralow-loss silicon nitride (Si 3 N 4 ) photonic integrated circuit[15]. This device achieves single-mode lasing with a freerunning intrinsic linewidth of 50 Hz, a relative intensity noise of <-150 dBc/Hz at >10 MHz offset, and an output power up to 17 mW, approaching the performance of fiber lasers [16] and stateof-the-art semiconductor extended cavity lasers[8,12,13,17]. An intra-cavity microring-based Vernier filter enables wavelength tunability of > 40 nm within the C-and L-bands while attaining side mode suppression ratio (SMSR) of > 70 dB, surpassing legacy fiber lasers in tuning and SMRS performance. This new class of lownoise, tuneable Erbium waveguide laser could find applications in LiDAR [18], microwave photonics[19,20], optical frequency synthesis[21], and freespace communications. Our approach is extendable to other wavelengths where rare-earth ions can provide gain, and more broadly, constitutes a novel way to photonic integrated circuit-based rare-earth-ion-doped lasers.Erbium-doped fiber lasers (EDFLs) [1-3] have become indispensable sources of high coherence laser light for distributed acoustic sensing[4,22], optical gyroscopes, free-space optical transmission [23], optical frequency metrology[5], and high-power laser machining[24,25]and are considered the 'gold standard' of laser phase noise. EDFLs exhibit many advantages such as all-fiberized cavities, alignment-free components, and benefit from the advantageous Erbium-based gain properties including slow gain dynamics, temperature insensitivity, low amplification related noise figure, lower spontaneous emission power coupled to oscillating modes than short semiconductor gain media[26,27], and excellent confinement of laser radiation for high beam quality. These properties along with low phase noise have led to wide proliferation of Erbium-based fiber lasers in industrial applications. Erbium ions can provide equally a basis for compact photonic integrated circuitbased lasers [28] that can benefit from manufacturing at lower cost, smaller form factor and reduced susceptibility to environmental vibrations compared to fiber lasers. Prior efforts have been made to implement chip-based waveguide lasers using Erbium-doped materials such as Al 2 O 3 [29, 30], TeO 2 [31], LiNbO 3 [32], and Erbium silicate compounds [33] as waveguide claddings or cores, but the demonstrated laser intrinsic linewidth remained at the level of MHz [34-37], far above the sub-100-Hz linewidth achieved in commercial fiber lasers and state-of-theart heterogeneously or hybrid integrated semiconductorbased lasers (Supplementary Note 1).One major obstacle to realizing narrow-linewidth Erbium waveguide lasers is the challenge of integrating long and low-loss active waveguides ranging from tens of centimeters to meters-the lengths routinely deployed in fiber lasers to ensure low phase noise, single-frequency operation, and sufficient round-trip gain[16].Here, we overcome this challenge and demonstrate hybrid integrated Erbium-doped waveguide lasers (ED-WLs) using Si 3 N 4 photonic integrated circuits that achieve narrow linewidth, frequency agility, high power, and the integration with pump lasers. Crucial to this advance are meter-scale-long Erbium-implanted silicon nitride (Er:Si 3 N 4 ) photonic integrated circuits that can provide > 30 dB net gain[15]with >100 mW output power. The Si 3 N 4 photonic integrated circuit moreover exhibits absence of two-photon absorption in telecommunication bands[38], radiation hardness for space compatibility, high power handling of up to tens of watts[39], a lower temperature sensitivity than silicon, and low Brillouin scattering (a power-limiting factor in silica-based fiber lasers)[40]. arXiv:2305.03652v1 [physics.optics] 5 May 2023	null	[ "https://export.arxiv.org/pdf/2305.03652v1.pdf" ]	258,546,726	2305.03652	71089c4a6c17c50cdd27d2851b585c50cdb563e8	A fully hybrid integrated Erbium-based laser Yang Liu Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Zheru Qiu Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Xinru Ji Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Andrea Bancora Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Grigory Lihachev Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Johann Riemensberger Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Rui Ning Wang Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Andrey Voloshin Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Tobias J Kippenberg Institute of Physics Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland Center for Quantum Science and Engineering Swiss Federal Institute of Technology Lausanne (EPFL) CH-1015LausanneSwitzerland A fully hybrid integrated Erbium-based laser Erbium-doped fiber lasers [1-3] exhibit high coherence and low noise as required for applications in fiber optic sensing[4], gyroscopes, Li-DAR, and optical frequency metrology[5]. Endowing Erbium-based gain in photonic integrated circuits can provide a basis for miniaturizing lownoise fiber lasers to chip-scale form factor, and enable large-volume applications. Yet, while major progress has been made in the last decade on integrated lasers based on silicon photonics with III-V gain media[6][7][8][9][10][11][12][13][14], the integration of Erbium lasers on chip has been compounded by large laser linewidth. Recent advances in photonic integrated circuit-based highpower Erbium-doped amplifiers make a new class of rare-earth-ion-based lasers possible[15]. Here, we demonstrate a fully integrated chip-scale Erbium laser that achieves high power, narrow linewidth, frequency agility and the integration of a III-V pump laser. The laser circuit is based on an Erbium-implanted ultralow-loss silicon nitride (Si 3 N 4 ) photonic integrated circuit[15]. This device achieves single-mode lasing with a freerunning intrinsic linewidth of 50 Hz, a relative intensity noise of <-150 dBc/Hz at >10 MHz offset, and an output power up to 17 mW, approaching the performance of fiber lasers [16] and stateof-the-art semiconductor extended cavity lasers[8,12,13,17]. An intra-cavity microring-based Vernier filter enables wavelength tunability of > 40 nm within the C-and L-bands while attaining side mode suppression ratio (SMSR) of > 70 dB, surpassing legacy fiber lasers in tuning and SMRS performance. This new class of lownoise, tuneable Erbium waveguide laser could find applications in LiDAR [18], microwave photonics[19,20], optical frequency synthesis[21], and freespace communications. Our approach is extendable to other wavelengths where rare-earth ions can provide gain, and more broadly, constitutes a novel way to photonic integrated circuit-based rare-earth-ion-doped lasers.Erbium-doped fiber lasers (EDFLs) [1-3] have become indispensable sources of high coherence laser light for distributed acoustic sensing[4,22], optical gyroscopes, free-space optical transmission [23], optical frequency metrology[5], and high-power laser machining[24,25]and are considered the 'gold standard' of laser phase noise. EDFLs exhibit many advantages such as all-fiberized cavities, alignment-free components, and benefit from the advantageous Erbium-based gain properties including slow gain dynamics, temperature insensitivity, low amplification related noise figure, lower spontaneous emission power coupled to oscillating modes than short semiconductor gain media[26,27], and excellent confinement of laser radiation for high beam quality. These properties along with low phase noise have led to wide proliferation of Erbium-based fiber lasers in industrial applications. Erbium ions can provide equally a basis for compact photonic integrated circuitbased lasers [28] that can benefit from manufacturing at lower cost, smaller form factor and reduced susceptibility to environmental vibrations compared to fiber lasers. Prior efforts have been made to implement chip-based waveguide lasers using Erbium-doped materials such as Al 2 O 3 [29, 30], TeO 2 [31], LiNbO 3 [32], and Erbium silicate compounds [33] as waveguide claddings or cores, but the demonstrated laser intrinsic linewidth remained at the level of MHz [34-37], far above the sub-100-Hz linewidth achieved in commercial fiber lasers and state-of-theart heterogeneously or hybrid integrated semiconductorbased lasers (Supplementary Note 1).One major obstacle to realizing narrow-linewidth Erbium waveguide lasers is the challenge of integrating long and low-loss active waveguides ranging from tens of centimeters to meters-the lengths routinely deployed in fiber lasers to ensure low phase noise, single-frequency operation, and sufficient round-trip gain[16].Here, we overcome this challenge and demonstrate hybrid integrated Erbium-doped waveguide lasers (ED-WLs) using Si 3 N 4 photonic integrated circuits that achieve narrow linewidth, frequency agility, high power, and the integration with pump lasers. Crucial to this advance are meter-scale-long Erbium-implanted silicon nitride (Er:Si 3 N 4 ) photonic integrated circuits that can provide > 30 dB net gain[15]with >100 mW output power. The Si 3 N 4 photonic integrated circuit moreover exhibits absence of two-photon absorption in telecommunication bands[38], radiation hardness for space compatibility, high power handling of up to tens of watts[39], a lower temperature sensitivity than silicon, and low Brillouin scattering (a power-limiting factor in silica-based fiber lasers)[40]. arXiv:2305.03652v1 [physics.optics] 5 May 2023 Erbium-doped fiber lasers [1][2][3] exhibit high coherence and low noise as required for applications in fiber optic sensing [4], gyroscopes, Li-DAR, and optical frequency metrology [5]. Endowing Erbium-based gain in photonic integrated circuits can provide a basis for miniaturizing lownoise fiber lasers to chip-scale form factor, and enable large-volume applications. Yet, while major progress has been made in the last decade on integrated lasers based on silicon photonics with III-V gain media [6][7][8][9][10][11][12][13][14], the integration of Erbium lasers on chip has been compounded by large laser linewidth. Recent advances in photonic integrated circuit-based highpower Erbium-doped amplifiers make a new class of rare-earth-ion-based lasers possible [15]. Here, we demonstrate a fully integrated chip-scale Erbium laser that achieves high power, narrow linewidth, frequency agility and the integration of a III-V pump laser. The laser circuit is based on an Erbium-implanted ultralow-loss silicon nitride (Si 3 N 4 ) photonic integrated circuit [15]. This device achieves single-mode lasing with a freerunning intrinsic linewidth of 50 Hz, a relative intensity noise of <-150 dBc/Hz at >10 MHz offset, and an output power up to 17 mW, approaching the performance of fiber lasers [16] and stateof-the-art semiconductor extended cavity lasers [8,12,13,17]. An intra-cavity microring-based Vernier filter enables wavelength tunability of > 40 nm within the C-and L-bands while attaining side mode suppression ratio (SMSR) of > 70 dB, surpassing legacy fiber lasers in tuning and SMRS performance. This new class of lownoise, tuneable Erbium waveguide laser could find applications in LiDAR [18], microwave photonics [19,20], optical frequency synthesis [21], and freespace communications. Our approach is extendable to other wavelengths where rare-earth ions can provide gain, and more broadly, constitutes a novel way to photonic integrated circuit-based rare-earth-ion-doped lasers. Erbium-doped fiber lasers (EDFLs) [1][2][3] have become indispensable sources of high coherence laser light for distributed acoustic sensing [4,22], optical gyroscopes, free-space optical transmission [23], optical frequency metrology [5], and high-power laser machining [24,25] and are considered the 'gold standard' of laser phase noise. EDFLs exhibit many advantages such as all-fiberized cavities, alignment-free components, and benefit from the advantageous Erbium-based gain properties including slow gain dynamics, temperature insensitivity, low amplification related noise figure, lower spontaneous emission power coupled to oscillating modes than short semiconductor gain media [26,27], and excellent confinement of laser radiation for high beam quality. These properties along with low phase noise have led to wide proliferation of Erbium-based fiber lasers in industrial applications. Erbium ions can provide equally a basis for compact photonic integrated circuitbased lasers [28] that can benefit from manufacturing at lower cost, smaller form factor and reduced susceptibility to environmental vibrations compared to fiber lasers. Prior efforts have been made to implement chip-based waveguide lasers using Erbium-doped materials such as Al 2 O 3 [29,30], TeO 2 [31], LiNbO 3 [32], and Erbium silicate compounds [33] as waveguide claddings or cores, but the demonstrated laser intrinsic linewidth remained at the level of MHz [34][35][36][37], far above the sub-100-Hz linewidth achieved in commercial fiber lasers and state-of-theart heterogeneously or hybrid integrated semiconductorbased lasers (Supplementary Note 1). One major obstacle to realizing narrow-linewidth Erbium waveguide lasers is the challenge of integrating long and low-loss active waveguides ranging from tens of centimeters to meters-the lengths routinely deployed in fiber lasers to ensure low phase noise, single-frequency operation, and sufficient round-trip gain [16]. Here, we overcome this challenge and demonstrate hybrid integrated Erbium-doped waveguide lasers (ED-WLs) using Si 3 N 4 photonic integrated circuits that achieve narrow linewidth, frequency agility, high power, and the integration with pump lasers. Crucial to this advance are meter-scale-long Erbium-implanted silicon nitride (Er:Si 3 N 4 ) photonic integrated circuits that can provide > 30 dB net gain [15] with >100 mW output power. The Si 3 N 4 photonic integrated circuit moreover exhibits absence of two-photon absorption in telecommunication bands [38], radiation hardness for space compatibility, high power handling of up to tens of watts [39], a lower temperature sensitivity than silicon, and low Brillouin scattering (a power-limiting factor in silica-based fiber lasers) [40]. RESULTS Hybrid integrated Erbium-based Vernier lasers The laser device is structured as a linear optical cavity with a spiral Erbium-doped gain waveguide and two reflectors formed by Sagnac loop mirrors at both ends (Fig.1A). One dichroic loop mirror that consists of a dichroic directional coupler allows for laser reflection near 1550 nm and optical pump transmission near 1480 nm, and the other reflector deploys a short waveguide splitter for broadband reflection. The optical pump can also be injected via a waveguide taper connected to a microring bus waveguide. The laser device (Fig.1B) exhibits a compact footprint of only 2 × 3 mm 2 with a densely-packed 0.2-m-long Erbium-doped Si 3 N 4 spiral waveguide (Fig.1C) with a cross section of 0.7 × 2.1 µm 2 . A narrow-band intra-cavity Vernier filter designed to achieve sub-GHz 3 dB bandwidth and 5 THz FSR using two cascaded add-drop microring resonators (100 GHz FSRs with 2 GHz difference) ( Fig.1D) is deployed to ensure single-mode lasing operation with a small laser cavity mode spacing of ca. 200 MHz (Supplementary Note 2). Integrated microheaters are used to align the Vernier filter peak transmission wavelength to a cavity longitudinal mode. This integrated laser circuit was fabricated using the photonic Damascene process [41], followed by selective Erbium ion implantation, post annealing, and heater fabrication ( Fig.1E; see Methods and Supplementary Note 3). To demonstrate a fully integrated EDWL, we performed photonic packaging via hybrid integration in a custom 14-pin butterfly package. The 1480 nm InP Fabry-Pérot (FP) laser diode (LD) was edge coupled to one of the laser cavities on an Er:Si 3 N 4 photonic integrated circuit ( Fig.2A), with simulated coupling loss of < 3 dB. The laser output waveguide was end-coupled and glued with a cleaved UHNA-7 optical fiber spliced to a SMF-28 optical fiber pigtail, exhibiting 2.7 dB coupling loss at 1550 nm. The pump LD, a Peltier element, a thermistor, and all microheaters are connected to butterfly pins using wire bonding. The integrated micro-heaters were used for the temperature control of the Vernier filter and the phase-shifter section to configure single-mode lasing and wavelength tuning. The Erbium ions can be optically excited by the pump light emitted from the multilongitudinal-mode pump LD (>4 nm spectral linewidth near 1480 nm), providing 1.9 dB/cm of measured net gain coefficient [15]. The optical spectrum of the collected laser output shows a single-mode lasing operation with > 70 dB of side mode suppression ratio (SMSR) at 0.1 nm resolution bandwidth (Fig.2B). This high 72-dB SMSR was made possible using the drop port of the the narrow passband intra-cavity Vernier filter, which can select the lasing mode and reject the broadband amplified spontaneous emission noise. This record high SMSR surpasses what has been reported in integrated Erbium lasers, fiber lasers, and integrated semiconductor-based lasers (Supplementary Note 1), typically below 60 dB that is usually limited by intra-cavity filtering performance. Conversely, this is challenging to implement in legacy fiber-based Erbium lasers where the filtering components based on long Bragg gratings can only offer several GHz wide passband with grating side lobes and lack of broadband wavelength tuning capability. We observed an off-chip lasing threshold pump power of ca. 20 mW and an on-chip slope efficiency of 6.7 % when sweeping the pump power (Fig.2B inset), which can be further optimized by reducing the coupling loss and the cavity loss. The fully packaged laser showed a frequency drift of < 20 MHz over 4 hours (Fig.2C) when performing a heterodyne beatnote measurement with a fully-stabilized optical frequency comb indicating a good frequency stability due the monolithic nature of the laser comprised of both cavity and gain medium. During a 24-hour test, this laser showed a frequency drift of < 140 MHz without mode hops (Supplementary Note 4), representing a comparable long-term frequency stability as a commercial diode laser (Toptica CTL). Single-mode lasing and wavelength tuning The use of photonic integrated circuits and Vernier structures (Fig.2) enables to endow the integrated Erbium laser with broad wavelength tuning, a capability that bulk fiber lasers lack. We investigated the intracavity filtering properties by characterizing the optical transmission of the middle bus waveguide (Fig.2D). The measured transmission of the individual resonators used for the Vernier filter is shown in Fig.2F and the designed 2 GHz FSR was experimentally attained (98 GHz and 100 GHz, respectively), leading to a measured Vernier filter FSR of 4.65 THz that corresponds to 37.1 nm span near 1550 nm wavelength (Fig.2G,H). Such a large Vernier FSR ensures the single-wavelength lasing within the Erbium emission wavelength range (Fig.2G). By overlapping the resonances from the two resonators, i.e. vanishing the frequency spacing (Fig.2H), the lasing wavelength is determined. By fitting the resonance linewidth near 194.8 THz (Fig.2I), we obtain an external coupling rate κ ex,0 /2π = 411 MHz (between the microring and the bus waveguide) and an intrinsic loss rate κ 0 /2π = 42.5 MHz. This strong over-coupled configuration ( κex κ0 > 10) can ensure that the Vernier filter simultaneously achieves a narrow 3-dB passband bandwidth of 636 MHz and in principle a low insertion loss. Such strong overcoupling can allow for low loss operation of the Vernier filter, which however in the current device was not attained. The Vernier filter exhibits an insertion loss of -3.2 dB due to the parasitic loss induced by the coupling from the fundamental waveguide mode to higher order modes, which leads to a suboptimal coupling ideality [42] of I = 0.87 (Supplementary Note 5). Next, we demonstrate the wavelength tunability (Fig. 3A). The coarse tuning of laser wavelength was carried out by switching the aligned resonance of two microresonators (Fig. 3B). The step size of ca. 0.8 nm was determined by the microring FSR. Fine tuning of the wavelength can be achieved by simultaneously shifting the two resonators in the same direction and adjusting the phase shifter to align the corresponding cavity longitudinal mode (164 MHz spacing) to the Vernier filter passband. Figure 3C shows the 2-dimensional (2D) laser wavelength tuning map when varying the electrical power applied to the microheaters. From the recorded entire 2D map of wavelengths we selected the settings marked in Fig.3C. This allowed for continuous and deterministic tuning over the entire wavelength band from 1548.1 nm to 1585.8 nm, maintaining power of > 4 mW and SMSR of > 70 dB (Fig.3D). Such wavelength tunability cannot be achieved in conventional rare-earth-ion-doped fiber lasers without the use of free space etalon filters. The wavelength tuning range was limited by the Vernier filter FSR and the wavelength-division multiplexing coupler transmission band (Supplementary Note 6). During heater power scanning, we note that a few of wavelength tuning steps were missed due to the misalignment of microring resonances of the Vernier filter . During tuning, the phase shifter was adjusted to maximize the output power at the desired mode. A maximum fibercoupled output power of ca. 17 mW were measured at 1585 nm with 219 mW pump power. Other competing lasing modes apart from the predominant lasing mode were observed when using high pump power, due to the fact that the large Si 3 N 4 waveguide cross section allows for multiple transversal optical modes that can coincidently satisfy the lasing condition (Supplementary Note 8). Frequency and intensity noise measurements To demonstrate the low noise features of the freerunning EDWLs, we characterized the frequency noise, the intrinsic laser linewidth, and the relative intensity noise (RIN), respectively (Fig. 4A). Firstly, a reference external cavity diode laser (free running Toptica CTL) was tuned close to the lasing wavelength near 1560 nm of an EDWL (not packaged) with ca. 3 mW output power for heterodyne photodetection. The in-phase and quadrature components of the sampled beatnote time trace was processed using Welch's method [43] to obtain the single-side power spectral density (PSD) of frequency noise S δv (f ). The frequency noise PSD (red line) reached a plateau of h 0 = 62.0 Hz 2 /Hz at the offset frequency of 6 MHz, corresponding to a Lorentzian linewidth of πh 0 = 194.8 Hz; this measured white noise floor was masked by the ECDL's white noise floor (Fig. 4C). We also applied the delayed self-heterodyne interferometric measurement [44] to validate the intrinsic linewidth (Fig. 4D), which generates a power spectrum of the autocorrelation of the laser line under sub-coherence condition (Supplementary Note 9). In the offset frequency range from 10 kHz to 2.5 MHz where a relaxation oscillation peak was observed, the Erbium laser shows a higher frequency noise due to the laser cavity fluctuation caused by the pump laser noise transduction and the thermorefractive noise in the microresonator [45]. The measured frequency noise at offset frequencies of <10 kHz was dominated by ECDL characteristic noise features [46]. We achieved a record low intrinsic linewidth (blue line) of πh 0 = 50.1 Hz (h 0 = 15.9 Hz 2 /Hz ) in an Erbium waveguide laser with a higher output power of 10 mW, when beating against a low-noise Erbium fiber laser (Koheras Adjustik). The fully packaged EDWL (Fig. 4B) with 2.8 mW output power shows a comparable intrinsic linewidth (purple line) and a lower frequency noise at the mid-range offset frequencies. Using laser cavity designs with reduced cold cavity losses and increased mode area, hertz-linewidth EDWL can be feasibly achieved (Supplementary Note 10). The full width at half maximum (FWHM) of the integral linewidth associated with Gaussian contribution was obtained by integrating the frequency noise PSD from the inverse of measurement time (1/T 0 ) up to the frequency where S δv (f ) intersects with the β-separation line S δv (f ) = 8 ln(2)f /π 2 (dashed line) [47]. With the integrated surface A, we obtained a minimum FWHM linewidth (8 ln(2)A 1/2 ) of the free-running EDWL is 82.2 kHz at 1 ms measurement time, which does not yet supersede a fiber laser, but is lower than 166.6 kHz of an ECDL (Toptica CTL) characterized as a reference The waveguide laser shows a RIN down to −130 dBc/Hz (yellow& purple) at mid-range offset frequencies between 10 kHz and 1 MHz, lower than the fiber laser RIN (grey) that has a PSD pole induced by relaxation oscillation (Fig. 4E). The mid-range RIN was mainly limited by the pump laser RIN transduction which even contributed to an increased RIN by 5 dB for the un-packaged EDWL. The pump RIN noise transduction at frequency above 20 MHz was suppressed due to the slow dynamics of Erbium ions. The relaxation oscillation frequency can be calculated by f r ≈ 1 2π Pcavκ Psatτ where P sat is saturation power of the gain medium, P cav is the laser cavity power,κ is the cold cavity loss rate, and τ is the Erbium ion upper-state lifetime (Supplementary Note 12). The waveguide laser RIN reduced to < −155 dBc/Hz at offset frequencies of > 10 MHz. We observed that the relaxation oscillation frequency the waveguide laser varied from 0.3 MHz to 2.4 MHz when increasing the optical pump power (Fig. 4F), which is higher than the case in the fiber laser (typically < 100 kHz ). This higher relaxation oscillation frequency originates from the smaller saturation power and the shorter Erbium upper-state lifetime of 3.4 ms [15]. Summary In summary, we have demonstrated a photonic integrated circuit-based Erbium laser that achievesd sub-100 Hz intrinsic linewidth, low RIN noise, > 72 dB SMSR, and 40 nm wide wavelength tunability with power exceeding 10 mW. The Erbium-doped waveguide lasers use foundry compatible silicon nitride waveguides, and have the potential to combine fiber-laser coherence with low size, weight, power and cost of integrated photonics. Such a laser may find application in existing applications such as coherent sensing, and may equally provide a disruptive solution for emerging applications that require high volumes, such as lasers for coherent FMCW LiDAR, or for coherent optical communications where iTLA (integrated tunable laser assembly) have been widely deployed, but fiber lasers' high coherence is increasingly demanded for advanced high-speed modulation formats while their use has been impeded by the high cost and large size. Co-doping other rare-earth ions such as ytterbium (emission at 1.1 µm) and thulium (0.8 µm, 1.45 µm and 2.0 µm) will moreover allow access to other wavelengths. Looking to the future, the compatibility of silicon nitride with heterogeneously integrated thinfilm lithium niobate [48], as well as piezoelectric thin films [11,49], and Erbium waveguide amplifiers [15] provides the capability to create fully-integrated highspeed, low-noise, high-power optical engines for LiDAR, long-haul optical coherent communications, and analog optical links. Author contributions: Y.L. and Z.Q. performed the experiments. Y.L. carried out data analysis and simulations. Y.L. and Z.Q. designed Si3N4 waveguide laser chips. X.J., G. L. and J.R. provided experimental supports. A. B. and A. V. designed and performed the device packaging. R.N.W., Z.Q. and X.J. fabricated the passive Si3N4 samples. Y.L. wrote the manuscript with the assistance from Z.Q. and the input from all co-authors. T.J.K supervised the project. Data Availability Statement: The code and data used to produce the plots within this work will be released on the repository Zenodo upon publication of this preprint. Competing interests T.J.K. is a cofounder and shareholder of LiGenTec SA, a start-up company offering Si3N4 photonic integrated circuits as a foundry service. * These authors contributed equally to this work. †[email protected] ‡[email protected] METHODS Device fabrication and ion implantation This ultralow loss Si 3 N 4 photonic integrated laser circuit is fabricated using the photonic Damascene process [41]. We applied selective Erbium ion implantation [50] (a total fluence of 1 × 10 16 ions cm −2 at a maximum beam energy of 2 MeV) to the pre-fabricated passive Si 3 N 4 photonic integrated circuits to endow the spiral waveguide with Erbium-based optical gain, while we kept other passive components un-doped by selectively masking a portion of the chip with photoresist ( Fig.1E and Supplementary Note 1). We achieved a high doping concentration of 3.25 × 10 20 ions cm −3 , more than one order of magnitude higher than that of conventional Erbiumdoped fibers. This allows for high roundtrip net gain of 1.9 dB/cm (characterized from a 4.5-mm-long Erbiumdoped waveguide) [15]. After ion implantation, we annealed the sample at 1000°C for one hour to activate the Erbium ions and heal implantation defects. The optical gain is provided by the stimulated emission of erbium ions excited by an optical pump (Fig.1A inset). Microheaters were subsequently added atop the silica upper cladding after the ion implantation and post annealing processes. Figure 1 . 1A hybrid integrated Er:Si3N4 laser. (A) Schematic of a hybrid integrated Vernier laser consisting of an Erbium-implanted silicon nitride Er:Si3N4 photonic integrated circuit and an edge-coupled III-V semiconductor pump laser diode. The intra-cavity microringbased Vernier filter enables single-mode lasing operation within the Erbium-based gain bandwidth. (B) Optical image of an Er:Si3N4 laser circuit integrated with micro heaters for wavelength and phase tuning. The green dashed circle indicates the Erbium-implanted gain spiral. (C) Optical images of the Erbium-implanted spiral waveguides and (D) the coupling regime of the Vernier filter indicated by coloured boxes in (B). (E) Fabrication process flow of the Er:Si3N4 photonic integrated circuit based on selective Erbium ion implantation. Figure 2 . 2A hybrid integrated Er:Si3N4 vernier laser operated at single-mode lasing.(A) Optical image of a hybrid integrated Er:Si3N4 Vernier laser edge-coupled with a pump laser diode chip (3SP Technologies, 1943 LCv1). Green luminescence was observed, stemming from the transition from higher-lying levels of excited Erbium ions to the ground state.(B) Measured optical spectrum of single-mode lasing. The inset shows the output power as a function of the pump power. (C) Measured time-frequency spectrogram of the heterodyne beatnote between the packaged EDWL and a fully-stabilized frequency comb (FC1500, Menlo Systems GmbH) over 4 hours. (D) Experimental setup for Vernier filter characterization (device ID: D85_04_F04_C15_V1). (E) Illustration of the Vernier effect by measuring the superposed resonances through the intermediate bus waveguide of the Vernier filter. (F) Zoomed-in range of the measured transmission. Colored circles indicate the resonances of each microring. (G) Wide-range transmission overlaid with the Erbium ion gain spectrum. (H) Frequency spacing variation between adjacent ring resonances, yielding a Vernier spacing of 4.65 THz, corresponding to 37.1 nm. (I) The curve fitting of the measured through port transmission of the resonance indicated in (F), and the calculated filtering response at the drop port. Figure 3 . 3Demonstration of wideband tuning of the laser wavelength.(A) Experimental setup for laser wavelength tuning demonstration (device ID: D85_04_F8_C508_VL2.0) (B) Operating principle of the wavelength tuning of the vernier laser. The traces indicate the transmission of each over-coupled microresonator and the entire vernier filter, respectively. (C) Two dimensional laser wavelength tuning map, showing the wavelength of the predominant lasing mode as a function of the electrical power applied to the two micro heaters. The dashed schematically indicates the approach to coarse wavelength tuning. The white regions indicate that the expected laser emission was missing due to the microring resonance misalignment or a competing lasing mode when approaching the edge of the WDM filter transmission band. (D) Measured optical spectra of single-mode lasing tuned over 40 nm wavelength range. The optical spectrum analyzer's resolution bandwidth is set to 0.1 nm. Figure 4 . 4Laser noise properties and the fully hybrid integration of an EDWL. (A) Experimental setups for the measurement of laser frequency noise, relative intensity noise, and intrinsic laser linewidth. (B) Optical image of a fully hybrid integrated EDWL assembly. (C) Measured laser frequency noise based on heterodyne detection with reference lasers. EDWL1 device ID: D85_04_F8_C508_VL2.0; EDWL2 device ID: D85_04_F8_C508_VL2.1; EDWL3 device ID: D85_04_F8_C508_VL2.1. (D) Measured and fitted spectra of delayed selfheterodyne interferometric measurement for intrinsic laser linewidth investigation. (E) Measured laser relative intensity noise (RIN) based on direct photodetetion. (F) Relaxation oscillation peaks under varied pump power. laser for comparison. For comparison, the commercial stabilized fiber-based laser shows 2.4 kHz of the FWHM linewidth at 1 ms measurement time.Next, we show that the Erbium waveguide laser features a lower RIN compared to a commercial fiber laser (Koheras Adjustik, KOH45) (Supplementary Note 11). Funding Information: This work was supported by the Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-19-1-0250, and by contract W911NF2120248 (NINJA) from the Defense Advanced Research Projects Agency (DARPA), Microsystems Technology Office (MTO). This work further supported by the EU H2020 research and innovation programme under grant No. 965124 (FEMTOCHIP), by the SNSF under grant no. 201923 (Ambizione), and by the Marie Sklodowska-Curie IF grant No. 898594 (CompADC) and grant No. 101033663 (RaMSoM). 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[ "Probing anomalous quartic gauge-boson couplings via e + e − → 4 fermions + γ", "Probing anomalous quartic gauge-boson couplings via e + e − → 4 fermions + γ" ]	[ "A Denner \nPaul Scherrer Institut CH-5232 Villigen PSI\nSwitzerland\n", "S Dittmaier \nDeutsches Elektronen-Synchrotron DESY\nD-22603HamburgGermany\n", "† ", "M Roth \nInstitut für Theoretische Physik\nUniversität Leipzig\nD-04109LeipzigGermany\n", "D Wackeroth \nDepartment of Physics and Astronomy\nUniversity of Rochester Rochester\n14627-0171NYUSA\n" ]	[ "Paul Scherrer Institut CH-5232 Villigen PSI\nSwitzerland", "Deutsches Elektronen-Synchrotron DESY\nD-22603HamburgGermany", "Institut für Theoretische Physik\nUniversität Leipzig\nD-04109LeipzigGermany", "Department of Physics and Astronomy\nUniversity of Rochester Rochester\n14627-0171NYUSA" ]	[]	All lowest-order amplitudes for e + e − → 4f γ are calculated including five anomalous quartic gauge-boson couplings that are allowed by electromagnetic gauge invariance and the custodial SU(2) c symmetry. Three of these anomalous couplings correspond to the operators L 0 , L c , and L n that have been constrained by the LEP collaborations in WWγ production. The anomalous couplings are incorporated in the Monte Carlo generator RacoonWW. ‡ Moreover, for the processes e + e − → 4f γ RacoonWW is improved upon including leading universal electroweak corrections such as initial-state radiation. The discussion of numerical results illustrates the size of the leading corrections as well as the impact of the anomalous quartic couplings for LEP2 energies and at 500 GeV.April 2001 † Heisenberg fellow of the Deutsche Forschungsgemeinschaft ‡ RacoonWW can be downloaded from	10.1007/s100520100678	[ "https://export.arxiv.org/pdf/hep-ph/0104057v2.pdf" ]	119,348,944	hep-ph/0104057	ff80614962f60ec5669130f75f9884dcc4b1899a	Probing anomalous quartic gauge-boson couplings via e + e − → 4 fermions + γ Apr 2001 A Denner Paul Scherrer Institut CH-5232 Villigen PSI Switzerland S Dittmaier Deutsches Elektronen-Synchrotron DESY D-22603HamburgGermany † M Roth Institut für Theoretische Physik Universität Leipzig D-04109LeipzigGermany D Wackeroth Department of Physics and Astronomy University of Rochester Rochester 14627-0171NYUSA Probing anomalous quartic gauge-boson couplings via e + e − → 4 fermions + γ Apr 2001arXiv:hep-ph/0104057v2 6 All lowest-order amplitudes for e + e − → 4f γ are calculated including five anomalous quartic gauge-boson couplings that are allowed by electromagnetic gauge invariance and the custodial SU(2) c symmetry. Three of these anomalous couplings correspond to the operators L 0 , L c , and L n that have been constrained by the LEP collaborations in WWγ production. The anomalous couplings are incorporated in the Monte Carlo generator RacoonWW. ‡ Moreover, for the processes e + e − → 4f γ RacoonWW is improved upon including leading universal electroweak corrections such as initial-state radiation. The discussion of numerical results illustrates the size of the leading corrections as well as the impact of the anomalous quartic couplings for LEP2 energies and at 500 GeV.April 2001 † Heisenberg fellow of the Deutsche Forschungsgemeinschaft ‡ RacoonWW can be downloaded from Introduction In recent years, the experiments at LEP and the Tevatron have established the existence of elementary self-interactions among three electroweak gauge bosons, mainly by analysing the reactions e + e − → W + W − and pp → Wγ + X. The empirical bounds (see e.g. Ref. [ 1]) on anomalous triple gauge-boson couplings confirm the Standard-Model (SM) couplings at the level of a few per cent. Recently, the LEP collaborations have started to put also bounds on anomalous quartic gauge-boson couplings (AQGC) upon studying the processes e + e − → W + W − γ, e + e − → Zγγ, and e + e − → ννγγ. The OPAL [ 2], L3 [ 3] and ALEPH collaborations have already presented first results, which have been combined by the LEPEWWG [ 4]. The experimental analysis of anomalous triple and quartic gauge-boson couplings requires precise predictions from Monte Carlo generators including these anomalous couplings. In particular, it is necessary to account for the instability of the produced weak bosons, which decay into fermion-antifermion pairs. While several generators including triple gauge-boson couplings in e + e − → WW → 4f exist for quite a long time [ 5], up to now no generator has been available that deals with the processes e + e − → WW(γ) → 4f γ in the presence of AQGC. 1 As a preliminary solution [ 2,3] a reweighting technique was used in existing programs for e + e − → WW(γ) → 4f γ in the SM where the SM matrix elements were reweighted with the anomalous effects deduced from the program EEWWG [ 7] for on-shell WWγ production. The aim of this paper is to improve on this situation by implementing the relevant AQGC in the Monte Carlo generator RacoonWW [ 8], which is at present the only generator for all processes e + e − → 4f γ. SM predictions for all 4f γ final states obtained with this generator were presented in Ref. [ 9]; further results for specific final states can be found in Refs. [ 6,10,11]. Here we supplement these numerical results by a study of AQGC effects at LEP2 and linear collider energies. Moreover, we include two quartic gauge-boson operators in the analysis that, to the best of our knowledge, have not yet been considered in the literature before. As a second topic, we improve the RacoonWW predictions for 4f γ production by including the dominant leading electroweak corrections. In particular, we take into account additional initial-state radiation (ISR) at the leading-logarithmic level in the structurefunction approach of Ref. [ 12], where soft-photon effects are exponentiated and collinear logarithms are included up to order O(α 3 ). Leading universal effects originating from the renormalization of the electroweak couplings are included by using the so-called G µ -inputparameter scheme. The singular part [ 13] of the Coulomb correction, which is relevant for intermediate W-boson pairs in e + e − → WW(γ) → 4f γ near their kinematical threshold, is also taken into account. The paper is organized as follows. In Section 2 we introduce the relevant AQGC and give the corresponding Feynman rules, which are used in Section 3 to calculate the AQGC contributions to all e + e − → 4f γ amplitudes. In Section 4 we improve the tree-level predictions by including leading universal electroweak corrections. Section 5 contains our discussion of numerical results, which illustrate the impact of the leading corrections to the SM predictions as well as the effects of the AQGC. A summary is given in Section 6. Since we consider the class of e + e − → 4f γ processes in this paper, we restrict our analysis to anomalous quartic gauge-boson couplings (AQGC) that involve at least one photon. Moreover, we consider only genuine AQGC, i.e. we omit all operators that contribute also to triple gauge-boson couplings, such as the quadrilinear part of the well-known operator F µν W +,ρ ν W − ρµ . Imposing in addition a custodial SU(2) c invariance [ 14] to keep the ρ parameter close to 1, we are left with operators of dimension 6 or higher. Following Refs. [ 7,15,16,17] we consider dimension-6 operators for genuine AQGC that respect local U(1) em invariance and global custodial SU(2) c invariance. These symmetries reduce the set of such operators to a phenomelogically accessible basis. More general AQGC were discussed in Ref. [ 18]. In order to construct the relevant AQGC, it is convenient to introduce the triplet of massive gauge bosons W µ = W 1 µ , W 2 µ , W 3 µ = 1 √ 2 (W + + W − ) µ , i √ 2 (W + − W − ) µ , 1 c w Z µ , (2.1) where W ± µ and Z µ are the fields of the W ± and Z bosons, and the (abelian) field-strength tensors F µν = ∂ µ A ν − ∂ ν A µ , W i,µν = ∂ µ W i,ν − ∂ ν W i,µ , (2.2) where A µ is the photon field. The parameter c w is the cosine of the electroweak mixing angle. The quartic dimension-6 operators are obtained upon contracting two factors of W µ with two field-strength tensors. Under the explained symmetry assumptions there are five independent AQGC operators of dimension 6. We choose the following basis: L 0 = − e 2 16Λ 2 a 0 F µν F µν W α W α , L c = − e 2 16Λ 2 a c F µα F µβ W β W α , L n = − e 2 16Λ 2 a n ε ijk F µν W i µα W j ν W k,α , L 0 = − e 2 16Λ 2ã 0 F µνF µν W α W α , L n = − e 2 16Λ 2ã n ε ijkF µν W i µα W j ν W k,α , (2.3) whereF µν = 1 2 ε µνρσ F ρσ (ε 0123 = +1) (2.4) is the dual electromagnetic field-strength tensor, and e is the electromagnetic coupling. The scale Λ is introduced to keep the coupling constants a i dimensionless. The operators L 0 and L c , which were introduced in Ref. [ 15], conserve the discrete symmetries 2 C, P, and CP, while the others respect only one of these symmetries. The operator L n , which was defined in Refs. [ 7,16,17], conserves only P, but violates C and CP. The P-violating operatorsL 0 andL n have to our knowledge not yet been considered in the literature. WhileL 0 conserves C and violates CP,L n conserves CP and violates C. We add some comments on the completeness of the set (2.3) of quartic couplings. At first sight, there are three more P-violating couplings of dimension 6 that can be constructed with the tensor ε µνρσ , namely ε ijk ε µνρσ W i,µα W j,ν W k,ρ F σ α , ε ijk ε µνρσ W i,µν W j,ρ W k,α F σ α , ε µνρσ F µν F ρα W σ W α . (2.5) These operators can be reduced toL 0 andL n by exploiting the Schouten identity g αβ ε µνρσ + g αµ ε νρσβ + g αν ε ρσβµ + g αρ ε σβµν + g ασ ε βµνρ = 0, (2.6) which is a consequence of the four-dimensionality of space-time. Moreover, we could have constructed also operators from ∂ µ W ν and ∂ ν W µ separately instead of taking W µν . However, the new operators obtained this way only lead to additional terms involving either ∂ µ W µ or ∂ µ F µν , which do not contribute to the amplitudes for e + e − → 4f γ for massless external fermions. In order to deduce the Feynman rules for the considered AQGC, we express the fields W i µ in terms of physical fields, W µ W ν = W + µ W − ν + W − µ W + ν + 1 c 2 w Z µ Z ν , ε ijk W i,µν W j,ρ W k,σ = i c w W + µν (W − σ Z ρ − W − ρ Z σ ) − W − µν (W + σ Z ρ − W + ρ Z σ ) + Z µν (W + σ W − ρ − W + ρ W − σ ) . (2.7) Taking all fields and momenta as incoming, we obtain the Feynman rules A µ , p 1 A ν , p 2 {W + α ; Z α } , p 3 W − β ; Z β , p 4 = i e 2 8Λ 2 1; 1 c 2 w ×{4 a 0 g αβ [(p 1 p 2 )g µν − p ν 1 p µ 2 ] + a c [(p α 1 p β 2 + p β 1 p α 2 )g µν + (p 1 p 2 )(g µα g νβ + g να g µβ ) − p ν 1 (p β 2 g µα + p α 2 g µβ ) − p µ 2 (p β 1 g να + p α 1 g νβ )] + 4ã 0 g αβ p 1ρ p 2σ ε µρνσ }, (2.8) A µ , p 1 Z ν , p 2 W + α , p 3 W − β , p 4 = − e 2 16Λ 2 c w ×{a n [−(p 1 p 2 )(g µα g βν − g µβ g να ) − (p 1 p 3 )(g µβ g να − g µν g αβ ) − (p 1 p 4 )(g µν g αβ − g µα g βν ) + p µ 2 (p α 1 g βν − p β 1 g να ) + p µ 3 (p β 1 g να − p ν 1 g αβ ) + p µ 4 (p ν 1 g αβ − p α 1 g βν ) − g µν (p β 1 p α 3 − p α 1 p β 4 ) − g µα (p ν 1 p β 4 − p β 1 p ν 2 ) − g µβ (p α 1 p ν 2 − p ν 1 p α 3 )] +ã n p 1ρ [(p 1 + p 2 ) ν ε αβµρ + (p 1 + p 3 ) α ε βνµρ + (p 1 + p 4 ) β ε ναµρ − (p 2 − p 3 ) σ g να ε σβµρ − (p 3 − p 4 ) σ g αβ ε σνµρ − (p 4 − p 2 ) σ g βν ε σαµρ ]}. (2.9) Note that the γZW + W − coupling is symmetric with respect to cyclic permutations of ZW + W − , i.e. of (p 2 , ν), (p 3 , α), (p 4 , β). In order to evaluate the diagrams with the P-violating couplings within the Weyl-van der Waerden spinor formalism (see Ref. [ 20] and references therein), which we use in the calculation of our amplitudes, the tensor ε µνρσ has to be translated into the spinor technique. Following the notation of Ref. [ 20] the tensor is substituted in the Feynman rules according to the identity (ε 0123 = +1) ε µνρσ 1 2 σȦ B µ 1 2 σĊ D ν 1 2 σĖ F ρ 1 2 σĠ H σ = i 4 ǫȦĖǫĊĠǫ BD ǫ F H − ǫȦĊǫĖĠǫ BF ǫ DH . (2.10) For the the Standard-Model (SM) parameters and fields, i.e. for the SM Feynman rules, we follow the conventions of Ref. [ 21]. 3 Amplitudes with anomalous quartic gauge-boson couplings In Ref. [ 9] we have presented the SM amplitudes for all e + e − → 4f γ processes with massless fermions in a generic way. The various channels have been classified into chargedcurrent (CC), neutral-current (NC), and mixed CC/NC reactions, and all amplitudes have been generated from the matrix elements M CCa and M NCa , which correspond to the simplest CC and NC reactions, called CCa and NCa, respectively: Here we supplement the SM amplitudes of Ref. [ 9] by the corresponding contributions resulting from the AQGC given in (2.3). We follow entirely the conventions of Ref. [ 9] and denote the external particles of the considered reaction according to CCa: e + e − → ff ′ FF ′ , NCa: e + e − → ff FF , V 1 V 2 V 3 f a (p a , σ a ) f b (p b , σ b )f c (p c , σ c ) f d (p d , σ d ) f e (p e , σ e ) f f (p f , σ f ) γ(k, λ)e + (p + , σ + ) + e − (p − , σ − ) → f 1 (k 1 , σ 1 ) +f 2 (k 2 , σ 2 ) + f 3 (k 3 , σ 3 ) +f 4 (k 4 , σ 4 ) + γ(k 5 , λ), (3.1) where the momenta and helicities are given in parentheses. We list the expressions for the contributions δM CCa,AQGC and δM NCa,AQGC of the anomalous couplings to the generic CC and NC matrix elements M CCa and M NCa , respectively, which have to be added to the SM contributions. From M CCa and M NCa the amplitudes for all other CC, NC, and CC/NC reactions are constructed as explained in Ref. [ 9]. Some minor corrections to this generic construction are given in App. A. We express the AQGC contributions δM CCa,AQGC and δM NCa,AQGC in terms of the two generic functions M γV V,AQGC and M ZW W,AQGC , which correspond to the γγV V and γZW W couplings, respectively, with V = W, Z, δM σ + ,σ − ,σ 1 ,σ 2 ,σ 3 ,σ 4 ,λ CCa,AQGC (p + , p − , k 1 , k 2 , k 3 , k 4 , k 5 ) = M σ + ,σ − ,−σ 1 ,−σ 2 ,−σ 3 ,−σ 4 ,λ γW W,AQGC (p + , p − , −k 1 , −k 2 , −k 3 , −k 4 , k 5 ) + M σ + ,σ − ,−σ 1 ,−σ 2 ,−σ 3 ,−σ 4 ,λ ZW W,AQGC (p + , p − , −k 1 , −k 2 , −k 3 , −k 4 , k 5 ), (3.2) δM σ + ,σ − ,σ 1 ,σ 2 ,σ 3 ,σ 4 ,λ NCa,AQGC (p + , p − , k 1 , k 2 , k 3 , k 4 , k 5 ) = M σ + ,σ − ,−σ 1 ,−σ 2 ,−σ 3 ,−σ 4 ,λ γZZ,AQGC (p + , p − , −k 1 , −k 2 , −k 3 , −k 4 , k 5 ) + M −σ 1 ,−σ 2 ,−σ 3 ,−σ 4 ,σ + ,σ − ,λ γZZ,AQGC (−k 1 , −k 2 , −k 3 , −k 4 , p + , p − , k 5 ) + M −σ 3 ,−σ 4 ,σ + ,σ − ,−σ 1 ,−σ 2 ,λ γZZ,AQGC (−k 3 , −k 4 , p + , p − , −k 1 , −k 2 , k 5 ). (3.3) The generic Feynman graph that corresponds to M V 1 V 2 V 3 ,AQGC is shown in Figure 1, where the fermions and antifermions are assumed as incoming and the photon as outgoing. Explicitly the generic functions read M σa,σ b ,σc,σ d ,σe,σ f ,λ γV V,AQGC (p a , p b , p c , p d , p e , p f , k) = − e 5 C γγV V 8 √ 2Λ 2 δ σa,−σ b δ σc,−σ d δ σe,−σ f g σ b γfaf b g σ d Vfcf d g σ f Vfef f P V (p c + p d )P V (p e + p f ) × 8A σa,σc,σe,λ a 0 (p a , p b , p c , p d , p e , p f , k) + A σa,σc,σe,λ ac (p a , p b , p c , p d , p e , p f , k) , with C γγZZ = 1/c 2 w , C γγW W = 1, (3.4) M σa,σ b ,σc,σ d ,σe,σ f ,λ ZW W,AQGC (p a , p b , p c , p d , p e , p f , k) = ie 5 8 √ 2c w Λ 2 δ σa,−σ b δ σc,+ δ σ d ,− δ σe,+ δ σ f ,− (Q c − Q d )g σ b Zfaf b g − Wfcf d g − Wfef f × P Z (p a + p b )P W (p c + p d )P W (p e + p f ) A σa,σc,σe,λ an (p a , p b , p c , p d , p e , p f , k), (3.5) where the propagator functions P V (p) and the fermion-gauge-boson couplings g σ Vf f ′ can be found in Ref. [ 9]. We have evaluated the auxiliary functions A σa,σc,σ d ,λ a k with k = 0, c, n in terms of Weyl-van der Waerden spinor products [ 20]: A ++++ a 0 (p a , p b , p c , p d , p e , p f , k) = (a 0 + iã 0 ) ( p b k * ) 2 p d p f * p c p e p a p b * , (3.6) A ++++ ac (p a , p b , p c , p d , p e , p f , k) = a c (p a · p b ) × 2 p a p b * p d k * p f k * p a p c p a p e + ( p b k * ) 2 p d p f * p a p b p c p e , (3.7) A ++++ an (p a , p b , p c , p d , p e , p f , k) = (a n + iã n ) × p b p f * p d k * p a p e p a k * p a p c + p b k * p b p c + p e k * p c p e + p f k * p c p f + p b p d * p f k * p a p c p c k * p c p e + p d k * p d p e − p a k * p a p e − p b k * p b p e − p d p f * p b k * p c p e p e k * p a p e + p f k * p a p f − p c k * p a p c − p d k * p a p d . (3.8) The remaining polarization combinations follow from crossing and discrete symmetries, A ++−+ a k (p a , p b , p c , p d , p e , p f , k) = A ++++ a k (p a , p b , p c , p d , p f , p e , k), A +−++ a k (p a , p b , p c , p d , p e , p f , k) = A ++++ a k (p a , p b , p d , p c , p e , p f , k), A +−−+ a k (p a , p b , p c , p d , p e , p f , k) = A ++++ a k (p a , p b , p d , p c , p f , p e , k), A −+++ a k (p a , p b , p c , p d , p e , p f , k) = A ++++ a k (p b , p a , p c , p d , p e , p f , k), A −+−+ a k (p a , p b , p c , p d , p e , p f , k) = A ++++ a k (p b , p a , p c , p d , p f , p e , k), A −−++ a k (p a , p b , p c , p d , p e , p f , k) = A ++++ a k (p b , p a , p d , p c , p e , p f , k), A −−−+ a k (p a , p b , p c , p d , p e , p f , k) = A ++++ a k (p b , p a , p d , p c , p f , p e , k), A σa,σc,σ d ,− a k (p a , p b , p c , p d , p e , p f , k) = A −σa,−σc,−σ d ,+ a k (p a , p b , p c , p d , p e , p f , k) * , k = 0, c, n. (3.9) It is interesting to observe that the helicity amplitudes for a 0 andã 0 , and similarly for a n andã n , differ only in factors ±i for equal coupling factors. These AQGC are the ones that are related by interchanging a field-strength tensor F with a dual field-strength tensorF in the corresponding operators in (2.3). As we had already done in Ref. [ 9] in the case of the SM amplitudes, we have numerically checked the amplitudes with the AQGC against an evaluation by Madgraph [ 22], 6 which we have extended by the anomalous couplings. We find numerical agreement for a set of representative 4f γ final states. Leading universal electroweak corrections Besides the genuine AQGC we have also included the dominant leading electroweak corrections to e + e − → 4f γ into RacoonWW, similar to our construction [ 23] of an improved Born approximation (IBA) for e + e − → WW → 4f . The dominant universal effects originating from the renormalization of the electroweak couplings are included by using the so-called G µ -input-parameter scheme. To this end, the global factor α 5 in the cross section is replaced by α 4 Gµ α(0) with α Gµ = √ 2G µ M 2 W s 2 w π . (4.1) While the fine-structure constant α(0) yields the correct coupling for the external on-shell photon, α Gµ takes into account the running of the electromagnetic coupling from zero to M 2 W and the leading universal m t -dependent corrections to CC processes correctly. The m t -dependent correction to NC processes are not included completely. These could be accounted for by introducing an appropriate effective weak mixing angle. However, we prefer to keep the weak mixing angle fixed by c 2 w = 1 − s 2 w = M 2 W /M 2 Z , in order to avoid potential problems with gauge invariance which may result by violating this condition. Initial-state radiation (ISR) to e + e − → 4f γ is implemented at the leading-logarithmic level in the structure-function approach of Ref. [ 12] as described for e + e − → 4f in Ref. [ 8] in equations (5.1)-(5.4), dσ e + e − →4f γ IBA = 1 0 dx 1 1 0 dx 2 Γ LL ee (x 1 , Q 2 )Γ LL ee (x 2 , Q 2 ) dσ e + e − →4f γ IBA (x 1 p + , x 2 p − ). (4.2) In the structure function Γ LL ee (x, Q 2 ) [ 8,24] soft-photon effects are exponentiated and collinear logarithms are included up to order O(α 3 ). The QED splitting scale Q 2 is a free parameter in leading-logarithmic approximation and has to be set to a typical momentum scale of the process. It is fixed as Q 2 = s by default but can be changed to any other scale in order to adjust the IBA to the full correction or to estimate the intrinsic uncertainty of the IBA by choosing different values for Q 2 . For processes with intermediate W-boson pairs, e + e − → WW(γ) → 4f γ, the singular part [ 13] of the Coulomb correction is taken into account, i.e. in this case we have dσ e + e − →4f γ IBA = dσ e + e − →4f γ Born 1 + δ Coul (s ′ , k 2 + , k 2 − )g(β) , s ′ = (k + + k − ) 2 . (4.3) The Coulomb singularity arises from diagrams where a soft photon is exchanged between two nearly on-shell W bosons close to their kinematical production threshold and results in a simple factor that depends on the momenta k ± of the W bosons [ 13,25], Figure 2: Generic diagrams contributing to the Coulomb singularity in e + e − → 4f γ δ Coul (s ′ , k 2 + , k 2 − ) = α(0) β Im ln β −β + ∆ M β +β + ∆ M , β = s ′2 + k 4 + + k 4 − − 2s ′ k 2 + − 2s ′ k 2 − − 2k 2 + k 2 − s ′ , (a) W W γ γ e − e + f 1 f 2 f 3 f 4 (b) W W γ γ e − e + f 1 f 2 f 3 f 4β = 1 − 4(M 2 W − iM W Γ W ) s ′ , ∆ M = \|k 2 + − k 2 − \| s ′ . (4.4) This correction factor is multiplied with the auxiliary function g(β) = 1 −β 2 2 ,(4.5) in order to restrict the impact of δ Coul to the threshold region where it is valid. For e + e − → 4f γ both diagrams where the real photon is emitted from the initial state (see Figure 2a) or from the final state (see Figure 2b) contribute to the Coulomb singularity. Therefore, it is not just given by a factor to the complete matrix element. However, applying different correction factors to different diagrams would violate gauge invariance. Therefore, we decided to use an effective treatment that takes into account the dominant effects of the Coulomb singularity. We actually implemented two different variants: 1. In the first variant we multiply the complete matrix element with the Coulomb correction factor with k + = k 1 + k 2 and k − = k 3 + k 4 . In this way we multiply the correct Coulomb correction to all diagrams with ISR ( Figure 2a). However, in this approach we do not treat the Coulomb singularity in diagrams with final-state radiation (Figure 2b) properly. Nevertheless, this recipe should yield a good description of the Coulomb singularity, since the diagrams with two resonant W bosons and photon emission from the initial state dominate the cross section. This expectation is confirmed by the numerical results presented below. 2. In the second variant we improve on this prescription by differentiating between initial-state and final-state radiation according to the invariant masses in the final state. To this end, the W-boson momenta entering the Coulomb correction factor are fixed as (k + , k − ) =                    (k 1 + k 2 , k 3 + k 4 ) for ∆ 12 < ∆ 125 , ∆ 34 < ∆ 345 , (k 1 + k 2 + k 5 , k 3 + k 4 ) for ∆ 12 > ∆ 125 , ∆ 34 < ∆ 345 or ∆ 12 > ∆ 125 , ∆ 34 > ∆ 345 , ∆ 125 < ∆ 345 , (k 1 + k 2 , k 3 + k 4 + k 5 ) for ∆ 12 < ∆ 125 , ∆ 34 > ∆ 345 or ∆ 12 > ∆ 125 , ∆ 34 > ∆ 345 , ∆ 125 > ∆ 345 ,(4.6) where ∆ ij = \|(k i + k j ) 2 − M 2 W \| and ∆ ijl = \|(k i + k j + k l ) 2 − M 2 W \|. In this way we effectively apply the correct Coulomb correction factor to all dominating doublyresonant contributions, shown in Figure 2. The Coulomb singularity is not included in processes that do not involve diagrams with two resonant W bosons. Finally, we optionally include the naive QCD correction factors (1 + α s /π) for each hadronically decaying W boson. In order to avoid any kind of mismatch with the decay, Γ W is calculated in lowest order using the G µ scheme. This choice guarantees that the "effective branching ratios", which result after integrating out the decay parts, add up to one when summing over all channels. Of course, if naive QCD corrections are taken into account, these are also included in the calculation of the total W-boson width. Numerical results For our numerical analysis we take the same SM input parameters as in Refs. [ 8,11]. We use the constant-width scheme, which has been shown to be practically equivalent to the complex-mass scheme for the considered processes in Ref. [ 9]. The errorbars shown in the plots for the relative corrections result from the statistical errors of the Monte-Carlo integration. Comparison with existing results We first compare our results for e + e − → 4f γ including leading corrections with results existing in the literature. Predictions for e + e − → 4f γ including ISR corrections have been provided with the program WRAP [ 6]. First results have been published in Ref. [ 11] where also a comparison with RacoonWW at tree level was performed. Here we present a comparison between WRAP and RacoonWW for the same set of input parameters and cuts as in Section 5.2. of Ref. [ 11] but including ISR. In this tuned comparison, the W-boson width is kept fixed at Γ W = 2.04277 GeV, and neither the Coulomb singularity nor naive QCD corrections are included. The results are shown in Figures 3 and 4. The absolute predictions on the left-hand sides are hardly distinguishable. The relative deviations shown on the right-hand sides reveal that the agreement between WRAP and RacoonWW is at the level of the statistical error of about 0.2%. The comparison has been made for collinear structure functions. Unlike p T -dependent structure functions, collinear structure functions do not allow to take into account the Bose symmetry of the final-state photons resulting in some double counting [ 26]. However, for not too small cuts on the photon energy and angle these effects are beyond the accuracy of the leading-logarithmic approximation. This has been confirmed by the numerical analysis in Ref. [ 6]. In Figures 5-7 we repeat the comparison between YFSWW3-1.14 (scheme A) [ 27] and RacoonWW given in Section 4.1. of Ref. [ 11] for the photonic distributions. But now we include besides the tree-level predictions of RacoonWW for e + e − → 4f γ also those including leading-logarithmic ISR. Note that unlike in all other distributions discussed here, a recombination of photons with fermions is performed for this comparison. Figure 3: Total cross section including leading-logarithmic ISR corrections for the process e + e − → udµ −ν µ γ as a function of the CM energy with minimal energy of the observed photon of 1 GeV. Absolute predictions from WRAP [ 6] and RacoonWW are shown on the left-hand side, the relative differences between the two programs are shown on the right-hand side. Figure 4: Total cross section including leading-logarithmic ISR corrections for the process e + e − → udµ −ν µ γ as a function of the minimal energy of the observed photon for √ s = 192 GeV. Absolute predictions from WRAP [ 6] and RacoonWW are shown on the lefthand side, the relative differences between the two programs are shown on the right-hand side. Figure 5: Distribution in the photon energy for the process e + e − → udµ −ν µ γ at √ s = ÏÊ È Ê ÓÓÒÏÏ Ô × Î℄ ℄ ¾½¼ ¾¼¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¿¼ ¾¼ ½¼ ¼ Ô × Î℄ ÏÊ È Ê ÓÓÒÏÏ ½ ±℄ ¾½¼ ¾¼¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ¾ ½ ½ ¼ ¼ ¼ ½ ½ ¾ÏÊ È Ê ÓÓÒÏÏ Ñ Ò Î℄ ℄ ¿ ¾ ½ ¼ ¼ ¼ ¼ ¿ ¿¼ Ñ Ò Î℄ ÏÊ È Ê ÓÓÒÏÏ ½ ±℄ ¿ ¾ ½ ¾ ½ ½ ¼ ¼ ¼ ½ ½ ¾Ê ÓÓÒÏÏ Á ËÏÏ Ê ÓÓÒÏÏ ÓÖÒ Î℄ Î ¼ ¿ ¿¼ ¾ ¾¼ ½ ½¼ ¼ ½¼ ½ ¼ ½ Ê ÓÓÒÏÏ Á Ê ÓÓÒÏÏ ÓÖÒ Î℄ ËÏÏ¿ Ê ÓÓÒÏÏ ½ ±℄ ¼ ¿ ¿¼ ¾ ¾¼ ½ ½¼ ¼ ½ ½¼ ¼ ½¼ ½ ¾¼ GeV We restrict ourselves to the "bare" recombination scheme (see Refs. [ 8,11] for details). Moreover, the W-boson width is calculated including the full O(α) electroweak corrections together with naive QCD corrections resulting in Γ W = 2.08699 GeV. We compare the distributions in the photon energy E γ , in the cosine of the polar angle θ γ of the photon w.r.t. the e + beam, and in the angle θ γf between the photon and the nearest charged final-state fermion for the process e + e − → udµ −ν µ γ at √ s = 200 GeV. The differences of 15-20% between YFSWW3 and the pure Born prediction of RacoonWW (RacoonWW Born), which have already been shown in Ref. [ 11], reduce to about 5% once the leading logarithmic ISR corrections are included in RacoonWW (RacoonWW IBA). The remaining differences should be due to the still quite different treatment of visible photon radiation in RacoonWW and YFSWW3: in contrast to RacoonWW, YFSWW does not include the complete lowest-order matrix elements for e + e − → 4f γ. Instead, the photon radiation from the final state is treated via PHOTOS [ 28]. In particular, for small photon energies, where the differences are largest, the non-factorizable contributions, which are not yet included in YFSWW3, might play a role. In Figures 8-10 we extend the comparison of the photonic distributions between YFSWW3 and RacoonWW to 500 GeV. Here the difference is typically at the level of 10% and in general not reduced by the inclusion of ISR for e + e − → 4f γ in RacoonWW, i.e. the agreement without ISR in RacoonWW was accidentally good. One should also recall that the diagrams without two resonant W bosons (background diagrams) become more and more important at higher energies. Thus, the increasing difference between YFSWW3 and RacoonWW for higher energies could be due to a less efficient description of final-state radiation by the effective treatment with PHOTOS. Figure 7: Distribution in the angle between the photon and the nearest charged final-state fermion for the process e + e − → udµ −ν µ γ at √ s = 200 GeV Figure 8: Distribution in the photon energy for the process e + e − → udµ −ν µ γ at √ s = 500 GeV Figure 9: Distribution in the cosine of the polar angle of the photon w.r.t. the e + beam for the process e + e − → udµ −ν µ γ at √ s = 500 GeV Figure 10: Distribution in the angle between the photon and the nearest charged finalstate fermion for the process e + e − → udµ −ν µ γ at √ s = 500 GeV Ê ÓÓÒÏÏ Á ËÏÏ Ê ÓÓÒÏÏ ÓÖÒ Ó× Ó× ℄ ½ ¼ ¼ ¼ ¼ ¾ ¼ ¼ ¾ ¼ ¼ ¼ ½ ½¼¼ ½¼ Ê ÓÓÒÏÏ Á Ê ÓÓÒÏÏ ÓÖÒ Ó× ËÏÏ¿ Ê ÓÓÒÏÏ ½ ±℄ ½ ¼ ¼ ¼ ¼ ¾ ¼ ¼ ¾ ¼ ¼ ¼ ½ ¾ ¼ ¾ ½¼ ½¾ ½ ½ ½+ e − → udµ −ν µ γ at √ s = 200 GeV Ê ÓÓÒÏÏ Á ËÏÏ Ê ÓÓÒÏÏ ÓÖÒ ℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ½ ¼ ½ ¼ ¼½ Ê ÓÓÒÏÏ Á Ê ÓÓÒÏÏ ÓÖÒ ℄ ËÏÏ¿ Ê ÓÓÒÏÏ ½ ±℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ½ ½¼ ¼ ½¼ ½ ¾¼ ¾Ê ÓÓÒÏÏ Á ËÏÏ Ê ÓÓÒÏÏ ÓÖÒ Î℄ Î ¼ ¿ ¿¼ ¾ ¾¼ ½ ½¼ ¼ ½ ¼ ½ Ê ÓÓÒÏÏ Á Ê ÓÓÒÏÏ ÓÖÒ Î℄ ËÏÏ¿ Ê ÓÓÒÏÏ ½ ±℄ ¼ ¿ ¿¼ ¾ ¾¼ ½ ½¼ ¼ ½¼ ¼ ½¼ ½ ¾¼Ê ÓÓÒÏÏ Á ËÏÏ Ê ÓÓÒÏÏ ÓÖÒ Ó× Ó× ℄ ½ ¼ ¼ ¼ ¼ ¾ ¼ ¼ ¾ ¼ ¼ ¼ ½ ½¼¼ ½¼ Ê ÓÓÒÏÏ Á Ê ÓÓÒÏÏ ÓÖÒ ½ ¼ ¼ ¼ ¼ ¾ ¼ ¼ ¾ ¼ ¼ ¼ ½ ¾ ¼ ¾ ½¼ ½¾Ê ÓÓÒÏÏ Á ËÏÏ Ê ÓÓÒÏÏ ÓÖÒ ℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ½ ¼ ½ Ê ÓÓÒÏÏ Á Ê ÓÓÒÏÏ ÓÖÒ ℄ ËÏÏ¿ Ê ÓÓÒÏÏ ½ ±℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ¾¼ ½ ½¼ ¼ ½¼ ½ Standard Model predictions We now discuss the predictions of RacoonWW for various observables in the SM. Here and in the following, the width is always calculated from the input parameters in lowest order in the G µ scheme including naive QCD corrections (Γ W = 2.09436 GeV). Naive QCD corrections are included in all results, in particular, also in the Born results. The results in the LEP2 energy range were obtained with the ADLO/TH cuts as defined in Ref. [ 9], those at √ s = 500 GeV with the cuts θ(l, beam) > 10 • , θ(l, l ′ ) > 5 • , θ(l, q) > 5 • , θ(γ, beam) > 1 • , θ(γ, l) > 5 • , θ(γ, q) > 5 • , E γ > 0.1 GeV, E l > 1 GeV, E q > 3 GeV, m(q, q ′ ) > 0.1 GeV, θ(q, beam) > 5 • ,(5.1) where θ(i, j) specifies the angle between the particles i and j in the LAB system, and l, q, γ, and "beam" denote charged final-state leptons, quarks, photons, and the beam electrons or positrons, respectively. The invariant mass of a quark pair qq ′ is denoted by m(q, q ′ ). The cuts (5.1) differ from the ADLO/TH cuts only in the looser cut on m(q, q ′ ) and in the additional cut on θ(q, beam). In Figure 11 we present the total cross section for the process e + e − → udµ −ν µ γ in the LEP2 energy range. On the left-hand side we show the absolute prediction in lowest order (Born), including ISR (ISR), and including in addition the Coulomb singularity according to variant 1) (IBA) discussed in Section 4. On the right-hand side we give the corrections relative to the lowest order including in addition a curve with the Coulomb singularity according to variant 2) (IBA2). The Coulomb singularity reaches about 5% at threshold and decreases with increasing energy. The effect is comparable to the one for the process without photon. The two variants for the implementation of the Coulomb Figure 11: Total cross section for the process e + e − → udµ −ν µ γ as a function of the CM energy. Á ÁËÊ ÓÖÒ Ô × Î℄ ℄ ¾½¼ ¾¼¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ¾ ¼ ¾¼¼ ½ ¼ ½¼¼ ¼ ¼ Á ¾ Á ÁËÊ Ô × Î℄ ÓÖÒ ½ ±℄ ¾½¼ ¾¼¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½¼ ½ ¾¼ ¾ ¿¼ singularity show hardly any difference. Consequently, we will always use variant 1) in the following. In Figures 12-14 we present the distributions in the photon energy E γ , in the cosine of the polar angle θ γ of the photon w.r.t. the e + beam, and in the angle θ γf between the photon and the nearest charged final-state fermion for √ s = 200 GeV. The left-hand sides contain the absolute prediction for the process e + e − → udµ −ν µ γ in lowest order (Born) and including the ISR corrections and the Coulomb singularity (IBA), and for the process e + e − → ude −ν e γ including these corrections. The relative corrections (righthand sides) are typically of the order of −10% wherever the cross sections are sizeable. Relative to the corresponding lowest-order results, the corrections to e + e − → ude −ν e γ would practically be indistinguishable from the relative corrections to e + e − → udµ −ν µ γ. We therefore prefer to plot the corrections to e + e − → ude −ν e γ normalized to the lowestorder of e + e − → udµ −ν µ γ in order to visualize the effect of the "background" diagrams contained in e + e − → ude −ν e γ. As can be seen, this effect is comparable to the radiative corrections but of opposite sign. In Figures 15-17 we show results for √ s = 500 GeV. Here, the left-hand sides contain the absolute prediction for the processes e + e − → udµ −ν µ γ and e + e − → ude −ν e γ in lowest order (Born) and including the ISR corrections and the Coulomb singularity (IBA). Note that here the distributions differ sizeably between the two processes. Therefore, on the right-hand sides, the IBA predictions for both processes are normalized to the corresponding lowest-order predictions. Where the cross sections are sizeable, the corrections are about +10% for e + e − → udµ −ν µ γ and +5% for e + e − → ude −ν e γ. They are larger where the cross sections are small. Figure 12: Distribution in the photon energy for the processes e + e − → udµ −ν µ γ and e + e − → ude −ν e γ at √ s = 200 GeV ("Born(µ)" indicates that the Born cross section for Figure 14: Distribution in the angle between the photon and the nearest charged finalstate fermion for the processes e + e − → udµ −ν µ γ and e + e − → ude −ν e γ at √ s = 200 GeV ("Born(µ)" indicates that the Born cross section for e + e − → udµ −ν µ γ is taken.) Figure 15: Distribution in the photon energy for the processes e + e − → udµ −ν µ γ and e + e − → ude −ν e γ at √ s = 500 GeV Figure 16: Distribution in the cosine of the polar angle of the photon w.r.t. the e + beam for the processes e + e − → udµ −ν µ γ and e + e − → ude −ν e γ at √ s = 500 GeV Figure 17: Distribution in the angle between the photon and the nearest charged finalstate fermion for the processes e + e − → udµ −ν µ γ and e + e − → ude −ν e γ at √ s = 500 GeV Ù Á Ù Á Ù ÓÖÒ Î℄ Î ¼ ¿¼ ¾¼ ½¼ ¼ ½¼¼ ½¼ ½ ¼ ½ Ù Ù Î℄ Á ÓÖÒ´ µ ½ ±℄ ¼ ¿¼ ¾¼ ½¼ ¼ ¾ ½¼ ½¾ ½ ½ ½ ¾¼e + e − → udµ −ν µ γ is taken.) Ù Á Ù Á Ù ÓÖÒ Ù Á Ù Á Ù ÓÖÒ ℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ¿ ¿ ¾ ¾ ½ ½ ¼ ¼ Ù Ù ℄ Á ÓÖÒ´ µ ½ ±℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ½ ½¼ ¼ ½¼ ½Ù Á Ù Á Ù ÓÖÒ Ù ÓÖÒ Î℄ Î ¾¼¼ ½ ¼ ½ ¼ ½ ¼ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ½¼ ½ ¼ ½ Ù Ù Î℄ Á ÓÖÒ ½ ±℄ ¾¼¼ ½ ¼ ½ ¼ ½ ¼ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ½¾ ½¼ ¾ ¼ ¾ ½¼ ½¾Ù Á Ù Á Ù ÓÖÒ ½ ¼ ¼ ¼ ¼ ¾ ¼ ¼ ¾ ¼ ¼ ¼ ½ ¾ ¾¼ ½ ½¼ ¼Ù Á Ù Á Ù ÓÖÒ Ù ÓÖÒ ℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ¿ ¾ ¾ ½ ½ ¼ ¼ Ù Ù ℄ Á ÓÖÒ ½ ±℄ ½¾¼ ½¼¼ ¼ ¼ ¼ ¾¼ ¼ ¿¼ ¾ ¾¼ ½ ½¼ ¼ Predictions with anomalous quartic couplings Since the matrix element depends linearly on the anomalous quartic couplings a i , the cross section is a quadratic form in the a i . Therefore, it is sufficient to evaluate the cross section for a finite set of sample values of the anomalous quartic couplings in order to get the cross section for arbitrary values of these couplings. We restrict ourselves here to the semileptonic process e + e − → udµ −ν µ γ and include ISR and the Coulomb singularity (variant 1). We use the cuts E γ > 5 GeV, \| cos θ γ \| < 0.95, \| cos θ γf \| < 0.90, m(f, f ′ ) = M W ± 2Γ W , (5.2) where E γ is the energy of the photon, θ γ the angle between the photon and the beam axis, θ γf the angle between the photon and any charged final-state fermion f , and m(f, f ′ ) the invariant mass of the fermion-antifermion pairs that result from W decay. In the computation of m(µ, ν µ ) the momentum of the neutrino is set equal to the missing momentum, since the neutrino is not detected, i.e. the energy loss in the ISR convolution (4.2) is implicitly included in the neutrino momentum. We first study the influence of the AQGC a 0 , a c , a n ,ã 0 , andã n on the cross section at √ s = 200 GeV and 500 GeV separately. Figure 18 shows the cross section normalized to the SM value as a function of each of these couplings for all the other a i 's equal to zero. The asymmetry results from the interference between the SM matrix element and the matrix element of the AQGC, which is suppressed for the CP-violating couplings a n andã 0 . The asymmetry is small for a 0 andã n and only visible at √ s = 500 GeV for a 0 in Figure 18, but sizeable for a c . 4 The cross section is most sensitive to a 0 andã 0 and least sensitive to a n andã n . In order to illustrate the potential of LEP2 and a linear e + e − collider in putting limits on the AQGC, we consider the following two scenarios: an integrated luminosity L = 320 pb −1 at √ s = 200 GeV and L = 50 fb −1 at √ s = 500 GeV. The corresponding total SM cross sections to e + e − → udµ −ν µ γ are 16.69 fb and 7.64 fb, respectively. Assuming that the measured number N of events is given by the SM cross section σ SM = σ(a i = 0) and the experimental errors by the corresponding square-root, we define χ 2 ≡ (N(a i ) − N) 2 N = σ(a i ) σ SM − 1 2 σ SM L,(5.3) where N(a i ) is the number of events that result from the cross section with anomalous couplings. Since the square-root of this χ 2 distribution is a quadratic form in the a i , the hypersurfaces of constant χ 2 form ellipsoids. The 1σ limits resulting from χ 2 = 1 on individual couplings can be illustrated by projecting the ellipsoids into the planes corresponding to pairs of couplings. Instead of the projections, often the sections of the planes with the ellipsoids are used. Note that the ellipses resulting from projections are in general larger and include those ellipses resulting from sections of the planes with the ellipsoids. Since the correlations are small for the cases under consideration, the difference Figure 18: Impact of the AQGC a 0 , a c , a n ,ã 0 , andã n on the cross section for e + e − → udµ −ν µ γ at √ s = 200 GeV and 500 GeV. Only one of the AQGC a i is varied while the others are kept zero. Ò Ò ¼ ¼ Ô × ¾¼¼ Î · ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¿ ¾ ¾ ½ ½ 20 between both types of ellipses is also small. In the following figures we include both the projections and the sections of the ellipsoids using the same type of lines. In Figures 19 and 20 we show some 1σ contours for various pairs of a i . In addition we list the 1σ limits derived from projecting the ellipsoids. Since the effects of a 0 andã 0 and of a n andã n on the cross section are equal up to relatively small interference terms, also the corresponding contours of these couplings with other couplings are of similar size. For transparency we omitted some contours involving a n ; for 200 GeV ( Figure 19) these contours practically coincide with the ones forã n , for 500 GeV ( Figure 20) the contours for a n are of the same size and shape as the ones forã n but shifted to become approximately symmetric w.r.t. a n → −a n . The best limits can be obtained for a 0 and a 0 . The correlations between the different couplings are in general small, and only a 0 and a c show a noticeable correlation. The limits obtainable at a linear collider are by about a factor of 200 better than those obtainable at LEP2. This improvement reflects the enhanced sensitivity of the cross section on the anomalous couplings at high energies, which can also be seen in Figure 18, and to a smaller part the higher luminosity. Summary We have calculated all lowest-order amplitudes for e + e − → 4f γ with five different genuine anomalous quartic gauge-boson couplings that are allowed by electromagnetic gauge invariance and the custodial SU(2) c symmetry. These couplings include the three operators L 0 , L c , and L n , which have been constrained by the LEP collaborations by analysing WWγ production, and two additional P-violating couplings, one of which conserves CP. The five anomalous couplings have been incorporated in the 4f (γ) Monte Carlo generator RacoonWW. We have calculated the dependence of the cross section for e + e − → 4f γ on the anomalous quartic couplings and illustrated the typical size of the limits that can be obtained for these couplings at LEP2 and a 500 GeV e + e − collider. Moreover, we have implemented the dominant leading electroweak corrections to e + e − → 4f γ into RacoonWW. These include initial-state radiation, the dominant universal effects originating from the running of the couplings, and the Coulomb singularity for processes involving W-boson pairs. We have compared the corresponding predictions with existing calculations, as far as possible, and investigated the numerical impact of the dominant corrections. With the additions described in this paper, RacoonWW is a state-of-the-art Monte Carlo generator for the classes of e + e − → 4f and e + e − → 4f γ processes with arbitrary massless four-fermion final states, both for the Standard Model and including anomalous quartic gauge-boson couplings. Figure 1 : 1Generic diagram for the AQGC contribution to e + e − → 4f γ where f and F are different fermions (f = F ) that are neither electrons nor electron neutrinos (f, F = e − , ν e ) and their weak-isospin partners are denoted by f ′ and F ′ , respectively. Figure 6 : 6Distribution in the cosine of the polar angle of the photon w.r.t. the e + beam for the process e Λ 2 17 Figure 19 : 21719GeV 2 < 0.032 −0.17 <ã n Λ 2 GeV 2 < 0.1σ contours in various (a i , a j ) planes for the process e + e − → udµ −ν µ γ at √ s = 200 GeV While finishing this paper a version of the Monte Carlo generator WRAP[ 6] became available that also includes AQGC but uses a different set of operators.1 We adopt the usual convention[ 19] that P V µ P −1 = V µ and CV µ C −1 = −V † µ for all electroweak gauge bosons. While for the photon (V = A) these transformations follow from the C and P invariance of the electromagnetic interaction, for the weak bosons (V = W ± , Z) they are mere definitions, which are, however, in agreement with the C and P invariance of the bosonic part of the electroweak interaction. The CP transformation, on the other hand, is well-defined for all electroweak gauge bosons. In this context it is important to recall that different conventions are used in the literature concerning the sign of the electroweak gauge coupling g ≡ e/s w , and thus of the sign of the sine of the weak mixing angle s w . Since the SM quartic coupling γZW + W − changes under the inversion of this sign, which of course can never affect physical quantities, the anomalous γZW + W − coupling also has to be reversed when switching from one convention to the other, although s w does not appear in (2.9) explicitly. The sign of the asymmetry differs from the results of Ref.[ 7], since the couplings a 0 and a c have been implemented[ 29] in EEWWG with a sign opposite to the definitions in Refs.[ 7,17], which agree with our choice. AcknowledgementWe are grateful to U. Parzefall and M. Thomson for providing us with suitable experimental cuts for AQGC studies. Moreover, we thank A. Werthenbach for discussions about Refs.[ 7,17]and for making the program EEWWG available to us. Finally, we thank the WRAP and YFSWW3 teams for their collaboration in preparing the comparisons with their results.Figure 13: Distribution in the cosine of the polar angle of the photon w.r.t. the e + beam for the processes e + e − → udµ −ν µ γ and e + e − → ude −ν e γ at √ s = 200 GeV ("Born(µ)"indicates that the Born cross section for e + e − → udµ −ν µ γ is taken.) In Ref.[ 9]we have constructed the amplitudes for all e + e − → 4f γ reactions from the two basic channels CCa and NCa, which are also specified in Section 3. Here we take the opportunity to correct two mistakes in the corresponding formulas:• Equation (2.24) of Ref.[ 9]is only correct for down-type fermions f , while some arguments have to be interchanged for up-type fermions. The correct formula isThe error affected the evaluation of the final states ν eνe ν µνµ and ν eνe uū inTable 1of Ref.[ 9]at the level of 0.2-0.4%. 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B 541 (1999) 31 [hep-ph/9807465]. . S Jadach, W Placzek, M Skrzypek, B F Ward, Z Was, hep-ph/9705429Phys. Lett. B. 417326S. Jadach, W. Placzek, M. Skrzypek, B. F. Ward and Z. Was, Phys. Lett. B 417 (1998) 326 [hep-ph/9705429]; . hep- ph/0007012 and hep-ph/0103163Phys. Rev. D. 61113010Phys. Rev. D 61 (2000) 113010 [hep-ph/9907436]; hep- ph/0007012 and hep-ph/0103163. . E Barberio, B Van Eijk, Z Was, Comput. Phys. Commun. 66115E. Barberio, B. van Eijk and Z. Was, Comput. Phys. Commun. 66 (1991) 115. . A Werthenbach, private communicationA. Werthenbach, private communication.	[]
[ "PERFORMANCE OF THE ATLAS-A SILICON DETECTOR WITH ANALOGUE READ OUT", "PERFORMANCE OF THE ATLAS-A SILICON DETECTOR WITH ANALOGUE READ OUT" ]	[ "P P Allport \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "P S L Booth \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "C Green \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "A Greenall \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "M Hanlon \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "J N Jackson \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "T J Jones \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "J D Richardson \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "S Martí I García \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "U Parzefall \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "A E Sheridan \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n", "N A Smith \nPhysics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K\n" ]	[ "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K", "Physics Department\nOliver Lodge laboratory\nUNIVERSITY OF LIVERPOOL\nUniversity of Liverpool\nOxford StreetL69 3BXLiverpoolU.K" ]	[]	The performance of an ATLAS-A silicon micro-strip detector prototype with FELIX 128 analogue read out chip has been studied. The noise level and the signal to noise ratio have been measured as a function of both detector bias and temperature. No evidence of micro-discharge was observed for detector bias voltages up to 300 V.	null	[ "https://arxiv.org/pdf/hep-ex/9606012v1.pdf" ]	109,568,603	hep-ex/9606012	e6ad14f9a1216270a8fd38ec4c85c6bb6471b02b	PERFORMANCE OF THE ATLAS-A SILICON DETECTOR WITH ANALOGUE READ OUT Jun 1996 June 21, 1996 P P Allport Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K P S L Booth Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K C Green Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K A Greenall Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K M Hanlon Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K J N Jackson Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K T J Jones Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K J D Richardson Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K S Martí I García Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K U Parzefall Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K A E Sheridan Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K N A Smith Physics Department Oliver Lodge laboratory UNIVERSITY OF LIVERPOOL University of Liverpool Oxford StreetL69 3BXLiverpoolU.K PERFORMANCE OF THE ATLAS-A SILICON DETECTOR WITH ANALOGUE READ OUT Jun 1996 June 21, 1996arXiv:hep-ex/9606012v1 19 ATLAS INDET-NO-xx arch-ive/9606012 The performance of an ATLAS-A silicon micro-strip detector prototype with FELIX 128 analogue read out chip has been studied. The noise level and the signal to noise ratio have been measured as a function of both detector bias and temperature. No evidence of micro-discharge was observed for detector bias voltages up to 300 V. Introduction The inner tracker of the ATLAS experiment at LHC will be built of silicon micro-strip detectors [?]. These detectors have to operate in a high radiation level environment [?]. The effect of the radiation damage is basically an increase of the leakage current and a change in the effective bulk doping concentration. Therefore, a reduction of the signal to noise ratio and detector efficiency could be expected. In order to recover the expected performance after the detector has been irradiated, the simplest solution is to increase the bias voltage. However, some phenomena can limit the maximum value of the bias voltage applied on the detector [?]. Then, the degraded performance of an irradiated detector could stand below the required minimum for a satisfactory operation of the ATLAS tracking system. In the case of AC-coupled silicon detectors, micro-discharges [?] at the edges of the strips are one of the possible causes limiting the maximum value of the bias voltage applied on the detector. The micro-discharges are manifested as a steep increase in the noise level with an associated increase in the detector current. The ATLAS-A detector design is based in n + -type strips on a n-type bulk, because after radiation damage the lightly doped n-type bulk inverts to a p-type bulk with an effective p doping that increases with dose. This leads to high values of the voltage needed to fully deplete the detector. This therefore could give a rise to micro-discharges. In this note, the possibility of producing micro-discharges on an ATLAS-A silicon detector prototype has been tested. Therefore, the prototype performance has been studied at different biasing voltages (from 0 up to 300 V) and at two different temperatures (±20 • C). The organization of this note is as follows: in section 2 the detector geometry and read out electronics are presented. Then the experimental set-up is described in section 3. In sections 4 and 5, the the experimental results in terms of the signal to noise ratio and the micro-discharge production are discussed. Finally, the conclusions are summarized in section 6. Prototype and Read Out Chip electronics The aim of this section is to describe the ATLAS-A silicon micro-strip detector, its technology and geometry, and the basic features of the FELIX 128 channel analogue read out chip. 1. The detector used in our studies is a 6 cm × 6 cm, with a 56.25 µm diode pitch (with an isolation strip) and 112.5 µm read out pitch (see figure 1). The strips are n + -type in a n-type bulk, AC coupled. These strips are connected together to a bias ring through a polysilicon resistor (600 KΩ). When biasing the detector the n + strips were directly grounded and the p-type backplane was connected to a negative bias voltage. Both sides of the detector are surrounded by multiple guard rings. The guard rings are used to shield the sensitive part of the detectors and to provide a gradual drop of the edge potential, even after irradiation. Between the n + strips, a p-stop isolation strip is used to avoid the short of the n + strips by accumulating positive charge in the SiO 2 layer. From the point of view of a possible production of micro-discharges, it is important to know the relative position of the implant and the metal contact [?]. In this case, the size of the metal contact is smaller by 2µm (see figure 1). − Front end pre-amplifier and shaper with a CR-RC characteristic of 75 ns peaking time. − An analogue pipeline, 1.6µs deep (Analogue Delay Buffer). − An Analogue Pulse Signal Processor (APSP). The chip has been designed to run in two modes: Deconvolution: the charge collected is obtained from a suitable weight [?] of three 25 ns consecutive points measurements. It allows to extract the charge deposition in a single event, even in the case of pile-up of events. Therefore it can operate at high luminosity running at LHC. − Peak: The charge collected is obtained for each event from the measurement of the height of the signal pulse. It requires a minimum distance of 75 ns between two consecutive pulses, in order to avoid the pulse pile-up. Therefore this mode is inadequate for high luminosity running where the associated trigger rate is 25 ns. Experimental Set-up In this section the experimental set-up used for this study is described. It consists of the following three main parts: The Trigger System The trigger system is based on the coincidences of two scintillators located above and below the silicon micro-strip detector (see figure 2). The coincidences can be triggered either by the ∼ 2 MeV electrons of a radioactive source or by cosmic rays. In the second case a lead shield (used to stop the soft component of the cosmic rays [?]) replaces the radioactive source. Then, mostly single muons (minimum ionizing particles, m.i.p.) trigger the scintillators coincidence. Data Acquisition System The FELIX read out uses a VME Test System which is based around an MVME 167 Processor, with OS9 operating system. Within the VME crate there is a VME sequencer (SEQSI), a SIROCCO card with a flash ADC and a Random Trigger Card. All clocks and control signals for the FELIX and the Analogue Multiplexor are generated by the sequencer. The Random Trigger Card is required to synchronise external triggers (e.g. the coincidence trigger from the scintillators) with the 40MHz BCO clock. Adjustment of the timing of this synchronised trigger relative to the data being collected is by means of a programmable delay unit. The data acquisition is by means of the VME SIROCCO where the analogue data is digitized and then written to disc. Detector and FELIX Biasing A Keithley 237 Source-Measure-Unit was used to supply a stable voltage for the detector biasing, and also allows the monitoring of the detector leakage current. Power supplies for the FELIX were provided by several low voltage bench power supplies, which also provided the biasing for the FELIX pre-amplifiers. It has been noticed that the performance of the FELIX 128 chip is strongly linked to variations on its bias power supplies. The main effects are: an increase in the noise level by a factor of ∼ 2, and/or a global shift of the pedestal values of all the channels. Detector Performance In this section the study of the detector performance is presented. Two types of particles were used to extract the signal to noise ratio. The first type is the muons from cosmic rays (considered as m.i.p). However the small trigger rate (∼ 15µ/hour) of these particles makes difficult to collect a large amount of events in order to perform precise measurements. Thus, a radioactive source emitting electrons of ∼ 2 MeV was used instead. A full range of bias voltages (from 30 V, just after depletion, up to 300 V) and two different temperatures (room ∼ +20 • C, and −20 • C) were used to determine the main factors affecting the resolution of a silicon micro-strip detector: • the noise level, • the signal to noise ratio, • the cluster size. Noise Studies Inter Strip Capacitance The inter strip capacitance is one of the most important parameters of the detector performance in terms of the noise level, and consequently in establishing the signal to noise ratio. In the cases of AC-coupled detectors, with integrated coupling capacitors, the measured capacitance is a convolution of both: the coupling capacitors and the direct capacitance between the implant layers. The measured values for the inter strip capacitance of one strip to its two nearest neighbours, as a function of the bias voltage, are shown in figure 3. It is clearly seen how the inter strip capacitance is drastically reduced when the detector is fully depleted (<25V). It falls to 5 pF, (equivalent to 0.85 pF/cm). Then, for higher bias voltages, it is almost flat, but the curves exhibit a small gradient when the voltage is increased. According to this observation, the noise is expected to be reduced when increasing the bias voltage. Any deviation with respect to this behaviour could indicate new phenomena affecting the detector operation, like micro-discharges for instance, as they manifest them self as a sudden onset on the noise. Effects of the Bias Voltage and Temperature on the Noise The noise level in the detector has been studied as a function of the bias voltage and the temperature. The obtained values for both data sampling modes (peak and deconvolution) are shown in figure 4. One can appreciate as the noise is drastically reduced when the detector is fully depleted at ∼ 25V (in both data sampling modes). This behaviour is in perfect agreement with the observed one for the inter strip capacitance (read section 4.1.1). The achieved value for the noise is ∼ 15 mV in peak mode. The Equivalent Noise Charge was (ENC) was ∼ 700 electrons. In figure 4 a slight decrease of the noise when increasing the bias voltage can be observed. The change is roughly a 10% in peak mode and a 12% in deconvolution mode. This can be understood as a consequence of the dependence shown by the inter strip capacitance on the detector bias voltage, and proves that our study is sensitive to small changes in the detector characteristics (section 4. 1.1 and figure 3). The noise of the detector exhibits a small dependence on the temperature. This effect can be observed in figure 4. As expected, the noise level is slightly reduced at lower temperatures. The reduction is a 7% for peak mode and a 4% for deconvolution mode. In the other hand, studies of the noise contribution from the read out electronics show that this is almost constant with the temperature. Consequently the noise reduction is due to the detector contribution, mainly due to a reduction of the leakage current (read section 5.1). No Evidence of Micro-Discharges From these results the production of micro-discharges in the ATLAS-A silicon detector prototype is discarded, as any onset on the noise in the detector have been observed, even at low temperatures where the effects are expected to be bigger if micro-discharges occur [?]. Hit and Cluster Search In each event the impact point of the incident particles on the detector is obtained from a strip cluster search that proceeds as follows: 1. the strip significance (s = charge collected in this event / noise) is computed for all the channels. 2. Then, the primary strip is defined as the channel with the highest strip significance if s > 3. 3. Those strips within an interval of ±5 strips around the primary strip with s > 3 are included in the cluster. 4. The hit significance is defined as the sum of the significance of all strips included in the cluster. 5. The search for more clusters in the same event continues until no strips with s > 3 remain. The signal to noise ratio is taken as the peak of the hit significance distribution, as it compares the charge collected in one strip with its noise and this is done for all the strips in a cluster. The cluster size is just given by the number of strips in each cluster. Signal to Noise Ratio The values obtained in this analysis for the signal to noise ratio depend basically on the following factors: • data sampling (peak or deconvolution mode), • bias voltage, Previous studies using the FELIX chip for analogue readout tests [?] have shown that its performance in peak mode is better than using deconvolution mode (section 2). For this reason, only the results obtained in peak mode are presented in this section. For a given bias voltage, temperature and read out scheme the only two magnitudes affecting the signal to noise ratio are: the collected charge (i.e. signal) and the noise level. The collected charge depends on the incident particle type. In the case of muons (m.i.p.) the collected charge was equivalent to ∼ 19, 800e − . In the second case, using the electrons from the radioactive source, the collected charge was slightly smaller, ∼ 18, 100e − . The errors associated to both figures are ∼ 10% mainly due to the calibration procedure. The ratio between muons and the ∼ 2 MeV electrons of the radioactive source is: 1.09 ± 0.03. In the ratio the systematic uncertainties due to the calibration are cancelled. The collected charge has been seen not to depend on the bias voltage (once the detector has been fully depleted). This can be understood in terms of a constant efficiency for collection of the electron-hole pairs that have been created, once the detector has been fully depleted. This reflects that no extra charge is lost in the recombination processes inside the depleted zone, and this is not expected to be dependent on the bias voltage. The noise level has been already studied in section 4.1, and as explained it is reduced when the bias voltage is increased. The signal to noise ratio at 30 V reverse bias voltage and −20 • C was 25.6 ± 0.6 for muons and 23.4 ± 0.2 for electrons. The amount of charge collected is the same for all bias voltage and the noise is lower for higher voltages, consequently the signal to noise ratio increases with the bias voltage (see figure 5.b). Using electrons as incident particles, the signal to noise ratio is ∼ 26 : 1. The Cluster Size In events with particles crossing the detector between two strips, only a fraction of the total produced charge is collected on each strip. Thus, the cluster size is in an important magnitude related with the spatial resolution of the detector. The worse resolution is obtained when all hits consist in a single strip cluster. Then the resolution is just given by: strip pitch/ √ 12. For multi-strip clusters, it could be improved, if the impact point is just given by weighting the charge collected in each strip in a suitable way [?]. At different bias voltages, the electric field varies inside the detector. This electric field accelerates the charges towards the collector strips. So, for high values of the electric field, the expected behaviour is to collect a bigger fraction of the charge in a single strip. Then the spatial resolution is degraded. The results obtained are presented in figure 6. The trend follows an increase of the fraction of clusters with a single strip when increasing the bias voltage. It varies from ∼ 16% of the events at low voltage, up to ∼ 19% at high biasing voltages. The absolute change is very small in a wide range of voltages. In consequence, the spatial resolution 7 is not dramatically affected when using high bias voltage values. The fraction of clusters with at least two strips remains above the 80%. The small dependence of the cluster size with the bias voltage is in part due to the charge sharing between the read out n + -strips propiciated by the intermediate strip. Discussion With the results presented in the previous section the micro-discharge production in the ATLAS-A prototype have been discarded. However, the leakage current on that device presents some peculiarities that need to be explained. Leakage Current The leakage current of the full detector has been studied in each step of the assembly process. The measured values of the leakage current are shown in the figure 7. In the case of the bare detector, two kinks can be appreciated in the I-V plot. These kinks are related with a change in the slope of the curve. These are reproducible phenomena, and the first kink (at ∼ 80V) matches quite well with the voltage for producing microdischarges according to the reference [?]. However, there is no agreement in the absolute change of the current. Actually in reference [?], the micro-discharges originate a change bigger than two orders of magnitude. Unfortunately, the wire-bonding process of the detector channels to the read out electronics had damaged the prototype due to the lack of a passivation layer. Therefore, after wire-bonding, the current is increased globally by a factor ∼ 100 and the kink structure is lost. In the other hand, the dependence of the leakage current on the temperature does not follow the expected one (approx a factor 1/2 each 8 • C) as shown in figure 7.a. As it is shown, the current is reduced just by a factor ∼ 1/3 by changing the temperature by ∼ 40 • C. The temperature dependence of the leakage current for a fixed bias voltage is almost linear (see figure 7.b). Neither breakdown nor punch-through in the leakage current is observed, and this is in agreement with a electric field being smaller than 30 V/µm, which is supported by simulation studies using the semiconductor simulation package TOSCA [?]. It is believed that: the damage of the prototype when wire-bonding can not be responsible of a micro-discharge inhibition in the module. 6 Conclusions The performance of an ATLAS-A silicon micro-strip has been studied using a FELIX chip with analogue read out. The signal to noise ratio (∼26:1) of the detector depends on the reverse bias voltage, as a consequence of the noise reduction. In the other hand, the cluster size shows that more than 80% of the cluster are multi-strip clusters. This two factors show that a good spatial resolution can be achieved with the ATLAS-A detector design. No evidence of micro-discharges has been observed in this device, neither by burst on the leakage current, nor by a onset of the noise, even working at high bias voltages (up to 300 V), and low temperatures (−25 • C). Noise vs bias voltage Figure 1 : 1Cross section view of the ATLAS-A prototype. - Data Acquisition System (DAS), -detector and FELIX biasing. 2 Figure 2 : 2Schematic view of the trigger system Figure 3 : 3Inter strip capacitance to the nearest neighbours. Figure 4 :Figure 5 :Figure 6 : 456Noise as a function of the detector bias voltage. Signal to noise ratio vs. the bias voltage. a) The hit significance distribution when applying a reverse bias voltage of 260 V. b) Evolution of the peak with the reverse bias voltage.11 Fraction of events with a single strip cluster as a function of the bias voltage. Figure 7 : 7Leakage current in the detector as a function of the bias voltage. The solid line corresponds to the characterization of the detector before bonding. The two others curves represent the measured current after bonding up the detector and at two different temperatures: room (≈ 25 o C) and cooled down (−20 o C). The bottom plot shows the current as a function of the temperature for a fix bias voltage. e-mail address for correspondence: [email protected]	[]
[ "Real-space mesh techniques in density functional theory", "Real-space mesh techniques in density functional theory" ]	[ "Thomas L Beck \nDepartment of Chemistry\nUniversity of Cincinnati\n45221-0172CincinnatiOH\n" ]	[ "Department of Chemistry\nUniversity of Cincinnati\n45221-0172CincinnatiOH" ]	[]	This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.	10.1103/revmodphys.72.1041	[ "https://export.arxiv.org/pdf/cond-mat/0006239v1.pdf" ]	119,094,089	cond-mat/0006239	7b90664205a496d7fee708cd2b56d258cb45e9ab	Real-space mesh techniques in density functional theory 14 Jun 2000 Thomas L Beck Department of Chemistry University of Cincinnati 45221-0172CincinnatiOH Real-space mesh techniques in density functional theory 14 Jun 2000to be published in Reviews of Modern Physics. This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies. I. INTRODUCTION The last decade has witnessed a proliferation in methodologies for numerically solving large-scale problems in electrostatics and electronic structure. The rapid growth has been driven by several factors. First, theoretical advances in the understanding of localization properties of electronic systems have justified at a fundamental level approaches which utilize localized density matrices or orbitals in their formulation (Kohn, 1996;Ismail-Beigi and Arias, 1998;Goedecker, 1999). Second, a wide variety of computational methods have exploited that physical locality, leading to linear scaling of the computing time with system size (Goedecker, 1999). Third, general algorithms for solving electrostatics and eigenvalue problems have been improved or newly developed including particle-mesh methods (Hockney and Eastwood, 1988;Darden et al., 1993;Pollock and Glosli, 1996), fast-multipole approaches (Greengard, 1994), multigrid techniques (Brandt, 1977(Brandt, , 1982(Brandt, , 1984Hackbusch, 1985), and Krylov subspace and related algorithms (Booten and van der Vorst, 1996). Last, and perhaps not least, the ready availability of very fast processors for low cost has allowed for quantum modeling of systems of unprecedented size. These calculations can be performed on workstations or workstation clusters, thus creating opportunities for a wide range of researchers in fields both inside and outside of computational physics and chemistry (Bernholc, 1999). Several monographs and collections of reviews illustrate the great variety of problems recently addressed with electrostatics and electronic structure methods (Gross and Dreizler, 1994;Bicout and Field, 1996;Seminario, 1996;Springborg, 1997;Banci and Comba, 1997;Von Rague Schleyer, 1998;Jensen, 1999;Hummer and Pratt, 1999). This review examines one subset of these new computational methods, namely realspace techniques. Real-space methods can loosely be categorized as one of three types: finite differences (FD), finite elements (FE), or wavelets. All three lead to structured, very sparse matrix representations of the underlying differential equations on meshes in real space. Applications of wavelets in electronic structure calculations have been thoroughly reviewed recently (Arias, 1999) and will therefore not be addressed here. This article discusses the fundamentals of FD and FE solutions of Poisson and nonlinear Poisson-Boltzmann equations in electrostatics and self-consistent eigenvalue problems in electronic structure. As implied in the title, the primary focus is on calculations in density functional theory (DFT); real-space methods are in no way limited to DFT, but since DFT calculations comprise a dominant theme in modern electrostatics and electronic structure, the discussion here will mainly be restricted to this particular theoretical approach. Consider a physical system for which local approaches such as real-space methods are appropriate: a transition metal ion bound to several ligands embedded in a protein. These systems are of significance in a wide range of biochemical mechanisms (Banci and Comba, 1997). Treating the entire system with ab initio methods is presently impossible. However, if the primary interest is in the nature of the bonding structure and electronic states of the transition metal ion, one can imagine a three-tier approach (Fig. 1). The central region, including the metal ion and the ligands, is treated with an accurate quantum method such as DFT. A second neighboring shell is represented quantum mechanically but is not allowed to change during self-consistency iterations. The wavefunctions in the central zone must be orthogonalized to the fixed orbitals in the second region. Finally, the very distant portions of the protein are fixed in space and treated classically; the main factors to include from the far locations are the electrostatic field from charged or partially charged groups on the protein and the response of the solvent (typically treated as a dielectric continuum). Realspace methods provide a helpful language for representing such a problem. The real-space grid can be refined to account for the high resolution necessary around the metal ion and can be adjusted for a coarser treatment further away. There is clearly no need to allow the metal and ligand orbitals to extend far from the central zone, so a localized representation is advantageous. Also, the electrostatic potential can be generated over the entire domain (quantum, classical, and solvent zones) with a single real-space solver without requiring special techniques for matching conditions in the various regions. The same ideas could be applied to defects in a covalent solid or impurity atoms in a cluster. In order to place the real-space methods in context, we first briefly examine other computational approaches. The plane-wave pseudopotential method has proven to be a powerful technique for locating the electronic ground state for many-particle systems in condensed phases (Payne et al., 1992). In this method the orbitals are expanded in the nonlocal plane-wave basis. The core states are removed via pseudopotential methods which allow for relatively smooth valence functions in the core region even for first-row and transition elements (Vanderbilt, 1990). Therefore, a reasonable number of plane-waves can be used to accurately represent most elements important for materials simulation. Strengths of this method include the use of efficient fast Fourier transform (FFT) techniques for updates of the orbitals and electrostatic potentials, lack of dependence of the basis on atom positions, and the rigorous control of numerical convergence of the approximation with decrease in wavelength of the highest Fourier mode. Algorithmic advances have led to excellent convergence characteristics of the method in terms of the number of required self-consistency steps (Payne et al., 1992;Hutter et al., 1994;Kresse and Furthmüller, 1996); only 5-10 selfconsistency iterations are required to obtain tight convergence of the total energy, even for metals. In spite of the numerous advantages of this approach, there are restrictions centered around the nonlocal basis set. Even with the advances in pseudopotential methods, strong variations in the potential occur in the core regions (especially for first-row and transition elements), and local refinements would allow for smaller effective energy cutoffs away from the nuclei. This issue has been addressed by the development of adaptive-coordinate plane-wave methods (Gygi, 1993). If any information is required concerning the inner-shell electrons, plane-wave methods suffer severe difficulties. Of course such states can be represented with a sufficient number of plane waves (Bellaiche and Kunc, 1997), but the short-wavelength modes required to build in the rapidly varying local structure extend over the entire domain to portions of the system where such resolution is not necessary. Also, for localized systems like molecules, clusters, or surfaces, nontrivial effort is expended to accurately reproduce the vacuum; the zero-density regions must be of significant size in order to minimize spurious effects in a supercell representation. In addition, charged systems create technical difficulties since a uniform neutralizing background needs to be properly added and subtracted in computations of total energies. Lastly, without special efforts to utilize localized-orbital representations, the wavefunction orthogonality step scales as N 3 , where N is the number of electrons. In quantum chemistry, localized basis sets built from either Slater-type orbitals (STOs) or Gaussian functions have predominated in the description of atoms and molecules (Szabo and Ostlund, 1989;Jensen, 1999). The molecular orbitals are constructed from linear combinations of the atomic orbitals (LCAO). An accurate representation can be obtained with less than thirty Gaussians for a first-row atom. In relation to plane-wave expansions, the localized nature of these basis functions is more in line with chemical concepts. With STOs or other numerical orbitals, the multicenter integrals in the Hamiltonian must be evaluated numerically, while with a Gaussian basis, the Coulomb integrals are available analytically. The 'price' for using Gaussians is that more basis functions are required to accurately describe the electron states, since they do not exhibit the correct behavior at either small or large distances from the nuclei. Techniques such as direct inversion in the iterative subspace (DIIS) have been developed to significantly accelerate the convergence behavior of basis-set self-consistent electronic structure methods (Pulay, 1980(Pulay, , 1982Hamilton and Pulay, 1986). 1 The LCAO methods have led to a dramatic growth in accurate calculations on molecules with up to tens of atoms. It is now common to see papers devoted to detailed comparisons of experimental results and electronic structure calculations on systems with more than one hundred electrons (Rodriguez et al., 1998). Often in basis-set calculations, care must be taken to account for basis-set superposition errors which arise due to overlap of nonorthogonal atom centered functions for composite systems. Also, linear dependence is a problem for very large basis sets chosen to minimize the errors. These factors lead to difficulty in obtaining the basis-set limit for a given level of theory. 2 The scaling of basis-set methods can be severe, but recent developments (see Section III) have brought the scaling down to linear for large systems. With the successes of plane-wave and quantum chemical basis functions, what is the motivation to search for alternative algorithms? Ten years ago, a review article discussing the relevance of Gaussian basis-set calculations for lattice gauge theories argued for the utilization of Gaussian basis sets in place of grids (Wilson, 1990). The author states (concerning the growth of quantum chemistry): "The most important algorithmic advance was the introduction of systematic algorithms using analytic basis functions in place of numerical grids, first proposed in the early 1950s." The point was illustrated by examination of core states for carbon: only a few Gaussians are required (with variable exponential parameters), while up to 8 × 10 6 grid points are necessary for the same energy resolution on a uniform mesh. What developments have occurred over the last decade which could begin to overcome such a large disparity in computational effort? This review seeks to answer the above question by summarizing recent research on realspace mesh techniques. To locate them in relation to plane-wave expansions and LCAO methods, some general features are introduced here and further developed throughout the article. The representation of the physical problems is simple: the potential operator is diagonal in coordinate space and the Laplacian is nearly local, depending on the order of the approximation. The near-locality makes real-space methods well suited for incorporation into linear-scaling approaches. It also allows for relatively straightforward domain-decomposition parallel implementation. Finite or charged systems are easily handled. With higher-order FD and FE approximations, the size of the overall domain is substantially reduced from the estimate above. Adaptive mesh refinements or coordinate transformations can be employed to gain resolution in local regions of space, further reducing the grid overhead. Real-space pseudopotentials result in smooth valence functions in the core region, again leading to smaller required grids. As mentioned above, the grid-based matrix representation produces structured and highly banded matrices, in contrast to plane-wave and LCAO expansions (Payne et al, 1992;Challacombe, 2000). These matrix equations can be rapidly solved with efficient multiscale (or other preconditioning) techniques. However, while more banded than LCAO representations, the overall dimension of the Hamiltonian is substantially higher. 3 In a sense, the real-space methods are closely linked to plane-wave approaches: they are both 'fully numerical' methods with one or at most a few parameters controlling the convergence of the approximation, for example the grid spacing h or the wavevector of the highest mode k. 4 On the other hand, the LCAO methods employ a better physical representation of the orbitals (thus requiring fewer basis functions); attached with this representation, however, are some of the problems discussed above related to the art of constructing nonorthogonal, atom-or bond-centered basis sets. The purposes of this paper are 1) to provide a basic introduction to real-space computational techniques, 2) to review their recent applications to chemical and physical problems, and 3) to relate the methods to other commonly used numerical approaches in electrostatics and electronic structure. The numerical problems addressed in this review can be categorized into four types in order of increasing complexity: ∇ 2 φ(r) = ? ; Real-Space Laplacian (1) ∇ 2 φ(r) = f (r) ; Poisson (2) ∇ 2 φ(r) = f (r, φ) ; Poisson-Boltzmann (3) ∇ 2 φ(r) + v(r, φ)φ(r) = λφ(r) ; Eigenvalue (4) The first expression symbolizes the generation of the Laplacian on the real-space grid. The second is the linear, elliptic Poisson equation. The third is the nonlinear Poisson-Boltzmann equation of electrostatics which describes the motion of small counterions in the field of fixed charges. The final equation is an eigenvalue equation such as the self-consistent Schrödinger equation occurring in electronic structure. Note that both the third and fourth equations are nonlinear. The Poisson-Boltzmann equation includes exponential driving terms on the rhs. The self-consistent eigenvalue problem is 'doubly nonlinear': one must solve for both the eigenvalues and eigenfunctions, and the potential generally depends nonlinearly on the eigenfunctions. The multigrid method allows for solution of both linear and nonlinear problems with similar efficiencies. The article is organized into several sections beginning with background discussion and then following the order of problems listed above. Section II introduces the central equations of DFT for electronic structure and charged classical systems. Section III reviews developments in linear-scaling computational algorithms and discusses their relationship to real-space methods. Section IV presents the fundamental aspects of representation in real space by examination of Poisson problems. Section V discusses multigrid methods for efficient solution of the resulting matrix representations. Section VI summarizes recent advances in electrostatics computations in real space including both Poisson and nonlinear Poisson-Boltzmann solvers. Applications in biophysics are illustrated with several examples. Section VII discusses real-space eigenvalue methods for self-consistent problems in electronic structure. Section VIII summarizes recent computations of optical response properties and excitation energies with real-space methods. The review concludes with a short summary and discussion of possible future directions for research. eigenvalue equations. These equations have provided a practical tool for realistic electronic structure computations on a vast array of atoms, molecules, and materials (Parr and Yang, 1989). The Hohenberg-Kohn theorems have been extended to finite-temperature quantum systems by Mermin (1965) and to purely classical fluids in subsequent work (Hansen and McDonald, 1986;Ichimaru, 1994). An integral formulation of electronic structure has also been discovered in which the one-electron density is obtained directly without the introduction of orbitals (Harris and Pratt, 1985;Parr and Yang, 1989). This approach is in the spirit of the original Hohenberg-Kohn theorems, but to date this promising theory has not been used extensively in numerical studies. This section reviews the basic equations of DFT for electronic structure and charged classical systems. These equations provide the background for discussion of the real-space numerical methods. A. Kohn-Sham equations The Kohn-Sham self-consistent eigenvalue equations for electronic structure can be written as follows (atomic units are assumed throughout): [− 1 2 ∇ 2 + v ef f (r)]ψ i (r) = ǫ i ψ i (r),(5) where the density-dependent effective potential is v ef f (r) = v es (r) + v xc ([ρ(r)]; r).(6) The classical electrostatic potential v es (r) is due to both the electrons and nuclei, and the (in principle) exact exchange-correlation potential v xc ([ρ(r)]; r) incorporates all nonclassical effects. The exchange-correlation potential includes a kinetic contribution since the expectation value of the Kohn-Sham kinetic energy is that for a set of non-interacting electrons moving in the one-electron effective potential. The electron density, ρ(r), is obtained from the occupied orbitals (double occupation is assumed here): ρ(r) = 2 Ne/2 i=1 \|ψ i (r)\| 2 .(7) The electrostatic portion of the potential for a system of electrons and nuclei (Hartree potential plus nuclear potential) is given by v es (r) = ρ(r ′ ) \|r − r ′ \| dr ′ − Nn i=1 Z i \|r − R i \| .(8) This potential can be obtained by numerical solution of the Poisson equation: ∇ 2 v es (r) = −4πρ tot (r),(9) where ρ tot (r) is the total charge density due to the electrons and nuclei. If the exchange-correlation potential is taken as a local function (as opposed to functional) of the density with the value the same as for a uniform electron gas, the approximation is termed the local density approximation (LDA). Ceperley and Alder (1980) determined the exchange-correlation energy for the uniform electron gas numerically via Monte Carlo simulation. The data have been parametrized in various ways for implementation in computational algorithms (see, for example, Vosko et al., 1980). The LDA theory has been extended to handle spin-polarized systems (Parr and Yang, 1989). The LDA yields results with accuracies comparable to or often superior to Hartree-Fock, but generally leads to overbinding in chemical bonds among other deficiencies. One obtains the Hartree-Fock approximation if the local exchange-correlation potential in Eq. (6) is replaced by the nonlocal exact exchange operator. In recent years, a great deal of effort has gone into developing more accurate exchangecorrelation potentials (see Jensen, 1999, for a review). These advances involve both gradient expansions which incorporate information from electron density derivatives and hybrid methods which include some degree of exact Hartree-Fock exchange. With the utilization of these modifications, results of chemical accuracy can be obtained. Since the main focus of this review is on numerical approaches for solving the self-consistent equations, we do not further examine these developments. Pseudopotential techniques allow for the removal of the core electrons. The valence electrons then move in a smoother (nonlocal) potential in the core region while exhibiting behavior the same as in an all-electron calculation outside the core. Recently developed real-space versions of the pseudopotentials allow for computations on meshes Martins, 1991a, 1991b;Goedecker et al., 1996;Briggs et al., 1996). Inclusion of the pseudopotentials substantially reduces the computational overhead since fewer orbitals are treated explicitly and the required mesh resolution can be coarser. However, truly local mesh-refinement techniques may allow for the efficient inclusion of core electrons when necessary (see Sections VI.A.2 and VII.D). Self-consistent solution of the Kohn-Sham equations [Eq. (5)] for fixed nuclear locations is conceptually straightforward. An initial guess is made for the orbitals. This yields an electron density from which the effective potential is constructed by solution of the Poisson equation and generation of the exchange-correlation potential. The eigenvalue equation is solved with the current effective potential [Eq. (6)], resulting in a new set of orbitals. The process is repeated until the density or total energy change only to within some desired tolerance. Alternatively, the total energy can be minimized variationally using a technique such as conjugate gradients (Payne et al., 1992); the orbitals at the minimum correspond to those from the iterative process described above. B. Classical DFT The ground-state theory discussed above has been extended to finite-temperature quantum and classical systems and has found wide use in the theory of fluids (Rowlinson and Widom, 1982;Hansen and McDonald, 1986;Ichimaru, 1994). Here I discuss the formulation for systems of charged point particles (mobile ions) moving in the external potential produced by other charged particles in the solution (for example, colloid spheres or cylinders). The solvent is assumed to be a uniform dielectric with dielectric constant ǫ in these equations. The free energy for an ion gas of counterions can be written as the sum of an ideal term, the energy of the mobile ions in the external field due to the fixed colloid particles (this term incorporates both the electrostatic potential from the fixed charges on the colloids and an excluded-volume potential), F ext = q drρ m (r)V ext (r, {R j }),(10) the Coulomb interaction potential energy of the mobile ions with each other, and a correlation free energy. The mobile-ion density ρ m (r) is the number density, not the charge density, in the solution. The charge on the counterions is q, and the approximate correlation free energy typically assumes a local density approximation for a one-component plasma. Thus the theory includes ion correlations, but the approximation is not systematically refineable, just as in the Kohn-Sham LDA equations. Löwen (1994) utilized this free energy functional in Car-Parrinello-type simulations (Car and Parrinello, 1985) of charged rods with surrounding counterions. The equilibrium distribution is obtained by taking the functional derivative of the free energy with respect to the mobile-ion density and setting it to zero. It is clear that, if the correlation term is set to zero, the equilibrium density of the mobile counterions is proportional to the Boltzmann factor of the sum of the external and mobile ion Coulomb potentials: ρ m (r) ∼ exp{−βq[V ext (r) + φ m (r)]}.(11) The potential φ m (r) is that due to the mobile ions only and β = (kT ) −1 . Since the total charge (fixed charges on colloid particles and mobile ion charges) at equilibrium must satisfy the Poisson equation, the following nonlinear differential equation results for the equilibrium distribution of the mobile ions in the absence of correlations. The treatment is generalized here to account for the possibility of additional salt in the solution and a dielectric constant that can vary in space (Coalson and Beck, 1998): ∇ · (ǫ(r)∇φ(r)) = −4π[ρ f (r) + qn + e −βqφ(r)−v(r) − qn − e βqφ(r)−v(r) ],(12) where φ(r) is the total potential due to the fixed colloid charges and mobile ions, ρ f (r) is the charge density of the fixed charges on the colloids,n + andn − are the bulk equilibrium ion densities at infinity (determined self-consistently so as to conserve charge in the region of interest), and v(r) is a very large positive excluded-volume potential which prevents penetration of the mobile ions into the colloids. Fushiki (1992) performed molecular dynamics simulations of charged colloidal dispersions at the Poisson-Boltzmann level; the nonlinear Poisson-Boltzmann equation was solved numerically at each time step with FD techniques. An alternative elegant statistical mechanical theory for the ion gas has been formulated (Coalson and Duncan, 1992). It uses field theoretic techniques to convert the Boltzmann factor for the ion interactions into a functional-integral representation of the partition function. The Poisson-Boltzmann-level theory results from a saddle-point approximation to the functional integral. The distinct advantage of this theory is that correlations can be systematically included by computing the corrections to the mean-field approximation via loop expansions. However, in practice the corrections are computationally costly for real-space grids of substantial size. This theory was used in simulations of colloids (Walsh and , and the deviations from mean-field theory were investigated. For realistic concentrations of monovalent background ions, the corrections are often small in magnitude, thus justifying the Poisson-Boltzmann-level treatment. Correlations must be considered, however, for accurate computations involving divalent ions (Guldbrand et al., 1984;Tomac and Gräslund, 1998;Patra and Yethiraj, 1999). III. LINEAR-SCALING CALCULATIONS Several new methods have appeared for computations involving systems with long-range interactions. In this section, developments in linear-scaling methods for classical and quantum systems are summarized. Goedecker (1999) has clearly reviewed applications in electronic structure, so the discussion of this topic will be limited. The purpose is to illustrate the context in which real-space methods can be utilized in linear-scaling solvers for electrostatics and electronic structure. A. Classical electrostatics Three algorithms have been most widely used in classical electrostatics calculations which require consideration of long-range forces. The first is the Ewald (1921) summation, which partitions the Coulomb interactions into a short-range sum handled in real space and a longrange contribution summed in reciprocal space. Both sums are convergent. The partitioning is effected by adding and subtracting localized Gaussian functions centered about the discrete charges (Pollock and Glosli, 1996). In the original Ewald method, the Coulomb interaction of the Gaussians is obtained analytically: E Gauss = 1 2 k =0 4π Ω exp(−k 2 /2G 2 ) k 2 \|S(k)\| 2 ,(13) once the charge structure factor, S(k) = N i=1 Z i exp(ik · r i ),(14) is computed. In Eq. (13), Ω is the cell volume and G is the Gaussian width. This method scales as N 3/2 (where N is the number of particles) so long as an optimal exponential factor is used in the Gaussians. Discussion of the optimization equation which yields the N 3/2 scaling is given in Pollock (1999). The Ewald technique has been used extensively in simulations of charged systems (Allen and Tildesley, 1987). An efficient alternative procedure for Madelung sums in electronic structure calculations on crystals was proposed by Harris and Monkhorst (1970). The scaling of the Ewald method has been reduced by an alternative treatment for the interaction energy of the Gaussians. Instead of solving the problem analytically, 1) the charge density is assigned to a mesh, 2) the Poisson equation is solved using FFT methods, 3) the potential is differentiated, and lastly 4) the forces are interpolated to the particles. These methods are termed particle-particle particle-mesh (Hockney and Eastwood, 1988) or particle-mesh Ewald (Darden et al., 1993) [or an improved version called smooth particlemesh Ewald (Essmann et al., 1995)]. Since the potential is generated numerically via FFT, the methods scale as N ln N (or N √ ln N if an optimal G is used; see Pollock, 1999). The above-mentioned methods differ in how the four steps in the force generation are performed, but all three center on the use of FFT algorithms for their efficiency. Comparative studies have suggested that the original particle-particle particle-mesh method is more accurate than the particle-mesh Ewald versions; Deserno and Holm (1998) recommend its use with modifications obtained from particle-mesh Ewald. See also Sagui and Darden (1999), where it is shown that similar accuracies can be obtained with particle-mesh Ewald as compared to the particle-particle particle-mesh method. The second algorithmic approach utilizes the fast multipole method (FMM) (Greengard, 1994) or related hierarchical techniques. In these methods, the near-field contributions are treated explicitly, while the far field is handled by clustering charges into spatial cells and representing the field with a multipole expansion. The methods are claimed to scale linearly with system size, but recent work contends the scaling is slightly higher (Pérez-Jordá and Yang, 1998). Fast multipole techniques and the quantum chemical tree code (QCTC) of Challacombe et al. (1996), have been widely applied in Gaussian-based electronic structure calculations. Since the classical Coulomb part of the problem is a significant or even dominant part of the overall computational effort, near linear scaling is required for an overall linear-scaling solver (Strain et al., 1996;. Pérez-Jordá and Yang (1997) have developed an alternative efficient recursive bisection method for obtaining the Coulomb energy from electron densities. The FMM has also been utilized extensively in particle simulations. In comparative studies of periodic systems, Pollock and Glosli (1996) and Challacombe et al. (1997), have shown that, for the case of discrete particles, the particlemesh related techniques are more efficient than the fast multipole method over a wide range of system sizes (up to 10 5 particles). However, for the case of continuous overlapping distributions, it is difficult to develop systematic ways in the particle-mesh approach to handle the charge penetration in large-scale Gaussian calculations (Challacombe, 1999a). Recently, Cheng et al. (1999) have developed a more efficient and adaptive version of the fast multipole method which will make the technique competitive with the particle-mesh method. Also, Greengard and Lee (1996) presented a method combining a local spectral approximation and the fast multipole method for the Poisson equation. A third set of linear-scaling algorithms for classical electrostatics employs real-space methods, which will be discussed in-depth in subsequent sections. The problem is represented with FD equations, FE methods, or wavelets, and solved iteratively on the mesh. Since all operations are near-local in space, the application of the Laplacian to the potential is strictly linear scaling. However, the iterative process on the fine mesh typically suffers from slowing down in the solution process, so efficient preconditioning techniques must be employed to obtain the linear scaling. The multigrid method (Brandt, 1977(Brandt, , 1982(Brandt, , 1984(Brandt, , 1999Hackbusch, 1985) is a particularly efficient method for solving the discrete equations. Linear-scaling real-space methods have been developed for solution of the Poisson problem in DFT (White et al., 1989;Merrick et al., 1995Merrick et al., , 1996Gygi and Galli, 1995;Briggs et al., 1995;Modine et al., 1997;Goedecker and Ivanov, 1998a). These studies have illustrated the accuracies and efficiencies of the real-space approach. One possible application of multigrid techniques which has not received attention is in solving for the Coulomb energy of the Gaussian charges in the particle-mesh algorithms. Since the multigrid techniques are highly efficient, scale linearly, and allow for variable resolution, they may provide a helpful counter-part to the FFT-based methods currently used. An advantageous feature of the multigrid solution during a charged-particle simulation is that, once the potential is generated for a given configuration, it can be saved for the next solution process for the updated positions which have changed only slightly. Thus the required number of iterations is likely to be low. Tsuchida and Tsukada (1998) utilized similar ideas in their FE method for electronic structure, where they employed MG acceleration for rapid solution of the Poisson equation and discussed the relation of their method to the particle-mesh approach. B. Electronic structure Electronic structure calculations involve computational complexities which go well beyond the necessity for efficient solution of the Poisson equation. In order to obtain linear scaling, physical localization properties must be exploited either for the range of the density matrix or the orbitals. Goedecker (1999) categorized the various linear-scaling electronic structure methods as follows: Fermi operator expansion (FOE), Fermi operator projection (FOP), divide and conquer (DC), density matrix minimization (DMM), orbital minimization (OM), and optimal-basis density matrix minimization (OBDMM). He further classified the algorithms into those which employ small basis sets (LCAO-type approaches) and ones which utilize large basis sets (FD or FE). 5 Clearly the methods most relevant to the present discussion are those which can be implemented with large basis sets (FOP, OM, and OBDMM). The two approaches most directly related to the FD and FE mesh techniques considered here are the OM and OBDMM methods, so we review their characteristics. The OM method obtains the localized Wannier functions by minimization of the functional: Ω = 2 n i,j c n i H ′ i,j c n j − n,m i,j c n i H ′ i,j c m j l c n l c m l .(15) The minimization is unconstrained in that no orthogonalization is required; the orthonormality condition is automatically satisfied at convergence. In Eq. (15), Ω is the 'grand potential', the c n i are the expansion coefficients for the Wannier function n with basis function i, and the H ′ i,j are the matrix elements of H − µI, where µ is the chemical potential controlling the number of electrons and I is the identity matrix. The functional can be derived by making a Taylor expansion of the inverse of the overlap matrix occurring in the total energy expression (Mauri et al., 1993). Ordejón et al. (1995) presented an alternative derivation and related the OM functionals to the DMM approach. Assuming no localization restriction on the orbitals, it can be shown that the functional Ω gives the correct ground state at its minimum. However, some problems arise when localization constraints are imposed: 1) the functional can have multiple minima, 2) the number of required iterations to reach the ground state can be quite large, 3) there may be runaway solutions depending on the initial guess, and 4) the total charge is not conserved for all stages of the minimization (although charge is accurately conserved close to the minimum). The original OM methods utilized underlying plane-wave (Mauri et al., 1993;Mauri and Galli, 1994) and tight-binding or LCAO-type bases (Kim, et al., 1995;Ordejón et al., 1995;Sánchez-Portal et al., 1997) for the representation of the localized orbitals. In the work of Sánchez-Portal et al. (1997) on very large systems, a fully numerical LCAO basis developed by Sankey and Niklewski (1989) was implemented for the orbitals, and the Hartree problem was solved via FFT techniques on a real-space grid. Lippert et al. (1997) developed a related hybrid Gaussian and plane-wave algorithm which uses Gaussians in place of the numerical atomic basis. Also, Haynes and Payne (1997) formulated a new localized spherical-wave basis which has features in common with plane waves in that a single parameter controls the convergence. Real-space formulations have also applied OM ideas; since the real-space approach is inherently local, it provides a natural representation for the linear-scaling algorithms. Tsuchida and Tsukada (1998) incorporated unconstrained minimization into their FE electronic structure method. Hoshi and Fujiwara (1997) also employed unconstrained minimization in their FD self-consistent electronic structure solver. Finally, Bernholc et al. (1997) utilized the original localized-orbital functional of Galli and Parrinello (1992) in their FD multigrid method to obtain linear scaling. They are also investigating other order N functionals. These real-space algorithms will be the subject of extensive discussion in Section VII. The OBDMM method is an efficient combination of density matrix and orbital-based methodologies. The optimization process to locate the ground state is divided into two minimization steps. In the inner loop, the usual DMM procedure is followed to obtain the density matrix for a fixed contracted basis. The density matrix F (r, r ′ ) is represented in terms of contracted basis functions ψ i and a matrix K which is a purified form from the DMM method: F (r, r ′ ) = i,j ψ * i (r)K i,j ψ j (r ′ ),(16) and K = 3LOL − 2LOLOL,(17) where L is the contracted basis density matrix and O the overlap matrix. The matrix K is 'purified' in that if the eigenvalues of L are close to zero or one, the eigenvalues of K will be even closer to those values. The outer loop searches for the optimal basis with fixed L. The OBDMM method was developed independently by Hierse and Stechel (1994) and Hernández and Gillan (1995). The two approaches differ in that the algorithm of Hernández and Gillan allows for a number of basis functions larger than the number of electrons. Also, Hierse and Stechel (1994) used tight-binding and Gaussian bases, while Hernández and Gillan (1994) employed a FD difference representation in their original work. Later, Hernández et al. (1997) developed a blip-function basis (a local basis of B-splines, see Strang and Fix, 1973), very closely related to FE methods. In the quantum chemistry literature, efforts have focused on Gaussian basis-function algorithms. As discussed above, the Coulomb problem is typically solved with the FMM or other hierarchical techniques Strain et al., 1996;Challacombe et al., 1996;Challacombe and Schwegler, 1997). Additional algorithmic advances include linear scaling for the exchange-correlation calculation in DFT (Stratmann et al., 1996), for the exact exchange matrix in Hartree-Fock theory (Schwegler and , and for the diagonalization operation (Millam and Scuseria, 1997;Challacombe, 1999b). Alternative linear-scaling algorithms include the early Green's function based FD method of Baroni and Giannozzi (1992) and the finite-temperature real-space method of Alavi et al. (1994). It is evident from the above discussion that real-space methods, in particular FD and FE approaches, 6 are well suited for linear-scaling algorithms. In classical electrostatics calculations, the multigrid method provides an efficient and linear-scaling technique for solution of Poisson problems given a charge distribution on a mesh (finite or periodic systems). In electronic structure, FD and FE representations have been extensively employed in the OM and OBDMM localized-orbital linear-scaling contexts. IV. REAL-SPACE REPRESENTATIONS The early development of FD and FE methods for solving partial differential equations stemmed from engineering problems involving complex geometries, where analytical approaches were not possible (Strang and Fix, 1973). Example applications include structural mechanics and fluid dynamics in complicated geometries. However, even in the early days of quantum mechanics, attention was paid to FD numerical solutions of the Schrödinger equation (Kimball and Shortley, 1934;Pauling and Wilson, 1935). Also, fully converged numerical solutions of self-consistent electronic structure calculations have played an important role in atomic physics (see Mahan and Subbaswamy, 1990, for a discussion of the methodology for spherically symmetric systems) and more recently in molecular physics (Laaksonen et al., 1985;Becke, 1989). Real-space calculations are performed on meshes; these meshes can be as simple as Cartesian grids or can be constructed to conform to the more demanding geometries arising in many applications. Finite-difference representations are most commonly constructed on regular Cartesian grids. They result from a Taylor series expansion of the desired function about the grid points. The advantages of FD methods lie in the simplicity of the representation and resulting ease of implementation in efficient solvers. Disadvantages are that the theory is not variational (in the sense of providing and upper bound, see below), and it is difficult to construct meshes flexible enough to conform with the physical geometry of many problems. Finite-element methods, on the other hand, have the advantages of significantly greater flexibility in the construction of the mesh and an underlying variational formulation. The cost of the flexibility is an increase in complexity and more difficulty in the implementation of multiscale or related solution methods. In this Section, we review the technical aspects of real-space FD and FE representations of differential equations by examination of Poisson problems. A. Finite differences Basic finite-difference representation The second-order FD representation of elliptic equations is very simple but serves to illustrate several key features. Consider the Poisson equation in one dimension (the 4π is left here since we will be considering three-dimensional problems): d 2 φ(x) dx 2 = −4πρ(x),(18) where φ(x) is the potential and ρ(x) the charge density. Expand the potential in the positive and negative directions about the grid point x i : φ(x i+1 ) = φ(x i ) + φ ′ (x i )h + 1 2 φ ′′ (x i )h 2 + 1 6 φ ′′′ (x i )h 3 + 1 24 φ (iv) (x i )h 4 . . . φ(x i−1 ) = φ(x i ) − φ ′ (x i )h + 1 2 φ ′′ (x i )h 2 − 1 6 φ ′′′ (x i )h 3 + 1 24 φ (iv) (x i )h 4 . . .(19) The grid spacing is h, here assumed uniform. If these two equations are added and the sum is solved for φ ′′ (x i ), the following approximation results: d 2 φ(x i ) dx 2 ≈ 1 h 2 [φ(x i−1 ) − 2φ(x i ) + φ(x i+1 )] − 1 12 φ (iv) (x i )h 2 + O(h 2 )(20) The first contribution to the truncation error is second order in h with a prefactor involving the fourth derivative of the potential. Depending on the nature of the function φ(x), the errors can be of either sign. When φ(x) is used to compute a physical quantity such as the total electrostatic energy, the net errors in the energy can be either positive or negative. In this sense, the FD approximation is not variational. As we will see below, the solution can be obtained by minimizing an energy (or action) functional, which is a variational process, but the solution does not necessarily satisfy the variational theorem obtained in a basis-set method. So the FD approach is not a basis-set method. In matrix form, the one-dimensional discrete Poisson equation is 1 h 2          −2 1 0 0 . . . 1 −2 1 0 . . . 0 1 −2 1 . . . 0 0 1 −2 . . . . . . . . . . . . . . . . . .                  φ(x 1 ) · · · φ(x N )         = −4π         ρ(x 1 ) · · · ρ(x N )         .(21) This equation can be expressed symbolically as L h u h ex = f h ,(22) where L h is the discrete Laplacian, u h ex is the exact solution on the grid, and f h is −4πρ. The operator −L is positive definite. An observation from the matrix form Eq. (21) is that the Laplacian is highly sparse and banded in the FD representation; its application to the potential is thus a linear-scaling step. In one dimension the matrix is tridiagonal, while in two or three dimensions it is no longer tridiagonal but is still extremely sparse with nonzero values only near the diagonal. This differs from the wavelet representation, which is sparse but includes several bands in the matrix (Goedecker and Ivanov, 1998b, Fig. 7;Arias, 1999, Fig. 10). In addition to the truncation error, t h = − 1 12 φ (iv) (x i )h 2 + O(h 4 ),(23) estimates can be made of the function error itself (see Strang and Fix, 1973, p. 19): e h a = u h ex − u a = e 2 h 2 + O(h 4 ),(24) where u a is the exact solution to the continuous differential equation and e 2 is proportional to the second derivative of the potential. Therefore, one can test the order of a given solver for a case with a known solution by computing errors over the domain and taking ratios for variable grid spacing h. For example, the ratio of the errors on a grid with spacing H = 2h to those on h for overlapping points should be close to 4.0 in a second-order calculation. The two-and three-dimensional representations are obtained by summing the onedimensional case along the two or three orthogonal coordinate axes (this holds for higherorder forms as well). Since the Laplacian is the dot product of two vector operators, offdiagonal terms are not necessary. The second-order two-dimensional Laplacian consists of five terms with a weight of -4 instead of -2 on the diagonal, and the three-dimensional case has seven terms with a weight of -6 along the diagonal. See Abramowitz and Stegun (1964, Sections 25.3.30 and 25.3.31), for the two-dimensional representation of the Laplacian. Solution by iterative techniques Consider the action functional: S[φ] = 1 2 \|∇φ\| 2 d 3 x − 4π ρφd 3 x.(25) If the first term on the rhs is integrated by parts (assuming the function and/or its derivative go to zero at infinity or are periodic), one obtains S[φ(r)] = − 1 2 φ∇ 2 φd 3 x − 4π ρφd 3 x.(26) Take the functional derivative of the action with respect to variations of the potential, and a 'force' term results, − δS δφ = ∇ 2 φ + 4πρ,(27) which can be employed in a steepest-descent minimization process: δφ δt = − δS δφ ,(28) where t is a fictitious time variable. Then discretize the problem in space and time, leading to (for simplicity of representation a one-dimensional form is given here) φ(x i ) t+1 = (1 − ω)φ(x i ) t + ω 2 [φ(x i−1 ) t + φ(x i+1 ) t + 4πρ(x i )h 2 ],(29) The parameter ω is 2δt/h 2 . The two-and three-dimensional expressions are easily obtained following the same procedure. Since the action as defined in Eq. (26) possesses only a single minimum, the iterative process eventually converges so long as a sufficiently small time step δt is chosen to satisfy the required stability criterion (below). Several relaxation strategies result from the steepest-descent scheme of Eq. (29). As it is written, the method is termed weighted-Jacobi iteration. If the previously updated value φ(x i−1 ) t+1 is used in place of φ(x i−1 ) t on the right hand side, the relaxation steps are called Successive Over-Relaxation or SOR. If the parameter in SOR is taken as ω = 1, the result is Gauss-Seidel iteration. Gauss-Seidel and SOR do not guarantee reduction in the action at each step since they use the previously updated value. Generally, Gauss-Seidel iteration is the best method for the smoothing steps in multigrid solvers (Brandt, 1984). If one cycles sequentially through the lattice points, the ordering is termed lexicographic. Higher efficiencies (and vectorization) can be obtained with red-black ordering schemes in which the grid points are partitioned into two interlinked sets and the red points are first updated, followed by the black (Brandt, 1984;Press et al., 1992). Similar techniques can be used for high orders with multicolor schemes. Conjugate-gradient methods (Press et al., 1992) significantly outperform the above relaxation methods when used on a single grid level. However, in a multigrid solver the main function of relaxation is only to smooth the high-frequency components of the errors on each level (see Section V.A), and simple relaxation procedures (especially Gauss-Seidel) do very well for less cost. An important issue in iterative relaxation steps relates to the eigenvalues of the update matrix defined by Eq. 29 (Briggs, 1987). Solution of the Laplace equation using weighted-Jacobi iteration illustrates the basic problem. For that particular case, the eigenvalues of the update matrix are λ k = 1 − 2ω sin 2 kπ 2N ; 1 ≤ k ≤ N − 1,(30) where ω is the relaxation parameter defined above, N + 1 is the number of grid points in the domain, and k labels the mode in the Fourier expansion of the function. Generally, the Fourier component of the error with wavevector k is reduced in magnitude by a factor proportional to λ t k in t iterations. First, it is easy to see that if too large an ω value (that is 'time' step for fixed h) is taken, the magnitude of some modes will exceed one, leading to instability. This shows up very quickly in a numerical solver! Second, for the longest-wavelength modes, the eigenvalues are of the form: λ = 1 − O(h 2 ).(31) As more grid points are used to obtain increased accuracy on a fixed domain, the eigenvalues of the longest-wavelength modes approach one. Therefore, these modes of the error are very slowly reduced. This fact leads to the phenomenon of critical slowing down in the iterative process ( Fig. 2), which motivated the development of multigrid techniques. Multigrid methods utilize information from multiple length scales to overcome the critical slowing down (Section V). Generation of high-order finite-difference formulas Mathematical arguments lead to the conclusion that the FD scheme discussed above is convergent in the sense that E h → 0 as h → 0 (Strang and Fix, 1973;Vichnevetsky, 1981). Therefore, one only needs to proceed to smaller grid spacings to obtain results with a desired accuracy. This neglects the practical issues of computer time and memory, however, and it has become apparent that orders higher than second are most often necessary to obtain sufficient accuracy in electronic structure calculations on reasonable-sized meshes . The higher-order difference formulas are well known (Hamming, 1962;Vichnevetsky, 1981), and can easily be generated using computer algebra programs (see Appendix A). Why does it pay to use high-order approximations? Consider the three-dimensional Poisson equation with a singular-source charge density: ∇ 2 φ(x) = −4πδ(x).(32) The Dirac delta function is approximated by a unit charge on a single grid point. Let us solve the FD version of Eq. (32) on a 65 3 domain using 2nd-and 4th-order Laplacians and compare the potential eight grid points away from the origin. 7 In order to obtain the same numerical accuracy with a 2nd-order Laplacian, a grid spacing with one third that for the 4th-order case is required. This implies a 27-fold increase in storage and roughly a 14-fold increase in computer time, since the application of the Laplacian contains 7(13) terms for the 2nd(4th)-order calculations. As a second example, we solve for five states of the hydrogen-atom eigenvalue problem using the fixed potential generated in the solution of Eq. (32). The grid parameters are the same as those used in the multigrid eigenvalue computations of Section VII.A.2. The variation of the eigenvalues, the first orbital moments, and the virial ratios with approximation order are presented in Figs. 3, 4, and 5. A possible accuracy target is the thermal energy at room temperature (kT ≈ 0.001 au); this accuracy is achieved at 12th order. Clearly the results at 2nd order are not physically reasonable, but accurate results can be obtained with the higher orders. Merrick et al. (1995) and Chelikowsky, Trouller, Wu, and Saad (1994) have presented analyses of the impact of order on accuracy in DFT electrostatics and Kohn-Sham calculations; in the Kohn-Sham calculations, 8th or 12th orders were required for adequate convergence. There exist alternative high-order discretizations such as the Mehrstellen form used in the work of Briggs et al. (1996). This discretization is 4th order and leads to terms which are off-diagonal in both the kinetic and potential operators. The advantage of the Mehrstellen approach is that both terms only require near-neighbor points on the lattice, while the highorder forms above include information from further points (which increases the communication overhead somewhat in parallel implementations). However, the 4th-order Mehrstellen operator involves 33 multiplies to apply the Hamiltonian to the wavefunction, while the standard 4th-order discretization requires only 14 (a 12th-order standard form uses 38 multiplies). Also, the Mehrstellen representation has only been applied to the 4th-order case, and for some applications higher orders may be required. The exact terms for the Mehrstellen representation of the real-space Hamiltonian are given in Briggs et al. (1996). B. Finite elements Variational formulation Consider again the action of Eq. (25) in one dimension: S[φ] = 1 2 dφ dx 2 dx − 4π ρφdx.(33) This form of the action proves useful since the appearance of the first derivative as opposed to the second expands the class of functions which may be used to represent the potential. Now, expand the potential in a basis: φ(x) = n i=1 u i ζ i (x),(34) where the u i are the expansion coefficients and ζ i the basis functions. The action is then S = 1 2 n i=1 dζ i dx u i 2 dx − 4π ρ n i=1 ζ i u i dx.(35) The variational calculation is performed by minimizing the action with respect to variations in the expansion coefficients (assuming the original differential operator is positive definite): ∂S ∂u i =   dζ i dx   n j=1 dζ j dx u j   − 4πρζ i   dx = 0.(36) The minimization equation leads to a matrix problem completely analogous to Eq. (22). In the present case, the grid index is replaced by the basis-function index. It is often necessary to perform the integral of the second term (which involves the charge density) numerically. A more general origin of the FE method is termed the Galerkin approach which takes as its starting point the "weak" formulation of the problem. This method allows one to handle problems which cannot be cast in the minimization format described above by requiring only an extremum of the action functional and not a minimum. Also, it does not require symmetric operators. Take the action functional of Eq. (33) and perturb it by the addition of a small term ǫv. 8 The action becomes S[φ + ǫv] = S[φ] + 1 2 ǫ 2 dv dx 2 dx + ǫ dφ dx dv dx dx − 4πǫ ρvdx.(37) By taking the derivative with respect to ǫ, making ǫ zero, and setting what remains to zero, the stationary point is obtained. This variational form results in the following integral equation: dφ dx dv dx dx = 4π ρvdx.(38) This equation is valid for any test function v; solution requires finding the function φ for which the equation holds for all v. Alternatively, Eq. 38 can be derived by simply left multiplying the differential equation by the test function v and integrating by parts. When the functions φ and v are represented in the ζ i basis, a matrix equation the same as Eq. (36) is obtained. This basis-set manifestation of the weak formulation is termed the Galerkin method. If the test function space for v is taken to include all Dirac delta functions, and the problem is cast in the strong form < v, Lφ+4πρ >= 0 (where L is the differential operator, in this case the Laplacian), the collocation (or pseudospectral) approximation results when the problem is discretized (Orszag, 1972;Vichnevetsky, 1981;Ringnalda et al., 1990). Excellent reviews of the theory and application of finite elements are given in Strang and Fix (1973), Vichnevetsky (1981), Brenner and Scott (1994), and Reddy (1998). Finite-element bases Any linearly independent basis may be used to expand the potential. One choice would be to expand in trigonometric functions which span the whole domain. Then Fourier transform techniques could be used to solve the equations. In the FE method, the basis functions are rather taken as piecewise polynomials which are nonzero only in a local region of space (that is, have small support). The simplest possible basis consists of piecewise-linear functions whose values are one at the grid point about which they are centered and zero everywhere beyond the nearest-neighbor grid points. Then the coefficients u i correspond to the actual function values on the mesh. With this basis and a basis-set representation of the charge density ρ(x), the resulting matrix representation of the one-dimensional Poisson equation is identical to Eq. (21), except the right hand side is replaced by terms which are local averages of the charge density over three points. The local average is idential to Simpson's rule integration. Therefore, for uniform meshes, there is a close correspondence between FD and FE representations. Relaxation methods similar to those described above can be used to solve the FE equations. Besides the variational foundation of the FE method, the key advantage over FD approaches is in the flexibility available to construct the mesh to conform to the physical geometry. This issue becomes particularly relevant for two-and three-dimensional problems. There is of course an immense literature on development of accurate and efficient basis sets for FE calculations in a wide variety of engineering and physical applications, and that topic cannot be covered properly here. Some representative bases are mentioned from recent three-dimensional electronic structure calculations. White et al. (1989) employed a cubic-polynomial basis and constructed an orthogonal basis from the nonorthogonal set. Ackermann et al. (1994) used a tetrahedral discretization with orders p = 1 − 5. Pask et al. (1999) utilized piecewise cubic functions (termed "serendipity" elements). Yu et al. (1994) employed a Lobatto-Gauss basis set with orders ranging from five to seven. Hernández et al. (1997) developed a B-spline basis which is closely related to traditional FE bases. Tsuchida and Tsukada (1998) used piecewise third-order polynomials in their self-consistent electronic structure calculations. In relating the FD and FE methods, two points are worth noting. First, the FE bases are typically nonorthogonal and this issue must be dealt with in the formulation. Second, since the basis is local, the representation is banded with the width depending on the degree of the polynomials. For the FD representation, the high-order Laplacian includes 3p + 1 terms in a row of L for three-dimensional calculations. Alternatively, the FE method requires O(p 3 ) terms along a row of L in the limit of high orders, although the exact number of terms depends on the particular elements (Pask, 1999). This issue of scaling of the bandwidth with order may become a significant one in development of efficient iterative solvers of the equations. Due to the relative merits of the two representations, there is no clear 'answer' as to which one is preferable; the key feature for this review is that both are near-local leading to structured and sparse matrix representations of the differential equations. The wavelet basis method is closer in form to the FE representation but, as mentioned above, leads to more complicated matrix structures than either the FD or FE cases (Goedecker and Ivanov, 1998b;Arias, 1999). V. MULTIGRID TECHNIQUES The previous section discussed the basics of real-space formulations. The representations are near-local in space, and this locality manifests itself in the stalling process of iterative solvers induced by Eq. (31). The finer the resolution of the mesh, the longer it takes to remove the long-wavelength modes of the error. The multigrid technique was developed in order to overcome this inherent difficulty in real-space methods. Multigrid methods provide the optimal solvers for problems represented in real space. A. Essential features of multigrid The asymptotic convergence of an iterative solver on a given scale is controlled by Eq. (31). However, for shorter-wavelength modes it is easy to show that the convergence factor µ = \|ẽ h \| \|e h \| ,(39) where \|e h \| is the norm of the difference vector between the exact grid solution u h ex and the current approximation u h , andẽ h is the vector for the next step of iteration, is of order 0.5 for Gauss-Seidel iteration on the Poisson equation (Brandt, 1984). Those components of the error are reduced by an order of magnitude in only three relaxation sweeps. Thus, relaxation steps on a given grid level are referred to as smoothing steps; the high-frequency components of the error are efficiently removed while the long-wavelength modes remain. Following the fine-scale smoothing, the key step of multigrid is then to pass the problem to a coarser level, say H = 2h (with appropriate rules for the construction of the problem on the coarse grid); smoothing steps on the coarse level efficiently remove errors of twice the wavelength. Finally, the fine-grid function is corrected with the error interpolated from the coarse level, and further iterations on the fine level remove remaining high-frequency components induced by the coarse-grid correction. When this process is recursively followed through several levels, the stalling behavior can be completely removed and the solution is obtained in O(N) operations, where N is the number of unknowns. Typically, the problem can be solved to within the truncation errors in roughly ten total smoothing steps on the finest level. The previous discussion rests on a local-mode analysis of the errors (Brandt, 1977(Brandt, , 1984; additional mathematical arguments confirm the excellent convergence rates and linear scaling of multigrid solvers (Hackbusch, 1985). B. Full approximation scheme multigrid V-cycle For linear problems, the algebraic Eq. (22) can be rewritten as L h e h = r h ,(40) where h is the finest grid spacing, e h = u h ex − u h (grid error), and r h = f h − L h u h (residual equation). During the multigrid correction cycles, the coarse-grid iterations only need to be performed on the error term e H which is subsequently interpolated to the fine grid to provide the correction. However, this rearrangement is not possible for nonlinear problems. Brandt (1977Brandt ( , 1984 developed the full approximation scheme (FAS) approach for handling such problems. Besides providing solutions to nonlinear differential equations like the Poisson-Boltzmann equation, the FAS strategy is well suited to handle eigenvalue problems and mesh-refinement approaches. The FAS form of multigrid is thus presented here due to its generality. In the case of linear problems, the FAS is equivalent to the error-iteration version mentioned above. Consider a Poisson problem discretized on a Cartesian lattice with a FD representation on a fine grid with spacing h (Eq. 22). Now construct a sequence of coarser grids each with grid spacing twice the previous finer value. For a 4-level problem in three dimensions, the sequence of grids will consist of 17 3 , 9 3 , 5 3 , and 3 3 points including the boundaries. If h = 1, the coarser grid spacings are 2, 4, and 8. The boundary values of the potential on each level are fixed based on the physics of the problem. For example, if there are a set of discrete charges inside the lattice, direct summation of the 1/r potential or a multipole expansion can be performed. Alternatively it is easy to apply periodic boundary conditions by wrapping the potential. On the coarsest grid, only the one central point is iterated during relaxation steps there. Assume there are l levels for the general case; each level is labeled by the index k which runs from 1 (coarsest level) to l (finest level). The operator L k is defined by the FD discretization on level k with grid spacing h k . The goal is to obtain the solution u l ex of Eq. 22 on the finest level. The equations to be iterated on level k take the form: L k u k = f k + τ k .(41) where one starts from a trial u k and improves it. The initial u k on coarse levels is obtained by applying the full-weighting restriction operator I k k+1 to u k+1 : u k = I k k+1 u k+1 .(42) The restriction operator takes a local average of the finer-grid function. The average is over all 27 fine grid points (in three dimensions) including the central point which coincides with the coarse grid and the 26 neighboring points. The weights are: 1/8 for the central point, 1/16 for the 6 faces, 1/32 for the 12 edges, and 1/64 for the 8 corners. The restriction operator is a rectangular matrix of size N k+1 g (columns) by N k g (rows) where N k g is the number of grid points on level k. Of course, only the weights need be stored. The coarsegrid charge density f k is obtained similary from f k+1 . The defect correction τ k is defined as τ k = L k I k k+1 u k+1 − I k k+1 L k+1 u k+1 + I k k+1 τ k+1 .(43) The defect correction is zero on the finest level l. Therefore the third term on the rhs is zero for the grid next-coarser to the fine scale. It is easy to show that if one had the exact grid solution u l ex on the finest level, the coarse-grid equations (Eq. 41) would also be satisfied on all levels, illustrating zero correction at convergence. Another point of view is that the defect correction modifies the coarse-grid equations to 'optimally mimic' the finer scales. The defect correction provides an approximate measure of the discretization errors and can be used in the construction of adaptive solvers (Brandt, 1984): higher resolution is placed in regions where the defect correction magnitude exceeds a prescribed value. The solver begins with initial iterations on the finest level (typically two or three relaxation steps are adequate on each level). The problem is restricted to the next coarser level as outlined above, and relaxation steps are performed there. This process is repeated until the coarsest grid is reached. The solver then returns to the fine level by providing corrections to each next-finer level and applying relaxation steps there. The correction equation for grid k + 1 is u k+1 ← u k+1 + I k+1 k (u k − I k k+1 u k+1 ).(44) The additional operator I k+1 k is the interpolation operator. Most often it is acceptable to use linear interpolation, and the easiest way to apply the operator in three dimensions is to interpolate along the lines in each plane and finally to interpolate along lines between the planes. That is, the operator can be applied by a sequence of one-dimensional interpolations. For linear interpolation, the coarse-grid points which coincide with the fine grid are placed directly into the fine-grid function, and the intermediate points get a weight of 1/2 from each neighboring coarse-grid point. In the same way, high-order interpolation operators can be applied as a sequence of one-dimensional operations; see Beck (1999b) for a listing of the high-order interpolation weights. The high-order weights are used in interpolating to new fine levels in full multigrid eigenvalue solvers (Brandt et al., 1993) and in high-order local mesh-refinement multigrid methods (Beck, 1999b). The interpolation operator is a rectangular matrix of size N k g (columns) by N k+1 g rows. Only the weights need be stored, just as for the restriction operator. All of the operators defined above can be initialized once and used repeatedly throughout the algorithm. The multigrid cycle defined by the above discussion is termed a V-cycle which is shown schematically in Fig. 6. Alternative cycling methods have been employed as well, such as W-cycles. Reductions in the norm of the residual in one V-cycle are generally an order of magnitude. The same set of operations is employed in a high-order solver; the second-order Laplacian is simply replaced by the highorder version. The form of the multigrid solver is quite flexible; for example, a lower-order representation could be used on coarse levels during the correction cycles. In our own work, we have observed similar optimal convergence rates for high-order solvers as for second-order ones, so there is no degradation in efficiency with order. Applications in electrostatics and extensions for eigenvalue problems are discussed in Sections VI and VII. C. Full multigrid The grid solution can be efficiently obtained with one or at most a few V-cycles described above. The process obeys linear scaling since the solution is obtained with a fixed number of multigrid cycles and each operation on the grid scales linearly with the number of grid points. In three dimensions, the total grid overhead is N tot = S l N f ine , where S l = 8 7 1 − 1 8 l ,(45) and l is the number of levels. In the limit of many levels, N tot thus approaches 1.143N f ine . Another development in the multigrid approach, full multigrid (FMG), can even further accelerate the solution process beyond the V-cycle algorithm. The idea of FMG is to begin iterations on the coarsest level. The initial approximation there is interpolated to the nextfiner level, iterated, and the new fine-grid approximation is corrected in a V-cycle on that level. This process is repeated until the finest scale is reached. The FMG solver for a Poisson problem is illustrated in Fig. 7. The advantage of this approach is that a good initial (or preconditioned) approximation to the fine-scale function is obtained on the left side of the final V-cycle. With this strategy, the solution to Poisson problems can be obtained with a single passage through the FMG solver. (Self-consistent problems may require two or more passages through the final V-cycle to obtain convergence.) Note that a direct passage via iterations and interpolation from coarse to fine scales without the correction cycles does not guarantee multigrid convergence behavior since residual long-wavelength errors can remain from coarser levels. The multigrid corrections on each level serve to remove those errors, leading to optimal convergence (Brandt, 1984;Hackbusch, 1985;Briggs, 1987;Wesseling, 1991). Multigrid solvers have been applied to many problems in fluid dynamics, structural mechanics, electrostatics, eigenvalue problems, etc. The majority of applications have utilized FD-type representations, but significant effort has gone into developing efficient solvers for FE representations as well (Brandt, 1980;Deconinck and Hirsch, 1982;Hackbusch, 1985;Braess and Verfurth, 1990;Brenner and Scott, 1994). An additional difficulty with FE multigrid methods is a proper representation of the problem on coarse levels: the more regular the fine-scale mesh, the easier is the coarsening process. VI. ELECTROSTATICS CALCULATIONS The original formulation of the multigrid method was directed at solution of linear elliptic equations like the Poisson equation. Subsequently, methods were developed to handle nonlinear problems such as the Poisson-Boltzmann equation of ionic solution theory. In this section, applications of real-space methods to electrostatics problems are discussed. First, the high efficiency of the multigrid method is demonstrated by examination of a Poisson problem. Then, new mesh-refinement techniques which allow for treatment of widely varying length scales are examined. Poisson-Boltzmann numerical solvers are discussed, with presentation of some representative applications in biophysics. A. Poisson solvers Illustration of multigrid efficiency We investigate a model atomic-like Poisson problem which has an analytic solution: ∇ 2 φ(r) = −4π δ(r) − 1 4π e −r r .(46) The analytic solution is φ(r) = e −r /r. The source singularity is modeled as a single discrete charge at the origin, and the neutralizing background charge value at the origin is set to give a net charge of zero summed over the whole domain. Here we discretize the problem with a 12th-order Laplacian on a 65 3 lattice with fine grid spacing h = 0.25. The problem was solved with the FAS-FMG technique with a single passage through the FMG process. Linear interpolation and full-weighting restriction were employed for the grid transfers. The potential was initially set to zero over the whole domain. Three Gauss-Seidel smoothing steps were performed on each level. Several additional smoothing steps were taken for points just surrounding the singularity to accelerate the convergence there (Bai and Brandt, 1987). This requires virtually no additional effort since only few grid points are involved. The solution is obtained to within the truncation errors with a total of six relaxation sweeps on the finest level (Fig. 8). Thus the entire solution process only requires roughly ten times the effort it takes to represent the differential equation on the grid. The total energy of the charge distribution is E = −S/4π, where S is the action of Eq. (26). After the single FMG cycle, the energy is converged to within 0.00029 au of the fully converged energy of 4.31800 au (obtained with repeated V-cycles on the finest level). The final residual (using the 1-norm divided by the total number of points, that is the average absolute value of the grid residuals) is 5 × 10 −6 . After 1200 Gauss-Seidel iterations on the finest level alone, the residual is still of magnitude 9 × 10 −6 . (With an optimal SOR parameter, the number of iterations to obtain a residual of 5 × 10 −6 can be reduced to 200 iterations.) Thus there is an enormous acceleration due to the multiscale processing. Similar efficiencies are observed for an FAS-FMG eigenvalue solver (Section VII.A). Since the number of iterations is independent of the number of fine grid points, these efficiencies are quite general and can be routinely expected from a correctly functioning multigrid solver. Next we compare the operations count for generating the solution to Eq. 46 from scratch using multigrid and FFT methods on the same 65 3 lattice. 9 The FFT solver required 33×10 6 floating point operations. The multigrid solver required 27 × 10 6 , 43 × 10 6 , 75 × 10 6 , and 106 × 10 6 operations for the 2nd, 4th, 8th, and 12th order solvers, respectively. Therefore, there appears to be no clear advantage in generating the Poisson potential from scratch with multigrid as opposed to FFT. However, there are some advantages to using the realspace multigrid approach: 1) finite and periodic systems are handled with equal ease, 2) in a quantum simulation where particles move only slightly from a previous configuration, the potential can be saved from the previous configuration, thus reducing the number of iterations, and 3) one can incorporate mesh refinements to reduce the computational overhead. For example, if the same problem is solved with three nested refinement patches centered on the singularity, the number of floating point operations is reduced by nearly two orders-of-magnitude while the accuracy is sufficient since the smooth parts of the potential away from the singularity can be well represented on coarser meshes. In addition, multigrid methods can be used to solve nonlinear problems such as the Poisson-Boltzmann equation with similar efficiencies. A situation that arises in many applied electrostatics computations is that of strongly varying dielectric profiles. Analogous problems occur in steady-state diffusion problems with widely varying diffusion coeffcients such as those encountered in neutron diffusion. If the coefficients vary by orders of magnitude, multigrid efficiency can be lost (Alcouffe et al., 1981). The reason is that the correct continuity condition across the boundaries is ǫ 1 ∇φ(r 1 ) = ǫ 2 ∇φ(r 2 ) rather than continuity of the gradients themselves. Thus the gradients vary widely across the boundaries, and the standard smoothing steps do not properly reduce the errors in the function. Alcouffe et al. (1981) developed procedures based on the above continuity condition which restore the standard multigrid convergence. In biophysical applications, the dielectric constant varies from one to eighty, so such modifications prove useful for that case (Holst and Saied, 1993). Mesh-refinement techniques Many physical problems require consideration of a wide range of length scales. One example given in the Introduction is a transition metal ion buried inside a protein. A protein interacting with a charged membrane surface is another example: particular charged groups near the interaction region must be treated accurately, but distant portions of the protein and membrane do not require high resolution to obtain reliable energetics. In electronic structure, the electron density is very large near the nucleus but is diffuse further away. A significant strength of real-space methods lies in the ability to place adaptive refinements in regions where the desired functions vary rapidly while treating the distant zones with a coarser description. Two approaches exist for such refinements in the FD method (FE methods allow quite easily for grid adaptation): grid curving and local mesh refinements. While grid curving is an elegant procedure for adapting higher resolution in certain regions of space, generally the coordinate transformations are global. Therefore, the higher resolution tends to spread some distance from where the refinement is necessary (Modine et al., 1997, Figs. 1, 4, and 5), and depending on the geometry of the problem it may be difficult or impossible to design an appropriate grid transformation. Also, the transformations can be quite complex leading to additional difficulties in the solution process. Finally, the grid-curving transformations alter the underlying spectral properties of the operators which can in principle lead to degradation of the multigrid efficiency in the solution process. However, this does not appear to have been a problem in the methods of Gygi and Galli (1995) and Modine et al. (1997), although the convergence behavior of their multigrid Poisson solvers was not extensively discussed in those works. Mesh-curving strategies for electronic structure calculations are discussed in Section VII.D. An alternative procedure is to place nested uniform patches of refinement locally in space (Fig. 9). Then the overall structure of the multigrid solver is the same, except fine-level iterations are performed only over the nested patches. The same forms for the Laplacian, restriction, interpolation, and smoothing operators are maintained. This procedure is highly flexible since the nested refinements can be centered about any locations of space and can move as the problem evolves. The placement of the refinements can be adaptively controlled by examination of the defect correction τ H ; higher resolution should be placed in regions where τ H is large. If an underlying FD representation is employed, it is relatively easy to extend the method to high-order solvers since the mesh of the refinement patch is uniform. Bai and Brandt (1987) developed an FAS multigrid mesh-refinement method for treating widely varying length-scale Poisson-type problems. They first developed a λ-FMG exchangerate algorithm which minimizes the error obtained for a given amount of computational work. Since the number of visits to the coarse levels (which extend over the whole domain) is proportional to the number of patches, direct application of the multigrid algorithm does not scale strictly linearly for many levels; the λ-FMG process restores the linear scaling for a solver including the mesh refinements. Second, they showed that extra local relaxations around structural singularities restore asymptotic convergence rates which can otherwise degrade. Third, they developed a conservative-differencing technique for handling source singularities. To motivate the need for conservative differencing in the FAS-FMG mesh-refinement solver, consider Eq. (41) and a two-level problem with one nested patch. The defect correction on the coarse level H is initially defined only over the interior region of the patch. However, if one examines the sum of τ H over the refinement, most of the terms in the interior cancel, but nonzero values remain near the boundaries. The remaining terms closely resemble flux operators at the boundary. The net effect is thus the introduction of additional sources in the Poisson equation, which pollutes the solution severely over the whole domain. By balancing the local fluxes with additional defect correction terms on the patch boundary, the correct source strength is restored. Bai and Brandt (1987) solved this problem for second-order equations and tested the method on a source-singularity problem in two dimensions. Recently the method has been extended to high-order FD approximations by Beck (1999bBeck ( , 2000. The boundary defect correction terms were determined by examination of the noncancelling terms for the high-order approximations. Without the conservative scheme, significant errors are apparent over the whole domain. With the inclusion of the boundary corrections, the sum of τ H over the patch is zero to machine precision and the correct high-order behavior is obtained over the whole domain. The method was tested on a source-singularity problem in three dimensions for multiple nested patches. Typical multigrid efficiencies were observed. Additional corrections will be necessary at the boundaries for continuous charge distributions which cover the refinement boundaries, but these are independent of the order of the Laplacian. These techniques are currently being included in high-order FD electronic structure calculations. They will significantly reduce the grid overhead in comparison to uniform-grid calculations while still maintaining the linear-scaling properties of the multigrid method. It is not possible to handle truly local refinements with the FFT approach. In related work, Goedecker and Ivanov (1998a) developed a linear-scaling multiresolution wavelet method for the Poisson equation which allows for treatment of widely varying length scales. They utilized second-generation interpolating wavelets since the mapping from grid values to expansion coefficients is easy for these functions, and they have a fast wavelet transform. They solved the Poisson equation for the challenging case of the all-electron uranium dimer. Their solver employed 22 hierarchical levels, and the potential was obtained to six significant digits. B. Poisson-Boltzmann solvers As discussed in Section II.B, the Poisson-Boltzmann equation arises from the assumption of no ion correlations. That is, it is a mean-field treatment. Onsager (1933) showed that there exists an inherent asymmetry at the Poisson-Boltzmann level. Nevertheless, calculations performed at this level of theory can yield accurate energetics for monovalent ions at moderate concentrations (Honig and Nicholls, 1995;Tomac and Gräslund, 1998;Patra and Yethiraj, 1999). Linearization of the Poisson-Boltzmann equation restores the symmetry, but for many cases of experimental interest the linearization assumption is too severe. Solution of the Poisson-Boltzmann equation produces the electrostatic potential throughout space, which in turn generates the equilibrium mobile-ion charge densities and the total free energy of the ion gas (below). By computing the total energies for several macroion configurations, the potential of mean force due to electrostatic effects can be approximated (Rice, 1959). In this section, we focus on real-space numerical methods for solution of the nonlinear Poisson-Boltzmann equation [Eq. (12)]. Numerical solution of nonlinear partial differential equations is problematic. For the Poisson-Boltzmann case, the nonlinearities can be severe near fixed charges since the ratio of the potential to kT can be large. Also, strong dielectric discontinuities at the boundary of a large molecular ion and the solution create technical difficulties. However, it is known that there is a single stable minimum of the action functional whose derivative yields the Poisson-Boltzmann equation (Coalson and Duncan, 1992; Ben-Tal and . Therefore, properly constructed iterative processes can be expected to locate that minimum. Early numerical work centered on FD representations. Nicholls and Honig (1991) developed an efficient single-level SOR method which included special techniques for memory allocation and for locating the optimal relaxation parameter. For the test cases considered, between 76 and 184 iterations were required for convergence. They also observed divergence for some highly nonlinear cases. Davis and McCammon (1989) and Luty et al. (1992) used instead a conjugate-gradients relaxation method. Between 90 and 118 iterations were required to obtain convergence. They observed a factor of at least two improvement in efficiency in comparison with SOR relaxation in their test calculations. After the development of multigrid methods for solving linear Poisson-type problems, efforts focused on nonlinear problems. The FAS algorithm presented above is well suited for solving nonlinear problems (Brandt, 1984). Two modifications are needed: the driving term f h on a given level now includes the nonlinear terms, and additional terms must be included in the defect correction to ensure zero correction at convergence. The defect correction for the Poisson-Boltzmann problem is of the form (a single monovalent positive ion component with uniform dielectric is considered here): τ H = L H I H h u h − I H h L h u h + 4π ǫ n H + e −βu H −v H −n h + I H h e −βu h −v h ,(47) where the additional terms reflect the differing representations of the nonlinear terms on the two levels. The concentration on a given level is given bȳ n h + = N + h 3 e −βu h −v h .(48) The sum is over the lattice and N + is the number of positive ions in the computation domain (Coalson and Duncan, 1992). This procedure for obtaining the bulk ion concentrations ensures charge conservation at all steps of iteration. Simple smoothing steps can be taken to relax on a given level, or Newton iterations (Press et al., 1992) may also be conducted on each level. Variable ω parameters may be required in the relaxation steps due to the differing degrees of nonlinearities on the respective levels. Related multigrid techniques for nonlinear problems are presented in Stüben and Trottenberg (1982) and Hackbusch (1985). Holst and Saied (1995) developed a highly efficient method which combines linear multilevel techniques with inexact-Newton iterations. They compared the convergence behavior of the inexact-Newton multigrid method with SOR and conjugate-gradients minimization on a single level. Their multilevel technique converged robustly and more efficiently than the relaxation methods on all problems investigated including challenging source problems with dielectric discontinuities. Conjugate gradients and SOR exhibited similar convergence rates when compared with each other. They also examined a standard nonlinear multigrid method similar to that outlined above. For some cases, the nonlinear multigrid technique gave good convergence, but under certain conditions it diverged. The authors thus recommended caution in applying the FAS multigrid method directly to the Poisson-Boltzmann equation. Coalson and Beck (1998) tested the FAS approach on model problems in the lattice field theory including source singularities and found convergence for each case. Oberei and Allewell (1993) have also developed a convergent multigrid solver for the Poisson-Boltzmann equation. It is not entirely clear at the present time whether differences in observed convergence are due to the model problems investigated or to differences in the algorithms. One issue that has not been addressed to date concerns charge conservation on the various levels. The standard form of the Poisson-Boltzmann equation (Honig and Nicholls, 1995) assumes fixed and equal concentrations of the mobile ions at infinity where the potential is zero. The ion charge density can then be expressed as the product of a constant term involving the Debye length of the ion gas and a sinh term involving the potential. Typically the boundary potential is fixed with the linearized Debye-Hückel value. This representation conserves charge if the system size is allowed to go to infinity due to the infinite extent of the bath. However, charge is not conserved for finite system sizes, and in a multilevel procedure differing charge states will be encountered on the various levels. In the lattice field theory of Coalson and Duncan (1992), on the other hand, the charge is naturally conserved (maintaining overall charge neutrality) by updating the parametersn + andn − during each step of iteration (see Eqs. 12 and 48). Lack of conservation of charge on the grid levels in the standard approach may impact the convergence behavior of a multilevel solver; this issue deserves further attention. In a recent study, Tomac and Gräslund (1998) extended the Poisson-Boltzmann level of theory to include ion correlations in an approximate way. They solved for the Kirkwood (1934) hierarchy of equations on a FD grid assuming a closure proposed by Loeb (1951). Multigrid techniques were used to solve the initial Poisson-Boltzmann equation and to implement the inclusion of ion correlations. Coarser grids were used to estimate the fluctuation term, and the impact of large grid spacing on the accuracy of the correlation term was examined. Excellent agreement with previous theoretical results and Monte Carlo simulation was obtained for divalent ion distributions around a central sphere. Test calculations were also performed on ion distributions around an ATP molecule. This work allows for the more accurate treatment of systems containing multivalent ions. The computational expense of obtaining the fluctuation contribution is extensive, however. What is clear from the multigrid studies to date is that multilevel methods can yield solutions to the Poisson-Boltzmann equation (and its modifications to include ion correlations) with efficiencies resembling those for linear problems and with linear-scaling behavior. Hence, they show a great deal of promise for large-scale colloid and biophysical applications. Under some circumstances, special measures may be necessary to obtain correct multigrid convergence efficiencies. To my knowledge, all FD Poisson-Boltzmann calculations so far have employed second-order Laplacians; going to higher orders improves accuracy for little additional cost, so higher-order solvers should be considered. However, high-order techniques near dielectric discontinuities introduce some additional complexity. In addition to FD-related methods for solving the Poisson-Boltzmann equation, FE solutions have appeared. The FE discretization leads to a more accurate physical representation of complex molecular surfaces at the expense of additional computational overhead. You and Harvey (1993) developed a three-dimensional FE method for solving the linearized Poisson-Boltzmann equation. More accurate results were obtained with the FE approach compared with FD solutions in model problems. Potential distributions were computed surrounding tRNA molecules and the enzyme superoxide dismutase. This was the first application of the FE method to large-scale biological macromolecular electrostatics. Cortis and Friesner (1997) formulated a method for constructing tetrahedral FE meshes around macromolecules. The authors discussed the relative merits of FD and FE representations including applications of multilevel methods in their solution. They used their discretization procedure to solve the linearized Poisson-Boltzmann equation. Bowen and Sharif (1997) presented a FE numerical method for solution of the nonlinear Poisson-Boltzmann equation in cylindrical coordinates. Adaptive mesh refinements were employed to gain accuracy near curved surfaces. They considered applications to membrane separation processes by examining the case of a charged spherical particle near a cylindrical pore. Alternative formulations of electrostatic problems include boundary element methods which reduce Poisson problems to calculations involving the molecular surface (Yoon and Lenhoff, 1990Lenhoff, , 1992Pratt et al., 1997). They lead to dense matrix representations of the problem; if nonlinear salt effects are to be included, volume integrals, in addition to surface integrals, must be incorporated. C. Computations of free energies Several proposals have appeared concerning computation of free energies of the ion gas once the solution of the Poisson-Boltzmann equation is obtained. The free energies are crucial for determining electrostatic interaction energies of charged macromolecules at the mean-field level. The energies can be obtained either by charging methods or volume/surface integrations (Verwey and Overbeek, 1948;Marcus, 1955;Rice, 1959;Reiner and Radke, 1990). The most commonly used volume integration approach stems from the variational formulation of Sharp and Honig (1990b). They postulated a form for the free energy which, when extremized, produces the Poisson-Boltzmann equation. Fogolari and critiqued this variational form, showing that the extremum in the free energy is a maximum, not a minimum, with respect to variations of the potential. They presented another form which is minimized. The lattice field theoretic free energy is derived from a rigorous representation of the grand partition function of the ion gas (Coalson and Duncan, 1992). In this section, we will derive the variational form from the lattice field theory formulation to illustrate differences between the two; the variational form is obtained from the infinite system size limit of the lattice field theory. We assume here the case of uniform dielectric and monovalent ions; the extensions for variable dielectric and higher valences follow the same arguments. The mean-field lattice field theory Helmholtz free energy is βF = −S LF T + N + ln(n + h 3 ) + N − ln(n − h 3 ),(49) where S LF T is an action term (defined below) and N + and N − are the total numbers of positive and negative mobile ions in the calculation domain. In order to handle periodic as well as finite domains, we assume that the total number of mobile and fixed charges is such that overall charge neutrality is maintained. The free energy Eq. (49) is invariant to a uniform shift of the potential, which is the correct physical result. The concentrationn + is given by Eq. (48), whilen − is obtained by the analogous formula for negative charges. Consider the action which, when minimized, results in the Poisson-Boltzmann equation: S = − 1 2 φ∇ 2 φd 3 x − 4π ǫ ρ f φ −n + β e −βφ−v −n − β e βφ−v d 3 x.(50) The action of Eq. (49) is related to S by S LF T = βǫ 4π S.(51) Then the total Helmholtz free energy on the lattice is βF = βǫh 3 8π φ h L h φ h + βh 3 ρ h f φ h + N + ln(n + h 3 /e) + N − ln(n − h 3 /e),(52) where the grid potential is φ h , and the sums are over the lattice points. Let us examine a process in which two macroions are moved relative to each other (Fig. 10). The macroions are assumed to reside in a large calculation domain which contains counterions plus perhaps salt ions, so the potential is screened at large distances. We assume that the potential decays effectively to zero some finite distance from the ions and is zero all the way to the boundaries. The mobile ions behave as an ideal gas where the potential is zero. The numbers of fixed and mobile ions is maintained constant throughout the process. Now, consider the free energy change from the 'activity' term for the positive mobile ions upon moving from configuration 1 to 2: β∆F a+ = N + ln e −βφ h 1 −v h 1 e −βφ h 2 −v h 2 .(53) Call the number of free sites in the domain where the potential is effectively zero N f 1 and N f 2 . The sums over regions where the potential is nonzero are labelled Σ 1 and Σ 2 . The free energy change ∆F a+ is then β∆F a+ = N + ln N f 1 + Σ 1 N f 2 + Σ 2 .(54) Factor out the N f terms: β∆F a+ = N + ln N f 1 1 + Σ 1 N f 1 N f 2 1 + Σ 2 N f 2 .(55) The term involving the ratio of the free sites can be represented as N + ln N f 1 N f 2 = N + ln N ′ tot 1 − Σ 1g N ′ tot N ′ tot 1 − Σ 2g N ′ tot ,(56) where N ′ tot is the total number of grid points outside any excluded volume regions and Σ 1g and Σ 2g count the numbers of grid points outside of excluded volume zones where the potential is nonzero. For very large system sizes, the above expressions can be approximated as β∆F a+ ≈ N + Σ 1 N f 1 − Σ 2 N f 2 + N + N ′ tot (−Σ 1g + Σ 2g ) .(57) Analogous terms are obtained for the negative ion case. As the system size approaches infinity, we can make the further approximations N + ≈ N − and N f 1 ≈ N f 2 ≈ N ′ tot . The resulting free energy change for both ionic species is then β∆F a = N + N ′ tot e −βφ h 1 + e βφ h 1 − 2 e −v h 1 − e −βφ h 2 + e βφ h 2 − 2 e −v h 2 .(58) If we call the grid concentration c g = N + /N ′ tot = N − /N ′ tot , then the free energy change can be written as β∆F a = −2c g cosh(βφ h 2 ) − 1 e −v h 2 − cosh(βφ h 1 ) − 1 e −v h 1 .(59) The grid 'activity coefficient' γ is c g /h 3 which can be assumed to be γ = √n +n− . In the continuum limit the free energy change due to the logarithmic terms in the total free energy is thus β∆F a = −2γ [cosh(βφ 2 ) − 1] e −v 2 d 3 x − [cosh(βφ 1 ) − 1] e −v 1 d 3 x .(60) The overall free energy change can then be written as the difference of two terms, one for each configuration: βF = β ρ f φd 3 x − 2γ [cosh(βφ) − 1] e −v d 3 x + βǫ 8π φ∇ 2 φd 3 x(61) So long as the potential and/or its derivative go to zero on the boundaries, Eq. (61) can be rewritten as βF = β ρ f φd 3 x − 2γ [cosh(βφ) − 1] e −v d 3 x − β 8π ǫ \|∇φ\| 2 d 3 x(62) which is identical to Eq. (13) in Sharp and Honig (1990b). Thus the variational free energy of Sharp and Honig is derived as the infinite system limit of the lattice field theory expression, where the potential is assumed to go to zero at the distant boundaries. Since the variational form is not invariant to a uniform shift of the potential, some arbitrariness is introduced. In addition, charge conservation is not maintained as discussed above. The issue of charge conservation is particularly relevant if one considers periodic boundary domains. Therefore, it is recommended to use the lattice field theoretic form for computations of free energies for cases where these considerations are deemed important. D. Biophysical applications One reason for a resurgence of interest in continuum models of solvation for large macromolecules is that, for many systems of interest, the total number of particles is simply too large to accurately model at the atomic level. For example, consider a protein interacting with a DNA strand: the atomistic treatment including solvent and salt effects would involve several tens of thousands of atoms, and the motions occur over time scales longer than nanoseconds. So long as the energetics are proven to be reasonable in testable model calculations, some confidence can be placed in the Poisson-Boltzmann calculations on larger systems. The number of applications of Poisson-Boltzmann-level theory to biological macromolecules is now very large. Previous reviews summarize progress in this area (Sharp and Honig, 1990a;Honig and Nicholls, 1995). A few representative studies from the main categories of application are presented here to give a flavor of the types of problems which are accessible. The first type of application concerns the computed average electrostatic potential and the resulting charge distributions. Haggerty and Lenhoff (1991) performed FD calculations to generate the electrostatic potential on the surfaces of proteins. They found a clear correlation between retention data in ion-exchange chromatography and the average protein surface potential. Ion-exchange chromatography is one of the important techniques for separating mixtures of proteins. Montoro and Abascal (1998) compared Monte Carlo simulations and FD Poisson-Boltzmann calculations on distributions of monovalent ions around a model of B-DNA. They found good agreement between the simulations and Poisson-Boltzmann calculations for low to moderate ion concentrations, but for concentrations above 1 M, the agreement deteriorates. Pettit and Valdeavella (1999) compared electrostatic potentials obtained from molecular dynamics simulations and Poisson-Boltzmann calculations for a tetra-peptide. They observed qualitative differences for the electrostatic potentials around the peptide. However, they argued that the free energies obtained by integration over the entire domain include cancellation of errors and yield more reliable results compared with the potential itself. Patra and Yethiraj (1999) developed a DFT method for the ion atmosphere around charged cylinders (a model for DNA or tobacco mosaic virus). Their theory includes contributions from finite ion size and ion correlations beyond the mean-field level. Their DFT approach gave good agreement with simulations for both monovalent and divalent ion atmospheres. The Poisson-Boltzmann level theory does well for low axial charge densities on the cylinder. Interesting charge inversion effects were seen for divalent salts which are entirely absent from the Poisson-Boltzmann calculations. Recently Baker et al. (1999) developed a highly adaptive multilevel FE method for solving the Poisson-Boltzmann equation. By placing adaptive meshes in the regions of the dielectric discontinuities, large reductions in overall computation cost were observed. Computations were performed to obtain the electrostatic potential around large protein and DNA systems. The second utility of Poisson-Boltzmann calculations lies in the computation of free energies and resulting interaction energies for variable macromolecule conformations. Yoon and Lenhoff (1992) used a boundary-elements method to compute interaction energies for a protein and a negatively charged surface at the linearized Poisson-Boltzmann level. They found the most favorable orientation with the protein active site facing the surface. Zacharias et al. (1992) investigated the interaction of a protein with DNA utilizing the FD Poisson-Boltzmann method. They studied the distribution of ions in the region between the two species and the energetics for protein binding. The interaction energy depends strongly on the charge distributions on the DNA and protein. The computed number of ions released upon complexation agreed well with experiment. Misra et al. (1994) performed FD Poisson-Boltzmann calculations to study the influence of added salt on protein-DNA interactions. Long-range salt effects play a significant role in relative stabilities of competing structures of protein-DNA complexes. Ben-Tal et al. (1997) examined electrostatic effects in the binding of proteins to biological membranes. The binding constant for the protein-membrane complex was successfully compared with experimental data. Chen and Honig (1997) extended their FD Poisson-Boltzmann method to mixed salts including both monovalent and divalent species. They found that, for pure salt cases, the electrostatic contribution to binding varies linearly with the logarithm of the ion concentration; for divalent salts, nonlinear effects were observed due to competitive binding of the two ionic species. A third type of problem addressed with Poisson-Boltzmann level computations is the determination of pH-dependent properties of proteins. Since the net charge of the protein is crucial in understanding its properties, a predictive method is desired for computing electrostatic effects (due to other charged groups) on the pK a 's of ionizable groups. Antosiewicz et al. (1994Antosiewicz et al. ( , 1996 presented extensive calculations on a large data set for several proteins. Somewhat surprisingly, they obtained the best agreement with experiment assuming an interior dielectric constant of 20 for the protein. Possible explanations of this effect were discussed, including approximate accounting for specific ion binding and conformational relaxation of the protein. They also found improvements if NMR structural sets were used as opposed to single crystal X-ray structures for the proteins. Vila et al. (1998) recently performed boundary-element multigrid calculations to determine pK a shifts; they obtained excellent agreement with experiment for polypentapeptides. Fourth, Poisson-Boltzmann methods have been incorporated into electronic structure calculations to study solvation effects. As an example, Fisher et al. (1996) performed DFT electronic structure calculations on a model for the manganese superoxide dismutase enzyme active site. The region treated explicitly included 37 or 38 atoms (115 valence electrons). The surrounding solvent was modeled as a dielectric continuum (water). The electronic structure was computed self-consistently by updating the reaction-field potential due to the solvent following calculations with fixed potential. Typically, the continuum solvation procedure converged within seven iterations. The authors computed redox potentials and pK a 's for the complex. Differences from measured redox potentials were observed, and the authors stressed the importance of explicitly including electrostatic effects from the rest of the protein in the calculations. As a final biophysical real-space application, a lattice relaxation algorithm has been developed by Kurnikova et al. (1999) to examine ion transport through membrane-bound proteins. The coupled Poisson and steady-state diffusion equations (Poisson-Nernst-Planck or PNP equations) were solved self-consistently on a FD real-space grid for motion through a membrane protein, the Gramicidin A dimer. The charges embedded in the channel interior had a large impact on computed diffusion rates. The computed current-voltage behavior agreed well with experimental findings. The accuracy of the continuum mean-field treatment is encouraging for the further study of ion transport through a wide range of membrane proteins. A recent study (Corry et al., 2000) has critiqued the mean-field approach for narrow ion channels, so some modifications in the PNP theory may be required for those cases. VII. SOLUTION OF SELF-CONSISTENT EIGENVALUE PROBLEMS Eigenvalue problems arise in a wide range of applications. Solution of the Schrödinger equation with fixed or self-consistent potential is of course a dominant one. However, eigenvalue problems occur in several other areas. Included are computation of modes and frequencies for molecular vibrations (Jensen, 1999) and optical modes of waveguides . Self-consistent eigenvalue problems also arise in polymer theory (Tsonchev et al., 1999). This section reviews recent research on real-space methods for fixed and selfconsistent potential eigenvalue problems. The main focus is on novel methods for solving the Kohn-Sham equations in electronic structure. Additional discussion concerns applications in semiconductor and polymer physics. A. Fixed-potential eigenvalue problems in real-space Algorithms Let us consider the problem of minimizing the total energy for a single quantum particle subject to the constraint that the wavefunction must be normalized. With the inclusion of a Lagrange multiplier term for the constraint, the energy functional reads E[ψ(r)] = − 1 2 ψ * ∇ 2 ψd 3 x + ψ * V ψd 3 x − λ ψ * ψd 3 x,(63) where λ is the Lagrange multiplier. If multiple states are desired, then the single Lagrange multiplier becomes a matrix of multipliers designed to enforce orthonormality of all the eigenfunctions. The 'force' analogous to Eq. (27) is then − δE δψ * = 1 2 ∇ 2 ψ − V ψ + λψ.(64) When the force term is set to zero indicating location of the minimum, the eigenvalue equation for the ground state results. Discretizing this equation on a one-dimensional grid leads to the second-order FD representation of the Schrödinger equation: − 1 2h 2       −2 1 0 . . . 1 −2 1 . . . 0 1 −2 . . . . . . . . . . . . . . .               ψ(x 1 ) · · · ψ(x N )         +       V (x 1 ) 0 0 . . . 0 V (x 2 ) 0 . . . 0 0 V (x 3 ) . . . . . . . . . . . . . . .               ψ(x 1 ) · · · ψ(x N )         = λ         ψ(x 1 ) · · · ψ(x N )        (65) Solution of this matrix equation with standard diagonalization routines (excluding the Lanczos and multigrid methods) results in an N 3 g scaling of the solution time, where N g is the number of grid points. Since the matrix is sparse, iterative techniques are expected to lead to increased efficiencies, just as for Poisson problems. We can note that the solution of Eq. (65) is a nonlinear problem since we seek both the eigenvalues and eigenvectors. In this section, we consider necessary extensions of the FAS-FMG method for the eigenvalue problem and discuss applications of FD and FE real-space methods for fixed-potential cases. Clear discussion of alternative Lanczos and related algorithms (such as the conjugate-gradient, GMRES, and Jacobi-Davidson algorithms) for handling sparse matrix diagonalization is given in Golub and van Loan (1996) and Booten and van der Vorst (1996). The derivation of the FD matrix eigenvalue equation above parallels that for the Poisson problem. The additional complexities introduced are: 1) the necessity of solving for multiple eigenfunctions, 2) computation of eigenvalues, and 3) enforcement of orthonormality related constraints. Brandt et al. (BMR, 1983) extended the FAS-FMG algorithm to eigenvalue problems. Hackbusch (1985) discussed related eigenvalue methods. The algorithm of BMR allows for fully nonlinear solution of the eigenvalue problem; due to the nonlinear treatment, the eigenvalues and constraint equations only need to be updated on the coarsest level where the computational expense is small. One exception to the previous statement in the original BMR algorithm is a Ritz projection (below) on the finest level at the end of each V-cycle, preceded by a Gram-Schmidt orthogonalization. Costiner and Ta'asan (1995a) have since extended the method to process the Ritz projection on coarse levels as well. The same basic FAS-FMG procedure is followed in the BMR eigenvalue algorithm as discussed above for Poisson problems. The Laplacian operator L h in Eq. (22) is replaced by the real-space Hamiltonian minus the eigenvalue λ i . There is no source term f h , and there are q equations, where q is the number of eigenfunctions. Since the orthogonalization constraints are global operations involving integrals over the whole domain, these processes can be performed on the coarse levels. The relaxation sweeps (two or three) on finer levels smooth the high-frequency errors and do not destroy the existing orthonormality of the functions; of course, if many unconstrained iterations were performed on fine levels, all wavefunctions would begin to collapse to the ground state. Linear interpolation and fullweighting restriction are sufficient, but use of cubic interpolation results in more accurate eigenfunctions upon entry to a new finer level. A direct Gram-Schmidt orthogonalization is not applicable on coarse levels; if the exact grid solution is restricted to the coarse levels, the resulting eigenfunctions are no longer orthonormal. Therefore, to satisfy the zero correction at convergence condition, a coarse grid matrix equation for the constraints is u H i , I H h u h j = I H h u h i , I H h u h j .(66) Solution requires inversion of a q × q matrix. The inversion can be effected by direct matrix methods if q is small or iterative procedures as performed by BMR in their solver. The grid overhead for the operation is very small since it is performed on the coarsest level; for example, if three levels were employed in the eigenvalue solver, the coarse grid operations would require 1/64 the effort compared to the fine scale in three dimensions. An additional consideration in the eigenvalue problem is that the coarse grid must contain enough points to 'properly resolve' the eigenfunctions; BMR give a criterion of N cg = 4q for the required number of points. The eigenvalues can also be updated on the coarse levels by inclusion of the defect correction: λ i = < H H u H i − τ H i , u H i > < u H i , u H i > .(67) The grid Hamiltonian on the coarse level is H H . The same set of eigenvalues applies on all levels with this formulation. Relaxation steps are performed on each level with Gauss-Seidel iterations. A final addition to the FAS-FMG technique in the eigenvalue algorithm of BMR is a Ritz projection performed at the conclusion of each V-cycle in the FMG solver. The purpose of this step is to improve the occupied subspace by making all residuals orthogonal to that subspace. The eigenfunctions are first orthogonalized with a Gram-Schmidt step and the q × q Hamiltonian matrix in the space of the occupied orbitals is diagonalized. The orbitals are then corrected. This step improves the convergence rate. The Ritz projection can be written as ω T H h ωz i − λ i z i = 0,(68) where ω is the q × N g (N g is the total number of grid points) matrix of the eigenfunctions, H h is the grid Hamiltonian, and the z i are the solved-for coefficients used to improve the occupied subspace. We have closely followed this algorithm in our own work with two changes: 1) we update the eigenfunctions simultaneously (as opposed to sequentially in the original algorithm) and 2) high-order approximations are used in the FD Hamiltonian. In the form presented above, the algorithm exhibits q 2 N g scaling due to the Ritz projection on the fine scale. The scaling of the relaxation steps is qN g so long as the orbitals span the entire grid. If a localized representation of the orbitals is possible (Fattebert and Bernholc, 2000), then linear scaling of each step in the algorithm results. Further discussion of the scaling of each operation is presented in Wang and Beck (2000). Costiner and Ta'asan (1995a) have generalized the BMR algorithm in several ways. They transferred the Ritz projection step to coarse grids and added a backrotation to prevent rotations of the solutions in subspaces of equal or close eigenvalues. They also developed an adaptive clustering algorithm for handling groups of eigenfunctions with near eigenvalues. The scaling of their algorithm is qN g when the eigenfunctions span the entire grid. Several numerical experiments in two and three dimensions demonstrated the high efficiency of their method, and the method was extended to handle self-consistency (Costiner and Ta'asan, 1995b). Applications To demonstrate the efficiency of the BMR FAS-FMG eigenvalue solver, consider the three-dimensional hydrogen atom. While this may seem a very simple case, it presents numerical difficulties for a real-space method due to the presence of the Coulomb singularity in the potential. In addition, the s-orbitals exhibit cusps at the singularity and the l > 0 angular momentum states are degenerate. Beck (1999a) presented numerical results for the hydrogen atom which exhibit the excellent convergence characteristics of the nonlinear FAS-FMG eigensolver. The potential was generated numerically with a 12th-order Poisson solver as described above. The grid was taken as a 65 3 Cartesian lattice. The boundary potentials were set to the analytical 1/r values. The fine grid uniform spacing was h = 0.5 au, and a 12th-order FD discretization was employed. Five eigenfunction/eigenvalue pairs were computed. The fully converged eigenvalues (obtained by repeated V-cycles on the finest scale) were -0.50050 for the 1s state, -0.12504 for the 2s state, and -0.12496 for the three 2p states (which are degenerate out to 10 decimal places when fully converged), so the results are accurate to better than kT . The eigenvalues were converged to five decimal places following one passage through the FAS-FMG solver with three relaxation sweeps on each level on each side of the V-cycles. Thus, only six fine-scale applications of the Hamiltonian to the wavefunctions were required to obtain the solution. The major computational cost for this system occurred during the relaxation steps on the fine scale. The total solution time was roughly 90 seconds on a 350 MHz Pentium II machine. These results show that similar convergence behavior can be expected for eigenvalue solvers as for Poisson solvers so long as the nonlinear FAS-FMG methodology is followed. Mesh refinements will yield comparable accuracies with much less numerical overhead. The required high-order methods are now in place (Beck, 1999b) and are being incorporated into the eigenvalue solver. We now consider related efforts at efficient solution of real-space fixed-potential eigenvalue problems. Grinstein et al. (1983) developed a second-order FD multigrid method to solve for a single eigenfunction. They employed an FAS-FMG approach and used a Gauss elimination method to exactly solve the equations on the coarsest level. Since they solved for single eigenfunctions, constraints were not necessary. The eigenvalue was fixed and not computed, so the problem was effectively linear. Seitsonen et al. (1995) solved fixed-potential eigenvalue problems using a high-order FD representation and a conjugate-gradient method for obtaining the eigenfunctions and eigenvalues. They tested their method on the P 2 dimer and obtained rapid convergence of the approximation with decreasing grid size. The representation of the wavefunctions was better than in corresponding plane-wave calculations. They also computed eigenfunctions for positron states centered at a Cd vacancy in CdTe. Extensive effort has also been applied to development of FE methodology for fixedpotential problems. Hackel et al. (1993) proposed a two-dimensional FE method in which Coulomb singularities were handled with condensed special elements around the nuclei. Test calculations were performed on the linear H 2+ 3 molecule, and highly accurate results (to 10 −7 au) were obtained. Ackerman and Roitzsch (1993) proposed an adaptive multilevel FE approach which utilized high-order shape functions. Inverse iteration was used to solve the large-dimension eigenvalue problem for the two-dimensional harmonic oscillator and the linear H 2+ 3 molecule; accuracies comparable or even superior to the previous study were reported. Subsequently, they extended their method to three dimensions (Ackerman et al., 1994); in this work, conjugate-gradient techniques were employed to solve the eigenproblem. Results were presented for the three-dimensional harmonic oscillator and H 2+ 3 in the equilateral triangle geometry. Sugawara (1998) presented a hierarchical FE method in which the mesh points and polynomial orders are generated adaptively to gain high accuracy. The method was tested on the one-dimensional harmonic oscillator. Batcho (1998) proposed a spectral element method which combines ideas from FE and collocation approaches. The Coulomb singularity was treated with a Duffy (1982) transformation. Pask et al. (1999) have recently developed a FE method for periodic solid-state computations. The method uses a flexible C 0 piecewise-cubic basis and incorporates general Bloch boundary conditions, thus allowing arbitrary sampling of the Brillouin zone. Band structure results were presented which illustrate the rapid convergence of the method with decreasing grid size. The authors emphasized the structured, banded, and variational properties of the FE basis. subsequently applied the method to large-scale ab initio positron calculations for systems of up to 863 atoms. B. Finite-difference methods for self-consistent problems In this section, we begin our examination of real-space methods for solving self-consistent eigenvalue problems with a discussion of FD methods. The focus here is mainly on the basic FD formulation and its relationship to other numerical methods in terms of accuracy. Later sections will discuss specialized techniques for solution in the real-space representation including multigrid, mesh refinements, FE formulations, and related LCAO methods. One direction has been to develop atom-centered numerical grids in order to obtain converged results independent of basis-set approximations. Becke (1989) presented a fully numerical FD method for performing molecular orbital calculations. In this method, the physical domain was partitioned into a collection of single-center components, with radial grids centered at each nucleus. A polyatomic numerical integration scheme was developed. This work was the first which extended the previous two-dimensional methods for diatomics (see, for example, Laaksonen et al., 1985). This numerical method has allowed for accurate computations to test various levels of DFT approximations on small molecules without concerns of basis-set linear-dependence effects. The main focus of this approach has been on numerically converged results and not on scaling and efficiency for large-scale problems. In contrast to the atom-centered grids discussed above, recent work has focused on development of high-order pseudopotential methods on uniform Cartesian grids. Chelikowsky, and Chelikowsky, proposed a FD pseudopotential method in which high-order forms were utilized for the Laplacian (Appendix). They employed the real-space pseudopotentials of Martins (1991a, 1991b). The simplicity of the FD method in relation to plane-wave approaches was highlighted. The Hartree potential was obtained either by a direct summation on the grid or by iterative subspace techniques. They also employed iterative subspace methods for the eigenvalue problem. A main emphasis was on the accuracy of the FD approximation in relation to plane-wave methods. A multipole expansion was used to generate the fixed potential on the boundaries. Three parameters determine the accuracy in their FD calculations: the grid spacing, the order of the Laplacian, and the overall domain size. Results were presented concerning the convergence of the eigenvalues with order and decreasing grid spacing. The 12th-order form of the Laplacian was found to be sufficient for well-converged results. Accurate eigenvalues (to 0.01 au) were obtained for atomic states. Extensive calculations on diatomic molecules were also presented. The high-order FD approximation gave good results for binding energies, bond lengths, and vibrational frequencies. Comparisons were made to plane-wave calculations with two supercell sizes, one with 12 au and one with 24 au on a side. The FD calculation box was 12 au on a side. The planewave energies were not converged with the smaller box size, but the plane-wave calculations approached the FD results when a supercell of 24 au was used, suggesting that quite large supercells must be employed (even for nonpolar molecules) for converged orbital energies in localized systems (see Table I). The authors obtained a dipole value of 0.10 D for the CO molecule (with the C − O + orientation). The experimental value is 0.1227 D with the same orientation, while Hartree-Fock theory yields the wrong sign for the dipole. However, the fully converged LDA dipole is 0.24 D (Laaksonen et al., 1985). 10 The error is most likely due to the restricted overall domain size in their calculation (see Kim, Städele, and Martin, 1999;Wang and Beck, 2000). To conclude, the authors emphasized that the FD method is ideal for localized and charged systems, is easy to implement, and is well suited for parallel computations. Related work which has analyzed the impact of FD order on accuracy for Poisson problems can be found in the multigrid papers of Merrick et al. (1995), Gupta et al. (1997), andZhang (1998). Also, see Section IV.A.3. Subsequently, Jing et al. (1994) extended the high-order FD method to compute forces and perform molecular dynamics simulations of Si clusters. For most of their work, they performed Langevin molecular dynamics simulations with a random force component to simulate a heat bath. The clusters were annealed from high temperature to room temperature and the cluster structures were examined; the FD method gave excellent agreement with other numerical methods. When the heat bath was turned off, the trajectory exhibited total energy fluctuations two orders of magnitude smaller than the potential energy fluctuations. The fluctuations agreed in magnitude with those in a plane-wave simulation to within a few percent. Vasiliev, et al. (1997) recently utilized the higher-order FD methods in computations of polarizabilities of semiconductor clusters with finite-field methods for the response. The results of the high-order FD method from Chelikowsky's group clearly show that the FD representation can yield results of comparable or superior accuracy compared with plane-wave calculations on similar-sized meshes. In related work, Hoshi et al. (1995) presented a supercell FD method in which they used an exact form of the FD Laplacian which spans the whole domain along each direction. Therefore, 3N 1/3 g points are necessary to apply the Laplacian to the wavefunction at each grid point; the method is equivalent to a very high-order representation. Fast Fourier transform routines were used to solve for the Hartree potential. A preconditioning technique similar to that of Payne et al. (1992) was employed to improve convergence. Pseudopotential results were presented for the He atom and the H 2 molecule. Their method required 45 steps of iteration to converge within 10 −5 au with the preconditioning. Subsequently, Hoshi and Fujiwara (1997) incorporated the unconstrained OM linear scaling scheme into their method. Windowing functions were employed to confine the orbitals to localized regions of space. Test calculations were performed on the diamond crystal with four localized orbitals per atom. They obtained a ground state energy of 5.602 au/atom which compared reasonably well with their previous result of 5.617 au/atom. As mentioned above, FD methods have found application in areas outside of traditional electronic structure theory. Abou-Elnour and Schuenemann (1993) developed a selfconsistent FD method for computing wave functions, carrier distributions, and sub-band energies in semiconductor heterostructures. Only one-dimensional problems were examined. They compared the FD method to a basis set calculation and found the FD approach to be faster. In polymer physics, self-consistent FD methods have also appeared. Tsonchev et al. (1999) derived a formal field theory for the statistical mechanics of charged polymers in electrolyte solution. The theoretical development parallels the earlier work of Coalson and Duncan (1992) for the ion gas. A functional-integral representation was derived for the partition function of the coupled polymer/ion system. The mean-field theory solution leads to coupled Poisson-Boltzmann (for the ion gas moving in the field of the other ions and the polymer charges) and eigenvalue (for the polymer chain distribution) equations. These equations were solved numerically with FD methods for polymers confined within spherical cavities. The three-dimensional eigenvalue problem was solved with the Lanczos technique. Electrostatics plays a key role in the chain structure for high chain charge densities and low salt concentrations in the cavities. C. Multigrid methods The finite-difference results of the previous section show that accurate results can be obtained on uniform grid domains with high-order approximations. Multiscale methods allow for accelerated solution of the grid-based equations. The first application of multigrid methods to self-consistent eigenvalue problems in electronic structure was by White et al. (1989). Many of the important issues related to real-space approaches were laid out in this early paper. The authors developed an orthogonal FE basis and solved the Poisson equation numerically with multigrid. Due to the orthogonal basis, a standard FD solver only required simple revisions to apply to the FE case. They also presented preliminary results of multigrid methods applied to the eigenvalue portion of the problem, but only single orbital cases were considered. They found that the multigrid solver was faster than a conjugate-gradient method (without preconditioning). Computations were performed on the hydgrogen atom, the H + 2 molecular ion, the He atom, and the H 2 molecule. More discussion of their method will be given below in Section VII.E on FE methods. Another early method by Davstad (1992) proposed a two-dimensional multigrid solver for diatomic molecules in the Hartree-Fock approximation. He combined multigrid and Krylov subspace methods in the solver. High-order FD discretization was employed. The Orthomin procedure (a Krylov subspace method) was used for iterations on all coarse levels, with Gauss-Seidel iteration as preconditioner. Computations were performed on the diatomics BH, HF, CO, CuH, and the Zn atom. Good convergence rates were observed (presented in terms of orbital residuals), and excellent agreement with previous numerical work was obtained for total energies and orbital eigenvalues. Since this early work, several groups have utilized multigrid solvers for many-orbital problems in three dimensions. Bernholc's group has developed a multigrid pseudopotential method for large systems. Preliminary calculations (Bernholc et al., 1991) were reported for the H atom and the H 2 molecule. A grid-refinement strategy for adding resolution around the nuclei was also presented. Subsequently, Briggs et al. (1995) included realspace pseudopotential techniques into their multigrid method and presented calculations for large condensed-phase systems on uniform grids. They introduced the FD Mehrstellen discretization which leads to a 4th-order representation. Variations of the total energy of atoms when moved in relation to the grid points were investigated. With increasing grid resolution, the errors decrease, so this criterion can be used to choose the necessary finegrid spacing for accurate dynamical simulations. The Hartree potential was also generated with a multigrid solver. In their method, the computation time to perform one multigrid step is comparable to a single propagation step in the Car-Parrinello method. Results were presented for a 64-atom diamond supercell, the C 60 molecule, and a 32-atom GaN cell. For large systems, the multigrid method was found to converge to the ground state an order-ofmagnitude faster than their Car-Parrinello code. For the GaN case, 240 multigrid iterations were required to reach the ground state from random initial wavefunctions, while for an 8-atom diamond cell roughly 20 iterations were necessary to converge the total energy to a tolerance of 10 −8 au. Their multigrid algorithm was further developed in Briggs et al. (1996), where extensive details of the solver were presented. Calculations were performed on a Si supercell, bulk Al, and an AlN supercell with comparisons made to Car-Parrinello calculations to test the accuracy of the approximations. Excellent agreement with the Car-Parrinello results was obtained. Their multigrid implementation for the eigenvalue problem utilized a doublediscretization scheme; on the fine level the Mehrstellen discretization was employed, while on the coarse grids a seven-point central-difference formula was used. Full-weighting restriction and trilinear interpolation were used for the grid transfers, and Jacobi iterations were performed for the smoothing steps. The eigenvalue problem was linearized by computing the eigenvalues only on the fine grid and performing coarse-grid corrections on each eigenvector. The constraints were imposed on the fine scale at the end of the double-discretization correction cycle. Subspace diagonalization was performed to accelerate convergence. Tests of the convergence were conducted on a 64-atom Si cell and a 64-atom diamond cell with a substitutional N impurity. Substantial accelerations were obtained with multigrid in comparison to steepest-descent iterations; roughly 20 self-consistency iterations were required in the multigrid solver to obtain 10 −3 Ry convergence in the total energy. While these convergence rates are a significant improvement over steepest-descent iterations, they are non-optimal due to the linearization in their method (see below). The overhead for implementing multigrid in addition to steepest-descent iterations was only 10% of the total computing time. The authors discussed extensions of the multigrid method for molecular dynamics (tested on a 64-atom Si supercell which exhibited good energy conservation). Applications to other large-scale systems appear in Bernholc et al. (1997). In an algorithm very similar to that described above, Ancilotto et al. (1999) developed a solver which included FMG processing to provide a good initial guess on the finest level. The Mehrstellen discretization was employed on all levels. With the FMG addition, the initial state of the orbitals is irrelevant since it takes very little numerical effort to obtain the initial fine-grid approximation during the preliminary coarse-grid cycles. They performed red-black Gauss-Seidel smoothing steps on each level and used full-weighting restriction and trilinear interpolation for grid transfers. Eigenvalues were computed only on the finest level, and Ritz projections were also performed to accelerate convergence. They also reported 20 self-consistency iterations to obtain convergence on several diatomic molecules (C 2 , O 2 , CO, and Si 2 ), and good agreement with plane-wave results was observed for equilibrium bond lengths and vibrational frequencies (both to within 1%). They presented numerical results for the C 2 dimer (pseudopotential calculations) which illustrated the convergence of their algorithm in comparison to a Car-Parrinello (damped molecular dynamics) plane-wave code. Superior convergence was found even in relation to state-of-the-art Car-Parrinello algorithms (Tassone et al., 1994), which exhibit performance similar to conjugate-gradient algorithms. The method was tested by using simulated annealing cycles to locate the most stable ground state of the Al 6 cluster. Then calculations were performed to find stable minima for charged Li clusters with sizes N = 9 − 11. The numerical results indicated that the fragmentation behavior observed in experiments likely has a strong non-statistical component. In addition to the two-dimensional solver of Davstad discussed above, all-electron multigrid methods in three dimensions have been developed. discussed a multigrid method for solving the Kohn-Sham equations in which the entire problem was discretized on a three-dimensional Cartesian lattice, including all electron orbitals and the nuclear charge densities. An eighth-order FD form for the Laplacian was used in this work. The nuclear charge densities were discretized as a single cube on the lattice, and the Poisson equation was solved with the standard multigrid technique. Since the total charge density included both the electron and nuclear densities, all the electrostatic interactions were handled in a single linear-scaling step, including the nucleus-nucleus term. A self energy must be subtracted from the total energy, but this is a one-time computation for each order of the Laplacian since the self energy scales as Z 2 /h. Computations were performed on hydrogenic atoms and the H + 2 molecule. A simple nested procedure was utilized for the Kohn-Sham solver in which an initial approximation was generated on a coarse level, smoothing steps were performed, and the problem was interpolated to the next finer grid followed by relaxations. This process significantly accelerated the convergence, but some critical slowing down remained due to incomplete decimation of long-wavelength modes on the coarse levels. These results show that a one-way multigrid procedure without coarse-grid corrections does not guarantee proper multigrid convergence. Various relaxation procedures were compared; conjugate gradients gave the best convergence per step but required more numerical effort than simple Gauss-Seidel iterations, so Gauss-Seidel is equally efficient. This result illustrates the important point that simple smoothing iterations are enough to decimate the errors with wavelength on the order of a few grid spacings on a given level, and special techniques are not necessary. Results were presented for the all-electron Ne atom which exhibited the significant speedup due to a multiscale treatment, but the residual stalling on the fine levels was used to motivate inclusion of the BMR FAS-FMG method for the eigenvalue problem. Beck et al. (1997) presented the first application of the BMR FAS-FMG algorithm (Section VII.A.1) to self-consistent electronic structure problems. In this initial effort, the BMR algorithm was followed, except the orbitals were updated simultaneously during the correction cycle as opposed to sequentially in the original method. Also, Gram-Schmidt orthogonalization steps were implemented on each level, so the constraint procedure outlined in Section VII.A.1 was not followed exactly. Convergence calculations were performed on the Ne atom on a 33 3 grid; the FAS-FMG approach led to faster convergence than the oneway multigrid calculations of . Beck (1997) extended these calculations to the CO molecule (all electrons and three dimensions) and developed an FAS solver for the Poisson-Boltzmann equation. The convergence of the CO molecular calculation was limited by the handling of the constraints discussed above. A relatively accurate dipole moment of 0.266 D (C − O + ) was obtained on a 33 3 mesh. Subsequently, Beck (1999a) and Wang and Beck (2000) developed a fully convergent FAS-FMG Kohn-Sham self-consistent all-electron solver. In this work, the eigenfunction constraint equations [Eq. (66)] were implemented on the coarsest grid only, and the eigenvalues were also updated on the coarsest level via Eq. (67). Ritz projection was performed on the finest level at the conclusion of each V-cycle. The effective potential was updated once upon entry to the next finest level and at the end of each V-cycle. Both sequential and simultaneous updates of the orbitals were examined to test the efficiency of each approach. The sequential method leads to slightly more rapid convergence to the ground state, but it results in a qN g scaling in a self-consistent method since the effective potential is updated following coarse-grid corrections on each orbital. The discretized problem was solved on a 65 3 grid domain with a 12th-order form for the Laplacian. Atomic ionization potential computations were performed to illustrate the ease of applicability to charged, finite systems. Numerical results were presented for the all-electron CO molecule. The CO eigenvalues were accurate to within 0.015 au for all states above the core, and the highest occupied π(2p) and σ(2p) states were accurate to within 0.006 au. The computed dipole was 0.25 D, in good agreement with previous fully numerical results on diatomics (Laaksonen et al., 1985). Convergence data was presented for the Be atom and the CO molecule (Fig. 11). Implementation of the nonlinear FAS-FMG strategy leads to order-of-magnitude efficiency improvement in relation to linearized versions of the multigrid algorithm (Ancilotto et al., 1999). The converged ground state was obtained in only two or three self-consistency cycles, with three orbital relaxation steps on each side of the V-cycle. Therefore, the entire selfconsistent solution process required a total of only 12-18 smoothing steps on the finest grid and a few updates of the effective potential. One self-consistency cycle for the 14 electron CO molecule on a 65 3 grid required roughly a minute of CPU time on a 350 MHz Pentium-II machine. The update of the Hartree potential involves the same effort as the update of a single eigenfunction; it is therefore a small contributor to the overall numerical effort. Due to the handling of the constraints and eigenvalues on the coarsest level, each self-consistency update requires less computation than the algorithms of Briggs et al. (1996) and Ancilotto et al. (1999). Since these FAS-FMG computations included all electrons and the nuclear singularities in three dimensions, the rapid convergence in relation to the pseudopotential computations of Ancilotto et al. (1999) is noteworthy (the total energy is nearly three orders-of-magnitude larger than in the pseudopotential calculation). These results are the first to exhibit the full power of the nonlinear BMR technique for solution of self-consistent electronic structure problems. The slightly slower convergence for the CO molecule (compared with the Be atom) is due to the relatively poor treatment of the core electrons on a uniform grid; with a finer grid, the convergence is even more rapid. Wang and Stuchebrukhov (1999) have applied the FAS-FMG algorithm described above to computation of tunneling currents in electron transfer; they found that real-space calculations give a significantly more accurate representation of current densities than Gaussian basis-set calculations. Some simple arguments can be made concerning the total number of operations for the multigrid solution vs. the conjugate-gradient plane-wave method. The present discussion assumes the orbitals span the entire physical domain. Payne et al. (1992) showed that the conjugate-gradient method requires 6qN F F T + 2q 2 N P W operations to update all the orbitals. The second term is for the orthogonalization constraints. The variable q is the number of orbitals and N F F T is 16N P W ln N P W where N P W is the number of plane waves. Thus N F F T is the number of operation counts for Fourier transformation on the real-space grid. The multigrid method requires qN mgop + 2q 2 N g + N mgop = (q + 1)N mgop + 2q 2 N g operations, where N mgop is the number of operations to update one orbital with the multigrid method and N g is the number of fine grid points. The q 2 dependent term is for the orthogonalization constraints (Gram-Schmidt followed by Ritz projection) which are performed once at the end of each correction cycle, and the second N mgop term is for the Poisson solver. Since a multigrid update of one eigenfunction (with say an 8th-order approximation) requires roughly four times the number of operations count of a single FFT (see Section VI.A.1), the net cost for the multigrid update (neglecting the relative constraint costs which are much smaller with multigrid, see below) is slightly less than that for the conjugate-gradient method. Figure 11 shows that the number of self-consistency iterations is also very low with the multigrid solver. The study of Ancilotto et al. (1999) compared damped molecular dynamics to their linearized multigrid method (on diatomic molecules). They also compared multigrid (favorably) with the optimized dynamics method of Tassone et al. (1994) which in turn exhibits convergence rates very similar to conjugate-gradients. Since the nonlinear FAS-FMG solver outperforms the linearized multigrid method by an order-of-magnitude, this suggests the multigrid solver is more efficient than the conjugate-gradient approach. The best available plane-wave techniques (see, for example, Kresse and Furthmüller, 1996) can reduce the number of self-consistency iterations to 5-10, so the multigrid solver is at least as efficient as the most efficient plane-wave techniques for uniform-domain problems where the orbitals span the whole domain. The major benefits of the multigrid approach in addition to the above discussion are: 1) all the constraint and subspace orthogonalization operations can be removed to coarse levels where the cost is minimal; for example if they are performed two levels removed from the fine level, the cost is 1/64 that on the fine level (Costiner and Ta'asan, 1995a), 2) it is quite easy to impose localization constraints in the real-space multigrid approach (Fattebert and Bernholc, 2000), and 3) mesh refinements can be incorporated while maintaining the same convergence rates (see Beck, 1999b, for the Poisson version). The mesh-refinement methods are in place and are currently being incorporated into Kohn-Sham solvers; they should lead to a further near order-of-magnitude reduction in computational cost. Finally, Costiner and Ta'asan (1995b) have shown that by updating the effective potential simultaneously with the eigenfunctions on coarse levels self-consistent solutions can be obtained in a single passage through the final V-cycle of the FMG process. Therefore, multiscale real-space approaches offer a promising alternative to plane-wave techniques. Recently, proposed a one-way multigrid method similar to that of . Initial approximations were obtained on coarse levels, and the solution was interpolated to the next finer level without multigrid correction cycles. High-order interpolation was used to proceed to the next finer grid. Conjugate-gradient techniques were employed to relax the orbitals on each level. The method led to a factor of five reduction in computation time compared to a single-grid calculation. Computations were performed on a 20-electron quantum dot and charged H clusters. Kim, Lee, and Martin (1999) developed an object-oriented code for implementation of the one-way multigrid algorithm. Several other groups have utilized multigrid solvers as components of real-space electronic structure algorithms; these will be discussed in the following sections on mesh-refinement techniques and FE methods. D. Finite-difference mesh-refinement techniques The previous sections have discussed FD methods for electronic structure; the calculations were performed primarily on uniform grids. With the incorporation of real-space pseudopotentials, results with accuracies comparable to plane-wave methods (with similar grid cutoffs) can be obtained with high-order FD techniques. The calculations of Beck (1999a) and Wang and Beck (2000) are instructive in that surprisingly accurate results are possible even in all-electron calculations on uniform grids; in addition, their work shows that multigrid efficiencies are obtainable for the challenging case of very harsh effective potentials which include the nuclear singularities. However, it is clear that increasing uniform grid resolution until acceptable accuracy is reached is a wasteful process since small grid spacings are only required in the neighborhood of the atomic cores. This section reviews recent work on development of FD mesh-refinement techniques which address this issue for the eigenvalue problem. As discussed in Section VI.A.2 which covered closely related methods for the Poisson equation, there are presently two strategies for mesh refinements: grid curving and local refinements which are included within a coarser mesh (Fig. 9). Gygi and Galli (1995) extended a previous plane-wave method of Gygi (1993) to adaptive-coordinate FD calculations. A curvilinear coordinate system was developed which focused resolution near the nuclei. The necessary extensions of the standard FD method to handle the curvilinear Laplacian were presented. FD forms of order 2 and 4 were utilized, and norm-conserving pseudopotentials were employed. The Poisson equation was solved with a multigrid method. The calculations were implemented on a Cray-T3D massively parallel machine. Test calculations were conducted on diatomics and the CO 2 molecule. The calculations of the total energy of the CO 2 molecule vs. internuclear distance exhibited a spurious double minimum with a uniform grid treatment (cutoff energy of 227 Ry). This double minimum is due to the numerical errors from a grid which is too coarse. When the adaptive coordinate transformation was included (with effective cutoffs of 360 Ry for carbon and 900 Ry for oxygen), a single minimum was observed near the correct experimental bond length. Modine et al. (1997) presented another adaptive-coordinate FD method which they termed ACRES (adaptive-coordinate real-space electronic structure). They first discussed the goals of their real-space method: 1) sparsity, 2) parallelizability, and 3) adaptability. The real-space approach satisfies these criteria, while the plane-wave method does not. Extensive details were given concerning the construction of their grid-curved meshes and the resulting Laplacian. One issue to note is that the FD Laplacian in curvilinear coordinates contains off-diagonal terms and the number of terms scales as 3((2n) 2 + 4n + 1), where n is the order. Therefore, high-order derivative forms add complexity to the adaptive-coordinate approach. Computations were performed on atoms and molecules at both the all-electron and pseudopotential levels. The authors discussed the limitations of the Lanczos method for the eigenvalue problem; the width of the real-space spectrum is dominated by the largest eigenvalue which in turn is determined by the minimum grid spacing, so the method slows with increasing resolution. Instead, they used a modified inverse iteration eigensolver. The equations were solved with a conjugate-gradient algorithm. Conjugate-gradient techniques were also employed for the Poisson equation, with multigrid used for preconditioning. Highly accurate all-electron results were obtained for the O atom and the H 2 and O 2 molecules; computed bond lengths for O 2 agreed with both the previous calculations of Chelikowsky, Trouller, Wu, and Saad (1994) and experiment to within 0.02Å. To conclude, they discussed the high efficiency of ACRES in relation to uniform grid computations. Two works have appeared which utilize nested mesh refinements as opposed to gridcurving techniques for increased resolution. Fattebert (1999) developed an algorithm to treat a single grid refinement placed inside a coarser-level grid domain. A FD Mehrstellen discretization was employed over the whole domain, with nonuniform difference stencils at the boundaries between the fine and coarse levels. The discretization is 4th order over the uniform regions and 2nd order at the boundaries. The impact of this nonuniformity of the representation order on the solution order was not examined. The eigenvalue problem was solved with a block Galerkin inverse iteration in which multigrid methods were used to solve the linear systems. Smoothing iterations were enacted with the GMRES algorithm (Golub and van Loan, 1996). Pseudopotential calculations were performed on the furan molecule which requires treatment of 13 eigenfunctions. Excellent convergence rates were observed, especially on the finer composite meshes; the coarse grid convergence was not as rapid. The author also presented results for the total energy of the CO molecule which are similar to those of Gygi and Galli (1995) described above. Incorporation of the grid refinements led to smooth variations of the energy, while the coarser-grid computation resulted in irregular variations. Ono and Hirose (1999) proposed another double-grid method in which the inner products of the wavefunctions and pseudopotentials are treated on a fine grid. The doublegrid treatment leads to smooth forces without the necessity of Pulay (1969) corrections (which are required in the adaptive-coordinate method). E. Finite-element solutions Just as for the FD formulation, the application of FE methods to self-consistent eigenvalue problems has followed two different tracks. In the first, the FE basis has been utilized to obtain highly accurate results for atoms and small molecules. The FE method can achieve very high accuracies since it does not suffer from the linear-dependence problems of LCAO approximations, and the mesh can be arbitrarily refined. The second type of application concerns development of efficient methods for large-scale electronic structure problems. We begin with methods designed to obtain high accuracies. Levin and Shertzer (1985) performed FE calculations on the He atom ground state. The problem reduces to three-dimensional for the s state. A basis of cubic Hermite polynomials was employed. They computed both the ground-state energy and the moments r n of the wavefunction. An energy within 0.0005 au of the numerically exact result was obtained. Also, the orbital moments were substantially more accurate than those computed in basis-set calculations. This occurs since the LCAO basis functions are global; if the functions are optimized to give a good wavefunction near the nucleus (where the largest contribution to the total energy occurs), they cannot be adjusted simultaneously to give a good representation far from the origin. The FE basis overcomes this difficulty. Heinemann et al. (1987) and Heinemann et al. (1988) developed a two-dimensional FE method and applied it to computations on the H 2 , N 2 , BH, and CO molecules. Using a 5th-order basis, accuracies to better than 10 −8 au for the total energies were observed, which exceeds by two orders the accuracy of the FD calculations by Laaksonen et al. (1985). Yu et al. (1994) implemented an order 5 or 6 Lobatto-Gauss FE basis and employed a block Lanczos algorithm to solve the eigenvalue problem. A Duffy (1982) transformation allowed for handling of the Coulomb singularity. Calculations were performed on diatomic and triatomic hydrogen molecules and ions; these three-dimensional results were not as accurate as in the two-dimensional study of Heinemann et al. (1987), differing by .00051 au in the total energy of H 2 . More recently, Kopylow et al. (1998) incorporated an FMG solver into their twodimensional method for diatomics. Conjugate-gradient smoothing steps were employed on each level. Excellent convergence rates were obtained for the solver which was tested on the Be 2 molecule; only 5 self-consistency iterations were required to obtain 10 −6 au convergence in the energy. Düsterhöft et al. (1998) combined the LCAO and FE methods in a defect correction approach which allowed for a more rapid attainment of the ground state due to a better representation around the nuclei. Next, we consider methods directed toward larger systems. The FE method of White et al. (1989) was discussed above related to multigrid methods for self-consistent problems. They utilized a high-order FE basis and constructed orthogonal functions from the nonorthogonal basis. The cost of this construction is the requirement of more functions to obtain the same level of completeness. The three-dimensional basis functions were products of the one-dimensional functions on a Cartesian grid. The Coulomb singularity was handled with an integral transform representation of 1/r. The Hamiltonian is sparse in their basis since only near-neighbor overlaps need to be considered. To solve the Poisson equation, multigrid techniques were employed with a double-discretization procedure similar to that of Briggs et al. (1996); on coarser levels, the problem was represented with a FD form rather than with a FE basis. As discussed above, multigrid solution of the eigenvalue problem was faster than conjugate gradients. To conclude, they emphasized the importance of developing new grid methods for refinements around the nuclei, where the largest errors occur. Gillan and coworkers have developed a general method for linear-scaling electronic structure (of the OBDMM form discussed in Section III.B). Closely related is the work of Hierse and Stechel (1994), which differs in the choice of basis and the number of basis functions. In their initial work (Hernández and Gillan, 1995), the OBDMM strategy was developed, and the calculations were performed directly on a real space grid with second-order FD techniques. The total energy was minimized with conjugate-gradients iterations. Typically, 50 iterations were required to obtain energy convergence to within 10 −4 eV/atom. Hernández et al. (1997) developed a blip-function basis instead of the previous FD representation. This method is general in the sense that any local function (that is, completely restricted to a finite volume) can be used for the basis; however, we examine this method in relation to FE bases since it is so closely related. The actual basis employed in their work is a set of B-splines (see, for example, Strang and Fix, 1973, p. 60). The basis was implemented on a Cartesian mesh as products of three one-dimensional functions. The kinetic and overlap terms were treated analytically, but the matrix elements of the potential were evaluated numerically on a grid different from the blip grid. The blip-function basis agreed very well with plane-wave results in calculations on Si solids; a discrepancy of only 0.1 eV/atom was observed between the two different approaches. Goringe et al. (1997) discussed implementation of the algorithm on very large systems (up to 6000 atoms) on parallel machines. The essential features of the OBDMM method were reviewed. Fast Fourier transform methods were used to solve for the electrostatic potential on a grid. Complete discussion was given of the steps in parallelizing every portion of the code using real-space domain decomposition. The numerical results on a Cray-T3D parallel machine exhibited linear scaling of CPU time with the number of atoms using between 32 and 512 processors. As discussed in Bowler et al. (1999) and reviewed in Goedecker (1999), three forms of ill-conditioning can lead to degradation of convergence to the ground state in the OBDMM method: length-scale, superposition, and redundancy ill-conditioning. The first is an inherent feature of any real-space solver (Section IV.A.2). The second form results from the localization constraints imposed in the method, and is similar to problems in OM methods. The third is related to the fact that their method includes more basis functions than occupied orbitals; the localization constraints lead to small but nonzero occupation numbers of the higher-lying states, and they have little influence on the total energy. Bowler and Gillan (1998) addressed the length-scale ill-conditioning problem. They developed a precondition-ing technique related to the plane-wave method of Payne et al. (1992). The blip-function preconditioning matrix only needs to be calculated once. Test calculations were performed for a Si crystal with significant accelerations of the convergence due to the preconditioning. However, the convergence efficiency of their method decreased both with decreasing grid spacing and with increasing localization radius. It is an interesting question whether multigrid methods might lead to higher efficiencies in the context of the OBDMM method. An alternative FE method for large-scale electronic structure has been developed by Tsuchida andTsukada (1995, 1998). In their original method, the authors utilized firstand second-order shape functions and derived the appropriate variational expression for the total energy at the LDA level. The Hartree potential was generated by conjugate-gradient iteration. They also implemented the OM linear-scaling method. Nonuniform meshes were employed to focus resolution around the nuclei in the H 2 molecule. Test calculations were also performed on an 8-atom Si solid with 16 3 uniform elements of the second order. Good agreement with plane-wave calculations and experiment were obtained for the lattice constant, cohesive energy, and bulk modulus. Due to the integral formulation of the total energy, the Coulomb singularity in the potential becomes finite. In Tsuchida and Tsukada (1998), the method was substantially extended for large-scale condensed-phase systems. Third-degree polynomials were used as FE basis functions. They implemented the gridcurving method of Gygi and Galli (1995) to adapt for higher resolution near the nuclei. Pulay (1969) corrections were computed to obtain accurate forces on the ions. A multigrid procedure was followed to solve the FE Poisson equation. The multigrid aspect was used as preconditioner to final conjugate-gradient iterations on the finest scale. Again, OM techniques were used to obtain linear scaling. They also utilized a one-way multigrid-type approach for the eigenvalue problem, where a good initial approximation was obtained on the finest level from previous iterations on a coarser level. With this approach, 20 to 30 self-consistency iterations were required for convergence on the fine level. Calculations were limited to the Γ point; for the treatment of general Bloch boundary conditions, see Pask et al. (1999). A parallel code was written using real-space domain decomposition. Many applications were considered in this work, including computations on diamond lattices, cubic BN, the C 60 molecule, molecular dynamics simulations, and parallel implementations. Pseudopotential calculations on systems with up to 512 carbon atoms were presented. The final statement from this paper captures well the rapid development of real-space methods in the last decade: "About ten years ago, the FE method was described to be in its infancy for electronic structure calculations (White et al., 1989). We have shown in this paper that it can be routinely used for large systems today." As a final application of FE methods to self-consistent eigenvalue problems, Lepaul et al. (1996) considered semiconductor quantum nanostructures. They solved the twodimensional Schrödinger equation self-consistently with updates of the Poisson equation (variable dielectric case) to obtain carrier densities, conduction bands, and the potential distribution at finite temperatures. The image potential and exchange-correlation energies were neglected. The carrier confinement was due to heterojunction discontinuities and the electrostatic potential. By varying the potential bias, a bidimensional quantum gas was observed. The real-space approach allowed for treatment of realistic device geometries. In the spin-unrestricted linear-response energy representation (Casida, 1996), the excitation energies are obtained from an eigenvalue equation: ΩF I = ω 2 I F I ,(70) where the excitation energy differences are ω I and the matrix Ω is Ω ijσ,klτ = δ σ,τ δ i,k δ j,l (ǫ lτ − ǫ kτ ) 2 + 2 (f iσ − f jσ )(ǫ jσ − ǫ iσ )K ijσ,klτ (f kτ − f lτ )(ǫ lτ − ǫ kτ ).(71) The (f iσ − f jσ ) terms are the occupation differences between the i and j σ-spin states, (ǫ jσ − ǫ iσ ) are the corresponding Kohn-Sham energy differences, and the response matrix is K ijσ,klτ = ∂v SCF ijσ ∂P klτ ,(72) where P klτ is the linear response of the Kohn-Sham density in the basis of the unperturbed orbitals. The resulting full expression for K ijσ,klτ in terms of the Kohn-Sham orbitals can be found in Casida (1996) and Vasiliev et al. (1999); it involves the unperturbed orbitals only and (in the adiabatic approximation) the second derivatives of the static exchangecorrelation functional E xc with respect to the spin densities. Therefore, at this level of theory K ijσ,klτ is time-and frequency-independent. However, it includes screening effects which alter the spectrum toward the correct physical result. The eigenvalues of Eq. (70) give the transition energies, and the eigenvectors yield the oscillator strengths from which the dynamic polarizability can be computed. The oscillator strengths in this formulation satisfy the same sum rule as for the real-time version presented above. The method scales as N 3 [where N is the number of electrons, see Casida (1996)]; however, linear-scaling methods can be applied just as for the ground-state Kohn-Sham theory. Vasiliev et al. (1999) utilized the high-order FD pseudopotential method of Chelikowsky, in solving Eq. (70) for excitation energies. They considered the exact form for Ω [Eq. (71)] and two approximate forms, one of which was employed by Petersilka et al. (1996) in their work. They first examined excitations in closed-shell atoms and found that the exact expression resulted in the highest accuracies. Errors for low-lying excitations attributed to the LDA exchange-correlation potential in Petersilka et al. (1996) were corrected by using the exact expression. Computed energies were in error by only a few tenths of an eV in comparison with experiment for singlet excitations. They also found that transition energies for singlet and triplet excitations computed with TDLDA theory are in better agreement with experiment than optimized effective potential (OEP, see Talman and Shadwick, 1976) or ordinary self-consistent field methods due to the approximate inclusion of correlation effects. The authors proceeded to apply the TDLDA method to computations of absorption spectra for Na clusters. Only computations using the exact formulation resulted in spectra that agreed with experiment (to within 0.2 eV). This indicates the importance of collective excitations since the approximate forms neglect these contributions. Finally, they computed the static polarizabilities of Na and Si clusters with the exact and approximate formulations and found that only the exact representation yielded good agreement with finite-field calculations. In related work, Ögüt et al. (1997) computed ab initio optical gaps for very large Si nanocrystals (up to Si 525 H 276 ) with high-order FD methods. Kim, Städele, and Martin (1999) have recently utilized the high-order FD pseudopotential method in calculations on small molecules at the Krieger-Li-Iagrate (KLI, 1992) level for the effective potential. This potential is an approximation to the OEP theory which is computationally tractable and has the correct −1/r tail in the effective potential. The calculations yielded better approximations to excited-state energies in relation to the Kohn-Sham LDA values, but the full TDDFT energy-representation method was not employed for corrections to the Kohn-Sham KLI levels. IX. SUMMARY Real-space methods for solving electrostatics and eigenvalue problems involve either local Taylor expansions of the desired functions about a point or localized basis-set representations. Higher accuracy is obtained by increasing the order of the approximation and/or the resolution of the mesh. However, standard iterative processes become less efficient on finer meshes due to the difficulty of reducing the long-wavelength modes of the errors. Multigrid methods provide a remedy for this slowing-down phenomenon inherent in real-space numerical methods. Many of the early limitations of real-space methods (such as very large required meshes) have been overcome in recent years with the development of efficient highorder finite-difference and finite-element methods. This review has surveyed a wide range of physical applications of real-space numerical techniques including biophysical electrostatics, ground-state electronic structure, and computations of electronic response and excitation energies. Recent real-space computations have tackled problems with hundreds to thousands of atoms at a realistic level of representation. The discussion presented in this review leads to several conclusions: • The underlying representation is relatively simple in real space. The finite-difference method is particulary straightforward, while the finite-element and wavelet methods involve some increased complexity. As an example, a self-consistent Kohn-Sham LDA multigrid program using the high-order finite-difference method requires less than 5000 lines of computer code. • With the incorporation of high-order methods, accuracies comparable to plane-wave calculations are obtained on similar-sized meshes. • The Laplacian and Hamiltonian operators require information only from close lattice points; that is, the operators are near-local in space. Therefore, the matrices are sparse, highly banded, and very structured. Each application of the operators scales linearly with system size, and the method is readily implemented on parallel computers by partitioning the problem in space. The locality also allows for incorporation into linear-scaling electronic structure methods. • Multigrid methods provide the optimal solvers for problems represented in real space. For Poisson problems, the multigrid method scales linearly with system size and requires only about 10 iterations on the finest level to obtain convergence. Eigenvalue solvers scale as q 2 N g (where q is the number of eigenfunctions and N g the number of fine grid points) if the eigenfunctions span the whole space. 12 If a localized orbital representation is possible, the multigrid eigenvalue methods scale linearly with size due to the locality of each operation in the algorithm. • Nonlinear multigrid methods require fewer operations per self-consistency update than plane-wave methods on uniform grids with orbitals that span the physical domain. In addition, the multigrid method is at least as efficient as the best plane-wave methods in terms of the number of self-consistency steps to reach the ground state. The multigrid solution requires at most a few self-consistency iterations. The solution involves 10-20 total applications of the Hamiltonian to the wavefunctions on the finest level and a few updates of the effective potential (one for each self-consistency cycle); each update of the Hartree potential requires the same effort as the update of one orbital. • Real-space methods allow for higher resolution in space without loss of efficiency. That is, they are readily adaptable and thus can handle problems with a wide range of length scales. • The eigenfunction constraint and subspace orthogonalization operations can be performed on coarse levels where the cost is very low. Also, the effective potential can be updated on coarse levels leading to the possibility of complete solution in a single selfconsistency cycle. These developments, along with the mesh-refinement techniques, will lead to reductions in computational cost of an order-of-magnitude compared with existing algorithms. • The flexibility of the representation has been utilized both in very high accuracy computations and in applications to large systems. The real-space methods do not suffer from linear dependence problems which occur in LCAO methods. Typically, the numerical convergence is controlled by a few parameters such as grid spacing, domain size, and order of the representation. • Real-space algorithms very similar to those for electrostatics and ground-state electronic structure can be employed to solve time-dependent problems. In the view of the author, the most promising areas for future work on real-space methods concern the development of highly adaptive and efficient numerical techniques which focus resolution in key regions of space as the iterative process moves towards the ground-state solution or evolves in real time. There will always exist a tradeoff between the simplicity of the representation (where finite differences are best) and the flexibility and accuracy of local basis functions (where finite element methods are superior). The related local LCAO methods allow for significantly smaller overall basis-set size in relation to real-space formulations, but the Laplacian and Hamiltonian operators are not as well structured and banded. The intersection between the simple structured approaches on the one hand and the more physical local bases on the other should provide for a fruitful growth of new ideas in computational materials science. Multiscale methods for solving the problems will figure prominently since they allow for flexibility in the representation while maintaining high efficiency. A brief survey of physical and chemical problems which have already been addressed serves to illustrate the wide range of length scales accessible with real-space techniques: electrostatics of proteins interacting with nucleic acids, charged polymers in confined geometries, largescale electronic structure of materials, and computation of spectroscopic quantities for large molecules in the gas phase. One can imagine a time in the not-too-distant future when it is possible to simulate the motion of a solute molecule in a liquid with the inclusion of all the electrons and model the solvent influence on the electronic absorption spectra. Real-space methods possess many of the features that would be required to address such a challenging problem. ACKNOWLEDGMENTS I would like to thank Matt Challacombe, Rob Coalson, and John Pask for helpful discussions, and Victoria Manea for a critical reading of the manuscript. I gratefully acknowledge the significant contributions from the members of my research group: Karthik Iyer, Michael Merrick, and Jian Wang. I especially thank Achi Brandt for his advice on multigrid methods. The research was partially supported by the National Science Foundation. APPENDIX A As an example of the ease of generating a high-order form for the Laplacian operator, the following Mathematica script for the 10th-order case is included: ym5},ym4},ym3},ym2},ym1},{x0,y0},{x0+1,yp1},{x0+2,yp2},{x0+3,yp3},{x0+4,yp4},{x0+5 OUTPUT: Out[1]=(-73766 y0 + 42000ym1 -6000ym2 + 1000ym3 -125ym4 + 8ym5 + 42000yp1 -6000yp2 + 1000yp3 -125yp4 + 8yp5)/25200 g[x_]:= Evaluate[ The weights obtained for the FD Laplacian up through 12th order are presented in Table II. For the three-dimensional case, the pth order approximation requires 3p + 1 terms. Hamming (1962) also discusses procedures for generating other high-order formulas such as interpolation and integration. . FD-12 refers to high-order FD calculations in a 12 au box. PW-12 and PW-24 refer to plane-wave calculations with supercells of 12 and 24 au on a side. Energies are in eV. The potential decays toward zero at locations distant from the colloids due to exponential screening. Ancilotto et al. (1999). The second curve is the MG result from that work. The next is the FAS-FMG result of Wang and Beck (1999) for the CO molecule with the FAS-FMG solver. The bottom curve is the FAS-FMG result for the Be atom. TABLES FIG. 1 . 1Schematic diagram for real-space treatment of a transition metal ion in a protein. The metal ion is labeled M, and the ligands are labeled L1-4. The electronic structure is treated self-consistently in the QM1 zone, while the orbitals are fixed in QM2. The fixed charges on the protein are located in the CM region. The solvent (typically water) may be included via a continuum dielectric model in the S zone. FIG. 3 .FIG. 4 .FIG. 5 .FIG. 6 .FIG. 7 .FIG. 8 .FIGFIG. 10 . 34567810Effect of order on the eigenvalues for the H atom. The (+) symbols are for the 1s orbital, (x) is for 2s, and the stars are for 2p. The analytical results are -0.5, -0.125, and -0.125 respectively. Effect of order on the orbital first moments for the H atom. The (+) symbols are for the 1s orbital, (x) is for 2s, and the stars are for 2p. The analytical results are 1.5, 6, and 5 respectively. Effect of order on the orbital virial ratios for the H atom. The (+) symbols are for the 1s orbital, (x) is for 2s, and the stars are for 2p. The analytical result is 2. A multigrid V-cycle. Iterations begin on the fine level on the left side of the diagram. R indicates restriction of the problem to the next coarser level. Corrections (C) begin as the computations move from the coarsest level to the finest level. Full multigrid cycle. Iterations begin on the left on the coarsest level. The solver proceeds sequentially down to the finest level, where a good initial approximation is generated from the coarse-level processing. The electrostatic potential for the screened atomic model. The analytic curve is the solid line, while the numerical results are the crosses. The numerical result deviates noticeably from the analytic values at points neighboring the origin due to the source singularity. The numerical result at the origin has been omitted. Schematic of two colloid particles located in a solution containing counterions and salt. SCF steps FIG. 11. Convergence behavior. The top curve is the Car-Parrinello (damped molecular dynamics) result of ,yp5}},x]] gp[x_]:=Evaluate[D[g[x],{x,2}]] r=Simplify[Expand[Collect[gp[x0+0],{ym5,ym4,ym3,ym2,ym1,y0,yp1,yp2, yp3,yp4,yp5}]]] TABLE I . IOrbital energies for the oxygen dimer, from Chelikowsky, TABLE II. Coefficients for the Laplacian. One side plus the central point are shown. Each coefficient term should be divided by the prefactor. The Laplacian is symmetric about the central point.Orbital FD-12 PW-12 PW-24 σ s -32.56 -32.09 -32.60 σ * s -19.62 -19.11 -19.57 σ p -13.63 -12.93 -13.37 π p -13.24 -12.54 -12.98 π * p -6.35 -5.53 -5.98 Points Order Prefactor Coefficients N=3 2nd 1 1 -2 N=5 4th 12 -1 16 -30 N=7 6th 180 2 -27 270 -490 N=9 8th 5040 -9 128 -1008 8064 -14350 N=11 10th 25200 8 -125 1000 -6000 42000 -73766 N=13 12th 831600 -50 864 -7425 44000 -222750 1425600 -2480478 One must be cautious, however, to properly initialize the orbitals in the DIIS procedure. SeeKresse and Furthmüller (1996). See Section IV.A.1 for discussion of the use of basis set terminology in reference to the FD method. SeeArias (1999) andGoedecker (1999) for discussion of linear-scaling applications of wavelets. The calculations in this section were performed with multigrid solvers discussed in Sections V and VII. The functions φ and v exist in a subspace of a Hilbert space which becomes a finite-dimensional subspace for any FE basis-set numerical computation. I thank Jeff Giansiracusa for providing the FFT results. This paper uses the Dirac-Slater Xα form for the exchange-correlation potential. However, the computed dipole is insensitive when that potential is changed to the LDA form. SeeJensen (1999) for converged basis set LDA results. Alternative accurate methods for propagating wavefunctions developed in the chemical physics community are discussed inLeforestier et al. (1991). See also Yu and Bandrauk (1995) which discusses a FE method for propagating wavefunctions in real time. The method was used to examine molecules in intense laser fields. With algorithmic improvements, this scaling can be reduced to qN g . See Costiner and Ta'asan(1995). F. Orbital-minimization methodsTo conclude our review of real-space self-consistent eigenvalue problems, we consider a related OM linear-scaling algorithm which uses LCAO bases(Sánchez-Portal et al., 1997). The reason for its inclusion here is that the bases(Sankey and Niklewski, 1989)are 1) numerical and 2) confined to a local region of space. Therefore, the method shares features in common with FD and FE approaches. The authors discussed construction of the Hamiltonian matrix elements and the total energy in the numerical bases. The total energy was re-expressed in a form which has terms involving only two-centered integrals which are interpolated from calculated tables (one-time calculation) and other terms computed entirely on a real-space grid (involving screened neutral-atom potentials and the Hartree and exchange-correlation potentials). The Hartree potential was computed via FFT methods. Rapid convergence of the approximations with decreasing grid spacing was observed. The OM functional ofKim et al. (1995)was employed to obtain linear scaling.The method was applied in calculations on several diatomics and triatomics where various-quality basis sets were tested in computations of bond lengths, bond angles, and binding energies. Gradient corrections to the LDA approximation were also considered. Finally, large-scale computations were performed on a turn of the DNA double helix consisting of ten guanine-cytosine base pairs in periodic boundaries (650 atoms). The equilibrium geometry was obtained in 200 minimization steps, requiring 5 days of computation on an HP C110 workstation. The number of self-consistency iterations required for each minimization step was not given. Also, it is not entirely clear what is the sparsity of the Hamiltonian in the numerical localized LCAO basis in relation to FD and FE methods. The spherical-wave basis set of Haynes and should also prove useful since it is localized in space and its truncation is controlled by a single parameter, the kinetic energy cutoff (similar to plane-wave methods). Hybrid basis-set/grid type methods such as discrete variable representations (DVR) and distributed approximating functionals (DAFs) also exist which yield accurate local representations(Light et al., 1985;Marchioro et al., 1994;Schneider and Feder, 1999).VIII. TIME-DEPENDENT DFT CALCULATIONS IN REAL SPACEThe Kohn-Sham method for electronic structure lies on solid theoretical ground due to the Hohenberg-Kohn theorems. Extensions of DFT to excited states and/or frequencydependent polarizabilities present more difficult challenges, but significant progress has been made in this area in the last few years. The developments include real-space computations of excitation energies and response properties(Yabana and Bertsch, 1996;Vasiliev et al., 1999;Kim, Städele, and Martin, 1999). Thorough reviews of the foundations of time-dependent DFT (TDDFT) methods are available (see, for example,Gross and Kohn, 1990;Casida, 1996). The starting point for practical computations is typically the solution of the timedependent LDA (TDLDA) equations:where the density-dependent effective potential is just the Kohn-Sham LDA potential (Eqs. 6 and 8) for the set of orbitals at time t. The TDLDA method includes dynamic screening effects which modify the excitation frequencies away from the Kohn-Sham LDA eigenvalue differences toward the physical ones. Inclusion of gradient corrections does not significantly improve the results(Bauernschmitt and Ahlrichs, 1996;Casida et al., 1998). There are two important approximations involved in Eq.(69): 1) the static LDA potential exhibits the incorrect asymptotic behavior at long range (exponential rather than −1/r), and 2) no time dependence is incorporated in the exchange-correlation potential (adiabatic approximation). It is generally recognized that the first approximation is the most severe(Van Gisbergen et al., 1998); with improvements to the LDA which yield the correct asymptotic larger behavior, quite accurate results are obtainable even for Rydberg states(Casida, 1996;Jamorski et al., 1996;Casida et al., 1998;Van Gisbergen et al., 1998;Tozer and Handy, 1998). The adiabatic approximation makes physical sense for slow processes, and integrals (over frequency) of the response in the mean-field theory obey rigorous sum rules which are satisfied for the small-amplitude TDLDA(Yabana and Bertsch, 1999). Proper modeling of the long-range behavior of the effective potential is important for the higher-lying Kohn-Sham states, which in turn are crucial for obtaining accurate excitation energies above the highest occupied Kohn-Sham LDA eigenvalue. These states are also important for computing accurate polarizabilities. For low-lying excitations, the TDLDA-level of theory is remarkably accurateYabana and Bertsch, 1999). Observed errors in excitation energies are on the order of one or a few tenths of an eV for small molecules in comparison with experiments. Two main approaches have been followed in development of the TDDFT method. In the first (Yabana and Bertsch, 1996), Eq. (69) is solved directly in real time by propagating the orbitals on a real-space grid. The frequency-dependent polarizability and the strength function are obtained by Fourier transformation of the time-dependent dipole moment computed on the grid. In the second approach(Petersilka et al., 1996;Casida, 1996), the problem is recast in the energy representation by calculating the response at the linear-response level. Solution of an eigenvalue problem involving the Kohn-Sham energy differences and a coupling matrix yields the excitation energies and oscillator strengths and from them the frequency-dependent polarizabilities. Applications of the second theoretical approach have employed both basis-set(Casida, 1996;Van Gisbergen et al., 1998;Tozer and Handy, 1998)and real-space(Vasiliev et al., 1999)formulations. The real-time and energy representations should give equivalent results for physical situations which allow a linear-response treatment. In this section, we review recent real-space computations in TDLDA theory.A. TDDFT in real time and optical responseThe real-time approach directly integrates Eq. (69) once an initial impulse has been given to the one-electron orbitals (obtained from a previous ground-state calculation).Yabana and Bertsch (1996)propagated the wavefunctions in time with a 4th-order Taylor expansion of the TDLDA equation. The procedure followed the previous time-dependent Hartree-Fock method ofFlocard et al. (1978)in nuclear physics. That method was shown to conserve the energy and wavefunction norms to high accuracy. 11 A predictor-corrector method was implemented to fix the density at times between successive wavefunction evaluations. The Hamiltonian was represented with a FD form on a uniform Cartesian mesh. An 8th-order expression was employed for the Laplacian operator, and the real-space pseudopotentials ofMartins (1991a, 1991b)were utilized to remove the core electrons. The method scales as qN g since it only requires repeated applications of the Kohn-Sham Hamiltonian to the wavefunctions, here assumed to cover the whole domain. If the orbitals could be confined to local regions of space, the method would scale linearly. Roughly 10 4 time-propagation steps are required to obtain the frequency-dependent response. As mentioned above, the physical quantities generated are the frequency-dependent polarizability and the closely related dipole strength function. The entire spectrum is produced in a single calculation without computations of excited-state Kohn-Sham orbitals, and the method is not restricted to the linear-response level of theory. In addition, the method only requires storage of the occupied states.Computations were first performed on the jellium model for Li 138 to compare with previous numerical results; the dipole strength function agreed well with that computed using a Green's function technique. Then calculations were performed on more physically realistic models of large charged Na clusters and C 60 . The strength function yields the polarizability; for the C 60 case, a value of α = 80Å 3 was computed compared with the experimental value of 85Å 3 obtained from the dielectric constant. A tight-binding model predicted a much lower polarizability of 45Å 3 . In a second study,Yabana and Bertsch (1997)applied the method to carbon chains and rings which are found in interstellar matter. For the C 7 chain, the lowest TDLDA mode occurs at roughly twice the frequency of the HOMO-LUMO gap in the Kohn-Sham LDA states. The size dependence of the transitions was modeled as the classical resonance of electrons in a conducting needle. The ring and chain geometries led to widely different frequencies for the lowest collective mode.Yabana and Bertsch (1999)presented further computations on conjugated hydrocarbons including polyenes, retinal (C 20 H 28 O), benzene, and C 60 . In this work, the scaling of the method was displayed vs. system size and was found to be even below N 2 . The TDLDA dipole strength was compared to precise experiments for the benzene molecule, and excellent agreement for the dipole strength was obtained. The computed lowest π → π * sharp transition was at 6.9 eV, the same as the experimental value, and a broad feature above 9 eV due to σ → σ * transitions was also relatively accurately reproduced. 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[ "Nano-artifact metrics based on random collapse of resist", "Nano-artifact metrics based on random collapse of resist" ]	[ "Tsutomu Matsumoto \nGraduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan\n", "Morihisa Hoga \nDai Nippon Printing Co. Ltd\n250-1 Wakashiba277-0871KashiwaChibaJapan\n", "Yasuyuki Ohyagi \nDai Nippon Printing Co. Ltd\n250-1 Wakashiba277-0871KashiwaChibaJapan\n", "Mikio Ishikawa \nDai Nippon Printing Co. Ltd\n250-1 Wakashiba277-0871KashiwaChibaJapan\n", "Makoto Naruse \nPhotonic Network Research Institute\nNational Institute of Information and Communications Technology\n4-2-1 Nukui-kita184-8795KoganeiTokyoJapan\n", "Kenta Hanaki \nGraduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan\n", "Ryosuke Suzuki \nGraduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan\n", "Daiki Sekiguchi \nGraduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan\n", "Naoya Tate \nDepartment of Electrical Engineering and Information Systems\nGraduate School of Engineering\nThe University of Tokyo\n2-11-16 Yayoi, Bunkyo-ku113-8656TokyoJapan\n", "Motoichi Ohtsu \nDepartment of Electrical Engineering and Information Systems\nGraduate School of Engineering\nThe University of Tokyo\n2-11-16 Yayoi, Bunkyo-ku113-8656TokyoJapan\n" ]	[ "Graduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan", "Dai Nippon Printing Co. Ltd\n250-1 Wakashiba277-0871KashiwaChibaJapan", "Dai Nippon Printing Co. Ltd\n250-1 Wakashiba277-0871KashiwaChibaJapan", "Dai Nippon Printing Co. Ltd\n250-1 Wakashiba277-0871KashiwaChibaJapan", "Photonic Network Research Institute\nNational Institute of Information and Communications Technology\n4-2-1 Nukui-kita184-8795KoganeiTokyoJapan", "Graduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan", "Graduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan", "Graduate School of Environment and Information Sciences\nYokohama National University\n240-8501Hodogaya, YokohamaKanagawaJapan", "Department of Electrical Engineering and Information Systems\nGraduate School of Engineering\nThe University of Tokyo\n2-11-16 Yayoi, Bunkyo-ku113-8656TokyoJapan", "Department of Electrical Engineering and Information Systems\nGraduate School of Engineering\nThe University of Tokyo\n2-11-16 Yayoi, Bunkyo-ku113-8656TokyoJapan" ]	[]	Artifact metrics is an information security technology that uses the intrinsic characteristics of a physical object for authentication and clone resistance. Here, we demonstrate nano-artifact metrics based on silicon nanostructures formed via an array of resist pillars that randomly collapse when exposed to electron-beam lithography. The proposed technique uses conventional and scalable lithography processes, and because of the random collapse of resist, the resultant structure has extremely fine-scale morphology with a minimum dimension below 10 nm, which is less than the resolution of current lithography capabilities. By evaluating false match, false non-match and clone-resistance rates, we clarify that the nanostructured patterns based on resist collapse satisfy the requirements for high-performance security applications.	10.1038/srep06142	[ "http://ftp.ncbi.nlm.nih.gov/pub/pmc/41/5e/srep06142.PMC4139945.pdf" ]	11,212,752	1412.6271	9a46740742652c6bbf68d17d2cb83a1c6c75b62f	Nano-artifact metrics based on random collapse of resist Tsutomu Matsumoto Graduate School of Environment and Information Sciences Yokohama National University 240-8501Hodogaya, YokohamaKanagawaJapan Morihisa Hoga Dai Nippon Printing Co. Ltd 250-1 Wakashiba277-0871KashiwaChibaJapan Yasuyuki Ohyagi Dai Nippon Printing Co. Ltd 250-1 Wakashiba277-0871KashiwaChibaJapan Mikio Ishikawa Dai Nippon Printing Co. Ltd 250-1 Wakashiba277-0871KashiwaChibaJapan Makoto Naruse Photonic Network Research Institute National Institute of Information and Communications Technology 4-2-1 Nukui-kita184-8795KoganeiTokyoJapan Kenta Hanaki Graduate School of Environment and Information Sciences Yokohama National University 240-8501Hodogaya, YokohamaKanagawaJapan Ryosuke Suzuki Graduate School of Environment and Information Sciences Yokohama National University 240-8501Hodogaya, YokohamaKanagawaJapan Daiki Sekiguchi Graduate School of Environment and Information Sciences Yokohama National University 240-8501Hodogaya, YokohamaKanagawaJapan Naoya Tate Department of Electrical Engineering and Information Systems Graduate School of Engineering The University of Tokyo 2-11-16 Yayoi, Bunkyo-ku113-8656TokyoJapan Motoichi Ohtsu Department of Electrical Engineering and Information Systems Graduate School of Engineering The University of Tokyo 2-11-16 Yayoi, Bunkyo-ku113-8656TokyoJapan Nano-artifact metrics based on random collapse of resist 10.1038/srep06142 Artifact metrics is an information security technology that uses the intrinsic characteristics of a physical object for authentication and clone resistance. Here, we demonstrate nano-artifact metrics based on silicon nanostructures formed via an array of resist pillars that randomly collapse when exposed to electron-beam lithography. The proposed technique uses conventional and scalable lithography processes, and because of the random collapse of resist, the resultant structure has extremely fine-scale morphology with a minimum dimension below 10 nm, which is less than the resolution of current lithography capabilities. By evaluating false match, false non-match and clone-resistance rates, we clarify that the nanostructured patterns based on resist collapse satisfy the requirements for high-performance security applications. A rtifact metrics 1 uses physical features unique to individual objects in terms of their physical properties or combinations of these properties, including electromagnetic 2,3 , mechanical and optical properties 4,5 . For an artifact metric to function, it should satisfy several conditions, such as (1) the extracted characters should vary between individual objects (individuality), (2) a given response should be consistently obtained for each measurement (measurement stability), (3) they should be robust against degradation caused by common use (durability) and (4) fabricated clones having an equivalent physical characteristic should be extremely difficult (clone resistance). Examples of existing artifact metrics include ordinary paper 5 , paper containing magnetic microfibers 6 , plastics and semiconductor chips. A physical unclonable function 7 is a type of artifact metrics that is essentially equivalent to what Matsumoto et al. examined under the name of 'clone-resistant modules' in 1997 8 . The critical-security battlefield in which artifact metrics are used is analogous to a defender and attacker relationship in which the former tries to produce patterns that are difficult copies, and the latter seeks to counterfeit these patters. In view of recent technological advancements in microfabrication and its strong demand in society and industry (e.g. optical document security 9 ), new technology must go beyond that developed so far, which has been limited to micrometre-scale precision, and be founded on the ultimate principles of physics. Here, we propose and demonstrate a nano-artifact metrics that is robust against cloning attacks. The proposed metric uses nanometre-scale structures obtained from the random collapse of resists induced by exposure to conventional electron-beam (e-beam) lithography. E-beam lithography is a mature and fundamental technology for prototyping fine structures. Minimum feature size is an important metric for lithography to produce the designated structures. To quantify the achievable minimum size, we use a two-dimensional array of pillars. The decrease in the minimum pitch of a pillar array over the last few years is summarized in Fig. 1a. The circles denoted (1), (2) and (3) in Fig. 1a are based on Refs. 10, 11 and 12, respectively and the dotted line, which represents an estimated pitch-resolution limit, is based on Refs. 13 and 14. Figure 1a also suggests that these feature sizes may be fabricated by attackers who use the available technology to make clones. Meanwhile, we must also consider the extent to which we can precisely measure fine structures with the available technology such as a scanning electron microscopy (SEM). Critical-dimension scanning electron microscopy (CD-SEM), which is specialized in measuring length and offers precision in the sub-nanometre scale, may be assumed 15 . For an artifact metrics to be made using silicon nanostructures fabricated based on conventional e-beam lithography, the defender, who wants to prevent counterfeiting, must fabricate finestructured patterns such that the attacker, who wants to copy the authentic device, will not be able to intentionally reproduce the pattern based on the information obtained by CD-SEM. However, this condition imposes a paradoxical requirement that fine structures which are smaller than the resolution limit of the state-of-the-art e-beam lithography should be fabricated. Otherwise, the authentic devices may be easily cloned. To overcome this paradox, we exploit the well-known phenomenon of the random collapse of resist 16 . Resist collapse may occur during the rinse process of lithography and depends on the pattern resolution, resist thickness and duration of e-beam exposure. The end result is the collapse of the intended pattern 16 . To produce a desired pattern, resist collapse must be suppressed in e-beam lithography, which can be achieved by deployment of 'anticollapse rinses'. However, from the standpoint of nano-artifact metrics, resist collapse occasionally provides structures finer than the original technological limitation. Furthermore, resist collapse occurs randomly. Therefore, as shown in Fig. 1a, we can use resist collapse to benefit from the uncertainty in position that is less than the resolution of nanofabrication and achieve nano-artifact-metric functionalities. To verify this notion, we fabricated an array of pillars from a layer of resist. The pillars had cross section area of 60 nm 3 60 nm, were 200 nm high and were positioned on a grid of 120 nm 3 120 nm squares that filled a 2 mm 3 2 mm square, as shown in Fig. 1b. As a guide for facilitating alignment, a 3 mm 3 3 mm square frame was drawn outside the pillar array area. We used a JEOL JBX-9300FS ebeam lithography system with the acceleration voltage set at 100 kV and with a dose of 37 mC/cm 2 . After post-exposure bake and resist development, the structure is rinsed, which is when the random collapse of resist pillars occurs. Figure 1c shows an SEM image of an array of collapsed resist pillars. The wafer is then etched with HBrbased gas using inductively coupled plasma (ICP)-type reactive ion etching (RIE), and the resist is stripped by oxygen ashing. The resulting nanostructured-silicon patterns were imaged by a CD-SEM (Hitachi High-Technologies CG4000). We fabricated 2401 samples on a single 200-mm-diamter wafer and used 2383 of these samples to evaluate their use for security applications. Figure 2a shows an image of a nanostructured-silicon pattern. The image contains 1024 3 1024 pixels, has eight-bit resolution (256 levels) and was obtained by averaging eight frames acquired by CD-SEM. A variety of different morphologies were obtained, as shown in Fig. 2b. Figures 2b.i and 2b.iii show that the structural details in the patterns are as small as 9.23 nm. Here, one minor remark is that the sizes and the layout of the original array of pillars have not been optimized so that the resultant security performances, described below, are maximized. Nevertheless, as shortly demonstrated, quite good properties have been obtained. This indicates that further advancements could be possible by engineering the original pillar (or not-like-a-pillar) structures to be collapsed, which could be an interesting future study. Meanwhile, we have experimentally confirmed that a proper dose of electron beam is necessary in order to induce versatile collapse of resist pillars; Figs. 2c and 2d show CD-SEM images when the dose was 30 and 40 mC/cm 2 , respectively, indicating that too low or too high doses do not yield versatile resultant patterns. To determine whether these patterns may be used as artifact metrics, we conducted the following analysis: A 512 pixel 3 512 pixel, 8 bit (256 levels) greyscale image was extracted from the centre of an image of a pillar array and smoothed by an 11 3 11 median filter. In comparing any two patterns, A (i, j) and B (i, j), we first created a 'mask' pattern defined by M(i,j)~1 , A(i, j)wT or B(i, j)wT 0 otherwise,ð1Þ where T is a given threshold value. Because the patterns are fabricated by conventional lithographic processes, they consist of areas of varying heights. Therefore, two peaks appear in the statistics of pixel values, with a valley between the two peaks. Specifically, the number of times higher and lower peak values occur was approximately 130 and 80, respectively, and the incidence of the valley (i.e. the threshold T) between the two peaks was 90. Here, a remark is that the pixel value in images is given by 8 bit (0-255), and the particular values of 130, 80 and 90 are related to the greyscale pixel values of the given images. As indicated by Fig. 1c and Fig. 2, the greyscale value is related to the height of the nanostructured pattern. We do not calibrate the pixel value to the actual height (i.e. A (i, j) and B (i, j) are dimensionless values), but it does not cause any problem in this particular study. Also, 1 pixel occupies approximately a 3.3 nm square area. By applying the mask, we obtain two images,Â(i, j)M (i, j)\|A(i, j) andB(i, j)~M(i, j)\|B(i, j), which means that we ignore regions where both patterns A (i, j) and B (i, j)are low (i.e. the pattern is not high). The correlation, or similarity, between the two patterns is evaluated by the Pearson correlation coefficient R~P i P jÂ (i, j){ A h iB (i, j){ B h i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P i P jÂ (i, j){ A h i 2B (i, j){ B h i 2 s ,ð2Þ where A and B indicate the average ofÂ(i, j) andB(i, j), respectively. R is a dimensionless value. If R is negative, it is set to zero. Furthermore, each value of A (i, j) and B (i, j) is shifted between one and five pixels to the upper, lower, left and right side, and R is calculated for each shifted position. The maximum R from these positions (no shift, left, right, up and down shifts) is used as the similarity between A (i, j) and B (i, j). We next calculate the false match rate (FMR) and false non-match rate (FNMR). FMR and FNMR are indicators of individuality and measurement stability, respectively. To calculate the FMR, all 2383 images were used. If the similarity given by Eq. (2) is greater than the given threshold, the two patterns are considered to be similar to each other, which is extremely likely to be a false decision. Since we had 2383 samples, we conducted 2383 3 2382 comparisons in calculating the FMR for threshold values between 0 and 1. To have an intuitive picture, suppose that only 1 case among the 2383 3 2382 comparisons resulted in a false decision. In this case, the 'error rate' would be 1/(2383 3 2383) < 1.76 3 10 27 . From such calculation, the logarithmic scale for the FMR in the y-axis of Fig. 3 is naturally recognized. From Fig. 3, the FMR drops below the error rate of 10 26 as the threshold is just above zero, indicating that the occurrence of false decisions among all comparisons are extremely small. To calculate the FNMR, we used images created from 100 images of each of the 74 samples. If the similarity is less than the given threshold, the two images are considered different from each other. In other words, identical samples are considered different, which is a www.nature.com/scientificreports false decision. The leftmost and rightmost curves in Fig. 3 show the FMR and FNMR, respectively. The FMR is substantially smaller than the FNMR, even with a smaller threshold value. In addition, the FMR and FNMR curves are well separated from each other, which means that it is possible to obtain sufficiently small FMR and FNMR by choosing adequate threshold values. In addition, using the following method, we examined the clone match rate (CMR). Suppose that attackers capture the authentic device pattern and precisely fabricate a pattern in such a manner that an average over every k 3 k pixel area is essentially equivalent to the authentic device pattern. To quantify such a cloning process, we transformed each of the 2383 images into 'virtually cloned images'. Let the pixel value for a k 3 k area be denoted by p (i, j). If the average value in this area is greater than the threshold T, we consider this area to have the higher average value (130). Thus, this pixel value of the virtually cloned image is p9(i, j)5 130. If the average value in the k 3 k area is less than or equal to the threshold T, we consider this area to have the lower average value (80). Thus, this pixel value of the virtually cloned image is p9(i, j) 5 80. The clone image thus obtained is then compared with the authentic image using Eq. (2). If the similarity exceeds the threshold value, the clone successfully mimics the original. The CMR is calculated by performing the above evaluation for all the 2383 samples. This scheme constitutes a very strict evaluation of the cloning resistance. Table 1 summarizes the assumed cloning technologies. For example, if k 5 3, a 3 3 3 pixel area (or 'unit tile size') corresponds to a 10 nm 3 10 nm square because, as mentioned earlier, 1 pixel occupies approximately a 3.3 nm square area, which may be regarded as state-of-the-art for the current nanofabrication technology. Nevertheless, the similarity cannot be greater than 0.4, as indicated by the calculated CMR shown by the purple dotted curve in Fig. 3. Considering that this value is considerably lower than the similarity between images of identical samples (i.e. the FNMR curve), such a clone does not pose a serious threat. In other words, the original authentic pattern is sufficiently random. Furthermore, were it possible to detect that a given pattern was formed from a combination of square units, it would presumably be determined that, based on such a feature, the pattern is a non-authentic device (or clone); this strategy is similar to liveness detection in biometrics 17 . Finally, note that the signal processing scheme described above is relatively simple yet requires highly skilled attackers. Based on these considerations and the results of the FMR, FNMR and CMR analysis, we conclude that the proposed nanostructured patterns based on the random collapse of resist could serve as superior nano-artifact metrics. Finally, we put forward few remarks on the demonstrated principles and technologies. First, the proposed principle is based on the 'uncertainty' inherent in conventional e-beam lithography technologies. Moreover, e-beam lithography is one of the widely spreading nanotechnologies including silicon fabrications. In this context, the proposed principle also has advantages in its general purpose properties or utilities. Second is a comment regarding optical lithography. Optical lithography in silicon nanostructring is based on the so-called reticle, which is subjected to reduced-projection exposure. A reticle is four times larger in scale than the intended nanostructured pattern and is fabricated by e-beam lithography. It is impossible to fabricate patterns by optical lithography in the same resolution of the randomly collapsed silicon nanostructures demonstrated in this study. In other words, optical lithography may not be useful for attackers in copying the demonstrated devices. The third remark concerns the term 'nano-artifact metrics'. In Ref. [1], Matsumoto et al. proposed 'artifact metrics'. The notion of artifact metrics is conceptually similar to 'biometrics', for which uniqueness in biological entities is utilized. Unlike biometrics, artifact metrics utilizes uniqueness in physical objects/things, physical processes or their combinations. We should emphasize that there are no implications such as 'defective patterns' in the word 'artifact'. One may imagine that 'nanoscale fingerprint' might be more appropriate than 'nano-artifact metrics'. However, the term 'fingerprint' implies information hiding, watermarking and their related technologies in the field of information security, which do not apply to our study. Furthermore, as discussed at the beginning of this paper, we showed four important requirements for artifact metrics to function (i.e. individuality, measurement stability, durability, clone resistance). Here, it should be noted that a total system as a whole is important, not just the elemental processes; this is another reason we describe the concept by 'nano-artifact metrics', which includes the notion of a total system while avoiding the use of 'nanoscale fingerprint'. In summary, with the goal of exploiting the fundamental laws of physics to produce nano-artifact metrics, we demonstrated nanoartifact metrics based on the random collapse of resist pillars in e-beam lithography. As qualitative significance for information security, this study opens new design principles and degrees-offreedom by exploiting uncertainty at the nanometre scale. Moreover, by developing sophisticated image preprocessing means and similarity indices for matching, the security performance of these metrics is further enhanced. This is also a qualitatively novel aspect for information security in the sense that the combinations of physical process and logical signal processing means provide new values. Note that our use of SEM technology is not a particularly important aspect of this study. To construct practical systems, many additional issues must be considered, such as reducing measurement cost and verifying interoperability for the case when different measurement devices are used for sample registration and authentication. Nevertheless, this study demonstrates that sufficiently good security performance can be achieved by the random collapse of resist in e-beam lithography; the enemy of silicon processing in previous studies turns out to be a strong enabler for information security in this study. Figure 1 \| 1Nano-artifact metrics based on random collapse of resist in electron-beam lithography. (a) Roadmap showing the minimum size of pillars formed by e-beam lithography. Using the phenomenon of randomly induced resist collapse, nano-artifact metrics contain length scales below the minimum dimension available in conventional lithography methods. (b) Schematic of array of pillars. (c) SEM image of collapsed resist. Figure 2 \| 2Versatile morphology in silicon nanostructures obtained from collapsed resist. (a) Example of entire region of fabricated silicon nanostructure. (b) Magnified view of several areas from the panel (a). The minimum feature size is indicated in each image. Note that feature sizes are smaller than the minimum feature size of the original array of pillars. In other words, the uncertainty obtained in this versatile morphology is less than that available directly by current technology. (c,d) Silicon nanostructure when the dose quantity in the e-beam lithography was (c) 30 and (d) 40 mC/cm 2 , respectively. Too low or too high doses do not yield versatile resultant patterns. Figure 3 \| 3Evaluation of security performance. Error rate as a function of the threshold. False match rate (FMR) and false non-match rate (FNMR) are labelled. The curves labelled 60, 50, 40, 30, 20 and 10 nm are the clone match rate (CMR) for the given minimum unit tile size. SCIENTIFIC REPORTS \| 4 : 6142 \| DOI: 10.1038/srep06142 AcknowledgmentsThe authors thank Hitachi High-Technologies Cooperation in operating the CD-SEM. The authors thank Mr. Nakagawa from JOEL Co. Ltd. for his helpful discussions regarding the resolution limits of e-beam lithography. This work was supported in part by the Strategic Information and Communications R&D Promotion Programme (SCOPE) of the Ministry of Internal Affairs and Communications and by the Core-to-Core Program of the Japan Society for the Promotion of Science.Additional informationCompeting financial interests: The authors declare no competing financial interests. Clone match rate evaluation for an artifactmetric system. H Matsumoto, T Matsumoto, IPSJ J. 44Matsumoto, H. & Matsumoto, T. Clone match rate evaluation for an artifact- metric system. IPSJ J. 44, 1991-2001 (2003). Extracting secret keys from integrated circuits. D Lim, J W Lee, B Gassend, G E Suh, M Van Dijk, S Devadas, IEEE T. VLSI Syst. 13Lim, D., Lee, J. W., Gassend, B., Suh, G. E., van Dijk, M. & Devadas, S. Extracting secret keys from integrated circuits. IEEE T. VLSI Syst. 13, 1200-1205 (2005). G Dejean, K Darko, Rf-Dna, Radio-frequency certificates of authenticity. Cryptographic Hardware and Embedded Systems-CHES 2007. ViennaDeJean, G. & Darko, K. RF-DNA: Radio-frequency certificates of authenticity. Cryptographic Hardware and Embedded Systems-CHES 2007 (Vienna, 10-13 . 10.1007/978-3-540-74735-2_24Lecture Notes in Computer Science. 4727September 2007) Lecture Notes in Computer Science 4727, 2007, pp. 346-363, DOI: 10.1007/978-3-540-74735-2_24. Fingerprinting documents and packaging. J D Buchanan, R P Cowburn, A V Jausovec, D Petit, P Seem, G Xiong, M T Bryan, Nature. 436475Buchanan, J. D., Cowburn, R. P., Jausovec, A. V., Petit, D., Seem, P., Xiong, G. & Bryan, M. T. Fingerprinting documents and packaging. Nature 436, 475 (2005). Individuality evaluation for paper based artifact-metrics using transmitted light image. M Yamakoshi, J Tanaka, M Furuie, M Hirabayashi, T Matsumoto, Proc. SPIE. SPIE6819Yamakoshi, M., Tanaka, J., Furuie, M., Hirabayashi, M. & Matsumoto, T. Individuality evaluation for paper based artifact-metrics using transmitted light image. Proc. SPIE 6819, 68190H-1-68190H-10 (2008). Evaluation of equal error rate in document authentication system using magnetic fiber. 3rd Intl. Conf. Anti-counterfeiting, Security, and Identification in Communication. T Ikeda, S Hiroe, T Yamada, T Matsumoto, Y Takemura, DOI: 10.1109/ ICASID.2009.5276959Hong KongIkeda, T., Hiroe, S., Yamada, T., Matsumoto, T. & Takemura, Y. Evaluation of equal error rate in document authentication system using magnetic fiber. 3rd Intl. Conf. Anti-counterfeiting, Security, and Identification in Communication 2009 (Hong Kong, 20-22 August 2009) pp. 382-385, 2009. DOI: 10.1109/ ICASID.2009.5276959 Physical one-way functions. R Pappu, B Recht, J Taylor, N Gershenfeld, Science. 297Pappu, R., Recht, B., Taylor, J. & Gershenfeld, N. Physical one-way functions. Science 297, 2026-2030 (2002). Proposal of authentication method based on cloning resistance. T Matsumoto, Proceedings of the IEICE Symposium on Cryptography and Information Security. the IEICE Symposium on Cryptography and Information SecurityFukuoka, 29The Institute of Electronics, Information and Communication EngineersSCIS 97-19C. in JapaneseMatsumoto, T. et al. Proposal of authentication method based on cloning resistance. Proceedings of the IEICE Symposium on Cryptography and Information Security 1997 (Fukuoka, 29 January-1 February 1997), SCIS 97-19C, The Institute of Electronics, Information and Communication Engineers (in Japanese). . 10.1038/srep06142SCIENTIFIC REPORTS \|. 46142SCIENTIFIC REPORTS \| 4 : 6142 \| DOI: 10.1038/srep06142 Optical document security (Artech House. R L Van Renesse, Bostonvan Renesse, R. L. Optical document security (Artech House, Boston, 2004). Electron beam lithography-resolution limits. A N Broers, A C F Hoole, J M Ryan, Microelectron. Eng. 32Broers, A. N., Hoole, A. C. F. & Ryan, J. M. Electron beam lithography-resolution limits. 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[ "Properties of the Anomalous States of Positronium", "Properties of the Anomalous States of Positronium", "Properties of the Anomalous States of Positronium", "Properties of the Anomalous States of Positronium" ]	[ "Chris W Patterson \nCentre for Quantum Dynamics\nGriffith University\n\n", "Nathan Qld \nCentre for Quantum Dynamics\nGriffith University\n\n", "Australia \nCentre for Quantum Dynamics\nGriffith University\n\n", "Chris W Patterson \nCentre for Quantum Dynamics\nGriffith University\n\n", "Nathan Qld \nCentre for Quantum Dynamics\nGriffith University\n\n", "Australia \nCentre for Quantum Dynamics\nGriffith University\n\n" ]	[ "Centre for Quantum Dynamics\nGriffith University\n", "Centre for Quantum Dynamics\nGriffith University\n", "Centre for Quantum Dynamics\nGriffith University\n", "Centre for Quantum Dynamics\nGriffith University\n", "Centre for Quantum Dynamics\nGriffith University\n", "Centre for Quantum Dynamics\nGriffith University\n" ]	[]	It is shown that there are anomalous bound-state solutions to the two-body Dirac equation for an electron and a positron interacting via an electromagnetic potential. These anomalous solutions have quantized coordinates at nuclear distances (fermi) and are orthogonal to the usual atomic positronium bound-states as shown by a simple extension of the Bethe-Salpeter equation. It is shown that the anomalous states have many properties which correspond to those of neutrinos.	10.1103/physreva.107.042816	[ "https://export.arxiv.org/pdf/2207.05725v2.pdf" ]	250,450,917	2207.05725	784beb1c3008b7ecf459816080c2be2541668e86	Properties of the Anomalous States of Positronium 11 May 2023 11 May 2023 Chris W Patterson Centre for Quantum Dynamics Griffith University Nathan Qld Centre for Quantum Dynamics Griffith University Australia Centre for Quantum Dynamics Griffith University Properties of the Anomalous States of Positronium 11 May 2023 11 May 2023arXiv:2207.05725v2 [quant-ph] * Guest Scientist, Theoretical Division, Los Alamos National Laboratory, NM 87545 USA 1 It is shown that there are anomalous bound-state solutions to the two-body Dirac equation for an electron and a positron interacting via an electromagnetic potential. These anomalous solutions have quantized coordinates at nuclear distances (fermi) and are orthogonal to the usual atomic positronium bound-states as shown by a simple extension of the Bethe-Salpeter equation. It is shown that the anomalous states have many properties which correspond to those of neutrinos. For clarity, only every fourth wavefunction is shown. Fig.2 The numerical energies E i are found by diagonalizing the potential matrix Φ nm in (27) with J=10 and N=40. The ρ i are zeros of j 10 (k 41 ρ). 1 I. INTRODUCTION As shown in previous papers [1] , [2], both the atomic states Ψ Atom and anomalous states Ψ An of positronium admit bound-state solutions to the two-body Dirac equation (TBDE), HΨ = (H 0 + Φ(ρ)Ψ = EΨ,(1) with an instantaneous Coulomb potential Φ(ρ) = Φ C = −e 2 /ρ, where H 0 is the free-particle Hamiltonian in the rest frame. By definition, the anomalous states Ψ An are solutions to (1) with Φ(ρ) = 0 such that H 0 Ψ An = 0.(2) Let the anomalous bound-states Ψ DV be linear combinations of Ψ An which are solutions to (1). From (1) and (2), the Ψ DV are then bound-state solutions to the simple equation Φ C (ρ)Ψ DV = EΨ DV .(3) It was shown in [2] that the bound-state solutions to (3) Ψ DV can be found by using the discrete variable (DV) representation [3]. That is, the solutions to (3) are the DV states Ψ DV in which the coordinate ρ is quantized at discrete separations or Ψ DV = Gδ(ρ−ρ i ) with ρ i at f ermi distances, where the factor G includes the dependence on the other coordinates as well as the spinors. Such DV states can be found by diagonalizing Φ C in the Ψ An bases only if the Ψ An bases is a complete set in the ρ coordinate. The use of the word 'discrete' in DV theory refers to the quantization of the radial coordinate ρ = ρ i , in contrast to the normal quantization of momentum for free-particles. Note that the free-particle anomalous states Ψ An in (2) are distinguished from the bound-states Ψ DV of (3) which are comprised of these free-particle states. This distinction has been made because not all anomalous states Ψ An can form a complete set in ρ and can thereby admit bound-states solutions Ψ DV of (1) or (3) using DV theory. The anomalous bound-states Ψ DV , with total angular momentum J = 0, were shown in [2] to have properties which were quite distinct from their atomic counterparts. Besides being bound at nuclear distances ρ i , it was found that these DV states are dark and stable. That is, they cannot absorb or emit light, nor can the electron and positron annihilate or dissociate. This unusual behavior was explained by solving both the TBDE and the Bethe-Salpeter equation (BSE) [4], [5] with a Coulomb potential. It was found that these DV states Ψ DV form doublets with spin S z = 0 and energy E i = −e 2 /ρ i However, in [2], it was also found that the Coulomb potential in the TBDE (1) erroneously mixes the anomalous bound-states with the atomic bound-states, resulting in the unusual behavior of the atomic ground-state wavefunction near the origin as found in [1]. This erroneous mixing, while small, was shown not to occur at all for the relativistically correct BSE because of the different time propagation for anomalous bound-states and atomic bound-states. In particular, it was shown, using the BSE, that the time dependence of the atomic states was determined by the two-body Feynman propagator K 2 F , whereas the time dependence of the anomalous states was determined by the two-body retarded propagator K 2 R . As a result, the anomalous bound-states for the instantaneous Coulomb potential were temporally orthogonal to the atomic bound-states and could not be mixed. It was further argued in [2] that the anomalous bound-states were, themselves, both mathematically viable and necessary for completeness in space-time when using the BSE. It seems appropriate that their properties should be investigated further to see if they could exist physically. In [2], only the DV states with total angular momentum J = 0 were considered in order to show their influence on the atomic J = 0 wavefunctions and energies when using the TBDE. In this paper, the DV states for all J are considered with the restriction to J z = 0. It is found that there are then four different possible DV states corresponding to a S = 0 doublet and a S = 1 doublet for each J > 0. Like the J = 0 DV states, these DV states for J = 0 are also dark and stable. Including the magnetic potential Φ M , the S = 0 DV doublet has energy E 0 i = 2e 2 /ρ i and the S = 1 DV doublet has energy E 1 i = 0. It is also shown that the Lorentz boost reduces the symmetry from spherical (ρ, θ) to cylindrical (r, z) where z is in the direction of motion. Because of this dynamical symmetry breaking, in the moving frame the DV doublet states with S z = ±1 can only occur in the z = 0 plane with θ = π/2, so that they are oriented perpendicular to the direction of motion. On the other hand the DV doublet states with S z = 0 can occur for any θ. Remarkably, using a Lorentz boost for the DV states, it is proven that doublets with θ = π/2 transform like Majorana fermions instead of bosons. It is then shown that the S z = ±1 fermions with mass M 1 i = 0 have well defined chirality and helicity as expected for a zero mass fermions. Finally, it is shown that all of the unusual properties of the anomalous bound-states Ψ DV are a result of the fact that either the electron or the positron (but not both) must be in a negative energy state. As a result, one's normal understanding of quantum and classical mechanics can be misleading. However, it is shown that these anomalous bound-states still have a simple classical correspondence which aids in understanding their novel properties. In order to make this paper reasonably self-contained, the notation and the salient developments of the previous work are briefly reviewed below so that the reader can better understand the three equations above. In this review, comparisons are made between the TBDE and the BSE which are especially important to the understanding of the anomalous states. (Note that the natural units c = 1 and = 1 are used below except for cases where clarity is needed.) II. REVIEW The Hamiltonian of the TBDE for a free electron and positron, H 0 , in (1) in the moving frame is the sum of the individual Dirac Hamiltonians, H 0 Ψ = {α e · p e + α p · p p + (γ e4 + γ p4 )m}Ψ = EΨ.(4) Transforming this equation using relative coordinates and their conjugate momenta, ρ = (r e − r p ), π = (p e − p p )/2,(5) R = (r e + r p )/2, P ′ = p e + p p , one finds [1], [6], H 0 Ψ = {K + M + (α e + α p ) 2 · P ′ }Ψ = EΨ,(6)K = (α e − α p ) · π, M = m(γ e4 + γ p4 ), where K is the two-body kinetic operator and M is the two-body mass operator. The prime is used for P ′ to indicate the frame is moving rather than at rest. The matrix elements of α k and γ 4 in the bases of the two Dirac-spinors e 1 and e 2 are given by α k = iγ 4 γ k = σ k   0 1 1 0   , (f or k = 1, 2, 3),(7)γ 4 =   1 0 0 −1   , so that α k e 1 = σ k e 2 , α k e 2 = σ k e 1 , γ 4 e 1 = e 1 , and γ 4 e 2 = −e 2 . One can then define the four Dirac-spinors e ij = e i × e j for the direct product wavefunctions Ψ = ψ(e) × ψ(p) such that Ψ ≡ ψ 1 (e)ψ 1 (p) ψ 1 (e)ψ 2 (p) ψ 2 (e)ψ 1 (p) ψ 2 (e)ψ 2 (p) ≡ Ψ 11 Ψ 12 Ψ 21 Ψ 22 ,(8) ≡ Ψ 11 e 11 + Ψ 12 e 12 + Ψ 21 e 21 + Ψ 22 e 22 . With this notation, one finds the simple equations for the mass operators γ e4 and γ p4 of M, It is useful to use a bases where Ψ g is symmetric and Ψ u is antisymmetric under the simultaneous exchange e 11 ↔ e 22 and e 12 ↔ e 21 , such that Ψ = Ψ g + Ψ u ,(9) Ψ g = 1 2 {(Ψ 11 + Ψ 22 )(e 11 + e 22 ) + (Ψ 12 + Ψ 21 )(e 12 + e 21 )}, Ψ u = 1 2 {(Ψ 11 − Ψ 22 )(e 11 − e 22 ) + (Ψ 12 − Ψ 21 )(e 12 − e 21 )}. The Ψ g and Ψ u states are only coupled by the mass operator M in (6) where M(e 11 ± e 22 ) = 2m(e 11 ∓ e 22 ).(10) One can also define the four Pauli-spinors χ ± 1 2 (e)χ ± 1 2 (p) in terms of the states Ω S Sz with total spin S and its z component S z , so that Ω 0 0 = 1 2 [χ1 2 (e)χ − 1 2 (p) − χ − 1 2 (e)χ1 2 (p)],(11)Ω 1 −1 = χ − 1 2 (e)χ − 1 2 (p) , Ω 1 0 = 1 2 [χ1 2 (e)χ − 1 2 (p) + χ − 1 2 (e)χ1 2 (p)] , Ω 1 1 = χ1 2 (e)χ1 2 (p). The singlet state Ω 0 0 is antisymmetric (X = −1) and triplet states (Ω 1 −1 , Ω 1 −0 , Ω 1 1 ) are symmetric (X = 1) under particle exchange X, where XΩ S Sz = XΩ S Sz . The Dirac wavefunctions Ψ include terms of the sixteen possible spinors e ij Ω S Sz for the four Dirac-spinors e ij and the four Pauli-spinors Ω S Sz . The free-particle solutions of (6) are direct products Ψ ±± = ψ ± (e) × ψ ± (p) of the singleparticle solutions for positive and negative energy states ψ + and ψ − with energies, E ±± = e e + e p = ± p 2 e + m 2 ± p 2 p + m 2 . Now let P ′ =0 in (6) so that H 0 = K + M. Using (5), one finds that p e = −p p and p 2 e = p 2 p = π 2 where π is the relative momentum. The energies of the free-particle states are then E ±± = ±e ± e,(12)e = √ π 2 + m 2 . When using the BSE for P ′ =0, it is useful to divide the free -particle solutions into either atomic states or anomalous states. The atomic free-particle states Ψ Atom have wavefunctions Ψ ++ and Ψ −− [4], [5] with energies E ++ = 2e, E −− = −2e.(13) The anomalous free -particle states Ψ An have wavefunctions Ψ +− and Ψ −+ [2] with energies as in (2) E +− = 0, E −+ = 0.(14) Both the atomic bound-states and anomalous bound-states are solutions to the TBDE (1) for Φ = Φ C . In the momentum representation with a Coulomb potential, one can write (1) (often called the Breit equation) as the integral equation, H 0 Ψ(π) − e 2 2π 2 d 3 k 1 k · k Ψ(k + π) = EΨ(π).(15) Note that this equation allows the Ψ Atom and Ψ An states to be mixed by the Coulomb potential. As shown in [2], the two different bound-state solutions to (1) or (15), Ψ Atom and Ψ DV , have different time behaviors. Together, these two solutions form a complete orthonormal set in space-time. The Ψ Atom and Ψ DV correspond to solutions to two different BSEs which are now described. These two BSEs demonstrate that the Coulomb potential cannot mix the Ψ Atom and Ψ An states. These two different BSEs are derived from the two-body Green's functions for the electron and positron propagators in the relative coordinates. Unlike the TBDE, the BSE is relativistically covariant and treats the relative time t = t e − t p and the relative energy ε = (e e − e p )/2 of the two particles correctly as the fourth component of the coordinates ρ = (ρ, it) and conjugate momenta π = (π, iε), respectively. One can define two propagators equations below, one needs to separate the atomic free -particle states from the anomalous free -particle states as in (13) and (14). For this purpose, the projection operators Λ ±± are defined by Λ ±± (π)Ψ(π) = Ψ ±± (π). K 2 F = K F (e) × K F (p) and K 2 R = K R (e) × K R (p) For atomic bound-states Ψ Atom , it is necessary to use the two-body Feynman propagator K 2 F [5] with the result that the atomic states are only comprised of the free -particle wavefunctions Ψ ++ (π) and Ψ −− (π) in (13). In the momentum representation, with a Coulomb potential Φ C , the BSE for Ψ Atom becomes, after some approximations [2], H 0 [Ψ ++ (π) + Ψ −− (π)] − e 2 2π 2 Λ Atom d 3 k 1 k · k Ψ(k + π) = E[Ψ ++ (π) + Ψ −− (π)],(16) with Λ Atom = Λ ++ (π) − Λ −− (π) and Ψ +− (π) = Ψ −+ (π) = 0. For the anomalous bound-states Ψ DV , it is necessary to use the two-body retarded prop- agator K 2 R [2] with the result that the Ψ DV states are only comprised of the anomalous states Ψ +− (π) and Ψ −+ (π) where H 0 Ψ +− (π) = H 0 Ψ −+ (π) = 0 in (14). Consequently, for a Coulomb potential, one finds the BSE for Ψ DV [2] in (1) and (3) is − e 2 2π 2 Λ DV d 3 k 1 k · k Ψ(k + π) = E[Ψ +− (π) + Ψ −+ (π)],(17) with Λ DV = Λ +− (π) + Λ −+ (π) and Ψ ++ (π) = Ψ −− (π) = 0. Note that the BSE (17) for Ψ DV is simply the equivalent of diagonalizing Φ C in the free-particle bases Ψ +− (π) and Ψ −+ (π) of Ψ An corresponding to (3). This equation is not correct when the masses of the two particles are unequal as in the case of Hydrogen where H 0 Ψ +− (π) = −H 0 Ψ −+ (π) = 0. Thus, it only applies to particles with equal masses which obey (2). One should compare the BSEs (16) and (17) with the TBDE or Breit equation in (15). A consequence of the two BSEs (16) and (17) is that the atomic states Ψ Atom and DV states Ψ DV cannot be mixed by the Coulomb potential in contrast to the TBDE equation (15). Summarizing, one must first eliminate the anomalous free-particle states Ψ +− and Ψ −+ in the TBDE (15) in order to calculate the atomic positronium bound-states. Conversely, one must first eliminate the atomic free-particle states Ψ ++ and Ψ −− in the TBDE (15) in order to calculate the positronium DV states. Mathematically, it must be emphasized that the BSE for Ψ Atom (16) automatically specifies the K F propagator for the atomic bound-states and the BSE for Ψ DV (17) automatically specifies the K R propagator for the anomalous bound-states. One has no choice for the temporal boundary conditions as they are determined by these equations unambiguously. The two different bases formed from atomic and DV states are each mathematically complete sets spatially and therefore overlap: they are not orthogonal spatially. However, they are orthogonal temporally because of their different time behaviors as determined by their different propagators. That is why atomic and DV states cannot be mixed by the instantaneous Coulomb potential. Physically, the temporal boundary condition arising from the K R propagator is incorrect for free-particle states. As shown by Feynman [7], the propagator K F must be used for free electrons and positrons in order to be consistent with the known behavior of these particles undergoing scattering or annihilation. Otherwise, a state ψ + can be scattered into a state ψ − and be lost. Furthermore, QED requires that a state ψ − going backward in time becomes the antiparticle state ψ + going forward in time. For this reason, anomalous states Ψ An , which are propagated by K R , can only exist as bound-states Ψ DV and not as free-particles. The anomalous bound-states Ψ DV cannot dissociate into free-particles due to the temporal boundary condition imposed by K R on the solutions to (17). III. DISCRETE VARIABLE (DV) REPRESENTATION In this section, it is shown explicitly that there are four different anomalous bound-state solutions Ψ DV to (3) or the equivalent (17), for any J > 0 corresponding to four orthogonal spinor combinations of e ij Ω S Sz . In this paper only J z = 0 states are considered. These four bound-states have radially quantized wavefunctions Ψ DV = Gδ(ρ − ρ i ) for an instantaneous Coulomb potential Φ(ρ) = Φ C (ρ) with energies E = Φ(ρ i ) = −e 2 /ρ i . These quantized wavefunctions Ψ DV correspond to the DV representation as reviewed in [3]. According to DV theory, DV states can be found either numerically, by diagonalizing the potential Φ C in a complete basis set, or analytically, by using the completeness relation for the bases. It will also be shown that, if one includes the appropriate instantaneous magnetic potential Φ M = e 2 (α e ·α p )/ρ, the DV states are also eigenstates of the resulting effective potential is Φ = Φ C + Φ M . With this potential, the same four bound-states Ψ DV have energies E 1 i = 0 for the S = 1 doublet and E 0 i = 2e 2 /ρ i for the S = 0 doublet. In this section DV states are in the rest frame where P ′ = 0. The case for the DV states boosted to the moving frame where P ′ = 0 will be considered in section V. In spherical coordinates ρ = (ρ, θ, φ), one can use the momentum representation with k = π for anomalous states, for a given total angular momentum J = L + S and projection J z = M + S z . The wavefunctions Ψ DV are then comprised of states j L (kρ)Y LM (θ, φ)e ij Ω S Sz which are products of spherical Bessel functions j L (kρ), spherical harmonics Y LM (θ, φ), and spinors e ij Ω S Sz . The DV representation allows us to form bound-states Ψ DV = Gδ(ρ − ρ i ) using the completeness condition on the spherical Bessel functions j L (kρ) for the radial wavefunctions. Using the boundary condition, j J (k n ρ 0 ) = 0,(18) with the normalization N Jn = 2 ρ 3 0 j 2 J+1 (k n ρ 0 ) ≃ 2 ρ 0 k n for k n >> 0, one has the orthonormality condition N 2 Jn ρ 0 0 ρ 2 j J (k n ρ)j J (k m ρ)dρ = δ nm ,(19) and the completeness relation ∞ n=1 N 2 Jn ρ 2 j J (k n ρ)j J (k n ρ i ) = δ(ρ − ρ i ).(20) For a given J and k = π, the DV states are denoted by \|Ψ S , Jk for total spin S = 0 and S = 1. The anomalous states with K = 0, which can form bound-states, are found to be \|Ψ 0 A , Jk = ϕ 0 (Jk)(e 11 − e 22 )/ √ 2, (21a) \|Ψ 0 S , Jk = ϕ 0 (Jk)(e 12 − e 21 )/ √ 2, \|Ψ 1 1S , Jk = ϕ 1 (Jk)(e 11 + e 22 )/ √ 2, (21b) \|Ψ 1 2S , Jk = ϕ 1 (Jk)(e 12 + e 21 )/ √ 2, with the normalized functions, ϕ 0 (Jk) = N Jk j J (kρ)[Y J Ω 0 ] J 0 ,(22a)ϕ 1 (Jk) = N Jk j J (kρ)[Y J Ω 1 ] J 0 ,(22b) and the Clebsch-Gordon LS coupling, [Y L Ω S ] J 0 = M C L S J M −M 0 Y L M Ω S −M . For the pair of equations (21b), which were not considered previously in [2], one must have J > 0. The subscripts S and A for these anomalous states indicate that these wavefunctions are composed of symmetric states (X = 1) and antisymmetric states (X = −1), respectively, under particle exchange XΨ = XΨ where Ψ S = 1 2 (Ψ +− + Ψ −+ ), Ψ A = 1 2 (Ψ +− − Ψ −+ ),(23) with X being the negative of the charge conjugation C or X = −C. The exchange symmetry X and inversion parity P are given in Table I for these four anomalous states. The labels S and A are for even J only. For odd J, the labels S and A will be reversed (S ↔ A). Table I. Exchange X = −C and Parity P \|Ψ 0 A , Jk \|Ψ 0 S , Jk \|Ψ 1 1S , Jk \|Ψ 1 2S , Jk X (−1) J+1 (−1) J (−1) J (−1) J P (−1) J+1 (−1) J (−1) J+1 (−1) J The labeling of these states agrees with that of Malenfant [6]. Note that, for a given J and k in Table I, the two states Ψ 0 A and Ψ 0 S for the S = 0 doublet have different X and P and the two states Ψ 1 1S and Ψ 1 2S for the S = 1 doublet have different P . That is, neither P nor C is conserved for a given J and k. Also note that the Ψ S and the Ψ A states are their own antiparticles if one lets Ψ +− ↔ Ψ −+ . It can be readily shown that the two pairs of states \|Ψ 0 , Jk and \|Ψ 1 , Jk are each anomalous states Ψ An which obey (2). To show this, one uses the relations (σ e · π )ϕ 0 (Jk) = −kϕ α (Jk), −(σ p · π )ϕ 0 (Jk) = −kϕ α (Jk), (σ e · π )ϕ 1 (Jk) = kϕ β (Jk), −(σ p · π )ϕ 1 (Jk) = −kϕ β (Jk), where ϕ α (Jk) = iN Jk {aj J+1 (kρ)[Y J+1 Ω 1 ] J 0 + bj J−1 (kρ)[Y J−1 Ω 1 ] J 0 }, ϕ β (Jk) = iN Jk {−bj J+1 (kρ)[Y J+1 Ω 1 ] J 0 + aj J−1 (kρ)[Y J−1 Ω 1 ] J 0 }, with recoupling coefficients a = J + 1 2J + 1 , b = J 2J + 1 . These relations result in the equations for K = (α e − α p ) · π in (6) such that Note that these four equations depend only on the Pauli-and Dirac-spinor exchange properties of (α e − α p ) for the four wavefunctions in (21). For the high k >> m needed to form the DV bound-states at f ermi distances, one may ignore the negligible mass term M in (6). As a result of (24) and (6) with P ′ =0 and M = 0, the four states \|Ψ, Jk above obey the equation for anomalous states in (2) or H 0 \|Ψ, Jk = K\|Ψ, Jk = 0.(25) Combining (21) and (22), one now has the four anomalous states, \|Ψ 0 A , Jk = N Jk j J (kρ)[Y J Ω 0 ] J 0 (e 11 − e 22 )/ √ 2, (26a) \|Ψ 0 S , Jk = N Jk j J (kρ)[Y J Ω 0 ] J 0 (e 12 − e 21 )/ √ 2, \|Ψ 1 1S , Jk = N Jk j J (kρ)[Y J Ω 1 ] J 0 (e 11 + e 22 )/ √ 2, J > 0, (26b) \|Ψ 1 2S , Jk = N Jk j J (kρ)[Y J Ω 1 ] J 0 (e 12 + e 21 )/ √ 2, J > 0, which form a bases for two pairs of anomalous bound-states or DV states labeled by the Pauli-spinors S = 0 in (26a) and S = 1 in (26b). These pairs are symmetrized functions Ψ u for S = 0 and Ψ g for S = 1 as defined in (9) and are only weakly coupled by the mass operator M. It is now possible to form four different DV bound-states from the anomalous states \|Ψ, Jk in (26) using the DV representation for any potential Φ(ρ). For each of these four states, one can diagonalize the potential matrix Φ nm = Ψ, Jk n \|Φ(ρ)\| Ψ, Jk m = N Jn N Jm ρ 0 0 ρ 2 j J (k n ρ)Φ(ρ)(j J (k m ρ)dρ,(27) to find the DV representation numerically. One may also find the DV states analytically by using the truncated completeness relation for a finite basis set. For a finite bases set with N different k n in (18), one finds, analytically, the approximate radial delta functions, R i (ρ) = D N n=1 N 2 Jn ρ 2 j J (k n ρ)j J (k n ρ i ) ≃ Dδ(ρ − ρ i ),(28) where the ρ i (i = 1, 2, ..., N) are at the N zeros of the Bessel function j J (k N +1 ρ). The normalization D depends on ρ i and the associated grid spacing ∆ρ i . Note for high N one has δ(0) ≃ 1/∆ρ and D 2 δ 2 (ρ − ρ i )dρ ≃ D 2 δ 2 (0)∆ρ ≃ D 2 /∆ρ = 1. One can then use the approximations, ∆ρ ∼ ρ 0 /N, ρ i ∼ i∆ρ, D ∼ ∆ρ.(29) From (3), the corresponding energies are E i ≃ Φ (ρ i ) .(30) For k >> m when ρ 0 ∼ f ermi, one can ignore the mass coupling term M in (6) and (10) which justify the assumption made previously to obtain (26). It will be convenient to define Ω 1 + = 1 2 (Ω 1 −1 e iφ + Ω 1 1 e −iφ ),(31)Ω 1 − = 1 2 (Ω 1 −1 e iφ − Ω 1 1 e −iφ ), and expand the LS coupling in terms of the (normalized) Associated Legendre Polynomials [8] A J 0 P J 0 (cos θ) and A J 1 P J 1 (cos θ), [Y J Ω 0 ] J 0 = A J 0 P J 0 (cos θ)Ω 0 0 / √ 2π, [Y J Ω 1 ] J 0 = A J 1 P J 1 (cos θ)Ω 1 + / √ 2π. For a given J, one obtains the DV bound-states Ψ DV from (26) and (28) in terms of the Pauli-and Dirac-spinors, \|Ψ 0 A , Jρ i = ∆ 0 i Ω 0 0 (e 11 − e 22 )/ √ 2, (32a) \|Ψ 0 S , Jρ i = ∆ 0 i Ω 0 0 (e 12 − e 21 )/ √ 2, \|Ψ 1 1S , Jρ i = ∆ 1 i Ω 1 + (e 11 + e 22 )/ √ 2,(32b)\|Ψ 1 2S , Jρ i = ∆ 1 i Ω 1 + (e 12 + e 21 )/ √ 2, for i = 1, 2, ..., N in (28). The factors ∆ i for the (ρ, θ) dependence, for a given J and ρ i , are ∆ 0 i ≃ D 0 i δ(ρ − ρ i )A J 0 P J 0 (cos θ)/ √ 2π,(33)∆ 1 i ≃ D 1 i δ(ρ − ρ i )A J 1 P J 1 (cos θ)/ √ 2π, with the normalization A J 0 and A J 1 determined on the interval [−1, 1] for cos θ, such that A J 0 = (2J + 1) 2 , A J 1 = − (2J + 1) 2J(J + 1) . The divisor √2π A. Coulomb Potential It is instructive to first use a Coulomb potential, Φ C = −e 2 /ρ, in order to examine the wavefunctions and energies of the four Ψ DV states using DV theory. An example using DV theory for J = 10 is shown in Fig. 1 for the DV normalized wavefunctions R 2 i (ρ) and in B. Magnetic Potential Using the Coulomb potential Φ C (ρ) = −e 2 /ρ in (1) for the DV states, one obtains the energies E i = −e 2 /ρ for all four Ψ DV in (32). However, one still needs to include the appropriate magnetic potential Φ M so that the total Coulomb potential Φ is then Φ=Φ C + Φ M .(34) The appropriate magnetic potential will depend on the gauge used. While, in theory, results should be gauge independent, in practice, different bases have different QED convergence characteristics which make some gauge choices impractical. Φ B = e 2 2ρ [α e · α p + (α e · ρ) (α p · ρ)], Φ Atom = Φ C + Φ B = Φ C {1 − 1 2 [α e · α p + (α e · ρ) (α p · ρ)]}, which results in fine structure corrections to the Coulomb energies for positronium. Using this magnetic potential and the Pauli approximation restriction to a Ψ ++ bases, the fine structure energies for positronium have been found analytically to order mα 4 by Ferrell [9], [10]. Also, Fulton and Martin [11] have used the BSE for the Ψ Atom bases with various two-body QED kernels in addition to the Coulomb potential to calculate the energies of positronium to order mα 5 . In agreement with the BSE (16) highly relativistic particles. Accordingly, one must use the DV potential Φ DV with the instantaneous magnetic potential of Gaunt Φ G [12], so that Φ G = e 2 ρ α e · α p ≡ e 2 ρ (α ex α px + α ey α py + α ez α pz ),(35)Φ DV = Φ C + Φ G = Φ C α 2 0 , α 2 0 = (1 − α e · α p ). This total potential has also been derived by Barut and Komy [13] using the action principal. The operator α e ·α p only acts on the Pauli-and Dirac-spinors of Ψ DV in (32) and it is shown below that the DV states are eigenstates of α 2 0 in (35) as well as the potential Φ C . Barut and Komy have shown that this instantaneous effective potential is appropriate for the retarded propagator K R which is the propagator used for the DV states in (17). As described in the Appendix, the DV potential Φ DV = Φ C α 2 0 is related to the Lorentz potential Φ L = Φ C γ 2 0 ,(36) where γ 2 0 = γ eu γ pu is the scalar product of γ eu and γ pu . The correct potential for the BSE equation in (17), which uses the K R propagator, should be multiplied on the left by the factor α 2 0 . One can evaluate the Pauli-spinor operator σ e · σ p in the magnetic potential for states in (32) using (σ e · σ p )Ω 0 0 = −3Ω 0 0 ,(37) (σ e · σ p )Ω 1 Sz = Ω 1 Sz , for S z = −1, 0, 1. Evaluating the operator α e · α p for Dirac-and Pauli-spinor states gives (α e · α p )(e 11 ± e 22 )Ω 0 0 = ∓3(e 11 ± e 22 )Ω 0 0 ,(38) (α e · α p )(e 12 ± e 21 )Ω 0 0 = ∓3(e 12 ± e 21 )Ω 0 0 , (α e · α p )(e 11 ± e 22 )Ω 1 Sz = ±(e 11 ± e 22 )Ω 1 Sz , (α e · α p )(e 12 ± e 21 )Ω 1 Sz = ±(e 11 ± e 22 )Ω 1 Sz , so that these spinor states are all eigenfunctions of α e · α p . For the DV states in (32), one then finds Φ DV \|Ψ 0 , Jρ i = E 0 i \|Ψ 0 , Jρ i ,(39)E 0 i = M 0 i = 2e 2 /ρ i , Φ DV \|Ψ 1 , Jρ i = E 1 i \|Ψ 1 , Jρ i , E 1 i = M 1 i = 0. These DV eigenstates of Φ DV no longer have negative energies and are, therefore, physically allowed. One can divide the magnetic Gaunt potential into its longitudinal and transverse components, Φ G = Φ GL + Φ GT ,(40) where Φ GL = e 2 ρ (α ez α pz ), Φ GT = e 2 ρ (α ex α px + α ey α py ). Interestingly, one has for all DV states in (32), α ez α pz \|Ψ, Jρ i = \|Ψ, Jρ i ,(42) so that (Φ C + Φ GL )\|Ψ, Jρ i = 0,(43)Φ DV \|Ψ, Jρ i = Φ GT \|Ψ, Jρ i . This means that, for the DV bound-states, only the transverse magnetic potential Φ GT of Gaunt determines the energies in (39). On the other hand, for the atomic states, it is the Coulomb potential which is dominant. Summarizing, the DV states \|Ψ 0 , Jρ i for Ω 0 0 form 'heavy' doublets with energy E 0 i = 2e 2 /ρ i and the DV states \|Ψ 1 , Jρ i for Ω 1 + form 'light' doublets with E 1 i = 0. The four degenerate states in (32) with Coulomb energies E i = −e 2 /ρ i are now split by the magnetic potential into two different doublets. IV. CLASSICAL CORRESPONDENCE For clarity, the constant c is now shown in this section. The different behaviors between the atomic and anomalous states all depend on the fact that, for anomalous states, either the electron or positron is in a negative energy state ψ − (e) or ψ − (p), respectively. It is possible to understand the important properties of the DV states by using both quantum and classical principals. In particular, it is now shown that the four anomalous bound-states Ψ DV are both stable and dark, as was the case for the J = 0 bound-states in [2]. For a free-particle with energy E = ± \|E\| and momentum p, one obtains the important result [14] for the expectation value of the Dirac matrices α k , α k = p k c/E = ±p k c/ \|E\| = v k /c.(44) In other words, the velocity v is in the direction opposite to the momentum p for negative energy states of free-particles. In the P ′ = 0 frame, where p e = −p p , one finds that v e = v p because either the electron or the positron is in a negative energy state. The difference in velocity components is then (α e − α p ) k = 0.(45) One can now understand why kinetic energy K is zero in (24) despite the fact that the relativistic momentum π is large, π = k >> mc. One has K = c (α e − α p ) · π , = c (α e − α p ) k π k = 0. Classically, (44) corresponds to m = − \|m\| when E = − \|E\| and for p e = −p p one also finds that v e = v p for both the anomalous states Ψ +− or Ψ −+ . With this in mind, one can also understand why the DV states are delta functions with Ψ DV = Gδ(ρ − ρ i ). From the Heisenberg uncertainty principal, these delta functions correspond to very high momentum states where π >> mc. Because the relative velocity is zero, v e − v p = 0, the classical particles will remain at the same arbitrary distance ρ = ρ i . Also, the electron and positron particles can never annihilate because they can never collide. Finally, two particles moving at the same velocity can neither radiate nor absorb light. This later property is also clear from quantum considerations, given that the expectation value of the Dirac transition operator W µ is W µ = −e (α e − α p ) · 1 µ = 0,(46) where µ = z for longitudinal and µ = x, y for transverse radiation. It is also apparent that an electron and positron moving with the same velocity can keep the same time so that one may let t e = t p = t. For this reason, there will be no difficulties with the DV solutions of the BSE in (17) that arise from the particles keeping different times. The total time T is the fourth component of R in (5) and is conjugate to the total energy E. One sees that the total time T and the relative time t are the same, T = t e + t p 2 = t.(47) In many respects, the solutions to the BSE (17) for anomalous bound-states Ψ DV are simpler than the solutions to the BSE (16) for the atomic states Ψ Atom . Finally, one can readily determine the classical instantaneous electromagnetic interaction between electron and positron point particles with angular momentum J = 0. One finds the magnetic force F M is F M = − e 2 ρ 2 c 2 [v e × (v p × ρ)](48) for velocities v ≪ c. In the center of mass frame with R = 0, let the classical velocities be perpendicular to r e = ρ/2 and r p = −ρ/2 for J = 0 so that [v e × (v p × ρ)] = (v e · ρ)v p − (v e ·v p ) ρ = −(v e ·v p ) ρ, and F M = e 2 ρ 2 c 2 (v e ·v p ) ρ.(49) One finds that the force F M is repulsive in the ρ direction for the electron and positron moving in the same direction as required. The electromagnetic potential for J = 0 is then Φ = Φ C + Φ M = − e 2 ρ (1 − v e ·v p c 2 ).(50) For relativistic velocities, the retarded Lienard-Wiechert potential preserves the ratio Consider a new frame in which the DV states are moving with velocity V ′ in the Z direction so that the DV states have total momentum P ′ = P ′ Z relative to the rest frame. One can also, without loss of generality, define the z direction to be in the direction of this momentum so that the components of spins s z will be defined in the z = Z direction. Φ C /Φ M = −v e ·v p /c 2 . Letting v e = v p = c where Φ C /Φ M = −1, One can transform the DV states Ψ i and potential Φ DV = Φ C α 2 0 (35) to the moving frame using both the two-body Lorentz boost L 2 of the Dirac spinors and the Lorentz contraction of the z coordinate. In the Appendix, some useful identities (A7) and (A10) for the conjugate expectations of α 2 0 , Φ C α 2 0 , and Φ ′ C α 2 0 are Ψ i α 2 0 Ψ i = Ψ ′ i α 2 0 Ψ ′ i ,(51)Ψ i \|Φ C α 2 0 \|Ψ i = M i , Ψ ′ i \|Φ ′ C α 2 0 \|Ψ ′ i = M ′ i , where M i is the binding energy of Ψ i in the rest frame and M ′ i is the binding energy of Ψ ′ i in the moving frame. Only if this binding energy remains constant, such that M ′ i = M i , can one demonstrate that the DV states transform like single-particle fermions. As shown below, the mass M ′ i is a constant in the moving frame for all Ψ ′1 i DV states in (32), but only for a special case of the DV states Ψ ′0 i which corresponds to the lowest energy state. The Lorentz boost has the properties Ψ ′ = L 2 Ψ, Φ ′ DV = L 2 Φ DV L −2 ,(52)L −2 = γ 2 4 L 2 γ 2 4 , γ 2 4 = γ e4 γ p4 , where L −2 is the inverse transform of L 2 . These transformations involve some difficulties arising from dynamical symmetry breaking and the fact that L 2 is not unitary, which are now addressed. As shown by (32) and (33), the DV wavefunctions Ψ 0 i and Ψ 1 i for a given J can be separated into two factors. One factor consist only of the Dirac spinors Ω S Sz e ij and the other factor consists only of the coordinate functions ∆ i (ρ, θ). The Lorentz boost operates on the Dirac spinors Ω S Sz e ij , whereas the Lorentz contraction operates on the functions ∆ i (ρ, θ). Similarly, for the potential Φ DV , the Lorentz boost operates on the factor α 2 0 and the Lorentz contraction operates on the factor Φ C = −e 2 /ρ. The Lorentz contraction of ρ in ∆ i (ρ, θ) and Φ C will be considered first. Because of the delta function factor δ(ρ − ρ i ) (33), the ∆ i (ρ, θ) in the rest frame are eigenstates of 1/ρ in Φ C such that 1 ρ ∆ i (ρ, θ) = 1 ρ i ∆ i (ρ, θ).(53) However, as a result of the Lorentz contraction of z i in the potential, z ′ i = z i /γ, the separations ρ ′ i are no longer spherically symmetric. Because of this dynamical symmetry breaking, the potential Φ C becomes cylindrically symmetric and ρ ′ i depends on θ ′ i . This means that ∆ ′ i (ρ ′ , θ ′ ) is no longer an eigenstate of the 1/ρ ′ . This problem can be remedied by using delta functions δ(cos θ − cos θ i ) in ∆ i formed from the degenerate P J 0 (cos θ) and P J 1 (cos θ) in (33) without any effect on the energies E i . Letting ζ = cos θ, the delta functions for the DV states directed at angle θ = θ i are given by    [δ(ζ − ζ i ) + δ(ζ + ζ i ) J = even [δ(ζ − ζ i ) − δ(ζ + ζ i ) J = odd    = J (2J + 1)P J 0 (ζ i )P J 0 (ζ),(54)   [δ(ζ − ζ i ) + δ(ζ + ζ i ) J = odd [δ(ζ − ζ i ) − δ(ζ + ζ i ) J = even    = J 2J + 1 J(J + 1) P J 1 (ζ i )P J 1 (ζ). One must keep in mind that the peaks of the delta functions ζ i are not the same for the symmetric and antisymmetric delta functions above because the peaks of P J 0 (ζ i ) are interleaved for J = even and J = odd and similarly for P J 1 (ζ i ). For example, the symmetric delta functions can have a peak at ζ i = 0 whereas the antisymmetric delta functions cannot. This means that when ζ i = 0, the Ψ 0 i DV state has J = even and the Ψ 1 i DV state has J = odd. In (33), one now has, approximately, For a given velocity V ′ with the Lorentz factor γ, one has the following relations for the Lorentz contraction of z, using the cylindrical coordinates (r i , z i ), ∆ 0 i ≃ D 0 i δ(ρ − ρ i )[δ(ζ − ζ i ) ± δ(ζ + ζ i )]/ √ 4π,(55)∆ 1 i ≃ D 1 i δ(ρ − ρ i )[δ(ζ − ζ i ) ± δ(ζ + ζ i )]/ρ ′ i cos θ ′ i = z ′ i = z i /γ = ρ i cos θ i /γ, ρ ′ i sin θ ′ i = r ′ i = r i = ρ i sin θ i . Solving these equations, one finds ρ ′ i = (ρ i /γ) γ 2 − cos 2 (θ i )(γ 2 − 1),(56)cos θ ′ i = ρ i cos θ i /(ρ ′ i γ). The new factors ∆ ′ i for Ψ ′ i , resulting from the Lorentz contraction, are then simply ∆ 0′ i ≃ D 0′ i δ(ρ ′ − ρ ′ i )[δ(ζ ′ − ζ ′ i ) ± δ(ζ ′ + ζ ′ i )]/ √ 4π,(57)∆ 1′ i ≃ D 1′ i δ(ρ ′ − ρ ′ i )[δ(ζ ′ − ζ ′ i ) ± δ(ζ ′ + ζ ′ i )]/ √ 4π, for DV states localized at coordinates (ρ ′ i , θ ′ i ) given by (56) where ζ ′ i = cos θ ′ i . With this simple procedure, the factors ∆ ′ i for Ψ ′ i are now eigenstates of 1/ρ ′ 1 ρ ′ ∆ ′ i (ρ ′ , θ ′ ) = 1 ρ ′ i ∆ ′ i (ρ ′ , θ ′ ).(58) Now consider the Lorentz boost of K, Φ DV , and the DV states Ψ i . In the Appendix it is shown, using adjoint spinors, that Ψ 0′ i \|K ′ \|Ψ 0′ i = 0, so the transformed wavefunctions are also DV states. For the potential, one can also verify from (39), (51), and (58) that Ψ 0′ i \|Φ ′ C α 2 0 \|Ψ 0′ i = M 0′ i = 2e 2 /ρ ′ i ,(59)Ψ 1′ i \|Φ ′ C α 2 0 \|Ψ 1′ i = M 1′ i = 0. For the Ψ 1′ i DV states, one has M 1′ i = M 1 i = 0 so that β ′ = 1 and the velocity V ′ will be the speed of light. For these light DV states, one has γ ′ → ∞ and z ′ i = 0 such that ρ ′ i = ρ i = r i and they form a disk perpendicular to the direction of motionẑ = Z. In this case the potential expectation value Φ DV is a Lorentz invariant as required for these DV states to transform like a single-particle fermion. For the Ψ 0′ i DV states, one has M 0′ i = 2e 2 /ρ ′ i = M 0 i when z i = 0 and the masses are not Lorentz invariant. In fact one finds from (56), for a given ρ i , that M 0′ i is a maximum when θ i = 0 or θ i = π where ρ ′ i = ρ i /γ and M 0′ i = γM 0 i . One also finds from (56), for a given ρ i , that M 0′ i is a minimum when θ i = π/2 (z i = 0) where ρ ′ i = ρ i and M 0′ i = M 0 i . In general one finds that γM 0 i ≥ M 0′ i ≥ M 0 i for a given ρ i . This is a consequence of the fact that, for a given ρ i , the Lorentz contraction will bring the electron and positron closer together for any angle θ i unless θ i = π/2. Because the potential is repulsive, this will then raise the binding energy and mass M 0′ i unless θ i = π/2. For the special case where θ i = π/2 (z i = 0), the DV states Ψ 0′ i form a disk perpendicular to the direction of motionẑ = Z such like the DV states Ψ 1′ i . One would expect that this particle would reside in the lowest energy DV state of Ψ 0′ i with θ i = π/2, where the mass is invariant. The fact that such DV doublets Ψ i with M 0′ i = M 0 i transform like single-particle fermions will now be shown. Having transformed the coordinate factors ∆ i (ρ, θ) for the DV states Ψ 0 i and Ψ 1 i for J = (even, odd) as shown in (57), it remains to transform their Dirac spinors Ω S Sz e ij to the frame moving with velocity V ′ . Brodsky and Primack [15] have shown, using the BSE for atomic hydrogen, that the Lorentz boost for P ′ mixes the four different Pauli-spinors Ω S Sz among themselves and the four different Dirac-spinors e ij among themselves. The Lorentz boost for a single-particle fermion in the z direction with velocity α ′ z = V ′ is L = γ ′ + 1 2 + γ ′ − 1 2 α ′ z .(60) For the DV states (32), with ∆ i given (55), including the plane-wave function f , the box-normalized wavefunction in the rest frame is \|Ψ, Jρ i f = \|Ψ, Jρ i e −iM i T / Z 0 . One can boost the DV state with velocity V ′ so that the total momentum is P ′ . The two-body Lorentz boost L 2 is the direct product of the Lorentz boosts L e × L p such that L 2 = L e × L p = γ + 1 2 + γ − 1 2 α ez × γ + 1 2 + γ − 1 2 α pz , where γ corresponds to the individual particle boosts ±β. Letting γ 2 − 1 = βγ and using (42) when operating on the DV states, one has L 2 = γ + 1 2 + γ − 1 2 α ez α pz + βγ (α ez + α pz ) 2 ,(61)= γ+βγα ′ z , where α ′ z = (α ez + α pz ) 2 ,(62) as in (6). In the frame where π = 0, one has p e = p p so that the velocity of the electron and positron are equal and opposite, v e = −v p , where either the electron or the positron is in a negative energy state. Transforming to either the electron or positron frame, one can add these velocities relativistically, so that V ′ = β ′ = ±2β/(1 + β 2 ). Then the two-body Lorentz boost (61) is equivalent to L 2 = γ ′ + 1 2 + γ ′ − 1 2 α ′ z ,(63) where γ ′ = (1 − β ′ 2 ) − 1 2 = (1 + β 2 )/(1 − β 2 ). But this is just the boost (60) with velocity V ′ = β ′ where the rest mass is now M ′ i in (59) such that V ′ = β ′ = P ′ /E ′ i ,(64)E ′ i = P ′2 + M ′2 i , γ ′ = E ′ i /M ′ i , β ′ γ ′ = P ′ /M ′ i . In general, because ρ ′ i = ρ i , the binding energy has changed in the moving frame so that M ′ i = M i and the particle cannot be considered a single-particle fermion. One can now use L 2 in (63) The transformed wavefunctions Ψ ′ i , after renormalization, are Ψ 0′ i,A = 1 2 ∆ 0′ i γ ′ + 1Ω 0 0 (e 11 − e 22 ) − γ ′ − 1Ω 1 0 (e 12 − e 21 ) / γ ′ ,(67)Ψ 0′ i,S = 1 2 ∆ 0′ i γ ′ + 1Ω 0 0 (e 12 − e 21 ) − γ ′ − 1)Ω 1 0 (e 11 − e 22 ) / γ ′ , Ψ 1′ i,1S = 1 2 ∆ 1′ i {Ω 1 + (e 11 + e 22 ) − Ω 1 − (e 12 + e 21 )}, Ψ 1′ i,2S = 1 2 ∆ 1′ i {Ω 1 + (e 12 + e 21 ) − Ω 1 − (e 11 + e 22 )}, where the ∆ ′ i are given by (57). The renormalization factor 1/γ ′ comes from the Lorentz contraction Z ′ 0 = Z 0 /γ ′ of the transformed plane-wavefunction, f ′ = e i(P ′ Z ′ −E ′ i T ′ ) / Z ′ 0 .(68) Using (39) and (6.20), one can readily verify (59). One can also verify directly that the DV states in (67) are solutions to the one-body Dirac equation (6), (α ′ z P ′ Z + M ′ γ 2 4 )f ′ \|Ψ ′ i = E ′ i f ′ \|Ψ ′ i ,(69) where M ′ γ 2 4 = Φ ′ C α 2 0 and E ′ i = P ′ Z + (M ′ i ) 2 as shown in the Appendix. To confirm that the DV states with z i = 0 are single-particle fermions, one has ρ ′ = ρ and Φ ′ C = Φ C for these special DV states, so that M ′ = M. For these 'pancake' wavefunctions with z i = z ′ i = 0, the DV states for S z = ±1 and S z = 0 then transform like fermion doublets in which the Lorentz contraction in z has no effect. Looking more closely at the spinor states of (67), one finds that, instead of the expected single-particle doublet Pauli-spinors (χ1 VI. CHIRALITY χ AND HELICITY h OF THE LIGHT FERMIONS The light DV fermions Ψ 1′ i = Ψ 1′ i,A with z i = 0 in (67) have A exchange symmetry for J = odd corresponding to the symmetric delta functions in ∆ 1′ i,S (57) and the e ±iφ dependence in (31). These Ψ 1′ i DV states have well defined chirality and helicity because the rest mass is M 1′ i = 0. The chirality operator χ = −γ 5 for a single-particle operates on the Dirac-spinors such that, for any i = 1, 2, χ e e 1i = e 2i , χ p e i1 = e i2 , and their Hermitian conjugates. When operating on these DV states in (67), one can define the chirality operator χ ′ as χ ′ = (χ e + χ p ) 2 ,(70) where χ ′ Ψ 1′ i = χ e Ψ 1′ i = χ p Ψ 1′ i . The helicity operator h = Σ· P ′ = Σ z for a single-particle operates on the Pauli-spinors such that h e Ω 1 1 = Ω 1 1 , h e Ω 1 −1 = −Ω 1 −1 , h p Ω 1 1 = Ω 1 1 , h p Ω 1 −1 = −Ω 1 −1 . Again, when operating on the DV states (67), one can define the helicity operator h ′ for the light fermions as h ′ = (h e + h p ) 2 ,(71)where h ′ Ψ 1′ i = h e Ψ 1′ i = h p Ψ 1′ i . The eigenfunctions of both χ ′ and h ′ are then Ψ 1′ i,+ = (Ψ 1′ i,1A + Ψ 1′ i,2A )/ √ 2 (72) = 1 2 ∆ 1′ i,S {(e 11 + e 22 + e 12 + e 21 )Ω 1 1 e −iφ }, Ψ 1′ i,− = (Ψ 1′ i,1A − Ψ 1′ i,2A )/ √ 2 = 1 2 ∆ 1′ i,S {(e 11 + e 22 − e 12 − e 21 )Ω 1 −1 e iφ },such that χ ′ Ψ 1′ i,± = χ ′ Ψ 1′ i,± and h ′ Ψ 1′ i,± = h ′ Ψ 1′ i,± . These states Ψ 1′ i,± have chirality χ ′ = ±1 and helicity h ′ = ±1, respectively, corresponding to right-and left-handed chirality and to right-and left-handed helicity. As in the case of a single-particle fermion with no mass, the chirality and helicity of these light DV Majorana fermions have the same signs. It has also been shown that the properties of the anomalous bound-states and the atomic bound-states differ radically because, for the former, either the electron or the positron must be in a negative energy state. It is instructive to compare and contrast the properties of the atomic and DV bound-state solutions for positronium, as they are complementary. The atomic bound-states in the rest frame have low relative momentum π << m, and can be treated using the Coulomb gauge, whereas the DV bound-states in the rest frame have very high relative momentum π >> m and must be treated relativistically with the fermions. It appears that the properties of the DV fermions are consistent with those of light and heavy neutrinos. One can then hypothesize that the DV states corresponding to the S z = ±1 light fermions are electron neutrinos and the DV states corresponding to the S z = 0 heavy fermions are 'sterile' neutrinos. The 'sterile' type of neutrino has been surmised by some particle physicists as an explanation for dark matter [16]. The sterile neutrino would then have a mass of ∼ MeV if the electron and positron were bound at nuclear distances This invariant form of the expectation should be used when transforming to the moving frame. For example, using (A3) for the kinetic operator K, one finds that, for the DV states, Ψ ′ \|K ′ \| Ψ ′ = Ψ \|K\| Ψ = 0,(A4) so that the transformed wavefunctions in (32) are still anomalous states with zero kinetic energy. The DV potential can be written Φ DV = − e 2 ρ (1 − α e · α p ) = Φ C α 2 0 , where Φ C is the Coulomb potential. The operators α 2 0 and γ 2 4 commute so that α 2 0 γ 2 4 = γ 2 4 α 2 0 . To find the transform properties of the DV potential, one must consider the Lorentz invariant operator γ 2 0 = γ eu γ pu , which is the scalar product of the two four-vectors γ eu and γ pu . This scalar operator transforms such that γ ′2 0 = γ ′ eu γ ′ pu = γ eu γ pu = γ 2 0 ,(A5) or equivalently, γ ′2 0 = γ ′2 4 α ′2 0 = γ 2 4 α 2 0 = γ 2 0 .(A6) Unlike the operator γ 2 0 , the operator α 2 0 is not a Lorentz invariant as seen in (A6). From the above equations one finds that Ψ γ 2 0 Ψ = Ψ γ 2 4 α 2 0 Ψ = Ψ α 2 0 Ψ , Ψ ′ γ ′2 0 Ψ ′ = Ψ ′ γ 2 4 α 2 0 Ψ ′ = Ψ ′ α 2 0 Ψ ′ . Letting Ψ \|γ 2 0 \| Ψ = Ψ ′ \|γ ′2 0 \| Ψ ′ , one finds the useful result Ψ α 2 0 Ψ = Ψ ′ α 2 0 Ψ ′ . The instantaneous Lorentz potential Φ L is given by Φ L = Φ C γ 2 0 = γ 2 4 Φ DV . This potential Φ L can be derived in the momentum representation from the invariant potential used by Salpeter [5], G(k u ) = −e 2 (γ 2 0 /k 2 0 ), k 2 0 = k · k − ̟ 2 . For the instantaneous Lorentz potential in the momentum representation, let ̟ 2 = 0. Now let the two-body mass operator for the DV states be defined by Mγ 2 4 = Φ L γ 2 4 = Φ C α 2 0 = Φ DV ,(A8) so that the two-body mass operator is the analogy of the one-body mass operator mγ 4 . One must then use the two-body mass operator, M ′ γ 2 4 = Φ ′ L γ 2 4 = Φ ′ C α 2 0 ,(A9) when finding the mass M ′ in the moving frame in analogy with the one-body case. From (A8) and (A9), one finds that the adjoint expectation of the Lorentz potential Φ L and Φ ′ L is M = Ψ \|Φ L \| Ψ = Ψ\|Φ C α 2 0 \|Ψ ,(A10)M ′ = Ψ ′ \|Φ ′ L \| Ψ ′ = Ψ ′ \|Φ ′ C α 2 0 \|Ψ ′ . The Lorentz potential is a Lorentz invariant when ρ ′ = ρ or when Φ ′ C = Φ C . Combining the result (A7) with the Lorentz contraction of ρ to ρ ′ in the moving frame allows one to find M and M ′ in (A10). Using M in (A10) and (53) confirms (39). Using M ′ in (A10) and (58) confirms (59), where M 0′ i = Ψ 0′ i \|Φ ′ C α 2 0 \|Ψ 0′ i = 2e 2 /ρ ′ i ,(A11)M 1′ i = Ψ 1′ i \|Φ ′ C α 2 0 \|Ψ 1′ i = 0. FigureFig. 1 1The R 2 i (ρ) DV wavefunctions for J=10 and N=40 with ρ 0 = 10 −4 bohr. The numerical wavefunctions are calculated by diagonalizing the potential matrix Φ nm in (27) and the analytic wavefunctions are found from (28) and (29). γ e4 Ψ = (Ψ 11 e 11 + Ψ 12 e 12 − Ψ 21 e 21 − Ψ 22 e 22 ), γ p4 Ψ = (Ψ 11 e 11 − Ψ 12 e 12 + Ψ 21 e 21 − Ψ 22 e 22 ). for the two-body Green's functions. For the Feynman propagator K F , the positive energy states are propagated forward in time, but the negative energy states are propagated backward in time. For the retarded propagator K R , both positive and negative energy states are propagated forward in time. For the two Kϕ 0 ( 0Jk)(e 11 − e 22 ) = 0, (24a) Kϕ 0 (Jk)(e 12 − e 21 ) = 0. Kϕ 1 (Jk)(e 11 + e 22 ) = 0, (24b) Kϕ 1 (Jk)(e 12 + e 21 ) = 0. FIG. 1 : 1is the normalization of the φ bases and the normalization D i depend on the width ∆ρ i of the delta function δ(ρ − ρ i ). For a finite basis set the delta functions δ(ρ − ρ i ) for ∆ i are only approximate, as indicated in (33). The R 2 i (ρ) DV wavefunctions for J=10 and N=40 with ρ 0 = 10 −4 bohr. The numerical wavefunctions are calculated by diagonalizing the potential matrix Φ nm in (27) and the analytic wavefunctions are found from (28) and (29). For clarity, only every fourth wavefunction is shown. Fig. 2 FIG. 2 : 22for the DV energies E i using a basis set with N = 40 and ρ 0 = 10 −4 bohr. The wavefunctions and energies can be found analytically from (28), (29), and (30), using the known zeros ρ i of j 10 (k 41 ρ), or can be found numerically by the diagonalization of the matrix Φ nm from (27). These figures show that the analytic and numerical results are in agreement except for low ρ i where the analytic approximations (29) fail for high J and low ρ i . The numerical energies E i are found by diagonalizing the potential matrix Φ nm in (27) with J=10 and N=40. The ρ i are zeros of j 10 (k 41 ρ). For the atomic positronium states where k << m, one finds that the expectation value of the Dirac operators α e = − α p is of order α = e 2 / c ∼ 1/137. Here the weak-weak to strong-strong component ratio Ψ 22 /Ψ 11 is of order α 4 . For the atomic calculations, it is most convenient to use the Coulomb gauge with the Breit magnetic potential Φ B (derived from second order perturbation theory), so that , they found it necessary to omit the Ψ +− and Ψ −+ anomalous states. Also, unlike the TBDE, it is necessary to use the negative of the Coulomb potential −Φ C in Λ Atom for the negative energy states Ψ −− . For the DV positronium states in (32), where m << k, one finds that the expectation value of the Dirac operators α e = α p → 1 when M → 0 in (6) and the components Ψ 11 and Ψ 22 are equal in magnitude as shown in (32). Because of these extremely relativistic states, QED covariant perturbation theory is not convergent and diagonalization of the potential is necessary. Indeed, one finds that the expectation values of the magnetic and electromagnetic potentials become comparable in magnitude as in the classical case for the classical and quantum potential is then Φ = 0 as in (39) for the S = 1 case. Note that the quantum potential (35) can be derived from (50) by replacing v e ·v p /c 2 with α e · α p . This quantum potential Φ DV = Φ C α 2 0 is valid for all velocities and is in agreement with Barut and Komy [13] using the retarded K R propagator. It's transformation to a moving frame is shown in the Appendix. V. LORENTZ BOOSTS, DYNAMICAL SYMMETRY BREAKING, AND MAJO-RANA FERMIONS 4π, from (54). These new functions ∆ i are localized at the coordinates (ρ i , θ i ) and remain eigenstates of 1/ρ (53) because ρ is independent of θ. to transform the DV states in (32) with ∆ i in (55) and confirm the results above in (59). Operating with α ez and α pz in (62), one finds thatα ez Ω 0 0 (e 11 − e 22 ) = α pz Ω 0 0 (e 11 − e 22 ) = −Ω 1 0 (e 12 − e 21 ),(65)α ez Ω 0 0 (e 12 − e 21 ) = α pz Ω 0 0 (e 12 − e 21 ) = −Ω 1 0 (e 11 − e 22 ), α ez Ω 1 + (e 11 + e 22 ) = α pz Ω 1 + (e 11 + e 22 ) = −Ω 1 − (e 12 + e 21 ), α ez Ω 1 + (e 12 + e 21 ) = α pz Ω 1 + (e 12 + e 21 ) = −Ω 1 − (e 11 + e 22 ), (and their Hermitian conjugates) so that α ′ z \|Ψ, Jρ i = α ez \|Ψ, Jρ i = α pz \|Ψ, Jρ i . one now has the doublet Diracspinors {(e 11 − e 22 ), (e 12 − e 21 )} for Ψ 0′ and {(e 11 + e 22 ), (e 12 + e 21 )} for Ψ 1′ , respectively.Also, one finds that, instead of the expected single-particle Dirac-spinors (e 1 , e 2 ), one now has the Pauli-spinors (Ω 0 0 , Ω 1 0 ) for Ψ 0′ and (Ω 1 + , Ω 1 − ) for Ψ 1′ . For a given DV bound-state doublet, the spin states s z (e) and s z (p) are completely correlated and act like a single-particle spin state s z . That is, for the Pauli-spinors (Ω 0 0 , Ω 1 0 ), one has s z (e) = −s z (p), whereas for the Pauli-spinors (Ω 1 + , Ω 1 − ), one has s z (e) = s z (p). Summarizing, the directed DV doublets with coordinates r = r i and z i = 0 in the rest frame transform like a single-particle fermion when undergoing a Lorentz boost because ofdynamical symmetry breaking. The rest mass for the heavy fermions is M 0′ i = M 0 i = 2e 2 /r i and rest mass for the light fermions is M 1′ i = M 1 i = 0. As shown by (54) for z i = 0, the Ψ 0′ DV states must have J = even and the Ψ 1′ DV states must have J = odd. However, for both of these fermions, the role of Pauli-spinors and Dirac-spinors has been reversed because either the electron or positron is in a negative energy state. The S z = 0 fermions in (67) have both S and A symmetry under exchange whereas the S z = ±1 fermions have only A symmetry. Most importantly, because these DV states are comprised of either S or A symmetry, they are their own antiparticles and the light and heavy DV doublets are Majorana fermions. been shown that the solutions of the Bethe-Salpeter equation (BSE) for the atomic bound-states of positronium in (16) and for the anomalous bound-states of positronium in (17), result in two entirely different forms of positronium. One is the normal atomic form of positronium in which the electron and positron are bound at atomic distances (bohr), and the other is the anomalous form of positronium in which the particles can be bound at nuclear distances (f ermi). Such anomalous bound-states are called discrete variable (DV) states because they form a bases for the DV representation in which the relative coordinates (ρ, θ) between the electron and positron are quantized at discrete values (ρ i , θ i ). For the atomic states, the BSE stipulates that the negative energy states must propagate backward in time with K F , whereas, for the DV states, the BSE stipulates that the negative energy states must propagate forward in time with K R . For the DV states, only bound-state solutions are allowed because of the time behavior of the negative energy states, and these bound-states cannot dissociate. Feynman gauge. For the atomic free-states, there is no potential energy and the momenta are quantized. For the DV bound-states, there is no relative kinetic energy and the relative coordinates are quantized. The atomic states are bound mainly by the Coulomb potential whereas the DV bound-states are bound only by the transverse magnetic potential. The atomic states of positronium are unstable and decay quickly into photons, whereas the DV states of positronium are stable and cannot decay. Also, the atomic states can emit and absorb light, whereas the DV states are dark. The atomic states of positronium are bosons with total spin S = 1 (corresponding to the triplets Ω 1 −1 , Ω 1 0 , Ω 1 1 ) and S = 0 (corresponding to the singlet Ω 0 0 ). The DV states of positronium are comprised of S z = 0 doublets (Ω 0 0 , Ω 1 0 ) which have opposite spins s z (e) = −s z (p), and S z = ±1 doublets (Ω 1 1 , Ω 1 −1 ) which have aligned spins s z (e) = s z (p). In both these cases the individual electron and positron spins are correlated and transform like single-particle fermions by a Lorentz boost when they are in the plane perpendicular to the direction of motion. The fermion nature of the DV states occurs because the wavefunctions are Lorentz contracted in the direction of motion z resulting in the spherical symmetry being reduced to cylindrical symmetry. The relative coordinates (r, z) can then be quantized in the z = 0 plane with r = r i . For such states the binding energy is independent of the motion and the DV doublet states behave like single-particle fermions. The S z = 0 fermions for (Ω 0 0 , Ω 1 0 ) are heavy particles with mass M 0 i = 2e 2 /r i and the S z = ±1 fermions for (Ω 1 −1 , Ω 1 1 ) are light particles with mass M 1 i = 0. Because the DV fermions are either symmetric Ψ S = 1 2 (Ψ +− +Ψ −+ ) or antisymmetric Ψ A = 1 2 (Ψ +− − Ψ −+ ), depending on their angular momentum J, they are their own antiparticle and are therefore Majorana fermions. However, for the light fermions, which can only occur in the z = 0 plane, only the Ψ A states are possible. They also have well defined helicity and chirality because their mass M i is zero. Thus, the solutions of the BSE equation for positronium result in two forms of matter with contrasting but complementary characteristics. It does not appear that the DV states are relevant, per se, to atomic physics. The question arises as to whether there are any particles which have the properties of the DV bound-state (f ermi). The fact that these DV bound-states are Majorana fermions and are not single point particles would then explain the violation of parity and charge conjugation by the weak force. These light Majorana fermions can have either left-or right-handed helicity (with the same chirality) which is then, presumably, selected by the weak force to give the observed handedness of neutrinos. The fact that these light and heavy Majorana fermions are dark, stable, and are composed of equal amounts of matter (electrons) and antimatter (positrons) could then simultaneously explain both the apparent absence of antimatter in the universe as well as the apparent presence of dark matter. Further investigation of this hypothesis is necessary to show that the light fermions have other properties of the electron neutrinos besides the ones shown here. One needs to show that the light fermions have the measured weak force cross-sections and are consistent with the proven theory of vector bosons. where Ψ\| is the conjugate spinor. There are two different expressions for the expectation value of an operator O which is Lorentz boosted. Define the adjoint expectation of operator O to be Ψ \|O\| Ψ and the conjugate expectation to be Ψ \|O\| Ψ . The adjoint expectation of O is a Lorentz invariant, Ψ ′ \|O ′ \| Ψ ′ = Ψ \|O\| Ψ (A3) which for O = 1 corresponds to the Lorentz scalar, Ψ ′ \|Ψ ′ = Ψ\|Ψ . AcknowledgmentsThe author is grateful to Dr. Tony Scott for his continued support and invaluable dis-where L −2 is the inverse transform. Define the adjoint spinor to beThis results in the two-body equation for the DV states in the moving frameas in(69). This equation is equivalent to the single-body Dirac equation for a fermion only if M ′ = M such that the rest mass is constant in the moving frame. . T C Scott, J Shertzer, R A Moore, Phys. Rev. A. 454393T. C. Scott, J. Shertzer, and R. A. Moore, Phys. Rev. A 45, 4393 (1992). . C W Patterson, Phys. Rev. A. 10062128C. W. Patterson, Phys. Rev. A 100, 062128 (2019). . J C Light, T Carrington, Adv. Chem. Phys. 114263J. C. Light and T. Carrington, Adv. Chem. Phys. 114, 263 (2000). . E E Salpeter, H A Bethe, Phys. Rev. 841232E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951). . E E Salpeter, Phys. Rev. 87328E. E. Salpeter, Phys. Rev. 87, 328 (1952). . J Malenfant, Phys. Rev. D. 383295J. Malenfant, Phys. Rev. D 38, 3295 (1988). . R P Feynman, Phys. Rev. 76749R. P. Feynman, Phys. Rev. 76, 749 (1949). M Abramowitz, I A Stegun, Handbook of Mathematical Functions. New YorkDover PublishingM. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publishing, New York, 1972). . R A Ferrell, Phys. Rev. 84858R. A. Ferrell, Phys. Rev. 84, 858 (1951). H A Bethe, E E Salpeter, Quantum Mechanics of One-and Two-Electron Atoms. New YorkAcademic PressH. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Aca- demic Press, New York, 1957). . T Fulton, P C Martin, Phys. Rev. 95811T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954). . P V Alstine, H W Crater, Found. Phys. 2767P. V. Alstine and H. W. Crater, Found. Phys. 27, 67 (1997). . A O Barut, S Komy, Fortschritte der Physik. 33309A. O. Barut and S. Komy, Fortschritte der Physik 33, 309 (1985). J J Sakurai, Advanced Quantum Mechanics. New YorkAddison-WesleyJ. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, New York, 1967). . S J Brodsky, J R Primack, Annals of Physics. 52315S. J. Brodsky and J. R. Primack, Annals of Physics 52, 315 (1969). . A Boyarsky, M Drewes, T Lasserre, S Merten, O Ruchayskiy, Prog. Part. Nucl. Phys. 1041A. Boyarsky, M. Drewes, T. Lasserre, S. Merten, and O. Ruchayskiy, Prog. Part. Nucl. Phys. 104, 1 (2019).	[]
[ "Untrapped HOM Radiation Absorption in the LCLS-II Cryomodules UNTRAPPED HOM RADIATION ABSORPTION IN THE LCLS-II CRYOMODULES", "Untrapped HOM Radiation Absorption in the LCLS-II Cryomodules UNTRAPPED HOM RADIATION ABSORPTION IN THE LCLS-II CRYOMODULES" ]	[ "K Bane ", "C Nantista ", "C Adolphsen ", "T Raubenheimer ", "K Bane ", "C Nantista ", "C Adolphsen ", "T Raubenheimer ", "\nSLAC\n94025Menlo ParkCA\n", "\nUSA A. Saini\nFNAL\n60510BataviaN. Solyak, V. Yakovlev, ILUSA\n", "\nA. Saini, N. Solyak and V. Yakovlev, FNAL\nSLAC\n94025, 60510Menlo Park, BataviaCA, ILUSA, USA\n" ]	[ "SLAC\n94025Menlo ParkCA", "USA A. Saini\nFNAL\n60510BataviaN. Solyak, V. Yakovlev, ILUSA", "A. Saini, N. Solyak and V. Yakovlev, FNAL\nSLAC\n94025, 60510Menlo Park, BataviaCA, ILUSA, USA" ]	[ "27 th Linear Accelerator Conference LINAC14 at" ]	The superconducting cavities in the continuous wave (CW) linacs of LCLS-II are designed to operate at 2 K, where cooling costs are very expensive. One source of heat is presented by the higher order mode (HOM) power deposited by the beam. Due to the very short bunch lengthespecially in L3 the final linacthe LCLS-II beam spectrum extends into the terahertz range. Ceramic absorbers, at 70 K and located between cryomodules, are meant to absorb much of this power. In this report we perform two kinds of calculations to estimate the effectiveness of the absorbers and the amount of beam power that needs to be removed at 2 K.	null	[ "https://export.arxiv.org/pdf/1411.1456v1.pdf" ]	56,156,257	1411.1456	b9cf1a9002b4e739b45155b1a6d2e12557a555af	Untrapped HOM Radiation Absorption in the LCLS-II Cryomodules UNTRAPPED HOM RADIATION ABSORPTION IN THE LCLS-II CRYOMODULES* 31 st August-5 th September, 2014 K Bane C Nantista C Adolphsen T Raubenheimer K Bane C Nantista C Adolphsen T Raubenheimer SLAC 94025Menlo ParkCA USA A. Saini FNAL 60510BataviaN. Solyak, V. Yakovlev, ILUSA A. Saini, N. Solyak and V. Yakovlev, FNAL SLAC 94025, 60510Menlo Park, BataviaCA, ILUSA, USA Untrapped HOM Radiation Absorption in the LCLS-II Cryomodules UNTRAPPED HOM RADIATION ABSORPTION IN THE LCLS-II CRYOMODULES* 27 th Linear Accelerator Conference LINAC14 at Geneva, Switzerland on31 st August-5 th September, 2014L C L S -II T E C H N I C A L N OTE Presented at the The superconducting cavities in the continuous wave (CW) linacs of LCLS-II are designed to operate at 2 K, where cooling costs are very expensive. One source of heat is presented by the higher order mode (HOM) power deposited by the beam. Due to the very short bunch lengthespecially in L3 the final linacthe LCLS-II beam spectrum extends into the terahertz range. Ceramic absorbers, at 70 K and located between cryomodules, are meant to absorb much of this power. In this report we perform two kinds of calculations to estimate the effectiveness of the absorbers and the amount of beam power that needs to be removed at 2 K. INTRODUCTION While the use of superconducting accelerating cavities in large particle accelerator facilities offers many advantages in areas such as RF efficiency and feasible beam parameter ranges, a major expense of operating such a machine is the power required of the cryogenic plant. Care must be taken to minimize both static and dynamic heat loads. One element of the latter, particularly relevant in a high-current, short bunch, CW facility like LCLS-II, is higher-order-mode (HOM) electromagnetic field power generated by the beam in passing through the cavities and beamline elements. LCLS-II will run with a CW megahertz bunch train of initial current 62 A, but eventually upgradable to 0.3 mA. HOM heat load is of particular concern in the 20 cryomodules of the L3 linac region, after the second bunch compressor, where the rms length of the 300 pC bunches will be only  z = 25 m. The main generators of HOM power are the 35 mm radius irises of the nine-cell periodic L-band accelerator cavities, though other features, such as inter-cavity bellows and beam pipe radius transitions between regions, play a role. For a quantitative treatment of generated HOM (or more accurately wakefield) power in LCLS-II see [1,2]. HOM coupler ports incorporated in the end pipes of the cavities provide damping against build-up of higher-order cavity mode fields. Since the spectrum of excited HOM power extends well beyond the beam pipe cutoff, annular ceramic RF absorbers are included in the drifts between the 8-cavity cryomodules in the hope of absorbing much of this untrapped wakefield radiation. In what follows, we describe and compare two attempts to theoretically characterize the relative HOM power lost in the different cryogenic environments (2 K, 70 K) by assessing the effectiveness of the HOM absorbers. We would like as much of the power as possible to be lost in the 70 K absorbers rather than in the NC beam pipes and bellows between the cavities, for which heat is removed by the 2 K cooling system. The first method uses a numerical, S-matrix approach, and the second involves an analytical diffusion-type calculation. This topic was previously addressed for the European XFEL project using a ray tracing method and the diffusion approach presented here [3]. Also, the S-matrix calculation has been previously applied to the ILC cryomodules [4]. We focus here on the maximum average beam current. S-MATRIX APPROACH Most of the LCLS-II linac can be seen as a periodically repeated sequence of cryomodule elements joined at their 39 mm-radius beam pipe ports. This periodic unit consists of eight nine-cell ILC-type cavities, each followed by a bellows, a long beam pipe drift (through the quad), an absorber and a shorter drift section. The absorber, a suspended ring of ceramic recessed in a pillbox diameter step, has an integrated bellows, which we treat as part of the same element. At a number of discrete frequencies, 4, 8, 12, 16, 20 and 40 GHz, we used the field solver HFSS [5] to calculate the scattering matrix for each element (cavity, bellows, drifts and absorber) for all TM 0n monopole modes propagating in the beam pipe at each respective frequency. The niobium cavities were modelled as perfect conductors. The bellows and drift pipes were assumed to be copper (or copper plated), for which an electrical conductivity in this temperature and frequency regime, ranging from 1.970.914×10 9 /m between 440 GHz, was calculated according to where Z 0 is the impedance of free space and Z Cu.eff is the surface impedance from Eq. (6) below for the extreme anomalous skin effect [6]. We also considered the case where the bellows and drifts are stainless steel, for which we took  ss = 1.85×10 6 /m. For the absorber element, only the ceramic material was given loss, to represent heat removed. We assumed here Re(/  = 15 and tan = 0.18, estimated from available data [7] out to 40 GHz. Fig. 1 shows geometries and sample field plots. Armed with these matrices for a set of M sequential elements, we can specify a set of equations relating the left (l) and right-going (r) waves of N propagating modes at the junctions as follows:     2 , 0 , ) ( Re 2 ) (       eff Cu eff Cu c Z Z ,(1) (l n,1 , l n,2 ,..., l n,M , r n+1,1 , r n+1,2 ,..., r n+1,M ) T = S n (r n,1 , r n,2 , ..., r n,M , l n+1,1 , l n+1,2 ,..., l n+1,M ) T , (2) With a constant driving vector d n,1 = after each cavity (z n being its downstream position) and 0 elsewhere, representing the wakes of a speed-of-light bunch, and speed-of-light phased periodic boundary conditions imposed at the ends of the sequence, determining the junction fields amounts to solving a matrix equation of the form Ay = d, where A is of order 2M(N+1). Now, correcting for the generated waves, we can solve for the relative power dissipated in each element as the difference between incoming and outgoing power, summed over the modes. That is, p n =  m=1,N (\|r n,m \| 2 + \|l n+1,m \| 2  \|l n,m \| 2  \|r n+1,m \| 2 ) + \|r n+1,1 \| 2  \|r n+1,1  d n+1,1 \| 2 .(3) This technique is described in more detail in [4]. The ratio of the p for the absorber element to the sum of all p's gives us a measure of the effectiveness of the absorber in reducing the losses at 2 K. As the number of modes for each case was the frequency in gigahertz divided by four, this approach is limited from going much higher than about 40 GHz by the S-matrix size. DIFFUSION APPROACH Another way to estimate the distribution of HOM power absorption is to use a diffusion-like model [3], in which the radiation fills the available volume like a gas. This is a good approximation well above cutoff and where the surface reflection coefficient is close to unity. The beam line elements can be grouped by type, for each of which, the power absorption can be characterized by ; )) ( Re( ) ( 0     d d dP S n I i i i abs i   Z(4) where i denotes a particular type of element (RF cavity, bellows, end pipe or absorber), n i is the element quantity, S i its surface area, Z i its surface impedance and dP 0 /d the HOM spectral density obtained using:          2 ) ( Re ) ( 2 z w 2 0 d e Z f q dP c r b   ,(5) where Z w is the cryomodule impedance. To obtain the spectrum, we begin with the point charge wake of a TESLA cryomodule: W(s) = 344 exp(-s/s 0 ) V/pC, with s 0 = 1.74 mm [7]. We then Fourier transform this to obtain Z w (). The resulting spectrum P 0 (f) (with f = /2i.e. the total power integrated from f to 2 THzis shown in Fig. 2 for the L3 linac parameters. Note that it is the derivative of this function divided by 2 that gives dP 0 /d of Eq. (4). With an rms bunch length of  z = 25 m, a bunch charge q b = 300 pC and a beam of repetition rate f r = 1 MHz, the total steady state power (P HOM ) deposited in a cryomodule is 13.8 W [1,2]. For comparison, the main RF cryomodule heat load is approximately 100 W. Here too, two cases were considered, in which the bellows and beam pipe surfaces were assumed to be either copper or stainless steel. For frequency > 1 GHz at 2 K, copper exhibits the extreme anomalous skin effect. Its effective impedance is expressed as [6]  , 3 1 ) ( 3 / 2 , i A eff Cu     Z (6) where A = 3.3×10 -10 ( s 2/3 ) is a material constant independent of RRR. The surface impedance of stainless steel is given by:   ) 2 ( 1 ) ( 0 ss SS c Z i      Z ,(7) where  ss is the electrical conductivity of stainless steel, for which we here used 10 6 /m. As before, the power absorbed in the superconducting niobium cavities was taken to be negligible. To estimate the surface impedance of the absorber material we use the following:   ); Im( , 1 , ) Re( ) Im( tan ) Re( 2 tan ) ( 0                   s c(8) where is the attenuation coefficient in the dielectric medium,  its permittivity, tan  the loss tangent, and  s the field penetration skin depth. (Here values of 16.5 and 0.2 were used respectively for Re(/  and tan .) The surface impedance of the absorber is given by:   s Absorber   1 ) (  Z (9) As for the n i S i area factors, the bellows and pipe total was estimated to exceed the ceramic by a factor of 57. Inserting the required parameters in Eq. (4) calculated from Eqs. (5), (6), (7) and (9) RESULTS Each of these methods yields an estimate for the fractional heating distribution among beamline elements of the high-frequency wakefield power generated by the LCLS-II beam. Table 1 tabulates side-by-side results at given frequencies for the fraction that goes into the 2 K cryogenics, i.e. is not extracted to 70 K by the beamline HOM absorber at the end of each cryomodule. The numbers are similar. The final row shows, for the diffusion model, the total integrated percent, weighted by the spectral density function, the derivative of Fig. 2. It suggests the absorber is considerably less effective at higher frequency. Stainless steel does not appear to be an acceptable beamline surface material for this machine, based on both the higher power absorption and consideration of thermal conduction to the liquid helium bath. Bellows and connecting drift pipes should be copper-plated. A sample plot is shown in Fig. 3 of the element power distribution from the S-matrix technique at 4 GHz. As this depends on mode phase lengths, we estimated the behaviour away from our discreet S-matrix frequencies by systematically extending cavity end pipes, analytically. Fig. 4 plots the effect on the 2 K power at 20 GHz. Statistical results over 2,000 steps are given in Table 2. Since driving only the lowest mode (d n, 1 ) is not what the beam actually does, we tested the sensitivity of our results to this assumption. At f = 16 GHz, we repeated the power vs. cavity pipe length calculation driving in turn each of the other three propagating monopole modes. The averages of p 2K /p tot resulting from driving modes 14, respectively, were 1.5%, 1.5%, 1.4% and 1.7%. It seems mixing makes the calculation fairly insensitive to this drive detail. CONCLUSION Two complementary approaches applied to the LCLS-II linac have helped to characterize the distribution of heating due to monopole wakefield power generated by the tightly-bunched CW beam. They provide us some confidence in the effectiveness of the beamline HOM absorbers, suggesting that no more than a few percent of this power will present added load to the 2 K cryogenics system. We should note, however, that the absorber material, for which we assume constant parameters, has not been characterized above 40 GHz. Figure 1 : 1Geometries and field plots from HFSS S-matrix calculations: cavity and absorber at 12 & 20 GHz. Figure 2 : 2Integrated steady state wakefield power from f up to 2 THz generated by 300 pC 25 m bunches at 1 MHz in L3. Of the 13.8 W total lost by beam, about 50% is lost above 20 GHz, and 25% above 100 GHz. Figure 3 : 3Relative power distribution at 4 GHz in the elements along the cryomodule. Blue dots are the cavities. The absorber, at > 99% at 12 m, is offscale. Figure 4 : 4Percentage of power deposited at 2 K, p 2K /p tot , vs. added cavity pipe length at f = 20 GHz. , one can estimate the I i abs for each element type and hence the power absorption for each in an L3 cryomodule from:     3 1 j abs j abs i HOM i I I P P (10) Table 1 : 1Percent of Untrapped HOM Power to 2 K. Table 2 : 2Element Spacing Statistics of Power to 2 K (Cu). ACKNOWLEDGMENTThe authors wish to thank Martin Dohlus for visiting us to explain the European XFEL effort on the topic of this report. Some Wakefield Effects in the Superconducting RF Cavities of LCLS-II. K Bane, A Romanenko, V Yakovlev, LCLS-II TN-13-04K. Bane, A. Romanenko and V. Yakovlev, "Some Wakefield Effects in the Superconducting RF Cavities of LCLS-II," LCLS-II TN-13-04, March, 2014. Wakefields in the Superconducting RF Cavities of LCLS-II. K Bane, THPP124, this conferenceK. Bane et al., "Wakefields in the Superconducting RF Cavities of LCLS-II," THPP124, this conference. . M Dohlus, M. Dohlus, http://www.desy.de/xfel-beam/data/talks /talks/dohlus_-_cryo_calc_20071112.pdf Higher Order Mode Heating Analysis for the ILC Superconducting Linacs. K L F Bane, C Nantista, C Adolphsen, at 25 th Int'l. Linear Accel. Conf. (LINAC10). Tsukuba, JapanK.L.F. Bane, C. Nantista and C. Adolphsen, "Higher Order Mode Heating Analysis for the ILC Superconducting Linacs," at 25 th Int'l. Linear Accel. Conf. (LINAC10), Tsukuba, Japan, Sept. 2010. . HFSS. Ansys, IncHFSS, Ansys, Inc. . B Podobedov, Phys Rev ST-AB. 1244401B. Podobedov, Phys Rev ST-AB 12, 044401 (2009). A Beam Line HOM Absorber for the European XFEL Linac. N Mildner, from 12 th International Workshop on Superconductivity (SRF2005). Ithaca, New YorkCornell UniversityN. Mildner, et al., "A Beam Line HOM Absorber for the European XFEL Linac," from 12 th International Workshop on Superconductivity (SRF2005), Cornell University, Ithaca, New York, July 10-15, 2005. . T Weiland, I Zagorodnov, Report, T. Weiland and I. Zagorodnov, TESLA Report 2003- 19 (2003).	[]
[ "HBReID: Harder Batch for Re-identification", "HBReID: Harder Batch for Re-identification" ]	[ "Wen Li \nANT GROUP\n\n", "Furong Xu \nANT GROUP\n\n", "Jianan Zhao [email protected] \nANT GROUP\n\n", "Ruobing Zheng \nANT GROUP\n\n", "Cheng Zou \nANT GROUP\n\n", "Meng Wang \nANT GROUP\n\n", "Yuan Cheng [email protected] \nANT GROUP\n\n" ]	[ "ANT GROUP\n", "ANT GROUP\n", "ANT GROUP\n", "ANT GROUP\n", "ANT GROUP\n", "ANT GROUP\n", "ANT GROUP\n" ]	[]	Triplet loss is a widely adopted loss function in ReID task which pulls the hardest positive pairs close and pushes the hardest negative pairs far away. However, the selected samples are not the hardest globally, but the hardest only in a mini-batch, which will affect the performance. In this report, a hard batch mining method is proposed to mine the hardest samples globally to make triplet harder. More specifically, the most similar classes are selected into a same minibatch so that the similar classes could be pushed further away. Besides, an adversarial scene removal module composed of a scene classifier and an adversarial loss is used to learn scene invariant feature representations. Experiments are conducted on dataset MSMT17 to prove the effectiveness, and our method surpasses all of the previous methods and sets stateof-the-art result.	null	[ "https://arxiv.org/pdf/2112.04761v1.pdf" ]	245,006,108	2112.04761	20d4d79c6b666d03c5913c23b693a50a89885d70	HBReID: Harder Batch for Re-identification Wen Li ANT GROUP Furong Xu ANT GROUP Jianan Zhao [email protected] ANT GROUP Ruobing Zheng ANT GROUP Cheng Zou ANT GROUP Meng Wang ANT GROUP Yuan Cheng [email protected] ANT GROUP HBReID: Harder Batch for Re-identification Triplet loss is a widely adopted loss function in ReID task which pulls the hardest positive pairs close and pushes the hardest negative pairs far away. However, the selected samples are not the hardest globally, but the hardest only in a mini-batch, which will affect the performance. In this report, a hard batch mining method is proposed to mine the hardest samples globally to make triplet harder. More specifically, the most similar classes are selected into a same minibatch so that the similar classes could be pushed further away. Besides, an adversarial scene removal module composed of a scene classifier and an adversarial loss is used to learn scene invariant feature representations. Experiments are conducted on dataset MSMT17 to prove the effectiveness, and our method surpasses all of the previous methods and sets stateof-the-art result. Introduction Object re-identification (ReID) aims to associate a particular object across different scenes and camera views, such as in the applications of person ReID and vehicle ReID. Academic research is progressing in four major directions: image preprocessing, feature representation learning, metric learning, and ranking optimization [33,32]. As for image preprocessing, data augmentation is crucial to ReID. Random cropping, padding, horizontal flipping and random erasing [34] are frequently-used methods which will increase the number and diversity of training samples. Grayscale patch replacement [7] is proposed to randomly replace a rectangular region of RGB pixels in the image with its corresponding grayscale counterpart, which has been proved effective. As for representation learning, CNN-based [31,13,19,24,11] architecture has dominated for a long time until transformer-based [16,3] methods appear. TransReID [9], a pure transformer method, has achieved state-of-the-art on many ReID benchmarks. Metric learning often concentrates on designing powerful loss functions to extract robust feature embeddings. In addition to Softmax-cross-entropy, proxy-based loss and pair-based loss [10,14,25,26,27] are used to pull the same classes close and push the different classes far away. Some methods also use domain adaption to reduce domain gap for retrieving under different cameras. In this report, we propose hard batch mining for ReID which mines the hardest samples globally to make triplet harder. Besides, an adversarial scene removal module composed of a scene classifier and an adversarial loss is used to learn scene invariant feature representations. The contributions are summarized as follows: • We propose a hard batch mining method to mine the hardest samples globally to make triplet harder during training. • We propose an adversarial scene removal module to learn scene invariant feature representations. Random Sampling Similar Sampling … Figure 1: The main framework of HBReID. In the first few steps, it randomly sample P person to construct mini-batches. And then in the following steps, it samples the most similar classes given a random class ID according to the class similarity measured by the learnt class weights, then updates the model. This process is called hard batch mining, and it will iteratively run until the training is finish. Note that there is an adversarial scene removal module on top of the global feature x. w C×L is the class weights of the last fully connected layer. • Our method surpasses all of the previous methods and sets state-of-the-art result on MSMT17. 2 Related Work Triplet Loss In classification task and metric learning, Softmaxcross-entropy is a mainstream loss function to split the different class into different feature space. But it can't handle far intra-class distance and close interclass distance very well. Therefore, proxy-based loss [20,14] and pair-based loss [28,23] are used to pull the samples of the same class closer and to push the samples of different classes far away. Triplet loss is the most prominent approach that uses pair-based distanced calculation for deep metric learning [28]. During training, a sample in a mini-batch is considered as an anchor, the distances of negative samples and positive samples with the anchor are calculated to select the closest negative samples and the furthest positive samples. Triplet loss minimizes the distance of the anchor-positive pairs and tries to maximize that of the anchor-negative pairs. But in most cases, the selected samples for triplet loss are not the hardest globally, but the hardest only in a mini-batch, which is suboptimal. Domain Adaption Domain adaption aims at learning from a source data distribution and well performing on a different (but related) target data distribution [15]. ReID can be regarded as a domain-variant problem because it retrieves the same person from diverse cam-eras with various illuminations and viewpoints which are different at training and test phase. Learning domain-invariant feature representations to reduce the distance of different domains is a common method [8,18,35,12]. Re-weighting the instances selected from a subset of two domains where they have smaller distance is also useful. Adversarial networks [5,1,2,22] or simply an adversarial loss [6,21] are also used to confuse networks to focus more on domain-invariant features. Proposed Method The proposed method is based on TransReID [9]. To further improve the performance, a hard batch mining method is proposed to sample closer classes to construct harder mini-batches during training. Meanwhile, an adversarial scene removal module is used to make the feature representations scene invariant. Figure. 1 shows the main framework of the proposed method. Hard Batch Mining PK sampling is an effective sampling strategy in ReID. P classes are randomly selected at first, and then K images are randomly selected from each class, so that there are PK images in a mini-batch. For each sample in the mini-batch, the hardest positive and the hardest negative samples can be selected for triplet loss. However, the similarities between classes can be quite different. Some classes are extremely similar while some are obviously dissimilar. Random sampling P classes per mini-batch directly is a limitation for further hard mining. In this report, we sample P classes with the highest similarity so that the hardest anchor-negative pairs can be mined for triplet loss to improve the discrimination of similar classes. Inspired by DAM [29], the weights w (w ∈ R C×L , where C is class number, and L is the length of classified embedding) of the last fully connected layer are used to get the pairwise similarity between classes. Mathematically, a feature x i is projected onto all weight vectors [w 1 , ..., w C ] to determine its class, where w i is a L-dim vector. Therefore, class weights can be used to represent the average feature of intraclass samples (class center), and the similarity between class weights can represent the distance between inter-classes. The class similarity s i , j between class c i and c j can be defined as a) b) c) d)s i , j = cos(w i , w j )(1) The higher the value is, the more similar two classes are. The common approach is to select the most difficult samples from the current remaining data in each iteration to form a batch, but multiple similarity calculations affect the training speed. Inspired by the equidistant constraint in EET [30], when some samples have the closed same distance from a certain sample, these samples are always distributed closer. Therefore, for each epoch of training, we randomly select a class to calculate the similarity with other classes. Then all classes are sorted according to the similarity, and the sorted classes are constructed batch in turn for training. This batch sampling operation allows classes that are closer to each other to be optimized in a mini-batch, thereby improving the distance distribution between classes. Adversarial Scene Removal ReID task is to retrieve the same person in diverse cameras which are placed at different scenes with various illuminations and viewpoints. To remove the influence of scene types, a classifier with adversarial loss is designed following [4]. The motivation behind this scene removal is that we want to learn feature representations for re-identification but invariant to different scene types. We learn a scene classifier f s on top of the global feature x in an adversarial way. The adversarial loss L adv is defined as L adv = − T t=1 y t log(f s (x))(2) where T is the number of scene types and y t is scene label. The loss is adversarial because the scene classifier aims to minimize it, while the feature extractor tries to maximize it to confuse the classifier. In the end, one can't tell the scene types from the feature, that is, the feature is scene invariant. The total loss is then defined as L total = L ReID − λL adv(3) where λ is a positive coefficient to balance the scene adversarial loss. Experiment Implementation Details The experiments are conducted on person ReID dataset MSMT17 [27]. All images are resized to 384×128. The batch size is set to 64 with 4 images per ID. SGD optimizer is employed with a momentum of 0.9 and the weight decay of 1e-4. The learning rate is initialized as 0.008 with cosine learning rate decay. All the experiments are performed with 4 Nvidia Tesla V100 GPUs. The initial weights of ViT are pre-trained on ImageNet-21K and then finetuned Table 2: Ablation studies and comparisons with SOTA methods. § indicates our re-implementation, † indicates test with re-ranking [33]. GS: grayscale patch replacement, HM: hard batch mining, SR: adversarial scene removal module. Method mAP cmc1 FlipReID [17] 68.0 85.6 TransReID [9] 69 Data Augmentation Data augmentation is crucial to ReID tasks. Proper augmentations can improve performance while improper ones could make it worse. We use the augmentations including horizontal flipping, padding, random crop and random erasing which are commonly used in ReID. Besides, another augmentation, grayscale patch replacement [7], is used to randomly select a rectangle region in an image and grayscale it, see Figure. 2. Experiments of other color distortion augmentations like color jitter, gaussian blur and gamma transform are reported in Table. 1, and the result indicates that grayscale patch replacement is effective. Comparison with SOTA The ablation studies and comparision with state-ofthe-art methods are shown in Table. 2. The baseline here is our re-implementation of TransReID. With grayscale patch replacement (GS), the mAP and cmc1 are improved by 0.8% and 0.5% respectively. Combined with GS, hard batch mining (HM) provides extra 0.5% mAP and 0.6% cmc1, and adversarial scene removal module (SR) provides extra 0.2% mAP and 0.6% cmc1 gains. Compared with SOTA methods, the proposed method provides overall 0.7% mAP and 0.8% cmc1 improvements. Further, with re-ranking, our method achieves 84.4% mAP and 89.9% cmc1, surpassing all of the previous methods and setting state-of-the-art result. Conclusion In this report, we propose hard batch mining for ReID to mine the globally similar classes to make harder mini-batches, which helps to improve the discrimination of similar classes. Besides, a scene removal module composed of a scene classifier and an adversarial loss is used to learn scene invariant feature representations. Experiments are conducted on dataset MSMT17 to prove the effectiveness, and our method surpasses all of the previous methods and sets state-of-the-art result. More datasets will be evaluated in the future. Figure 2 : 2Different augmentation examples. a) Original. b) Grayscale patch replacement. c) Gaussian blur. d) Gamma transformation. Table 1 : 1Performance on MSMT17 dataset with different data augmentations. CJ: color jitter, GB: gaussian blur, GT: gamma transformation, GS: grayscale patch replacement.Default +CJ +GB +GT +GS 68.8 -3.5 -0.6 -0.3 +0.8 Unsupervised pixel-level domain adaptation with generative adversarial networks. Konstantinos Bousmalis, Nathan Silberman, David Dohan, Dumitru Erhan, Dilip Krishnan, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)Konstantinos Bousmalis, Nathan Silberman, David Dohan, Dumitru Erhan, and Dilip Krishnan. Un- supervised pixel-level domain adaptation with gen- erative adversarial networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. 3 . 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[]	[]	[]	[]	In pixelized detectors, reducing power consumption in the front end ASIC chips becomes a crucial demand. Optimization based on mature pre-amplifier schemes today is unlikely to bring sufficient improvements. A new CMOS frontend gain stage topology with very low power consumption called regenerative current mirror is developed to fulfill the demand. The circuit takes advantage of high speed performance of current amplification while operating with relatively low bias current. The regenerative current mirror uses a NMOS current mirror and a PMOS current mirror, both with nominal gain of 1, to form a loopback topology that provides a positive feedback. An NMOS FET with an external adjustable voltage applied to it gate terminal is used to limit open loop gain of the current mirrors to be slightly lower than 1. This yields a net gain of the positive feedback loop to be much larger than 1 while operating the current mirrors under relatively low bias currents. Simulation shows that in 65 nm fabrication process, the power consumption of the gain stages suitable for silicon pixel detectors can be controlled < 10 micro-Watts per channel.	null	[ "https://export.arxiv.org/pdf/2305.10208v1.pdf" ]	258,741,127	2305.10208	8e20a8bba4ca2236376563a46e20db1a19a78941	Index Terms-Front-end electronicsCMOS ASICLow Power Amplifier In pixelized detectors, reducing power consumption in the front end ASIC chips becomes a crucial demand. Optimization based on mature pre-amplifier schemes today is unlikely to bring sufficient improvements. A new CMOS frontend gain stage topology with very low power consumption called regenerative current mirror is developed to fulfill the demand. The circuit takes advantage of high speed performance of current amplification while operating with relatively low bias current. The regenerative current mirror uses a NMOS current mirror and a PMOS current mirror, both with nominal gain of 1, to form a loopback topology that provides a positive feedback. An NMOS FET with an external adjustable voltage applied to it gate terminal is used to limit open loop gain of the current mirrors to be slightly lower than 1. This yields a net gain of the positive feedback loop to be much larger than 1 while operating the current mirrors under relatively low bias currents. Simulation shows that in 65 nm fabrication process, the power consumption of the gain stages suitable for silicon pixel detectors can be controlled < 10 micro-Watts per channel. I. INTRODUCTION N pixelized detector used in high energy physics experiments, power consumption in the front-end ASIC becomes a critical issue. Since the number of detector elements becomes so large, very low power consumption of the front-end circuits is required. In today's mature pre-amplifier scheme, the power consumption is around 800 micro-Watts per channel in 65 nm CMOS fabrication technology, while the desired power consumption in future ASIC for pixelized detectors can be as low as 10 micro-Watts. In typical small signal voltage amplifiers, transistors must operate with a sufficiently high bias current so that the transistors are in high conductivity region to follow the input pulse at high speed. Also, the topologies used in amplifiers usually employs deep negative feedback to improve bandwidth and linearity. Given these basic principle, it is unlikely to achieve demanded improvements by optimizing on the existing topologies. As an attempt of significantly reducing power consumption of the front-end gain stage, a new CMOS front-end topology with very low power consumption called regenerative current mirror is developed. The simplified schematics of the circuit is shown in Fig. 1. In this circuit, MOS FET pairs (M11, M22) and (M13, M14) form two current mirrors. The two current mirrors are cross connected to form a positive feedback loop. The transistors in each current mirror are identical so that the nominal gain of each current mirror is 1. Obviously, should the total gain of the cross connected current mirrors be larger than 1, the circuit would be locked to the short circuit condition in which all the FETs would be conductive. To prevent the locking condition from happening, the NMOS FET M15 is used to limit the gain of the NMOS current mirror so that it is <1. With the external voltage VB1 adjusted appropriately, the total open loop gain A of the two current mirrors can be set to be slightly <1. In this case, the close loop gain of the whole circuit G = A/(1-A). If A=0.8, for example, G = 4. The current, primarily bias current, flowing through M11, M15 and M14 path is the same as the M12 and M13 path, rather than G times higher in regular current mirror structure. In this scheme, the bias currents in both paths remain unchanged and hence reduces power consumption. the simulation. To conveniently study the power consumption of the circuit, a small internal resistance is introduced in the power supply V1 and a bypass capacitor C2 is also included. When VB1=0.42V, the close-loop gain of the current mirror is ~5. The bias current in both columns are about 1uA (rather than 1uA and 5uA). Small current pulses I1 are send into PMOSG1 net. As shown in the lower pane of Fig. 3, the current pulses are amplified by about factor of 5 in the drain currents of M13 and M14, respectively. The bias currents flowing through M11 and M12 are about 1µA each, while the total current flowing out the power supply V1 is about 2µA as shown in the top pane of Fig. 3. At 1.2V power supply voltage, the power consumption is around 2.4 µW for a single gain stage. III. AN APPLICATION For applications with very weak input signal such as silicon pixel detectors, multiple gain stages are needed. A circuit with two regenerative current mirror stages AC coupled together is shown in Fig. 4. When a small current pulse in I1 is fed into the VPG1 net, it is amplified with two gain stages as well as the AC coupling stages, and a logic level output is anticipated at net VW5. The simulation results of the signal gain and power consumption are shown in Fig. 5 The total charge of the input pulse is 10ns x 10nA = 0.1fC. (~625 e-). After two stages (with AC coupling), a logic level (1.2V) can be generated. The total power consumption is ~9.6uW (8uA x 1.2V). IV. CONCLUSIONS A low power amplifier suitable for weak signals from silicon pixel detectors is design and simulated with the regenerative current mirror topology. The low power feature is realized due to two characters of this design: current amplification and positive feedback. The amplifier can be coupled to the pseudo-thyristor a novel circuit we developed replacing conventional discriminator, (which will be described in another document). This provides a full chain solution for pixel detector with power consumption at the level of 10µW per channel. II. SIMULATION OF THE GAIN STAGEThe actual circuit diagram of a single stage of the regenerative current mirror simulated with the Spice software is shown inFig. 2. The 65 ns CMOS device model is used inThe author is with Fermi National Accelerator Laboratory, Batavia, IL 60510 USA (phone: 630-840-8911; fax: 630-840-2950; e-mail: jywu168@ fnal.gov). Fig. 2 .Fig. 5 .Fig. 4 . 254Circuit schematics of the regenerative current mirrorFig. 3. Simulation results of the gain stage Simulation results of the two-stage amplifier Two AC coupled stages of regenerative current mirror with a logic level output buffer Manuscript submitted Aug. 12, 2021, revised May 02, 2022. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. J Wu, Register-like Block RAM: Implementation, Testing in FPGA and Applications for High Energy Physics Trigger Systems, Presented at 2016 IEEE Real Time Conference. J. Wu. (June 2016). Register-like Block RAM: Implementation, Testing in FPGA and Applications for High Energy Physics Trigger Systems, Presented at 2016 IEEE Real Time Conference. Accessed: Mar. 19, 2022. [Online]. Available: https://indico.cern.ch/event/390748/contributions/1825169/ J Wu, Register-Like Block RAM with Boundary Coverage and Its Applications for High Energy Physics Trigger Systems. Presented at 2018 IEEE Real Time Conference. J. Wu. (June 2018). Register-Like Block RAM with Boundary Coverage and Its Applications for High Energy Physics Trigger Systems. Presented at 2018 IEEE Real Time Conference. Accessed: Mar. 19, 2022. [Online]. Available: https://lss.fnal.gov/archive/2018/conf/fermilab-conf-18-822-e.pdf Examples of an FPGA in daily design jobs. H Sadrozinski, & J Wu, Applications of Field-Programmable Gate Arrays in Scientific Research. New York, NY, USA, Taylor & FrancisH. Sadrozinski & J. Wu, "Examples of an FPGA in daily design jobs," in Applications of Field-Programmable Gate Arrays in Scientific Research, New York, NY, USA, Taylor & Francis, December 2010, pp. 31-35. Fast pattern recognition with the ATLAS L1Track trigger for the HL-LHC. Mikael Martensson, the 25th International workshop on vertex detectors. Mikael Martensson, "Fast pattern recognition with the ATLAS L1Track trigger for the HL-LHC," in the 25th International workshop on vertex detectors 2016.	[]
[ "Two-dimensional Chiral Anomaly in Differential Regularization", "Two-dimensional Chiral Anomaly in Differential Regularization" ]	[ "W F Chen \nDepartment of Physics\nWinnipeg Institute for Theoretical Physics\nUniversity of Winnipeg\nR3B 2E9Winnipeg, WinnipegManitoba, ManitobaCanada\n" ]	[ "Department of Physics\nWinnipeg Institute for Theoretical Physics\nUniversity of Winnipeg\nR3B 2E9Winnipeg, WinnipegManitoba, ManitobaCanada" ]	[]	The two-dimensional chiral anomaly is calculated using differential regularization. It is shown that the anomaly emerges naturally in the vector and axial Ward identities on the same footing as the four-dimensional case. The vector gauge symmetry can be achieved by an appropriate choice of the mass scales without introducing the seagull term. We have analyzed the reason why such a universal result can be obtained in differential regularization.	10.1016/s0370-2693(99)00691-7	[ "https://export.arxiv.org/pdf/hep-th/9902199v6.pdf" ]	2,536,900	hep-th/9902199	a64424270845924c43683f4cc21e959610d3fa25	Two-dimensional Chiral Anomaly in Differential Regularization 4 Jun 1999 W F Chen Department of Physics Winnipeg Institute for Theoretical Physics University of Winnipeg R3B 2E9Winnipeg, WinnipegManitoba, ManitobaCanada Two-dimensional Chiral Anomaly in Differential Regularization 4 Jun 1999Two-dimensional chiral anomalyDifferential regularizationarbi- trary local termWard identity The two-dimensional chiral anomaly is calculated using differential regularization. It is shown that the anomaly emerges naturally in the vector and axial Ward identities on the same footing as the four-dimensional case. The vector gauge symmetry can be achieved by an appropriate choice of the mass scales without introducing the seagull term. We have analyzed the reason why such a universal result can be obtained in differential regularization. One of the most remarkable dynamical phenomena in two-dimensional massless Quantum Electrodynamics (QED 2 ) is that the photon field becomes massive through a dynamical Higgs mechanism induced by the fermion [1]. The essence for its occurrence lies in the spontaneous breaking of the global chiral symmetry U(1)× U (1), U(1) and U (1) denoting the Abelian group associated with the charge and chirality of the fermion, respectively. This fact makes QED 2 served as a laboratory for understanding the vacuum structure of QCD 4 [2]. The two-dimensional chiral anomaly plays a crucial role in revealing above dynamical phenomena [3,4]. In fact, the two-dimensional chiral anomaly was first found by Johnson [5] prior to the discovery of the famous ABJ chiral anomaly in four dimensions [6]. Later an explicit perturbative calculation was also carried out in Ref. [7]. In the last decade this anomaly has been re-investigated by various perturbative and non-perturbative calculation techniques such as dimensional, Pauli-Villars and zeta function regularization schemes as well as a dispersion relation approach [8] and topological analysis [9]. Recently, it is calculated again by a quite old method called analytic regularization [10]. However, we think that it is still worthwhile to use a newly developed quantum field theory methods to gain insight. This may not only reveal some of the new features of the two-dimensional anomaly, but also test the feasibility of the modern computation techniques. In view of the above considerations, in this letter we shall use a relatively new regularization scheme called differential regularization [11] to calculate the two-dimensional chiral anomaly. The basic idea of this regularization is quite simple. It works in coordinate space for Euclidean field theory and is based on the observation that the higher order amplitude cannot have a Fourier transform into momentum space due to the short-distance singularity. Thus one can regulate such an amplitude by first writing its singular parts as the derivatives of the less singular functions, which have a well defined Fourier transform, and then performing Fourier transformation in partial integration and discarding the surface term. In this way one can directly get a renormalized result. Up to now this method and its modified version have been applied successfully to almost every field theories, including the supersymmetric ones [11][12][13][14][15][16]. The main motivation for us to choose this regularization scheme to study the twodimensional chiral anomaly is its great advantage over other regularization schemes in preserving gauge symmetry. The maintenance of the gauge symmetry in differential regularization is achieved by appropriately choosing the indefinite renormalization scales at the final stage of the calculation. Therefore, this regularization method can lead to a most democratic expression for the chiral anomaly. On the other hand, in Pauli-Villars, dimensional and Zeta function regularization, a renormalization condition is automatically imposed on the vector and axial Ward identities when calculating the chiral anomaly. That is, these regularization schemes themselves are compatible with the conservation of vector gauge symmetry and violate the axial vector symmetry, thus one can only observe an anomalous axial vector Ward identity. Differential regularization is different. The study of four-dimensional chiral anomaly in terms of differential regularization was initially carried out in Ref. [11] and [13], and it was shown that the anomalous term can naturally arise in the VWI and AWI on the same footing. Furthermore, combined with the conformal symmetry of the correlate function, as initially proposed by Baker and Johnson [17], differential regularization was also applied to investigate the possible two-loop radiative correction to the chiral anomaly and the axial gravitational anomaly [18]. Therefore, the investigation of the chiral anomaly in terms of differential regularization will definitely be more powerful in revealing the quantum structure of the theory than other regularization schemes. In addition, differential regularization does not modify the original Lagrangian at all, hence it never shifts the value of a primitively divergent Feynman diagram away from its singularities, and neither does it extend the dimensionality of the space-time or introduce a regulator [13]. Thus this regularization method is free from the ambiguity in defining the dimensional continuation of γ 5 , which can occur in dimensional regularization. The details of the calculation in the differential regularization are also simpler than in Pauli-Villars and Zeta function regularization. We now set up the framework for calculating this anomaly. The Lagrangian of QED 2 with one flavour fermion in Euclidean space is read as follows [4]: L =ψ(i/ ∂ − e/ A − m)ψ − 1 4 F µν F µν , µ, ν = 1, 2,(1) where the γ-matrices are chosen as the two-component form γ 1 = σ 2 , γ 2 = −σ 1 , γ 5 = −iγ 1 γ 2 = σ 3 .(2) Classically, the following vector and axial currents j µ =ψγ µ ψ, j 5 µ = iψγ µ γ 5 ψ (3) satisfy the relations ∂ µ j µ = 0, ∂ µ j 5 µ = 2imj 5(4) with j 5 being the pseudo-scalar current, j 5 ≡ψγ 5 ψ. In QED 2 the chiral anomaly comes from the two-point function [9] Π 5 µν (x − y) = T [j µ (x)j 5 ν (y)] .(5) This is contrary to the four-dimensional case, where the chiral anomaly comes from the triangle composed of the axial and vector currents. Due to the explicit relation iγ µ γ 5 = ǫ µν γ ν , T 5 µν is relevant to the vacuum polarization tensor Π 5 µν (x − y) = ǫ νρ Π ρµ (x − y).(6) Thus Π 5 µν can be calculated via the vacuum polarization tensor. If the classical symmetries of the theory were preserved at the quantum level there would exist the following vector Ward identity (VWI), ∂ x µ Π 5 µν (x − y) = ∂ x µ Π µν (x − y) = ∂ y ν Π µν (x − y) = 0(7) and the corresponding axial vector one (AWI), ∂ y ν Π 5 µν (x − y) = 2imΠ 5 µ (x − y) (8) with that Π 5 µ (x − y)≡ T [j µ (x)j 5 (y)] . Later it can be seen that the above Ward identities will be violated due to the chiral anomaly. For massless QED 2 , the propagator of the fermion in Euclidean space is S(x) = − 1 2π / ∂ ln 1 x ,(9) where and later we denote x≡\|x\|; While for the massive case, we have S(x) = 1 2π (/ ∂ − m)K 0 (mx),(10) where K is the modified Bessel function of the second kind. Let us first consider the massless case. With the propagator (9) we write down the vacuum polarization tensor Π µν (x) = −e 2 Tr [γ µ S(x)γ ν S(−x)] = e 2 4π 2 Tr(γ µ γ α γ ν γ β ) x α x β x 4 = e 2 2π 2 2x µ x ν x 4 − δ µν x 2 ,(11) where we have used the relation ∂ µ f (x) = x µ /x[d/dxf (x)] and the two-dimensional γ-matrix trace formula Tr(γ µ γ α γ ν γ β ) = 2 (δ µα δ νβ − δ µν δ αβ + δ µβ δ να ). The term with the tensor structure x µ x ν can be rewritten as x µ x ν x 4 = 1 2 ∂ µ ∂ ν ln 1 x + δ µν x 2 .(12) Upon substitution of (12) into the vacuum polarization tensor (11), it seems that the term with the tensor structure δ µν could cancel. In fact, this is not allowed since in two dimension the term ∼ 1/x 2 is obviously singular. It is analogous to the fact that two divergent terms with the same form but opposite sign in momentum space cannot be canceled, only after a regularization procedure is implemented so that they become well defined, the substraction operation can work safely. Otherwise, a finite term will probably lost since the difference of two infinite quantities is generally not zero. We therefore use differential regularization schemes to make 1/x 2 well defined. Writing 1 x 2 = ∂ 2 f (x) = 1 x d dx x d dx f (x) ,(13) we get f (x) = − 1 2 ln(Mx) ln 1 x ,(14) and hence the regulated version of 1/x 2 is 1 x 2 R = − 1 2 ∂ 2 ln(Mx) ln 1 x ,(15) where M can be explained as the renormalization scale, which is necessary for the differential regularization. (15) are exactly the analogue of four-dimensional case [11], 1/ x 4 = −1/4∂ 2 [ln(M 2 x 2 )/x 2 ]. With (12) and (15) the differential regulated vacuum polarization tensor is obtained Π µν (x) = e 2 2π 2 ∂ µ ∂ ν ln 1 x − δ µν 1 2 ∂ 2 ln(M 1 x) ln 1 x + δ µν 1 2 ∂ 2 ln(M 2 x) ln 1 x = e 2 2π 2 ∂ µ ∂ ν − δ µν 1 2 ln M 1 M 2 ∂ 2 ln 1 x .(16) Eq.(6) gives the two-point function between the vector and axial vector currents Π 5 µν (x) = ǫ νρ Π ρµ (x) = e 2 2π 2 ǫ νρ ∂ ρ ∂ µ − δ ρµ 1 2 ∂ 2 ln M 1 M 2 ln 1 x .(17) Just like the four-dimensional case, the results (16) and (17) have displayed a most democratic expression for the two-dimensional chiral anomaly. With a general choice M 1 = e n M 2 ,(18) we obtain the two-point functions Π µν (x) = e 2 2π 2 ∂ µ ∂ ν − n 2 δ µν ∂ 2 ln 1 x , Π 5 µν (x) = e 2 2π 2 ǫ νρ ∂ ρ ∂ µ − n 2 δ ρµ ∂ 2 ln 1 x .(19) and the corresponding vector and axial vector Ward identities, ∂ µ Π 5 µν (x) = n 2 − 1 e 2 π ǫ νµ ∂ µ δ (2) (x), ∂ ν Π 5 µν (x) = n 2 e 2 π ǫ νµ ∂ ν δ (2) (x).(20) The choice n = 2 is the case we are familiar with, where the VWI is satisfied, while AWI becomes anomalous, ∂ µ Π µν (x) = ∂ ν Π µν (x) = ∂ µ Π 5 µν (x) = 0, ∂ ν Π 5 µν (x) = − e 2 2π ǫ νµ ∂ ν ∂ 2 ln 1 x = e 2 π ǫ νµ ∂ ν δ (2) (x).(21) The n = 0 choice just corresponds to the conservation of axial vector current but an anomalous vector current. In any choices, both of the vector and axial Ward identities cannot be fulfilled simultaneously. The massive case can be calculated in a similar way. The vacuum polarization tensor is Π µν (x) = − e 2 4π 2 Tr [γ µ (/ ∂ − m) K 0 (mx)γ ν (−/ ∂ − m) K 0 (mx)] = e 2 4π 2    Tr (γ µ γ α γ ν γ β ) x α x β 1 x d dx K 0 (mx) 2 − m 2 Tr(γ µ γ ν ) [K 0 (mx)] 2    = e 2 2π 2   2x µ x ν mK 1 (mx) x 2 − m 2 δ µν [K 0 (mx)] 2 + [K 1 (mx)] 2   .(22) Making use of the following two-dimensional differential operations, 2x µ x ν m 2 [K 1 (mx)] 2 x 2 = ∂ µ ∂ ν m 2 x 2 [K 0 (mx)] 2 − [K 1 (mx)] 2 + mxK 0 (mx)K 1 (mx) −δ µν m 2 [K 0 (mx)] 2 − [K 1 (mx)] 2 ; 2m 2 [K 0 (mx)] 2 = ∂ 2 m 2 x 2 [K 0 (mx)] 2 − [K 1 (mx)] 2 + mxK 0 (mx)K 1 (mx) ,(23) we can write the vacuum polarization tensor of the massive QED 2 as the following form, Π µν (x) = e 2 2π 2 ∂ µ ∂ ν m 2 x 2 [K 0 (mx)] 2 − m 2 x 2 [K 1 (mx)] 2 + mxK 0 (mx)K 1 (mx) −δ µν m 2 [K 0 (mx)] 2 − [K 1 (mx)] 2 − δ µν m 2 [K 0 (mx)] 2 + [K 1 (mx)] 2 = e 2 2π 2 ∂ µ ∂ ν − δ µν ∂ 2 m 2 x 2 [K 0 (mx)] 2 − [K 1 (mx)] 2 −mxK 0 (mx)K 1 (mx)] + δ µν m 2 [K 1 (mx)] 2 − m 2 [K 1 (mx)] 2 .(24) The last two m 2 [K 1 (mx)] 2 terms of Eq.(24) cannot be canceled naively since they are singular as x∼0. Considering the short-distance expansion of K 1 (mx), K 1 (mx) x∼0 −→ 1 mx + mx 1 2 ln mx 2 + 1 2 γ − 1 4 + O(x 2 )(25) with γ being the Euler constant, we can see that in Eq.(24) the short-distance singularity is only carried by the leading term 1/x 2 , the other terms are finite and they are exactly canceled. Therefore, employing Eq.(15) again, we finally get the differential regulated form of the vacuum polarization tensor for the massive case, Π µν (x) = e 2 2π 2 ∂ µ ∂ ν − δ µν ∂ 2 m 2 x 2 [K 0 (mx)] 2 − [K 1 (mx)] 2 −mxK 0 (mx)K 1 (mx)] + δ µν 1 x 2 − 1 x 2 = e 2 2π 2 ∂ µ ∂ ν − δ µν ∂ 2 m 2 x 2 [K 0 (mx)] 2 − [K 1 (mx)] 2 −mxK 0 (mx)K 1 (mx)] + δ µν 1 2 ln M 1 M 2 ∂ 2 ln 1 x .(26) To check the Ward identities of the massive case, we need the two-point function (5), Π 5 µ (x) = − e 2 4π 2 Tr [γ µ (/ ∂ − m) K 0 (mx)γ 5 (−/ ∂ − m) K 0 (mx\|)] = ie 2 π 2 mǫ µν K 0 (mx)∂ ν K 0 (mx) = ie 2 2π 2 mǫ µν ∂ ν [K 0 (mx)] 2 .(27) In order to satisfy the VWI, we must impose M 1 = M 2 . Then using the identity for the two-dimensional Euclidean scalar propagator, ∂ 2 K 0 (mx) = m 2 K 0 (mx) − 2πδ (2) (x).(28) one can check that the VWI is satisfied and the AWI becomes anomalous, ∂ µ Π µν (x) = ∂ ν Π µν (x) = ∂ µ Π 5 µν (x) = 0; ∂ ν Π 5 µν = 2imΠ 5 µ (x) + e 2 π ǫ νµ ∂ ν δ (2) (x).(29) On the other hand, if we choose M 1 = e 2 M 2 , the AWI will be satisfied while the VWI is violated. Note that the mass scale choices for carrying the Ward identities out in the massive case are different from those in the massless case. Eqs. (16), (17), (26) and the calculation deriving them show that the origin of the twodimensional anomaly in the differential regularization is the same as in the four-dimensional case [13]. To regulate two different singular pieces, one has to introduce two different mass scales. The terms with these two mass scales actually can combine into a finite amplitude with a continuous one-parameter shift degree of freedom given by the quotient of these two scales. This degree of freedom cannot accommodate the vector and axial vector Ward identities simultaneously so that the two-dimensional chiral anomaly has to emerge and manifests itself in the vector and axial vector Ward identities on the same footing. In summary, we have obtained two-dimensional anomaly neatly in terms of differential regularization. In comparison with other approaches such as dimensional and Pauli-Villars regularization, this regularization scheme has clearly exhibited the nature of the two-dimensional chiral anomaly and especially, the anomaly has manifested itself in the vector and axial Ward identities impartially. Furthermore, the impartiality is achieved automatically in differential regularization and does not require an explicit particular choice of the renormalization conditions on the physical amplitude as other regularization schemes [19]. This has not only demonstrate the applicability of differential regularization to two-dimensional gauge theory, but also has revealed an intrinsic anomaly structure of the theory. Another point which should be emphasized is that we have overcome the difficulty of the gauge non-invariance of the two-point function of QED 2 in coordinate space. It is known that the vacuum polarization tensor calculated before in coordinate space is not gauge invariant, a covariant seagull term has to be added to gain the gauge symmetry [20]. Of course, the reason for this is that the singularity is not properly regulated. Here in the framework of differential regularization we have achieved the gauge invariance by an appropriate choice of the undetermined mass scales. The above result obtained in differential regularization is not accidental and the profound reason lies in the nice features presented by differential regularization itself [21] As it is shown above, the basic operation in differential regularization is replacing a singular term by the derivative of another less singular function. This operation has provided a possibility to add arbitrary local terms into the amplitude. Whenever performing such a operation, we are introducing into a new arbitrary local term into the quantum effective action. For the issue we are considering, the vacuum polarization tensor in QED 2 , this arbitrariness is parameterized by the ratio of the two renormalization scales, M 1 /M 2 . In fact, this case is exactly the phenomenon emphasized by Jackiw [22] recently. According to renormalization theory, the introduction of an arbitrary local term in the amplitude is equivalent to the addition of a finite counterterm to the Lagrangian. Therefore, differential regularization can yield the most general quantum effective action. In particular, as stated in the introduction, this local parameterized ambiguity in differential regularization can be fixed at the final stage of the calculation by inputing some physical requirements. This special feature of differential regularization has formed a sharp contrast to other regularization schemes such as dimensional, Pauli-Villars and cut-off regularization etc. These regularization methods, together with given renormalization prescription, can fix the arbitrary local terms automatically. Therefore, differential regularization can give a more universal result than any other regularization method, since it does not impose any preferred choice on the amplitude at the beginning of implementing the regularization. This is just the reason why the chiral anomaly can emerge in both vector and axial vector Ward identities on the same footing, while in dimensional and Pauli-Villars regularization method, the anomaly only reflects in the axial Ward identity since the vector gauge symmetry has already been fixed in these regularization schemes. ACKNOWLEDGMENTSThis work is supported by the Natural Sciences and Engineering Research Council of Canada. I am greatly indebted to Prof. G. Kunstatter for his important remarks and improvements on this manuscript. I would like to thank Dr. M. Carrington and Prof. R. Kobes for their encouragements and help. I am also grateful to Prof. R. Jackiw for his comments and especially informing us Ref.[22]. I am especially obliged to Dr. M. Perez-Victoria for his comments and enlightening discussions on differential regularization. Last but not the least, I would like to thank the referee for his pointing out a big mistake in the first submitted version. . 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[ "Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off", "Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off" ]	[ "Bharath Sriperumbudur \nDepartment of Statistics\nPennsylvania State University University Park\n16802PAUSA\n", "Nicholas Sterge [email protected] \nDepartment of Statistics\nPennsylvania State University University Park\n16802PAUSA\n" ]	[ "Department of Statistics\nPennsylvania State University University Park\n16802PAUSA", "Department of Statistics\nPennsylvania State University University Park\n16802PAUSA" ]	[]	Kernel methods are powerful learning methodologies that allow to perform non-linear data analysis. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature approximation, have been proposed to alleviate the problem. However, the statistical consistency of most of these approximate kernel methods is not well understood except for kernel ridge regression wherein it has been shown that the random feature approximation is not only computationally efficient but also statistically consistent with a minimax optimal rate of convergence. In this paper, we investigate the efficacy of random feature approximation in the context of kernel principal component analysis (KPCA) by studying the trade-off between computational and statistical behaviors of approximate KPCA. We show that the approximate KPCA is both computationally and statistically efficient compared to KPCA in terms of the error associated with reconstructing a kernel function based on its projection onto the corresponding eigenspaces. The analysis hinges on Bernstein-type inequalities for the operator and Hilbert-Schmidt norms of a selfadjoint Hilbert-Schmidt operator-valued U-statistics, which are of independent interest. MSC 2010 subject classification: Primary: 62H25; Secondary: 62G05.	null	[ "https://arxiv.org/pdf/1706.06296v4.pdf" ]	88,515,097	1706.06296	5126d33f4e370442b36ec41a30dd63a5d1d68609	Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off 11 Jun 2022 Bharath Sriperumbudur Department of Statistics Pennsylvania State University University Park 16802PAUSA Nicholas Sterge [email protected] Department of Statistics Pennsylvania State University University Park 16802PAUSA Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off 11 Jun 2022arXiv:1706.06296v4 [stat.ML]and phrases: Principal component analysiskernel PCArandom feature approxi- mationreproducing kernel Hilbert spacecovariance operatorU-statisticsBernstein's inequality Kernel methods are powerful learning methodologies that allow to perform non-linear data analysis. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature approximation, have been proposed to alleviate the problem. However, the statistical consistency of most of these approximate kernel methods is not well understood except for kernel ridge regression wherein it has been shown that the random feature approximation is not only computationally efficient but also statistically consistent with a minimax optimal rate of convergence. In this paper, we investigate the efficacy of random feature approximation in the context of kernel principal component analysis (KPCA) by studying the trade-off between computational and statistical behaviors of approximate KPCA. We show that the approximate KPCA is both computationally and statistically efficient compared to KPCA in terms of the error associated with reconstructing a kernel function based on its projection onto the corresponding eigenspaces. The analysis hinges on Bernstein-type inequalities for the operator and Hilbert-Schmidt norms of a selfadjoint Hilbert-Schmidt operator-valued U-statistics, which are of independent interest. MSC 2010 subject classification: Primary: 62H25; Secondary: 62G05. Introduction Let X be a random variable distributed according to a probability measure P defined on a measurable space X . Principal component analysis (PCA) (Jolliffe, 1986) deals with finding a direction a ∈ X (= R d ) with a 2 = 1 such that Var[a ⊤ X] is maximized. More generally, it provides a low-dimensional representation that retains as much variance as possible of X and is used as a popular statistical methodology for dimensionality reduction and feature extraction. In fact, the lowdimensional representation is the orthogonal projection of X onto the ℓ-eigenspace, i.e., the span of eigenvectors associated with top ℓ eigenvalues of the covariance matrix EXX ⊤ − EXEX ⊤ where ℓ < d, resulting in a ℓ-dimensional representation. A non-linear generalization of PCA (called kernel PCA) was proposed by Schölkopf et al. (1998) which solves sup{Var[f (X)] : f H = 1}, where H is a reproducing kernel Hilbert space (RKHS) (Aronszajn, 1950), with reproducing kernel k : X × X → R (see Section 2 for definition). Similar to linear PCA, the solution turns out to be the eigenfunction corresponding to the top eigenvalue of the covariance operator, Σ = EΦ(X) ⊗ H Φ(X) − EΦ(X) ⊗ H EΦ(X), where Φ(x) := k(·, x) is called the feature map. More generally, kernel PCA provides a Euclidean representation for X by projecting Φ(X) onto the ℓ-eigenspace of Σ. Clearly, if Φ(x) = x, x ∈ X = R d (which corresponds to the linear kernel, k(x, y) = x, y 2 , x, y ∈ R d ), then kernel PCA reduces to linear PCA. On the other hand, depending on the choice of k, higher order moments of X are considered through the second order moment of Φ(X) to compute the ℓ-dimensional representation for X, resulting in a non-linear interpretation of dimensionality reduction. Due to this, KPCA is popular in applications such as image denoising (Mika et al., 1999;Jade et al., 2003;Teixeira et al., 2008;Phophalia and Mitra, 2017), image/systems modeling (Kim et al., 2005;Li et al., 2015), novelty/fault detection (Hoffmann, 2007;Samuel and Cao, 2016;de Moura and de Seixas, 2017), feature extraction (Chang and Wu, 2015), and computer vision (Lampert, 2009;Peter et al., 2019). We refer the reader to Section 2 for notation and Section 3 for preliminaries on KPCA and its variants. Given X 1 , . . . , X n i.i.d. ∼ P with P being unknown, empirical version of linear PCA computes the eigenvectors of the empirical covariance matrix onto which (X i ) n i=1 are projected to obtain a low-dimensional representation. Similarly, the empirical version of kernel PCA (we refer to it as EKPCA) involves finding the eigenfunctions of the empirical covariance operator Σ = 1 2n(n − 1) n i =j (Φ(X i ) − Φ(X j )) ⊗ H (Φ(X i ) − Φ(X j )) . While this requires solving a possibly infinite dimensional eigenvalue problem, it can be shown that these eigenfunctions can be computed by only solving a finite dimensional eigenvalue problem (see Proposition 1). In particular, it involves finding the eigenvectors of the Gram matrix, [k(X i , X j )] n i,j=1 which has a computational requirement of O(n 2 ℓ) where ℓ is the number of eigenvectors of interest. In addition to KPCA, more generally, most of the kernel algorithms (see Schölkopf and Smola, 2002) have a space complexity requirement of O(n 2 ) and time complexity requirement of O(n 3 ) as in some sense, all of them involve an eigen decomposition of the Gram matrix. However, in big data scenarios where n is large, the kernel methods including KPCA suffer from large space and time complexities. An elegant approach to address this computational issue is to approximate the feature map Φ by a finite-dimensional map Φ m , i.e., Φ m (x) ∈ R m so that Σ is approximated as Σ m = 1 2n(n − 1) n i =j (Φ m (X i ) − Φ m (X j )) (Φ m (X i ) − Φ m (X j )) ⊤ . Clearly, this is equivalent to performing linear PCA on the mapped data (Φ m (X i )) n i=1 , which involves finding the eigensystem of Σ m . Since this has a computational complexity of O(m 2 ℓ+m 2 n), the computational burden is reduced from O(n 2 ℓ) if m < √ nℓ. However, since this computational gain may be achieved at the cost of statistical performance, the goal of this paper is to investigate the trade-off between computational and statistical efficiency of approximate empirical KPCA (we refer to it as RF-EKPCA) using a random finite dimensional approximation of Φ(X). In the following, we briefly introduce the idea of random feature approximation introduced by Rahimi and Recht (2008), which involves computing a finite dimensional feature map that approximates the kernel function. Suppose say k is a continuous translation invariant kernel on R d , i.e., k(x, y) = ψ(x − y), x, y ∈ R d where ψ is a continuous positive definite function on R d . Bochner's theorem (Wendland, 2005, Theorem 6.6) states that ψ is the Fourier transform of a finite non-negative Borel measure Λ on R d , i.e., k(x, y) = R d e − √ −1 x−y,ω 2 dΛ(ω) (⋆) = R d cos( x − y, ω 2 ) dΛ(ω),(1) where ·, · 2 denotes the usual Euclidean inner product and (⋆) follows from the fact that ψ is real-valued and symmetric. Since Λ(R d ) = ψ(0), we can write (1) as k(x, y) = ψ(0) R d cos( x − y, ω 2 ) d Λ ψ(0) (ω) where Λ ψ(0) is a probability measure on R d . Therefore, without loss of generality, throughout the paper we assume that Λ is a probability measure. Rahimi and Recht (2008) proposed a random approximation to k by replacing the integral with Monte Carlo sums constructed from (ω i ) m i=1 i.i.d. ∼ Λ, i.e., k m (x, y) = ψ m (x − y) = 1 m m i=1 cos( x − y, ω i 2 ) ( †) = Φ m (x), Φ m (y) 2 , where Φ m = 1 √ m (cos ·, ω 1 2 , . . . , cos ·, ω m 2 , sin ·, ω 1 2 , . . . , sin ·, ω m 2 ) ⊤ and ( †) holds based on the trigonometric identity: cos(a−b) = cos a cos b+sin a sin b. This kind of random approximation to k can be constructed for a more general class of kernels of the form k(x, y) = Θ ϕ(x, θ)ϕ(y, θ) dΛ(θ) by using k m (x, y) = 1 m m i=1 ϕ(x, θ i )ϕ(y, θ i ) = Φ m (x), Φ m (y) 2 , where Φ m = 1 √ m (ϕ(·, θ 1 ), . . . , ϕ(·, θ m )) ⊤ , ϕ(x, ·) ∈ L 2 (Θ, Λ) for all x ∈ X , (θ i ) m i=1 i.i.d. ∼ Λ, with X and Θ being measurable spaces. Based on this approximation, the question of interest is whether RF-EKPCA consistent and how should m depend on n for RF-EKPCA to have similar statistical behavior to that of EKPCA, while still maintaining the computational edge. The goal of this paper is to address these questions. Contributions The main contributions of the paper are as follows: (i) In Section 4, we compare the performance of RF-EKPCA with EKPCA in terms of the reconstruction error of the associated ℓ-eigenspace, i.e., the error involved in reconstructing Φ(X) based on its projections onto the corresponding ℓ-eigenspace. Since the ℓ-eigenspace associated with RF-EKPCA is a subspace of R m in contrast to H as is the case with EKPCA, the notion of projecting Φ(X) ∈ H onto a subspace of R m is vacuous. To alleviate the problem, we define inclusion and approximation operators that embed both H and R m as subspaces in L 2 (P). This, however, results in two different notions of reconstruction error: First reconstructing in H and R m and then embedding the reconstructed functions in L 2 (P), which we refer to as Reconstruct and Embed (R-E), in contrast to first embedding the functions into L 2 (P) and then reconstructing in L 2 (P), which we refer to as Embed and Reconstruct (E-R). In Propositions 2, 3, and 4, we provide a new reformulation of KPCA, EKPCA, and RF-EKPCA as minimizers of appropriate E-R and R-E reconstruction errors. This reformulation provides a generalization error type interpretation which can be used to investigate the statistical behavior of EKPCA and RF-EKPCA. Since PCA is a special case of KPCA, this reformulation also provides a novel interpretation for classical PCA as a minimizer of covariance matrix weighted reconstruction error. (ii) In Sections 4.2 and 4.3, we show that RF-EKPCA has better computational complexity with no loss in statistical performance than EKPCA as long as m, which grows monotonically with ℓ, is large enough with ℓ not being too large (Theorems 6 and 8). In other words, the number of eigen functions, ℓ used in the reconstruction cannot grow too fast with the sample size n while requiring enough number of random features m so that the approximation error does not dominate the estimation error. By specializing Theorems 6 and 8 to the cases of polynomial and exponential decay rates of the eigenvalues of Σ, in Corollaries 7, A.1 and 10, A.2, respectively, we show R-E and E-R reconstruction errors to have different statistical behaviors. However, under each of these reconstruction errors, as mentioned above, RF-EKPCA matches the statistical performance of EKPCA at better computational complexity. (iii) In Section 4.4, we investigate a generalization of R-E (similar generalization also holds for E-R) based on certain weighted L 2 (P)-norms that are weighted by (II * ) −s/2 , s ≤ 1 with I being the inclusion operator, wherein the choice of s = 1 yields a reconstruction error that matches with the reconstruction error for KPCA in the H-norm and s = 0 matches with the R-E reconstruction error considered in Section 4.2. In Proposition 11, we again provide a new reformulation of KPCA, EKPCA, and RF-EKPCA as minimizers of the generalized R-E reconstruction errors, using which we establish a similar result as aforementioned that RF-EKPCA has same statistical complexity and better computational complexity than EKPCA as long as m is sufficiently large with ℓ being sufficiently small with respect to the growth of n (see Theorem 12 and Remark 7). All these results hinge on Bernstein-type inequalities for the operator and Hilbert-Schmidt norms of a self-adjoint Hilbert-Schmidt operator-valued U-statistics, which are of independent interest (see Theorem D.3). Related work To the best of our knowledge, not much investigation has been carried out on the statistical analysis of RF-EKPCA. Lopez-Paz et al. (2014) studied the quality of approximation of the Gram matrix by the approximate Gram matrix (using random Fourier features) in operator norm and showed a convergence rate of n( (log n)/m + (log n)/m). This approximation bound is too loose as we require m to grow faster than n to achieve convergence to zero, which defeats the purpose of random feature approximation. More recently, based on (Blanchard et al., 2007) and an earlier version of this work (Sriperumbudur and Sterge, 2017), using inclusion and approximation operators, Ullah et al. (2018) compared the ℓ-eigenspaces (with ℓ fixed) of EKPCA and RF-EKPCA by comparing certain inner product of the uncentered covariance operator with the difference between the projection operators associated with ℓ-eigenspaces of KPCA and EKPCA (resp. RF-EKPCA), after embedding them all as Hilbert-Schmidt operators on L 2 (P). Through upper bounds on these differences of inner products, they argued that m = √ n random features are sufficient for RF-EKPCA to have similar statistical behavior to that of EKPCA, thereby guaranteeing better computational complexity at no statistical loss. However, the work lacks on two fronts: (i) The comparison is made using only upper bounds on the performance criterion (i.e., difference of inner products) and no matching lower bounds are provided to establish their sharpness, which means the sufficiency of √ n random features is inconclusive, and (ii) the criterion used for comparison has no clear interpretation. In contrast, in this work, we use performance criteria which have a clear interpretation and establish matching upper and lower bounds on their statistical behavior. On the other hand, statistical behavior of EKPCA is well understood. Shawe-Taylor et al. (2005) studied the statistical consistency of EKPCA in terms of the reconstruction error of the estimated ℓ-eigenspace and obtained a convergence rate of n −1/2 . By taking into account the decay rate of the eigenvalues of the covariance operator, improved rates are obtained by Blanchard et al. (2007) and Rudi et al. (2013). However, unlike in this paper where the reconstruction error is defined in terms of convergence in L 2 (P), these works consider convergence in H. The question of convergence of ℓ-eigenspaces associated with EKPCA was considered by Zwald and Blanchard (2006) as convergence of orthogonal projection operators on H in Hilbert-Schmidt norm and obtained a convergence rate of n −1/2 . In the discussion so far, we only considered random feature approximation to Φ. At a broader level, to address the computational issues, various other approximation methods have been proposed and investigated in the kernel methods literature. Some of the popular approximation strategies include the incomplete Cholesky factorization (Fine and Scheinberg, 2001;Bach and Jordan, 2005), Nyström method (e.g., see Williams and Seeger, 2001;Drineas and Mahoney, 2005), sketching (Yang et al., 2017), sparse greedy approximation (Smola and Schölkopf, 2000), etc. While it has been widely accepted that these approximate methods including random feature approximation provide significant computational advantages and has been empirically shown to provide learning algorithms or solutions that do not suffer from significant deterioration in performance compared to those without approximation (Rahimi and Recht, 2008;Kumar et al., 2009;Yang et al., 2012Yang et al., , 2017, until recently, the statistical consistency of these approximate methods is not well understood. In fact, over the last few years, the statistical behavior of these approximation schemes have been investigated only in the context of kernel ridge regression, wherein it has been shown (Bach, 2013;Alaoui and Mahoney, 2015;Rudi et al., 2015;Yang et al., 2017;Rudi and Rosasco, 2017) that Nyström, random feature and sketching based approximate kernel ridge regression are consistent and achieve minimax rates of convergence as achieved by the exact methods but using fewer features than the sample size. This means, these approximate kernel ridge regression algorithms are not only computationally efficient compared to their exact counterpart but also statistically efficient, i.e., achieve the best possible convergence rate. On the other hand, the theoretical behavior of approximate kernel algorithms other than approximate kernel ridge regression is not well understood. This paper provides a theoretical understanding on the question of computational vs. statistical trade-off in random feature based approximate kernel PCA. Definitions & Notation Define a 2 := d i=1 a 2 i and a, b 2 := d i=1 a i b i , where a := (a 1 , . . . , a d ) ∈ R d and b := (b 1 , . . . , b d ) ∈ R d . a ⊗ 2 b := ab ⊤ denotes the tensor product of a and b. I n denotes an n × n identity matrix. We define 1 n := (1, n . . ., 1) ⊤ and H n := I n − 1 n 1 n ⊗ 2 1 n . δ ij denotes the Kronecker delta. a ∧ b := min(a, b) and a ∨ b := max(a, b). [n] := {1, . . . , n} for n ∈ N. For constants a and b, a b (resp. a b) denotes that there exists a positive constant c (resp. c ′ ) such that a ≤ cb (resp. a ≥ c ′ b). For a random variable A with law P and a constant b, A P b denotes that for any δ > 0, there exists a positive constant c δ < ∞ such that P (A ≤ c δ b) ≥ δ. For a topological space X , M b + (X ) denotes the set of all finite non-negative Borel measures on X . For µ ∈ M b + (X ), L r (X , µ) denotes the Banach space of r-power (r ≥ 1) µ-integrable functions. For f ∈ L r (X , µ), f L r (µ) := X \|f \| r dµ 1/r denotes the L r -norm of f for 1 ≤ r < ∞. µ n := µ× n . . . ×µ is the n-fold product measure. H denotes a reproducing kernel Hilbert space with a reproducing kernel k : X × X → R. S ∈ L(H) is called self-adjoint if S * = S and is called positive if Sx, x H ≥ 0 for all x ∈ H, where L(H) is the space of bounded linear operators on a Hilbert space H and S * denotes the adjoint of S. α ∈ R is called an eigenvalue of S ∈ L(H) if there exists an x = 0 such that Sx = αx and such an x is called the eigenvector /eigenfunction of S and α. An eigenvalue is said to be simple if it has multiplicity one. S L r (H) denotes the trace, Hilbert-Schmidt and operator norms of a self-adjoint operator S ∈ L(H) when r = 1, 2 and ∞, respectively. For x, y ∈ H, x ⊗ H y is an element of the tensor product space H ⊗ H which can also be seen as an operator from H to H as (x ⊗ H y)z = x y, z H for any z ∈ H. Variants of Kernel PCA: Population, Empirical and Approximate In this section, we review kernel PCA (Schölkopf et al., 1998) in population and empirical settings and introduce approximate kernel PCA based on random features. This section not only provides preliminaries on kernel PCA but also fixes some notation that will be used throughout the paper. To start with, we assume the following for the rest of the paper: (A 1 ) (X , B) is a second countable (i.e., completely separable) space endowed with Borel σ-algebra B. (H, k) is an RKHS of real-valued functions on X with a bounded continuous strictly positive definite kernel k satisfying sup x∈X k(x, x) =: κ < ∞. The second countability of X and B being countably generated ensure that for any σ-finite measure µ defined on B, L r (X , µ) is separable for any r ∈ [1, ∞) (Cohn, 2013, Proposition 3.4.5). The second countability of X and continuity of k ensures H is separable (Steinwart and Christmann, 2008, Lemma 4.33). The separability of H and k being bounded continuous ensures that k(·, x) : X → H is Bochner-measurable for all x ∈ X (Dinculeanu, 2000, Theorem 8 on p.5). The separability of L r (X , µ) and Bochner-measurability of k(·, x) will be crucial in our analysis. PCA in Reproducing Kernel Hilbert Space As mentioned in Section 1, kernel PCA extends the idea of PCA in R d to an RKHS by finding a function f ∈ H such that Var[f (X)] is maximized, i.e., sup {Var[f (X)] : f H = 1} = sup E [f (X) − E [f (X)]] 2 : f H = 1 . Since f ∈ H, using the reproducing property f (X) = f, k(·, X) H , we have Var[f (X)] = E [ f, k(·, X) H − f, m P H ] 2 where m P := X k(·, x) dP(x) ∈ H is the mean element of P, de- fined as: for all f ∈ H, f, m P H = E [f (X)]. The boundedness of k guarantees that m P is well-defined as it ensures k(·, X) is P-integrable in the Bochner sense (see Diestel and Uhl, 1977, Definition 1 and Theorem 2). Therefore, Var[f (X)] = E f, k(·, X) − m P 2 H (⋆) = f, Σf H , where (⋆) follows from the Riesz representation theorem and the boundedness of k, which combinedly guarantee the Bochner P-integrability of k(·, X) ⊗ H k(·, X). Here Σ := X (k(·, x) − m P ) ⊗ H (k(·, x) − m P ) dP(x)(2) is the covariance operator on H whose action on f ∈ H is defined as Σf = X k(·, x)f (x) dP(x) − m P X f (x) dP(x) . Therefore, the kernel PCA problem exactly resembles classical PCA where the goal is to find f ∈ H that solves sup { f, Σf H : f H = 1} with Σ being defined as in (2). Since k is bounded, it can be shown (see Proposition C.2(iii)) that Σ is a trace-class operator and therefore Hilbert-Schmidt and compact. Also it is obvious that Σ is self-adjoint and positive and therefore by spectral theorem (Reed and Simon, 1980, Theorems VI.16, VI.17), Σ can be written as Σ = i∈I λ i φ i ⊗ H φ i ,(3) where (λ i ) i∈I ⊂ R + are the eigenvalues and (φ i ) i∈I are the orthonormal system of eigenfunctions of Σ that span Ran(Σ) with the index set I being either countable in which case λ i → 0 as i → ∞ or finite. It is therefore obvious that the solution to KPCA is an eigenfunction of Σ corresponding to the largest eigenvalue. Throughout the paper, we assume that (A 2 ) The eigenvalues (λ i ) i∈I of Σ in (2) are simple, positive and without any loss of generality, they satisfy a decreasing rearrangement, i.e., λ 1 > λ 2 > · · · . (A 2 ) ensures that (φ i ) i∈I form an orthonormal basis and the eigenspace corresponding to λ i for any i ∈ I is one-dimensional. This means, the orthogonal projection operator onto span{ (φ i ) ℓ i=1 } is given by P ℓ (Σ) = ℓ i=1 φ i ⊗ H φ i . Empirical Kernel PCA In practice, P is unknown and the knowledge of P is available only through random samples (X i ) n i=1 drawn i.i.d. from it. The goal of empirical kernel PCA (EKPCA) is therefore to find f ∈ H such that Var[f (X)] := 1 2n(n − 1) n i =j (f (X i ) − f (X j )) 2 , i.e., the empirical variance, is maximized. Note that this is an estimate of Var[f (X)] = E[f 2 (X)] − E 2 [f (X)] = f 2 (x) dP(x) − f, m P 2 H based on the U -statistic representation, although in the literature (e.g., Schölkopf et al. 1998), assuming m P = 0, a V -statistic form, i.e., 1 n n i=1 f 2 (X i ) is used. However, it is important to note that the assumption of m P = 0 is not satisfied by many kernels including the Gaussian kernel, and if this assumption is relaxed, the corresponding V -statistic form is not unbiased. Since unbiasedness turns to be crucial in our analysis, we choose the above U -statistic form though from the point of view of methodology alone, the V -statistic can be equally used. Using the reproducing property, it is easy to show that Var[f (X)] = f, Σf H where Σ : H → H, Σ := 1 2n(n − 1) n i =j (k(·, X i ) − k(·, X j )) ⊗ H (k(·, X i ) − k(·, X j ))(4) is an unbiased estimator (U -statistic) of Σ, referred to as the empirical covariance operator. Since Σ is a self-adjoint operator on (a possibly infinite dimensional) H with rank at most n−1 (therefore, compact), it follows from the spectral theorem (Reed and Simon, 1980, Theorems VI.16, VI.17) that Σ = n−1 i=1 λ i φ i ⊗ H φ i ,(5) where ( λ i ) n−1 i=1 and ( φ i ) n−1 i=1 are the eigenvalues and eigenfunctions of Σ. In fact, since k is strictly positive definite, it can be shown that rank( Σ) = n − 1 P-a.s., and therefore, similar to (A 2 ), we assume the following: (4) are simple P-a.s. and without loss of generality, they satisfy a decreasing rearrangement, i.e., λ 1 > λ 2 > · · · P-a.s. (A 3 ) The eigenvalues ( λ i ) n−1 i=1 of Σ in We would like to mention that the simplicity of the eigenvalues of Σ is not really required for the results of this paper to hold. However, this assumption simplifies the notation and proofs, and therefore for the sake of simplicity and clarity, we resort to the above assumption. Based on (A 3 ), a low-dimensional Euclidean representation of X i ∈ X can be obtained as k(·, X i ), φ 1 H , . . . , k(·, X i ), φ ℓ H ⊤ = φ 1 (X i ), . . . , φ ℓ (X i ) ⊤ ,(6) where ℓ < n − 1 and i ∈ [n]. Clearly, the choice of k(·, x) = ·, x 2 for x ∈ R d in (6) reduces to the usual low-dimensional representation using linear PCA. Under (A 3 ), we denote the orthogonal projection operator onto span{( φ i ) ℓ i=1 } as P ℓ ( Σ), which is given by P ℓ ( Σ) = ℓ i=1 φ i ⊗ H φ i . Note that the Euclidean representation in (6) requires the knowledge of ( φ i ) n−1 i=1 , which are not obvious to compute even though Σ has finite rank, as they are solution to a possibly infinite dimensional eigen problem. The following result (proved in Section 6.1) shows that the eigensystem ( λ i , φ i ) n−1 i=1 of Σ can be obtained by finding the eigensystem of a n × n matrix. This means the computation of ( λ i , φ i ) ℓ i=1 for ℓ ≤ n has a space complexity of O(n 2 ) and a time complexity of O(n 2 ℓ)-e.g., by partial SVD methods such as Krylov subspace method (see Halko et al., 2011, Sections 3.3.2 & 3.3.3). j∈[n] and H n := I n − 1 n 1 n ⊗ 2 1 n . Then ( λ i , α i ) i are the eigenvalues and eigenvectors of 1 n−1 KH n with Proposition 1. Let ( λ i , φ i ) i be the eigensystem of Σ in (5). Define K = [k(X i , X j )] i,φ i = 1 λ i √ n n j=1 γ i,j k(·, X j ), where γ i := (γ i,1 , . . . , γ i,n ) ⊤ = n n−1 H n α i with α i / ∈ N (H n ). Using representer theorem (Kimeldorf and Wahba, 1971), Schölkopf et al. (1998) have shown a similar result for EKPCA but with uncentered covariance operator (i.e., Σ with m P = 0) when K is invertible. Since m P = 0 is not a valid assumption for many kernels, Proposition 1 handles the U-statistic version of the centered covariance operator without using representer theorem and without requiring K to be invertible. Approximate Kernel PCA using Random Features In this section, we present approximate kernel PCA using random features, which we call as RF-KPCA. Throughout this section, we assume the following: (A 4 ) H is an RKHS with reproducing kernel k of the form k(x, y) = Θ ϕ(x, θ)ϕ(y, θ) dΛ(θ) = ϕ(x, ·), ϕ(y, ·) L 2 (Λ) , where ϕ : X × Θ → R is continuous, sup θ∈Θ,x∈X \|ϕ(x, θ)\| ≤ √ κ and Λ is a probability measure on a second countable space (Θ, A) endowed with Borel σ-algebra A. The assumption of Λ being a probability measure on Θ is not restrictive as any Λ ∈ M b + (Θ) can be normalized to a probability measure. However, the uniform boundedness of ϕ over X × Θ is somewhat restrictive as it is sufficient to assume ϕ(x, ·) ∈ L 2 (X , Λ), ∀ x ∈ X for k to be welldefined. But the uniform boundedness of ϕ ensures that k is bounded, as assumed in (A 1 ). By sampling (θ i ) m i=1 i.i.d. ∼ Λ, an approximation to k can be constructed as k m (x, y) = 1 m m i=1 ϕ(x, θ i )ϕ(y, θ i ) =: m i=1 ϕ i (x)ϕ i (y) = Φ m (x), Φ m (y) 2 , where ϕ i := 1 √ m ϕ(·, θ i ) and Φ m (x) := (ϕ 1 (x), . . . , ϕ m (x)) ⊤ ∈ R m is the random feature map. It is easy to verify that k m is the reproducing kernel of the RKHS H m = f : f = m i=1 β i ϕ i , (β i ) m i=1 ⊂ R w.r.t. ·, · Hm defined as f, g Hm := m i=1 α i β i where g = m i=1 α i ϕ i . Therefore H m is isometrically isomorphic to R m . We refer the reader to (Rudi and Rosasco, 2017, Appendix E) for examples of ϕ that yield some widely used reproducing kernels. Having obtained a random feature map, the idea of RF-KPCA is to perform linear PCA on Φ m (X) where X ∼ P, i.e., RF-KPCA involves finding a direction β ∈ R m such that the variance of β, Φ m (X) 2 is maximized: sup {Var[ β, Φ m (X) 2 ] : β 2 = 1} = sup { β, Ω m β 2 : β 2 = 1} ,(7) where Ω m := Cov[Φ m (X)] = E[(Φ m (X) − E[Φ m (X)]) ⊗ 2 (Φ m (X) − E[Φ m (X)]) ] is a self-adjoint positive definite matrix. In fact, it is easy to verify that performing linear PCA on Φ m (X) is same as performing KPCA in H m since sup {Var[f (X)] : f Hm = 1} = sup {Var[ β, Φ m (X) 2 ] : β 2 = 1} ,(8) which follows from H m being isometrically isomorphic to R m and f ∈ H m has the form f ( x) = β, Φ m (x) 2 . Note that sup {Var[f (X)] : f Hm = 1} = sup { f, Σ m f Hm : f Hm = 1} ,(9) where Σ m = X k m (·, x) ⊗ Hm k m (·, x) dP(x) − m P,m ⊗ Hm m P,m , and m P,m := k m (·, x) dP(x). It therefore follows from (7)-(9) that the eigenvalues of Σ m and Ω m coincide and the eigenfunctions, (φ m,i ) m i=1 of Σ m and eigenvectors, (β m,i ) m i=1 of Ω m are related as φ m,i (x) = β m,i , Φ m (x) 2 . The empirical counterpart of RF-KPCA (we call it as RF-EKPCA) is obtained by solving sup β 2 =1 Var[ β, Φ m (X) 2 ] = sup β 2 =1 β, Ω m β 2 = sup f Hm =1 f, Σ m f Hm , where Σ m = 1 2n(n − 1) n i =j (k m (·, X i ) − k m (·, X j )) ⊗ Hm (k m (·, X i ) − k m (·, X j )) is a self-adjoint positive definite operator on H m that is equivalent (in the above mentioned sense) to Ω m , which is a U -statistic estimator of Ω m . Since Σ m and Σ m are trace-class (see Proposition C.4(iii)) and self-adjoint, spectral theorem (Reed and Simon, 1980, Theorems VI.16, VI.17 ) yields that Σ m = m i=1 λ m,i φ m,i ⊗ Hm φ m,i and Σ m = m i=1 λ m,i φ m,i ⊗ Hm φ m,i , where (λ m,i ) m i=1 ⊂ R + (resp. ( λ m,i ) m i=1 ⊂ R + ) and (φ m,i ) m i=1 (resp. ( φ m,i ) m i=1 ) are the eigenvalues and eigenvectors of Σ m (resp. Σ m ). We will assume that (A 5 ) The eigenvalues (λ m,i ) m i=1 (resp. ( λ m,i ) m i=1 ) of Σ m (resp. Σ m ) are simple, positive and without any loss of generality, they satisfy a decreasing rearrangement, i.e., λ m,1 > λ m,2 > · · · (resp. λ m,1 > λ m,2 > · · · ) Λ-a.s. (resp. Λ × P-a.s.). Based on (A 5 ), a low-dimensional representation of X i ∈ X can be obtained as ( φ m,1 (X i ), . . . , φ m,ℓ (X i )) ⊤ ∈ R ℓ , ℓ ≤ m, i ∈ [n] . The orthogonal projection operators onto the ℓ-eigenspaces of Σ m and Σ m are given by P ℓ (Σ m ) = ℓ i=1 φ m,i ⊗ Hm φ m,i and P ℓ ( Σ m ) = ℓ i=1 φ m,i ⊗ Hm φ m,i , respectively. Since ( λ m,i , φ m,i ) ℓ i=1 for ℓ ≤ m is a subset of the eigensystem of Σ m (which is equivalent to the m × m matrix Ω m ), the associated time complexity of finding this set scales as O(m 2 ℓ + m 2 n), where O(m 2 n) is the complexity of computing Ω m . This implies that RF-EKPCA is computationally cheaper than EKPCA if m < √ nℓ for ℓ ≤ n, i.e., m = o( √ nℓ) as n, ℓ → ∞. Computational vs. Statistical Trade-off The main goal of this paper is to investigate whether the above mentioned computational saving achieved by RF-EKPCA is obtained at the cost of statistical "efficiency" or not. To this end, we investigate this question by using the reconstruction error as a measure of statistical performance. To elaborate, in linear PCA, the quality of reconstruction after projecting a random variable X ∈ R d onto the span of top ℓ eigenvectors of Σ is captured by the reconstruction error, given by E X∼P (X − µ) − ℓ i=1 (X − µ), φ i 2 φ i 2 2 , where (φ i ) i are the eigenvectors of Σ = E[XX ⊤ ] − µµ ⊤ with µ := E[X] . Since (φ i ) i form an orthonormal basis in R d , the above consideration makes sense and clearly, the choice of ℓ = d yields zero error. Since KPCA generalizes linear PCA-the choice of k(x, y) = x, y 2 reduces kernel PCA to linear PCA-, it is natural to consider the reconstruction error in KPCA and EKPCA to be E X∼P k(·, X) − ℓ i=1 ξ(X), ζ i H ζ i 2 H(10) with ξ(X) = k(·, X), ζ i = φ i and ξ(X) = k(·, X), ζ i = φ i respectively, where (φ i ) i and ( φ i ) i are the orthonormal eigenfunctions of Σ and Σ given in (3) and (5), corresponding to the eigenvalues (λ i ) i and ( λ i ) i satisfying (A 2 ) and (A 3 ) respectively. Here for any x ∈ X , k(·, x) = k(·, x) − X k(·, x) dP(x) and k(·, x) = k(·, x) − 1 n n i=1 k(·, X i ). Since empirical mean is used in empirical linear PCA to find principal components, we used k in (10) to measure the performance of EKPCA. However, similar performance measure as in (10) is not possible for RF-KPCA and RF-EKPCA as the orthonormal eigenvectors of Σ m and Σ m belong to H m (isometrically isomorphic to R m ) while k(·, X) and k(·, X) belong to H, which means the notion of respectively projecting k(·, X) and k(·, X) onto (φ m,i ) i and ( φ m,i ) i is vacuous. However, since both H and H m are subspaces of L 2 (P), it is natural to consider the reconstruction error in L 2 (P)-norm so that the behaviors of EKPCA and RF-EKPCA can be compared. To this end, define an inclusion operator (up to a constant) I : H → L 2 (P), f → f − f P , where f P := X f (x) dP(x). It can be shown (see Proposition C.2) that I * : L 2 (P) → H, f → X k(·, x)f (x) dP(x) − m P f P and Σ = I * I. Similarly, we define an approximation operator A : H m → L 2 (P), f = m i=1 β i ϕ i → m i=1 β i (ϕ i − ϕ i,P ) = f − f P , where ϕ i,P := X ϕ i (x) dP(x). It can be shown (see Proposition C.4) that A * : L 2 (P) → H m , f → m i=1 ( f, ϕ i L 2 (P) − f P ϕ i,P )ϕ i and Σ m = A * A. Based on these operators, we first consider alternate notions of reconstruction error in L 2 (P) for KPCA, EKPCA, RF-KPCA and RF-EKPCA in Section 4.1 and then present results comparing the statistical behavior of EKPCA and RF-EKPCA in Sections 4.2 and 4.3. Alternate Notions of Reconstruction Error Let (ψ i ) i ⊂ H and (µ i ) i ⊂ L 2 (P) be arbitrary collections of functions. We now define two notions of reconstruction error in L 2 (P) as follows: Reconstruct and Embed (R-E) R(ψ 1 , . . . , ψ ℓ ) = E X∼P Ik(·, X) − I ℓ i=1 k(·, X), ψ i H ψ i 2 L 2 (P) = E X∼P Ik(·, X) − I ℓ i=1 ψ i ⊗ H ψ i k(·, X) 2 L 2 (P) , where k(·, X) is first projected 1 and reconstructed along (ψ i ) i∈ [ℓ] in H, and then embedded into L 2 (P) through I; and Embed and Reconstruct (E-R) S(µ 1 , . . . , µ ℓ ) = E X∼P Ik(·, X) − ℓ i=1 Ik(·, X), µ i L 2 (P) µ i 2 L 2 (P) , where k(·, X) is first embedded into L 2 (P) through I and then projected and reconstructed along (µ i ) i∈ [ℓ] in L 2 (P). The following result (proved in Section 6.2) shows that the minimizers of R and S over any (ψ i ) i∈ [ℓ] in H and (µ i ) i∈ [ℓ] in L 2 (P) are precisely the PCA solutions in H and L 2 (P), respectively. Proposition 2. Suppose (A 1 ) and (A 2 ) hold. Then the following hold: (i) (φ 1 , . . . , φ ℓ ) = arg inf{R(ψ 1 , . . . , ψ ℓ ) : (ψ i ) ℓ i=1 ⊂ H}; (ii) Iφ 1 √ λ 1 , . . . , Iφ ℓ √ λ ℓ = arg inf{S(µ 1 , . . . , µ ℓ ) : (µ i ) ℓ i=1 ⊂ L 2 (P)}; (iii) R Σ,ℓ := R(φ 1 , . . . , φ ℓ ) = S Iφ 1 √ λ i , . . . , Iφ ℓ √ λ ℓ := S Σ,ℓ ; (iv) R Σ,ℓ ≤ Σ L ∞ (H) E X∼P k(·, X) − ℓ i=1 k(·, X), φ i H φ i 2 H . Remark 1. (i) Proposition 2(i) shows that the minimizer of the R-E reconstruction error is precisely the KPCA solution which is obtained by minimizing E k(·, X) − ℓ i=1 k(·, X), ψ i H ψ i 2 H over (ψ i ) i∈[ℓ] ⊂ H, i.e., R-E provides an alternate interpretation for KPCA. (ii) Since minimizing S is equivalent to performing PCA in L 2 (P), it follows that minimizers of S are the eigenfunctions of II * , which are precisely ( Iφ i √ λ i ) i∈[ℓ] with (λ i , φ i ) i∈[ℓ] being the eigenpairs of Σ = I * I (see the proof for details). (iii) Proposition 2(iv) implies that R Σ,ℓ is a weaker measure of reconstruction error than the one defined in (10), with the latter matching with the reconstruction error of linear PCA when k(x, y) = x, y 2 . More precisely, in Theorems 6 and 8, we will show that R Σ,ℓ = S Σ,ℓ = i>ℓ λ 2 i while the reconstruction error in (10) behaves as i>ℓ λ i , which clearly establishes R Σ,ℓ to be weaker than the one in (10). Later, in Section 4.4, we will generalize R such that KPCA's reconstruction error also behaves as i>ℓ λ i . Similar to KPCA, in the following result (proved in Section 6.3), we present alternate interpretations for EKPCA, RF-KPCA and RF-EKPCA as minimization of appropriate R-E reconstruction errors. Proposition 3. Suppose (A 1 ) − (A 5 ) hold. Define T A,H (P ) := A 1/2 (I − P )A 1/2 2 L 2 (H) where A : H → H is a positive self-adjoint Hilbert-Schmidt operator on H with H being a separable Hilbert space. Then the following hold. (i) R(ψ 1 , . . . , ψ ℓ ) = T Σ,H (P ψ ) where P ψ := ℓ i=1 ψ i ⊗ H ψ i ; (ii) (KPCA) (φ 1 , . . . , φ ℓ ) = arg inf{T Σ,H (P ψ ) : (ψ i ) i∈[ℓ] ⊂ H}; (iii) (EKPCA) ( φ 1 , . . . , φ ℓ ) = arg inf{T Σ,H (P ψ ) : (ψ i ) i∈[ℓ] ⊂ H}; (iv) (RF-KPCA) (φ m,1 , . . . , φ m,ℓ ) = arg inf{T Σm,Hm (P τ ) : (τ i ) i∈[ℓ] ⊂ H m } where P τ = ℓ i=1 τ i ⊗ Hm τ i ; (v) (RF-EKPCA) ( φ m,1 , . . . , φ m,ℓ ) = arg inf{T Σm,Hm (P τ ) : (τ i ) i∈[ℓ] ⊂ H m }. Remark 2. (i) Proposition 3(i) provides an alternate expression for the R-E error in terms of the Hilbert-Schmidt norm of a certain self-adjoint operator on H and Proposition 3(ii) is obvious from Proposition 2(i). Since this alternate expression depends on Σ, by replacing Σ with Σ (resp. Σ m , Σ m ), alternate formulation for EKPCA (resp. RF-KPCA, RF-EKPCA) can be provided as in Proposition 3(iii)-(v). (ii) Based on Proposition 3(iii), EKPCA can be interpreted as the minimizer of the following empirical R-E reconstruction error, R(ψ 1 , . . . , ψ ℓ ) = 1 n n j=1 I k(·, X j ) − I ℓ i=1 k(·, X j ), ψ i H ψ i 2 L 2 (Pn) = T Σ,H (P ψ ), where I and L 2 (P n ) are defined in Proposition 4(iii). Similar interpretation can be provided for RF-EKPCA (resp. RF-KPCA) by considering approximate empirical (resp. population) R-E reconstruction error which is defined by replacing I by A (resp. A) and k by k m (resp. k m ), respectively. (iii) The minimal value of T A,H for (A, H) = ( Σ, H), (Σ m , H m ) and ( Σ m , H m ) can be shown to be i>ℓ λ 2 i , i>ℓ λ 2 m,i , and i>ℓ λ 2 m,i , respectively, all of which are closely related to i>ℓ λ 2 i which is the minimum value of T Σ,H . The following result (proved in Section 6.4) is similar to Proposition 3 and shows the relation between the EKPCA (resp. RF-KPCA, RF-EKPCA) solution and the minimizer of appropriate empirical versions of the E-R reconstruction error. Proposition 4. Suppose (A 1 ) − (A 5 ) hold. Define V A,H (P ) := (I − P ) 2 A 2 L 2 (H) where A : H → H is a positive self-adjoint Hilbert-Schmidt operator on H with H being a separable Hilbert space. Then the following hold. (i) S(µ 1 , . . . , µ ℓ ) = V II * ,L 2 (P) (P µ ) where P µ := ℓ i=1 µ i ⊗ L 2 (P) µ i ; (ii) (KPCA) Iφ 1 √ λ 1 , . . . , Iφ ℓ √ λ ℓ = arg inf{V II * ,L 2 (P) (P µ ) : (µ i ) i∈[ℓ] ⊂ L 2 (P)}; (iii) (EKPCA) I φ 1 / λ 1 , . . . , I φ ℓ / λ ℓ = arg inf V I I * ,L 2 (Pn) (Q µ,n ) : (µ i ) i∈[ℓ] ⊂ L 2 (P n ) , where Q µ,n := ℓ i=1 µ i ⊗ L 2 (Pn) µ i , L 2 (P n ) := f : 1 n n i=1 f 2 (X i ) < ∞ , and I : H → L 2 (P n ), f → n n−1 f − 1 n n i=1 f (X i ) ; (iv) (RF-KPCA) Aφ m,1 / λ m,1 , . . . , Aφ m,ℓ / λ m,ℓ = arg inf (µ i ) i∈[ℓ] ⊂L 2 (P) V AA * ,L 2 (P) (P µ ) ; (v) (RF-EKPCA) A φ m,1 / λ m,1 , . . . , A φ m,ℓ / λ m,ℓ = arg inf (µ i ) i∈[ℓ] ⊂L 2 (Pn) V A A * ,L 2 (Pn) (Q µ,n ) ; where A : H m → L 2 (P n ), f → n n−1 f − 1 n n i=1 f (X i ) . Remark 3. (i) Proposition 4(ii) is obvious from Proposition 4(i) and Proposition 2(ii). Using Proposition 4(ii), alternate interpretation for EKPCA (resp. RF-EKPCA) can be obtained by replacing I with its empirical version I (resp. A) and P with its empirical version P n . (ii) V I I * ,L 2 (Pn) ( ℓ i=1 µ i ⊗ L 2 (Pn) µ i ) in Proposition 4(iii) can be shown to be equal to the following empirical E-R reconstruction error, S(µ 1 , . . . , µ ℓ ) = 1 n n j=1 I k(·, X j ) − ℓ i=1 I k(·, X j ), µ i L 2 (Pn) µ i 2 L 2 (Pn) . Similar interpretation can be provided for RF-EKPCA by replacing I and k by A and k m , respectively. (iii) Similar to T A,H , the minimal value of V A,H for (A, H) = ( I I * , L 2 (P n )), (AA * , L 2 (P)) and ( A A * , L 2 (P n )) can be shown to be i>ℓ λ 2 i , i>ℓ λ 2 m,i , and i>ℓ λ 2 m,i , respectively, which matches with their R-E counterparts. Using these alternate interpretations of KPCA and its variants, in the following sections, we will investigate and compare the performances of EKPCA and RF-EKPCA in R-E and E-R settings. Reconstruct and Embed (R-E) The performance of EKPCA and RF-EKPCA can be measured by how well the output of these algorithms (i.e., the reconstructed functions) approximate Ik(·, X) in L 2 (P) in expectation. To elaborate, since the principal components outputted by EKPCA and RF-EKPCA are k(·, X), φ i H and k m (·, X), φ m,i Hm , i ∈ [ℓ] along the directions ( φ i ) ℓ i=1 and ( φ m,i ) ℓ i=1 respectively, we compare their L 2 (P)-embedded versions to Ik(·, X), as defined below: R Σ,ℓ = E X∼P Ik(·, X) − I ℓ i=1 k(·, X), φ i H φ i 2 L 2 (P) ,(11)R Σm,ℓ = E X∼P Ik(·, X) − A ℓ i=1 k m (·, X), φ m,i Hm φ m,i 2 L 2 (P) ,(12) and R Σm,ℓ = E X∼P Ik(·, X) − A ℓ i=1 k m (·, X), φ m,i Hm φ m,i 2 L 2 (P) .(13)Note that T Σ,H (P ℓ (Σ)) measures the performance of KPCA where P ℓ (Σ) = ℓ i=1 φ i ⊗ H φ i . Therefore T Σ,H (P ℓ ( Σ)) can be considered as a performance measure of EKPCA with P ℓ ( Σ) = ℓ i=1 φ i ⊗ H φ i . Similarly, T Σm,Hm (P ℓ (Σ m )) and T Σm,Hm (P ℓ ( Σ m )) can be used as performance measures of RF-KPCA and RF-EKPCA, respectively. However, in T Σm,Hm , the reconstructed function is Ak m (·, X) in contrast to Ik(·, X) as in T Σ,H . In order to reconstruct the same function in all the algorithms, we consider the reconstruction errors defined in (11)-(13). The following result (proved in Section 6.5) shows that the reconstruction errors defined in (11)-(13) are statistically equivalent to T Σ,H (P ℓ ( Σ)), T Σm,Hm (P ℓ (Σ m )) and T Σm,Hm (P ℓ ( Σ m )). Theorem 5. Under the assumptions (A 1 ) − (A 5 ), the following hold: (i) R Σ,ℓ P n T Σ,H (P ℓ ( Σ)) P n R Σ,ℓ + 1 n ; (ii) R Σm,ℓ − 1 √ m 2 Λ m T Σm,Hm (P ℓ (Σ m )) Λ m R Σm,ℓ + 1 m ; (ii) R Σm,ℓ − 1 √ m 2 P n ×Λ m T Σm,Hm (P ℓ ( Σ m )) P n ×Λ m R Σm,ℓ + 1 n + 1 m . In the above result, the term 1 n is the error incurred by centering k(·, X) (resp. k m (·, X)) around m P (resp. m P,m ) instead of m P (resp. m P,m ). The term 1 m is the approximation error incurred by approximating Ik(·, X) by Ak m (·, X), i.e., E Ik(·, X) − Ak m (·, X) 2 L 2 (P) . The following result, which is proved in Section 6.6, provides a finite-sample bound on (11)-(13), using which convergence rates can be obtained. Theorem 6. Suppose (A 1 ) − (A 5 ) hold. For any t > 0, define N Σ (t) = tr(Σ(Σ + tI) −1 ) . Then the following hold: (i) R Σ,ℓ = i>ℓ λ 2 i . (ii) For any δ > 0 with n ≥ 2 log 2 δ and 140κ n log 16κn δ ≤ t ≤ Σ L ∞ (H) , P n (X i ) n i=1 : i>ℓ λ 2 i ≤ R Σ,ℓ ≤ 9N Σ (t)(λ ℓ+1 + t) 2 + 64κ 2 log 2 δ n ≥ 1 − 3δ. (iii) For any δ > 0 with m ≥ 2 ∨ 1024κ 2 i>ℓ λ 2 i log 2 δ , Λ m (θ i ) m i=1 : 1 4 i>ℓ λ 2 i ≤ R Σm,ℓ ≤ 4 i>ℓ λ 2 i + 256κ 2 log 2 δ m ≥ 1 − 6δ. (iv) For any δ > 0 with n ≥ 2 log 2 δ , m ≥ 2 ∨ 1024κ 2 i>ℓ λ 2 i log 2 δ and 140κ n log 16κn δ ∨ 86κ m log 16κm δ ≤ t ≤ Σ L ∞ (H) 3 , with probability at least 1 − 12δ over the choice of (X i ) n i=1 , (θ j ) m j=1 : 1 4 i>ℓ λ 2 i ≤ R Σm,ℓ ≤ 162A 1 (t)(λ ℓ+1 + t) 2 + 640κ 2 log 2 δ 3n + 256κ 2 log 2 δ m , where A 1 (t) := N Σ (t) + 16κ log 2 δ tm + 8κN Σ (t) log 2 δ tm . Since Σ is trace class it is obvious that λ ℓ → 0 and i>ℓ λ 2 i → 0 as ℓ → ∞. It therefore follows that R Σ,ℓ → 0 and R Σm,ℓ → 0 as ℓ, m → ∞. Further, by assuming a decay rate on (λ i ) i , a convergence rate for R Σ,ℓ and R Σm,ℓ may be obtained. Note that up to constants, R Σ,ℓ and R Σm,ℓ will have the same statistical behavior if m is chosen to be large enough that i>ℓ λ 2 i dominates 1 m . As in Theorem 6, the behavior of the empirical varieties depend on t and N Σ (t). N Σ (t) is referred to as the effective dimension or degrees of freedom (Caponnetto and Vito, 2007), and captures the complexity of H. Since N Σ (t) 1 t (better bound can be obtained if a certain decay rate for (λ i ) i is assumed), it is easy to see that R Σ,ℓ → 0 if ℓ, n → 0 and nλ 2 ℓ → ∞, and R Σm,ℓ → 0 if ℓ, m, n → 0 and λ 2 ℓ (m∧n) → ∞. However, in order to properly compare the behavior of EKPCA and RF-EKPCA to each other, as well as to their population counterparts, an assumption on the decay rate of (λ i ) i must be made, and the trade-off between t, λ ℓ and N Σ (t) must be explored. The following corollary (proved in Section 6.7) to Theorem 6 investigates the statistical behavior of EKPCA and RF-EKPCA in detail under the polynomial decay condition (exponential decay condition is analyzed in Corollary A.1) on the eigenvalues of Σ. Corollary 7 (Polynomial decay of eigenvalues). Suppose Ai −α ≤ λ i ≤Āi −α for α > 1 and A,Ā ∈ (0, ∞). Let ℓ = n θ α , 0 < θ ≤ α. Then (i) n −2θ(1− 1 2α ) R Σ,ℓ n −2θ(1− 1 2α ) . There existsñ ∈ N such that for all n >ñ, the following hold: (ii) n −2θ(1− 1 2α ) R Σ,ℓ P n n −2θ(1− 1 2α ) , θ ≤ α 2α−1 1 n , θ ≥ α 2α−1 ; (iii) For 0 < γ ≤ 1 and m = n γ , n −2θ(1− 1 2α ) 1 {γ≥θ(2− 1 α )} Λ m R Σm,ℓ Λ m n −2θ(1− 1 2α ) , γ ≥ θ 2 − 1 α , θ ≤ α 2α−1 n −γ , γ ≤ 1 ∧ θ 2 − 1 α ; (iv) For 0 < γ ≤ 1 and m = n γ , n −2θ(1− 1 2α ) 1 {γ≥θ(2− 1 α )} Λ m R Σm,ℓ P n ×Λ m n −2θ(1− 1 2α ) , γ ≥ θ 2 − 1 α , θ ≤ α 2α−1 n −γ , γ ≤ 1 ∧ θ 2 − 1 α . Remark 4. (i) The condition α > 1 is required to ensure that Σ is trace class. Comparing the behavior of R Σ,ℓ to that of R Σ,ℓ it is clear that EKPCA recovers optimal convergence rates (compared to that of KPCA) if ℓ grows to infinity not faster than n 1/(2α−1) . Since the reconstruction error is based on ℓ eigenfunctions, the computational complexity of EKPCA behaves as O(n 2 ℓ) = O(n 2+ θ α ). It is important to note that 0 < θ ≤ α 2α−1 is the only useful region both computationally and statistically as θ > α 2α−1 does not improve the statistical rates of EKPCA (than that achieved at θ = α 2α−1 ) but increases its computational complexity. (ii) Comparing R Σm,ℓ with R Σ,ℓ it is clear that if ℓ grows to infinity not faster than n 1/(2α−1) and the number of random features m grows sufficiently fast, then RF-EKPCA and EKPCA enjoy the same statistical behavior. The rate at which the number of random features must grow depends on the growth of ℓ through θ and α; the choice of 1 ≥ γ ≥ θ 2 − 1 α yields the same statistical behavior for RF-EKPCA, EKPCA, and KPCA. (iii) The computational complexity of RF-EKPCA is given by O(m 2 ℓ + m 2 n) = O(n 2γ+1 ) which is better than that of EKPCA if γ < 1 2 + θ 2α . This means, RF-EKPCA has a lower computational complexity with similar statistical behavior to that of EKPCA if 2θ − θ α ≤ γ < 1 2 + θ 2α and θ ≤ α 2α−1 respectively, which implies θ < α 4α−3 . In other words, if ℓ grows at a lower order than n 1/(4α−3) and the number of random features are larger than n θ(2− 1 α ) , then RF-EKPCA enjoys computational superiority with no loss in statistical performance over that of EKPCA. On the other hand, if ℓ grows at an order faster than n 1/(4α−3) but not faster than n 1/(2α−1) , it results in loss of computational advantage for RF-EKPCA while retaining the same statistical behavior to that of EKPCA-in fact, the rate in this regime is faster than in the previous regime of θ < α 4α−3 . Embed and Reconstruct (E-R) In this section, we compare EKPCA and RF-EKPCA in terms of E-R reconstruction error. Since ( I φ i / λ i ) ℓ i=1 and ( A φ m,i / λ m,i ) ℓ i=1 are the minimizers of the empirical E-R reconstruction error (see Proposition 4(iii),(v)), it is natural to compare (⋆) S(I φ 1 / λ 1 , . . . , I φ ℓ / λ ℓ ) and S(A φ m,1 / λ m,1 , . . . , A φ m,ℓ / λ m,ℓ ). Instead, like in Section 4.2, we consider the following reconstruction errors since both EKPCA and RF-EKPCA provide the principal components for the empirical kernel functions, i.e., k(·, X) and k m (·, X): S Σ,ℓ = E X∼P Ik(·, X) − ℓ i=1 I k(·, X), I φ i λ i L 2 (P) I φ i λ i 2 L 2 (P) ,(14)S Σm,ℓ = E X∼P Ik(·, X) − ℓ i=1 Ak m (·, X), Aφ m,i λ m,i L 2 (P) Aφ m,i λ m,i 2 L 2 (P) ,(15) and S Σm,ℓ = E X∼P Ik(·, X) − ℓ i=1 A k m (·, X), A φ m,i λ m,i L 2 (P) A φ m,i λ m,i 2 L 2 (P) .(16) Note that unlike in R-E where the reconstructed functions (before being embedded) are computable (see (11)-(13)), in (14)- (16) and (⋆), the reconstructed functions are not computable because of their dependence on unknown P. First, in Theorem 8 (proved in Section 6.8), we provide a finite-sample bound on the behavior of (14)- (16), which is then used in Theorem 9 (proved in Section 6.9), to show that (⋆) is statistically equivalent to (14) and (16). Theorem 8. Suppose (A 1 ) − (A 5 ) hold. For any t > 0, define N Σ (t) = tr(Σ(Σ + tI) −1 ). Then the following hold: (i) S Σ,ℓ = i>ℓ λ 2 i . (ii) For any δ > 0 with n ≥ 2 log 2 δ and 140κ n log 16κn δ ≤ t ≤ λ ℓ 3 , with probability at least 1 − 11δ over the choice of (X i ) n i=1 , i>ℓ λ 2 i ≤ S Σ,ℓ N Σ (t)(λ ℓ+1 + t) 2 + κ 5/2 log 2 δ N Σ (t) n √ t ∧ κ 3/2 nt + κ 3 (κ ∧ 1) log 2 3 δ n 2 t + κ 2 log 2 δ n . (iii) For any δ > 0 with m ≥ 2 ∨ 1024κ 2 i>ℓ λ 2 i log 2 δ , Λ m (θ i ) m i=1 : 1 4 i>ℓ λ 2 i ≤ S Σm,ℓ ≤ 4 i>ℓ λ 2 i + 256κ 2 log 2 δ m ≥ 1 − 6δ. (iv) For any δ > 0 with n ≥ 2 log 2 δ , m ≥ 2 ∨ 1024κ 2 i>ℓ λ 2 i log 2 δ and 140κ n log 16κn δ ∨ 86κ m log 16κm δ ≤ t ≤ λ ℓ 9 , with probability at least 1 − 26δ over the choice of (X i ) n i=1 , (θ j ) m j=1 : 1 4 i>ℓ λ 2 i ≤ S Σm,ℓ A 2 (t)(λ ℓ+1 + t) 2 + κ 5/2 log 2 δ A 2 (t) n √ t ∧ κ 3/2 nt + κ 3 (1 ∧ κ) log 2 3 δ n 2 t + κ 2 log 2 δ n + κ 2 log 2 δ m , where A 2 (t) := κ log 2 δ tm + κN Σ (t) log 2 δ tm + N Σ (t). Theorem 9. Under the assumptions (A 1 ) − (A 5 ), the following hold: (i) i>ℓ λ 2 i ≤ S I φ 1 √ λ 1 , . . . , I φ ℓ √ λ ℓ P n S Σ,ℓ + 1 n ; (ii) For m 1 i>ℓ λ 2 i , i>ℓ λ 2 i Λ m S Aφ m,1 √ λ m,1 , . . . , Aφ m,ℓ √ λ m,ℓ Λ m S Σm,ℓ + 1 m ; (iii) For m 1 i>ℓ λ 2 i , i>ℓ λ 2 i Λ m S A φ m,1 λ m,1 , . . . , A φ m,ℓ λ m,ℓ P n ×Λ m S Σm,ℓ + 1 n + 1 m . Remark 5. (i) A key difference (which may be an artifact of the proof technique) between Theorems 6 and 8 is the upper bound on t. While t is upper bounded by a constant in Theorem 6, t is upper bounded by λ ℓ (up to constants) in Theorem 8, which enforces a lower bound on λ ℓ . Since λ ℓ → 0 as ℓ → ∞ and the lower bound on λ ℓ converges to zero as n → ∞, this enforces a constraint on λ ℓ to not converge to zero too fast. In other words, it imposes a condition on ℓ to not grow too fast with n. (ii) The explicit universal constants, which are suppressed in (ii) and (iv) of Theorem 8 for brevity, are provided in the proof. It is clear from (ii) and (iv) of Theorem 8 that for m large enough, both EKPCA and RF-EKPCA have similar statistical behavior-a similar observation was made in Theorem 6. The following corollary (proved in Sections 6.10) to Theorem 8 investigates the statistical behavior of EKPCA and RF-EKPCA under the polynomial decay condition (exponential decay condition is analyzed in Corollary A.2) on the eigenvalues of Σ. Corollary 10 (Polynomial decay of eigenvalues). Suppose Ai −α ≤ λ i ≤Āi −α for α > 1 and A,Ā ∈ (0, ∞). Let ℓ = n θ α , 0 < θ ≤ α. Define 1 α ′ := 1 α + 1 2 ∧ 1 and β := 1 2+ 1 α ′ − 1 α . Then (i) n −2θ(1− 1 2α ) S Σ,ℓ n −2θ(1− 1 2α ) . There existsñ ∈ N such that for all n >ñ, the following hold: (ii) n −2θ(1− 1 2α ) S Σ,ℓ P n n −2θ(1− 1 2α ) , θ ≤ β n −(1− θ α ′ ) , β ≤ θ < 1 ; (iii) For 0 < γ ≤ 1 and m = n γ , n −2θ(1− 1 2α ) 1 {γ≥θ(2− 1 α )} Λ m S Σm,ℓ Λ m n −2θ(1− 1 2α ) , γ ≥ θ 2 − 1 α , θ ≤ α 2α−1 n −γ , γ ≤ 1 ∧ θ 2 − 1 α ; (iv) For 0 < γ ≤ 1 and m = n γ , n −2θ(1− 1 2α ) 1 {γ≥θ(2− 1 α )} Λ m S Σm,ℓ P n ×Λ m      n −2θ(1− 1 2α ) , γ ≥ θ 2 − 1 α , θ ≤ β n −(1− θ α ′ ) , γ ≥ 1 − θ α ′ , γ > θ, β ≤ θ < 1 n −γ , θ < γ ≤ 1 ∧ θ 2 − 1 α ∧ 1 − θ α ′ . Remark 6. (i) In Corollary 10, note that β = α 3α−1 for 1 < α ≤ 2 and β = 2 5 for α ≥ 2 with the best convergence rate for S Σ,ℓ being attained at θ = β as the rate is a convex function of θ. Clearly, from both computational and statistical view points, only the range of 0 < θ ≤ β is interesting and useful as θ > β yields similar/slower convergence rates with more computational complexity. This means, optimal convergence rates are obtained for EKPCA and RF-EKPCA for ℓ not growing faster than n θ/α , θ ≤ β and m ≥ n θ(2− 1 α ) . Arguing as in Remark 4(iii), it can be shown that the computational complexity of RF-EKPCA is better than of EKPCA and with no loss in statistical performance if θ < α 4α−3 ∧ β and γ ≥ θ(2 − 1 α ). (ii) More interesting observations can be made by comparing Corollaries 7 (resp. Corollary A.1) and 10 (resp. Corollary A.2). First, EKPCA and RF-EKPCA have different upper asymptotic behaviors in R-E (Corollary 7 and Corollary A.1) and E-R (Corollary 10 and Corollary A.2). Particularly, while the reconstruction error rate improves with increase in θ in both the cases, it saturates beyond a certain θ in the case of R-E while it decreases in the case of E-R. This latter behavior is due to the inverse of empirical eigenvalues that appear in S Σ,ℓ and S Σm,ℓ -as ℓ becomes large, then inverse of the empirical eigenvalues makes large contributions to the error, resulting in slower convergence rates. In the regimes of θ where R-E and E-R behave similarly (for both EKPCA and RF-EKPCA), we note that θ has a larger upper bound (i.e., ℓ can have faster growth) in R-E than in E-R, which again relates to the above mentioned issue of the inverse of empirical eigenvalues. Particularly, in the case of E-R, for α ≤ 2, RF-EKPCA has better computational behavior than EKPCA if θ < β and γ ≥ θ(2 − 1 α ) while such a result holds for R-E for a wider range of θ, i.e., θ < α 4α−3 , which means faster growth for ℓ is allowable for R-E without losing computational or statistical efficiency. On the other hand, for α ≥ 2, R-E and E-R behave similarly for 0 < θ ≤ α 4α−3 . Finally, we would like to highlight that the number of random features (m) needed in E-R and R-E so that their statistical behavior match that of KPCA is only a sufficient condition based on the upper bounds in Theorems 6 and 8. It is not clear whether this requirement on m is sharp. Schatten norms So far, we have seen that the population reconstruction error in E-R and R-E, i.e., R Σ,ℓ and S Σ,ℓ behave as i>ℓ λ 2 i , which is the squared ℓ 2 -norm of λ ℓ := (λ ℓ+1 , λ ℓ+2 , . . .). Of course, if we use the population reconstruction error defined in (10), it is easy to show that it behaves as i>ℓ λ i , which is the ℓ 1 -norm of λ ℓ . But the reconstruction error defined in (10) is not useful for our purpose because of the aforementioned technical issues and that is why we introduced E-R and R-E in Sections 4.2 and 4.3. In this section, we explore an extension of R Σ,ℓ (similar extension holds for S Σ,ℓ as well) which yields different norms of λ ℓ . To this end, define R s (ψ 1 , . . . , ψ ℓ ) = E (II * ) −s/2 Ik(·, X) − I ℓ i=1 k(·, X), ψ i H ψ i 2 L 2 (P) for any (ψ 1 , . . . , ψ ℓ ) ⊂ H and s ≤ 1, which is a weighted version of R with the error being weighted by (II * ) −s/2 . Here (II * ) −1 is treated as the inverse of II * restricted to L 2 (P)\Ker(II * ) where Ker(II * ) = Ker(I * ) = {f ∈ L 2 (P) : f is a constant a.s.-P}. The following result (similar to Propositions 2 and 3), which is proved in Section 6.11 shows that KPCA solution is the minimizer of R s with the minimum value being i>ℓ λ 2−s i , i.e., (2−s)-Schatten norm of λ ℓ and that EKPCA, RF-KPCA and RF-EKPCA solutions are minimizers of appropriate empirical versions of R s . Proposition 11. Suppose (A 1 ) − (A 5 ) hold. For s ≤ 1, define T s,A,H (P ) := A (1−s)/2 (I − P )A 1/2 2 L 2 (H) , where A : H → H is a positive self-adjoint Hilbert-Schmidt operator on H with H being a separable Hilbert space. Then the following hold. (i) R s (ψ 1 , . . . , ψ ℓ ) = T s,Σ,H (P ψ ) where P ψ := ℓ i=1 ψ i ⊗ H ψ i ; (ii) (KPCA) (φ 1 , . . . , φ ℓ ) = arg inf{T s,Σ,H (P ψ ) : (ψ i ) i∈[ℓ] ⊂ H}; (iii) (EKPCA) ( φ 1 , . . . , φ ℓ ) = arg inf{T s, Σ,H (P ψ ) : (ψ i ) i∈[ℓ] ⊂ H}; (iv) (RF-KPCA) (φ m,1 , . . . , φ m,ℓ ) = arg inf{T s,Σm,Hm (P τ ) : (τ i ) i∈[ℓ] ⊂ H m } where P τ = ℓ i=1 τ i ⊗ Hm τ i ; (v) (RF-EKPCA) ( φ m,1 , . . . , φ m,ℓ ) = arg inf{T s, Σm,Hm (P τ ) : (τ i ) i∈[ℓ] ⊂ H m }. Proposition 11 implies that the case of s = 1 exactly recovers the original KPCA problem (see (10)) and it follows from Lemma B.1 that R s (φ 1 , . . . , φ ℓ ) = i>ℓ λ 2−s i . With the intuition gained from Proposition 11 and for the same reasons mentioned in Section 4.2, we measure the performance of EKPCA, RF-KPCA and RF-EKPCA as follows: Define R Σ,ℓ,s := R s (φ 1 , . . . , φ ℓ ). R Σ,ℓ,s is the performance measure of EKPCA which is defined by replacing k and ψ i with k and φ i respectively in R s . Similarly, the reconstruction error of RF-KPCA can be defined as R Σm,ℓ,s = E X∼P (II * ) −s/2 Ik(·, X) − (AA * ) −s/2 AP ℓ (Σ m )k m (·, X) 2 L 2 (P) with the performance of RF-EKPCA being measured by R Σm,ℓ,s , which is defined by replacing k m with k m and P ℓ (Σ m ) with P ℓ ( Σ m ), where P ℓ (Σ m ) = ℓ i=1 φ m,i ⊗ Hm φ m,i and P ℓ ( Σ m ) = ℓ i=1 φ m,i ⊗ Hm φ m,i . The following result provides the probabilistic behavior of these generalized reconstruction errors. Theorem 12. Suppose (A 1 ) − (A 5 ) hold. For any t > 0, define N Σ (t) = tr(Σ(Σ + tI) −1 ). Then the following hold: (i) For any s ≤ 1, R Σ,ℓ,s = i>ℓ λ 2−s i . (ii) For log n n t Σ L ∞ (H) and any s ≤ 1, i>ℓ λ 2−s i ≤ R Σ,ℓ,s P n N Σ (t)(λ ℓ+1 + t) 2 t s + 1 n . (iii) For m i>ℓ λ 2−s i 2 s−2 ∨ i>ℓ λ 2−s i 2 s 1 [−2,0) (s) + 1 {0} (s) i>ℓ λ 2 i , i>ℓ λ 2−s i Λ m R Σm,ℓ,s Λ m i>ℓ λ 2−s i + m s/2 1 [−2,0) (s) + 1 m 1 {0} (s). (iv) For m i>ℓ λ 2−s i 2 s−2 ∨ i>ℓ λ 2−s i 2 s 1 [−2,0) (s) + 1 {0} (s) i>ℓ λ 2 i and log n n ∨ log m m t Σ L ∞ (H) , i>ℓ λ 2−s i Λ m R Σm,ℓ,s Λ m ×P n A 1 (t)(λ ℓ+1 + t) 2 t s + 1 n + 1 [−2,0) (s) m −s/2 + 1 {0} (s) m , where A 1 (t) := N Σ (t) + 1 tm + N Σ (t) tm . Remark 7. (i) The restriction of s ≤ 1 for R Σ,ℓ,s appears because Σ is a trace class. On the other hand, the bounds for R Σm,ℓ,s and R Σm,ℓ,s hold only for s ∈ [−2, 0]. This could be an artifact of the analysis as the proof of these bounds involve bounding (II * ) −s/2 − (AA * ) −s/2 L ∞ (L 2 (P)) by II * − AA * −s/2 L ∞ (L 2 (P)) by using the operator monotonicity of the map x → x −s/2 for 0 ≤ −s/2 ≤ 1, i.e., s ∈ [−2, 0]. Note that this range of s yields weaker notions of reconstruction error as it corresponds to Schatten norms of order greater than 2, with the more interesting range being [0, 1]. However, the current bounds for R Σm,ℓ,s and R Σm,ℓ,s do not hold for s taking values in this interesting range. Also note that s = 0 exactly reduces to Theorem 6. (ii) As before, assuming i −α λ i i −α for α > 1 and ℓ = n θ α , 0 < θ ≤ α, it follows that n −2θ(1− 1 2α − s 2 ) R Σ,ℓ,s n −2θ(1− 1 2α − s 2 ) and n −2θ(1− 1 2α − s 2 ) R Σ,ℓ,s P n n −2θ(1− 1 2α − s 2 ) , θ ≤ α 2α−1−αs 1 n , θ ≥ α 2α−1−αs , which matches with Corollary 7(i,ii) for s = 0. These show the convergence rate to be slower for 0 < s ≤ 1 compared to s = 0, which is expected as the former corresponds to a stronger notion of reconstruction error. Also 0 < θ ≤ α 2α−1−αs is the only useful range both computationally and statistically as θ > α 2α−1−αs does not improve the statistical rates but increases the computational complexity. On the other hand for −2 ≤ s ≤ 0, it follows that n −2θ(1− 1 2α − s 2 ) Λ m R Σm,ℓ,s Λ m ×P n n −2θ(1− 1 2α − s 2 ) for m n 4θ 2−s (1− 1 2α − s 2 ) and θ ≤ α 2α−1−αs , which implies that for sufficiently large m, EKPCA and RF-EKPCA have similar statistical behavior as long as θ ≤ α 2α−1−αs . However, RF-EKPCA is computationally better than EKPCA only when 0 < θ < (2−s)α 8α−6+s−4sα . Also note that for θ = α 2α−1−αs , which is where the best rate of 1 n is achieved for any s ∈ [−2, 0], we obtain m n 2 2−s , i.e., the requirement on the number of random features is monotonically increasing w.r.t. s ∈ [−2, 0]. Since θ is an increasing function of s, it implies fewer ℓ is sufficient for optimal rates for smaller s. To elaborate, at the chosen value of θ, statistical optimality is conserved for RF-EKPCA at s = 0 if m n while only m √ n is required at s = −2. This is understandable as smaller values of s result in weaker notions of reconstruction error as explained above. Discussion To summarize, we investigated the computational vs. statistical trade-off in the problem of approximating kernel PCA using random features. While it is obvious that approximate kernel PCA using m random features has lower computational complexity than kernel PCA when m < n with n being the number of samples, it is not obvious that this computational gain is not achieved at the cost of statistical efficiency. Through inclusion and approximation operators, we explored various notions of reconstructing a kernel function using ℓ eigenfunctions, wherein we showed that approximate kernel PCA has computational advantage with no loss in statistical optimality as long as m is large enough (but still m < n) and ℓ is small enough with m depending on the number of eigenfunctions ℓ being considered. If ℓ is large, then more features are needed to maintain the statistical behavior, thereby resulting in the loss of computational advantage. There are few open questions in this topic which may be of interest to address. (i) In contrast to the setting of this paper where ℓ grows with n, it may be of interest to consider asymptotics when ℓ is fixed but n → ∞. In such a setting, one may investigate E-R, R-E and their variations/generalizations. For example, in R-E, we can compare EKPCA and RF-EKPCA by comparing R Σ,ℓ − R Σ,ℓ and R Σm,ℓ − R Σ,ℓ . While Theorems 6, 8 and 12 do not directly specialize to the setting of fixed ℓ, using ideas employed in their proofs, upper bounds can be derived on R Σ,ℓ − R Σ,ℓ and R Σm,ℓ − R Σ,ℓ . However, lower bounds are needed to establish the sharpness of these upper bounds so that these excess errors can be matched for a certain choice of m. (ii) Apart from reconstruction error, one may compare ℓ-eigenspaces (for fixed ℓ) associated with EKPCA and RF-EKPCA by comparing the corresponding projection operators through their embeddings as bounded operators on L 2 (P). Ullah et al. (2018) investigated this direction by comparing certain inner product of the uncentered covariance operator with the difference between the projection operators associated with ℓ-eigenspaces of KPCA and EKPCA (resp. RF-EKPCA). Different meaningful notions of comparing the projection operators can be explored and upper convergence rates can be derived using the perturbation theory for self-adjoint operators (see Sriperumbudur and Sterge, 2018 for some preliminary results). However, as above, developing lower bounds will be critical to establish the sharpness of the upper bounds, thereby facilitating a meaningful comparison of the statistical performances of EKPCA and RF-EKPCA. Proofs In this section we present the proofs of the results in Sections 3 and 4. Proof of Proposition 1 Define the sampling operator S : H → R n , f → 1 √ n (f (X 1 ), . . . , f (X n )) ⊤ whose adjoint, called the reconstruction operator can be shown (see Proposition C.1(i)) to be S * : R n → H, α → 1 √ n n i=1 α i k(·, X i ), where α := (α 1 , . . . , α n ) ⊤ . DefineH n = n n−1 H n . It follows from Proposition C.1(ii) that Σ = S * H n S, which implies ( φ i ) i satisfy S * H n S φ i = λ i φ i ,(17) where λ i ≥ 0. Multiplying both sides of (17) on the left by S, we obtain that ( α i ) i , α i := S φ i , i ∈ [n] are eigenvectors of SS * H n = 1 n KH n , i.e., they satisfy the finite dimensional linear system, KH n α i = n λ i α i ,(18) where K is the Gram matrix, i.e., (K) ij = k(X i , X j ), i, j ∈ [n] and the fact that K = nSS * follows from Proposition C.1(iii). It is important to note that ( α i ) i do not form an orthogonal system in the usual Euclidean inner product but in the weighted inner product where the weighting matrix isH n . Indeed, it is easy to verify that α i ,H n α j 2 = S φ i ,H n S φ j 2 = φ i , Σ φ j H = λ j φ i , φ j H = λ j δ ij , where δ ij is the Kronecker delta. Having obtained ( α i ) i from (18), the eigenfunctions of Σ are obtained from (17) as φ i = 1 λ i S * H n α i , and the result follows. Proof of Proposition 2 (i) Define P ψ := ℓ i=1 ψ i ⊗ H ψ i . Therefore, R((ψ i ) i∈[ℓ] ) = E I(I − P ψ )k(·, X) 2 L 2 (P) ( * ) = Σ 1/2 (I − P ψ )Σ 1/2 2 L 2 (H) ( †) = R Σ 0,1,1 (P ψ ), where we used Lemma B.7 in ( * ) since Σ = I * I and k is continuous and bounded (therefore, Bochner integrable), and the definition in (B.1) in ( †). The result therefore follows from Lemma B.1 that (φ i ) i∈ [ℓ] is the unique minimizer of R. (ii) Define P µ := ℓ i=1 µ i ⊗ L 2 (P) µ i . Therefore, S((µ i ) i∈[ℓ] ) = E (I − P µ )Ik(·, X) 2 L 2 (P) = I * (I − P µ ) 2 I, Σ L 2 (H) = Tr I * (I − P µ ) 2 II * I = Tr[T (I − P µ ) 2 T ] = (I − P µ )T 2 L 2 (L 2 (P)) , where T := II * . It follows from Lemma B.1(ii) that (χ i ) i∈[ℓ] is the minimizer of S where (χ i ) i∈[ℓ] are the eigenfunctions of T that correspond to the eigenvalues (λ i ) i∈ [ℓ] . We now show that χ i = Iφ i √ λ i . Since Σφ i = λ i φ i , we have I * Iφ i = λ i φ i , which implies II * Iφ i = λ i Iφ i , i.e., T Iφ i √ λ i = λ i Iφ i √ λ i . Therefore Iφ i √ λ i i are eigenfunctions of T and so Iφ i √ λ i i∈[ℓ] is the minimizer of S. (iii) It follows from (i), (ii) and Lemma B.1 that R Σ,ℓ = R Σ 0,1,1 (P ℓ (Σ)) = i>ℓ λ 2 i and S Σ,ℓ = R T 0,0,2 ( ℓ i=1 χ i ⊗ L 2 (P) χ i ) = i>ℓ λ 2 i . (iv) Note that for any (ψ i ) i∈[ℓ] , R((ψ i ) i∈[ℓ] ) = E I(I − P ψ )k(·, X) 2 L 2 (P) = E Σ 1/2 (I − P ψ )k(·, X) 2 H ≤ Σ 1/2 2 L ∞ (H) E (I − P ψ )k(·, X) 2 H . The result therefore follows by applying the above inequality for (φ i ) i∈[ℓ] . Proof of Proposition 3 (i), (ii) Refer to the proof of Proposition 2(i). (iii), (iv), (v) Note that T Σ,H (P ψ ) = R Σ 0,1,1 (P ψ ), T Σm,Hm (P τ ) = R Σm 0,1,1 (P τ ) and T Σm,Hm (P τ ) = R Σm 0,1,1 (P τ ), following the notation defined in (B.1). The result therefore follows from Lemma B.1. Proof of Proposition 4 (i), (ii) Refer to the proof of Proposition 2(ii). (iii) It follows from Lemma B.1 that the eigenfunctions corresponding to the top ℓ eigenvalues of I I * is the minimizer of V I I * ,L 2 (Pn) . In the following, we will show that I * I = Σ which will then imply I I * I φ i = I Σ φ i = λ i I φ i . The result is completed by noting that I φ i λ i , I φ j λ j L 2 (Pn) = 1 λ i λ j Σ φ i , φ j H = δ ij and ( λ i , I φ i / λ i ) i is the eigensystem of I I * . To show I * I = Σ, for any f ∈ H, consider (iv), (v) The proof is exactly same as that of (iii) by replacing I, Σ, φ i and λ i by A, Σ m , φ m,i and λ m,i (resp. A, Σ m , φ m,i and λ m,i ), respectively. k m (·, X i ). f, I * If H = If 2 L 2 (Pn) = 1 n n i=1 ( If ) 2 (X i ) = 1 n − 1 n i=1   f (X i ) − 1 n n j=1 f (X j )   2 = 1 n − 1 n i=1 f 2 (X i ) − 1 n(n − 1)   n j=1 f (X j )   2 (C.1) = f, Σf Proof of Theorem 5 (i) From Proposition 3(i), we have T Σ,H (P ℓ ( Σ)) = E Ik(·, X) − IP ℓ ( Σ))k(·, X) 2 L 2 (P) ( †) = E Ik(·, X) − IP ℓ ( Σ) k(·, X) + IP ℓ ( Σ)(m P − m P ) 2 L 2 (P) ( * ) = R Σ,ℓ + IP ℓ ( Σ)(m P − m P ) 2 L 2 (P) ( ‡) P n R Σ,ℓ + 1 n , where expanding the squares in ( †) and noting that the expectation of the inner product is zero, yields ( * ). ( ‡) follows from (22) and Lemma B.5. The lower bound is obtained by noting that IP ℓ ( Σ)(m P − m P ) 2 L 2 (P) ≥ 0. (ii) From Proposition 3(iv), we have T Σm,Hm (P ℓ (Σ m )) = E Ak m (·, X) − AP ℓ (Σ m ))k m (·, X) 2 L 2 (P) = E Ak m (·, X) − Ik(·, X) + Ik(·, X) − AP ℓ (Σ m )k m (·, X) 2 L 2 (P) , and the result follows from Lemmas B.6 and B.10. (ii) From Proposition 3(v), we have T Σm,Hm (P ℓ ( Σ m )) = E Ak m (·, X) − AP ℓ ( Σ m ))k m (·, X) 2 L 2 (P) = E Ak m (·, X) − AP ℓ ( Σ m ) k m (·, X) + AP ℓ ( Σ m )(m P,m − m P,m ) 2 L 2 (P) = E Ak m (·, X) − AP ℓ ( Σ m ) k m (·, X) 2 L 2 (P) + E AP ℓ ( Σ m )(m P,m − m P,m ) 2 L 2 (P) = E Ak m (·, X) − Ik(·, X) + Ik(·, X) − AP ℓ ( Σ m ) k m (·, X) 2 L 2 (P) +E AP ℓ ( Σ m )(m P,m − m P,m ) 2 L 2 (P) and the result follows from Lemmas B.5, B.6 and B.10. Proof of Theorem 6 (i) The result follows from the proof of Proposition 2(iii). (ii) Upper bound: Note that R Σ,ℓ = E Ik(·, X) − IP ℓ ( Σ)(k(·, X) − m P ) 2 L 2 (P) . Therefore, adding and subtracting IP ℓ ( Σ)m P and expanding squares, we obtain R Σ,ℓ = 1 + 2 with the inner product being zero since E[k(·, X)] = 0. Here 1 := E I(I − P ℓ ( Σ))k(·, X) 2 L 2 (P) , and 2 := IP ℓ ( Σ)(m P − m P ) 2 L 2 (P) . It follows from Lemma B.7 that 1 = E I(I − P ℓ ( Σ))k(·, X) 2 L 2 (P) = Σ 1/2 (I − P ℓ ( Σ))Σ 1/2 2 L 2 (H) . For any t > 0, we have Σ 1/2 (I − P ℓ ( Σ))Σ 1/2 2 L 2 (H) = Σ 1/2 (Σ + tI) −1/2 (Σ + tI) 1/2 (I − P ℓ ( Σ))(Σ + tI) 1/2 (Σ + tI) −1/2 Σ 1/2 2 L 2 (H) ≤ Σ 1/2 (Σ + tI) −1/2 2 L 2 (H) Σ 1/2 (Σ + tI) −1/2 2 L ∞ (H) (Σ + tI) 1/2 (I − P ℓ ( Σ))(Σ + tI) 1/2 2 L ∞ (H) ( * ) ≤ N Σ (t) (Σ + tI) 1/2 (I − P ℓ ( Σ))(Σ + tI) 1/2 2 L ∞ (H) ≤ N Σ (t) (Σ + tI) 1/2 ( Σ + tI) −1/2 4 L ∞ (H) ( Σ + tI) 1/2 (I − P ℓ ( Σ))( Σ + tI) 1/2 2 L ∞ (H) , ≤ N Σ (t)( λ ℓ+1 + t) 2 (Σ + tI) 1/2 ( Σ + tI) −1/2 4 L ∞ (H) ,(20) where in ( * ), we have used Σ 1/2 (Σ + tI) −1/2 2 L 2 (H) = tr((Σ + tI) −1/2 Σ(Σ + tI) −1/2 ) = tr(Σ(Σ + tI) −1 ) = N Σ (t) and Σ 1/2 (Σ + tI) −1/2 2 L ∞ (H) ≤ 1. Applying Lemma B.2(ii, iv) to (20), we obtain that for any δ > 0 such that 140κ n log 16κn δ ≤ t ≤ Σ L ∞ (H) , with probability at least 1 − 2δ over the choice of (X i ) n i=1 , 1 = Σ 1/2 (I − P ℓ ( Σ))Σ 1/2 2 L 2 (H) ≤ 4N Σ (t)( λ ℓ+1 + t) 2 ≤ 9N Σ (t)(λ ℓ+1 + t) 2 .(21) We now bound 2 as follows. 2 = IP ℓ ( Σ)(m P − m P ), IP ℓ ( Σ)(m P − m P ) L 2 (P) = Σ 1/2 P ℓ ( Σ)(m P − m P ), Σ 1/2 P ℓ ( Σ)(m P − m P ) H = Σ 1/2 P ℓ ( Σ)(m P − m P ) 2 H ≤ Σ 1/2 2 L ∞ (H) P ℓ ( Σ) 2 L ∞ (H) m P − m P 2 H = Σ L ∞ (H) m P − m P 2 H ,(22) where the last equality uses P ℓ ( Σ) (21) in (19), and using Σ L ∞ (H) ≤ 2κ. Lower bound: It follows from (19) that R Σ,ℓ ≥ 1 . Lemma B.1 implies that 1 = R Σ 0,1,1 (P ℓ ( Σ)) ≥ i>ℓ λ 2 i = R Σ,ℓ and the result follows by combining these bounds. (iii) Upper bound: Since R Σm,ℓ = E Ik(·, X) − AP ℓ (Σ m )k m (·, X) 2 L 2 (P) , by (B.21), we have R Σm,ℓ ≤ 2 3 + 4 , where 3 := E Ik(·, X) − Ak m (·, X) 2 L 2 (P) and 4 := E A(I − P ℓ (Σ m ))k m (·, X) 2 L 2 (P) . It follows from Lemma B.7 that 4 = Σ 1/2 m (I − P ℓ (Σ m ))Σ 1/2 m 2 L 2 (Hm) = m i=ℓ+1 λ 2 m,i = m i=ℓ+1 (λ m,i − λ i + λ i ) 2 ≤ 2 m i=ℓ+1 (λ m,i − λ i ) 2 + 2 i>ℓ λ 2 i ( * * ) ≤ 2 II * − AA * 2 L 2 (L 2 (P)) + 2 i>ℓ λ 2 i ,(23) where in ( * * ), we used Hoffman-Wielandt inequality (Bhatia, 1997) along with the fact that Σ = I * I and II * have same eigenvalues, and similarly Σ m = A * A and AA * . The result follows by applying Lemma B.4 to (23) and Lemma B.6 to 3 . Lower bound: Note that R Σm,ℓ (B.21) (24) holding with probability at least 1 − 3δ over the choice of (θ i ) m i=1 . (iv) Upper bound: R Σm,ℓ can be alternately written as ≥ 4 − 3 2 , where 4 − 3 = i>ℓ λ 2 m,i − 3 ≥ i>ℓ λ 2 i − i>ℓ (λ i − λ m,i ) 2 − 3 ≥ i>ℓ λ 2 i − II * − AA * L 2 (L 2 (P)) − 3 ≥ i>ℓ λ 2 i − 4κ 2 log 2 δ m − 8κ log 2 δ m ≥ i>ℓ λ 2 i − 16κ log 2 δ m ≥ 1 2 i>ℓ λ 2 i (24) as 1 2 i>ℓ λ 2 i ≥ 16κ log 2 δ m , withR Σm,ℓ = E Ik(·, X) − AP ℓ ( Σ m )(k m (·, X) − m P,m ) 2 L 2 (P) , which can be bounded as R Σm,ℓ ≤ 2E Ik(·, X) − AP ℓ ( Σ m )k m (·, X) 2 L 2 (P) + 2E AP ℓ ( Σ m )(m P,m − m P,m ) 2 L 2 (P) ≤ 4 3 + 5 + 2 6 ,(25)where 5 := E A(I − P ℓ ( Σ m )k m (·, X) 2 L 2 (P) and 6 := AP ℓ ( Σ m )(m P,m − m P,m ) 2 L 2 (P) . Using Lemma B.7, for t > 0, we bound 5 as 5 = Σ 1/2 m (I − P ℓ ( Σ m ))Σ 1/2 m 2 L 2 (Hm) = Σ 1/2 m (Σ m + tI) −1/2 (Σ m + tI) 1/2 (I − P ℓ ( Σ m ))(Σ m + tI) 1/2 (Σ m + tI) −1/2 Σ 1/2 m 2 L 2 (Hm) ≤ Σ 1/2 m (Σ m + tI) −1/2 2 L 2 (Hm) (Σ m + tI) −1/2 Σ 1/2 m 2 L ∞ (Hm) × (Σ m + tI) 1/2 (I − P ℓ ( Σ m ))(Σ m + tI) 1/2 2 L ∞ (Hm) ( * ) ≤ N Σm (t) (Σ m + tI) 1/2 (I − P ℓ ( Σ m ))(Σ m + tI) 1/2 2 L ∞ (Hm) ≤ N Σm (t) (Σ m + tI) 1/2 ( Σ m + tI) −1/2 4 L ∞ (Hm) × ( Σ m + tI) 1/2 (I − P ℓ ( Σ m ))( Σ m + tI) 1/2 2 L ∞ (Hm) ≤ N Σm (t)( λ m,ℓ+1 + t) 2 (Σ m + tI) 1/2 ( Σ m + tI) −1/2 4 L ∞ (Hm) ,(26) where we used Σ ≤ 1 in ( * ). Conditioning on (θ i ) m i=1 and applying Lemma B.2(ii, iv) to (26), we obtain that for any δ > 0 and 140κ n log 16κn δ ≤ t ≤ Σ m L ∞ (H) ,(27)P n \|(θ i ) m i=1 (X i ) n i=1 : 5 ≤ 9N Σm (t)(λ m,ℓ+1 + t) 2 ≥ 1 − 2δ.(28) Now, unconditioning w.r.t. (θ i ) m i=1 and applying Lemma B.3(ii, iv) in (28), we obtain that for any δ > 0 and 86κ m log 16κm δ ≤ t ≤ Σ L ∞ (H) , Λ m × P n    ((θ i ) m i=1 , (X i ) n i=1 ) : 5 ≤ 81 4   32κ log 2 δ tm + 32κN Σ (t) log 2 δ tm + 2N Σ (t)   (λ ℓ+1 + t) 2    ≥ 1 − 5δ.(29) Note that the upper bound in (27) holds because we assumed that t ≤ 1 3 Σ L ∞ (H) which is equivalent to t ≤ 1 2 ( Σ L ∞ (H) − t) ≤ Σ m L ∞ (H) where we used 1 2 (λ 1 + t) ≤ λ m,1 + t from Lemma B.3(iii). We now bound 6 as 6 = AP ℓ ( Σ m )(m P,m − m P,m ), AP ℓ ( Σ m )(m P,m − m P,m ) L 2 (P) = Σ 1/2 m P ℓ ( Σ m )(m P,m − m P,m ), Σ 1/2 m P ℓ ( Σ m )(m P,m − m P,m ) Hm = Σ 1/2 m P ℓ ( Σ m )(m P,m − m P,m ) 2 Hm ≤ Σ 1/2 m 2 L ∞ (Hm) P ℓ ( Σ m ) 2 L ∞ (Hm) m P,m − m P,m 2 Hm ≤ λ m,1 m P,m − m P,m 2 Hm .(30) Applying Lemma B.5(ii) (by conditioning w. (30), for any δ > 0 and n ≥ 2 log 2 δ , we obtain r.t. (θ i ) m i=1 ) and Lemma B.3(ii) (to uncondition (θ i ) m i=1 ) toΛ m × P n ((θ i ) m i=1 , (X i ) n i=1 ) : 6 ≤ 320κ 2 log 2 δ 3n ≥ 1 − 2δ,(31) where we used λ m,1 ≤ 3λ 1 +t 2 from Lemma B.3(ii) and t ≤ λ 1 3 (as per our assumption), resulting in λ m,1 ≤ 5 3 Σ L ∞ (H) ≤ 10κ 3 . The result therefore follows by applying Lemma B.6 to 3 and combining it with (29) and (31) in (25). Lower bound: As carried out in the proof of the lower bound of (iii), it can be shown that R Σm,ℓ = E Ik(·, X) − AP ℓ ( Σ m ) k m (·, X) 2 L 2 (P) ≥ 7 − 3 2 , where 7 := E Ak m (·, X) − AP ℓ ( Σ m ) k m (·, X) 2 L 2 (P) = E A(I − P ℓ ( Σ m ))k m (·, X) 2 L 2 (P) + AP ℓ ( Σ m )(m P,m − m P,m ) 2 L 2 (P) −2E A(I − P ℓ ( Σ m ))k m (·, X), AP ℓ ( Σ m )(m P,m − m P,m ) L 2 (P) ( †) = 5 + 6 ( ‡) ≥ i>ℓ λ 2 m,i + 6 ≥ i>ℓ λ 2 m,i , where ( †) follows by noting that the term involving the inner product is zero and ( ‡) follows by applying Lemma B.1 to 5 . The result, therefore, follows from (24). Proof of Corollary 7 (i) From Theorem 6(i) we have R Σ,ℓ = i>ℓ λ 2 i i>ℓ i −2α ∞ ℓ x −2α dx ℓ 1−2α = n −2θ(1− 1 2α ) . Similarly, R Σ,ℓ = i>ℓ λ 2 i i>ℓ i −2α ∞ ℓ+1 x −2α dx (ℓ + 1) 1−2α n −2θ(1− 1 2α ) . (ii) From Theorem 6(ii) we have R Σ,ℓ P n 1 n + N Σ (t)(λ ℓ + t) 2 , with log n n t ≤ λ 1 3 . Using N Σ (t) t −1/α from Lemma B.9(i), it follows that R Σ,ℓ P n inf t −1/α (n −θ + t) 2 + n −1 : log n n t ≤ λ 1 3    n −2θ(1− 1 2α ) + 1 n , θ < 1 log n n 2− 1 α + 1 n , θ ≥ 1 . Of course, log n n 2− 1 α ≤ 1 n always holds, and n −2θ(1− 1 2α ) ≤ 1 n for θ ≥ α 2α−1 , yielding the result. (iii) From Theorem 6(iii) we have R Σm,ℓ Λm 1 m + i>ℓ λ 2 i . From (i) we have i>ℓ λ 2 i n −2θ(1− 1 2α ) , and the result follows. (iv) From Theorem 6(iv) we have R Σm,ℓ P n ×Λ m 1 n + 1 m + N Σ (t) + N Σ (t) tm + 1 tm (λ ℓ + t) 2 , for log n n ∨ log m m t λ 1 . Note that N Σ (t)+ 1 tm + N Σ (t) tm t −1/α , which follows from N Σ (t) t −1/α (Lemma B.9(i)), 1 tm < t −1/α and t −(1+1/α) m t −1/α since 1 m < log n n ∨ log m m 1− 1 α t 1− 1 α . Therefore, we have R Σm,ℓ P n ×Λ m n −γ + t −1/α (n −θ + t) 2 , for log n n γ t λ 1 , where we have used m = n γ with γ < 1 and ℓ = n θ α . This implies R Σm,ℓ P n ×Λ m inf n −γ + t −1/α (n −θ + t) 2 : log n n γ t λ 1    n −γ + n θ α −2θ , θ < γ n −γ + log n n 2− 1 α , θ ≥ γ and the result follows by considering the cases of γ ≥ θ 2 − 1 α and γ < θ 2 − 1 α . Proof of Theorem 8 (i) The result follows from the proof of Proposition 2(iii). (ii) Upper bound: By defining Σ −1 ℓ := ℓ i=1 1 λ i φ i ⊗ H φ i , we have, S Σ,ℓ = E Ik(·, X) − I Σ −1 ℓ I * I(k(·, X) − m P ) 2 L 2 (P) (⋆) = E I(I − Σ −1 ℓ Σ)k(·, X) 2 L 2 (P) + I Σ −1 ℓ Σ(m P − m P ) 2 L 2 (P) ( †) = Σ 1/2 (I − Σ −1 ℓ Σ)Σ 1/2 2 L 2 (H) + Σ 1/2 Σ −1 ℓ Σ(m P − m P ) 2 H ,(32) where we have used I * I = Σ (see Proposition C.2(iii)) in (⋆) and Lemma B.7 in ( †). We can decompose the first term of (32) as Σ 1/2 (I − Σ −1 ℓ Σ)Σ 1/2 2 L 2 (H) ≤ 2 1 + b ,(33)where b := Σ 1/2 (P ℓ ( Σ) − Σ −1 ℓ Σ)Σ 1/2 2 L 2 (H) . Therefore, P n (X i ) n i=1 : a ≤ 9N Σ (t)(λ ℓ+1 + t) 2 ≥ 1 − 2δ,(34) where 140κ n log 16κn δ ≤ t ≤ Σ L ∞ (H) . For b we write, b = Σ 1/2 Σ −1 ℓ ( Σ − Σ)Σ 1/2 2 L 2 (H) ≤ Σ 1/2 (Σ + tI) −1/2 2 L ∞ (H) (Σ + tI) 1/2 ( Σ + tI) −1/2 2 L ∞ (H) × ( Σ + tI) 1/2 Σ −1 ℓ ( Σ + tI) 1/2 2 L ∞ (H) ( Σ + tI) −1/2 (Σ + tI) 1/2 2 L ∞ (H) × (Σ + tI) −1/2 ( Σ − Σ)Σ 1/2 2 L 2 (H) ( * ) ≤ 4 sup i≤ℓ λ i + t λ i 2 × c = 4 λ ℓ + t λ ℓ 2 × c ( ‡) ≤ 36 λ ℓ + t λ ℓ − t 2 × c (⋆) ≤ 144 × c ,(35) which holds with probability at least 1−2δ over the choice of (X i ) n i=1 , where we used Lemma B.2(ii) in ( * ), Lemma B.2(iv, v) in ( ‡) and the assumption that t ≤ 1 3 λ ℓ in (⋆), with c := (Σ + tI) −1/2 ( Σ − Σ)Σ 1/2 2 L 2 (H) . In the following, we will obtain two different bounds for c based on different decompositions, which we then combine by choosing the minimum of them. Applying (B.9) to c yields P n (X i ) n i=1 : c ≤ 128κ 5/2 N Σ (t) log 2 δ n √ t + 4096κ 3 log 2 3 δ n 2 t ≥ 1 − 2δ.(36) c can be alternately bounded as c ≤ (Σ + tI) −1/2 2 L ∞ (H) Σ − Σ 2 L 2 (H) Σ 2 L ∞ (H) , yielding P n (X i ) n i=1 : c ≤ 256κ 4 log 2 δ nt + 8192κ 4 log 2 3 δ n 2 t ≥ 1 − 2δ(37) through an application of Theorem D.3(ii). Combining (36) and (37) provides P n (X i ) n i=1 : c ≤ 256κ 5/2 log 2 δ N Σ (t) n √ t ∧ κ 3/2 nt + 8192κ 3 (κ ∧ 1) log 2 3 δ n 2 t ≥ 1 − 4δ, using which in (35) yields P n (X i ) n i=1 : b ≤ 144 256κ 5/2 log 2 δ N Σ (t) n √ t ∧ κ 3/2 nt + 8192κ 3 (κ ∧ 1) log 2 3 δ n 2 t ≥ 1 − 6δ.(38) To bound the second term of (32) we have Σ 1/2 Σ −1 ℓ Σ(m P − m P ) 2 H ≤ Σ 1/2 Σ −1 ℓ Σ 1/2 2 L ∞ (H) Σ 1/2 (m P − m P ) 2 H ≤ Σ 1/2 (Σ + tI) −1/2 4 L ∞ (H) (Σ + tI) 1/2 ( Σ + tI) −1/2 4 L ∞ (H) × ( Σ + tI) 1/2 Σ −1 ℓ ( Σ + tI) 1/2 2 L ∞ (H) Σ 1/2 (m P − m P ) 2 H (⋄) ≤ 144 Σ L ∞ (H) m P − m P 2 H ≤ 288κ m P − m P 2 H ,(39) which holds with probability at least 1 − 2δ over the choice of (X i ) n i=1 , wherein we have used Lemma B.2(ii) and the bound in (35) on ( Σ + tI) 1/2 Σ −1 ℓ ( Σ + tI) 1/2 2 L ∞ (H) in (⋄). Applying Lemma B.5(i) to (39), we obtain P n (X i ) n i=1 : Σ 1/2 Σ −1 ℓ Σ(m P − m P ) 2 H ≤ 9216κ 2 log 2 δ n ≥ 1 − 3δ.(40) Combining (32)-(34), (38) and (40), yields the result. Lower bound: It follows from (32) that S Σ,ℓ ≥ Σ 1/2 (I − Σ −1 ℓ Σ)Σ 1/2 2 L 2 (H) = R Σ 1,1,1 ( Σ −1 ℓ ), where the equality follows from the definition in Lemma B.1. The result follows from Lemma B.1 by noting that R Σ 1,1, 1 ( Σ −1 ℓ ) ≥ R Σ 1,1,1 (Σ −1 ℓ ) = S Σ,ℓ = i>ℓ λ 2 i . (iii) Define Σ −1 m,ℓ = ℓ i=1 1 λ i,m φ i,m ⊗ Hm φ i,m . Then S Σm,ℓ = E Ik(·, X) − AΣ −1 m,ℓ A * Ak m (·, X) 2 L 2 (P) = E Ik(·, X) − AΣ −1 m,ℓ Σ m k m (·, X) 2 L 2 (P) = E Ik(·, X) − AP ℓ (Σ m )k m (·, X) 2 L 2 (P) = R Σm,ℓ and the result follows from Theorem 6(iii). (iv) Upper bound: Define Σ −1 m,ℓ = ℓ i=1 1 λ i,m φ i,m ⊗ Hm φ i,m . Then S Σm,ℓ = E Ik(·, X) − A Σ −1 m,ℓ A * A(k m (·, X) − m P,m ) 2 L 2 (P) ≤ 3 3 + e + f ,(41)where e := E (I − A Σ −1 m,ℓ A * )Ak m (·, X) 2 L 2 (P) and f := A Σ −1 m,ℓ A * A(m P,m − m P,m ) 2 L 2 (P) . Note that 3 which can be bounded using Lemma B.6 and e = E A(I − Σ −1 m,ℓ Σ m )k m (·, X) 2 L 2 (P) (♣) = Σ 1/2 m (I − Σ −1 m,ℓ Σ m )Σ 1/2 m 2 L 2 (Hm) ≤ 2 5 + e2 ,(42) where we have used Lemma B.7 in (♣) and e2 := Σ 1/2 Hm) . Note that the bound on 5 , follows from (29). By handling e2 in a similar manner as b in (ii), by conditioning on Hm) , with probability at least 1 − 4δ over the choice of (X i ) n i=1 , we obtain m (P ℓ ( Σ m ) − Σ −1 m,ℓ Σ m )Σ 1/2 m 2 L 2 ((θ i ) m i=1 , for 140κ n log 16κn δ ≤ t ≤ Σ m L ∞ (e2 = Σ 1/2 m Σ −1 m,ℓ ( Σ m − Σ m )Σ 1/2 m 2 L 2 (Hm) ≤ 36 λ m,ℓ + t λ m,ℓ − t 2 128κ 5/2 N Σm (t) log 2 δ n √ t + 4096κ 3 log 2 3 δ n 2 t .(43) By unconditioning w.r.t. (θ i ) m i=1 in the above inequality, for 86κ m log 16κm δ ≤ t ≤ Σ L ∞ (H) , with probability at least 1 − 7δ jointly over the choice of ( (θ i ) m i=1 , (X i ) n i=1 ), we obtain e2 ≤ 36 3λ ℓ + 3t λ ℓ − 3t 2 4096κ 3 log 2 3 δ n 2 t + 256κ 5/2 D(t) log 2 δ n √ t ≤ 900 4096κ 3 log 2 3 δ n 2 t + 256κ 5/2 D(t) log 2 δ n √ t ,(44) where the first inequality follows from applying Lemma B.3(ii)-(iv) to (43), the second inequality follows by using t ≤ λ ℓ 9 and we used D(t) := 16κ log 2 δ tm + 8κN Σ (t) log 2 δ tm + N Σ (t) . e2 can be alternately bounded as follows. By conditioning on (θ i ) m i=1 , for 140κ n log 16κn Hm) , with probability at least 1 − 4δ over the choice of ( δ ≤ t ≤ Σ m L ∞ (X i ) n i=1 , we obtain e2 ≤ 36λ 2 m,1 t λ m,ℓ + t λ m,ℓ − t 2 64κ 2 log 2 δ n + 2048κ 2 log 2 3 δ n 2 . By unconditioning w.r.t. (θ i ) m i=1 in the above inequality, for 86κ m log 16κm H) , with probability at least 1 − 5δ jointly over the choice of ( δ ≤ t ≤ Σ L ∞ ((θ i ) m i=1 , (X i ) n i=1 ), we obtain e2 ≤ 400κ 4 t 3λ ℓ + 3t λ ℓ − 3t Combining (44) and (45), with probability at least 1−12δ jointly over the choice of ( (θ i ) m i=1 , (X i ) n i=1 ), we have e2 ≤ 24 × 10 4 κ 4 log 2 δ nt ∧ κ 5/2 D(t) log 2 δ n √ t + 8 × 10 6 κ 3 (1 ∧ k) log 2 3 δ n 2 t . (46) f can be bounded as Hm) , with probability at least 1 − 3δ over the choice of (X f = Σ 1/2 m Σ −1 m,ℓ Σ m (m P,m − m P,m ) 2 Hm ≤ Σ 1/2 m Σ −1 m,ℓ Σ 1/2 m 2 L ∞ (Hm) Σ 1/2 m (m P,m − m P,m ) 2 Hm ≤ λ m,1 Σ 1/2 m (Σ m + tI) −1/2 4 L ∞ (Hm) m P,m − m P,m 2 Hm (Σ m + tI) 1/2 ( Σ m + tI) −1/2 4 L ∞ (Hm) × ( Σ m + tI) 1/2 Σ −1 m,ℓ ( Σ m + tI) 1/2 2 L ∞ (Hm) .(47)By conditioning on (θ i ) m i=1 , for 140κ n log 16κn δ ≤ t ≤ Σ m L ∞ (i ) n i=1 , we obtain f ≤ 3840κ 2 log 2 δ n λ m,ℓ + t λ m,ℓ − t 2 , where we used the fact that λ m,1 ≤ 10κ 3 (see the proof of Theorem 6(iv)) and employed Lemmas B.2(ii, iv, v) and B.5(ii). By unconditioning w.r.t. (θ i ) m i=1 in the above inequality, with probability at least 1 − 4δ jointly over the choice of ( (θ i ) m i=1 , (X i ) n i=1 ), we obtain f ≤ 3840κ 2 log 2 δ n 3λ ℓ + 3t λ ℓ − 3t 2 ≤ 96000κ 2 log 2 δ n ,(48) where we applied Lemma B.3(ii, iii) and 86κ m log 16κm H) . The result therefore follows by combining (41), (42), (46) and (48) under the condition that 140κ n log 8n δ ∨ 86κ m log 8m δ ≤ t ≤ λ ℓ 9 and n ∧ m ≥ 8 log 1 δ . Lower bound: As carried out in the proof of the lower bound of (iii), it can be shown that δ ≤ t ≤ Σ L ∞ (S Σm,ℓ = E Ik(·, X) − A Σ −1 m,ℓ Σ m k m (·, X) 2 L 2 (P) ≥ e + f − 3 2 ≥ e − 3 2 ( * ) ≥   i>ℓ λ 2 m,i − 3   2 , where ( * ) follows by applying Lemma B.1 to e . The result, therefore, follows from (24). Proof of Theorem 9 Define (i) By adding and subtracting I k(·, X) to the first argument of the inner product in T Σ,ℓ , we obtain T Σ,ℓ = S I φ 1 √ λ 1 , . . . , I φ ℓ √ λ ℓ , T Σm,ℓ = S Aφ m,1 √ λ m,1 , . . . ,T Σ,ℓ ≤ 2S Σ,ℓ + 2E ℓ i=1 Ik(·, X) − I k(·, X), I φ i λ i L 2 (P) I φ i λ i 2 L 2 (P) = 2S Σ,ℓ + 2 I Σ −1 ℓ Σ(m P − m P ) 2 L 2 (P) = 2S Σ,ℓ + 2 Σ 1/2 Σ −1 ℓ Σ(m P − m P ) 2 H and the result follows from (40). Also note that T Σ,ℓ = Σ 1/2 (I − Σ −1 ℓ Σ)Σ 1/2 2 L 2 (H) and therefore the lower bound follows from the proof of the lower bound of S Σ,ℓ . (ii) As above, adding and subtracting Ak m (·, X) to the first argument of the inner product in T Σm,ℓ , we obtain ) and the result follows by noting that Σ m Σ −1 m,ℓ = P ℓ (Σ m ), P ℓ (Σ m ) L ∞ (Hm) = 1 and applying Lemma B.6. For the lower bound, note that T Σm,ℓ ≤ 2S Σm,ℓ + 2E ℓ i=1 Ik(·, X) − Ak m (·, X), Aφ m,i λ m,i L 2 (P) Aφ m,i λ m,i 2 L 2 (P) = 2S Σm,ℓ + 2E AΣ −1 m,ℓ A * (Ik(·, X) − Ak m (·, X)) 2 L 2 (P) ≤ 2S Σm,ℓ + 2 Σ m Σ −1 m,ℓ 2 L ∞ (Hm) E Ik(·, X) − Ak m (·, X) 2 L 2 (PT Σm,ℓ = E Ik(·, X) − AΣ −1 m,ℓ A * Ik(·, X) 2 L 2 (P) = S Σm,ℓ + E AΣ −1 m,ℓ A * Ik(·, X) − Ak m (·, X) 2 L 2 (P) −2E Ik(·, X) − AΣ −1 m,ℓ A * Ak m (·, X), AΣ −1 m,ℓ A * Ik(·, X) − Ak m (·, X) L 2 (P) ≥ S Σm,ℓ + g − 2 S Σm,ℓ g = S Σm,ℓ − g 2 ≥ S Σm,ℓ − 3 2 , where we used g := E AΣ −1 m,ℓ A * Ik(·, X) − Ak m (·, X) 2 L 2 (P) ≤ 3 × AΣ −1 m,ℓ A * 2 L ∞ (L 2 (P)) = 3 × Σ m Σ −1 m,ℓ 2 L ∞ (Hm) = 3 . Since S Σm,ℓ ≥ 4 − 3 , we have T Σm,ℓ ≥ 4 − 2 × 3 2 . Based on the calculation we made for the lower bound of R Σm,ℓ , for 1 2 i>ℓ λ 2 i ≥ 20κ log 2 δ m , we obtain T Σm,ℓ ≥ 1 4 i>ℓ λ 2 i , which holds with probability at least 1 − 3δ over the choice of (θ i ) m i=1 . (iii) Doing as above, we obtain T Σm,ℓ ≤ 2S Σm,ℓ + 2E ℓ i=1 Ik(·, X) − A k m (·, X), Aφ m,i λ m,i L 2 (P) Aφ m,i λ m,i 2 L 2 (P) = 2S Σm,ℓ + 2E A Σ −1 m,ℓ A * (Ik(·, X) − A k m (·, X)) 2 L 2 (P) ≤ 2S Σm,ℓ + 4 A Σ −1 m,ℓ A * 2 L ∞ (L 2 (P)) E Ik(·, X) − Ak m (·, X) 2 L 2 (P) +4E A Σ −1 m,ℓ A * (Ak(·, X) − A k(·, X)) 2 L 2 (P) ≤ 2S Σm,ℓ + 4 Σ 1/2 m Σ −1 m,ℓ Σ 1/2 m 2 L ∞ (Hm) E Ik(·, X) − Ak m (·, X) 2 L 2 (P) +4 Σ 1/2 m Σ −1 m,ℓ Σ m (m P,m − m P,m ) 2 Hm . By applying Lemma B.6, the result follows from (47) T Σm,ℓ ≥ S Σm,ℓ − h 2 ≥ i>ℓ λ 2 m,i − 3 − h 2 , where h := E A Σ −1 m,ℓ A * Ik(·, X) − Ak m (·, X) 2 L 2 (P) ≤ A Σ −1 m,ℓ A * 2 L ∞ (L 2 (P)) × 3 = Σ 1/2 m Σ −1 m,ℓ Σ 1/2 m 2 L ∞ (Hm) × 3 Λ m 3 . The result therefore follows by choosing m sufficiently larger than 1 i>ℓ λ 2 i . Proof of Corollary 10 (i) and (iii) are exactly same as that of the proof of Corollary 7. (ii) From Theorem 8(ii) we have S Σ,ℓ P n 1 n + 1 n 2 t + N Σ (t) n √ t ∧ 1 nt + N Σ (t)(λ ℓ + t) 2 , with log n n t ≤ λ ℓ 3 . Clearly 1 n 2 t 1 n . Using N Σ (t) t −1/α from Lemma B.9(i), it follows that S Σ,ℓ P n inf t −1/α (n −θ + t) 2 + t −( 1 α + 1 2 ) n ∧ 1 nt + 1 n : log n n t n −θ . It is clear that both t −( 1 α + 1 2 ) n and 1 nt dominate n −1 and using t n −θ in the first term, we obtain S Σ,ℓ P n inf t −1/α n −2θ + t − 1 α ′ n : log n n t n −θ = n −2θ(1− 1 2α ) + n −(1− θ α ′ ) , where 1 α ′ := 1 α + 1 2 ∧ 1 and the result follows. (iv) From Theorem 8(iv) we have S Σm,ℓ P n ×Λ m 1 n + 1 m + 1 n 2 t + A(t) n √ t ∧ 1 nt + A(t)(λ ℓ + t) 2 for log n n ∨ log m m t λ ℓ , where A(t) = N Σ (t) + N Σ (t) tm + 1 tm . From the proof of Corollary 7(iv), we have A(t) t −1/α . Also it is obvious that 1 n 2 t 1 n . Therefore, S Σm,ℓ P n ×Λ m inf 1 n γ + t −1/α n −2θ + t − 1 α ′ n : log n n γ t n −θ = n −γ + n −2θ(1− 1 2α ) + n −(1− θ α ′ ) and the result follows by imposing θ < γ to ensure the constraint log n n γ t n −θ is satisfied. Proof of Proposition 11 (i) Note that for any a ≥ 1, (II * ) a I = I(I * I) a . This follows by observing that for any f ∈ H, (II * ) a If = i λ a i Iφ i √ λ i ⊗ L 2 (P) Iφ i √ λ i If = I i λ a−1 i (φ i ⊗ H φ i )Σf = I i λ a−1 i φ i φ i , Σf H = I i λ a i φ i φ i , f H = IΣ a f = I(I * I) a f.(49) Therefore, R s (ψ 1 , . . . , ψ ℓ ) = E (II * ) −s/2 I (I − P ψ ) k(·, X) 2 L 2 (P) = E IΣ −s/2 (I − P ψ ) k(·, X) 2 L 2 (P) ( †) = Σ (1−s)/2 (I − P ψ ) Σ 1/2 2 L 2 (H) ,(49) where ( †) follows from Lemma B.7. (ii)-(v) These exactly follow the proof of Proposition 3(ii)-(v) by using δ = 1 − s in Lemma B.1. Proof of Theorem 12 (i) The result follows from Proposition 11(ii). (ii) Along the lines of the proof of Theorem 6(ii) and using (49), it is easy to show that R Σ,ℓ,s = Σ (1−s)/2 (I − P ℓ ( Σ))Σ 1/2 2 L 2 (H) + Σ (1−s)/2 P ℓ ( Σ)(m P − m P ) 2 H , where the second term can be bounded as Σ 1−s L ∞ (H) P ℓ ( Σ) 2 L ∞ (H) m P − m P 2 H P n 1 n , through an application of Lemma B.5(i). For the first term, employing the strategy used for bounding 1 , for any t > 0, we obtain Σ (1−s)/2 (I − P ℓ ( Σ))Σ 1/2 2 L 2 (H) = Σ (1−s)/2 (Σ + tI) −1/2 (Σ + tI) 1/2 (I − P ℓ ( Σ))(Σ + tI) 1/2 (Σ + tI) −1/2 Σ 1/2 2 L 2 (H) ≤ (Σ + tI) −1/2 Σ 1/2 2 L 2 (H) Σ (1−s)/2 (Σ + tI) −1/2 2 L ∞ (H) × (Σ + tI) 1/2 (I − P ℓ ( Σ))(Σ + tI) 1/2 2 L ∞ (H) ≤ N Σ (t)( λ ℓ+1 + t) 2 Σ (1−s)/2 (Σ + tI) −1/2 2 L ∞ (H) , where Σ (1−s)/2 (Σ + tI) −1/2 2 L ∞ (H) = sup i λ 1−s i λ i + t = sup i λ i λ i + t 1−s 1 (λ i + t) s ≤ 1 t s for s ≤ 1. The result is completed by bounding ( λ ℓ+1 + t) 2 as in the proof of the upper bound in Theorem 6(ii). The lower bound follows from Lemma B.1 by noting that R Σ,ℓ,s ≥ R Σ 0,1−s,1 ( Σ) ≥ R Σ 0,1−s,1 (Σ) = R Σ,ℓ,s . (iii) Note that R Σm,ℓ,s = E X∼P (II * ) −s/2 Ik(·, X) − (AA * ) −s/2 AP ℓ (Σ m )k m (·, X) 2 L 2 (P) ≤ 2 8 + 9 ,(50) where 8 := E X∼P (II * ) −s/2 Ik(·, X) − (AA * ) −s/2 Ak m (·, X) 2 L 2 (P) and 9 := E X∼P (AA * ) −s/2 A(I − P ℓ (Σ m ))k m (·, X) 2 L 2 (P) = i>ℓ λ 2−s m,i ,(51) which follows by replicating the analysis in (i) for AA * . To bound 8 , we adapt the proof idea of Lemma B.6. Similar to (B.15), it can be shown that 8 = (II * ) (2−s)/2 − (AA * ) (2−s)/2 2 L 2 (L 2 (P)) + 2tr (II * ) −s/2 II * AA * (AA * ) −s/2 −2E (II * ) −s/2 Ik, (AA * ) −s/2 Ak m (·, X) L 2 (P) .(52) Similar to (B.16) and (B.17), we obtain E (II * ) −s/2 Ik(·, X), (AA * ) −s/2 Ak m (·, X) L 2 (P) = Θ m i=1 B i (θ) ϕ(·, θ), ϕ i L 2 (P) dΛ(θ) and tr (II * ) −s/2 II * AA * ( AA * ) −s/2 = Θ m i=1 B i (θ) ϕ(·, θ), ϕ i L 2 (P) − ϕ P (θ)ϕ i,P dΛ(θ), where B i (θ) := (II * ) −s/2 (ϕ(·, θ) − ϕ P (θ)) , (AA * ) −s/2 (ϕ i − ϕ i,P ) L 2 (P) . This implies tr (II * ) −s/2 II * AA * (AA * ) −s/2 − E (II * ) −s/2 Ik(·, X), (AA * ) −s/2 Ak m (·, X) L 2 (P) = − (II * ) −s/2 Θ A(θ) dΛ(θ), (AA * ) −s/2 1 m m i=1 A(θ i ) L 2 (P) ( * ) ≤ 1 4 (II * ) −s/2 Θ A(θ) dΛ(θ) − (AA * ) −s/2 1 m m i=1 A(θ i ) 2 L 2 (P) ≤ 1 2 (II * ) −s/2 − (AA * ) −s/2 2 L ∞ (L 2 (P)) Θ A(θ) dΛ(θ) 2 L 2 (P) + (II * ) −s/2 − (AA * ) −s/2 2 L ∞ (L 2 (P)) + (II * ) −s/2 2 L ∞ (L 2 (P)) × Θ A(θ) dΛ(θ) − 1 m m i=1 A(θ i ) 2 L 2 (P) where ( * ) follows from the parallelogram identity with A(θ) := ϕ(·, θ)ϕ P (θ) − ϕ 2 P (θ). It therefore follows from Theorem D.1 that tr (II * ) −s/2 II * AA * (AA * ) −s/2 − E (II * ) −s/2 Ik(·, X), (AA * ) −s/2 Ak m (·, X) L 2 (P) Λ m (II * ) −s/2 − (AA * ) −s/2 2 L ∞ (L 2 (P)) + 1 m (II * ) −s/2 2 L ∞ (L 2 (P)) .(53) Combining (50) + (II * ) −s/2 − (AA * ) −s/2 2 L ∞ (L 2 (P)) ,(54) where for s ≤ 1, i>ℓ λ 2−s m,i i>ℓ \|λ m,i − λ i \| 2−s + i>ℓ λ 2−s i ( * ) II * − AA * 2−s 2−s + i>ℓ λ 2−s i , with ( * ) following from (Kato, 1987, Theorem II). Here · 2−s denotes the (2 − s)-Schatten norm. Since t → t α is Lipschitz on a bounded subset of (0, ∞) for α ≥ 1, it follows from (De Vito et al., 2014, Lemma 7) that the second term of (54) is bounded (up to constants) by II * −AA * 2 L 2 (L 2 (P)) for s ≤ 0. Using the fact that t → t α is operator monotone on (0, ∞) for 0 ≤ α ≤ 1, the third term in (54) is bounded by II * − AA * −s L ∞ (L 2 (P)) for −2 ≤ s ≤ 0 (follows from Bhatia, 1997, Theorem X.1.1). Therefore for −2 ≤ s ≤ 0, (54) reduces to R Σm,ℓ,s Λ m i>ℓ λ 2−s i + II * − AA * 2−s 2−s + II * − AA * 2 L 2 (L 2 (P)) + II * − AA * −s L ∞ (L 2 (P)) + 1 m . The result follows by applying Lemma B.4 to II * −AA * L 2 (L 2 (P)) and noting that II * − AA * 2−s ≤ II * − AA * L 2 (L 2 (P)) since −2 ≤ s ≤ 0. The lower bound follows the idea in the proof of lower bound on R Σm,ℓ by noticing that R Σm,ℓ,s ≥ 9 − 8 2 where 8 m s/2 for s ∈ [−2, 0) and 8 1 m for s = 0. Considering (iv) We skip the proof as it follows the ideas in the proof of R Σ,ℓ combined with the bounds on 8 and 9 . 9 = i>ℓ λ 2−s m,i , we have i>ℓ λ 2−s m,i 1 2−s ≥ i>ℓ λ 2−s i 1 2−s − i>ℓ \|λ m,i − λ i \| 2−s 1 2−s ≥ i>ℓ λ 2−s i 1 2−s − II * − AA * 2−s Λ m i>ℓ λ 2−s i 1 2−s − 1 √ m i>ℓ λ 2−s i 1 2−s , since m i>ℓ λ 2−s i 2 s−2 . Since m i>ℓ λ 2−s i 2 s for s ∈ [−2, 0), (i) n −2θ R Σ,ℓ n −2θ . There existsñ ∈ N such that for all n >ñ, the following hold: (ii) n −2θ RΣ ,ℓ P n n −2θ log n, θ ≤ 1 2 1 n , θ > 1 2 ; (iii) For 0 < γ ≤ 1 and m = n γ , n −2θ 1 {γ≥2θ} Λ m R Σm,ℓ Λ m n −2θ , γ ≥ 2θ, θ ≤ 1 2 n −γ , γ ≤ 1 ∧ 2θ ; (iv) For 0 < γ ≤ 1 and m = n γ , n −2θ 1 {γ≥2θ} Λ m RΣ m,ℓ P n ×Λ m n −2θ log n, γ ≥ 2θ, θ ≤ 1 2 n −γ , γ < 2θ, γ ≤ 1 . We may draw conclusions similar to Remark 4 from Corollary A.1. The behavior of R Σ,ℓ matches that of R Σ,ℓ , up to a log n factor, if ℓ grows slower than log √ n. If m ≥ n 2θ with θ ≤ 1 2 , then RF-EKPCA and EKPCA have similar statistical convergence behavior (i.e., no statistical loss) but with RF-EKPCA enjoying a computational edge if m < n log n θ , i.e., θ ≤ 1 4 . Proof. (i) From Theorem 6(i) we have R Σ,ℓ = i>ℓ λ 2 i i>ℓ e −2τ i ∞ ℓ e −2τ x dx e −2τ ℓ = n −2θ and R Σ,ℓ = i>ℓ λ 2 i i>ℓ e −2τ i ∞ ℓ+1 e −2τ x dx e −2τ (ℓ+1) = e −2τ n −2θ . (ii) Using N Σ (t) log 1 t from Lemma B.9(ii) in Theorem 6(ii), we have RΣ ,ℓ P n inf (n −θ + t) 2 log 1 t + n −1 : log n n t ≤ λ 1 3 n −2θ log n + 1 n , θ < 1 log 3 n n 2 + 1 n , θ ≥ 1 , and the result follows. (iii) We obtain R Σm,ℓ Λm 1 m + n −2θ and the result follows. (iv) Theorem 6 (iv) yields R Σm,ℓ P n ×Λ m n −γ + N Σ (t) + N Σ (t) tn γ + 1 tn γ (λ ℓ + t) 2 , for log n n γ t λ 1 3 . Using N Σ (t) log 1 t , it is clear that 1 tn γ log 1 t and N Σ (t)n −γ /t log 1 t which follows from the constraint on t. Therefore, R Σm,ℓ P n ×Λ m inf (n −θ + t) 2 log 1 t + n −γ : log n n γ t ≤ λ 1 3 n −2θ log n + n −γ , θ < γ log 3 n n 2 + n −γ , θ ≥ γ , and the result follows. Corollary A.2 (Exponential decay of eigenvalues). Suppose Be −τ i ≤ λ i ≤Be −τ i for τ > 0 and B,B ∈ (0, ∞). Let ℓ = 1 τ log n θ for θ > 0. Then (i) n −2θ S Σ,ℓ n −2θ . There existsñ ∈ N such that for all n >ñ, the following hold: (ii) n −2θ S Σ,ℓ P n n −2θ log n, θ ≤ 2 5 n −(1− θ 2 ) log n, 2 5 ≤ θ < 1 ; (iii) For 0 < γ ≤ 1 and m = n γ , n −2θ 1 {γ≥2θ} Λ m S Σm,ℓ Λ m n −2θ , γ ≥ 2θ, θ ≤ 1 2 n −γ , γ ≤ 1 ∧ 2θ ; (iv) For 0 < γ ≤ 1 and m = n γ , n −2θ 1 {γ≥2θ} Λ m S Σm,ℓ P n ×Λ m      n −2θ log n, γ ≥ 2θ, θ ≤ 2 5 n −(1− θ 2 ) log n, γ ≥ 1 − θ 2 , γ > θ, 2 5 ≤ θ < 1 n −γ , θ < γ ≤ 1 ∧ 2θ ∧ 1 − θ 2 . Proof. (i) and (iii) are exactly same as that of the proof of Corollary A.1. (ii) Using N Σ (t) log 1 t from Lemma B.9(ii) in Theorem 8(ii), we have N Σ (t) n √ t ∧ 1 nt N Σ (t) n √ t log 1 t n √ t , which implies SΣ ,ℓ P n inf n −2θ log 1 t + log 1 t n √ t : log n n t n −θ n −2θ + n −(1− θ 2 ) log n and the result follows. (iv) By noting that A(t) N Σ (t) log 1 t , we obtain SΣ ,ℓ P n inf n −γ + n −2θ log 1 t + log 1 t n √ t : log n n t n −θ n −γ + n −2θ + n −(1− θ 2 ) log n, which yields the result. B Technical Results In this section, we collect important technical results used to prove the main results of this paper. Lemma B.1. Let A : H → H be a positive self-adjoint Hilbert-Schmidt operator on a separable Hilbert space H with (λ i , ψ i ) i being its eigenvalues and eigenfunctions. Suppose the eigenvalues are simple and satisfy λ 1 > λ 2 > · · · . Define Q ℓ = ℓ i=1 τ i ⊗ H τ i : (τ i ) i∈[ℓ] ⊂ H , R A α,δ,θ (Q) = A δ/2 (I − QA α ) A θ/2 2 L 2 (H) , Q ∈ Q ℓ , (B.1) and S A ρ (Q) = (I − Q) A ρ/2 2 L 2 (H) , Q ∈ Q ℓ , where α, δ ≥ 0 and θ, ρ > 0. Then, (i) A −α ℓ = arg inf Q∈Q ℓ R A α,δ,θ (Q), where A ℓ = ℓ i=1 λ i ψ i ⊗ H ψ i and R A α,δ,θ (A −α ℓ ) = i>ℓ λ θ+δ i ; (ii) Q ℓ = arg inf Q∈Q ℓ S A ρ (Q), where Q ℓ = ℓ i=1 ψ i ⊗ H ψ i and S A ρ (Q ℓ ) = i>ℓ λ ρ i . Proof. (i) Define A ≤ = ℓ i=1 λ i ψ i ⊗ H ψ i and A > = i>ℓ λ i ψ i ⊗ H ψ i so that A = A ≤ + A > . Also since A δ/2 = i λ δ/2 i ψ i ⊗ H ψ i , we have A δ/2 = A δ/2 ≤ + A δ/2 > . Consider R A α,δ,θ (Q) = A (θ+δ)/2 − A δ/2 QA α+θ/2 2 L 2 (H) = A (θ+δ)/2 ≤ + A (θ+δ)/2 > − A δ/2 ≤ + A δ/2 > Q A α+θ/2 ≤ + A α+θ/2 > 2 L 2 (H) ( * ) = A (θ+δ)/2 ≤ − A δ/2 ≤ QA α+θ/2 ≤ + A (θ+δ)/2 > − A δ/2 > QA α+θ/2 > − A δ/2 > QA α+θ/2 ≤ + A δ/2 ≤ QA α+θ/2 > 2 L 2 (H) = A (θ+δ)/2 ≤ − A δ/2 ≤ QA α+θ/2 ≤ 2 L 2 (H) + A (θ+δ)/2 > − A δ/2 > QA α+θ/2 > 2 L 2 (H) + A δ/2 > QA α+θ/2 ≤ + A δ/2 ≤ QA α+θ/2 > 2 L 2 (H) , (B.2) where (B.2) is obtained by expanding the square in ( * ) and noting that all the inner products are zero since Tr(A β ≤ BA γ > ) = 0 for any β, γ ≥ 0 and any operator B : H → H. Again expanding the square in the last term of (B.2) and noting that the inner product is zero, we obtain R A α,δ,θ (Q) = A (θ+δ)/2 ≤ − A δ/2 ≤ QA α+θ/2 ≤ 2 L 2 (H) + A (θ+δ)/2 > − A δ/2 > QA α+θ/2 > 2 L 2 (H) + A δ/2 > QA α+θ/2 ≤ 2 L 2 (H) + A δ/2 ≤ QA α+θ/2 > 2 L 2 (H) . (B.3) We now decompose Q ∈ Q ℓ as Q = Q 1 + Q 2 + Q 3 where Q j = i∈A j τ i ⊗ H τ i with A j ⊂ {1, . . . , ℓ}, j ∈ [3], ∪ j A j = {1, . . . , ℓ}, A i ∩ A j = ∅ for all i, j ∈ [3] such that (τ i ) i∈A 1 ⊂ Ran(A ≤ ), (τ i ) i∈A 2 ⊂ Ran(A > ) , and (τ i ) i∈A 3 ⊂ Ker(A). This means optimizing over Q ∈ Q ℓ is equivalent to optimizing over Q j ∈ Q \|A j \| and A j , j ∈ [3]. Using A β > QA γ ≤ = A β > (Q 1 + Q 2 + Q 3 )A γ ≤ = 0 and A β ≤ QA γ > = A β ≤ (Q 1 + Q 2 + Q 3 )A γ > = 0 for any β, γ ≥ 0 in (B.3), we obtain R A α,δ,θ (Q) = A (θ+δ)/2 ≤ − A δ/2 ≤ QA α+θ/2 ≤ 2 L 2 (H) + A (θ+δ)/2 > − A δ/2 > QA α+θ/2 > 2 L 2 (H) = A δ/2 ≤ P ≤ − QA α ≤ A θ/2 ≤ 2 L 2 (H) + A δ/2 > (P > − QA α > ) A θ/2 > 2 L 2 (H) = A δ/2 ≤ P ≤ − Q 1 A α ≤ A θ/2 ≤ 2 L 2 (H) + A δ/2 > (P > − Q 2 A α > ) A θ/2 > 2 L 2 (H) , (B.4) where P ≤ := ℓ i=1 ψ i ⊗ H ψ i and P > := i>ℓ ψ i ⊗ H ψ i . Let B ⊆ {1, . . . , ℓ}, B c := {1, . . . , ℓ}\B, C ⊆ {ℓ + 1, ℓ + 2, . . .}, C c := {ℓ + 1, ℓ + 2, . . .}\C such that span{(ψ i ) i∈B } = span((τ i ) i∈A 1 ) and span{(ψ i ) i∈C } = span((τ i ) i∈A 2 ). This means \|B\| ≤ \|A 1 \| and \|C\| ≤ \|A 2 \| and \|B\| + \|C\| ≤ ℓ. Note that A ≤ = A ≤,B + A ≤,B c and A > = A >.C + A >,C c , where A ≤,• := i∈• λ i ψ i ⊗ H ψ i , P ≤,• = i∈• ψ i ⊗ H ψ i , • ∈ {B, B c }, A >, := i∈ λ i ψ i ⊗ H ψ i , and P >, = i∈ ψ i ⊗ H ψ i , ∈ {C, C c }. Consider the first term in the r.h.s. of (B.4), i.e., A δ/2 ≤ P ≤ − Q 1 A α ≤ A θ/2 ≤ 2 L 2 (H) = A δ/2 ≤,B + A δ/2 ≤,B c P ≤,B + P ≤,B c − Q 1 A α ≤ A θ/2 ≤ 2 L 2 (H) ( †) = A δ/2 ≤,B P ≤,B − Q 1 A α ≤ A θ/2 ≤ 2 L 2 (H) + A δ/2 ≤,B c P ≤,B c − Q 1 A α ≤ A θ/2 ≤ 2 L 2 (H) ( * ) = A δ/2 ≤,B P ≤,B − Q 1 A α ≤,B − Q 1 A α ≤,B c A θ/2 ≤,B + A θ/2 ≤,B c 2 L 2 (H) + A δ/2 ≤,B c P ≤,B c − Q 1 A α ≤,B − Q 1 A α ≤,B c A θ/2 ≤,B + A θ/2 ≤,B c 2 L 2 (H) ( ‡) = A δ/2 ≤,B P ≤,B − Q 1 A α ≤,B A θ/2 ≤,B − A δ/2 ≤,B Q 1 A α+θ/2 ≤,B c 2 L 2 (H) + A δ/2 ≤,B c P ≤,B c − Q 1 A α ≤,B c A θ/2 ≤,B c − A δ/2 ≤,B c Q 1 A α+θ/2 ≤,B 2 L 2 (H) , (B.5) where ( †) is obtained by expanding the square in the previous line and using A β ≤,B A γ ≤,B c = A γ ≤,B c A β ≤,B = 0 for any β, γ ≥ 0. (B.6) Again using (B.6), ( * ) reduces to ( ‡). By noting that A δ/2 ≤,B Q 1 A α+θ/2 ≤,B c = A δ/2 ≤,B c Q 1 A α+θ/2 ≤,B = A δ/2 ≤,B c Q 1 A α+θ/2 ≤,B c = 0, (B.5) reduces to A δ/2 ≤ P ≤ − Q 1 A α ≤ A θ/2 ≤ 2 L 2 (H) = A δ/2 ≤,B P ≤,B − Q 1 A α ≤,B A θ/2 ≤,B 2 L 2 (H) + A (δ+θ)/2 ≤,B c 2 L 2 (H) . (B.7) Carrying out similar calculation for the second term of (B.4), and combining the result along with (B.7) in (B.4) yields R A α,δ,θ (Q) = A δ/2 ≤,B P ≤,B − Q 1 A α ≤,B A θ/2 ≤,B 2 L 2 (H) + A (δ+θ)/2 ≤,B c 2 L 2 (H) + A δ/2 >,C P >,C − Q 2 A α >,C A θ/2 >,C 2 L 2 (H) + A (δ+θ)/2 >,C c 2 L 2 (H) = A δ/2 ≤,B P ≤,B − Q 1 A α ≤,B A θ/2 ≤,B 2 L 2 (H) + A δ/2 >,C P >,C − Q 2 A α >,C A θ/2 >,C 2 L 2 (H) + ∞ i=1 λ δ+θ i − i∈B∪C λ δ+θ i , (B.8) where we used A (δ+θ)/2 ≤,B c 2 L 2 (H) + A (δ+θ)/2 >,C c 2 L 2 (H) = i∈B c ∪C c λ δ+θ i = ∞ i=1 λ δ+θ i − i∈B∪C λ δ+θ i . It follows from (B.8) that R A α,δ,θ is minimized only when Q 1 and Q 2 satisfy P ≤,B = Q 1 A α ≤,B (i.e., Q 1 = A −α ≤,B ), P >,C = Q 2 A α >,C (i.e., Q 2 = A −α >,C ) for B and C such that i∈B∪C λ δ+θ i is maximized. Subject to the constraint \|B\|+\|C\| ≤ ℓ, clearly i∈B∪C λ δ+θ i is maximized only when B = {1, . . . , ℓ}, C = ∅. This yields Q 1 = A −α ≤ , Q 2 = Q 3 = 0 and the result follows by noting that Q = Q 1 +Q 2 +Q 3 and R A α,δ,θ (A −α ≤ ) = i>ℓ λ δ+θ i . (ii) Let P Ran(A) := i ψ i ⊗ H ψ i and P ⊥ denote the orthogonal projection operators that project onto Ran(A) and Ker(A) respectively. Then I = P Ran(A) + P ⊥ . Therefore, (I − Q)A ρ/2 2 L 2 (H) = (P Ran(A) + P ⊥ )(I − Q)A ρ/2 2 L 2 (H) = P Ran(A) (I − QP Ran(A) )A ρ/2 + P ⊥ (I − Q)A ρ/2 2 L 2 (H) = P Ran(A) (I − QP Ran(A) )A ρ/2 2 L 2 (H) + P ⊥ (I − Q)A ρ/2 2 L 2 (H) ( * ) = R A 0,0,ρ (Q) + P ⊥ (I − Q)A ρ/2 2 L 2 (H) ≥ R A 0,0,ρ (Q) ( †) ≥ R A 0,0,ρ (Q ℓ ), where ( †) follows from Lemma B.1(i). Note that, at the minimizer of R A 0,0,ρ , which is Q ℓ , the second term in ( * ) is zero, which implies Q ℓ is the minimizer of S A ρ (Q) over Q ∈ Q ℓ . Therefore S A ρ (Q ℓ ) = R A 0,0,ρ (Q ℓ ) = i>ℓ λ ρ i . The following result extends Lemma 3.6 of (Rudi et al., 2013), which holds for uncentered covariance operators, to centered covariance operators that are estimated using a U -statistic. Lemma B.2. Let H be a separable Hilbert space and Y be a separable topological space. Define C = 1 2 Y Y (s(x) − s(y)) ⊗ H (s(x) − s(y)) dP (x) dP (y) where s : Y → H is a Bochner-measurable function with sup x∈Y s(x) 2 H = κ. Given (Y i ) r i=1 i.i.d. ∼ P with r ≥ 2, define C = 1 2r(r − 1) r i =j (s(Y i ) − s(Y j )) ⊗ H (s(Y i ) − s(Y j )). Then for any 0 ≤ δ ≤ 1 2 and 140κ r log 16κr δ ≤ t ≤ C L ∞ (H) , the following hold: (i) P r (Y i ) r i=1 : (C + tI) −1/2 ( C − C)(C + tI) −1/2 L ∞ (H) ≤ 1 2 ≥ 1 − 2δ; (ii) P r (Y i ) r i=1 : 2 3 ≤ (C + tI) 1/2 ( C + tI) −1/2 L ∞ (H) ≤ √ 2 ≥ 1 − 2δ; (iii) P r (Y i ) r i=1 : (C + tI) −1/2 ( C + tI) 1/2 L ∞ (H) ≤ 3 2 ≥ 1 − 2δ; (iv) P r (Y i ) r i=1 : λ ℓ ( C) + t ≤ 3 2 (λ ℓ (C) + t) ≥ 1 − 2δ for all ℓ ≥ 1; (v) P r (Y i ) r i=1 : λ ℓ (C) + t ≤ 2(λ ℓ ( C) + t) ≥ 1 − 2δ for all ℓ ≥ 1. In addition, for any 0 < t ≤ C L ∞ (H) , P r    (Y i ) r i=1 : C −1/2 t ( C − C)C 1/2 L 2 (H) ≤ 64κ 5/2 N C (t) log 2 δ r √ t + 32 √ 2κ 3/2 log 3 δ r √ t    ≥ 1 − 2δ, (B.9) where C t := (C + tI) and N C (t) = tr(C −1 t C). Proof. (i) Define A(x, y) := 1 √ 2 (s(x) − s(y)), U (x, y) := (C + tI) −1/2 A(x, y) ∈ H and Z(x, y) := U (x, y) ⊗ H U (x, y). Clearly Z(x, y) = Z(y, x) and (C + tI) −1/2 ( C − C)(C + tI) −1/2 = 1 r(r − 1) r i =j Z(Y i , Y j ) − E[Z(X, Y )]. Also sup x,y∈Y Z(x, y) L 2 (H) ( * ) = sup x,y∈Y U (x, y) 2 H ≤ 1 2 sup x,y∈Y (C + tI) −1/2 (s(x) − s(y)) 2 H = 2κ t , where ( * ) follows from Lemma B.8. Define ψ(x) := E Y [Z(x, Y )] = E Y [U (x, Y ) ⊗ H U (x, Y )]. Clearly, sup x∈Y ψ(x) L ∞ (H) ≤ sup x,y∈Y U (x, y) ⊗ H U (x, y) L ∞ (H) ( †) = sup x,y∈Y U (x, y) 2 H ≤ 2κ t , where ( †) follows from Lemma B.8. Since E[ψ(X)] = E[Z(X, Y )], E[(ψ(X) − E[Z(X, Y )]) 2 ] = E[ψ 2 (X)] − E 2 [Z(X, Y )] E[ψ 2 (X)]. By defining C t = C + tI, we have E[ψ 2 (X)] = E[E 2 Y [U (X, Y ) ⊗ H U (X, Y )]] = E C −1/2 t E Y [A(X, Y ) ⊗ H A(X, Y )]C −1 t E Y [A(X, Y ) ⊗ H A(X, Y )]C −1/2 t sup x∈Y C −1/2 t E Y [A(x, Y ) ⊗ H A(x, Y )]C −1/2 t L ∞ (H) ×E C −1/2 t E Y [A(X, Y ) ⊗ H A(X, Y )]C −1/2 t 2κ t (C + tI) −1/2 C(C + tI) −1/2 =: S. Note that S L ∞ (H) ≤ 2κ t and d := S L 1 (H) S L ∞ (H) = tr(C −1 t C) C −1 t C L ∞ (H) ≤ ( C L ∞ (H) + t)tr(C −1 t C) C L ∞ (H) . Therefore, applying Theorem D.3 yields that for 0 < δ ≤ d with probability at least 1 − 2δ, (C + tI) −1/2 ( C − C)(C + tI) −1/2 L ∞ (H) ≤ 4κβ rt + 24κβ rt + 16κ log 3 δ rt ≤ 4κβ rt + 24κβ rt + 24κβ rt = 28κβ rt + 24κβ rt , (B.10) where β = 2 3 log 4d δ and we used that fact that d > 1 in the second line. Since t ≥ 140κ r log 16κr δ , it follows that t ≥ 140κ r log 4d δ as 140κ r log 16κr δ ≥ 140κ r log 16κ tδ ≥ 140κ r log 4d δ where we use the fact that d ≤ 4κ t which follows from t ≤ C L ∞ (H) and tr( C −1 t C) ≤ tr(C) t ≤ 2κ t . This implies t ≥ 210κβ r or κβ rt ≤ 1 210 . Using this in (B.10) yields the result. (ii) By defining B n = (C + tI) −1/2 (C − C)(C + tI) −1/2 , we have (C + tI) 1/2 ( C + tI) −1/2 L ∞ (H) = ( C + tI) −1/2 (C + tI)( C + tI) −1/2 1/2 L ∞ (H) = (C + tI) 1/2 ( C + tI) −1 (C + tI) 1/2 1/2 L ∞ (H) = (I − B n ) −1 1/2 L ∞ (H) ≤ (1 − B n L ∞ (H) ) −1/2 , where the last inequality holds whenever B n L ∞ (H) < 1. Similarly, (C + tI) 1/2 ( C + tI) −1/2 L ∞ (H) = (I + (−B n )) −1 1/2 L ∞ (H) ≥ (1 + B n L ∞ (H) ) −1/2 . The result therefore follows from (i). (iii) Since (C + tI) −1/2 ( C + tI) 1/2 L ∞ (H) = I − B n 1/2 L ∞ (H) ≤ (1 + B n L ∞ (H) ) 1/2 , the result follows from (i). (iv) Since 2 3 ≤ (C + tI) 1/2 ( C + tI) −1/2 L ∞ (H) ≤ √ 2 as obtained in (i), it follows that C + tI 3 2 (C + tI) (see Rudi et al., 2013, Lemmas B.2 and 3.5). This implies (see Gohberg and Goldberg, 2003) that λ ℓ ( C) + t ≤ 3 2 (λ ℓ (C) + t) for all ℓ ≥ 1. (v) follows similarly. Proof of (B.9): Define Z(x, y) := C −1/2 t (A(x, y) ⊗ H A(x, y))C 1/2 so that Z(x, y) = Z(y, x) and C −1/2 t (Ĉ − C)C 1/2 = 1 r(r − 1) r i =j Z(X i , X j ) − E[Z(X, Y )]. We have sup x,y∈X Z(x, y) L 2 (H) ≤ C −1/2 t L ∞ (H) C 1/2 L ∞ (H) A(x, y) 2 H ≤ (2κ) 3/2 √ t := M. By defining ψ(x) := E Y [Z(x, Y )], we have E ψ(X) − C 2 L 2 (H) = E ψ(X) 2 L 2 (H) − C 2 L 2 (H) ≤ E ψ(X) 2 L 2 (H) = E C −1/2 t E Y [A(X, Y ) ⊗ H A(X, Y )]C 1/2 2 L 2 (H) = E tr C 1/2 E Y [A(X, Y ) ⊗ H A(X, Y )]C −1 t E Y [A(X, Y ) ⊗ H A(X, Y )]C 1/2 = E tr C −1/2 t E Y [A(X, Y ) ⊗ H A(X, Y )]C −1 t E Y [A(X, Y ) ⊗ H A(X, Y )]CC 1/2 t ≤ sup x∈X C −1/2 t E Y [A(X, Y ) ⊗ H A(X, Y )]CC 1/2 t L ∞ (H) ×E tr C −1/2 t E Y [A(X, Y ) ⊗ H A(X, Y )]C −1/2 t ≤ C −1/2 t L ∞ (H) C L ∞ (H) C 1/2 t L ∞ (H) tr C −1 t C sup x,y∈X A(x, y) ⊗ H A(x, y) L ∞ (H) ≤ √ 2κ + t(2κ) 2 N C (t) √ t ≤ √ 2(2κ) 5/2 N C (t) √ t , where we used t ≤ C L ∞ (H) ≤ 2κ in the last inequality. The result follows by applying Theorem D.3 (ii). Lemma B.3. Suppose (A 1 ), (A 2 ), (A 4 ) and (A 5 ) hold. For t > 0, define N Σ (t) = tr(Σ(Σ + tI) −1 ) and N Σm (t) = tr(Σ m (Σ m + tI) −1 ). For δ > 0 and 86κ m log 16κm δ ≤ t ≤ Σ L ∞ (H) , the following hold: (i) Λ m (θ i ) m i=1 : 2 3 ≤ (II * + tI) 1/2 (AA * + tI) −1/2 L ∞ (L 2 (P)) ≤ √ 2 ≥ 1 − δ; (ii) Λ m (θ i ) m i=1 : λ m,j + t ≤ 3 2 (λ j + t) ≥ 1 − δ for all j ≥ 1; (iii) Λ m (θ i ) m i=1 : 1 2 (λ j + t) ≤ λ m,j + t ≥ 1 − δ for all j ≥ 1; (iv) Λ m (θ i ) m i=1 : N Σm (t) ≤ 32κ log 2 δ tm + 32κN Σ (t) log 2 δ tm + 2N Σ (t) ≥ 1 − 2δ. Proof. (i, ii, iii) Define A i := ϕ(·, θ i ) − (1 ⊗ L 2 (P) 1)ϕ(·, θ i ) and D i := A i ⊗ L 2 (P) A i . Then it follows from Propositions C.2 and C.4 that AA * = 1 m m i=1 D i and II * = E[AA * ]. Define E m := (II * + tI) −1/2 (II * − AA * )(II * + tI) −1/2 . By mimicking the strategy of Lemma B.2(ii, iii), we obtain (1 + E m L ∞ (L 2 (P)) ) −1/2 ≤ (II * + tI) 1/2 (AA * + tI) −1/2 L ∞ (L 2 (P)) (B.11) ≤ (1 − E m L ∞ (L 2 (P)) ) −1/2 provided E m L ∞ (L 2 (P)) < 1. We will now apply Theorem D.2 to bound E m L ∞ (L 2 (P)) . By defining Z i := (II * + tI) −1/2 A i and U i : = Z i ⊗ L 2 (P) Z i , we obtain E m = 1 m m i=1 U i − E Λ [U i ]. Note that U i L ∞ (L 2 (P)) = Z i 2 L 2 (P) ≤ 1 t A i 2 L 2 (P) ≤ 2 ϕ(·, θ i ) 2 L 2 (P) t 1 + 1 ⊗ L 2 (P) 1 L ∞ (L 2 (P)) ≤ 4κ t . Define T := E Λ [U i ]. Then E Λ [(U i − T ) 2 ] = E Λ [ Z i 2 L 2 (P) U i − T 2 ] E Λ [ Z i 2 L 2 (P) U i ] 4κ t T. Now we set σ 2 = 4κ t T L ∞ (L 2 (P)) ≤ 4κ t and d = T L 1 (L 2 (P)) T L ∞ (L 2 (P)) ≤ (λ 1 + t) T L 1 (L 2 (P)) λ 1 , where λ 1 = Σ L ∞ (H) = II * L ∞ (L 2 (P)) . Then Theorem D.2 yields Λ m B m L ∞ (L 2 (P)) ≤ 8βκ 3tm + 8κβ tm ≤ 1 − δ, (B.12) where β = log 4d δ . Since t ≥ 86κ m log 16κm δ , it follows that t ≥ 86κ m log 4d δ as 86κ m log 16κm δ ≥ 86κ m log 16κ tδ ≥ 86κ m log 4d δ , where we have used d ≤ 4κ t which follows from t ≤ Σ L ∞ (H) and tr(T ) ≤ tr(II * ) t = tr(Σ) t ≤ 2κ t . This implies t ≥ 86βκ m . Combining this with (B.12) yields that with probability at least 1 − δ, B m L ∞ (L 2 (P)) ≤ 1 2 . (i) follows by using this in (B.11). (ii), (iii) are implied as in (iv), (v) of Lemma B.2. (iv) Observe that N Σm (t) = tr(Σ m (Σ m + tI) −1 ) = tr(A * A(A * A + tI) −1 ) = tr(A(A * A + tI) −1 A * ) = tr((AA * + tI) −1 AA * ), where we have used the fact that A(A * A + tI) −1 = (AA * + tI) −1 A and the invariance of trace under cyclic permutations. Similarly, it can be shown that N Σ (t) = tr((II * + tI) −1 II * ). For the ease of notation, define A := AA * , B := II * , A t := A + tI and B t := B + tI. Then A −1 t = B −1/2 t I + B −1/2 t (A − B)B −1/2 t −1 B −1/2 t . Therefore, N m (t) = tr(AA −1 t ) = tr AB −1/2 t I + B −1/2 t (A − B)B −1/2 t −1 B −1/2 t = tr B −1/2 t AB −1/2 t I + B −1/2 t (A − B)B −1/2 t −1 ≤ (I + B −1/2 t (A − B)B −1/2 t ) −1 L ∞ (L 2 (P)) tr(B −1/2 t AB −1/2 t ) = (I − E m ) −1 L ∞ (L 2 (P)) tr(B −1/2 t AB −1/2 t ), where E m := B −1/2 t (B − A)B −1/2 t = (II * + tI) −1/2 (II * − AA * )(II * + tI) −1/2 . Since we showed in the proof of (i) that with probability at least 1 − δ, E m L ∞ (L 2 (P)) ≤ 1 2 , we obtain N m (t) ≤ 2 tr(B −1/2 t AB −1/2 t ), (B.13) where we use (I − E m ) −1 L ∞ (L 2 (P)) ≤ 1 1− Em L ∞ (L 2 (P)) . Next, consider tr(B −1/2 t AB −1/2 t ) = tr(B −1 t (A − B + B)) = B −1 t , A − B L 2 (L 2 (P)) + N Σ (t), (B.14) where B −1 t , A − B L 2 (L 2 (P)) = (II * + tI) −1 , AA * − II * L 2 (L 2 (P)) . We now bound this term as follows. Let ζ i = ϕ(·, θ i ) − (1 ⊗ L 2 (P) 1)ϕ(·, θ i ) ⊗ L 2 (P) ϕ(·, θ i ) − (1 ⊗ L 2 (P) 1)ϕ(·, θ i ) so that E Λ [ζ 1 ] = II * , 1 m m i=1 ζ i = AA * and B −1 t , AA * − II * L 2 (L 2 (P)) = 1 m m i=1 (II * + tI) −1 , (ζ i − II * ) L 2 (L 2 (P)) . We will now apply Bernstein's inequality (Theorem D.1). To this end, note that (II * + tI) −1 , ζ 1 − II * L 2 (L 2 (P)) ≤ (II * + tI) −1 , II * L 2 (L 2 (P)) + (II * + tI) −1 , ζ i L 2 (L 2 (P)) = N Σ (t) + 1 t tr τ i ⊗ L 2 (P) τ i = N Σ (t) + 1 t τ i 2 L 2 (P) ≤ Σ L 1 (H) + 4κ t ≤ 8κ t , where we use Σ L 1 (H) ≤ E k(·, X) ⊗ H k(·, X) L 1 (H) = E k(·, X) 2 H ≤ 4κ and τ i := ϕ(·, θ i ) − (1 ⊗ L 2 (P) 1)ϕ(·, θ i ). Also E Λ (II * + tI) −1 , ζ 1 − II * 2 L 2 (L 2 (P)) = E Λ (II * + tI) −1 , ζ 1 L 2 (L 2 (P)) − N Σ (t) L 2 (L 2 (P)) − N 2 Σ (t) ≤ E Λ (II * + tI) −1 , ζ 1 2 L 2 (L 2 (P)) = E Λ tr (II * + tI) −1 ζ 1 (II * + tI) −1 ζ 1 ≤ sup θ 1 (II * + tI) −1/2 ζ 1 (II * + tI) −1/2 L ∞ (L 2 (P)) E Λ tr (II * + tI) −1/2 ζ 1 (II * + tI) −1/2 ≤ N Σ (t) t sup θ 1 ϕ(·, θ 1 ) − (1 ⊗ L 2 (P) 1)ϕ(·, θ 1 ) 2 L 2 (P) ≤ 4κN Σ (t) t . The result follows by applying Theorem D.1 to B −1 t , (A − B) L 2 (L 2 (P)) and combining (B.13) and (B.14). Lemma B.4. Suppose (A 1 ) and (A 4 ) hold. Then for any 0 < δ < 1 and m ≥ 2 log 2 δ , Λ m    (θ i ) m i=1 : AA * − II * L 2 (L 2 (P)) ≤ 4κ 2 log 2 δ m    ≥ 1 − δ. Proof. From Proposition C.2(iv), Lemma C.3 and Proposition C.4(iv), we have II * = Υ − (1 ⊗ L 2 (P) 1)Υ − Υ(1 ⊗ L 2 (P) 1) + (1 ⊗ L 2 (P) 1)Υ(1 ⊗ L 2 (P) 1) and AA * = Π − (1 ⊗ L 2 (P) 1)Π − Π(1 ⊗ L 2 (P) 1) + (1 ⊗ L 2 (P) 1)Π(1 ⊗ L 2 (P) 1) where Υ := Θ ϕ(·, θ) ⊗ L 2 (P) ϕ(·, θ) dΛ(θ) and Π := m i=1 ϕ i ⊗ L 2 (P) ϕ i = 1 m m i=1 ϕ(·, θ i ) ⊗ L 2 (P) ϕ(·, θ i ). Define A i := ϕ(·, θ i ) − (1 ⊗ L 2 (P) 1)ϕ(·, θ i ) and D i := A i ⊗ L 2 (P) A i . Then it follows that AA * = 1 m m i=1 D i and II * = E[AA * ]. The result follows by applying Theorem D.1 with B = θ = sup θ 1 A 1 ⊗ L 2 (P) A 1 L 2 (L 2 (P)) = sup θ 1 A 1 2 L 2 (P) ≤ 2 sup θ 1 ϕ(·, θ 1 ) 2 L 2 (P) ≤ 2κ and noting that L 2 (L 2 (P)) is a separable Hilbert space since L 2 (P) is separable. Lemma B.5. Suppose (A 1 ) and (A 4 ) hold. For any 0 < δ < 1 with n ≥ 2 log 2 δ , then the following hold: (i) P n (X i ) n i=1 : m P − m P 2 H ≤ 32κ log 2 δ n ≥ 1 − δ; (ii) P n (X i ) n i=1 : m P,m − m P,m 2 Hm ≤ 32κ log 2 δ n (θ i ) m i=1 ≥ 1 − δ. Proof. Define ξ i = k(·, X i ) − X k(·, x)dP(x). Clearly 1 n n i=1 ξ i = m P − m P . Note that ξ i H ≤ 2 √ κ for all i. The result therefore follows by applying Theorem D.1 with B = θ = 2 √ κ. Conditioned on (θ i ) m i=1 , the second result follows exactly the first one with k replaced by k m . Lemma B.6. Suppose (A 1 ) and (A 4 ) hold. For any δ > 0 with m ≥ 2 log 2 δ , Λ m E Ik(·, X) − Ak m (·, X) 2 L 2 (P) ≤ 64κ 2 log 2 δ m ≥ 1 − 2δ. Proof. Note that E Ik(·, X) − Ak m (·, X) 2 L 2 (P) = E Ik(·, X) 2 L 2 (P) + E Ak m (·, X) 2 L 2 (P) − 2E Ik(·, X), Ak m (·, X) L 2 (P) ( †) = Σ 2 L 2 (H) + Σ m 2 L 2 (Hm) − 2E Ik(·, X) , Ak m (·, X) L 2 (P) = II * 2 L 2 (L 2 (P)) + AA * 2 L 2 (L 2 (P)) − 2E Ik(·, X), Ak m (·, X) L 2 (P) = II * − AA * 2 L 2 (L 2 (P)) + 2 tr(II * AA * ) − E Ik(·, X), Ak m (·, X) L 2 (P) , (B.15) where we used Lemma B.7 in ( †). We will now focus on computing E Ik(·, X), Ak m (·, X) L 2 (P) and tr(II * AA * ). Note that k(·, x) = Θ ϕ(·, θ)ϕ(x, θ) dΛ(θ) and k m (·, x) = m i=1 ϕ i (x)ϕ i . Define ϕ P (θ) := X ϕ(x, θ) dP(x), ϕ i,P := X ϕ i (x) dP(x), µ(·, θ) = ϕ(·, θ) − ϕ P (θ) and µ i := ϕ i − ϕ i,P . Therefore, E Ik(·, X), Ak m (·, X) L 2 (P) = X Ik(·, x), Ak m (·, x) L 2 (P) dP(x) ( * ) = X Θ µ(·, θ)ϕ(x, θ) dΛ(θ), m i=1 ϕ i (x)µ i L 2 (P) dP(x) = X Θ m i=1 µ(·, θ), µ i L 2 (P) ϕ i (x)ϕ(x, θ) dΛ(θ) dP(x) = Θ m i=1 µ(·, θ), µ i L 2 (P) ϕ i , ϕ(·, θ) L 2 (P) dΛ(θ), (B.16) where the penultimate and last equalities follow by employing Fubini's theorem and ( * ) follows from Propositions C.2 and C.4. On the other hand, by defining τ (·, θ) := ϕ(·, θ)− 1 ⊗ L 2 (P) 1 ϕ(·, θ) and τ i := ϕ i − 1 ⊗ L 2 (P) 1 ϕ i , we have tr(II * AA * ) ( ‡) = tr Θ τ (·, θ) ⊗ L 2 (P) τ (·, θ) dΛ(θ) m i=1 τ i ⊗ L 2 (P) τ i = tr Θ m i=1 µ(·, θ), µ i L 2 (P) µ(·, θ) ⊗ L 2 (P) µ i dΛ(θ) = Θ m i=1 µ(·, θ), µ i L 2 (P) µ(·, θ), µ i L 2 (P) dΛ(θ) = Θ m i=1 µ(·, θ), µ i L 2 (P) ϕ(·, θ), ϕ i L 2 (P) − ϕ P (θ)ϕ i,P dΛ(θ), (B.17) where we used Propositions C.2(iv) and C.4(iv) in ( ‡). It follows from (B.16) and (B.17) that tr(II * AA * ) = E Ik(·, X), Ak m (·, X) L 2 (P) − Θ A(θ) dΛ(θ), 1 m m i=1 A(θ i ) L 2 (P) , (B.18) where A(θ) = ϕ(·, θ)ϕ P (θ)− ϕ 2 P (θ). We remind the reader that ϕ i = 1 √ m ϕ(·, θ i ) with (θ i ) m i=1 i.i.d. ∼ Λ. Define Λ m to be the empirical measure based on (θ i ) m i=1 . Then, (B.18) can be written as tr(II * AA * ) = E Ik(·, X), Ak m (·, X) L 2 (P) + 1 2 Θ A(θ) d(Λ m − Λ)(θ) 2 L 2 (P) − 1 2 Θ A(θ) dΛ(θ) 2 L 2 (P) − 1 2 Θ A(θ) dΛ m (θ) 2 L 2 (P) ≤ E Ik(·, X), Ak m (·, X) L 2 (P) + 1 2 Θ A(θ) d(Λ m − Λ)(θ) 2 L 2 (P) ,(B.E Ik(·, X) − Ak m (·, X) 2 L 2 (P) ≤ II * − AA * 2 L 2 (L 2 (P)) + Θ A(θ) d(Λ m − Λ)(θ) 2 L 2 (P) , (B.20) which holds Λ-a.s. The result follows by applying Lemma B.4 and Theorem D.1 to (B.20) by noting that sup θ∈Θ A(θ) L 2 (P) ≤ 2κ and E θ∼Λ A(θ) 2 L 2 (P) ≤ 4κ 2 . Lemma B.7. Let X be a separable topological space, H be a separable Hilbert space and ρ be a probability measure on X. Suppose v : X → H is Bochner-measurable and E ρ v 2 H := X v(x) 2 H dρ(x) < ∞. Define A = B * B = X v(x)⊗ H v(x) dρ(x) =: E ρ [v⊗ H v] where B : H → G and G is a separable Hilbert space. Then for any Q : H → H, E ρ BQv 2 G = A 1/2 QA 1/2 2 L 2 (H) . Proof. Note that E ρ BQv 2 G = E ρ BQv, BQv G = E ρ Q * AQv, v H = E ρ Q * AQ, v ⊗ H v L 2 (H) . Since v is Bochner-measurable and E ρ v 2 H < ∞, it is Bochner integrable, which yields E ρ Q * AQ, v ⊗ H v L 2 (H) = Q * AQ, E ρ [v ⊗ H v] L 2 (H) = Q * AQ, A L 2 (H) . The result follows by noting that Q * AQ, A L 2 (H) = tr (Q * AQA) = tr A 1/2 Q * A 1/2 A 1/2 QA 1/2 = A 1/2 QA 1/2 2 L 2 (H) , where we have used invariance of the trace under cyclic permutations. Lemma B.8. Define B = f ⊗ H f where H is a separable Hilbert space and f ∈ H. Then B L ∞ (H) = B L 2 (H) = B L 1 (H) = f 2 H . Proof. Since B is self-adjoint, B L ∞ (H) = λ 1 (B) = sup g H =1 g, Bg H = sup g H =1 f, g 2 H = f 2 H . Note that B L 1 (H) = j e j , (f ⊗ H f )e j H = j f, e j 2 H = f 2 H for any orthonormal basis (e j ) j in H. Lemma B.9. For any trace class self-adjoint operator C, the following hold: (i) Suppose ai −α ≤ λ i (C) ≤ Ai −α for α > 1 and a, A ∈ (0, ∞). Then t −1/α N C (t) t −1/α . (ii) Suppose be −τ i ≤ λ i (C) ≤ Be −τ i for τ > 0 and b, B ∈ (0, ∞). Then log 1 t N C (t) log 1 t . Proof. (i) Define λ i := λ i (C). We have N C (t) = tr (C + tI) −1 C = i≥1 λ i λ i + t ≤ i≥1 Ai −α ai −α + t = A a i≥1 i −α i −α + ta −1 ≤ A a ∞ 0 x −α x −α + ta −1 dx ≤ A a a t 1/α ∞ 0 1 1 + x α dx, where clearly the integral is finite for α > 1, thereby yielding N C (t) t −1/α . (ii) follows by carrying out a similar calculation as in (i). Lemma B.10. Let X and Y be H-valued random elements where H is a separable Hilbert space. Then, (ii) For any f ∈ H, f, Σf H = 1 2n(n − 1) n i =j (f (X i ) − f (X j )) 2 = 1 n n i=1 f 2 (X i ) − 1 n(n − 1) i =j f (X i )f (X j ) = 1 n − 1 n i=1 f 2 (X i ) − 1 n − 1 1 √ n n i=1 f (X i ) 2 (C.1) = n n − 1 Sf, Sf 2 − 1 n − 1 1 n , Sf 2 2 = n n − 1 f, S * Sf H − 1 n − 1 S * 1 n , f 2 H = n n − 1 f, S * Sf H − 1 n − 1 f, S * (1 n ⊗ 2 1 n )Sf H = n n − 1 f, S * H n Sf H . (iii) For any α ∈ R n , SS * α = S 1 √ n n i=1 α i k(·, X i ) = 1 √ n n i=1 α i Sk(·, X i ) = 1 n Kα, Clearly f P is well defined as for any f ∈ L 2 (P), f P ≤ \|f (x)\| dP(x) ≤ f L 2 (P) < ∞ and for f ∈ H, f P = f, m P H ≤ f H k(x, x) dP(x) < ∞ and the result therefore follows. (ii) For any orthonormal basis (e j ) j in H, I 2 L 2 (H,L 2 (P)) = j Ie j 2 L 2 (P) = j e j − e j,P 2 L 2 (P) = j e j 2 L 2 (P) − e 2 j,P ≤ j e j 2 L 2 (P) = j X e j , k(·, x) 2 H dP(x) (⋆) = X j e j , k(·, x) 2 H dP(x) = X k(x, x) dP(x) < ∞, where (⋆) follows from monotone convergence theorem. Since I L 2 (H,L 2 (P)) = I * L 2 (L 2 (P),H) , the result follows. (iii) For any f ∈ H, (I * I)f = I * (f − f P ) = I * f − I * f P = I * f , where we use the fact that I * f P = 0 since f P is a constant function. By using the reproducing property, I * If = I * f = X f (x)k(·, x) dP(x) − m P f P = X k(·, x) k(·, x), f H dP − m P m P , f H = X (k(·, x) ⊗ H k(·, x))f dP(x) − (m P ⊗ H m P )f = Σf and the result follows. Since I 2 L 2 (H,L 2 (P)) = I * I L 1 (H) , Σ is trace-class. (iv) For any f ∈ L 2 (P), (II * )f = I(I * f ) = I X k(·, x)f (x) dP(x) − m P f P = X k(·, x)f (x) dP(x) − m P f P − X X k(y, x)f (x) dP(x) dP(y) +f P X X k(y, x) dP(x) dP(y) = Υf − Υ1 1, f L 2 (P) − 1 Υ1, f L 2 (P) + 1 1, Υ1 L 2 (P) 1, f L 2 (P) = Υf − Υ(1 ⊗ L 2 (P) 1)f − (1 ⊗ L 2 (P) 1)Υf + (1 ⊗ L 2 (P) 1)Υ(1 ⊗ L 2 (P) 1)f and the result follows, where in the last line we use the fact that Υ is self-adjoint, which follows from (Steinwart and Christmann, 2008, Theorem 4.27). Since I * 2 L 2 (L 2 (P),H) = II * L 1 (L 2 (P)) , it follows that II * is trace-class. The following result presents a representation for Υ if k satisfies (A 4 ). Lemma C.3. Suppose (A 4 ) holds. Then Υ = Θ ϕ(·, θ) ⊗ L 2 (P) ϕ(·, θ) dΛ(θ). Proof. Since k(x, y) = Θ ϕ(x, θ)ϕ(y, θ) dΛ(θ), for any f ∈ L 2 (P), Υf = X k(·, x)f (x) dP(x) = X Θ ϕ(·, θ)ϕ(x, θ) dΛ(θ)f (x) dP(x) ( * ) = Θ ϕ(·, θ) X ϕ(x, θ)f (x) dP(x) dΛ(θ) = Θ ϕ(·, θ) ϕ(·, θ), f L 2 (P) dΛ(θ) = Θ ϕ(·, θ) ⊗ L 2 (P) ϕ(·, θ) f dΛ(θ) = Θ ϕ(·, θ) ⊗ L 2 (P) ϕ(·, θ) dΛ(θ) f, where Fubini's theorem is applied in ( * ). C.3 Properties of the approximation operator The following result presents the properties of the approximation operator, A. Proposition C.4. Define A : H m → L 2 (P), f = m i=1 β i ϕ i → m i=1 β i (ϕ i − ϕ i,P ) where ϕ i,P := X ϕ i (x) dP(x) and sup x∈X \|ϕ i (x)\| ≤ κ m for all i ∈ [m] with κ < ∞. Then the following hold: (i) A * : L 2 (P) → H m , f → m i=1 f, ϕ i L 2 (P) − f P ϕ i,P ϕ i . (ii) A and A * are Hilbert-Schmidt. (iii) Σ m = A * A is trace-class. (iv) AA * = Π − (1 ⊗ L 2 (P) 1)Π − Π(1 ⊗ L 2 (P) 1) + (1 ⊗ L 2 (P) 1)Π(1 ⊗ L 2 (P) 1) is trace-class where Π := m i=1 ϕ i ⊗ L 2 (P) ϕ i : L 2 (P) → L 2 (P). Proof. The proof is similar to that of Proposition C.2. (i) For any g = m i=1 β i ϕ i ∈ H m and f ∈ L 2 (P), A * f, g Hm = f, Ag L 2 (P) = X m i=1 β i (ϕ i (x) − ϕ i,P ) f (x)dP(x) = m i=1 β i ( f, ϕ i L 2 (P) − f P ϕ i,P ), and the result follows from the definition of ·, · Hm . (ii) For any orthonormal basis (e j ) j in L 2 (P), (iii) For any f = m i=1 β i ϕ i ∈ H m , A * Af = A * m i=1 β i (ϕ i − ϕ i,P ) = m i=1 β i A * (ϕ i − ϕ i,P ) = m i=1 β i m j=1 ( ϕ i , ϕ j L 2 (P) − ϕ i,P ϕ j,P )ϕ j = m j=1 m i=1 β i ϕ i , ϕ j L 2 (P) ϕ j − X m i=1 β i ϕ i (x)dP(x)   X m j=1 ϕ j (x)ϕ j dP(x)   = X m i=1 β i ϕ i (x)   m j=1 ϕ j (x)ϕ j   dP(x) − X f (x) dP(x) X k m (·, x) dP(x) = X f (x)k m (·, x) dP(x) − X f (x) dP(x) X k m (·, x) dP(x) = Σ m f. That Σ m is trace class is implied by (ii). (iv) For any f ∈ L 2 (P), AA * f = m i=1 ( f, ϕ i L 2 (P) − f P ϕ i,P )(ϕ i − ϕ i,P ) = m i=1 ( f, ϕ i L 2 (P) − f, 1 L 2 (P) ϕ i , 1 L 2 (P) )(ϕ i − ϕ i , 1 L 2 (P) ) = Πf − Π1, f L 2 (P) − Π(1 ⊗ L 2 (P) 1)f + (1 ⊗ L 2 (P) 1)Π1, f L 2 (P) = Πf − (1 ⊗ L 2 (P) 1)Πf − Π(1 ⊗ L 2 (P) 1)f + (1 ⊗ L 2 (P) 1)Π(1 ⊗ L 2 (P) 1)f and the result follows. AA * is trace-class since A * is Hilbert-Schmidt. D Supplementary Results In this appendix, we collect Bernstein's inequality for Hilbert-valued random elements (quoted from Yurinsky, 1995) and Tropp's inequality for operator-valued random elements (quoted from Rudi et al., 2013, Theorem A.1), that are used to prove the results of this paper. Based on these two results, Theorem D.3 presents a Bernstein-type inequality for the operator and Hilbert-Schmidt norms of a operator-valued U-statistics. Theorem D.1 (Bernstein's inequality in separable Hilbert spaces). Let (Ω, A, P ) be a probability space, H be a separable Hilbert space, B > 0 and θ > 0. Furthermore, let ξ 1 , . . . , ξ n : Ω → H be zero mean i.i.d. random variables satisfying E ξ 1 r H ≤ r! 2 θ 2 B r−2 , ∀ r > 2. Then for any 0 < δ < 1, P n    (ξ i ) n i=1 : 1 n n i=1 ξ i H ≥ 2B log 2 δ n + 2θ 2 log 2 δ n    ≤ δ. Theorem D.2 (Tropp's inequality for operators). Let (Z i ) n i=1 be independent copies of the random variable Z with law P taking values in the space of bounded self-adjoint operators for a separable Hilbert space H. 0 < δ ≤ d, P n    (Z i ) n i=1 : 1 n n i=1 Z i − E[Z] L ∞ (H) ≥ βM n + 3βσ 2 n    ≤ δ, where β := 2 3 log 4d δ . Theorem D.3. Let (X , P ) be a measurable space and Z : X × X → L 2 (H) with Z(x, y) = Z(y, x) for all x, y ∈ H, where H is a separable Hilbert space. Let D = Z(x, y) dP (x) dP (y) with D = 1 n(n − 1) Z(X i , X j ) − ψ(X i ) − ψ(X j ) + D L ∞ (H) +2 1 n n i=1 ψ(X i ) − D L ∞ (H) . (D.1) The first term can be bounded by applying Theorem D.2 since E[ψ(X)] = D. We now bound the second term as follows. Define h(X i , X j ) := Z(X i , X j ) − ψ(X i ) − ψ(X j ) + D. Applying Markov's inequality to the second term, we obtain that for any ǫ > 0 and t > 0, P n      (X i ) n i=1 : 1 n(n − 1) n i =j h(X i , X j ) L ∞ (H) ≥ ǫ      ≤ e −tǫ E exp t ′ n i =j h(X i , X j ) L ∞ (H) , (D.2) where t ′ := t n(n−1) . Consider E exp t ′ n i =j h(X i , X j ) L ∞ (H) = E exp t ′ n i =j Z(X i , X j ) − E X ′ j Z(X i , X ′ j ) − E X ′ i Z(X j , X ′ i ) + E X ′ i ,X ′ j Z(X ′ i , X ′ j ) L ∞ (H) = E exp t ′ E (X ′ i ) n i=1 \|(X i ) n i=1 n i =j Z(X i , X j ) − Z(X i , X ′ j ) − Z(X j , X ′ i ) + Z(X ′ i , X ′ j ) L ∞ (H) ≤ E exp    E (X ′ i ) n i=1 \|(X i ) n i=1 t ′ n i =j Z(X i , X j ) − Z(X i , X ′ j ) − Z(X ′ i , X j ) + Z(X ′ i , X ′ j ) Dirac measure. In ( †), the expectation is jointly over (X i , X ′ i ) n i=1 which is obtained through an application of Jensen's inequality. Therefore, E exp t ′ n i =j h(X i , X j ) L ∞ (H) ≤ E exp A + B + C + D , where A := E ǫ (1) E ǫ (2) t ′ n i =j ǫ (1) i ǫ (2) j Z(X i , X j ) L ∞ (H) , B := E ǫ (1) E ǫ (2) t ′ n i =j ǫ (1) i ǫ (2) j Z(X i , X ′ j ) L ∞ (H) , C := E ǫ (1) E ǫ (2) t ′ n i =j ǫ (1) i ǫ (2) j Z(X ′ i , X j ) L ∞ (H) and D := E ǫ (1) E ǫ (2) t ′ n i =j ǫ (1) i ǫ (2) j Z(X ′ i , X ′ j ) L ∞ (H) . Since (X i ) n i=1 and (X ′ i ) n i=1 are i.i.d., we have E exp t ′ n i =j h(X i , X j ) L ∞ (H) ≤ E exp   4t ′ E ǫ (1) E ǫ (2) n i =j ǫ (1) i ǫ (2) j Z(X i , X j ) L ∞ (H)    (D.3) ≤ E exp   4t ′ E ǫ (1) E ǫ (2) n i =j ǫ (1) i ǫ (2) j Z(X i , X j ) L 2 (H)    ≤ E exp   4t ′ E ǫ (1) E ǫ (2) n i =j ǫ (1) i ǫ (2) j Z(X i , X j ) 2 L 2 (H)    , where the last inequality follows from Jensen's inequality. We will now bound E ǫ (1) E ǫ (2) n i =j ǫ (1) i ǫ (2) j Z(X i , X j ) 2 L 2 (H) = E ǫ (1) E ǫ (2) n i =j n k =l ǫ (1) i ǫ (2) j ǫ (1) k ǫ (2) l Z(X i , X j ), Z(X k , X l ) L 2 (H) . We consider the following cases. (2) l Z(X i , X j ), Z(X i , X l ) L 2 (H) = 0. Case 3: i = k, j = l E ǫ (1) E ǫ (2) n i =j ǫ (1) i ǫ (2) j Z(X i , X j ) 2 L 2 (H) = E ǫ (1) n i =j =k ǫ (1) i ǫ(1) k Z(X i , X j ), Z(X k , X j ) L 2 (H) = 0. Case 4: i = k, j = l E ǫ (1) E ǫ (2) n i =j ǫ (1) i ǫ (2) j Z(X i , X j ) 2 L 2 (H) = 0. Therefore, E exp t ′ n i =j h(X i , X j ) L ∞ (H) ≤ exp 4tM n(n − 1) ≤ exp 8M t n for n ≥ 2 as n − 1 ≥ n 4 . Using this in (D.2) and choosing t = n 8M , we obtain P n      (X i ) n i=1 : 1 n(n − 1) n i =j h(X i , X j ) L ∞ (H) ≥ ǫ      ≤ 3 exp − nǫ 8M , which is equivalent to P n      (X i ) n i=1 : 1 n(n − 1) n i =j h(X i , X j ) L ∞ (H) ≥ 8M n log 3 δ      ≤ δ. (D.4) Combining (D.4) with the bound on the first term in (D.1) yields the result. (ii) As in (i), we first write D − D L 2 (H) ≤ 1 n(n − 1) i =j Z(X i , X j ) − ψ(X i ) − ψ(X j ) + D L 2 (H) +2 1 n n i=1 ψ(X i ) − D L 2 (H) . (D.5) The first term in (D.5) is bounded through an application of Theorem D.1. For the second term, we replicate the analysis between (D.2) and (D.3) with the operator norm being replaced by the Hilbert-Schmidt norm. Since the analysis between (D.3) and (D.4) anyway relies only on the Hilbert-Schmidt norm, the result follows by employing that log 2 δ < log 3 δ . T Σm,ℓ = S   A φ m,1 λ m,1 , . . . , A φ m,ℓ λ m,ℓ   . -see the second line in the chain of equations leading to (47)-and (48). For the lower bound, note that i + (II * ) (2−s)/2 − (AA * ) A + tI = (A − B) + (B + tI) = (B + tI) 1/2 I + (B + tI) −1/2 (A − B)(B + tI) −1/2 (B + tI) 1/2 implying, , ϕ i L 2 (P) − e j,P ϕ iL 2(P) + e 2 j,P ϕ 2 i,P − 2e j,P ϕ i,P e j , ϕ i L 2 j , (ϕ i ⊗ L 2 (P) 1 + 1 ⊗ L 2 (P) ϕ i )e j L 2 (P) i,P ϕ i , 1 L 2 (P) 2 (P) ≤ κ < ∞, and so A and A * are Hilbert-Schmidt. Suppose there exists S ∈ L 2 (H) such that E[(Z − E[Z]) 2 ] S and 0 < M < ∞ such that Z L ∞ (H) ≤ M almost everywhere. Let d := S L 1 (H) S L ∞ (H)and σ 2 := S L ∞ (H) . Then for X i , X j ) being its U -statistic estimator, where (X i ) n i=1 i.i.d.∼ P and n ≥ 2. Define ψ(x) = E Y [Z(x, Y )] and let sup x,y∈X Z(x, y) L 2 (H) ≤ M . Then the following hold: (i) Suppose Z : X × X → S(H), where S(H) is the space of self-adjoint Hilbert-Schmidt operators on H, E[(ψ(X) − D) 2 ] S, σ 2 := S L ∞ (H) , and sup x∈X ψ(x) L ∞ (H) ≤ R. Then for 0 < δ ≤ d,P n (X i ) n i=1 : D − D Z(X i , X j ) − ψ(X i ) − ψ(X j ) + D Case 1 :Z 1i = k, j = l E ǫ (1) E ǫ (2) (X i , X j ) 2 L 2 (H) ≤ n(n − 1)M 2 . 19 ) 19which holds Λ-a.s. Using (B.19) in (B.15), we obtain In general, P ψ := ℓ i=1 ψi ⊗H ψi is not a projection operator since P 2 ψ = P ψ . AcknowledgmentsBKS is supported by National Science Foundation (NSF) award DMS-1713011 and CAREER award DMS-1945396.A Additional ResultsIn this section, we present corollaries of Theorems 6 and 8 assuming exponential decay rates for the eigenvalues of Σ.Corollary A.1 (Exponential decay of eigenvalues). Suppose Be −τ i ≤ λ i ≤Be −τ i for τ > 0 and B,B ∈ (0, ∞). Let ℓ = 1 τ log n θ for θ > 0. Thengives the lower bonud.C Sampling, Inclusion and Approximation OperatorsIn this appendix, we present some technical results related to the properties of sampling, inclusion and approximation operators.C.1 Properties of the sampling operatorThe following result presents the properties of the sampling operator, S and its adjoint. While these results are known in the literature (e.g., seeSmale and Zhou, 2007), we present it here for completeness.Proposition C.1. Let H be an RKHS of real-valued functions on a non-empty set X with k as the reproducing kernel. Define S :Then the following hold:(ii) Σ = n n−1 S * H n S where Σ is defined in (5);Proof. (i) For any g ∈ H and α ∈ R n , we have S * α, g H = α,, where the last equality follows from the reproducing property.where in the second equality, we used the fact S is a linear operator.C.2 Properties of the inclusion operatorThe following result captures the properties of the inclusion operator I. A variation of the result is known in the literature (e.g., seeSteinwart and Christmann, 2008, Theorem 4.26).. Then the following hold:(ii) I and I * are Hilbert-Schmidt.Proof. (i) For any f ∈ L 2 (P) and g ∈ H,f (x) k(·, x), g H dP(x) − m P , g H f P = X k(·, x)f (x) dP(x), g H − m P f P , g H . Fast randomized kernel ridge regression with statistical guarantees. A Alaoui, M Mahoney, Advances in Neural Information Processing Systems. C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. GarnettCurran Associates, Inc28A. Alaoui and M. Mahoney. Fast randomized kernel ridge regression with statistical guarantees. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, editors, Advances in Neural Information Processing Systems 28, pages 775-783. Curran Associates, Inc., 2015. Theory of reproducing kernels. N Aronszajn, Trans. Amer. Math. Soc. 68N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337-404, 1950. 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[ "A C O Survey ofG ravi tati onal l y Lensed Q uasars w i th the IR A M Interferom eter ?", "A C O Survey ofG ravi tati onal l y Lensed Q uasars w i th the IR A M Interferom eter ?" ]	[ "R B Arvai Ni S ", "D A L L Oi N ", "M B Rem Er " ]	[]	[]	A bstract. W e present the resul ts ofa C O survey of gravi tati onal l y l ensed quasars,conducted w i th thePl ateau de B ure Interferom eter over the l ast three years.A m ong the 18 objectssurveyed,onewasdetected i n C O l i neem i ssi on, w hi l e si x were detected i n the conti nuum at 3m m and three i n the conti nuum at 1m m .T he l ow C O detecti on rate m ay at l east i n part be due to uncertai nti es i n the redshi fts deri ved from quasar broad em i ssi on l i nes. T he detected C O source,the z = 3: 2 radi o qui et quasar M G 0751+ 2716,i squi te strong i n the C O (4 3)l i ne and i n the m i l l i m eter/subm i l l i m eter conti nuum ,the l atter bei ng em i ssi on from cooldust. T he i ntegrated C O l i ne ux i s 5: 96 0: 45 Jy km s 1 ,and the totalm ol ecul argasm assi s esti m ated to be i n the range M H 2 = 1: 6 3: 1 10 9 M . K ey w ords. quasars: general -gravi tati onal l ensi ngquasars: i ndi vi dual :M G 0751+ 2716 1.Introducti on T he m easurem entofC O l i ne em i ssi on i n hi gh redshi ftobjects has proven to be a frui tfulavenue for i nvesti gati ng the properti es ofdi stant quasars and gal axi es.M ol ecul ar observati onsaddressi nteresti ng i ssuessuch asthatofstar form ati on i n the earl y uni verse and how the presence of a central m assi ve engi ne can a ect the i nterstel l ar m ateri al i n i ts host gal axy. D etecti ons of C O have show n Send o print requests to:D .A l l oi n dal l oi n@ eso. org ? B ased on observati ons obtai ned w i th the Pl ateau de B ure Interferom eter of the Insti tut de R adi o A stronom i e M i l l i m etri que, supported by IN SU /C N R S (France), M PG (G erm any) and IG N (Spai n).thatquasarssharesom eproperti esw i th l um i nousi nfrared gal axi es,both l ocal l y and at hi gh redshi ft.For exam pl e, com pari son of the m ol ecul ar,i nfrared and opti cal properti es of the C l overl eaf quasar and the i nfrared gal axy IR A S F10214+ 4724 (know n to harbor a buri ed quasar seen i n pol ari zed l i ght) has dem onstrated that these two objects are nearl y i denti cal , except i n the opti cal range w here the di erences can probabl y be attri buted to obscurati on/ori entati on e ects (B arvai ni s et al .1995).Such ndi ngs l end support to theori es uni fyi ng l um i nous i nfrared gal axi es and quasars vi a ori entati on e ects, w i th hi gh redshi fti nfrared gal axi esbei ng thel um i nouscounterpartsofl ocalSeyfert2' s.H owever,on physi calgrounds,i t al so seem sl i kel y thatIR -sel ected gal axi esand U V -sel ected quasars m ay di er i n thei r stage ofevol uti on.T here are currentl y about 15 wel l -docum ented detecti ons ofm ol ecul ar gas at hi gh redshi ft (e. g. ,C om bes 2001),ofw hi ch at l east 9 are gravi tati onal l y l ensed system s.T he advantagesofusi ng an i nterveni ng \gravi tati onal tel escope" to boostthe uxesare obvi ous,w i th esti m ated m agni cati on factorsofup to 100 i n theopti cal .M oreover, di erenti al gravi tati onal e ects provi de an el egant tool to probe the si ze and structure of the m ol ecul ar m ateri alw i thi n the quasar.For exam pl e,a poi nt-l i ke em i tti ng regi on (rest-fram e U V and opti calconti nua from the i nner accreti on zone),and an extended dusty m ol ecul ar regi on (the \torus") i n the quasar w i l lproduce,after gravi tati onal e ects from the i nterveni ng l ens, i m ages w i th di erent m orphol ogi es.M ol ecul ar l i ne pro l es,re ecti ng i ntri nsi c geom etri caland ki nem ati cal properti es, can be parti cul arl y usefuli n understandi ng the extended structure. It shoul d be noted however that a detai l ed m odel	10.1051/0004-6361:20020121	[ "https://export.arxiv.org/pdf/astro-ph/0201429v1.pdf" ]	18,376,602	astro-ph/0201429	b750e7bd5c78c695a7444aa348e5c424545e6afb	A C O Survey ofG ravi tati onal l y Lensed Q uasars w i th the IR A M Interferom eter ? 25 Jan 2002 ctober 1,2001,A ccepted January 22,2002 R B Arvai Ni S D A L L Oi N M B Rem Er A C O Survey ofG ravi tati onal l y Lensed Q uasars w i th the IR A M Interferom eter ? 25 Jan 2002 ctober 1,2001,A ccepted January 22,2002A stronom y & A strophysi cs m anuscri pt no. (w i l lbe i nserted by hand l ater) 1 N ati onalSci ence Foundati on,4201 W i l son B oul evard,A rl i ngton VA 22230,U SA 2 European Southern O bservatory,C asi l l a 19001,Santi ago 19,C hi l e 3 IR A M ,300 R ue de l a Pi sci ne,38406 St M arti n d' H eres,France R ecei ved O A bstract. W e present the resul ts ofa C O survey of gravi tati onal l y l ensed quasars,conducted w i th thePl ateau de B ure Interferom eter over the l ast three years.A m ong the 18 objectssurveyed,onewasdetected i n C O l i neem i ssi on, w hi l e si x were detected i n the conti nuum at 3m m and three i n the conti nuum at 1m m .T he l ow C O detecti on rate m ay at l east i n part be due to uncertai nti es i n the redshi fts deri ved from quasar broad em i ssi on l i nes. T he detected C O source,the z = 3: 2 radi o qui et quasar M G 0751+ 2716,i squi te strong i n the C O (4 3)l i ne and i n the m i l l i m eter/subm i l l i m eter conti nuum ,the l atter bei ng em i ssi on from cooldust. T he i ntegrated C O l i ne ux i s 5: 96 0: 45 Jy km s 1 ,and the totalm ol ecul argasm assi s esti m ated to be i n the range M H 2 = 1: 6 3: 1 10 9 M . K ey w ords. quasars: general -gravi tati onal l ensi ngquasars: i ndi vi dual :M G 0751+ 2716 1.Introducti on T he m easurem entofC O l i ne em i ssi on i n hi gh redshi ftobjects has proven to be a frui tfulavenue for i nvesti gati ng the properti es ofdi stant quasars and gal axi es.M ol ecul ar observati onsaddressi nteresti ng i ssuessuch asthatofstar form ati on i n the earl y uni verse and how the presence of a central m assi ve engi ne can a ect the i nterstel l ar m ateri al i n i ts host gal axy. D etecti ons of C O have show n Send o print requests to:D .A l l oi n dal l oi n@ eso. org ? B ased on observati ons obtai ned w i th the Pl ateau de B ure Interferom eter of the Insti tut de R adi o A stronom i e M i l l i m etri que, supported by IN SU /C N R S (France), M PG (G erm any) and IG N (Spai n).thatquasarssharesom eproperti esw i th l um i nousi nfrared gal axi es,both l ocal l y and at hi gh redshi ft.For exam pl e, com pari son of the m ol ecul ar,i nfrared and opti cal properti es of the C l overl eaf quasar and the i nfrared gal axy IR A S F10214+ 4724 (know n to harbor a buri ed quasar seen i n pol ari zed l i ght) has dem onstrated that these two objects are nearl y i denti cal , except i n the opti cal range w here the di erences can probabl y be attri buted to obscurati on/ori entati on e ects (B arvai ni s et al .1995).Such ndi ngs l end support to theori es uni fyi ng l um i nous i nfrared gal axi es and quasars vi a ori entati on e ects, w i th hi gh redshi fti nfrared gal axi esbei ng thel um i nouscounterpartsofl ocalSeyfert2' s.H owever,on physi calgrounds,i t al so seem sl i kel y thatIR -sel ected gal axi esand U V -sel ected quasars m ay di er i n thei r stage ofevol uti on.T here are currentl y about 15 wel l -docum ented detecti ons ofm ol ecul ar gas at hi gh redshi ft (e. g. ,C om bes 2001),ofw hi ch at l east 9 are gravi tati onal l y l ensed system s.T he advantagesofusi ng an i nterveni ng \gravi tati onal tel escope" to boostthe uxesare obvi ous,w i th esti m ated m agni cati on factorsofup to 100 i n theopti cal .M oreover, di erenti al gravi tati onal e ects provi de an el egant tool to probe the si ze and structure of the m ol ecul ar m ateri alw i thi n the quasar.For exam pl e,a poi nt-l i ke em i tti ng regi on (rest-fram e U V and opti calconti nua from the i nner accreti on zone),and an extended dusty m ol ecul ar regi on (the \torus") i n the quasar w i l lproduce,after gravi tati onal e ects from the i nterveni ng l ens, i m ages w i th di erent m orphol ogi es.M ol ecul ar l i ne pro l es,re ecti ng i ntri nsi c geom etri caland ki nem ati cal properti es, can be parti cul arl y usefuli n understandi ng the extended structure. It shoul d be noted however that a detai l ed m odel 1.Introducti on thatquasarssharesom eproperti esw i th l um i nousi nfrared gal axi es,both l ocal l y and at hi gh redshi ft.For exam pl e, com pari son of the m ol ecul ar,i nfrared and opti cal properti es of the C l overl eaf quasar and the i nfrared gal axy IR A S F10214+ 4724 (know n to harbor a buri ed quasar seen i n pol ari zed l i ght) has dem onstrated that these two objects are nearl y i denti cal , except i n the opti cal range w here the di erences can probabl y be attri buted to obscurati on/ori entati on e ects (B arvai ni s et al .1995).Such ndi ngs l end support to theori es uni fyi ng l um i nous i nfrared gal axi es and quasars vi a ori entati on e ects, w i th hi gh redshi fti nfrared gal axi esbei ng thel um i nouscounterpartsofl ocalSeyfert2' s.H owever,on physi calgrounds,i t al so seem sl i kel y thatIR -sel ected gal axi esand U V -sel ected quasars m ay di er i n thei r stage ofevol uti on.T here are currentl y about 15 wel l -docum ented detecti ons ofm ol ecul ar gas at hi gh redshi ft (e. g. ,C om bes 2001),ofw hi ch at l east 9 are gravi tati onal l y l ensed system s. T he advantagesofusi ng an i nterveni ng \gravi tati onal tel escope" to boostthe uxesare obvi ous,w i th esti m ated m agni cati on factorsofup to 100 i n theopti cal .M oreover, di erenti al gravi tati onal e ects provi de an el egant tool to probe the si ze and structure of the m ol ecul ar m ateri alw i thi n the quasar.For exam pl e,a poi nt-l i ke em i tti ng regi on (rest-fram e U V and opti calconti nua from the i nner accreti on zone),and an extended dusty m ol ecul ar regi on (the \torus") i n the quasar w i l lproduce,after gravi tati onal e ects from the i nterveni ng l ens, i m ages w i th di erent m orphol ogi es.M ol ecul ar l i ne pro l es,re ecti ng i ntri nsi c geom etri caland ki nem ati cal properti es, can be parti cul arl y usefuli n understandi ng the extended structure. It shoul d be noted however that a detai l ed m odel ofthe i nterveni ng l ens m ust be avai l abl e to perform the transfer from the i m age pl ane (observati onaldata) to the source pl ane (i ntri nsi c properti es ofthe quasar).W e appl i ed thi s techni que forthe rstti m e to recoverthe properti es of the m ol ecul ar torus i n the C l overl eaf,a quasar at z = 2: 56 (K nei b et al .1998),com pari ng H ST i m ages and IR A M i nterferom eter C O m aps (A l l oi n et al .1997). T he C O -em i tti ng regi on i n the quasar was found to be a di sk or ri ng-l i ke structure orbi ti ng the centralengi ne at a radi us between 75 and 100 pc, w i th K epl eri an vel oci ty around 100 km s 1 .T he e ecti ve resol uti on resul ti ng from thi stechni que turned outi n thi scase to be about20 ti m es sm al l er than the synthesi zed beam si ze ofthe C O i nterferom eter data. Such si gni cantbene ts { ux boosti ng and i ncreased e ecti ve angul ar resol uti on { have l ed us to focus our attenti on on gravi tati onal l y l ensed system s and to conduct a C O survey ofthese objects.A nother possi bl e bene t of l ensed versus unl ensed objects i s the potenti alfor hi gher ux boosti ng for hi gher-J transi ti ons (di erenti al m agni cati on).Si nce experi ence has show n that the best sel ecti on cri teri on for C O detecti on i s the presence of detectabl e far-IR or subm m /m m dust conti nuum em i ssi on, we started w i th a conti nuum survey ofthe know n l ensed quasars usi ng the IR A M 30m radi o tel escope and the JC M T (B arvai ni s& Ivi son 2002).A hi gh dustconti nuum detecti on rate encouraged us to pursue a C O search w i th the IR A M i nterferom eter (G ui l l oteau et al .1992). Si nce thi s project was started earl y i n the conti nuum survey,we were not at that ti m e abl e to m ake a general sel ecti on based upon subm i l l i m eter ux.Instead,the sampl e consi sted of m ost of the then-know n l ensed quasars havi ng opti calredshi fts m easured to good accuracy.W e observed 18 gravi tati onal l y l ensed quasars,w i th redshi fts i n the range 1: 375 3: 595. H owever, rel i abl e system i c redshi fts rem ai n a m ajor di cul ty for C O searches at hi gh z because currentl y avai l abl e redshi fts are m ostl y deri ved from hi ghl y i oni zed speci es i n the quasar broad l i ne regi on.A s thi s regi on i s often coupl ed to a hi gh vel oci ty w i nd, redshi fts deri ved thi s way have been found to be bl ueshi fted up to 1200 km s 1 w i th respect to the system i c vel oci ty of the host gal axy and the m ol ecul ar envi ronm ent of the quasar probed by C O m easurem ents.A typi cal o set i s 600 km s 1 , but there i s w i de di spersi on from one object to another.M eanw hi l e,spectrom eter bandw i dths i n the m i l l i m eter dom ai n are too narrow ( 1500 km s 1 at 3m m )to ful l y span thi sredshi ftuncertai nty usi ng a si ngl e centralfrequency setti ng.T he com bi nati on of these two facts m akes i t l i kel y that som e C O l i nes w i l l be m i ssed i n the course ofa survey.In the case ofthe present survey,w heneverthe quasarredshi ftwasfrom hi ghl y i oni zed speci es we appl i ed a 600 km s 1 redshi ft i ncrem ent to search for i ts C O em i ssi on.W e are ful l y aware that thi s o set,al though stati sti cal l y m eani ngful ,m ay be just i ncorrectfor som e i ndi vi dualquasars. In Sect. 2 we descri be the sam pl e of gravi tati onal l y l ensed quasars and the acqui si ti on and reducti on of the i nterferom eterdata set.R esul ts,both i n C O l i ne em i ssi on and i n the 1m m and 3m m conti nua,are al so presented i n Sect. 2 for the enti re sam pl e.In Sect. 3,we di scuss the generalresul ts ofthe C O survey,and i n Sect.4 consi der the detecti on ofM G 0751+ 2716 i n the C O (4 3)transi ti on i n m ore detai l .C oncl usi onsand future prospectsare gi ven i n Sect.5. A l lofthe conti nuum detecti ons,and the one l i ne detecti on, are w i thi n 1 00 of the opti cal posi ti ons l i sted i n Tabl e 1,except for two cases.For 2016+ 112,the oset i s R A = 1: 00 5 and D ec = 3: 00 5 from the observed coordi nates.H owever,checki ng the N A SA Extragal acti c D atabase (N ED ) we found i m proved coordi nates w hi ch are w i thi n about 1 00 ofthe conti nuum source i n both R A and D ec.In the case of R X J0911+ 055,the o set of the conti nuum source (a m ean ofthe 3m m and 1m m conti nuum sources) i s R A = 2: 00 0 and D ec = 2: 00 1.T he ori gi n ofthi s o set,w hi ch i s l argerthan typi calopti calposi ti on errors,i s unknow n at present. W e bel i eve som e C O l i nes, such as the strong one expected from R X J0911+ 055, and som e am ong the i nterm edi ate-strength subm m sources, m ay have been m i ssed because of the uncertai nty i n the redshi ft of the m ol ecul arl i nescom bi ned w i th thenarrow observi ng bandw i dth. In fact, duri ng our 1999 rst search for C O i n M G 0751+ 2716,the C O (4 3) transi ti on was detected on the edge of the bandpass, at a vel oci ty o set of 600 km s 1 w i th respect to our ori gi nalguess for m ol ecul ar em i ssi on.T hanksto the strength ofthi sl i ne we were abl e to i denti fy i ts presence and obtai n new observati ons at the appropri ate frequency,w hi ch ful l y con rm ed the 1999 m easurem ent.H owever,thi swoul d notwork i n the case of fai nter C O l i ne em i tters.T hi s di cul ty w i l lonl y be ful l y resol ved by usi ng broadband backends i n the future. In them eanti m e,wepl an to reobserveatl eastR X J0911+ 055 w i th anki ng bandpasses to coverm ore redshi ft space. 2.T he sam 3.R esul ts of the C O survey C onti nuum em i ssi on wasdetected at3m m i n 6 targets, w i th 3 ofthose al so detected at 1m m (one bei ng parti cul arl y strong,the radi o l oud quasar B 1030+ 074). 4.T he C O l i ne i n M G 0751+ 2716 A strong l i ne i n the C O (4 3) transi ti on was detected i n M G 0751+ 2761.A set ofthe channelm aps,w i th vel oci ty steps of100 km s 1 ,i s show n i n Fi g.1 for the C O (4 3) transi ti on.It convi nci ngl y reveal s the C O (4 3) em i tti ng regi on at a l ocati on very cl ose to the quasaropti calcoordi nates(o setby R A = 0: 00 6, D ec = 0: 00 5).T he C O (4 3) l i ne pro l e i s di spl ayed i n Fi g.2.A G aussi an t provi des thefol l ow i ng param eters:FW H M of390 38km s 1 ,peak frequency at109: 778 0: 005 G H z (correspondi ng to a redshi ftof3. 200),and i ntensi ty of5: 96 0: 45 Jy km s 1 .T he C O (9 8)transi ti on wasnotdetected i n the 1m m w i ndow . A G aussi an t w i th xed l i ne w i dth and centralposi ti on anal ogousto the C O (4 3)transi ti on showed onl y conti nuum at the posi ti on.A l ocalpeak cl ose to 100 km s 1 i n the object' s vel oci ty fram e i s bel ow the 3 l evel .B y combi ni ng both si debands of the 1m m recei ver,we obtai n a conti nuum detecti on of6: 75 1: 32 m Jy. In order to com pute the C O l i ne l um i nosi ty and total gas m ass, a l ensi ng correcti on m ust be appl i ed. In spi te ofseverale orts to m odelthe l ens system towards M G 0751+ 2716 (Leh ar et al . 1997;Tonry & K ochanek 1999),addi ti onalwork rem ai ns to be done:the l ens appears to be a qui te com pl ex system w hi ch requi res m ore shear than accounted for by the l ensi ng gal axy (i denti - For the deconvol uti on ofeach channel ,the appropri ate com bi ned U V coverage wastaken i nto account.T he conti nuum ux has been subtracted. on H ST opti cal i m agi ng, provi des an esti m ated opti cal m agni cati on of 16. 6. W hi l e di erences between opti cal and C O /subm i l l i m eternetm agni cati onscan beexpected i n som e cases because of the di erence i n source si zes (subpc versus tens to hundreds of pc), m odel s suggest that for opti calm agni cati onsl ess than about20 the di fferences are general l y not expected to be l arge (see Fi g. 1 of B arvai ni s and Ivi son 2002, and associ ated di scussi on).T herefore,we correct the C O em i ssi on for a m agni cati on of factor of 16. 6 and deri ve a l i ne l um i nosi ty L 0 C O = 3: 9 10 9 K km s 1 pc 2 (H 0 = 75 km s 1 M pc 1 , q 0 = 0: 5). T he m ol ecul arm ateri alem i tti ng i n the C O (4 3)transi ti on i s m ost l i kel y cl ose to the quasar,as i n the case of the C l overLeaf (K nei b etal .1998)and A PM 08279+ 5255 (D ow nes et al .1999),and i t i s al so probabl y dense and warm .In order to cal cul ate the totalm ol ecul ar gas m ass we have consi dered two val ues for the rati o M H 2 =L 0 C O : 0: 8M (K km s 1 pc 2 ) 1 , fol l ow i ng D ow nes & Sol om on (1998) for nucl ear ri ngs i n ul tral um i nous gal axi es, and 0: 4M (K km s 1 pc 2 ) 1 fol l ow i ng B arvai ni setal .(1997) forthe m ol ecul ar\torus" i n the C l overl eaf.T hese conver- It i s of i nterest as wel l to com pare the gas m ass to the dynam i cal m ass deri ved from the observed C O l i ne w i dth.W i th a deri ved K epl eri an vel oci ty of400/si n(i)km s 1 ,and assum i ng that the m ol ecul ar gas i s l ocated at a radi us of200 pc (see above references for the C l overLeaf and A PM 08279+ 5255),weobtai n a l owerl i m i tofM dyn > 1: 8 10 9 M ,consi stent w i th the deri ved val ue ofM H 2 . 5.C oncl udi ng rem arks and future prospects T hough the present survey yi el ded a l ow detecti on rate i n C O , there are several new l ensed quasar candi dates yet to be observed based on thei r strong 850 m conti nua,recentl y di scovered i n the course ofthe subm i l l i m eter survey by B arvai ni s & Ivi son (2002).Suppl em entary, expanded-frequency observati ons of som e sources (m ost notabl y R X J0911+ 055)m ay turn up m ore C O detecti ons from the present source l i st. A sforM G 0751+ 2716,the centi m eterradi o source has four com ponents connected by arcs (Leh ar et al . 1997), and i n the opti cal i t appears as a 1 00 di am eter Ei nstei n R i ng.A pri m ary dri verfor thi s projectwas to nd l ensed sources that coul d be spati al l y resol ved i n C O l i ne em i ssi on. T hi s i s currentl y possi bl e for M G 0751+ 2716 usi ng the PdB I,and,l i ke the C l overl eaf,reconstructi on ofthe m ol ecul ar source structure and ki nem ati cs on very sm al l angul ar scal es usi ng the l ensi ng properti es m ay prove to be qui te i nteresti ng. A cknow l edgem ents. W e w arm l y thank al l the IR A M sta w ho perform ed i n servi ce m ode, w i th the Pl ateau de B ure Interferom eter,al lof the observati ons rel ated to thi s project. W e al so thank Ski A ntonucci for i m portant earl y contri buti ons. Joseph Leh ar and B ri an M cLeod generousl y provi ded an esti m ate ofthe m agni cati on ofM G 0751+ 2716 i n advance of publ i cati on. D A and R B w i sh to thank the IR A M i nstitute i n G renobl e for hospi tal i ty duri ng vi si tsthere.T he N A SA Extragal acti c D atabase (N ED )and the C A ST LeS com pi l ati on of l enses (http: //cfa-w w w . harvard. edu/castl es) w ere used extensi vel y i n the course ofthi s w ork. R eferences T he m easurem entofC O l i ne em i ssi on i n hi gh redshi ftobjects has proven to be a frui tfulavenue for i nvesti gati ng the properti es ofdi stant quasars and gal axi es.M ol ecul ar observati onsaddressi nteresti ng i ssuessuch asthatofstar form ati on i n the earl y uni verse and how the presence of a central m assi ve engi ne can a ect the i nterstel l ar m ateri al i n i ts host gal axy. D etecti ons of C O have show n Send o print requests to:D .A l l oi n dal l oi n@ eso. org ? B ased on observati ons obtai ned w i th the Pl ateau de B ure Interferom eter of the Insti tut de R adi o A stronom i e M i l l i m etri que, supported by IN SU /C N R S (France), M PG (G erm any) and IG N (Spai n). pl e:acqui si ti on and reducti on of the C O i nterferom etri c data set T he gravi tati onal l y l ensed quasar sam pl e i s presented i n Tabl e 1.T he coordi nates general l y refer to the bri ghtest quasari m agei n the opti cal .R edshi ftshavebeen corrected i n som ecasesusi ng thetechni quedescri bed i n theprevi ous secti on,except for M G 0751+ 2716 w here an i ni ti aldetecti on al l owed re nem ent ofthe val ue to the center ofthe l i ne i n fol l ow up observati ons. T he centeri ng frequenci es gi ven for the 3m m and 1m m w i ndow s correspond to vari ous C O transi ti ons from C O (2 1) to C O (9 8),dependi ng on the quasarredshi ftand the w i ndow consi dered:the targetted C O transi ti onsi n the 3m m w i ndow are speci ed for each source i n Tabl e 1.T he \seei ng" esti m ates correspond to the 3m m data set.Som e targets were observed on severaldi erentobservi ng runs;seei ng val ues,num bers ofantennas,and hours spent per exposure are provi ded.T he observed spectralbandw i dth was ei ther 560 M H z or 595 M H z. Tabl e 2 l i sts the C O transi ti on observed (3m m ), the \channelrm s" (rm s/beam /100 km s 1 ,i n m Jy),the l i ne ux (for M G 0751+ 2716),and the 3m m and 1m m conti nuum resul ts.Su ci ent sensi ti vi ty and bandw i dth for l i ne m easurem ents were onl y avai l abl e at 3m m .Tabl e 2 al so l i sts850 m conti nuum uxesorupperl i m i tsderi ved from SC U B A observati ons at the JC M T (B arvai ni s & Ivi son 2002).N ote that B 1600+ 434 and B 1030+ 074 are strong radi o l oud quasars w hose m i l l i m eter/subm i l l i m eter conti nua are consi stentw i th pure synchrotron radi ati on.T he othersubm i l l i m eter-detected sourcesarel i kel y to be domi nated by dustem i ssi on.T heconti nuum ux rati o between 3m m and 1m m for M G 0751+ 2716 i s not typi calof dust em i ssi on,though the rati o between 1m m and 850 m i s. T hi s i s probabl y caused by som e resi dualsteep-spectrum synchrotron em i ssi on contri buti ng to the 3m m (and to a l esser extent the 1m m ) ux. O ut of18 sources observed,we obtai ned:(a) one strong C O l i neem i tter,M G 0751+ 2716,detected i n C O (4 3),(b) three m argi naldetecti ons(to be i nvesti gated further)and (c) 14 non-detecti ons.G eneral l y speaki ng,strong subm m em i ssi on from dusti sa good predi ctorofstrong C O em i ssi on. O f the two strongest dust sources, R X J0911+ 055 (S 850 = 26: 7 m Jy)and M G 0751+ 2716 (S 850 = 25: 8 m Jy), onl y the l atter wasdetected i n C O .T he l i ne,w i th a peak ux of18 m Jy,i soneofthestrongestknow n am ong hi gh-z sources.Seven sourceshad ei therno m easurem entsoronl y upperl i m i tsat850 m ,but6 addi ti onalsourceshad m oderatel y strong subm m conti nuum detecti ons i n the 8 15 m Jy range. ed as G 3 i n Leh ar et al .1997).T hi s extra shear m i ght com e from w hatappearsto be a group orcl usterofgal axi es,i ndi cated by the l arge num ber ofgal axi es i n the el d around the quasar. Very recent m odel i ng by J. Leh ar and B . M cLeod (2001, pri vate com m uni cati on), based F ig.1. C hannelm aps of the C O (4 3) l i ne em i ssi on i n M G 0751+ 2716 w i th a contour spaci ng of2. 5 m Jy/beam (the zero contour i s not show n).T he cross i ndi cates the opti calposi ti on ofthe quasar.T wo di erent spectralsetups were m erged w hi ch overl ap i n the [ 75, 725]km s 1 channel s,resul ti ng i n three di erent rm s l evel s over the spectralrange (see Fi g.2 error bars). F ig. 2 . 2Spectra i n C O (4 3)(upper panel )and C O (9 8) (l ower panel )forM G 0751+ 2716,bi nned i nto 100 km s 1 w i de channel s. si on factors yi el d a totalm ol ecul ar gas m ass i n the range M H 2 = 1: 6 3: 1 10 9 M . n, D . , G ui l l oteau, S. , B arvai ni s, R . , A ntonucci , R . , & Tacconi ,L.1997,A & A ,321,24. B arvai ni s,R . ,A ntonucci ,R . ,H urt,T . ,C ol em an,P. ,& R euter, H . P.1995,A pJ,451,L9 B arvai ni s R . , M al oney P. , A ntonucci R . , & A l l oi n D . 1997, A pJ,484,695 B arvai ni s,R . ,& Ivi son,R .2002,A pJ,i n press C om bes F.2001,astro-ph/0008456 D ow nes D . ,& Sol om on P.1998,A pJ,507,615 D ow nes D . ,N eriR . ,W i kl i nd T . ,W i l ner D . ,& Shaver P.1999, A pJ,513,L1 G ui l l oteau S. ,D el annoy J. ,D ow nes D .et al .1992,A & A ,329, 827 K nei b J. P. ,A l l oi n D . ,M el l i er Y . ,G ui l l oteau S. ,B arvai ni s R . , & A ntonucciR .1998,A & A ,329,827 Leh ar J. ,B urke B . ,C onner S.et al .1997,A J,114,48 Tonry J. ,& K ochanek C .1999,A J,117,2034 Table 1 . 1Targetl i st and observi ng param etersofthe C O surveyO b ject R A (2000) D ec(2000) z 3m m Seeing B eam size/P A 1m m B eam size/P A A ntennas E xp osure (G H z) ( 00 ) ( 00 00 = ) (G H z) ( 00 00 = ) (H rs) 0047-2808 00:49:41.87 27:52:25.7 3.595 100.335 1.6;1.0 14:1 3:5=11 225.661 4;5 3.5;1.6 U M 673 01:45:17.22 09:45:12.3 2.730 92.706 0.75 4:5 1:9=0 216.260 1:9 0:7=199 5 4.2 M G 0751+ 2716 07:51:41.46 + 27:16:31.4 3.200 109.778 1.0;1.1;0.6 9:4 5:6=32 246.898 3:1 2:1=14 4;3;4 3.4;3.6;2.0 SB S0909+ 523 09:13:00.76 + 52:59:31.5 1.375 97.068 0.2 2:4 1:7=155 242.639 0:9 0:7=147 5 4.7 R X J0911+ 055 09:11:27.50 + 05:50:52.0 2.807 90.831 0.7 6:5 6:0=53 211.886 3:0 2:0=127 4 6.3 Q 1009-0252 10:12:16.09 03:07:03.0 2.746 92.311 0.7 13:0 6:2=9 215.337 5:5 2:9=15 4 3.1 J0313 10:17:24.13 20:47:00.4 2.552 97.352 0.9 17:4 6:3=7 227.097 8:0 2:4=7 4 4.0 B 1030+ 074 10:33:34.0 + 07:11:26.1 1.535 90.942 1.1;2.3 9:9 5:8=49 227.325 5;5 1.1;3.2 H E 1104-1805 11:06:33.45 18:21:24.2 2.326 103.967 1.7 13:5 4:5=8 242.529 3 3.2 P G 1115+ 080 11:18:16.96 + 07:45:59.3 1.723 84.663 0.7 7:8 5:5=53 211.630 2:8 2:2=60 4 5.8 1208+ 1011 12:10:57.16 + 09:54:25.6 3.831 95.434 0.5 2:5 1:4=23 214.637 1:1 0:6=24 4 6.3 H ST 14176+ 530 14:17:36.51 + 52:26:40.4 3.403 104.711 0.8;2.0 6:6 4:8=50 235.501 2:4 1:7=98 4;5 3.7;5.9 SB S1520+ 530 15:21:44.83 + 52:54:48.6 1.860 80.608 1.1;0.7;0.7;1.2 10 5:7=123 241.773 5:3 2:5=63 4;3;5;5 4.4;3.8;0.33;2.3 B 1600+ 434 16:01:40.45 + 43:16:47.8 1.589 89.045 0.2;1.4;2.0 8:3 7:6=138 222.583 1:0 0:6=46 5;4;4 1.9;6.1;2.9 B 1608+ 656 16:09:13.96 + 65:32:29.0 1.394 96.298 0.4 3:0 1:7=83 240.713 1:2 0:7=75 5 3.9 2016+ 112 20:19:18.15 + 11:27:08.3 3.282 107.670 0.7;0.6;1.3 7:8 5:0=45 242.156 3:0 2:4=21 5;5;4 4.9;2.6;4.9 H E 2149-2745 21:52:07.44 27:31:50.2 2.033 114.011 3.6 24:1 4:6=7 227.938 5 3.0 Q 2237+ 0305 22:40:30.14 + 03:21:31.0 1.696 85.511 1.0 6:5 4:1=2 213.749 2:6 1:6=176 4 5.1 Table 2.R esul ts ofthe C O survey O bject Target l i ne Li ne rm s Li ne ux 3m m Scont 1m m Scont 850 m Scont (m Jy/beam ) (m Jy/100km s 1 ) (Jy km s 1 ) (m Jy/beam ) (m Jy/beam ) (m Jy) 0047-2808 C O (4 3) 3. 3 0. 9 0. 9 na < 7: 0 U M 673 C O (3 2) 2. 0 0. 7 0. 5 na 12: 0 2: 2 M G 0751+ 2716 C O (4 3) 1. 9 5: 96 0: 45 4. 1 0. 5 6. 7 1. 3 25: 8 1: 3 SB S0909+ 523 C O (2 1) 1. 1 0. 5 0. 3 0. 0 1. 3 < 5: 5 R X J0911+ 055 C O (3 2) 1. 2 1. 7 0. 3 10. 2 1. 8 26: 7 1: 4 Q 1009-0252 C O (3 2) 1. 8 0. 1 0. 5 0. 3 4. 0 na J0313 C O (3 2) 1. 8 0. 0 0. 5 -6 4 na B 1030+ 074 C O (2 1) 3. 3 184 1. 9 na na H E1104-1805 C O (3 2) 6. 8 1. 9 2. 0 na 14: 8 3: 4 PG 1115+ 080 C O (2 1) 1. 3 -0. 3 0. 3 -0. 9 1. 2 3: 7 1: 3 1208+ 1011 C O (4 3) 1. 3 0. 2 0. 3 4. 2 1. 9 8: 1 2: 0 H ST 14176+ 530 C O (4 3) 1. 2 0. 1 0. 3 -5 3 < 3: 5 SB S1520+ 530 C O (2 1) 1. 5 hi nt 0. 0 0. 2 -0. 5 1. 4 9: 4 2: 6 B 1600+ 434 C O (2 1) 1. 1 hi nt 25 0. 3 12. 6 2. 3 7: 3 1: 8 B 1608+ 656 C O (2 1) 1. 5 hi nt 8. 1 0. 4 0. 8 2. 2 8: 1 1: 7 2016+ 112 C O (4 3) 0. 8 1. 8 0. 2 1. 1 1. 0 < 4: 8 H E2149-2745 C O (3 2) 10. -2. 8 2. 7 na 8: 0 1: 9 Q 2237+ 0305 C O (2 1) 1. 4 0. 1 0. 35 -1. 9 2. 1 3: 9 1: 2	[]
[ "Semileptonic Hyperon Decays and CKM Unitarity ", "Semileptonic Hyperon Decays and CKM Unitarity " ]	[ "Nicola Cabibbo [email protected] ", "Earl C Swallow [email protected] ", "Roland Winston [email protected] ", "\nDepartment of Physics\nDepartment of Physics\nUniversity of Rome\nLa Sapienza and INFN, Sezione di Roma 1 Piazzale A. Moro 500185RomeItaly\n", "\nElmhurst College Elmhurst\n60126Illinois\n", "\nDivision of Natural Sciences\nEnrico Fermi Institute\nThe University of Chicago\nChicagoIllinois\n", "\nThe University of California -Merced\n95344MercedCalifornia\n" ]	[ "Department of Physics\nDepartment of Physics\nUniversity of Rome\nLa Sapienza and INFN, Sezione di Roma 1 Piazzale A. Moro 500185RomeItaly", "Elmhurst College Elmhurst\n60126Illinois", "Division of Natural Sciences\nEnrico Fermi Institute\nThe University of Chicago\nChicagoIllinois", "The University of California -Merced\n95344MercedCalifornia" ]	[]	Using a technique that is not subject to first-order SU (3) symmetry breaking effects, we determine the Vus element of the CKM matrix from data on semileptonic hyperon decays. We obtain Vus = 0.2250(27), where the quoted uncertainty is purely experimental. This value is of similar experimental precision to the one derived from K l3 , but it is higher and thus in better agreement with the unitarity requirement, \|V ud \| 2 + \|Vus\| 2 + \|V ub \| 2 = 1. An overall fit including the axial contributions, and neglecting SU (3) breaking corrections, yields F + D = 1.2670 ± 0.0035 and F − D = −0.341 ± 0.016 with χ 2 = 2.96/3 d.f.The determination of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] is one of the main ingredients for evaluating the solidity of the standard model of elementary particles. This is a vast subject which has seen important progress with the determination [3, 4] of ǫ ′ /ǫ and the observation [5, 6] of CP violation in B decays.While a lot of attention has recently been justly devoted to the higher mass sector of the CKM matrix, it is the low mass sector, in particular V ud and V us where the highest precision can be attained. The most sensitive test of the unitarity of the CKM matrix is provided by the relation \|V ud \| 2 + \|V us \| 2 + \|V ub \| 2 = 1 − ∆. Clearly the unitarity condition is ∆ = 0. The \|V ub \| 2 contribution [7] is negligible ( 10 −5 ) at the current level of precision. The value V ud = 0.9740 ± 0.0005 is obtained from superallowed pure Fermi nuclear decays[8]. In combination with V us = 0.2196 ± 0.0023, derived from K e3 decay [9, 10], this yields ∆ = 0.0032 ± 0.0014. On its face, this represents a 2.3 standard deviation departure from unitarity[8].In this communication we reconsider the contribution that the hyperon beta decays can give to the determination of V us . The conventional analysis of hyperon beta decay in terms of the parameters F, D and V us is marred * To be published in Physical Review Letters.by the expectation of first order SU (3) breaking effects in the axial-vector contribution. The situation is only made worse if one introduces adjustable SU (3) breaking parameters as this increases the number of degrees of freedom and degrades the precision. If on the contrary, as we do here, one focuses the analysis on the vector form factors, treating the rates and g 1 /f 1 [11] as the basic experimental data, one has directly access to the f 1 form factor for each decay, and this in turn allows for a redundant determination of V us . The consistency of the values of V us determined from the different decays is a first confirmation of the overall consistency of the model. A more detailed version of this work will be published in the Annual Reviews of Nuclear and Particle Sciences[12].In 1964 Ademollo and Gatto proved [13] that there is no first-order correction to the vector form factor, ∆ 1 f 1 (0) = 0. This is an important result: since experiments can measure V us f 1 (0), knowing the value of f 1 (0) in ∆S = 1 decays is essential for determining V us .The Ademollo-Gatto Theorem suggests an analytic approach to the available data that first examines the vector form factor f 1 because it is not subject to first-order SU (3) symmetry breaking effects. An elegant way to do this is to use the measured value of g 1 /f 1 along with the predicted values of f 1 and f 2 to extract a V us value from the decay rate for each decay. If the theory is correct, these should coincide within errors, and can be combined to obtain a best value of V us . This consistency of the V us	10.1103/physrevlett.92.251803	[ "https://export.arxiv.org/pdf/hep-ph/0307214v2.pdf" ]	119,377,379	hep-ph/0307214	bba08c52590ac0cd7f2b2af3066410a0154a3191	Semileptonic Hyperon Decays and CKM Unitarity * May 2004 Nicola Cabibbo [email protected] Earl C Swallow [email protected] Roland Winston [email protected] Department of Physics Department of Physics University of Rome La Sapienza and INFN, Sezione di Roma 1 Piazzale A. Moro 500185RomeItaly Elmhurst College Elmhurst 60126Illinois Division of Natural Sciences Enrico Fermi Institute The University of Chicago ChicagoIllinois The University of California -Merced 95344MercedCalifornia Semileptonic Hyperon Decays and CKM Unitarity * May 2004(Dated: March 25, 2022)arXiv:hep-ph/0307214v2 7 APS/123-QEDnumbers: 1215Hh1330Ce1420Jn Using a technique that is not subject to first-order SU (3) symmetry breaking effects, we determine the Vus element of the CKM matrix from data on semileptonic hyperon decays. We obtain Vus = 0.2250(27), where the quoted uncertainty is purely experimental. This value is of similar experimental precision to the one derived from K l3 , but it is higher and thus in better agreement with the unitarity requirement, \|V ud \| 2 + \|Vus\| 2 + \|V ub \| 2 = 1. An overall fit including the axial contributions, and neglecting SU (3) breaking corrections, yields F + D = 1.2670 ± 0.0035 and F − D = −0.341 ± 0.016 with χ 2 = 2.96/3 d.f.The determination of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] is one of the main ingredients for evaluating the solidity of the standard model of elementary particles. This is a vast subject which has seen important progress with the determination [3, 4] of ǫ ′ /ǫ and the observation [5, 6] of CP violation in B decays.While a lot of attention has recently been justly devoted to the higher mass sector of the CKM matrix, it is the low mass sector, in particular V ud and V us where the highest precision can be attained. The most sensitive test of the unitarity of the CKM matrix is provided by the relation \|V ud \| 2 + \|V us \| 2 + \|V ub \| 2 = 1 − ∆. Clearly the unitarity condition is ∆ = 0. The \|V ub \| 2 contribution [7] is negligible ( 10 −5 ) at the current level of precision. The value V ud = 0.9740 ± 0.0005 is obtained from superallowed pure Fermi nuclear decays[8]. In combination with V us = 0.2196 ± 0.0023, derived from K e3 decay [9, 10], this yields ∆ = 0.0032 ± 0.0014. On its face, this represents a 2.3 standard deviation departure from unitarity[8].In this communication we reconsider the contribution that the hyperon beta decays can give to the determination of V us . The conventional analysis of hyperon beta decay in terms of the parameters F, D and V us is marred * To be published in Physical Review Letters.by the expectation of first order SU (3) breaking effects in the axial-vector contribution. The situation is only made worse if one introduces adjustable SU (3) breaking parameters as this increases the number of degrees of freedom and degrades the precision. If on the contrary, as we do here, one focuses the analysis on the vector form factors, treating the rates and g 1 /f 1 [11] as the basic experimental data, one has directly access to the f 1 form factor for each decay, and this in turn allows for a redundant determination of V us . The consistency of the values of V us determined from the different decays is a first confirmation of the overall consistency of the model. A more detailed version of this work will be published in the Annual Reviews of Nuclear and Particle Sciences[12].In 1964 Ademollo and Gatto proved [13] that there is no first-order correction to the vector form factor, ∆ 1 f 1 (0) = 0. This is an important result: since experiments can measure V us f 1 (0), knowing the value of f 1 (0) in ∆S = 1 decays is essential for determining V us .The Ademollo-Gatto Theorem suggests an analytic approach to the available data that first examines the vector form factor f 1 because it is not subject to first-order SU (3) symmetry breaking effects. An elegant way to do this is to use the measured value of g 1 /f 1 along with the predicted values of f 1 and f 2 to extract a V us value from the decay rate for each decay. If the theory is correct, these should coincide within errors, and can be combined to obtain a best value of V us . This consistency of the V us Using a technique that is not subject to first-order SU (3) symmetry breaking effects, we determine the Vus element of the CKM matrix from data on semileptonic hyperon decays. We obtain Vus = 0.2250 (27), where the quoted uncertainty is purely experimental. This value is of similar experimental precision to the one derived from K l3 , but it is higher and thus in better agreement with the unitarity requirement, \|V ud \| 2 + \|Vus\| 2 + \|V ub \| 2 = 1. An overall fit including the axial contributions, and neglecting SU (3) breaking corrections, yields F + D = 1.2670 ± 0.0035 and F − D = −0.341 ± 0.016 with χ 2 = 2.96/3 d.f. The determination of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is one of the main ingredients for evaluating the solidity of the standard model of elementary particles. This is a vast subject which has seen important progress with the determination [3,4] of ǫ ′ /ǫ and the observation [5,6] of CP violation in B decays. While a lot of attention has recently been justly devoted to the higher mass sector of the CKM matrix, it is the low mass sector, in particular V ud and V us where the highest precision can be attained. The most sensitive test of the unitarity of the CKM matrix is provided by the relation \|V ud \| 2 + \|V us \| 2 + \|V ub \| 2 = 1 − ∆. Clearly the unitarity condition is ∆ = 0. The \|V ub \| 2 contribution [7] is negligible ( 10 −5 ) at the current level of precision. The value V ud = 0.9740 ± 0.0005 is obtained from superallowed pure Fermi nuclear decays [8]. In combination with V us = 0.2196 ± 0.0023, derived from K e3 decay [9,10], this yields ∆ = 0.0032 ± 0.0014. On its face, this represents a 2.3 standard deviation departure from unitarity [8]. In this communication we reconsider the contribution that the hyperon beta decays can give to the determination of V us . The conventional analysis of hyperon beta decay in terms of the parameters F, D and V us is marred * To be published in Physical Review Letters. by the expectation of first order SU (3) breaking effects in the axial-vector contribution. The situation is only made worse if one introduces adjustable SU (3) breaking parameters as this increases the number of degrees of freedom and degrades the precision. If on the contrary, as we do here, one focuses the analysis on the vector form factors, treating the rates and g 1 /f 1 [11] as the basic experimental data, one has directly access to the f 1 form factor for each decay, and this in turn allows for a redundant determination of V us . The consistency of the values of V us determined from the different decays is a first confirmation of the overall consistency of the model. A more detailed version of this work will be published in the Annual Reviews of Nuclear and Particle Sciences [12]. In 1964 Ademollo and Gatto proved [13] that there is no first-order correction to the vector form factor, ∆ 1 f 1 (0) = 0. This is an important result: since experiments can measure V us f 1 (0), knowing the value of f 1 (0) in ∆S = 1 decays is essential for determining V us . The Ademollo-Gatto Theorem suggests an analytic approach to the available data that first examines the vector form factor f 1 because it is not subject to first-order SU (3) symmetry breaking effects. An elegant way to do this is to use the measured value of g 1 /f 1 along with the predicted values of f 1 and f 2 to extract a V us value from the decay rate for each decay. If the theory is correct, these should coincide within errors, and can be combined to obtain a best value of V us . This consistency of the V us values obtained from different decays then indicates the success of the Cabibbo model. A similar approach appears to have been taken in Ref. [14]. Four hyperon beta decays have sufficient data to perform this analysis: [9]. Table I shows the results for them. In this analysis, both model-independent and model-dependent radiative corrections [11] are applied and q 2 variation of f 1 and g 1 is included. Also SU (3) values of g 2 = 0 and f 2 are used along with the numerical rate expressions tabulated in Ref. [11]. We have not however included SU (3)-breaking corrections to the f 1 form factor, which will be discussed in the next section. The stated V us errors are purely experimental, coming from experimental uncertainties in the hyperon lifetimes, branching ratios, and form factor ratios. Λ → p e −ν , Σ − → n e −ν , Ξ − → Λ e −ν , Ξ 0 → Σ + e −ν The four values are clearly consistent (χ 2 = 2.26/3d.f.) with the combined value of V us = 0.2250 ± 0.0027. This value is nearly as precise as that obtained from kaon decay (V us = 0.2196 ± 0.0023) and, as observed in previous analyses [15,16,26], is somewhat larger. In combination with V ud = 0.9740 ± 0.0005 obtained from superallowed pure Fermi nuclear decays [8], the larger V us value from hyperon decays beautifully satisfies the unitarity constraint \|V ud \| 2 + \|V us \| 2 + \|V ub \| 2 = 1. We will limit our discussion to the effects that are most relevant for the determination of V us . Turning our attention first to SU (3)-breaking corrections to the f 1 form factor, we find in the literature computations that use some version of the quark model, as in [17,18], or some version of chiral perturbation theory, as in [15,19,20]. The quark-model computations find that the f 1 form factors for the different ∆S = 1 decays are reduced by a factor, the same for all decays, given as 0.987 in [17], and 0.975 in [18], a decrease respectively of 1.3% or 2.5%. This is a very reasonable result, the decrease arising from the mismatch of the wave functions of baryons containing different numbers of the heavier s quarks. Evaluations of f 1 in chiral perturbation theory range from small negative corrections in [19] to larger positive corrections in [15,20]. Positive corrections in f 1 for all hyperon beta decays cannot be excluded, but are certainly not expected in view of an argument [21] according to which one expects a negative correction to f 1 at least in the Σ − → n e −ν case. This result follows from the observation that the intermediate states that con-tribute to the positive second-order terms in the Ademollo and Gatto sum rule have, in this case, quantum numbers S = −2, I = 3/2; no resonant baryonic state is known with these quantum numbers. If we accept the hypothesis that the contribution of resonant hadronic states dominate, we can conclude that the correction to f 1 in Σ − beta decay should be negative. We note that this argument also applies to K l3 decays, and that the corrections to these decays, computed with chiral perturbation theory, are, as expected, negative. A modern revisitation of the quark-model computations will be feasible in the near future with the technologies of lattice QCD, and we would expect that a small negative correction would be obtained in quenched lattice QCD, an approximation that consists in neglecting components in the wave function of the baryons with extra quark-antiquark pairs. This is known to be an excellent approximation in low-energy hadron phenomenology [22]. Multiquark effects can be included in lattice QCD by forsaking the quenched approximation for a full simulation. Alternatively one could resort to chiral perturbation theory to capture the major part of the multiquark contributions which will be dominated by virtual π, K, η states. Early results of a similar strategy applied to the K e3 decays [23] indicate that in that case a 1% determination of the f + (0) form factor is within reach, and we expect that a similar precision can be obtained in the case of hyperon decays. In the present situation we consider it best not to include any SU (3) breaking corrections in our evaluation, nor to include an evaluation of a theoretical error. Our expectation that the corrections to f + (0) will be small and negative can only be substantiated by further work. We next turn our attention to the possible effect of ignoring the g 2 form factor. In the absence of second class currents [24] the form factor g 2 can be seen to vanish in the SU (3) symmetry limit. The argument is very straightforward: the neutral currents A 3 α =qλ 3 γ α γ 5 q and A 8 α =qλ 8 γ α γ 5 q that belong to the same octet as the weak axial current are even under charge conjugation, so that their matrix elements cannot contain a weak -electricity term, which is C-odd. The vanishing of the weak electricity in the proton and neutron matrix elements of A 3 α , A 8 α implies the vanishing of the D and F coefficients for g 2 (0), so that, in the SU (3) limit, the g 2 (0) form factor vanishes for any current in the octet. In hyperon decays a nonvanishing g 2 (0) form factor can arise from the breaking of SU (3) symmetry. Theoretical estimates [25] indicate a value for g 2 (0)/g 1 (0) in the −0.2 to −0.5 range. In determining the axial-vector form factor g 1 from the Dalitz Plot -or, equivalently, the electron-neutrino correlation -one is actually measuringg 1 , a linear combination of g 1 and g 2 (g 1 ≈ g 1 − δg 2 up to first order in δ = ∆M/M ). This has already been noticed in past experiments and is well summarized in Gaillard and Sauvage [26], Table 8. Therefore, in deriving V 2 us f 2 1 (hence V us ) from the beta decay rate, there is in fact a small sensitivity to g 2 . To first order, the rate is proportional to V 2 us [f 2 1 + 3g 2 1 − 4δ g 1 g 2 ] ≈ V 2 us [f 2 1 + 3g 2 1 + 2δg 1 g 2 ]. In fact, this is a second order correction to the value of V us , potentially of the same order of magnitude as the corrections to f 1 . Experiments that measure correlations with polarization -in addition to the electron-neutrino correlationare sensitive to g 2 . While the data are not yet sufficiently precise to yield good quantitative information, one may nevertheless look for trends. In polarized Σ − → p e −ν [27], negative values of g 2 /f 1 are clearly disfavored (a positive value is preferred by 1.5σ). Since the same experiment unambiguously established that g 1 /f 1 is negative one concludes that allowing for nonvanishing g 2 would increase the derived value of V 2 us f 2 1 . In polarized Λ → p e −ν the data favor [28] negative values of g 2 /f 1 (by about 2σ). In this decay, g 1 /f 1 is positive so that again, allowing for the presence of nonvanishing g 2 would increase the derived value of V 2 us f 2 1 . In either case, we may conclude that making the conventional assumption of neglecting the g 2 form factor tends to underestimate the derived value of V us . A more quantitative conclusion must await more precise experiments. We consider it to be of the highest priority to determine the g 2 form-factor (or a stringent limit on its value) in at least one of the hyperon decays, ideally in Λ semileptonic decay which at the moment seems to offer the single most precise determination of V us . The excellent agreement with the unitarity condition of our determination of V us , which neglects SU (3)-breaking effects, seems to indicate that such effects were overestimated in the past, probably as a consequence of the uncertainties of the early experimental results. We also find [12] that the g 1 form factor of the different decays, which is subject to first order corrections, is well fitted by the F, D parameters [1], with F + D = 1.2670 ± 0.0035 and F − D = −0.341 ± 0.016 with χ 2 = 2.96/3 d.f. The value of V us obtained from hyperon decays is of comparable experimental precision with that obtained from K l3 decays, and is in better agreement with the value of θ C obtained from nuclear beta decay. While a discrepancy between V us and V ud could be seen as a portent of exciting new physics, a discrepancy between the two different determinations of V us can only be taken as an indication that more work remains to be done both on the theoretical and the experimental side. On the theoretical side, renewed efforts are needed for the determination of SU (3)-breaking effects in hyperon beta decays as well as in K l3 decays. While it is quite possible to improve the present situation on the quark-model front, the best hopes lie in lattice QCD simulations, perhaps combined with chiral perturbation theory for the evaluation of large-distance multiquark contributions. We have given some indication that the trouble could arise from the K l3 determination of V us , and we would like to encourage further experimental work in this field [29]. We are however convinced of the importance of renewed experimental work on hyperon decays, of the kind now in progress at the CERN SPS. The interest of this work goes beyond the determination of V us , as it involves the intricate and elegant relationships that the model predicts. The continuing intellectual stimulation provided by colleagues in the Fermilab KTeV Collaboration, particularly members of the hyperon working group, is gratefully acknowledged. This work was supported in part by the U.S. Department of Energy under grant DE-FG02-90ER40560 (Task B). PACS numbers: 12.15.Hh, 13.30.Ce, 14.20.Jn TABLE I : IResults from Vus analysis using measured g1/f1 → Σ + e − ν 0.876(71) 1.32(+.22/ − .18) 0.209 ± 0.values Decay Rate g1/f1 Vus Process (µsec −1 ) Λ → pe − ν 3.161(58) 0.718(15) 0.2224 ± 0.0034 Σ − → ne − ν 6.88(24) −0.340(17) 0.2282 ± 0.0049 Ξ − → Λe − ν 3.44(19) 0.25(5) 0.2367 ± 0.0099 Ξ 0 027 Combined - - 0.2250 ± 0.0027 . N Cabibbo, Phys. Rev. Lett. 10531N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963) . M Kobayashi, T Maskawa, Prog. Theor. Phys. 49652M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973). . A Alavi-Harati, KTeV CollaborationPhys. Rev. D. 6712005KTeV Collaboration, A. Alavi-Harati et al., Phys. Rev. D 67:012005 (2003) . 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C 25, 91 (1984) With the metric and γ matrix conventions used here, g1/f1 is positive for neutron beta decay. A Garcia, P Kielanowski, Lecture Notes in Physics. 222Springer-VerlagThe Beta Decay of HyperonsA. Garcia and P. Kielanowski, The Beta Decay of Hy- perons, Lecture Notes in Physics 222, Springer-Verlag, Berlin, 1985. With the metric and γ matrix conventions used here, g1/f1 is positive for neutron beta decay. . N Cabibbo, E C Swallow, Winston, Ann. Rev. Nucl. Part. Sci. 53N. Cabibbo, E. C. Swallow, R Winston, Ann. Rev. Nucl. Part. Sci. Dec. 2003, 53: 39-75. . M Ademollo, R Gatto, Phys. Rev. Lett. 13264M. Ademollo and R. Gatto, Phys. Rev. Lett. 13, 264 (1964). . P M Gensini, G Violini, hep-ph/9311270P. M. Gensini and G. Violini , hep-ph/9311270. SU(3) symmetry breaking in hyperon semileptonic decays. R Flores-Mendieta, E Jenkins, A V Manohar, arXiv:hep-ph/9805416Phys. Rev. D. 5894028R. Flores-Mendieta, E. Jenkins and A. V. Manohar, "SU(3) symmetry breaking in hyperon semilep- tonic decays," Phys. Rev. D 58, 094028 (1998) [arXiv:hep-ph/9805416]. . R Flores-Mendieta, A Garcia, G Sanchez-Colon, Phys. Rev. 546855R. Flores-Mendieta, A. Garcia, G. Sanchez-Colon Phys. Rev. D54:6855 (1996); K-M Angles And SU(3) Breaking In Hyperon Beta Decay. J F Donoghue, B R Holstein, S W Klimt, Phys. Rev. D. 35934J. F. Donoghue, B. R. Holstein and S. W. Klimt, "K-M Angles And SU(3) Breaking In Hyperon Beta Decay," Phys. Rev. D 35, 934 (1987). Beta decay of hyperons in a relativistic quark model. F Schlumpf, arXiv:hep-ph/9409272Phys. Rev. D. 512262F. Schlumpf, "Beta decay of hyperons in a relativis- tic quark model," Phys. Rev. D 51, 2262 (1995) [arXiv:hep-ph/9409272]. Baryon Matrix Elements Of The Vector Current In Chiral Perturbation Theory. A Krause, Helv. Phys. Acta. 633A. Krause, "Baryon Matrix Elements Of The Vector Cur- rent In Chiral Perturbation Theory," Helv. Phys. Acta 63, 3 (1990). Chiral corrections to hyperon vector form-factors. J Anderson, M A Luty, arXiv:hep-ph/9301219Phys. Rev. D. 474975J. Anderson and M. A. Luty, "Chiral corrections to hy- peron vector form-factors," Phys. Rev. D 47, 4975 (1993) [arXiv:hep-ph/9301219]. Renormalization Of Weak Form-Factors, And The Cabibbo Angle. H R Quinn, J D Bjorken, Phys. Rev. 1711660H. R. Quinn and J. D. Bjorken, "Renormalization Of Weak Form-Factors, And The Cabibbo Angle," Phys. Rev. 171, 1660 (1968). . J Sexton, D Weingarten, Phys. Rev. 554025J. Sexton, D. Weingarten, Phys. Rev. D55:4025 (1997) We are grateful to Guido Martinelli for a discussion on this point, see. D Becirevic, arXiv:hep-ph/0403217We are grateful to Guido Martinelli for a dis- cussion on this point, see D. Becirevic et al., [arXiv:hep-ph/0403217]. . S Weinberg, Phys. Rev. 1121375S. Weinberg, Phys. Rev. 112, 1375 (1958). Hyperon 99. R Barry, Holstein, Proceedings of the Hyperon Physics Symposium, pgs. 4-9. D.A. Jensen, E. Monnierthe Hyperon Physics Symposium, pgs. 4-9FermilabBarry R. Holstein, "Hyperon 99", Proceedings of the Hy- peron Physics Symposium, pgs. 4-9, Fermilab, September 27-29 1999, D.A. Jensen, E. Monnier, Editors. . J M Gaillard, G Sauvage, Ann. Rev. Nucl. Part. Sci. 34351J. M. Gaillard and G. Sauvage, Ann. Rev. Nucl. Part. Sci. 34, 351 (1984). . S Y Hsueh, Phys. Rev. D. 382056S. Y. Hsueh, et al., Phys. Rev. D 38, 2056 (1988). . R Oehme, R Winston, A Garcia, Phys. Rev. D. 31618R. Oehme, R. Winston, A. Garcia, Phys. Rev. D 3, 1618 (1971). hep-ex/0305042fact, a recently reported result from the Brookhave E865 Collaboration indicates a higher Ke3 decay rate than that previously used to determine Vus. See A. Sher. In fact, a recently reported result from the Brookhave E865 Collaboration indicates a higher Ke3 decay rate than that previously used to determine Vus. See A. Sher, et al., hep-ex/0305042 (June 24, 2003).	[]
[ "Observation of femto-joule optical bistability involving Fano resonances in high-Q/V m silicon photonic crystal nanocavities", "Observation of femto-joule optical bistability involving Fano resonances in high-Q/V m silicon photonic crystal nanocavities" ]	[ "Xiaodong Yang \nOptical Nanostructures Laboratory\nColumbia University\n10027New YorkNY\n\nElectronic\n\n", "Chad Husko \nOptical Nanostructures Laboratory\nColumbia University\n10027New YorkNY\n", "Mingbin Yu \nScience Park II\nThe Institute of Microelectronics\n11 Science Park Road117685SingaporeSingapore, Singapore\n", "Dim-Lee Kwong \nOptical Nanostructures Laboratory\nColumbia University\n10027New YorkNY\n\nScience Park II\nThe Institute of Microelectronics\n11 Science Park Road117685SingaporeSingapore, Singapore\n\nElectronic\n\n", "Chee Wei Wong " ]	[ "Optical Nanostructures Laboratory\nColumbia University\n10027New YorkNY", "Electronic\n", "Optical Nanostructures Laboratory\nColumbia University\n10027New YorkNY", "Science Park II\nThe Institute of Microelectronics\n11 Science Park Road117685SingaporeSingapore, Singapore", "Optical Nanostructures Laboratory\nColumbia University\n10027New YorkNY", "Science Park II\nThe Institute of Microelectronics\n11 Science Park Road117685SingaporeSingapore, Singapore", "Electronic\n" ]	[]	We observe experimentally optical bistability enhanced through Fano interferences in high-Q localized silicon photonic crystal resonances (Q ~ 30,000 and modal volume ~ 0.98 cubic wavelengths). This phenomenon is analyzed through nonlinear coupled-mode formalism, including the interplay of χ (3) effects such as two-photon absorption and related free-carrier dynamics, and optical Kerr as well as thermal effects and linear losses. Our experimental and theoretical results demonstrate Fano-resonance based bistable states with switching thresholds of 185 µW and 4.5 fJ internally stored cavity energy (~ 540 fJ consumed energy) in silicon for scalable optical buffering and logic.	10.1063/1.2757607	[ "https://export.arxiv.org/pdf/physics/0703132v3.pdf" ]	119,380,460	physics/0703132	52b9747e025f46541353f63619ff87b1cda01d8b	Observation of femto-joule optical bistability involving Fano resonances in high-Q/V m silicon photonic crystal nanocavities Xiaodong Yang Optical Nanostructures Laboratory Columbia University 10027New YorkNY Electronic Chad Husko Optical Nanostructures Laboratory Columbia University 10027New YorkNY Mingbin Yu Science Park II The Institute of Microelectronics 11 Science Park Road117685SingaporeSingapore, Singapore Dim-Lee Kwong Optical Nanostructures Laboratory Columbia University 10027New YorkNY Science Park II The Institute of Microelectronics 11 Science Park Road117685SingaporeSingapore, Singapore Electronic Chee Wei Wong Observation of femto-joule optical bistability involving Fano resonances in high-Q/V m silicon photonic crystal nanocavities 1 We observe experimentally optical bistability enhanced through Fano interferences in high-Q localized silicon photonic crystal resonances (Q ~ 30,000 and modal volume ~ 0.98 cubic wavelengths). This phenomenon is analyzed through nonlinear coupled-mode formalism, including the interplay of χ (3) effects such as two-photon absorption and related free-carrier dynamics, and optical Kerr as well as thermal effects and linear losses. Our experimental and theoretical results demonstrate Fano-resonance based bistable states with switching thresholds of 185 µW and 4.5 fJ internally stored cavity energy (~ 540 fJ consumed energy) in silicon for scalable optical buffering and logic. 2 Two-dimensional photonic crystal (2D PhC) slabs confine light by Bragg reflection in-plane and total internal reflection in the third dimension. Introduction of point and line defects into 2D PhC slabs create localized resonant cavities and PhC waveguides respectively, with ab initio arbitrary dispersion control. Such defect cavities in high-index contrast materials possess strong confinement with subwavelength modal volumes (V m ) at ~ (λ/n) 3 , corresponding to high field intensities per photon for increased nonlinear interaction. Moreover, cavities with remarkable high quality factors (Q) [1,2] have been achieved recently, now permitting nanosecond photon lifetimes for enhanced light-matter interactions. The strong localization and long photon lifetimes in these high-Q/V m photonic crystal nanocavities point to enhanced nonlinear optical physics, such as Lorentzian-cavity-based bistability [3][4][5][6], Raman lasing [7,8] and cavity QED [9] in silicon photonics. The interference of a discrete energy state with a continuum can give rise to sharp and asymmetric lineshapes, referred to Fano resonances [10,11]. Compared to a Lorentzian resonance, these lineshapes arise from temporal pathways that involve direct and indirect (for example a resonance) transmission, with a reduced frequency shift required for nonlinear switching due to its sharper lineshape. If the indirect pathway can be further strongly localized (such as in a high-Q/V m 3D cavity instead of a 1D Fabry-Perot cavity), the nonlinear characteristic switching thresholds can be further reduced. Optical bistability involving Fano resonances due to Kerr effect in photonic crystal cavities has been theoretically studied based on Green's function solution of Maxwell's equations [12]. Fano resonances have also been studied by transfer matrix technique [11,13], and coupled-mode equations [14]. In this Letter, we present our measurements on Fano-based optical bistability as well as a temporal nonlinear coupled-mode framework for numerical analysis. Figure 1(a) shows a schematic of the theoretical model. A waveguide with two partially reflecting elements is side-coupled to a cavity. a is the amplitude of the cavity mode which is normalized to represent the energy of the cavity mode U = \|a\| 2 and s is the amplitude of the waveguide mode which is normalized to represent the power of the waveguide mode P = \|s\| 2 . With the coupled-mode formalism [15,16], the dynamic equation for the amplitude a(t) of the resonance mode is [14,17] 3 + + + +         − ∆ + + − = 2 1 0 ) ( 2 1 a a wg total s s a i dt da κ κ ω ω ω τ (1) As shown in Figure 1( + − = + − 1 2 ) exp( . c L n eff wg / ω φ = is the phase shift. κ is the coupling coefficient between the waveguide mode s(t) and a(t), and [17]. For a lossy partially reflecting element with the amplitude reflectivity r and transmissivity t, ( in i i τ φ κ 2 / ) 2 / exp(− =1 2 2 ≤ + t r ) [15]                 − + − =         − + − + j j aj aj s s r r t r it s s 1 ) ( 1 2 2 , j = 1, 2(2) In equation (1), the total loss rate for the resonance mode total τ 1 [3,5,7] is FCA TPA lin v in total τ τ τ τ τ τ 1 1 1 1 1 1 + + + + =(3) where in in Ref. 7), the coupled nonlinear dynamical behavior of the Fano optical system is numerically integrated. The optical system consisting of a photonic crystal waveguide side coupled to a high-Q/V m nanocavity with five linearly aligned missing air holes (L5) in an air-bridge triangular lattice photonic crystal slab with thickness of 0.6a and the radius of air holes is 0.29a, where the lattice period a = 420 nm, as shown in Figure 1(b). The shift S 1 of two air-holes at cavity edge is 0.02a to tune the radiation mode pattern for increasing the Q factors. The waveguide-to-cavity separation is five layers of holes. The index contrast at the waveguide input and output facets act as partially reflecting elements with distance L of around 1.9 mm to form a Fabry-Perot resonator and perturb the phase of waveguide 4 mode. Figure 1(c) shows the E y field of the resonance mode mid-slab from 3D FDTD simulations. The devices were fabricated with the standard integrated circuit techniques in a silicon-on-insulator substrate. A polarization controller and a lensed fiber are used to couple transverse-electric polarization light from tunable laser source into the waveguide. A second lensed fiber collects the transmission from the waveguide output that is sent to the photodetector and lock-in amplifier. The input power coupled to the waveguide is estimated from the transmitted power through the input lensed fiber, waveguide and the output lensed fiber [5]. The total transmission loss of the whole system is around 24.8 dB at wavelength of 1555 nm. At low input power of 20 µW, the measured resonant wavelength λ 0 is 1556.805 nm. To measure the Q factor, the vertical radiation from the top of only L5 nanocavity is collected by a 40X objective lens and a 4X telescope through an iris (spatial filter), which will isolate the cavity region only so that there is no any other influence other than the radiation of the cavity mode. The estimated Q, based on the full-width at half maximum (FWHM) ∆λ of 52 pm is around 30,000. From 3D FDTD method, the vertical Q factor Q v is around 100,000 and the in-plane Q factor Q in is around 45,000 so that the total Q factor Q tot = 1/(1/Q v +1/Q in ) = ~ 31,000. spacing dλ is around 230 pm, which corresponds to the distance between two waveguide facets d = 1.902 mm (d = λ 2 /(2dλn eff ) and effective index of 2.77 from FDTD simulations). As the input power increases, the Fano lineshapes were red-shifted due to two-photon-absorption induced thermo-optic nonlinearities in silicon [3][4][5]. Figure 2 (b) shows the calculated transmission spectrum from nonlinear coupled-mode model with the input powers used in the experiment. All parameters used in calculation are from either reference papers or FDTD results [7]. When the input power is 1 µW or less, the cavity response is in the linear regime. As the input power increases, the Fano lineshapes were red-shifted. should be expected for increasing input power [18][19][20]. Thirdly, the dip in the transmission (as indicted by the dotted red circle in Figure 3 Now we examine parametrically the dependence of the Fano-type bistability against achievable device characteristics, with our developed nonlinear model. Figure 4 summarizes the extensive numerically-calculated effects of normalized-detuning (δ/∆λ), mirror reflectivity r, cavity Q, and the position of cavity resonance on the characteristic threshold power p on , and switching contrast. A baseline Q of 30,000, a r of 0.5 with 11% mirror loss, a λ 0 of 1556.805 nm, and a detuning of δ/∆λ=22pm/52pm=0.423 is used, which correspond to the current experimental parameters and are represented as the redfilled symbols in Figure 4. In Figure 4 (a), for both Fano bistability (solid lines) and Lorentzian bistability (r=0, dashed lines), the threshold power increases for increasing normalized detuning (further normalized shift of incident laser frequency from the cavity resonance) due to the larger shift in resonance needed for bistable switching. The minimum detuning required for Lorentzian bistability is δ/∆λ~0.7, which agrees well with the theoretical threshold detuning δ/∆λ= 2 / 3 [21]. However, the threshold detuning for Fano bistability to appear is δ/∆λ~0.3, which is much smaller than Lorentzian case. The threshold power is similar for both cases (around 140 µW) at the threshold detuning. For both cases, the switching contrast decreases with increasing detuning due to the reduced contrast in the transmission at the higher input powers needed for the bistable operation. Compared to Lorentzian bistability, Fano bistability has higher switching contrast (9.87 dB at threshold detuning) and it decreases more slowly with increased wavelength detuning. The low threshold detuning and high switching contrast for Fano bistability is due to the sharp and asymmetric Fano lineshapes. The inset of Figure 2 (Figure 4(b)). The increase in p on is due to higher mirror reflectivity, resulting in lower power coupled into the Fano system. A limit of 0.35 is used because for smaller r, a combination of both Lorentzian and ascending Fano resonance starts to appear. For r greater than 0.8, the threshold power will go up even higher. dependence. For cavity Q factor of half a million, the Fano threshold power is estimated at 2.4 µW, which corresponds to the stored cavity energy of 0.55 fJ. This stored cavity energy is much lower than a Fano resonance with cavity Q of 30,000 (4.5 fJ). The comparison between Fano bistability and Lorentzian bistability at threshold detuning shows that Fano system has similar threshold power and stored cavity energy as Lorentzian system. We also note that direct comparisons between an ascending Fano-type bistability and a Lorentzian-type bistability in term of threshold power are difficult because the Fano system has different threshold detuning from Lorentzian system and Fano system also depends on additional parameters such as mirror reflectivity r. Perot background maximum to its minimum. The switching contrast has a maximum at a region close to the minimum Fabry-Perot background, illustrating an interesting trade-off 8 when selecting an optimum set of Fano-type bistable operating parameters. The Fano resonance has little control in the current configuration, which depends on the Fabry-Perot cavity formed by two waveguide facets. For different devices, either ascending or descending Fano resonance can be obtained, depending on the position of cavity resonance in the Fabry-Perot background. In the future, integrated partially reflecting mirrors embedded inside photonic crystal waveguide can be adopted [22]. By tuning the distance between the air holes, the Fabry-Perot background can be tuned to achieve the designed Fano lineshapes. In this work we demonstrate experimentally all-optical bistability arising from sharp Fano resonances in high-Q/V m silicon photonic crystal nanocavities. Using the twophoton-absorption induced thermo-optic nonlinearity, an "on"-state threshold of 189 µW and stored cavity energy of 4.5 fJ is observed, and in good agreement with the nonlinear coupled-mode formalism. Although the thermo-optic is slow (on order of µs), other nonlinear mechanisms such as two-photon-absorption induced free-carrier dispersion [3,4,6] can remarkably achieve ~ 50 ps switching in silicon. The threshold power can be further reduced to the µW level (or sub-fJ of stored cavity energy) with higher-Q/V m nanocavities or further optimization of the detuning for reduced threshold and large contrast ratio. Our observations of Fano-type bistability highlight the feasibility of an ultra-low energy and high contrast switching mechanism in monolithic silicon benefiting from the sharp Fano lineshapes, for scalable functionalities such as all-optical switching, memory, and logic for information processing. This work was partially supported by DARPA and the National Science Foundation (ECCS-0622069). X. Yang acknowledges the support of an Intel Fellowship. (c) E y -field of the resonance mode mid-slab from 3D FDTD simulations. Figure captions since operation is within the bandgap of the silicon material. loss rates due to two-photon absorption (TPA) and free-carrier absorption (FCA) respectively. The ω ∆ detuning of the cavity resonance from 0 ω is modeled due to the Kerr effect, free-carrier dispersion (FCD), and thermal dispersion effects under firstshifted resonant frequency of the cavity and wg ω is the input light frequency in the waveguide. With the modeled TPA generated carrier dynamics and thermal transients due to total absorbed optical power (Eq.(31) and Eq.(42) Figure 2 ( 2a) shows the measured transmission spectrum of the waveguide with different input powers. Each transmission shown is repeated over multiple scans. Sharp and asymmetric Fano lineshapes are observed. The spectral lineshapes depend on the position of cavity resonance in a Fabry-Perot background, highlighting Fano interference pathways. Here the spectra show ascending Fano resonances. The Fabry-Perot fringe 5 Figure 3 53(a) shows the observed hysteresis loop of Fano resonance at red detuning δ of 22 pm (δ/∆λ=0.423). For comparison, the inset of Figure 3(a) shows the measured hysteresis loop for Lorentzian resonance at the detuning of 25 pm for a L5 nanocavity, where the resonance wavelength is λ 0 = 1535.95 nm, and Q ~ 80,000, the detuning δ/∆λ The bistable loops of ascending Fano lineshapes are very distinct from Lorentzian lineshapes. Firstly, one suggestive indication is the asymmetry in the hysteresis loop, with sharp increase (gentle decrease) with increasing (decreasing) power for lower (upper) branch, resulting from the asymmetric Fano lineshape. Secondly, for ascending Fano resonances, an important indication is the upward slope (increase in transmission) for increasing input power for a side-coupled cavity. For a symmetric Lorentzian in a side-coupled drop cavity, a downward slope (or decrease in transmission) (a)) is another signature of the Fano resonance. This feature is not observable with a symmetric Lorentzian and in fact is an aggregate result of the three self-consistent solutions of the nonlinear Fano system, such as predicted using Green's function method in Ref. 12. Our nonlinear coupled-mode theory framework cannot trace out the individual solutions [12] but show the aggregate behavior, and is in remarkable agreement with our experimental measurements and the Green's function predictions. The Fano bistable "off" power (p off ) is estimated at 147 µW and the "on" power (p on ) at 189 µW for a 22 pm detuning, as shown in Figure 3(a). These threshold powers are determined experimentally from half the total system transmission losses. From the 189 µW (147 µW) p on (p off ) thresholds, this corresponds to an estimated internally stored cavity energy [3] of 4.5 fJ (1.5 fJ) based on a numerical estimate of waveguide-to-cavity coupling coefficient (κ 2 ) of 13.3 GHz. The consumed energy, in terms of definition used in Ref. 19, is ~ 540 fJ (60 fJ) based on the numerical estimated thermal relaxation time of 25 ns and 11.4 % (1.6 %) of input power absorbed by TPA process for "on" ("off") state,although this could be much lower with minimum detuning to observe bistability. The femto-joule level switching in the stored cavity energy is due to the lowered threshold from the sharp Fano interference lineshape, the small mode volume and high-Q photonic crystal cavities. For the 22 pm detuning, the switching intensity contrast ratio is estimated 6 at 8.5 dB (from the regions with sharp discrete bistable "jumps") with a p on /p off ratio of 1.286.Figure 3(b)shows the calculated Fano bistable hysteresis at the detuning of 22 pm from nonlinear coupled-mode theory. The calculated p off and p on thresholds are 151 µW (with the stored cavity energy of 1.5 fJ) and 186 µW (4.5 fJ) respectively, with a switching contrast of 9.3 dB and p on /p off ratio of 1.232, in excellent agreement with the experimental results. (b) plots the transmission spectrums of Fano lineshapes and Lorentzian lineshapes with input power of 1 µW and 230 µW respectively. The sharp transition right before the cavity resonance makes Fano bistability to occur with much lower wavelength detuning, while Lorentzian bistability needs a larger detuning to reach the multivalue transmission regime. For the 7 detuning shown in the figure, the switching contrast of Fano bistability (marked as a red arrow) is higher than Lorentzian bistability (marked as a blue arrow). There is significant difference in term of wavelength detuning required for bistability to occur and the bistable switching contrast between Fano bistability and Lorentzian bistability. This switching contrast can significantly increase when the mirror reflectivity r increases from 0.35 to 0.8 (at detuning of δ/∆λ = 0.423) at the expense of increasing p on Figure 4 ( 4c) plots the threshold power and the stored cavity energy with different cavity Q factors at detuning of δ/∆λ=0.423 and r=0.5. Note that p on shows a (1/Q 1.569 )dependence, while the stored cavity energy needed for bistable shows a (1/Q 0.689 )- Figure 4 ( 4d) illustrates the influence of different position of cavity resonance λ relative to experimental λ 0 within the half period of Fabry-Perot background dλ/2 with Q=30,000, r=0.5, and δ/∆λ=0.423, where the limits of (λ-λ 0 )/(dλ/2) are from -0.8 to 0.2 for ascending Fano resonances. These limits are chosen because they cover the region where the ascending Fano resonances are dominant over the Lorentzian or the descending Fano resonances. For ascending Fano resonances at detuning of δ/∆λ=0.423, both threshold power and switching contrast increase as the cavity resonance λ shifts from the Fabry- Fig. 1 . 1(a) Schematic of optical system including a waveguide side-coupled to a cavity.Two partially reflecting elements are placed in the waveguide. (b) SEM of photonic crystal L5 point-defect cavity side-coupled to line-defect waveguide. The input and output facets of the high-index-contrast waveguide form the partially reflecting elements. Fig. 2 . 2Measured (a) and CMT-calculated (b) transmission spectrum at different input powers, illustrating the asymmetric lineshapes. The side-coupled L5 cavity has a total Q of ~ 30,000. The inset of (b) plots the calculated transmission spectrums of Fano lineshapes (red) and Lorentzian lineshapes (blue) with input power of 1 µW (solid line) and 230 µW (dashed line) respectively. Fig. 3 . 3(a) Measured and (b) CMT-calculated asymmetric hysteresis loops for Fano resonance at a detuning of 22 pm. The red-circled region in panel (a) highlights a dip in transmission with increasing input power, a signature not present in Lorentzian-type resonances, and indicative of nonlinear Fano-type solutions. The inset of (a) shows the measured hysteresis loop for Lorentzian resonance. The arrows depict ascending and descending input powers to the Fano system. Fig. 4 .Fig. 2 .Fig. 3 .Fig. 4 . 4234CMT calculated effects of (a) the wavelength detuning δ/∆λ. Dashed lines(--) represent Lorentzian bistability which, compared to this particular Fano-type bistability, requires higher detuning to observe bistability and results in lower switching contrast.Effects of (b) mirror reflectivity r, (c) cavity Q factor, and (d) the position of cavity resonance on the switching threshold power p on and switching contrast. (The red-filled symbols correspond to the experimental parameters.) 12 Fig. 1. (a) Schematic of optical system including a waveguide side-coupled to a cavity. Two partially reflecting elements are placed in the waveguide. (b) SEM of photonic crystal L5 point-defect cavity side-coupled to line-defect waveguide. The input and output facets of the high-index-contrast waveguide form the partially reflecting elements. (c) E y -field of the resonance mode mid-slab from 3D FDTD simulations. Measured (a) and CMT-calculated (b) transmission spectrum at different input powers, illustrating the asymmetric lineshapes. The side-coupled L5 cavity has a total Q of ~ 30,000. The inset of (b) plots the calculated transmission spectrums of Fano lineshapes (red) and Lorentzian lineshapes (blue) with input power of 1 µW (solid line) and 230 µW (dashed line) respectively. (a) Measured and (b) CMT-calculated asymmetric hysteresis loops for Fano resonance at a detuning of 22 pm. The red-circled region in panel (a) highlights a dip in transmission with increasing input power, a signature not present in Lorentzian-type resonances, and indicative of nonlinear Fano-type solutions. The inset of (a) shows the measured hysteresis loop for Lorentzian resonance. The arrows depict ascending and descending input powers to the Fano system. CMT calculated effects of (a) the wavelength detuning δ/∆λ. Dashed lines (--) represent Lorentzian bistability which, compared to this particular Fano-type bistability, requires higher detuning to observe bistability and results in lower switching contrast. Effects of (b) mirror reflectivity r, (c) cavity Q factor, and (d) the position of cavity resonance on the switching threshold power p on and switching contrast. (The red-filled symbols correspond to the experimental parameters.) . B S Song, S Noda, T Asano, Y Akahane, Nat. Mater. 4207B. S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005). . 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[ "The electromagnetic Sigma-to-Lambda transition form factors with coupled-channel effects in the space-like region", "The electromagnetic Sigma-to-Lambda transition form factors with coupled-channel effects in the space-like region" ]	[ "Yong-Hui Lin \nHelmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n", "Hans-Werner Hammer \nDepartment of Physics\nInstitut für Kernphysik\nTechnische Universität Darmstadt\n64289DarmstadtGermany\n\nExtreMe Matter Institute EMMI and Helmholtz Forschungsakademie Hessen für FAIR (HFHF)\nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "Ulf-G Meißner \nHelmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n\nInstitut für Kernphysik\nInstitute for Advanced Simulation and Jülich Center for Hadron Physics\nForschungszentrum Jülich\nD-52425JülichGermany\n\nTbilisi State University\n0186TbilisiGeorgia\n" ]	[ "Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany", "Department of Physics\nInstitut für Kernphysik\nTechnische Universität Darmstadt\n64289DarmstadtGermany", "ExtreMe Matter Institute EMMI and Helmholtz Forschungsakademie Hessen für FAIR (HFHF)\nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany", "Institut für Kernphysik\nInstitute for Advanced Simulation and Jülich Center for Hadron Physics\nForschungszentrum Jülich\nD-52425JülichGermany", "Tbilisi State University\n0186TbilisiGeorgia" ]	[]	Using dispersion theory, the electromagnetic Sigma-to-Lambda transition form factors are expressed as the product of the pion electromagnetic form factor and the ΣΛ → ππ scattering amplitudes with the latter estimated from SU(3) chiral perturbation theory including the baryon decuplet as explicit degrees of freedom. The contribution of the KK channel is also taken into account and the ππ-KK coupledchannel effect is included by means of a two-channel Muskhelishvili-Omnès representation. It is found that the electric transition form factor shows a significant shift after the inclusion of the KK channel, while the magnetic transition form factor is only weakly affected. However, the KK effect on the electric form factor is obscured by the undetermined coupling hA in the three-flavor chiral Lagrangian. The error bands of the Sigma-to-Lambda transition form factors from the uncertainties of the couplings and low-energy constant in three-flavor chiral perturbation theory are estimated by a bootstrap sampling method.	10.1140/epja/s10050-023-00973-1	[ "https://export.arxiv.org/pdf/2205.00850v2.pdf" ]	248,496,021	2205.00850	8461c98d49fd8b67dc4a09568ef2250e0aac22e8	The electromagnetic Sigma-to-Lambda transition form factors with coupled-channel effects in the space-like region Yong-Hui Lin Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany Hans-Werner Hammer Department of Physics Institut für Kernphysik Technische Universität Darmstadt 64289DarmstadtGermany ExtreMe Matter Institute EMMI and Helmholtz Forschungsakademie Hessen für FAIR (HFHF) GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany Ulf-G Meißner Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany Institut für Kernphysik Institute for Advanced Simulation and Jülich Center for Hadron Physics Forschungszentrum Jülich D-52425JülichGermany Tbilisi State University 0186TbilisiGeorgia The electromagnetic Sigma-to-Lambda transition form factors with coupled-channel effects in the space-like region Received: date / Revised version: dateEPJ manuscript No. (will be inserted by the editor)1340Gp1155Fv1375Gx1130Rd Using dispersion theory, the electromagnetic Sigma-to-Lambda transition form factors are expressed as the product of the pion electromagnetic form factor and the ΣΛ → ππ scattering amplitudes with the latter estimated from SU(3) chiral perturbation theory including the baryon decuplet as explicit degrees of freedom. The contribution of the KK channel is also taken into account and the ππ-KK coupledchannel effect is included by means of a two-channel Muskhelishvili-Omnès representation. It is found that the electric transition form factor shows a significant shift after the inclusion of the KK channel, while the magnetic transition form factor is only weakly affected. However, the KK effect on the electric form factor is obscured by the undetermined coupling hA in the three-flavor chiral Lagrangian. The error bands of the Sigma-to-Lambda transition form factors from the uncertainties of the couplings and low-energy constant in three-flavor chiral perturbation theory are estimated by a bootstrap sampling method. Introduction Electromagnetic form factors (EMFFs) give access to the strong interaction, which provides one of the most notorious challenges in the Standard Model due to the nonperturbative nature of Quantum Chromodynamics (QCD) at the low energy scale. On the one hand, the EMFFs can be extracted from a variety of experimental processes, such as lepton-hadron scattering, lepton-antilepton annihilation or radiative hadron decays. These EMFFs can be measured over a large energy range. On the other hand, dispersion theory, which is a powerful nonperturbative approach, allows for a theoretical description of the EMFFs. Consequently, the EMFFs are an ideal bridge between experimental measurements and theoretical studies of the low-energy strong interaction. In the last decade, much research effort both in experiment and theory was focused on the nucleon EMFFs, largely triggered by the so-called proton radius puzzle [1]. For recent reviews, see e.g. Refs. [2,3,4,5,6]. In the process of unravelling this puzzle, dispersion theory has played and is playing a crucial role in the theoretical description of the nucleon EMFFs [7,8,9,10]. The dispersion theo-retical parametrization of the nucleon EMFFs, first proposed in the early works [11,12,13] and further developed in Refs. [14,15,9], incorporates all constraints from unitarity, analyticity, and crossing symmetry, as well as the constraints on the asymptotic behavior of the form factors from perturbative QCD [16]. The state of the art of dispersive analyses of the nucleon EMFFs is reviewed in Ref. [17]. Very recently, all current measurements on electron-proton scattering, electron-positron annihilation, muonic hydrogen spectroscopy, and polarization measurements from Jefferson Laboratory could be consistently described in a dispersion theoretical analysis of the nucleon EMFFs [18]. The dispersive prescription of parameterizing the nucleon EMFFs can also be applied to other hadron states. The first two straightforward extensions concern the Delta baryon and the hyperon states, with the former obtained by flipping the spin of one of the quarks inside the nucleon and the latter by replacing one or several up or down quarks with one or more strange quarks. The EMFFs of the Delta and the hyperons provide complementary information about the intrinsic structure of the nucleon [19]. The electromagnetic properties of the Delta baryon have been studied in detail in Ref. [20]. Recent investigations of the hyperon EM structure are given in Refs. [21,22,19,23,24,25,26]. Ref. [19] considered once-subtracted dispersion relations for the electromagnetic Sigma-to-Lambda transition form factors (TFFs) and expressed these in terms of the pion EMFF and the two-pion-Sigma-Lambda scattering amplitudes. Using an Omnès representation, the pion EMFF could be expressed as the Omnès function of the pion P -wave phase shift which has been well determined from the Roy-type analyses of the pion-pion scattering amplitude [27]. An improved parameterization of the pion EMFF is also available, which includes further inelasticities and is applicable at higher energies [28]. Moreover, the two-pion-Sigma-Lambda scattering amplitudes could be calculated in a model-independent way by using three-flavor chiral perturbation theory (ChPT) [21]. Combining these studies and taking some reasonable values for couplings and the low-energy constants in three-flavor ChPT, the electromagnetic Sigma-to-Lambda TFFs were predicted in Ref. [19] where the pion rescattering and the role of the explicit inclusion of the decuplet baryons in three-flavor ChPT were also investigated. In the present work, we extend the theoretical framework used in Ref. [19] to explore the effect of the KK inelasticity on the electromagnetic Sigma-to-Lambda transition form factors. This is performed by considering the two-channel Muskhelishvili-Omnès representation when introducing the pion rescattering effects. In principle, one should include even more inelasticities when implementing the dispersion theoretical parameterization for the Sigmato-Lambda TFFs, as done in our previous work on the nucleon EMFFs [18]. However, it is difficult in the current case due to the poor data base which is required when constructing reliable inelasticities in the higher energy region, that is above the KK threshold at ∼ 1 GeV. Note that the four-pion channel has negligible effects in the energy region around 1 GeV, see Ref. [29]. It is also known that the contribution of the four-pion channel to the pion and kaon form factors below 1 GeV is a three-loop effect in ChPT [30] and thus is heavily suppressed. We remark that the 4π channel was shown to play an important role starting from 1.4 GeV in the S-wave case [31]. This is caused by the presence of the nearby scalar resonances f 0 (1370) and f 0 (1500) which were both observed to have a sizable coupling to four-pion states [32,33,34]. There is no evidence, however, for the presence of corresponding 1 − isovector states in the energy region of 1...2 GeV in the P -wave case. Moreover, another experimental finding of these references is that the 4π system likes to cluster into two resonances in the energy region above 1 GeV [31]. The lowest candidate is supposed to be the ρρ channel for the P -wave isovector problem. From the phenomenological point of view, the inelasticity around 1 GeV should be saturated to a good approximation by the ππ and KK coupled-channel treatments in the Pwave case. Moreover, as we will show later, the effect of KK inelasticity is small. Thus, the relative ratio between the effects of the KK and four-pion channel could be enhanced. To investigate this relative ratio, a sophisticated calculation on the four-pion inelasticity is needed which goes beyond the present work. In the present work, the KK inelasticity is implemented using SU(3) ChPT. The inclusion of the KK channel allows one to construct the Sigma-to-Lambda transition form factors up to 1 GeV precisely. In addition, the estimation of the theoretical uncertainties is improved by using the bootstrap approach [35]. The paper is organized as follows: In Sect. 2 we introduce the dispersion theoretical description of the electromagnetic Sigma-to-Lambda transition form factors and present the coupled-channel Muskhelishvili-Omnès representation for the inclusion of the KK inelasticity. Numerical results are collected in Sect. 3. The paper closes with a summary. Some technicalities are relegated to the appendix. Formalism Here, we discuss the basic formalism underlying our calculations. We first write down once-subtracted dispersion relations for the electric and magnetic Sigma-to-Lambda transition form factor and then discuss in detail their various ingredients, namely the vector form factor of the pion and the kaon and the amplitudes for Σ 0Λ → ππ and Σ 0Λ → KK, in order. Dispersion relations for the Sigma-to-Lambda TFFs The electromagnetic Sigma-to-Lambda TFFs are defined as in Refs. [21,19], Σ 0 (p )\|j µ \|Λ(p) = eū(p ) γ µ + m Λ − m Σ 0 t q µ F 1 (t) + iσ µν q ν m Λ + m Σ 0 F 2 (t) u(p) ,(1) with t = (p − p) 2 = q 2 the four-momentum transfer squared. The scalar functions F 1 (t) and F 2 (t) are called the Dirac and Pauli transition form factors, respectively. One also writes the electric and magnetic Sachs transition form factors, given by the following linear combinations, G E (t) = F 1 (t) + t (m Λ + m Σ 0 ) 2 F 2 (t), G M (t) = F 1 (t) + F 2 (t),(2) with the normalizations F 1 (0) = G E (0) = 0 and F 2 (0) = G M (0) = κ ≈ 1.98. Here, κ is estimated from the experimental width of the decay Σ 0 → Λγ, see Ref. [19] for details. Unlike the nucleon case where one constructs dispersion relations for F 1 and F 2 [17], we work with the electric and magnetic Sachs form factors, i.e. G E and G M , for the Sigma-to-Lambda TFFs as in Ref. [19] since the Sigma-to-Lambda TFFs are of pure isovector type and the helicity decomposition used in Ref. [19] can easier be applied to the Sachs FFs. In order to apply the spectral decomposition to estimate the imaginary part Im G E/M , we consider the matrix element of the electromagnetic current Eq. (1) in the time-like region (t > 0), which is obtained via crossing symmetry, 0\|j µ \|Σ 0 (p 3 )Λ(p 4 ) = ev(p 4 ) γ µ + m Λ − m Σ 0 t (p 3 + p 4 ) µ F 1 (t) − iσ µν (p 3 + p 4 ) ν m Λ + m Σ 0 F 2 (t) u(p 3 )(3) where p 3 and p 4 are the momenta of the Σ 0 andΛ created by the electromagnetic current, respectively. The fourmomentum transfer squared in the time-like region is then t = (p 3 + p 4 ) 2 . With the ππ and KK inelasticities taken into account as depicted in Fig. 1, the unitarity relations for the Sigma-to-Lambda TFFs read [36,37,19,24], 1 2i disc unit G E/M (t) = 1 12π √ t × q 3 π (t) F V π (t) * T ππ E/M (t) θ t − 4M 2 π + 2q 3 K (t) F V K (t) * T KK E/M (t) θ t − 4M 2 K ,(4) where q π/K (t) = λ(M 2 π/K , M 2 π/K , t) 4t(5) is the center-of-mass momentum of the ππ/KK two-body continuum with λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + yz + zx) the Källén function. F V π/K (t) is the vector-isovector form factor (J = I = 1) of the pion/kaon and T ππ E (t) and T ππ M (t) are two independent reduced P -wave Σ 0Λ → π + π − amplitudes in the helicity basis. Similarly, T KK E (t) and T KK M (t) denote the corresponding reduced amplitudes for Σ 0Λ → K + K − (K 0K 0 ). Then the once-subtracted dispersion relations for the Sigma-to-Lambda TFFs are written as, G E/M (t) = G E/M (0) + t 2πi ∞ 4M 2 π dt disc unit G E/M (t ) t (t − t − i ) + G anom E/M (t) ,(6) where the last term G anom E/M (t) denotes the contribution of the anomalous cut which is non-zero when there exists an anomalous threshold in the involved processes [38,39,40,41,24]. This does happen when the KK channel is taken into account, see Appendix A for detailed discussions and the explicit expressions of the anomalous part. Next, we need to consider the various factors contributing to Eq. (6). 2.2 The Σ 0Λ -ππ and Σ 0Λ -KK amplitudes in the two-channel Muskhelishvili-Omnès representation We start with the four-point amplitudes Σ 0Λ → π + π − and Σ 0Λ → K + K − . Note that the matrix element Eq. (3), and also the four-point function Σ 0Λ → ππ, can be written in the general formv Λ (−p z , λ)Γ u Σ 0 (p z , σ) when one works in the center-of-mass frame and chooses the z-axis along the direction of motion of the Σ 0 . Here, σ and λ are the helicities of the Σ 0 andΛ baryons, respectively. Due to parity invariance, there are only two non-vanishing terms, σ = λ = +1/2 and σ = −λ = +1/2. Concerning the matrix element Eq. (3), all components except for µ = 3 vanish in the case of σ = λ = +1/2, that is, 0\|j 3 \|Σ 0 (p z , 1 2 )Λ(−p z ,1 2 ) =v Λ (−p z , +1/2)γ 3 u Σ (p z , +1/2)G E (t). (7) For σ = −λ = +1/2, only components related to µ = 1, 2 survive: 0\|j 1 \|Σ 0 (p z , 1 2 )Λ(−p z , − 1 2 ) =v Λ (−p z , −1/2)γ 1 u Σ (p z , +1/2)G M (t),(8) and the matrix element for µ = 2 differs from µ = 1 only by a factor i. Eqs. (7) and (8) show that T E in the imaginary part of G E is only related to the amplitude component M ΣΛ→ππ/KK (+1/2, +1/2) while T M comes from M ΣΛ→ππ/KK (+1/2, −1/2). Then we define the reduced amplitudes T E/M as [42,19] M ΣΛ→ππ/KK (t, θ, +1/2, +1/2) = v Λ (−p z , +1/2)γ 3 u Σ (p z , +1/2)q π/K T π/K E (t)d 1 0,0 (θ) + other partial waves with J = 1 ,(9)M ΣΛ→ππ/KK (t, θ, +1/2, −1/2) = − √ 2v Λ (−p z , −1/2)γ 1 u Σ (p z , +1/2)q π/K T π/K M (t)d 1 1,0 (θ) + other partial waves with J = 1 ,(10) where with M ΣΛ→ππ/KK (t, θ) = M pole + M contact . The magnetic parts are derived equivalently from Eq. (12), d 1 1/2±1/2,0 (θ) is the Wigner d-matrix. Finally, we obtain T π/K √ 2KK θ(t − 4M 2 K ) T is implicit in the integrand [Im Ω −1 (t )] K(t ).K M (t) = 3 4 π 0 dθ sin θ M pole (t, θ, +1/2, −1/2) v Λ (−p z , −1/2) γ 1 u Σ (p z , +1/2) q π/K sin θ ,(16)P M 0 (t) = 3 4 π 0 dθ sin θ M contact (t, θ, +1/2, −1/2) v Λ (−p z , −1/2) γ 1 u Σ (p z , +1/2) q π/K sin θ .(17) All the reduced amplitudes K π/K and P 0,π/K are calculated up to next-to-leading order (NLO) within the framework of the three-flavor baryon ChPT that includes the decuplet baryon as explicit degrees of freedom. Their explicit expressions are derived in detail in Appendix B. The P -wave Omnès matrix and π, K vector form factors In this subsection, we derive the P -wave isovector Omnès matrix and solve for the pion and kaon EMFFs in the coupled-channel formalism. Ω satisfies the unitarity rela- tion [46] 1 2i disc Ω ij = (t 1 1 ) * im Σ m Ω mj ,(18) where Σ(t) = diag σ π q 2 π θ(t − 4M 2 π ), σ K q 2 K θ(t − 4M 2 K )(19) with σ π/K (t) = 1 − 4M 2 π/K t(20) is the diagonal phase space matrix. The J = I = 1 ππ-KK coupled-channel T -matrix t 1 1 is parameterized as t 1 1 =     ηe 2iδ 1 1 − 1 2iσ π q 2 π ge iψ ge iψ ηe 2i(ψ−δ 1 1 ) − 1 2iσ K q 2 K     .(21) where g and ψ are the modulus and phase of the P -wave isovector ππ → KK scattering amplitude, respectively. The inelasticity η is defined by η(t) = 1 − 4σ π σ K (q π q K ) 2 g 2 θ(t − 4M 2 K ).(22) Then we can write the dispersion relation for the Omnès matrix Ω as Ω ij (t) = 1 2πi ∞ 4M 2 π dz disc Ω ij (z) z − t − i .(23) The analytic solution of the integral equation Eq. (23) was given in Ref. [47] for the single-channel problem. However, there are no known analytic solutions for two or more channel cases where one has to construct the solutions numerically, either by an iterative procedure [48] or a discretization method [31]. Here, we adopt the iterative approach to solve the P -wave ππ-KK coupled-channel Omnès matrix. Substituting Eq. (18) into Eq. (23), one obtains a two-dimensional system of integral equations        Re χ 1 (t) = 1 π P ∞ 4M 2 π dz Imχ 1 (z) z − t , Re χ 2 (t) = 1 π P ∞ 4M 2 π dz Imχ 2 (z) z − t ,(24) where Imχ 1 (z) = Re (t 1 1 ) * 11 Σ 1 χ 1 + Re (t 1 1 ) * 12 Σ 2 χ 2 , Imχ 2 (z) = Re (t 1 1 ) * 21 Σ 1 χ 1 + Re (t 1 1 ) * 22 Σ 2 χ 2 ,(25) and P denotes the principal value. Searching for solutions of Ω(t) is equivalent to searching for two independent solutions of the integral equation set for the two-dimensional array (χ 1 , χ 2 ) T . Using the iterative procedure, one can obtain a series of solutions (χ λ 1 , χ λ 2 ) T starting with various initial inputs χ 1 (t) = 1, χ 2 (t) = λ, where λ is a real parameter. Note that the iterative process is linear and the results of the iteration is therefore a linear function of λ [48]. Then the solution family {(χ λ 1 , χ λ 2 ) T } contains only two linearly independent members. Here, we take the same convention as Ref. [46] to construct two independent solutions, (Ω 11 , Ω 21 ) T and (Ω 12 , Ω 22 ) T , that satisfy the normalizations Ω 11 (0) = Ω 22 (0) = 1 and Ω 12 (0) = Ω 21 (0) = 0 , from two arbitrary solutions (χ λ1 1 , χ λ1 2 ) T and (χ λ2 1 , χ λ2 2 ) T . With the two-channel Muskhelishvili-Omnès representation, the binary function composed of the vector FFs of the pion and the kaon fulfills the same unitarity relation Eq. (18). Then one can solve the pion and kaon vector form factors F V π (t) √ 2F V K (t) = Ω 11 (t) Ω 12 (t) Ω 21 (t) Ω 22 (t) F V π (0) √ 2F V K (0) ,(26) which are normalized as F V π (0) = 1 and F V K (0) = 1/2. To solve the J = I = 1 ππ-KK Omnès matrix, the required input is the P -wave isovector ππ-KK scattering matrix t 1 1 , i.e. Eq. (21), that is constructed from the ππ Pwave isovector phase shift δ 1 1 , the modulus g and phase ψ of the P -wave isovector ππ → KK amplitude. The phase shift δ 1 1 up to 1.4 GeV was extracted precisely from the Roy-type analyses of the pion-pion scattering amplitude in Ref. [27]. We take the same prescription as in Ref. [28] to extrapolate it smoothly to reach π at infinity. Then δ 1 1 (t) is given by δ 1 1 (t) =          0, 0 ≤ √ t ≤ 2M π , δ f1 (t), 2M π < √ t ≤ 2M K , δ f2 (t), 2M K < √ t ≤ √ t 0 , δ f3 (t), √ t 0 < √ t,(27) where δ f1 (t) = cot −1 √ t 2q 3 π (M 2 ρ − t) 2M 3 π M 2 ρ √ t + 1.043 + 0.19 √ t − √ t 1 − t √ t + √ t 1 − t , δ f2 (t) = δ f1 (4M 2 K ) + 1.39 √ t 2M K − 1 − 1.7 √ t 2M K − 1 2 , δ f3 (t) = π + (δ f2 (t 0 ) − π) t 2 + t 0 t 2 + t .(28) Here, t 0 = (1.4 GeV) 2 , t 1 = (1.05 GeV) 2 and t 2 = (10 GeV) 2 . The P -wave ππ → KK amplitude up to √ t 3 = 1.57 GeV is taken from Ref. [49] where the modulus g in the region of 4M 2 π ...4M 2 K was solved from the Roy-Steiner equation with the experimental data of P -wave ππ → KK scattering [50,51] above the KK threshold as input, while the phase ψ was fitted to experimental data [50,51]. Note that the two-channel Muskhelishvili-Omnès representation in terms of ππ and KK intermediate states should only work well in the lower energy region [46]. Further, the asymptotic values of phase shifts in the coupled-channel systems have to satisfy lim t→∞ δ I l (t) ≥ nπ,(29) to ensure that the system of integral equations, Eq. (24), has a unique solution [31,52]. n is the number of channels that are considered in the formalism. It requires ψ = δ 1 1,ππ + δ 1 1,KK ≥ 2π in Eq. (21). g and ψ are extrapolated smoothly to 0 and 2π by means of [31] ψ(t) = 2π + (ψ(t 4 ) − 2π)f t t 4 , g(t) = g(t 3 )f t t 3 , withf (x) = 2 1 + x 3/2 .(30) where the extrapolation point t 4 of ψ should be far away from 1.5 GeV since there is a structure located around 1.5 GeV in the phase of the P -wave ππ → KK amplitude. Here we take the value √ t 4 = 5 GeV for ψ. Such a structure should also leave trails in the modulus g. However, only g up to √ 2 GeV is estimated in Ref. [49] and a small bump around 1.5 GeV in g is only reflected roughly by several data points above 1.4 GeV measured by Ref. [50], see Fig. 9 in Ref. [49]. The modulus used in our work is presented in Fig. 3, while the δ 1 1 and ψ are presented when we show the solved pion and kaon vector form factors. The obtained Ω matrix elements are presented in Fig. 4. The pion and kaon vector form factors calculated from Eq. (26) are then given in Fig. 5. Clearly, one can see from Fig. 5 that the phase of F V π and F V K are consistent with the input ππ phase shift δ 1 1 and the phase ψ of the Pwave ππ → KK scattering amplitude respectively, which is similar with the finding for the S-wave case by Ref. [46]. Results Using the reduced amplitudes P [21]. In SU(3) ChPT, F Φ can take three different values at LO, namely F π = 92.4 MeV, F K = 113.0 MeV and F η = 120.1 MeV [53]. Often, one chooses the average of these, that is, F Φ = (F π + F K + F η )/3. Here, we take F Φ = 100 ± 10 MeV to cover mainly the π and K contributions. h A can be determined from the experimental widths of either Σ * → Λπ or Σ * → Σπ. We take the value h A = 2.3 ± 0.3 [19], here an additional 10% error is added to account for the SU(3) flavor symmetry breaking effect when applied to the vertices involving a Ξ * . The low-energy constant b 10 was estimated in Ref. [54] based on the resonance saturation hypothesis as b 10 = 0.95 GeV −1 . A larger value b 10 = 1.24 GeV −1 is used in Ref. [21]. A very recent determination based on the ChPT fits to lattice data of the axial-vector currents of the octet baryons gives b 10 = 0.76 GeV −1 [55]. Taking all these determinations into account, b 10 = (1.0±0.3) GeV −1 is used here. Second, we introduce an energy cutoff Λ in the integration along the unitarity cut in Eq. (6) and Eq. (13). We consider two values for the cutoff, Λ = 1.5 and 2.0 GeV, to check the sensitivity of our results to it. Now we are in the position to present our numerical results for the electromagnetic Sigma-to-Lambda transition form factors. First, we present the electric transition form factor G E obtained with the radius-adjusted parameters given in Ref. [19], i.e. F Φ = 100 MeV, b 10 = 1.06 GeV −1 and h A = 2.22 where the radius is adjusted to the fourthorder ChPT result from Ref. [21], in Fig. 6. Note that Λ = 1.5 GeV is used in these calculations. The result from the single ππ channel consideration is also plotted for an intuitive comparison. Taking the same parameter values, we find good agreement with Ref. [19]. After the inclusion of the KK inelasticity, a logarithmic singularity located at the anomalous threshold t − = 0.935 GeV in the unphysical area of the time-like region is introduced into the TFF G E . Moreover, additional nonzero imaginary parts along the anomalous cut are produced for the TFFs by Eq. (42) and Eq. (43). This is similar to the triangle singularity mechanism that leads to a quasi-state phenomenon in the physical observables [56], except the anomalous threshold here can not be accessed directly by the experiments. The imaginary parts of G E in the space-like region, however, are still zero since the nonzero contributions from Eq. (42) are exactly canceled by those from the unitarity integral of Eq. (43). A similar plot for the magnetic TFF G M is shown in Fig. 7 where there is a cusp-like structure rather than a logarithmic singularity in G E located at the anomalous threshold since the coefficient f in Eq. (39) which is proportional to (Y 2 − κ 2 ) does vanish at the anomalous threshold for G M . Note that such cusp-like structure is almost invisible due to the large scale variation of the magnitude of G M . With that set of parameters, a 52% decrease is produced by the KK channel for G E at t = −1 GeV 2 , 2 while only a 3% decrease happens for G M . One should be aware, however, of the large difference between the effects of KK channel in G E and G M is the result of the much larger magnitude that G M has overall than G E . The absolute effect of the KK inelasticity in G M is actually of compatible size as in G E (sometimes even larger). In Fig. 8, we show the electric transition form factor G E between the estimation including only the ππ intermediate state and the ππ-KK coupled-channel determination with errors. Note that the TFFs are real-valued in the space-like region. The solid curves are calculated again with the radius-adjusted parameters. The error bands in Fig. 8 are estimated by the bootstrap sampling over the three-dimensional parameter space that is spanned by F Φ , b 10 and h A . Note that the electric form factor is independent of the low-energy constant b 10 , see the expressions in Appendix B. As in Ref. [19], the uncertainty in h A gives the dominant contribution. The effect on G E introduced by the inclusion of the KK inelasticity is heav-ily intertwined with the large uncertainties from the variation of h A and Λ. Overall, the role of the cutoff is a bit more complicated than in the single ππ channel case. The situation is different for G M which is displayed in Fig. 9. The magnetic Sigma-to-Lambda transition form factor G M is almost unchanged after including the KK inelasticity. Moreover, G M has much larger absolute errors from the bootstrap method. At t = −1 GeV 2 , the bootstrap uncertainty from F Φ , h A and b 10 is already of order ±1, dominated by the uncertainty in b 10 . As in Ref. [19], we find a very small sensitivity of G M to the variation of the cutoff Λ. In addition to providing valuable insights into the electromagnetic structure of hyperons, experimental data for the transition form factors may thus also help to constrain these parameters. Summary In this paper, we extended the dispersion theoretical determination of the electromagnetic Sigma-to-Lambda transition form factors presented in Ref. [19] from the ππ intermediate state to the ππ-KK coupled-channel configuration within the SU(3) ChPT framework. After in-cluding the KK channel, a shift of the electric Sigma-to-Lambda transition form factor G E is presented, while the magnetic form factor G M stays essentially unchanged. At present, the dispersion theoretical determination of electromagnetic Sigma-to-Lambda transition form factors suffers from sizeable uncertainties due to the poor knowledge of the LEC b 10 and coupling h A . The precise determination of this three-flavor LEC from the future experiments will be helpful to pin down the hyperon TFFs. In a next step, it will be of interest to explore the elastic hyperon electromagnetic form factors based on the theoretical framework that combines dispersion theory and three-flavor chiral perturbation theory. A Unitarity relations and the anomalous pieces Let us start from the single channel case. The unitarity relations for the Σ-to-Λ TFFs G E/M (in the followings we drop the index E/M ) within the single ππ channel assumption read [19,24] 1 2i disc unit G(t) = 1 24π T π Σ π F V * π ,(31) where Σ π = σ π q 2 π with σ and q defined by Eq. (20) and Eq. (5) respectively, and q = √ tσ/2. Moving to the ππ-KK coupled-channel case, one first considers the vector pion and kaon form factors; they satisfy the unitarity relations [46,57], 1 2i disc F V (t) = t 1 * 1 Σ F V , F V = F V π , √ 2F V K T . (32) Similarly, the Σ 0Λ → ππ and Σ 0Λ → KK P -wave amplitudes fulfill the unitarity relations 1 2i disc T (t) = t 1 * 1 Σ T , T = T π , √ 2T K T .(33) The key information that the above two equations provide us is the relative ratio between the ππ and KK channels in the J = I = 1 coupled-channel problem. Then with the single-ππ unitarity relations at hand already, that is, Eq. (31), one can easily extend to the two-channel case: 1 2i disc unit G(t) = 1 24π T T Σ F V * = 1 24π T π , √ 2T K . Σ π 0 0 Σ K . F V * π √ 2F V * π = 1 24π T π Σ π F V * π θ t − 4M 2 π + 2T K Σ K F V * K θ t − 4M 2 K .(34) That becomes Eq. (4) after substituting the identity q = √ tσ/2. Recalling that all the left-hand cut (LHC) part of T is included in K, then T − K only contains the right-hand cut (RHC) and its unitarity relation is given by Eq. (33) for the two-channel assumption. One can also write [46] 1 2i disc Ω −1 ( T − K) = − Im Ω −1 K,(35) which leads to Eq. (13). When m 2 Σ +m 2 Λ −2M 2 i > 2m 2 exch and λ(m 2 Λ , m 2 exch , M 2 i ) < 0 with M i = M π (M K ) for the process ΣΛ → ππ (ΣΛ → KK) , the LHC and RHC will overlap, leading to the non-zero anomalous terms G anom and T anom in Eq. (4) and Eq. (13), respectively [38,39,40,41,24]. This indeed happens in the proton exchange diagram for the process ΣΛ → KK. Such anomalous contributions are estimated by the dispersive integrals of the discontinuity along the cut that connects the anomalous threshold to the starting point of the RHC (the physical threshold of the two-body intermediate state). The anomalous threshold t − is defined by [39] t − = 1 2 (m 2 Σ + m 2 Λ + 2M 2 K − m 2 N ) − 1 2m 2 N (m 2 Σ − M 2 K )(m 2 Λ − M 2 K ) + λ 1/2 (m 2 Σ , m 2 N , M 2 K )λ 1/2 (m 2 Λ , m 2 N , M 2 K ) .(36) Numerically, t − = 0.935 GeV located at the real axis of t just below the KK threshold. To go further, one first has to derive the discontinuity along the anomalous cut for the TFFs G and the scattering amplitudes T . After implementing the partial-wave projection, namely the integration in Eq. (14) and Eq. (16), one obtains K N = f κ 3 log Y + κ Y − κ + remainder,(37) where Y = −(m 2 Σ + m 2 Λ + 2M 2 K − t − 2m 2 N ), κ = λ 1/2 (t, m 2 Σ , m 2 Λ )σ K (t).(38) f is the coefficient of the logarithm which is a smooth function over the transferred momentum square t without any cut. The anomalous threshold is generated by the logarithm function. As illustrated in Refs. [39,24], the discontinuity of K N along the anomalous cut reads 1 2i disc anom K N = f κ 2 2π (−λ(t, m 2 Σ , m 2 Λ )) 1/2 σ K .(39) Note that the argument of √ z is defined in the range of [0, π) in the present work. Regarding T , one can rewrite Eq. (35) into 1 2i disc anom Ω −1 ( T − K) = − Im Ω −1 1 2i disc anom K = Ω −1 t 1 * 1 Σ 1 2i disc anom K, where we replace (− Im Ω −1 ) with ( Ω −1 t 1 * 1 Σ) in the second line since ImΩ −1 12 = ImΩ −1 22 = 0 below the KK threshold. 3 Finally, the discontinuity of the TFFs G along the anomalous cut can be read off straightforwardly in terms of that of T , 1 2i disc anom G = 1 24π ( t 1 * 1 ) −1 1 2i disc anom ( T − K) T F V * = 1 24π ( t 1 * 1 ) −1 Ω Ω −1 t 1 * 1 Σ 1 2i disc anom K T F V * = 1 24π 1 2i (disc anom K) T Σ F V * .(40) Substituting Eq. (39) into the above equation, one obtains 1 2i disc anom G = 1 24 −f F V * K t (−λ(t, m 2 Σ , m 2 Λ )) 3/2 .(41) Then we arrive at the expressions for G anom and T anom . They are G anom (t) = t 24π 1 0 dx dt (x) dx 1 t (x) − t × −f (t (x))F V * K (t (x)) (−λ(t (x), m 2 Σ , m 2 Λ )) 3/2 ,(42)T anom (t) = Ω(t) t π 1 0 dx dt (x) dx 1 t (x) − t × Ω −1 t 1 * 1 Σ 1 2i disc anom K t (t − t − i ) ,(43) 3 This replacement is necessary since Im Ω −1 is solved numerically in our calculation and ImΩ −1 12 = ImΩ −1 22 = 0 always holds in the unphysical region. The combined quantity Ω −1 t 1 * 1 Σ can be simplified analytically when multiplied to discanom K. Then it turns out that the products Ω −1 t 1 * 1 and Σ discanom K, respectively, are finite along the anomalous cut. Moreover, the identity − Im Ω −1 = Ω −1 t 1 * 1 Σ is checked numerically and does hold near the KK threshold. with t (x) = (1 − x) t − + x 4M 2 K . To cross-check whether this prescription is correct, we present the calculation of a scalar triangle loop function Fig. 10. The exact agreement is achieved only when the anomalous contribution is taken into account. B The reduced amplitudes K π/K and P 0,π/K The four-point amplitudes M ΣΛ→ππ/KK (t, θ) are calculated up to next-to leading order within the framework of SU(3) chiral perturbation theory. It turns out that the explicit inclusion of the decuplet baryon in the three-flavor ChPT Lagrangian is important to reproduce the correct G E/M (0) 4 and reasonable electric and magnetic transition radii, r 2 E and r 2 M [19]. We use the same Lagragians as in Ref. [19]. To be specific, the relevant interaction part of the leading order (LO) chiral Lagrangian that contains both the octet and decuplet states as active degrees of freedom for the reactions of interest is given by [21,58] L (1) (44) and the relevant NLO Lagrangian reads [59,60] L (2) C 0 (m 2 Σ , m 2 Λ , s, M 2 K , m 2 N , M 2 K ) in8+10 = D 2 B γ µ γ 5 {u µ , B} + F 2 B γ µ γ 5 [u µ , B] + 1 2 √ 2 h A ade g µν (T µ abc u ν bd B ce +B ec u ν db T µ abc ),8 = i 2 b 10 B {[u µ , u ν ], σ µν B} ,(45) where · · · denotes a flavor trace. The chirally covariant derivatives are defined by D µ B := ∂ µ B + [Γ µ , B](46) with Γ µ = 1 2 u † (∂ µ − i(v µ + a µ )) u + u (∂ µ − i(v µ − a µ )) u † .(47) Here, v and a are external sources and u 2 = U = exp(iΦ/F Φ ) with the Goldstone bosons encoded in the matrix Φ =    π 0 + 1 √ 3 η √ 2 π + √ 2 K + √ 2 π − −π 0 + 1 √ 3 η √ 2 K 0 √ 2 K − √ 2K 0 − 2 √ 3 η    .(48) The octet baryons also make up a 3×3 matrix in the flavor space that is given by B =    1 √ 2 Σ 0 + 1 √ 6 Λ Σ + p Σ − − 1 √ 2 Σ 0 + 1 √ 6 Λ n Ξ − Ξ 0 − 2 √ 6 Λ    . (49) Finally, T abc is a totally symmetric flavor tensor that denotes the decuplet baryons, T 111 = ∆ ++ , T 112 = 1 √ 3 ∆ + , T 122 = 1 √ 3 ∆ 0 , T 222 = ∆ − , T 113 = 1 √ 3 Σ * + , T 123 = 1 √ 6 Σ * 0 , T 223 = 1 √ 3 Σ * − , T 133 = 1 √ 3 Ξ * 0 , T 233 = 1 √ 3 Ξ * − , T 333 = Ω .(50) The amplitudes M ΣΛ→ππ/KK are described as a Born term in the LO plus a contact term in the NLO within the three-flavor ChPT, see Fig. 11 and Fig. 12. From above Lagrangians, one obtains the Σ-exchange Born term for Σ 0 (p 1 ) +Λ(p 2 ) → π − (p 3 ) + π + (p 4 ), iM Σ = i(M t + M u ) iM t = DF √ 3F 2 Φ v Λ γ µ γ 5 p 3,µ S Σ − ,t γ ν γ 5 p 4,ν u Σ , iM u = −DF √ 3F 2 Φ v Λ γ µ γ 5 p 4,µ S Σ + ,u γ ν γ 5 p 3,ν u Σ , (51) with S Σ − ,t = i((p 1 − p 4 ) µ γ µ + m Σ )/(t − m 2 Σ ) and S Σ + ,u = i((p 1 − p 3 ) µ γ µ + m Σ )/(u − m 2 Σ ) the propagator of the exchanged Σ in the t-and u-channel respectively. And the Σ * -exchange Born term, iM Σ * = i(M t + M u ) = −h A 2 √ 2F Φ 2v Λ g µν p ν 3 ∆ µα t ( −1 √ 3 )g αβ p β 4 u Σ , + −h A 2 √ 2F Φ 2v Λ (−1)g µν p ν 4 ∆ µα u ( −1 √ 3 )g αβ p β 3 u Σ ,(52) with the spin-3/2 Rarita-Schwinger propagator [61] i∆ µν (p) = γ α p α + m p 2 − m 2 g µν − 1 3 γ µ γ ν − 1 3p 2 γ α γ β p ρ p λ (g µβ g νλ g αρ + g να g µρ g βλ ) − 2 3 m 2 p µ p ν p 2 (γ α p α + m) + −i 3 m p 2 (g µρ g νβ g αλ + g µα g νλ g βρ )σ αβ p ρ p λ , and t = (p 1 − p 4 ) 2 , u = (p 1 − p 3 ) 2 . Here, m denotes the mass of the exchanged spin-3/2 resonance. The NLO contact term for the reaction Σ 0 (p 1 ) +Λ(p 2 ) → π − (p 3 ) + π + (p 4 ) is given by M NLO = b 10 1 F 2 Φ 4 √ 3 1 2 × (m Σ + m Λ ) (−v Λ γ µ (p 4 − p 3 ) µ u Σ ) + (u − t)v Λ u Σ .(53) The corresponding expressions for the Σ 0 (p 1 ) +Λ( p 2 ) → K − (p 3 )+K + (p 4 ) (M Σ 0Λ →K 0K0 = −M Σ 0Λ →K + K − in the isospin limit) read iM born = i(M u + M t + M Ξ * ) , iM u = 1 F 2 Φ −D 2 √ 3 + − √ 3F 2 D − F 2 × v Λ γ µ γ 5 p 4,µ S p,u γ ν γ 5 p 3,ν u Σ , iM t = 1 F 2 Φ −D 2 √ 3 + √ 3F 2 D + F 2 × v Λ γ µ γ 5 p 3,µ S Ξ,t γ ν γ 5 p 4,ν u Σ , iM Ξ * = −h A 2 √ 2F Φ 2v Λ (+1)g µν p ν 3 ∆ µα t ( −1 √ 3 )g αβ p β 4 u Σ . M NLO = b 10 1 F 2 Φ 2 √ 3 1 2 × (m Σ + m Λ ) (−v Λ γ µ (p 4 − p 3 ) µ u Σ ) + (u − t)v Λ u Σ ,(54)with S p,u = i((p 1 − p 3 ) µ γ µ + m p )/(u − m 2 p ) and S Ξ,t = i((p 1 − p 4 ) µ γ µ + m Ξ )/(t − m 2 Ξ ) the propagator of the exchanged proton and Ξ baryon, respectively. To proceed, it is helpful to introduce the following equivalents, E1 ≡v 1/2,Λ γ µ (p 1 − p 2 ) µ u 1/2,Σ v 1/2,Λ γ 3 u 1/2,Σ =v 1/2,Λ u 1/2,Σ (m Λ + m Σ ) v 1/2,Λ γ 3 u 1/2,Σ = (m Σ + m Λ ) 2 − s 2p z , E2 ≡v 1/2,Λ γ µ (p 4 − p 3 ) µ u 1/2,Σ v 1/2,Λ γ 3 u 1/2,Σ = −2p c.m. cos θ , M 1 ≡v −1/2,Λ u 1/2,Σ (m Λ + m Σ ) v −1/2,Λ γ 1 u 1/2,Σ = 0 , M 2 ≡v −1/2,Λ γ µ (p 4 − p 3 ) µ u 1/2,Σ v −1/2,Λ γ 1 u 1/2,Σ = −2p c.m. sin θ ,(55) where s = (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 is the center-of-mass energy. p z and p c.m. denote the modulus of the threedimensional center-of-mass momenta of the ΣΛ and ππ/KK two-body systems, respectively, i.e. p c.m. = q π/K . The equations (55) are calculated in the center-of-mass frame with the p z the modulus of the three-momentum along the direction of the z-axis and θ is the scattering angle of π or K. Substitute Eqs. (51), (52), (53), (54) into Eqs. (14), (15), (16), (17), we obtain P E 0,π , P M 0,π , K E π and K M π for the ππ inelasticity, P E 0,π = P E Σ + P E Σ * ,(56)P E Σ = 3 2 π 0 dθ sin θ cos θ DF √ 3F 2 Φ E2 p c.m. = − 2 √ 3 DF F 2 Φ , P E Σ * = 3 2 π 0 dθ sin θ cos θ −h A 2 √ 2F π 2 1 √ 3 t − u 12m 2 Σ * E1 p c.m. + E2 p c.m. 1 12m 2 Σ * (−2m 2 Σ * − 2m Σ * (m Σ + m Λ ) + m 2 Σ + m 2 Λ + s − 6M 2 π ) = h 2 A 24 √ 3F 2 Φ (m Λ + m Σ * )(m Σ + m Σ * ) m 2 Σ * + O(M 2 π , s) . K E π = K E Σ + K E Σ * ,(57)K E Σ = 3 2 π 0 dθ sin θ cos θ DF √ 3F 2 Φ × E1 p c.m. m Σ (m Σ − m Λ ) 1 t − m 2 Σ − 1 u − m 2 Σ + E2 p c.m. m Σ (m Σ + m Λ ) 1 t − m 2 Σ + 1 u − m 2 Σ , K E Σ * = 3 2 π 0 dθ sin θ cos θ −h A 2 √ 2F π 2 1 √ 3 × + F (s) m Σ + m Λ E1 p c.m. 1 u − m 2 Σ * − 1 t − m 2 Σ * + E2 p c.m. 1 u − m 2 Σ * + 1 t − m 2 Σ * G(s) 2 , where F (s) = m Σ + m Λ 2 + m Σ * H 1 (s) + m Σ + m Λ 2 − m Σ * H 2 , G(s) = H 1 (s) + H 2 , H 1 (s) = m 2 Σ + m 2 Λ − s 2 − (m 2 Λ + m 2 Σ * − M 2 π )(m 2 Σ + m 2 Σ * − M 2 π ) 4m 2 Σ * , H 2 = 1 3 m Λ + m 2 Λ + m 2 Σ * − M 2 π 2m Σ * × m Σ + m 2 Σ + m 2 Σ * − M 2 π 2m Σ * . P M 0,π = P M Σ + P M NLO − K M Σ * ,low ,(58)P M Σ = 3 4 π 0 dθ sin θ sin θ DF √ 3F 2 Φ M 2 p c.m. = − 2 √ 3 DF F 2 Φ , P M NLO = 3 4 π 0 dθ sin θ sin θ b 10 1 F 2 Φ 4 √ 3 (m Σ + m Λ ) 2 −M 2 p c.m. = 4 √ 3 b 10 F 2 Φ (m Λ + m Σ ) . K M π = K M Σ + K M Σ * ,(59)K M Σ = 3 4 π 0 dθ sin θ sin θ DF √ 3F 2 Φ M 2 p c.m. × m Σ (m Σ + m Λ ) 1 t − m 2 Σ + 1 u − m 2 Σ , K M Σ * = 3 4 π 0 dθ sin θ sin θ −h A 2 √ 2F π 2 1 √ 3 × + M 2 p c.m. 1 u − m 2 Σ * + 1 t − m 2 Σ * G(s) 2 . Note that we subtract a term K M Σ * ,low in the polynomial part of the magnetic amplitude P M 0,π , which denotes the low-energy limit of the LHC contribution of the decupletexchanged magnetic amplitude. It is proposed to remove the doubly counted decuplet baryon contribution caused by the using of the resonance saturation assumption for the estimation of b 10 in the present ChPT framework. A similar term K E Σ * ,low should be subtracted in P E 0,π . However, it belongs to a higher chiral order and is dropped here. Note that P E NLO belongs to P 1 (s) that is beyond the accuracy of Eq. (13) and is also dropped. Taking the same convention with Ref. [19], K M Σ * ,low is given by K M Σ * ,low = lim s→0 lim m Λ →m Σ lim Mπ→0 K M Σ * (s) = h 2 A 24 √ 3F 2 Φ (−m 2 Σ * + 4m Σ * m Σ − m 2 Σ ) (m Σ * + m Σ ) m 2 Σ * (m Σ * − m Σ ) . And similarly, the P E 0,K , P M 0,K , K E K and K M K for the KK inelasticity read P E 0,K = P E born + P E Ξ * ,(60) with P E born = 3 2 π 0 dθ sin θ cos θ 1 2 g A (m Λ + m Σ + 2m N ) + g B (m Λ + m Σ + 2m Ξ ) E1 (m Λ + m Σ )p c.m. + g B − g A 2 E2 p c.m. = g A − g B , P E Ξ * = 3 2 π 0 dθ sin θ cos θ h 2 A 8 √ 3F 2 Φ 1 12m 2 Ξ * × (m Λ + m Σ )(t − m 2 Ξ * ) + 1 12m 2 Ξ * (m Λ + m Σ + 2m Ξ * ) × (−m 2 Λ − m 2 Σ + 2M 2 K + 2m 2 Ξ * + m Ξ * (m Λ + m Σ )) E1 (m Λ + m Σ )p c.m. + 1 12 (1 − t m 2 Ξ * ) + 1 12m 2 Ξ * (m 2 Λ + m 2 Σ − 2M 2 K − 2m 2 Ξ * − m Ξ * (m Λ + m Σ )) E2 p c.m. = h 2 A (m Λ + m Ξ * )(m Σ + m Ξ * ) 48 √ 3F 2 Φ m 2 Ξ * + O(s, M 2 K ) . K E K = K E N + K E Ξ + K E Ξ * ,(61)K E N = 3 2 π 0 dθ sin θ cos θ E1 (m Λ + m Σ )p c.m. × g A (m Λ + m N )(m Σ + m N )(m Λ + m Σ − 2m N ) 2(m 2 N − u) + E2 p c.m. 1 2 g A (m Λ + m N )(m Σ + m N ) m 2 N − u , K E Ξ = 3 2 π 0 dθ sin θ cos θ E1 (m Λ + m Σ )p c.m. × g B (m Λ + m Ξ )(m Σ + m Ξ )(m Λ + m Σ − 2m Ξ ) 2(m 2 Ξ − t) + E2 p c.m. 1 2 − g B (m Λ + m Ξ )(m Σ + m Ξ ) m 2 Ξ − t , K E Ξ * = 3 2 π 0 dθ sin θ cos θ h 2 A 8 √ 3F 2 Φ −F (s) 12m 2 Ξ * (m 2 Ξ * − t) × E1 (m Λ + m Σ )p c.m. + E2 p c.m. 1 12m 2 Ξ * (m 2 Ξ * − t)G (s) , wherẽ F (s) = −M 4 K (m Λ + m Σ + 4m Ξ * ) + M 2 K (m 3 Λ + 8m 3 Ξ * + 4m 2 Ξ * m Σ + 3m Ξ * m 2 Σ + m 3 Σ + m 2 Λ (3m Ξ * + m Σ ) + m Λ (4m 2 Ξ * − 2m Ξ * m Σ + m 2 Σ )) + (m Λ + m Ξ * )(m Σ + m Ξ * )(m 2 Λ (2m Ξ * − m Σ ) + m Λ (m 2 Ξ * − m 2 Σ ) + m Ξ * (−4m 2 Ξ * + 2m 2 Σ + m Ξ * m Σ )) − 3m 2 Ξ * (m Λ + 2m Ξ * + m Σ )s , G(s) = 3m 2 Ξ * s + M 4 K + (m Λ + m Ξ * )(m Σ + m Ξ * ) × (m Ξ * (m Ξ * − 2m Σ ) + m Λ (−2m Ξ * + m Σ )) − M 2 K (m 2 Λ + m 2 Σ + 2m 2 Ξ * − m Ξ * (m Λ + m Σ )) . Note that the Pascalutsa prescription of the spin-3/2 particle will bring an ambiguity in the P E Σ * and P E Ξ * while it keeps K E Σ * and K E Ξ * consistent with the interaction between the decuplet and octet states listed in Eq. (44), see Ref. [19] for the details. The uncertainties on the TFFs originating from such ambiguity, however, are negligible when compared with the parameter errors. And we take he same convention with Ref. [19] where the O(M 2 π , s) and O(M 2 K , s) terms are dropped in the P E Σ * and P E Ξ * . Further, P M 0,K = P M born + P M NLO − K M Ξ * ,low ,(62)P M born = 3 4 π 0 dθ sin θ sin θ g B − g A 2 M 2 p c.m. = g A − g B , P M NLO = 3 4 π 0 dθ sin θ sin θ b 10 1 F 2 Φ 2 √ 3 (m Σ + m Λ ) 2 −M 2 p c.m. = 2 √ 3 b 10 F 2 Φ (m Λ + m Σ ) . K M K = K M N + K M Ξ + K M Ξ * ,(63)K M N = 3 4 π 0 dθ sin θ sin θ − M 2 p c.m. 1 2 × g A (m Λ + m N )(m Σ + m N ) u − m 2 Fig. 1 . 1The spectral decomposition of the matrix element of the electromagnetic current jµ in Eq. (3). Fig. 2 . 2The four-point function ππ → Σ 0Λ including twopion rescattering. The hatched circle is the part containing only left-hand cuts and a polynomial. Fig. 3 . 3The modulus g of the P -wave ππ → KK amplitude given by Ref.[49]. Fig. 4 . 4Real (solid line) and imaginary (dashed line) parts of the Omnès matrix elements Ω. Fig. 5 . 5Modulus (left) and phase (right) of the vector pion (upper) and kaon (lower) form factors given by Eq.(26). The input ππ phase shift δ 1 1 (upper) and phase ψ (lower) of the P -wave isovector ππ → KK amplitude are also presented as the red-dashed lines for comparison. Note that the asymptotic values of δ 1 1 and ψ are π and 2π respectively. The latter is invisible in the plot since its extrapolation point is set as 5 GeV. Eq. (4) including ππ/KK rescattering effects from Eq.(13). Finally, we calculate the Sigma-to-Lambda transition form factors G E/M from the dispersion relations Eq.(6). Two issues remain to be clarified. First, we have to fix all the couplings in the expressions of P K . These are D, F , F Φ for the LO octet-to-octet interactions, h A for the LO decuplet-to-octet interaction, and b 10 for the NLO octet-to-octet interaction. In ChPT, D and F are well constrained around 0.8 and 0.5, respectively. Here we use D = 0.80, F = 0.46 Fig. 6 . 6The imaginary (red) and real (blue) part of the electric transition form factor GE. The dash-dotted, dotted, solid lines denote the results within the single ππ channel, ππ-KK coupled channel without and with the anomalous contribution scenarios, respectively, when FΦ = 100 MeV, b10 = 1.06 GeV −1 , hA = 2.22 and Λ = 1.5 GeV. The vertical dashed and solid lines represent respectively the anomalous threshold (Eq. (36)) and the KK threshold. Fig. 7 . 7The imaginary (red) and real (blue) part of the magnetic transition form factor GM . For notations, seeFig. 6. Fig. 8 .Fig. 9 . 89The electric transition form factor GE obtained from the once-subtracted dispersion relation Eq. (6) with an energy cutoff Λ = 1.5 GeV (left) and 2.0 GeV (right). The blue lines denote the results from the single ππ channel consideration as in Ref.[19] and the red lines are those after including the KK channel. The error bands are estimated based on bootstrap sampling. The magnetic transition form factor GM obtained from the once-subtracted dispersion relation Eq. (6) with an energy cutoff Λ = 1.5 GeV (left) and 2.0 GeV (right). For notations, seeFig. 8. Acknowledgements YHL is grateful to Meng-Lin Du, De-Liang Yao and Feng-Kun Guo for many valuable discussions. YHL thanks also Yu-Ji Shi for some discussions on the kaon vector form factors. This work of UGM and YHL is supported in part by the DFG (Project number 196253076 -TRR 110) and the NSFC (Grant No. 11621131001) through the funds provided to the Sino-German CRC 110 "Symmetries and the Emergence of Structure in QCD", by the Chinese Academy of Sciences (CAS) through a President's International Fellowship Initiative (PIFI) (Grant No. 2018DM0034), by the VolkswagenStiftung (Grant No. 93562), and by the EU Horizon 2020 research and innovation programme, STRONG-2020 project under grant agreement No 824093. HWH was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Projektnummer 279384907 -CRC 1245 and by the German Federal Ministry of Education and Research (BMBF) (Grant no. 05P21RDFNB). Fig. 10 . 10The absolute values of the scalar triangle loop function C0(m 2 Σ , m 2 Λ , s, M 2 K , m 2 N , M 2 K ) calculated numerically using Feynman parameters (solid black line) as well as dispersively with (dashed red line) and without the anomalous contribution (dot-dashed blue line). Note that the solid black and the dashed red line coincide. Fig. 11 . 11Pictorial representation of the bare input of the fourpoint amplitude ππ → Σ 0Λ obtained up to NLO. Fig. 12 . 12Pictorial representation of the bare input of the fourpoint amplitude KK → Σ 0Λ obtained up to NLO. K = Kπθ(t − 4M 2 π ), Note that GE is overall very small, as is expected due to the vanishing overall charge of the Λ and Σ 0 . Here, the normalization of electromagnetic Sigma-to-Lambda TFFs is estimated with the unsubtract dispersion relations, see Ref.[19] for more details. dθ sin θ sin θ h 2.Here, g A and g B are defined asThere are only three different kinds of integration over angle involved in the K E/M . 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[ "B → K ( * ) νν in covariant confined quark model", "B → K ( * ) νν in covariant confined quark model" ]	[ "Aidos Issadykov \nBogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nInstitute of Nuclear Physics\nMinistry of Energy\n\n", "Mikhail A Ivanov \nBogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980DubnaRussia\n", "Almaty Kazakhstan " ]	[ "Bogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980DubnaRussia", "Institute of Nuclear Physics\nMinistry of Energy\n", "Bogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980DubnaRussia" ]	[]	We study the B → K ( * ) νν decays within the Standard Model (SM) by using the relevant transition form factors obtained from the covariant confined quark model (CCQM) developed by us. The B → K and B → K * transition form factors are calculated in the full kinematic q 2 range. The branching fractions are then calculated. It is shown that our results are in an agreement with those obtained in other theoretical approaches. Currently, the Babar and Belle collaborations provide us by the upper limits at 90% confidence limit. The obtained bounds are roughly an order of magnitude larger than the SM predictions. This should stimulate experimental collaborations to set up experiments that allow one to obtain more accurate branching values, which is quite achievable on the updated LHCb and Belle machines. If the discrepancies between theory and experiment are confirmed, this will open up opportunities for constructing models with new particles and interactions leading to an extension of the SM. * Electronic address: [email protected] † Electronic address: [email protected] 1 arXiv:2211.10683v2 [hep-ph] 12 Dec 2022	10.1142/s0217732323500062	[ "https://export.arxiv.org/pdf/2211.10683v2.pdf" ]	253,735,063	2211.10683	a1de4fb8389b73eb6eda2dcb5e2286d976bdf80d	B → K ( * ) νν in covariant confined quark model Aidos Issadykov Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research 141980DubnaRussia Institute of Nuclear Physics Ministry of Energy Mikhail A Ivanov Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research 141980DubnaRussia Almaty Kazakhstan B → K ( * ) νν in covariant confined quark model We study the B → K ( * ) νν decays within the Standard Model (SM) by using the relevant transition form factors obtained from the covariant confined quark model (CCQM) developed by us. The B → K and B → K * transition form factors are calculated in the full kinematic q 2 range. The branching fractions are then calculated. It is shown that our results are in an agreement with those obtained in other theoretical approaches. Currently, the Babar and Belle collaborations provide us by the upper limits at 90% confidence limit. The obtained bounds are roughly an order of magnitude larger than the SM predictions. This should stimulate experimental collaborations to set up experiments that allow one to obtain more accurate branching values, which is quite achievable on the updated LHCb and Belle machines. If the discrepancies between theory and experiment are confirmed, this will open up opportunities for constructing models with new particles and interactions leading to an extension of the SM. * Electronic address: [email protected] † Electronic address: [email protected] 1 arXiv:2211.10683v2 [hep-ph] 12 Dec 2022 I. INTRODUCTION The rare weak decays B → K ( * ) νν (K ( * ) = K or K * ) proceed via a b − s flavor changing neutral current (FCNC), i.e. at loop level in the Standard Model (SM). Such transitions are significantly suppressed in such a way that the branching fractions are of 10 −6 order. Nevertheless, the decays with a lepton pair in the final state are observed and studied by BaBar and Belle collaborations. The decays with di-neutriono in the final state have been only observed and the upper limits established [1][2][3][4][5][6][7]. Theoretically, the rare B-meson decays to final states, containing a pair of neutrinos are among the most "clean" processes with neutral currents that change the flavors of quarks. Since the neutrino is electrically neutral, the factorization of hadron and lepton currents in this decay is exact, in contrast to other decays of B mesons. For this reason, precise measurement of the B → K ( * ) νν processes should allow one to extract the B → K ( * ) transition form factors with high accuracy. Another advantage of such decays is that they are free of charmonium resonance contributions. A detailed analysis of these decays has been performed in the series of publications by A. Buras and his collaborators [8][9][10][11], recently the lattice QCD determination of B −K transition form factors have been used to determine SM differential branching fractions for B → Kνν decay [12]. There are some other theoretical approaches to this activity, see, for instance, [13][14][15][16][17][18][19]. The B → K ( * ) νν processes can also be associated with other decays of the B-meson, which proceed through the formation of some exotic state that decays into a pair of neutrinos. Such signals are interesting in the context of the problem of searching for dark matter and may allow us to investigate the relationship between the Standard Model and the so-called dark sector of the Universe [20]. The interplay between dineutrino modes and semileptonic rare B-decays has been elaborated in Ref. [21]. Current experimental bounds on the B → Kνν branching fraction are roughly an order of magnitude larger than the SM predictions. Thus, increasing the accuracy of measuring B → K ( * ) νν processes is an extremely important task. In this paper we focus on these decays within the SM framework using the relevant form factors calculated in the covariant constituent quark model (CCQM) previously developed by us. II. A GENERAL INFORMATION ON THE DECAY OF B → K ( * ) νν The effective Hamiltonian for b → sνν transitions in the SM can be written in the form, see, for instance Ref. [9] and other references therein H SM eff = G F √ 2 V tb V * ts α em 2π X t sin 2 Θ W (s O µ b)(νO µ ν) + h.c.(1) Here G F is the Fermi constant, V tb and V * ts are the matrix elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, α em is the fine structure constant calculated at the electroweak scale of Z-boson mass, θ W is the Weinberg angle, O µ = γ µ (1 − γ 5 ) is the weak Dirac matrix with the left chirality. The Wilson coefficient X t is known with a high accuracy, including NLO QCD corrections [22,23] and two-loop electroweak contributions [24], resulting in X t = 1.469 ± 0.017.(2) Matrix elements K ( * ) \|sO µ b)\|B can be written in terms of scalar functions -form factors depending on the transferred momentum squared K(p 2 ) \|s O µ b \|B(p 1 ) = F + (q 2 ) P µ + F − (q 2 ) q µ ,(3)K * (p 2 , 2 ) \|s O µ b \| B(p 1 ) = † 2 α m 1 + m 2 − g µα P q A 0 (q 2 ) + P µ P α A + (q 2 ) + q µ P α A − (q 2 ) + i ε µαP q V (q 2 ) .(4) Here P = p 1 +p 2 , q = p 1 −p 2 , and 2 is the polarization vector of the K * meson, so † 2 ·p 2 = 0 . The abbreviation ε µαP q = ε µανδ P ν q δ is adopted, where the absolutely antisymmetric Levi-Civita tensor is defined as ε 0123 = −ε 0123 = −1. The particles are on the mass shell, i.e. p 2 1 = m 2 1 = m 2 B and p 2 2 = m 2 2 = m 2 K ( * ) . It should be noted that the calculation of form factors requires methods outside the framework of perturbation theory, such as, for example, various quark models, QCD sum rules, lattice QCD, etc. Taking into account the above definitions of form factors, the differential branching fraction is written as follows: dB(B → K ( * ) + νν) dq 2 = 3τ B (G F λ t α em ) 2 3(2π) 5 X t sin 2 θ W 2 × \|p 2 \| 4m 2 1 H 2 + +H 2 − +H 2 0 .(5) The factor 3 at the beginning of the formula comes from the summation over the neutrino flavors: ν e , ν µ and ν τ . In the Standard Model, they are assumed to be massless and therefore do not interfere with each other. Further, τ B is the B-meson lifetime, λ t ≡ \|V tb V * ts \|, \|p 2 \| = λ 1/2 (m 2 1 , m 2 2 , q 2 )/2m 1 is the daughter meson momentum in the rest frame of the initial B meson. Scaled helicity amplitudesH are related to the helicity amplitudes defined in [25] asH = q 2 H. As a result, we have B → K transition:H ± = 0,H 0 = 2 m 1 \|p 2 \| F + .(6) B → K * transition: H ± = q 2 m 1 + m 2 − P q A 0 ± 2 m 1 \|p 2 \| V , H 0 = 1 m 1 + m 2 1 2 m 2 − P q (P q − q 2 ) A 0 + 4 m 2 1 \|p 2 \| 2 A + .(7) The scaling is done in order to avoid uncertainties at the point q 2 = 0 in numerical calculations. We note that in these decays the physical range of the variable q 2 is 0 ≤ q 2 ≤ q 2 max = (m 1 − m 2 ) 2 . It is interesting to note that in the case of a B → K transition, differential branching fraction behaves like \|p 2 \| 3 near the end point of the spectrum dB(B → K + νν) dq 2 = τ B (G F λ t α em ) 2 (2π) 5 X t sin 2 θ W 2 · \|p 2 \| 3 \|F + (q 2 )\| 2 ,(8) and disappears as (q 2 − q 2 max ) 3/2 . Note that in contrast to B → K ( * ) + − decays, the isospin asymmetries of the B → K ( * ) νν vanish identically, so the branching fractions of the B 0 and B ± decays only differ due to the lifetime difference [9]: τ B ± = 1.638 ± 0.004 ps and τ B 0 = 1.519 ± 0.004 ps. In this paper we will consider the charged B-meson only. In the CCQM the nonlocal quark interpolating currents are used to describe the internal structure of a hadron J M (x) = dx 1 dx 2 F M (x; x 1 , x 2 ) ·q a f 1 (x 1 ) Γ M q a f 2 (x 2 ) Meson J B (x) = dx 1 dx 2 dx 3 F B (x; x 1 , x 2 , x 3 ) Baryon × Γ 1 q a 1 f 1 (x 1 ) ε a 1 a 2 a 3 q T a 2 f 2 (x 2 )C Γ 2 q a 3 f 3 (x 3 ) J T (x) = dx 1 . . . dx 4 F T (x; x 1 , . . . , x 4 ) Tetraquark × ε a 1 a 2 c q T a 1 f 1 (x 1 ) CΓ 1 q a 2 f 2 (x 2 ) · ε a 3 a 4 cqT a 3 f 3 (x 3 ) Γ 2 Cq a 4 f 4 (x 4 )(9) The vertex functions F H are chosen in the translational invariant form F H (x; x 1 , . . . , x n ) = δ x − n i=1 w i x i Φ H i<j (x i − x j ) 2 ,(10) where w i = m i / n j=1 m j . The function Φ H is taken Gaussian in such a way that its Fouriertransform decreases quite rapidly in the Euclidean direction and provides the ultraviolet convergence of the Feynman diagrams. The matrix elements of the rare decay B → K ( * ) νν are described by the diagram shown in Fig. 1. The invariant matrix elements of the weak current between the initial and final meson states are written down K(p 2 ) \|s O µ b \|B(p 1 ) = = N c g B g K d 4 k (2π) 4 i Φ B − (k + w 13 p 1 ) 2 Φ K − (k + w 23 p 2 ) 2 × tr O µ S 1 (k + p 1 ) γ 5 S 3 (k) γ 5 S 2 (k + p 2 ) = F + (q 2 ) P µ + F − (q 2 ) q µ ,(11)K * (p 2 , 2 ) \|s O µ b \| B(p 1 ) = = N c g B g K * d 4 k (2π) 4 i Φ B − (k + w 13 p 1 ) 2 Φ K * − (k + w 23 p 2 ) 2 × tr O µ S 1 (k + p 1 ) γ 5 S 3 (k) † 2 S 2 (k + p 2 ) = † 2 α m 1 + m 2 −g µα P q A 0 (q 2 ) + P µ P α A + (q 2 ) + q µ P α A − (q 2 ) +i ε µαP q V (q 2 ) .(12) We introduce 2-index notation for the reduced masses w ij = m q j /(m q i + m q j ) (i, j = 1, 2, 3), so that w ij + w ji = 1. In our case we have q 1 = b, q 2 = s and q 3 = u. Using the technique described in our previous papers, see, for example, [26][27][28][29][30], the final expressions for the form factors are represented as two-fold parametric integrals. Numerical results for the form factors F (q 2 ) can be approximated by the dipole formula with high accuracy F (q 2 ) = F (0) 1 − as + bs 2 , s = q 2 m 2 1 .(13) The relative error of such an approximation is less than 1% in the entire kinematic region. The parameters F (0), a, b are given by Eq. (14). F + F − A 0 A + A − V F( The behavior of the form factors F + , F − and A 0 , A + , A − , V in the entire kinematic region 0 ≤ q 2 ≤ q 2 max , where q 2 max = (m 1 − m 2 ) 2 , are shown in Fig. 2 and 3. IV. NUMERICAL RESULTS In this section, we present numerical results for both differential and integrated branching fractions of the B → K ( * ) νν decays. In Table I we present numerical values of parameters required for calculations [31]. G F 1.166 × 10 −5 GeV −2 λ t 0.0401(10) The behavior of differential branching fractions of B → K ( * ) νν decays in the entire kinematic region of the momentum transfer squared are shown in Fig. 4. It should be noted that in the case of a B → K transition, differential branching fraction behaves as (q 2 − q 2 max ) 3/2 near the end point of the spectrum, in contrast to B → K * transition, where the behavior is (q 2 − q 2 max ) 1/2 . FIG. 1 : 1Diagram describing the decay of B → K ( * ) νν in the CCQM. FIG. 2 : 2Form factors of B → K transition. FIG. 3: Form factors of B → K * transition. FIG. 4 : 4Behavior of differential branchings of B → K ( * ) νν decays over the entire kinematic region of the squared momentum transfer. provided a thorough analysis of the semileptonic decays B → K ( * ) νν in the SM by using the relevant form factors obtained from our covariant confined quark model. The decay branching fractions have been calculated and the following results obtained B(B + → K + νν) = 4.96(0.74) × 10 −6 , B(B + → K * + νν) = 9.57(1.43) × 10 −6 . The results are in an agreement with those obtained in other theoretical approaches. Currently, the Babar and Belle collaboratiions provide us by the upper limits at 90% confidence limit. The obtained bounds are roughly an order of magnitude larger than the SM predictions. This stimulates experimental collaborations to set up experiments that allow one to obtain more accurate branching values, which is quite achievable on the updated LHCb and Belle machines. If the discrepancies between theory and experiment are confirmed, this will open up opportunities for constructing models with new particles and interactions leading to an extension of the Standard Model. TABLE I : IInput parameters used in calculations.m B + 5279.25(26) MeV τ B + 1.638(4) ps m K + 493.677(13) MeV sin 2 θ W 0.23126(5) m K * + 891.67(26) MeV α −1 em 127.925(16) Table II IIshows the results of calculations of the total branching fractions of B → K ( * ) νν, obtained in our work, and, for comparison, the results of the experimental measurements and theoretical predictions obtained in variiuos approaches. We estimate the errors of the results of our model at 15 %. TABLE II : IISummary of experimental and theoretical results for the branching fractions B(B → K ( * ) νν). Experimental upper limits are given at 90% confidence level. 10 6 B(B + → K + νν) 10 6 B(B → K * νν)Exp. (90% CL) Ref. Exp. (90% CL) Ref. < 14 [1] < 140 [1] < 13 [3] < 80 [2] < 55 [4] < 40 [4] < 16 [5] < 64 [5] < 19 [6] < 61 [6] < 41 [7] Theory Ref. Theory Ref. 2.4(0.6) [14] 5.1(0.8) [14] 5.2(1.1) [15] 13(5) [15] 4.19(42) [16] 4.5(0.7) [8] 6.8 +1.0 −1.1 [8] 5.1(0.8) [13] 8.4(1.4) [13] 4.23(56) [21] 8.93(1.07) [21] 4.4 +1.4 −1.1 [17] 4.0(0.5) [9] 9.2(1.0) [9] 4.94(52) [18] 4.45(62) [10] 9.70(92) [10] 4.65(62) [11] 10.13(92) [11] 5.67(38) [12] 4.96(0.74) this work 9.57(1.43) this work AcknowledgmentsWe thank Egor Tretyakov who participated at the early steps of this work. This research has been funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09057862). . K F Chen, BellearXiv:0707.0138Phys. Rev. Lett. 99221802hep-exK. F. Chen et al. [Belle], Phys. Rev. Lett. 99, 221802 (2007) [arXiv:0707.0138 [hep-ex]]. . B Aubert, BaBararXiv:0808.1338Phys. Rev. D. 7872007hep-exB. Aubert et al. [BaBar], Phys. Rev. D 78, 072007 (2008) [arXiv:0808.1338 [hep-ex]]. . P Del Amo, BaBarSanchez, BaBararXiv:1009.1529Phys. Rev. D. 82112002hep-exP. del Amo Sanchez et al. [BaBar], Phys. Rev. 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[ "Unfolded Fierz-Pauli Equations in Three-Dimensional Asymptotically Flat Spacetimes", "Unfolded Fierz-Pauli Equations in Three-Dimensional Asymptotically Flat Spacetimes", "Unfolded Fierz-Pauli Equations in Three-Dimensional Asymptotically Flat Spacetimes", "Unfolded Fierz-Pauli Equations in Three-Dimensional Asymptotically Flat Spacetimes" ]	[ "Martin Ammon [email protected]†[email protected] \nTheoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n", "Michel Pannier \nTheoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n", "Martin Ammon [email protected]†[email protected] \nTheoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n", "Michel Pannier \nTheoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n" ]	[ "Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany", "Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany", "Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany", "Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany" ]	[]	We utilise a quotient of the universal enveloping algebra of the Poincaré algebra in three spacetime dimensions, on which we formulate a covariant constancy condition. The equations so obtained contain the Fierz-Pauli equations for non-interacting, massive higher-spin fields, and can thus be regarded as an unfolding of the Fierz-Pauli system. All fundamental fields completely decouple from each other. In the non-truncated case, the field content includes infinitely many copies of each field at fixed spin.	10.1007/jhep02(2023)161	[ "https://export.arxiv.org/pdf/2211.12530v1.pdf" ]	253,801,935	2211.12530	aa392a967a91d59b48756331654e013c6cb72233	Unfolded Fierz-Pauli Equations in Three-Dimensional Asymptotically Flat Spacetimes 22 Nov 2022 Martin Ammon *[email protected]†[email protected] Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Michel Pannier Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Unfolded Fierz-Pauli Equations in Three-Dimensional Asymptotically Flat Spacetimes 22 Nov 2022 We utilise a quotient of the universal enveloping algebra of the Poincaré algebra in three spacetime dimensions, on which we formulate a covariant constancy condition. The equations so obtained contain the Fierz-Pauli equations for non-interacting, massive higher-spin fields, and can thus be regarded as an unfolding of the Fierz-Pauli system. All fundamental fields completely decouple from each other. In the non-truncated case, the field content includes infinitely many copies of each field at fixed spin. Introduction In recent years there has been revived interest in theories of massive and massless higher spins in flat spacetimes, both in the context of holography, such as in [1][2][3][4][5][6][7][8][9][10][11][12][13], and from various other points of view, such as in [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Here, we are particularly interested in the introduction of propagating (massive) degrees of freedom to an otherwise purely topological gravitational theory in the case of three spacetime dimensions, an objective that is largely motivated by the successful utilisation of such additional fields in examples of higher-spin AdS 3 /CFT 2 dualities [28][29][30][31][32][33]. Our proposal relies on the construction of the associative higher-spin algebra established in [34] that allows the introduction of massive scalar fields and, as will be shown here, massive fields up to any spin. The attempt to formulate wave equations describing massive or massless fields of any spin dates back already to early works in the thirties of the last century [35,36]. It was argued by Fierz and Pauli [37,38] that the dynamics of massive, freely propagating fields of arbitrary integer spin are captured by traceless tensor fields that obey a natural generalisation of the standard wave equation and a divergence-free condition, − m 2 φ µ 1 ...µ σ = 0 , ∇ µ φ µµ 2 ...µ σ = 0 , g µν φ µνµ 3 ...µ σ = 0 . (1.1) Here, φ µ 1 ...µ σ is a totally symmetric tensor of rank σ. Despite the discovery of various no-go theorems concerning the consistent interaction of higher-spin fields [39][40][41][42][43][44], a variety of working examples could be established [45][46][47][48] (for more comprehensive literature references see, e.g., [49][50][51]) and, eventually, work by Vasiliev and collaborators [52][53][54] provided a consistent, fully interacting theory of massless higherspin fields as well as their interaction with massive matter fields in the case of a negative cosmological constant. At the heart of Vasiliev's construction lies the unfolded formulation of the equations of motion. The introduction of an infinite tower of auxiliary fields that obey first-order equations coupling them together is well suited in circumstances where one is dealing with a large symmetry group and allows one to switch on interactions in a manifestly Lorentz covariant manner. However, as of today, no fully interacting theory analogous to Vasiliev theory is known in flat spacetimes. In the present work we focus on the case of 2+1 spacetime dimensions, which brings with it several simplifications. The purely topological nature of three-dimensional Einstein gravity allows one to recast it in form of a Chern-Simons gauge theory and provides a particularly simple introduction of massless higher-spin gauge fields, just by replacing the underlying isometry algebra by a suitable higher-spin extension. The goal is to add additional degrees of freedom in the form of massive scalar or higher-spin fields linearly coupled to the gravitational background. Higher-Spin Gravity and Scalar Fields in Flat Space We briefly review some important aspects of three-dimensional (higher-spin) gravity in asymptotically flat spacetimes, thereby fixing our notation. As far as the higher-spin generalisation is concerned, we will be following the proposal of [34]. Chern-Simons Formulation It is a well known fact that three-dimensional gravity can be formulated as a Chern-Simons gauge theory [55]. The gauge field A is a flat one-form, dA + A ∧ A = 0, and it takes values in an isometry algebra that depends on the sign of the cosmological constant. In the case of asymptotically flat spacetimes the latter is the Poincaré algebra iso(2,1) ≃ isl(2,R) ≃ sl(2,R) R 3 , spanned by generators J m , P m (m ∈ {0, ±1}) with Lie brackets [J m , J n ] = (m − n)J m+n , (2.1a) [J m , P n ] = (m − n)P m+n , (2.1b) [P m , P n ] = 0 . (2.1c) It is convenient to split the gauge field into spin connection and vielbein, A = ω + e, the first being an element of the sl(2,R)-subalgebra, ω = ω m J m , the latter being an element of the subalgebra of translations, e = e m P m . A suitable framework for studying flat-space holography is provided by Einstein gravity in Bondi gauge, 1 in which the most general asymptotically flat solution is given by the spin con-nection and vielbein gauge fields [6,7,57] ω = J 1 − M (φ) 4 J −1 dφ , (2.2a) e = P 1 − M (φ) 4 P −1 du + 1 2 P −1 dr + r P 0 − N (u, φ) 2 P −1 dφ , (2.2b) where the mass aspect M (φ) and the angular momentum aspect N (u, φ) are related by the integrability condition ∂ φ M (φ) = 2∂ u N (u, φ), but otherwise arbitrary functions. These fields correspond to the metric ds 2 = M (φ)du 2 − 2dudr + 2N (u, φ)dudφ + r 2 dφ 2 (2.3) in outgoing Eddington-Finkelstein coordinates. Special cases include [1,58] Minkowski space- time (M (φ) = −1, N (u, φ) = 0) and flat-space cosmologies (M (φ) = M > 0, N (u, φ) = N = 0). An intriguing feature of the Chern-Simons formulation of gravity is its straightforward generalisation to higher-spin gravity, simply by letting the gauge field take values in some suitable higher-spin algebra, which is a well-known procedure both in AdS [59][60][61][62][63][64] and flat spacetimes [2,6,7,11]. One candidate for such an algebra will be reviewed in the following subsection. The choice of that particular algebra is determined by its being the only known model that allows the introduction of matter coupling. Flat-Space Higher-Spin Gravity In a recent proposal [34] it was suggested to use a quotient of the universal enveloping algebra (UEA) of isl(2,R) to introduce higher-spin charges to asymptotically flat spacetimes. Let us summarise the set-up. The associative algebra U (isl(2, R)) is spanned by (ordered) formal products of the generators J m , P m and contains two second-order Casimir elements, M 2 = P 0 P 0 − P 1 P −1 , (2.4a) S = J 0 P 0 − 1 2 (J 1 P −1 + J −1 P 1 ) , (2.4b) which generate an ideal that can be quotiented out by setting them to a multiple of the unit element. 2 The resulting quotient is an associative algebra, which we will denote by ihs(M 2 , S) = U (isl(2, R)) M 2 , S , (2.5) defines a Lie algebra by the usual identification of the commutator as Lie bracket. In order to find a convenient basis for the algebra we classify its elements with respect to their behaviour under the adjoint action of sl(2,R)-generators J m . Any element that commutes with J 1 we call highest weight and one easily sees that any highest-weight element can be built from powers of J 1 , powers of P 1 , powers of C ≡ J 0 J 0 − J 1 J −1 + J 0 , (2.6) which commutes with any J m , since it is the Casimir element of the sl(2,R)-subalgebra, and single factors of J 0 P 1 − J 1 P 0 , thus forcing the introduction of three parameters (l , s, ξ) to label a complete set of highest-weight generators; we define Q l s ξ s−1−ξ :=    (J 1 ) l −ξ C ξ 2 (P 1 ) s−1−l , ξ even, (J 1 ) l −ξ C ξ 2 (J 0 P 1 − J 1 P 0 ) (P 1 ) s−2−l , ξ odd. (2.7) A complete set of algebra generators can then be defined by repeated adjoint action of J −1 on highest-weight generators, thereby introducing a mode index m, Q l s ξ m := (−1) s−1−ξ−m (s + m − ξ − 1)! (2s − 2ξ − 2)! ad s−1−ξ−m J −1 Q l s ξ s−1−ξ . (2.8) Values of the indices s ∈ N, l , ξ ∈ N 0 , m ∈ Z are by construction restricted to s ≥ 1 , 0 ≤ ξ ≤ 2 s − 1 2 , \|m\| ≤ s − 1 − ξ , ξ ≤ l ≤ s − 1 , ξ even s − 2 , ξ odd (2.9) and the definition implies the standard commutation relation Q l s ξ m , J n = (m − (s − ξ − 1)n) Q l s ξ m+n . (2.10) We will denote the associative product of ihs(M 2 , S) in this basis as ⋆-product. A number of results on products and commutators of the above defined generators as well as remarks on the relation of ihs(M 2 , S) to anİnönü-Wigner contraction from the AdS case can be found in [34]. In appendix B.1 we provide the product rules that are needed for the present considerations. Of particular interest will be the Lie-subalgebra spanned by generators of indices l ∈ {0, 1}, ξ = 0. Let ⌊ s+t−4 2 ⌋ u=0 g st u (m, n) J s+t −2u−2 m+n + S M 2 ⌊ s+t−3 2 ⌋ u=0 ug st u (m, n) P s+t −2u−2 m+n , (2.11a) J s m , P t n = 1 2 ⌊ s+t−3 2 ⌋ u=0 g st u (m, n) P s+t −2u−2 m+n , (2.11b) P s m , P t n = 0 , (2.11c) with structure constants (2.12) and the mode functions g st u (m, n) ≡ (−1) u M 2u 4 2u u! N st 2u+1 (m, n) (s − 3 /2) u (t − 3 /2) u (s + t − u − 5 /2) uN st u (m, n) = u k=0 (−1) k u k (s − 1 + m) u−k (s − 1 − m) k (t − 1 + n) k (t − 1 − n) u−k . (2.13) The symbol a k = a(a − 1) . . . (a − k + 1) denotes the descending factorial. It is possible to write down a simple expression for the product of two (l = 0)-generators; it reads P s m ⋆ P t n = ⌊ s+t−2 2 ⌋ u=0 (−1) u M 2u 4 2u u! N st 2u (m, n) (s − 3 /2) u (t − 3 /2) u (s + t − u − 3 /2) u P s+t −2u−1 m+n . (2.14) Note that, while the purely translational generators P s m span an ideal of the Lie algebra, the Lorentz-like generators J s m do not span a Lie-subalgebra unless we set S = 0. This case was further discussed in [4], there denoted ihs 3 [∞], and shown to be a realisation of the Schouten bracket of Killing tensors of arbitrary rank. Note that the authors of [4] also discuss an alternative higher-spin Lie algebra, which can be obtained as another quotient of U (isl(2, R)) by identification of J m P n ∼ P m J n and P m P n ∼ 0 (first discussed in [34] as "left slice"). With the above construction at hand it is possible to introduce higher-spin charges Z (s) (φ) and W (s) (u, φ), where s ≥ 2 upon including the classical functions mass aspect Z (2) (φ) ≡ M (φ) and angular momentum aspect W (2) (u, φ) ≡ N (u, φ). In a Drinfeld-Sokolov-like gauge we may consider the connections ω = J 1 − 1 4 ∞ s=2 Z (s) (φ) J s −s+1 dφ , (2.15a) e = P 1 − 1 4 ∞ s=2 Z (s) (φ) P s −s+1 du + 1 2 P −1 dr + r P 0 − 1 2 ∞ s=2 W (s) (u, φ) P s −s+1 dφ , (2.15b) which imply vanishing curvature and torsion upon imposing ∂ φ Z (s) (φ) = 2∂ u W (s) (u, φ). Matter Coupling in (Higher-Spin) Gravity In this section we intend to review and clarify a prescription that allows to introduce propagating massive matter fields to a classical or higher-spin background, which was first introduced in [34]; some of the results presented here are more general. Introducing a master field C as an ihs(M 2 , S)-valued zero-form, the proposed coupling equation takes on the form of a covariant constancy condition (see [33] for a related construction in the case of asymptotically AdS spacetimes) and reads DC ≡ dC + [ω ,C ] ⋆ + e ⋆C = 0 . (2.16) The derivative so defined obeys D 2 = 0, given flatness and vanishing torsion of the gauge fields. Gauge Invariance A necessary requirement on the above equation is its invariance under finite Poincaré trans- formations, i.e. under gauge transformations of the group ISO(2, 1) ≃ ISL(2, R). The trans- formation behaviour of a Chern-Simons gauge field A → g −1 Ag + g −1 dg , with g ∈ ISL(2, R), upon splitting the group element into a Lorentz and a translation part, g = g T g L , g L = exp ξ m L J m ≡ e ξ L , g T = exp ξ m T P m ≡ e ξ T ,(2.17) implies the following transformation behaviour of spin connection and vielbein: ω → g −1 L (ω + d) g L , (2.18a) e → g −1 L (e + [ω, ξ T ] + dξ T ) g L . (2.18b) Then equation (2.16) remains invariant under ISL(2, R)-transformations, if we prescribe to the master field the transformation behaviour C → g −1 L g −1 T C g L . (2.19) Unfolded Klein-Gordon Equation A simple starting point for the following considerations is to restrict the master field to be purely translational, i.e. to expand C = ∞ s=1 \|m\|≤s−1 c s m P s m , (2.20) where the coefficients depend on the spacetime coordinates, c s m = c s m (u, r, φ). Using this expansion as well as the classical gauge fields (2.2) and the product rules (2.14), the master equation results in the set of equations on the coefficients 0 = ∂ u c s m + c s−1 m−1 − (s + 1 − m) 2 M 2 4(s + 1 /2) 2 c s+1 m−1 − M 4 c s−1 m+1 − (s + 1 + m) 2 M 2 4(s + 1 /2) 2 c s+1 m+1 , (2.21a) 0 = ∂ r c s m + 1 2 c s−1 m+1 − 1 2 (s + 1 + m) 2 M 2 4(s + 1 /2) 2 c s+1 m+1 , (2.21b) 0 = ∂ φ c s m + (s − m)c s m−1 + M 4 (s + m)c s m+1 + r c s−1 m + (s + m)(s − m)M 2 4(s + 1 /2) 2 c s+1 m − N 2 c s−1 m+1 − (s + 1 + m) 2 M 2 4(s + 1 /2) 2 c s+1 m+1 , (2.21c) where the coordinate dependence of all functions is suppressed in the notation. This set can be reduced to a second-order equation for the lowest-spin component φ ≡ c 1 0 , which reads (0) − M 2 φ = 0 , (2.22) with the operator (0) ≡ −M + N 2 r 2 ∂ 2 r − 2∂ u ∂ r + 2N r 2 ∂ r ∂ φ + ∂ 2 φ r 2 − ∂ u r + −M − N 2 r 2 + ∂ φ N r ∂ r r − N r 3 ∂ φ , (2.23) where M = M (φ) and N = N (u, φ), being the d'Alembert operator in the metric (2.3). This consideration shows that, first, the master equation (2.16) indeed provides an unfolded formulation of the Klein-Gordon equation (2.22) and, second, that the parametrisation of the Casimir element M 2 is to be identified with the mass squared of the Klein-Gordon field (in close analogy to the AdS case [65]). Generalised Klein-Gordon Equation We may use the higher-spin gauge fields (2.15), while still keeping the ansatz (2.20). First, writing out the matter-coupling equation in spacetime components on the level of the master field, we have 0 = ∂ u C + P 1 ⋆C − 1 4 ∞ s=2 Z (s) (φ) P s −s+1 ⋆C , (2.24a) 0 = ∂ r C + 1 2 P −1 ⋆C , (2.24b) 0 = ∂ φ C + [J 1 ,C ] + r P 0 ⋆C − 1 4 ∞ s=2 Z (s) (φ) J s −s+1 ,C ⋆ + 2W (s) (u, φ) P s −s+1 ⋆C . (2.24c) Using the commutation relations (2.11) and the product rules (2.14) this yields the set of firstorder differential equations that we noted down in (B.3) of the appendix. However, a significantly easier way to proceed is to stay at the level of the master field and to try and assemble the charge-free part of the scalar Klein-Gordon operator −2∂ u ∂ r − ∂ u /r + ∂ 2 φ /r 2 acting on C from equations (2.24). Comparing coefficients of the unit element, we arrive at the generalised Klein-Gordon equation (hs) − M 2 φ = 0 (2.25) for φ ≡ c 1 0 with the higher-spin Klein-Gordon operator (hs) ≡ ∞ s=2 (−1) s−1 2 s−2 Z (s) r ∂ r r ∂ s−1 r − 1 r ∂ φ W (s) ∂ s−1 r − W (s) r ∂ s−1 r ∂ φ r + 1 r ∞ s,s ′ =2 (−1) s+s ′ 2 s+s ′ −4 W (s) W (s ′ ) ∂ s−1 r ∂ s ′ −1 r r − 2∂ u ∂ r − ∂ u r + ∂ 2 φ r 2 , (2.26) where Z (s) = Z (s) (φ) and W (s) = W (s) (u, φ). All in all, it appears reasonable to assume that the dynamics of a scalar field, both on a classical and a higher-spin background, are accurately captured by equation (2.16) if the master field is restricted to the (l = 0)-part of ihs(M 2 , S); its lowest component c 1 0 is to be identified with the physical scalar field φ, while all higher components are auxiliary fields that can be expressed through derivatives of c 1 0 . Massive Fields of Arbitrary Spin Unfolded In this section we will show how the inclusion of higher-l generators into the expansion of the master field C allows the description of massive fields of spin l . As it turns out, it will in general be possible to to truncate the master field to some finite ℓ, thus including fields of spin 0, 1, . . . , ℓ. Letting ℓ → ∞ one obtains the complete Fierz-Pauli system. In this section we will use our master equation with a slight modification, where the vielbein is acting multiplicatively from the right, dC + [ω ,C ] ⋆ +C ⋆ e = 0 . (3.1) This will not alter the physical content of the equation and is only due to the significantly easier form of the product rules Q l s ξ m ⋆ P n compared to left multiplication. 3 Recovering the Proca Equation in Minkowski Spacetime To illustrate the working mechanism we start with the case of the spin-1 massive field, using Minkowski spacetime as background geometry for simplicity. We expand the master field into generators of l = 0 and l = 1: C = ∞ s=1 \|m\|≤s−1 c 0 s 0 m Q 0 s 0 m + ∞ s=2 \|m\|≤s−1 c 1 s 0 m Q 1 s 0 m + ∞ s=3 \|m\|≤s−2 c 1 s 1 m Q 1 s 1 m , (3.2) such that the lowest components, respectively the fundamental fields, are the scalar field c 0 1 0 0 at l = 0 and c 1 2 0 m with m ∈ {0, ±1} at l = 1. Fields of index s ≥ 2 at l = 0 or s ≥ 3 at l = 1 as well as fields of index ξ = 1 are generally considered auxiliary. We will argue below that this identification is justified. The gauge background in the case of Minkowski spacetime is given by (2.2) with M = −1 and N = 0, such that, in components, equation (3.1) reads 0 = ∂ u C +C ⋆ P 1 + 1 4 P −1 , (3.3a) 0 = ∂ r C + 1 2 C ⋆ P −1 , (3.3b) 0 = ∂ φ C + J 1 + 1 4 J −1 ,C + r C ⋆ P 0 . (3.3c) Using the commutation relations (2.10) and the product rules (B.1) we can write out these expressions in algebra components and find the following set of first-order differential equations (unfolded equations): l = 0, ξ = 0 : 0 = ∂ u c 0 s 0 m − 1 2 ∂ r c 0 s 0 m + c 0 s−1 0 m−1 − (s + 1 − m) 2 4s 2 (s + 1 /2) 2 s 2 M 2 c 0 s+1 0 m−1 + S c 1 s+1 0 m−1 − (s − m)S s 2 c 1 s+1 1 m−1 , (3.4a) 0 = ∂ r c 0 s 0 m + 1 2 c 0 s−1 0 m+1 − (s + 1 + m) 2 8s 2 (s + 1 /2) 2 s 2 M 2 c 0 s+1 0 m+1 + S c 1 s+1 0 m+1 + (s + m)S 2s 2 c 1 s+1 1 m+1 , (3.4b) 0 = ∂ φ c 0 s 0 m + (s − m) c 0 s 0 m−1 − s + m 4 c 0 s 0 m+1 + r c 0 s−1 0 m + (s + m)(s − m)r 4s 2 (s + 1 /2) 2 s 2 M 2 c 0 s+1 0 m + S c 1 s+1 0 m + mr S s 2 c 1 s+1 1 m ; (3.4c) l = 1, ξ = 0 : 0 = ∂ u c 1 s 0 m − 1 2 ∂ r c 1 s 0 m + c 1 s−1 0 m−1 − (s 2 − 1)(s + 1 − m) 2 M 2 4s 2 (s + 1 /2) 2 c 1 s+1 0 m−1 + (s − m)M 2 s 2 c 1 s+1 1 m−1 ,(3. 4d) 3 Note that, in the case of right multiplication, the gauge transformation behaviour (2.19) of the master field must be modified to be C → g −1 1 m , only, one may combine various equations in such a way that all higher-spin auxiliary fields get eliminated. A linear combination of the first-order equations yields L C g −1 T g L . 0 = ∂ r c 1 s 0 m + 1 2 c 1 s−1 0 m+1 − (s 2 − 1)(s + 1 + m) 2 M 2 8s 2 (s + 1 /2) 2 c 1 s+1 0 m+1 − (s + m)M 2 2s 2 c 1 s+1 1 m+1 , (3.4e) 0 = ∂ φ c 1 s 0 m + (s − m) c 1 s 0 m−1 − s + m 4 c 1 s 0 m+1 + r c 1 s−1 0 m + (s 2 − 1)(s + m)(s − m)r M 2 4s 2 (s + 1 /2) 2 c 1 s+1 0 m − mr M 2 s 2 c 1 s+1 1 m ; (3.4f) l = 1, ξ = 1 : 0 = ∂ u c 1 s 1 m − 1 2 ∂ r c 1 s 1 m + c 1 s−1 1 m−1 − s(s − 2)(s − m) 2 M 2 4(s − 1) 2 (s − 1 /2) 2 c 1 s+1 1 m−1 + s − 1 − m (s − 1) 2 c 1 s−1 0 m−1 , (3.4g) 0 = ∂ r c 1 s 1 m + 1 2 c 1 s−1 1 m+1 − s(s − 2)(s + m) 2 M 2 8(s − 1) 2 (s − 1 /2) 2 c 1 s+1 1 m+1 − s − 1 + m 2(s − 1) 2 c 1 s−1 0 m+1 , (3.4h) 0 = ∂ φ c 1 s 1 m + (s − 1 − m) c 1 s 1 m−1 − s − 1 + m 4 c 1 s 1 m+1 + r c 1 s−1 1 m + s(s − 2)(s − 1 + m)(s − 1 − m)r M 2 4(s − 1) 2 (s − 1 /2) 2 c 1 s+1 1 m − mr (s − 1) 2 c 1 s−1 0 m .∂ u − 1 2 ∂ r − 1 2r c 1 2 0 1 + 1 r ∂ φ c 1 2 0 0 + 2 ∂ r + 1 r c 1 2 0 −1 = 0 ,(3.− M 2 φ µ = 0 , ∇ µ φ µ = 0 ,(3.7) when the field is written in the basis given by the vielbein (2.2b) for M = −1 and N = 0, i.e. φ u = − 1 2 φ 1 + 4φ −1 , φ r = −φ 1 , φ φ = r φ 0 ,(3.8) and, finally, using the identification c 1 2 0 m = φ m . To conclude this example, we have to address the question of the actual independence of fundamental fields of different spin from each other, since the appearance of higher-spin terms in lower-spin equations, particularly (3.4a), (3.4b) and (3.4c), may lead to the suspicion that the fundamental fields are not independent after all. For example, it might be the case that the Proca field can be directly expressed in terms of derivatives of the scalar field, thus rendering it auxiliary, or that the ubiquitous auxiliary fields give rise to a non-trivial coupling in the form of higher-order differential equations. This is, however, not the case. 0 m . Only in the case of highest-or lowest-weight components one has to resort to φ-equations. This uses up a significant number of equations already. For the moment, let us argue only through explicit calculation: one may use the recurrence relations to evaluate all remaining first-order equations at lower orders of the index s and find that all these are satisfied automatically, once the equations of motion (3.5) and (3.6) are employed. This is not a proof, of course, but we will strengthen this observation in the general case below. Recovering the Fierz-Pauli Equations in Asymptotically Flat Spacetimes The same procedure that was used in the previous subsection can be applied in the case of a master field that is truncated to l ≤ 2, which should then result in the equations of linearised topologically massive gravity in addition to the Proca and Klein-Gordon equations. This is indeed the case, only with the peculiarity the an additional scalar field shows up in the form of c 2 3 2 0 , giving rise to the question whether or not the Fierz-Pauli fields here described are actually traceless. This new scalar field, however, completely decouples from the rest and could, in principle, be set to zero in the end. We will not present the details of the spin-two case but rather step forward to the general case, immediately. where the sum runs over all index combinations (l , s, ξ, m) allowed by (2.9). If necessary, one may still think of the expansion as being truncated to some finite index ℓ, which is simply achieved by setting components of l > ℓ to zero. Though the lowest components at fixed l are c s−1 s 0 m with \|m\| ≤ s − 1, there is more to take care about: our notion of spin is tied to the behaviour of algebra generators under the commutation relation (2.10) and translates to the spin we associate to any particular field. It may therefore not come as a surprise that we actually have to take into account all fields c s−1 s ξ m with \|m\| ≤ s − ξ − 1 as being fundamental fields of spin σ = s − ξ − 1, where 0 ≤ ξ ≤ 2⌊(s − 1)/2⌋ and ξ is an even number. That is, for any spin σ we get an infinite number of massive fields, as long as no truncation is made. The gauge background in the case of a general asymptotically flat spacetime is given by (2.2), such that equation (3.1) reads 0 = ∂ u C +C ⋆ P 1 − M 4 P −1 , (3.10a) 0 = ∂ r C + 1 2 C ⋆ P −1 , (3.10b) 0 = ∂ φ C + J 1 − M 4 J −1 ,C +C ⋆ r P 0 − N 2 P −1 . (3.10c) Coordinate dependencies are again suppressed in the notation. Using the commutation relations (2.10) and the product rules (B.1) we get three sets of first-order differential equations on c l s ξ m , which can be found in the appendix as equations (B.6), and we are interested in a linear combination of these equations that de-couples the field components c s−1 s ξ m from any other fields. Such a linear combination can be found and it results in the first-order equations 0 = ∂ u + M 2 ∂ r + M 2r (s − ξ − m) c s−1 s ξ m + 2 r s − ξ − m s − ξ + m − 1 ∂ φ + N ∂ r c s−1 s ξ m−1 + 2 (s − ξ − m + 1) 2 (s − ξ + m − 1) 2 ∂ r + s − ξ + m − 2 r c s−1 s ξ m−2 . (3.11) Here −(s − ξ − 3) ≤ m ≤ s − ξ − 1, such that there are 2s − 2ξ − 3 equations at fixed s − ξ. In order to find a set of second-order partial differential equations for the set of field components c s−1 s ξ m it is convenient to first assemble the scalar Klein-Gordon operator (0) − M 2 from the equations (3.10), which results in (0) − M 2 C = 1 r 2 ad 2 ω φ (C ) + 2 r ad ω φ C ⋆ P 0 + N 2r 2 C + ∂ φ M 4r 2 [J −1 ,C ] ,(3.r 2 (s − ξ − m) ∂ φ + N ∂ r − N 2r c s−1 s ξ m−1 + M 2r 2 (s − ξ + m) ∂ φ + N ∂ r − N 2r + ∂ φ M 2M c s−1 s ξ m+1 + 1 r 2 (s − ξ − m + 1) 2 c s−1 s ξ m−2 + M 4r 2 (s − ξ + m + 1) 2 c s−1 s ξ m+2 . (3.13) We will in the following show that equations (3.11) and ( φ (1) s−1+m 2 (−1) s−1−m 2 = (s − 1 − m)!(s − 1 + m)! (2s − 2)! c s−1 s 0 m , (3.18a) φ (1) s−1+m 2 0 (−1) s−1−m 2 = 2(s − 1 − m)!(s − 1 + m)! (2s − 2)! c s−1 s 0 m (3.18b) and m 2 = M 2 we find the wave equation in (1.1) to be precisely (3.13) for ξ = 0 and the divergence equation to be (3.11) for ξ = 0. Obviously, the basis change works along the same lines if we label the spin by σ = s − ξ − 1 and include this additional index in (3.18), such that we arrive at equations (3.13) and (3.11) for ξ = 0. A few comments regarding the field content are in order. The number of additional fields introduced by even values of the index ξ exactly matches the number of additional fields that would be introduced if the tracelessness-condition of the Fierz-Pauli system was not imposed. In the latter case, however, a field of spin σ would be coupled to a field of spin σ − 2, which is not the case in our system -all fundamental fields are completely decoupled (see also the next subsection). In particular, we could set all fundamental higher-ξ fields to zero without any obstacles. Decoupling of Fundamental Fields So far we did not make sure that the divergence equations (3.11) and the wave equations (3.13) are the only constraints on the fundamental fields c s−1 s ξ m and that there are no additional couplings between these fields, e.g. via higher-derivative equations. The rather involved form of the unfolded equations (B.6), in which fields of different l -slices are mixed together, makes it hard to see if there are any such constraints introduced through the interplay of auxiliary fields. We will in the following argue that this is not the case. Replacing the derivative therms in this equations using the first-order equations one can find conditions on the coefficients α µ (s, ξ, m) and α(s, ξ, m). These are 0 = 1 n=−1 N s−ξ,2 1 (m, n)e n µ α µ (s, ξ, m + n) , (3.22a) 0 = 1 n=−1 N s−ξ+1,2 2 (m, n)e n µ α µ (s, ξ, m + n) , (3.22b) as well as α(s, ξ, m) = (s − ξ − m − 1)α φ (s, ξ, m + 1) + (s − ξ + m − 1)M /4α φ (s, ξ, m − 1) . The important point is that coefficients of different parameters s or ξ do not mix, such that there is one set of relations for every fundamental field. Accordingly, the divergence equations we presented in (3.11) are the only first-order differential equations of the general form (3.21) allowed by the unfolded equations (as long as no purely algebraic relations between fields are imposed). We repeat the same procedure for the case of second-order differential equations. The most general expression involving only fundamental fields in this case is (m + n, n ′ ) e n µ e n ′ ν α µν (s, ξ, m + n + n ′ ) , (3.24b) as well as equations for α µ (s, ξ, m), 2 2 (m, n) e n µ α µ (s, ξ, m + n) + ∂ µ e n ν α µν (s, ξ, m + n) 2 2 (m, n) 2 1 (m, n) e n µ α µ (s, ξ, m + n) + ∂ µ e n ν α µν (s, ξ, m + n) 1 n=−1 N s−ξ+1,= 1 n=−1 e n µ (s − ξ − m − n − 1)N s−ξ+1,2 2 (m, n) +(s − ξ − m)N s−ξ+1 2 (m + 1, n) α µφ (s, ξ, m + n + 1) + M 4 1 n=−1 e n µ (s − ξ + m + n − 1)N s−ξ+1,+(s − ξ + m)N s−ξ+1,2 2 (m − 1, n) α µφ (s, ξ, m + n − 1) , (3.25a) 1 n=−1 N s−ξ,= 1 n=−1 e n µ (s − ξ − m − n − 1)N s−ξ,2 1 (m, n) +(s − ξ − m − 1)N s−ξ 1 (m + 1, n) α µφ (s, ξ, m + n + 1) + M 4 1 n=−1 e n µ (s − ξ + m + n − 1)N s−ξ,2 1 (m, n) +(s − ξ + m − 1)N s−ξ,2 1 (m − 1, n) α µφ (s, ξ, m + n − 1) , (3.25b) and for α(s, ξ, m), In principle, one could carry on with this kind of calculation to higher-order differential equations, the repeated insertion of ∂ µ c-equations, however, introduces more and more auxiliary fields, and makes the calculation increasingly involved. To convince us of the decoupling of field we instead considered the TMG case, i.e. the case in which the fundamental field components present are two scalars, c We speculate that a general proof of the decoupling of fundamental equations is possible on a purely algebraic level, only by utilising the equation of motion (3.20). Since the commutators involved do not change the indices l , s or ξ of a generator, the repeated action of differential operators D m can be traced back to right multiplication with translational generators, as far as operations are concerned that do change these indices. This observation may be a starting point for a general argumentation in favour of decoupling, however, we were so far not able to construct an adequate proof. α(s, ξ, m) = (s − ξ − m − 1)α φ (s, ξ, m + 1) + (s − ξ + m − 1) M 4 α φ (s, ξ, m − 1) + (s − ξ + m − 1) M ′ 4 α φφ (s, ξ, m − 1) − (s − ξ − m − 1) 2 α φφ (s, ξ, m + 2) − (s − ξ) 2 − m 2 M 2 α φφ (s, ξ, m) − (s − ξ + m − 1) 2 M Summary and Outlook The main objective of this work was to shed some light on the linear coupling of massive matter fields to a gravitational (higher-spin) background, here in the case of three-dimensional asymptotically flat space-times. We showed that a covariant constancy condition evaluated on a quotient of the universal enveloping algebra of the isometry algebra, here the threedimensional Poincaré algebra, gives rise to a set of unfolded equations that can ultimately be reduced to a system of Fierz-Pauli equations for free massive higher-spin fields, all possessing the same mass. The theory contains an infinite number of copies of each higher-spin field, but no coupling between these fields is imposed. The picture that emerges here gives an appealing interpretation to the structure of the algebra ihs(M 2 , S), namely to the infinite wedge spanned due to the appearance of different powers of Lorentz and translation generators in the UEA. We expect that a similar result can be achieved in the case of asymptotically AdS spacetimes by considering the larger higherspin algebra that is obtained as a quotient of U (sl(2, R)⊕sl(2,R)). Furthermore, we speculate that the algebra obtained from the additional quotient P m P n ∼ 0 of ihs(M 2 , S) plays a role in the case of massless higher-spin fields. While in the first part of the paper we reviewed and generalised a possible formulation of higher-spin gravity, i.e. the introduction of an infinite tower of massless higher-spin gauge fields in the Chern-Simons formalism together with suitable boundary conditions, in the second part we discussed the propagation of massive higher-spin fields in a classical, i.e. spin 2, background, only. Though we expect the basic idea to remain valid in the case of a higherspin gauge background, explicit calculations would require full knowledge of the structure constants of the associative product of ihs(M 2 , S), which are currently not at our disposal. It is however clear that no truncation in the spin of the massive fields will be possible anymore, since the commutator of the matter field with a higher-spin deformed spin connection will produce higher-l terms. The proposal here described is so far free of any interactions, be it of the massive higherspin fields amongst each other or a back-reaction to the gauge background, which, in a sense, still renders the theory trivial in the sense that no known no-go theorems apply. A natural next step that will be undertaken is the introduction of interactions via suitable gauge potentials. Possibly, the infinite number of copies of higher-spin fields we found in our approach may play a role there. In this context it would also be useful to find connections to other unfolding proposals, such as [66]. In a broader context, our proposal may be viewed as a necessary first step in a bottom-up approach to the construction of a flat-space counterpart to Vasiliev theory; if existing, such a model should certainly contain the unfolded equations here presented as linearised version. As an application of our approach in the realm of holography it could be interesting to try and derive three-point correlation functions involving one scalar or arbitrary-spin current on the dual field-theory side (which is expected to be a Carrollian field theory). A further interesting question concerns the applicability of the method here described to 3+1 or, more general, d + 1 spacetime dimensions. Though the higher-dimensional gravitational theories are not topological theories anymore, it should still be possible to implement (linear) matter coupling through a suitable covariant-constancy condition on the universal enveloping algebra of the respective Poincaré algebra. A Metric Quantities and Spin Connection In the case of asymptotically flat spacetimes the metric and its inverse read g µν =   M (φ) −1 N (u, φ) −1 0 0 N (u, φ) 0 r 2   , g µν =    0 −1 0 −1 −M (φ) + N (u,φ) 2 r 2 N (u,φ) r 2 0 N (u,φ) r 2 1 r 2    . (A.1) The non-vanishing Christoffel symbols are Γ u φφ = r , Γ r uφ = Γ r φu = − M ′ (φ) 2 , Γ r r φ = Γ r φr = N (u, φ) r , Γ r φφ = r M (φ) − N (u, φ) 2 r − ∂ φ N (u, φ) , Γ φ r φ = Γ φ φr = 1 r , Γ φ φφ = − N (u, φ) r . (A.2) In isl(2,R)-components we have for the spin connection ω mn = −ε mnk ω k , where ε −101 = 1. Accordingly, for asymptotically flat spacetimes ω φ 1 −1 = 0 = ω φ −1 1 , ω φ 1 0 = 1 , ω φ 0 1 = M (φ) 2 , ω φ −1 0 = M (φ) 4 , B Collection of Lengthy Expressions In order to spare the reader the trouble of leafing through a whole lot of less handsome equations but still provide a complete communication of our results, we decided to put some of these lengthy expressions into this appendix. B.1 Product Rules of the Higher-Spin Algebra The calculations in the main part of this paper require some knowledge about the ihs(M 2 , S)product rules. In particular, one needs the following spin-s-spin-2 products: The product rules were taken from [34], where a couple of further identities can be found. B.2 Unfolded Scalar Field The case of a scalar field of mass M propagating in a higher-spin deformed gauge background, where Z (s) = Z (s) (φ) and W (s) = W (s) (u, φ) with s ≥ 2 are higher-spin charges (generalising the classical charges M (φ) ≡ Z (2) (φ) and N (u, φ) ≡ W (2) (u, φ)) is discussed in section 2.3. We derived the following first-order differential equations for components of the master field: us clear up notation by defining J s m := (s − 1) Q 1 s 0 m and P s m := Q 0 s 0 m . The brackets of this Lie algebra read the (l = 0)-equations are now coupled to components of l = 1, both through auxiliary fields and on the level of the fundamental fields. However, we can still reduce this set of equations to the Klein-Gordon equation(2.22), only containing the scalar field.Considering the above expressions for c d'Alembert operator (0) is defined in (2.23). These equations are precisely the Proca equations for a spin-one field of mass M , and similarly one of the equations (3.4a) or (3.4b) as recurrence relations for c 0 s The master field is expanded into the complete set of ihs(M 2 , S)-generators like 12) where the scalar d'Alembert operator (0) was defined in (2.23). Finally, by using the firstorder equations for ∂ from any other fields one arrives at the set of equations0 = (0) + M 2r 2 (s − ξ) 2 − m 2 − M 3.13) are indeed the Fierz-Pauli equations (1.1) for a field of mass M and spin s − ξ − 1 that freely propagates on an asymptotically flat spacetime.Let us start by re-writing the Fierz-Pauli equations (1.1) for a field of spin σ = s−1. Changing from spacetime indices to flat indices by using the vielbein,φ µ 1 ...µ s−1 −1 η m 1 n 1 . . . η m s−1 n s−1 φ m 1 ...m s−1 ∇ µ − m 2 φ m 1 ...m s−1 = 0 , e µ m ∇ µ φ mm 2 ...m s−1 = 0 ,(3.15) where the covariant derivative now acts on φ m 1 ...m s−1 through the spin connection ω m n (noted down in (A.3) of the appendix). The irreducibility condition g µν φ µνµ 3 ...µ s−1 = 0 reduces to φ 00m 3 ...m s−1 = 4φ 1−1m 3 ...m s−1 . (3.16) Due to this condition and the fact that all fields are symmetric in their indices, we only need to consider fields of the form φ 1...1−1...−1 and φ 1...10−1...−1 ; let us introduce the short-hand (±1) k ≡ ±1 . . . action of the spin connection in the isl(2,R)-basis can then be obtained and we noted it down in section A of the appendix (as well as the necessary metric quantities). The final ingredient is the explicit identification of these fields with the fields c probably the most compact form that can be achieved. From this one may extract an equation for ∂ µ c l s ξ m . We start at the simplest case under question, namely coupling equations of the fundamental fields up to first order in derivatives. The most general form of such an (m + n, n ′ )e n µ e n ′ ν α µν (s, ξ, m + n + n ′ (s, ξ, m)∂ µ ∂ ν c s−1 s ξ m + α µ (s, ξ, m)∂ µ c s−1 s ξ m + α(s, ξ, m) c s−1 s ξ m = 0 . (3.23)Then we get a set of recurrence relations for α µν (s, ξ, m), + n, n ′ ) α µν (s, ξ, m + n + n ′ ) .(3.26) These relations show that the independence of different fundamental fields c s−1 s ξ m (i.e., fields of different s and ξ) holds up to second order in derivatives as well. . We evaluated the general ansatz of the form (3.23) up to third and fourth order and found explicit solutions for the α-coefficients using Mathematica. Indeed, there is no mixture in these cases of coefficients of different indices s and ξ, thus showing the decoupling of the scalar, Proca and TMG fields up to fourth order in derivatives. ωω moved using the form η mn = (−1) m (1 + m)!(1 − m)! δ m+n,0 . The action of the spin connection in the isl(2,R)-basis for the relevant cases of ordered indices with at most one zero can be determined as (m 1 , . . . , m s−1 ) = (1) · ω · φ m 1 ...m s−1 = (s − 1 + m) · ω · φ m 1 ...m s−1 = (s − 1 + m) also give the inverse components of the vielbein, e µ m = η mn g µν e n ν , which are e = e µ m P m ∂ µ = P 1 ∂ u + ∂ r + ∂ φ , e µ −1 ∂ µ = 2∂ r . (A.6) +− (ξ − 2⌊ ξ /2⌋) (l − ξ + 1m+n − (l − ξ) 2 (l − ξ − 1 /2) Q l −1 s ξ+1 m+n − (s − ξ − 1) 2 − (l − 2⌊ ξ /2⌋) 2 M 2 Q l s−1 ξ m+n − 2(l − ξ)(l − 2⌊ ξ /2⌋ − 1 /2)S Q l −1 s−1 ξ m+n + (l − 2⌊ ξ /2⌋ − 1)(l − 2⌊ ξ /2⌋(ξ − 2⌊ ξ /2⌋) (l − ξ + 1) 2 (l − ξ + 1 /2)M 2 Q l −1 s−2 ξ−1 m+n − 2(l − ξ) (l − ξ) of the mode functions N st u (m, n) is equation (2.13) of the main part and the special cases u = 1 and u = 2 with t = 2 appearing here readN s2 1 (m, n) = 2(m − (s − 1)n) , (B.2a) N s2 2 (m, n) = 2 m 2 + 2(s − 1)(s − 3 /2)n 2 − 2(s − 3 /2)mn − (s − 1) 2 . (B.2b) ′ + m ′ − 1) s ′ −s+m ′ −m M s ′ −s+m ′ −m Z (m ′ −m+1) 2 s ′ −s+m ′ −m+2 s ′ −s+m ′ For more general boundary conditions see, e.g.,[56]. Our notation does not distinguish between the number a Casimir element is parameterised with and the actual element of the UEA. AcknowledgementsThe authors would like to thank Xavier BekaertNon-existing index combinations in c s m or the charges are to be identified with zeros. 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[ "A framework for simulating systemic risk and its application to the South African banking sector", "A framework for simulating systemic risk and its application to the South African banking sector" ]	[ "N M Walters [email protected] ", "Fjc Beyers ", "A J Van Zyl ", "R J Van Den Heever ", "\nDepartment of Actuarial Science\nNM Walters\nUniversity of Pretoria\nHatfield\n", "\nDepartment of Actuarial Science\nDepartment of Mathematics and Applied Mathematics\nFJC Beyers\nUniversity of Pretoria\nHatfield AJ van Zyl\n", "\nUniversity of Pretoria\nHatfield\n" ]	[ "Department of Actuarial Science\nNM Walters\nUniversity of Pretoria\nHatfield", "Department of Actuarial Science\nDepartment of Mathematics and Applied Mathematics\nFJC Beyers\nUniversity of Pretoria\nHatfield AJ van Zyl", "University of Pretoria\nHatfield" ]	[ "SOUTH AFRICAN ACTUARIAL JOURNAL SAAJ" ]	We present a network-based framework for simulating systemic risk that considers shock propagation in banking systems. In particular, the framework allows the modeller to reflect a top-down framework where a shock to one bank in the system affects the solvency and liquidity position of other banks, through systemic market risks and consequential liquidity strains. We illustrate the framework with an application using South African bank balance sheet data. Spikes in simulated assessments of systemic risk agree closely with spikes in documented subjective assessments of this risk. This indicates that network models can be useful for monitoring systemic risk levels. The model results are sensitive to liquidity risk and market sentiment and therefore the related parameters are important considerations when using a network approach to systemic risk modelling.	10.4314/saaj.v18i1.5	null	169,918,648	1811.04223	47016e691729540454d39b1e35ad4c7d94f4fa10	A framework for simulating systemic risk and its application to the South African banking sector 2018 N M Walters [email protected] Fjc Beyers A J Van Zyl R J Van Den Heever Department of Actuarial Science NM Walters University of Pretoria Hatfield Department of Actuarial Science Department of Mathematics and Applied Mathematics FJC Beyers University of Pretoria Hatfield AJ van Zyl University of Pretoria Hatfield A framework for simulating systemic risk and its application to the South African banking sector SOUTH AFRICAN ACTUARIAL JOURNAL SAAJ 18201810.4314/saaj.v18i1.5Submission date 14 December 2017 Acceptance date 8 October 2018CONTACT DETAILSSystemic riskbanking networksnetwork structuremarket sentiment We present a network-based framework for simulating systemic risk that considers shock propagation in banking systems. In particular, the framework allows the modeller to reflect a top-down framework where a shock to one bank in the system affects the solvency and liquidity position of other banks, through systemic market risks and consequential liquidity strains. We illustrate the framework with an application using South African bank balance sheet data. Spikes in simulated assessments of systemic risk agree closely with spikes in documented subjective assessments of this risk. This indicates that network models can be useful for monitoring systemic risk levels. The model results are sensitive to liquidity risk and market sentiment and therefore the related parameters are important considerations when using a network approach to systemic risk modelling. INTRODUCTION 1.1 Systemic risk and the spread of financial contagion are important considerations for regulators tasked with overseeing stability of banking systems. Banking systems are at the core of a well-functioning financial system. A breakdown of the system would hinder economic growth, which in turn may cause permanent damage to the economy (Cerra & Saxena, 2017). Therefore, it is important for regulators to prevent such a breakdown from being triggered. Regulatory intervention at a late stage could prove to be costlier than intervention at an earlier stage. The burden of costly bailouts by the regulator are ultimately borne by the taxpayers, which negatively affects the economy. On the other hand, if banks are allowed to fail without any intervention, the economy can be strained by losses on investors' deposits, rising interest rates, possible bank runs etc. Monitoring the level of systemic risk in a financial system is therefore crucial for ensuring long-term stability and growth of an economy. 1.2 Liquidity and market sentiment are two key requirements for a working banking system that are also closely related. During times of economic distress, a lack of trust translates into a reluctance of non-bank financial institutions to renew funding to banks. They then impose more stringent lending requirements, which leads to increased risk premia on loans and debentures thereby increasing banks' wholesale funding costs. The higher interest rates charged on servicing new debt means that additional assets may need to be liquidated to service the debt or a reduction in asset origination, reducing (shrinking) the balance sheet sizes of the affected banks. This puts a strain on those banks' liquidity positions as the maturity mismatch between short-term liabilities and assets increases. Ultimately, when the funding costs become unsustainably high the bank may be forced to call in loans or liquidate assets prematurely. This, together with the increased funding costs can substantially reduce the bank's profitability and hence its retained earnings. This in turn reduces its Tier I capital, which may lead to solvency problems (Furceri & Mourougane, 2009). This creates a spiral of distrust. 1.3 The complex nature of banking systems remains difficult to replicate and model precisely. Bottom-up approaches using integrated modelling frameworks are very useful, yet they are difficult to calibrate, expensive and not readily available. This is because, in practice, such an approach would involve the regulator providing a specified scenario to all banks, after which the banks quantify their own risk position so that the regulator can then aggregate the risk positions (Borio et al., 2014). It is therefore of interest to find simplified models that consider the entire system from the start and can detect changes in systemic risk. We contribute to this by showing that network models of systemic risk can satisfy this requirement to a large extent. We illustrate how such a top-down model can be used, by applying it to real-world balance sheet data and showing that changes in risk are detected under times of market stress for various network structures. We turn our attention to problems in rolling forward short-term debt that are caused by frictions such as a lack of trust in the system. 1.4 The chain of events that we aim to model is as follows: One bank in the system experiences solvency problems, which may arise because of a significant increase in impairments from non-performing loans. This could be because of a number of causes such as unsustainable lending practices or a disruption in its target market (such as the mine closures experienced in South Africa). It is important to note that the applied model does not require us to specify the event that leads to the initial bank's default, nor do we attempt to model it. The equity of the aforementioned bank then declines, and shareholders need to absorb the losses (followed by other subordinated creditors). Now there are two key potential effects on the banking system. Firstly, other banks' balance sheets may be affected through a revaluation of assets and impairment provisions and they may need to raise additional impairment provisions (e.g. if the initial bank's troubles were due to increased impairments on a specific type of loan book, other banks may need to raise their impairment provisions for similar books to account for an anticipated rise in impairments). Another possibility is that the bank may ultimately need to resort to forced sales to generate liquidity. The increased supply of those assets in the market may depress their market value, leading to some (albeit limited) mark-to-market losses for other banks holding similar assets. For this study, the distinction between these possibilities (and hence the effect of the initial default on the banking or trading books of other banks) is not explicitly made. Here, we assume a net reduction in the balance sheets of other banks takes place, which could be due to an increase in required reserves or to a combination of this and mark-to-market losses. This approach is adopted to keep the model simple and consistent with existing models in the literature (see for example Nier et al. (2008), Gai & Kapadia (2010), May & Arinaminpathy (2010) Arinaminpathy et al. (2012)). It is worthwhile to note that the narrative of these papers focuses on the fire-sale aspect of this contagion mechanism. However, as this is a practical application within a specific financial environment, we include losses due to raised provisions on the banking book in this mechanism as well. The second effect of the initially troubled bank affects the liability side of other banks' balance sheets and is more likely to lead to contagion. Funders' trust in the ability of banks to service their debt may decline as they become incapable of distinguishing between financially sound and troubled banks. This leads to liquidity issues, as the cost of rolling forward short-term debt increases for the affected banks as non-bank financial institutions are reluctant to renew their loans. Banks need to roll forward their short-term debt as they usually invest in long-term assets and take short-term deposits from funders. This gives them the needed liquidity at a low funding cost under normal circumstances. Banks may then be forced to sell assets below their market value to generate liquid funds and avoid maturity mismatches on their balance sheets. 1.5 As a top-down model, we propose the application of network theory. It has been applied in a wide variety of disciplines including sociology, computer science, epidemiology, biology, economics and finance. Network models of systemic risk have been developed in the literature over the past decade (see Upper & Worms (2004) and Chinazzi & Fagiolo (2013) for surveys). It involves representing the banking system as a network of interconnected agents, where interactions between banks are modelled explicitly. Allen & Babus (2008) explain why network theory is a useful tool for understanding and analysing systemic risk in financial systems and provides a meaningful way of analysing connections between them. They argue that network theory is instrumental in investigating financial stability by considering how a single institution can cause risk to the entire system. As shown by Georg (2011) and Ladley (2013), network theory can also be used to investigate the effect of common system-wide shocks. 2. RELATED LITERATURE 2.1 Background to Network Models 2.1.1 A network is a system of N interacting agents (called nodes), where the inter actions between them form links between the nodes (called edges). It can be represented as a graph, which is a pair ( ) , G V E = , where { } 1, 2, , V N = … denotes the collection of nodes and { } { } , E i j = is the collection of edges. An edge { } , i j is a collection of two nodes , i j V ∈ that are connected via a link in the network. In this application, we use directed graphs. Here, the collection { } , i j is ordered and each edge starts at the first node i and ends at the second node j. Figures 1 and 2 illustrate the difference between undirected and directed graphs. 2.1.2 A commonly used method of modelling the edges between nodes is based on one of the earliest probability models of a graph, studied by Erdӧs & Rényi (1959). Here, each edge in the network is present with a fixed probability p, referred to as the Erdӧs-Rényi probability. The resulting graph is called an Erdӧs-Rényi network. It can be used for both directed and undirected graphs. We compare this model with extensions of it where the probability that a directed edge exists from node i to j, ( ) , p i j , is dependent on the nodes i and j. Note that for this application, ( ) , 0 p i i = for all i V ∈ , and it is said that the graph does not contain any loops (i.e. an edge cannot start and end at the same node). 2.1.3 The banks are represented by nodes and the paths through which the problems of one bank spill over to others are represented by the edges. The edges need to be directed to take account of the direction in which the losses propagate through the system. The interpretation of the edges as mechanisms through which uncertainty is channelled is discussed further in Section 3.2. The average probability that any node i is connected to another node j i ≠ is given by ( ) ( ) 1 , 1 i j p p i j n n ≠ = − ∑ . It can be used as a measure of the interconnectedness of a network. We also use it to standardise different extensions of the Erdӧs-Rényi network as described in Appendix C. The transmission of losses in the network as described in Section 3.2 makes use of nodes' shortest paths. The shortest path from a node i to another node j, say ( ) , d i j , is the smallest number of edges that can be used to travel from node i to node j. It is used as a measure of distance between two nodes in a network and takes account of the edge directions. 2.1.5 The way that banks are connected to one another via edges in a network is referred to as the structure of the network. Real-world systems are often more complex than purely random networks such as the standard Erdӧs-Rényi network (Kim & Wilhelm, 2008). They often exhibit characteristics such as low average shortest paths and nodes that are clustered together (called the small-world property). The Watts-Strogatz model (Watts & Strogatz, 1998) can be used to generate such a network. Other commonly used network models include those where nodes have a power-law distribution of the number of edges connected to them (referred to as a power-law degree distribution). The Barabási-Albert model (Albert & Barabási, 2002) can be used to generate networks with this property. 2.1.6 Such power-law distributions are of interest to large networks which may exhibit this behaviour (Boss et al.;2004, Gabrieli, 2011Santos & Cont, 2010;Cont et al., 2012). For smaller networks such as the South African system, a power-law distribution is theoretically unsuitable and difficult to test for. This paper addresses this shortcoming by investigating network structures that are applicable to smaller networks. We restrict ourselves to models where the probabilities ( ) , p i j are functions of properties of the nodes i and j. This is done firstly to enable us to test a wide variety of structures that are comparable to one another, since they are simulated in a similar way. Secondly, since the ( ) , p i j probabilities are dependent on the nodes i and j, it enables us to explicitly take account of differences between banks. Finally, models with power-law degree distributions do not make much sense for networks with few nodes such the one we consider here. 2.2 Relevant Literature 2.2.1 Methods for measuring systemic risk can generally be categorised according to whether they are based on market information (i.e. asset prices) or balance sheet information. The studies by He & Krishnamurthy (2014) and Hautsch et al. (2015) are examples of the former, whilst network models usually make use of balance sheet information. As methods based on market information rely on assumptions of market efficiency, this must be taken into account when interpreting the results, which is not the case for balance sheet-driven models. Another advantage of using balance sheet data is that it is possible to separate the effects of systemic risk from any mitigating actions by regulators, whereas asset prices usually implicitly take account of the possibility of regulatory intervention (Birchler & Facchinetti, 2007. Network models have a further advantage of emphasising the ways in which banks influence one another. Such models provide a clear distinction between individual entities and the financial network as a whole (Bisias et al., 2012). 2.2.2 It is common for network models of systemic risk to have the edges between the nodes represent interbank assets and liabilities. Such models assume that whenever a bank in the system defaults, it cannot honour its commitments to its creditors, and hence defaults on its interbank commitments. We note that in the South African banking environment, the hierarchy of interbank loans compared to other unsecured debt is not well-defined. Therefore, in the event of a bank's default, other unsecured liabilities may be subjected to bail-in before interbank liabilities. That is why we take a different approach in this study, and do not model the interbank lending relationships. Instead, we consider contagion mechanisms applicable to any jurisdiction, namely, how a loss of trust in the market may spill over to other banks, creating uncertainty and a difficulty to roll forward short-term debt. 2.2.3 Regardless of what the edges represent, network models are flexible enough to be used in a range of different circumstances. For example, Markose et al. (2012) investigate the network of exposures generated by credit default swaps of the United States of America, Battiston et al. (2012) study the loans generated by the US Federal Reserve Bank emergency loans programme, and Garratt et al. (2011) consider international banking exposures. 2.2.4 When modelling systemic risk in financial systems, it is important to include other channels of contagion such as market liquidity risk (Upper, 2011). Chen et al. (2016) find that market liquidity risk has a significant effect on systemic risk in a network setting. Network studies of systemic risk that incorporate market liquidity risk include those by Gai & Kapadia (2010), Roukny et al. (2013) and May & Arinaminpathy (2010), while Gai et al. (2011) focus on funding liquidity risk by incorporating haircuts to short-term debt. 2.2.5 We explicitly account for this by including factors representing market liquidity risk for short-, medium-and long-term assets. Short-term assets are defined to have a maturity of less than a month, and medium-term assets a maturity of more than a month and less than a year. Assets with a maturity of more than a year are deemed long-term. Network models generally assume simplified balance-sheet structures to facilitate the modelling of key balance sheet components (see for example Nier et al. (2008), Gai & Kapadia (2010), Arinaminpathy et al. (2012) and Cont et al. (2012)). A similar approach is taken by this study, with the simplified balance sheet illustrated in Table 1. The modelling of all channels of contagion is explained in Section 3.2. 2.2.6 One difficulty arising from network models is the specification of the links between banks, since it is not possible to determine the paths through which losses will spread. Empirical network studies that focus on interbank lending as the direct contagion channel can use maximum-entropy estimation techniques to estimate connections between banks (Upper & Worms, 2004). However, this estimation method can lead to inaccuracies when assessing systemic risk (Mistrulli, 2011) and makes use of each bank's total interbank assets and loans. It is therefore not appropriate to use this for our contagion mechanism. 2.2.7 The process of deciding which banks to connect to each other should consider the definition of the edges in the network. For this to make sense, the formation of edges should ideally be consistent with the event that initiated the contagion in the first place. For example, banks that are heavily exposed to the mining sector may experience a loss of investor sentiment as a result of sudden mine closures. From a modelling perspective, it is impractical to try to account for all possible bottom-up contagion events, seeing that this would need to cover a wide range of exposures and their associated risks. For this reason, we consider a range of different top-down network structures when modelling systemic risk. Furthermore, we note that different network structures may exhibit different levels of risk and the effect of changes to network characteristics is dependent on the chosen structure (Gai et al., 2011;Georg, 2011;Krause & Giansante, 2012). As the true network structure is unknown, it is important to investigate how changes in the structure can affect the modelling of systemic risk. 2.2.8 The structures investigated here are explained in Section 3.3, some of which facilitate the modelling of core-peripheral networks. These are networks that consist of a small number of tightly connected 'core' banks and numerous sparsely connected 'peripheral' banks. Iori et al. (2008) and Fricke & Lux (2015) find evidence that real-world interbank networks exhibit this behaviour (Hüser, 2015;Glasserman & Young, 2016). Even though we do not model interbank lending relationships, it is possible that losses due to investor sentiment follow a similar pattern. Therefore, such structures are included for consideration in this study. 2.2.9 We contribute to the growing body of empirical analyses of banking networks (see e.g. the work by Georg & Brink (2011), Huang et al. (2012, Vallascas & Keasey (2012), Upper & Worms (2004), Martinez-Jaramillo et al. (2014) and Boss et al. (2004) to name but a few) by applying a network model of market sentiment to South African bank balance-sheet data. The results of the model are used to assess whether the model can be used to monitor systemic risk levels by capturing increases in systemic risk during stressed market conditions. This is done by considering systemic risk at different points in time during which incidents occurred that adversely affected the local economy. 2.2.10 As the focus of this paper is on network mechanisms alone, it does not incorporate central bank activity or macro-economic factors. This is because their influence on the results would obscure effects relating to the network model itself and the differences in network structure. Central bank activity needs to be excluded as this research is done from the regulator's point of view and its reaction to bank failures is impossible to predict beforehand. The results would therefore be skewed by any assumptions made regarding central bank responses. From a practical perspective, regulators assessing policy responses to banking crises should compare the cost of intervention to the cost of not intervening, since both options can bear a high cost (Furceri & Mourougane, 2009). A useful area of future research would be to consider a set of policy responses and investigate how the network structure affects the risks borne by various stakeholders such as depositors, taxpayers and other banks. One can then consider the risk borne by each stakeholder when the risk is ignored and when the regulator intervenes. While it is possible and desirable to embed network models into larger macro-economic models (see for example Georg & Poschmann (2010) and Aikman et al. (2009)), it is specifically excluded here since one aspect of the study is to assess the performance of pure network models. Therefore, as this is a gross risk assessment, the inclusion of macro-economic effects may influence the conclusions unduly. 2.2.11 To summarise, this paper aims to make the following contributions: -A top-down network approach is used to model systemic risk over time in the South African banking system. We investigate whether this model is capable of monitoring systemic risk by detecting instances of market turmoil. -We introduce a novel contagion mechanism that focuses on market sentiment. -The effect of the network structure on the results is investigated. Since the actual network structure is unknown, it is important to obtain insight into how sensitive the results are to the choice of network structure. 2.2.12 The remainder of this paper is structured as follows: Section 3 explains how the balance sheets are constructed, discusses the modelling procedure, and presents the different network structures. The results obtained by applying the model to the South African system are presented in Section 4, after which Section 5 concludes. DATA AND NETWORK DESCRIPTION 3.1 Data and Balance Sheet Construction 3.1.1 Standardised monthly bank balance sheet data 1 of South African banks are used from April 2015 to March 2017. The BA900 returns are not granular enough to allow the extraction of CET1 capital data, which was instead obtained from banks' annual statements, Pillar III capital disclosures and the Orbis Bank Focus database. 2 It is not sensible to use a period dating back further since the capital data becomes too scarce. Capital data before 2015 are difficult to obtain for all banks, since numerous small banks either did not exist, or their capital data for earlier periods are simply not available in the public domain. 3.1.2 As at March 2017 there were ten locally-controlled banks, three mutual banks, six foreign-controlled banks and fifteen branches of foreign banks, making up a total of 34 registered banks. For the purpose of this investigation, we do not consider the parent companies of the foreign branches. Firstly, subsidiaries may not be supported by the parent company. Secondly, while the assumption may under-or overestimate systemic risk in the local banking sector if subsidiaries were supported by the parent (depending on the solvency position of the parent company), it is necessary in order to keep the system closed. In other words, one must ensure that risk levels within the system are not influenced by external market players, any actions that they may take or any regulations that may apply to them. This banking system can be considered as a typical candidate for a core-peripheral structure, as it consists of five large, 'core', banks and 29 smaller banks. To illustrate this, the total asset values of the banks are shown graphically in Appendix A, Figure A.1. For this study, eight banks are excluded because of a lack of capital data (more detail is given in ¶ ¶3.1.7 and 3.1.9 below), leaving 26 that are included in the analysis. The process for composing the simplified balance sheets as illustrated in Table 1 is explained below. 3.1.3 On the asset side, items are categorised according to whether they have a short, medium or long time to maturity at inception. Recall that short-term assets have a maturity of less than a month, medium-term assets have a maturity of more than a month and less than a year, and long-term assets have a maturity of more than a year. Assets that do not have a contractual maturity date are categorised according to their expected holding period, for example remittances in transit which are categorised as medium-term. Not all balance sheet items fall distinctly into only one category. Most of these items are placed into the category in which most individual assets are expected to fall. For example, the local Treasury Bills can have maturities ranging from one day to twelve months, but normally have an unexpired maturity of 91 days or 182 days. Therefore, these are categorised as medium-term assets for our purpose. There are two exceptions to this rule: -Marketable government stock on the BA900 forms (line item 198) is only given with a maturity of up to three years, and a maturity of over three years. Marketable government stock with a maturity of over three years are included in the long-term asset category. Marketable government bonds with a maturity of up to three years are assumed to be equally distributed across short-, medium-and long-term assets as all three of these maturity categories are included in this line item. -Derivatives are divided according to term on the liability side of the BA900 forms, but not on the asset side. The assumption is therefore made that on the asset side of each bank, the proportion of short-term derivatives to total derivatives is the same as on the liability side. The same assumption holds for the medium-and long-term derivative instruments. If there are no derivatives on the liability side, the derivatives on the asset side are divided equally among the short-, medium-and long-term assets. 3.1.4 Derivative exposures constitute an important source of systemic risk because increased margining requirements during stress scenarios can place excessive strain on banks' liquidity positions. While this can be modelled using a network approach (see e.g. the study by Markose et al. (2012)), we do not explicitly model these exposures, but include such effects indirectly via the trust mechanism. This is because counterparty relationships are not publicly available, and it avoids complicating a model that is meant to remain simple. 3.1.5 Credit impairments with respect to loans and advances are deducted from the medium-term assets. This is because private-sector loans and advances (that are categorised as medium-term assets) generally make up a large portion of total loans and advances and should also contain the majority of impaired accounts. Any impairments in respect of investments are deducted from the long-term assets since investments are generally regarded as long-term assets. The categorisation of assets is illustrated in Table B.1 in Appendix B. 3.1.6 Note that with more granular data the categorisation of assets according to maturity can be done more precisely. In this case it is necessary to aggregate the balance sheet items at this level since the available detail does not allow for a finer categorisation according to term. Regulators with more detailed information could use a larger number of categories so that assets can be grouped according to more time horizons and other characteristics as well. 3.1.7 A bank's Common Equity Tier I (CET1) capital represents the capital part of the balance sheet for the purpose of this investigation. This is because problems in financial systems can spread rapidly, and the CET1 capital can quickly be converted into cash (Gai & Kapadia, 2010). (The same approach is used by Wells (2004), Mistrulli (2011) and Cont et al. (2012).) Additional Tier 1 capital is excluded since this must first be converted in the event of a crisis. The equity side of the BA900 balance sheets is not sufficiently granular to allow for calculation of the banks' CET1 capital. For this reason, data from financial statements, published Pillar III capital disclosures and Orbis Bank Focus is used to supplement the primary balance sheet information. However, the data obtained via these sources are quarterly at best (in some cases only annually) and not all banks publish these on the same dates. Furthermore, some banks publish only risk-weighted CET1 ratios and do not necessarily include a monetary amount for this type of capital. The available data for this part of the balance sheet must therefore be used to estimate the missing datapoints where possible. 3.1.8 To estimate a bank's monthly CET1 capital, the available CET1 amounts are divided by the corresponding total asset values of the respective bank at the available points in time. In other words, if t K and t A denote a bank's CET1 capital and total asset value at time t respectively, we calculate t t t K C A = for all months t. This gives an unweighted ratio of CET1 to total assets at selected points in time. There are two main reasons for using this ratio. It firstly strips out any inflationary effects over time and secondly removes the effect of significant increases or decreases in banks' growth rates. Where available, the unweighted ratio of CET1 to total assets are very stable for all banks over the period considered (the maximum variance for this ratio for over all banks is 0,00344). Therefore, for most banks the missing unweighted CET1 ratios could easily be estimated. 3.1.9 Table B.2 in Appendix B shows all the unweighted CET1 ratios for registered local banks that could be obtained from the available data. For each bank that has at least three CET1 datapoints available between February 2015 and May 2017, the remaining ratios are estimated for the outstanding months. Banks that have less than three CET1 datapoints are excluded from the analysis, reducing the total number of banks from 34 to 26. The total assets of all excluded banks make up less than 3% of all banks' assets as at May 2017. 3.1.10 For the remaining banks, the available unweighted CET1 capital ratios are used to estimate the unknown CET1 ratios as follows: -Where missing datapoints fall in-between two known datapoints, linear interpolation between these points is used to estimate the missing values. For example, if the ratios C τ and 3 C τ + are available for months τ and τ + 3, but ratios for months τ + 1 and τ + 2 are not, we use the estimates ( ) 3 3 k k C C C C τ τ τ τ + + − = + for k = 1,2. -Where a missing datapoint does not lie between two known datapoints, the average unweighted CET1 ratio for the associated bank is taken. For example, if no CET1 data is available for month t = 1, then 1 1 C C m τ τ = ∑ , where m is the number of months τ for which C τ is available and the sum is taken over all available ratios C τ . 3.1.11 Once the estimates ˆt C for the CET1 capital are determined, all the required balance sheet entries are known. The next step is then to specify the interactions between the banks that are represented by the edges, where different assumptions regarding these interactions lead to different network structures. Notation and Default Cascades 3.2.1 The modelling procedure is based on the work of May & Arinaminpathy (2010). In this section the month t is fixed, and therefore subscripts relating to the month are not included as in Section 3.1. Suppose a network consists of N banks, where each bank i's total assets are denoted by i a . The short-, medium-and long-term assets of a bank i are denoted by ( ) For ease of reference, the terms CET1 capital and capital will be used interchangeably for the remainder of the paper. 3.2.2 We choose an initial bank n and suppose that it suffers an initial loss. For the purpose of this paper, such an event is called an 'initial shock' since we assume that it was a significant and unexpected event. In this event bank n loses a fraction, say s, of its assets. If n n s a c > , it fails, and the shortfall causes friction in the market. It is noted that the bank might not technically be insolvent at this point but might rather be in liquidation. However, for our purposes it is excluded from the network, and hence the distinction between liquidation and insolvency is not required. 3.2.3 Now three effects come into play. Firstly, the regulator may require other banks to assist with capitalisation in order to limit the spread of losses to other parts of the economy by making whole the retail and institutional creditors' unsecured loans. We assume that a proportion, say u, of this shortfall must be covered by the remaining banks. The remaining proportion 1 -u is absorbed by the Total Loss Absorbing Capacity (TLAC) part of the troubled bank's balance sheet, after which unsecured creditors bear the loss. The resulting funding requirement is spread over all banks in the system in proportion to their asset sizes. In other words, if bank n experiences an initial loss event, its capital is reduced by n n S s a = . If n n S c ≥ , then n defaults and each bank i suffers a loss of ( ) ( ) 1 . i i n n k a L u S c a = − ∑  3.2.4 Secondly, we include losses due to raised provisions and mark-to-market effects. For each remaining bank in the system, the reduced value of the short-term assets is given by ( ) ( ) ( ) exp s s i a g − ,where ( ) s g is a parameter associated with the reduction of value for the short-term assets. This is a commonly used method in the systemic risk literature for modelling changes in asset prices due to changes in supply and demand (Cifuentes et al., 2005, May & Arinaminpathy, 2010, Gai & Kapadia, 2010, Nier et al., 2008. 3.2.5 The medium-and long-term assets are reduced in the same way, where the associated parameters are given by ( ) m g and ( ) l g . This implicitly assumes that all banks in the system hold similar classes of assets, which is generally not the case. However, to avoid over-complicating the model we make the simplifying assumption that all banks will be affected to the same degree. Each of these parameters represents the expected effect that the insolvency of a bank would have on the assets in the system. The resulting losses are referred to as liquidity losses or liquidity shocks for the remainder of this paper as the methodology is similar to the liquidity shocks presented by e.g. Nier et al., (2008). The associated parameters are referred to as liquidity reduction parameters. At this stage each bank i in the system experiences a liquidity loss of Finally, we include losses due to a deterioration in market sentiment. The perceived exposure of other banks to the problems faced by bank n determines the edges in the network. An edge starting at a bank j and pointing towards bank n means that the market believes j may be exposed to similar difficulties as n, or may be adversely affected by the default of n. The edges in the network are assumed to be random, and the different structures are discussed further in Section 3.3. Recall that the shortest distance between two nodes is the smallest number of edges that can be used to travel from the one to the other. The shortest distance ( ) , d i n in the network from any bank i to the failing bank n determines the degree to which i is affected by a loss of trust. Small values of ( ) , d i n indicate 'closeness' in the network, which represents a perceived tendency for a bank i to experience similar problems as n. In order to reflect the shrinkage of a bank i's balance sheet due to increased funding costs and any resulting forced sales, each asset class of bank i is reduced by a factor ( ) exp , d i n δ   −       . Therefore, each remaining bank i in the system experiences a further loss of    = − −             ∑(1) where δ is the associated reduction factor. Similar to the parameters ( ) g η for { } , , s m l η ∈ , different proximity factors should be assigned to different types of assets when applying this model in practice. However, we avoid introducing too many parameters for the purpose of illustration by using the same parameter for all asset classes. 3.2.7 This type of loss is called a proximity shock for the remainder of the paper, as it is related to the distance between banks in the network and avoids confusion with losses arising from the devaluation of assets following the default of a bank. Proximity shocks aim to capture the consequences of the market's sentiment following a bank's default. The reaction of the market will typically depend on the circumstances surrounding the default and therefore it is preferred to take a generalised modelling approach to capture the effects of market sentiment. For example, following the curatorship of African Bank, some of the larger banks received credit downgrades from Moody's which increased their cost of borrowing. The reason given for the downgrade was that while the reserve bank did mitigate contagion risk by issuing a bailout, some creditors were allowed to suffer losses and hence Moody's was of the view that there is a "lower likelihood of systemic support from South African authorities to fully protect creditors in the event of need". 3 Other examples include the banking crisis in Greece which saw a run on the banks that negatively affected banks' liquidity positions, and the European sovereign debt crisis which resulted in credit downgrades and increased costs of borrowing. Instead of restricting ourselves to particular scenarios, we consider a wide range of possibilities regarding the spread of distrust in the system. This is done by simulating network paths according to the structures presented in Section 3.3, which avoids the need to consider the circumstances surrounding the initial default. This approach implicitly assumes that distrust in the system is only initiated after a default, and that banks of equal distance to the defaulting bank will experience losses of a similar degree. The appropriateness of the first assumption depends on the circumstances surrounding the default, and whether the market was aware of any friction within the system beforehand. The second assumption is unlikely to hold in practice but is required to keep the model simple and tractable. Finally, the model makes the underlying assumption that the loss of market sentiment due to a bank's failure is independent of the failed bank's size. In practice, it is expected that the failure of bigger banks will affect market sentiment more adversely than that of smaller banks. However, it is not straightforward to determine the magnitude of such differences, and the resulting effects may obscure the network implications that we aim to investigate in this paper. 3.2.8 The way that proximity shocks are modelled in the network accounts for the fact that some banks will experience a worse loss of confidence than others. From equation (1) it is seen that banks with smaller shortest distances to the failing bank will experience worse losses than those with greater shortest distances. This is illustrated in Figure 3 below, where the failing bank is indicated by the cross. The darker nodes experience greater losses than the lighter node, since they have a smaller shortest distance to the failing bank. The edges in the network are directed to take account of a wide range of possibilities without over complicating the model. In some circumstances the default of one bank may lead to distrust in another bank, but not the other way around. For example, the default of a large, systemically important bank may affect the market's perception of small banks, but the default of a small bank may not necessarily affect the perceived financial positions of much larger banks. Note that the direction of the edges does not represent the direction in which the losses spread but rather represents the similarity between banks. For example, if a directed edge exists from bank i to bank j, the interpretation is that bank i is similar to bank j in the sense that the market perceives i to be exposed to similar difficulties as bank j in the event of bank j's default. 3.2.9 The total loss to each remaining bank i is given by ( ) ( ) ( ) 1 2 3 , , i n i i i n L L L L = + + . The losses for each bank are subtracted from their respective capital amounts, leading to further bank failures whenever , i n i L c ≥ . For each further bank failure, losses due to funding requirements, liquidity shocks and proximity shocks are calculated. For each remaining bank in the system, all losses are added together to determine the next round of failures. This is repeated until the default cascade stops, i.e. until either all banks have defaulted or the remaining banks in the system have absorbed all losses. Let n θ denote the total number of banks that had defaulted because of the initial shock of bank n (including bank n). The proportion of banks that have defaulted, say n n N θ α = , is then calculated. 3.2.10 The above procedure is repeated for 1, 2, , n N = … . The average proportion of defaulted banks over the N repetitions of the cascade is then calculated, i.e. we calculate Network Structures 3.3.1 Recall that the network structure is determined by the way that banks are connected to one another in the network. We construct a network of trust deterioration, where the edges represent paths through which trust is lost in the system. 3.3.2 As it is not possible to know beforehand which banks will be perceived as being affected by another bank's failure, the edges in the network are assumed to be random and various network structures are considered. Even though some structures may be unrealistic, it is of interest to include them to consider a wider range of outcomes. This allows for a better understanding of the relevance of network structure in a network model based on trust deterioration. 3.3.3 Six network structures are investigated based on the probability ( ) , p i j that a connecting edge from bank i to bank j is present. To take account of heterogeneity between banks, we let this probability be dependent on the relative asset sizes of banks. The structures described in this section are selected as they are either well-known structures found in the network theory literature and capture a range of possibilities or facilitate the modelling of core-peripheral structures in networks that are not necessarily scale-free. 3.3.4 Figures 4 to 9 illustrate the behaviour of each structure. The size of a node is indicative of the bank's total asset value. A solid line represents a high probability that the associated edge is present, and a dashed, transparent line indicates a lower probability. The formulae for determining and standardising the connection probabilities ( ) , p i j are included in Appendix C. 3.3.5 Figure 4 contains the first structure, which is an Erdös-Rényi network and is the simplest of all the structures considered. The probability that an edge exists between two nodes is independent of the asset sizes and is the same between all banks. 3.3.6 Figure 5 illustrates the second structure. Here, it is assumed that the probability that a large bank causes a loss of trust in any other bank is high. Shocks experienced by smaller banks have a small probability of affecting other banks. This structure is termed 'flight to quality'. Here, the market assumed before the shock event that the bigger banks were the most financially sound. The failure of a big bank therefore causes widespread panic, affecting most other banks in the system. 3.3.7 A disassortative network structure is included, where banks of dissimilar size are more likely to have connecting edges between them. This is illustrated in Figure 6. The assortative structure shown in Figure 7 exhibits the opposite behaviour, where banks of similar size are more likely to have connections between them. Such structures may not be realistic for banking networks. They are included to widen the range of structures and because of the prominent role that these networks play in related fields of study such as social networks or ecosystems where, for example, 'opposites attract'. 3.3.8 The final two structures represent core-peripheral networks, where a network has a small, highly connected core (the top tier), with a larger, sparsely connected peripheral (the bottom tier). As the South African system consists of a small number of big banks and several small banks, it is reasonable to include core-peripheral structures into our range of networks. The Tiered type I network is illustrated by Figure 8. Large banks have FIGURE 8. Illustration of a Tiered type I structure's connection probabilities FIGURE 9. Illustration of a Tiered type II structure's connection probabilities high probabilities of being linked to one another, and small banks lower probabilities of being connected to one another. The probabilities of large and small banks being connected to one another lie in between. 3.3.9 The final structure is termed Tiered type II and is more refined than the previous structures. As shown by Figure 9, the probability that: -a small bank connects to another small bank is low; -a small bank connects to a large bank is also relatively low; -a large bank connects to a small bank is high; and -a large bank connects to another large bank is also high. IMPLICATIONS FOR REAL WORLD BANKING SYSTEMS 4.1 Applying the Model to South African Balance Sheet Data 4.1.1 The different network structures discussed in Section 3.3 are compared to one another over time. The combined effect of network structure, the system's inter connectedness and the consequences of liquidity shortages and a deterioration of market sentiment on systemic risk is investigated. Recall that systemic risk is measured by calculating the probability that a bank defaults as a result of a shock to the system. For ease of reference, liquidity risk and the risk of loss due to a deterioration of market sentiment are referred to as indirect risk from here onwards. This is because losses resulting from these risks are not directly attributable to exposures between banks. 4.1.2 We illustrate how the systemic default indicator changes over time by calculating a point in time probability at each month during the investigation period. At each time interval, an initial shock of 0,4 is applied to the system. In other words, the bank that suffers the initial loss as explained in Section 3.2 experiences a loss equal to 40% of its total asset value. Whenever a bank defaults, it is assumed that 30% of the shortfall must be covered by the remaining banks, i.e. we assume that u = 0,3. Four scenarios are considered regarding the interconnectedness and the effect of indirect risk factors on systemic risk: -Low-risk parameters The values for ( ) ( ) ( ) , , and s m l g g g δ for the high and low indirect risk scenarios are simply a scaling of one another. Note that the parameter associated with the longterm assets is higher than for the other maturities. This is done to account for the illiquidity of these assets. It is important to note that different results may be obtained with different combinations of these parameters. However, it is impractical to consider an arbitrary number of combinations without more information regarding realistic values. Therefore only a few combinations are considered in this study. 4.1.4 For the low-risk scenario, the chosen parameter values imply that after each default, the short-and medium-term assets of banks are decreased by approximately 1% and the long-term assets by 2%. The proximity shock parameter reduces all asset values of banks with a shortest distance of 1 to the failing bank by 1%. For the high-risk scenarios, the short-and medium-term assets are reduced by 1,5% and the long-term assets by 3%. The proximity shocks decrease all assets of banks directly connected to the failing bank by 1,5%. 4.1.5 Figures 10 to 13 show the relative levels of systemic risk over time for all network structures considered in Section 3.3. Where the graphs reach a flat baseline just below 0,04, the system did not experience any additional defaults over and above the initial default. In those cases, the average fraction of defaults experienced in the system is one out of 26. It is noted that the systemic default indicators shown by the graphs are based on hypothetical values of the parameters associated with indirect risks and are not necessarily accurate. This is because the focus of this study is on the relative risk levels associated with different network structures, and not to calculate actual probabilities for these events. 4.1.6 As expected, higher levels of interconnectedness result in lower levels of discrimination between the different structures. The lines in Figures 11 and 13 are closer to one another when compared to Figures 10 and 12 respectively. This is because the higher value of p pushes the probabilities ( ) , p i j towards 1 for all structures and hence they become more representative of fully connected systems. The levels of risk over time do not change significantly when interconnectedness is increased, and the overall shapes of the graphs are preserved when increasing the level of connectedness of the system. From all four scenarios, it is seen that there is a spike in systemic risk around December 2015. This corresponds to the month during which former South African finance minister Nhlanhla Nene was replaced, which was an unexpected and controversial political event in South Africa. The local financial market reacted negatively, and the local currency depreciated significantly during that period. 4.1.8 A second spike in systemic risk is observed around June 2016. This increase is less prominent in Figures 12 and 13 (where higher risk parameters are used) than in Figures 10 and 11 (where lower risk parameters are used). This was also a time during which the rand depreciated steeply against the US dollar. This was due a combination of factors, namely a weak economic growth outlook, rumours that the former finance minister FIGURE 12. High indirect risk, moderate interconnectedness FIGURE 11. Low indirect risk, high interconnectedness was to be arrested, and an approaching credit review by Standard and Poor to decide whether they would downgrade South Africa's sovereign rating to junk status. During March 2017, former finance minister Pravin Gordhan was also replaced during another controversial political event. This coincides with a sudden increase in systemic risk in Figures 10 to 13. 4.1.9 The prominence of the December 2015 spike may be explained by looking at the average balance sheet items over time (see Figure A.2 in Appendix A). At December 2015, the relative increase in the average for the short-and long-term assets is much greater compared to the CET1 capital. Because this is defined as a proportion of assets, this could, on average, lead to relatively larger losses that need to be absorbed by the capital of the initially shocked bank. However, the average asset values at June 2016 and March 2017 do not show the same extreme behaviour, which could explain why these spikes are less prominent. 4.1.10 However, Figures 12 and 13 with high-risk parameters show a smaller increase in systemic risk. This may be because overall risk levels are higher in these scenarios, thereby decreasing the prominence of the spikes. This shows that the level of indirect risk influences which events lead to an increase in systemic risk. 4.1.11 In general, it appears that the importance of the network structure is to a large extent influenced by the values chosen for the risk parameters. In the low indirect-risk scenarios (Figures 10 and 11) at times when systemic risk levels are relatively high, the network structures exhibit small differences. Otherwise they are virtually indistinguishable from one another. 4.1.12 For higher-risk parameters (Figures 12 and 13), the network structure plays a greater role in the level of systemic risk. Overall, the flight-to-quality structure exhibits the most risk when differences between the structures can be seen. From all four scenarios it is seen that the structures are mostly consistent regarding directional changes, i.e. the structures' risk levels move in the same direction at each time step, albeit at different rates. The only exception is around September 2016 in Figures 12 and 13, where the Erdös-Rényi 4.1.13 The above results show that the indirect risk parameters can affect how systemic risk changes over time. To further illustrate this point, we consider the effect of changing the relative values of the indirect risk parameters. A base parameter value of 0,015 is used for all indirect risk parameters. These parameter values will be referred to as the base parameters for the remainder of the section. The resultant graph of systemic risk over time is shown in Figure 14. 4.1.14 The effect of increasing any one risk parameter is considered. The liquidity risk parameters are each increased to 0,03, where the proximity shock parameter is increased to 0,025. This is because the results are very sensitive to this parameter, which is reasonable as it affects assets of all maturities. Therefore, increasing it to 0,03 increases the risk levels too much. The level of interconnectedness is kept at 0,5, since it was seen above that increasing the interconnectedness does not significantly influence the shape of the graphs. Instead, it brings the structures closer to one another. 4.1.15 Figure 15 shows the effect of increasing only the parameter associated with the short-term liquidity losses from 0,015 to 0,03. Figures 16 to 18 show the same results for the medium-term, long-term and proximity shock parameters, respectively. Note that Figure 17 is the same as Figure 12 but is included again and scaled to facilitate the comparison between graphs. 4.1.16 From Figures 15 to 18 it is seen that the parameters don't have the same effect on systemic risk. By increasing only the short-term liquidity parameter in Figure 15, the peaks in systemic risk are more pronounced than in Figure 14 for the base parameters. Differences between the network structures are decreased at the peaks but are more When only the long-term liquidity risk parameter is increased in Figure 17, the graph again flattens out to an extent, but the dip in risk levels during December 2016 is preserved. Differences between the network structures are generally more pronounced than in Figures 15 and 17. By increasing only the parameter associated with market sentiment, the general level of risk increases more quickly than for the other parameters. The December 2015 peak becomes much more pronounced than in Figures 15 to 17. Despite this parameter being directly related to the network structure, the differences between the structures become less pronounced. This suggests that the effect of the increased emphasis on the parameter related to the network structure is diminished by the increase in defaults experienced by all structures. 4.1.19 The differences between Figures 15, 16 and 17 are likely because of differences in asset values for different maturities between banks and within each bank. This is because the three liquidity parameters enter the model in the same way via reductions in the associated asset values. Therefore, networks derived from different countries' banking systems will likely differ in the way that they react to changes in network structure and liquidity risk parameters. For regulators, it is important to note that conclusions reached for one banking system will not necessarily hold for another. 4.1.20 The results show that both network structure and indirect risk are important in determining the level of risk present in the system. Network structure can affect the degree to which market disturbances fuel systemic risk. Determining parameters associated with liquidity risk for different asset types and market sentiment is important for network models of systemic risk, since these can significantly influence results. Figure 19 shows the average value of n α for large, medium, small and very small banks. Figure A.1 in Appendix A was used to determine the groups. The four largest banks in the system are included in the 'large' group, the fifth largest bank in the 'medium' group, the sixth to thirteenth largest banks (Capitec Bank to African Bank) in the 'small' group, and the remainder in the 'very small' group. Figure 20 shows a scatterplot of n α against the logarithm of n's asset value. Both figures support the expectation that large banks have a greater knock-on effect when they default. It is interesting to note that all the other structures lead to the same conclusions (the graphs are omitted to avoid repetitiveness). This shows that the model tends to behave as expected in this regard irrespective of the network structure. Implications of Results 4.2.1 The network structures behaved similarly for most of the cases considered here. The levels of risk were generally similar for high levels of connectivity and for lower levels of indirect risk. However, the differences in risk levels between the structures were not consistent over time. There were many time periods when the risk levels were very close to one another. Where the structures' risk levels did differ from one another, they mostly followed similar trends over time. This suggests that the changes in systemic risk detected by the model are not highly dependent on the network structure. The general relationship between a bank's size and its contribution to systemic risk was also the same for the different structures. These observations have the following implications: -The materiality of the network structure is firstly influenced by the objective of the network model. If the objective is to accurately determine the level of risk in the system, then the network structure does not make a significant difference for highly interconnected systems. For lower levels of interconnectedness, there are time periods when the network structures exhibit similar risk levels. However, this is highly dependent on the risk parameter values. -If the objective is to detect changes in systemic risk, the materiality of network structure decreases. This means that the uncertainty around the initial shock to the system (and hence the resulting path through which losses spread) is less problematic. -Network models are capable of capturing the intuitive relationship between a bank's size and the consequences of its default for a wide range of network structures. 4.2.2 The results show that the risk parameters significantly influence the extent to which network structure affects systemic risk. The liquidity risk mechanism employed by the model affects the assets of all banks and therefore is not directly related to the network FIGURE 20. Scatterplot of n α against the natural logarithm of bank n's assets for the Erdӧs-Rényi network structure. Nevertheless, small changes in the liquidity risk parameters have a non-trivial influence on the relative differences in risk exhibited by the structures. 4.2.3 Furthermore, each risk parameter influences the results in its own way. For example, increasing the short-term liquidity parameter emphasised the December 2015 and June 2016 increases in risk, whereas the other liquidity parameters reduced the significance of these spikes. This either means that the model is not able to detect increases in risk for certain parameter values, or that the system does not experience a significant increase in systemic risk during times of market turmoil for some liquidity scenarios. For example, the high liquidity risk scenarios (see Figures 12 and 13) might increase the risk levels during all months to such an extent that the effect of weak economic conditions are diminished. 4.2.4 The above observations have the following implications for the modelling of systemic risk using a network approach: -Empirical studies that aim to determine the level of systemic risk should take care to calibrate the liquidity risk parameters to levels appropriate for the system being considered. The difficulty of calibrating these parameters is a drawback of such a network approach to modelling systemic risk. As incidents of bank closure/liquidation can be scarce, one may have to work with few datapoints. As such it would not be possible to calibrate the parameters precisely, but it could be possible to determine a realistic range for the required parameters. One can consider the balance sheet positions of all banks before and after each closure/liquidation incident to measure the size of any shrinkages in the remaining banks' balance sheets. -A finer division of assets is recommended. The fact that the liquidity risk parameters each had a different effect on the model output suggests that the classification of assets can be a material aspect of such a study. -Since the model showed increases in systemic risk during times of market turmoil, it shows that network models of systemic risk may be valuable modelling tools. A great advantage of this is that publicly available balance sheet information can be used to model systemic risk, thereby avoiding the need to obtain confidential trading information. It may be by chance that the model detected increases in systemic risk due to balance sheet fluctuations. This warrants further investigation to determine with greater certainty whether the model can accurately identify potential crises. CONCLUSION 5.1 We use a novel network approach to model systemic risk in South Africa by considering how the liquidity problems and default of one bank can lead to market frictions such as losing trust in the financial wellbeing of other banks. Here, the type of event that leads to the default of the first bank is likely to infer the network structure. As this cannot be determined beforehand, and a lack of past systemic crises make liquidity parameters difficult to determine, we consider the effect of network structure and liquidity risk on the results. 5.2 The network structure's influence on systemic risk was considered under different circumstances and over time. The general trend is the same for all network structures, showing that the model may detect fluctuations in systemic risk even if the true network structure is unknown. The differences between the network structures are influenced by the effect of liquidity risk and the losses due to negative investor sentiment. The trends in systemic risk over time are sensitive to changes in the parameters associated with these risks. This shows that any investigation regarding systemic risk in banking networks must incorporate indirect losses, such as losses due to liquidity problems and a deterioration of market sentiment, since these significantly influence the results. 5.3 Indirect losses have a significant effect on how the system reacts to changes in structure and interconnectedness. This indicates that the calibration of these para meters is important when making decisions based on network models of systemic risk. The importance of this is emphasised by the fact that systemic risk levels over time behave significantly differently depending on the combination of all the parameters used. It is imperative for regulators to incorporate and accurately model these effects when assessing the effect of proposed regulatory changes. 5.4 Despite the problems associated with determining the correct network structure and liquidity risk parameters, such models can be useful. These models are simple, easy to understand and make use of publicly available balance sheet data. The framework presented here can be useful to answer 'what-if' questions that arise in practice and to give insight into what might happen to the system given an appropriate network. The framework in itself can be used to generate a wide range of output; for example, one can investigate a range of different risk measures (average capital lost, average proportion of asset value lost by the system etc.), and test for correlations between this and the size of the initially defaulted bank. 5.5 During the time frame considered here, the network model detected increases in systemic risk at times when the economy experienced unexpected market disturbances. An important avenue for future research is to determine whether this is by chance, or whether the model accurately predicts the probability that a crisis can occur. Furthermore, it is important to note that the model does not forecast times of distress, but instead provides a proxy for the level of risk at a point in time. In other words, the true proportion of banks defaulting following a shock to the system is not determined, but rather a value that increases or decreases along with it. 5.6 Note that this investigation solely focused on relative changes in systemic risk and is not meant to provide an accurate estimate of the probability that a bank defaults following an initial shock to the system. Important avenues for future research include assigning banks different probabilities of receiving an initial shock and including the possibility that such a shock may affect more than one bank at the same time. This should be done parallel to modelling the macro-economic environment over time. In this case, a lender of last resort and the cost of regulation can be included. Furthermore, the effect of leverage can be modelled more explicitly in the presence of a macro-economic environment. Finally, the interaction between the banking sector and other financial institutions (e.g. insurance and investment companies) should be included in future work. For further work, it is important to consider a range of different risk measures and to test whether the same conclusions hold. Other important directions for future work include considering a wider range of structures, a finer division of assets and using a networks-on-networks approach to incorporate a range of contagion mechanisms. Short-term assets -Central bank money and gold -Deposits with, and loans and advances to banks -Loans granted to the SARB and other institutions under resale agreements -Foreign currency deposits, loans and advances -One third of marketable government stock that have an unexpired maturity of less than three years -Derivative instruments assigned to the short-term asset category according to the rules in Section 3.1 APPENDIX A Additional Figures APPENDIX B Balance Sheet Information Medium-term assets -Instalment sales -Credit-card debtors -Overdrafts, loans and advances to the private sector -Bankers' acceptances (Treasury bills, SARB bills, promissory notes, commercial paper and Land Bank bills) -Clients' liabilities per contra -Remittances in transit -Current income tax receivables and deferred income tax assets -One third of marketable government stock that have an unexpired maturity of less than three years -Derivative instruments assigned to the medium-term asset category according to the rules in Section 3.1 Long-term Assets -Redeemable preference shares -Leasing transactions -Mortgage advances -Overdrafts, loans and advances to the public sector -Non-marketable government stock -All marketable government stock excluding two thirds of those stock that have an unexpired maturity of less than three years -Debentures and other interest-bearing security investments of private sector -All equity investments -Derivative instruments assigned to the long-term asset category according to the rules in Section 3.1 -Other investments -Non-financial assets -Retirement benefit assets -Assets acquired or bought to protect an advance or investment FIGURE 1 . 1Illustration of an undirected graph FIGURE 2. Illustration of a directed graph 2.1.4 . Finally, bank i's CET1 capital is denoted by i c . . This is repeated m times, with the edges simulated each time. The average defaulted fraction over all the simulations is then denoted by α . For the remainder of the paper, we refer to α as the systemic risk indicator. FIGURE 3 . 3Illustration of a proximity shock following the default of a bank © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 113 FIGURE 4 .FIGURE 5 . 45Illustration of an Erdös-Rényi structure's connection probabilities Illustration of a flight-to-quality structure's connection probabilities FIGURE 6 . 6Illustration of a disassortative structure's connection probabilitiesFIGURE 7. Illustration of an assortative structure's connection probabilities FIGURE 10 . 10Low indirect risk, moderate interconnectedness 4.1.7 FIGURE 13 . 13High indirect risk, high interconnectedness and disassortativeness structures show a slight decrease in risk, whereas the other structures show an increase. FIGURE 14 . 14Systemic risk over time for base parameters of the troughs. Increasing only the medium-term liquidity parameter (Figure 16) flattens out the graph to such an extent that the December 2015 and June 2016 spikes are not distinguishable from other peaks in the graph. Only the December 2016 decrease in risk is preserved. Once again, the differences between the network structures become greater during the December 2016 dip in risk but are less during other months when compared to Figure 14. 4.1.17 FIGURE 15 .FIGURE 16 .\| 1516Systemic risk over time for base parameters, but with ( ) Systemic risk over time for base parameters, but with ( ) SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 121 4.1.18 FIGURE 17 .FIGURE 18 . 1718Systemic risk over time for base parameters, but with ( ) Systemic risk over time for base parameters, but with 0,025 δ = understand how the network structure affects how the different banks in the system contribute to systemic risk, it is useful to consider how n α (defined in ¶3.2.10 as the proportion of nodes that default if n is the initially shocked bank) varies with the asset value of bank n. Let n α denote the average of n α over 2000 simulations. It is reasonable to expect the default of larger banks to have a greater knock-on effect on the system compared to smaller banks and hence n α is expected to be higher for larger banks. This is confirmed by Figures 19 and 20, which are based on the Erdӧs-Rényi network structure, with Figure 20 based on balance sheet data as at March 2017. FIGURE 19 . 19Systemic risk indicator by bank size for the Erdӧs-Rényi network FIGURE A. 1 . 1The distribution of assets in the banking sector as at 31 March 2017 FIGURE A.2. Relative increases of the average balance sheet items in the system TABLE 1 . 1Illustration of a simplified balance sheetAssets Equity & liabilities Short-term assets Capital Medium-term assets Other equity & liabilities Long-term assets TABLE B . B1. Division of assets according to term TABLE B . 2 . B2Available unweighted ratios of CET1 to total assets at month-end for all registered banks www.resbank.co.za/RegulationAndSupervision/BankSupervision/Banking%20sector%20ata/ Pages/Banks-BA900-Returns.aspx 2 https://orbisbanks.bvdinfo.com/version-2017713/home.serv?product=OrbisBanks © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 107 © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 111 www.moodys.com/research/Moodys-downgrades-four-South-African-banks-on-review-forfurther--PR_306571 \| NM WALTERS, FJC BEYERS, AJ VAN ZYL & RJ VAN DEN HEEVER © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 115 © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 119 \| NM WALTERS, FJC BEYERS, AJ VAN ZYL & RJ VAN DEN HEEVER © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 127 © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 129 © ASSA licensed under 3.0 \| SAAJ 18 (2018) FRAMEWORK FOR SIMULATING SYSTEMIC RISK AND ITS APPLICATION TO SA BANKING SECTOR \| 133 ACKNOWLEDGEMENTSThe financial assistance of the National Research Foundation (NRF) and the Absa chair in Actuarial Science towards this research is hereby is hereby acknowledged. Opinions expressed, and conclusions arrived at, are those of the authors and are not necessarily to be attributed to the NRF or Absa. The authors further acknowledge the valuable advice and input given by Jeannette de Beer of Absa Capital.APPENDIX C Connection Probabilities associated with Different Network StructuresC.1 Let ( ) , p i j be the probability that bank i has an outgoing edge to bank j and let j a denote the asset value of bank j. As banks cannot be connected to themselves, ( )For the rest of this discussion, assume thatThe different network structures are then specified as follows:1.Erdös-Rénji:Flight to quality:3.Disassortativeness:5.Tiered type I:For a given set of asset values, the average number of edges will differ between the different structures. For comparative purposes, it is important to work with similar levels of connectedness between the different structures. In addition to this, it is important to have a systematic way of varying the level of connectedness in the network to investigate its effect on systemic risk.C.3For these reasons, a method for scaling the probabilities is required. Such a method will need to provide scaled probabilities that remain in the range [ ] 0,1 . It must further allow one to standardise the different structures to represent the same level of connectivity between them. The level of connectivity is measured by means of the average probability that an edge exists in the system (i.e. the average probability that one bank is exposed to a loss of trust in case of default of another).C.4Let 0 p be the average probability that an edge exists in the system based on any one of the above six structures. Then The new connection probabilities ( ) , p i j will then have the required average p. Funding liquidity risk in a quantitative model of systemic stability. Central Bank of Chile. 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[ "Qualitative Analysis and Optimal Control Strategy of an SIR Model with Saturated Incidence and Treatment", "Qualitative Analysis and Optimal Control Strategy of an SIR Model with Saturated Incidence and Treatment" ]	[ "Jayanta Kumar Ghosh \nBoalia Junior High School\nNadiaWest BengalIndia\n", "Uttam Ghosh \nDepartment of Applied Mathematics\nUniversity of Calcutta\nKolkataIndia\n", "M H A Biswas \nMathematics Discipline\nKhulna University\nKhulna-9208Bangladesh\n", "Susmita Sarkar \nDepartment of Applied Mathematics\nUniversity of Calcutta\nKolkataIndia\n" ]	[ "Boalia Junior High School\nNadiaWest BengalIndia", "Department of Applied Mathematics\nUniversity of Calcutta\nKolkataIndia", "Mathematics Discipline\nKhulna University\nKhulna-9208Bangladesh", "Department of Applied Mathematics\nUniversity of Calcutta\nKolkataIndia" ]	[]	This paper deals with an SIR model with saturated incidence rate affected by inhibitory effect and saturated treatment function. Two control functions have been used, one for vaccinating the susceptible population and other for the treatment control of infected population. We have analysed the existence and stability of equilibrium points and investigated the transcritical and backward bifurcation. The stability analysis of non-hyperbolic equilibrium point has been performed by using Centre manifold theory. The Pontryagin's maximum principle has been used to characterize the optimal control whose numerical results show the positive impact of two controls mentioned above for controlling the disease. Efficiency analysis is also done to determine the best control strategy among vaccination and treatment.Mathematics Subject Classification: 37N25.34C23.49J15.92D30	10.1007/s12591-019-00486-8	[ "https://arxiv.org/pdf/1807.05954v1.pdf" ]	59,406,081	1807.05954	e1297bb5614a7260d82e97956d3a18b49889c2c8	Qualitative Analysis and Optimal Control Strategy of an SIR Model with Saturated Incidence and Treatment July 17, 2018 Jayanta Kumar Ghosh Boalia Junior High School NadiaWest BengalIndia Uttam Ghosh Department of Applied Mathematics University of Calcutta KolkataIndia M H A Biswas Mathematics Discipline Khulna University Khulna-9208Bangladesh Susmita Sarkar Department of Applied Mathematics University of Calcutta KolkataIndia Qualitative Analysis and Optimal Control Strategy of an SIR Model with Saturated Incidence and Treatment July 17, 2018Inhibitory factorsNon-hyperbolic equilibrium pointCentre manifold theoryTranscritical bifurca- tionBackward bifurcationOptimal controlEfficiency analysis This paper deals with an SIR model with saturated incidence rate affected by inhibitory effect and saturated treatment function. Two control functions have been used, one for vaccinating the susceptible population and other for the treatment control of infected population. We have analysed the existence and stability of equilibrium points and investigated the transcritical and backward bifurcation. The stability analysis of non-hyperbolic equilibrium point has been performed by using Centre manifold theory. The Pontryagin's maximum principle has been used to characterize the optimal control whose numerical results show the positive impact of two controls mentioned above for controlling the disease. Efficiency analysis is also done to determine the best control strategy among vaccination and treatment.Mathematics Subject Classification: 37N25.34C23.49J15.92D30 Introduction Mathematical modelling in epidemiology have become powerful and important tool to understand the infectious disease dynamics and to improve control of infection in the population. A good epidemic model is an intelligent model which is able to predict any possible outbreak of the disease and is effective in reducing the transmission of the disease. It is a simplest version of reality in Biology [1][2][3]. In mathematical epidemiology, the incidence rate as well as treatment rate plays a crucial role while analysing the transmission of infectious diseases. The researchers in this field consider different type of incidence rate depending on character of disease spreading. Firstly, the bi-linear incidence rate [4] βSI, (where the parameter β is transmission rate of infection and the variables S, I are respectively the number of susceptible and infected population) is based on the law of mass action which is not realistic because, at the initial stage of spreading of disease, the number of infected population is low and so people do not take care at this stage but at the later stage when the number of infected population becomes large, people take more care to protect them from the infection of the disease. The saturated incidence rate βSI 1+αI was introduced by Anderson and May in 1978 [5], where α is defined as inhibitory coefficient. Clearly, this incidence rate is an increasing function of S as well as I and by this incidence rate the total growth of infected population is less compared to the standard incidence. This type of infection sometimes named as 'incidence rate with psychological effect' [6], because the effect of α stems from epidemic control (taking appropriate preventive measures and awareness) and the rate of infection decreases as the inhibitory coefficient α increases. Again, we are aware of the fact that the treatment is an important method to control diseases. The treatment rate of infected individuals is considered to be either constant or proportional to the number of infected individuals. Wang and Ruan [7] . Again, we know that there are limited treatment resources or limited capacity for treatment in every community. To include this type of limitations in treatment Zhang and Liu [8] introduced a new continuously differentiable treatment function T (I) = rI 1+αI to characterize the saturation phenomenon of the limited medical resources. Here T (I) is an increasing function of I and r α is the maximal supply of medical resource per unit time. On the other hand, optimal control theory is a powerful mathematical tool that is used extensively to control the spread of infectious diseases. It is often used in the control of the spread of most infectious diseases for which either vaccine or treatment is available. Some researchers considered only vaccination control to their models [9] and some of them used only treatment control [9][10]. The author in [11] have used both the controls in their models. The purpose of considering both vaccination and treatment in finding optimal control in epidemiological models is to minimize the susceptible and infected individuals as well as the cost of implementing these two controls. In this paper, we have considered an SIR model with saturated incidence rate βSI 1+αI affected by the inhibitory effect α and saturated treatment function. Both vaccination control u 1 and treatment control u 2 have been used to address the question of how to optimally combine the vaccination and treatment strategies for minimizing the susceptible and infected individuals as well as the cost of the implementation of the two interventions. Here, we have used the treatment function T (I, u 2 ) = ru 2 I 1+bu 2 I , which is clearly the increasing function of I and u 2 and the maximal supply of medical resource is r b , where b is the delayed parameter of treatment because T decreases as b increases and r is the cure rate. Here, we have analysed the stability of equilibrium points using eigen analysis method. The stability analysis of non-hyperbolic equilibrium point has been carried out by using Center manifold theory. Exhibition of transcritical and backward bifurcation have been analysed in our work. It is important to mention here that our work is different from some of the other related works cited in this paper [10][11] because the stability or instability of endemic equilibrium point(s) is analysed by applying different techniques described in [12]. It should also be noted that in this paper, we shall deal with the qualitative analysis of the model as well as the optimal control of the disease. Numerical simulations and efficiency analysis are performed to understand the positive impact of controls and to determine the best strategy among vaccination and treatment. Organization of the paper is as follows. We have formulated the model in Section 2 and discussed the boundedness of the solutions, existence of equilibria and basic reproduction number R 0 in Section 3. The Section 4 is devoted to the stability and bifurcation analysis about disease free equilibrium point and Section 5 is devoted to the backward bifurcation and stability analysis of endemic equilibrium points. Section 6 gives detailed description about the characterization of optimal control. The numerical simulations and efficiency analysis are given in Section 7 and final section gives the conclusions. Model Formulation Let the total population be divided into three classes, namely susceptible population S(t), infected population I(t) and recovered population R(t) at time t. Here, we have considered an epidemic model in which the birth rate of susceptible class is constant, the incidence and treatment rate are of saturated type, susceptible class is vaccinated, the normal and disease induced death are also taken into consideration. It is also assumed that some of the infected individuals who are physically strong enough can recover themselves without treatment. Incorporating all the assumptions the governing differential equations of the model can be written in the following form              dS dt = A − βSI 1+αI − dS − u 1 S dI dt = βSI 1+αI − (d + δ + γ)I − ru 2 I 1+bu 2 I dR dt = ru 2 I 1+bu 2 I + γI + u 1 S − dR(1) with initial conditions S(0) ≥ 0, I(0) ≥ 0, R(0) ≥ 0. Parameters used in the system (1) are non-negative and listed in Table 1. Parameters Interpretations A Recruitment rate of the population. β Transmission rate. α The parameter that measures the inhibitory factors. d The natural mortality rate of the populations. δ Disease induced death rate. γ The natural recovery rate of the infected individuals. r Cure rate. b Delayed parameter of treatment. u 1 The control variable, be the percentage of susceptible individuals being vaccinated per unit of time. u 2 The treatment control parameter. Since the exact solution of the non linear autonomous system (1) is impossible to find, so we are analysing the qualitative behaviour of the solutions in the neighbourhood of the equilibrium points. Boundedness of Solutions, Existence of the Equilibria and the Basic Reproduction Number In this section we shall discuss the boundedness of the solutions and existence of the equilibrium points of system (1) for fixed value of control parameters u 1 and u 2 . We shall derive the basic reproduction number when the control parameters are taken as fixed. Lemma: 1. The region D = (S, I, R) ∈ R 3 + /S + I + R ≤ A d is a positively invariant set for the model (1). Proof. : Let N = S + I + R. So, dN dt = A − dN − δI ≤ A − dN , integrating and taking limsup as t → ∞ we get lim sup t→∞ N (t) ≤ A d . Hence the lemma is proved. The system (1) has always the disease free equilibrium point (DFE) A 1 (S 1 , 0, R 1 ) = A d+u 1 , 0, u 1 A d(d+u 1 ) at which the population remains in the absence of disease. Therefore, the model (1) has a threshold parameter R 0 , known as the basic reproduction number, which is defined as the number of secondary infection produced by a single infection in a completely susceptible population. Lemma: 2. The basic reproduction number for the model (1) is R 0 = βA (d+u 1 )(d+δ+γ+ru 2 ) . Proof. : Here is only one infected compartment, that is, the variable I and the disease free equilibrium point is A 1 . The basic reproduction number R 0 is defined as the spectral radius of the next generation matrix F V −1 with small domain [13], where F = βS (1+αI) 2 1×1 DF E = βA d+u 1 1×1 and V = d + δ + γ + ru 2 (1+bu 2 I) 2 1×1 DF E = d + δ + γ + ru 2 1×1 . Thus, R 0 of the model is βA (d+u 1 )(d+δ+γ+ru 2 ) . Hence the lemma is proved. The existence of the endemic equilibrium point(s) can be determined by the relation S = ru 2 (1+αI) β(1+bu 2 I) + (d+δ+γ)(1+αI) β = A(1+αI) βI+(d+u 1 )(1+αI) . Thus, the compartment I of the equilibrium point (S, I, R) must satisfy the equation H(I) = 0,(2) where H(I) ≡ A βI+(d+u 1 )(1+αI) − ru 2 β(1+bu 2 I) − d+δ+γ β . By simplifying the equation (2), we get C 1 I 2 + C 2 I + C 3 = 0,(3) where C 1 = bu 2 (d + δ + γ) β + α(d + u 1 ) , C 2 = bu 2 (d + u 1 )(d + δ + γ) − βA + (d + δ + γ + ru 2 ) β + α(d + u 1 ) , C 3 = (d + u 1 )(d + δ + γ + ru 2 )(1 − R 0 ). Here, the coefficient C 1 is always positive and the sign of C 3 depends only on the value of R 0 . Thus, we have (1) If R 0 > 1 , then only one endemic equilibrium point (S * , I * , R * ) exists. (2) If C 2 > 0 and R 0 < 1 , then there is no endemic equilibrium point. (3) If C 2 < 0, C 2 2 − 4C 1 C 3 > 0 and R 0 < 1 , then two endemic equilibrium points (S * 1 , I * 1 , R * 1 ) and (S * 2 , I * 2 , R * 2 ) exist with I * 1 < I * 2 . Stability and Bifurcation Analysis at DFE In this section we shall investigate the stability and transcritical bifurcation at the disease free equilibrium point(DFE) for fixed vaccination and treatment control. Here, the variational matrix corresponding to the system (1) is J(S, I, R) =      − βI 1+αI − d − u 1 − βS (1+αI) 2 0 βI 1+αI βS (1+αI) 2 − (d + δ + γ) − ru 2 (1+bu 2 I) 2 0 u 1 ru 2 (1+bu 2 I) 2 + γ −d      .(4) Theorem 1. If R 0 < 1 then the disease free equilibrium point A 1 is asymptotically stable and if R 0 > 1 then it is unstable. Proof. : The characteristic roots of the variational matrix (4) at the disease free equilibrium point A 1 are −d, −(d + u 1 ) and (d + δ + γ + ru 2 )(R 0 − 1) . Therefore, A 1 is asymptotically stable when R 0 < 1 and is unstable when R 0 > 1. Hence the theorem is proved. Theorem 2. For R 0 = 1, A 1 is asymptotically stable if (d + δ + γ + ru 2 ) β + (d + u 1 )α < (d + u 1 )rbu 2 2 and is unstable if (d + δ + γ + ru 2 ) β + (d + u 1 )α > (d + u 1 )rbu 2 2 . Proof. : For R 0 = 1, the eigen values of the variational matrix (4) at the equilibrium point A 1 (S 1 , 0, R 1 ) are 0, −(d + u 1 ), −d. So, A 1 is a non-hyperbolic equilibrium point and Centre manifold theory will be applied to determine its stability. Putting S = S − S 1 , I = I, R = R − R 1 in the system (1) and using Taylor's expansion we get (omitting the 'dash' sign) dX dt = BX + F (S, I, R),(5)where B =      −(d + u 1 ) − βA d+u 1 0 0 0 0 u 1 (ru 2 + γ) −d      , X =      S I R      , F (S, I, R) =      −βSI + αβS 1 I 2 βSI + (rbu 2 2 − αβS 1 )I 2 −rbu 2 2 I 2      . (Neglecting the terms of order ≥ 3) Now, we construct a matrix P =      βAd 1 0 −d(d + u 1 ) 2 0 0 u 1 βA − (ru 2 + γ)(d + u 1 ) 2 −1 1      so that P −1 BP = diag 0, −(d + u 1 ), −d . By using the transformation X = P Y ,where Y =      S I R      , the system (5) can be transformed into the form (omitting the 'dash' sign)              dS dt = 0 + g 11 (S, I, R) dI dt = −(d + u 1 )I + g 22 (S, I, R) dR dt = −dR + g 33 (S, I, R) (6) where g 11 (S, I, R) = A 11 S 2 + B 11 SI, g 22 (S, I, R) = A 22 S 2 + B 22 SI, g 33 (S, I, R) = A 33 S 2 + B 33 SI, with A 11 = β 2 A + (d + u 1 )αβA − (d + u 1 ) 2 rbu 2 2 d. By the Centre manifold theory [14], there exists a centre manifold of the system (6) which can be expressed by W c (0) = (S, I, R)/I = h 1 (S), R = h 2 (S)f orS < δ , where δ(> 0) is some number and h 1 (0) = h 2 (0) = 0, Dh 1 (0) = Dh 2 (0) = 0. To compute the centre manifold W c (0), we assume that I = h 1 (S) = h 11 S 2 + h 12 S 3 + ..........., R = h 2 (S) = h 21 S 2 + h 22 S 3 + ........... . So from the Local Centre manifold theorem we have the flow on the centre manifold W c (0) defined by the differential equation dS dt = A 11 S 2 + A 11 B 11 (d + u 1 ) S 3 + A 22 B 11 (B 22 − 2A 11 ) (d + u 1 ) 2 S 4 + .......(7) Now, by the condition R 0 = 1, we get after simplification that A 11 = d(d + u 1 ) (d + δ + γ + ru 2 ) β + (d + u 1 )α − (d + u 1 )rbu 2 2 . The system will be stable and unstable according as A 11 < 0 and A 11 > 0 respectively. Hence the theorem is proved. Note: Other components B 11 , A 22 , B 22 , A 33 and B 33 are not derived here as they are not in use. Theorem 3. If R 0 < 1 and α ≥ bu 2 then the disease free equilibrium point A 1 is globally asymptotically stable. Proof. : We rewrite the system (1) in (S, I) plane as given below      dS dt = A − βSI 1+αI − dS − u 1 S ≡ F (S, I) dI dt = βSI 1+αI − (d + δ + γ)I − ru 2 I 1+bu 2 I ≡ G(S, I).(8) Now considering the Dulac function B(S, I) = 1+bu 2 I SI , we get ∂(BF ) ∂S + ∂(BG) ∂I = − A(1 + bu 2 I) IS 2 − (d + δ + γ)bu 2 S − β(α − bu 2 ) (1 + αI) 2 . Thus the DFE is globally asymptotically stable if α ≥ bu 2 . Hence the theorem is proved. In other words DFE is globally asymptotically stable if the inhibitory coefficient exceeds a value that is the product of delayed parameter of treatment and the treatment control. Theorem 4. If βA > (d + u 1 )(d + δ + γ), then the system (1) experiences a transcritical bifurcation at A 1 as u 2 varies through the bifurcation value u 0 2 = βA r(d+u 1 ) − d+δ+γ r . Proof. Let f (S, I, R; u 2 ) =      A − βSI 1+αI − dS − u 1 S βSI 1+αI − (d + δ + γ)I − ru 2 I 1+bu 2 I ru 2 I 1+bu 2 I + γI + u 1 S − dR      , u 0 2 = βA r(d+u 1 ) − d+δ+γ r . So, Df (A 1 , u 0 2 ) =      −(d + u 1 ) − βA d+u 1 0 0 0 0 u 1 βA d+u 1 − (d + δ) −d      . Clearly, f (A 1 , u 0 2 ) = 0 and Df (A 1 , u 0 2 ) has a simple eigen value λ = 0. Thus, we shall use Sotomayor theorem [14] to establish the existence of transcritical bifurcation. Now, a eigen vector of Df (A 1 , u 0 2 ) corresponding to the eigen value λ = 0 is V =      1 − (d+u 1 ) 2 βA (d+δ)(d+u 1 ) 2 βAd − 1     f u 2 =      0 − rI (1+bu 2 I) 2 rI (1+bu 2 I) 2      and so f u 2 (A 1 , u 0 2 ) =      0 0 0      . Therefore, W T f u 2 (A 1 , u 0 2 ) = 0, W T Df u 2 (A 1 , u 0 2 )V = r(d+u 1 ) 2 βA = 0 and W T D 2 f (A 1 , u 0 2 )(V, V ) = 2(d + u 1 ) 2 − 1 A − α(d+u 1 ) βA + b r + b(d+δ+γ) 2 (d+u 1 ) 2 rA 2 β 2 − 2b(d+δ+γ)(d+u 1 ) rAβ = 0. Therefore, all the conditions for transcritical bifurcation in Sotomayor theorem are satisfied. Hence, the system (1) experiences a transcritical bifurcation at the equilibrium point A 1 as the parameter u 2 varies through the bifurcation value u 2 = u 0 2 . Hence the theorem is proved. Backward Bifurcation and Stability Analysis of Endemic Equilibria In this section, we shall analyse the stability and the bifurcation behaviour at endemic equilibrium point by assuming that two controls u 1 and u 2 are constant. We have already proved that DFE is stable if R 0 < 1 and is unstable if R 0 > 1. Here, we shall establish that the bifurcating endemic equilibrium exists for R 0 < 1, which implies that the backward bifurcation occurs. Now, we shall obtain the necessary and sufficient condition on model parameters for the existence of backward bifurcation. Theorem 5. The system (1) has a backward bifurcation at R 0 = 1 if and only if ( ru 2 + d + δ + γ)(ru 2 + d + δ + γ + αA) < bru 2 2 A. Proof. : In Section 3, we have seen that the infected component I of endemic equilibrium points are the roots of the equation (3). Again, from Lemma 2 we can express β as R 0 (d+u 1 )(d+δ+γ+ru 2 ) A . Now, we substitute β in the coefficients of equation (3) and rewrite equation (3) as C 1 I 2 + C 2 I + C 3 = 0, where C 1 = bu 2 (d+δ+γ)(d+u 1 ){R 0 (d+δ+γ+ru 2 )+αA} A , C 2 = bu 2 (d + u 1 ) (d + δ + γ) − R 0 (d + δ + γ + ru 2 ) + (d+δ+γ+ru 2 )(d+u 1 ){R 0 (d+δ+γ+ru 2 )+αA} A and C 3 = (d + u 1 )(d + δ + γ + ru 2 )(1 − R 0 ). To obtain a necessary and sufficient condition on the model parameters such that backward bifurcation occurs we have to compute the value of ∂I ∂R 0 R 0 =1,I=0 . Now, differentiating the equation (3) implicitly with respect to R 0 we obtain ∂I ∂R 0 R 0 =1,I=0 = A(d + δ + γ + ru 2 ) (d + δ + γ + ru 2 )(d + δ + γ + ru 2 + αA) − bru 2 2 A . The system (1) has a backward bifurcation at R 0 = 1 if and only if the value of the slope ∂I ∂R 0 R 0 =1,I=0 of the curve I = I(R 0 ) is less than zero. Hence we obtain the necessary and sufficient condition for backward bifurcation in the form (ru 2 + d + δ + γ)(ru 2 + d + δ + γ + αA) < bru 2 2 A. Hence the theorem is proved. So, there is a real number R * 0 < 1 for which two endemic equilibria exist for R * 0 < R 0 < 1 if the condition in Theorem 5 holds. Now, we shall focus on the stability analysis of the endemic equilibrium point(s) for different values of R 0 and we shall prove in the following theorem that the locally asymptotically stable DFE co-exists with a locally asymptotically stable endemic equilibrium point when R 0 < 1 . Theorem 6. If R 0 > 1 and β ≥ max{rbu 2 2 , rαu 2 }, then the system (1) has a unique endemic equilibrium point (S * , I * , R * ) that is locally asymptotically stable. On the other hand if R * 0 < R 0 < 1 and (ru 2 + d + δ + γ)(ru 2 + d + δ + γ + αA) < bru 2 2 A , then the system (1) has two endemic equilibrium points. The one with the smaller number of infecteds, (S * 1 , I * 1 , R * 1 ), is unstable, while the other , with a higher number of infecteds, (S * 2 , I * 2 , R * 2 ), is locally asymptotically stable if β ≥ max{rbu 2 2 , rαu 2 }. Proof. We have the following characteristic equation of the variational matrix (4) at the endemic equilibrium point (S, I, R). − βI 1+αI − d − u 1 − λ − βS (1+αI) 2 0 βI 1+αI βS (1+αI) 2 − (d + δ + γ) − ru 2 (1+bu 2 I) 2 − λ 0 u 1 ru 2 (1+bu 2 I) 2 + γ −d − λ = 0, or equivalently, where (λ + d) − βI 1+αI − d − u 1 − λ − βS (1+αI) 2 βI 1+αI βS (1+αI) 2 − (d + δ + γ) − ru 2 (1+bu 2 I) 2 − λ = 0.G(λ) ≡ − βI 1+αI − d − u 1 − λ − βS (1+αI) 2 βI 1+αI βS (1+αI) 2 − (d + δ + γ) − ru 2 (1+bu 2 I) 2 − λ . So, G(0) = − βI 1+αI − d − u 1 − βS (1+αI) 2 βI 1+αI βS (1+αI) 2 − (d + δ + γ) − ru 2 (1+bu 2 I) 2 . Again, we know from (2) 2 and K 2 = G(0). By using the relation S = ru 2 (1+αI) β(1+bu 2 I) + (d+δ+γ)(1+αI) β , K 1 is simplified as β(1+bu 2 I) 2 − A{β+α(d+u 1 )} {βI+(1+αI)(d+u 1 )} 2 . Now, G(0)= − βI 1+αI − d − u 1 − βS (1+αI) 2 βI 1+αI rbu 2 2 I (1+bu 2 I) 2 − αβSI (1+αI) 2 = − βI 1+αI − d − u 1 − βS (1+αI) 2 −d − u 1 rbu 2 2 I (1+bu 2 I) 2 − βS 1+αI = (− A S ) 1 βA {βI+(d+u 1 )(1+αI)} 2 −(d + u 1 ) rbu 2 2 I (1+bu 2 I) 2 − βS 1+αI =(− A S ) 1 βA {βI+(d+u 1 )(1+αI)} 2 −(d + u 1 )λ 2 + K 1 λ + K 2 = 0, where K 1 = 2d + δ + γ + u 1 + ru 2 (1+bu 2 I) 2 + βI 1+αI − βS (1+αI)K 1 = d + u 1 + αβSI (1 + αI) 2 + βI 1 + αI − rbu 2 2 I (1 + bu 2 I) 2 ≥ d + u 1 + αβSI (1 + αI) 2 + βI 1 + αI − rbu 2 2 I 1 + bu 2 I = d + u 1 + αβSI (1 + αI) 2 + I (1 + αI)(1 + bu 2 I) {(β − rbu 2 2 ) + (βbu 2 − αrbu 2 2 )I}. Thus, K 1 is positive if the condition β ≥ max{rbu 2 2 , rαu 2 } holds and K 2 = G(0) > 0. Hence, all the eigen values of the variational matrix have negative real part. Therefore, (S * , I * , R * ) is asymptotically stable if β ≥ max{rbu 2 2 , rαu 2 }. Case II: Suppose R * 0 < R 0 < 1. Then H(0) < 0. Again, we have already proved that two endemic equilibria (S * 1 , I * 1 , R * 1 ) and (S * 2 , I * 2 , R * 2 ) (with I * 1 < I * 2 ) exist for R * 0 < R 0 < 1 if the condition (ru 2 + d + δ + γ)(ru 2 + d + δ + γ + αA) < bru 2 2 A holds. So, the function H(I) must increase in some neighbourhood of I * 1 and decrease in some neighbourhood of I * 2 . Therefore, H (I * 1 ) > 0 and H (I * 2 ) < 0. In this case, we have reached following two conclusions. (1) For the equilibrium point (S * 1 , I * 1 , R * 1 ), we have H (I * 1 ) > 0. So, G(0) < 0. Again, lim G(λ) = ∞ as λ → ∞. Thus, G(λ i ) = 0 for some λ i > 0. So, at least one eigen value of the variational matrix is positive. Therefore, (S * 1 , I * 1 , R * 1 ) is unstable. (2) For the equilibrium point (S * 2 , I * 2 , R * 2 ) , we have H (I * 2 ) < 0 and so G(0) > 0. Thus, we proceed same as case I and derive that (S * 2 , I * 2 , R * 2 ) is asymptotically stable if β ≥ max{rbu 2 2 , rαu 2 }. Hence the theorem is proved. In Figure 1, we have plotted backward bifurcation curve where blue and red lines represent the lines of stable and unstable equilibrium points respectively. Therefore, Theorem 6 is justified by Figure 1. Characterization of the Optimal Control In this model, we have considered two controls, one control variable u 1 is used for vaccinating the susceptible populations and other control variable u 2 is used for treatment efforts for infected individuals. We assume that both vaccination and treatment controls are the functions of time t as they are applied according to the necessity. Our main objective is to minimize the total loss occurs due to the presence of infection and the cost due to vaccination of susceptible individuals and treatment of infected individuals. Thus, the strategy of the optimal control is to minimize the susceptible and infected individuals as well as the cost of implementing the two controls. Thus, we construct the objective functional to be minimized as follows : J(u 1 , u 2 ) = T 0 (A 1 S + A 2 I + B 1 u 2 1 + B 2 u 2 2 )dt where the constants A 1 and A 2 are respectively the per capita loss due to presence of susceptible and infected population at any time instant. Also, the constants B 1 and B 2 respectively represent the costs associated with vaccination of susceptible and treatment of infected individuals. We also assume that the time interval is [0, T ]. The problem is to find optimal functions (u * 1 (t), u * 2 (t)) such that J(u * 1 , u * 2 ) = min{J(u 1 , u 2 ), (u 1 , u 2 ) ∈ U }, where the control set is defined as U = {(u 1 , u 2 )/u i (t) is Lebesgue measurable on [0, 1], 0 ≤ u 1 (t), u 2 (t) ≤ 1, t ∈ [0, T ]}. Theorem 7. There are optimal controls u * 1 and u * 2 such that J(u * 1 , u * 2 ) = min J(u 1 , u 2 ), (u 1 , u 2 ) ∈ U . Proof. : The integrand of the objective functional J(u 1 , u 2 ) is a convex function of u 1 and u 2 . Since the solution of the system (1) is bounded, hence the system satisfies the Lipshitz property with respect to the variables S, I and R. Therefore, there exists an optimal pair (u * 1 , u * 2 ) . Hence the theorem is proved. The Lagrangian of the problem is given by L = A 1 S + A 2 I + B 1 u 2 1 + B 2 u 2 2 . Now, we form the Hamiltonian H for the problem given by, H(S, I, R, u 1 , u 2 , λ 1 , λ 2 , λ 3 ) = A 1 S + A 2 I + B 1 u 2 1 + B 2 u 2 2 + λ 1 (t){A − βSI 1+αI − dS − u 1 S} + λ 2 (t){ βSI 1+αI − (d + δ + γ)I − ru 2 I 1+bu 2 I } + λ 3 (t){ ru 2 I 1+bu 2 I + γI + u 1 S − dR}. In order to determine the adjoint equations and transversality conditions, we use Pontryagin's Maximum Principle [15][16] which gives dλ 1 (t) dt = − ∂H ∂S , dλ 2 (t) dt = − ∂H ∂I , dλ 3 (t) dt = − ∂H ∂R , with the transversality conditions λ i (T ) = 0, i = 1, 2, 3. Thus, we have              dλ 1 dt = −A 1 + (λ 1 −λ 2 )βI 1+αI + dλ 1 + u 1 (λ 1 − λ 3 ) dλ 2 dt = −A 2 + (λ 1 −λ 2 )βS (1+αI) 2 + (λ 2 −λ 3 )ru 2 (1+bu 2 I) 2 + (d + δ)λ 2 + γ(λ 2 − λ 3 ) dλ 3 dt = dλ 3(9) with the transversality conditions λ 1 (T ) = 0, λ 2 (T ) = 0, λ 3 (T ) = 0. Now, using the optimality conditions ∂H ∂u 1 = 0 and ∂H ∂u 2 = 0 we get u 1 = (λ 1 −λ 3 )S 2B 1 and u 2 (1 + bu 2 I) 2 = (λ 2 −λ 3 )rI 2B 2 . Clearly, ∂ 2 H ∂u 2 1 > 0, ∂ 2 H ∂u 2 2 > 0 and ∂ 2 H ∂u 2 1 ∂ 2 H ∂u 2 2 − ( ∂ 2 H ∂u 1 ∂u 2 ) 2 > 0. Therefore, the optimal problem is minimum at controls u * 1 and u * 2 where u * 1 = max 0, min (λ * 1 −λ * 3 )S * 2B 1 , 1 and u * 2 = max 0, min u 2 , 1 , where u 2 is the non-negative root of the equation u 2 (1 + bu 2 I * ) 2 = (λ * 2 −λ * 3 )rI * 2B 2 . Here, S * , I * , R * are respectively the optimum values of S, I, R and (λ * 1 , λ * 2 , λ * 3 ) is the solution of the system (9) with the condition (10). Thus, we summarize the details in the following: Theorem 8. The optimal controls u * 1 and u * 2 which minimize J over the region U are given by u * 1 = max 0, min (λ * 1 −λ * 3 )S * 2B 1 , 1 and u * 2 = max 0, min u 2 , 1 , where u 2 is the non-negative root of the equation u 2 (1 + bu 2 I * ) 2 = (λ * 2 −λ * 3 )rI * 2B 2 . Numerical Simulations and Efficiency Analysis To justify the impact of optimal control, we have used the forward-backward sweep method to solve the optimality system numerically. This method combines the forward application of a fourth order Runge-Kutta method for the state system (1) with the backward application of a fourth order Runge-Kutta method for the adjoint system (9) and the transversality conditions (10). Here, we fixed up our problem for 20 months and assume that the vaccination and treatment are stopped after 20 months. The simulation which we carried out by using the parametric values given in Table 2 with the initial conditions S(0) = 50, I(0) = 4 and R(0) = 0.01. In this paper, we have considered two controls, one is vaccination control u 1 and other is treatment control u 2 . Parameters A β α d δ γ r b A 1 A 2 B 1 B 2 Values But, if we use only one control among u 1 and u 2 then one question may arise 'which control is more efficient to reduce infection ?' To answer this question we will perform an efficiency analysis [17] which will allow us to determine the best control strategy. Here, we distinguish two control strategies STR-1 and STR-2 where STR-1 is the strategy where u 1 = 0 , u 2 = 0 and STR-2 is the strategy where u 1 = 0 , u 2 = 0. To determine the best control strategy among these two, we have to calculate the efficiency index (E. I.) = (1 − A c A o ) × 100, where A c and A o are the cumulated number of infected individuals with and without control, respectively. The best strategy will be the one whom efficiency index will be bigger [17]. It can be noted that the cumulated number of infected individuals Table 3. Strategy A c E.I. Table 3, it follows that STR-1 is the best strategy among STR-1 and STR-2 which permits to reduce the number of incident cases. Thus, vaccination is more effective than treatment. Conclusions In this paper, we have analysed the qualitative behaviour and optimal control strategy of an SIR model. We have introduced a saturated incidence rate which is affected by inhibitory factors and considered a saturated treatment function which characterizes the effect of limited treatment capacity on the spread of infection. Two control functions have been used, one for vaccinating the susceptible populations and other for controlling the treatment efforts to the infected populations. To describe the complex dynamics of the solutions for constant controls, we have obtained the basic reproduction number R 0 which plays a crucial role for the study of stability analysis of both disease free equilibrium point and endemic equilibrium points as well as backward bifurcation analysis. We have established that DFE is locally asymptotically stable for R 0 < 1 and in addition, if inhibitory coefficient is greater than some quantity (α ≥ bu 2 ) then DFE is globally asymptotically stable which is very significant at the biological point of view. If R 0 = 1, DFE is a non-hyperbolic equilibrium point and the stability analysis of this point has been investigated by using Centre manifold theory. We have also used Sotomayor theorem to show transcritical bifurcation at DFE with respect to the treatment control. We have obtained a necessary and sufficient condition on the model parameters such that backward bifurcation occurs. Moreover, stability analysis of endemic equilibrium points is discussed analytically for the different values of R 0 . We have also studied and determined the optimal vaccination and treatment to minimize the number of infective and susceptible populations as well as the cost due to vaccination and treatment. A comparative study between the system with controls and without control has been presented to realize the positive impact of vaccination and treatment in controlling the infectious diseases. Finally, efficiency analysis has been performed to determine that the vaccinating to the susceptible populations is better than treatment control to infected populations in order to minimize the infected individuals. The entire study of this paper is mainly based on the deterministic framework and our proposed model is valid for large population only. The work is a theoretical modelling and it can be further justified using experimental results. introduced a constant treatment function T (I) in an SIR model, where T (I . of Df (A 1 , u 0 2 ) T corresponding to the eigen value λ = 0 is W = Let f u 2 denote the vector of partial derivatives of the components of f with respect of u 2 . Thus Figure 1 : 1Backward bifurcation curve for the parametric values A = 11, α = 0.5, d = 0.000039, γ = 0.08, δ = 0.02, r = 0.4, b = 2.21, u 1 = 0.5, u 2 = 0.5. So, one of the three eigen values of the variational matrix is −d. The remaining eigen values are the solutions of the equation G(λ) = 0, that the component I of the endemic equilibrium point(s) are the solutions of the equation H(I) = 0 and H(0) = ( d+δ+γ+ru 2 β )(R 0 − 1). Thus, R 0 > 1 if and only if H(0) > 0 and R 0 < 1 if and only if H(0) < 0. Now, we shall derive the relation between G(0) and H (I). Differentiating H(I) with respect to I, So, we have G(0) > 0 if and only if H (I) < 0 and G(0) < 0 if and only if H (I) > 0. Now, we shall discuss two cases. Case I: Suppose R 0 > 1. Then H(0) > 0. We have already proved in Section 3 that when R 0 > 1 then only one endemic equilibrium point (S * , I * , R * ) exists. Since H(0) > 0, hence H(I) should decrease in some neighbourhood of I * . Thus, in this case H (I * ) < 0 and so G(0) > 0. Again, we know that one of the eigen values of the variational matrix is −d and the remaining eigen values are the solutions of the equation G(λ) = 0 i.e. the equation Figure 2 Figure 2 :Figure 3 : 223(a)-(c) show the time series of the susceptible (S), infected (I) and recovered (R) individuals both with and without control.Figure 3(a)-(b) represent the optimal control u * 1 and u * 2 respectively for the time interval [0, 20]. From Figure 2(a)-(c), we see that optimal controls due to vaccination and treatment are very effective for reducing the number of susceptible and infected individuals and so enhancing the number of recovered Time series of the populations with control and without control:(a) susceptible individuals (b) infected individuals ,(c) recovered individuals . Time series of control variables: (a) Optimal control u 1 , (b) Optimal control u 2 . during the time interval [0, 20] is defined by A = 20 0 I(t)dt. 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[ "Ancillary Gaussian modes activate the potential to witness non-Markovianity", "Ancillary Gaussian modes activate the potential to witness non-Markovianity" ]	[ "Dario De Santis \nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain\n", "Donato Farina \nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain\n", "Mohammad Mehboudi \nDépartement de Physique Appliquée\nUniversité de Genève\n1211GenèveSwitzerland\n", "Antonio Acín \nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain\n\nICREA -Institució Catalana de Recerca i Estudis Avançats\n08010BarcelonaSpain\n" ]	[ "ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain", "ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain", "Département de Physique Appliquée\nUniversité de Genève\n1211GenèveSwitzerland", "ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain", "ICREA -Institució Catalana de Recerca i Estudis Avançats\n08010BarcelonaSpain" ]	[]	We study how the number of employed modes impacts the ability to witness non-Markovian evolutions via correlation backflows in continuous-variable quantum dynamics. We first prove the existence of non-Markovian Gaussian evolutions that do not show any revivals in the correlations between the mode evolving through the dynamics and a single ancillary mode. We then demonstrate how this scenario radically changes when two ancillary modes are considered. Indeed, we show that the same evolutions can show correlation backflows along a specific bipartition when three-mode states are employed, and where only one mode is subjected to the evolution. These results can be interpreted as a form of activation phenomenon in non-Markovianity detection and are proven for two types of correlations, entanglement and steering, and two classes of Gaussian evolutions, a classical noise model and the quantum Brownian motion model. arXiv:2207.01649v2 [quant-ph] 9 Aug 2022	10.1088/1367-2630/acba3a	[ "https://export.arxiv.org/pdf/2207.01649v2.pdf" ]	250,280,206	2207.01649	e685a3cdb6b1c2d4ab6db8a31ae1a0540eb1fc6b	Ancillary Gaussian modes activate the potential to witness non-Markovianity (Dated: August 10, 2022) Dario De Santis ICFO-Institut de Ciencies Fotoniques The Barcelona Institute of Science and Technology 08860CastelldefelsBarcelonaSpain Donato Farina ICFO-Institut de Ciencies Fotoniques The Barcelona Institute of Science and Technology 08860CastelldefelsBarcelonaSpain Mohammad Mehboudi Département de Physique Appliquée Université de Genève 1211GenèveSwitzerland Antonio Acín ICFO-Institut de Ciencies Fotoniques The Barcelona Institute of Science and Technology 08860CastelldefelsBarcelonaSpain ICREA -Institució Catalana de Recerca i Estudis Avançats 08010BarcelonaSpain Ancillary Gaussian modes activate the potential to witness non-Markovianity (Dated: August 10, 2022) We study how the number of employed modes impacts the ability to witness non-Markovian evolutions via correlation backflows in continuous-variable quantum dynamics. We first prove the existence of non-Markovian Gaussian evolutions that do not show any revivals in the correlations between the mode evolving through the dynamics and a single ancillary mode. We then demonstrate how this scenario radically changes when two ancillary modes are considered. Indeed, we show that the same evolutions can show correlation backflows along a specific bipartition when three-mode states are employed, and where only one mode is subjected to the evolution. These results can be interpreted as a form of activation phenomenon in non-Markovianity detection and are proven for two types of correlations, entanglement and steering, and two classes of Gaussian evolutions, a classical noise model and the quantum Brownian motion model. arXiv:2207.01649v2 [quant-ph] 9 Aug 2022 I. INTRODUCTION The interaction between any given quantum system and the surrounding environment can never be completely avoided; the theory of open quantum systems is an indispensable framework to describe the realistic dynamics of quantum systems [1,2]. The interaction with the environment is usually detrimental for quantum resources, like quantum coherence or entanglement, and in general makes the resulting map on the system no longer unitary, but by a quantum channel, described by a completely-positive trace-preserving (CPTP) map. The evolution in time is then described by a continuous family of quantum channels, which can be classified as Markovian or non-Markovian. While the former class is characterized by the continuous degradation of any type of information encoded in the system, in the latter the decoherence process is not monotonic in time-these recoherences are often called backflows of information. Non-Markovian evolutions have attracted much interest, not only because of their fundamental interest, but also because the associated backflows can have a positive effect in various quantum information tasks, such as metrology [3], quantum key distribution [4], quantum teleportation [5], entanglement generation [6], quantum communication [7], information screening [8] and quantum thermodynamics [9][10][11][12]. The mathematical property used to define Markovianity is called CP-divisibility; the dynamics is Markovian if and only if it is possible to describe the evolution between any two times through the action of a physical quantum channel, that is, a CPTP map (for reviews on this topic see Refs. [13][14][15]). Various strategies have been adopted to connect this mathematical definition to more physically motivated ones. This has lead to the development of witnesses of non-Markovianity through backflows of different quantities, such as the error probability in state discrimination [16][17][18], channel capacity [7], Fisher information [19,20], the volume of accessible states [21] and correlations [22][23][24][25][26][27][28]. In the case of correlations, the standard method for witnessing non-Markovianity works as follows: (i) prepare an initial state of two particles, which we name system and ancilla; (ii) apply the considered evolution to one of the two particles, the system, while the ancilla remains untouched and (iii) monitor how the correlations between the two particles change during the evolution. If the correlations do not decrease monotonically with time, the dynamics gives rise to a correlation backflow and therefore is non-Markovian. Beyond this recipe, in general it is not known whether and how to construct the initial two-particle state for a given non-Markovian dynamics or, even simpler, what is the minimal dimension of the ancillary system that is needed for this task. Even less is known for continuousvariable systems and, in particular, for Gaussian dynamics, despite their prominent role in many physically relevant scenarios. While for finite-dimensional, several works have studied correlation backflows in quantum evolutions [25,26,29], continuous-variable settings have not been explored beyond the use of a single ancillary mode [27,28,30]. In this work, we firstly ask ourselves whether considering a single ancillary mode is sufficient to witness quantum correlation backflows for arbitrary non-Markovian dynamics. Particularly, we use both entanglement and Gaussian steerability as correlations. Our results show that indeed using a single ancillary mode is not always sufficient to witness backflows. Secondly, and motivated by this shortcoming, we ask ourselves whether deploying a secondary ancillary system would be advantageous. We show through two examples, namely the dynamics of a single mode under (i) a classical noise model and (ii) the quantum Brownian motion model, that the secondary ancillary mode allows witnessing non-Markovian evolutions that are impossible to detect with any possible single ancillary mode initialization. Finally, we show that, while for some open dynamics two ancillary modes are sufficient for witnessing non-Markovianity, for some other dynamics one may need even a higher number of ancillary modes. The article is structured as follows. In Sec. II we briefly introduce Gaussian states and the entangled initializations of interest. Such initializations are assumed to undergo local Gaussian dynamics, characterized in Sec. III, and their quantum correlations are quantified through Gaussian steerability and entanglement, Sec. IV. The advantage stemming from the use of more than one ancillary mode is presented in Sec. V, through paradigmatic examples. Finally, we summarize our results in Sec. VI. II. PRELIMINARIES In this Section we set the notation and introduce the adopted formalism to describe quantum Gaussian systems. An nmode continuous variable quantum system is defined through states over the Hilbert space H (n) = ⊗ n i=1 H i , where H i is the Hilbert space of a bosonic harmonic oscillator corresponding to the i-th mode of the system. We call S (H (n) ) the state space of density operatorsρ associated to H (n) . The quadrature operators of the i-th mode areq i = (â i +â † i ) and p i = −i(â i −â † i ), whereâ i (â † i ) is the annihilation (creation) operator for the i-th mode. By grouping these operators in the vectorX = (q 1 ,p 1 ,q 2 ,p 2 , . . . ), we can write the canonical commutation relations as [X i ,X j ] = 2iΩ (n) i j , where Ω (n) = n i=1 Ω (1) , Ω (1) = 0 1 −1 0 ,(1) the 2n × 2n matrix Ω (n) being the n-mode symplectic form and Ω (1) the corresponding single-mode form. A quantum stateρ ∈ S (H (n) ) is called Gaussian when the first and second moment of the quadrature vectorX, namely d i = X i ρ and σ i j = 1 2 {X i ,X j } ρ − X i ρ X j ρ ,(2) are sufficient to fully describeρ, where Ô ρ = Tr[ρÔ] is the expectation value of the operatorÔ on the stateρ. The 2n × 2n real symmetric matrix σ is called the covariance matrix of the system. Two Gaussian states with different first moments and same covariance matrix can be mapped one into the other by a displacement unitary transformation. In the following we are interested on the information contained in the covariance matrix only and therefore we ignore d i . In case of a bipartite scenario, where Alice owns the first n A modes and Bob owns the last n B , the covariance matrix σ AB of a shared Gaussian state can be written as follows: σ AB = A C C T B ,(3) where the 2n A × 2n A matrix A (2n B × 2n B matrix B) is the covariance matrix of Alice's (Bob's) system and the correlation matrix C is 2n A × 2n B . In order for σ AB to correspond to a physical quantum state, namely to satisfy the uncertainty principle, the following condition has to be satisfied: σ AB + iΩ (n) 0 ,(4) where n = n A + n B and the inequality means that the matrix in the l.h.s. is positive semi-definite. Two-mode entangled states In the following we consider two main classes of Gaussian states: the two-mode squeezed states [31] and the three-mode GHZ/W states [32], the corresponding covariance matrices being indicated, respectively, as σ 2,r and σ 3,r . First, in order to define σ 2,r , consider a two-mode scenario, where Alice and Bob own each a single mode (n A = n B = 1). The covariance matrix σ 2,r corresponding to a two-mode squeezed state is given by Eq. (3), with A = B = cosh(2r) I and C = sinh(2r) Z, where I is the identity matrix and Z = diag(1, −1), namely: σ 2,r = cosh(2r) I sinh(2r) Z sinh(2r) Z cosh(2r) I ,(5) As the squeezing parameter r 0 increases, the two modes become more and more correlated (entangled) [31], where r = 0 corresponds to a separable state, namely the two-mode vacuum state. The maximally entangled EPR state corresponds to lim r→∞ σ 2,r . It must be noticed that σ 2,∞ corresponds to an infinite energy state [31] and therefore it cannot be realized experimentally. Three-mode entangled states The GHZ/W state is a three-mode Gaussian state which is entangled among each mode. It is realized by three squeezed beams mixed in a tritter [32]. In case of equal squeezings, the corresponding covariance matrix is given by σ 3,r =          σ(r) (r) (r) (r) σ(r) (r) (r) (r) σ(r)          ,(6) where σ(r) = diag((e 2r + 2e −2r )/3, (e −2r + 2e 2r )/3) and (r) = (2/3) sinh(2r)Z. The parameter r 0 is the global squeezing parameter of the state, where r = 0 corresponds to the separable case and r → ∞ provides maximal entanglement. Consider the bipartite scenario where Alice owns the first two modes of σ 3,r , while Bob owns the last mode. The covariance matrix σ 3,r can be divided into blocks as in Eq. (3), where A = σ(r) (r) (r) σ(r) , B = σ(r) , C = (r) (r) . III. GAUSSIAN CHANNELS AND EVOLUTIONS Gaussian channels are those that preserve Gaussianity of quantum states, that is, they map Gaussian states into Gaussian states. They can be fully characterised by their application on the displacement vector and the covariance matrix. Nonetheless, since we are interested only in the information contained in the covariance matrix of Gaussian states, we represent the action of a generic Gaussian transformation as [31] Λ : σ → σ = T σT T + N , where T and N are 2n × 2n matrices of reals. Moreover, N must be symmetric to preserve the symmetry of the covariance matrices. Thus, any Gaussian channel can be represented by the pair Λ ≡ (T, N), such that σ = (N, T )σ. It is clear from Eq. (7) that the identity map corresponds to (T, N) = (I, 0), where, again, I is the identity matrix and 0 is the null matrix. The channel (T, N) CPTP if and only if [33] N − iT ΩT T + iΩ 0 . For single-mode channels, condition (8) reduces to the following two conditions N = N T 0 (9) det N ≥ (det T − 1) 2 .(10) A Gaussian dynamical evolution can be denoted by the time-parametrised family {Λ t } t≥0 = {T t , N t } t≥0 , where the channel (T t , N t ) that represents the evolution at time t is called dynamical map. One expects that, naturally, at t = 0 the dynamics is given by the identity channel, i.e., (T 0 , N 0 ) = (I, 0). We further assume the evolution to be divisible, i.e., for any arbitrary times 0 s t one can write Notice that N t,s has to be symmetric. Importantly, for a general evolution the intermediate map (T t,s , N t,s ) could be non-CPTP for some 0 < s t. This fact can be used to define Markovian Gaussian evolutions as the CP-divisible family of Gaussian channels {T t , N t } t 0 . In other words, Markovian evolutions are those with CPTP intermediate maps (T t,s , N t,s ) for all 0 < s t. In case the evolution is not CP-divisible, we call it non-Markovian. (T t , N t ) = (T t,s , N t,s ) • (T s , N s ) = (T t,s T s , T t,s N s T T t,s + N t,s ) ,(11) In the following we consider Gaussian evolutions that are applied only to one mode of a multimode Gaussian system. Accordingly, the covariance matrix of such multimode Gaussian state evolves as σ(t) = (T (1) t ⊕ I, N (1) t ⊕ 0)σ(0) = (T (1) t ⊕ I)σ(0)(T (1) t ⊕ I) T + N (1) t ⊕ 0 ,(12) with (T (1) t , N (1) t ) being a single-mode dynamical Gaussian channel. In the remainder of this text, we drop the label (1) to lighten our notation. We follow by describing how information quantifiers can be used to witness non-Markovianity of evolutions through their non-monotonic behaviors, namely backflows. IV. NON-MARKOVIANITY WITNESSES Given a functional I : S (H) → R + that maps quantum states into non-negative real numbers, we call it an information quantifier if it is non-increasing under CPTP maps, namely if I(ρ) I(Λ(ρ)) for all CPTP maps Λ : S (H) → S (H) and statesρ ∈ S (H). The minimum value I = 0 is interpreted as the absence of the considered information in the state. It follows that all evolutions {Λ t } t≥0 cannot increase the amount of information contained in the initial stateρ(0), namely I(ρ(0)) I(ρ(t)) for all t 0 andρ(0) ∈ S (H), whereρ(t) := Λ t (ρ(0)). Nonetheless, it could be the case that an intermediate map between two times s < t is not CPTP and that we obtain the increase I(ρ(s)) < I(ρ(t)). Hence, since Markovian evolutions are characterized by having CPTP intermediate maps, any increase, or backflow, of I witnesses non-Markovianity. Notice that, in general, it does not suffice to consider the evolution of a single initial stateρ(0) in order to state that an evolution is Markovian due to the monotonicity of I(ρ(t)). Indeed, even if a stateρ(0) does not allow observing backflows of I, there may be a different stateρ (0) for which an increase of I(ρ (t)) can be observed. The same is true for I: some quantifiers are not able to witness certain types of non-Markovian evolutions. In this context, a key ingredient is the use of ancillary systems. Indeed, we can use initial statesρ(0) ∈ S (H ⊗ H ) such thatρ(t) = Λ t ⊗ I (ρ(0)), where the non-evolving ancillary system is defined over the Hilbert space H and I is the identity map on S (H ). In general, initializations that make use of ancillas allow witnessing non-Markovianity with higher precision. Indeed, for some evolutions and information quantifiers, we can obtain backflows if and only if particular systemancilla initializations are considered [17,29,35]. Moreover, the dimension of the ancilla is also important: depending on the case, a minimal ancillary size could be required to obtain backflows. In the following, we exploit two information quantifiers as non-Markovianity witnesses: Gaussian steerability and entanglement. Our results reveal that there exist some non-Markovian Gaussian evolutions that (i) cannot be witnessed by means of the aforementioned correlations with any twomode Gaussian initialization, but (ii) can be witnessed by using three-mode initial Gaussian states, where in (i) and (ii) we respectively consider one and two ancillary modes. Thus, we highlight the crucial role that ancillary modes can play. More in details, we consider the scenario where Alice and Bob share a Gaussian correlated system A 1 A 2 B, where A 1 is Alice's evolving system, A 2 is Alice's ancillary system (in case there is one) and B is Bob's ancillary system. Hence, we compare the potentials of the settings A 1 \|B and A 1 A 2 \|B to provide correlation backflows, where A 1 , A 2 and B are onemode systems. As described before, the aim of this work is to describe the advantages of using the three-mode setup A 1 A 2 \|B. Finally, we propose the following analogy between finite and infinite dimensional evolutions to discuss the minimal ancillary sizes of A 2 and B needed to observe correlation backflows when A 1 is a generic n-mode Gaussian evolving system. There exists a hierarchy for the degree of non-Markovianity of d-dimensional evolutions called k-divisibility [36], which is based on the minimal ancillary dimension needed to obtain information backflows [35]. The backflows considered here correspond to increases in distinguishability of two states given with a-priori probabilities p and 1 − p, which are defined over the evolving system and an ancilla. If an invertible evolution is k-divisible but not k + 1-divisible, we can obtain backflows if and only if k + 1-dimensional (or larger) ancillas are considered. In this framework, Markovianity cor-responds to d-divisibility. Hence, d is the largest ancillary dimension needed to witness non-Markovianity, which is required for d − 1-divisible evolutions. In terms of correlation backflows, by considering the setting A 1 A 2 \|B explained above, d + 1 and 2 are, respectively, the minimum dimensions of A 2 and B that have been proven to be sufficient to witness any invertible non-Markovian evolution [24]. Similarly, in order to observe correlation backflows from n-mode non-Markovian Gaussian evolutions, we may expect to need n + 1-mode A 2 ancillas. Nonetheless, Gaussian non-Markovianity follows a simpler hierarchy [37]: intermediate maps are either CP, positive or non-positive, where Markovianity corresponds to 1-divisibility. Therefore, we expect that, given a generic invertible non-Markovian Gaussian evolution, a minimal requirement for a bipartite system A 1 A 2 \|B to provide correlation backflows is that A 2 and B are respectively (at most) two-mode and one-mode Gaussian systems, no matter the number of evolving modes of A 1 . A. Gaussian steerability Gaussian steerability is a form of quantum correlations and, similarly to other correlation measures, is non-increasing under the action of CPTP local maps. For instance, it means that if one is interested in a single mode dynamical channel (T t , N t ), one can construct a local evolution as in (12) and Gaussian steerability can be considered as an information quantifier and used to witness non-Markovianity through backflows, as suggested in Ref. [28]. There, the authors show that σ AB (0) = σ 2,r can be deployed as initial state to witness non-Markovianity for the quantum Brownian motion modelsee the description in Section V B. In what follows, we first present the mathematical description of Gaussian steerability. Then, we provide examples of dynamics where this correlation cannot witness non-Markovianity when using any twomode Gaussian system. We then proceed by showing that using three-modes one can witness non-Markovianity in many cases where two modes fail. We furthermore show that even with three entangled modes it can happen that some non-Markovian evolutions cannot be witnessed using Gaussian steerability. Consider a bipartite scenario where Alice and Bob share an n A + n B -mode Gaussian state with covariance matrix σ AB , where Alice holds the first n A modes and Bob holds the last n B modes-see Eq. (3). A quantifier for the potential of Alice to steer Bob's share through Gaussian measurements has been introduced in Ref. [38]. It turns out that σ AB is Gaussian steerable from Alice to Bob, or A → B steerable with Gaussian measurements, if and only if the following condition is violated [39]: σ AB + i(0 A ⊕ Ω B ) 0 ,(13) where 0 A is the 2n A × 2n A null matrix and Ω B = Ω (n B ) . This condition is equivalent to the Schur complement of B M σ B = B − C T A −1 C ,(14) not being a physical covariance matrix; that is a violation of the following inequality M σ B + iΩ B 0 .(15) Therefore, A → B Gaussian steerability can be verified by studying the set {ν i } n B i=1 of symplectic eigenvalues of M σ B . Recall that these are associated to the absolute value of the eigenvalues 40]. It can be shown that Eq. (15) is violated if and only if ν i < 1 for one or more i [31]. Hence, following Ref. [38], one can quantify A → B Gaussian steerability as: {±ν i } n B i=1 of the matrix iΩ B M σ B [G A→B (σ AB ) = max        0, − ν i <1 log ν i        .(16) In case we want to evaluate B → A Gaussian steerability, we replace M σ B with the Schur complement of A, namely M σ A = A − CB −1 C T , and evaluate its symplectic eigenvalues. Notice that in general steering is not symmetric, i.e., G A→B (σ AB ) G B→A (σ AB ). Measurement incompatibility and steering A necessary condition for Gaussian steerability is given by Gaussian measurement incompatibility. Imagine an A → B Gaussian steering scenario, where Alice owns n A Gaussian modes which are transformed by the Gaussian channel (T, N). In case the action of the (dual) channel (T, N) * makes the set of Alice's Gaussian measurements compatible, no A → B Gaussian steering can be performed -that is G A→B ((T ⊕ I B , N ⊕ 0 B )σ AB ) = 0 for all initializations σ AB . In turn, a Gaussian channel breaks incompatibility of all Gaussian measurements if and only if [41][42][43] N − iT ΩT T 0 .(17) We call a channel (T, N) Gaussian incompatibility breaking (GIB) in case it satisfies Eq. (17). This immediately leads to the following observation: Observation 1.-Consider a dynamics that breaks incompatibility of all Gaussian measurements on Alice's side within some time interval t ∈ (t 1 , t 2 ). Any non-Markovian behaviour, namely the violation of the CP-divisibility condition, of the dynamics within this interval cannot be witnessed by Gaussian steerability from Alice to Bob. Indeed, steering is equal to zero in the time interval (t 1 , t 2 ). On the other hand, Alice can always extend her system to include one or more new Gaussian modes which do not undergo the Gaussian channel. Such an extension leads to our second observation: Observation 2. -If one or more of Alice's modes do not undergo the Gaussian dynamics, namely if Alice extends her modes by at least one such that her total share undergoes the Gaussian dynamics (T t ⊕ I, N t ⊕ 0), the criterion (17) is always violated, i.e., the dynamics on Alice is never GIB. Note that Observation 2 does not imply that Gaussian steerability from Alice's extended system to Bob can witness non-Markovianity. Indeed, on the one hand measurement incompatibility is a necessary but not sufficient condition for Gaussian steerability. On the other hand, provided that one has nonzero steering, it is not guaranteed that steering backflows are always observed when non-Markovianity is at play. Nonetheless, we can increase the chance to witness a bigger class of non-Markovian dynamics by simply extending the number of modes. We showcase this through some examples in Section Sections V A 1 and V B 1. B. Entanglement A necessary condition for the separability of a bipartite n A + n B -mode Gaussian stateρ AB with covariance matrix σ AB , is given by the positivity of the partial transposition of the density matrix, namely the PPT condition [44,45], which states that separable states satisfy the condition: σ AB + iΩ A ⊕ Ω T B 0 ,(18) where Ω A = Ω (n A ) and Ω B = Ω (n B ) . Hence, a quantifier for the entanglement in σ AB can be defined as: E PPT (σ AB ) = max          0, − µ i <0 µ i          ,(19) where µ i is the i-th eigenvalue of σ AB + iΩ A ⊕ Ω T B . The PPT condition (18) is a necessary separability condition in general, but turns out to be sufficient for any 1 + n B -mode and n A + 1-mode Gaussian state, namely when at least one of the two parties is single mode (the case we will be considering in Sec. V). A Gaussian channel (T, N) applied to Alice's share is entanglement breaking (EB), i.e., nullifies the entanglement content of any bipartite input state, if and only if the matrix N admits [46] N = N 1 + N 2 , where N 1 iΩ (n A ) , N 2 iT Ω (n A ) T T . (20) Remark.-Any EB channel is also GIB. To see this for Gaussian channels considered here, note that (20) implies that a necessary condition for EB is to have N 1 0. When we add this to the condition for N 2 , we revive (17). Also, notice that the reverse is not necessarily true. We can now make the following two observations analogous to Observations 1 and 2. Observation 3.-Consider a dynamics on Alice's side that is EB within some time interval t ∈ (t 1 , t 2 ). Any non-Markovian behaviour, namely the violation of the CPdivisibility condition, of the dynamics within this interval cannot be witnessed by entanglement between Alice's and Bob's system. Indeed, entanglement is always zero in (t 1 , t 2 ). Observation 4.-If one or more of Alice's modes do not undergo the Gaussian dynamics-i.e., if Alice extends her modes by at least one, such that her total share undergoes the Schematic setup for the detection of non-Markovianity with (a) two-mode squeezed input state and with (b) GHZ/W threemode input state. Notice the selected bipartition concerning Alice's and Bob's shares (black horizontal line) where only Alice is allowed to own more than one mode. We consider Gaussian evolutions (T t , N t ) that are applied only to the first mode of Alice's share. Non-monotonic behaviors of Gaussian steering/entanglement quantifiers as function of time (backflows) are used as non-Markovianity witnesses. The GHZ/W three-mode configuration (b) can activate the potential to witness non-Markovian behaviors which in the twomode scenario (a) do not imply any revivals of quantum correlations. Gaussian channel (T t ⊕ I, N t ⊕ 0)-the criterion (20) is always violated, i.e., the dynamics on Alice is never EB. It turns out that for a generic one-mode Gaussian channel (T, N), the EB character can be tested by applying the channel locally over the maximally entangled two-mode squeezed state σ 2,∞ = lim r→∞ σ 2,r , i.e., the separability condition E PPT ((T ⊕ I, N ⊕ 0)σ 2,∞ ) = 0 is a necessary and sufficient condition for the EB character of the channel [40]. V. PARADIGMATIC EXAMPLES Here, we demonstrate how by using an extra auxiliary mode on Alice's share, one can witness non-Markovianity through correlation backflows within a bigger class of Gaussian dynamics, if compared to the one-ancillary mode scenario. The potential of the method was anticipated in Observations 2 and 4: since some of the modes owned by Alice do not undergo the channel and can maintain their correlations with Bob's side the correlation quantifier does not nullify implying more chances of observing its backflow. A schematic representation of our settings is reported in Fig. 1. A. Classical noise channel We start by probably the simplest example, i.e., a classical noise channel applied to a single mode. It is given by [31] (T t , N t ) = (I, η(t)I) , where η(t) 0 is a time-dependent continuous function such that η(0) = 0. The form for the intermediate map of this evolution can be derived using Eq. (11), obtaining, for any s, t such that 0 < s < t, (T t,s , N t,s ) = (I, (η(t) − η(s))I) . It follows that this Gaussian evolution is Markovian if and only if η(t) is monotonically increasing. Indeed, from Eq. (9), as soon as η(t)−η(s) < 0 the corresponding intermediate channel is non-CPTP. Gaussian steerability Consider a two-mode Gaussian state shared between Alice and Bob described by the covariance matrix σ AB . When acting on Alice's mode, the dynamical map (I, η(t)I) makes the set of all Gaussian measurements compatible if and only if η(t) 1 (see Eq. (17) or [43]). Therefore, according to Observation 1, it is not possible to obtain information backflows through Gaussian steerability G A→B at times { t \| η(t) 1}. However, inspired by Observation 2, Alice can extend her system to contain an auxiliary mode, such that her share undergoes a local evolution on the first mode, see Eq. (12). For instance, if we take σ AB = σ 3,r , where Alice holds the first two modes, there will be a backflow in G A→B at anytime that η(t) decreases, i.e., whenever the dynamics is not CP-divisible. Interestingly, this is true for any squeezing r > 0. We provide the proof of this result in Appendix B, where we obtain the analytical forms of G A→B (σ 2,r (t)) and G A→B (σ 3,r (t)) for any r and η(t). As an example, let the noise assume the following time dependence η(t) = t 2 /(t 2 − 2t + 2).(23) Since η(t) is monotonically decreasing for t ≥ 2, it follows that the evolution is non-Markovian and the intermediate map (T t,s , N t,s ) is not CPTP for any 2 s < t. Nonetheless, (T t , N t ) breaks the incompatibility of all Gaussian measurements for t 1 and therefore no backflow of G A→B can be observed when Alice holds only one mode. This is showcased in Fig. 2(a) where we take σ AB = σ 2,r . This is contrary to when we take σ AB = σ 3,r , where Alice holds the first two modes. As seen from Fig. 2(a), in this case G A→B increases whenever the dynamics is not CP-divisible. Entanglement The dynamical map (I, η(t)I) breaks the entanglement of all states if and only if η(t) 2, see Eq. (20). Accordingly, we can choose η(t) to be the following time-dependent function, η(t) = 2t 2 /(t 2 − 2t + 2) ,(24) namely the function (23) rescaled by a factor 2, implying that the non-Markovian interval is the same as before, i.e., [2, ∞] where the function is decreasing. For t ≥ t EB = 1 we have the EB property (η(1) = 2 and η(t) ≥ 2 if and only if t ≥ 1). Analogously to what we observed for steering, in Fig. 2(b) we show that whether we cannot observe any backflow of E PPT (σ 2,r (t)) (this quantity nullifies for t ≥ t EB ), we do observe, instead, a backflow of E PPT (σ 3,r (t)) as soon as the evolution becomes non-Markovian. B. Lossy channel and the quantum Brownian motion The second example we consider is the wider class of lossy channels, which are the evolutions with dynamical maps (T t , N t ) = (τ(t)I, η(t)I) ,(25) including the classical noise channel (21) in the particular cases τ(t) = 1. The intermediate maps of these evolutions assume the form (T t,s , N t,s ) = (τ(t, s)I, η(t, s)I) ,(26) where τ(t, s) = τ(t)/τ(s) and η(t, s) = η(t) − η(s)(τ(t)/τ(s)) 2 . As we show in Appendix C, the lossy channel is not CPdivisible, namely the intermediate map (T t+ ,t , N t+ ,t ) is not CPTP, at times t if and only if either one or both of the following inequalities are violateḋ η(t) − 2(η(t) ± 1)τ (t) τ(t) 0.(27) The class of lossy channels is relevant in the description of quantum Brownian motion, where a harmonic oscillator with frequency ω 0 undergoes a dissipative dynamics by interacting with a bosonic bath at temperature T . The total Hamiltonian is quadratic, which guarantees that the system undergoes a Gaussian dynamics-details of the interaction and the derivation of the dynamics are given in the Appendix D. The two dynamical parameters η(t) and τ(t) are connected to the physical parameters describing the system and the bath as follows τ(t) = exp − t 0 ds γ(s)/2 , η(t) = τ(t) 2 t 0 ds ∆(s)/τ(s) 2 , γ(t) = α 2 t 0 dτ ∞ 0 dωJ(ω) sin(ωτ) sin(ω 0 τ) ,(28)∆(t) = α 2 t 0 dτ ∞ 0 dωJ(ω) coth ω 2T cos(ωτ) cos(ω 0 τ) . Here, ∆(t) is the so called diffusion coefficient and γ(t) is the damping coefficient. The parameter α quantifies the systembath interaction strength. 2. (a) Gaussian steerability G A→B as function of time for the three-mode squeezed state initialization σ 3,r (0) and the two-mode squeezed state initialization σ 2,r (0) and Alice's first mode subjected to the classical noise evolution with η(t) = t 2 /(t 2 − 2t + 2) (black line, Eq. (23)), i.e. such that η(0) = 0, η(1) = 1 (horizontal dashed line, level of noise for which we have the GIB property), η(2) = 2, η(∞) = 1, it is increasing in [0, 2] and decreasing in [2, ∞] (non-Markovian interval, blue shadow region). We cannot observe any backflow of G A→B (σ 2,r (t)) (blue line) in the time interval when the evolution is non-Markovian, because the dynamical maps (T t , N t ) of the evolution are GIB for t ≥ 1. We do observe, instead, a backflow of G A→B (σ 3,r (t)) (red line) in the time interval when the evolution is non-Markovian. The plots are made for r = 2. (b) Same as in (a) but for E PPT and for η(t) = 2t 2 /(t 2 − 2t + 2) (Eq. (24)), such that η(0) = 0, η(1) = 2 (horizontal dashed line, level of noise for which we have the EB property), η(2) = 4, η(∞) = 2, it is increasing in [0, 2] and decreasing in [2, ∞]. For convenience we report now the rescaled quantity η(t)/4, black line. While we cannot observe any backflow of E PPT (σ 2,r (t)) (blue line) in the time interval when the evolution is non-Markovian, namely, again, for t 2 (because the dynamical maps (T t , N t ) of the evolution are EB for t ≥ 1), we do observe, instead, a backflow of E PPT (σ 3,r (t)) (red line). For our simulations, we follow Ref. [28] and choose a spectral density J(ω) of the bath with a Lorentz-Drude cutoff J(ω) = 2ω s π ω 3−s c ω 2 c + ω 2 ,(29) where ω c is the cutoff frequency. The parameter s defines the ohmicity of the spectral density: s < 1 corresponds to the sub-Ohmic regime, s = 1 to the Ohmic regime, and s > 1 the super-Ohmic regime. This model is non-Markovian, i.e., the infinitesimal intermediate map (T t+ ,t , N t+ ,t ) is not CPTP, for those times t such that one or both of the following inequalities are violated ∆(t) ± γ(t) 0,(30) a situation that is typically encountered when the cutoff frequency is smaller than the characteristic frequency of the oscillator, i.e., ω c < ω 0 (see, e.g., [28]). The criterion (30) above can be also found directly by substituting (28) in (27). Gaussian steerability Analogously to the classical noise channel, we first consider a two-mode state σ AB , where Alice's share undergoes the dynamical map (τ(t)I, η(t)I). Ref. [28] uses such a setting to witness non-Markovianity of the channel in the context of quantum Brownian motion-where the initial state is σ AB = σ 2,r . In several scenarios this technique allows to witness non-Markovianity via Gaussian steerability backflows. However, under such a channel, all Gaussian measurements on Alice's mode become compatible if and only if η(t) τ 2 (t) (see Eq. (17) or Ref. [43]). Therefore, it is not possible to witness non-Markovianity through Gaussian steerability at times {t \| η(t) τ(t) 2 }. In Appendix C we derive the analytical form of G A→B (σ 2,r (t)) for any r, η(t) and τ(t). Moreover, we show that a Gaussian steerability backflow is obtained, more precisely ∂ t G A→B > 0, if and only if the inequalitieṡ η(t) − 2η(t)τ (t) τ(t) < 0, η(t) − τ(t) 2 < 0,(31) are satisfied simultaneously (the latter inequality being the aforementioned non-GIB condition). Notice that the first inequality is the arithmetic average of the two possible violations of (27), and hence more restrictive than (27). This implies that some non-Markovian evolution cannot be witnessed. Nonetheless, when Alice's share is extended to include an auxiliary mode that does not undergo the channel, i.e., the dynamical map is (τ(t)I ⊕ I, η(t)I ⊕ 0)-the set of Gaussian measurements will remain incompatible at all times. In particular, let Alice and Bob share the three-mode squeezed state σ AB = σ 3,r , where Alice holds the first two modes. In Appendix C we derive the analytical form of G A→B (σ 3,r (t)) for any r, η(t) and τ(t). Moreover, we show that we have a backflow ∂ t G A→B > 0 if and only iḟ , threshold value for the E PPT (σ 2,r (t)) sensitivity; 0.7 (orange). Both G A→B (σ 3,r (t)) and E PPT (σ 3,r (t)) show backflow whenever non-Markovianity is present (shadow blue regions, according to condition (30)). All the plots are made in the Ohmic regime (s = 1), setting r = 2, ω 0 = 7, ω c = 1, T = 100 (high temperatures). η(t) − 2η(t)τ (t) τ(t) < 0,(32) which for the quantum Brownian motion is equivalent to ∆(t) < 0. This criterion is clearly less restrictive than (31)-in that it does not require η(t) < τ 2 (t). Thus, using the extra mode one can witness information backflow for a bigger class of dynamics. Notice that, the criterion (32) is still more restrictive than (27)-except in caseτ(t) = 0, where τ(t) = 1 for all t reduces the problem to the classical noise channel (for quantum Brownian motion, instead, an analogous situation is realized in the limit of high temperatures in which \|∆(t)\| \|γ(t)\| ∀t, see discussion in Appendix D 2). In order to witness all non-Markovian Gaussian evolutions one might consider a different three-mode initialization or increase the number of ancillary modes. Yet, it is not guaranteed that by doing so one can witness all non-Markovian dynamics with Gaussian steerability. In Fig. 3 (a), we plot Gaussian steerability G A→B for the quantum Brownian motion model at high temperatures. As a benchmark, we also depict the breakdown of CP-divisibility (shaded regions), i.e., the breakdown of inequality (30). We compare two cases with σ AB = σ 2,r and σ AB = σ 3,r . As the inset shows, using the two-mode state is not successful for some of the parameter regimes that we consider. In particular, if the interaction strength α is too large, the evolution of two-mode initializations cannot show any backflow of A → B Gaussian steerability because the evolution breaks the incompatibility of Alice's measurements. Furthermore, as one increases α, the time intervals during which one cannot witness backflow of Gaussian steerability in the two-mode configuration increases. On the contrary, in the three-mode scenario, if Alice owns the first two modes of σ 3,r , the incompatibility of Alice's measurements is not broken and we can witness non-Markovianity via backflows for all α. Entanglement Firstly, notice that the dynamical map (τ(t)I, η(t)I) is EB iff η(t) τ 2 (t) + 1 (see Eq. (20)). When we use the two-mode squeezed state for small r, one can show that ∂ t E PPT > 0 if the following two inequalities are satisfied simultaneously η(t) − τ 2 (t) − 1 < 0,(33)η(t) − 2(η(t) − 1)τ (t) τ(t) < 0.(34) The first inequality expresses the fact that steerable states are entangled but in general not vice-versa, i.e., entanglement is more robust to noise. Interestingly, provided steerability is non-zero, the second inequality, being one of the two possible violations of (27), when compared with (31) suggests that, steerability (entanglement) could be preferable for observing backflows ifτ(t)/τ(t) > (<) 0, the two figures of merit yielding the same performance in the caseτ(t) = 0 While we are not able to retrieve analytical conclusions about entanglement backflows in the case of the three-mode initialization σ 3,r , the potential of this configuration can be shown numerically for the quantum Brownian motion model at high temperatures. In Fig. 3 (b), we plot E PPT as function of time comparing the two cases σ 2,r (t) (inset) and σ 3,r (t). Again, using the two-mode state is not successful for large values of α. On the contrary, in the three-mode scenario, if Alice owns the first two modes of σ 3,r we can witness non-Markovianity via backflows of entanglement for all α. VI. DISCUSSION We considered Gaussian steerability and entanglement as quantifiers of the correlations contained in a two-party Gaussian system and at the same time their backflows as non-Markovianity witnesses. We were interested in those non-Markovian evolutions that cannot be witnessed by backflows of these correlations when two-mode Gaussian initializations are considered. This happens on time intervals where the considered one-mode evolutions are incompatibility and entanglement breaking, nullifying Gaussian steerability and entanglement, respectively. Therefore, we considered a strategy that makes use of three-mode Gaussian states, namely the GHZ/W three-mode squeezed states, where Alice owns the first two modes which allows to overcome the problem of Gaussian incompatibility breaking and entanglement breaking. For classical noise evolutions, we also showed that our use of the GHZ/W three-mode squeezed states allows to witness any non-Markovian evolution of this kind. However, in the more general case of evolutions characterized by lossy channels, our results show that, while the GHZ/W three-mode squeezed states allows witnessing non-Markovian evolutions that cannot be witnessed with two-mode initializations, this three-mode state could not be enough for detecting all non-Markovian dynamics. In order to increase the potential to witness non-Markovianity via correlation backflows, it would be interesting to consider three-mode initializations different from the GHZ/W states or to increase the size of the ancillas. Nonetheless, for the quantum Brownian motion, a particular instance of the lossy channels of experimental interest (but still more sophisticated than the classical noise channels), the GHZ/W three-mode squeezed state strategy accomplishes the task of detecting non-Markovianity through correlation backflows for values of the parameters where any two-mode initialization fails. An interesting open question is to identify a mode geometry, and corresponding state, able to display a correlation backflow for all non-Markovian evolutions. Our results have shown that two modes are not enough for Gaussian steerability and entanglement, while three modes do provide an improvement. Are three modes enough? If not, is a finite num-ber of modes enough? Note that in the finite dimensional case, solutions to this question were derived in [24,25], and [26]. In the first work, a correlation measure C was introduced based on state distinguishability and it was shown, for an evolution acting on a system A 1 of dimension d, how to construct an initial state defined on systems A 1 A 2 B with respective dimensions d, d + 1, and 2, that displays a backflow in the correlation measure C along the bipartition A 1 A 2 \|B for all invertible non-Markovian evolutions. This approach was generalized in [25], where the authors considered a slightly more complex arrangement that made use of larger ancillas A 2 and B to prove that for all non-Markovian evolutions there always exist a correlation and an initial state that provide a backflow. In [26], again for an evolution acting on a system S of dimension d, it was provided an initial state consisting of systems A 1 A 2 A 3 B with respective dimensions d, d + 1, 2, and 2 such that an entanglement negativity backflow along the bipartition A 1 A 2 A 3 \|B could be detected for all invertible non-Markovian evolutions. Whether a similar arrangement is possible in the Gaussian case remains an open question. Nonetheless, as discussed in Sec. IV, we expect that any invertible n-mode non-Markovian Gaussian evolution on A 1 can be witnessed with a correlation backflow along the bipartition A 1 A 2 \|B, where A 2 and B are respectively (at most) two and one mode ancillary Gaussian systems, no matter the number n of evolving modes. It would be interesting to start this study by checking whether it is always possible to obtain correlation backflows along the bipartition A 1 A 2 \|B when a generic one-mode non-Markovian Gaussian evolution is applied on A 1 and A 2 (B) is a two-mode (one-mode) ancilla. σ 3,r (t) =          σ(r) + η t I (1) (r) (r) (r) σ(r) (r) (r) (r) σ(r)          . (B1) Whenever ∂ t η t 0 the dynamics is non-Markovian (see Eq. (22) and related discussion). We want to see it via steering, in particular we use the first two modes (including the noisy one) in order to steer the third mode. According to (13) the state is steerable from the first two modes to the third one if and only if the following is violated σ 3,r (t) + i0 (2) ⊕ Ω (1) 0,(B2) which is equivalent to the non-physicality of the covariance matrix M B = B − C T A −1 C ,(B3) with A ≡ σ 3,r (t)(1 : 4, 1 : 4), C ≡ σ 3,r (t)(1 : 4, 5 : 6), and B ≡ σ 3,r (t)(5 : 6, 5 : 6)-note that Eq. (14) corresponds to Eq. (B3) for σ = σ 3,r . This gives the matrix M B =         η t +3 e 2 r +2η t e 4r 2 e 4 r +2 η t e 2 r +η t e 6 r +1 0 0 e 2 r (2 η t +3 e 2 r +η t e 4 r ) η t +2 e 2 r +e 6 r +2η t e 4r         .(B4) The symplectic eigenvalue of the matrix above, ν − = e 2r e 4r + 2 η t + 3e 2r [e 2r (2e 2r η t + 3) + η t ] e 4r + 2 e 2r η t + 2e 4r + 1 e 2r e 4r + 2e 2r η t + 2 + η t ,(B5) is always lower than 1, since, considering the expression inside the square root, the denominator exceeds the numerator by the positive quantity (−1 + e 4r ) 2 [η t + e 2r (2 + e 2r η t )] and both the numerator and the denominator are positive. Furthermore, the derivative with respect to η t , ∂ν − ∂η t = e 2r e 4r − 1 2 4e 2r 2e 8r + 5e 4r + 2 η t + 9 e 8r + e 4r + 2e 12r + 7e 8r + 7e 4r + 2 η 2 t 2 e 4r + 2 e 2r η t + 2e 4r + 1 2 e 2r [(e 4r +2)η t +3e 2r ][e 2r (2e 2r η t +3)+η t ] [(e 4r +2)e 2r η t +2e 4r +1][e 2r (e 4r +2e 2r η t +2)+η t ] e 2r e 4r + 2e 2r η t + 2 + η t 2 ,(B6) is also positive. This shows that for all values of r and η t , one can always detect non-Markovianity. In particular, in the cases of small and large squeezing parameter r, the Gaussian steerability function is less cumbersome. The symplectic eigenvalue around r = 0 + , i.e., for epsilon-entangled states, reads ν − = 1 − 16 9(η t + 1) r 2 + O(r 3 ).(B7) which is always smaller than one. Gaussian steerability from Alice, who owns the first two modes, to Bob, who owns the third mode, reads G A→B (σ 3,r (t)) = − log(ν − ) ≈ 16 9(η t + 1) r 2 + O(r 3 ),(B8) which is a monotonically decreasing function of η t . Hence, whenever the dynamics is non-Markovian, i.e., ∂ t η t 0, the Gaussian steerability increases and one can detect non-Markovianity. For large squeezing parameter r, instead, we get G A→B (σ 3,r (t)) = 1 2 ln e 2r 2η t + O(e −2r ),(B9) leading to the same conclusion as before. These results show that the state remains always steerable from the first two modes to the third mode, no matter how large η t is. Two-mode scenario For a scenario with two modes and input σ 2,r (Eq. (5)) one can find that the matrix M B (see Eq. (14)) reads        η t cosh(2 r)+1 η t +cosh(2 r) 0 0 η t cosh(2 r)+1 η t +cosh(2 r)        . (B10) We have ν − = η t cosh (2 r) + 1 η t + cosh (2 r) ,(B11) which is smaller than 1 iff η t < 1, and ∂ν − ∂η t = cosh 2 (2r) − 1 (η t + cosh(2r)) 2 (B12) is always positive. This implies that for any r > 0, the two-mode initialization setup cannot detect non-Markovianity if η t 1, but it does if η t < 1 since the function G A→B (σ 2,r (t)) = max{0, − ln(ν − )} is monotonically decreasing with η t . Again, behaviors for small and large squeezing parameter r are the following. For small r = 0 + , we have ν − = 1 − 2 1 − η t 1 + η t r 2 + O(r 3 ),(B13) and therefore, at leading order, the Gaussian steerability from Alice, who owns the first mode, to Bob, who owns the second mode, reads G A→B (σ 2,r (t)) =        2 1−η t 1+η t r 2 , η t < 1 0, η t 1 (B14) which, consistently, cannot detect non-Markovianity if η t 1, but it does if η t < 1, since the function is monotonically decreasing with η t . For large r we finally obtain G A→B (σ 2,r (t)) = max 0, ln 1 η t + O(e −2r ) . (B15) Classical noise: oscillating noise We conclude the discussion on the classical noise channel of Sec. V A by considering an oscillating noise that didactically shows when the two-mode configuration (5) fails in detecting non-Markovianity. We hence set the noise as η(t) = η 0 (1 − cos(2πt))/2, with η 0 being the constant gauging its intensity. If the constant η 0 1, an initialization in σ 2,r (0) implies backflows of G A→B (σ 2,r (t)) if and only if η(t) decreases, see Fig. 4(a). In this case two modes are enough to detect any non-Markovian character. On the contrary, if η 0 > 1 we observe backflows of G A→B (σ 2,r (t)) if and only if η(t) decreases and η(t) < 1, see Fig. 4(b). As analytically shown, the three-mode initialization σ 3,r (0) does not suffer of this limitation providing steering backflow as soon as the map shows non-Markovian behavior and for any value of η(t). Fig. 4(c) and (d) concern instead entanglement E PPT , again, for different values of η 0 . The core message does not change by considering this other correlation quantifier. The dynamical maps of this evolution are periodically EB in finite time intervals for η 0 > 2. For this reason, whether some time intervals of non-Markovianity cannot be witnessed by backflows of entanglement in the two-mode configuration (5) (see Fig. 4(c)), the three-mode configuration (6) never fails in detecting non-Markovianity (see Fig. 4(d)). (a) Gaussian steerability G A→B (σ 2,r (t)) (blue) and G A→B (σ 3,r (t)) (red) with Alice's first mode evolving through the classical noise channel. Setting η(t) = η 0 (1 − cos(2πt))/2 and η 0 = 0.8 (black), G A→B (σ 2,r (t)) provides backflows whenever η(t) is decreasing, namely for any non-Markovian behavior (blue shadow regions). In this case the dynamical maps are not GIB. (b) Same as in (a) but for η 0 = 2: G A→B (σ 2,r (t)) provides backflows whenever η(t) is decreasing and is smaller than 1 (dashed line), but does not witness non-Markovianity when η(t) is decreasing and above 1 (GIB threshold, dashed line). (c) Same as in (a) and (b), but for entanglement E PPT (σ 2,r (t)) and for values of η 0 : 1 (purple), 2 (orange) and 4 (red). The dynamical maps of this evolution are periodically EB in finite time intervals for η 0 > 2. For this reason, some time intervals of non-Markovianity cannot be witnessed by backflows of entanglement. (d) Same as in (c) but for the three-mode squeezed state initialization σ 3,r (0) and for values of η 0 : 1 (purple), 2 (orange), 4 (red) and 30 (pink). Since now Alice owns two modes but only the first is subjected to the noise, the dynamical maps of this evolution are never EB and all the non-Markovian behavior of the dynamics can be witnessed with entanglement backflows. All the plots in this Figure where we used the composition law for Gaussian channels [34] (T , N ) • (T , N ) = (T T , T N T T + N ) and (T t,s , N t,s ) is called the intermediate map of the evolution for the time interval [s, t]. FIG. 1. Schematic setup for the detection of non-Markovianity with (a) two-mode squeezed input state and with (b) GHZ/W threemode input state. Notice the selected bipartition concerning Alice's and Bob's shares (black horizontal line) where only Alice is allowed to own more than one mode. We consider Gaussian evolutions (T t , N t ) that are applied only to the first mode of Alice's share. Non-monotonic behaviors of Gaussian steering/entanglement quantifiers as function of time (backflows) are used as non-Markovianity witnesses. The GHZ/W three-mode configuration (b) can activate the potential to witness non-Markovian behaviors which in the twomode scenario (a) do not imply any revivals of quantum correlations. FIG. 2. (a) Gaussian steerability G A→B as function of time for the three-mode squeezed state initialization σ 3,r (0) and the two-mode squeezed state initialization σ 2,r (0) and Alice's first mode subjected to the classical noise evolution with η(t) = t 2 /(t 2 − 2t + 2) (black line, Eq. (23)), i.e. such that η(0) = 0, η(1) = 1 (horizontal dashed line, level of noise for which we have the GIB property), η(2) = 2, η(∞) = 1, it is increasing in [0, 2] and decreasing in [2, ∞] (non-Markovian interval, blue shadow region). We cannot observe any backflow of G A→B (σ 2,r (t)) (blue line) in the time interval when the evolution is non-Markovian, because the dynamical maps (T t , N t ) of the evolution are GIB for t ≥ 1. We do observe, instead, a backflow of G A→B (σ 3,r (t)) (red line) in the time interval when the evolution is non-Markovian. The plots are made for r = 2. (b) Same as in (a) but for E PPT and for η(t) = 2t 2 /(t 2 − 2t + 2) (Eq. (24)), such that η(0) = 0, η(1) = 2 (horizontal dashed line, level of noise for which we have the EB property), η(2) = 4, η(∞) = 2, it is increasing in [0, 2] and decreasing in [2, ∞]. For convenience we report now the rescaled quantity η(t)/4, black line. While we cannot observe any backflow of E PPT (σ 2,r (t)) (blue line) in the time interval when the evolution is non-Markovian, namely, again, for t 2 (because the dynamical maps (T t , N t ) of the evolution are EB for t ≥ 1), we do observe, instead, a backflow of E PPT (σ 3,r (t)) (red line). FIG. 3 . 3(a) Gaussian steerability G A→B as function of time for the three-mode squeezed state initialization σ 3,r (0) and Alice's first mode subjected to the quantum Brownian motion, for different values of the coupling parameter α: 0.25 (purple), 0.35 (blue), 0.42 (red) and 0.7 (orange). Inset: G A→B for the two-mode squeezed state initialization σ 2,r (0) with the same parameters. Here for α greater than ≈ 0.42 (red), we cannot witness non-Markovianity as the dynamical maps of the evolution become GIB before any non-Markovian behavior. (b) Same as in (a) but for entanglement E PPT and for the following values of α: 0.25 (purple); 0.35 (blue); 0.595(red) FIG. 4. (a) Gaussian steerability G A→B (σ 2,r (t)) (blue) and G A→B (σ 3,r (t)) (red) with Alice's first mode evolving through the classical noise channel. Setting η(t) = η 0 (1 − cos(2πt))/2 and η 0 = 0.8 (black), G A→B (σ 2,r (t)) provides backflows whenever η(t) is decreasing, namely for any non-Markovian behavior (blue shadow regions). In this case the dynamical maps are not GIB. (b) Same as in (a) but for η 0 = 2: G A→B (σ 2,r (t)) provides backflows whenever η(t) is decreasing and is smaller than 1 (dashed line), but does not witness non-Markovianity when η(t) is decreasing and above 1 (GIB threshold, dashed line). (c) Same as in (a) and (b), but for entanglement E PPT (σ 2,r (t)) and for values of η 0 : 1 (purple), 2 (orange) and 4 (red). The dynamical maps of this evolution are periodically EB in finite time intervals for η 0 > 2. For this reason, some time intervals of non-Markovianity cannot be witnessed by backflows of entanglement. (d) Same as in (c) but for the three-mode squeezed state initialization σ 3,r (0) and for values of η 0 : 1 (purple), 2 (orange), 4 (red) and 30 (pink). Since now Alice owns two modes but only the first is subjected to the noise, the dynamical maps of this evolution are never EB and all the non-Markovian behavior of the dynamics can be witnessed with entanglement backflows. All the plots in this Figure are made for r = 2. are made for r =FIG. 5 . 5(a) Gaussian steerability G A→B as function of time for the three-mode squeezed state initialization σ 3,r (0) and Alice's first mode subjected to the quantum Brownian motion, for coupling parameter α = 0.7, Ohmic regime (s = 1), r = 2, ω 0 = 7, ω c = 1, T = 0.5 (low temperatures). Inset: G A→B for the two-mode squeezed state initialization σ 2,r (0) with the same parameters. (b) Same as in (a) but for entanglement E PPT . Both G A→B and E PPT do not show backflow in the non-Markovian region (shadow blue regions, identified by condition(30)), this happening for both the considered initializations. VII. ACKNOWLEDGEMENTSThe authors would like to thank Saleh Rahimi-Keshari for illuminating discussions. This work is supported by the Spanish Government (FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), the ERC AdG CERQUTE, the AXA Chair in Quantum Information Science, Fundacio Cellex, Fundacio Mir-Puig and Generalitat de Catalunya (CERCA, AGAUR SGR 1381) and the Swiss National Science Foundation NCCR SwissMAP.Appendix A: CP-divisibility criterion for smoothGaussian dynamicsIn the main text we introduced non-Markovianity as violation of CP-divisibility of the intermediate map. For a smooth dynamics, CP-divisibility is equivalent to having an infinitesimally CP-divisible dynamics. In other words, at any time t and for infinitesimally small > 0 we should havewhere the intermediate map(T t,, N t, ) is CP. On the one hand, since is small we expect the map to take the formwhich means any covariance matrix σ 0 evolves as followsOn the other hand, we should haveBy comparing Eqs. (A3) and (A4) we should have the following two equivalenceswhere we use the conventionFurthermore, the positivity of the intermediate map implieswhich by using Eqs. (A5) and (A6)-and keeping the leading order in -is equivalent to the following criterionThe above condition is a necessary and sufficient condition for the intermediate map being non-Markovian, i.e., if it holds at all times, the dynamics is Markovian, otherwise, it is not. Furthermore, it can be deduced directly from Eq. (8) of Ref.[47], substituting in that equation at first order in , X(t + , 0) →Ẋ(t, 0) + X(t, 0) and Y(t + , 0) →Ẏ(t, 0) + Y(t, 0), where X(t, 0) and Y(t, 0) of Ref.[47]are the matrices T t and N t , respectively.Appendix B: Classical noise channel: potential of the GHZ/W three-mode squeezed state setupHere we show that using the initialization σ 3,r (6) one can witness any non-Markovianity of the classical noise evolution (21) by means of steering backflow. Interestingly, this happens for any value of the squeezing parameter r. The impossibility of achieving the same result through the two-mode initialization σ 2,r (5) is also shown analytically. Finally, we conclude the section with an explanatory example of oscillating noise.Three-mode scenarioConsider the state given by σ 3,r with only the first mode undergoing the classical noise channel. The total map is given by the pair (I(3), η t I (1) ⊕ 0 (2) ), where we keep using the notation where I (n) (0 (n) ) is the 2n × 2n identity (null) matrix. The covariance matrix after the channel readsAppendix C: Lossy channelThe lossy channel is given by T t = τ t I and N t = η t I, see Eq.(25). First of all, from Eqs.(9)and(10), the complete positivity of the channel imposeswhile τ t can also be considered positive in full generality.The CP-divisibility of the intermediate map breaks down if and only if Eq. (A8) is violated. For the lossy channel, the matrix on the l.h.s. of (A8) has the following eigenvaluesand therefore when the sign of at least one eigenvalue is negative we have non-Markovianity, formallygetting the value 1 iff we are in the Markovian case. This proves inequality(27)of Sec. V B.Gaussian steerabilityThe two-mode Gaussian steerability, where Alice and Bob own one mode, readsfor small r and G A→B (σ 2,r (t)) = max 0, lnfor large r. As a consequence, one can check that, both for the limits of small and large r,This condition remains valid for any r > 0. Indeed, we havei.e. nothing but Eq. (B11) with the substitution η t → η t /τ 2 t . Hence, (C7) is smaller than 1 iff η t < τ 2 t and its timeis always positive (to see it one can directly use the results from Sec. B 2), sign[This proves condition (C6) and (31) for any r > 0.On the other hand, the three-mode Gaussian steerability (Alice owns the first two modes and Bob owns the third mode) for small r and large r reads, respectively, aswhich is always positive. Therefore, Alice can (somewhat obviously) always steer the third mode. One can easily check that, both for the limits of small and large r,and therefore, one can detect non-Markovianity if the sign is positive, proving condition(32). However, it is a stricter constraint than (C3), and thus we may not detect some non-Markovian dynamics. One can finally analytically check that condition (C10) is valid for any r > 0. In this case, analogously to what we observed for expression (C7), ν − is nothing but expression (B5) with the substitution η t → η t /τ 2 t . Therefore, ν − < 1 and the sign[ν − ] = sign[η t − 2η tτt /τ t ], proving (C10) for any r > 0.Appendix D: Quantum Brownian motionThe quantum Brownian motion is a particular example of the lossy channel (25) admitting a microscopic derivation.Microscopic derivation and master equationWe consider the following Hamiltonian H for the whole system-environment compound,whereĤ S ,Ĥ E are the local terms on the system and on the environment, respectively,Ĥ I is the system-environment interaction term andq =â +â † andQ k =b k +b † k are the position operators of the system (bosonic ladder operatorsâ,â † ) and environment (bosonic ladder operatorsb k ,b † k ).Ĥ I is a dipole-like interaction having coupling constant α controlling its strength. Performing a second order expansion on the exact dynamics in interaction picture and enforcing Born (weak-coupling) and first Markov approximations one arrives to the Redfield equation for the reduced system dynamics in interaction picture[1]where . . . T denotes the average over the bath thermal state at temperature T and we indicated operators in interaction pictureIn the secular approximation, i.e., canceling the fast oscillating counter-rotating terms exp(±i2ω 0 t), neglecting the Lamb shift and assuming the environment to be in the thermal state with temperature T , the above equation reduces to the quantum Brownian motion master equation (see e.g.[28,48])with ∆(t) and γ(t) being the diffusion and damping coefficients, respectively, defined in(28)and J(ω) = k g 2 k δ(ω − ω k ) is the spectral density (in the main text, assumed to be of the form(29)). This implies the following master equation for the (1 + n)-mode covariance matrixwithwhere we assumed only the first mode to be affected from the channel. In order to get the dynamics of second order moments we solved numerically the integrals (28) and the master equation (D5) for the covariance matrix. An equivalent approach would be to consider the integrated lossy channel expression (see, e.g.,[28])with τ(t) and η(t) given in (28).2. Low temperature regime: failure of the GHZ/W three-mode squeezed state setup.Comparing the violation of the Markovian condition(27)with the backflow conditions for Gaussian steerability (31) and (32), we infer that any non-Markovianity can be detected in the limit η(t) 1 via steering backflow, at least for the three-mode initialization σ 3,r (0). For quantum Brownian motion this limit is achieved at high temperatures, seeFig. 3. Nonetheless, by lowering the temperature one expects the limitation in sensitivity to manifest. InFig. 5(a), we plot G A→B as a function of time for inputs σ 3,r (0) and σ 2,r (0) (inset) at low temperatures. We observe that G A→B does not show backflow in the non-Markovian region, for both the considered initializations. InFig. 5(b)we also observe that the same insensitivity is obtained by considering entanglement E PPT . 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[ "Neural Oscillators are Universal", "Neural Oscillators are Universal" ]	[ "Samuel Lanthaler \nCalifornia Institute of Technology\n\n", "T Konstantin Rusch \nCalifornia Institute of Technology\n\n", "Eth Zurich \nCalifornia Institute of Technology\n\n", "Siddhartha Mishra \nCalifornia Institute of Technology\n\n", "Eth Zurich \nCalifornia Institute of Technology\n\n" ]	[ "California Institute of Technology\n", "California Institute of Technology\n", "California Institute of Technology\n", "California Institute of Technology\n", "California Institute of Technology\n" ]	[]	Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We introduce an abstract class of neural oscillators that encompasses these architectures and prove that neural oscillators are universal, i.e, they can approximate any continuous and casual operator mapping between time-varying functions, to desired accuracy. This universality result provides theoretical justification for the use of oscillator based ML systems. The proof builds on a fundamental result of independent interest, which shows that a combination of forced harmonic oscillators with a nonlinear read-out suffices to approximate the underlying operators.Preprint. Under review.	10.48550/arxiv.2305.08753	[ "https://export.arxiv.org/pdf/2305.08753v1.pdf" ]	258,686,762	2305.08753	e4a5bfebc5c26c432809d5c564024f789c58eeb8	Neural Oscillators are Universal Samuel Lanthaler California Institute of Technology T Konstantin Rusch California Institute of Technology Eth Zurich California Institute of Technology Siddhartha Mishra California Institute of Technology Eth Zurich California Institute of Technology Neural Oscillators are Universal Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We introduce an abstract class of neural oscillators that encompasses these architectures and prove that neural oscillators are universal, i.e, they can approximate any continuous and casual operator mapping between time-varying functions, to desired accuracy. This universality result provides theoretical justification for the use of oscillator based ML systems. The proof builds on a fundamental result of independent interest, which shows that a combination of forced harmonic oscillators with a nonlinear read-out suffices to approximate the underlying operators.Preprint. Under review. Introduction Oscillators are ubiquitous in the sciences and engineering [11,29]. Prototypical examples include pendulums in mechanics, feedback and relaxation oscillators in electronics, business cycles in economics and heart beat and circadian rhythms in biology. Particularly relevant to our context is the fact that the neurons in our brain can be thought of as oscillators on account of the periodic spiking and firing of the action potential [28,10]. Consequently, functional brain circuits such as cortical columns are being increasingly analyzed in terms of networks of coupled oscillators [28]. Given this wide prevalence of (networks of) oscillators in nature and man-made devices, it is not surprising that oscillators have inspired various machine learning architectures in recent years. Prominent examples include the CoRNN [26] and UnICORNN [27] recurrent neural networks for sequence modeling. CoRNN is based on a network of coupled, forced and damped oscillators, whereas UnICORNN is a multi-layer sequence model that stacks networks of independent undamped oscillators as hidden layers within an RNN. Both these architectures were rigorously shown to mitigate the exploding and vanishing gradient problem [19] that plagues RNNs. Hence, both CoRNN and UnICORNN performed very well on sequence learning tasks with long-term dependencies. Another example of the use of oscillators in machine learning is provided by GraphCON [25], a framework for designing graph neural networks (GNNs) [3], that is based on coupled oscillators. GraphCON was also shown to ameliorate the oversmoothing problem [24] and allow for the deployment of multi-layer deep GNNs. Other examples include Second Order Neural ODEs (SONODEs) [18], which can be interpreted as oscillatory neural ODEs, locally coupled oscillatory recurrent networks (LocoRNN) [16], and Oscillatory Fourier Neural Network (O-FNN) [12]. Another avenue where ML models based on oscillators arise is that of physical neural networks (PNNs) [34] i.e., physical devices that perform machine learning on analog (beyond digital) systems. Such analog systems have been proposed as alternatives or accelerators to the current paradigm of machine learning on conventional electronics, allowing us to significantly reduce the prohibitive energy costs of training state-of-the-art ML models. In [34], the authors propose a variety of physical neural networks which include a mechanical network of multi-mode oscillations on a plate and electronic circuits of oscillators as well as a network of nonlinear oscillators. Coupled with a novel physics aware training (PAT) algorithm, the authors of [34] demonstrated that their nonlinear oscillatory PNN achieved very good performance on challenging benchmarks such as Fashion-MNIST [35]. Moreover, other oscillatory systems such as coupled lasers and spintronic nano-oscillators have also been proposed as possible PNNs, see [32] as an example of the use of thermally coupled vanadium dioxide oscillators for image recognition and [23,31] for the use of spin-torque nano-oscillators for speech recognition and for neuromorphic computing, respectively. What is the rationale behind the successful use of (networks of) oscillators in many different contexts in machine learning? The authors of [26] attribute it to the inherent stability of oscillatory dynamics, as the state (and its gradients) of an oscillatory system remain within reasonable bounds throughout the time-evolution of the system. However, this is at best a partial explanation, as it does not demonstrate why oscillatory dynamics can learn (approximate) mappings between inputs and outputs rather than bias the learned states towards oscillatory functions. As an example, consider the problem of classification of MNIST [17] (or Fashion-MNIST) images. It is completely unclear if the inputs (vectors of pixel values), outputs (class probabilities) and the underlying mapping possess any (periodic) oscillatory structure. Consequently, how can oscillatory RNNs (such an CoRNN and UnICORNN) or a network of oscillatory PNNs learn the underlying mapping? Our main aim in this paper is to provide an answer to this very question on the ability of neural networks, based on oscillators, to express (to approximate) arbitrary mappings. To this end, • We introduce an abstract framework of neural oscillators that encompasses both sequence models such as CoRNN and UnICORNN, as well as variants of physical neural networks as the ones proposed in [34]. These neural oscillators are defined in terms of second-order versions of neural ODEs [4], and combine nonlinear dynamics with a linear read-out. • We prove a Universality theorem for neural oscillators by showing that they can approximate, to any given tolerance, continuous operators between appropriate function spaces. • Our proof of universality is based on a novel theoretical result of independent interest, termed the fundamental Lemma, which implies that a suitable combination of linear oscillator dynamics with nonlinear read-out suffices for universality. Such universality results, [1,5,13,20] and references therein, have underpinned the widespread use of traditional neural networks (such as multi-layer perceptrons and convolutional neural networks). Hence, our universality result establishes a firm mathematical foundation for the deployment of neural networks, based on oscillators, in myriad applications. Moreover, our constructive proof provides insight into how networks of oscillators can approximate a large class of mappings. Neural Oscillators General Form of Neural Oscillators. Given u : [0, T ] → R p as an input signal, for any final time T ∈ R + , we consider the following system of neural ODEs for the evolution of dynamic hidden variables y ∈ R m , coupled to a linear read-out to yield the output z ∈ R q ,     ÿ (t) = σ (W y(t) + V u(t) + b) , y(0) =ẏ(0) = 0, z(t) = Ay(t) + c. (2.1a) (2.1b) (2.1c) Equation (2.1) defines an input-/output-mapping u(t) → z(t), with time-dependent output z : [0, T ] → R q . Specification of this system requires a choice of the hidden variable dimension m and the activation function σ. The resulting mapping u → z depends on tunable weight matrices W ∈ R m×m , V ∈ R m×p , A ∈ R q×m and bias vectors b ∈ R m , c ∈ R q . For simplicity of the exposition, we consider only activation functions σ ∈ C ∞ (R), with σ(0) = 0 and σ (0) = 1, such as tanh or sin, although more general activation functions can be readily considered. This general second-order neural ODE system (2.1) will be referred to as a neural oscillator. Multi-layer neural oscillators. As a special case of neural oscillators, we consider the following much sparser class of second-order neural ODEs,            y 0 (t) := u(t), y (t) = σ w y (t) + V y −1 (t) + b , ( = 1, . . . , L), y (0) =ẏ (0) = 0, z(t) = Ay L (t) + c. (2.2a) (2.2b) (2.2c) (2.2d) In contrast to the general neural oscillator (2.1), the above multi-layer neural oscillator (2.2) defines a hierarchical structure; The solution y ∈ R m at level solves a second-order ODE with driving force y −1 , and the lowest level, y 0 = u, is the input signal. Here, the layer dimensions m 1 , . . . , m L can vary across layers, the weights w ∈ R m are given by vectors, with componentwise multiplication, V ∈ R m ×m −1 is a weight matrix, and b ∈ R m the bias. Given the result of the final layer, y L , the output signal is finally obtained by an affine output layer z(t) = Ay L (t) + c. In the multilayer neural oscillator, the matrices V , A and vectors w , b and c represent the trainable hidden parameters. The system (2.2) is a special case of (2.1), since it can be written in the form (2.1), with y := [y L , y L−1 , . . . , y 1 ] T , b := [b L , . . . , b 1 ] T , and a (upper-diagonal) block-matrix structure for W : W :=         w L I V L 0 . . . 0 0 w L−1 I V L−1 . . . . . . . . . . . . . . . 0 0 . . . 0 w 2 I V 2 0 . . . 0 0 w 1 I         , V :=        0 . . . . . . 0 V 1        (2.3) Given the block-diagonal structure of the underlying weight matrices, it is clear that the multi-layer neural oscillator (2.2) is a much sparser representation of the general neural operator (2.1). Moreover, one can observe from the structure of the neural ODE (2.2) that within each layer, the individual neurons are independent of each other. Assuming that w i = 0, for all 1 ≤ i ≤ m and all 1 ≤ ≤ L, we further highlight that the multi-layer neural oscillator (2.2) is a Hamiltonian system, y = ∂H ∂ẏ ,ÿ = − ∂H ∂y , (2.4) with the layer-wise time-dependent Hamiltonian, H(y ,ẏ , t) = 1 2 ẏ 2 − m i=1 1 w i σ(w i y i + (V y −1 ) i + b i ),(2.5) with σ being the antiderivative of σ, and x 2 = x, x denoting the Euclidean norm of the vector x ∈ R m and ·, · the corresponding inner product. Hence, any symplectic discretization of the multi-layer neural oscillator (2.2) will result in a fully reversible model, which can first be leveraged in the context of normalizing flows [22], and second leads to a memory-efficient training, as the intermediate states (i.e., y (t 0 ),ẏ (t 0 ), y (t 1 ),ẏ (t 1 ), . . . , y (t N ),ẏ (t N ), for some time discretization t 0 , t 1 , . . . , t N of length N ) do not need to be stored and can be reconstructed during the backward pass. This potentially leads to a drastic memory saving of O(N ) during training. Examples of Neural Oscillators (Forced) harmonic oscillator. Let p = m = q = 1 and we set W = −ω 2 , for some ω ∈ R, V = 1, b = 0 and the activation function to be identity σ(x) = x. In this case, the neural ODE (2.1) reduces to the ODE modeling the dynamics of a forced simple harmonic oscillator [11] of the form, y = −ω 2 y + u, y(0) =ẏ(0) = 0. (2.6) Here, y is the displacement of the oscillator, ω the frequency of oscillation and u is a forcing term that forces the motion of the oscillator. Note that (2.6) is also a particular example of the multi-layer oscillator (2.2) with L = 1. This simple example provides justification for our terminology of neural oscillators, as in general, the hidden state y can be thought of as the vector of displacements of m-coupled oscillators, which are coupled together through the weight matrix W and are forced through a forcing term u, whose effect is modulated via V and a bias term b. The nonlinear activation function mediates possible nonlinear feedback to the system on account of large displacements. CoRNN. The Coupled oscillatory RNN (CoRNN) architecture [26] is given by the neural ODE: y = σ (W y + Wẏ + V u + b) − γy − ẏ. We can recover the neural oscillator (2.1) as a special case of CoRNN by setting W = 0, γ = = 0; thus, a universality theorem for neural oscillators immediately implies a corresponding universality result for the CoRNN architecture. Nonlinear oscillatory PNN of [34]. In [34,SM,Sect. 4.A], the authors propose an analog machine learning device that simulates a network of nonlinear oscillators, for instance realized through coupled pendula. The resulting mathematical model is the so-called simplified Frenkel-Kontorova model [2] given by the ODE system, Mθ = −K sin(θ) − C sin(θ) + F, where θ = (θ 1 , . . . , θ N ) is the vector of angles across all coupled pendula, M = diag(µ 1 , . . . , µ N ) is a diagonal mass matrix, F an external forcing, K = diag(k 1 , . . . , k N ) the "spring constant" for pendula, given by k i = µ i g/ with the pendulum length and g the gravitational acceleration, and where C = C T is a symmetric matrix, with C = − = C , so that [C sin(θ)] = = C (sin(θ ) − sin(θ )), (2.7) which quantifies the coupling between different pendula. We note that this simplified Frenkel-Kontorova system can also model other coupled nonlinear oscillators, such as coupled lasers or spintronic oscillators [34]. We can bring the above system into a more familiar form by introducing the variable y according to the relationship P y = θ for a matrix P . Substitution of this ansatz then yields M Pÿ = −(K + C) sin(P y) + F ; choosing P = M −1 (K + C), we find y = − sin(M −1 (K + C)y) + F,(2.8) which can be written in the formÿ = σ(W y) + F for σ = − sin( · ) and W = M −1 (K + C). If we now take C in a block-matrix form C :=          γ L I C L 0 . . . 0 C L,T γ L−1 I . . . . . . 0 . . . . . . 0 . . . . . . C 3,T γ 2 I C 2 0 . . . 0 C 2,T γ 1 I          , and with corresponding mass matrix M in block-matrix form M = diag(µ L I, µ L−1 I, . . . , µ 1 I), then with ρ := γ /µ , we have M −1 C :=          ρ L I C L /µ L 0 . . . 0 C L,T /µ L−1 ρ L−1 I . . . . . . 0 . . . . . . 0 . . . . . . ρ 2 I C 2 /µ 2 0 . . . 0 C 2,T /µ 1 ρ 1 I          , Introducing an ordering parameter > 0, and choosing γ , C , µ ∼ , it follows that ρ , C µ = O(1), and C µ −1 = O( ). Hence, with a suitable ordering of the masses across the different layers, one can introduce an effective one-way coupling, making M −1 C =         ρ L I V L 0 . . . 0 0 ρ L−1 I V L−1 . . . . . . . . . . . . 0 0 . . . 0 ρ 2 I V 2 0 . . . 0 0 ρ 1 I         + O( ), upper triangular, up to small terms of order . We note that the diagonal entries ρ in M −1 C are determined by the off-diagonal terms through the identity (2.7). The additional degrees of freedom in the (diagonal) K-matrix in (2.8) can be used to tune the diagonal weights of the resulting weight matrix W = M −1 (K + C). Thus, physical systems such as the Frankel-Kontorova system of nonlinear oscillators can be approximated (to leading order) by multi-layer systems of the form y = σ w y + V y −1 + F , (2.9) with F an external forcing, representing a tunable linear transformation of the external input to the system. The only formal difference between (2.9) and (2.2) is (i) the absence of a bias term in (2.9) and (ii) the fact that the external forcing appears outside of the nonlinear activation function σ in (2.9). A bias term could readily be introduced by measuring the angles represented by y in a suitably shifted reference frame; physically, this corresponds to tuning the initial position y (0) of the pendula, with y (0) also serving as the reference value. Furthermore, in our proof of universality for (2.2), it makes very little difference whether the external forcing F is applied inside the activation function, as in (2.2b) resp. (2.1a), or outside as in (2.9); indeed, the first layer in our proof of universality will in fact approximate the linearized dynamics of (2.2b), i.e. a forced harmonic oscillator (2.6). Consequently, a universality result for the multi-layer neural oscillator (2.2) also implies universality of variants of nonlinear oscillator-based physical neural networks, such as those considered in [34]. Universality of Neural Oscillators In this section, we state and sketch the proof for our main result regarding the universality of neural oscillators (2.1) or, more specifically, multi-layer oscillators (2.2). To this end, we start with some mathematical preliminaries to set the stage for the main theorem. Setting Input signal. We want to approximate operators Φ : u → Φ(u), where u = u(t) is a time- dependent input signal over a time-interval t ∈ [0, T ], and Φ(u)(t) is a time-dependent output signal. We will assume that the input signal t → u(t) is continuous, and that u(0) = 0. To this end, we introduce the space C 0 ([0, T ]; R p ) := {u : [0, T ] → R p \| t → u(t) is continuous and u(0) = 0}. We will assume that the underlying operator defines a mapping Φ : C 0 ([0, T ]; R p ) → C 0 ([0, T ]; R q ). The approximation we discuss in this work are based on oscillatory systems starting from rest. These oscillators are forced by the input signal u. For such systems the assumption that u(0) = 0 is necessary, because the oscillator starting from rest takes a (arbitrarily small) time-interval to synchronize with the input signal (to "warm up"); If u(0) = 0, then the oscillator cannot accurately approximate the output during this warm-up phase. This intuitive fact is also implicit in our proofs. We will provide a further comment on this issue in Remark 3.2, below. Operators of interest. We consider the approximation of an operator Φ : C 0 ([0, T ]; R p ) → C 0 ([0, T ]; R q ) , mapping a continuous input signal u(t) to a continuous output signal Φ(u)(t). We will restrict attention to the uniform approximation of Φ over a compact set of input functions K ⊂ C 0 ([0, T ]; R p ). We will assume that Φ satisfies the following properties: • Φ is causal: For any t ∈ [0, T ], if u, v ∈ C 0 ([0, T ]; R p ) are two input signals, such that u\| [0,t] ≡ v\| [0,t] , then Φ(u)(t) = Φ(v)(t), i.e. the value of Φ(u)(t) at time t does not depend on future values {u(τ ) \| τ > t}. • Φ is continuous as an operator Φ : (C 0 ([0, T ]; R p ), · L ∞ ) → (C 0 ([0, T ]; R q ), · L ∞ ), with respect to the L ∞ -norm on the input-/output-signals. Note that the class of Continuous and Causal operators are very general and natural in the contexts of mapping between sequence spaces or time-varying function spaces, see [7,6] and references therein. Universal approximation Theorem The universality of neural oscillators is summarized in the following theorem: Theorem 3.1. [Universality of the multi-layer neural oscillator] Let Φ : C 0 ([0, T ]; R p ) → C 0 ([0, T ]; R q ) be a causal and continuous operator. Let K ⊂ C 0 ([0, T ]; R p ) be compact. Then for any > 0, there exist hyperparameters L, m 1 , . . . , m L , weights w ∈ R m , V ∈ R m ×m −1 , A ∈ R q×m L and bias vectors b ∈ R m , c ∈ R q , for = 1, . . . , L, such that the output z : [0, T ] → R q of the multi-layer neural oscillator (2.2) satisfies sup t∈[0,T ] \|Φ(u)(t) − z(t)\| ≤ , ∀ u ∈ K. It is important to observe that the sparse, independent multi-layer neural oscillator (2.2) suffices for universality in the considered class. Thus, there is no need to consider the wider class of neural oscillators (2.1), at least in this respect. We remark in passing that Theorem 3.1 immediately implies another universality result for neural oscillators, showing that they can also be used to approximate arbitrary continuous functions F : R p → R q . This extension is explained in detail in SM A. [0, T ] → R p to a function E(u) ∈ C 0 ([−t 0 , T ]; R p ), by E(u)(t) := (t0+t) t0 u(0), t ∈ [−t 0 , 0), u(t), t ∈ [0, T ]. Our proof of Theorem 3.1 can readily be used to show that the oscillator system with forcing E(u), and initialized at time −t 0 < 0, can uniformly approximate Φ(u) over the entire time interval [0, T ], without requiring that u(0) = 0, or Φ(u)(0) = 0. In this case, the initial time interval [−t 0 , 0] provides the required "warm-up phase" for the neural oscillator. Remark 3.3. In practice, neural ODEs such as (2.2) need to be discretized via suitable numerical schemes. As examples, CoRNN and UnICORNN were implemented in [26] and [27], respectively, with implicit-explicit time discretizations. Nevertheless, universality also applies for such discretizations as long as the time-step is small enough, as the underlying discretization is going to be a sufficiently accurate approximation of (2.2) and Theorem 3.1 can be used for showing universality of the discretized version of the multi-layer neural oscillator (2.2). Outline of the Proof In the following, we outline the proof of the universality Theorem 3.1, while postponing the technical details to the SM. For a given tolerance , we will explicitly construct the weights and biases of the multi-layer neural oscillator (2.2) such that the underlying operator can be approximated within the given tolerance. This construction takes place in the following steps: Figure 1: Illustration of the universal 3layer neural oscillator architecture constructed in the proof of Theorem 3.1. (Forced) Harmonic Oscillators compute a timewindowed sine transform. Recall that the forced harmonic oscillator (2.6) is the simplest example of a neural oscillator (2.1). The following lemma, proved by direct calculation in SM B.1, shows that this forced harmonic oscillator actually computes a time-windowed variant of the sine transform at the corresponding frequency: Lemma 3.4. Assume that ω = 0. Then the solution of (2.6) is given by y(t) = 1 ωˆt 0 u(t − τ ) sin(ωτ ) dτ. (3.1) Given the last result, for a function u, we define its timewindowed sine transform as follows, L t u(ω) :=ˆt 0 u(t − τ ) sin(ωτ ) dτ. (3.2) Lemma 3.4 shows that a forced harmonic oscillator computes (3.2) up to a constant. Approximation of causal operators from finite realizations of time-windowed sine transforms. The following novel result, termed the fundamental Lemma, shows that the time-windowed sine transform (3.2) composed with a suitable nonlinear function can approximate causal operators Φ to desired accuracy; as a consequence, one can conclude that forced harmonic oscillators combined with a nonlinear read-out defines a universal architecture in the sense of Theorem 3.1. The proof of this fundamental Lemma, detailed in SM B.2, is based on first showing that any continuous function can be reconstructed to desired accuracy, in terms of realizations of its timewindowed sine transform (3.2) at finitely many frequencies ω 1 , . . . , ω N (see SM Lemma B.1). Then, we leverage the continuity of the underlying operator Φ to approximate it with a finite-dimensional function Ψ, which takes the time-windowed sine transforms as its arguments. Lemma 3.5 (Fundamental Lemma). Let Φ : K ⊂ C 0 ([0, T ]; R p ) → C 0 ([0, T ]; R q ) Given these two results, we can discern a clear strategy to prove the universality Theorem 3.1. First, we will show that a general nonlinear form of the neural oscillator (2.2) can also compute the time-windowed sine transform at arbitrary frequencies. Then, these outputs need to be processed in order to apply the fundamental Lemma 3.5 and approximate the underlying operator Φ. To this end, we will also approximate the function Ψ (mapping finite-dimensional inputs to finite-dimensional outputs) by oscillatory layers. The concrete steps in this strategy are outlined below. Nonlinear Oscillators approximate the time-windowed sine transform. The building block of multi-layer neural oscillators (2.2) is the nonlinear oscillator of the form, y = σ(w y + V u + b). Coupled Nonlinear Oscillators approximate time-delays. The next step in the proof is to show that coupled oscillators can approximate time-delays in the continuous input signal. This fact will be of crucial importance in subsequent arguments. We have the following Lemma (proved in SM B.4), Lemma 3.7. Let K ⊂ C 0 ([0, T ]; R p ) be a compact subset. For every > 0, and ∆t ≥ 0, there exist m ∈ N, w ∈ R m , V ∈ R m×p , b ∈ R m and A ∈ R p×m , such that the oscillator (3.3), initialized at y(0) =ẏ(0) = 0, has output sup t∈[0,T ] \|u(t − ∆t) − Ay(t)\| ≤ , ∀ u ∈ K, where u(t) is extended to negative values t < 0 by zero. Two-layer neural oscillators approximate neural networks pointwise. As in the strategy outlined above, the final ingredient in our proof of the universality theorem 3.1 is to show that neural oscillators can approximate continuous functions, such as the Ψ in the fundamental lemma 3.5, to desired accuracy. To this end, we will first show that neural oscillators can approximate general neural networks (perceptrons) and then use the universality of neural networks in the class of continuous functions to prove the desired result. We have the following lemma, Ay 2 (t) + c − Σσ(Λu(t) + γ) ≤ , ∀u ∈ K. The proof, detailed in SM B.5, is constructive and the neural oscillator that we construct has two layers. The first layer just processes a nonlinear input function through a nonlinear oscillator and the second layer, approximates the second-derivative (in time) from time-delayed versions of the input signal that were constructed in Lemma 3.7. Combining the ingredients to prove the universality theorem 3.1. The afore-constructed ingredients are combined in SM B.6 to prove the universality theorem. In this proof, we explicitly construct a three-layer neural oscillator (2.2) which approximates the underlying operator Φ. The first layer follows the construction of Lemma 3.6, to approximate the time-windowed sine transform (3.2), for as many frequencies as are required in the fundamental Lemma 3.5. The second-and third-layers imitate the construction of Lemma 3.8 to approximate a neural network (perception), which in turn by the universal approximation of neural networks, approximates the function Ψ in Lemma 3.5 to desired accuracy. Putting the network together leads to a three-layer oscillator that approximates the continuous and casual operator Φ. This construction is depicted in Figure 1. Discussion Machine learning architectures, based on networks of coupled oscillators, for instance sequence models such as CoRNN [26] and UnICORNN [27], graph neural networks such as GraphCON [25] and increasingly, the so-called physical neural networks (PNNs) such as linear and nonlinear mechanical oscillators [34] and spintronic oscillators [23,31], are being increasingly used. A priori, it is unclear why ML systems based on oscillators can provide competitive performance on a variety of learning benchmarks, e.g. [26,27,25,34], rather than biasing their outputs towards oscillatory functions. In order to address these concerns about their expressivity, we have investigated the theoretical properties of machine learning systems based on oscillators. Our main aim was to answer a fundamental question: "are coupled oscillator based machine learning architectures universal?". In other words, can these architectures, in principle, approximate a large class of input-output maps to desired accuracy. To answer this fundamental question, we introduced an abstract framework of neural oscillators (2.1) and its particular instantiation, the multi-layer neural oscillators (2.2). This abstract class of second-order neural ODEs encompasses both sequence models such as CoRNN and UnICORNN, as well as a very general and representative PNN, based on the so-called Frenkel-Kontorova model. The main contribution of this paper was to prove the universality theorem 3.1 on the ability of multi-layer neural oscillators (2.2) to approximate a large class of operators, namely causal and continuous maps between spaces of continuous functions, to desired accuracy. Despite the fact that the considered neural oscillators possess a very specific and constrained structure, not even encompassing general Hamiltonian systems, the approximated class of operators is nevertheless very general, including solution operators of general ordinary and even time-delay differential equations. The crucial theoretical ingredient in our proof was the fundamental Lemma 3.5, which implies that linear oscillator dynamics combined with a pointwise nonlinear read-out suffices for universal operator approximation; our construction can correspondingly be thought of as a large number of linear processors, coupled with nonlinear readouts. This construction could have implications for other models such as structured state space models [9,8] which follow a similar paradigm, and the extension of our universality results to such models could be of great interest. Our universality result has many interesting implications. To start with, we rigorously prove that an ML architecture based on coupled oscillators can approximate a very large class of operators. This provides theoretical support to many widely used sequence models and PNNs based on oscillators. Moreover, given the generality of our result, we hope that such a universality result can spur the design of innovative architectures based on oscillators, particularly in the realm of analog devices as ML inference systems or ML accelerators [34]. It is also instructive to lay out some of the limitations of the current article and point to avenues for future work. In this context, our setup currently only considers time-varying functions as inputs and outputs. Roughly speaking, these inputs and outputs have the structure of (infinite) sequences. However, a large class of learning tasks can be reconfigured to take sequential inputs and outputs. These include text (as evident from the tremendous success of large language models [21]), DNA sequences, images [15], timeseries and (offline) reinforcement learning [14]. Nevertheless, a next step would be to extend such universality results to inputs (and outputs) which have some spatial or relational structure, for instance by considering functions which have a spatial dependence or which are defined on graphs. On the other hand, the class of operators that we consider, i.e., casual and continuous, is not only natural in this setting but very general [7,6]. Another limitation lies in the feed forward structure of the multi-layer neural oscillator (2.2). As mentioned before, most physical (and neurobiological) systems exhibit feedback loops between their constituents. However, this is not common in ML systems. In fact, we had to use a mass ordering in the Frenkel-Kontorova system of coupled pendula (2.8) in order to recast it in the form of the multi-layer neural oscillator (2.2). Such asymptotic ordering may not be possible for arbitrary physical neural networks. Exploring how such ordering mechanisms might arise in physical and biological systems in order to effectively give rise to a feed forward system could be very interesting. One possible mechanism for coupled oscillators that can lead to a hierarchical structure is that of synchronization [33,30] and references therein. How such synchronization interacts with universality is a very interesting question and will serve as an avenue for future work. Finally, universality is arguably necessary but far from sufficient to analyze the performance of any ML architecture. Other aspects such as trainability and generalization are equally important, and we do not address these issues here. We do mention that trainability of oscillatory systems would profit from the fact that oscillatory dynamics is (gradient) stable and this formed the basis of the proofs of mitigation of the exploding and vanishing gradient problem for CoRNN in [26] and UnICORNN in [27] as well as GraphCON in [25]. Extending these results to the general second-order neural ODE (2.2), for instance through an analysis of the associated adjoint system, is left for future work. Supplementary Material for: Neural Oscillators are Universal A Another universality result for neural oscillators The universal approximation Theorem 3.1 immediately implies another universal approximation results for neural oscillators, as explained next. We consider a continuous map F : R p → R q ; our goal is to show that F can be approximated to given accuracy by suitably defined neural oscillators. Fix a time interval [0, T ] for (an arbitrary choice) T = 2. Let K 0 ⊂ R p be a compact set. Given ξ ∈ R p , we associate with it a function u ξ (t) ∈ C 0 ([0, T ]; R p ), by setting u ξ (t) := tξ. (A.1) Clearly, the set K := {u ξ \| ξ ∈ K 0 } is compact in C 0 ([0, T ]; R p ) . Furthermore, we can define an operator Φ : C 0 ([0, T ]; R p ) → C 0 ([0, T ]; R q ), by Φ(u)(t) := 0, t ∈ [0, 1), (t − 1)F (u(1)), t ∈ [1, T ]. (A.2) where F : R p → R q is the given continuous function that we wish to approximate. One readily checks that Φ defines a causal and continuous operator. Note, in particular, that Φ(u ξ )(T ) = (T − 1)F (u ξ (1)) = F (ξ), is just the evaluation of F at ξ, for any ξ ∈ K 0 . Since neural oscillators can uniformly approximate the operator Φ for inputs u ξ ∈ K, then as a consequence of Theorem 3.1 and (2.3), it follows that, for any > 0 there exists m ∈ N, matrices W ∈ R m×m , V ∈ R m×p and A ∈ R q×m , and bias vectors b ∈ R m , c ∈ R q , such that for any ξ ∈ K 0 , the neural oscillator system,     ÿ ξ (t) = σ (W y ξ (t) + tV ξ + b) , y ξ (0) =ẏ ξ (0) = 0, z ξ (t) = Ay ξ (t) + c, (A.3) (A.4) (A.5) satisfies \|z ξ (T ) − F (ξ)\| = \|z ξ (T ) − Φ(u ξ )(T )\| ≤ sup t∈[0,T ] \|z ξ (t) − Φ(u ξ )(t)\| ≤ , uniformly for all ξ ∈ K 0 . Hence, neural oscillators can be used to approximate an arbitrary continuous function F : R p → R q , uniformly over compact sets. Thus, neural oscillators also provide universal function approximation. Proof. We can rewrite y(t) = 1 ω´t 0 u(τ ) sin(ω(t − τ )) dτ . By direct differentiation, one readily verifies that y(t) so defined, satisfieṡ y(t) =ˆt 0 u(τ ) cos(ω(t − τ )) dτ + [u(τ ) sin(ω(t − τ ))] τ =t =ˆt 0 u(τ ) cos(ω(t − τ )) dτ, in account of the fact that sin(0) = 0. Differentiating once more, we find thaẗ y(t) = −ωˆt 0 u(τ ) sin(ω(t − τ )) dτ + [u(τ ) cos(ω(t − τ ))] τ =t = −ω 2 y(t) + u(t). Thus y(t) solves the ODE (2.6), with initial condition y(0) =ẏ(0) = 0. B.2 Proof of Fundamental Lemma 3.5 Reconstruction of a continuous signal from its sine transform. Let [0, T ] ⊂ R be an interval. We recall that we define the windowed sine transform L t u(ω) of a function u : [0, T ] → R p , by L t u(ω) =ˆt 0 u(t − τ ) sin(ωτ ) dτ, ω ∈ R. In the following, we fix a compact set K ⊂ C 0 ([0, T ]; R p ). Note that for any u ∈ K, we have u(0) = 0, and hence K can be identified with a subset of C((−∞, T ]; R p ), consisting of functions with supp(u) ⊂ [0, T ]. We consider the reconstruction of continuous functions u ∈ K. We will show that u can be approximately reconstructed from knowledge of L t (ω). More precisely, we provide a detailed proof of the following result: Lemma B.1. Let K ⊂ C((−∞, T ]; R p ) be compact, such that supp(u) ⊂ [0, T ] for all u ∈ K. For any , ∆t > 0, there exists N ∈ N, frequencies ω 1 , . . . , ω N ∈ R \ {0}, phase-shifts ϑ 1 , . . . , ϑ N ∈ R and weights α 1 , . . . , α N ∈ R, such that sup τ ∈[0,∆t] u(t − τ ) − N j=1 α j L t u(ω j ) sin(ω j τ − ϑ j ) ≤ , for all u ∈ K and for all t ∈ [0, T ]. Proof. Step 0: (Equicontinuity) We recall the following fact from topology. If K ⊂ C((−∞, T ]; R p ) is compact, then it is equicontinuous; i.e. there exists a continuous modulus of continuity φ : [0, ∞) → [0, ∞) with φ(r) → 0 as r → 0, such that \|u(t − τ ) − u(t)\| ≤ φ(τ ), ∀ τ ≥ 0, t ∈ [0, T ], ∀ u ∈ K. (B.1) Step 1: (Connection to Fourier transform) Fix t 0 ∈ [0, T ] and u ∈ K for the moment. Define f (τ ) = u(t 0 − τ ). Note that f ∈ C([0, ∞); R p ), and f has compact support supp(f ) ⊂ [0, T ]. We also note that, by (B.1), we have \|f (t + τ ) − f (t)\| ≤ φ(τ ), ∀ τ ≥ 0, t ∈ [0, T ]. We now consider the following odd extension of f to all of R: F (τ ) := f (τ ), for τ ≥ 0, −f (−τ ), for τ ≤ 0. Since F is odd, the Fourier transform of F is given by F (ω) :=ˆ∞ −∞ F (τ )e −iωτ dτ = iˆ∞ −∞ F (τ ) sin(ωτ ) dτ = 2iˆT 0 f (τ ) sin(ωτ ) dτ = 2iL t0 u(ω). Let > 0 be arbitrary. Our goal is to uniformly approximate F (τ ) on the interval [0, ∆t]. The main complication here is that F lacks regularity (is discontinuous), and hence the inverse Fourier transform of F does not converge to F uniformly over this interval; instead, a more careful reconstruction based on mollification of F is needed. We provide the details below. Step 2: (Mollification) We now fix a smooth, non-negative and compactly supported function ρ : R → R, such that supp(ρ) ⊂ [0, 1], ρ ≥ 0,´R ρ(t) dt = 1, and we define a mollifier ρ (t) := 1 ρ(t/ ). In the following, we will assume throughout that ≤ T . We point out that supp(ρ ) ⊂ [0, ], and hence, the mollification F (t) = (F * ρ )(t) satisfies, for t ≥ 0: \|F (t) − F (t)\| = ˆ 0 (F (t) − F (t + τ ))ρ (τ ) dτ = ˆ 0 (f (t) − f (t + τ ))ρ (τ ) dτ ≤ sup τ ∈[0, ] \|f (t) − f (t + τ )\| ˆ 0 ρ (τ ) dτ ≤ φ( ). In particular, this shows that sup t∈[0,T ] \|F (t) − F (t)\| ≤ φ( ), can be made arbitrarily small, with an error that depends only on the modulus of continuity φ. Step 3: (Fourier inverse) Let F (ω) denote the Fourier transform of F . Since F is smooth and compactly supported, it is well-known that we have the identity F (τ ) = 1 2πˆ∞ −∞ F (ω)e −iωτ dω, ∀ t ∈ R, where ω → F (ω) decays to zero very quickly (almost exponentially) as \|ω\| → ∞. In fact, since F = F * ρ is a convolution, we have F (ω) = F (ω) ρ (ω), where \| F (ω)\| ≤ 2 f L ∞ T is uniformly bounded, and ρ (ω) decays quickly. In particular, this implies that there exists a L = L( , T ) > 0 independent of f , such that F (τ ) − 1 2πˆL −L F (ω) ρ (ω)e −iωτ dω ≤ 2T f L ∞ˆ\| ω\|>L \| ρ (ω)\| dω ≤ f L ∞ , ∀ τ ∈ R. (B.2) Step 4: (Quadrature) Next, we observe that, since F and ρ are compactly supported, their Fourier transform ω → F (ω) ρ (ω)e −iωτ is smooth; in fact, for \|τ \| ≤ T , the Lipschitz constant of this mapping can be explicitly estimated by noting that ∂ ∂ω F (ω) ρ (ω)e −iωτ = ∂ ∂ωˆs upp(F ) (F * ρ )(t)e iω(t−τ ) dt =ˆs upp(F ) i(t − τ )(F * ρ )(t)e iω(t−τ ) dt. We next take absolute values, and note that any t in the support of F obeys the bound \|t\| ≤ T + ≤ 2T , while \|τ \| ≤ T by assumption; it follows that Lip ω → F (ω) ρ (ω)e −iωτ ≤ (2T + T ) F L ∞ ρ L 1 = 3T F L ∞ , ∀ τ ∈ [0, T ]. It thus follows from basic results on quadrature that for an equidistant choice of frequencies ω 1 < · · · < ω N , with spacing ∆ω = 2L/(N − 1), we have 1 2πˆL −L F (ω) ρ (ω)e −iωτ dω − ∆ω 2π N j=1 F (ω j ) ρ (ω j )e −iωj τ ≤ CL 2 3T F L ∞ N , ∀ τ ∈ [0, T ], for an absolute constant C > 0, independent of F , T and N . By choosing N to be even, we can ensure that ω j = 0 for all j. In particular, recalling that L = L(T, ) depends only on and T , and choosing N = N (T, ) sufficiently large, we can combine the above estimate with (B.2) to ensure that F (τ ) − ∆ω 2π N j=1 F (ω j ) ρ (ω j )e −iωj τ ≤ 2 f L ∞ , ∀ τ ∈ [0, T ], where we have taken into account that F L ∞ = f L ∞ . Step 5: (Conclusion) To conclude the proof, we recall that F (ω) = 2iL t0 u(ω) can be expressed in terms of the sine transform L t u of the function u which was fixed at the beginning of Step 1. Recall also that f (τ ) = u(t 0 − τ ), so that f L ∞ = u L ∞ . Hence, we can write the real part of ∆ω 2π F (ω j ) ρ (ω j )e −iωj τ = ∆ω 2π 2iL t0 u(ω j ) ρ (ω j )e −iωj τ , in the form α j L t0 (ω j ) sin(ω j τ − ϑ j ) for coefficients α j ∈ R and θ j ∈ R which depend only on ∆ω and ρ (ω j ), but are independent of u. In particular, it follows that sup τ ∈[0,∆t] u(t 0 − τ ) − N j=1 α j L t0 u(ω j ) sin(ω j τ − ϑ j ) = sup t∈[0,∆t] F (τ ) − Re   ∆ω 2π N j=1 F (ω j ) ρ (ω j )e −iωj τ   ≤ sup τ ∈[0,∆t] F (τ ) − ∆ω 2π N j=1 F (ω j ) ρ (ω j )e −iωj τ ≤ sup τ ∈[0,∆t] \|F (τ ) − F (τ )\| + sup τ ∈[0,∆t] F (τ ) − ∆ω 2π N j=1 F (ω j ) ρ (ω j )e −iωj τ . By Steps 1 and 3, the first term on the right-hand side is bounded by ≤ φ( ), while the second one is bounded by ≤ 2 sup u∈K u L ∞ ≤ C , where C = C(K) < ∞ depends only on the compact set K ⊂ C([0, T ]; R p ). Hence, we have sup τ ∈[0,∆t] u(t 0 − τ ) − N j=1 α j L t0 u(ω j ) sin(ω j τ − ϑ j ) ≤ φ( ) + C . In this estimate, the function u ∈ K and t 0 ∈ [0, T ] were arbitrary, and the modulus of continuity φ as well as the constant C on the right-hand side depend only on the set K. it thus follows that for this choice of α j , ω j and ϑ j , we have sup u∈K sup t∈[0,T ] sup τ ∈[0,∆t] u(t − τ ) − N j=1 α j L t u(ω j ) sin(ω j τ − ϑ j ) ≤ φ( ) + C . Since > 0 was arbitrary, the right-hand side can be made arbitrarily small. The claim then readily follows. The next step in the proof of the fundamental Lemma 3.5 needs the following preliminary result in functional analysis, Lemma B.2. Let X , Y be Banach spaces, and let K ⊂ X be a compact subset. Assume that Φ : X → Y is continous. Then for any > 0, there exists a δ > 0, such that if u − u K X ≤ δ with u ∈ X , u K ∈ K, then Φ(u) − Φ(u K ) Y ≤ . Proof. Suppose not. Then there exists 0 > 0 and a sequence u j , u K j , (j ∈ N), such that u j − u K j X ≤ j −1 , while Φ(u j ) − Φ(u K j ) Y ≥ 0 . By the compactness of K, we can extract a subsequence j k → ∞, such that u K j k → u K converges to some u K ∈ K. By assumption on u j , this implies that u j k − u K X ≤ u j k − u K j k X + u K j k − u K X (k→∞) −→ 0, which, by the assumed continuity of Φ, leads to the contradiction that 0 < 0 ≤ Φ(u j k ) − Φ(u K ) Y → 0, as k → ∞. Proof of Lemma 3.5. Now, we can prove the fundamental Lemma in the following, Proof. Let > 0 be given. We can identify K ⊂ C 0 ([0, T ]; R p ) with a compact subset of C((−∞, T ]; R p ), by extending all u ∈ K by zero for negative times, i.e. we set u(t) = 0 for t < 0. Applying Lemma B.2, with X = C 0 ([0, T ]; R p ) and Y = C 0 ([0, T ]; R q ), we can find a δ > 0, such that for any u ∈ C 0 ([0, T ]; R p ) and u K ∈ K, we have u − u K L ∞ ≤ δ ⇒ Φ(u) − Φ(u K ) L ∞ ≤ . (B.3) By the inverse sine transform Lemma B.1, there exist N ∈ N, frequencies ω 1 , . . . , ω N = 0, phaseshifts ϑ 1 , . . . , ϑ N and coefficients α 1 , . . . , α N , such that for any u ∈ K and t ∈ [0, T ]: sup τ ∈[0,T ] u(t − τ ) − N j=1 α j L t u(ω j ) sin(ω j τ − ϑ j ) ≤ δ. Given L t u(ω 1 ), . . . , L t u(ω N ), we can thus define a reconstruction mapping R : R N × [0, T ] → C([0, T ]; R p ) by R(β 1 , . . . , β N ; t)(τ ) := N j=1 α j β j sin(ω j (t − τ ) − ϑ j ). Then, for τ ∈ [0, t], we have \|u(τ ) − R(L t u(ω 1 ), . . . , L t u(ω N ); t)(τ )\| ≤ δ. We can now uniquely define Ψ : R N × [0, T 2 /4] → C 0 ([0, T ]; R p ), by the identity Ψ(L t u(ω 1 ), . . . , L t u(ω N ); t 2 /4) = Φ (R(L t u(ω 1 ), . . . , L t u(ω N ); t)) . Using the short-hand notation R t u = R(L t u(ω 1 ), . . . , L t u(ω N ); t), we have sup τ ∈[0,t] \|u(τ ) − R t u(τ )\| ≤ δ, for all t ∈ [0, T ]. By (B.3), this implies that Φ(u)(t) − Ψ(L t u(ω 1 ), . . . , L t u(ω N ); t 2 /4) = \|Φ(u)(t) − Φ(R t u)(t)\| ≤ . B.3 Proof of Lemma 3.6 Proof. Let ω = 0 be given. For a (small) parameter s > 0, we consider y s = 1 s σ(−sω 2 y s + su), y s (0) =ẏ s (0) = 0. Let Y be the solution ofŸ = −ω 2 Y + u, Y (0) =Ẏ (0) = 0. Then we have, on account of σ(0) = 0 and σ (0) = 1, s −1 σ(−sω 2 Y + su) − [−ω 2 Y + u] = σ(−sω 2 Y + su) − σ(0) s − σ (0)[−ω 2 Y + u] = 1 sˆs 0 ∂ ∂ζ σ(−ζω 2 Y + ζu) dζ − σ (0)[−ω 2 Y + u] = 1 s ˆs 0 σ (−ζω 2 Y + ζu) − σ (0) dζ −ω 2 Y + u . It follows from Lemma 3.4 that for any input u ∈ K, with sup u∈K u L ∞ =: B < ∞, we have a uniform bound Y L ∞ ≤ BT /ω, hence we can estimate \| − ω 2 Y + u\| ≤ B(ωT + 1), uniformly for all such u. In particular, it follows that s −1 σ(−sω 2 Y + su) − [−ω 2 Y + u] ≤ B(T ω + 1) sup \|x\|≤sB(T ω+1) \|σ (x) − σ (0)\|. Clearly, for any δ > 0, we can choose s ∈ (0, 1] sufficiently small, such that the right hand-side is bounded by δ, i.e. with this choice of s, s −1 σ(−sω 2 Y (t) + su(t)) − [−ω 2 Y (t) + u(t)] ≤ δ, ∀ t ∈ [0, T ], holds for any choice of u ∈ K. We will fix this choice of s in the following, and write g(y, u) := s −1 σ(−sω 2 y + su). We note that g is Lipschitz continuous in y, for all \|y\| ≤ BT /ω and \|u\| ≤ B, with Lip y (g) ≤ ω 2 sup \|ξ\|≤B(ωT +1) \|σ (ξ)\|. To summarize, we have shown that Y solves Y = g(Y, u) + f, Y (0) =Ẏ (0) = 0, where f L ∞ ≤ δ. By definition, y s solves y s = g(y s , u), y s (0) =ẏ s (0) = 0. It follows from this that \|y s (t) − Y (t)\| ≤ˆt 0ˆτ 0 {\|g(y s (θ), u(θ)) − g(Y (θ), u(θ))\| + \|f (θ)\|} dθ dτ ≤ˆt 0ˆτ 0 Lip y (g)\|y s (θ) − Y (θ)\| + δ dθ dτ ≤ T ω 2 sup \|ξ\|≤B(ωT +1) \|σ (ξ)\|ˆt 0 \|y s (τ ) − Y (τ )\| dτ + T 2 δ. Recalling that Y (t) = L t u(ω), then by Gronwall's inequality, the last estimate implies that sup t∈[0,T ] \|y s (t) − L t u(ω)\| = sup t∈[0,T ] \|y s − Y \| ≤ Cδ, for a constant C = C(T, ω, sup \|ξ\|≤B(ωT +1) \|σ (ξ)\|) > 0, depending only on T , ω, B and σ . Since δ > 0 was arbitrary, we can ensure that Cδ ≤ . Thus, we have shown that a suitably rescaled nonlinear oscillator approximates the harmonic oscillator to any desired degree of accuracy, and uniformly for all u ∈ K. To finish the proof, we observe that y solves Proof. Let , ∆t be given. By the sine transform reconstruction Lemma B.1, there exists N ∈ N, frequencies ω 1 , . . . , ω N , weights α 1 , . . . , α N and phase-shifts ϑ 1 , . . . , ϑ N , such that y = σ(−ω 2 y + su), y(0) =ẏ(0) = 0,sup τ ∈[0,∆t] u(t − τ ) − N j=1 α j L t u(ω j ) sin(ω j τ − ϑ j ) ≤ 2 , ∀ t ∈ [0, T ], ∀ u ∈ K, (B.4) where any u ∈ K is extended by zero to negative times. It follows from Lemma 3.6, that there exists a coupled oscillator network, y = σ(w y + V u + b), y(0) =ẏ(0) = 0, with dimension m = pN , and w ∈ R m , V ∈ R m×p , and a linear output layer y → Ay, A ∈ R m×m , such that [ Ay(t)] j ≈ L t u(ω j ) for j = 1, . . . , N ; more precisely, such that sup t∈[0,T ] N j=1 \|α j \| L t u(ω j ) − [ Ay] j (t) ≤ 2 , ∀ u ∈ K.\|u(t − ∆t) − Ay(t)\| ≤ sup t∈[0,T ] u(t − ∆t) − N j=1 α j L t u(ω j ) sin(ω j ∆t − ϑ j ) + sup t∈[0,T ] N j=1 \|α j \| L t u(ω j ) − [ Ay] j (t) \| sin(ω j ∆t − ϑ j )\| ≤ . B.5 Proof of Lemma 3.8 Proof. Fix Σ, Λ, γ as in the statement of the lemma. Our goal is to approximate u → Σσ(Λu + γ). Step 1: (nonlinear layer) We consider a first layer for a hidden state y = [y 1 , y 2 ] T ∈ R p+p , given by ÿ 1 (t) = σ(Λu(t) + γ) y 2 (t) = σ(γ) , y(0) =ẏ(0) = 0. This layer evidently does not approximate σ(Λu(t) + γ); however, it does encode this value in the second derivative of the hidden variable y 1 . The main objective of the following analysis is to approximately computeÿ 1 (t) through a suitably defined additional layer. Step 2: (Second-derivative layer) To obtain an approximation of σ(Λu(t) + γ), we first note that the solution operator S : u(t) → η(t), whereη(t) = σ(Λu(t) + γ) − σ(γ), η(0) =η(0) = 0, defines a continuous mapping S : C 0 ([0, T ]; R p ) → C 2 0 ([0, T ]; R p ), with η(0) =η(0) =η(0) = 0. Note that η is very closely related to y 1 . The fact thatη = 0 is important to us, because it allows us to smoothly extend η to negative times by setting η(t) := 0 for t < 0 (which would not be true for y 1 (t)). The resulting extension defines a compactly supported function η : (−∞, 0] → R p , with η ∈ C 2 ((−∞, T ]; R p ). Furthermore, by continuity of the operator S, the image S(K) of the compact set K under S is compact in C 2 ((−∞, T ]; R p ). From this, it follows that for small ∆t > 0, the second-order backward finite difference formula converges, where the bound on the right-hand side is uniform in u ∈ K, due to equicontinuity of {η \| η = S(u), u ∈ K}. In particular, the second derivative of η can be approximated through linear combinations of time-delays of η. We can now choose ∆t > 0 sufficiently small so that [η(t) − 2η(t − ∆t) + η(t − 2∆t)] − Az(t) ≤ ∆t 2 2 Σ , ∀η = S(u), u ∈ K. Indeed, Lemma 3.7 shows that time-delays of any given input signal can be approximated with any desired accuracy, and η(t) − 2η(t − ∆) − η(t − 2∆) is simply a linear combination of time-delays of the input signal η in (B.6). To connect η(t) back to the y(t) = [y 1 (t), y 2 (t)] T constructed in Step 1, we note thaẗ η = σ(Au(t) + b) − σ(b) =ÿ 1 −ÿ 2 , and hence, taking into account the initial values, we must have η ≡ y 1 − y 2 by ODE uniqueness. In particular, upon defining a matrix V such that V y := V y 1 − V y 2 ≡ V η, we can equivalently write (B.6) in the form,z (t) = σ(w z(t) + V y(t) + b), z(0) =ż(0) = 0. (B.7) Step 3: (Conclusion) Composing the layers from Step 1 and 2, we obtain a coupled oscillator y = σ(w y + V y −1 + b ), ( = 1, 2), initialized at rest, with y 1 = y, y 2 = z, such that for A := Σ A and c := Σσ(γ), we obtain Az(t) − η(t) − 2η(t − ∆t) + η(t − 2∆t) ∆t 2 + Σ sup t∈[0,T ] η(t) − 2η(t − ∆t) + η(t − 2∆t) ∆t 2 −η(t) ≤ 2 + 2 = . This concludes the proof. B.6 Proof of Theorem 3.1 Proof. Step 1: By the Fundamental Lemma 3.5, there exist N , a continuous mapping Ψ, and frequencies ω 1 , . . . , ω N , such that Step 2: Fix 1 ≤ 1 sufficiently small, such that also Σ Λ Lip(σ) 1 ≤ , where Lip(σ) := sup \|ξ\|≤ Λ M +\|γ\|+1 \|σ (ξ)\| denotes an upper bound on the Lipschitz constant of the activation function over the relevant range of input values. It follows from Lemma 3.6, that there exists an oscillator network,ÿ 1 = σ(w 1 y 1 + V 1 u + b 1 ), y 1 (0) =ẏ 1 (0) = 0, (B.8) of depth 1, such that sup t∈ [0,T ] \|[L t u(ω 1 ), . . . , L t u(ω N ); t 2 /4] T − A 1 y 1 (t)\| ≤ 1 , for all u ∈ K. Step 3: Finally, by Lemma 3.8, there exists an oscillator network, y 2 = σ(w 2 y 2 + V 2 y 1 + b 1 ), of depth 2, such that sup t∈[0,T ] \|A 2 y 2 (t) − Σσ(ΛA 1 y 1 (t) + γ)\| ≤ , Remark 3. 2 . 2We note that the theorem can be readily extended to remove the requirement on u(0) = 0 and Φ(u)(0) = 0. To this end, let Φ : C([0, T ]; R p ) → C([0, T ]; R q ) be an operator between spaces of continuous functions, u → Φ(u) on [0, T ]. Fix a t 0 > 0, and extend any input function u : be a causal and continuous operator, with K ⊂ C 0 ([0, T ]; R p ) compact. Then for any > 0, there exists N ∈ N, frequencies ω 1 , . . . , ω N and a continuous mapping Ψ : R p×N × [0, T 2 /4] → R q , such that \|Φ(u)(t) − Ψ(L t u(ω 1 ), . . . , L t u(ω N ); t 2 /4)\| ≤ , for all u ∈ K. following Lemma (proved in SM B.3), we show that even for a nonlinear activation function σ such as tanh or sin, the nonlinear oscillator (3.3) can approximate the time-windowed sine transform. Lemma 3. 6 . 6Fix ω = 0. Assume that σ(0) = 0, σ (0) = 1. For any > 0, there exist w, V, b, A ∈ R, such that the nonlinear oscillator (3.3), initialized at y(0) =ẏ(0) = 0, has output \|Ay(t) − L t u(ω)\| ≤ , ∀u ∈ K, t ∈ [0, T ], with L t u(ω) being the time-windowed sine transform(3.2). Lemma 3. 8 . 8Let K ⊂ C 0 ([0, T ]; R p ) be compact. For matrices Σ, Λ and bias γ, and any > 0, there exists a two-layer (L = 2) oscillator (2.2), initialized at y (0) =ẏ (0) = 0, = 1, 2, such that sup t∈[0,T ] if, and only if, y s = y/s solvesy s = s −1 σ(−sω 2 y s + su), y s (0) =ẏ s (0) = 0.Hence, with W = −ω 2 , V = s, b = 0 and A = s −1 , we have sup t∈[0,T ] \|Ay(t) − L t u(ω)\| = sup t∈[0,T ] \|y s (t) − L t u(ω)\| ≤ .This concludes the proof.B.4 Proof of Lemma 3.7 another linear layer B : R m R p×N → R p , which maps β = [β 1 , . . . , β N ] to Bβ := N j=1α j β j sin(ω j ∆t − ϑ j ) ∈ R p , we define A := B A, and observe that from (B. t) − 2η(t − ∆t) + η(t − 2∆t) ∆t 2 −η(t) = o ∆t→0 (1), ∀η = S(u), u ∈ K, t) − 2η(t − ∆t) + η(t − 2∆t) ∆t 2 −η(t) ≤ 2 Σ , ∀y = S(u), u ∈ K,where Σ denotes the operator norm of the matrix Σ. By Lemma 3.7, applied to the input set K = S(K) ⊂ C 0 ([0, T ]; R p ), there exists a coupled oscillatorz(t) = σ(w z(t) + V η(t) + b), z(0) =ż(0) = 0, (B.6)and a linear output layer z → Az, such that sup t∈[0,T ] \|Φ(u)(t) − Ψ(L t u(ω 1 ), . . . , L t u(ω N ); t 2 /4)\| ≤ , for all u ∈ K, and t ∈ [0, T ]. Let M be a constant such that\|L t u(ω 1 )\|, . . . , \|L t u(ω N )\|, t 2 4 ≤ M,for all u ∈ K and t ∈ [0, T ]. By the universal approximation theorem for ordinary neural networks, there exist weight matrices Σ, Λ and bias γ, such that\|Ψ(β 1 , . . . , β N ; t 2 /4) − Σσ(Λβ + γ)\| ≤ , β := [β 1 , . . . , β N ; t 2 /4] T ,holds for all t ∈ [0, T ], \|β 1 \|, . . . , \|β N \| ≤ M . Universal approximation bounds for superpositions of a sigmoidal function. A R Barron, IEEE Transcations on Information Theory. 39A . R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transcations on Information Theory, 39, 1993. 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Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017. holds for all y 1 belonging to the compact set K 1 := S(K) ⊂ C 0 ([0, T ]; where S denotes the solution operator of (B.8). R N +1, R N +1 ), where S denotes the solution operator of (B.8). Step 4: Thus, we have for any u ∈ K, and with short-hand L. t u(ω) := (L t u(ω 1 ), . . . , L t u(ω N )), Φ(u)(t) − A 2 y 2 (t) ≤ Φ(u)(t) − Ψ(L t u(ω)Step 4: Thus, we have for any u ∈ K, and with short-hand L t u(ω) := (L t u(ω 1 ), . . . , L t u(ω N )), Φ(u)(t) − A 2 y 2 (t) ≤ Φ(u)(t) − Ψ(L t u(ω); . + , L t u(ω)+ Ψ(L t u(ω); . + Σσ, L t u(ω)+ Σσ(Λ[L t u(ω); . − Σσ, ΛA 1 y 1 (t) + γt 2 /4] + γ) − Σσ(ΛA 1 y 1 (t) + γ) ΛA 1 y 1 (t) + γ) − A 2 y 2 (t). + Σσ, + Σσ(ΛA 1 y 1 (t) + γ) − A 2 y 2 (t) . . Σσ, L t u(ω)Σσ(Λ[L t u(ω); . − Σσ, ΛA 1 y 1 (t) + γt 2 /4] + γ) − Σσ(ΛA 1 y 1 (t) + γ) . ≤ Σ Lip, L t u(ω≤ Σ Lip(σ) Λ [L t u(ω); . Σσ, ΛA 1 y 1 (t) + γ) − A 2 y 2 (t) ≤ , ∀ t ∈ [0, T ], u ∈ KΣσ(ΛA 1 y 1 (t) + γ) − A 2 y 2 (t) ≤ , ∀ t ∈ [0, T ], u ∈ K. . Thus, we conclude that \|Φ(u)(t) − A 2 y 2 (t)\| ≤ 4 , for all t ∈ [0, T ] and u ∈ K. Since > 0 was arbitrary, we conclude that for any causal and continuous operator Φ : C 0 ([0, TThus, we conclude that \|Φ(u)(t) − A 2 y 2 (t)\| ≤ 4 , for all t ∈ [0, T ] and u ∈ K. Since > 0 was arbitrary, we conclude that for any causal and continuous operator Φ : C 0 ([0, T ]; . R , 3which uniformly approximates Φ to accuracy for all u ∈ K. This completes the proofR p ) and > 0, there exists a coupled oscillator of depth 3, which uniformly approximates Φ to accuracy for all u ∈ K. This completes the proof.	[]
[ "GANterfactual-RL: Understanding Reinforcement Learning Agents' Strategies through Visual Counterfactual Explanations Explainable Deep Reinforcement Learning; Explainable Artificial Intelligence; Interpretable Machine Learning ACM Reference Format: GANterfactual-RL: Understanding Reinforce- ment Learning Agents' Strategies through Visual Counterfactual Explana- tions . In", "GANterfactual-RL: Understanding Reinforcement Learning Agents' Strategies through Visual Counterfactual Explanations Explainable Deep Reinforcement Learning; Explainable Artificial Intelligence; Interpretable Machine Learning ACM Reference Format: GANterfactual-RL: Understanding Reinforce- ment Learning Agents' Strategies through Visual Counterfactual Explana- tions . In" ]	[ "Tobias Huber [email protected] ", "Maximilian Demmler [email protected] ", "Silvan Mertes [email protected] ", "Matthew L Olson ", "Elisabeth André ", "Tobias Huber ", "Maximilian Demmler ", "Silvan Mertes ", "Matthew L Olson ", "Elisabeth André ", "\nUniversity of Augsburg\nAugsburgGermany\n", "\nUniversity of Augsburg\nAugsburgGermany\n", "\nUniversity of Augsburg\nAugsburgGermany\n", "\nOregon State University Corvallis\nORUnited States\n", "\nUniversity of Augsburg\nAugsburgGermany\n" ]	[ "University of Augsburg\nAugsburgGermany", "University of Augsburg\nAugsburgGermany", "University of Augsburg\nAugsburgGermany", "Oregon State University Corvallis\nORUnited States", "University of Augsburg\nAugsburgGermany" ]	[ "IFAAMAS" ]	Counterfactual explanations are a common tool to explain artificial intelligence models. For Reinforcement Learning (RL) agents, they answer "Why not?" or "What if?" questions by illustrating what minimal change to a state is needed such that an agent chooses a different action. Generating counterfactual explanations for RL agents with visual input is especially challenging because of their large state spaces and because their decisions are part of an overarching policy, which includes long-term decision-making. However, research focusing on counterfactual explanations, specifically for RL agents with visual input, is scarce and does not go beyond identifying defective agents. It is unclear whether counterfactual explanations are still helpful for more complex tasks like analyzing the learned strategies of different agents or choosing a fitting agent for a specific task. We propose a novel but simple method to generate counterfactual explanations for RL agents by formulating the problem as a domain transfer problem which allows the use of adversarial learning techniques like StarGAN. Our method is fully model-agnostic and we demonstrate that it outperforms the only previous method in several computational metrics. Furthermore, we show in a user study that our method performs best when analyzing which strategies different agents pursue.	10.48550/arxiv.2302.12689	[ "https://export.arxiv.org/pdf/2302.12689v1.pdf" ]	257,205,945	2302.12689	cfeefcc801fd0dd1b16eb31e87f728db1abadb59	GANterfactual-RL: Understanding Reinforcement Learning Agents' Strategies through Visual Counterfactual Explanations Explainable Deep Reinforcement Learning; Explainable Artificial Intelligence; Interpretable Machine Learning ACM Reference Format: GANterfactual-RL: Understanding Reinforce- ment Learning Agents' Strategies through Visual Counterfactual Explana- tions . In May 29 -June 2, 2023 Tobias Huber [email protected] Maximilian Demmler [email protected] Silvan Mertes [email protected] Matthew L Olson Elisabeth André Tobias Huber Maximilian Demmler Silvan Mertes Matthew L Olson Elisabeth André University of Augsburg AugsburgGermany University of Augsburg AugsburgGermany University of Augsburg AugsburgGermany Oregon State University Corvallis ORUnited States University of Augsburg AugsburgGermany GANterfactual-RL: Understanding Reinforcement Learning Agents' Strategies through Visual Counterfactual Explanations Explainable Deep Reinforcement Learning; Explainable Artificial Intelligence; Interpretable Machine Learning ACM Reference Format: GANterfactual-RL: Understanding Reinforce- ment Learning Agents' Strategies through Visual Counterfactual Explana- tions . In IFAAMAS . of the 22nd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023)London, United Kingdom29May 29 -June 2, 2023KEYWORDS Counterfactual explanations are a common tool to explain artificial intelligence models. For Reinforcement Learning (RL) agents, they answer "Why not?" or "What if?" questions by illustrating what minimal change to a state is needed such that an agent chooses a different action. Generating counterfactual explanations for RL agents with visual input is especially challenging because of their large state spaces and because their decisions are part of an overarching policy, which includes long-term decision-making. However, research focusing on counterfactual explanations, specifically for RL agents with visual input, is scarce and does not go beyond identifying defective agents. It is unclear whether counterfactual explanations are still helpful for more complex tasks like analyzing the learned strategies of different agents or choosing a fitting agent for a specific task. We propose a novel but simple method to generate counterfactual explanations for RL agents by formulating the problem as a domain transfer problem which allows the use of adversarial learning techniques like StarGAN. Our method is fully model-agnostic and we demonstrate that it outperforms the only previous method in several computational metrics. Furthermore, we show in a user study that our method performs best when analyzing which strategies different agents pursue. same time, these deep RL agents are being deployed into increasingly high-risk domains like healthcare, autonomous driving, and robotic navigation [11,20,43]. In such domains, it is crucial to be able to understand the agents to enable appropriate use of them and to facilitate human-agent cooperation [37]. One prominent paradigm to make the decisions of intelligent agents transparent and comprehensible are so-called Counterfactual Explanations. By providing an alternative reality where the agent would have made a different decision, these explanations follow a rather human way of describing decisions [4,26]. For example, if a person would have to explain why a warehouse robot took a detour instead of directly moving to its desired target, they would probably give an explanation similar to If there was no production worker in the way, the robot would have moved straight to its target -and, by doing so, give a counterfactual explanation of the warehouse robot's behavior. Figure 1 shows a similar situation from the Atari game Pacman. In other machine learning domains, such as image classification, counterfactual explanations are already frequently used. However, this is not the case for RL, as several factors make explaining the decisions of RL agents more challenging. For one, RL agents are used for sequential decision-making tasks: their actions are not isolated. These actions are part of a long-term strategy that might be influenced by delayed rewards. Secondly, RL agents are not trained on a given ground truth strategy. The reward function only indirectly specifies the agent's goals [10]. The emerging strategies might not be what humans would expect, even if the strategy is optimal for the reward function. Finally, for RL agents, there is no direct counterpart to the training datasets used by supervised models. Therefore, counterfactual explanation approaches for supervised models that utilize the training data cannot be applied to RL agents without adjustment [41]. Due to the difficulties mentioned above, there is only one approach that focuses on creating counterfactual explanations for deep RL agents with visual input [31]. This approach utilizes a complex combination of models where the final generator is only indirectly trained to change the action. Olson et al. [31] show that their approach can be applied to a variety of RL environments and helps users identify a flawed agent. With the help of their counterfactual explanations, users were able to differentiate between a normal RL agent for the Atari game Space Invaders and a flawed agent that did not see a specific in-game object. For this task, it is sufficient for the counterfactual explanation to not change the particular object at all while other objects frequently change. This clearly communicates that the unchanged object is irrelevant and ignored by the agent, implying that it is not seen at all. But for counterfactual explanations to be employed more widely, they also have to be useful for more complex tasks. According to Hoffman et al. [14], one of the main goals of a good explanation is to refine the user's mental model of the agent. For RL agents, this includes understanding what strategy and intentions an agent pursues. Another critical goal for explanations is that they should help users to calibrate their trust in different agents [14]. For RL agents, this entails that users should be able to choose fitting agents for specific problems, which is more complex than simply identifying defective agents. The two aforementioned challenges require counterfactual explanations to not only convey what objects need to change but also how the objects need to be altered to change the agents' policy. To tackle these challenges, this paper proposes a novel method for generating counterfactual explanations for RL agents with visual input. We do so by formulating the generation problem as a domain transfer problem where the domains are represented by sets of states that lead the agent to different actions. Our approach is fully model-agnostic, easier to train than the approach presented by Olson et al., and includes the counterfactual actions more directly into the training routine by solving an action-to-action domain transfer problem. We evaluate our approach with computational metrics (e.g., how often do the counterfactuals change the agent's decision) and a user study using the Atari Learning Environment (ALE) [3], a common benchmark for RL agents with visual input. In our user study, we present participants with different kinds of counterfactual explanations and investigate whether this helps them to understand the strategies of Pacman agents. Furthermore, we investigate if the counterfactuals help them to calibrate their trust, so they can choose fitting agents for specific tasks (surviving or receiving points). As such, the contributions of this paper are as follows: We formulate a novel, model-agnostic approach for generating counterfactual explanations for RL agents. We demonstrate that our approach outperforms the previous method in several computational metrics. Furthermore, we conduct a user study that shows, for the first time, that counterfactual explanations can help to understand the strategies of RL agents. This user study also identifies current deficiencies of counterfactual explanations for RL agents that point the way for future work. RELATED WORK Our work deals with post-hoc explanations that are generated for fully trained black-box agents. Recent years saw a plethora of work on such explanations for (deep) RL agents. The literature often divides them by scope into global and local explanations. Global explanations try to explain the agent's overall strategy. This can be done by picking a subset of important state-action pairs that summarize the agent's strategy [1,15] or by distilling the agent's policy into a simpler model like a finite state machine [8] or a soft decision tree [7]. In this paper, we focus on local explanations that explain a specific decision of an agent. The most common approach to local explanations for RL agents are Feature Attribution or Saliency Map methods [17,33,44]. These methods try to identify the most important input features for a specific decision and highlight them, for example in a heatmap. However, recent work questioned whether one can rely on post-hoc feature attribution to faithfully represent the agent's internal reasoning [2,16]. Furthermore, previous studies showed that saliency maps for visual RL agents are hard to understand for end-users [18]. Counterfactual explanations are another type of local explanation. Since they follow the human thinking paradigm of counterfactual reasoning, it is often argued that they are easier to interpret than feature attribution methods [4,26]. For classification models, there is a growing body of work on counterfactual explanations. In 2017, Wachter et al. [39] were the first to introduce counterfactual explanations into the XAI domain by defining them as an optimization problem. Since then, various approaches to generate such counterfactuals were proposed, e.g., van Looveren and Klaise [22], and [12]. As various research has observed that generating counterfactual explanations is, at its core, a generative problem, the use of generative models like Generative Adversarial Networks (GANs) quickly became prevailing in state-of-the-art counterfactual explanation generation algorithms. E.g., Nemirovsky et al. [30] proposed Coun-terGAN, a framework to build highly realistic and actionable counterfactual explanations. Zhao et al. [46] propose an approach for generating counterfactual image explanations by using text descriptions of relevant features of an image to be explained. Furthermore, various specialized GAN-based algorithms were introduced to generate counterfactual explanations in the medical domain [23,24,38]. More recent frameworks for counterfactual explanation generation make use of the StyleGAN architecture, which implicitly models style-related aspects of an image, which makes it perfectly suitable for a whole range of image classification tasks [21,35]. As for a broad range of use cases, it is essential to be able to provide explanations for multiple counter-classes, various approaches have focused on that particular capability by using architectures based on StarGAN, an adversarial framework that was specifically designed for image translation between multiple domains [42,47]. One drawback of the aforementioned approaches for supervised learning is that their GANs are trained to transfer between domains given by the labeled classes from the classifier's training dataset. Then they add additional measures (e.g., loss functions [24,47]), to ensure that the generated counterfactuals are actually classified as the desired class by the classifier that is to be explained. This is not possible for RL agents that do not have a training set. Furthermore, the additional measures are often not model-agnostic. RL is often used to create counterfactual explanations for other models (for example in [5]). However, to the best of the authors' knowledge, there is only one previous work on generating visual counterfactual explanations for RL agents [31]. Olson et al. [31] train an encoder that creates an action-invariant latent representation of the agent's latent space. This is achieved by adversarially training in tandem with a discriminator , where tries to predict the agent's action and aims to make the decision of as uncertain as possible. In addition, they train a generative model to replicate states based on the action-invariant latent representation ( ) and the agent's action probability distribution ( ) for this state. By providing with a counterfactual action distribution ( ) ′ , they obtain a state that is similar to but brings the agent's action distribution closer to the desired counterfactual distribution. However, Olson et al. argue that an arbitrary counterfactual action distribution does not represent a realistic agent output and thus leads to unrealistic counterfactual states. To avoid this, they train an additional Wasserstein Auto Encoder and use it to perform gradient descent in the latent space of the agent towards an agent output that resembles the desired counterfactual action. Olson et al. refer to their approach as Counterfactual State Explanations (CSE), therefore we will also refer to it as CSE in this paper. The CSE approach is fairly complex and requires extensive access to the agent's inner workings. Furthermore, as Olson et al. mention themselves, the loss function of the generator does not directly force the resulting state ( ( ), ( ) ′ ) to be classified as the counterfactual action distribution ( ) ′ . This is only learned indirectly by replicating states based on the action-invariant latent space and the desired action distribution ( ) ′ . As we show in Section 4, this does not seem to be enough to change the agent's decision correctly. To solve those problems, we formulate a simpler counterfactual generation method that uses the counterfactual actions in a more direct way. APPROACH 3.1 GANterfactual-RL RL agents are usually employed in a Markov Decision Process (MDP) which consists of states ∈ , actions ∈ , and rewards . Given a state , the goal of an RL agent : → is to choose an action ( ) that maximizes its cumulative future rewards. To explain such an agent, the objective of a counterfactual explanation approach for RL agents is defined as follows. Given an original state and a desired counterfactual action ′ , we want a counterfactual state ′ that makes the agent choose the desired action ( ′ ) = ′ . Hereby, the original state should be altered as little as possible. On an abstract level, the action ( ) that the agent chooses for a state can be seen as a top-level feature that describes a combination of several underlying features which the agent considers to be relevant for its decision. Thus, the counterfactual state ′ should only change the features that are relevant to the agent's decision, while maintaining all other features not relevant to the decision. This is similar to image-to-image translation, where features that are relevant for a certain image domain should be transformed into features leading to another image domain, while all other features have to be maintained (e.g., the background should remain constant when transforming horses to zebras). Taken together, we can formulate the generation of counterfactual states for RL agents as a domain transfer problem similar to image-to-image translation: The agent's action space defines the different domains = { ∈ \| ( ) = }, where each state belongs to the domain that corresponds to the action that the agent chooses for this state (see Figure 2). To solve the reformulated domain transfer problem, we base our system on the StarGAN architecture [6], since RL agents usually use more than two actions. The StarGAN architecture incorporates multiple loss components that can be reformulated to be applicable to the RL domain. The first component, the so-called adversarial loss, leads the network to produce highly realistic states that look like states from the original environment. Reformulated for the task of generating RL states, we define it as follows (following Choi et al. [6] we use a Wasserstein objective with gradient penalty): L = E [ ( )] − E , ′ [ ( ( , ′ ))] − Eˆ (\|\|∇ˆ(ˆ)\|\| 2 − 1) 2 , where is the StarGAN's discriminator network and its generator network. The second loss component, which is specific to the StarGAN architecture, guides the generator network to produce states that lead to the desired counterfactual actions. It consists of two sub-objectives, one that is applied while the network is fed with original (real) states from the training set (Eq. 1), and the other while the network is generating counterfactual states (Eq. 2): L = E , [− log ( \| )],(1)L ′ = E , ′ [− log ( ′ \| ( , ′ ))],(2) where refers to the StarGAN discriminator's classification output, which learns to approximate the action that the agent is performing in a particular state. Further, as counterfactual states should be as close to the original states as possible, a Reconstruction Loss is used. This loss forces the network to only change features that are relevant to the agent's choice of action: L = E , , ′ [\|\| − ( ( , ′ ), )\|\| 1 ] Taken together, the whole loss of the StarGAN architecture, reformulated for RL counterfactual explanations, is defined as follows: L = −L + L , L = L + L ′ + L , where and are weights controlling the corresponding loss component's relevance. Since our approach utilizes a GAN architecture to generate counterfactuals for RL agents, we refer to it as GANterfactual-RL. Dataset Generation As described above, our GANterfactual-RL approach relies on training data in the form of state-action pairs. Olson et al. [31] train their CSE approach on state-action pairs generated by concurrently running an MDP with a trained agent. This strategy is simple but allows for little control over the training data, which can lead to the following complications: Frames extracted from a running MDP contain a temporal pattern since consecutive states typically have a high correlation. Such correlations and patterns can lead to bias and sub-optimal convergence during training. For episodic MDPs, there is a high probability of reaching the same state throughout several episodes. This is amplified by the fact that RL agents often learn to execute only a few optimal trajectories. This results in duplicate samples that are effectively over-sampled during training. RL agents generally do not execute each action equally frequently, since most environments contain actions that are useful more often than others. This leads to an imbalanced amount of training samples per domain. To mitigate the aforementioned issues, we propose to generate datasets as follows: Data is gathered by running a trained agent in an MDP. Each state corresponds to one dataset sample and is labeled with the action that the agent chooses to execute in this state. An -greedy policy ( =0.2 in our case) is used to increase the diversity of states reached over multiple episodes. State-action pairs with an explored (randomly chosen) action are not added to the dataset. After the data is gathered, duplicates are removed. Then, a class balancing technique (under-sampling in our case) is used to account for over-or underrepresented actions. Finally, the dataset is split into a training set, a test set, and potentially a validation set. Most of these techniques are commonly used in other application domains of machine learning. However, to our best knowledge, this is the first work to generate and preprocess datasets for generating counterfactual explanations for RL agents. Application to Atari Domain Environment. The environments we use for our experiments are the Atari 2600 games MsPacman (henceforth referred to as Pacman) and Space Invaders, included in the Arcade Learning Environment (ALE) [3]. The ALE states are based on the raw pixel values of the game. Each input frame is cropped so that only the actual playing field remains. This removes components such as the score and life indicators which would allow participants to easily see which agent receives higher scores. After that, we use the same preprocessing as Mnih et al. [28]. Two steps from this preprocessing are particularly important for us. First, the frames are gray-scaled and downsized. Second, in addition to the current frame, the agent receives the last three preprocessed frames as input. This allows the agent to detect temporal relations. The ALE actions normally correspond to the meaningful actions achieved with an Atari 2600 controller (e.g. six actions for Space Invaders). Since we wanted to use our Pacman agents in a user study we removed 4 redundant actions (e.g., Up & Right) whose effect differs between situations and is therefore hard to convey to participants. This left us with 5 actions for Pacman (Do nothing, Up, Down, Left, Right). Agent Training. To evaluate participants' ability to differentiate between alternative agents and analyze their strategies, we modified the reward function of three Pacman agents. This is a more natural method of obtaining different agents compared to withholding information from the agent as Olson et al. [31] did. Furthermore, it results in agents that behave qualitatively differently. Therefore participants have to actually analyze the agents' strategies instead of simply looking for objects that the agents ignore. Blue-Ghost Agent: This agent was trained using the default reward function of the ALE, where blue ghosts get the highest reward. Power Pill Agent: This agent only received positive rewards for eating power pills. Fear-Ghost Agent: This agent got a small positive reward of 1 for every step in which it did not die to ghosts. For training the first two Pacman agents, we use the DQN algorithm [28]. Each agent was trained for 5 Million steps. The fear-ghosts agent was trained using the ACER algorithm [40] for 10M steps. At the end of the training period, the best-performing policy is restored. For all three agents, we build upon the OpenAI baselines [9] repository. For Space Invaders, we used the two Asynchronous Advantage Actor-Critic (A3C) agents trained by Olson et al. [31]. For training details, we refer to their paper. One agent is trained normally, while the other agent is flawed and does not see the laser cannon at the bottom of the screen. GANterfactual-RL on Atari. To generate human-understandable counterfactual explanations for our Atari agents, the generated counterfactual states should represent the frames that humans see during gameplay. That means we cannot train our GANterfactual-RL model on the preprocessed and stacked frames that the Atari agents use. Instead, we train it on the cropped RGB frames before preprocessing. The only preprocessing we still use on those frames is a countermeasure against flickering objects in Atari games, which was proposed by Mnih et al. [28]. While generating the dataset, we only save the most recent of the four stacked frames for each state . This frame generally influences the agent's decision the most. For feeding the counterfactual frame back into the agent (e.g., to evaluate the approach), we stack it four times and then apply preprocessing. Our implementation details can be found in the appendix. The full code is available online. 1 COMPUTATIONAL EVALUATION 4.1 Used Metrics We evaluate our approach using the metrics validity (or success rate), proximity (or cost), sparsity, and generation time. We consider these metrics to be the most suitable and widely used metrics for image-based counterfactual explanations [5,19,29,32]. Validity captures the rate of CounterFactuals (CFs) that actually evoke the targeted action when fed to the agent. With being true CFs (correctly changing the agent's action), being false CFs, and the total amount of evaluated CFs, this metric is defined as: = + = Proximity measures the similarity between an original state image and its CF via the 1-norm. We normalize the metric to measure the proximity in the range [0, 1]. ( , ) = 1 − 1 255 · S \|\| − ( , )\|\| 1 where is the original state image, ( , ) is the generated CF for an arbitrary target action domain and S is the domain of color values of (S = 3 · ℎ · ℎ for RGB-encoded images). The normalization with 255 · S assumes an 8-bit color encoding with color values in range [0, 255]. High proximity values are desirable since they indicate small adjustments to the original state. Sparsity quantifies the number of unmodified pixel values between an original state image and its CF via the 0-norm (a pseudonorm that counts the number of non-zero entries of a vector/matrix). The sparsity is normalized to the range [0, 1] as well. ( , ) = 1 − 1 S \|\| − ( , )\|\| 0 A completely altered image has a sparsity of 0, an unmodified image has a sparsity of 1. High sparsity values are thus desirable. Generation Time determines the time it takes to generate one CF with a trained generator, not including pre-or post-processing. Computational Results The computational results for the three Pacman agents are shown in Table 1 and the results for the two Space Invaders agents in Table 2. For the Pacman agents, we generated fully cleaned datasets (Section 3.2) and sampled 10% of each action for the evaluation test set. To show the contribution of our proposed dataset generation, we additionally trained a GANterfactual-RL model for the blueghost agent without the steps proposed in Section 3.2 and evaluated it on the test set from the clean dataset. This dropped the validity to 0.45 and sparsity to 0.50 ± 0.01 while the other values stayed comparable. To be more comparable to the results by Olson et al. [31], we do not remove duplicates from the Space Invaders datasets and do not apply class balancing. Here we create the test set by sampling 500 states for each action and removing all duplicates of these states from the training set. Our GANterfactual-RL approach outperforms the CSE counterfactuals in every single metric. The research question for our study was which counterfactual explanations help users to understand the strategies of RL agents and help them to choose fitting agents for a specific task. We hypothesized that our GANterfactual-RL method is more useful than the CSE method and is more useful than a presentation of the original states without counterfactuals. Further, we thought that the counterfactuals generated by the CSE approach might mislead participants due to the low validity of the generated counterfactual explanations (see Section 4). Therefore, we hypothesized that only providing the original states is more useful than adding CSE counterfactuals. Dependent Variables and Main Tasks. Agent Understanding Task. To measure whether participants understand the strategies of different agents and build a correct mental model of them, we used an agent understanding task inspired by Hoffman et al. [14] and Huber et al. [18]. Here, participants were presented with five states and the actions that the agent chooses in these states. This was done for each of the three Pacman agents described in Section 3.3 (one agent at a time). The states were selected by the HIGHLIGHTS-Div algorithm [1]. To this end, we let each trained agent play for additional 50 episodes and chose the most important states according to HIGHLIGHTS-Div. The resulting states show gameplay that is typical for the agent, without the need to manually select states that might be biased toward our approach. Based on these states (and additional explanations depending on the condition), participants had to select up to two in-game objects that were most important for the agent's strategy from a list of objects (Pacman, normal pills, power pills, ghosts, blue ghosts, or cherries). As described in Section 3.3, each agent, strongly focuses on a different single in-game object depending on their reward function (e.g., the fear-ghosts agent focuses on normal ghosts). If the participants select this object and none of the other objects, they receive a point. The only exception is Pacman. Every agent heavily relies on the position of Pacman as a source of information. Therefore, participants receive the point whether they select Pacman or not. Agent Comparison Task. To measure how well the participants' trust is calibrated, we used an agent comparison task inspired by Amir and Amir [1] and Miller [27]. Here, we implicitly measure if the participants' trust is appropriate by asking them, for each possible pair of the three Pacman agents, which agent they would like to play on their behalf to obtain certain goals. Since a single agent can be good for one goal but bad for another, this requires a deeper analysis than the distinction between a normal and a defective agent. For each pair, the participants are shown their own descriptions of each agent from the agent understanding task and the same states and explanations that they saw during the agent understanding task. Then they have to decide which agent should play on their behalf to achieve more points and which agent should play on their behalf to survive longer. We know the ground truth for this by measuring the agents' average score and amount of steps for the 50 episodes used to find the HIGHLIGHTS states. The amount of steps that the blue-ghost agent and the power pill agent survive is so close that we do not include this specific comparison in the evaluation. Explanation Satisfaction. To measure the participant's subjective satisfaction, we use statements adapted from the Explanation Satisfaction Scale by Hoffman et al. [14]. Participants have to rate their agreement with each statement on a 5-point Likert scale. Participants' final rating was averaged over all those ratings, reversing the rating of negative statements. We do this once after the agent understanding task and once after the agent comparison task in case there are satisfaction differences between the tasks. Conditions and Explanation Presentation. We used three independent conditions, one Control condition without explanations and two conditions where the states during the agent understanding task and the agent comparison task are accompanied by counterfactual explanations. In the CSE condition, the counterfactuals are generated by the approach from Olson et al. [31], and in the GANterfactual-RL condition the counterfactuals are generated by our proposed method. The presentation of the counterfactual explanations is designed as follows. For each state, we generate a single counterfactual state. We were concerned that too many counterfactual states would cause too much cognitive load. The way that MsPacman is implemented, actions that do nothing or move directly into a wall are ignored. To generate meaningful counterfactual states, we limited the counterfactual action to turning around in a corridor and randomly selecting a new direction at an intersection (do not turn around). The counterfactual states are presented by a slider under each state. Moving the slider from left to right linearly interpolates the original state to the counterfactual state (per-pixel interpolation). The original and counterfactual actions are written above the state. Figure 4 shows a simplified version of the beginning of our agent understanding task. Procedure and Compensation. After completing a consent form, participants were asked to answer demographic questions (age and gender) and questions regarding their experience with Pacman and their views on AI. Then, they were shown a tutorial explaining the rules of the game Pacman and were asked to play the game to familiarize themselves with it. To verify that participants understood the rules, they were asked to complete a quiz and were only allowed to proceed with the survey after answering all questions correctly. Afterward, participants in the counterfactual conditions received additional information and another quiz regarding the counterfactual explanations. Then, they proceeded to the agent understanding task which was repeated three times, once for each agent. The order of the agents was randomized. After that, participants filled the explanation satisfaction scale and continued to the agent comparison task. Again, this task was repeated three times, once for each possible agent pair, and the order was randomized. Finally, participants had to complete another satisfaction scale for the agent comparison task. Participants got a compensation of 5$ for participating in the study. As an incentive to do the tasks properly, they received a bonus payment of 10 cents for each point they get in the agent understanding task and 5 cents for each point in the agent comparison task. The complete questionnaire can be seen in the appendix. We preregistered our study online. 2 5.1.5 Participants. We recruited participants through Amazon Mechanical Turk. Participation was limited to Mechanical Turk Masters from the US, UK, or Canada (to ensure a sufficient English level) with a task approval rate greater than 95% and without color vision impairment. We conducted a power analysis with an estimated medium effect size of 0.7 based on previous similar experiments [18,24,25]. This determined that we need 28 participants per condition to achieve a power of 0.8. and a significance level of 0.05. To account for participant exclusions, we recruited 30 participants per condition. Participants were excluded if they did not look at any of the counterfactual explanations for any of the agents during the agent understanding task, if their textual answers were nonsensical or if they took considerably less time than the average. This left us with 30 participants in the Control condition, 28 participants in the CSE condition, and 23 in the GANterfactual-RL condition. The distribution of age, AI experience, and Pacman experience was similar between the conditions (see the appendix). There was a difference in the gender distribution and the attitude towards AI between the conditions. The Control condition had 40% female participants, the CSE condition had 32% and the GANterfactual-RL condition had 26%. The mean attitude towards AI was the highest in the GANterfactual-RL condition and the lowest in the Control condition (see the appendix). Results The results for the participants' scores during the main tasks can be seen in Figure 5, while their explanation satisfaction values are shown in Figure 6. In the following, we will summarize the results of our main hypotheses, which we analyzed using non-parametric one-tailed Mann-Whitney U tests. Counterfactuals helped participants to understand the agents' strategies. In the agent understanding task, there was a significant difference between the Control condition ( =0.8) and the GANterfactual-RL condition ( =1.65), =181, =0.001, =0.477. 3 Contrary to our hypothesis, the Control condition got lower scores than the CSE condition ( =0.8 vs =1.18), =0.953. Our GANterfactual-RL explanations were significantly more useful than the CSE approach for understanding the agents' strategies. In the agent understanding task, the CSE condition got 2 https://aspredicted.org/m9fi5.pdf 3 is the mean and is rank biserial correlation. (a) Total score (summed over all three agents) for the agent understanding task. (b) Number of correct agent selections in the agent comparison task (Out of five). Counterfactuals did not increase explanation satisfaction. Even though participants objectively had a better understanding of the agents' strategies, they did not feel more satisfied with them. Participants in the Control condition were significantly more satisfied than participants in the CSE condition in both the agent understanding task (Control: =3.77, CSE: =3.20; =249, =0.004, =0.4071) and the agent comparison task (Control: =3.75, CSE: =3.14; =267, =0.008, =0.3643). Contrary to our expectations, the participants in the GANterfactual-RL condition were not more satisfied than the participants in the Control condition or the CSE condition in both the agent understanding task (Control vs. GANterfactual-RL: =0.996, CSE vs GANterfactual-RL: =0.546) or the agent comparison task (Control vs. GANterfactual-RL: =0.967, CSE vs GANterfactual-RL: =0.334). DISCUSSION 6.1 Computational Evaluation Our computational evaluation shows that our proposed approach is correctly changing the agents' actions in 46% to 70% of the cases depending on the agent. While this is not perfect, one has to consider that this is not a binary task but that the agents have 5 or 6 different actions. Furthermore, CSE [31], the only previous method that focuses on generating counterfactual explanations for RL agents, only successfully changed the agent's decision in 17% to 28% of the cases. We can think of two reasons for the low validity values for the CSE approach. First, they only incorporate the agent's action in their loss functions related to the latent space (where their discriminator and WAE were trained). The generation of the final pixels did not include constraints to faithfully ensure that a specific action was taken by the agent. Second, their loss functions for the latent space focus on creating action-invariant states. Olson et al. [31] showed that their CSE approach was useful for differentiating between a normal agent and a flawed agent. We think this is due to the fact that CSE is good at generating action-invariant states. This can help to identify the object that the flawed agent did not see since irrelevant objects are not changed for action-invariant states. We found that our approach also does not change the irrelevant object for the flawed agent (illustrated in Figure 3). This demonstrates that the counterfactuals generated by our approach are similarly effective for identifying the flawed agent. Looking at the distance between the original and the counterfactual states in pixel-space, we see that counterfactual states generated by our GANterfactual-RL approach on average have less distance to the original states and change fewer pixel values compared to the counterfactuals generated by the previous CSE method by Olson et al. [31]. This indicates that our GANterfactual-RL method is better at achieving the goal of finding the smallest possible modification of the original state to change the agent's decision. Since our method only requires a single forward pass to generate a counterfactual state, it is faster than the CSE method, which relies on potentially time-consuming gradient descent for the counterfactual generation. User Study Our user study showed that counterfactual explanations help users to understand which strategies different agents pursue. In particular, our method was significantly more useful than both the CSE method and not providing counterfactuals. Contrary to our hypothesis, even the counterfactuals generated by the CSE method resulted in a better understanding of the agents than not providing counterfactual explanations. This demonstrates the usefulness of counterfactual explanations for RL agents even in more complex tasks than identifying defective agents. Two recent studies evaluated the usefulness of other explanation techniques for understanding the strategies of RL agents in a similar way to our study. Huber et al. [18] looked at saliency map explanations and found that they did not help more than showing HIGHLIGHTS states without saliency maps. Their participants achieved 37% of the maximum possible score in their agent understanding task, while the participants with our counterfactual explanations obtained 50%. Septon et al. [36] investigated so-called reward decomposition explanations and found that they helped participants to achieve 60% of the maximum score in their agent understanding task. However, reward decomposition is an intrinsic explanation method which the agent and the reward function have to be specifically designed for. Our counterfactual explanations resulted in only 10% less average score even though they are post-hoc explanations that can be generated for already trained black-box agents. Our agent comparison task showed that the increased understanding of the agent's strategies through both counterfactual explanation methods did not help participants choose fitting agents for specific tasks. For choosing the correct agent for a given problem, it is not enough to identify the strategies of the agents. It also requires enough expertise in the environment (e.g., Pacman) to judge which strategy is better suited for the problem at hand. For example, in Pacman, humans often assume that an agent that survives longer will accumulate more points in the long run. However, this is not necessarily the case since an aggressive agent can better exploit the very high rewards of eating blue ghosts. Our results for this task are in line with the results of the agent comparison task for saliency maps by Huber et al. [18]. Finally, our study showed that participants subjectively were not satisfied with the counterfactual explanations even though they objectively helped them to understand the agents. This might be due to the additional cognitive load of interpreting the explanations. The two aforementioned studies [18,36] also did not find a significant difference in user satisfaction for their local explanation techniques. Only the choice of states, which does not provide additional information, influenced the satisfaction in [18]. However, our study is the first to see significantly higher satisfaction for the no-explanation condition than one of the two explanation conditions. This indicates that counterfactuals are subjectively less satisfying than saliency maps or reward decomposition. One possible explanation for this is the visual quality of the counterfactuals. Some participants from both counterfactual conditions commented that the counterfactuals had too many artifacts. One participant from the GANterfactual-RL condition for example wrote that "the counterfactuals were somewhat helpful, but they would have worked better if there were fewer or no artifacts". Another possible reason for the low satisfaction is the presentation of the explanation. Because our study primarily aimed at investigating the benefits and drawbacks of our specific counterfactual approach, we did not use a user-friendly explanatory system where different types of explanations are provided according to the requests of the explainee. CONCLUSION AND FUTURE WORK In this work, we formulated a novel method for generating counterfactual explanations for RL agents. This GANterfactual-RL method is fully model-agnostic, which we demonstrate by applying it to three RL algorithms, two actor-critic methods, and one deep Qlearning method. Using computational metrics, we show that our proposed method is better at correctly changing the agent's decision while modifying less of the original input and taking less time than the only previous method that focuses on generating visual counterfactuals for RL. Furthermore, it significantly improved users' understanding of the strategies of different agents in a user study. Our user study also identified two remaining deficiencies of counterfactual explanations. First, participants were subjectively not satisfied with the explanations, which might be due to unnatural artifacts in some counterfactuals. Second, the counterfactuals did not help them to calibrate their trust in the agents. Future work should try to improve counterfactual explanations in these directions. While there is still room for improvement, we can confidently say that our approach can be considered the current state of the art for counterfactual explanations for RL agents with visual input. ACKNOWLEDGMENTS This paper was partially funded by the DFG through the Leibniz award of Elisabeth André (AN 559/10-1). A IMPLEMENTATION DETAILS In this section, we provide implementation details regarding the training of our counterfactual generation methods. Our full implementation can be found online. 4 A.1 Training Data For the size of our training data sets, we aimed for around 200000 states, since the StarGAN architecture from Choi et al. [6] that we use in our approach was fine-tuned for the CelebA dataset, which contains around 200000 images. To this end, we started by sampling 400000 states for each game and each RL agent. For the Pacman agents, after duplicate removal and under-sampling (see Section 3.2), this leaves us with 230450 states for the blue-ghost agent, 277045 states for the power pill agent and 40580 states for the fear-ghosts agent. For Space Invaders, the dataset size is only slightly reduced due to the removal of training samples that are duplicates of test samples. For the normal agent, this resulted in 382989 states and for the flawed agent it resulted in 376711 states. As is custom for the Atari environment [3,28], we use a random amount (in the range [0, 30]) of initial Do Nothing actions for each episode to make the games less deterministic. A.2 Training GANterfactual-RL For training the StarGAN within our GANterfactual-RL approach, we tried to stay as close to Choi et al. [6] as possible. We built our implementation upon the published source code 5 of Choi et al. [6] and used their original settings. The network architecture is the same as in Choi et al. [6]. For the loss functions specified in the main paper, we use = 1, = 10, and = 10. For training, we use an ADAM optimizer with = 0.0001, 1 = 0.5 and 2 = 0.999. The model is trained for 200, 000 batch iterations with a batch size of 16. The learning rate linearly decays after half of the batch iterations are finished. The Critic is updated 5 times per generator update during training. One thing we change compared to Choi et al. [6] is that we do not flip images horizontally during training. This is an augmentation step that improves the generalization on datasets of face images. However, it is counterproductive for Atari frames since flipped frames would often leave the space of possible Atari states or change the action that the agent would select. A.3 Training the Counterfactual State Explanations Model B USER STUDY DEMOGRAPHICS In this section, we provide more details regarding the demographic of the participants in our user study. As Fig. 7 shows, the mean age for each condition was around 40. We verified that participants in different conditions did not differ much in their AI experience and views and their Pacman experience. To this end, we asked them when they played Pacman for the last time. The results are shown in Figure 8. From left to right the bars represent: "more than 5 years ago", "less than 5 years ago" and "less than 1 year ago". For the AI experience we adapted a description of AI from Zhang et al. [45] and Russel [34] to "The following questions ask about Artificial Intelligence (AI). Colloquially, the term 'artificial intelligence' is often used to describe machines (or computers) that mimic 'cognitive' functions that humans associate with the human mind, such as 'learning' and 'problem solving'." After this description, participants had to select one or more of the following items: 1: I know AI from the media. Other: The distribution of the items for each condition is shown in Fig. 9. The option Other was never chosen. To measure the participants' attitude towards AI we adapted a question from Zhang et al [45] and asked them to rate their answer to the question "Suppose that AI agents would achieve high-level performance in more areas one day. How positive or negative do Control CSE GANterfactual-RL Figure 9: Distribution of the chosen AI experience items for each condition. The x-axis depicts the items described above. you expect the overall impact of such AI agents to be on humanity in the long run?" on a 5-point Likert scale from "Extremely negative" to "Extremely positive". The results are shown in Fig. 10. C SUPPLEMENTARY RESULTS In this section, we present additional information about the results of the user study that did not fit the scope of the main paper. During our agent understanding task participants got the point for each agent independent of whether they selected Pacman as important since Pacman is the main source of information for all our agents. We still wanted to exploratively look at how often the participants picked Pacman in each condition. Figure 11 shows that the results for picking Pacman are similar to the results for the participants' scores in this task. Another value we exploratively looked at is how long the participants in each condition spent on doing the two main tasks (excluding the time they spent on the instructions, quizzes, and satisfaction questions). Figure 12 shows that the time that participants spend on the main tasks does not differ much between conditions. Finally, we want to report the average in-game score and survival time of our Pacman agents since we used this as ground truth for the agent comparison task. The blue-ghost agent got a mean score of 2035.6 and survived for 708.36 steps on average. The power pill agent got a mean score of 1488 and survived for 696.4 steps on average. The fear-ghosts agent got a mean score of 944.4 and survived for 6490.16 steps on average. Figure 13: Example counterfactual states for the blue-ghost agent. The first row shows the original states and the second and third rows show counterfactual states by Olson et al. [31] and our GANterfactual-RL approach respectively. The states and actions are the same states that were used during our user study and were chosen by the HIGHLIGHTS-Div algorithm [1]. Figure 14: Example counterfactual states for the power pill agent. The first row shows the original states and the second and third rows show counterfactual states by Olson et al. [31] and our GANterfactual-RL approach respectively. The states and actions are the same states that were used during our user study and were chosen by the HIGHLIGHTS-Div algorithm [1]. Figure 15: Example counterfactual states for the fear-ghosts agent. The first row shows the original states and the second and third rows show counterfactual states by Olson et al. [31] and our GANterfactual-RL approach respectively. The states and actions are the same states that were used during our user study and were chosen by the HIGHLIGHTS-Div algorithm [1]. Figure 16: Example counterfactual states for the flawed Space Invader agent. The first row shows the original states and the second and third rows show counterfactual states by Olson et al. [31] and our GANterfactual approach respectively. The states were chosen by the HIGHLIGHTS-Div algorithm [1]. The counterfactual actions were chosen to be complete opposites of the original action. Despite this big difference in the action, both approaches do not move the laser cannon (highlighted with a blue circle in the original frames) that the flawed agent does not see. : The first part of the agent comparison task. This task was repeated for all three agent pairs. The order of the pairs was randomized. Figure 28: The second part of the agent comparison task. This task was repeated for all three agent pairs. The order of the pairs was randomized. D EXAMPLE COUNTERFACTUALS Figure 1 : 1Example for a counterfactual explanation: In the original situation (a), the agent does not take the fastest path to the pill in the top right corner. It is unclear if the agent is afraid of the ghost or does not recognize the shortest path. The counterfactual state (b) shows that the agent would have taken the fastest path to the pill if the ghost was not there. This indicates that the agent is afraid of the ghost. Figure 2 : 2Schematic of our counterfactual generation approach. We formulate the problem as domain transfer where each domain represents an action. States are assigned to domains based on the action that the agent chooses for them. 1 https://github.com/hcmlab/GANterfactual-RL Figure 3 : 3Example counterfactual states. Our approach does not change the Laser Cannon (marked in blue) for the flawed agent, who does not see it, but changes it for the normal agent. Figure 3 3shows example counterfactuals generated for the Pacman fear-ghosts agent and the two Space Invaders agents. Additional examples for all our agents can be seen in the appendix. Figure 4 : 4A simplified scheme of the beginning of our agent understanding task with a single example state. Figure 5 : 5Comparison of participants' average performance in each task, by condition. Error bars show the 95% CI. Figure 6 : 6Comparison of participants' average explanation satisfaction in each task, by condition. a mean score of 1.18, while the GANterfactual-RL condition got a mean score of 1.65 ( =232, =0.038, =0.2795).The increased understanding of the agents' strategies did not result in a more calibrated trust. Contrary to our hypothesis, there were no significant differences in the trust task (Control vs CSE: =0.536, Control vs. GANterfactual-RL: =0.852, CSE vs GANterfactual-RL: =0.876). Figure 7 : 7The participants' age per condition. Figure 8 : 8The Pacman experience across all conditions where the bars depict when the participants played Pacman the last time. Figure 10 : 10The average attitude towards AI, rated on a 5point Likert scale. Figure 11 : 11How often the participants in each condition selected Pacman as important for the agent's strategy. Figure 12 : 12How much time (in seconds) the participants spent on our two main tasks. Figure 17 : 17Example counterfactual states for the normal Space Invader agent. The first row shows the original states and the second and third rows show counterfactual states by Olson et al.[31] and our GANterfactual approach respectively. The states are chosen by the HIGHLIGHTS-Div algorithm[1]. The counterfactual actions are chosen to be complete opposites of the original action. In contrast to the counterfactuals for the flawed Space Invaders Agent, both approaches sometimes modify the laser cannon. Figure 18 : 18Demographic information. Figure 19 : 19The Pacman tutorial. Figure 20 : 20The Pacman quiz. Figure 21 : 21The first part of the counterfactual tutorial, which is built upon the tutorial by Olson et al.[31]. Figure 22 : 22The second part of the counterfactual tutorial, which is built upon the tutorial by Olson et al.[31]. Figure 23 : 23The quiz about counterfactual explanations. Figure 24 : 24The first part of the agent understanding task. This task was repeated for all three agents. The order of the agents was randomized. Figure 25 : 25The second part of the agent understanding task. This task was repeated for all three agents. The order of the agents was randomized. Figure 26 : 26Explanation Satisfaction for the agent understanding task. Figure 27 27Figure 27: The first part of the agent comparison task. This task was repeated for all three agent pairs. The order of the pairs was randomized. Figure 29 : 29Explanation Satisfaction for the agent comparison task. Table 1 : 1Computational evaluation results for the Pacman agents. Proximity, sparsity and generation time are specified by mean ± standard deviation. Validity (↑) Proximity (↑) Sparsity (↑) Gen. Time [s] (↓)Approach Blue-Ghost Agent Ours 0.59 0.997 ± 0.001 0.73 ± 0.02 0.011 ± 0.012 CSE 0.28 0.992 ± 0.002 0.33 ± 0.03 0.085 ± 0.021 Power-Pill Agent Ours 0.49 0.997 ± 0.001 0.70 ± 0.02 0.011 ± 0.008 CSE 0.20 0.993 ± 0.002 0.32 ± 0.02 0.566 ± 0.731 Fear-Ghost Agent Ours 0.46 0.995 ± 0.001 0.45 ± 0.01 0.013 ± 0.014 CSE 0.20 0.992 ± 0.002 0.32 ± 0.04 0.020 ± 0.017 Table 2 : 2Computational evaluation results for the Space Invaders agents. Proximity, sparsity and generation time are specified by mean ± standard deviation.Approach Validity (↑) Proximity (↑) Sparsity (↑) Gen. Time [s] (↓) Normal Agent Ours 0.70 0.998 ± 0.002 0.97 ± 0.02 0.011 ± 0.013 CSE 0.18 0.995 ± 0.003 0.89 ± 0.05 6.180 ± 9.727 Flawed Agent Ours 0.53 0.998 ± 0.002 0.96 ± 0.01 0.011 ± 0.015 CSE 0.17 0.995 ± 0.004 0.94 ± 0.01 0.020 ± 0.035 Agent: Original Action: Pacman Fear Ghosts Move Down Space Invader Flawed Right & Fire Space Invader Normal Right & Fire Original State: Target Action: Move Up Move Left Move Left CSE Counterfac- tual State: GANterfactual- RL Counterfac- tual State: For training the counterfactual state explanation model proposed by Olson et al.[31], we reuse their published source code 6 to ensure comparability and reproducibility. For this reason, we also use the same Training parameters and network architecture. The only change we had to make to the network architecture is that the size of the latent space of our DQN Pacman agents is 256 compared to the Space Invaders agents in Olson et al.[31] that have a latent space size of 32. 4 https://github.com/hcmlab/GANterfactual-RL 5 https://github.com/yunjey/stargan 6 https://github.com/mattolson93/counterfactual-state-explanations/ 2: I use AI technology in my private life. 3: I use AI technology in my work. 4: I took at least one AI-related course. 5: I do research on AI-related topics. Figures 13, 14, 15, 16, and 17 show example counterfactuals for both approaches tested in this work. E FULL USER STUDY Figures 18 to 29 present screenshots of our user study. 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[ "PairRE: Knowledge Graph Embeddings via Paired Relation Vectors", "PairRE: Knowledge Graph Embeddings via Paired Relation Vectors", "PairRE: Knowledge Graph Embeddings via Paired Relation Vectors", "PairRE: Knowledge Graph Embeddings via Paired Relation Vectors" ]	[ "Linlin Chao ", "Jianshan He ", "Taifeng Wang [email protected] ", "Wei Chu Antgroup ", "Linlin Chao ", "Jianshan He ", "Taifeng Wang [email protected] ", "Wei Chu Antgroup " ]	[]	[ "Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing", "Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing" ]	Distance based knowledge graph embedding methods show promising results on link prediction task, on which two topics have been widely studied: one is the ability to handle complex relations, such as N-to-1, 1-to-N and N-to-N, the other is to encode various relation patterns, such as symmetry/antisymmetry. However, the existing methods fail to solve these two problems at the same time, which leads to unsatisfactory results. To mitigate this problem, we propose PairRE, a model with paired vectors for each relation representation. The paired vectors enable an adaptive adjustment of the margin in loss function to fit for complex relations. Besides, PairRE is capable of encoding three important relation patterns, symmetry/antisymmetry, inverse and composition. Given simple constraints on relation representations, PairRE can encode subrelation further. Experiments on link prediction benchmarks demonstrate the proposed key capabilities of PairRE. Moreover, We set a new stateof-the-art on two knowledge graph datasets of the challenging Open Graph Benchmark.	10.18653/v1/2021.acl-long.336	[ "https://www.aclanthology.org/2021.acl-long.336.pdf" ]	226,281,660	2011.03798	312143958930d7c045feee2eb847cae9da9d59e2	PairRE: Knowledge Graph Embeddings via Paired Relation Vectors August 1-6, 2021 Linlin Chao Jianshan He Taifeng Wang [email protected] Wei Chu Antgroup PairRE: Knowledge Graph Embeddings via Paired Relation Vectors Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language ProcessingAugust 1-6, 20214360 Distance based knowledge graph embedding methods show promising results on link prediction task, on which two topics have been widely studied: one is the ability to handle complex relations, such as N-to-1, 1-to-N and N-to-N, the other is to encode various relation patterns, such as symmetry/antisymmetry. However, the existing methods fail to solve these two problems at the same time, which leads to unsatisfactory results. To mitigate this problem, we propose PairRE, a model with paired vectors for each relation representation. The paired vectors enable an adaptive adjustment of the margin in loss function to fit for complex relations. Besides, PairRE is capable of encoding three important relation patterns, symmetry/antisymmetry, inverse and composition. Given simple constraints on relation representations, PairRE can encode subrelation further. Experiments on link prediction benchmarks demonstrate the proposed key capabilities of PairRE. Moreover, We set a new stateof-the-art on two knowledge graph datasets of the challenging Open Graph Benchmark. Introduction Knowledge graphs store huge amounts of structured data in the form of triples, with projects such as WordNet (Miller, 1995), Freebase (Bollacker et al., 2008), YAGO (Suchanek et al., 2007) and DBpedia (Lehmann et al., 2015). They have gained widespread attraction from their successful use in tasks such as question answering (Bordes et al., 2014), semantic parsing (Berant et al., 2013), and named entity disambiguation (Zheng et al., 2012) and so on. Since most knowledge graphs suffer from incompleteness, predicting missing links between entities has been a fundamental problem. This problem is named as link prediction or knowledge graph completion. Knowledge graph embedding methods, which embed all entities and relations into a low dimensional space, have been proposed for this problem. Distance based embedding methods from TransE (Bordes et al., 2013) to the recent state-of-the-art RotatE (Sun et al., 2019) have shown substantial improvements on knowledge graph completion task. Two major problems have been widely studied. The first one refers to handling of 1-to-N, N-to-1, and N-to-N complex relations (Bordes et al., 2013;Lin et al., 2015). In case of the 1-to-N relations, given triples like (StevenSpielberg, DirectorOf , ?), distance based models should make all the corresponding entities about film name like Jaws and JurassicP ark have closer distance to entity StevenSpielberg after transformation via relation DirectorOf . The difficulty is that all these entities should have different representations. Same issue happens in cases of N-to-N and N-to-1 relations. The latter is learning and inferring relation patterns according to observed triples, as the success of knowledge graph completion heavily relies on this ability (Bordes et al., 2013;Sun et al., 2019). There are various types of relation patterns: symmetry (e.g., IsSimilarT o), antisymmetry (e.g., F atherOf ), inverse (e.g., P eopleBornHere and P laceOf Birth), composition (e.g., my mother's father is my grandpa) and so on. Previous methods solve these two problems separately. TransH (Wang et al., 2014), TransR (Lin et al., 2015), TransD all focus on ways to solve complex relations. However, these methods can only encode symmetry/antisymmetry relations. The recent state-ofthe-art RotatE shows promising results to encode symmetry/antisymmetry, inverse and composition relations. However, complex relations remain challenging to predict. Here we present PairRE, an embedding method that is capable of encoding complex relations and multiple relation patterns simultaneously. The proposed model uses two vectors for relation representation. These vectors project the corresponding head and tail entities to Euclidean space, where the distance between the projected vectors is minimized. This provides three important benefits: • The paired relation representations enable an adaptive adjustment of the margin in loss function to fit for different complex relations; • Semantic connection among relation vectors can be well captured, which enables the model to encode three important relation patterns, symmetry/antisymmetry, inverse and composition; • Adding simple constraints on relation representations, PairRE can encode subrelation further. Besides, PairRE is a highly efficient model, which contributes to large scale datasets. We evaluate PairRE on six standard knowledge graph benchmarks. The experiment results show PairRE can achieve either state-of-the-art or highly competitive performance. Further analysis also proves that PairRE can better handle complex relations and encode symmetry/antisymmetry, inverse, composition and subrelation relations. Background and Notation Given a knowledge graph that is represented as a list of fact triples, knowledge graph embedding methods define scoring function to measure the plausibility of these triples. We denote a triple by (h, r, t), where h represents head entity, r represents relation and t represents tail entity. The column vectors of entities and relations are represented by bold lower case letters, which belong to set E and R respectively. We denote the set of all triples that are true in a world as T . f r (h, t) represents the scoring function. We take the definition of complex relations from (Wang et al., 2014). For each relation r, we compute average number of tails per head (tphr) and average number of heads per tail (hptr). If tphr < 1.5 and hptr < 1.5, r is treated as 1-to-1; if tphr > 1.5 and hptr > 1.5, r is treated as a N-to-N; if tphr > 1.5 and hptr < 1.5, r is treated as 1-to-N. We focus on four important relation patterns, which includes: (1) Symmetry/antisymmetry. A relation r is symmetric if ∀e 1 , e 2 ∈ E, (e 1 , r, e 2 ) ∈ T ⇐⇒ (e 2 , r, e 1 ) ∈ T and is antisymmetric if (e 1 , r, e 2 ) ∈ T ⇒ (e 2 , r, e 1 ) / ∈ T ; (2) Inverse. If ∀e 1 , e 2 ∈ E, (e 1 , r 1 , e 2 ) ∈ T ⇐⇒ (e 2 , r 2 , e 1 ) ∈ T , then r 1 and r 2 are inverse relations; (3) Composition. If ∀e 1 , e 2 , e 3 ∈ E, (e 1 , r 1 , e 2 ) ∈ T ∧ (e 2 , r 2 , e 3 ) ∈ T ⇒ (e 1 , r 3 , e 3 ) ∈ T , then r 3 can be seen as the composition of r 1 and r 2 ; (4) Subrelation (Qu and Tang, 2019). If ∀e 1 , e 2 ∈ E, (e 1 , r 1 , e 2 ) ∈ T ⇒ (e 1 , r 2 , e 2 ) ∈ T , then r 2 can be seen as a subrelation of r 1 . Related Work Distance based models. Distance based models measure plausibility of fact triples as distance between entities. TransE interprets relation as a translation vector r so that entities can be connected, i.e., h + r ≈ t. TransE is efficient, though cannot model symmetry relations and have difficulty in modeling complex relations. Several models are proposed for improving TransE to deal with complex relations, including TransH, TransR, TransD, TranSparse (Ji et al., 2016) and so on. All these methods project the entities to relation specific hyperplanes or spaces first, then translate projected entities with relation vectors. By projecting entities to different spaces or hyperplanes, the ability to handle complex relations is improved. However, with the added projection parameters, these models are unable to encode inverse and composition relations. The recent state-of-the-art, RotatE, which can encode symmetry/antisymmetry, inverse and composition relation patterns, utilizes rotation based translational method in a complex space. Although expressiveness for different relation patterns, complex relations remain challenging. GC-OTE (Tang et al., 2020) proposes to improve complex relation modeling ability of RotatE by introducing graph context to entity embedding. However, the calculation of graph contexts for head and tail entities is time consuming, which is inefficient for large scale knowledge graphs, e.g. ogbl-wikikg (Hu et al., 2020). Another related work is SE (Bordes et al., 2011), which utilizes two separate relation matrices to project head and tail entities. As pointed out by (Sun et al., 2019), this model is not able to encode symmetry/antisymmetry, inverse and composition relations. Table 1 shows comparison between our method and some representative distance based methods. As the table shows, our model is the most expressive one, with the ability to handle complex relations and encode four key relation patterns. Semantic matching models. Semantic matching models exploit similarity-based scoring functions, which can be divided into bilinear models and neural network based models. As the models have been developed, such as RESCAL (Nickel et al., 2011), DistMult (Yang et al., 2014), HolE (Nickel et al., 2016), ComplEx (Trouillon et al., 2016) and QuatE (Zhang et al., 2019), the key relation encoding abilities are enriched. However, all these models have the flaw in encoding composition relations (Sun et al., 2019). RESCAL, ComplEx and SimplE (Kazemi and Poole, 2018) are all proved to be fully expressive when embedding dimensions fulfill some requirements Trouillon et al., 2016;Kazemi and Poole, 2018). The fully expressiveness means these models can express all the ground truth which exists in the data, including complex relations. However, these requirements are hardly fulfilled in practical use. It is proved by ) that, to achieve complete expressiveness, the embedding dimension should be greater than N /32, where N is the number of entities in dataset. Neural networks based methods, e.g., convolution neural networks (Dettmers et al., 2018), graph convolutional networks (Schlichtkrull et al., 2018) show promising performances. However, they are difficult to analyze as they work as a black box. Encoding Subrelation. Existing methods encode subrelation by utilizing first order logic rules. One way is to augment knowledge graphs with grounding of rules, including subrelation rules Qu and Tang, 2019). The other way is adding constraints on entity and relation representations, e.g., ComplEx-NNE-AER and SimplE + . The second way enriches the models' expressiveness with relatively low cost. In this paper, we show that PairRE can encode subrelation with constraints on relation representations while keeping the ability to encode symmetry/antisymmetry, inverse and composition relations. Methodology To overcome the problem of modeling 1-to-N/Nto-1/N-to-N complex relations and enrich the capabilities for different relation patterns, we propose a model with paired vectors for each relation. Given a training triple (h, r, t), our model learns vector embeddings of entities and relation in real space. Specially, PairRE takes relation embedding as paired vectors, which is represented as [r H , r T ]. r H and r T project head entity h and tail entity t to Euclidean space respectively. The projection operation is the Hadamard product 1 between these two vectors. PairRE then computes distance of the two projected vectors as plausibility of the triple . We want that h • r H ≈ t • r T when (h, r, t) holds, while h • r H should be far away from t • r T otherwise. In this paper, we take the L 1 -norm to measure the distance. In order to remove scaling freedoms, we also add constraint on embeddings similar to previous distance based models (Bordes et al., 2013;Wang et al., 2014;Lin et al., 2015). And the constraint is only added on entity embeddings. We want relation embeddings to capture semantic connection among relation vectors (e.g., P eopleBornHere and P laceOf Birth) and complex characteristic (e.g., 1-N) easily and sufficiently. For entity embedding, the L 2 -norm is set to be 1. The scoring function is defined as follows: f r (h, t) = −\|\|h • r H − t • r T \|\|,(1) where h, r H , r T , t ∈ R d and \|\|h\|\| 2 = \|\|t\|\| 2 = 1. The model parameters are, all the entities' embed- dings, {e j } E j=1 and all the relations' embeddings, {r j } R j=1 . Illustration of the proposed PairRE is shown in Figure 1. Compared to TransE/RotatE, PairRE enables an entity to have distributed representations when involved in different relations. We also find the paired relation vectors enable an adaptive adjustment of the margin in loss function, which alleviates the modeling problem for complex relations. Let's take a 1-to-N relation as an example. We set the embedding dimension to one and remove the constraint on entity embeddings for better illustration. Given triples (h, r, ?), where the correct tail entities belong to set S = {t 1 , t 2 , ..., t N }, PairRE predicts tail entities by letting \|\|h • r H − t i • r T \|\| < γ, where γ is a fixed margin for distance based embedding models and t i ∈ S. The value of t i should stay in the following range: t i ∈      ((h • r H − γ)/r T , (h • r H + γ)/r T ), if r T > 0, ((h • r H + γ)/r T , (h • r H − γ)/r T ), if r T < 0, (−∞, +∞), otherwise. The above analysis shows PairRE can adjust the value of r T to fit the entities in S. The larger the size of S, the smaller the absolute value r T . While models like TransE or RotatE have a fixed margin for all complex relation types. When the size of S is large enough, these models will be difficult to fit the data. For N-to-1 relations, PairRE can also adjust the value of r H adaptively to fit the data. Meanwhile, not adding a relation specific translational vector enables the model to encode several key relation patterns. We show these capabilities below. Proposition 1. PairRE can encode symmetry/antisymmetry relation pattern. Proof. If (e 1 , r 1 , e 2 ) ∈ T and (e 2 , r 1 , e 1 ) ∈ T , we have e 1 • r H 1 = e 2 • r T 1 ∧ e 2 • r H 1 = e 1 • r T 1 ⇒ r H 1 2 = r T 1 2(2) if (e 1 , r 1 , e 2 ) ∈ T and (e 2 , r 1 , e 1 ) / ∈ T , we have e 1 • r H 1 = e 2 • r T 1 ∧ e 2 • r H 1 = e 1 • r T 1 ⇒ r H 1 2 = r T 1 2(3) Proposition 2. PairRE can encode inverse relation pattern. Proof. If (e 1 , r 1 , e 2 ) ∈ T and (e 2 , r 2 , e 1 ) ∈ T , we have e 1 • r H 1 = e 2 • r T 1 ∧ e 2 • r H 2 = e 1 • r T 2 ⇒ r H 1 • r H 2 = r T 1 • r T 2(4) Proposition 3. PairRE can encode composition relation pattern. Proof. If (e 1 , r 1 , e 2 ) ∈ T , (e 2 , r 2 , e 3 ) ∈ T and (e 1 , r 3 , e 3 ) ∈ T , we have e 1 • r H 1 = e 2 • r T 1 ∧ e 2 • r H 2 = e 3 • r T 2 ∧ e 1 • r H 3 = e 3 • r T 3 ⇒ r T 1 • r T 2 • r H 3 = r H 1 • r H 2 • r T 3(5) Moreover, with some constraint, PairRE can also encode subrelations. For a subrelation pair, ∀h, t ∈ E : (h, r 1 , t) → (h, r 2 , t), it suggests triple (h, r 2 , t) should be always more plausible than triple (h, r 1 , t). In order to encode this pattern, PairRE should have the capability to enforce f r 2 (h, r 2 , t) ≥ f r 1 (h, r 1 , t). Proposition 4. PairRE can encode subrelation relation pattern using inequality constraint. Proof. Assume a subrelation pair r 1 and r 2 that ∀h, t ∈ E: (h, r 1 , t)→(h, r 2 , t). We impose the following constraints: r H 2,i r H 1,i = r T 2,i r T 1,i = α i , \|α i \| ≤ 1,(6) where α ∈ R d . Then we can get f r 2 (h, t) − f r 1 (h, t) = \|\|h • r H 1 − t • r T 1 \|\| − \|\|h • r H 2 − t • r T 2 \|\| = \|\|h • r H 1 − t • r T 1 \|\| − \|\|α • (h • r H 1 − t • r T 1 )\|\| ≥ 0. (7) When the constraints are satisfied, PairRE forces triple (h, r 2 , t) to be more plausible than triple (h, r 1 , t). Optimization. To optimize the model, we utilize the self-adversarial negative sampling loss (Sun et al., 2019) as objective for training: L = − log σ(γ − f r (h, t)) − n i=1 p(h i , r, t i ) log σ(f r (h i , t i ) − γ),(8) where γ is a fixed margin and σ is the sigmoid function. (h i , r, t i ) is the i th negative triple and p(h i , r, t i ) represents the weight of this negative sample. p(h i , r, t i ) is defined as follows: p((h i , r, t i )\|(h, r, t)) = exp f r (h i , t i ) j exp f r (h j , t j ) . (9) 5 Experimental results Experimental setup We evaluate the proposed method on link prediction tasks. At first, we validate the ability to deal with complex relations and symmetry/antisymmetry, inverse and composition relations on four benchmarks. Then we validate our model on two subrelation specific benchmarks. Statistics of these benchmarks are shown in Table 2. ogbl-wikikg2 2 (Hu et al., 2020) is extracted from Wikidata knowledge base (Vrandečić and Krötzsch, 2014). One of the main challenges for this dataset is complex relations. ogbl-biokg FB15k 13k 15k 483k 50k 59k FB15k-237 237 15k 272k 18k 20k DB100k 470 100k 598k 50k 50k Sports 4 1039 1312 -307 Table 2: Number of entities, relations, and observed triples in each split for the six benchmarks. (Hu et al., 2020) contains data from a large number of biomedical data repositories. One of the main challenges for this dataset is symmetry relations. Evaluation protocol. Following the state-ofthe-art methods, we measure the quality of the ranking of each test triple among all possible head entity and tail entity substitutions: (h , r , t) and (h, r, t ), ∀h , ∀t ∈ E. Three evaluation metrics, including Mean Rank(MR), Mean Reciprocal Rank (MRR) and Hit ratio with cut-off values n = 1, 3, 10, are utilized. MR measures the average rank of all correct entities. MRR is the average inverse rank for correct entities with higher value representing better performance. Hit@n measures the percentage of correct entities in the top n predictions. The rankings of triples are computed after removing all the other observed triples that appear in either training, validation or test set. For experiments on ogbl-wikikg2 and ogbl-biokg, we follow the evaluation protocol of these two benchmarks (Hu et al., 2020). Implementation. We utilize the official implementations of benchmarks ogbl-wikikg2 and ogblbiokg (Hu et al., 2020) for the corresponding experiments 3 . Only the hypeparameter γ and embedding dimension are tuned. The other settings are kept the same with baselines. For the rest experiments, we implement our models based on the implementation of RotatE (Sun et al., 2019). All hypeparam- (Nickel et al., 2016); Results of [3] are taken from (Kadlec et al., 2017). Other results are taken from the corresponding papers. GC-OTE adds graph context to OTE (Tang et al., 2020). eters except γ and embedding dimension are kept the same with RotatE. Subrelation (h, CoachesTeam, t) → (h, PersonBelongsToOrganization, t) (h, AthleteLedSportsTeam, t) → (h, AtheletePlaysForTeam, t) Main results Comparisons for ogbl-wikikg2 and ogbl-biokg are shown in Table 3. On these two large scale datasets, PairRE achieves state-of-the-art performances. For ogbl-wikikg2 dataset, PairRE performs best on both limited embedding dimension and increased embedding dimension. With the same number of parameters to ComplEx (dimension 100), PairRE improves Test MRR close to 10%. With increased dimension, all models are able to achieve higher MRR on validation and test sets. Due to the limitation of hardware, we only increase embedding dimension to 200 for PairRE. PairRE also outperforms all baselines and improves Test MRR 8.7%. Based on performances of baselines, the performance of PairRE may be improved further if embedding dimension is increased to 500. Under the same experiment setting and the same number of parameters, PairRE also outperforms all baselines on ogbl-biokg dataset. It improves Test MRR by 0.69%, which proves the superior ability to encode symmetry relations. Comparisons for FB15k and FB15k-237 datasets are shown in Table 4. Since our model shares the same hyper-parameter settings and implementation with RotatE, comparing with this state-of-the-art model is fair to show the advantage and disadvantage of the proposed model. Besides, the comparisons also include several leading methods, such as TransE (Bordes et al., 2013), DistMult (Yang et al., 2014), HolE (Nickel et al., 2016), ConvE (Dettmers et al., 2018), ComplEx (Trouillon et al., 2016), SimplE (Kazemi andPoole, 2018), SeeK (Xu et al., 2020) and OTE (Tang et al., 2020). Compared with RotatE, PairRE shows clear improvements on FB15k and FB15k-237 for all evaluation metrics. For MRR metric, the improvements are 1.4% and 1.3% respectively. Compared with the other leading methods, PairRE also shows highly competitive performances. All these comparisons prove the effectiveness of PairRE to encode inverse and composition relations. Further experiments on subrelation We further compare our method with two of the leading methods ComplEx-NNE-AER and SimplE + , which focus on encoding subrelation. These two methods add subrelation rules to semantic matching models. We utilize these rules as constraints on relation representations for PairRE. Two ways are validated. We first test the performance of weight tying for subrelation rules on Sports dataset. The rules (r 1 −→r 2 ) are added as follows: r H 2 = r H 1 • cosine(θ), r T 2 = r T 1 • cosine(θ),(10) where θ ∈ R d . The added rules are shown in Table 5. The experiments results in Table 6 show effectiveness of the proposed method. Weight tying on relation representation is a way to incorporate hard rules. The soft rules can also be incorporated into PairRE by approximate entailment constraints on relation representations. In this section, we add the same rules from ComplEx-NNE-AER, which includes subrelation and inverse rules. We denote by r 1 λ −→ r 2 the approximate entailment between relations r 1 and r 2 , with confidence level λ. The objective for training is then changed to: L rule = L + µ τ subrelation λ1 T (r H 1 • r T 2 − r T 1 • r H 2 ) 2 + µ τ inverse λ1 T (r H 1 • r H 2 − r T 1 • r T 2 ) 2 ,(11) where L is calculated from Equation 8, µ is loss weight for added constraints, τ subrelation and τ inverse are the sets of subrelation rules and inverse rules respectively. Following (Ding et al., 2018), we take the corresponding two relations from subrelation rules as equivalence. Because τ subrelation contains both rule r 1 →r 2 and rule r 2 →r 1 . We validate our method on DB100k dataset. The results are shown in Table 7. We can see PairRE outperforms the recent state-of-the-art SeeK and ComplEx based models with large margins on all evaluation metrics. With added constraints, the performance of PairRE is improved further. The improvements for the added rules are 0.7%, 1.2% for MRR and Hit@1 metrics respectively. Model analysis Analysis on complex relations We analyze the performances of PairRE for complex relations. The results of PairRE on different relation categories on FB15k and ogbl-wikikg2 are summarized into Table 8. We can see PairRE performs quite well on N-to-N and N-to-1 relations. It has a significant lead over baselines. We also notice that performance of 1-to-N relations on ogbl-wikikg2 dataset is not as strong as the other relation categories. One of the reasons is that only 2.2% of test triples belong to the 1-to-N relation category. In order to further test the performance of paired relation vectors, we change the relation vector in RotatE to paired vectors. In the modified Ro-tatE model, both head and tail entities are rotated with different angles based on the paired Figure 2: Performance comparison between RotatE and RotatE+PairRelation on ogbl-wikikg2 dataset. -FB15k(Hits@10) ogbl-wikikg2(Hits@10) Model 1-to-1 1-to-N N-to-1 N-to-N 1-to-1 1-to-N N-to-1 N-to-N KGE2E KL (Sun et al., 2019). The embedding dimensions for models on ogbl-wikikg2 are same to the experiments in Table 3, which is 100 for real space models and 50 for complex value based models. Figure 3: Histograms of relation embeddings for different relation patterns. r 1 is relation spouse. r 2 is relation /broadcast/tv station/owner. r 3 is relation /broadcast/tv station owner/tv stations. r 4 is relation /location/administrative division/capital/location/administrative divisioncapital relationship/capital. r 5 is relation /location/hud county place/place. r 6 is relation base/areas/schema/administrative area/capital. relation vectors. This model can also be seen as complex value based PairRE. We name this model as RotatE+PairRelation. The experiment results are shown in Figure 2. With the same embedding dimension (50 in the experiments), Ro-tatE+PairRelation improves performance of RotatE with 20.8%, 27.5%, 14.4% and 39.1% on 1-to-1, 1-to-N, N-to-1 and N-to-N relation categories respectively. These significant improvements prove the superior ability of paired relation vectors to handle complex relations. (a) r1 (b) r H 1 2 − r T 1 2 (c) r2 (d) r H 2 2 − r T 2 2 (e) r3 (f) r H 2 • r H 3 − r T 2 • r T 3 (g) r4 (h) r5 (i) r6 (j) r H 4 • r H 5 • r T 6 − r T 4 • r T 5 • r H 6 Analysis on relation patterns To further verify the learned relation patterns, we visualize some examples. Histograms of the learned relation embeddings are shown in Figure 3 . Symmetry/AntiSymmetry. Figure 3a shows a symmetry relation spouse from DB100k. The embedding dimension is 500. For PairRE, symmetry relation pattern can be encoded when embedding r satisfies r H 2 = r T 2 . Figure 3b shows most of the paired elements in r H and r T have the same absolute value. Figure 3c shows a antisymmetry relation tv station owner, where most of the paired elements do not have the same absolute value as shown in Figure 3d. Inverse. Figure 3c and Figure 3e show an example of inverse relations from FB15k. As the histogram in Figure 3f shows these two inverse relations tv station owner (r 2 ) and tv station owner tv stations (r 3 ) close to satisfy r H 3 • r H 2 = r T 3 • r T 2 . Composition. Figures 3g, 3h, 3i show an example of composition relation pattern from FB15k, where the third relation r 6 can be seen as the composition of the first relation r 4 and the second relation r 5 . As Figure 3j shows these three relations close to satisfy r H 4 • r H 5 • r T 6 − r T 4 • r T 5 • r H 6 . Conclusion To better handle complex relations and tackle more relation patterns, we proposed PairRE, which represents each relation with paired vectors. With a slight increase in complexity, PairRE can solve the aforementioned two problems efficiently. Beyond the symmetry/antisymmetry, inverse and composition relations, PairRE can further encode subrelation with simple constraint on relation representations. On large scale benchmark ogbl-wikikg2 an ogbl-biokg, PairRE outperforms all the state-of-theart baselines. Experiments on other well designed benchmarks also demonstrate the effectiveness of the focused key abilities. Figure 1 : 1Illustration of TransE, RotatE and PairRE when the entities stay in a plane. For PairRE, all entities are on the unit circle. The relation vectors project entities to different locations. FB15k (Bordes et al., 2013) contains triples from Freebase. The main relation patterns are inverse and symmetry/antisymmetry. FB15k-237 (Toutanova and Chen, 2015) is a subset of FB15k, with inverse relations removed. The main relation patterns are antisymmetry and composition. DB100k (Ding et al., 2018) is a subset of DBpedia.The main relation patterns are composition, inverse and subrelation. Sports(Wang et al., 2015) is a subset of NELL(Mitchell et al., 2018). The main relation patterns are antisymmetry and subrelation. Table 3 : 3Link prediction results on ogbl-wikikg2 and ogbl-biokg. Best results are in bold. ±0.4979 ±0.00077 ±0.00071 ±0.0011 ±0.0012 ±0.9949 ±0.00066 ±0.00093 ±0.00079 ±0.00097All the results except Table 4 : 4Link prediction results on FB15k and FB15k-237. Results of [ †] are taken from Table 5 : 5The added subrelation rules for Sports dataset.Model MRR hit@1 SimplE 0.230 0.184 SimplE + 0.404 0.349 PairRE 0.468 ± 0.003 0.416 ± 0.005 PairRE+Rule 0.475 ± 0.003 0.432 ± 0.004 Table 6 : 6Link prediction results on Sports dataset. Other results are taken from(Fatemi et al., 2019). 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[ "Dual-band coupling between nanoscale polaritons and vibrational and electronic excitations in molecules Main text", "Dual-band coupling between nanoscale polaritons and vibrational and electronic excitations in molecules Main text" ]	[ "A Bylinkin \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n\nDonostia International Physics Center (DIPC)\n20018Donostia-San SebastiánSpain\n", "F Calavalle \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n", "M Barra-Burillo \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n", "R V Kirtaev \nDonostia International Physics Center (DIPC)\n20018Donostia-San SebastiánSpain\n", "E Nikulina \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n", "E B Modin \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n", "E Janzen \nDepartment of Chemical Engineering\nTim Taylor\nKansas State University Manhattan\n66506KSUSA\n", "J H Edgar \nDepartment of Chemical Engineering\nTim Taylor\nKansas State University Manhattan\n66506KSUSA\n", "F Casanova \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n\nBasque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain\n", "L E Hueso \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n\nBasque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain\n", "V S Volkov \nDIP\nXPANCEO\n607-0406Bayan Business Center, DubaiUAE\n", "P Vavassori \nCIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain\n\nBasque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain\n", "I Aharonovich \nSchool of Mathematical and Physical Sciences\nUniversity of Technology Sydney\n2007UltimoNew South WalesAustralia\n\nFaculty of Science\nARC Centre of Excellence for Transformative Meta-Optical Systems\nUniversity of Technology Sydney\n2007UltimoNew South WalesAustralia\n", "P Alonso-Gonzalez \nDepartamento de Fisica\nUniversidad de Oviedo\n33006OviedoSpain\n\nNanomaterials and Nanotechnology Research Center (CINN)\n33940 El EntegoSpain\n", "R Hillenbrand [email protected] \nCIC nanoGUNE BRTA and EHU/UPV\n20018Donostia-San SebastiánSpain\n\nBasque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain\n", "A Y Nikitin \nDonostia International Physics Center (DIPC)\n20018Donostia-San SebastiánSpain\n\nBasque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain\n" ]	[ "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "Donostia International Physics Center (DIPC)\n20018Donostia-San SebastiánSpain", "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "Donostia International Physics Center (DIPC)\n20018Donostia-San SebastiánSpain", "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "Department of Chemical Engineering\nTim Taylor\nKansas State University Manhattan\n66506KSUSA", "Department of Chemical Engineering\nTim Taylor\nKansas State University Manhattan\n66506KSUSA", "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "Basque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain", "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "Basque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain", "DIP\nXPANCEO\n607-0406Bayan Business Center, DubaiUAE", "CIC nanoGUNE BRTA\n20018Donostia -San SebastianSpain", "Basque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain", "School of Mathematical and Physical Sciences\nUniversity of Technology Sydney\n2007UltimoNew South WalesAustralia", "Faculty of Science\nARC Centre of Excellence for Transformative Meta-Optical Systems\nUniversity of Technology Sydney\n2007UltimoNew South WalesAustralia", "Departamento de Fisica\nUniversidad de Oviedo\n33006OviedoSpain", "Nanomaterials and Nanotechnology Research Center (CINN)\n33940 El EntegoSpain", "CIC nanoGUNE BRTA and EHU/UPV\n20018Donostia-San SebastiánSpain", "Basque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain", "Donostia International Physics Center (DIPC)\n20018Donostia-San SebastiánSpain", "Basque Foundation for Science\nIKERBASQUE\n48009BilbaoSpain" ]	[]	Strong coupling (SC) between light and matter excitations such as excitons and molecular vibrations bear intriguing potential for controlling chemical reactivity, conductivity or photoluminescence. So far, SC has been typically achieved either between mid-infrared (mid-IR) light and molecular vibrations or between visible light and excitons. Achieving SC simultaneously in both frequency bands may open unexplored pathways for manipulating material properties. Here, we introduce a polaritonic nanoresonator (formed by h-BN layers placed on Al ribbons) hosting surface plasmon polaritons (SPPs) at visible frequencies and phonon polaritons (PhPs) at mid-IR frequencies, which simultaneously couple to excitons and atomic vibration in an adjacent molecular layer (CoPc). Employing near-field optical nanoscopy, we first demonstrate the co-localization of strongly confined near-fields at both visible and mid-IR frequencies. After covering the nanoresonator structure with a layer of CoPc molecules, we observe clear mode splittings in both frequency ranges by far-field transmission spectroscopy, unambiguously revealing simultaneous SPP-exciton and PhP-vibron coupling. Dual-band SC may be exploited for manipulating the coupling between excitons and molecular vibrations in future optoelectronics, nanophotonics, and quantum information applications.	10.1021/acs.nanolett.3c00768	[ "https://export.arxiv.org/pdf/2302.14814v1.pdf" ]	257,232,458	2302.14814	72638611847fb9bf0ba45beb3b6a8061c7cb10a7	Dual-band coupling between nanoscale polaritons and vibrational and electronic excitations in molecules Main text A Bylinkin CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain Donostia International Physics Center (DIPC) 20018Donostia-San SebastiánSpain F Calavalle CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain M Barra-Burillo CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain R V Kirtaev Donostia International Physics Center (DIPC) 20018Donostia-San SebastiánSpain E Nikulina CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain E B Modin CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain E Janzen Department of Chemical Engineering Tim Taylor Kansas State University Manhattan 66506KSUSA J H Edgar Department of Chemical Engineering Tim Taylor Kansas State University Manhattan 66506KSUSA F Casanova CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain Basque Foundation for Science IKERBASQUE 48009BilbaoSpain L E Hueso CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain Basque Foundation for Science IKERBASQUE 48009BilbaoSpain V S Volkov DIP XPANCEO 607-0406Bayan Business Center, DubaiUAE P Vavassori CIC nanoGUNE BRTA 20018Donostia -San SebastianSpain Basque Foundation for Science IKERBASQUE 48009BilbaoSpain I Aharonovich School of Mathematical and Physical Sciences University of Technology Sydney 2007UltimoNew South WalesAustralia Faculty of Science ARC Centre of Excellence for Transformative Meta-Optical Systems University of Technology Sydney 2007UltimoNew South WalesAustralia P Alonso-Gonzalez Departamento de Fisica Universidad de Oviedo 33006OviedoSpain Nanomaterials and Nanotechnology Research Center (CINN) 33940 El EntegoSpain R Hillenbrand [email protected] CIC nanoGUNE BRTA and EHU/UPV 20018Donostia-San SebastiánSpain Basque Foundation for Science IKERBASQUE 48009BilbaoSpain A Y Nikitin Donostia International Physics Center (DIPC) 20018Donostia-San SebastiánSpain Basque Foundation for Science IKERBASQUE 48009BilbaoSpain Dual-band coupling between nanoscale polaritons and vibrational and electronic excitations in molecules Main text 2plasmon polaritonphonon polaritonexcitonstrong couplingvan der Waals crystal Strong coupling (SC) between light and matter excitations such as excitons and molecular vibrations bear intriguing potential for controlling chemical reactivity, conductivity or photoluminescence. So far, SC has been typically achieved either between mid-infrared (mid-IR) light and molecular vibrations or between visible light and excitons. Achieving SC simultaneously in both frequency bands may open unexplored pathways for manipulating material properties. Here, we introduce a polaritonic nanoresonator (formed by h-BN layers placed on Al ribbons) hosting surface plasmon polaritons (SPPs) at visible frequencies and phonon polaritons (PhPs) at mid-IR frequencies, which simultaneously couple to excitons and atomic vibration in an adjacent molecular layer (CoPc). Employing near-field optical nanoscopy, we first demonstrate the co-localization of strongly confined near-fields at both visible and mid-IR frequencies. After covering the nanoresonator structure with a layer of CoPc molecules, we observe clear mode splittings in both frequency ranges by far-field transmission spectroscopy, unambiguously revealing simultaneous SPP-exciton and PhP-vibron coupling. Dual-band SC may be exploited for manipulating the coupling between excitons and molecular vibrations in future optoelectronics, nanophotonics, and quantum information applications. Main text Strong coupling (SC) -light-matter interaction leading to the formation of new hybrid modes whose separation of energy levels is larger than the sum of their average linewidths -provides intriguing possibilities for controlling various material properties, such as e.g. carrier transport in organic semiconductors 1,2 , magnetotransport in two-dimensional electron gases 3 , or chemical reactivity changes in microcavities [4][5][6][7] . Conventional optical resonators, such as Fabry-Perot microcavities, are typically used to couple matter excitations with light. An approach to explore SC at nanoscale is to use polariton resonators, enabling SC between excitons and surface plasmon polaritons (SPPs) at visible frequencies 16,1 , or between molecular vibrations and phonon polaritons (PhPs) at mid-infrared (mid-IR) 10,11 to THz frequencies 12,13 . Interestingly, the interaction between excitons and molecular vibrations plays an important role in the optical properties of organic semiconductors [14][15][16] , particularly in the singlet fission. Achieving dual-band SC, i.e. SC between polaritons and both excitons and molecular vibrations simultaneously could provide new opportunities for controlling the state of matter at the nanoscale. However, excitonic and vibrational resonances emerge at significantly different energy domains separated by orders of magnitude, thus challenging the creation of strongly confined electromagnetic fields (hot spots) at visible and infrared frequencies -an essential ingredient for achieving SC at the nanoscale -at the same spatial position and with a similar size. Metallic antennas have already been used to enhance the light-matter interaction in both visible and mid-IR frequency bands 17 , in particular, to combine surface-enhanced Raman and infrared spectroscopy 18,19 . However, although metallic antennas work relatively well in the visible range, it is desirable to find alternatives for achieving SC in the mid-IR frequency range due to their low quality factor at these frequencies. Van der Waals (vdW) materials have recently emerged as a promising platform for exploring enhanced light-matter interactions at the nanoscale, as they support a large family of ultra-confined polaritons 24,2 from visible to THz frequencies. In addition, these materials can be engineered with nanoscale precision, allowing precise control of light-matter interactions at subwavelength scales 22 . Furthermore, the combination of metal slabs or antennas with vdW materials can lead to hybrid heterostructures that support electromagnetic modes at different frequency ranges 23 and potentially allow the creation of co-located visible and infrared hotspots to achieve dual-band SC. Here we demonstrate by numerical simulations that SC can be achieved simultaneously between SPPs and electronic transitions at visible frequencies and between PhPs and molecular vibration at mid-IR frequencies. For an experimental study, we employed nanoresonators based on a heterostructures composed of metal (Al) ribbons and monoisotopic hexagonal boron nitride (h-BN) flakes, which support both SPPs at visible frequencies and PhPs at infrared frequencies. PhP resonances offer the advantage of being stronger and narrower as compared to SPP resonators 2 , thus facilitating the achievement of nanoscale vibrational SC 10,11 . We verify the two hotspots and their co-localization by scattering-type scanning nearfield optical microscopy (s-SNOM). By placing molecules on the nanoresonators within the corresponding hotspots and performing visible and mid-IR far-field spectroscopy, we provide direct experimental evidence for large mode splitting in both frequency ranges. SC can be directly observed by visualizing the propagation of PhPs in a cavity-free (unpatterned) slab in contact with organic molecules 25,26 . To explore the possibility of observing dual-band SC, we thus first perform a theoretical study considering a cavity-free heterostructure formed by a 50 nm-thick layer of CoPc between a 50 nm-thick Al layer and a 75 nm-thick h-BN slab (Fig 1a). CoPc molecules support both a vibrational resonance at mid-IR frequencies (1525 cm -1 ) and two excitonic resonances at visible frequencies (14400 cm -1 and 16200 cm -1 ) (Fig 1c,d). By placing the molecules between the two polaritonic materials (h-BN and Al) we expect to achieve highly confined mid-IR and visible electromagnetic fields inside the molecular layer (dashed lines in the left panel of Fig. 1a), thus guaranteeing a strong overlap of both fields and the molecules at the same spatial location. Since the plasma frequency of Al is p = 10.83 eV  8.710 4 cm -1 27,28 , the Al slab supports SPPs below /√2 = 5.4 eV  1.410 4 cm -1 , i.e. in the whole visible range. The field of these SPPs is vertically confined on a length scale of 1/kSPP,z ~ 100 nm in the frequency range corresponding to the exciton resonances of CoPc. On the other hand, h-BN exhibits two mid-IR Reststrahlen bands -defined by the transverse (TO) and longitudinal (LO) optical phonons (785 to 845 cm -1 and 1394 to 1650 cm -1 ) -where PhPs are supported 29 . Interestingly, since the out-of-plane and in-plane dielectric permittivities of h-BN differ in sign in both RBs, the h-BN slab supports a set of PhPs modes that are typically denoted M0, M1, etc. (so-called hyperbolic polaritons) 30 . The fundamental M0 mode exhibits the longest wavelength and propagation length and thus is typically the dominating mode in h-BN nanoresonators 33,3 . In the chosen structure, the momentum of the M0 mode at the molecular-vibrational resonance of CoPc is comparable to that of SPPs in the Al layer at the exciton resonance of CoPc. As a result, the vertical field confinement of the fundamental M0-PhP mode outside of the slab (1/kM0,z ~ 80 nm) is comparable to that of the SPPs, indicating that a molecular layer with a thickness of a few dozens of nm should be sufficient to achieve SC in both frequency bands. To explore the concept of simultaneous dual-band SC, we calculated the dispersion of the polaritons in the cavity-free heterostructure, Fig 1a. To that end, we plot the imaginary part of the Fresnel reflection coefficient, Im[rp(q, )], at both visible and mid-IR frequencies (colour plots in Fig. 1e and 1f, respectively). The dashed red curves represent the dispersions of the quasi-normal polaritonic modes calculated from the poles of rp(q,  c) in the plane of the complex-valued frequency c=-iΓ/2, with Γ being the mode linewidth (see Methods). Both colour plots and dispersion curves clearly reveal an anticrossing (mode splitting) in both visible and mid-IR frequency ranges. To characterize the anti-crossing, we determine the mode splitting, Ω, as the minimum vertical distance between the real part of the complex frequencies. The comparison of Ω with half the sum of the linewidths of the upper, Γ+, and lower, Γ-, coupled states determines whether the coupling between polaritons and molecular excitations is weak, Ω < (Γ+ +Γ-)/2, or strong, Ω > (Γ+ +Γ-)/2 (ref 33 ). In the mid-IR frequency range, our calculation yields ΩIR = 6.8 cm -1  8.410 -4 eV and ΓIR+,-= 4 cm -1  510 -4 eV, such that the SC condition is well fulfilled. In the visible range, the three initial excitations (SPP and two excitons) couple, leading to the formation of three hybrid states (red dashed lines in Fig.1e). By extracting the mode splitting as the minimum vertical distance between the real frequencies of the adjacent hybrid states, we obtain Ωvis,1 = 1.610 3 cm -1  0.2 eV (between the low "-" and middle "0" branches) and Ωvis,2 = 3.910 3 cm -1  0.48 eV (between the middle "0" and upper "+" branches). We found that the SC criterion is fulfilled for the split branches "0" and "+", since Ωvis,2 > (Γvis,2,+ +Γvis,2,0 )/2, where Γvis,2,+,0 ⋍ 1.610 3 cm -1  0.2 eV is the linewidth of the middle and upper hybrid states, respectively. On the other hand, the SC criterion is not fulfilled for the split branches "-" and "0". Altogether, our calculations reveal that the combination of different polaritonic materials in a single heterostructure can be used to achieve dual-band SC. To demonstrate experimentally the dual-band SC between polaritons and CoPc molecular excitations, we first have to consider the need to access the large intrinsic momenta of polaritons, which in the case of farfield illumination requires the presence of leaky modes in the heterostructure. The latter can arise in open Fabry-Perot (FP) nanoresonators, which can be fabricated by (i) nanostructuring the polaritonic slab, e.g. in the form of ribbons 1,3 , or (ii) nanostructuring the substrate below the h-BN slab 35 , i.e. by refractive index engineering. Importantly, the latter option allows the use of a pristine h-BN slab, thus preserving its crystal quality. Interestingly, by placing a 75 nm-thick h-BN flake on an Al grating (Fig 2a), we can engineer FP nanoresonators by simultaneously applying concepts (i) and (ii). Indeed, while SPPs propagating across the Al ribbons (along the x-axis) in the visible range reflect directly at the Al edges (concept i), PhPs in the h-BN slab are reflected between the h-BN/air and h-BN/Al boundaries due to a refractive index step at mid-IR frequencies (concept ii). Fig. 2b,c show the normalised measured far-field extinction spectra of the nanoresonators in both frequency ranges. We can clearly recognize asymmetric peaks around = 1510 cm -1 and = 1.610 4 cm -1 , which are identified (see Supplementary information SIII) as Fano-type FP resonances and, partially, as Bragg resonances arising from the overlap between the electromagnetic fields of adjacent cavities. As the inverse width of the Al ribbons (w -1 ) corresponds to an effective momentum of the resonating polaritons, we fabricated a set of structures with different w, in order to cover a wide range of momenta (Supplementary information SII). To corroborate our far-field spectroscopy experiments and to better understand the distribution of the near-field amplitude at the frequencies of the extinction peaks in Fig.2b,c, we performed s-SNOM nanoimaging of the nanoresonator heterostructure (see schematics in Fig. 2g,h) at mid-IR and visible frequencies (see Methods). In these nanoimaging experiments, the nanoresonators were illuminated with s-polarised light, whose electric field is perpendicular to the metal ribbons. The top panels in Fig. 2e,f show the resulting s-SNOM images for a nanoresonator when recording tip-scattered p-polarized light, which yields the z-component of the real part of the electric field at mid-IR (mid-IR = 1510 cm -1 , mid-IR= 6.62 m) and visible (vis = 15798 cm -1 , vis= 0.632 m) frequencies (see Methods and Supplementary Information SIV). Interestingly, despite the large difference in the wavelength of the incident light, we observe two bright areas with opposite polarity above the Al ribbons for both frequencies, revealing the excitation of transverse Fabry-Perot modes. To support these observations, we performed numerical simulations considering a nanoresonator with the experimental parameters and illuminating conditions. As can be clearly seen in the lower panels of Fig. 2e,f, excellent agreement is obtained between the simulated and experimental near-field distributions. In the normalized distributions of the vertical component of the electric fields (Fig. 2g,h), we identify strongly enhanced field amplitudes compared to the incident electric field, Einc, thus confirming the formation of the hot spots. Therefore, we can conclude that the peaks observed in the far-field extinction spectra correspond to transverse Fabry-Perot resonances of PhPs at mid-IR frequencies and SPPs at visible frequencies. More importantly, the s-SNOM images in combination with the simulations confirm that both mid-IR and visible hot spots are spatially co-localized on the metal ribbons, thus potentially allowing for dual-band light-matter coupling involving the same molecules. Figure 3. Experimental demonstration of dual-band coupling between polaritons and vibrational and electronic molecular excitations. a, d, Imaginary part of the permittivity of CoPc molecules (Im(εCoPc)) experimentally extracted in the visible and mid-IR frequency ranges, respectively. b, e, Experimental extinction spectra of the polaritonic nanoresonators loaded with the molecular layer (see schematic) in the visible and infrared frequency ranges, respectively. Green dashed lines indicate the resonance frequencies of the molecular excitations. Black dashed lines are guides to the eyes indicating the resonance features in the extinction spectra corresponding to the coupled polaritonic modes. Thick grey lines in b,e represent fits using a three and two coupled harmonic oscillators model, respectively. c, f, Colour plots showing the simulated extinction of the nanoresonators in the visible and infrared frequency ranges, respectively. Red lines indicate the calculated dispersion assuming complex-valued frequency where w -1 = 2q/(-vis,IR) and vis= -0.12, IR = -0.3. The white and black squares in f represent the dispersion of the quasi-normal modes of the nanoresonators calculated using parameters obtained from the coupled oscillators fit in e. To study experimentally the frequency splitting of the hybridized modes, we evaporated an 80 nm thick layer of CoPc molecules -for practical reasons -on Al/h-BN nanoresonators with different ribbon widths, w (inset of Fig. 3), and measured the far-field extinction spectra in the visible and mid-IR frequency ranges (Fig 3b,e). We clearly observe peaks in the spectra whose positions depend on w, which can be attributed to the transverse polaritonic FP resonances that couple with the molecular excitations. More importantly, we observe an anti-crossing behaviour of the peaks (indicated by dashed black lines in Fig. 3b,e). These anti-crossings reveal coupling between SPPs and molecular excitons in the visible frequency range and coupling between PhPs and molecular vibrations in the mid-IR frequency range. To corroborate the observed coupling between polaritons and molecular excitations, we simulated extinction spectra of nanoresonators with different ribbon widths (from 125 nm to 550 nm, colour plots in Fig. 3c,f) coated with a 80 nm thick CoPc layer. The colour plots in Fig. 3c,f show bright maxima, corresponding to polaritonic resonances, which shift to higher frequencies as w -1 increases. We find good agreement between the trends in the frequency position of the maxima in the colour plot and the eye guides marking the maxima in the experimental extinction spectra (dashed black lines, Fig. 3b, e). In particular, in both the visible and mid-IR frequency ranges, the maxima in the simulated extinction spectra show anti-crossing behaviour around the frequencies of the molecular excitations (horizontal green dashed lines, Fig. 3), supporting our experimental observations. To theoretically quantify the mode splitting, we performed a quasi-normal polaritonic mode analysis in a threelayer CoPc/Al/h-BN continuous heterostructure (see Methods). The calculated dispersions of the quasi-normal modes, (q), are shown on top of the simulated extinction spectra in Fig. 3c, f (dashed red lines). To compare the mode dispersion and the simulated extinction spectra, we related the real-valued in-plane polariton momenta, q, with the width of the grating, w, for each frequency range, w -1 = 2q/(-vis,IR), where vis= -0.12, IR = -0.3 are fit constants for the visible and mid-IR ranges, respectively. According to a simple FP model, the extracted values of the parameter vis,IR can be interpreted as the phase acquired by the polaritonic modes under reflection from the Al ribbon edges (SPP modes) and refraction index step in h-BN slab (PhP modes). We find excellent agreement between the calculated dispersions of the polaritons and the positions of the peaks in the extinction spectra in both spectral ranges. This agreement demonstrates that the interaction between polaritons and matter excitations in the continuous heterostructure is approximately equivalent to that in the nanoresonator heterostructure, which justifies that the analysis of a continuous heterostructure can be used to characterize the coupling parameters in the nanoresonator heterostructure. From the quasi-normal mode analysis we extract a mode splitting at mid-IR frequencies of ΩIR = 5.9 cm -1  7.310 -4 eV, and a mode splitting at visible frequencies of Ωvis,1 = 1.110 3 cm -1  0.14 eV (between the lower and middle polariton branches) and Ωvis,2 = 310 3 cm -1  0.37 eV (between the middle and upper polariton branches). Considering the linewidths of the coupled states in the mid-IR frequency range, ΓIR+,-= 4 cm -1  510 -4 eV, and the linewidths in the visible frequency range, Γvis1,-,0 = 1.510 3 cm -1  0.19 eV and Γvis2,0,+ = 1.710 3 cm -1  0.21 eV, we find that the strong coupling criterion, Ω > (Γ+ +Γ-)/2, is fulfilled both at mid-IR frequencies and for the "0" and "+" branches at visible frequencies. To determine the dispersion of the quasi-normal modes and extract the value of the mode splitting from the experimental data, we fitted our extinction spectra in Fig. 3b,e using a classical model of coupled harmonic oscillators: two coupled oscillators in the mid-IR frequency range and three coupled oscillators in the visible frequency range, respectively (see Supplementary information SV). The coupled harmonic oscillator models allows us to reproduce the experimental extinction spectra by fitting the parameters of the uncoupled oscillators and the coupling strengths for each nanoresonator structure. With the parameters extracted from the fits we can calculate the dispersion of the quasi-normal modes of the coupled system, according to the dispersion equations in Supplementary information SV. In both spectral ranges, we find that the dispersion of the quasi-normal modes exhibits anti-crossing, as clearly shown by the squares in Fig.3cf. In the IR range, the minimum vertical distance between the dispersion branches of the modes yields a mode splitting of ΩIR = 6.0 cm -1  7.310 -4 eV (Supplementary Information SV). In the visible spectral range, we find two mode splittings of Ωvis,1 = 1.110 3 cm -1  0.14 eV (between the lower and middle polariton branches) and Ωvis,2 = 2.110 3 cm -1  0.26 eV (between the middle and upper polariton branches). Thus, the extracted dispersions of the quasi-normal modes and the values of the mode splittings are in good agreement with the corresponding theoretical values for the continuous heterostructure in the mid-IR and visible spectral ranges (dashed red lines in Fig.3c,f) at the values of w -1 corresponding to the fabricated structures (white vertical dashed lines in Fig. 3c,f). We note that the linewidths of the quasi-normal modes extracted from the fits in both frequency ranges are about 2-6 times larger than those calculated theoretically and vary for different structures, so that only the SC onset is reached. We explain this discrepancy by fabrication uncertainties and width variation along the Al ribbons, fabrication-induced roughness, defects, and the presence of higher order PhP modes in h-BN slab. Finally, we analyze theoretically the coupling strength between polaritons and matter excitations. To do so, we extracted the mode splitting for continuous heterostructures with different molecular layer thickness dCoPc from the previously developed quasi-normal analysis (see Methods). We find that the mode splitting, and thus the coupling strength between polaritons and molecular excitations, increases with dCoPc in both visible and mid-IR frequency ranges (Fig. 4b, c). This result is explained by a larger portion of the electromagnetic field of the polaritonic modes inside the molecular layer for larger thicknesses (Fig. 4a) 38,3 . Furthermore, we find that the mode splitting reaches saturation for a thickness of 60 nm, which is due to the full confinement of the PhP field inside the molecular layer (the field does not reach the air region). Interestingly, in the visible frequency range, the SC criterion is not fulfilled for the first exciton excitation even for a molecular layer thickness of 100 nm, while for the second excitonic excitation, the SC criterion is fulfilled for molecule layers thicker than 20 nm. In contrast, in the mid-IR range, the numerical calculations predict the observation of SC already for molecule layers as thin as 15 nm. Our work demonstrates that engineering a heterostructure composed of plasmonic and phononic materials allows simultaneous access to light-matter interactions in the visible and mid-IR frequencies. Such heterostructure can be exploited to achieve dual-band strong light matter coupling, namely between nanoscale confined polaritons (SPPs and PhPs) and electronic or vibrational excitations of molecules. Momentum-energy coupling between excitations and polaritons can be achieved by tuning the dispersion of the latter through the thickness of the layers in the heterostructure. Future dual-band SC experiments may offer novel opportunities for manipulating chemical reactions, advanced optical imaging and sensing, or optomechanical up-conversion. Methods Sample preparation Cobalt(II) Phthalocyanine, CoPc with sublimed quality (99.9%) (Sigma-Aldrich, Saint Louis, MO, USA) was thermally evaporated in an ultra-high-vacuum evaporator chamber (base pressure <10 −9 mbar) at a rate of 0.2 nm s −1 using a Knudsen cell. The h-BN crystal flake was grown from a metal flux at atmospheric pressure as described previously 38 . The thin layer used in this study was prepared by mechanical exfoliation with blue Nitto tape. Then, we performed a second exfoliation of the h-BN flakes from the tape onto a transparent polydimethylsiloxane (PDMS) stamp. Using optical inspection of the h-BN flake on the stamp, we identified a high-quality flake with appropriate thickness. This flake was transferred onto a CaF2/Al gratings substrate using the deterministic dry transfer technique. To pick up and transfer this flake to another set of Al gratings we used a PDMS stamp with polycarbonate (PC) film, following the procedure described in ref 39 . Aluminium (Al) metal ribbon arrays (size of each array is 20 m20 m) with different width of ribbons are fabricated using high-resolution electron beam (e-beam) lithography. The 50 nm of Al layer was e-beam evaporated onto the CaF2 substrate. Then negative resist (MA-N2401, 90 nm) was spin-coated followed by ebeam lithography of gratings (50 keV, 200 pA, dose 225 ÷ 375 C/cm 2 ) and resist development in AZ726 and reactive ion etching (RIE) of Al in BCl3/Cl2 plasma (pressure 40 mT, RIE power 100W). The resist was finally removed in O2 plasma. Visible spectroscopy measurements Transmission spectra in the visible (VIS) range were recorded using a wide-field optical microscope (Zeiss Axio). A broadband light source (400-1800 nm) is linearly polarised along the desired direction through a rotatable polarizer and used to illuminate the sample from the substrate in the Koehler configuration. After the polarizer, the light passes through an adjustable 4-blades slit and a condenser (NA 0.9) that projects the image of the slit on the sample surface, which is imaged by a CCD camera through a 50x polarisation-maintaining objective. By adjusting the slit blades, only the light transmitted by selected rectangular portions of the sample surface, as small as 20×20 m 2 , reaches the CCD detector for imaging. Once the desired area is selected, the transmitted light is diverted from the CCD and focused into a multicores optical fiber that convey the transmitted light to the VIS spectrometer (Ocean Optics USB2000+). VIS spectra are taken from the bare substrate, T0, and from the area of interest, T, and the resulting normalised extinction spectra are obtained as 1-T/T0. Fourier transform infrared spectroscopy measurements Infrared transmission spectra of the molecules, bare and molecule-coated heterostructure were recorded with a Bruker Hyperion 2000 infrared microscope (Bruker Optics GmbH, Ettlingen, Germany) coupled to a Bruker Vertex 70 FTIR spectrometer (Bruker Optics GmbH, Ettlingen, Germany). The normal-incidence infrared radiation from a thermal source (globar) was linearly polarised via a wire grid polarizer. The spectral resolution was 1 cm −1 . Eigenmode analysis We used the transfer matrix approach to calculate the quasi-normal modes 40 . They can be found by determining the poles in the Fresnel reflectivity of the layered sample for p-polarised light, rp. To determine the poles, we numerically solved the equation 1/Abs(rp) = 0. We considered complex frequencies ωc = ω − iΓ/2 and real-valued momenta q, and determined the poles of rp(q, ω− iΓ/2), yielding ω(q), mode linewidth Γ. The dielectric permittivity of h-BN, CoPc, Al and CaF2 were modelled as described in Supplementary Section I. Numerical simulation Full-wave numerical simulations using the finite-element method in the frequency domain (COMSOL) were performed to simulate the extinction spectra and study the electric field distribution around the heterostructure on top of a CaF2 substrate. The dielectric permittivity of h-BN, CoPc, Al and CaF2 were modelled as described in Supplementary Section I. Infrared and visible nanoimaging by s-SNOM We used a commercial s-SNOM set-up (Neaspec GmbH, Martinsried, Germany), in which the oscillating (at a frequency Ωtip ≅ 270 kHz) metal-coated (Pt/Ir) atomic force microscope tip (Arrow-NCPt-50, Nanoworld, Nano-World AG, Neuchâtel, Switzerland) was illuminated by s-polarised mid-IR or visible radiations. We used tunable quantum cascade laser and He-Ne laser in the mid-IR and visible frequency ranges, respectively. The p-polarised backscattered light is recorded with a pseudoheterodyne Michelson interferometer. To suppress background scattering from the tip shaft and sample, the detector signal was demodulated at a frequency 3Ωtip. We note that our imaging procedure (illuminating with s-polarised light and recording p-polarised light) allows substantially suppressing the excitation of polariton modes via the metallic tip. I. Dielectric functions of materials A. CoPc dielectric function in the infrared frequency range We measured the relative infrared transmission spectrum of a 100 nm thick Cobalt(II) Phthalocyanine (CoPc) layer evaporated on top of a CaF2 substrate. To extract the dielectric function of the CoPc molecules we calculate the relative transmission spectra, / 0 , for the three layer system, using Fresnel coefficients [1], where is transmission through CaF2/CoPc and 0 is transmission through the CaF2 substrate ( CaF 2 = 1.37 in the considered infrared frequency range). We modeled the dielectric function of the CoPc molecules by the Drude−Lorentz model assuming one classical harmonic oscillator to describe molecular vibrations in the considered infrared frequency range, as follows: CoPc ( ) = ∞,IR + 0 2 − 2 − Γ CoPc ,(S1) where ∞,inf is a high-frequency dielectric constant, is a constant that is proportional to the effective strength of the Lorentz oscillator, 0 and Γ CoPc represent the central frequency and the linewidth of the Lorentz oscillator, respectively. Fit yields ∞,IR = 2.8 cm -1 , = 3600 cm -2 , Figure S1. The dielectric function of the CoPc molecules in the infrared frequency range. Black and blue curves represent the real and imaginary parts of extracted CoPc dielectric function, respectively. B. CoPc dielectric function in the visible frequency range Following the same procedure as in the infrared frequency range, we measured the relative transmission spectrum for 20, 30, 50 and 100 nm thick Cobalt(II) Phthalocyanine (CoPc) layers evaporated on top of a CaF2 substrate ( CaF 2 = 1.43 in the considered visible frequency range). To extract the dielectric function of the CoPc molecules we used relative transmission spectra, / 0 calculated with the help of Fresnel coefficients. We fit the dielectric function of the CoPc molecules by the Drude−Lorentz model assuming two classical harmonic oscillators to describe electronic transitions in visible frequency range, as follows: CoPc ( ) = ∞,vis + 1 0,1 2 − 2 − Γ CoPc,1 + 2 0,2 2 − 2 − Γ CoPc,2 ,(S2) where ∞,vis is a high-frequency dielectric constant, (k = 1,2) are constants proportional to the effective strength of the kth Lorentz oscillator, 0, and Γ CoPc, represent the central frequency and the linewidth of the kth Lorentz oscillator, respectively. We fitted the relative transmission spectrum for each thickness of the molecular layer independently and then averaged the extracted fit parameters. The averaged fit parameters: ∞,vis = 1.6 cm -1 , 0,1 = 14402 cm -1 , 1 = 31539456 cm -2 , Γ CoPc,1 = 1739 cm -1 , 0,2 = 16264 cm -1 , 2 = 65755881 cm -2 , Γ CoPc,2 = 2369 cm -1 . Figure S2 shows the dielectric permittivity of CoPc molecules in the visible frequency range calculated according to Equation S2 with the parameters extracted from the fit. C. h-BN dielectric function We used the isotopically ( 10 B) enriched h-BN [2]. The dielectric permittivity tensor of h-BN is modeled according to the following formula: h−BN, ( ) = ∞, ( LO, 2 − 2 − Γ TO, 2 − 2 − Γ ),(S3) where j =,  indicates the component of the tensor parallel and perpendicular to the crystal axis, respectively. We took the parameters for the dielectric function of h-BN from ref. [2], except of ∞, . For the best matching of our near-field and far-field experiments we took ∞, = 4 instead of ∞, = 5.1 used in ref. 3. We attribute this discrepancy to uncertainties introduced by the fabrication. All parameters for the dielectric function, which were used in the simulation, are presented in the Table S1. D. Al dielectric function We modelled the Al dielectric function as a sum of Drude and Lorentz terms using parameters from the ref. [3]. II. Parameters of the arrays of Al ribbons We fabricated the arrays of Al ribbons with different widths of the ribbons, w, and the periods of the structures, p, the latter designed to be twice the width of the ribbons, p=2·w. Figure S3 shows scanning electron microscope (SEM) images of the fabricated ribbon arrays. We extracted p and w from the measured SEM images and made sure that the filling factors, f=w/p, were indeed approximately 1/2 for all the fabricated structures. It is important to note that in Figure 3c,f of the main text and Figure S7, S8 of the supplementary information, to plot the quasi-normal modes of nanoresonator heterostructures we used doubled inverse period, 2·p -1 , instead of the inverse ribbon width, w -1 . The extracted p, w and calculated f, p -1 parameters of the ribbon arrays are presented in Table S2 Figure S3. SEM images of the fabricated arrays of Al ribbons, with the corresponding names of the arrays above the images. Table S2. Parameters of the fabricated arrays of Al ribbons, which are shown in Figure S3. III. Analysis of the polaritonic modes in the nanoresonator heterostructure To study polaritonic modes in the nanoresonator heterostructure we performed the full-wave numerical simulation of electromagnetic fields using the finite-element method in frequency domain (COMSOL). We assumed a two dimensional (2D) geometry, namely an infinite number of infinitely long Al ribbons below the h-BN slab. We simulated transmission, reflection and scattering of a plane monochromatic wave normally incident onto the periodic array of nanoresonators. Figure S4a shows the schematics of one period of the simulated structure. The right panels of Figure S4b,c show the calculated far-field extinction spectra, 1 -T, where the T is the power transmission coefficient in the visible and mid-IR frequency ranges, respectively. The calculated extinction spectra in both frequency ranges reveal numerous peaks. In the visible frequency range, we can clearly recognize a sharp peak in the extinction spectrum around  = 1.410 4 cm -1 followed by a broad peak around  = 1.6510 4 cm -1 ( Figure S4b, right panel). Both peaks form the so-called Wood-Rayleigh anomaly. Namely, the sharp peak (Rayleigh point) corresponds to the zero value of the z-component of the wavevector of the 1 st order diffracted wave in the CaF2 substrate. It takes place directly at the boundary between frequency regions where the 1 st order diffracted wave has evanescent and propagating character. In contrast, the second peak (Wood anomaly) represents the first-order SPP Bragg resonance, corresponding to the pole in the transmission and reflection coefficients, and partially the Fabry-Perot (FP) resonance appearing as a result of multiple reflection of SPP mode (along the x-axis) from the edges of Al ribbons. In the mid-IR frequency range, the extinction spectrum manifests multiple peaks ( Figure S4c, right panel). We assume that the latter emerge due to FP resonances appearing as a result of multiple reflection of PhP waveguiding modes from their refractive index steps defined along the -axis by Al ribbons. To analyze and interpret these peaks we generate the color plots (left panels of Figure S4b,c), representing the z-component of the electric field above the h-BN slab as a function of the frequency, , and coordinate, x. In the visible frequency range, we observe the periodic field pattern along both the frequency and coordinate axes ( Figure S4b, left panel). We see two bright localized areas ("hot spots") of different polarity along the coordinate axis, x, which can be explained by the presence of the transverse FP mode in the considered frequency range. We note that this FP mode can also be recognized in the simulated field distribution, Re(Ez)/\|Ei\|, in the x-z plane at vis = 15798 cm -1 (Figure 2h of the main text). We speculate that the periodicity of the field pattern along the frequency axis arises due to the complex interference between the electromagnetic fields of the FP mode and the incident and reflected waves. In the mid-IR frequency range, we find two frequency regions around  = 1460 cm -1 and  = 1510 cm -1 with the bright localized areas of the different polarity along the coordinate axis. These bright areas arise due to the presence of transverse FP resonances of PhP waveguiding modes at corresponding frequencies. These FP resonances can be characterized as "bright" modes since they have a nonzero in-plane dipole moment which can couple with propagating waves in free-space. As a result, these resonances appear as peaks in the far-field extinction spectrum (see the right panel of Figure S4с). In the color plot shown in the left panel of Figure S4c, we see that at  = 1460 cm -1 , a large portion of the mode volume is localized above the air region. In contrast, at  = 1510 cm -1 the electric field of the mode is mainly localized above the Al ribbon. These observations allow us to assign the resonance around  = 1460 cm -1 to the FP resonance of the PhP mode in h-BN slab above the air region. In contrast, a multi-peak in the extinction spectrum around the  = 1510 cm -1 can be attributed to the FP resonances of the PhP modes above the Al ribbons. We explain the multi-peak structure in the extinction spectrum of the resonance around  = 1510 cm -1 by the presence of higher-order PhP waveguiding modes in the h-BN layer. We finally identify the multi-peak resonance around  = 1510 cm -1 as FP resonances and, partially, as Bragg resonance arising from the overlap between the electromagnetic fields of adjacent air and Al regions. Note that, in contrast to the simulation, the experimental extinction spectrum in Figure 2b of the main text shows only the single resonant peak around  = 1510 cm -1 . This discrepancy can be explained by the fabrication uncertainties of the Al ribbon width throughout the array and by quality of the Al edges that play a crucial role in the far-field excitation of the higher-order PhP waveguiding modes in the h-BN slab. IV. Data processing of infrared nanoimaging experiments A. Mid-IR frequency range Figure S5c,e,g show the raw amplitude, phase and real part (calculated using the amplitude and phase) of the complex-valued s-SNOM signal, 3 . The data is represented as near-field imagesthe signal as a function of the tip position above the sample-of the set of 4 nanoresonators, which is schematically shown in Figure S5a,b. In order to reveal the mode field pattern above the nanoresonators, we subtracted the mean value of both the real and imaginary parts of signal for each horizontal line profile at the fixed y coordinate. Figure S5d Figure S6c,e,g show the raw amplitude, phase and real part (calculated using the amplitude and phase images) of the complex-valued s-SNOM signal, 3 . Analogously to the mid-IR range, the near-field images are shown for the set of 4 nanoresonators, which is schematically shown in Figure S6a,b. We also analogously subtracted the mean value of both the real and imaginary parts of signal for each horizontal line profile at the fixed y coordinate. The resulting amplitude, phase and real part of the near-field images, respectively. B. Visible frequency range V. Coupled classical harmonic oscillators A. Mid-IR frequency range In order to analyze the extinction spectra shown in Figure 3e of the main text, we phenomenologically described the coupling between the molecular vibration and the phonon polaritons via a model of classical coupled harmonic oscillators [4], [5]. The equations describing the motion of two coupled harmonic oscillators are given by [6]: where PhP , PhP and Γ PhP represent the displacement, frequency and linewidth of the PhP mode, respectively. 1 , 1 and Γ 1 represent the displacement, frequency and linewidth of the molecular vibration of CoPc, respectively. PhP and 1 represent the effective external forces that drive the motion of the oscillators. In the realistic electromagnetic problem the external electromagnetic field is the analog of the effective forces. represents the coupling strength between PhP mode and CoPc molecular vibration. From the oscillators model we can construct a quantity equivalent to the extinction, ext , of the analogous electromagnetic problem. It can be calculated as the work done by the external forces according to ext  〈 PhP ( )̇P hP ( ) + 1 ( )̇1( )〉 [4]. We fit ext to the measuared extinction spectra of nanoresonator heterostructure with different ribbon width. Fits were performed according to 1 − 0 = ext + . In the fitting procedure, we take the same value of Γ 1 = 6 cm -1 as we used in the CoPc dielectric function. 1 was limited within a few wavenumbers from its initial value according to the CoPc dielectric function ( 1 = CoPc =1524.8 cm −1 ), to allow for an eventual Lamb shift of the molecular vibration [7], [8]. PhP , , 1,PhP and Γ PhP were considered as free parameters in all fits. The extracted values of the coupling strength for each fits are plotted as black symbols in the Figure S7b. With the parameters extracted from the fits we calculated the frequencies of the quasi-normal modes of the coupled system. The eigenfrequencies can be found from the dispersion relation, which arises from equaling the determinant of the system (S4) to zero [9] (assuming the harmonic time-dependence of the displacements, − ): ± + Γ ± 2 = PhP + 1 2 − Γ PhP + Γ 1 4 ± 1 2 √ 4 2 + ( PhP − 1 − Γ PhP − Γ 1 2 ) 2 (S5) We have used the approximation -j << j, so that  2 -j 2 = 2j (-j), with j=1, PhP. Figure S7a shows the calculated frequencies quasi-normal modes ± as a function of inverse width of nanoresonators, 1/w. To find the mode splitting we first splined the real part of the calculated frequency of the quasi-normal modes (gray lines in Figure S7a) and then found the smallest vertical separation between the splined branches, = 5.9 cm -1 . All the fitting parameters are presented in Table S3. Table S3. Parameters of the coupled oscillators model to fit the experimental extinctions spectra in the mid-IR frequency range. B. Visible frequency range In order to analyze the extinction spectra in the visible frequency range in Figure 3b of the main text, we phenomenologically described the coupling of the excitons and the surface plasmon polaritons (SPPs) via a classical model of three coupled harmonic oscillators, where we considered the coupling only between the SPP and two excitons. The equations of motion for the three coupled harmonic oscillators are given by: where SPP , SPP and Γ SPP represent the displacement, frequency and linewidth of the SPP mode, respectively. 1,2 , 1,2 and Γ 1,2 represent the displacement, frequency and linewidth of the "1" and "2" excitons of CoPc, respectively. SPP and 1,2 represent the effective forces that drive their motions. 1 represents the coupling strength between SPP mode and CoPc exciton "1", 2 represents the coupling strength between SPP mode and CoPc exciton "2". Analogously to the mid-IR range, from the oscillators model we can construct a quantity equivalent to the extinction, ext , that can be calculated according to ext  〈 SPP ( )̇S PP ( ) + 1 ( )̇1( ) + 2 ( )̇2( )〉. We fit ext to the measured extinction spectra of nanoresonator heterostructures with different ribbon width. Fits were performed according to 1 − 0 = ext + . In the fitting procedure, we take the same value of Γ CoPc,1 = 1739 cm -1 and Γ CoPc,2 = 2369 cm -1 as we used in the CoPc dielectric function. To minimize the number of fitting parameters in the visible frequency range, we fix the uncoupled frequencies of excitons 1 = 0,1 = 14402 cm -1 and 2 = 0,2 = 16264 cm -1 according to the CoPc dielectric function. SPP , 1,2 , 1,2,SPP and Γ SPP were considered as free parameters in all fits. The extracted values of the coupling strengths for each fits are plotted as black and red symbols in the Figure S8b. With the parameters extracted from the fits we numerically calculated the three frequencies of the quasi-normal modes of the coupled system. The eigenfrequencies can be found from the dispersion relation, which arises from equaling the determinant of the system (S6) to zero (assuming the harmonic time-dependence of the displacements, − ). Figure S8a shows the three branches (red circles) of the real part of calculated frequencies of quasi-normal modes as a function of inverse width of the nanoresonators, 1/w. To find the mode splitting, analogously to the mid-IR range, we first splined the real part of calculated frequencies of quasi-normal modes (gray lines in Figure S8a) and then extracted the smallest vertical separation between the adjacent splined branches Ωvis,1 = 1.110 3 cm -1 (between the lower and middle polariton branches) and Ωvis,2 = 310 3 cm -1 (between the middle and upper polariton branches). All the fitting parameters are presented in Table S4. Table S4. Parameters of the coupled oscillators model to fit the experimental extinctions spectra in the visible frequency range. Figure 1 . 1Dual-band SC between polaritons and molecular excitations at both visible and mid-IR frequencies. a, Schematic of the three-layer structure considered (50nm-Al/50nm-CoPc/75nm-h-BN). Left panel: blue and red dashed lines show the mode profile of the polaritonic modes in the mid-IR and visible frequency ranges, respectively (blue line for  = 1530 cm -1 , red line for  = 19500 cm -1 ). b, Schematic of the atomic structure of the molecule CoPc together with both the vibrational and excitonic resonances in it. c, d, Experimentally extracted Im(εCoPc) in the visible and mid-IR frequency ranges, respectively. e, f, Colour plots showing the calculated imaginary part of the Fresnel reflection coefficient of the structure in both frequency ranges. Red dashed lines show the calculated dispersion of the quasi-normal modes, assuming complex-valued frequency and real-valued momentum. Horizontal green dashed lines indicate the resonance frequencies of the molecular excitations. Figure 2 . 2Experimental demonstration of dual-band polaritonic nanoresonators. a,d Schematics of the farfield experiments and s-SNOM optical characterization of the polaritonic nanoresonators. b, c, Experimental extinction spectrum of the nanoresonators created by employing an Al grating with period p = 500 nm in the mid-IR and visible frequency ranges, respectively. e, f, (top panels) Near-field images of the nanoresonators in the mid-IR and visible frequency ranges, measured at mid-IR = 1510 cm -1 and vis = 15798 cm -1 as indicated by the blue and red dashed lines in b,c, respectively. A clear dipolar fundamental mode is observed in both cases. (Bottom panels) Simulated Re(Ez) in the x-y plane, extracted 80 and 20 nm above the structure as indicated by the vertical dashed lines in g and h, respectively. Vertical dashed lines indicate the edges of Al ribbon. g, h, Simulated Re(Ez)/\|Ei\| in the x-z plane at the same frequencies. The period of the Al grating, the width of the Al ribbon and the thickness of the h-BN slab employed in the bottom panels e, f and g, h are p = 500 nm, w = 250 nm, and dh-BN = 75 nm, respectively. Figure 4 . 4Mode splitting dependence on molecular layer thickness. a. Left: Schematic illustration of the Al/h-BN/CoPc heterostructure. Right: calculated electric field distribution in the heterostructure at  = 1530 cm -1 (blue dashed curve) and  = 19500 cm -1 (red dashed curve). b,c, Calculated mode splitting as a function of the molecular layer thickness in the visible and mid-IR frequency ranges. The square symbols in c are mode splitting extracted from the measured spectra. The Al and h-BN thicknesses are dAl = 50 nm, and dh-BN = 75. Horizontal dashed lines separate the regions of the strong and weak coupling. SC in both bands is observed for CoPc thicknesses larger than about 15 nm. heterostructures of bilayer graphene and hexagonal boron nitride. Appl. Phys. Lett. 105, formalism for light propagation in anisotropic stratified media: study of surface phonon polaritons in polar dielectric heterostructures. J. Opt. Soc. Am. B 34, Dielectric functions of materials.............................................................................................. 2 A. CoPc dielectric function in the infrared frequency range ................................................ 2 B. CoPc dielectric function in the visible frequency range .................................................. 3 C. h-BN dielectric function ................................................................................................... 4 D. Al dielectric function ........................................................................................................ 4 II. Parameters of the arrays of Al ribbons ................................................................................ 5 III. Analysis of the polaritonic modes in the nanoresonator heterostructure ............................. 6 IV. Data processing of infrared nanoimaging experiments ....................................................... 9 A. Mid-IR frequency range ................................................................................................... 9 B. Visible frequency range ................................................................................................. 10 V. Coupled classical harmonic oscillators .............................................................................. 11 A. Mid-IR frequency range ................................................................................................. 11 B. Visible frequency range ................................................................................................. 13 References ..................................................................................................................................... 15 Figure S2 . S2The dielectric function of the CoPc molecules in visible frequency range. Black and blue curves represent the real and imaginary parts of extracted CoPc dielectric function, respectively. Figure S4 . S4Analysis of the polaritonic resonances in the visible and mid-IR frequency ranges. a. Schematics of one period of the nanoresonator heterostructure with the period, p = 500 nm, and Al ribbon width, w = 250 nm. b,c, (left panel) Simulated electric field at the height of z0 = 80 nm above the h-BN layer as a function of  and x coordinate in the visible and mid-IR frequency ranges, respectively. (right panel) Simulated extinction spectrum of the nanoresonator heterostructure in the visible and mid-IR frequency ranges, respectively ,f,g show the final data set of amplitude, phase and real part of the near-field images, 3 * ( , ), after the subtraction of the mean values. The top panel in Figure 2e of the main text shows the final data of real part of the first resonator of Figure. S5h, for ∈ [0 ∶ 0.5] . Figure S5 . S5Mean value subtraction from the near-field data in the mid-IR frequency range. a,b, Schematics of the measured nanoresonator heterostructure. c,e,g, Raw amplitude, phase and real part of the near-field images of 4 nanoresonators, as indicated in the panels a,b in the mid-IR frequency range, measured at mid-IR = 1510 cm -1 , respectively. d,f,h, The resulting amplitude, phase and real part of the near-field images after the mean value subtraction, respectively. Then we removed the propagating SPP Bloch mode by subtraction the complex signal B ( ) = B + B for each y coordinate, where B = 0.075, B = 0.25 are real-valued fitting parameters and = 2 is the Bragg vector of the array of Al ribbons with the period = 500 nm. Figure S6d,f,g show the final data set of the amplitude, phase and real part of the near-field images, 3 * ( , ), after the subtraction of the mean values and propagating SPP Bloch mode. The top panel in Figure 2f of the main text shows the final data of the real part of the first resonator of Figure S6h, for ∈ [0 ∶ 0.5] . Figure S6 . S6Mean value and SPP Bloch mode subtraction from the near-field data in the visible frequency range. a,b, Schematics of the measured nanoresonator heterostructure. c,e,g, Raw amplitude, phase and real part of the near-field images of 4 nanoresonators, as indicated in the panels a,b in the visible frequency range, measured at vis = 15798 cm -1 , respectively. d,f,h, Figure S7 . S7Fitting the extinction spectra by the coupled oscillators model: mid-IR range. a, Uncoupled frequencies of PhPs: PhP , and CoPc: 1 (blue triangles and black squares, respectively), as extracted from the fits. Frequencies of the quasi-normal modes, ± , calculated according to Eq. S6 (red circles). Grey solid lines show the spline of the eigenmodes frequencies as a function of inverse width of ribbons b, Black symbols show the coupling strength from the fits. c, The linewidth of uncoupled PhPs from the fits. Figure S8 . S8Fitting the extinction spectra by the coupled oscillators model: visible range. Uncoupled frequencies of SPPs, SPP , and CoPc excitons, 1,2 , (black squares and blue triangles, respectively) as extracted from the fits. Red circles show the eigenmode frequencies. Grey solid lines show the spline of the eigenmode frequencies as a function of inverse width of ribbons b, Black symbols show the coupling strength from the fits. c, The linewidth of uncoupled SPPs from the fits. Table S1. Parameters for the dielectric function of h-BN.j ∞ TO , cm -1 LO , cm -1 , cm -1  4 1394.5 1650 1.8  2.5 785 845 1 = 1524.8 cm -1 , Γ CoPc = 6 cm -1 . With the parameters extracted from the fit, we are able to calculate the dielectric permittivity of CoPc molecules in the infrared frequency range, according to Equation S1,Figure S1. AcknowledgementsWe acknowledge the Spanish Ministry of Science, Innovation and Universities(national projects PID2021-123949OB-I00, PID2021-122511OB-I00, PID2021-123943NB-I00, RTI2018-094861-B-I00, and the Conductivity in organic semiconductors hybridized with the vacuum field. E Orgiu, Nat. Mater. 14Orgiu, E. et al. Conductivity in organic semiconductors hybridized with the vacuum field. Nat. Mater. 14, 1123-1129 (2015). 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[ "Clustered Planarity = Flat Clustered Planarity", "Clustered Planarity = Flat Clustered Planarity", "Clustered Planarity = Flat Clustered Planarity", "Clustered Planarity = Flat Clustered Planarity" ]	[ "Pier Francesco Cortese \nRoma Tre University\nRomeItaly\n", "Maurizio Patrignani [email protected] \nRoma Tre University\nRomeItaly\n", "Pier Francesco Cortese \nRoma Tre University\nRomeItaly\n", "Maurizio Patrignani [email protected] \nRoma Tre University\nRomeItaly\n" ]	[ "Roma Tre University\nRomeItaly", "Roma Tre University\nRomeItaly", "Roma Tre University\nRomeItaly", "Roma Tre University\nRomeItaly" ]	[]	The complexity of deciding whether a clustered graph admits a clustered planar drawing is a long-standing open problem in the graph drawing research area. Several research efforts focus on a restricted version of this problem where the hierarchy of the clusters is 'flat', i.e., no cluster different from the root contains other clusters. We prove that this restricted problem, that we call Flat Clustered Planarity, retains the same complexity of the general Clustered Planarity problem, where the clusters are allowed to form arbitrary hierarchies. We strengthen this result by showing that Flat Clustered Planarity is polynomial-time equivalent to Independent Flat Clustered Planarity, where each cluster induces an independent set. We discuss the consequences of these results. This research was partially supported by MIUR project "MODE -MOrphing graph Drawings Efficiently", prot. 20157EFM5C 001. arXiv:1808.07437v2 [cs.DM] 31 Aug 2018 2 P. F. Cortese and M. Patrignaniis mapped to the same point in Γ 1 and in Γ 2 and (ii) every edge e ∈ E 1 ∩ E 2 is mapped to the same Jordan curve in Γ 1 and in Γ 2 .However, the polynomial-time equivalence of the two problems is open and the reverse reduction of SEFE to Clustered Planarity is known only for the case when the intersection graph G ∩ = (V, E 1 ∩ E 2 ) of the instance of SEFE is connected[4]. Also in this special case, the complexity of the problem is unknown, with the exception of the case when G ∩ is a star, which produces a c-graph with only two clusters, a known polynomial case for Clustered Planarity[10,46].Since the general Clustered Planarity problem appears to be elusive, several authors focused on a restricted version of it where the hierarchy of the clusters is 'flat', i.e., only the root cluster contains other clusters and it does not directly contain vertices of the underlying graph [2,3,5-7,9,16,20,21,24,28,36,38-40, 46, 50]. This restricted problem, that we call Flat Clustered Planarity, is expressive enough to be useful in several applicative domains, as for example in computer networks where routers are grouped into Autonomous Systems [15], or social networks where people are grouped into communities[13,29], or software diagrams where classes are grouped into packages[51]. Also, several hybrid representations have been proposed for the visual analysis of (not necessarily planar) flat clustered graphs, such as mixed matrix and node-link representations[13,22,23,30,45], mixed intersection and node-link representations[8], and mixed space-filling and node-link representations[1,47,53].Unfortunately, the complexity of Flat Clustered Planarity is open as the complexity of the general problem. The authors of[14], after recasting Flat Clustered Planarity as an embedding problem on planar multi-graphs, conclude that we are still far away from solving it. The authors of [4] wonder whether Flat Clustered Planarity retains the same complexity of Clustered Planarity. In this paper we answer this question in the affirmative. Obviously, a reduction of Flat Clustered Planarity to Clustered Planarity is trivial, since the instances of Flat Clustered Planarity are simply a subset of those of Clustered Planarity. The reverse reduction is the subject of Section 3, that proves the following theorem.	10.1007/978-3-030-04414-5_2	[ "https://arxiv.org/pdf/1808.07437v2.pdf" ]	52,070,116	1808.07437	799a0b5ed243f242b16c94d8b4617cb4e576f9f9	Clustered Planarity = Flat Clustered Planarity Pier Francesco Cortese Roma Tre University RomeItaly Maurizio Patrignani [email protected] Roma Tre University RomeItaly Clustered Planarity = Flat Clustered Planarity The complexity of deciding whether a clustered graph admits a clustered planar drawing is a long-standing open problem in the graph drawing research area. Several research efforts focus on a restricted version of this problem where the hierarchy of the clusters is 'flat', i.e., no cluster different from the root contains other clusters. We prove that this restricted problem, that we call Flat Clustered Planarity, retains the same complexity of the general Clustered Planarity problem, where the clusters are allowed to form arbitrary hierarchies. We strengthen this result by showing that Flat Clustered Planarity is polynomial-time equivalent to Independent Flat Clustered Planarity, where each cluster induces an independent set. We discuss the consequences of these results. This research was partially supported by MIUR project "MODE -MOrphing graph Drawings Efficiently", prot. 20157EFM5C 001. arXiv:1808.07437v2 [cs.DM] 31 Aug 2018 2 P. F. Cortese and M. Patrignaniis mapped to the same point in Γ 1 and in Γ 2 and (ii) every edge e ∈ E 1 ∩ E 2 is mapped to the same Jordan curve in Γ 1 and in Γ 2 .However, the polynomial-time equivalence of the two problems is open and the reverse reduction of SEFE to Clustered Planarity is known only for the case when the intersection graph G ∩ = (V, E 1 ∩ E 2 ) of the instance of SEFE is connected[4]. Also in this special case, the complexity of the problem is unknown, with the exception of the case when G ∩ is a star, which produces a c-graph with only two clusters, a known polynomial case for Clustered Planarity[10,46].Since the general Clustered Planarity problem appears to be elusive, several authors focused on a restricted version of it where the hierarchy of the clusters is 'flat', i.e., only the root cluster contains other clusters and it does not directly contain vertices of the underlying graph [2,3,5-7,9,16,20,21,24,28,36,38-40, 46, 50]. This restricted problem, that we call Flat Clustered Planarity, is expressive enough to be useful in several applicative domains, as for example in computer networks where routers are grouped into Autonomous Systems [15], or social networks where people are grouped into communities[13,29], or software diagrams where classes are grouped into packages[51]. Also, several hybrid representations have been proposed for the visual analysis of (not necessarily planar) flat clustered graphs, such as mixed matrix and node-link representations[13,22,23,30,45], mixed intersection and node-link representations[8], and mixed space-filling and node-link representations[1,47,53].Unfortunately, the complexity of Flat Clustered Planarity is open as the complexity of the general problem. The authors of[14], after recasting Flat Clustered Planarity as an embedding problem on planar multi-graphs, conclude that we are still far away from solving it. The authors of [4] wonder whether Flat Clustered Planarity retains the same complexity of Clustered Planarity. In this paper we answer this question in the affirmative. Obviously, a reduction of Flat Clustered Planarity to Clustered Planarity is trivial, since the instances of Flat Clustered Planarity are simply a subset of those of Clustered Planarity. The reverse reduction is the subject of Section 3, that proves the following theorem. Introduction A clustered graph (c-graph) is a planar graph with a recursive hierarchy defined on its vertices. A clustered planar (c-planar) drawing of a c-graph is a planar drawing of the underlying graph where: (i) each cluster is represented by a simple closed region of the plane containing only the vertices of the corresponding cluster, (ii) cluster borders never intersect, and (iii) any edge and any cluster border intersect at most once (more formal definitions are given in Section 2). The complexity of deciding whether a c-graph admits a c-planar drawing is still an open problem after more than 20 years of intense research [12, 14, 17-19, 25, 31, 33-35, 37, 41-44, 48, 49, 52]. If we had an efficient c-planarity testing and embedding algorithm we could produce straight-line drawings of clustered trees [27] and straight-line drawings [11,32] and orthogonal drawings [26] of c-planar c-graphs with rectangular regions for the clusters. In order to shed light on the complexity of Clustered Planarity, this problem has been compared with other problems whose complexity is likewise challenging. This line of investigation was opened by Marcus Schaefer's polynomial-time reduction of Clustered Planarity to SEFE [52]. Simultaneous Embedding with Fixed Edges (SEFE) takes as input two planar graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ) and asks whether a planar drawing Γ 1 (G 1 ) and a planar drawing Γ 2 (G 2 ) exist such that: (i) each vertex v ∈ V Theorem 1. There exists a quadratic-time transformation that maps an instance of Clustered Planarity to an equivalent instance of Flat Clustered Planarity. With very similar techniques we are able to prove also a stronger result. Theorem 2. There exists a linear-time transformation that maps an instance of Flat Clustered Planarity to an equivalent instance of Independent Flat Clustered Planarity. Here, by Independent Flat Clustered Planarity we mean the restriction of Flat Clustered Planarity to instances where each non-root cluster induces an independent set. The paper is structured as follows. Section 2 contains basic definitions. Section 3 contains the proof of Theorem 1 under some simplifying hypotheses (which Preliminaries Let T be a rooted tree. We denote by r(T ) the root of T and by T [µ] the subtree of T rooted at one of its nodes µ. The depth of a node µ of T is the length (number of edges) of the path from r(T ) to µ. The height h(T ) of a tree T is the maximum depth of its nodes. The nodes of a tree can be partitioned into leaves, that do not have children, and internal nodes. In turn, the internal nodes can be partitioned into two sets: lower nodes, whose children are all leaves, and higher nodes, that have at least one internal-node child. We say that a node is homogeneous if its children are either all leaves or all internal nodes. A tree is homogeneous if all its nodes are homogeneous. We say that a tree is flat if all its leaves have depth 2. A flat tree is homogeneous. Figure 1 shows a non-homogeneous tree ( Fig. 1(a)), a homogeneous tree ( Fig. 1(b)), and a flat tree ( Fig. 1(c)). We also need a special notion of size: the size of a tree T , denoted by S(T ), is the number of higher nodes of T different from the root of T . Observe that a homogeneous tree T is flat if and only if S(T ) = 0. For example, the sizes of the trees represented in Figs. 1(a), 1(b), and 1(c) are 2, 2, and 0, respectively (filled gray nodes in Fig. 1). The proof of the following lemma can be found in Appendix A. Lemma 1. A homogeneous tree T of height h(T ) ≥ 2 and size S(T ) > 0 contains at least one node µ * = r(T ) such that T [µ * ] is flat. A graph G = (V, E) is a set V of vertices and a set E of edges, where each edge is an unordered pair of vertices. A drawing Γ (G) of G is a mapping of its vertices to distinct points on the plane and of its edges to Jordan curves joining the incident vertices. Drawing Γ (G) is planar if no two edges intersect except at common end-vertices. A graph is planar if it admits a planar drawing. A clustered graph (or c-graph) C is a pair (G, T ) where G = (V, E) is a planar graph, called the underlying graph of C, and T , called the inclusion tree of C, is a rooted tree such that the set of leaves of T coincides with V . A cluster µ is an internal node of T . When it is not ambiguous we also identify a cluster with the respective subset of the vertex set. An inter-cluster edge of a cluster µ of T is an edge of G that has one end-vertex inside µ and the other endvertex outside µ. An independent set of vertices is a set of pairwise non-adjacent vertices. A cluster µ of T is independent if its vertices form an independent set. A c-graph is independent if all its clusters, with the exception of the root, are independent clusters. A cluster µ of T is a lower cluster (higher cluster ) of C if µ is a lower node (higher node) of T . A c-graph is flat if its inclusion tree is flat. The clusters of a flat c-graph are all lower clusters with the exception of the root cluster. A cluster is called singleton if it contains a single cluster or a single vertex. A drawing Γ (C) of a c-graph C(G, T ) is a mapping of vertices and edges of G to points and to Jordan curves joining their incident vertices, respectively, and of each internal node µ of T to a simple closed region R(µ) containing exactly the vertices of µ. Drawing Γ (C) is c-planar if: (i) curves representing edges of G do not intersect except at common end-points; (ii) the boundaries of the regions representing clusters do not intersect; and (iii) each edge intersects the boundary of a region at most one time. A c-graph is c-planar if it admits a c-planar drawing. Problem Clustered Planarity is the problem of deciding whether a cgraph is c-planar. Problem Flat Clustered Planarity is the restriction of Clustered Planarity to flat c-graphs. Problem Independent Flat Clustered Planarity is the restriction of Clustered Planarity to independent flat c-graphs. The proof of the following lemmas can be found in Appendix A. Lemma 2. An instance C(G, T ) of Clustered Planarity with n vertices and c clusters can be reduced in time O(n+c) to an equivalent instance such that: (1) T is homogeneous, (2) r(T ) has at least two children, and (3) h(T ) ≤ n − 1. Proof of Theorem 1 We describe a polynomial-time reduction of Clustered Planarity to Flat Clustered Planarity. Let C(G, T ) be a clustered graph, let n be the number of vertices of G, and let c be the number of clusters of C. Due to Lemma 2 we can achieve in O(n + c) time that T is homogeneous and S(T ) ∈ O(n). We reduce C to an equivalent instance C f (G f , T f ) where T f is flat. The reduction consists of a sequence of transformations of C = C 0 into C 1 , C 2 , . . . , C S(T ) = C f , where each C i (G i , T i ), i = 0, 1, . . . , S(T ) , has an homogeneous inclusion tree T i and each transformation takes O(n) time. Consider any C i (G i , T i ), with i = 0, . . . , S(T ) − 1, where T i is a homogeneous, non-flat tree of height h(T i ) ≥ 2 (refer to Fig. 2(a)). By Lemma 1, T i has at least one node µ * = r(T i ) such that T i [µ * ] is flat. Since µ * = r(T i ), node µ * has a parent ν. Also, denote by ν 1 , ν 2 , . . . , ν h the children of µ * and by µ 1 , µ 2 , . . . , µ k the siblings of µ * in T i . We construct C i+1 (G i+1 , T i+1 ) as follows (refer to Fig. 2(b)). Graph G i+1 is obtained from G i by introducing, for each inter-cluster edge e = (u, v) of µ * , two new vertices e χ and e ϕ and by replacing e with a path (u, e χ )(e χ , e ϕ )(e ϕ , v). Tree T i+1 is obtained from T i by removing node µ * , attaching its children ν 1 , ν 2 , . . . , ν h directly to ν and adding to ν two new children χ and ϕ, where cluster χ (cluster ϕ, respectively) contains all vertices e χ (e ϕ , respectively) introduced when replacing each inter-cluster edge e of µ * with a path. The proof of the following lemmas can be found in Appendix B. χ (b)Lemma 3. If T i is homogeneous then T i+1 is homogeneous. Lemma 4. We have that S(T i+1 ) = S(T i ) − 1. Lemma 5. The c-graph C f = C S(T ) is flat. The proof of the following lemma is given here under two simplifying hypotheses (the proof of the general case can be found in Appendix B): H-conn: The underlying graph G i is connected H-not-root: Cluster ν is not the root of T Observe that Hypothesis H-conn implies that also G i+1 is connected. Observe, also, that Hypothesis H-not-root and Property 2 of Lemma 2 imply that there is at least one vertex of G i that is not part of ν (this hypothesis is not satisfied, for example, by the c-graph depicted in Fig. 2(a)). Lemma 6. C i (G i , T i ) is c-planar if and only if C i+1 (G i+1 , T i+1 ) is c-planar. Proof sketch. The first direction of the proof is straightforward. Let Γ (C i ) be a c-planar drawing of C i (refer to Fig. 3(a)). We show how to construct a c-planar drawing of C i+1 (refer to Fig. 3(b)). Consider the region R(µ * ) that contains R(ν i ), with i = 1, . . . , h. The boundary of R(µ * ) is crossed exactly once by each inter-cluster edge of µ * . Identify outside the boundary of R(µ * ) two arbitrarily thin regions R(χ) and R(ϕ) that turn around R(µ * ) and that intersect exactly once all and only the inter-cluster edges of µ * . Insert into each inter-cluster edge e of µ * two vertices e χ and e ϕ , placing e χ inside R(χ) and e ϕ inside R(ϕ). By ignoring R(µ * ) you have a c-planar drawing Γ (C i+1 ) of C i+1 . Suppose now to have a c-planar drawing Γ (C i+1 ) of C i+1 . We show how to construct a c-planar drawing Γ (C i ) of C i under the Hypotheses H-conn and H-not-root. Consider the regions R(χ) and R(ϕ) inside R(ν) (refer to Fig. 4). Regions R(χ) and R(ϕ) are joined by the p inter-cluster edges introduced when replacing each inter-cluster edge e i of µ * , where i = 1, . . . , p, with a path (red edges of Fig. 4). Such inter-cluster edges of χ and ϕ partition R(ν) into p regions that have to host the remaining children of ν and the inter-cluster edges among them. In particular, p − 1 of these regions are bounded by two inter-cluster edges and two portions of the boundaries of R(χ) and R(ϕ). One of such regions, instead, is also externally bounded by the boundary of R(ν). Now consider the regions R(ν i ) corresponding to the children ν i of ν, with i = 1, . . . , h, that were originally children of µ * . These regions (filled white in Fig. 4) may have inter-cluster edges among them and may be connected to χ, but by construction cannot have inter-cluster edges connecting them to ϕ, or connecting them to the original children µ i = µ * of ν, or exiting the border of R(ν). In particular, due to Hypothesis H-conn, these regions must be directly or indirectly connected to χ. Finally, consider the regions R(µ i ) corresponding to the original children µ i = µ * of ν (filled gray in Fig. 4). These regions may have inter-cluster edges among them, connecting them to ϕ, or connecting them to the rest of the graph outside ν. In particular, due to Hypotheses H-conn and H-not-root, each µ i (and also ϕ) must be directly or indirectly connected to the border of R(ν). It follows that the drawing in Γ (C i+1 ) of the subgraph G µ * composed by the regions of χ, ν 1 , ν 2 , . . . , ν h and their inter-cluster edges cannot contain in one of its internal faces any other cluster of ν. Hence, the sub-region R(µ * ) of R(ν) that is the union of R(χ) and the region enclosed by G µ * is a closed and simple region that only contains the regions R(ν 1 ), . . . R(ν h ) plus the region R(χ) and all the inter-cluster edges among them (see Fig. 5). By ignoring R(χ) and R(ϕ) and by removing vertices e χ and e ϕ and joining their incident edges we obtain a c-planar drawing Γ (C i ). The proof of Theorem 1 descends from Lemmas 5 and 6 and from the consideration that each construction of C i+1 from C i takes at most O(n) time and, hence, the time needed to construct C f is O(n 2 ). Due to the O(n + c)-time preprocessing (Lemma 2), the overall time complexity of the reduction is O(n 2 + c). Remarks about Theorem 1 In this section we discuss some consequences of Theorem 1 that descend from the properties of the reduction described in Section 3. Such properties are summarized in the following lemma. Lemma 7. Let C(G, T ) be an n-vertex clustered graph with c clusters. The flat clustered graph C f (G f , T f ) equivalent to C built as described in the proof of Theorem 1 has the following properties: 1. Graph G f is a subdivision of G 2. Each edge of G is replaced by a path of length at most 4h(T ) − 8 3. The number of vertices of G f is n f ∈ O(n · h(T )) 4. The number of clusters of C f is c f = c + S(T ) Proof. Regarding Property 1, observe that, for i = 1, . . . , S(T ), each G i is ob- tained from G i−1 by replacing edges with paths. Hence G S(T ) = G f is a sub- division of G 0 = G. To prove Property 2 observe that each time an edge e is subdivided, a pair of vertices e χ and e ϕ is inserted and that edges are subdivided when the boundary of a higher cluster is removed. Edges that traverse more boundaries are those that link two vertices whose lowest common ancestor is the root of T . These edges traverse 2h(T ) − 4 higher-cluster boundaries in C. Hence, the number of vertices inserted into these edges is 4h(T ) − 8. Property 3 can be proved by considering that G has O(n) edges and each edge, by Property 2, is replaced by a path of length at most O(h(T )). Finally, Property 4 descends from the fact that at each step C i+1 has exactly one cluster more than C i , since new clusters χ and ϕ are inserted but cluster µ * is removed. An immediate consequence of Property 1 of Lemma 7 is that the number of faces of G f is equal to the number of faces of G. Also, if G is connected, biconnected, or a subdivision of a triconnected graph, G f is also connected, biconnected, or a subdivision of a triconnected graph, respectively. If G is a cycle or a tree, G f is also a cycle or a tree, respectively. Hence, the complexity of Clustered Planarity restricted to these kinds of graphs can be related to the complexity of Flat Clustered Planarity restricted to the same kinds of graphs. Further, since a subdivision preserves the embedding of the original graph, the problem of deciding whether a c-graph C(G, T ) admits a c-planar drawing where G has a fixed embedding is polynomially equivalent to deciding whether a flat c-graph C f (G f , T f ) admits a c-planar drawing where G f has a fixed embedding. By the above observations some results on flat clustered graphs can be immediately exported to general c-graphs. Consider for example the following. We generalize Theorem 3 to non-flat c-graphs. Theorem 4. Let C(G, T ) be an n-vertex c-graph where G has a fixed embedding. There exists an O(n 3 · h(T ) 3 )-time algorithm to test the c-planarity of C if each lower cluster has at most two vertices on the same face of G and each higher cluster has at most two inter-cluster edges on the same face of G. Proof sketch. The proof is based on showing that, starting from a c-graph C(G, T ) that satisfies the hypotheses of the statement, the equivalent flat c-graph C f (G f , T f ) built as described in the proof of Theorem 1 satisfies the hypotheses of Theorem 3. Hence, we first transform C(G, T ) into C f (G f , T f ) in O(n 2 ) time and then apply Theorem 3 to C f (G f , T f ), which gives an answer to the c-planarity test in O(n 3 f ) time, which is, by Property 3 of Lemma 7, O(n 3 · h(T ) 3 ) time. In [24] it has been proven that Flat Clustered Planarity admits a subexponential-time algorithm when the underlying graph has a fixed embedding and its maximum face size belongs to o(n). The authors of [24] ask whether their results can be generalized to non-flat c-graphs. We give an affirmative answer with the following theorem. Proof sketch. The proof is based on applying Theorem 5 to the equivalent flat c-graph C f (G f , T f ) built as described in the proof of Theorem 1. Observe that Theorem 6 gives a subexponential-time upper bound for Clustered Planarity whenever · h(T ) 2 ∈ o(n). Also observe that Theorems 4 and 6 are actual generalizations of the corresponding Theorems 3 and 5, respectively, as they yield the same bounds when applied to flat clustered graphs. Proof of Theorem 2 In this section we reduce Flat Clustered Planarity to Independent Flat Clustered Planarity by applying a transformation very similar to the one described in Section 3 to each non-independent cluster. Let C(G, T ) be a flat c-graph. Let k be the number of lower clusters of C that are not independent. The reduction consists of a sequence of transformations of ρ µ * µ 4 µ 5 µ 3 e f g C i µ 2 µ 1 (a) ρ µ 4 µ 5 µ 3 C i+1 µ 2 µ 1 χ ϕ eχ fχ gχ eϕ fϕ gϕ fχ ν 1 ν 2 ν 3 ν 4 ν 5 (b)C = C 0 into C 1 , C 2 , . . . , C k where each C i , i = 0, . . . , k, is a flat c-graph with k − i non-independent lower clusters. Consider a flat c-graph C i (G i , T i ), with i = 0, . . . , k − 1, such that C i has k − i non-independent clusters and let µ * be a non-independent cluster of C. We show how to construct an flat c-graph C i+1 (G i+1 , T i+1 ) equivalent to C i and such that C i+1 has k − i − 1 non-independent clusters (refer to Fig. 6). Denote by µ j , with j = 1, 2, . . . , l, those children of r(T i ) such that µ j = µ * . Suppose that µ * has children v 1 , v 2 , . . . , v h , which are vertices of G i . The underlying graph G i+1 of C i+1 is obtained from G i by introducing, for each inter-cluster edge e = (u, v) of µ * , two new vertices e χ and e ϕ and replacing e with a path (u, e χ )(e χ , e ϕ )(e ϕ , v). The inclusion tree T i+1 of C i+1 is obtained from T i by removing cluster µ * and introducing, for each j = 1, 2, . . . , h, a lower cluster ν j child of r(T i+1 ) containing only v j . We also introduce two lower clusters χ and ϕ as children of r(T i+1 ) that contain all the vertices e χ and e ϕ , respectively, introduced when replacing each inter-cluster edge e of µ * with a path. It is easy to see that C i+1 is a flat clustered graph and that it has one non-independent cluster less than C i . We prove the following lemma assuming that Hypothesis H-conn holds. The complete proof is in Appendix D. Lemma 8. C i (G i , T i ) is c-planar if and only if C i+1 (G i+1 , T i+1 ) is c-planar. Proof sketch. The proof is very similar to the proof of Lemma 6. First, we show that, given a c-planar drawing Γ (C i ) of the flat c-graph C i it is easy to construct a c-planar drawing Γ (C i+1 ) of C i+1 (see, as an example, Fig. 7). Second we show that, given a c-planar drawing Γ (C i+1 ) of the flat c-graph C i+1 it is possible to construct a c-planar drawing Γ (C i ) of C i . This second part of the proof is complicated by the fact that, since in this case Hypothesis H-not-root does not apply, we may have that in Γ (C i+1 ) the region R(ϕ) is embraced by inter-cluster edges and region boundaries of R(ν 1 ), R(ν 2 ), . . . R(ν l ), and R(χ)(see Fig. 10(a) in Appendix D). Hence, before identifying the region R(µ * ) the drawing Γ (C i+1 ) needs to be modified so that the external face touches R(ϕ). This can be easily done by rerouting edges (see the example in Fig. 10(b)). µ * µ 2 µ 1 µ 3 µ 4 µ 5 e f g (a) µ 2 µ 1 µ 3 µ 4 µ 5 χ ϕ ν 5 ν 2 ν 4 ν 3 ν 1 (b) Remarks about Theorem 2 Starting from a flat c-graph, the reduction described in Section 5 allows us to find an equivalent independent flat c-graph with the properties stated in the following lemma (the proof is in Appendix E). Observation 1. At the same asymptotic cost of the reduction described in the proof of Theorem 2 it can be achieved that non-root clusters are of two types: ( Type 1) clusters containing a single vertex of arbitrary degree or ( Type 2) clusters containing multiple vertices of degree two. All observations of Section 4 regarding the consequences of Property 1 of Lemma 7 apply here to of Property 1 of Lemma 9. Further, the two reductions can be concatenated yielding the following. Conclusions and Open Problems We showed that Clustered Planarity can be reduced to Flat Clustered Planarity and that this problem, in turn, can be reduced to Independent Flat Clustered Planarity. The consequences of these results are twofold: on one side the investigations about the complexity of Clustered Planarity could legitimately be restricted to (independent) flat clustered graphs, neglecting more complex hierarchies of the inclusion tree; on the other side some polynomial-time results on flat clustered graphs could be easily exported to general c-graphs (we gave some examples in Section 4). We remark that while Theorems 1 and 2 are formulated in terms of decision problems, their proofs offer a solution of the corresponding search problems, meaning that they actually describe a polynomial-time algorithm to compute a c-planar drawing of a c-graph, provided to have a c-planar drawing of the corresponding flat c-graph or a c-planar drawing of the corresponding independent flat c-graph. Several interesting questions are left open: -Can the reduction presented in this paper be used to generalize some other polynomial-time testing algorithm for Flat Clustered Planarity to plain Clustered Planarity? -What is the complexity of Independent Flat Clustered Planarity when the underlying graph is a cycle? We know that this problem is polynomial only for constrained drawings of the inter-cluster edges [20, 21]. Proof. Let v be a leaf of T whose depth is h(T ). Since S(T ) > 0, T is not flat and h(T ) ≥ 3. Consider the parent µ * of the parent µ of v. The subtree T [µ * ] of T has h(T [µ * ]) = 2. Since T is homogeneous and µ * has a non-leaf child µ, all the children of µ * are internal nodes. Hence, T [µ * ] is flat. Lemma 2. An instance C(G, T ) of Clustered Planarity with n vertices and c clusters can be reduced in time O(n + c) to an equivalent instance such that: (1) T is homogeneous, (2) r(T ) has at least two children, and (3) h(T ) ≤ n − 1. Proof. First we prove Property (1). Suppose that the inclusion tree T of a c-graph C(G, T ) is not homogeneous. We transform C into an equivalent c-graph C h (G, T h ) such that T h is homogeneous. Consider a node µ * that has both internal node children and leaf children v 1 , v 2 , . . . , v k . For each such µ * and for each child v i of µ * , we insert between µ * and v i a lower node µ i that is child of µ * and parent of v i . The obtained c-graph C h (G, T h ) is homogeneous and may be constructed in time O(n + c). Also, given a c-planar drawing Γ h (C h ) of C h one can immediately obtain a c-planar drawing Γ (C) of C simply by ignoring the boundaries of the regions R(µ i ), where µ i is a cluster introduced by the above described transformation. Conversely, given a c-planar drawing Γ (C) of C one can obtain a c-planar drawing Γ h (C h ) of C h by inserting a small boundary around the vertices v i that changed their parent in the above transformation. Hence, C(G, T ) is c-planar if and only if C h (G, T h ) is c-planar. Finally, we prove Properties (2) and (3). Suppose to have an instance C(G, T ) with h(T ) > n−1 or such that r(T ) has a single child. We traverse T and recursively replace each cluster that has a single child with its child. The obtained instance C (G, T ) is equivalent to original one since from a c-planar drawing of C one can obtain a c-planar drawing of C simply by ignoring the boundaries of the removed clusters and from a c-planar drawing of C one can obtain a c-planar drawing of C by suitably adding the boundary of each removed parent cluster around the boundary of its child cluster (or around the child vertex leaf). Since all clusters of C have at least two children Property (2) is satisfied. We claim that h(T ) ≤ n. In fact, since all the n i internal nodes of T have degree at least two, we have that the number n of leaves of T is at least n ≥ n i + 1. Hence, h(T ) ≤ n i ≤ n − 1. This proves Property (3). Appendix B -Proofs of Lemmas 3-6 of Section 3. Let C i (G i , T i ) be a flat c-planar c-graph and let µ * = r(T i ) be a node of T i such that T i [µ * ] is flat. Denote by ν 1 , ν 2 , . . . , ν h the children of µ * and by µ 1 , µ 2 , . . . , µ k the siblings of µ * in T i . Let C i+1 (G i+1 , T i+1 ) be the flat c-graph constructed as described in Section 3. We prove the following lemmas. Lemma 3. If T i is homogeneous then T i+1 is homogeneous. Proof. Suppose T i is homogeneous. The only part of T i that is changed in T i+1 is the subtree T i [ν]. In particular, ν has all cluster children, while the newly introduced clusters χ and ϕ have all vertex children. Hence T i+1 is homogeneous. Lemma 4. We have that S(T i+1 ) = S(T i ) − 1. Proof. Consider a node µ = r(T i+1 ) of T i+1 for which h(T i+1 [µ]) > 1. Since the transformation of C i into C i+1 only reduces the height of some subtree of T i , such a node was also the root of a subtree of height greater than 1 in T i . Conversely, consider a node µ = r(T i ) of T i for which h(T i+1 [µ]) > 1. If µ = µ * then µ is not present in T i+1 , otherwise it is still the root of a subtree of height greater than one in T i+1 . Hence, the number of nodes that are root of subtrees of height greater than one of T i+1 is reduced by one with respect to the same number in T i . Lemma 5. The c-graph C f = C S(T ) is flat. Proof. By Property 1 of Lemma 2 we can assume that T is homogeneous. Lemma 3 ensures that all C i , with i = 1, . . . , S(T ) are also homogeneous. By Lemma 4 we have that the sizes of the trees T i are decreasing and, in particular, that the size of T S(T ) is S(T ) − S(T ) = 0. Therefore, T S(T ) is a homogeneous tree that has size 0 and, hence, is a flat tree. Now, we provide the proof of Lemma 6 in the general case, i.e., without leveraging on Hypotheses H-conn and H-not-root. Lemma 6. C i (G i , T i ) is c-planar if and only if C i+1 (G i+1 , T i+1 ) is c-planar. Proof. The first direction of the proof is straightforward. Let Γ (C i ) be a c-planar drawing of C i . We show how to construct a c-planar drawing Γ (C i+1 ) of C i+1 . Consider the region R(µ * ) that contains R(ν i ), with i = 1, . . . , h (refer to Fig. 3(a)). The boundary of R(µ * ) is crossed exactly once by each inter-cluster edge of µ * . Identify outside the boundary of R(µ * ) two arbitrarily thin regions R(χ) and R(ϕ) that follow the boundary of R(µ * ) and that intersect all and only the inter-cluster edges of µ * exactly once (see Fig. 3(b)). Insert into each inter-cluster edge e of µ * two vertices e χ and e ϕ , placing e χ inside R(χ) and e ϕ inside R(ϕ). By ignoring R(µ * ) you have a c-planar drawing Γ (C i+1 ) of C i+1 . Conversely, suppose to have a c-planar drawing Γ (C i+1 ) of C i+1 . We show how to construct a c-planar drawing Γ (C i ) of C i . Consider the regions R(χ) and R(ϕ) inside R(ν) (refer to Fig. 8(a)). Regions R(χ) and R(ϕ) are joined by the p inter-cluster edges (drawn red in Fig. 8(a)) introduced when replacing each inter-cluster edge e i of µ * , where i = 1, . . . , p, with a path. Such inter-cluster edges of χ and ϕ partition R(ν) into p regions that have to host the remaining children of ν and the inter-cluster edges among them. In particular, p − 1 of these regions are simple and bounded by two inter-cluster edges and two portions of the boundaries of R(χ) and R(ϕ). One of such regions, instead, is also externally bounded by the boundary of R(ν). Now, consider the regions corresponding to the children ν i of ν, with i = 1, . . . , h, that were originally children of µ * . These regions (filled white in Fig. 8(a)) may be without any inter-cluster edge (as, for example, R(ν 11 ) in Fig. 8(a)); may have inter-cluster edges among themselves (as, for example, R(ν 2 ), R(ν 3 ), R(ν 4 ), R(ν 5 ), R(ν 6 ), R(ν 12 ), and R(ν 13 ) in Fig. 8(a)); and may be connected to R(χ) (as, for example, R(ν 1 ), R(ν 2 ), R(ν 3 ), R(ν 4 ), R(ν 6 ), R(ν 7 ), R(ν 8 ), R(ν 9 ), and R(ν 10 ) in Fig. 8(a)). However, by construction these regions cannot have inter-cluster edges connecting them to R(ϕ), or connecting them to the regions of the original children µ i of ν, or exiting the border of R(ν). Hence, the regions corresponding to ν 1 , . . . , ν h can be classified into two sets, denoted A χ and F χ , of 'anchored regions' and 'floating regions' of χ, respectively, where an anchored region of χ is a region R(ν a ) whose cluster ν a contains at least one vertex of G i+1 that is connected (via a path) to a vertex in χ and a floating region of χ is a region R(ν f ) whose cluster ν f contains all vertices not connected to vertices in χ. For example, in Fig. 8(a), F χ contains R(ν 11 ), R(ν 12 ), and R(ν 13 ), while A χ contains all the other white-filled regions. Analogously, consider the regions R(µ j ), with j = 1, . . . , k, corresponding to the original children µ j = µ * of ν (filled gray in Fig. 8(a)). These regions may be without any inter-cluster edge (as, for example, R(µ 11 ), R(µ 14 ), and R(µ 15 ) in Fig. 8(a)); may have inter-cluster edges among themselves (as, for example, R(µ 1 ), R(µ 3 ), R(µ 5 ), R(µ 6 ), R(µ 7 ), R(µ 12 ), and R(ν 13 ) in Fig. 8(a)); may have inter-cluster edges connecting them to R(ϕ) (as, for example, R(µ 2 ), R(µ 3 ), R(µ 4 ), R(µ 6 ), R(µ 7 ), R(µ 9 ), and R(µ 10 ) in Fig. 8(a)); or may have inter-cluster edges connecting them the rest of the graph outside R(ν) (as, for example, R(µ 8 ) in Fig. 8(a)). However, by construction these regions cannot have inter-cluster edges connecting them to R(χ), or connecting them to the the regions in F χ or A χ . Hence, we can classify the regions corresponding to µ 1 , . . . , µ k into two sets, denoted A ϕ and F ϕ , of 'anchored regions' and 'floating regions' of ϕ, where an anchored region of ϕ is a region R(µ a ) whose cluster µ a contains at least one vertex of G i+1 that is connected to a vertex in ϕ or to a vertex outside ν and a floating region of ϕ is a region R(µ f ) whose cluster µ f contains all vertices not connected to vertices in ϕ nor outside ν. For example, in Fig. 8(a), set F ϕ contains R(µ 11 ), R(µ 12 ), R(µ 13 ), R(µ 14 ), and R(µ 15 ), while A ϕ contains all the other gray-filled regions. Our strategy will be that of removing altogether from Γ (C i+1 ) the drawings of the floating regions (and all their content), possibly modifying the drawing of the remaining graph, and then suitably reinserting the drawing of the floating regions. Suppose now to have temporarily removed from Γ (C i+1 ) the drawings of the floating regions in F χ and F ϕ (see, for example, Fig. 8(b)). We define an auxiliary multigraph H that has one vertex v χ representing χ and one vertex v νi for each child ν i of µ * such that R(ν i ) ∈ A χ . For each inter-cluster edge between two clusters λ 1 and λ 2 corresponding to the vertices v λ1 and v λ2 of H χ , respectively, we add an edge (v λ1 , v λ2 ) to H. Observe that H, by the definition of the anchored regions in A χ , is connected. Drawing Γ (C i+1 ) induces a drawing Γ (H) of the multigraph H, where each vertex v λ of H is represented by the region R(λ) of the cluster λ corresponding to v λ and each edge (v λ1 , v λ2 ) of H is represented as the corresponding inter-cluster edge of λ 1 and λ 2 restricted to the portion that is drawn outside the boundaries of R(λ 1 ) and R(λ 2 ). There are two cases: either Γ (H) does not contain in one of its internal faces R(ϕ) (Case 1, depicted in Fig. 8(b)) or it contains R(ϕ) (Case 2, depicted in Fig. 9(a)). In Case 1 no change has to be done to Γ (C i+1 ). In Case 2 we modify Γ (H) and, consequently, Γ (C i+1 ) so to fall again into Case 1. Namely, we identify a minimal set {e 1 , e 2 , . . . , e q } of edges of H that, if removed, would bring R(ϕ) on the external face of Γ (H) (for example in Fig 9(a) this set contains only edge e 1 ). Starting from edge e 1 , that is incident to the external face of Γ (H), we redraw each e i , with i = 1, . . . , q, as follows. Suppose that the curve for e i = (v λ1 , v λ2 ) in Γ (H) starts from a point p 1 on the boundary of R(λ 1 ) and ends with a point p 2 on the boundary of R(λ 2 ). We arbitrarily choose two distinct points p 3 and p 4 , encountered in this order when traversing e i from p 1 to p 2 . We remove the portion of e i between p 3 and p 4 and we redraw it by returning back from p 3 towards p 1 on the external face of Γ (H) and then moving along the external face of Γ (H) until we reach p 4 (see, for example, Fig. 9(b)). Observe that this corresponds to moving the external face of Γ (H) to a face that was previously an internal face of Γ (H) enclosed by e i . We carry on doing the same operation for each e i , with i = 1, . . . , q, until the external face of Γ (H) is incident on the boundary of R(ϕ). At this point we are in Case 1. Observe that since Γ (H) does not contain in one of its internal faces R(ϕ), then it cannot contain any region in A ϕ either, as, by definition, these regions are either connected to R(ϕ) or to the boundary of R(ν). Hence, the internal faces of Γ (H) only contain vertices and edges that in C i belong to µ * . Now we reinsert the drawings of the floating regions. We identify an arbitrarily small empty disk F χ inside R(χ) and move inside F χ the (suitably scaled down) drawings of the floating regions in F χ . Analogously, we identify an arbitrarily small empty disk F ϕ inside R(ϕ) and move inside F ϕ the (suitably scaled down) drawings of the floating regions in F ϕ . Consider the region R(µ * ) that is the region covered by Γ (H). Such a region is connected, is simple, contains only vertices and nodes of µ * , and its boundary is a simple curve (see Fig. 9(b)). Therefore, by neglecting the boundaries of R(χ) and R(ϕ) and by removing their internal vertices and joining their incident edges we obtain a c-planar drawing Γ (C i ) of C i . The transformation of G i into G i+1 described in the proof of Theorem 1 removes one higher cluster µ * and introduces two lower clusters χ and ϕ. Each inter-cluster edge e = (u, v) of µ * is subdivided into three edges (u, e χ ), (e χ , e ϕ ), and (e ϕ , v), where e χ ∈ χ and e ϕ ∈ ϕ. Since at most two inter-cluster edges of µ * belong to the same face of G i we have that the lower clusters χ and ϕ have at most two vertices on the same face of G i+1 . Any other higher or lower cluster of T i+1 is not modified by the transformation. It follows that C f (G f , T f ), which, with the exception of the root cluster, has only lower clusters, satisfies the conditions of Theorem 3. Hence, we first transform C(G, T ) into C f (G f , T f ) in O(n 2 ) time (Theorem 1) and then apply Theorem 3 to C f (G f , T f ), which gives an answer to the c-planarity test in O(n 3 f ) time, which is, by Property 3 of Lemma 7, O(n 3 · h(T ) 3 ) time. Proof. The proof is based on applying Theorem 5 to the flat c-graph C f (G f , T f ) built as described in the proof of Theorem 1 and equivalent to C(G, T ). By Property 2 of Lemma 7 each edge of G is replaced by a path of length at most 2h(T ) − 2. Hence, each face of G f has a maximum size f = · O(h(T )). Also, by Property 3 of Lemma 7 we have that the number of vertices of G f is n f ∈ O(n · h(T )). Theorem 5 guarantees that we can test for c-planarity in 2 O( √ f n f ·log n f ) time, which gives the statement. Appendix D -Proof of Lemma 8 of Section 5. In this section, we provide the proof of Lemma 8 in the general case, i.e., without leveraging on Hypothesis H-conn. Let C i (G i , T i ) be a flat c-planar c-graph and let µ * be a non-independent cluster of C i containing vertices v 1 , v 2 , . . . , v h of G i . Also, denote by ν j , with j = 1, 2, . . . , l, those children of r(T i ) such that ν j = µ. Let C i+1 (G i+1 , T i+1 ) be the flat c-graph constructed as described in Section 5. We have the following. Lemma 8. C i (G i , T i ) is c-planar if and only if C i+1 (G i+1 , T i+1 ) is c-planar. Proof. The proof is similar to the proof of Lemma 6. Given a c-planar drawing Γ (C i ) of the flat c-graph C i , we show how to construct a c-planar drawing Γ (C i+1 ) of C i+1 (refer to Fig. 7). The construction is based on identifying two arbitrary thin regions R(χ) and R(ϕ) outside the border of R(µ * ) such that R(χ) and R(ϕ) intersect exactly once all and only the inter-cluster edges of µ * . By ignoring R(µ * ), by inserting for each intercluster edge e of µ * vertices e χ and e ϕ in R(χ) and R(ϕ), respectively, and by adding a boundary to each vertex v 1 , . . . , v h , we obtain Γ (C i+1 ). Converserly, given a c-planar drawing Γ (C i+1 ) of the flat c-graph C i+1 , we show how to construct a c-planar drawing Γ (C i ) of C i (refer to Fig. 10). Consider the regions corresponding to the independent clusters ν i , with i = 1, . . . , h, containing the nodes that were originally children of µ * . These regions (filled white in Fig. 10(a)) 20 P. F. Cortese and M. Patrignani Fig. 6(b). (b) The corresponding c-planar drawing the flat c-graph of Fig. 6(a). The gray region is R(µ * ). may be without any inter-cluster edge; may have inter-cluster edges among themselves; and may be connected to R(χ). However, by construction these regions cannot have inter-cluster edges connecting them to R(ϕ), or connecting them to the regions of the original children µ i of ρ. Hence, the regions corresponding to ν 1 , . . . , ν h can be classified into two sets, denoted A χ and F χ , of 'anchored regions' and 'floating regions' of χ, respectively, where an anchored region of χ is a region R(ν a ) containing a vertex of G i+1 that is connected (via a path) to a vertex in χ and a floating region of χ is a region R(ν f ) containing a vertex that is not connected to vertices in χ. Analogously, consider the regions R(µ j ), with j = 1, . . . , l, corresponding to the original children µ j = µ * of ρ (filled gray in Fig. 10(a)). These regions may be without any inter-cluster edge; may have inter-cluster edges among them; or may have inter-cluster edges connecting them to R(ϕ). However, by construction these regions cannot have inter-cluster edges connecting them to R(χ), or connecting them to the the regions in F χ or A χ . Hence, we can classify the regions corresponding to µ 1 , . . . , µ l into two sets, denoted A ϕ and F ϕ , of 'anchored regions' and 'floating regions' of ϕ, where an anchored region of ϕ is a region R(µ a ) whose cluster µ a contains at least one vertex of G i+1 that is connected to a vertex in ϕ and a floating region of ϕ is a region R(µ f ) whose cluster µ f contains all vertices not connected to vertices in ϕ. Our strategy will be that of removing altogether from Γ (C i+1 ) the drawings of the floating regions (and all their content), possibly modifying the drawing of the remaining graph, and then suitably reinserting the drawing of the floating regions. Suppose now to have temporarily removed from Γ (C i+1 ) the drawings of the floating regions in F χ and F ϕ . We define an auxiliary multigraph H that has one vertex v χ representing χ and one vertex v νi for each singleton ν i introduced when removing µ * such that R(ν i ) ∈ A χ . For each inter-cluster edge between two clusters λ 1 and λ 2 corresponding to the vertices v λ1 and v λ2 of H χ , respectively, we add an edge (v λ1 , v λ2 ) to H. Observe that H, by the definition of the anchored regions in A χ , is connected. Drawing Γ (C i+1 ) induces a drawing Γ (H) of the multigraph H, where each vertex v λ of H is represented by the region R(λ) of the cluster λ corresponding to v λ and each edge (v λ1 , v λ2 ) of H is represented as the corresponding inter-cluster edge of λ 1 and λ 2 restricted to the portion that is drawn outside the boundaries of R(λ 1 ) and R(λ 2 ). Two are the cases: either Γ (H) does not contain in one of its internal faces R(ϕ) (Case 1) or it contains R(ϕ) (Case 2, depicted in Fig. 10(a)). In Case 1 no change has to be done to Γ (C i+1 ). In Case 2 we modify Γ (H) and, consequently, Γ (C i+1 ) so to fall again into Case 1. Namely, we identify a minimal set {e 1 , e 2 , . . . , e q } of edges of H that, if removed, would bring R(ϕ) on the external face of Γ (H) (for example in Fig 10(a) this set contains only edge e 1 ). Starting from edge e 1 , that is incident to the external face of Γ (H), we redraw each e i , with i = 1, . . . , q, as follows. Suppose that the curve for e i = (v λ1 , v λ2 ) in Γ (H) starts from a point p 1 on the boundary of R(λ 1 ) and ends with a point p 2 on the boundary of R(λ 2 ). We arbitrarily choose two distinct points p 3 and p 4 , encountered in this order when traversing e i from p 1 to p 2 . We remove the portion of e i between p 3 and p 4 and we redraw it by returning back from p 3 towards p 1 on the external face of Γ (H) and then moving along the external face of Γ (H) until we reach p 4 (see, for example, Fig. 10(b)). Observe that this corresponds to moving the external face of Γ (H) to a face that was previously an internal face of Γ (H) enclosed by e i . We carry on doing the same operation for each e i , with i = 1, . . . , q, until the external face of Γ (H) is incident on the boundary of R(ϕ). At this point we are in Case 1. Observe that since Γ (H) does not contain in one of its internal faces R(ϕ), then it cannot contain any region in A ϕ either, as, by definition, these regions are connected to R(ϕ). Hence, the internal faces of Γ (H) only contain vertices and edges that in C i belong to µ * . Now we reinsert the drawings of the floating regions. We identify an arbitrarily small empty disk F χ inside R(χ) and move inside F χ the (suitably scaled down) drawings of the floating regions in F χ . Analogously, we identify an arbitrarily small empty disk F ϕ inside R(ϕ) and move inside F ϕ the (suitably scaled down) drawings of the floating regions in F ϕ . Consider the region R(µ * ) that is the region covered by Γ H. Such a region is connected, is simple, contains only vertices and nodes of µ * , and its boundary is a simple curve (see Fig. 10(b)). Therefore, by neglecting the boundaries of R(χ), R(ϕ), ν 1 , ν 2 , . . . , ν h and by removing the internal vertices of R(χ) and R(ϕ) and joining their incident edges we obtain a c-planar drawing Γ (C i ) of C i . in Appendix B). Some immediate consequences of Theorem 1 are discussed in Section 4. The proof of Theorem 2 and some remarks about it are in Sections 5 and 6, respectively. Conclusions and open problems are in Section 7. For space reasons some proofs are moved to the appendix. Fig. 1 . 1(a) A tree that is not homogeneous. (b) A homogeneous tree. (c) A flat tree. Fig. 3 . 3(a) A c-planar drawing Γ (Ci) of c-graph Ci. (b) The construction of a c-planar drawing Γ (Ci+1). Fig. 4 . 4A c-planar drawing of clusters ν, χ, and ϕ in Γ (Ci+1). Fig. 5 . 5The drawing of cluster µ in Γ (Ci) corresponding to the drawing Γ (Ci+1) ofFig. 4. Theorem 3 . 3([16, Theorem 1]). There exists an O(n 3 )-time algorithm to test the c-planarity of an n-vertex embedded flat c-graph C with at most two vertices per cluster on each face. Theorem 5 . 5([24, Theorem 3]). Flat Clustered Planarity can be solved in 2 O( √ n·log n) time for n-vertex embedded flat c-graphs with maximum face size . Theorem 6 . 6Clustered Planarity can be solved in 2 O(h(T )· √ n·log(n·h(T )) time for n-vertex embedded c-graphs with maximum face size and height h(T ) of the inclusion tree. Fig. 6 . 6(a) A flat c-graph Ci with a non-independent cluster µ * . (b) The construction of Ci+1 where µ * is replaced by independent clusters ν1, . . . , ν5, χ, and ϕ. Fig. 7 . 7(a) A c-planar drawing of the flat c-graph of Fig. 6(a). (b) The corresponding c-planar drawing the flat c-graph of Fig. 6(b) where the non-independent cluster µ * is replaced by independent clusters ν1, . . . , ν5, χ, and ϕ.The proof of Theorem 2 is concluded by showing that each G i+1 can be obtained from G i in time proportional to the number of vertices and inter-cluster edges of µ * , which gives an overall O(n) time for the reduction. Lemma 9 . 9Let C f (G f , T f ) be an n f -vertex flat clustered graph with c f clusters. The independent flat clustered graph C if (G if , T if ) equivalent to C f built as described in the proof of Theorem 2 has the following properties:1. Graph G if is a subdivision of G f 2. Each inter-cluster edge of G f is replaced by a path of length at most 4.3. The number of vertices ofG if is O(n f ) 4. The number of clusters of C if (including the root) is c if ≤ 2c f + n f − 1Also, a further property can be pursued. Lemma 10 . 10Let C(G, T ) be an n-vertex clustered graph with c clusters. The independent flat clustered graph C if (G if , T if ) equivalent to C built by concatenating the reduction of Theorem 1 and the reduction of Theorem 2, as modified by Observation 1, has the following properties:1. Graph G if is a subdivision of G 2. Each inter-cluster edge of G f is replaced by a path of length at most 4h(T )−4 3. The number of vertices of G if is O(n 2 ) 4. The number of clusters of C if is O(n · h(T )) 5.Non-root clusters are of two types: ( Type 1) clusters containing a single vertex of arbitrary degree or ( Type 2) clusters containing multiple vertices of degree two Lemma 10 describes the most constrained version of Clustered Planarity that is known to be polynomially equivalent to the general problem. Observe that if all non-root clusters of a c-graph C(G, T ) are of Type 1 then Independent Flat Clustered Planarity is linear, since C is c-planar if and only if G is planar. Conversely, if all clusters are of Type 2 then the underlying graph is a collection of cycles, and the problem has unknown complexity[20, 21]. - What is the complexity of Independent Flat Clustered Planarity when the number of Type 2 clusters is bounded? 18. Cortese, P.F., Di Battista, G.: Clustered planarity (invited lecture). In: SoCG 05. pp. 30-32. ACM (2005) 19. Cortese, P.F., Di Battista, G., Frati, F., Patrignani, M., Pizzonia, M.: C-planarity of c-connected clustered graphs. J. Graph Algorithms Appl. 12(2), 225-262 (2008) 20. Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: Clustering cycles into cycles of clusters. J. Graph Algorithms Appl. 9(3), 391-413 (2005) 21. Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: On embedding a cycle in a plane graph. Discrete Mathematics 309(7), 1856-1869 (2009), doi:10.1016/j.disc.2007.12.090 22. Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M.: Computing NodeTrix representations of clustered graphs. 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(b) The same drawing after the removal of the floating regions. Fig. 9 . 9(a) A possible drawing of cluster ν in Γ (Ci+1) in the case of non-connected Gi+1. (b) The corresponding drawing of ν in Γ (Ci). Theorem 6 . 6Clustered Planarity can be solved in 2 O(h(T )· √ n·log(n·h(T )) time for n-vertex embedded cgraphs with maximum face size and height h(T ) of the inclusion tree. Fig. 10 . 10(a) A c-planar drawing of the flat c-graph of Fig. 11 . 11A figure for the proof of Lemma 10. (a) An example of a c-graph where h(T ) = 5. Edges connecting vertices in two lower clusters µ1 and µ2 traverse at most 2h(T ) − 4 = 6 boundaries of higher clusters. (b) The corresponding flat c-graph obtained as described in the proof of Theorem 1 replaces each edge with a path of at most 4h(T ) − 8 = 12. (c) The final independent flat c-graph obtained as described in the proof of Theorem 2 replaces each original edge with a path of length at most 4h(T ) − 4 = 16. Appendix C -Proof of Theorems 4 and 6 of Section 4. Theorem 4. Let C(G, T ) be an n-vertex c-graph where G has a fixed embedding. There exists an O(n 3 · h(T ) 3 )time algorithm to test the c-planarity of C if each lower cluster has at most two vertices on the same face of G and each higher cluster has at most two inter-cluster edges on the same face of G.Proof. The proof is based on showing that, starting from a c-graph C(G, T ) that satisfies the hypotheses of the statement, the equivalent flat c-graph C f (G f , T f ) built as described in the proof of Theorem 1 satisfies the hypotheses of Theorem 3. By Property 1 of Lemma 2 we can assume that T is homogeneous. 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Springer (2008), doi:10.1007/978- 3-642-00219-9 13 Clustered planarity: Clusters with few outgoing edges. V Jelínek, O Suchý, M Tesar, T Vyskocil, GD 2008. Tollis, I.G., Patrignani, M.5417Jelínek, V., Suchý, O., Tesar, M., Vyskocil, T.: Clustered planarity: Clusters with few outgoing edges. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 102-113 (2009) Clustered planarity: Small clusters in cycles and eulerian graphs. E Jelínková, J Kára, J Kratochvíl, M Pergel, O Suchý, T Vyskocil, 10.7155/jgaa.00192J. Graph Algorithms Appl. 133Jelínková, E., Kára, J., Kratochvíl, J., Pergel, M., Suchý, O., Vyskocil, T.: Clus- tered planarity: Small clusters in cycles and eulerian graphs. J. Graph Algorithms Appl. 13(3), 379-422 (2009), doi:10.7155/jgaa.00192 Visualisierungstechniken für den compilerbau. G Sander, GermanyUniversität SaarbrückenPhD ThesisSander, G.: Visualisierungstechniken für den compilerbau. PhD Thesis, Universität Saarbrücken, Germany (1996) Toward a theory of planarity: Hanani-Tutte and planarity variants. M Schaefer, 10.7155/jgaa.00298J. Graph Algorithms Appl. 174Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algorithms Appl. 17(4), 367-440 (2013), doi:10.7155/jgaa.00298 This proves Property 2. Since by Property 2 each edge is replaced by a path of bounded length and G f has O(n f ) edges, the number of vertices of G if is O(n f ) (Property 3). all vertices of µ * are enclosed into new singleton clusters. Therefore, the number c if of clusters of C if is at most 2c f +. S Zhao, M J Mcguffin, M H Chignell, 10.1109/INFOVIS.2005.12Elastic hierarchies: Combining treemaps and node-link diagrams. Stasko, J.T., Ward, M.O.8particular, every inter-cluster edge of µ * is subdivided twice when removing the nonindependent cluster µ * . It follows that inter-cluster edges are replaced by paths of length at most 4 (exactly 4 if the edge links two non-independent clusters). n f − 1 (the minus 1 is due to the fact that the root cluster does not need to be removedZhao, S., McGuffin, M.J., Chignell, M.H.: Elastic hierarchies: Combining treemaps and node-link diagrams. In: Stasko, J.T., Ward, M.O. (eds.) IEEE InfoVis 2005. p. 8 (2005), doi:10.1109/INFOVIS.2005.12 particular, every inter-cluster edge of µ * is subdivided twice when removing the non- independent cluster µ * . It follows that inter-cluster edges are replaced by paths of length at most 4 (exactly 4 if the edge links two non-independent clusters). This proves Property 2. Since by Property 2 each edge is replaced by a path of bounded length and G f has O(n f ) edges, the number of vertices of G if is O(n f ) (Property 3). all vertices of µ * are enclosed into new singleton clusters. Therefore, the number c if of clusters of C if is at most 2c f + n f − 1 (the minus 1 is due to the fact that the root cluster does not need to be removed). At the same asymptotic cost of the reduction described in the proof of Theorem 2 it can be achieved that non-root clusters are of two types: ( Type 1) clusters containing a single vertex of arbitrary degree or ( Type 2) clusters containing multiple vertices of degree two. Observation 1Observation 1. At the same asymptotic cost of the reduction described in the proof of Theorem 2 it can be achieved that non-root clusters are of two types: ( Type 1) clusters containing a single vertex of arbitrary degree or ( Type 2) clusters containing multiple vertices of degree two. The property is achieved if, in addition to removing non-independent clusters of the instace C f (G f , T f ), we also use the same technique described in the proof of Theorem 2 to remove those independent clusters of C f that contain at least one vertex of degree greater than 2. In this case all clusters of C if that contain more than one vertex are guaranteed to have all degree-two vertices. The cost of the reduction is still linear for the same reasons discussed in the proof of Theorem 2Proof. The property is achieved if, in addition to removing non-independent clusters of the instace C f (G f , T f ), we also use the same technique described in the proof of Theorem 2 to remove those independent clusters of C f that contain at least one vertex of degree greater than 2. In this case all clusters of C if that contain more than one vertex are guaranteed to have all degree-two vertices. The cost of the reduction is still linear for the same reasons discussed in the proof of Theorem 2. T ) be an n-vertex clustered graph with c clusters. The independent flat clustered graph C if (G if , T if ) equivalent to C built by concatenating the reduction of Theorem 1 and the reduction of Theorem 2, as modified by Observation 1. Lemma 10. Let C(Ghas the following propertiesLemma 10. Let C(G, T ) be an n-vertex clustered graph with c clusters. The independent flat clustered graph C if (G if , T if ) equivalent to C built by concatenating the reduction of Theorem 1 and the reduction of Theorem 2, as modified by Observation 1, has the following properties: Non-root clusters are of two types: ( Type 1) clusters containing a single vertex of arbitrary degree or ( Type 2) clusters containing multiple vertices of degree two. Non-root clusters are of two types: ( Type 1) clusters containing a single vertex of arbitrary degree or ( Type 2) clusters containing multiple vertices of degree two T ) into the flat c-graph C f (G f , T f ) replaces an edge with a path of length at most 4h(T ) − 8 (see Figs. 11(a) and 11(b)). When transforming C f (G f , T f ) into the independent flat c-graph C if (G if , T if ) only the original lower clusters of C (µ 1 and µ 2 in the example of Fig. 11) need to be replaced, since the cluster introduced by the first transformation are already independent and of Type 2. By Property 2 of Lemma 9 this adds 4 more internal vertices to each replaced edge (see Fig. 11(c)). Hence. 1Proof. Properties 1, 3, and 4 directly descends by concatenating the analogous Properties. each edge of G is replaced by a path of length at most 4h(T ) − 4 in G ifProof. Properties 1, 3, and 4 directly descends by concatenating the analogous Properties 1, 3, and 4 of Lemmas 7 of C(G, T ) into the flat c-graph C f (G f , T f ) replaces an edge with a path of length at most 4h(T ) − 8 (see Figs. 11(a) and 11(b)). When transforming C f (G f , T f ) into the independent flat c-graph C if (G if , T if ) only the original lower clusters of C (µ 1 and µ 2 in the example of Fig. 11) need to be replaced, since the cluster introduced by the first transformation are already independent and of Type 2. By Property 2 of Lemma 9 this adds 4 more internal vertices to each replaced edge (see Fig. 11(c)). Hence, each edge of G is replaced by a path of length at most 4h(T ) − 4 in G if .	[]
[ "ONLINE SIMULATOR-BASED EXPERIMENTAL DESIGN FOR COGNITIVE MODEL SELECTION", "ONLINE SIMULATOR-BASED EXPERIMENTAL DESIGN FOR COGNITIVE MODEL SELECTION" ]	[ "Alexander Aushev [email protected] ", "Aini Putkonen [email protected] ", "Gregoire Clarte [email protected] ", "Suyog Chandramouli [email protected] ", "Luigi Acerbi [email protected] ", "Samuel Kaski [email protected] ", "Andrew Howes [email protected] ", "\nDepartment of Computer Science\nDepartment of Communications and Networking\nAalto University\nFinland\n", "\nDepartment of Computer Science\nAalto University\nFinland\n", "\nDepartment of Communications and Networking\nUniversity of Helsinki & FCAI\nFinland\n", "\nDepartment of Computer Science\nAalto University\nFinland\n", "\nDepartment of Computer Science\nUniversity of Helsinki & FCAI\nFinland\n", "\nDepartment of Computer Science\nAalto University\nFinland\n", "\nSchool of Computer Science\nUniversity of Manchester\nUK\n", "\nUniversity of Birmingham\nUK\n" ]	[ "Department of Computer Science\nDepartment of Communications and Networking\nAalto University\nFinland", "Department of Computer Science\nAalto University\nFinland", "Department of Communications and Networking\nUniversity of Helsinki & FCAI\nFinland", "Department of Computer Science\nAalto University\nFinland", "Department of Computer Science\nUniversity of Helsinki & FCAI\nFinland", "Department of Computer Science\nAalto University\nFinland", "School of Computer Science\nUniversity of Manchester\nUK", "University of Birmingham\nUK" ]	[]	The problem of model selection with a limited number of experimental trials has received considerable attention in cognitive science, where the role of experiments is to discriminate between theories expressed as computational models. Research on this subject has mostly been restricted to optimal experiment design with analytically tractable models. However, cognitive models of increasing complexity, with intractable likelihoods, are becoming more commonplace. In this paper, we propose BOSMOS: an approach to experimental design that can select between computational models without tractable likelihoods. It does so in a data-efficient manner, by sequentially and adaptively generating informative experiments. In contrast to previous approaches, we introduce a novel simulator-based utility objective for design selection, and a new approximation of the model likelihood for model selection. In simulated experiments, we demonstrate that the proposed BOSMOS technique can accurately select models in up to 2 orders of magnitude less time than existing LFI alternatives for three cognitive science tasks: memory retention, sequential signal detection and risky choice.	10.48550/arxiv.2303.02227	[ "https://export.arxiv.org/pdf/2303.02227v1.pdf" ]	257,365,234	2303.02227	289e743d9fb3ac687ce6ca6f681febe3bc06fe79	ONLINE SIMULATOR-BASED EXPERIMENTAL DESIGN FOR COGNITIVE MODEL SELECTION March 7, 2023 Alexander Aushev [email protected] Aini Putkonen [email protected] Gregoire Clarte [email protected] Suyog Chandramouli [email protected] Luigi Acerbi [email protected] Samuel Kaski [email protected] Andrew Howes [email protected] Department of Computer Science Department of Communications and Networking Aalto University Finland Department of Computer Science Aalto University Finland Department of Communications and Networking University of Helsinki & FCAI Finland Department of Computer Science Aalto University Finland Department of Computer Science University of Helsinki & FCAI Finland Department of Computer Science Aalto University Finland School of Computer Science University of Manchester UK University of Birmingham UK ONLINE SIMULATOR-BASED EXPERIMENTAL DESIGN FOR COGNITIVE MODEL SELECTION March 7, 2023 The problem of model selection with a limited number of experimental trials has received considerable attention in cognitive science, where the role of experiments is to discriminate between theories expressed as computational models. Research on this subject has mostly been restricted to optimal experiment design with analytically tractable models. However, cognitive models of increasing complexity, with intractable likelihoods, are becoming more commonplace. In this paper, we propose BOSMOS: an approach to experimental design that can select between computational models without tractable likelihoods. It does so in a data-efficient manner, by sequentially and adaptively generating informative experiments. In contrast to previous approaches, we introduce a novel simulator-based utility objective for design selection, and a new approximation of the model likelihood for model selection. In simulated experiments, we demonstrate that the proposed BOSMOS technique can accurately select models in up to 2 orders of magnitude less time than existing LFI alternatives for three cognitive science tasks: memory retention, sequential signal detection and risky choice. Introduction The problem of selecting between competing models of cognition is critical to progress in cognitive science. The goal of model selection is to choose the model that most closely represents the cognitive process which generated the observed behavioural data. Typically, model selection involves maximizing the fit of each model's parameters to data and balancing the quality of the model-fit and its complexity. It is crucial that any model selection method used is robust and sample-efficient, and that it correctly measures how well each model approximates the data-generating cognitive process. It is also crucial that any model selection process is provided with high quality data from well-designed experiments, and that these data are sufficiently informative to support efficient selection. Research on optimal experimental design arXiv:2303.02227v1 [cs.LG] 3 Mar 2023 (OED) addresses this problem by focusing on how to design experiments that support parameter estimation of single models and, in some cases, maximize information for model selection [Cavagnaro et al., 2010, Moon et al., 2022, Blau et al., 2022. However, one outstanding difficulty for model selection is that many models do not have tractable likelihoods. The model likelihoods represent the probability of observed data being produced by model parameters and simplify tractable inference [van Opheusden et al., 2020]. In their absence, likelihood-free inference (LFI) methods can be used, which rely on forward simulations (or samples from the model) to replace the likelihood. Another difficulty is that existing methods for OED are slow -very slow -which makes them impractical for many applications. In this paper, we address these problems by investigating a new algorithm that automatically designs experiments for likelihood-free models much more quickly than previous approaches. The new algorithm is called Bayesian optimization for simulator-based model selection (BOSMOS). In BOSMOS, model selection is conducted in a Bayesian framework. In this setting, inference is carried out using marginal likelihood, which incorporate, by definition, a penalty for model complexity, i.e., Occam's Razor. Additionally, the Bayesian framework allows getting Bayesian posteriors over all possible values, rather than point estimates; this is crucial for quantifying uncertainty, for instance, when multiple models can explain the data similarly well (non-identifiability or poor identifiability; Anderson, 1978, Acerbi et al., 2014, or when some of the models are misspecified (e.g. the behaviour cannot be reproduced by the model due to non-independence of the experimental trials; Lee et al., 2019). These problems are compounded in computational cognitive modeling where non-identifiability also arises due to human strategic flexibility , Madsen et al., 2019, Kangasrääsiö et al., 2019. For these reasons, there is an interest in Bayesian approaches in computational cognitive science [Madsen et al., 2018]. As we have said, a key problem for model selection is selection of the design variables that define an experiment. When resources are limited, experimental designs can be carefully selected to yield as much information about the models as possible. Adaptive Design Optimization (ADO) [Cavagnaro et al., 2010[Cavagnaro et al., , 2013b is one influential approach to selecting experimental designs. ADO proposes designs by maximizing the so-called utility objective, which measures the amount of information about the candidate models and their quality. Unfortunately, common utility objectives, such as mutual information [Shannon, 1948, Cavagnaro et al., 2010 or expected entropy [Yang and Qiu, 2005], cannot be applied when computational models lack a tractable likelihood. In such cases, research suggests adopting LFI methods, in which the computational model generates synthetic observations for inference [Gutmann and Corander, 2016, Sisson et al., 2018, Papamakarios et al., 2019. This broad family of methods is also known as approximate Bayesian computation (ABC) [Beaumont et al., 2002, Kangasrääsiö et al., 2019 and simulator-or simulation-based inference [Cranmer et al., 2020]. To date, LFI methods for ADO have focused on parameter inference for a single model rather than model selection. Model selection with limited design iterations requires a choice of design variables that optimize model discrimination, as well as improving parameter estimation. The complexity of this task is compounded in the context of LFI, where expensive samples from the model are required. We aim at reducing the number of model simulations. For this reason, in our approach, called BOSMOS, we use Bayesian Optimization (BO) [Frazier, 2018, Greenhill et al., 2020 for both design selection and model selection. The advantage of BO is that it is highly sample-efficient and therefore has a direct impact on reducing the need for model simulation. BOSMOS combines the ADO approach with LFI techniques in a novel way, resulting in a faster method to carry out optimal design of experiments to discriminate between computational cognitive models, with a minimal number of trials. The main contributions of the paper are as follows: • A novel approach to simulator-based model selection that casts LFI for multiple models under the Bayesian framework through the approximation of the model likelihood. As a result, the approach provides a full joint Bayesian posterior for models and their parameters given collected experimental data. • A novel simulator-based utility objective for choosing experimental designs that maximizes the behavioural variation in current beliefs about model configurations. Along with the sample-efficient LFI procedure, it reduces the time cost from one hour, for competitor methods, to less than a minute in the majority of case studies, bringing the method closer to enabling real-time cognitive model testing with human subjects. • Close integration of the two above contributions yields the first online, sample-efficient, simulation-based, and fully Bayesian experimental design approach to model selection. • The new approach was tested on three well-known paradigms in psychology -memory retention, sequential signal detection and risky choice -and, despite not requiring likelihoods, reaches similar accuracy to the existing methods which do require them. Background In this article, we are concerned with situations where the purpose of experiments is to gather data that can discriminate between models. The traditional approach in such a context begins with the collection of large amounts of data from a large number of participants on a design that is fixed based on intuition; this is followed by evaluation of the model fits using a desired model selection criteria such as as AIC, BIC, cross-validation, etc. This is an inefficient approach -the informativeness of the collected data for choosing models is unknown in advance, and collecting large amounts of data may often prove expensive in terms of time and monetary resources (for instance, cases that involve expensive equipment, such as fMRI, or in clinical settings). These issues have been addressed by modern optimal experimental design methods which we consider in this section and summarize in Table 1. Optimal experimental design. Optimal experiment design (OED) is a classic problem in statistics [Lindley, 1956, Kiefer, 1959, which saw a resurgence in the last decade due to improvements in computational methods and availability of computational resources. Specifically, adaptive design optimization (ADO) Cavagnaro et al. [2010, 2013b] was proposed for cognitive science models, which has been successfully applied in different experimental settings including memory and decision-making. In ADO, the designs are selected according to a global utility objective, which is an average value of the local utility over all possible data (behavioural responses) and model parameters, weighted by the likelihood and priors [Myung et al., 2013]. More general approaches, such as Kim et al. [2014], improve upon ADO by combining it with hierarchical modelling, which allow them to form richer priors over the model parameters. While useful, the main drawback of these methods is that they work only with tractable (or analytical) parametric models, that is models whose likelihood is explicitly available and whose evaluation is feasible. Model selection for simulator-based models. In the LFI setting, a critical feature of many cognitive models is that they lack a closed-form solution, but allow forward simulations for a given set of model parameters. A few approaches have made advances in tackling the problem of intractability of these models. For instance, Kleinegesse and Gutmann [2020] and Valentin et al. [2021] proposed a method which combines Bayesian optimal experimental design (BOED) and approximate inference of simulator-based models. The Mutual Information Neural Estimation for Bayesian Experimental Design (MINEBED) method performs BOED by maximizing a lower bound on the expected information gain for a particular experimental design, which is estimated by training a neural network on synthetic data generated by the computational model. By estimating mutual information, the trained neural network no longer needs to model the likelihood directly for selecting designs and doing the Bayesian update. Similarly, Mixed Neural Likelihood Estimation (MNLE) by Boelts et al. [2022] trains neural density estimators on model simulations to emulate the simulator. Pudlo et al. [2016] proposed an LFI approach to model selection, which uses random forests to approximate the marginal likelihood of the models. Despite these advances, these methods have not been designed for model selection in an adaptive experimental design setting. Table 1 summarizes the main differences between modern approaches and the method proposed in this paper. Cognitive models increasingly operate in an agent-based paradigm [Madsen et al., 2019], where the model is a reinforcement learning (RL) policy [Kaelbling et al., 1996, Sutton andBarto, 2018]. The main problem with these agent-based models is that they need retraining if any of their parameters are altered, which introduces a prohibitive computational overhead when doing model selection. Recently, Moon et al. [2022] proposed a generalized model parameterized by cognitive parameters, which can quickly adapt to multiple behaviours, theoretically bypassing the need for model selection altogether and replacing it with parameter inference. Although the cost of evaluating these models is low in general, they lack the interpretability necessary for cognitive theory development. Therefore, training a parameterized policy within a single RL model family may be preferable: this would still require model selection but would avoid the need for retraining when parameters change (see Section 4.4 for a concrete example). Amortized approaches to OED. Recently proposed amortized approaches to OED [Blau et al., 2022] -i.e., flexible machine learning models trained upfront on a large set of problems, with the goal of making fast design selection at runtime -allow more efficient selection of experimental designs by introducing an RL policy that generates design proposals. This policy provides a better exploration of the design space, does not require access to a differentiable probabilistic model and can handle both continuous and discrete design spaces, unlike previous amortized approaches . These amortized methods are yet to be applied to model selection. Even though OED is a classical problem in statistics, its application has mostly been relegated to discriminating between simple tractable models. Modern methods such as likelihood-free inference and amortized inference can however make it more feasible to develop OED methods that can work with complex simulator models. In the next sections, we elaborate on our LFI-based method BOSMOS, and demonstrate its working using three classical cognitive science tasks: memory retention, sequential signal detection and risky choice. RF-ABC This work BOSMOS Table 1: Comparison of experimental design approaches to parameter inference (Par. inf.) and model selection (Model sel.) with the references to the selected representative works. Here, we emphasize LFI methods, as they do not need tractable model likelihoods, and amortized methods since they are the fastest to propose designs. The amortized approaches, however, need to be retrained when the population distributions (i.e. priors over models or parameters) change, as in the setting such as ours where beliefs are updated sequentially as new data are collected. Methods Our method carries out optimal experiment design for model selection and parameter estimation involving three main stages as shown in Figure 1: selecting the experimental design d, collecting new data x at the design d chosen from a design space, and, finally, updating current beliefs about the models and their parameters. The process continues until the allocated budget for design iterations T is exhausted, and the preferred cognitive model m est ∈ M, which explains the subject behaviour the best, and its parameters θ est ∈ Θ est are extracted. While the method is rooted in Bayesian inference and thus builds a full joint posterior over models and parameters, we also consider that ultimately the experimenter may want to report the single 'best' model and parameter setting, and we use this decision-making objective to guide the choices of our algorithm. The definition of what 'best' here means depends on a cost function chosen by the user [Robert et al., 2007]. In this paper, for the sake of simplicity, we choose the most common Bayesian estimator, the maximum a posteriori (MAP), of the full posterior computed by the method: m est = arg max m p(m \| D 1:t ), (1) θ est = arg max θm p(θ m \| m, D 1:t ), where m ∈ M, θ m ∈ Θ m and D 1:t = ((d 1 , x 1 ), ...(d t , x t )) is a sequence of experimental designs d (e.g. shown stimulus) and the corresponding behavioural data x (e.g. the response of the subject to the stimuli) pairs. In our usage context, it is important to make a few reasonable assumptions. First, we assume that the prior over the models p(m) and their parameters p(θ m \| m), as well as the domain of the design space, have been specified using sufficient prior knowledge; they may be given by expert psychologists or previous empirical work. This guarantees that the space of the problem is well-defined. Notice that this also implies that the set of candidate models M = (m 1 , . . . , m k ) is known, and each model is defined, for any design, by its own parameters. Second, we assume that the computational models that we consider may not necessarily have a closed-form solution, in case their likelihoods p(x \| d, θ m , m) are intractable, but it is possible to sample from the forward model m, given parameter setting θ m , and design d. In other words, we operate in a simulator-based inference setting. Please note that this likelihood depends only on the current design and parameters, as assumed in our setting. The third assumption is that each subject's dataset is analyzed separately: we consider single subjects with fixed parameters undergoing the whole set of experiments, as opposed to the statistical setting where information about one dataset may impact the whole population such as, for instance, in hierarchical modelling or pooled models. As evidenced by Equations (1) and (2), the sequential choice of the designs at any point depends on the current posterior over the models and parameters p(θ m , m \| D 1:t ) = p(θ m \| D 1:t , m) · p(m \| D 1:t ), which needs to be approximated and updated at each iteration step of the main loop in Figure 1. This problem can be formulated through sequential importance sampling methods, such as Sequential Monte Carlo (SMC; Del Moral et al., 2006). Thus, the resulting parameter posteriors can be approximated, up to resampling, in the form of equally weighted particle sets: q t (θ m , m \| D 1:t ) = N1 i=1 N −1 1 δ θ (i) m ,m i , with θ (i) m , m (i) the parameters and models associated with the particle i, as an approximation of p(θ m \| m, D 1:t ). These particle sets are later sampled to select designs and update parameter posteriors. In the following sections, we take a closer look at the design selection and belief update stages. Figure 1: Components of the model selection approach. Main loop continues until the experimental design budget is depleted. Input panel: the experimenter defines a design policy (e.g. random choice of designs), as well as the models and their parameter priors. Middle panel: (i) the next experimental design is selected based on the design policy and current beliefs about models and their parameters (initially sampled from model and parameter priors); (ii) the experiment is carried out using the chosen design, and the observed response-design pair is stored; (iii) current beliefs are updated (e.g. resampled) based on experimental evidence acquired thus far. Output panel: the model and parameters that are most consistent with the collected data are selected by applying one of the well-established decision rules to the final beliefs about models and their parameters. Selecting experimental designs Traditionally, in the experimental design literature, the designs are selected at each iteration t by maximizing the reduction of the expected entropy H(·) of the posterior p(m, θ m \| D 1:t ). By definition of conditional probability we have: d t = argmin dt E xt\|D1:t−1 H(θ m , m \| D 1:t−1 ∪ (d t , x t )) (3) = argmin dt E xt\|D1:t−1 E p(θm,m\|D1:t) [− log p(θ m , m \| D 1:t ∪ (d t , x t ))] = argmin dt E xt\|D1:t−1 E p(θm,m\|D1:t−1) [− log(p(x t \| d t , θ m , m))] + E xt\|D1:t−1 log p(x t \| d t , D 1:t−1 ),(4) where x t is the response predicted by the model. The first equality comes from the definition of entropy and the second from Bayes rule, where we removed the prior, as this term is a constant term in d t . Here, lower entropy corresponds to a narrower, more concentrated, posterior -with maximal information about models and parameters. Since neither p(x t \| d t , θ m , m) nor, by extension, Equation (4) are tractable in our setting, we propose a simulatorbased utility objective d t = arg min dt E qt(θm,m\|D1:t−1) [Ĥ(x t \| d t , θ m , m)] −Ĥ(x t \| D 1:t−1 , d t ),(5) where q t is a particle approximation of the posterior at time t, andĤ is a kernel-based Monte Carlo approximation of the entropy H. The intuition behind this utility objective is that we choose such designs d t that would maximize identifiability (minimize the entropy) between N responses x simulated from different computational models p(· \| d t , θ m , m). The models m as well as their parameters θ m are sampled from the current beliefs q t (θ m , m \| D 1:t−1 ). The full asymptotic validity of the Monte Carlo approximation of the decision rule in Equation (5) can be found in Appendix A. The utility objective in (5) allows us to use Bayesian Optimization (BO) to find the design d t and then run the experiment with the selected design. In the next section, we discuss how to update beliefs about the models m and their parameters θ m based on the data collected from the experiment. Likelihood-free posterior updates The response x t from the experiment with the design d t is used to update approximations of the posterior q t (m \| D t ) and q t (θ m \| m, D t ), obtained via marginalization and conditioning, respectively, from q t (θ m , m \| D t ). We use LFI with synthetic responses x θm simulated by the behavioural model m to perform the approximate Bayesian update. Parameter estimation conditioned on the model. We start with parameter estimation for each of the candidate models using Bayesian Optimization for Likelihood-Free Inference (BOLFI; Gutmann and Corander, 2016). In BOLFI, a Gaussian process (GP) [Rasmussen, 2003] surrogate for the discrepancy function between the observed and simulated data, ρ(x θm , x t ) (e.g., Euclidean distance), serves as a base to an unnormalized approximation of the intractable likelihood p(x t \| d t , θ m , m). Thus, the posterior can be approximated through the following approximation of the likelihood function L m (·) and the prior over model parameters p(θ m ): p(θ m \| x t ) ∝ L m (x t \| θ m ) · p(θ m ), (6) L m (x t \| θ m ) ≈ E x θm [κ m (ρ m (x θm , x t ))].(7) Here, following Section 6.3 of [Gutmann and Corander, 2016], we choose κ m (·) = 1 [0, m] (·), where the bandwidth m takes the role of a acceptance/rejection threshold. Using a Gaussian likelihood for the GP, this leads to: E x θm [κ m (ρ(x θm , x t ))] = Φ(( m − µ(θ m ))/ ν(θ m ) + σ 2 ), where Φ(·) denotes the standard Gaussian cumulative distribution function (cdf). Note that µ(θ m ) and ν(θ m ) + σ 2 are the posterior predictive mean and variance of the GP surrogate at θ m . Model estimation. A principled way of performing model selection is via the marginal likelihood, that is p(x t \| m) = p(x t \| θ m , m) · p(θ m \| m)dθ m , which is proportional to the posterior over models assuming an equal prior for each model. Unfortunately, a direct computation of the marginal likelihood is not possible with Equation (7), since it only allows us to compute a likelihood approximation up to a scaling factor that implicitly depends on . For instance, when calculating a Bayes factor (ratio of marginal likelihoods) for models m 1 and m 2 p(x t \| m 1 ) p(x t \| m 2 ) = E θm1 [p(x t \| θ m1 , m 1 )] E θm2 [p(x t \| θ m2 , m 2 )] = E θm1 [L m1 (x t \| θ m1 )] E θm2 [L m2 (x t \| θ m2 )] ,(8) their respective m1 and m2 , chosen independently, may potentially bias the marginal likelihood ratio in favour of one of the models, rendering it unsuitable for model selection. Choosing the same for each model is not possible either, as it would lead to numerical instability due to the shape of the kernel. To approximate the marginal likelihood p(x t \| m), we adopt a similar approach as in Equation (7), by reframing the marginal likelihood computation as a distinct LFI problem. In ABC for parameter estimation, we would generate pseudo-observations from the prior predictive distribution of each model, and compare the discrepancy with the true observations on a scale common to all models. This comparison involves a kernel that maps the discrepancy into a likelihood approximation. For example, in rejection ABC [Tavaré et al., 1997, Marin et al., 2012 this kernel is uniform. In our case, we will generate samples from the joint prior predictive distribution on both models and parameters, and we use a Gaussian kernel κ η (·) = N (· \| 0, η 2 ), chosen to satisfy all of the requirements from Gutmann and Corander [2016]; in particular, this kernel is non-negative, non-concave and has a maximum at 0. The parameter η > 0 serves as the kernel bandwidth, similarly to m in Equation (7). The value of κ η (·) monotonically increases as the model m produces smaller discrepancy values. This kernel leads to the following approximation of the marginal likelihood: L(x t \| m, D t−1 ) ∝ E x θ ∼p(·\|θm,m)·q(θm\|m,Dt−1) κ η (ρ(x θ , x t )),(9) where κ η (·) = N (· \| 0, η 2 ), andρ is the GP surrogate for the discrepancy. Eq. 9 is a direct equivalent of Eq. 7, but here we integrate (marginalize) over both θ and x θ . Here we used the Gaussian kernel, instead of the uniform kernel used in Eq. 7, as it produced better results for model selection in preliminary numerical experiments. Note that in Eq. 9 we have two approximations, the first one from κ η , stating that the likelihood is approximated from the discrepancy, and the second from the use of a GP surrogate for the discrepancy. The choice of η is a complex problem, and in this paper we propose the simple solution of setting η as the minimum value of E x θ ∼p(·\|θm,m)·q(θm\|m,Dt−1)ρ (x θ , x t ) across all models m ∈ M. This value has the advantage of giving non extreme values to the estimations of the marginal likelihood, which should in principle avoid over confidence. Posterior update. The resulting marginal likelihood approximation in Equation (9) can then be used in posterior updates for new design trials as follows: q(m \| D t ) ∝ L(x t \| m, D t−1 ) · q(m \| D t−1 ) ≈ κ η (ω m ) · q(m \| D t−1 ), (10) q(θ m \| m, D t ) ∝ L m (x t \| θ m , m) · q(θ m \| D t−1 , m).(11) Which is equivalent to: q(θ m , m \| D t ) ∝ L m (x t \| θ m , m) · L(x t \| m, D t−1 ) · q(θ m , m \| D t−1 ).(12) Once we updated the joint posterior of models and parameters, it is straightforward to obtain the model and parameter posterior through marginalization and apply a decision rule (e.g. MAP) to choose the estimate. The entire algorithm for BOSMOS can be found in Appendix B. Experiments In the experiments, our goal was to evaluate how well the proposed method described in Section 3 discriminated between different computational models in a series of cognitive tasks: memory retention, signal detection and risky choice. Specifically, we measured how well the method chooses designs which help the estimated model imitate the behaviour of the target model, discriminate between models, and correctly estimate their ground-truth parameters. In our simulated experimental setup, we created 100 synthetic participants by sampling the ground-truth model and its parameters (not available in the real world) through priors p(m) and p(θ m \| m). Then, we ran the sequential experimental design procedure for a range of methods described in Section 4.1, and recorded four main performance metrics shown in Figure 3 for 20 design trials (results analysed further later in the section): the behavioural fitness error η b , defined below, the parameter estimation error η p , the accuracy of the model prediction η m and the empirical time cost of running the methods. Furthermore, we evaluated the methods at different stages of design iterations in Figure 3 for the convergence analysis. The complete experiments with additional evaluation points can be found in Appendix C. We compute η b , η p and η m for a single synthetic participant using the known ground truth model m true and parameters θ true . The behavioural fitness error η b = X true − X est 2 is calculated as the Euclidean distance between the groundtruth model (X true ) and synthetic (X est ) behavioural datasets, which consist of means µ(·) of 100 responses evaluated Figure 2: An overview of the performance of the methods, compared with the prior predictive with random design, (rows) after 20 design trials across four different cognitive modelling tasks (columns): demonstrative example, memory retention, signal detection and risky choice. While requiring 10 times fewer simulations and 60-100 times less time, the proposed BOSMOS method (red) shows consistent improvement over the alternative LFI method, MINEBED (green), in terms of behavioural fitness error η b , parameter estimation error η p , model predictive accuracy η m and empirical time cost t log (here, for 100 designs, in minutes on a log scale). The model accuracy bars indicate the proportion of correct prediction of models across 100 simulated participants. The error bars show the mean (marker) and std. (caps) of the error by the respective methods. at the same 100 random designs T generated from a proposal distributions p(d), defined for each model: T = {d i ∼ p(d)} 100 i=1 ,(13)X true = {µ({x s : x ∼ p(· \| d i , θ true , m true )} 100 s=1 ) : d i ∈ T } 100 i=1 ,(14)X est = {µ({x s : x ∼ p(· \| d i , θ est , m est )} 100 s=1 ) : d i ∈ T } 100 i=1 .(15) Here, m est and θ est are, respectively, the model and parameter values estimated via the MAP rule (unless specified otherwise). m est is also used to calculate the predictive model accuracy η m as a proportion of correct model predictions for the total number of synthetic-participants, while θ est is used to calculate the averaged Euclidean distance θ true − θ est 2 across all synthetic participants, which constitutes the parameter estimation error η p . Comparison methods Throughout the experiments, we compare several strategies for experimental design selection and parameter inference, where prior predictive distribution (evaluation of the prior without any collected data) with random design choice from the proposal distribution of each model is used as a baseline (we call this method results Prior in the results). The explanations of these methodologies, as well as the exact setup parameters, are provided below. Likelihood-based inference with random design "Likelihood-based" inference with random design (LBIRD) applies the ground-truth likelihood, where it is possible, to conduct Bayesian inference and samples the design from the proposal distribution p(d) instead of design selection: Figure 3: Evaluation of three performance measures (rows) after 1, 4 and 20 design trials for BOSMOS (solid red) and two alternative best methods, ADO (blue) and MINEBED (green), in four cognitive tasks (columns). As the number of design trials grows, the methods accumulate more observed data from subjects' behaviour and, hence, should reduce behavioural fitness error η b , parameter estimation error η p , and increase model predictive accuracy η m . Since η b is the performance metric MINBED and BOSMOS optimize, its convergence is the most prominent. The lack of convergence for the other two metrics in the memory retention and signal detection tasks is likely due to the possibility of the same behavioural data being produced by models and parameters that are different from the ground-truth (i.e., non-identifiability of these models). D t = (x t , d t ), x t ∼ π(· \| θ, m, d t ), d t ∼ p(·) . This procedure serves as a baseline by providing unbiased estimates of models and parameters. As other methods in this section, LBIRD uses 5000 particles (empirical samples) to approximate the joint posterior of models and parameters for each model. The Bayesian updates are conducted through importance-weighted sampling with added Gaussian noise applied to the current belief distribution. ADO ADO requires a tractable likelihood of the models, and hence is used as an upper bound of performance in cases where the likelihood is available. ADO [Cavagnaro et al., 2010] employs BO for the mutual information utility objective: U (d) = K m=1 p(m) y p(x \| m, d) · log p(x \| m, d) K m=1 p(m)p(x \| m, d) ,(16) where we used 500 parameters sampled from the current beliefs to integrate p(x \| m, d) = p(x \| θ m , m, d) · p(θ m \| m)dθ.(17) Similarly to other approaches below which also use BO, the BO procedure is initialized with 10 evaluations of the utility objective with d sampled from the design proposal distribution p(d), while the next 5 design locations are determined by the Monte-Carlo-based noisy expected improvement objective. The GP surrogate for the utility uses a constant mean function, a Gaussian likelihood and the Matern kernel with zero mean and unit variance. All these components of the design selection procedure were implemented using the BOTorch package [Balandat et al., 2020]. MINEBED MINEBED [Kleinegesse and Gutmann, 2020] focuses on design selection for parameter inference with a single model. Since our setting requires model selections and by extension working with multiple models, we compensate for that by having a separate MINEBED instance for each of the models and then assigning a single model (sampled from the current beliefs) for design optimization at each trial. The model is assigned by the MAP rule over the current beliefs about models q(m \| D 1:t ), and the data from conducting the experiment with the selected design are used to update all MINEBED instances. We used the original implementation of the MINEBED method by Kleinegesse and Gutmann [2020], which uses a neural surrogate for mutual information consisting of two fully connected layers with 64 neurons. This configuration was optimized using Adam optimizer [Kingma and Ba, 2014] with initial learning rate of 0.001, 5000 simulations per training at each new design trial and 5000 epochs. BOSMOS BOSMOS is the method proposed in this paper and described in Section 3. It uses the simulator-based utility objective from Equation (5) in BO to select the design and BO for LFI, along with the marginal likelihood approximation from Equation (9) to conduct inference. The objective for design selection is calculated with the same 10 models (a higher number increases belief representation at the cost of more computations) sampled from the current belief over models (i.e. particle set q t (m \| D 1:t ) at each time t), where each model is simulated 10 times to get one evaluation point of the utility (100 simulations per point). In total, in each iteration, we spent 1500 simulations to select the design and additional 100 simulations to conduct parameter inference. As for parameter inference in BOSMOS, BO was initialized with 50 parameter points randomly sampled from the current beliefs about model parameters (i.e. the particle set q t (θ m \| m, D 1:t )), the other 50 points were selected for simulation in batches of 5 through the Lower Confidence Bound Selection Criteria [Srinivas et al., 2009] acquisition function. Once again, a GP is used as a surrogate, with the constant mean function and the radial basis function [Seeger, 2004] kernel with zero mean and unit variance. Once the simulation budget of 100 is exhausted, the parameter posterior is extracted through an importance-weight sampling procedure, where the GP surrogate with the tolerance threshold set at a minimum of the GP mean function [Gutmann and Corander, 2016] acts as a base for the simulator parameter likelihood. Demonstrative example The demonstrative example serves to highlight the significance of design optimization for model selection with a simple toy scenario. We consider two normal distribution models with either positive (PM) or negative (NM) mean. Responses are produced according to the experimental design d ∈ [0.001, 5] which determines the quantity of observational noise variance: (PM) x ∼ N (θ µ , d 2 ),(18)(NM) x ∼ N (−θ µ , d 2 ).(19) These two models have the same prior over parameters θ µ ∈ [0, 5] and may be clearly distinguished when the optimal design value is d = 0.001. We choose a uniform prior over models. Results As shown in the first set of analyses in Figure 2, selecting informative designs can be crucial. When compared to the LBIRD method, which picked designs at random, all the design optimization approaches performed exceedingly well. This highlights the significance of design selection, as random designs produce uninformative results and impede the inference procedure. Figure 3 illustrates the convergence of the key performance measures, demonstrating that the design optimization methods had nearly perfect estimates of ground-truths after only one design trial. This indicates that the PM and NM models are easily separable, provided informative designs. In terms of the model predictive accuracy, MINEBED outperformed BOSMOS after the first trial, however BOSMOS rapidly caught up as trials proceeded. This is most likely because our technique employs fewer simulations per trial but a more efficient LFI surrogate than MINEBED. As a result, our method has the second-best time cost not only for the demonstrative example but also across all cognitive tasks. The only method that was faster is the LBIRD method, which skips the design optimization procedure entirely and avoids lengthy computations related to LFI by accessing the ground-truth likelihood. Memory retention Studies of memory are a fundamental research area in experimental psychology. Memory can be viewed functionally as a capability to encode, store and remember, and neurologically as a collection of neural connections [Amin and Malik, 2013]. Studies of memory retention have a long history in psychological research, in particular in relation to the shape of the retention function [Rubin and Wenzel, 1996]. These studies on functional forms of memory retention seek to quantitatively answer how long a learned skill or material is available [Rubin et al., 1999], or how quickly it is forgotten. Distinguishing retention functions may be a challenge [Rubin et al., 1999], and Cavagnaro et al. [2010] showed that employing an ADO approach can be advantageous. Specifically, studies of memory retention typically consist of a 'study phase' (for memorizing) followed by a 'test phase' (for recalling), and the time interval between the two is called a 'lag time'. Varying the lag time by means of ADO allowed more efficient differentiation of the candidate models [Cavagnaro et al., 2010]. To demonstrate our approach with the classic memory retention task, we consider the case of distinguishing two functional forms, or models, of memory retention, defined as follows. Power and exponential models of memory retention. In the classic memory retention task, the subject recalls a stimulus (e.g. a word) at a time d ∈ [0, 100], which is modelled by two Bernoulli models B(1, p): the power (POW) and exponential (EXP) models. The samples from these models are the responses to the task x, which can be interpreted as 'stimulus forgotten' in case x = 0 and x = 1 otherwise. We follow the definition of these models by Cavagnaro et al. [2010], where p = θ a · (d + 1) −θPOW in POW and p = θ a · e −θEXP·d in EXP, as well as the same priors: θ a ∼ Beta(2, 1), (20) θ POW ∼ Beta(1, 4),(21)θ EXP ∼ Beta(1, 8).(22) Similarly to the previous demonstrative example and the rest of the experiments, we use equal prior probabilities for the models. Results Studies on the memory task show that the performance gap between LFI approaches and methods that use groundtruth likelihood grows as the number of design trials increases (Figure 2). This is expected, since doing LFI introduces an approximation error, which becomes more difficult to decrease when the most uncertainty around the models and their parameters has been already removed by previous trials. Unlike in the demonstrative example, where design selection was critical, the ground-truth likelihood appears to have a larger influence than design selection for this task, as evidenced by the similar performance of the LBIRD and ADO approaches. In regard to LFI techniques, BOSMOS outperforms MINEBED in terms of behavioural fitness and parameter estimation, as shown in Figure 3, but only marginally better for model selection. Moreover, both approaches seem to converge to the wrong solutions (unlike ADO), as evidenced by their lack of convergence in the parameter estimation and model accuracy plots. Interestingly, both techniques continued improving behavioural fitness, implying that behavioural data of the models can be reproduced by several parameters that are different from the ground-truth, and LFI methods fail to distinguish them. A deeper examination of the parameter posterior can reveal this issue, which can be likely alleviated by adding new features for observations and designs that can assist in capturing the intricacies within the behavioural data. Sequential signal detection Signal detection theory (SDT) focuses on perceptual uncertainty, presenting a framework for studying decisions under such ambiguity [Tanner and Swets, 1954, Peterson et al., 1954, Swets et al., 1961, Wickens, 2002. SDT is an influential developing model stemming from mathematical psychology and psychophysics, providing an analytical framework for assessing optimal decision-making in the presence of ambiguous and noisy signals. The origins of SDT can be traced to the 1800s, but its modern form emerged in the latter half of the 20th century, with the realization that sensory noise is consciously accessible [Wixted, 2020]. Example of a signal detection task could be a doctor making a diagnosis: they have to make a decision based on a (noisy) signal of different symptoms [Wickens, 2002]. SDT is largely considered a normative approach, assuming that a decision-maker is bounded rational [Swets et al., 1961]. We will consider a sequential signal detection task and two models, Proximal Policy Optimization (PPO) and Probability Ratio (PR), implemented as follows. SDT. In the signal detection task, the subject needs to correctly discriminate the presence of the signal o sign ∈ {present, absent} in a sensory input o in ∈ R. The sensory input is corrupted with sensory noise σ sens ∈ R: o in = 1 present (o sign ) · d str + γ, γ ∝ N (0, σ sens ). Due to the noise in the observations, the task may require several consecutive actions to finish. At every time-step, the subject has three actions a ∈ {present, absent, look} at their disposal: to make a decision that the signal is present or absent, and to take another look at the signal. The role of the experimenter is to adjust the signal strength d str ∼ Unif(0, 4) and discrete number of observations d obs ∼ Unif discr (2, 10) the subject can make such that the experiment will reveal characteristics of human behaviour. In particular, our goal is to identify the hit value parameter of the subject, which determines how much reward r(a, s) the subject receives, in case the signal is both present and identified correctly. Hence, we have that r(a, s) = r a (s) + r step , r a (s) = θ hit , when the signal is present, and the action is present. r a (s) = 2, when the signal is absent, and the action is absent. r a (s) = 0, when the action is look. r a (s) = −1, in other cases. where r step = −0.05 is the constant cost of every consecutive action. PPO. We implement the SDT task as an RL model due to the sequential nature of the task. In particular, the look action will postpone the signal detection decision to the next observation. The model assumes that the subject acts according to the current observation o in and an internal state β: π(a \| o in , β). The internal state β is updated over trials by aggregating observations o in using a Kalman Filter, and after each trial, the agent chooses a new action. As we have briefly discussed in Section 2, the RL policies need to be retrained when their parameters change. To address this issue, the policy was parameterized and trained using a wide range of model parameters as policy inputs. The resulting model was implemented using the PPO algorithm [Schulman et al., 2017]. PR. An alternative to the RL model is a PR model. It also assumes sequential observations: a hypothesis test as to whether the signal is present is performed after every observation, and the sequence of observations is called evidence [Griffith et al., 2021]. A likelihood for the evidence (sequence of observations) is the product of likelihoods of each observation. A likelihood ratio is used as a decision variable (denoted f t here). Specifically, f t is evaluated against a threshold, which determines which action a t to take as follows: a t = present, if f t ≤ θ low , (23) a t = absent, if f t ≥ θ low + θ len , (24) a t = look, if θ low ≤ f t ≤ θ low + θ len .(25) where f t = dobs i=1 ω 1 ω 2 , ω 1 ∼ N CDF 1 θ hit − 1 ; d str , θ sens ,(26)ω 2 ∼ N CDF 1 θ hit − 1 ; 0, θ sens .(27) Here, N CDF (·; µ, ν) is the Gaussian cumulative distribution function (CDF) with the mean µ and standard deviation ν. For more information about the PR model, we refer the reader to Griffith et al. [2021]. For both models, we used the following priors for their parameters and design values: θ sens ∼ Unif(0.1, 1), θ hit ∼ Unif(1, 7), (28) θ low ∼ Unif(0, 5), θ len ∼ Unif(0, 5).(29) Results BOSMOS and MINEBED are the only methodologies capable of performing model selection in sequential signal detection models, as specified in Section 4.4, due to the intractability of their likelihoods. The experimental conditions are therefore very close to those in which these LFI approaches are usually applied, with the exception that we now know the ground-truth of synthetic participants for performance assessments. BOSMOS showed a faster convergence of the estimates than MINEBED requiring only 4 design trails to reduce the majority of the uncertainty associated with model prediction accuracy and behaviour fitness error, as demonstrated in Figure 3. In contrast, it took 20 design trials for MINEBED to converge, and extending it beyond 20 trials provided very little benefit. Similarly as in the memory retention task from Section 4.3, error in BOSMOS parameter estimates did not converge to zero, showing difficulty in predicting model parameters for PPO and PR models. Improving parameter inference may require modifying priors to encourage more diverse behaviours and selecting more descriptive experimental responses. Finally, BOSMOS outperformed MINEBED across all performance metrics after only one design trial, with the model predictive accuracy showing a large difference, establishing BOSMOS as a clear favourite approach for this task. An example of posterior distributions returned by BOSMOS is demonstrated in Figure 4. Despite overall positive results, there are occasional cases in a population of synthetic participants, where BOSMOS fails to converge to the ground-truth. The same problem can be observed with MINEBED, as demonstrated in Appendix D. These findings may be attributed to poor identifiability of the signal detection models, suggested earlier in the memory task, but also due to the approximation inaccuracies accumulated over numerous trials. Since both methods operate in a LFI setting, some inconsistency between replicating the target behaviour and converging to the ground-truth parameters is to be expected when the models are poorly identifiable. Risky choice Risky choice problems are typical tasks used in psychology, cognitive science and economics to study attitudes towards uncertainty. Specifically, risk refers to 'quantifiable' uncertainty, where a decision-maker is aware of probabilities associated with different outcomes [Knight, 1985]. In risky choice problems, individuals are presented with options that are lotteries (i.e., probability distributions of outcomes). For example, a risky choice problem could be a decision between winning 100 euros with a chance of 25%, or getting 25 euros with a chance of 99%. The choice is between two lotteries (100, 0.25; 0, 0.75) and (25, 0.99; 0, 0.01). The goal of the participant is to maximize the subjective reward of their single choice, so they need to assess the risk associated with outcomes in each lottery. Several models have been proposed to explain tendencies in these tasks, including normative approaches derived from logic to descriptive approaches based on empirical findings [Johnson and Busemeyer, 2010]. In this paper, we will consider four classic models (following Cavagnaro et al., 2013b): expected utility theory (EU) [Von Neumann and Morgenstern, 1990], weighted expected utility theory (WEU) [Hong, 1983], original prospect theory (OPT) [Kahneman and Tversky, 1979] and cumulative prospect theory (CPT) [Tversky and Kahneman, 1992]. The risky choice models we consider consist of a subjective utility objective (characterizing the amount of value an individual attaches to an outcome) and possibly a probability weighting function (reflecting the tendency for non-linear weighting of probabilities). Despite the long history of development, risky choice is still a focus of the ongoing research [Begenau, 2020, Gächter et al., 2022, Frydman and Jin, 2022. The objective is to maximize reward from risky choices. Risky choice problems consist of 2 or more options, each of which is described by a set of probability and outcome pairs. For each option, the probabilities sum to 1. Problems may also have an endowment and/or have multiple stages. These variants are not modelled in this version. We will use similar implementations as Cavagnaro et al. [2013b] to test four models M with our method: EU, WEU, OPT and CPT. Each model has its own corresponding parameters θ m . We consider choice problems where individuals choose between two lotteries A and B. The design space for the risky-choice problems is a combination of designs for lottery A and B. The design space for lottery A is defined as the probabilities of the high and low outcome (d phA and d plA ) in this lottery. The design space for lottery B is analogous to lottery A (d phB and d plB ). We assume that there the decisions contain choice stochasticity, which serves as a likelihood for the ADO and LBIRD methods. The models are implemented as follows. Choice stochasticity. It is typical to assume that individual choices in risky choice problems are not deterministic (i.e., there is choice stochasticity). We use the following definition for probability of choosing lottery A over B in a choice problem i [Cavagnaro et al., 2013a]: φ i (A i \|θ m , ) =      , if A i ≺ B i (30) 1 2 , if A i ∼ B i (31) 1 − , if A i B i(32) where θ m denotes the model parameters and is a value in range [0,0.5] quantifying stochasticity of the choice (with = 0 corresponding to a deterministic choice). Whether lottery A is preferred is determined using the utilities defined for each model separately. Following Cavagnaro et al. [2013b], we specify EU using indifference curves on the Marschak-Machina (MM) probability triangle. Lottery A consists of three outcomes (x lA , x mA , x hA ), and associated probabilities (p lA , p mA , p hA ). Lottery A can be represented using a right triangle (MM) with two of the probabilities as the plane (p lA and p hA as Figure 4: An example of evolution of the posterior approximation in each of the models tested resulting from BOSMOS in the signal detection task. The last bottom row panels are empty as in both cases the posterior probability of the PR model becomes negligible, so that the particle approximation of this posterior does not contain any more particle. The true value of the parameters is indicated by the cross and the true model is POW in both cases. BOSMOS successfully identified the ground-truth model in both cases: all posterior density (shaded area) has concentrated there by 20 trials, and no more particle exists in the other model. However, only in the first example (top panel) did the ground-truth parameter values (cross) fall inside the 90% confidence interval, indicating some inconsistency in terms of the posterior convergence towards the ground-truth. The axes correspond to the model parameters: sensor-noise (x-axis) and hit value (y-axis); θ low and θ len of the PR model are omitted to simplify visualization. EU. x and y axes, respectively). Hence, the design space for lottery A consists of only the high and low probability (d plA and d phA ). Lottery B can be represented on the triangle similarly (using d plB and d phB ). Then, indifference curves can be drawn on this triangle, as their slope represents the marginal rate of substitution between the two probabilities. EU is defined using indifference curves that all have the same slope θ a ∈ θ EU . If lottery B is riskier, A B, if \| d phB − d phA \| / \| d plB − d plA \|< θ a . We ask to turn to Cavagnaro et al. [2013b] for a more comprehensive explanation of this modelling approach. WEU. WEU is also defined using the MM-triangle, as per Cavagnaro et al. [2013b]. In contrast to EU, the slope of the indifference curves varies across the MM-triangle for WEU. This is achieved by assuming that all the indifference curves intersect at a point (θ x , θ y ) outside the MM-triangle, where [θ x , θ y ] ∈ θ WEU . Then, A B, if \| d phA − θ y \| / \| d plA − θ x \|>\| d phB − θ y \| / \| d plB − θ x \|. OPT. In contrast to EU and WEU, OPT assumes that both the outcomes x and probabilities p have specific editing functions v and w, respectively. Assuming that for lottery A, v(x A low ) = 0 and v(x A high ) = 1, the utility objectives in OPT can be defined using v(x A middle ) as a parameter θ v u(A) = w(d phA ) · 1 + θ v · (1 − w(d phA )), if d plA = 0 (33) w(d phA ) · 1 + w(1 − d phA − d plA ) · θ v , otherwise.(34) Utility u(B) for lottery B can be calculated analogously, and A i B i if u(A) > u(B). The probability weighting function w(·) used is the original work by Tversky and Kahneman [1992] is w(p) = p θr (p θr + (1 − p) θr ) (1/θr) ,(35) where θ r is a parameter describing the shape of the function. Thus, OPT has two parameters [θ v , θ r ] ∈ θ OPT , describing the subjective utility of the middle outcome and the shape of the probability weighting function, respectively. CPT. CPT is defined similarly to OPT, however, the subjective utilities u for lottery A are calculated using u(A) = w(d phA ) · 1 + (w(1 − d plA ) − w(d phA )) · θ v .(36) Utility u(B) for lottery B is calculated similarly and [θ v , θ r ] ∈ θ CPT . We use the following priors for the parameters of models θ a ∼ Unif(0, 10), θ v ∼ Unif(0, 1), (37) θ r ∼ Unif(0.01, 1), θ x ∼ Unif(−100, 0),(38)θ y ∼ Unif(−100, 0), θ ∼ Unif(0, 0.5),(39) with the design proposal distributions d plA ∼ Unif(0, 1), d phA ∼ Unif(0, 1), (40) d plB ∼ Unif(0, 1), d phB ∼ Unif(0, 1).(41) Please note that d pmA and d pmB can be calculated analytically from d pmA = 2 − d plA − d phA , after which the designs for the same lottery (d plA , d pmA , d phA ) are normalized, so they are summed to 1 (and similar for the lottery B). Results The risky choice task comprises four computational models, which significantly expand the space of models and makes it much more computationally costly than the memory task. Despite the larger model space, BOSMOS maintains its position as a preferred LFI approach to model selection, most notably when compared to the parameter estimation error of MINEBED from Figure 2. With more models, BOSMOS's performance advantage over MINEBED grows, with BOSMOS exhibiting higher scalability for larger model spaces. It is crucial to note that having several candidate models reduces model prediction accuracy by the LFI approaches, thus we recommend reducing the number of candidate models as low as feasible. In terms of performance, BOSMOS is comparable to ground-truth likelihood approaches during the first four design trials, as shown in Figure 3, since it is significantly easier to minimize uncertainty early in the trials. Similarly to the memory task, the error of LFI approximation becomes more apparent as the number of trials rises, as evidenced by comparing BOSMOS to ADO for the behavioural fitness error and model predictive accuracy. In terms of the parameter estimate error, BOSMOS performs marginally better than ADO. Finally, BOSMOS has a relatively low runtime cost, especially compared to other methods (about one minute per design trial). This brings adaptive model selection closer to being applicable to real-world experiments in risky choice. The proposed method can be useful in online experiments that include lag times between trials, for instance, in assessing investment decisions (e.g., Camerer, 2004, Gneezy andPotters, 1997) or game-like settings (e.g., Bauckhage et al., 2012, Putkonen et al., 2022, Viljanen et al., 2017 where the participant waits between events. Discussion In this paper, we proposed a simulator-based experimental design method for model selection, BOSMOS, that does design selection for model and parameter inference at a speed orders of magnitude higher than other methods, bringing the method closer to online design selection. This was made possible with newly proposed approximation of the model likelihood and simulator-based utility objective. Despite needing orders of magnitude fewer simulations, BOSMOS significantly outperformed LFI alternatives in the majority of cases, while being orders of magnitude faster, bringing the method closer to an online inference tool. Crucially, the time between experiment trials was reduced to less than a minute. Whereas in some settings this time between trials may be too long, BOSMOS is a viable tool in experiments where the tasks include a lag time, for instance, in studies of language learning (e.g., Gardner et al., 1997, Nioche et al., 2021 and task interleaving (e.g., Payne et al., 2007, Brumby et al., 2009, Gebhardt et al., 2021, Katidioti et al., 2014. Moreover, our code implementation represents a proof of concept and was not fully optimized for maximal efficiency: in particular, a parallel implementation that exploits multiple cores and batches of simulated experiments would enable additional speedups [Wu and Frazier, 2016]. As an interactive and sample-efficient method, BOSMOS can help reduce the number of required experiments. This can be of interest to both the subject and the experimenter. In human trials it allows for faster interventions (e.g. adjusting the treatment plan) in critical settings such as ICUs or RCTs. However, it can also have detrimental applications, such as targeted advertising and collecting personal data, therefore the principles and practices of responsible AI [Dignum, 2019, Arrieta et al., 2020 also have to be taken into account in applying our methodology. There are at least two remaining issues left for future work. The first issue we witnessed in our experiments is that the accuracy of behaviour imitation does not necessarily correlate with the convergence to ground-truth models. This usually happens due to poor identifiability in the model-parameter space, which may be quite prevalent in current and future computational cognitive models, since they are all designed to explain the same behaviour. Currently, the only way to address this problem is to use Bayesian approaches, such as BOSMOS, that quantify the uncertainty over the models and their parameters. The second issue is the consistency of the method: in selecting only the most informative designs, the methods may misrepresent the posterior and return an overconfident posterior. This bias may occur, for example, due to a poor choice of priors or summary statistics [Nunes andBalding, 2010, Fearnhead andPrangle, 2012] for the collected data (when the data is high-dimensional). Ultimately, these issues do not hinder the goal of automating experimental designs, but introduce the necessity for a human expert, who would ensure that the uncertainty around estimated models is acceptable, and the design space is sufficiently explored to make final decisions. Future work for simulator-based model selection in computational cognitive science needs to consider adopting hierarchical models, accounting for the subjects' ability to adapt or change throughout the experiments, and incorporating amortized non-myopic design selection. A first step in this direction would be to study hierarchical models [Kim et al., 2014] which would allow adjusting prior knowledge for populations and expanding the theory development capabilities of model selection methods from a single individual to a group level. We could also remove the assumption on the stationarity of the model by proposing a dynamic model of subjects' responses which adapts to the history of previous responses and previous designs, which is more reasonable in longer settings of several dozens of trials. Lastly, amortized non-myopic design selections [Blau et al., 2022] would even further reduce the wait time between design proposals, as the model can be pre-trained before experiments, and would also improve design exploration by encouraging long-term planning of the experiments. Addressing these three potential directions may have a synergistic effect on each other, thus expanding the application of simulator-based model selection in cognitive science even further. Supplementary information The article has the following accompanying supplementary materials: • Appendix A shows the validity of the approximation of the entropy gain for the design selection rule; • Appendix B details the algorithm for the proposed BOSMOS method; • Appendix C contains tables with full experimental results, which shows additional design evaluation points; • Appendix D showcases a side-by-side comparison of the posterior evolution resulting from BOSMOS and MINEBED for the signal detection task. A Approximation of the entropy gain for design selection Since the expected values in the entropy gain from Equation (3) are not tractable, we rely on a Monte-Carlo estimation of these quantities. We focus on the first term, as the second one only features one of the approximations. Following the SMC framework [Del Moral et al., 2006], we propose to sequentially update a particle population between trials. This allows the online update of the posterior along the experiment. At each step we estimate the distribution p(m, θ m \| D 1:t ) with a population of N 1 particles (m i , θ i m ). Following importance sampling, we know that N1 i=1 w i δ (m i ,θ i m ) converges to the distribution associated with density p(m, θ m \| D t ). This population of particles then evolves according to the SMC algorithm [Del Moral et al., 2006]. To estimate the expected value in Equation (3), we use a standard Monte-Carlo estimation, making use of the particles at time t as an estimation of the posterior at time t. For each particle, we simulate N 2 vectors x i t (j) ∼ p(· \| d t , θ i m , m i ). The convergence to prove is then: i,j (N 2 ) −1 w iĤ (x i t (j) \| m i , θ i m ) → N1,N2→∞ E q(m,θm\|Dt−1) [H(x t \| m, θ m )],(42) whereĤ is a modified version of the entropy. Note that it is also possible to estimate the gradient of this quantity with respect to the design. If we truly computed H(x t ) as the entropy with respect to the measure i δ x i , it would lead to constantly null value. Thus, we decide to use a kernel approximation of the distribution:Ĥ( x i t (j) \| m i , θ i m ) = H( i N (· \| x i (j), σ N2 ) \| m i , θ i m ), with σ N2 → N2→∞ 0. The convergence of the estimator in Equation 42 to the true entropy requires two results. First, the convergence in N 2 : (5); collect the data x at the design location d and store it in D i ; for m ∈ M do get the likelihood L m (x i \| θ m ) with Equation (7); end for get the marginal likelihood L(x i \| m, D i−1 ) with Equation (9); update q(θ m , m \| D i ) with Equation (12); end for apply the decision rule (e.g. MAP): m = arg max m θm q(θ m , m \| D N d ); θ m = arg max θm q(θ m \| m, D N d ); N1 i=1 N2 j=1 (N 2 ) −1 w iĤ (x i t (j) \| m i , θ i m ) → N2→∞ N1 i=1 w iĤ (p(· \| d t , θ i m , m i ) \| m i , θ i m ), C Full experimental results We provide full experimental results data in Tables 2-4, where evaluations of performance metrics were made after different numbers of design iterations. Moreover, we report time costs of running 100 design trials for each method in Table 5, where our method was 80-100 times faster than the other LFI method, MINEBED. The rest of the section discusses three additional minor points with relation to the performance of ADO for the demonstrative example, the bias in the model space for the memory task, and results of testing two decision rules in the risky choice. As we have seen in the main text, MINEBED had the fastest behavioural fitness convergence rate in the demonstrative example, while BOSMOS was the close second. Hence, ADO had the slowest convergence rate among design optimization methods for behavioural fitness (Table 2) and also for parameter estimation (Table 3). This result is somewhat counterintuitive, as we expected ADO, with its ground-truth likelihood and design optimization, to be the fastest to converge. Since the only other factor influencing this outcome, Bayesian updates, had access to the ground-truth and avoided LFI approximations, suboptimal designs are likely to blame for the poor performance. This problem is likely to be mitigated by expanding the size of the grid used by ADO to calculate the utility objective. However, expanding it would likely increase ADO's convergence at the expense of more calculations; therefore we aimed to get its running time closer to that of BOSMOS, so both methods could be fairly compared. In the results of model selection for the memory retention task discussed in the main text, MINEBED showed a marginally better average model accuracy than BOSMOS. However, upon closer inspection (Table 4), this accuracy can be solely attributed to the strong bias towards the POW model; the other approaches show it as well, albeit less dramatically. This suggests that the two models in the memory task are separable, but the EXP model cannot likely replicate parts of the behaviour space that the POW model can, resulting in this skewness towards the more flexible model. This is also more broadly related to non-identifiability: since these cognitive models were designed to explain the same target behaviour, it is inevitable that there will be an overlap in their response (or behavioural data) space, complicating model selection. Since the risky choice model had four models of varied complexity, we experimented with two distinct decisionmaking rules for estimating the models and parameters: the default MAP and Bayesian information criterion (BIC) [Schwarz, 1978, Vrieze, 2012. Both decision-making rules include a penalty for the size of the model (artificially for BIC, and by definition for MAP). Interestingly, the results are the same for both decision-making rule, indicating that the EU model cannot be completely replaced by a more flexible model. BIC's slightly superior parameter estimates for the BOSMOS technique is most likely explained by the poorer model prediction accuracy. Nevertheless, the BIC rule remains a viable option for model selection in situations when there is a risk of having a more flexible and complex model alongside few-parameter alternatives, despite being less supported theoretically. D Posterior evolution examples for BOSMOS and MINEBED In Figures 5 and 6, we compare posterior distributions returned by MINEBED and BOSMOS for two synthetic participants in the signal detection task. In both examples, the methods have successfully identified the ground-truth POW model, as the majority of the posterior density (shaded area in the figure) has moved to the correct model. Nevertheless, BOSMOS and MINEBED have quite different posteriors, which emphasizes the influence of the design selection strategy on the resulting convergence, as one of the method performs better than the other in each of the provided examples. Figure 5: The first example of evolution of the posterior distribution resulting from MINEBED (green; rows 1 and 2) and BOSMOS (red; rows 3 and 4) for the signal detection task. For each method, the first row shows the distributions of parameters of the POW model (ground-truth), followed by the PR model parameter distributions in the second row. The axes correspond to the model parameters: sensor-noise (x-axis) and hit value (y-axis); θ low and θ len of the PR model are omitted to simplify visualization. The last bottom row panels are empty as in both cases the posterior probability of the PR model becomes negligible, so that the particle approximation of this posterior does not contain any more particle. Figure 6: The second example of evolution of the posterior distribution resulting from MINEBED (green; rows 1 and 2) and BOSMOS (red; rows 3 and 4) for the signal detection task. For each method, the first row shows the distributions of parameters of the POW model (ground-truth), followed by the PR model parameter distributions in the second row. The axes correspond to the model parameters: sensor-noise (x-axis) and hit value (y-axis); θ low and θ len of the PR model are omitted to simplify visualization. The last bottom row panels are empty as in both cases the posterior probability of the PR model becomes negligible, so that the particle approximation of this posterior does not contain any more particle. Methods Tasks: number of design trials Demonstrative example 1 trial 2 trials 4 trials 20 trials 100 trials ADO 0.03 ± 0.03 0.02 ± 0.02 0.02 ± 0.02 0.01 ± 0.01 -MINEBED 0.02 ± 0.07 0.01 ± 0.00 0.01 ± 0.00 0.01 ± 0.00 -BOSMOS 0.05 ± 0.07 0.01 ± 0.01 0.01 ± 0.00 0.01 ± 0.00 -LBIRD 0.36 ± 0.24 0.33 ± 0.25 0.29 ± 0.24 0.14 ± 0.18 -Prior Baseline for 0 trials: 0.33 ± 0. 0.32 ± 0.11 0.30 ± 0.13 0.27 ± 0.12 0.14 ± 0.08 0.07 ± 0.04 MINEBED 0.30 ± 0.11 0.31 ± 0.12 0.26 ± 0.12 0.21 ± 0.12 0.22 ± 0.13 BOSMOS 0.26 ± 0.11 0.23 ± 0.12 0.24 ± 0.13 0.18 ± 0.11 0.14 ± 0.08 MINEBED (BIC) 0.25 ± 0.11 0.26 ± 0.13 0.23 ± 0.11 0.21 ± 0.12 0.22 ± 0.13 BOSMOS (BIC) 0.24 ± 0.11 0.24 ± 0.13 0.23 ± 0.11 0.19 ± 0.12 0.15 ± 0.10 LBIRD 0.31 ± 0.12 0.30 ± 0.13 0.26 ± 0.13 0.14 ± 0.07 0.08 ± 0.03 Prior Baseline for 0 trials: 0.44 ± 0.50 Table 2: Convergence of behavioural fitness error η b (mean ± std. across 100 simulated participants) for comparison methods (rows) with increased number of trials (columns). Methods Tasks: number of design trials Demonstrative example 1 trial 2 trials 4 trials 20 trials 100 trials ADO 0.05 ± 0.06 0.04 ± 0.05 0.03 ± 0.04 0.01 ± 0.01 -MINEBED 0.00 ± 0.02 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 -BOSMOS 0.05 ± 0.07 0.01 ± 0.02 0.00 ± 0.00 0.00 ± 0.00 -LBIRD 0.34 ± 0.22 0.32 ± 0.24 0.29 ± 0.25 0.17 ± 0.18 -Prior Baseline for 0 trials: 0.33 ± 0. Table 3: Convergence of parameter estimation error η p (mean ± std. across 100 simulated participants) when the model is predicted correctly for comparison methods (rows) with increased number of trials (columns). using the convergence of N (· \| x i (j), σ N2 ) to δ x i (j) in distribution and the law of large numbers. Second, the convergence for N 1 → ∞ comes from the results on SMC [Del Moral et al., 2006]. Bayesian optimization for simulator-based model selection Input: prior over models p(m) and parameters p(θ m \| m) m ; set of all models M = {p(x \| θ m , m, d)}; design budget N d ; total number of particles N q ; Output: selected model m and its parameters θ m ; initialize current beliefs from the priors: q(m, θ m ) := {(m , θ m ) : θ m ∼ p(θ m \| m ), m ∼ p(m)} Nq i=0 ; initialize an empty set for the collected data: D 0 = {}; for i := 1 : N d do get the design d with Equation 20 ± 0.16 0.17 ± 0.14 0.15 ± 0.10 0.07 ± 0.06 0.05 ± 0.03 MINEBED 0.27 ± 0.22 0.24 ± 0.19 0.24 ± 0.19 0.23 ± 0.15 0.23 ± 0.17 BOSMOS 0.24 ± 0.19 0.19 ± 0.16 0.17 ± 0.14 0.15 ± 0.13 0.13 ± 0.11 LBIRD 0.20 ± 0.17 0.17 ± 0.15 0.14 ± 0.11 0.08 ± 0.06 0.05 ± 0.03 Prior Baseline for 0 trials: 0.33 ± 0.473 Memory retention 1 trial 2 trials 4 trials 20 trials 100 trials ADO 0.Signal detection 1 trial 2 trials 4 trials 20 trials 100 trials MINEBED 0.27 ± 0.24 0.24 ± 0.20 0.23 ± 0.17 0.21 ± 0.18 0.20 ± 0.17 BOSMOS 0.25 ± 0.21 0.20 ± 0.17 0.17 ± 0.15 0.17 ± 0.15 0.15 ± 0.12 Prior Baseline for 0 trials: 0.40 ± 0.49 Risky choice 1 trial 2 trials 4 trials 20 trials 100 trials ADO 25 ± 0.21 0.25 ± 0.20 0.23 ± 0.19 0.19 ± 0.14 0.14 ± 0.12 MINEBED 0.47 ± 0.38 0.46 ± 0.38 0.46 ± 0.39 0.48 ± 0.40 0.48 ± 0.43 BOSMOS 0.29 ± 0.21 0.27 ± 0.22 0.28 ± 0.20 0.27 ± 0.22 0.29 ± 0.22 LBIRD 0.25 ± 0.21 0.25 ± 0.20 0.22 ± 0.18 0.22 ± 0.20 0.15 ± 0.13 Prior Baseline for 0 trials: 0.33 ± 0.20 42 ± 0.22 0.44 ± 0.23 0.43 ± 0.22 0.33 ± 0.24 0.26 ± 0.23 MINEBED 0.86 ± 0.28 0.87 ± 0.27 0.81 ± 0.32 0.76 ± 0.36 0.86 ± 0.29 BOSMOS 0.41 ± 0.25 0.40 ± 0.21 0.45 ± 0.26 0.29 ± 0.25 0.21 ± 0.23 MINEBED (BIC) 0.85 ± 0.33 0.83 ± 0.31 0.87 ± 0.30 0.86 ± 0.31 0.84 ± 0.35 BOSMOS (BIC) 0.27 ± 0.23 0.26 ± 0.19 0.40 ± 0.26 0.23 ± 0.22 0.21 ± 0.22 LBIRD 0.51 ± 0.25 0.48 ± 0.24 0.40 ± 0.24 0.35 ± 0.22 0.24 ± 0.19 Prior Baseline for 0 trials: 0.48 ± 0.2523 Memory retention 1 trial 2 trials 4 trials 20 trials 100 trials ADO 0.Signal detection 1 trial 2 trials 4 trials 20 trials 100 trials MINEBED 0.60 ± 0.24 0.56 ± 0.28 0.43 ± 0.34 0.45 ± 0.35 0.49 ± 0.34 BOSMOS 0.37 ± 0.22 0.35 ± 0.19 0.36 ± 0.21 0.35 ± 0.20 0.35 ± 0.19 Prior Baseline for 0 trials: 0.35 ± 0.20 Risky choice 1 trial 2 trials 4 trials 20 trials 100 trials ADO 0. AcknowledgementsThis work was supported by the Academy of Finland (Flagship programme: Finnish Center for Artificial Intelligence FCAI; grants 328400, 345604, 328400, 320181). AP and SC were funded by the Academy of Finland projects BAD (Project ID: 318559) and Human Automata (Project ID: 328813). AP was additionally funded by Aalto University School of Electrical Engineering. SC was also funded by Interactive Artificial Intelligence for Research and Development (AIRD) grant by Future Makers. SK was supported by the Engineering and Physical Sciences Research Council (EPSRC; Project ID: EP-W002973-1). Computational resources were provided by the Aalto Science-IT Project.Table 5: Empirical time cost (mean ± std. in minutes across 100 simulated participants) of applying comparison methods (rows) in cognitive tasks (columns) with 100 sequential designs. ADO and LBIRD for the signal detection task need available likelihoods and therefore cannot be used for this task. 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[ "Atom Nano-lithography with Multi-layer Light Masks: Particle Optics Analysis", "Atom Nano-lithography with Multi-layer Light Masks: Particle Optics Analysis" ]	[ "R Arun \nDepartment of Chemical Physics\nThe Weizmann Institute of Science\nRehovotIsrael\n", "I Sh ", "Averbukh \nDepartment of Chemical Physics\nThe Weizmann Institute of Science\nRehovotIsrael\n", "T Pfau \n5th Institute of Physics\nUniversity of Stuttgart\nGermany\n" ]	[ "Department of Chemical Physics\nThe Weizmann Institute of Science\nRehovotIsrael", "Department of Chemical Physics\nThe Weizmann Institute of Science\nRehovotIsrael", "5th Institute of Physics\nUniversity of Stuttgart\nGermany" ]	[]	We study the focusing of atoms by multiple layers of standing light waves in the context of atom lithography. In particular, atomic localization by a double-layer light mask is examined using the optimal squeezing approach. Operation of the focusing setup is analyzed both in the paraxial approximation and in the regime of nonlinear spatial squeezing for the thin-thin as well as thinthick atom lens combinations. It is shown that the optimized double light mask may considerably reduce the imaging problems, improve the quality of focusing and enhance the contrast ratio of the deposited structures.	10.1103/physreva.72.023417	[ "https://export.arxiv.org/pdf/quant-ph/0503181v1.pdf" ]	56,200,096	quant-ph/0503181	d86fb55b6b5faf07ac2e45229ae3dd5c51ea4c03	Atom Nano-lithography with Multi-layer Light Masks: Particle Optics Analysis 22 Mar 2005 (April 1, 2022) R Arun Department of Chemical Physics The Weizmann Institute of Science RehovotIsrael I Sh Averbukh Department of Chemical Physics The Weizmann Institute of Science RehovotIsrael T Pfau 5th Institute of Physics University of Stuttgart Germany Atom Nano-lithography with Multi-layer Light Masks: Particle Optics Analysis 22 Mar 2005 (April 1, 2022)arXiv:quant-ph/0503181v1number(s): 0375Be3280Lg4282Cr0365Sq We study the focusing of atoms by multiple layers of standing light waves in the context of atom lithography. In particular, atomic localization by a double-layer light mask is examined using the optimal squeezing approach. Operation of the focusing setup is analyzed both in the paraxial approximation and in the regime of nonlinear spatial squeezing for the thin-thin as well as thinthick atom lens combinations. It is shown that the optimized double light mask may considerably reduce the imaging problems, improve the quality of focusing and enhance the contrast ratio of the deposited structures. I. INTRODUCTION Since the early realization of sub-micron atom lithography [1], the subject of focusing neutral atoms by use of light fields continues to attract a great deal of attention. The basic principle of atom lithography relies on the possibility of concentrating the atomic flux in space utilizing a spatially modulated atom-light interaction. In the conventional atom-lithographic schemes, a standing wave (SW) of light is used as a mask on atoms to concentrate the atomic flux periodically and create desired patterns at the nanometer scale [2]. The technique has been applied to many atomic-species in one [3][4][5][6][7][8][9][10][11] as well as two-dimensional [12] pattern formations. There are two ways to focus a parallel beam of atoms by light masks in close correspondence with conventional optics. In the thin-lens approach, atoms are focused outside the region of light field which happens for low intensity light beams. On the other hand, the atoms can be focused within the light beam when its intensity is high. This is known as thick-lens regime and is very similar to the graded-index lens of traditional optics. The laser focusing of atoms depends on parameters such as thickness of light beam, velocity spread of atoms, detuning of laser frequency from the atomic transition frequencies, etc,. Experimentally, atomic nanostructures have been reported with sodium [1,5], chromium [6,7], aluminium [8], cesium [9], ytterbium [10], and iron [11] atoms. Most of the theoretical studies on atom lithography employ a particle optics approach to laser focusing of atoms [3,4,13]. The classical trajectories of atoms in the potentials induced by light fields suffice to study the focal properties of light lens. In the case of direct laserguided atom deposition, the diffraction resolution limit will be ultimately determined by the de Broglie wavelength of atoms, and may reach several picometers for typical atomic beams [14]. In practice, however, this limit has never been relevant because of the surface diffusion process, the quality of the atomic beam, and severe aberrations due to anharmonicity of the sinusoidal dipole potential. As a result, all current atom lithography schemes suffer from a considerable background in the deposited structures. A possible way to overcome the aberration problem was suggested in [15], by using nanofabricated mechanical masks that block atoms passing far from the minima of the dipole potential. However, this complicates considerably the setup and reduces the deposition rate. Therefore, there is a considerable need in a pure atom optics solution for the enhanced focusing of an atomic beam having a significant angular spread. In paraxial approximation, the steady-state propagation of an atomic beam through a standing light wave is closely connected to the problem of the time-dependent lateral motion of atoms subject to a spatially periodic potential of an optical lattice. From this point of view, enhanced focusing of the atomic beam can be considered as a squeezing process on atoms in the optical lattice. In recent work [16], novel squeezing technique has been introduced for atoms in a pulsed optical lattice. The approach considered a time modulation of the SW with a series of short laser pulses. Based on specially designed aperiodic sequence of pulses, it has been shown that atoms can be squeezed to the minima of the light-induced potential with reduced background level. Oskay et al. [17] have verified this proposal experimentally using Cs atoms in an optical lattice. In Refs. [16,17], the atoms were loaded into the optical lattice and the dynamics of atoms along the direction of SW was studied as a time-dependent problem. The aim of the present work is to extend the focusing scenario of Ref. [16] to the beam configuration employed for atomic nanofabrication. We generalize the results on atomic squeezing in the pulsed SW to a system involving the atomic-beam traversing several layers of light masks. In particular, we will investigate prospects for reducing spherical and chromatic aberrations in atom focusing with double-layer light masks. High-resolution deposition of chromium atoms will be considered as an example. The plan of the paper is as follows. In Sec. II, the basic framework of the problem is defined and the linear focusing of atoms by a double-layer light mask is studied using the particle optics approach in paraxial approximation. In Sec. III, we examine the optimal squeezing scheme of [16] in application to the atomic-beam traversing two layers of light masks. The effects of beam collimation and chromatic aberrations are considered in Sec. IV. Here, we optimize the double lens performance and give parameters for the minimum spot-size in the atom deposition. Finally, in Sec. V, we summarize our main results. II. SQUEEZING OF ATOMS BY MULTI-LAYER LIGHT MASKS -CLASSICAL TREATMENT The focusing property of a single SW light has been studied in great details by McClelland et al. [4,6]. The light acts like an array of cylindrical lenses for the incident atomic beam, focusing the atoms into a grating on the substrate. However, because of the non-parabolic nature of the light-induced potential, the focusing of atoms is subject to spherical aberrations giving a finite width to the deposited features [4]. A doublet of light masks made from two standing light waves may, in principle, reduce the focusing imperfections due to a clear physical mechanism. In this configuration, the first SW prefocuses the atoms towards the minima of the sinusoidal potential. When the pre-focused atoms cross the second SW, they see closely the parabolic part of the potential which should result in a reduction of the over-all spherical aberrations. To test this scheme, we consider the propagation of an atomic-beam through a combination of two SWs formed by counter-propagating laser beams. The two SWs are identical except for their intensities and are assumed to be formed along the x-direction. Atoms are described as two-level systems with transition frequency ω o . We take the direction of propagation of atoms through the SW fields along the z-direction. If the atoms move sufficiently slow (adiabatic conditions) through the light fields, the internal variables of atoms maintain a steady state during propagation [18]. In this approximation, the atoms can be described as point-like particles moving under the influence of an average dipole-force. The potential energy of interaction is given by [13,19] U (x, z) =h ∆ 2 ln[1 + p(x, z)] ,(1) where p(x, z) = γ 2 γ 2 + 4∆ 2 I(x, z) I s .(2) In Eq. (2), ∆ is the detuning of the laser frequency from the atomic resonance, I(x, z) is the light intensity, γ is the spontaneous decay rate of excited level, and I s is the saturation intensity associated with the atomic transition. For the arrangement of two SW light masks (denoted by 1 and 2) with separation S between them, the net intensity profile of light is given by I(x, z) = I 1 exp(−2z 2 /σ 2 z ) + I 2 exp(−2(z − S) 2 /σ 2 z ) × sin 2 (kx) .(3) Here, σ z is the 1/e 2 radius and λ = 2π/k is the wavelength of laser beams forming the SWs. We consider Gaussian intensity profiles and ignore any y-dependence of laser intensities as the force on atoms along the ydirection is negligible compared to that along the direction of SW (x-axis). I 1 and I 2 denote the maximum intensity of the standing light waves 1 and 2, respectively. We neglect the overlap and interference between two SWs. The intensity profile of light and the focusing of atoms by light fields are shown schematically in Fig. 1. The classical trajectories of atoms in the potential (1) induced by the double-layer light masks obey the Newton's equations of motions : d 2 x dt 2 + 1 m ∂U (x, z) ∂x = 0 , d 2 z dt 2 + 1 m ∂U (x, z) ∂z = 0 . (4) Using the conservation of energy, we can combine the above two equations and solve for x as a function of z. This results in two first-order coupled differential equations for x(z), α ≡ dx(z)/dz : dx(z) dz = α ,(5)dα(z) dz = 1 + α 2 2(E − U ) α dU dz − (1 + α 2 ) dU dx . Here, E represents the total energy of the incoming atoms (the kinetic energy in the field-free region) and α gives the slope of the trajectory x(z). Table I. The solid (dashed) curve in graph (c) shows the probability density of atoms at the focal point z = z f ≈ 650 (700) of the double (single) light lens. The region to the right of origin (x = 0) in graph (c) is zoomed and shown in the inset. We first study the focal properties of the light fields, and solve numerically Eq. (5) for an atomic beam that is initially parallel to the z-axis. The linear focal points and principal-plane locations can be obtained by tracing paraxial trajectories as discussed in [4]. Some typical results are shown in Fig. 2, where we present the numerical calculation of a series of atomic trajectories entering the nodal region of both single (I 1 = 0, I 2 ≡ 0) and double (I 1 , I 2 = 0) light masks. Table I lists the parameters used in dimensionless units, in which length is expressed in units of λ, and frequency is in units ω r ≡hk 2 /2m corresponding to the recoil energy. We have considered the intensities of light SWs to be equal in the case of double light masks. For the other variables, the values close to the experimental parameters of the chromium atom-deposition [20] are taken as an example, though the general conclusions to be drawn should apply to other atoms. It is seen from Fig. 2, that a sharp focal spot appears in the flux of focused atoms [21]. Despite the small size of the focal spot, the overall localization of atoms in the focal plane is not very marked. Atomic background in the focal plane indeed gets reduced with double light masks as shown in the inset of Fig. 2(c), however this effect is not very pronounced. To take full advantage from double-mask arrangement, we have to replace the concept of linear focusing (useful for paraxial trajectories only) by the notion of optimized nonlinear spatial squeezing. III. OPTIMAL SQUEEZING THEORY -APPLICATION TO ATOM NANOLITHOGRAPHY We have seen that the double light lens leads to some improvement in feature contrast in the focal plane in comparison to the single light lens. However, even for a single SW, the best squeezing of atoms (maximal spatial compression) is achieved not at the focal plane, but after the linear focusing phenomenon takes place. To characterize the spatial localization of atoms we use a convenient figure of merit, the localization factor [16]: L(z) = 1− < cos (2kx(z, x o )) > ≡ 2 λ λ/4 −λ/4 dx o [1 − cos (2kx(z, x o ))] ,(6) where x(z, x o ) is the solution of the differential equations (5) satisfying the initial condition x → x 0 at z → −∞. The average in Eq. (6) is taken over the random initial positions of atoms and the localization factor is measured as a function of distance z from the center (z = 0) of the first SW. The localization factor equals zero for an ideally localized atomic ensemble, and is proportional to the mean-square variation of the x coordinate (modulo standing wave period) in the case of well-localized distribution (L << 1). Ref. [16] considered the squeezing process in the timedomain by analyzing the action of pulsed SWs on atoms. In the Raman-Nath approximation, this corresponds to the thin-lens regime (in space domain) for interaction of a propagating atomic-beam with multiple layers of light masks. According to the optimal squeezing strategy [16], the time sequence of pulses applied to the atomic system is determined by minimizing the localization factor. To apply this procedure to the atom squeezing by multilayer light masks, we should minimize the localization factor (6) in the parameter space: the separations between the light SWs, their intensities, and the relative distance of substrate surface with respect to the layers of light masks. This optimization can be done numerically using the established simplex-search method. Our numerical analysis shows that the localization factor exhibits multiple local minima even for the simplest case of double light masks. In Fig. 3, we plot the localization factor as a function of distance z both for single and double light masks around its global minimum (z m , S m ). The intensities of SWs have been chosen to be equal and satisfy the thin-lens condition of atom-light interaction [22]. The graph shows that the localization factor gets a sizable reduction with double light masks indicating for an enhanced focusing of atoms. The minimum values of L(z) in Fig. 3 are in conformity with the values obtained for optimal squeezing of atoms with single and double pulses in the time-dependent problem [16]. We emphasize that the best squeezing (localization) of atoms does not occur at the focal point. Figure 4 displays the spatial distribution of atoms at the point of best localization. Instead of a single focal peak, a two-peaked spatial distribution of atoms near the potential minima is observed in Fig. 4. The origin of these peaks can be related to the formation of rainbows in the wave optics and quantum mechanics, and it is discussed in detail in [16,23]. Moreover, on comparing the inset of Figs. 2(c) and (4), it is seen that the optimized separation between layers of the double light mask results in a considerable reduction of atomic deposition in the background. This also leads to an overall increased concentration of atomic flux near the potential minima. We note, that according to [16,17], further squeezing of atoms can be achieved by increasing the number of identical SWs in the multi-layer light masks. For the best localization, again the optimized values for the separations between light masks should be used. Table I. The region to the right of origin (x = 0) is zoomed and shown in the inset. In the above analysis, we have considered the case of equal intensities for the light lenses and the problem has been studied in the thin-lens [24] regime of atom focusing by light masks. However, in many current atom-lithographic schemes, focusing of atoms is generally achieved using an intense SW light. This corresponds to the thick-lens regime of atom-light interaction. In this limit, the focal point is within or close to the region of laser fields and hence a detailed information on atomic motion within the light is required for a full description [4]. For the chromium atoms deposition, the focusing of atoms to the center of an intense SW has been extensively studied both theoretically [4] and experimentally [6]. We show here that a combination of a thin and thick lenses can result in the enhanced localization of atoms Table I with I 1 /I s = 1500. with minimal background structures. For illustration, we consider the focusing of atoms by a doublet of light masks made of a thin and a thick lens. We fix the intensity of the first SW light mask to satisfy the thin-lens limit and study the best localization of atoms that can be achieved by varying the intensity of the second SW. A plot of the minimal value of the localization factor versus the relative intensity of the second light mask is shown in Fig. 5. The graph shows that the localization factor becomes almost insensitive to the variation in relative intensity after the intensity ratio reaches the value of 5, and it approaches a small value of L = 0.15. This result is to be compared with the value of L = 0.31 for the optimal squeezing by two thin lenses, and L = 0.42 achievable by a single thin lens. Fig. 6 shows the corresponding trajectories of atoms and a plot of atomic distribution at the point of the best localization. Note that the optimized double light mask reduces the atomic background by a factor of three in the midpoint (x = 0.25) between two deposition peaks (see the inset of Fig. 6). Moreover, the background in the optimized double mask scheme is five times smaller compared to the usual atom deposition in the focal plane (graph not shown) of a single thick lens. IV. PARAMETERS FOR OPTIMAL SQUEEZING OF A THERMAL ATOMIC BEAM The effects that have been discussed so far assume an initially collimated (α = 0) beam of atoms with fixed velocity (or energy). However, in atom optics experiments involving thermal atomic beams, the atoms possess a wide range of velocities along the longitudinal (z-axis) and transverse (x-axis) directions. In order to characterize the atom spatial squeezing under such conditions, we need to average the localization factor Eq. (6) over the random initial velocities and angles of the beam. The averaging can be done by using the normalized probability density [4] P (v, α) = 1 2 √ 2π 1 α o v 5 o v 4 exp − v 2 2v 2 o 1 + α 2 α 2 o ,(7) where v o is the root mean square speed of atoms with average energyĒ ≡ mv 2 o /2. In the above equation, the term proportional to v 3 exp(−v 2 /2v 2 o )dv represents the thermal flux probability of having a longitudinal velocity v along the z-direction. The probability of having a transverse velocity v x = αv along the x-direction is proportional to the Gaussian distribution exp(−v 2 x /2v 2 o α 2 o )dv x , where α o is the FWHM of the angular distribution. Using the probability density (7), the averaged localization factor is thus given by L(z) = 2 λ α=∞ α=−∞ v=∞ v=0 xo=λ/4 xo=−λ/4 P (v, α) × [1 − cos (2kx(z, x o ))] dx o dvdα .(8) Here, x(z, x o ) represents the solution of differential equations (5) for varying initial conditions (x o , v, α) at z → −∞ of atoms. Note that the solution of Eq. (5) depends on the initial conditions (v, α) through the energy term E ≡ mv 2 (1 + α 2 )/2 as well. Since the focal length of light masks depends on velocity of the incoming atoms, the velocity spread in the atomic beam leads to the broadening of the deposited feature size. In the particle optics context of atom focusing, this is referred to as chromatic aberration. In addition, the initial angular divergence (α = 0) of the atomic beam degrades greatly the focusing of atoms. We are interested in the extent to which the velocity and angular spreads degrade the optimal squeezing of atoms. The best feature contrast in the presence of aberrations is again defined by minimizing the localization factor, Eq. (8). We have carried out the triple integration in Eq. (8) numerically and optimized the localization factor L(z) in the parameter space (z, S) for the case of the double-layer light masks. Figures 7 and 8 display atomic distribution at the point of the best squeezing by thin-thin and thinthick lenses configurations. On comparing the results with those ones for a single thin or thick lens, it is seen that the thin-thick lens combination provides the smallest feature size for the atom deposition. In the case of thin-thin lenses, the effects of chromatic aberrations are greater because of the strong dependence of focal length on the atomic velocity. We note that, though the initial velocity and angular spread of thermal beam worsen the optimal squeezing of atoms, the effects may become less important with increasing the number of layers in the multi-layer light masks. Further, chromatic aberrations can be greatly reduced by employing low-temperature supersonic beams of highly collimated atoms. V. SUMMARY In this paper, we presented the particle-optics analysis for atom lithography using multiple layers of SW light masks. In particular, we studied the spatial squeezing of atoms by a double layer of standing light waves with particular reference to minimizing the feature size of atom deposition. At first, linear focusing of atoms using paraxial approximation was considered. This showed an improvement in feature contrast at the focal plane, but the effect was rather modest. We then applied the approach of optimal squeezing that was suggested recently for the enhanced localization of atoms in a pulsed SW [16]. We showed that this approach works effectively for atomic nanofabrication and can considerably reduce the background in the atom deposition. Based on the optimal squeezing approach, a new figure of merit, the localization factor, was introduced to characterize the atomic localization. Both, thin-thin and thin-thick lens regimes of atom focusing were considered for monoenergetic as well as thermal beams of atoms. The parameters for the smallest feature size were found by minimizing the localization factor. We have shown that using a proper choice of lens parameters, it is possible to narrow considerably the atomic spatial distribution using the doublelayer light mask instead of the single-layer one. Finally, we note that our model calculations neglect the effects of atomic recoil due to spontaneous emission and the dipole force fluctuations. These effects are generally beyond the scope of the classical particle optics analysis and can be treated by means of a fully quantum approach. A detailed quantum mechanical study of the optimal atomic squeezing in application to nanofabrication will be published elsewhere. ACKNOWLEDGMENTS This work was supported by German -Israeli Foundation for Scientific Research and Development. FIG. 1 . 1Schematic representation of the laser focusing of atoms by a double layer of Gaussian standing waves. The intensity profile shows the Gaussian envelopes along the z-axis and the sinusoidal variations along the x-axis. FIG. 2 . 2Numerically calculated trajectories of atoms for laser focusing by a single-(a) and double-layer (b) light masks. The parameters used are I 1 /I s = 1000, I 2 /I s = 0 (a) and I 1 /I s = 1000, I 2 = I 1 , S = 500 (b). All other parameters are the same as in FIG. 3 . 3Localization factor of the atomic distribution for squeezing by a single-(dashed curve) and doublelayer (solid curve) light masks. The parameters used are I 1 /I s = 1000, I 2 /I s = 0 (dashed curve) and I 1 /I s = 1000, I 2 = I 1 , S = S m ≈ 1000 (solid curve). All other parameters are the same as inTable I. The minimal value of L(z) is 0.31 (0.42) and it occurs at z = z m ≈ 1450 (1300) for the solid (dashed) curve. In the case of the double light mask, the point (z m , S m ) corresponds to the numerically found global minimum of the localization factor. FIG. 4 . 4Probability density of atoms at the point of the best squeezing by a single-(dashed curve) and doublelayer (solid curve) light masks. The parameters used are I 1 /I s = 1000, I 2 /I s = 0, z = z m ≈ 1300 (dashed curve) and I 1 /I s = 1000, I 2 = I 1 , S = S m ≈ 1000, z = z m ≈ 1450 (solid curve). All other parameters are the same as in FIG. 5 . 5Minimal localization factor (maximal squeezing) of the atomic distribution as a function of the relative intensity I r ≡ I 2 /I 1 of standing light waves in a double light mask. The parameters used are same as in FIG. 6 . 6(a) Numerical trajectory calculation for laser focusing by a double light mask. The parameters used for the calculation are I 1 /I s = 1500, I 2 = 25I 1 , and S = S m ≈ 1500. All other parameters are the same as inTable I. (b) Probability density of the atomic distribution at the point (z m , S m ) of maximal squeezing by the double light mask. The parameters used are the same as those of (a) with z = z m ≈ 1550. The point (z m , S m ) is the numerically found global minimum of the function L(z) with respect to the variables (z, S). The dashed curve in graph (b) shows the atomic distribution at the point z = z m of the best squeezing by a single thick light lens with parameters I 1 /I s = 37500, I 2 /I s = 0, z m ≈ 90. The region to the right of origin (x = 0) is zoomed and shown in the inset. FIG. 7 . 7Localization factor of the atomic distribution for squeezing by a single-(a) and double-layer (b) light masks. The parameters used areĒ = 3×10 9 , α o = 10 −4 , I 1 /I s = 1500, and (a) I 2 /I s = 0, (b) I 2 = I 1 , S = S m ≈ 800. All other parameters used are the same as in Table I. The minimal value of L(z) is 0.67 [0.8] and it occurs at z = z m ≈ 1350 [975] in the graph (b) [(a)]. The dashed and solid curves in graph (c) give the atomic distribution at the point (z m , S m ) of best squeezing by the single-and double-layer light masks with the parameters of (a) and (b). The region to the right of origin (x = 0) in graph (c) is zoomed and shown in the inset. FIG. 8 . 8Localization factor of the atomic distribution for squeezing by a single-(a) and double-layer (b) light masks. The parameters used areĒ = 3×10 9 , α o = 10 −4 , and (a) I 1 /I s = 37500, I 2 /I s = 0, (b) I 1 /I s = 1500, I 2 = 25I 1 , S = S m ≈ 1200. All other parameters used are the same as in Table I. The minimal value of L(z) is 0.5 [0.66] and it occurs at z = z m ≈ 1350 [160] in the graph (b) [(a)]. The dashed and solid curves in graph (c) give the atomic distribution at the point (z m , S m ) of best squeezing by the single-and double-layer light masks with the parameters of (a) and (b). The region to the right of origin (x = 0) in graph (c) is zoomed and shown in the inset. TABLE I . IParameters in scaled units. Frequency is measured in recoil units, and length in units of the optical wavelength. 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Dirk Jürgens, GermanyUniversität KonstanzPhD dissertationDirk Jürgens, PhD dissertation, 2004, Universität Kon- stanz, Germany In all figures, we normalize the flux such that its integral over the range. −λ/4, λ/4] has a value of 1In all figures, we normalize the flux such that its integral over the range [−λ/4, λ/4] has a value of 1. . I Sh, R Averbukh, Arvieu, Phys. Rev. Lett. 87163601I.Sh. Averbukh and R. Arvieu, Phys. Rev. Lett. 87, 163601 (2001). We note that the first experimental demonstration of atom focusing using a thin SW lens was reported by. T Sleator, T Pfau, V Balykin, J Mlynek, Appl. Phys. B. 54375We note that the first experimental demonstration of atom focusing using a thin SW lens was reported by T. Sleator, T. Pfau, V. Balykin, and J. Mlynek [Appl. Phys. B 54, 375 (1992)].	[]

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